what is mathematics essay

“What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it

• Conference paper
• Open Access
• First Online: 02 November 2017
• Cite this conference paper

You have full access to this open access conference paper

• Günter M. Ziegler 3 &
• Andreas Loos 4  

Part of the book series: ICME-13 Monographs ((ICME13Mo))

110k Accesses

9 Citations

3 Altmetric

“What is Mathematics?” [with a question mark!] is the title of a famous book by Courant and Robbins, first published in 1941, which does not answer the question. The question is, however, essential: The public image of the subject (of the science, and of the profession) is not only relevant for the support and funding it can get, but it is also crucial for the talent it manages to attract—and thus ultimately determines what mathematics can achieve, as a science, as a part of human culture, but also as a substantial component of economy and technology. In this lecture we thus

discuss the image of mathematics (where “image” might be taken literally!),

sketch a multi-facetted answer to the question “What is Mathematics?,”

stress the importance of learning “What is Mathematics” in view of Klein’s “double discontinuity” in mathematics teacher education,

present the “Panorama project” as our response to this challenge,

stress the importance of telling stories in addition to teaching mathematics, and finally,

suggest that the mathematics curricula at schools and at universities should correspondingly have space and time for at least three different subjects called Mathematics.

This paper is a slightly updated reprint of: Günter M. Ziegler and Andreas Loos, Learning and Teaching “ What is Mathematics ”, Proc. International Congress of Mathematicians, Seoul 2014, pp. 1201–1215; reprinted with kind permission by Prof. Hyungju Park, the chairman of ICM 2014 Organizing Committee.

You have full access to this open access chapter,  Download conference paper PDF

Similar content being viewed by others

The Missing Element for Teachers: Learning What Mathematics Is

Educational Paths to Mathematics: Which Paths Forward to What Mathematics?

Mathematics at the Center of Distinct Fields: A Response to Michael and Ted

What is mathematics.

Defining mathematics. According to Wikipedia in English, in the March 2014 version, the answer to “What is Mathematics?” is

Mathematics is the abstract study of topics such as quantity (numbers), [2] structure, [3] space, [2] and change. [4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. [7][8] Mathematicians seek out patterns (Highland & Highland, 1961 , 1963 ) and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

None of this is entirely wrong, but it is also not satisfactory. Let us just point out that the fact that there is no agreement about the definition of mathematics, given as part of a definition of mathematics, puts us into logical difficulties that might have made Gödel smile. Footnote 1

The answer given by Wikipedia in the current German version, reads (in our translation):

Mathematics […] is a science that developed from the investigation of geometric figures and the computing with numbers. For mathematics , there is no commonly accepted definition; today it is usually described as a science that investigates abstract structures that it created itself by logical definitions using logic for their properties and patterns.

This is much worse, as it portrays mathematics as a subject without any contact to, or interest from, a real world.

The borders of mathematics. Is mathematics “stand-alone”? Could it be defined without reference to “neighboring” subjects, such as physics (which does appear in the English Wikipedia description)? Indeed, one possibility to characterize mathematics describes the borders/boundaries that separate it from its neighbors. Even humorous versions of such “distinguishing statements” such as

“Mathematics is the part of physics where the experiments are cheap.”

“Mathematics is the part of philosophy where (some) statements are true—without debate or discussion.”

“Mathematics is computer science without electricity.” (So “Computer science is mathematics with electricity.”)

contain a lot of truth and possibly tell us a lot of “characteristics” of our subject. None of these is, of course, completely true or completely false, but they present opportunities for discussion.

What we do in mathematics . We could also try to define mathematics by “what we do in mathematics”: This is much more diverse and much more interesting than the Wikipedia descriptions! Could/should we describe mathematics not only as a research discipline and as a subject taught and learned at school, but also as a playground for pupils, amateurs, and professionals, as a subject that presents challenges (not only for pupils, but also for professionals as well as for amateurs), as an arena for competitions, as a source of problems, small and large, including some of the hardest problems that science has to offer, at all levels from elementary school to the millennium problems (Csicsery, 2008 ; Ziegler, 2011 )?

What we teach in mathematics classes . Education bureaucrats might (and probably should) believe that the question “What is Mathematics?” is answered by high school curricula. But what answers do these give?

This takes us back to the nineteenth century controversies about what mathematics should be taught at school and at the Universities. In the German version this was a fierce debate. On the one side it saw the classical educational ideal as formulated by Wilhelm von Humboldt (who was involved in the concept for and the foundation 1806 of the Berlin University, now named Humboldt Universität, and to a certain amount shaped the modern concept of a university); here mathematics had a central role, but this was the classical “Greek” mathematics, starting from Euclid’s axiomatic development of geometry, the theory of conics, and the algebra of solving polynomial equations, not only as cultural heritage, but also as a training arena for logical thinking and problem solving. On the other side of the fight were the proponents of “Realbildung”: Realgymnasien and the technical universities that were started at that time tried to teach what was needed in commerce and industry: calculation and accounting, as well as the mathematics that could be useful for mechanical and electrical engineering—second rate education in the view of the classical German Gymnasium.

This nineteenth century debate rests on an unnatural separation into the classical, pure mathematics, and the useful, applied mathematics; a division that should have been overcome a long time ago (perhaps since the times of Archimedes), as it is unnatural as a classification tool and it is also a major obstacle to progress both in theory and in practice. Nevertheless the division into “classical” and “current” material might be useful in discussing curriculum contents—and the question for what purpose it should be taught; see our discussion in the Section “ Three Times Mathematics at School? ”.

The Courant–Robbins answer . The title of the present paper is, of course, borrowed from the famous and very successful book by Richard Courant and Herbert Robbins. However, this title is a question—what is Courant and Robbins’ answer? Indeed, the book does not give an explicit definition of “What is Mathematics,” but the reader is supposed to get an idea from the presentation of a diverse collection of mathematical investigations. Mathematics is much bigger and much more diverse than the picture given by the Courant–Robbins exposition. The presentation in this section was also meant to demonstrate that we need a multi-facetted picture of mathematics: One answer is not enough, we need many.

Why Should We Care?

The question “What is Mathematics?” probably does not need to be answered to motivate why mathematics should be taught, as long as we agree that mathematics is important.

However, a one-sided answer to the question leads to one-sided concepts of what mathematics should be taught.

At the same time a one-dimensional picture of “What is Mathematics” will fail to motivate kids at school to do mathematics, it will fail to motivate enough pupils to study mathematics, or even to think about mathematics studies as a possible career choice, and it will fail to motivate the right students to go into mathematics studies, or into mathematics teaching. If the answer to the question “What is Mathematics”, or the implicit answer given by the public/prevailing image of the subject, is not attractive, then it will be very difficult to motivate why mathematics should be learned—and it will lead to the wrong offers and the wrong choices as to what mathematics should be learned.

Indeed, would anyone consider a science that studies “abstract” structures that it created itself (see the German Wikipedia definition quoted above) interesting? Could it be relevant? If this is what mathematics is, why would or should anyone want to study this, get into this for a career? Could it be interesting and meaningful and satisfying to teach this?

Also in view of the diversity of the students’ expectations and talents, we believe that one answer is plainly not enough. Some students might be motivated to learn mathematics because it is beautiful, because it is so logical, because it is sometimes surprising. Or because it is part of our cultural heritage. Others might be motivated, and not deterred, by the fact that mathematics is difficult. Others might be motivated by the fact that mathematics is useful, it is needed—in everyday life, for technology and commerce, etc. But indeed, it is not true that “the same” mathematics is needed in everyday life, for university studies, or in commerce and industry. To other students, the motivation that “it is useful” or “it is needed” will not be sufficient. All these motivations are valid, and good—and it is also totally valid and acceptable that no single one of these possible types of arguments will reach and motivate all these students.

Why do so many pupils and students fail in mathematics, both at school and at universities? There are certainly many reasons, but we believe that motivation is a key factor. Mathematics is hard. It is abstract (that is, most of it is not directly connected to everyday-life experiences). It is not considered worth-while. But a lot of the insufficient motivation comes from the fact that students and their teachers do not know “What is Mathematics.”

Thus a multi-facetted image of mathematics as a coherent subject, all of whose many aspects are well connected, is important for a successful teaching of mathematics to students with diverse (possible) motivations.

This leads, in turn, to two crucial aspects, to be discussed here next: What image do students have of mathematics? And then, what should teachers answer when asked “What is Mathematics”? And where and how and when could they learn that?

The Image of Mathematics

A 2008 study by Mendick, Epstein, and Moreau ( 2008 ), which was based on an extensive survey among British students, was summarized as follows:

Many students and undergraduates seem to think of mathematicians as old, white, middle-class men who are obsessed with their subject, lack social skills and have no personal life outside maths. The student’s views of maths itself included narrow and inaccurate images that are often limited to numbers and basic arithmetic.

The students’ image of what mathematicians are like is very relevant and turns out to be a massive problem, as it defines possible (anti-)role models, which are crucial for any decision in the direction of “I want to be a mathematician.” If the typical mathematician is viewed as an “old, white, male, middle-class nerd,” then why should a gifted 16-year old girl come to think “that’s what I want to be when I grow up”? Mathematics as a science, and as a profession, looses (or fails to attract) a lot of talent this way! However, this is not the topic of this presentation.

On the other hand the first and the second diagnosis of the quote from Mendick et al. ( 2008 ) belong together: The mathematicians are part of “What is Mathematics”!

And indeed, looking at the second diagnosis, if for the key word “mathematics” the images that spring to mind don’t go beyond a per se meaningless “ \( a^{2} + b^{2} = c^{2} \) ” scribbled in chalk on a blackboard—then again, why should mathematics be attractive, as a subject, as a science, or as a profession?

We think that we have to look for, and work on, multi-facetted and attractive representations of mathematics by images. This could be many different, separate images, but this could also be images for “mathematics as a whole.”

Four Images for “What Is Mathematics?”

Striking pictorial representations of mathematics as a whole (as well as of other sciences!) and of their change over time can be seen on the covers of the German “Was ist was” books. The history of these books starts with the series of “How and why” Wonder books published by Grosset and Dunlop, New York, since 1961, which was to present interesting subjects (starting with “Dinosaurs,” “Weather,” and “Electricity”) to children and younger teenagers. The series was published in the US and in Great Britain in the 1960s and 1970s, but it was and is much more successful in Germany, where it was published (first in translation, then in volumes written in German) by Ragnar Tessloff since 1961. Volume 18 in the US/UK version and Volume 12 in the German version treats “Mathematics”, first published in 1963 (Highland & Highland, 1963 ), but then republished with the same title but a new author and contents in 2001 (Blum, 2001 ). While it is worthwhile to study the contents and presentation of mathematics in these volumes, we here focus on the cover illustrations (see Fig.  1 ), which for the German edition exist in four entirely different versions, the first one being an adaption of the original US cover of (Highland & Highland, 1961 ).

The four covers of “Was ist was. Band 12: Mathematik” (Highland & Highland, 1963 ; Blum, 2001 )

All four covers represent a view of “What is Mathematics” in a collage mode, where the first one represents mathematics as a mostly historical discipline (starting with the ancient Egyptians), while the others all contain a historical allusion (such as pyramids, Gauß, etc.) alongside with objects of mathematics (such as prime numbers or \( \pi \) , dices to illustrate probability, geometric shapes). One notable object is the oddly “two-colored” Möbius band on the 1983 cover, which was changed to an entirely green version in a later reprint.

One can discuss these covers with respect to their contents and their styles, and in particular in terms of attractiveness to the intended buyers/readers. What is over-emphasized? What is missing? It seems more important to us to

think of our own images/representations for “What is Mathematics”,

think about how to present a multi-facetted image of “What is Mathematics” when we teach.

Indeed, the topics on the covers of the “Was ist was” volumes of course represent interesting (?) topics and items discussed in the books. But what do they add up to? We should compare this to the image of mathematics as represented by school curricula, or by the university curricula for teacher students.

In the context of mathematics images, let us mention two substantial initiatives to collect and provide images from current mathematics research, and make them available on internet platforms, thus providing fascinating, multi-facetted images of mathematics as a whole discipline:

Guy Métivier et al.: “Image des Maths. La recherche mathématique en mots et en images” [“Images of Maths. Mathematical research in words and images”], CNRS, France, at images.math.cnrs.fr (texts in French)

Andreas D. Matt, Gert-Martin Greuel et al.: “IMAGINARY. open mathematics,” Mathematisches Forschungsinstitut Oberwolfach, at imaginary.org (texts in German, English, and Spanish).

The latter has developed from a very successful travelling exhibition of mathematics images, “IMAGINARY—through the eyes of mathematics,” originally created on occasion of and for the German national science year 2008 “Jahr der Mathematik. Alles was zählt” [“Year of Mathematics 2008. Everything that counts”], see www.jahr-der-mathematik.de , which was highly successful in communicating a current, attractive image of mathematics to the German public—where initiatives such as the IMAGINARY exhibition had a great part in the success.

Teaching “What Is Mathematics” to Teachers

More than 100 years ago, in 1908, Felix Klein analyzed the education of teachers. In the introduction to the first volume of his “Elementary Mathematics from a Higher Standpoint” he wrote (our translation):

At the beginning of his university studies, the young student is confronted with problems that do not remind him at all of what he has dealt with up to then, and of course, he forgets all these things immediately and thoroughly. When after graduation he becomes a teacher, he has to teach exactly this traditional elementary mathematics, and since he can hardly link it with his university mathematics, he soon readopts the former teaching tradition and his studies at the university become a more or less pleasant reminiscence which has no influence on his teaching (Klein, 1908 ).

This phenomenon—which Klein calls the double discontinuity —can still be observed. In effect, the teacher students “tunnel” through university: They study at university in order to get a degree, but nevertheless they afterwards teach the mathematics that they had learned in school, and possibly with the didactics they remember from their own school education. This problem observed and characterized by Klein gets even worse in a situation (which we currently observe in Germany) where there is a grave shortage of Mathematics teachers, so university students are invited to teach at high school long before graduating from university, so they have much less university education to tunnel at the time when they start to teach in school. It may also strengthen their conviction that University Mathematics is not needed in order to teach.

How to avoid the double discontinuity is, of course, a major challenge for the design of university curricula for mathematics teachers. One important aspect however, is tied to the question of “What is Mathematics?”: A very common highschool image/concept of mathematics, as represented by curricula, is that mathematics consists of the subjects presented by highschool curricula, that is, (elementary) geometry, algebra (in the form of arithmetic, and perhaps polynomials), plus perhaps elementary probability, calculus (differentiation and integration) in one variable—that’s the mathematics highschool students get to see, so they might think that this is all of it! Could their teachers present them a broader picture? The teachers after their highschool experience studied at university, where they probably took courses in calculus/analysis, linear algebra, classical algebra, plus some discrete mathematics, stochastics/probability, and/or numerical analysis/differential equations, perhaps a programming or “computer-oriented mathematics” course. Altogether they have seen a scope of university mathematics where no current research becomes visible, and where most of the contents is from the nineteenth century, at best. The ideal is, of course, that every teacher student at university has at least once experienced how “doing research on your own” feels like, but realistically this rarely happens. Indeed, teacher students would have to work and study and struggle a lot to see the fascination of mathematics on their own by doing mathematics; in reality they often do not even seriously start the tour and certainly most of them never see the “glimpse of heaven.” So even if the teacher student seriously immerges into all the mathematics on the university curriculum, he/she will not get any broader image of “What is Mathematics?”. Thus, even if he/she does not tunnel his university studies due to the double discontinuity, he/she will not come back to school with a concept that is much broader than that he/she originally gained from his/her highschool times.

Our experience is that many students (teacher students as well as classical mathematics majors) cannot name a single open problem in mathematics when graduating the university. They have no idea of what “doing mathematics” means—for example, that part of this is a struggle to find and shape the “right” concepts/definitions and in posing/developing the “right” questions and problems.

And, moreover, also the impressions and experiences from university times will get old and outdated some day: a teacher might be active at a school for several decades—while mathematics changes! Whatever is proved in mathematics does stay true, of course, and indeed standards of rigor don’t change any more as much as they did in the nineteenth century, say. However, styles of proof do change (see: computer-assisted proofs, computer-checkable proofs, etc.). Also, it would be good if a teacher could name “current research focus topics”: These do change over ten or twenty years. Moreover, the relevance of mathematics in “real life” has changed dramatically over the last thirty years.

The Panorama Project

For several years, the present authors have been working on developing a course [and eventually a book (Loos & Ziegler, 2017 )] called “Panorama der Mathematik” [“Panorama of Mathematics”]. It primarily addresses mathematics teacher students, and is trying to give them a panoramic view on mathematics: We try to teach an overview of the subject, how mathematics is done, who has been and is doing it, including a sketch of main developments over the last few centuries up to the present—altogether this is supposed to amount to a comprehensive (but not very detailed) outline of “What is Mathematics.” This, of course, turns out to be not an easy task, since it often tends to feel like reading/teaching poetry without mastering the language. However, the approach of Panorama is complementing mathematics education in an orthogonal direction to the classic university courses, as we do not teach mathematics but present (and encourage to explore ); according to the response we get from students they seem to feel themselves that this is valuable.

Our course has many different components and facets, which we here cast into questions about mathematics. All these questions (even the ones that “sound funny”) should and can be taken seriously, and answered as well as possible. For each of them, let us here just provide at most one line with key words for answers:

When did mathematics start?

Numbers and geometric figures start in stone age; the science starts with Euclid?

How large is mathematics? How many Mathematicians are there?

The Mathematics Genealogy Project had 178854 records as of 12 April 2014.

How is mathematics done, what is doing research like?

Collect (auto)biographical evidence! Recent examples: Frenkel ( 2013 ) , Villani ( 2012 ).

What does mathematics research do today? What are the Grand Challenges?

The Clay Millennium problems might serve as a starting point.

What and how many subjects and subdisciplines are there in mathematics?

See the Mathematics Subject Classification for an overview!

Why is there no “Mathematical Industry”, as there is e.g. Chemical Industry?

There is! See e.g. Telecommunications, Financial Industry, etc.

What are the “key concepts” in mathematics? Do they still “drive research”?

Numbers, shapes, dimensions, infinity, change, abstraction, …; they do.

What is mathematics “good for”?

It is a basis for understanding the world, but also for technological progress.

Where do we do mathematics in everyday life?

Not only where we compute, but also where we read maps, plan trips, etc.

Where do we see mathematics in everyday life?

There is more maths in every smart phone than anyone learns in school.

What are the greatest achievements of mathematics through history?

Make your own list!

An additional question is how to make university mathematics more “sticky” for the tunneling teacher students, how to encourage or how to force them to really connect to the subject as a science. Certainly there is no single, simple, answer for this!

Telling Stories About Mathematics

How can mathematics be made more concrete? How can we help students to connect to the subject? How can mathematics be connected to the so-called real world?

Showing applications of mathematics is a good way (and a quite beaten path). Real applications can be very difficult to teach since in most advanced, realistic situation a lot of different mathematical disciplines, theories and types of expertise have to come together. Nevertheless, applications give the opportunity to demonstrate the relevance and importance of mathematics. Here we want to emphasize the difference between teaching a topic and telling about it. To name a few concrete topics, the mathematics behind weather reports and climate modelling is extremely difficult and complex and advanced, but the “basic ideas” and simplified models can profitably be demonstrated in highschool, and made plausible in highschool level mathematical terms. Also success stories like the formula for the Google patent for PageRank (Page, 2001 ), see Langville and Meyer ( 2006 ), the race for the solution of larger and larger instances of the Travelling Salesman Problem (Cook, 2011 ), or the mathematics of chip design lend themselves to “telling the story” and “showing some of the maths” at a highschool level; these are among the topics presented in the first author’s recent book (Ziegler, 2013b ), where he takes 24 images as the starting points for telling stories—and thus developing a broader multi-facetted picture of mathematics.

Another way to bring maths in contact with non-mathematicians is the human level. Telling stories about how maths is done and by whom is a tricky way, as can be seen from the sometimes harsh reactions on www.mathoverflow.net to postings that try to excavate the truth behind anecdotes and legends. Most mathematicians see mathematics as completely independent from the persons who explored it. History of mathematics has the tendency to become gossip , as Gian-Carlo Rota once put it (Rota, 1996 ). The idea seems to be: As mathematics stands for itself, it has also to be taught that way.

This may be true for higher mathematics. However, for pupils (and therefore, also for teachers), transforming mathematicians into humans can make science more tangible, it can make research interesting as a process (and a job?), and it can be a starting/entry point for real mathematics. Therefore, stories can make mathematics more sticky. Stories cannot replace the classical approaches to teaching mathematics. But they can enhance it.

Stories are the way by which knowledge has been transferred between humans for thousands of years. (Even mathematical work can be seen as a very abstract form of storytelling from a structuralist point of view.) Why don’t we try to tell more stories about mathematics, both at university and in school—not legends, not fairy tales, but meta-information on mathematics—in order to transport mathematics itself? See (Ziegler, 2013a ) for an attempt by the first author in this direction.

By stories, we do not only mean something like biographies, but also the way of how mathematics is created or discovered: Jack Edmonds’ account (Edmonds, 1991 ) of how he found the blossom shrink algorithm is a great story about how mathematics is actually done . Think of Thomas Harriot’s problem about stacking cannon balls into a storage space and what Kepler made out of it: the genesis of a mathematical problem. Sometimes scientists even wrap their work into stories by their own: see e.g. Leslie Lamport’s Byzantine Generals (Lamport, Shostak, & Pease, 1982 ).

Telling how research is done opens another issue. At school, mathematics is traditionally taught as a closed science. Even touching open questions from research is out of question, for many good and mainly pedagogical reasons. However, this fosters the image of a perfect science where all results are available and all problems are solved—which is of course completely wrong (and moreover also a source for a faulty image of mathematics among undergraduates).

Of course, working with open questions in school is a difficult task. None of the big open questions can be solved with an elementary mathematical toolbox; many of them are not even accessible as questions. So the big fear of discouraging pupils is well justified. On the other hand, why not explore mathematics by showing how questions often pop up on the way? Posing questions in and about mathematics could lead to interesting answers—in particular to the question of “What is Mathematics, Really?”

Three Times Mathematics at School?

So, what is mathematics? With school education in mind, the first author has argued in Ziegler ( 2012 ) that we are trying cover three aspects the same time, which one should consider separately and to a certain extent also teach separately:

A collection of basic tools, part of everyone’s survival kit for modern-day life—this includes everything, but actually not much more than, what was covered by Adam Ries’ “Rechenbüchlein” [“Little Book on Computing”] first published in 1522, nearly 500 years ago;

A field of knowledge with a long history, which is a part of our culture and an art, but also a very productive basis (indeed a production factor) for all modern key technologies. This is a “story-telling” subject.

An introduction to mathematics as a science—an important, highly developed, active, huge research field.

Looking at current highschool instruction, there is still a huge emphasis on Mathematics I, with a rather mechanical instruction on arithmetic, “how to compute correctly,” and basic problem solving, plus a rather formal way of teaching Mathematics III as a preparation for possible university studies in mathematics, sciences or engineering. Mathematics II, which should provide a major component of teaching “What is Mathematics,” is largely missing. However, this part also could and must provide motivation for studying Mathematics I or III!

What Is Mathematics, Really?

There are many, and many different, valid answers to the Courant-Robbins question “What is Mathematics?”

A more philosophical one is given by Reuben Hersh’s book “What is Mathematics, Really?” Hersh ( 1997 ), and there are more psychological ones, on the working level. Classics include Jacques Hadamard’s “Essay on the Psychology of Invention in the Mathematical Field” and Henri Poincaré’s essays on methodology; a more recent approach is Devlin’s “Introduction to Mathematical Thinking” Devlin ( 2012 ), or Villani’s book ( 2012 ).

And there have been many attempts to describe mathematics in encyclopedic form over the last few centuries. Probably the most recent one is the gargantuan “Princeton Companion to Mathematics”, edited by Gowers et al. ( 2008 ), which indeed is a “Princeton Companion to Pure Mathematics.”

However, at a time where ZBMath counts more than 100,000 papers and books per year, and 29,953 submissions to the math and math-ph sections of arXiv.org in 2016, it is hopeless to give a compact and simple description of what mathematics really is, even if we had only the “current research discipline” in mind. The discussions about the classification of mathematics show how difficult it is to cut the science into slices, and it is even debatable whether there is any meaningful way to separate applied research from pure mathematics.

Probably the most diplomatic way is to acknowledge that there are “many mathematics.” Some years ago Tao ( 2007 ) gave an open list of mathematics that is/are good for different purposes—from “problem-solving mathematics” and “useful mathematics” to “definitive mathematics”, and wrote:

As the above list demonstrates, the concept of mathematical quality is a high-dimensional one, and lacks an obvious canonical total ordering. I believe this is because mathematics is itself complex and high-dimensional, and evolves in unexpected and adaptive ways; each of the above qualities represents a different way in which we as a community improve our understanding and usage of the subject.

In this sense, many answers to “What is Mathematics?” probably show as much about the persons who give the answers as they manage to characterize the subject.

According to Wikipedia , the same version, the answer to “Who is Mathematics” should be:

Mathematics , also known as Allah Mathematics , (born: Ronald Maurice Bean [1] ) is a hip hop producer and DJ for the Wu-Tang Clan and its solo and affiliate projects. This is not the mathematics we deal with here.

Blum, W. (2001). Was ist was. Band 12: Mathematik , Tessloff Verlag, Nürnberg. Revised version, with new cover, 2010.

Google Scholar  

Cook, W. (2011). In pursuit of the traveling salesman: Mathematics at the limits of computation . Princeton NJ: Princeton University Press.

Courant, R., & Robbins, H. (1941). What is mathematics? an elementary approach to ideas and methods (2nd ed.), Oxford: Oxford University Press. Stewart, I (ed), 1996.

Csicsery, G. (2008). Hard problems. the road to the world’s toughest math contest , Documentary film, 82 minutes (feature)/45 minutes (classroom version), Washington, DC: Mathematical Association of America.

Devlin, K. J. (2012). Introduction to mathematical thinking , published by Keith Devlin, Palo Alto CA.

Edmonds, J. (1991). A glimpse of heaven, In: J. K. Lenstra, A. Schrijver, & A. Rinnooy Kan (eds.) History of mathematical programming—A collection of personal reminiscences (pp. 32–54). Amsterdam: CWI and North-Holland.

Frenkel, E. (2013). Love & math. The heart of hidden reality . Philadelphia PA: Basic Books/Perseus Books.

Gowers, Timothy, Leader, Imre, & Barrow-Green, June (Eds.). (2008). The princeton companion to mathematics . Princeton NJ: Princeton University Press.

Highland, E. H., & Highland, H. J. (1961). The how and why wonder book of mathematics . New York: Grosset & Dunlop.

Highland, E. H., & Highland, H. J. (1963). Was ist was. Band 12: Mathematik , Neuer Tessloff Verlag, Hamburg, 1963. Revised edition 1969. New cover 1983.

Hersh, R. (1997). What is mathematics, really? . Oxford: Oxford University Press.

Klein, F.(1933). Elementarmathematik vom höheren Standpunkte aus. Teil I: Arithmetik, Algebra, Analysis , B. G. Teubner, Leipzig, 1908. Vierte Auflage. Heidelberg: Springer.

Lamport, L., Shostak, R., & Pease, M. (1982). The byzantine generals problem. ACM Transactions on Programming Languages and Systems, 4, 382–401.

Article   Google Scholar  

Langville, A. N., & Meyer, C. D. (2006). Google’s pagerank and beyond. The science of search engine rankings . Princeton and Oxford: Princeton University Press.

Loos, A., & Ziegler, G. M. (2017). Panorama der Mathematik . Heidelberg: Springer Spectrum, to appear.

Mendick, H., Epstein, D., & Moreau, M.-P. (2008). Mathematical images and identities: Education, entertainment, social justice . London: Institute for Policy Studies in Education, London Metropolitan University.

Page, L. (2001) Method for node ranking in a linked database , United States Patent No. US 6,285,999 B1, (submitted: January 9, 1998), http://www.google.com/patents/US6285999

Rota, G.-C. (1996). Indiscrete thoughts . Basel: Birkhäuser.

Tao, T. (2007). What is good mathematics? Bulletin of the American Mathematical Society, 44 (4), 623–634.

Villani, C. (2012). Théorème vivant . Paris: Bernard Grasset. (in French).

Ziegler, G. M. (2011). Three competitions. In D. Schleicher & M. Lackmann (Eds.), Invitation to mathematics. From competition to research (pp. 195–205). Berlin: Springer.

Chapter   Google Scholar  

Ziegler, G. M. (2012). Mathematics school education provides answers—To which questions? EMS Newsletter (84), 8–11.

Ziegler, G. M.(2013a). Do I count? stories from mathematics , Boca Raton FL: CRC Press/Taylor & Francis. English translation of “Darf ich Zahlen? Geschichten aus der Mathematik”, Piper, München, 2010.

Ziegler, G. M. (2013b). Mathematik—Das ist doch keine Kunst! . München: Knaus.

Download references

Acknowledgment

The authors’ work has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 247029, the DFG Research Center Matheon, and the the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.

Author information

Authors and affiliations.

Institut Für Mathematik, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

Günter M. Ziegler

Zeit Online, Askanischer Platz 1, 10963, Berlin, Germany

Andreas Loos

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Günter M. Ziegler .

Editor information

Editors and affiliations.

Faculty of Education, Universität Hamburg, Hamburg, Hamburg, Germany

Gabriele Kaiser

Rights and permissions

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ .

Reprints and permissions

Copyright information

© 2017 The Author(s)

About this paper

Cite this paper.

Ziegler, G.M., Loos, A. (2017). “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. In: Kaiser, G. (eds) Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-62597-3_5

Download citation

DOI : https://doi.org/10.1007/978-3-319-62597-3_5

Published : 02 November 2017

Publisher Name : Springer, Cham

Print ISBN : 978-3-319-62596-6

Online ISBN : 978-3-319-62597-3

eBook Packages : Education Education (R0)

Share this paper

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

What is mathematics?

Mathematics is at the heart of science and our daily lives.

• Inventor of mathematics
• Ancient Greek Mathematics
• Importance of mathematics

Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, computers , software, architecture (ancient and modern), art, money, engineering and even sports.

Since the beginning of recorded history, mathematical discovery has been at the forefront of every civilized society, and math has been used by even the most primitive and earliest cultures . The need for math arose because of the increasingly complex demands from societies around the world, which required more advanced mathematical solutions, as outlined by mathematician Raymond L. Wilder in his book " Evolution of Mathematical Concepts " (Dover Publications, 2013). 

The more complex a society, the more complex the mathematical needs. Primitive tribes needed little more than the ability to count, but also used math to calculate the position of the sun and the physics of hunting. "All the records — anthropological and historical — show that counting and, ultimately, numeral systems as a device for counting form the inception of the mathematical element in all cultures," Wilder wrote in 1968.

Who invented mathematics?

Several civilizations — in China , India, Egypt , Central America and Mesopotamia — contributed to mathematics as we know it today. The Sumerians, who lived in the region that is now southern Iraq, were the first people to develop a counting system with a base 60 system, according to Wilder. 

This was based on using the bones in the fingers to count and then use as sets, according to Georges Ifrah in his book " The Universal History Of Numbers " (John Wiley & Sons, 2000). From these systems we have the basis of arithmetic, which includes basic operations of addition, multiplication, division, fractions and square roots. Wilder explained that the Sumerians' system passed through the Akkadian Empire to the Babylonians around 300 B.C. Six hundred years later, in Central America, the Maya developed elaborate calendar systems and were skilled astronomers. About this time, the concept of zero was developed in India.

As civilizations developed, mathematicians began to work with geometry, which computes areas, volumes and angles, and has many practical applications. Geometry is used in everything from home construction to fashion and interior design. As Richard J. Gillings wrote in his book " Mathematics in the Time of the Pharaohs " (Dover Publications, 1982), the pyramids of Giza in Egypt are stunning examples of ancient civilizations' advanced use of geometry.

Geometry went hand in hand with algebra . Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī authored the earliest recorded work on algebra called "The Compendious Book on Calculation by Completion and Balancing" around 820 A.D., according to Philip K. Hitti , a history professor at Princeton and Harvard University. Al-Khwārizmī also developed quick methods for multiplying and dividing numbers, which are known as algorithms — a corruption of his name, which in Latin was translated to Algorithmi.

Algebra offered civilizations a way to divide inheritances and allocate resources. The study of algebra meant mathematicians could solve linear equations and systems, as well as quadratics , and delve into positive and negative solutions. Mathematicians in ancient times also began to look at number theory, which "deals with properties of the whole numbers, 1, 2, 3, 4, 5, …," Tom M. Apostol, a professor at the California Institute of Technology, wrote in " Introduction to Analytic Number Theory " (Springer, 1976). With origins in the construction of shape, number theory looks at figurate numbers, the characterization of numbers, and theorems.

Mathematics in ancient Greece

The word mathematics comes from the ancient Greeks and is derived from the word máthēma, meaning "that which is learnt," according to Douglas R. Harper, author of the " Online Etymology Dictionary ." The ancient Greeks built on other ancient civilizations’ mathematical studies, and they developed the model of abstract mathematics through geometry. 

Greek mathematicians were divided into several schools, as outlined by G. Donald Allen, professor of Mathematics at Texas A&M University in his paper, " The Origins of Greek Mathematics ":

In addition to the Greek mathematicians listed above, a number of other ancient Greeks made an indelible mark on the history of mathematics, including Archimedes , most famous for the Archimedes' principle around the buoyant force; Apollonius, who did important work with parabolas ; Diophantus, the first Greek mathematician to recognize fractions as numbers; Pappus, known for his hexagon theorem; and Euclid, who first described the golden ratio .

During this time, mathematicians began working with trigonometry , which studies relationships between the sides and angles of triangles and computes trigonometric functions, including sine, cosine, tangent and their reciprocals. Trigonometry relies on the synthetic geometry developed by Greek mathematicians like Euclid. In past cultures, trigonometry was applied to astronomy and the computation of angles in the celestial sphere.

The development of mathematics was taken on by the Islamic empires, then concurrently in Europe and China, according to Wilder. Leonardo Fibonacci was a medieval European mathematician and was famous for his theories on arithmetic, algebra and geometry. The Renaissance led to advances that included decimal fractions, logarithms and projective geometry. Number theory was greatly expanded upon, and theories like probability and analytic geometry ushered in a new age of mathematics, with calculus at the forefront.

Development of calculus

In the 17th century, Isaac Newton in England and Gottfried Leibniz in Germany independently developed the foundations for calculus, Carl B. Boyer, a science historian, explained in " The History of the Calculus and Its Conceptual Development " (Dover Publications, 1959). Calculus development went through three periods: anticipation, development and rigorization. 

In the anticipation stage, mathematicians attempted to use techniques that involved infinite processes to find areas under curves or maximize certain qualities. In the development stage, Newton and Leibniz brought these techniques together through the derivative (the curve of mathematical function) and integral (the area under the curve). Though their methods were not always logically sound, mathematicians in the 18th century took on the rigorization stage and were able to justify their methods and create the final stage of calculus. Today, we define the derivative and integral in terms of limits.

In contrast to calculus, which is a type of continuous mathematics (dealing with real numbers), other mathematicians have taken a more theoretical approach. Discrete mathematics is the branch of math that deals with objects that can assume only distinct, separated value, as mathematician and computer scientist Richard Johnsonbaugh explained in " Discrete Mathematics " (Pearson, 2017). Discrete objects can be characterized by integers, rather than real numbers. Discrete mathematics is the mathematical language of computer science, as it includes the study of algorithms. Fields of discrete mathematics include combinatorics, graph theory and the theory of computation.

Why mathematics is important

It's not uncommon for people to wonder what relevance mathematics serves in their daily lives. In the modern world, math such as applied mathematics is not only relevant, it's crucial. Applied mathematics covers the branches that study the physical, biological or sociological world. 

"The goal of applied mathematics is to establish the connections between separate academic fields," wrote Alain Goriely in " Applied Mathematics: A Very Short Introduction " (Oxford University Press, 2018). Modern areas of applied math include mathematical physics, mathematical biology, control theory, aerospace engineering and math finance. Not only does applied math solve problems, but it also discovers new problems or develops new engineering disciplines, Goriely added. The common approach in applied math is to build a mathematical model of a phenomenon, solve the model and develop recommendations for performance improvement.

While not necessarily an opposite to applied mathematics, pure mathematics is driven by abstract problems, rather than real-world problems. Much of the subjects that are pursued by pure mathematicians have their roots in concrete physical problems, but a deeper understanding of these phenomena brings about problems and technicalities. 

These abstract problems and technicalities are what pure mathematics attempts to solve, and these attempts have led to major discoveries for humankind, including the universal Turing machine, theorized by Alan Turing in 1937. This machine, which began as an abstract idea, later laid the groundwork for the development of modern computers. Pure mathematics is abstract and based in theory, and is thus not constrained by the limitations of the physical world.

According to Goriely, "Applied mathematics is to pure mathematics, what pop music is to classical music." Pure and applied are not mutually exclusive, but they are rooted in different areas of math and problem solving. Though the complex math involved in pure and applied mathematics is beyond the understanding of most people, the solutions developed from the processes have affected and improved the lives of many.

Originally published on Live Science .

Sign up for the Live Science daily newsletter now

Get the world’s most fascinating discoveries delivered straight to your inbox.

Avi Wigderson wins $1 million Turing Award for using randomness to change computer science

Pi calculated to 105 trillion digits, smashing world record

Suspected thieves nearly swipe pre-Hispanic artifacts from an archaeological site in Peru

Most Popular

• 2 'Hostilities began in an extremely violent way': How chimp wars taught us murder and cruelty aren't just human traits
• 3 EV batteries could last much longer thanks to new capacitor with 19-times higher energy density that scientists created by mistake
• 4 2,500-year-old Illyrian helmet found in burial mound likely caused 'awe in the enemy'
• 5 Record-shattering Tonga volcanic eruption wasn't triggered by what we thought, new study suggests
• 2 30,000 years of history reveals that hard times boost human societies' resilience
• 3 New invention transforms any smartphone or TV display into a holographic projector
• 4 Roman-era skeletons buried in embrace, on top of a horse, weren't lovers, DNA analysis shows
• 5 Gargantuan sunspot 15-Earths wide shoots powerful X-class flare toward Earth, triggering radio blackouts

Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

• The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

• The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

• The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

• Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

• The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

• Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

• The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

Writing a math essay doesn't have to be a daunting task. By choosing a captivating topic and exploring the various mathematical concepts, you can turn your essay into a fascinating journey of discovery. Whether you're uncovering the beauty of the Fibonacci sequence, exploring the mathematical underpinnings of music, or delving into the game theory revolution, there's a world of possibilities waiting to be explored. So embrace the power of mathematics and let your creativity shine through your words!

Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

Tobi Columb, a math expert, is a dedicated educator and explorer. He is deeply fascinated by the infinite possibilities of mathematics. Tobi's mission is to equip his students with the tools needed to excel in the realm of numbers. He also advocates for the benefits of a gluten-free lifestyle for students and people of all ages. Join Tobi on his transformative journey of mathematical mastery and holistic well-being.

Related Calculators

In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us .

log in to your account

Reset password.

• What is Mathematics?

4th ed. Jan 2024; 3rd ed. May 2023; 2nd ed. Dec 2009; 1st ed. Sep 2004

“It is not philosophy but active experience in mathematics itself that alone can answer the question: `What is Mathematics?'” – Richard Courant & Herbert Robbins, 1941, What is Mathematics?, Oxford University Press) “An adequate presentation of any science cannot consist of detailed information alone, however extensive. It must also provide a proper view of the essential nature of the science as a whole.” – Aleksandrov, 1956, Mathematics: Its Content, Methods, and Meaning

‘What is mathematics?’ Much ink has been spilled over this question, as can be seen from the selection of ten respected responses provided in the footnote 1 , with seven book-length answers, and three written in the current millenium. One might well ask, is there anything new that can be said, that should be said? We’ll start by clarifying what a good answer should look like, and then explore the answer proposed.

The rest of the paper follows the structure below:

1. Criteria for a ‘Satisfactory’ Definition of Mathematics

A satisfactory answer to the question ‘What is Mathematics?’ should, in my view, hold up well against the following three criteria:

Accordingly, it is not just ‘What is Mathematics today ?’ that we should be answering, but the broader scoped and more fundamental question: ‘Can we find a definition that adequately describes mathematics across the more than 5000 years of its written history ?’ In other words ‘Are there common threads that unite the practice of mathematics today with its long history and pre-history of mathematical practice ?’

The difficulty with the above criteria is that 1) our understanding of number, shape, and change stretches back into pre-history, 2) the concepts and applications that have emerged out of this understanding, what we may call mathematical practice, have seen profound evolution over the subsequent periods of mathematical history, in scope, in outlook, and in the organization of this mathematical knowledge, and 3) mathematicians today are investigating an enormous breadth of material, and the range of applications of mathematics is vast (to see this one need only glance at the subject classification of the American Mathematical Society , or the topics contained in the Princeton Companion to Pure Mathematics [Gowers, 2008] and Princeton Companion to Applied Mathematics [Higham, 2015]).

On the principle of requiring more than a current definition, we must therefore reject at the outset any definition of mathematics that is extracted from contemporary pure mathematics. Such definitions are often built around the abstract, deductive presentation of contemporary mathematics, organized axiomatically around a “sets with structure” theme. The problem with these definitions is that they would exclude vibrant, highly productive episodes in the history of mathematics up to the 19th century. They would exclude the pre-modern work of the Old Babylonians on pure mathematics within the scribal schools, the Neo-Babylonians on mathematical astronomy including a precursor of the differential calculus to interpolate the position of Jupiter, Omar Khayyam on the classification of the 19 classes of cubic equations and their complete solution using the intersection of at most two conics, and Fibonacci’s work. They would also exclude the work of Leibniz, Euler, the Bernoullis, Pascal on probability and combinatorics, Fermat on number theory, Lagrange, Laplace on the 3-body problem, and the work of other mathematicians, whose path-breaking advances were was NOT obtained axiomatically, coming after the adoption of symbolism in 1590 (Viete) but before the contemporary reformulation of mathematics axiomatically that began in the mid-1800s and accelerated in the mid-1900s. (For other limited answers also rejected, see Appendix 1.)

In what follows, we will attempt to build a definition of mathematics that covers earlier periods in the history of mathematics and also applies across the enormous range of pure and applied mathematics today. For the purpose of this article, we will consciously set aside additional, not inconsiderable, philosophical questions , such as ‘What is Truth?’, ‘What constitutes Proof?’, and ‘On what foundations does Mathematics build, and with what certainty?’ We’ll discuss a definition of Mathematics that is consistent with mathematical practice across its entire recorded history, from the first written account-keeping in Uruk by the Sumerians (c.3200 BCE), over the ensuing five millenia , to the present era with its its broad scope and modern practice in the present time.

We will proceed iteratively, in two passes. The first pass covers mathematics through to the end of the 18th century; the second pass extends the definition to cover mathematics through to the present.

2. First Pass: A definition covering all mathematics up end of the 18th century

“There are two fundamental sources of ‘bare facts’ for the mathematician, that is, there are some real things out there to which we can confront our understanding. These are, on the one hand the physical world which is the source of geometry, and on the other hand the arithmetic of numbers which is the source of number theory. Any theory concerning either of these subjects can be tested by performing experiments either in the physical world or with numbers.” – Alain Connes (mathematician, Fields Medalist 1982) from Non-Commutative Geometry, 2000

Definition 1: Mathematics is a subject concerned with number, shape, and change.

These first three elements adequately cover pre-modern mathematics and modern mathematics upto the end of the 18th century including the mathematics of ancient Babylonia, Egypt, Greece 2 , China, India, Arabia and Central Asia 3 This includes numeration systems, integer arithmetic, division of property, taxation, and arithmetic of fractions, Euclidean geometry and its offshoots astronomy and trigonometry, solution of algebraic equations lower than degree five, mensuration, the properties of various two and three dimensional figures, the fundamentals of statics and mechanics, dynamics, infinite series, and even the differential and integral calculus, and physics. In particular, this definition covers the works of Euler, Laplace, Taylor, Newton, etc. Most importantly, the mathematics up to the start of the 19th century, was about the real, physical world, and was a language used to investigate its notions. Mathematics for a long time had been considered either part of accounting, a part of geometry, or a branch of natural philosophy. The mathematics up to the end of the 18th century was concerned with the “bare facts” of this physical and arithmetically coherent world:

To cover contemporary mathematics from the 19th century onward requires further elements, the notion of mathematical structure and the study of relations between mathematical objects and mathematical structures. It requires a deeper understanding of how mathematics has and continues to develop.

Let’s start with the word itself. “ Matema” is the ancient Greek term for “that which is learnt,” or “what one gets to know.” So we get the first pillar: Mathematics is a body of knowledge, and area of human understanding. But knowledge and understanding of what?

There are two compelling perspectives: the dialetic perspective of the philosopher-mathematican Imre Lakatos and the anthropology of mathematics perspective of mathematical historian Jens Hoyrup.

3. Mathematics as dialectic and an Anthropology of Mathematics

“In the history of the development of mathematics, three different processes of growth now change places, now run side by side independent of one another, now finally mingle. Plan A is based upon a more particularistic conception of science which divides the total field into a series of mutually separated parts and attempts to develop each part for itself, with a minimum of resources and with all possible avoidance of borrowing from neighboring fields. Its idea is to crystallize out each of the partial fields into a logically closed system. Plan B lays the chief stress upon the organic combination of the partial fields, and upon the stimulation which these exert one upon another. Plan B prefers, therefore, the methods which open an understanding of several fields under a uniform point of point of view. Its ideal is the comprehension of the sum total of mathematical science as a great connected whole. There is still a third Plan C, algorithmic, which, along side of and within the processes of development A and B, often plays an important role as a quasi-independent, onward-driving force, inherent in the formulas, operating apart from the intention and insight of the mathematician, at the time , often indeed in opposition to them.” – Felix Klein, 1908, Elementary Mathematics from an Advanced Perspective, pp.77-85

1. Mathematics as dialectic . In this perspective, mathematics is a great conversation, happening through time, and across traditions, a conversation about ideas, refining, reworking, testing and objecting, ultimately uncovering new and deeper understanding. This is the perspective of [Lakatos, 1976, Proofs and Refutations] and of [Aleksandrov, 1956, A General View of Mathematics]. Lakatos illustrates this with a facinating dialogue between a group of students debating the proof of the Euler characteristic of the polyhedron, artfully compressing into a single dialogue understanding that evolved over the course of approx. 200 years. Aleksandrov focuses on the results of the dialectic, on the steady progress of mathematical concepts: “they are brought into being by a series of successive abstractions and generalizations, each resting on a combination of experience with preceding abstract concepts.[Aleksandrov, 1956, p.17]

There is another aspect of dialectic, a dialectic tension of ideas, concepts, and perspectives, which Felix Klein brings out clearly in the quote displayed at the start of this section [Klein, 1908, 77-85].

In it, Klein describes how in the history of mathematics, the interplay between these three Plans for mathematics moved back and forth, unifying, specializing, unifying again. What drove the impulses between them are both impulses within the individual mathematician and anthropological (social and culture) influences, mainly from the requirement to teach comprehensibly, which ties to a second important perspective.

For an equally fascinating didactic exploration of the nature of mathematics, one is recommended to read Renyi’s “Dialogues on Mathematics” [Renyi, 1967], in which a Socrates discusses the nature of mathematics, an Archimedes discusses the applications of mathematics, and Galileo discusses the ability of mathematics to assist in uncovering the workings of nature. (The first of these 3 dialogues is reproduced in [Hersh, 2006, Ch.1]).

2. The anthropology of mathematics takes the perspective that “the character of mathematical thinking and argument is strongly affected — indeed is almost essentially determined — by the dynamics of the specific social, mostly professional environments in which it is carried” [Hoyrup, 2017], i.e. mathematics is shaped by the interplay between the characteristics of a society and, in particular, the institutions for teaching which influence mathematical thought, research directions, and determine the limits of mathematical practice. The term “anthropology of mathematics” 4 is due to Jens Hoyrup [Hoyrup, 1980, 1994, 2017, 2019] with the catalyzing idea coming from a paper by Judith Grabiner [Grabiner, 1974]. Grabiner showed how the rise of university teaching of mathematics in France and Germany in the 1800s created pressure to make the mathematics of Newton, Leibniz, Euler, and the Bernoullis more easily accessible to students, which in turn accelerated the re-emergence of deductive mathematics. Taking this as a launching point, Jens Hoyrup in ground-breaking researches from 1980 onward, meticulously investigated and detailed the same phenomenon in the major pre-modern mathematical centres (Babylonia, Greece, Islamic Spain to Afghanistan, and mercantile Italy). Hoyrup showed [Hoyrup, 1994] how the temple culture in the early city-states of Sumeria and the rise of scribal schools shaped mathematical development over the entire Sumero-Akkadian-Old Babylonian period (3200-1600 BCE). He shows how in ancient Greece the decentralization of teaching ( sophists were the early itinerant teachers) and the philosophical pre-occupation of aristocratic Greek society, shaped the lens through which the Greeks approached the corpus of 2000 years of Babylonian and Egyptian mathematics and created a new, philosophical, deductive science. [Hoyrup, 2019] With the exception of the Greek experience, a sort of practitioner’s, or utilitarian mathematics (what Hoyrup calls subscientific mathematics ).

Perhaps the strongest case for the anthropology of mathematics (culture as a carrier and the socially specific institutions of teaching) is the fact that the uniquely Greek creation of a deductive reasoning did not persist beyond Alexandria, neither in the era of mathematics in the Islamic period, nor in the pre-Renaissance Abacus period, both of which were dominated by the subscientific, practitioner approach to mathematics which also under-pinned both Babylonian and Egyptian mathematics. [Hoyrup, 1990, 1994, 2003].

Hoyrup’s researches since 1980 have shown that: “Old Babylonian ‘algebra’ and Euclidean ‘geometric algebra’ were connected. The geometric riddles of Arabic misaha treatises as well as al-Khwarizmi’s geometric proofs for the basic al-jabr (algebra) procedures belonged within the same network. The Old Babylonian ‘algebraic’ school discipline built upon original borrowings from the ‘neck riddles’ of a lay surveyor’s environment, and that this environment and its riddles, not the tradition of scholar-scribes, was responsible for the transmission of the inspiration to later times [both to the Greeks as well as to the Islamic scientists].” [Hoyrup, 2003, 9] He points out that there is a continuity of problem classes all the way through to the 1200s CE in Jacopo’s Algebra, and Fibonacci (Leonardo of Pisa)’s Liber Abaci. “These belong to a cluster of problems that are found in ancient and medieval sources from Ireland to India. This cluster of problems that usually go together was apparently carried by the community of merchants travelling along the Silk Road and adopted as ‘recreational problems’ by the literate in many places; it is thus a good example of a body of sub-scientific knowledge influencing school knowledge in many places and an illustration of the principle that it is impossible to trace the ‘source’ for a particular trick or problem in a situation where ‘the ground was wet everywhere’.

Hoyrup’s anthropology of mathematics perspective also addresses the WHEN aspect of mathematics:

[We may say that] transition[s] to [M]athematics occurred [in history] when pre-existent and previously independent mathematical practices and techniques were wielded by specialist practitioners who were organized professionally and linked in a network of communication. Such professional groups fall into two main types: in one type knowledge is transmitted within an apprenticeship-system of ‘learning by doing under supervision’ [sub-scientific mathematics]. The other type involves some kind of school [in which] teaching is separate from actual work. In the former type, those who transmit are actively involved in the practical activities of their trade; they will tend to train exactly what is needed, and the understanding they will try to communicate will be that of practical procedures. [In the latter type, the] school teaching of mathematical skill is bound to a writing system extensive enough to carry a literate culture. [While] teachers in the school type may well have as their aim to impart knowledge for practice, the mathematical understanding that they teach will concentrate on inner connections of the topic, i.e. on mathematical explanations.” – Jens Hoyrup, 2017, Perspectives on an Anthropology of Mathematics

Subscientific mathematical culture and pre-modern rhetorical mathematical practice does not mean that the mathematics was primitive, nor that there was no pure mathematics. “On the contrary, at various points in its history, in particular during the Old Babylonian period, mathematical activity turned toward the pure and systematic, pursued for supra-utilitarian reasons, what we might call scientific, or perhaps better systematic.” [Hoyrup, 1980]. In the past 40 years, examples have been discovered of creative and advanced approaches, including a pre-cursor of our Calculus already being used by the Neo-Babylonians of c.500 BCE to predict the position of Jupiter in the night-sky .

Creative progress in European mathematics before the Renaissance (pre-1300s) was constrained by Aristotelian philosophy on the one hand, Euclidean geometry on the other, and the monastic preservation of classical knowledge as a received but not an indiginous intellectual activity. Even when this influenced finally lifted with the rise of the Abacus schools in Italy (1300s-1400s), and when mathematical practice shifted away from rhetorical to symbolic around Francois Viete (1591) who began the extensive use of symbols in calculation, even then the form of mathematical creative culture was not deductive, even if the results were presented that way (see e.g. Newton’s Principia). Only from the mid-1800s did creative mathematical advancement occur again in the deductive style of the classical Greece, this time in the axiomatic works of Boole (logic), Cayley, Weber, Dedekind, Noether (abstract algebra), Cauchy, Bolzano, Weierstrass, Dedekind, Cantor (analysis), culminating in the the formalist program of Hilbert and the structuralist organization of all of mathematics from the 1950s.

We see the similar effects of culture on contemporary mathematics (New Math in the 1960s and its pushback and subsequent splintering of approaches in the 2000s).

But while the dialectic perspective and the anthropology of mathematics perspective do not tell us WHAT this great conversation is about, they do tell us a lot about HOW it happens. And that HOW provides the missing fifth element to complete our definition.

We now have what we need to answer directly:

4. Second Pass: A definition covering all mathematics, including contemporary mathematics

Definition 2: Mathematics is a body of knowledge, built up over time in a dialetic process, on matters whose origin lies in the three natural phenomena: quantity (number, measurement, scale), space (shape, configuration, arrangement, symmetry, perspective), transformation (change, variation), shaped by the prevailing social culture, and in turn influencing it (often profoundly) through its diverse, and often ingenious, applications.

Deepening understanding of the three natural phenomena lead to the development of a chain of evolving conceptual abstractions, to greater generalization, and to correspondingly broader areas of investigation. Observation of relations (association, comparison, similarity, equivalence) between diverse mathematical phenomenona has created a fifth, humanistic area of mathematical activity: the rational structuring of its accumulated body knowledge through the development of axiomatic (deductively structured) mathematical systems that generalize and extend empirical concepts, introduce fundamentally new theoretical concepts, and examine the laws that govern their structure, properties, and the relationships between them.

Let’s look in turn at the elements.

The first three are elaborated versions of what we had before:

The fifth element brings it all together:

Also worth a further comment is the remark on the influence of mathematics on society through applications. I rather hope that this statement is not controversial. The applications of mathematics are everywhere, amplified through science and engineering. To take but one example, the development of operations research to optimize military supply chains and bombing patterns against U-boats in WWII.

5. Testing the expanded definition against contemporary mathematical practice

We have seen how the first three elements were sufficient to cover mathematics up to the start of the 19th century.

The last two elements (relation and structure) are INTROSPECTIVE views. They are about mathematicians looking at the mathematics that they know and asking how/why they know it and how best to capture/catalogue that knowledge. With the addition of “structure”, we cover the axiomatic mathematics that returned again to the 19th century universities in France and Germany. We are able to cover the development of abstraction in the notions of algebra and number and an abstract algebraic mathematical logic in the early part of the 1800s, led by the English mathematicians George Peacock, George Boole, Augustus De Morgan, and then William Rowan Hamilton, Cayley, and then picked up in Germany by Weber, Noether, Klein, Lie, and others developing abstract algebra, and discovering its power to unlock additional questions.

We are also able to cover the transition from the intuitive mathematics of the 18th-century to the formalist mathematics of the 19th- and 20th-centuries, including the development of set theory as the foundation of the real numbers and analysis (Cauchy, Weierstrass, Dedekind), and the higher cardinalities (Cantor), the rise of point-set topology, functional analysis, linear algebra (geometry in n dimensions), measure theory and the axiomatic foundations of probability (Kolmogorov), and much more.

Is the Fifth Element really new? What is fascinating in this story is that when one takes Hoyrup’s viewpoint of “anthropology of mathematics”, then this fifth element, introspection on its content, has been with mathematics throughout its known/recorded history, influencing how knowledge is organized, arranged, simplified, rigorized. Hoyrup’s research shows that it appears in every culture and context through the pre-modern mathematical period. It appears differently depending on the social and institutional context of the time (this is the anthropological perspective), but it appears nonetheless. So it is not the case that the current modern axiomatic style is somehow a natural endpoint in a linear mathematical development. On the contrary, there have been periods of more or less rigour, more or less scientific vs. subscientific (utilitarian) styles. Indeed the two high points of the rigorous axiomatic approach have been the Greek period 500 BCE-200 BCE and from the 19th-century onward. Both have seen the primacy of developing logically structured (axiomatic) mathematical systems that systematize, generalize and extend empirical concepts.

Why is this important? Because it is introspection, the fifth element, I claim, that gives mathematics its essential restlessness, its continual development, resifting, refinement of mathematical knowledge, and the exploration of other subjects using a mathematical lens. It has always been the source of advancement in every age of mathematics, and this continues in contemporary mathematics. Because it is the element that introduces new theoretical concepts, and simplifies and clarifies existing concepts by examining the laws that govern their structure, finding deep structural analogies between superficially dissimilar context.

The systematization of mathematics as a science has inevitably led to deeper understanding of the three natural phenomena. The breadth of modern mathematics is organized within these mathematical systems, and it is from within these systems that they find application in areas beyond the historical core. Where conceptual abstractions have evolved, and effective generalizations have been found, these have in turn led to correspondingly broader areas of investigation. For example, modern mathematical physics (gauge theories, string theories, quantum field theories), has advanced materially through the contributions of abstract algebra, continuous group theory, and the theory of representations. So also have several other fields, for example computer science (computability, recursion, mathematical linguistics), biology (chaos theory, self-organizing systems), economics, finance , linguistics and fuzzy logic , and others.

6. A Deeper Look: Three Facets of Mathematics

To properly understand Mathematics in a way that is consistent with its history, evolution, and its many diverse applications, as well as with its contemporary, abstract, and highly specialized state, we need to go beyond the definition. Here it is helpful to identify three co-existing facets of Mathematics:

These facets of Mathematics explain both the historical development, maturity, and modern separation between theoretical and applied considerations. Let’s look at each in turn.

6.1. Mathematics as an Empirical Science  

Mathematics originates out of science, i.e. out of human interest in the surrounding world, its careful observation, and the empirical verification of mathematical fact. In fact, the world and its patterns are consistently present in the inspiration of all mathematical studies. Even the abstract, abstruse, and seemingly detached topics of advanced higher mathematics are generalizations of patterns observed in the layers of less abstract mathematics, that are themselves an attempt to capture patterns observed in the real world itself.

Thus, the essence of a mathematical concept can always be related back to an original proto-concept that has its roots in empirical observations and the patterns arising out of these.

6.2. Mathematics as a Modeling Art  

Mathematics as a modeling art involves an effort to develop, maintain, and perfect models of perceived or envisioned reality. Mathematics has always involved, and continues to involve, the exploration, explanation and modeling of phenomena.

At the root of this facet of mathematics is the intimate relationship between the physical world and the world of mathematical ideas. Most of the major laws of mathematics are modeled on actual physical occurrences, suitably abstracted. Thus, the origin of most of mathematics is a model of something real that has been experienced. Indeed, typical applied mathematics proceeds from a physical context to the context of a mathematical model, performs computations and analysis using mathematical reasoning within the domain of this model, and then finally brings the result back to the physical context for interpretation.

The success of Mathematics in keeping ever-improving mathematical models of many and various phenomena, and the fact that the methods behind these models are often applicable in widely different areas and contexts, often lead Mathematics to be viewed enthusiastically (though incorrectly) as the key to the knowledge of all things. 5

6.3. Mathematics as an Axiomatic Arrangement of Knowledge  

The exploration of the logical structure of mathematical knowledge is a relatively recent development, beginning with the ancient Greeks circa 800 BCE. Comparatively, this phenomenon has occupied less than 3 millenia, or less than 10% of the documented history of mathematical knowledge of humankind (30,000 years).

Rapid progress in understanding the logical structure of mathematics occurred since the 1800s CE,and has led to the flowering of a wide variety of modern mathematical systems and theories whose areas of interests and domains of application go far beyond the historical core of mathematics. To put it in context, the past 200 years of mathematics since 1800 CE is less than 5% of the documented history of mathematics.

Today, the vast scope of modern mathematical knowledge is organized within structured mathematical systems from within which it finds wide application.

Mathematical structures distill informal mathematical knowledge, identify the important concepts out of the body of informal mathematical knowledge, and provide streamlined logical models to underpin these areas.

7. Mathematics “from the inside”

I will close with a few observations of mathematics ‘from the inside’, i.e. mathematicians writing about mathematics :

“Mathematics is the backbone of modern science and a remarkably efficient source of new concepts and tools to understand the ‘reality’ in which we participate. The new concepts themselves are the result of a long process of ‘distillation’ in the alembic 6 of human thought.” – Alain Connes, (mathematician, Fields Medalist 1982), from Advice to the Beginner, 2006

The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. Any one or another of these aspects of mathematics can be at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the “concrete” ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual “reality”. In brief, the flight into abstract generality must start from and return again to the concrete and specific. — Richard Courant (mathematician), from Mathematics in the Modern World, Scientific American Vol.211 No.3, pp.41-49, 1964

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them. — Heinrich Hertz (physicist), from Men of Mathematics, Vol 2, p.16, 1937

Very often in mathematics the crucial problem is to recognize and to discover what are the relevant concepts; once this is accomplished the job may be more than half done. — I.N. Herstein (mathematician), Topics in Algebra

It looked absolutely impossible. But it so happens that you go on worrying away at a problem in science and it seems to get tired, and lies down and lets you catch it. — William Lawrence Bragg \footnote{Bragg, at age 24, won the Nobel Prize for the invention of x-ray crystallography. He remains the youngest person ever to receive the Nobel Prize.}

(For more quotes on Mathematics and the process of doing mathematics , see here.)

8. Continue Reading

Appendix 1: answers deemed unsatisfactory to the criteria outlined above.

Several authors attempt to answer the title question by providing a survey of mathematics but this in my view is as unhelpful as stating that ‘mathematics is the sum of its contents’. An author should not have to take an intelligent reader through 300+ pages of technical material before providing an clear answer. The caveat of course is that that one must have actively experienced enough mathematics (and for our purposes, enough exposure to its history) to find the answer satisfying.

A second frequently given, but also limiting answer, is that mathematics is the exploration through deductive reasoning of mathematical structures whose properties are abstracted into axioms from objects of practical experience. Simplified, this boils down to mathematics as the study of necessary deductions, which is problematic. It covers much of contemporary mathematics which is built around abstraction and deduction and organized into areas defined by the structures they study (groups, rings, fields, lattices, manifolds, functions, sets, logic structural presentation of contemporary mathematics). While it roots mathematics in practical experience, it denies that there can be mathematics that is not deductive, which is problematic given the long periods of history (before classical Greece and between classical Greece and the 1800s) when much mathematical knowledge was discovered without the use of a formal deductive style of arrangement of that knowledge. [See Hoyrup, 1980, 1994, 2019] To use a metaphor, if the deductive structure of mathematics is like varnish on wood, then we cannot accept that it is only the varnished variety of wood that is mathematics, but not the wood itself. The appeal of varnished wood is aesthetic, its longevity perhaps more assured, but the essential element is the wood, much of which is not easy to discover (as anyone who has tried will find). Hoyrup identifies the separate traditions as “scientific” and “sub-scientific”. Of the sub-scientific tradition, it is rooted in problem-solving and historically persisted even when the scientific tradition waned, as it did after Old Babylonian high point and after classical Greece. (Hoyrup, 1980, 1994, 2001), (Grabiner, 1976). In our view, both are entitled to be called mathematics.

9. References

Group 2: Expository content that supports the views expressed in this article

Group 3: Surveys of Mathematics

Group 4: Additional Recommended Reading

• Responses from 1941 to 2017: (Courant, Robbins, 1941), (Alexandrov, Kolmogorov, Lavrentiv, 1963), (Renyi, 1967), (Halmos, 1973), (Lakatos, 1976), (Davis, Hersh, 1981), (MacLane, 1986), (Hersh, 2006), (Zeilberger, 2017), (Hoyrup, 2017), 7 books, 3 articles. ↩
• The Greeks had already encountered the paradoxes of the real numbers, though not their resolution, and were already using the method of exhaustion and converging upper and lower bounds, a precursor to the methods of the integral calculus, apart from a rigorous passing to the limit. ↩
• The Arab and Central Asians and Indians advanced algebra to an abstract science, had resolved the solution of algebraic equations including most instances of the general cubic, had developed expansion by Taylor series and precursors of the calculus, had developed the trigonometric identities, applied algebra and trigonometry to astronomical problems, and had fully developed systems of computation for interest rates, taxation, and other numerical calculations using decimal digits including zero, and a place decimal system. ↩
• “(paraphrased) The actual content of mathematics, the changing mode of mathematical thought, the functions mathematics can fulfill, and the pursuit and development of mathematics (are) conditioned by the wider social and cultural context.” “Individual contributors are seen as members of one or perhaps several intersecting subgroups within a general cultural matrix.” “The character and substance of scientific thinking and the aims pursued by the sciences, as well as insitutions, ideologies, and general social need, change when mathematical development is considered over longer periods or when comparing between different cultures, which requires dialectical synthesis.” The content of mathematics (and) its interaction with the sociocultural setting (Hoyrup, 1994, p.xii-xiii). ↩

“a thorough study of all things, insight into all that exists, knowledge of all obscure secrets.” ( Bur , p.38.)

• an “alembic” is an obsolete apparatus used by alchemists for the distillation of liquids – it’s a suggestive metaphor. ↩

3 comments to What is Mathematics?

Thanks, I was interested in your perspective about what mathematics is and the nature of mathematics.

I saw this really good post today.

[…] to strengthen the understanding of technically important developments in a way that is memorable, i.e. that can be understood fully and reproduced at will. (For the first three objectives, see What is Mathematics?) […]

Leave a Reply

You can use these HTML tags

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

   Name (required)

   Email (will not be published) (required)

   Website

Save my name, email, and website in this browser for the next time I comment.

Your comments are valued! (Please indulge the gatekeeping question as spam-bots cannot (yet) do simple arithmetic...) - required In order to pass the CAPTCHA please enable JavaScript

Upload attachment   (Allowed file types: jpg, gif, png, pdf, doc, docx, ppt, pptx, xls, xlsx, avi , maximum file size: 6MB.

Optionally add an image (JPEG only)

  Notify me when new comments are added to this post.

Stats : 1,072,680 article views since 2010 (May '24 update)

Dear Readers:

Welcome to the conversation!  We publish long-form pieces as well as a curated collection of spotlighted articles covering a broader range of topics.   Notifications for new long-form articles are through the feeds (you can join below).  We love hearing from you.  Feel free to leave your thoughts in comments, or use the contact information to reach us!

Reading List…

Looking for the best long-form articles on this site? Below is a curated list by the main topics covered.

Mathematics-History & Philosophy

• Prehistoric Origins of Mathematics
• The Mathematics of Uruk & Susa (3500-3000 BCE)
• How Algebra Became Abstract: George Peacock & the Birth of Modern Algebra (England, 1830)
• The Rise of Mathematical Logic: from Laws of Thoughts to Foundations for Mathematics
• Mathematical Finance and The Rise of the Modern Financial Marketplace
• A Course in the Philosophy and Foundations of Mathematics
• The Development of Mathematics
• Catalysts in the Development of Mathematics
• Characteristics of Modern Mathematics

Electronic & Software Engineering

• Electronics in the Junior School - Gateway to Technology
• Coding for Pre-Schoolers - A Turtle Logo in Forth
• Experimenting with Microcontrollers - an Arduino development kit for under £12
• Making Sensors Talk for under £5, and Voice Controlled Hardware
• Forth, Lisp, & Ruby: languages that make it easy to write your own domain specific language (DSL)
• Programming Microcontrollers: Low Power, Small Footprints & Fast Prototypes
• Building a 13-key pure analog electronic piano.
• TinyPhoto: Embedded Graphics and Low-Fat Computing
• Computing / Software Toolkits
• Assembly Language programming ( Part 1 | Part 2 | Part 3 )
• Bare Bones Programming: The C Language

Pure & Applied Mathematics

• Fuzzy Classifiers & Quantile Statistics Techniques in Continuous Data Monitoring
• LOGIC in a Nutshell: Theory & Applications (including a FORTH simulator and digital circuit design)
• Finite Summation of Integer Powers: ( Part 1 | Part 2 | Part 3 )
• The Mathematics of Duelling
• A Radar Tracking Approach to Data Mining
• Analysis of Visitor Statistics: Data Mining in-the-Small
• Why Zero Raised to the Zero Power IS One

Technology: Sensors & Intelligent Systems

• Knowledge Engineering & the Emerging Technologies of the Next Decade
• Sensors and Systems
• Unmanned Autonomous Systems & Networks of Sensors
• The Advance of Marine Micro-ROVs

Math Education

• Teaching Enriched Mathematics, Part 1
• Teaching Enriched Mathematics, Part 2: Levelling Student Success Factors
• Logic, Proof, and Professional Communication: five reflections
• Good mathematical technique and the case for mathematical insight

Browse…

Explore….

Mathematics: Discovered or Created? Essay (Critical Writing)

Mathematics is a branch of science that has had far-reaching impacts on many spheres of life. Through mathematics, man has made remarkable advances in technology and other fields of life. Mathematics also provides us with a logical order for describing the various prototypes and structures that comprise nature. Mathematics is also responsible for some of the greatest breakthroughs that have been made by humanity so far.

For instance, mathematics has played a hand in humanity’s foray into the cosmos and it has been responsible for the modern internet advancements. Albert Einstein once asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” (Ernest 9). This question is part of a big debate on whether math is a product of human creation or human discovery. Consequently, if math is part of human discovery what are some of the laws and notions that are subject to this discovery? This paper explores the issues surrounding the debate on whether math is a product of human creation or human discovery.

First, it is important to note that math acts like the outline to our universe. Many mathematicians agree that the universe is governed by a singular order that is defined using mathematical principles. Consequently, even if the universe ceased to exist, all mathematical principles would still be true. Therefore, like other aspects of human nature, mathematics is part of human discovery. Furthermore, there are several mathematical principles that are yet to be discovered. When these principles are discovered, they “will then assist us in building models that will give us predictive power and understanding of the physical phenomena we seek to control” (Ernest 10). Therefore, math is a natural concept that is to be discovered and used by humanity. This argument is common among lovers of mathematics.

Another viable explanation of the existence of mathematics is that it is merely part of the human creation. The argument about math being part of the intricate web of nature could be easily refuted by the view that human beings invented mathematics as a tool that could aid in the description of the physical world. Therefore, mathematics is only popular among human beings because it suits their needs when they are exploring the world.

It is also true that some mathematical concepts have been changed and altered for them to be palatable to human beings. Furthermore, if the universe ceased to exist, there would be no need for mathematics and it would not exist. Mathematics has been made possible by geography, astronomy, and physics among other areas of universal studies. Mathematics exists solely to satisfy the needs of studying and understanding the universe but it is not part of these studies. Therefore, mathematics is not something that is discovered but it is a human creation.

These two arguments form the basis of our understanding of the institution of mathematics. However, in my understanding, mathematics is a human creation. The argument for mathematics being part of human discovery is far-fetched and fanatical. For instance, mathematics only describes certain variables of the physical universe. There are several other factors of the universe that cannot be defined or explained by mathematical concepts. Therefore, the argument about mathematics being part of human discovery can be nullified by the idea that there are discoveries that are outside of the mathematical realm. In my view, when discoveries about the physical world are made, man proceeds to create mathematical concepts that can help him analyze and explain these new discoveries.

Works Cited

Ernest, Paul. “Is mathematics discovered or invented.” Philosophy of Mathematics Education Journal 12.1 (2009): 9-13. Print.

• Chicago (A-D)
• Chicago (N-B)

IvyPanda. (2023, November 1). Mathematics: Discovered or Created? https://ivypanda.com/essays/mathematics-discovered-or-created/

"Mathematics: Discovered or Created?" IvyPanda , 1 Nov. 2023, ivypanda.com/essays/mathematics-discovered-or-created/.

IvyPanda . (2023) 'Mathematics: Discovered or Created'. 1 November.

IvyPanda . 2023. "Mathematics: Discovered or Created?" November 1, 2023. https://ivypanda.com/essays/mathematics-discovered-or-created/.

1. IvyPanda . "Mathematics: Discovered or Created?" November 1, 2023. https://ivypanda.com/essays/mathematics-discovered-or-created/.

Bibliography

IvyPanda . "Mathematics: Discovered or Created?" November 1, 2023. https://ivypanda.com/essays/mathematics-discovered-or-created/.

• Albert Einstein’s Contributions to Science
• Albert Einstein, His Life and Career
• Women Mathematicians: Maria Agnesi, Sophie Germain
• Statistical Thinking and Collection of Information
• Quantitative Approach to Research
• Statistics: Chi-Square Test and Regression Analysis
• P-Value Definition and Role
• Statistical Process in Data Analysis

Extended Essay: Group 5: Mathematics

• General Timeline
• Group 1: English Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts
• Interdisciplinary essays
• Six sub-categories for WSEE
• IB Interdisciplinary EE Assessment Guide
• Brainstorming
• Pre-Writing
• Research Techniques
• The Research Question
• Paraphrasing, Summarising and Quotations
• Writing an EE Introduction
• Writing the main body of your EE
• Writing your EE Conclusion
• Sources: Finding, Organising and Evaluating Them
• Conducting Interviews and Surveys
• Citing and Referencing
• Check-in Sessions
• First Formal Reflection
• Second Formal Reflection
• Final Reflection (Viva Voce)
• Researcher's Reflection Space (RRS) Examples
• Information for Supervisors
• How is the EE Graded?
• EE Online Resources
• Stavanger Public Library
• Exemplar Essays
• Extended Essay Presentations
• ISS High School Academic Honesty Policy

Mathematics

An extended essay (EE) in mathematics is intended for students who are writing on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Essays in this group are divided into six categories:

• the applicability of mathematics to solve both real and abstract problems
• the beauty of mathematics—eg geometry or fractal theory
• the elegance of mathematics in the proving of theorems—eg number theory
• the history of mathematics: the origin and subsequent development of a branch of mathematics over a period of time, measured in tens, hundreds or thousands of years
• the effect of technology on mathematics:
• in forging links between different branches of mathematics,
• or in bringing about a new branch of mathematics, or causing a particular branch to flourish.

These are just some of the many different ways that mathematics can be enjoyable or useful, or, as in many cases, both.

For an Introduction in a Mathematics EE look HERE . 

Choice of topic

The EE may be written on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Students may choose mathematical topics from fields such as engineering, the sciences or the social sciences, as well as from mathematics itself.

Statistical analyses of experimental results taken from other subject areas are also acceptable, provided that they focus on the modeling process and discuss the limitations of the results; such essays should not include extensive non-mathematical detail.

A topic selected from the history of mathematics may also be appropriate, provided that a clear line of mathematical development is demonstrated. Concentration on the lives of, or personal rivalries between, mathematicians would be irrelevant and would not score highly on the assessment criteria.

It should be noted that the assessment criteria give credit for the nature of the investigation and for the extent that reasoned arguments are applied to an appropriate research question.

Students should avoid choosing a topic that gives rise to a trivial research question or one that is not sufficiently focused to allow appropriate treatment within the requirements of the EE.

Students will normally be expected either to extend their knowledge beyond that encountered in the Diploma Programme mathematics course they are studying or to apply techniques used in their mathematics course to modeling in an appropriately chosen topic.

However, it is very important to remember that it is an essay that is being written, not a research paper for a journal of advanced mathematics, and no result, however impressive, should be quoted without evidence of the student’s real understanding of it.

Example and Treatment of Topic

Examples of topics

These examples are just for guidance. Students must ensure their choice of topic is focused (left-hand column) rather than broad (right-hand column

Treatment of the topic

Whatever the title of the EE, students must apply good mathematical practice that is relevant to the

chosen topic, including:

• data analysed using appropriate techniques

• arguments correctly reasoned

• situations modeled using correct methodology

• problems clearly stated and techniques at the correct level of sophistication applied to their solution.

Research methods

Students must be advised that mathematical research is a long-term and open-ended exploration of a set of related mathematical problems that are based on personal observations. 

The answers to these problems connect to and build upon each other over time.

Students’ research should be guided by analysis of primary and secondary sources.

A primary source for research in mathematics involves:

• data-gathering

• visualization

• abstraction

• conjecturing

• proof.

A secondary source of research refers to a comprehensive review of scholarly work, including books, journal articles or essays in an edited collection.

A literature review for mathematics might not be as extensive as in other subjects, but students are expected to demonstrate their knowledge and understanding of the mathematics they are using in the context of the broader discipline, for example how the mathematics they are using has been applied before, or in a different area to the one they are investigating.

Writing the essay

Throughout the EE students should communicate mathematically:

• describing their way of thinking

• writing definitions and conjectures

• using symbols, theorems, graphs and diagrams

• justifying their conclusions.

There must be sufficient explanation and commentary throughout the essay to ensure that the reader does not lose sight of its purpose in a mass of mathematical symbols, formulae and analysis.

The unique disciplines of mathematics must be respected throughout. Relevant graphs and diagrams are often important and should be incorporated in the body of the essay, not relegated to an appendix.

However, lengthy printouts, tables of results and computer programs should not be allowed to interrupt the development of the essay, and should appear separately as footnotes or in an appendix. Proofs of key results may be included, but proofs of standard results should be either omitted or, if they illustrate an important point, included in an appendix.

Examples of topics, research questions and suggested approaches

Once students have identified their topic and written their research question, they can decide how to

research their answer. They may find it helpful to write a statement outlining their broad approach. These

examples are for guidance only.

An important note on “double-dipping”

Students must ensure that their EE does not duplicate other work they are submitting for the Diploma Programme. For example, students are not permitted to repeat any of the mathematics in their IA in their EE, or vice versa.

The mathematics EE and internal assessment

An EE in mathematics is not an extension of the internal assessment (IA) task. Students must ensure that they understand the differences between the two.

• The EE is a more substantial piece of work that requires formal research
• The IA is an exploration of an idea in mathematics.

It is not appropriate for a student to choose the same topic for an EE as the IA. There would be too much danger of duplication and it must therefore be discouraged.

• << Previous: Group 4: Sciences
• Next: Group 6: The Arts >>
• Last Updated: May 2, 2024 8:11 AM
• URL: https://isstavanger.libguides.com/c.php?g=695888

Find anything you save across the site in your account

What Is Mathematics?

By Alec Wilkinson

A while ago, I got interested in mathematics, mostly because I had done so poorly at it in school. I’m being coy. I didn’t do poorly; I pretty much failed. I only passed by cheating. Anyway, I bought a copy of “ Algebra for Dummies ” to see whether I could improve, but it turned out that I didn’t like algebra as an adult any more than I had as a boy. Even so, I was determined to see whether I could understand why I hadn’t been able to learn it. Doing teen-age mathematics as an older person, though, was harder than I had expected it to be, and I’m not sure how long I could have kept going if I hadn’t become aware, mostly from reading books about mathematics and talking to mathematicians, that outside my overheated room at the Algebra Hotel, mathematics had a grandeur and reach that I hadn’t even suspected. I then spent more of my time trying to learn what I could of its qualities.

Mathematicians know what mathematics is but have difficulty saying it. I have heard: Mathematics is the craft of creating new knowledge from old, using deductive logic and abstraction. The theory of formal patterns. Mathematics is the study of quantity. A discipline that includes the natural numbers and plane and solid geometry. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. Statements related by very strict rules of deduction. A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A science involving things you can’t see, whose presence is confined to the imagination. A proto-text whose existence is only postulated. A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined. The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. The longest continuous human thought. Laboriously constructed intuition. The thing that scientific ideas, as they grow toward perfection, become. An ideal reality. A story that has been written for thousands of years, is always being added to, and might never be finished. The largest coherent artifact that’s been built by civilization. Only a formal game. What mathematicians do, the way musicians do music.

Bertrand Russell said that mathematics, by its nature as an explorative art, is “the subject in which we never know what we are talking about, nor whether what we are saying is true.” Darwin tried studying mathematics with a tutor when he was nineteen and hated it, mainly from “not being able to see any meaning in the early steps in algebra.” He is supposed to have concluded that “a mathematician is a blind man in a dark room looking for a black cat which isn’t there.” In “ Alice’s Adventures in Wonderland ,” Lewis Carroll has the Mock Turtle say that the four operations of arithmetic (addition, subtraction, multiplication, and division) are ambition, distraction, uglification, and derision. A complicating circumstance is that mathematics, especially in its higher ranges, is hard to understand. It begins as simple, shared speech (everyone can count) and becomes specialized into dialects so arcane that some of them are spoken by only a few hundred people in the world. Other fields haven’t even been discovered yet.

No scripture is as old as mathematics is. All the other sciences are younger, most by thousands of years. More than history, mathematics is the record that humanity is keeping of itself. History can be revised or manipulated or erased or lost. Mathematics is permanent. A² + B² = C² was true before Pythagoras had his name attached to it, and will be true when the sun goes out and no one is left to think of it. It is true for any alien life that might think of it, and true whether they think of it or not. It cannot be changed. So long as there is a world with a horizontal and a vertical axis, a sky and a horizon, it is inviolable and as true as anything that can be thought.

Mathematicians live within a world that is essentially certain. The rest of us, even other scientists, live within one where what represents certainty is so-far-as-we-can-tell-this-result-occurs-almost-all-of-the-time. Because of mathematics’ insistence on proof, it can tell us, within the range of what it knows, what happens time after time.

As precise as mathematics is, it is also the most explicit language we have for the description of mysteries. Being the language of physics, it describes actual mysteries—things we can’t see clearly in the natural world but suspect are true and later confirm—and imaginary mysteries, things that exist only in the minds of mathematicians. A question is where these abstract mysteries exist, what their home range is. Some people would say that they reside in the human mind, that only the human mind has the capacity to conceive of what are called mathematical objects, meaning numbers and equations and formulas and so on—the whole glossary and apparatus of mathematics—and to bring these into being, and that such things arrive as they do because of the way our minds are structured. We are led to examine the world in a way that agrees with the tools that we have for examining it. (We see colors as we do, for example, because of how our brains are structured to receive the reflection of light from surfaces.) This is a minority view, held mainly by neuroscientists and a certain number of mathematicians disinclined toward speculation. The more widely held view is that no one knows where math resides. There is no mathematician/naturalist who can point somewhere and say, “That is where math comes from” or “Mathematics lives over there,” say, while maybe gesturing toward magnetic north and the Arctic, which I think would suit such a contrary and coldly specifying discipline.

The belief that mathematics exists somewhere else than within us, that it is discovered more than created, is called Platonism, after Plato’s belief in a non-spatiotemporal realm that is the region of the perfect forms of which the objects on earth are imperfect reproductions. By definition, the non-spatiotemporal realm is outside time and space. It is not the creation of any deity; it simply is. To say that it is eternal or that it has always existed is to make a temporal remark, which does not apply. It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is. The physical world is temporal and declines; the non-spatiotemporal one is ideal and doesn’t.

A third point of view, historically and presently, for a small but not inconsequential number of mathematicians, is that the home of mathematics is in the mind of a higher being and that mathematicians are somehow engaged with Their thoughts. Georg Cantor, the creator of set theory—which in my childhood was taught as a part of the “new math”—said, “The highest perfection of God lies in the ability to create an infinite set, and its immense goodness leads Him to create it.” And the wildly inventive and self-taught mathematician Srinivasa Ramanujan, about whom the movie “ The Man Who Knew Infinity ” was made, in 2015, said, “An equation for me has no meaning unless it expresses a thought of God.”

In Book 7 of the Republic, Plato has Socrates say that mathematicians are people who dream that they are awake. I partly understand this, and I partly don’t.

By signing up, you agree to our User Agreement and Privacy Policy & Cookie Statement . This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

By Paul Grimstad

By Dan Rockmore

By Manvir Singh

Essays About Math: Top 10 Examples and Writing Prompts 

Love it or hate it, an understanding of math is said to be crucial to success. So, if you are writing essays about math, read our top essay examples.  

Mathematics is the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them . It can be used for a variety of purposes, from calculating a business’s profit to estimating the mass of a black hole. However, it can be considered “controversial” to an extent.

Most students adore math or regard it as their least favorite. No other core subject has the same infamy as math for generating passionate reactions both for and against it. It has applications in every field, whether basic operations or complex calculus problems. Knowing the basics of math is necessary to do any work properly. 

If you are writing essays about Math, we have compiled some essay examples for you to get started. 

1. Mathematics: Problem Solving and Ideal Math Classroom by Darlene Gregory 

2. math essay by prasanna, 3. short essay on the importance of mathematics by jay prakash.

• 4.  Math Anxiety by Elias Wong

5. Why Math Isn’t as Useless as We Think by Murtaza Ali

1. mathematics – do you love or hate it, 2. why do many people despise math, 3. how does math prepare you for the future, 4. is mathematics an essential skill, 5. mathematics in the modern world.

“The trait of the teacher that is being strict is we know that will really help the students to change. But it will give a stress and pressure to students and that is one of the causes why students begin to dislike math. As a student I want a teacher that is not so much strict and giving considerations to his students. A teacher that is not giving loads of things to do and must know how to understand the reasons of his students.”

Gregory discusses the reasons for most students’ hatred of math and how teachers handle the subject in class. She says that math teachers do not explain the topics well, give too much work, and demand nothing less than perfection. To her, the ideal math class would involve teachers being more considerate and giving less work. 

You might also be interested in our ordinal number explainer.

“Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.”

In her essay, Prasanna gives readers a basic idea of what math is and its importance. She additionally lists down some of the many uses of mathematics in different career paths, namely managing finances, cooking, home modeling and construction, and traveling. Math may seem “useless” and “annoying” to many, but the essay gives readers a clear message: we need math to succeed. 

“In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ‘Mathematics is a Science of all Sciences and art of all arts.’”

As its title suggests, Prakash’s essay briefly explains why math is vital to human nature. As the world continues to advance and modernize, society emphasizes sciences such as medicine, chemistry, and physics. All sciences employ math; it cannot be studied without math. It also helps us better our reasoning skills and maximizes the human mind. It is not only necessary but beneficial to our everyday lives. 

4.   Math Anxiety by Elias Wong

“Math anxiety affects different not only students but also people in different ways. It’s important to be familiar with the thoughts you have about yourself and the situation when you encounter math. If you are aware of unrealistic or irrational thoughts you can work to replace those thoughts with more positive and realistic ones.”

Wong writes about the phenomenon known as “math anxiety.” This term is used to describe many people’s hatred or fear of math- they feel that they are incapable of doing it. This anxiety is caused mainly by students’ negative experiences in math class, which makes them believe they cannot do well. Wong explains that some people have brains geared towards math and others do not, but this should not stop people from trying to overcome their math anxiety. Through review and practice of basic mathematical skills, students can overcome them and even excel at math. 

“We see that math is not an obscure subject reserved for some pretentious intellectual nobility. Though we may not be aware of it, mathematics is embedded into many different aspects of our lives and our world — and by understanding it deeply, we may just gain a greater understanding of ourselves.”

Similar to some of the previous essays, Ali’s essay explains the importance of math. Interestingly, he tells a story of the life of a person name Kyle. He goes through the typical stages of life and enjoys typical human hobbies, including Rubik’s cube solving. Throughout this “Kyle’s” entire life, he performed the role of a mathematician in various ways. Ali explains that math is much more prevalent in our lives than we think, and by understanding it, we can better understand ourselves. 

Writing Prompts on Essays about Math

Math is a controversial subject that many people either passionately adore or despise. In this essay, reflect on your feelings towards math, and state your position on the topic. Then, give insights and reasons as to why you feel this way. Perhaps this subject comes easily to you, or perhaps it’s a subject that you find pretty challenging. For an insightful and compelling essay, you can include personal anecdotes to relate to your argument. 

It is well-known that many people despise math. In this essay, discuss why so many people do not enjoy maths and struggle with this subject in school. For a compelling essay, gather interview data and statistics to support your arguments. You could include different sections correlating to why people do not enjoy this subject.

In this essay, begin by reading articles and essays about the importance of studying math. Then, write about the different ways that having proficient math skills can help you later in life. Next, use real-life examples of where maths is necessary, such as banking, shopping, planning holidays, and more! For an engaging essay, use some anecdotes from your experiences of using math in your daily life.

Many people have said that math is essential for the future and that you shouldn’t take a math class for granted. However, many also say that only a basic understanding of math is essential; the rest depends on one’s career. Is it essential to learn calculus and trigonometry? Choose your position and back up your claim with evidence. 

Prasanna’s essay lists down just a few applications math has in our daily lives. For this essay, you can choose any activity, whether running, painting, or playing video games, and explain how math is used there. Then, write about mathematical concepts related to your chosen activity and explain how they are used. Finally, be sure to link it back to the importance of math, as this is essentially the topic around which your essay is based. 

If you are interested in learning more, check out our essay writing tips !

For help with your essays, check out our round-up of the best essay checkers

Martin is an avid writer specializing in editing and proofreading. He also enjoys literary analysis and writing about food and travel.

View all posts

• Math Article
• What is Mathematics

What is Mathematics?

Mathematics is one of the most important subjects. Mathematics is a subject of numbers, shapes, data, measurements and also logical activities. It has a huge scope in every field of our life, such as medicine, engineering, finance, natural science, economics, etc. We are all surrounded by a mathematical world.

The concepts, theories and formulas that we learn in Maths books have huge applications in real-life. To find the solutions for various problems we need to learn the formulas and concepts. Therefore, it is important to learn this subject to understand its various applications and significance.

What Is The Definition of Mathematics?

Mathematics simply means to learn or to study or gain knowledge. The theories and concepts given in mathematics help us understand and solve various types of problems in academic as well as in real life situations.

Mathematics is a subject of logic. Learning mathematics will help students to grow their problem-solving and logical reasoning skills. Solving mathematical problems is one of the best brain exercises.

Basic Mathematics

The fundamentals of mathematics begin with arithmetic operations such as addition, subtraction, multiplication and division. These are the basics that every student learns in their elementary school. Here is a brief of these operations.

• Addition: Sum of numbers (Eg. 1 + 2 = 3)
• Subtraction: Difference between two or more numbers (Eg. 5 – 4 = 1)
• Multiplication: Product of two or more numbers (Eg. 3 x 9 = 27)
• Division: Dividing a number into equal parts (Eg. 10 ÷ 2 = 5, 10 is divided in 2 equal parts)

History of Mathematics

Mathematics is a historical subject. It has been explored by various mathematicians across the world since centuries, in different civilizations. Archimedes, from the BC century is known to be the Father of Mathematics. He introduced formulas to calculate surface area and volume of solids. Whereas, Aryabhatt, born in 476 CE, is known as the Father of Indian Mathematics.

In the 6th century BC, the study of mathematics began with the Pythagoreans, as a “demonstrative discipline”. The word mathematics originated from the Greek word “mathema”, which means “subject of instruction”.

Another mathematician, named Euclid, introduced the axiom, postulates, theorems and proofs, which are also used in today’s mathematics.

History of Mathematics has been an ancient study and is described by each part of the world, in a varying method. There were many mathematicians who have given different theories for many concepts, which we are applying in modern mathematics.

Numbers, which we use for calculations, had variations in the medieval period. The Romans introduced the Roman numerals that uses English alphabets to represent a number, such as:

Branches of Mathematics

The main branches of mathematics are:

• Number System
• Trigonometry
• Probability and Statistics

These mathematical concepts fall under pure mathematics . These form the base of mathematics. In our academics we will come across all these theories and fundamentals to solve questions based on them.

Applied mathematics is another form, where mathematicians, scientists or technicians use mathematical concepts to solve practical problems. It describes the professional use of mathematics.

Symbols in Mathematics

Some of the basic and most important symbols, used in mathematics, are listed below in the table.

These are the most common symbols used in basic mathematical calculations. To get more maths symbols click here.

Properties in Mathematics

In mathematics, we learn about four major properties of numbers. They are:

• Commutative Property
• Associative property
• Distributive Property
• Identity Property

These are the four basic properties of numbers. These properties are also applicable to some other mathematical concepts such as algebra.

Rules in Mathematics

The most common rule used in mathematics is the BODMAS rule. As per this rule, the arithmetic operations are performed based on the brackets and order of operations. By the full form of BODMAS, we can easily understand this logic.

BODMAS – Brackets Orders Division Multiplication Addition and Subtraction

Therefore, the first priority here is given to the brackets then division>multiplication>addition>subtraction.

For example, if we have to solve [5+(3 x 5)÷2], then using the BODMAS rule, first multiply 3 and 5, within the brackets.

→ 5+(3 x 5)÷2 = 5 + 15÷2

Now divide 15 by 2

Formulas in Mathematics

Here are some common formulas used in mathematics to solve multiple problems.

• Area and Perimeter Formula
• Coordinate Geometry Formulas
• Heron’s Formula
• Quadratic Formula
• Differentiation Formulas
• Distance Formula
• Section Formula & Conic Sections
• Standard Deviation Formula
• Trigonometry Formulas

Topics in Mathematics

Let us see some important topics for each Class (from 1 to 12) that are covered under mathematics.

Class 1 Mathematics

• Numbers In Words
• Addition And Subtraction Of Integers

Class 2 Mathematics

• Counting Numbers
• Place Value

Class 3 Mathematics

• Multiplication Tables
• Multiplication And Division Of Integers
• Comparing Fractions
• Introduction To Data

Class 4 Mathematics

• Factors And Multiples
• Multiplication And Division Of Decimals
• Multiplying Fractions
• Introduction to Large Numbers

Class 5 Mathematics

• Dividing Fractions
• Addition and Subtraction of Decimals
• Lines and Angles Introduction
• Area Of A Square – Introduction To Area

Class 6 Mathematics

• Whole Numbers

Class 7 Mathematics

• Lines And Angles
• Percentage: Means Of Comparing Quantities
• Visualising Solid Shapes

Class 8 Mathematics

• Rational Numbers
• Mensuration
• Squares and Square Roots
• Exponents And Powers

Class 9 Mathematics

• Polynomials
• Quadrilateral
• Surface Areas and Volume

Class 10 Mathematics

• Arithmetic Progression
• Co-ordinate Geometry
• Constructions
• Probability And Statistics

Class 11 Mathematics

• Relations and Functions
• Trigonometric Functions
• Linear Inequalities
• Permutation And Combination
• Conic Sections
• Limits and Derivatives

Class 12 Mathematics

• Inverse Trigonometric Functions
• Determinants
• Application of Integrals
• Vector algebra
• Linear Programming
• Continuity And Differentiability

Frequently Asked Questions on Mathematics

Define mathematics..

Mathematics is a subject that deals with numbers, shapes, logic, quantity and arrangements. Mathematics teaches to solve problems based on numerical calculations and find the solutions.

Why is Mathematics an important subject for students?

Learning mathematics will help students to build their logical thinking and problem solving skills. It has huge applications in day to day life. The basic arithmetic operations such as addition, subtraction, multiplication and division are the most important part of our lives. Based on these operations, we do numerous calculations.

Who is the Father of Mathematics?

Archimedes, (287-212 BC) is known to be the Father of Mathematics.

Which part of mathematics does Trigonometry belong to?

Geometry is one of the most important branches of mathematics that includes trigonometry, where we deal with sides and angles of a right triangle. It has huge applications in the fields of construction and architecture.

What are the two forms of Mathematics?

Mathematics is described in two forms:

Pure mathematics and Applied mathematics

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

Chapter: part one: connecting mathematics with work and life, part one— connecting mathematics with work and life.

Mathematics is the key to opportunity. No longer just the language of science, mathematics now contributes in direct and fundamental ways to business, finance, health, and defense. For students, it opens doors to careers. For citizens, it enables informed decisions. For nations, it provides knowledge to compete in a technological community. To participate fully in the world of the future, America must tap the power of mathematics. (NRC, 1989, p. 1)

The above statement remains true today, although it was written almost ten years ago in the Mathematical Sciences Education Board's (MSEB) report Everybody Counts (NRC, 1989). In envisioning a future in which all students will be afforded such opportunities, the MSEB acknowledges the crucial role played by formulae and algorithms, and suggests that algorithmic skills are more flexible, powerful, and enduring when they come from a place of meaning and understanding. This volume takes as a premise that all students can develop mathematical understanding by working with mathematical tasks from workplace and everyday contexts . The essays in this report provide some rationale for this premise and discuss some of the issues and questions that follow. The tasks in this report illuminate some of the possibilities provided by the workplace and everyday life.

Contexts from within mathematics also can be powerful sites for the development of mathematical understanding, as professional and amateur mathematicians will attest. There are many good sources of compelling problems from within mathematics, and a broad mathematics education will include experience with problems from contexts both within and outside mathematics. The inclusion of tasks in this volume is intended to highlight particularly compelling problems whose context lies outside of mathematics, not to suggest a curriculum.

The operative word in the above premise is "can." The understandings that students develop from any encounter with mathematics depend not only on the context, but also on the students' prior experience and skills, their ways of thinking, their engagement with the task, the environment in which they explore the task—including the teacher, the students, and the tools—the kinds of interactions that occur in that environment, and the system of internal and external incentives that might be associated with the activity. Teaching and learning are complex activities that depend upon evolving and rarely articulated interrelationships among teachers, students, materials, and ideas. No prescription for their improvement can be simple.

This volume may be beneficially seen as a rearticulation and elaboration of a principle put forward in Reshaping School Mathematics :

Principle 3: Relevant Applications Should be an Integral Part of the Curriculum.

Students need to experience mathematical ideas in the context in which they naturally arise—from simple counting and measurement to applications in business and science. Calculators and computers make it possible now to introduce realistic applications throughout the curriculum.

The significant criterion for the suitability of an application is whether it has the potential to engage students' interests and stimulate their mathematical thinking. (NRC, 1990, p. 38)

Mathematical problems can serve as a source of motivation for students if the problems engage students' interests and aspirations. Mathematical problems also can serve as sources of meaning and understanding if the problems stimulate students' thinking. Of course, a mathematical task that is meaningful to a student will provide more motivation than a task that does not make sense. The rationale behind the criterion above is that both meaning and motivation are required. The motivational benefits that can be provided by workplace and everyday problems are worth mentioning, for although some students are aware that certain mathematics courses are necessary in order to gain entry into particular career paths, many students are unaware of how particular topics or problem-solving approaches will have relevance in any workplace. The power of using workplace and everyday problems to teach mathematics lies not so much in motivation, however, for no con-

text by itself will motivate all students. The real power is in connecting to students' thinking.

There is growing evidence in the literature that problem-centered approaches—including mathematical contexts, "real world" contexts, or both—can promote learning of both skills and concepts. In one comparative study, for example, with a high school curriculum that included rich applied problem situations, students scored somewhat better than comparison students on algebraic procedures and significantly better on conceptual and problem-solving tasks (Schoen & Ziebarth, 1998). This finding was further verified through task-based interviews. Studies that show superior performance of students in problem-centered classrooms are not limited to high schools. Wood and Sellers (1996), for example, found similar results with second and third graders.

Research with adult learners seems to indicate that "variation of contexts (as well as the whole task approach) tends to encourage the development of general understanding in a way which concentrating on repeated routine applications of algorithms does not and cannot" (Strässer, Barr, Evans, & Wolf, 1991, p. 163). This conclusion is consistent with the notion that using a variety of contexts can increase the chance that students can show what they know. By increasing the number of potential links to the diverse knowledge and experience of the students, more students have opportunities to excel, which is to say that the above premise can promote equity in mathematics education.

There is also evidence that learning mathematics through applications can lead to exceptional achievement. For example, with a curriculum that emphasizes modeling and applications, high school students at the North Carolina School of Science and Mathematics have repeatedly submitted winning papers in the annual college competition, Mathematical Contest in Modeling (Cronin, 1988; Miller, 1995).

The relationships among teachers, students, curricular materials, and pedagogical approaches are complex. Nonetheless, the literature does supports the premise that workplace and everyday problems can enhance mathematical learning, and suggests that if students engage in mathematical thinking, they will be afforded opportunities for building connections, and therefore meaning and understanding.

In the opening essay, Dale Parnell argues that traditional teaching has been missing opportunities for connections: between subject-matter and context, between academic and vocational education, between school and life, between knowledge and application, and between subject-matter disciplines. He suggests that teaching must change if more students are to learn mathematics. The question, then, is how to exploit opportunities for connections between high school mathematics and the workplace and everyday life.

Rol Fessenden shows by example the importance of mathematics in business, specifically in making marketing decisions. His essay opens with a dialogue among employees of a company that intends to expand its business into

Japan, and then goes on to point out many of the uses of mathematics, data collection, analysis, and non-mathematical judgment that are required in making such business decisions.

In his essay, Thomas Bailey suggests that vocational and academic education both might benefit from integration, and cites several trends to support this suggestion: change and uncertainty in the workplace, an increased need for workers to understand the conceptual foundations of key academic subjects, and a trend in pedagogy toward collaborative, open-ended projects. Further-more, he observes that School-to-Work experiences, first intended for students who were not planning to attend a four-year college, are increasingly being seen as useful in preparing students for such colleges. He discusses several such programs that use work-related applications to teach academic skills and to prepare students for college. Integration of academic and vocational education, he argues, can serve the dual goals of "grounding academic standards in the realistic context of workplace requirements and introducing a broader view of the potential usefulness of academic skills even for entry level workers."

Noting the importance and utility of mathematics for jobs in science, health, and business, Jean Taylor argues for continued emphasis in high school of topics such as algebra, estimation, and trigonometry. She suggests that workplace and everyday problems can be useful ways of teaching these ideas for all students.

There are too many different kinds of workplaces to represent even most of them in the classrooms. Furthermore, solving mathematics problems from some workplace contexts requires more contextual knowledge than is reasonable when the goal is to learn mathematics. (Solving some other workplace problems requires more mathematical knowledge than is reasonable in high school.) Thus, contexts must be chosen carefully for their opportunities for sense making. But for students who have knowledge of a workplace, there are opportunities for mathematical connections as well. In their essay, Daniel Chazan and Sandra Callis Bethell describe an approach that creates such opportunities for students in an algebra course for 10th through 12th graders, many of whom carried with them a "heavy burden of negative experiences" about mathematics. Because the traditional Algebra I curriculum had been extremely unsuccessful with these students, Chazan and Bethell chose to do something different. One goal was to help students see mathematics in the world around them. With the help of community sponsors, Chazen and Bethell asked students to look for mathematics in the workplace and then describe that mathematics and its applications to their classmates.

The tasks in Part One complement the points made in the essays by making direct connections to the workplace and everyday life. Emergency Calls (p. 42) illustrates some possibilities for data analysis and representation by discussing the response times of two ambulance companies. Back-of-the-Envelope Estimates (p. 45) shows how quick, rough estimates and calculations

are useful for making business decisions. Scheduling Elevators (p. 49) shows how a few simplifying assumptions and some careful reasoning can be brought together to understand the difficult problem of optimally scheduling elevators in a large office building. Finally, in the context of a discussion with a client of an energy consulting firm, Heating-Degree-Days (p. 54) illuminates the mathematics behind a common model of energy consumption in home heating.

Cronin, T. P. (1988). High school students win "college" competition. Consortium: The Newsletter of the Consortium for Mathematics and Its Applications , 26 , 3, 12.

Miller, D. E. (1995). North Carolina sweeps MCM '94. SIAM News , 28 (2).

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education . Washington, DC: National Academy Press.

National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum . Washington, DC: National Academy Press.

Schoen, H. L. & Ziebarth, S. W. (1998). Assessment of students' mathematical performance (A Core-Plus Mathematics Project Field Test Progress Report). Iowa City: Core Plus Mathematics Project Evaluation Site, University of Iowa.

Strässer, R., Barr, G. Evans, J. & Wolf, A. (1991). Skills versus understanding. In M. Harris (Ed.), Schools, mathematics, and work (pp. 158-168). London: The Falmer Press.

Wood, T. & Sellers, P. (1996). Assessment of a problem-centered mathematics program: Third grade. Journal for Research in Mathematics Education , 27 (3), 337-353.

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

The study of mathematics stands, in many ways, as a gateway to student success in education. This is becoming particularly true as our society moves inexorably into the technological age. Therefore, it is vital that more students develop higher levels of competency in mathematics. 1

The standards and expectations for students must be high, but that is only half of the equation. The more important half is the development of teaching techniques and methods that will help all students (rather than just some students) reach those higher expectations and standards. This will require some changes in how mathematics is taught.

Effective education must give clear focus to connecting real life context with subject-matter content for the student, and this requires a more ''connected" mathematics program. In many of today's classrooms, especially in secondary school and college, teaching is a matter of putting students in classrooms marked "English," "history," or "mathematics," and then attempting to fill their heads with facts through lectures, textbooks, and the like. Aside from an occasional lab, workbook, or "story problem," the element of contextual teaching and learning is absent, and little attempt is made to connect what students are learning with the world in which they will be expected to work and spend their lives. Often the frag-

mented information offered to students is of little use or application except to pass a test.

What we do in most traditional classrooms is require students to commit bits of knowledge to memory in isolation from any practical application—to simply take our word that they "might need it later." For many students, "later" never arrives. This might well be called the freezer approach to teaching and learning. In effect, we are handing out information to our students and saying, "Just put this in your mental freezer; you can thaw it out later should you need it." With the exception of a minority of students who do well in mastering abstractions with little contextual experience, students aren't buying that offer. The neglected majority of students see little personal meaning in what they are asked to learn, and they just don't learn it.

I recently had occasion to interview 75 students representing seven different high schools in the Northwest. In nearly all cases, the students were juniors identified as vocational or general education students. The comment of one student stands out as representative of what most of these students told me in one way or another: "I know it's up to me to get an education, but a lot of times school is just so dull and boring. … You go to this class, go to that class, study a little of this and a little of that, and nothing connects. … I would like to really understand and know the application for what I am learning." Time and again, students were asking, "Why do I have to learn this?" with few sensible answers coming from the teachers.

My own long experience as a community college president confirms the thoughts of these students. In most community colleges today, one-third to one-half of the entering students are enrolled in developmental (remedial) education, trying to make up for what they did not learn in earlier education experiences. A large majority of these students come to the community college with limited mathematical skills and abilities that hardly go beyond adding, subtracting, and multiplying with whole numbers. In addition, the need for remediation is also experienced, in varying degrees, at four-year colleges and universities.

What is the greatest sin committed in the teaching of mathematics today? It is the failure to help students use the magnificent power of the brain to make connections between the following:

• subject-matter content and the context of use;
• academic and vocational education;
• school and other life experiences;
• knowledge and application of knowledge; and
• one subject-matter discipline and another.

Why is such failure so critical? Because understanding the idea of making the connection between subject-matter content and the context of application

is what students, at all levels of education, desperately require to survive and succeed in our high-speed, high-challenge, rapidly changing world.

Educational policy makers and leaders can issue reams of position papers on longer school days and years, site-based management, more achievement tests and better assessment practices, and other "hot" topics of the moment, but such papers alone will not make the crucial difference in what students know and can do. The difference will be made when classroom teachers begin to connect learning with real-life experiences in new, applied ways, and when education reformers begin to focus upon learning for meaning.

A student may memorize formulas for determining surface area and measuring angles and use those formulas correctly on a test, thereby achieving the behavioral objectives set by the teacher. But when confronted with the need to construct a building or repair a car, the same student may well be left at sea because he or she hasn't made the connection between the formulas and their real-life application. When students are asked to consider the Pythagorean Theorem, why not make the lesson active, where students actually lay out the foundation for a small building like a storage shed?

What a difference mathematics instruction could make for students if it were to stress the context of application—as well as the content of knowledge—using the problem-solving model over the freezer model. Teaching conducted upon the connected model would help more students learn with their thinking brain, as well as with their memory brain, developing the competencies and tools they need to survive and succeed in our complex, interconnected society.

One step toward this goal is to develop mathematical tasks that integrate subject-matter content with the context of application and that are aimed at preparing individuals for the world of work as well as for post-secondary education. Since many mathematics teachers have had limited workplace experience, they need many good examples of how knowledge of mathematics can be applied to real life situations. The trick in developing mathematical tasks for use in classrooms will be to keep the tasks connected to real life situations that the student will recognize. The tasks should not be just a contrived exercise but should stay as close to solving common problems as possible.

As an example, why not ask students to compute the cost of 12 years of schooling in a public school? It is a sad irony that after 12 years of schooling most students who attend the public schools have no idea of the cost of their schooling or how their education was financed. No wonder that some public schools have difficulty gaining financial support! The individuals being served by the schools have never been exposed to the real life context of who pays for the schools and why. Somewhere along the line in the teaching of mathematics, this real life learning opportunity has been missed, along with many other similar contextual examples.

The mathematical tasks in High School Mathematics at Work provide students (and teachers) with a plethora of real life mathematics problems and

challenges to be faced in everyday life and work. The challenge for teachers will be to develop these tasks so they relate as close as possible to where students live and work every day.

Parnell, D. (1985). The neglected majority . Washington, DC: Community College Press.

Parnell, D. (1995). Why do I have to learn this ? Waco, TX: CORD Communications.

D ALE P ARNELL is Professor Emeritus of the School of Education at Oregon State University. He has served as a University Professor, College President, and for ten years as the President and Chief Executive Officer of the American Association of Community Colleges. He has served as a consultant to the National Science Foundation and has served on many national commissions, such as the Secretary of Labor's Commission on Achieving Necessary Skills (SCANS). He is the author of the book The Neglected Majority which provided the foundation for the federally-funded Tech Prep Associate Degree Program.

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

"OK, the agenda of the meeting is to review the status of our launch into Japan. You can see the topics and presenters on the list in front of you. Gregg, can you kick it off with a strategy review?"

"Happy to, Bob. We have assessed the possibilities, costs, and return on investment of opening up both store and catalog businesses in other countries. Early research has shown that both Japan and Germany are good candidates. Specifically, data show high preference for good quality merchandise, and a higher-than-average propensity for an active outdoor lifestyle in both countries. Education, age, and income data are quite different from our target market in the U.S., but we do not believe that will be relevant because the cultures are so different. In addition, the Japanese data show that they have a high preference for things American, and, as you know, we are a classic American company. Name recognition for our company is 14%, far higher than any of our American competition in Japan. European competitors are virtually unrecognized, and other Far Eastern competitors are perceived to be of lower quality than us. The data on these issues are quite clear.

"Nevertheless, you must understand that there is a lot of judgment involved in the decision to focus on Japan. The analyses are limited because the cultures are different and we expect different behavioral drivers. Also,

much of the data we need in Japan are simply not available because the Japanese marketplace is less well developed than in the U.S. Drivers' license data, income data, lifestyle data, are all commonplace here and unavailable there. There is little prior penetration in either country by American retailers, so there is no experience we can draw upon. We have all heard how difficult it will be to open up sales operations in Japan, but recent sales trends among computer sellers and auto parts sales hint at an easing of the difficulties.

"The plan is to open three stores a year, 5,000 square feet each. We expect to do $700/square foot, which is more than double the experience of American retailers in the U.S. but 45% less than our stores. In addition, pricing will be 20% higher to offset the cost of land and buildings. Asset costs are approximately twice their rate in the U.S., but labor is slightly less. Benefits are more thoroughly covered by the government. Of course, there is a lot of uncertainty in the sales volumes we are planning. The pricing will cover some of the uncertainty but is still less than comparable quality goods already being offered in Japan.

"Let me shift over to the competition and tell you what we have learned. We have established long-term relationships with 500 to 1000 families in each country. This is comparable to our practice in the U.S. These families do not know they are working specifically with our company, as this would skew their reporting. They keep us appraised of their catalog and shopping experiences, regardless of the company they purchase from. The sample size is large enough to be significant, but, of course, you have to be careful about small differences.

"All the families receive our catalog and catalogs from several of our competitors. They match the lifestyle, income, and education demographic profiles of the people we want to have as customers. They are experienced catalog shoppers, and this will skew their feedback as compared to new catalog shoppers.

"One competitor is sending one 100-page catalog per quarter. The product line is quite narrow—200 products out of a domestic line of 3,000. They have selected items that are not likely to pose fit problems: primarily outerwear and knit shirts, not many pants, mostly men's goods, not women's. Their catalog copy is in Kanji, but the style is a bit stilted we are told, probably because it was written in English and translated, but we need to test this hypothesis. By contrast, we have simply mailed them the same catalog we use in the U.S., even written in English.

"Customer feedback has been quite clear. They prefer our broader assortment by a ratio of 3:1, even though they don't buy most of the products. As the competitors figured, sales are focused on outerwear and knits, but we are getting more sales, apparently because they like looking at the catalog and spend more time with it. Again, we need further testing. Another hypothesis is that our brand name is simply better known.

"Interestingly, they prefer our English-language version because they find it more of an adventure to read the catalog in another language. This is probably

a built-in bias of our sampling technique because we specifically selected people who speak English. We do not expect this trend to hold in a general mailing.

"The English language causes an 8% error rate in orders, but orders are 25% larger, and 4% more frequent. If we can get them to order by phone, we can correct the errors immediately during the call.

"The broader assortment, as I mentioned, is resulting in a significantly higher propensity to order, more units per order, and the same average unit cost. Of course, paper and postage costs increase as a consequence of the larger format catalog. On the other hand, there are production efficiencies from using the same version as the domestic catalog. Net impact, even factoring in the error rate, is a significant sales increase. On the other hand, most of the time, the errors cause us to ship the wrong item which then needs to be mailed back at our expense, creating an impression in the customers that we are not well organized even though the original error was theirs.

"Final point: The larger catalog is being kept by the customer an average of 70 days, while the smaller format is only kept on average for 40 days. Assuming—we need to test this—that the length of time they keep the catalog is proportional to sales volumes, this is good news. We need to assess the overall impact carefully, but it appears that there is a significant population for which an English-language version would be very profitable."

"Thanks, Gregg, good update. Jennifer, what do you have on customer research?"

"Bob, there's far more that we need to know than we have been able to find out. We have learned that Japan is very fad-driven in apparel tastes and fascinated by American goods. We expect sales initially to sky-rocket, then drop like a stone. Later on, demand will level out at a profitable level. The graphs on page 3 [ Figure 2-1 ] show demand by week for 104 weeks, and we have assessed several scenarios. They all show a good underlying business, but the uncertainty is in the initial take-off. The best data are based on the Italian fashion boom which Japan experienced in the late 80s. It is not strictly analogous because it revolved around dress apparel instead of our casual and weekend wear. It is, however, the best information available.

FIGURE 2-1: Sales projections by week, Scenario A

FIGURE 2-2: Size distributions, U.S. vs. Japan

"Our effectiveness in positioning inventory for that initial surge will be critical to our long-term success. There are excellent data—supplied by MITI, I might add—that show that Japanese customers can be intensely loyal to companies that meet their high service expectations. That is why we prepared several scenarios. Of course, if we position inventory for the high scenario, and we experience the low one, we will experience a significant loss due to liquidations. We are still analyzing the long-term impact, however. It may still be worthwhile to take the risk if the 2-year ROI 1 is sufficient.

"We have solid information on their size scales [ Figure 2-2 ]. Seventy percent are small and medium. By comparison, 70% of Americans are large and extra large. This will be a challenge to manage but will save a few bucks on fabric.

"We also know their color preferences, and they are very different than Americans. Our domestic customers are very diverse in their tastes, but 80% of Japanese customers will buy one or two colors out of an offering of 15. We are still researching color choices, but it varies greatly for pants versus shirts, and for men versus women. We are confident we can find patterns, but we also know that it is easy to guess wrong in that market. If we guess wrong, the liquidation costs will be very high.

"Bad news on the order-taking front, however. They don't like to order by phone. …"

In this very brief exchange among decision-makers we observe the use of many critically important skills that were originally learned in public schools. Perhaps the most important is one not often mentioned, and that is the ability to convert an important business question into an appropriate mathematical one, to solve the mathematical problem, and then to explain the implications of the solution for the original business problem. This ability to inhabit simultaneously the business world and the mathematical world, to translate between the two, and, as a consequence, to bring clarity to complex, real-world issues is of extraordinary importance.

In addition, the participants in this conversation understood and interpreted graphs and tables, computed, approximated, estimated, interpolated, extrapolated, used probabilistic concepts to draw conclusions, generalized from

small samples to large populations, identified the limits of their analyses, discovered relationships, recognized and used variables and functions, analyzed and compared data sets, and created and interpreted models. Another very important aspect of their work was that they identified additional questions, and they suggested ways to shed light on those questions through additional analysis.

There were two broad issues in this conversation that required mathematical perspectives. The first was to develop as rigorous and cost effective a data collection and analysis process as was practical. It involved perhaps 10 different analysts who attacked the problem from different viewpoints. The process also required integration of the mathematical learnings of all 10 analysts and translation of the results into business language that could be understood by non-mathematicians.

The second broad issue was to understand from the perspective of the decision-makers who were listening to the presentation which results were most reliable, which were subject to reinterpretation, which were actually judgments not supported by appropriate analysis, and which were hypotheses that truly required more research. In addition, these business people would likely identify synergies in the research that were not contemplated by the analysts. These synergies need to be analyzed to determine if—mathematically—they were real. The most obvious one was where the inventory analysts said that the customers don't like to use the phone to place orders. This is bad news for the sales analysts who are counting on phone data collection to correct errors caused by language problems. Of course, we need more information to know the magnitude—or even the existance—of the problem.

In brief, the analyses that preceded the dialogue might each be considered a mathematical task in the business world:

• A cost analysis of store operations and catalogs was conducted using data from existing American and possibly other operations.
• Customer preferences research was analyzed to determine preferences in quality and life-style. The data collection itself could not be carried out by a high school graduate without guidance, but 80% of the analysis could.
• Cultural differences were recognized as a causes of analytical error. Careful analysis required judgment. In addition, sources of data were identified in the U.S., and comparable sources were found lacking in Japan. A search was conducted for other comparable retail experience, but none was found. On the other hand, sales data from car parts and computers were assessed for relevance.
• Rates of change are important in understanding how Japanese and American stores differ. Sales per square foot, price increases,
• asset costs, labor costs and so forth were compared to American standards to determine whether a store based in Japan would be a viable business.
• "Nielsen" style ratings of 1000 families were used to collect data. Sample size and error estimates were mentioned. Key drivers of behavior (lifestyle, income, education) were mentioned, but this list may not be complete. What needs to be known about these families to predict their buying behavior? What does "lifestyle" include? How would we quantify some of these variables?
• A hypothesis was presented that catalog size and product diversity drive higher sales. What do we need to know to assess the validity of this hypothesis? Another hypothesis was presented about the quality of the translation. What was the evidence for this hypothesis? Is this a mathematical question? Sales may also be proportional to the amount of time a potential customer retains the catalog. How could one ascertain this?
• Despite the abundance of data, much uncertainty remains about what to expect from sales over the first two years. Analysis could be conducted with the data about the possible inventory consequences of choosing the wrong scenario.
• One might wonder about the uncertainty in size scales. What is so difficult about identifying the colors that Japanese people prefer? Can these preferences be predicted? Will this increase the complexity of the inventory management task?
• Can we predict how many people will not use phones? What do they use instead?

As seen through a mathematical lens, the business world can be a rich, complex, and essentially limitless source of fascinating questions.

R OL F ESSENDEN is Vice-President of Inventory Planning and Control at L. L. Bean, Inc. He is also Co-Principal Investigator and Vice-Chair of Maine's State Systemic Initiative and Chair of the Strategic Planning Committee. He has previously served on the Mathematical Science Education Board, and on the National Alliance for State Science and Mathematics Coalitions (NASSMC).

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

In high school education, preparation for work immediately after high school and preparation for post-secondary education have traditionally been viewed as incompatible. Work-bound high-school students end up in vocational education tracks, where courses usually emphasize specific skills with little attention to underlying theoretical and conceptual foundations. 1 College-bound students proceed through traditional academic discipline-based courses, where they learn English, history, science, mathematics, and foreign languages, with only weak and often contrived references to applications of these skills in the workplace or in the community outside the school. To be sure, many vocational teachers do teach underlying concepts, and many academic teachers motivate their lessons with examples and references to the world outside the classroom. But these enrichments are mostly frills, not central to either the content or pedagogy of secondary school education.

Rethinking Vocational and Academic Education

Educational thinking in the United States has traditionally placed priority on college preparation. Thus the distinct track of vocational education has been seen as an option for those students who are deemed not capable of success in the more desirable academic track. As vocational programs acquired a reputation

as a ''dumping ground," a strong background in vocational courses (especially if they reduced credits in the core academic courses) has been viewed as a threat to the college aspirations of secondary school students.

This notion was further reinforced by the very influential 1983 report entitled A Nation at Risk (National Commission on Excellence in Education, 1983), which excoriated the U.S. educational system for moving away from an emphasis on core academic subjects that, according to the report, had been the basis of a previously successful American education system. Vocational courses were seen as diverting high school students from core academic activities. Despite the dubious empirical foundation of the report's conclusions, subsequent reforms in most states increased the number of academic courses required for graduation and reduced opportunities for students to take vocational courses.

The distinction between vocational students and college-bound students has always had a conceptual flaw. The large majority of students who go to four-year colleges are motivated, at least to a significant extent, by vocational objectives. In 1994, almost 247,000 bachelors degrees were conferred in business administration. That was only 30,000 less than the total number (277,500) of 1994 bachelor degree conferred in English, mathematics, philosophy, religion, physical sciences and science technologies, biological and life sciences, social sciences, and history combined . Furthermore, these "academic" fields are also vocational since many students who graduate with these degrees intend to make their living working in those fields.

Several recent economic, technological, and educational trends challenge this sharp distinction between preparation for college and for immediate post-high-school work, or, more specifically, challenge the notion that students planning to work after high school have little need for academic skills while college-bound students are best served by an abstract education with only tenuous contact with the world of work:

• First, many employers and analysts are arguing that, due to changes in the nature of work, traditional approaches to teaching vocational skills may not be effective in the future. Given the increasing pace of change and uncertainty in the workplace, young people will be better prepared, even for entry level positions and certainly for subsequent positions, if they have an underlying understanding of the scientific, mathematical, social, and even cultural aspects of the work that they will do. This has led to a growing emphasis on integrating academic and vocational education. 2
• Views about teaching and pedagogy have increasingly moved toward a more open and collaborative "student-centered" or "constructivist" teaching style that puts a great deal of emphasis on having students work together on complex, open-ended projects. This reform strategy is now widely implemented through the efforts of organizations such as the Coalition of Essential Schools, the National Center for Restructuring Education, Schools, and Teaching at
• Teachers College, and the Center for Education Research at the University of Wisconsin at Madison. Advocates of this approach have not had much interaction with vocational educators and have certainly not advocated any emphasis on directly preparing high school students for work. Nevertheless, the approach fits well with a reformed education that integrates vocational and academic skills through authentic applications. Such applications offer opportunities to explore and combine mathematical, scientific, historical, literary, sociological, economic, and cultural issues.
• In a related trend, the federal School-to-Work Opportunities Act of 1994 defines an educational strategy that combines constructivist pedagogical reforms with guided experiences in the workplace or other non-work settings. At its best, school-to-work could further integrate academic and vocational learning through appropriately designed experiences at work.
• The integration of vocational and academic education and the initiatives funded by the School-to-Work Opportunities Act were originally seen as strategies for preparing students for work after high school or community college. Some educators and policy makers are becoming convinced that these approaches can also be effective for teaching academic skills and preparing students for four-year college. Teaching academic skills in the context of realistic and complex applications from the workplace and community can provide motivational benefits and may impart a deeper understanding of the material by showing students how the academic skills are actually used. Retention may also be enhanced by giving students a chance to apply the knowledge that they often learn only in the abstract. 3
• During the last twenty years, the real wages of high school graduates have fallen and the gap between the wages earned by high school and college graduates has grown significantly. Adults with no education beyond high school have very little chance of earning enough money to support a family with a moderate lifestyle. 4 Given these wage trends, it seems appropriate and just that every high school student at least be prepared for college, even if some choose to work immediately after high school.

Innovative Examples

There are many examples of programs that use work-related applications both to teach academic skills and to prepare students for college. One approach is to organize high school programs around broad industrial or occupational areas, such as health, agriculture, hospitality, manufacturing, transportation, or the arts. These broad areas offer many opportunities for wide-ranging curricula in all academic disciplines. They also offer opportunities for collaborative work among teachers from different disciplines. Specific skills can still be taught in this format but in such a way as to motivate broader academic and theoretical themes. Innovative programs can now be found in many vocational

high schools in large cities, such as Aviation High School in New York City and the High School of Agricultural Science and Technology in Chicago. Other schools have organized schools-within-schools based on broad industry areas.

Agriculturally based activities, such as 4H and Future Farmers of America, have for many years used the farm setting and students' interest in farming to teach a variety of skills. It takes only a little imagination to think of how to use the social, economic, and scientific bases of agriculture to motivate and illustrate skills and knowledge from all of the academic disciplines. Many schools are now using internships and projects based on local business activities as teaching tools. One example among many is the integrated program offered by the Thomas Jefferson High School for Science and Technology in Virginia, linking biology, English, and technology through an environmental issues forum. Students work as partners with resource managers at the Mason Neck National Wildlife Refuge and the Mason Neck State Park to collect data and monitor the daily activities of various species that inhabit the region. They search current literature to establish a hypothesis related to a real world problem, design an experiment to test their hypothesis, run the experiment, collect and analyze data, draw conclusions, and produce a written document that communicates the results of the experiment. The students are even responsible for determining what information and resources are needed and how to access them. Student projects have included making plans for public education programs dealing with environmental matters, finding solutions to problems caused by encroaching land development, and making suggestions for how to handle the overabundance of deer in the region.

These examples suggest the potential that a more integrated education could have for all students. Thus continuing to maintain a sharp distinction between vocational and academic instruction in high school does not serve the interests of many of those students headed for four-year or two-year college or of those who expect to work after high school. Work-bound students will be better prepared for work if they have stronger academic skills, and a high-quality curriculum that integrates school-based learning into work and community applications is an effective way to teach academic skills for many students.

Despite the many examples of innovative initiatives that suggest the potential for an integrated view, the legacy of the duality between vocational and academic education and the low status of work-related studies in high school continue to influence education and education reform. In general, programs that deviate from traditional college-prep organization and format are still viewed with suspicion by parents and teachers focused on four-year college. Indeed, college admissions practices still very much favor the traditional approaches. Interdisciplinary courses, "applied" courses, internships, and other types of work experience that characterize the school-to-work strategy or programs that integrate academic and vocational education often do not fit well into college admissions requirements.

Joining Work and Learning

What implications does this have for the mathematics standards developed by the National Council of Teachers of Mathematics (NCTM)? The general principle should be to try to design standards that challenge rather than reinforce the distinction between vocational and academic instruction. Academic teachers of mathematics and those working to set academic standards need to continue to try to understand the use of mathematics in the workplace and in everyday life. Such understandings would offer insights that could suggest reform of the traditional curriculum, but they would also provide a better foundation for teaching mathematics using realistic applications. The examples in this volume are particularly instructive because they suggest the importance of problem solving, logic, and imagination and show that these are all important parts of mathematical applications in realistic work settings. But these are only a beginning.

In order to develop this approach, it would be helpful if the NCTM standards writers worked closely with groups that are setting industry standards. 5 This would allow both groups to develop a deeper understanding of the mathematics content of work.

The NCTM's Curriculum Standards for Grades 9-12 include both core standards for all students and additional standards for "college-intending" students. The argument presented in this essay suggests that the NCTM should dispense with the distinction between college intending and non-college intending students. Most of the additional standards, those intended only for the "college intending" students, provide background that is necessary or beneficial for the calculus sequence. A re-evaluation of the role of calculus in the high school curriculum may be appropriate, but calculus should not serve as a wedge to separate college-bound from non-college-bound students. Clearly, some high school students will take calculus, although many college-bound students will not take calculus either in high school or in college. Thus in practice, calculus is not a characteristic that distinguishes between those who are or are not headed for college. Perhaps standards for a variety of options beyond the core might be offered. Mathematics standards should be set to encourage stronger skills for all students and to illustrate the power and usefulness of mathematics in many settings. They should not be used to institutionalize dubious distinctions between groups of students.

Bailey, T. & Merritt, D. (1997). School-to-work for the collegebound . Berkeley, CA: National Center for Research in Vocational Education.

Hoachlander, G . (1997) . Organizing mathematics education around work . In L.A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow's America , (pp. 122-136). New York: College Entrance Examination Board.

Levy, F. & Murnane, R. (1992). U.S. earnings levels and earnings inequality: A review of recent trends and proposed explanations. Journal of Economic Literature , 30 , 1333-1381.

National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform . Washington, DC: Author.

T HOMAS B AILEY is an Associate Professor of Economics Education at Teachers College, Columbia University. He is also Director of the Institute on Education and the Economy and Director of the Community College Research Center, both at Teachers College. He is also on the board of the National Center for Research in Vocational Education.

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

For decades our industrial society has been based on fossil fuels. In today's knowledge-based society, mathematics is the energy that drives the system. In the words of the new WQED television series, Life by the Numbers , to create knowledge we "burn mathematics." Mathematics is more than a fixed tool applied in known ways. New mathematical techniques and analyses and even conceptual frameworks are continually required in economics, in finance, in materials science, in physics, in biology, in medicine.

Just as all scientific and health-service careers are mathematically based, so are many others. Interaction with computers has become a part of more and more jobs, and good analytical skills enhance computer use and troubleshooting. In addition, virtually all levels of management and many support positions in business and industry require some mathematical understanding, including an ability to read graphs and interpret other information presented visually, to use estimation effectively, and to apply mathematical reasoning.

What Should Students Learn for Today's World?

Education in mathematics and the ability to communicate its predictions is more important than ever for moving from low-paying jobs into better-paying ones. For example, my local paper, The Times of Trenton , had a section "Focus

on Careers" on October 5, 1997 in which the majority of the ads were for high technology careers (many more than for sales and marketing, for example).

But precisely what mathematics should students learn in school? Mathematicians and mathematics educators have been discussing this question for decades. This essay presents some thoughts about three areas of mathematics—estimation, trigonometry, and algebra—and then some thoughts about teaching and learning.

Estimation is one of the harder skills for students to learn, even if they experience relatively little difficulty with other aspects of mathematics. Many students think of mathematics as a set of precise rules yielding exact answers and are uncomfortable with the idea of imprecise answers, especially when the degree of precision in the estimate depends on the context and is not itself given by a rule. Yet it is very important to be able to get an approximate sense of the size an answer should be, as a way to get a rough check on the accuracy of a calculation (I've personally used it in stores to detect that I've been charged twice for the same item, as well as often in my own mathematical work), a feasibility estimate, or as an estimation for tips.

Trigonometry plays a significant role in the sciences and can help us understand phenomena in everyday life. Often introduced as a study of triangle measurement, trigonometry may be used for surveying and for determining heights of trees, but its utility extends vastly beyond these triangular applications. Students can experience the power of mathematics by using sine and cosine to model periodic phenomena such as going around and around a circle, going in and out with tides, monitoring temperature or smog components changing on a 24-hour cycle, or the cycling of predator-prey populations.

No educator argues the importance of algebra for students aiming for mathematically-based careers because of the foundation it provides for the more specialized education they will need later. Yet, algebra is also important for those students who do not currently aspire to mathematics-based careers, in part because a lack of algebraic skills puts an upper bound on the types of careers to which a student can aspire. Former civil rights leader Robert Moses makes a good case for every student learning algebra, as a means of empowering students and providing goals, skills, and opportunities. The same idea was applied to learning calculus in the movie Stand and Deliver . How, then, can we help all students learn algebra?

For me personally, the impetus to learn algebra was at least in part to learn methods of solution for puzzles. Suppose you have 39 jars on three shelves. There are twice as many jars on the second shelf as the first, and four more jars on the third shelf than on the second shelf. How many jars are there on each shelf? Such problems are not important by themselves, but if they show the students the power of an idea by enabling them to solve puzzles that they'd like to solve, then they have value. We can't expect such problems to interest all students. How then can we reach more students?

Workplace and Everyday Settings as a Way of Making Sense

One of the common tools in business and industry for investigating mathematical issues is the spreadsheet, which is closely related to algebra. Writing a rule to combine the elements of certain cells to produce the quantity that goes into another cell is doing algebra, although the variables names are cell names rather than x or y . Therefore, setting up spreadsheet analyses requires some of the thinking that algebra requires.

By exploring mathematics via tasks which come from workplace and everyday settings, and with the aid of common tools like spreadsheets, students are more likely to see the relevance of the mathematics and are more likely to learn it in ways that are personally meaningful than when it is presented abstractly and applied later only if time permits. Thus, this essay argues that workplace and everyday tasks should be used for teaching mathematics and, in particular, for teaching algebra. It would be a mistake, however, to rely exclusively on such tasks, just as it would be a mistake to teach only spreadsheets in place of algebra.

Communicating the results of an analysis is a fundamental part of any use of mathematics on a job. There is a growing emphasis in the workplace on group work and on the skills of communicating ideas to colleagues and clients. But communicating mathematical ideas is also a powerful tool for learning, for it requires the student to sharpen often fuzzy ideas.

Some of the tasks in this volume can provide the kinds of opportunities I am talking about. Another problem, with clear connections to the real world, is the following, taken from the book entitled Consider a Spherical Cow: A Course in Environmental Problem Solving , by John Harte (1988). The question posed is: How does biomagnification of a trace substance occur? For example, how do pesticides accumulate in the food chain, becoming concentrated in predators such as condors? Specifically, identify the critical ecological and chemical parameters determining bioconcentrations in a food chain, and in terms of these parameters, derive a formula for the concentration of a trace substance in each link of a food chain. This task can be undertaken at several different levels. The analysis in Harte's book is at a fairly high level, although it still involves only algebra as a mathematical tool. The task could be undertaken at a more simple level or, on the other hand, it could be elaborated upon as suggested by further exercises given in that book. And the students could then present the results of their analyses to each other as well as the teacher, in oral or written form.

Concepts or Procedures?

When teaching mathematics, it is easy to spend so much time and energy focusing on the procedures that the concepts receive little if any attention. When teaching algebra, students often learn the procedures for using the quadratic formula or for solving simultaneous equations without thinking of intersections of curves and lines and without being able to apply the procedures in unfamiliar settings. Even

when concentrating on word problems, students often learn the procedures for solving "coin problems" and "train problems" but don't see the larger algebraic context. The formulas and procedures are important, but are not enough.

When using workplace and everyday tasks for teaching mathematics, we must avoid falling into the same trap of focusing on the procedures at the expense of the concepts. Avoiding the trap is not easy, however, because just like many tasks in school algebra, mathematically based workplace tasks often have standard procedures that can be used without an understanding of the underlying mathematics. To change a procedure to accommodate a changing business climate, to respond to changes in the tax laws, or to apply or modify a procedure to accommodate a similar situation, however, requires an understanding of the mathematical ideas behind the procedures. In particular, a student should be able to modify the procedures for assessing energy usage for heating (as in Heating-Degree-Days, p. 54) in order to assess energy usage for cooling in the summer.

To prepare our students to make such modifications on their own, it is important to focus on the concepts as well as the procedures. Workplace and everyday tasks can provide opportunities for students to attach meaning to the mathematical calculations and procedures. If a student initially solves a problem without algebra, then the thinking that went into his or her solution can help him or her make sense out of algebraic approaches that are later presented by the teacher or by other students. Such an approach is especially appropriate for teaching algebra, because our teaching of algebra needs to reach more students (too often it is seen by students as meaningless symbol manipulation) and because algebraic thinking is increasingly important in the workplace.

An Example: The Student/Professor Problem

To illustrate the complexity of learning algebra meaningfully, consider the following problem from a study by Clement, Lockhead, & Monk (1981):

Write an equation for the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (p. 288)

The authors found that of 47 nonscience majors taking college algebra, 57% got it wrong. What is more surprising, however, is that of 150 calculus-level students, 37% missed the problem.

A first reaction to the most common wrong answer, 6 S = P , is that the students simply translated the words of the problems into mathematical symbols without thinking more deeply about the situation or the variables. (The authors note that some textbooks instruct students to use such translation.)

By analyzing transcripts of interviews with students, the authors found this approach and another (faulty) approach, as well. These students often drew a diagram showing six students and one professor. (Note that we often instruct students to draw diagrams when solving word problems.) Reasoning

from the diagram, and regarding S and P as units, the student may write 6 S = P , just as we would correctly write 12 in. = 1 ft. Such reasoning is quite sensible, though it misses the fundamental intent in the problem statement that S is to represent the number of students, not a student.

Thus, two common suggestions for students—word-for-word translation and drawing a diagram—can lead to an incorrect answer to this apparently simple problem, if the students do not more deeply contemplate what the variables are intended to represent. The authors found that students who wrote and could explain the correct answer, S = 6 P , drew upon a richer understanding of what the equation and the variables represent.

Clearly, then, we must encourage students to contemplate the meanings of variables. Yet, part of the power and efficiency of algebra is precisely that one can manipulate symbols independently of what they mean and then draw meaning out of the conclusions to which the symbolic manipulations lead. Thus, stable, long-term learning of algebraic thinking requires both mastery of procedures and also deeper analytical thinking.

Paradoxically, the need for sharper analytical thinking occurs alongside a decreased need for routine arithmetic calculation. Calculators and computers make routine calculation easier to do quickly and accurately; cash registers used in fast food restaurants sometimes return change; checkout counters have bar code readers and payment takes place by credit cards or money-access cards.

So it is education in mathematical thinking, in applying mathematical computation, in assessing whether an answer is reasonable, and in communicating the results that is essential. Teaching mathematics via workplace and everyday problems is an approach that can make mathematics more meaningful for all students. It is important, however, to go beyond the specific details of a task in order to teach mathematical ideas. While this approach is particularly crucial for those students intending to pursue careers in the mathematical sciences, it will also lead to deeper mathematical understanding for all students.

Clement, J., Lockhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly , 88 , 286-290.

Harte, J. (1988). Consider a spherical cow: A course in environmental problem solving . York, PA: University Science Books.

J EAN E. T AYLOR is Professor of Mathematics at Rutgers, the State University of New Jersey. She is currently a member of the Board of Directors of the American Association for the Advancement of Science and formerly chaired its Section A Nominating Committee. She has served as Vice President and as a Member-at-Large of the Council of the American Mathematical Society, and served on its Executive Committee and its Nominating Committee. She has also been a member of the Joint Policy Board for Mathematics, and a member of the Board of Advisors to The Geometry Forum (now The Mathematics Forum) and to the WQED television series, Life by the Numbers .

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

Teaching a mathematics class in which few of the students have demonstrated success is a difficult assignment. Many teachers avoid such assignments, when possible. On the one hand, high school mathematics teachers, like Bertrand Russell, might love mathematics and believe something like the following:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. … Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its nature home, and where one, at least, of our nobler impulses can escape from the dreary exile of the natural world. (Russell, 1910, p. 73)

But, on the other hand, students may not have the luxury, in their circumstances, of appreciating this beauty. Many of them may not see themselves as thinkers because contemplation would take them away from their primary

focus: how to get by in a world that was not created for them. Instead, like Jamaica Kincaid, they may be asking:

What makes the world turn against me and all who look like me? I won nothing, I survey nothing, when I ask this question, the luxury of an answer that will fill volumes does not stretch out before me. When I ask this question, my voice is filled with despair. (Kincaid, 1996, pp. 131-132)

Our Teaching and Issues it Raised

During the 1991-92 and 1992-93 school years, we (a high school teacher and a university teacher educator) team taught a lower track Algebra I class for 10th through 12th grade students. 1 Most of our students had failed mathematics before, and many needed to pass Algebra I in order to complete their high school mathematics requirement for graduation. For our students, mathematics had become a charged subject; it carried a heavy burden of negative experiences. Many of our students were convinced that neither they nor their peers could be successful in mathematics.

Few of our students did well in other academic subjects, and few were headed on to two- or four-year colleges. But the students differed in their affiliation with the high school. Some, called ''preppies" or "jocks" by others, were active participants in the school's activities. Others, "smokers" or "stoners," were rebelling to differing degrees against school and more broadly against society. There were strong tensions between members of these groups. 2

Teaching in this setting gives added importance and urgency to the typical questions of curriculum and motivation common to most algebra classes. In our teaching, we explored questions such as the following:

• What is it that we really want high school students, especially those who are not college-intending, to study in algebra and why?
• What is the role of algebra's manipulative skills in a world with graphing calculators and computers? How do the manipulative skills taught in the traditional curriculum give students a new perspective on, and insight into, our world?
• If our teaching efforts depend on students' investment in learning, on what grounds can we appeal to them, implicitly or explicitly, for energy and effort? In a tracked, compulsory setting, how can we help students, with broad interests and talents and many of whom are not college-intending, see value in a shared exploration of algebra?

An Approach to School Algebra

As a result of thinking about these questions, in our teaching we wanted to avoid being in the position of exhorting students to appreciate the beauty or utility of algebra. Our students were frankly skeptical of arguments based on

utility. They saw few people in their community using algebra. We had also lost faith in the power of extrinsic rewards and punishments, like failing grades. Many of our students were skeptical of the power of the high school diploma to alter fundamentally their life circumstances. We wanted students to find the mathematical objects we were discussing in the world around them and thus learn to value the perspective that this mathematics might give them on their world.

To help us in this task, we found it useful to take what we call a "relationships between quantities" approach to school algebra. In this approach, the fundamental mathematical objects of study in school algebra are functions that can be represented by inputs and outputs listed in tables or sketched or plotted on graphs, as well as calculation procedures that can be written with algebraic symbols. 3 Stimulated, in part, by the following quote from August Comte, we viewed these functions as mathematical representations of theories people have developed for explaining relationships between quantities.

In the light of previous experience, we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. (Quoted in Serres, 1982, p. 85)

The "Sponsor" Project

Using this approach to the concept of function, during the 1992-93 school year, we designed a year-long project for our students. The project asked pairs of students to find the mathematical objects we were studying in the workplace of a community sponsor. Students visited the sponsor's workplace four times during the year—three after-school visits and one day-long excused absence from school. In these visits, the students came to know the workplace and learned about the sponsor's work. We then asked students to write a report describing the sponsor's workplace and answering questions about the nature of the mathematical activity embedded in the workplace. The questions are organized in Table 5-1 .

Using These Questions

In order to determine how the interviews could be structured and to provide students with a model, we chose to interview Sandra's husband, John Bethell, who is a coatings inspector for an engineering firm. When asked about his job, John responded, "I argue for a living." He went on to describe his daily work inspecting contractors painting water towers. Since most municipalities contract with the lowest bidder when a water tower needs to be painted, they will often hire an engineering firm to make sure that the contractor works according to specification. Since the contractor has made a low bid, there are strong

TABLE 5-1: Questions to ask in the workplace

financial incentives for the contractor to compromise on quality in order to make a profit.

In his work John does different kinds of inspections. For example, he has a magnetic instrument to check the thickness of the paint once it has been applied to the tower. When it gives a "thin" reading, contractors often question the technology. To argue for the reading, John uses the surface area of the tank, the number of paint cans used, the volume of paint in the can, and an understanding of the percentage of this volume that evaporates to calculate the average thickness of the dry coating. Other examples from his workplace involve the use of tables and measuring instruments of different kinds.

Some Examples of Students' Work

When school started, students began working on their projects. Although many of the sponsors initially indicated that there were no mathematical dimensions to their work, students often were able to show sponsors places where the mathematics we were studying was to be found. For example, Jackie worked with a crop and soil scientist. She was intrigued by the way in which measurement of weight is used to count seeds. First, her sponsor would weigh a test batch of 100 seeds to generate a benchmark weight. Then, instead of counting a large number of seeds, the scientist would weigh an amount of seeds and compute the number of seeds such a weight would contain.

Rebecca worked with a carpeting contractor who, in estimating costs, read the dimensions of rectangular rooms off an architect's blueprint, multiplied to find the area of the room in square feet (doing conversions where necessary), then multiplied by a cost per square foot (which depended on the type of carpet) to compute the cost of the carpet. The purpose of these estimates was to prepare a bid for the architect where the bid had to be as low as possible without making the job unprofitable. Rebecca used a chart ( Table 5-2 ) to explain this procedure to the class.

Joe and Mick, also working in construction, found out that in laying pipes, there is a "one by one" rule of thumb. When digging a trench for the placement of the pipe, the non-parallel sides of the trapezoidal cross section must have a slope of 1 foot down for every one foot across. This ratio guarantees that the dirt in the hole will not slide down on itself. Thus, if at the bottom of the hole, the trapezoid must have a certain width in order to fit the pipe, then on ground level the hole must be this width plus twice the depth of the hole. Knowing in advance how wide the hole must be avoids lengthy and costly trial and error.

Other students found that functions were often embedded in cultural artifacts found in the workplace. For example, a student who visited a doctor's office brought in an instrument for predicting the due dates of pregnant women, as well as providing information about average fetal weight and length ( Figure 5-1 ).

TABLE 5-2: Cost of carpet worksheet

FIGURE 5-1: Pregnancy wheel

While the complexities of organizing this sort of project should not be minimized—arranging sponsors, securing parental permission, and meeting administrators and parent concerns about the requirement of off-campus, after-school work—we remain intrigued by the potential of such projects for helping students see mathematics in the world around them. The notions of identifying central mathematical objects for a course and then developing ways of identifying those objects in students' experience seems like an important alternative to the use of application-based materials written by developers whose lives and social worlds may be quite different from those of students.

Chazen, D. (1996). Algebra for all students? Journal of Mathematical Behavior , 15 (4), 455-477.

Eckert, P. (1989). Jocks and burnouts: Social categories and identity in the high school . New York: Teachers College Press.

Fey, J. T., Heid, M. K., et al. (1995). Concepts in algebra: A technological approach . Dedham, MA: Janson Publications.

Kieran, C., Boileau, A., & Garancon, M. (1996). Introducing algebra by mean of a technology-supported, functional approach. In N. Bednarz et al. (Eds.), Approaches to algebra , (pp. 257-293). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Kincaid, J. (1996). The autobiography of my mother . New York: Farrar, Straus, Giroux.

Nemirovsky, R. (1996). Mathematical narratives, modeling and algebra. In N. Bednarz et al. (Eds.) Approaches to algebra , (pp. 197-220). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Russell, B. (1910). Philosophical Essays . London: Longmans, Green.

Schwartz, J. & Yerushalmy, M. (1992). Getting students to function in and with algebra. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy , (MAA Notes, Vol. 25, pp. 261-289). Washington, DC: Mathematical Association of America.

Serres, M. (1982). Mathematics and philosophy: What Thales saw … In J. Harari & D. Bell (Eds.), Hermes: Literature, science, philosophy , (pp. 84-97). Baltimore, MD: Johns Hopkins.

Thompson, P. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics , 25 , 165-208.

Yerushalmy, M. & Schwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions , (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates.

D ANIEL C HAZAN is an Associate Professor of Teacher Education at Michigan State University. To assist his research in mathematics teaching and learning, he has taught algebra at the high school level. His interests include teaching mathematics by examining student ideas, using computers to support student exploration, and the potential for the history and philosophy of mathematics to inform teaching.

S ANDRA C ALLIS B ETHELL has taught mathematics and Spanish at Holt High School for 10 years. She has also completed graduate work at Michigan State University and Western Michigan University. She has interest in mathematics reform, particularly in meeting the needs of diverse learners in algebra courses.

Emergency Calls

A city is served by two different ambulance companies. City logs record the date, the time of the call, the ambulance company, and the response time for each 911 call ( Table 1 ). Analyze these data and write a report to the City Council (with supporting charts and graphs) advising it on which ambulance company the 911 operators should choose to dispatch for calls from this region.

TABLE 1: Ambulance dispatch log sheet, May 1–30

This problem confronts the student with a realistic situation and a body of data regarding two ambulance companies' response times to emergency calls. The data the student is provided are typically "messy"—just a log of calls and response times, ordered chronologically. The question is how to make sense of them. Finding patterns in data such as these requires a productive mixture of mathematics common sense, and intellectual detective work. It's the kind of reasoning that students should be able to do—the kind of reasoning that will pay off in the real world.

Mathematical Analysis

In this case, a numerical analysis is not especially informative. On average, the companies are about the same: Arrow has a mean response time of 11.4 minutes compared to 11.6 minutes for Metro. The spread of the data is also not very helpful. The ranges of their distributions are exactly the same: from 6 minutes to 19 minutes. The standard deviation of Arrow's response time is a little longer—4.3 minutes versus 3.4 minutes for Metro—indicating that Arrow's response times fluctuate a bit more.

Graphs of the response times (Figures 1 and 2 ) reveal interesting features. Both companies, especially Arrow, seem to have bimodal distributions, which is to say that there are two clusters of data without much data in between.

FIGURE 1: Distribution of Arrow's response times

FIGURE 2: Distribution of Metro's response times

The distributions for both companies suggest that there are some other factors at work. Might a particular driver be the problem? Might the slow response times for either company be on particular days of the week or at particular times of day? Graphs of the response time versus the time of day (Figures 3 and 4 ) shed some light on these questions.

FIGURE 3: Arrow response times by time of day

FIGURE 4: Metro response times by time of day

These graphs show that Arrow's response times were fast except between 5:30 AM and 9:00 AM, when they were about 9 minutes slower on average. Similarly, Metro's response times were fast except between about 3:30 PM and 6:30 PM, when they were about 5 minutes slower. Perhaps the locations of the companies make Arrow more susceptible to the morning rush hour and Metro more susceptible to the afternoon rush hour. On the other hand, the employees on Arrow's morning shift or Metro's afternoon shift may not be efficient. To avoid slow responses, one could recommend to the City Council that Metro be called during the morning and that Arrow be called during the afternoon. A little detective work into the sources of the differences between the companies may yield a better recommendation.

Comparisons may be drawn between two samples in various contexts—response times for various services (taxis, computer-help desks, 24-hour hot lines at automobile manufacturers) being one class among many. Depending upon the circumstances, the data may tell very different stories. Even in the situation above, if the second pair of graphs hadn't offered such clear explanations, one might have argued that although the response times for Arrow were better on average the spread was larger, thus making their "extremes" more risky. The fundamental idea is using various analysis and representation techniques to make sense of data when the important factors are not necessarily known ahead of time.

Back-of-the-Envelope Estimates

Practice "back-of-the-envelope" estimates based on rough approximations that can be derived from common sense or everyday observations. Examples:

• Consider a public high school mathematics teacher who feels that students should work five nights a week, averaging about 35 minutes a night, doing focused on-task work and who intends to grade all homework with comments and corrections. What is a reasonable number of hours per week that such a teacher should allocate for grading homework?
• How much paper does The New York Times use in a week? A paper company that wishes to make a bid to become their sole supplier needs to know whether they have enough current capacity. If the company were to store a two-week supply of newspaper, will their empty 14,000 square foot warehouse be big enough?

Some 50 years ago, physicist Enrico Fermi asked his students at the University of Chicago, "How many piano tuners are there in Chicago?" By asking such questions, Fermi wanted his students to make estimates that involved rough approximations so that their goal would be not precision but the order of magnitude of their result. Thus, many people today call these kinds of questions "Fermi questions." These generally rough calculations often require little more than common sense, everyday observations, and a scrap of paper, such as the back of a used envelope.

Scientists and mathematicians use the idea of order of magnitude , usually expressed as the closest power of ten, to give a rough sense of the size of a quantity. In everyday conversation, people use a similar idea when they talk about "being in the right ballpark." For example, a full-time job at minimum wage yields an annual income on the order of magnitude of $10,000 or 10 4 dollars. Some corporate executives and professional athletes make annual salaries on the order of magnitude of $10,000,000 or 10 7 dollars. To say that these salaries differ by a factor of 1000 or 10 3 , one can say that they differ by three orders of magnitude. Such a lack of precision might seem unscientific or unmathematical, but such approximations are quite useful in determining whether a more precise measurement is feasible or necessary, what sort of action might be required, or whether the result of a calculation is "in the right ballpark." In choosing a strategy to protect an endangered species, for example, scientists plan differently if there are 500 animals remaining than if there are 5,000. On the other hand, determining whether 5,200 or 6,300 is a better estimate is not necessary, as the strategies will probably be the same.

Careful reasoning with everyday observations can usually produce Fermi estimates that are within an order of magnitude of the exact answer (if there is one). Fermi estimates encourage students to reason creatively with approximate quantities and uncertain information. Experiences with such a process can help an individual function in daily life to determine the reasonableness of numerical calculations, of situations or ideas in the workplace, or of a proposed tax cut. A quick estimate of some revenue- or profit-enhancing scheme may show that the idea is comparable to suggesting that General Motors enter the summer sidewalk lemonade market in your neighborhood. A quick estimate could encourage further investigation or provide the rationale to dismiss the idea.

Almost any numerical claim may be treated as a Fermi question when the problem solver does not have access to all necessary background information. In such a situation, one may make rough guesses about relevant numbers, do a few calculations, and then produce estimates.

The examples are solved separately below.

Grading Homework

Although many component factors vary greatly from teacher to teacher or even from week to week, rough calculations are not hard to make. Some important factors to consider for the teacher are: how many classes he or she teaches, how many students are in each of the classes, how much experience has the teacher had in general and has the teacher previously taught the classes, and certainly, as part of teaching style, the kind of homework the teacher assigns, not to mention the teacher's efficiency in grading.

Suppose the teacher has 5 classes averaging 25 students per class. Because the teacher plans to write corrections and comments, assume that the students' papers contain more than a list of answers—they show some student work and, perhaps, explain some of the solutions. Grading such papers might take as long as 10 minutes each, or perhaps even longer. Assuming that the teacher can grade them as quickly as 3 minutes each, on average, the teacher's grading time is:

This is an impressively large number, especially for a teacher who already spends almost 25 hours/week in class, some additional time in preparation, and some time meeting with individual students. Is it reasonable to expect teachers to put in that kind of time? What compromises or other changes might the teacher make to reduce the amount of time? The calculation above offers four possibilities: Reduce the time spent on each homework paper, reduce the number of students per class, reduce the number of classes taught each day, or reduce the number of days per week that homework will be collected. If the teacher decides to spend at most 2 hours grading each night, what is the total number of students for which the teacher should have responsibility? This calculation is a partial reverse of the one above:

If the teacher still has 5 classes, that would mean 8 students per class!

The New York Times

Answering this question requires two preliminary estimates: the circulation of The New York Times and the size of the newspaper. The answers will probably be different on Sundays. Though The New York Times is a national newspaper, the number of subscribers outside the New York metropolitan area is probably small compared to the number inside. The population of the New York metropolitan area is roughly ten million people. Since most families buy at most one copy, and not all families buy The New York Times , the circulation might be about 1 million newspapers each day. (A circulation of 500,000 seems too small and 2 million seems too big.) The Sunday and weekday editions probably have different

circulations, but assume that they are the same since they probably differ by less than a factor of two—much less than an order of magnitude. When folded, a weekday edition of the paper measures about 1/2 inch thick, a little more than 1 foot long, and about 1 foot wide. A Sunday edition of the paper is the same width and length, but perhaps 2 inches thick. For a week, then, the papers would stack 6 × 1/2 + 2 = 5 inches thick, for a total volume of about 1 ft × 1 ft × 5/12 ft = 0.5 ft 3 .

The whole circulation, then, would require about 1/2 million cubic feet of paper per week, or about 1 million cubic feet for a two-week supply.

Is the company's warehouse big enough? The paper will come on rolls, but to make the estimates easy, assume it is stacked. If it were stacked 10 feet high, the supply would require 100,000 square feet of floor space. The company's 14,000 square foot storage facility will probably not be big enough as its size differs by almost an order of magnitude from the estimate. The circulation estimate and the size of the newspaper estimate should each be within a factor of 2, implying that the 100,000 square foot estimate is off by at most a factor of 4—less than an order of magnitude.

How big a warehouse is needed? An acre is 43,560 square feet so about two acres of land is needed. Alternatively, a warehouse measuring 300 ft × 300 ft (the length of a football field in both directions) would contain 90,000 square feet of floor space, giving a rough idea of the size.

After gaining some experience with these types of problems, students can be encouraged to pay close attention to the units and to be ready to make and support claims about the accuracy of their estimates. Paying attention to units and including units as algebraic quantities in calculations is a common technique in engineering and the sciences. Reasoning about a formula by paying attention only to the units is called dimensional analysis.

Sometimes, rather than a single estimate, it is helpful to make estimates of upper and lower bounds. Such an approach reinforces the idea that an exact answer is not the goal. In many situations, students could first estimate upper and lower bounds, and then collect some real data to determine whether the answer lies between those bounds. In the traditional game of guessing the number of jelly beans in a jar, for example, all students should be able to estimate within an order of magnitude, or perhaps within a factor of two. Making the closest guess, however, involves some chance.

Fermi questions are useful outside the workplace. Some Fermi questions have political ramifications:

• How many miles of streets are in your city or town? The police chief is considering increasing police presence so that every street is patrolled by car at least once every 4 hours.
• When will your town fill up its landfill? Is this a very urgent matter for the town's waste management personnel to assess in depth?
• In his 1997 State of the Union address, President Clinton renewed his call for a tax deduction of up to $10,000 for the cost of college tuition. He estimates that 16.5 million students stand to benefit. Is this a reasonable estimate of the number who might take advantage of the tax deduction? How much will the deduction cost in lost federal revenue?

Creating Fermi problems is easy. Simply ask quantitative questions for which there is no practical way to determine exact values. Students could be encouraged to make up their own. Examples are: ''How many oak trees are there in Illinois?" or "How many people in the U.S. ate chicken for dinner last night?" "If all the people in the world were to jump in the ocean, how much would it raise the water level?" Give students the opportunity to develop their own Fermi problems and to share them with each other. It can stimulate some real mathematical thinking.

Scheduling Elevators

In some buildings, all of the elevators can travel to all of the floors, while in others the elevators are restricted to stopping only on certain floors. What is the advantage of having elevators that travel only to certain floors? When is this worth instituting?

Scheduling elevators is a common example of an optimization problem that has applications in all aspects of business and industry. Optimal scheduling in general not only can save time and money, but it can contribute to safety (e.g., in the airline industry). The elevator problem further illustrates an important feature of many economic and political arguments—the dilemma of trying simultaneously to optimize several different needs.

Politicians often promise policies that will be the least expensive, save the most lives, and be best for the environment. Think of flood control or occupational safety rules, for example. When we are lucky, we can perhaps find a strategy of least cost, a strategy that saves the most lives, or a strategy that damages the environment least. But these might not be the same strategies: generally one cannot simultaneously satisfy two or more independent optimization conditions. This is an important message for students to learn, in order to become better educated and more critical consumers and citizens.

In the elevator problem, customer satisfaction can be emphasized by minimizing the average elevator time (waiting plus riding) for employees in an office building. Minimizing wait-time during rush hours means delivering many people quickly, which might be accomplished by filling the elevators and making few stops. During off-peak hours, however, minimizing wait-time means maximizing the availability of the elevators. There is no reason to believe that these two goals will yield the same strategy. Finding the best strategy for each is a mathematical problem; choosing one of the two strategies or a compromise strategy is a management decision, not a mathematical deduction.

This example serves to introduce a complex topic whose analysis is well within the range of high school students. Though the calculations require little more than arithmetic, the task puts a premium on the creation of reasonable alternative strategies. Students should recognize that some configurations (e.g., all but one elevator going to the top floor and the one going to all the others) do not merit consideration, while others are plausible. A systematic evaluation of all possible configurations is usually required to find the optimal solution. Such a systematic search of the possible solution space is important in many modeling situations where a formal optimal strategy is not known. Creating and evaluating reasonable strategies for the elevators is quite appropriate for high school student mathematics and lends itself well to thoughtful group effort. How do you invent new strategies? How do you know that you have considered all plausible strategies? These are mathematical questions, and they are especially amenable to group discussion.

Students should be able to use the techniques first developed in solving a simple case with only a few stories and a few elevators to address more realistic situations (e.g., 50 stories, five elevators). Using the results of a similar but simpler problem to model a more complicated problem is an important way to reason in mathematics. Students

need to determine what data and variables are relevant. Start by establishing the kind of building—a hotel, an office building, an apartment building? How many people are on the different floors? What are their normal destinations (e.g., primarily the ground floor or, perhaps, a roof-top restaurant). What happens during rush hours?

To be successful at the elevator task, students must first develop a mathematical model of the problem. The model might be a graphical representation for each elevator, with time on the horizontal axis and the floors represented on the vertical axis, or a tabular representation indicating the time spent on each floor. Students must identify the pertinent variables and make simplifying assumptions about which of the possible floors an elevator will visit.

This section works through some of the details in a particularly simple case. Consider an office building with six occupied floors, employing 240 people, and a ground floor that is not used for business. Suppose there are three elevators, each of which can hold 10 people. Further suppose that each elevator takes approximately 25 seconds to fill on the ground floor, then takes 5 seconds to move between floors and 15 seconds to open and close at each floor on which it stops.

Scenario One

What happens in the morning when everyone arrives for work? Assume that everyone arrives at approximately the same time and enters the elevators on the ground floor. If all elevators go to all floors and if the 240 people are evenly divided among all three elevators, each elevator will have to make 8 trips of 10 people each.

When considering a single trip of one elevator, assume for simplicity that 10 people get on the elevator at the ground floor and that it stops at each floor on the way up, because there may be an occupant heading to each floor. Adding 5 seconds to move to each floor and 15 seconds to stop yields 20 seconds for each of the six floors. On the way down, since no one is being picked up or let off, the elevator does not stop, taking 5 seconds for each of six floors for a total of 30 seconds. This round-trip is represented in Table 1 .

TABLE 1: Elevator round-trip time, Scenario one

Since each elevator makes 8 trips, the total time will be 1,400 seconds or 23 minutes, 20 seconds.

Scenario Two

Now suppose that one elevator serves floors 1–3 and, because of the longer trip, two elevators are assigned to floors 4–6. The elevators serving the top

TABLE 2: Elevator round-trip times, Scenario two

floors will save 15 seconds for each of floors 1–3 by not stopping. The elevator serving the bottom floors will save 20 seconds for each of the top floors and will save time on the return trip as well. The times for these trips are shown in Table 2 .

Assuming the employees are evenly distributed among the floors (40 people per floor), elevator A will transport 120 people, requiring 12 trips, and elevators B and C will transport 120 people, requiring 6 trips each. These trips will take 1200 seconds (20 minutes) for elevator A and 780 seconds (13 minutes) for elevators B and C, resulting in a small time savings (about 3 minutes) over the first scenario. Because elevators B and C are finished so much sooner than elevator A, there is likely a more efficient solution.

Scenario Three

The two round-trip times in Table 2 do not differ by much because the elevators move quickly between floors but stop at floors relatively slowly. This observation suggests that a more efficient arrangement might be to assign each elevator to a pair of floors. The times for such a scenario are listed in Table 3 .

Again assuming 40 employees per floor, each elevator will deliver 80 people, requiring 8 trips, taking at most a total of 920 seconds. Thus this assignment of elevators results in a time savings of almost 35% when compared with the 1400 seconds it would take to deliver all employees via unassigned elevators.

TABLE 3: Elevator round-trip times, Scenario three

Perhaps this is the optimal solution. If so, then the above analysis of this simple case suggests two hypotheses:

• The optimal solution assigns each floor to a single elevator.
• If the time for stopping is sufficiently larger than the time for moving between floors, each elevator should serve the same number of floors.

Mathematically, one could try to show that this solution is optimal by trying all possible elevator assignments or by carefully reasoning, perhaps by showing that the above hypotheses are correct. Practically, however, it doesn't matter because this solution considers only the morning rush hour and ignores periods of low use.

The assignment is clearly not optimal during periods of low use, and much of the inefficiency is related to the first hypothesis for rush hour optimization: that each floor is served by a single elevator. With this condition, if an employee on floor 6 arrives at the ground floor just after elevator C has departed, for example, she or he will have to wait nearly two minutes for elevator C to return, even if elevators A and B are idle. There are other inefficiencies that are not considered by focusing on the rush hour. Because each floor is served by a single elevator, an employee who wishes to travel from floor 3 to floor 6, for example, must go via the ground floor and switch elevators. Most employees would prefer more flexibility than a single elevator serving each floor.

At times when the elevators are not all busy, unassigned elevators will provide the quickest response and the greatest flexibility.

Because this optimal solution conflicts with the optimal rush hour solution, some compromise is necessary. In this simple case, perhaps elevator A could serve all floors, elevator B could serve floors 1-3, and elevator C could serve floors 4-6.

The second hypothesis, above, deserves some further thought. The efficiency of the rush hour solution Table 3 is due in part to the even division of employees among the floors. If employees were unevenly distributed with, say, 120 of the 240 people working on the top two floors, then elevator C would need to make 12 trips, taking a total of 1380 seconds, resulting in almost no benefit over unassigned elevators. Thus, an efficient solution in an actual building must take into account the distribution of the employees among the floors.

Because the stopping time on each floor is three times as large as the traveling time between floors (15 seconds versus 5 seconds), this solution effectively ignores the traveling time by assigning the same number of employees to each elevator. For taller buildings, the traveling time will become more significant. In those cases fewer employees should be assigned to the elevators that serve the upper floors than are assigned to the elevators that serve the lower floors.

The problem can be made more challenging by altering the number of elevators, the number of floors, and the number of individuals working on each floor. The rate of movement of elevators can be determined by observing buildings in the local area. Some elevators move more quickly than others. Entrance and exit times could also be measured by students collecting

data on local elevators. In a similar manner, the number of workers, elevators, and floors could be taken from local contexts.

A related question is, where should the elevators go when not in use? Is it best for them to return to the ground floor? Should they remain where they were last sent? Should they distribute themselves evenly among the floors? Or should they go to floors of anticipated heavy traffic? The answers will depend on the nature of the building and the time of day. Without analysis, it will not be at all clear which strategy is best under specific conditions. In some buildings, the elevators are controlled by computer programs that "learn" and then anticipate the traffic patterns in the building.

A different example that students can easily explore in detail is the problem of situating a fire station or an emergency room in a city. Here the key issue concerns travel times to the region being served, with conflicting optimization goals: average time vs. maximum time. A location that minimizes the maximum time of response may not produce the least average time of response. Commuters often face similar choices in selecting routes to work. They may want to minimize the average time, the maximum time, or perhaps the variance, so that their departure and arrival times are more predictable.

Most of the optimization conditions discussed so far have been expressed in units of time. Sometimes, however, two optimization conditions yield strategies whose outcomes are expressed in different (and sometimes incompatible) units of measurement. In many public policy issues (e.g., health insurance) the units are lives and money. For environmental issues, sometimes the units themselves are difficult to identify (e.g., quality of life).

When one of the units is money, it is easy to find expensive strategies but impossible to find ones that have virtually no cost. In some situations, such as airline safety, which balances lives versus dollars, there is no strategy that minimize lives lost (since additional dollars always produce slight increases in safety), and the strategy that minimizes dollars will be at $0. Clearly some compromise is necessary. Working with models of different solutions can help students understand the consequences of some of the compromises.

Heating-Degree-Days

An energy consulting firm that recommends and installs insulation and similar energy saving devices has received a complaint from a customer. Last summer she paid $540 to insulate her attic on the prediction that it would save 10% on her natural gas bills. Her gas bills have been higher than the previous winter, however, and now she wants a refund on the cost of the insulation. She admits that this winter has been colder than the last, but she had expected still to see some savings.

The facts: This winter the customer has used 1,102 therms, whereas last winter she used only 1,054 therms. This winter has been colder: 5,101 heating-degree-days this winter compared to 4,201 heating-degree-days last winter. (See explanation below.) How does a representative of the energy consulting firm explain to this customer that the accumulated heating-degree-days measure how much colder this winter has been, and then explain how to calculate her anticipated versus her actual savings.

Explaining the mathematics behind a situation can be challenging and requires a real knowledge of the context, the procedures, and the underlying mathematical concepts. Such communication of mathematical ideas is a powerful learning device for students of mathematics as well as an important skill for the workplace. Though the procedure for this problem involves only proportions, a thorough explanation of the mathematics behind the procedure requires understanding of linear modeling and related algebraic reasoning, accumulation and other precursors of calculus, as well as an understanding of energy usage in home heating.

The customer seems to understand that a straight comparison of gas usage does not take into account the added costs of colder weather, which can be significant. But before calculating any anticipated or actual savings, the customer needs some understanding of heating-degree-days. For many years, weather services and oil and gas companies have been using heating-degree-days to explain and predict energy usage and to measure energy savings of insulation and other devices. Similar degree-day units are also used in studying insect populations and crop growth. The concept provides a simple measure of the accumulated amount of cold or warm weather over time. In the discussion that follows, all temperatures are given in degrees Fahrenheit, although the process is equally workable using degrees Celsius.

Suppose, for example, that the minimum temperature in a city on a given day is 52 degrees and the maximum temperature is 64 degrees. The average temperature for the day is then taken to be 58 degrees. Subtracting that result from 65 degrees (the cutoff point for heating), yields 7 heating-degree-days for the day. By recording high and low temperatures and computing their average each day, heating-degree-days can be accumulated over the course of a month, a winter, or any period of time as a measure of the coldness of that period.

Over five consecutive days, for example, if the average temperatures were 58, 50, 60, 67, and 56 degrees Fahrenheit, the calculation yields 7, 15, 5, 0, and 9 heating-degree-days respectively, for a total accumulation of 36 heating-degree-days for the five days. Note that the fourth day contributes 0 heating-degree-days to the total because the temperature was above 65 degrees.

The relationship between average temperatures and heating-degree-days is represented graphically in Figure 1 . The average temperatures are shown along the solid line graph. The area of each shaded rectangle represents the number of heating-degree-days for that day, because the width of each rectangle is one day and the height of each rectangle is the number of degrees below 65 degrees. Over time, the sum of the areas of the rectangles represents the number of heating-degree-days accumulated during the period. (Teachers of calculus will recognize connections between these ideas and integral calculus.)

The statement that accumulated heating-degree-days should be proportional to gas or heating oil usage is based primarily on two assumptions: first, on a day for which the average temperature is above 65 degrees, no heating should be required, and therefore there should be no gas or heating oil usage; second, a day for which the average temperature is 25 degrees (40 heating-degree-days) should require twice as much heating as a day for which the average temperature is 45

FIGURE 1: Daily heating-degree-days

degrees (20 heating-degree-days) because there is twice the temperature difference from the 65 degree cutoff.

The first assumption is reasonable because most people would not turn on their heat if the temperature outside is above 65 degrees. The second assumption is consistent with Newton's law of cooling, which states that the rate at which an object cools is proportional to the difference in temperature between the object and its environment. That is, a house which is 40 degrees warmer than its environment will cool at twice the rate (and therefore consume energy at twice the rate to keep warm) of a house which is 20 degrees warmer than its environment.

The customer who accepts the heating-degree-day model as a measure of energy usage can compare this winter's usage with that of last winter. Because 5,101/4,201 = 1.21, this winter has been 21% colder than last winter, and therefore each house should require 21% more heat than last winter. If this customer hadn't installed the insulation, she would have required 21% more heat than last year, or about 1,275 therms. Instead, she has required only 5% more heat (1,102/1,054 = 1.05), yielding a savings of 14% off what would have been required (1,102/1,275 = .86).

Another approach to this would be to note that last year the customer used 1,054 therms/4,201 heating-degree-days = .251 therms/heating-degree-day, whereas this year she has used 1,102 therms/5,101 heating-degree-days = .216 therms/heating-degree-day, a savings of 14%, as before.

How good is the heating-degree-day model in predicting energy usage? In a home that has a thermometer and a gas meter or a gauge on a tank, students could record daily data for gas usage and high and low temperature to test the accuracy of the model. Data collection would require only a few minutes per day for students using an electronic indoor/outdoor thermometer that tracks high and low temperatures. Of course, gas used for cooking and heating water needs to be taken into account. For homes in which the gas tank has no gauge or doesn't provide accurate enough data, a similar experiment could be performed relating accumulated heating-degree-days to gas or oil usage between fill-ups.

It turns out that in well-sealed modern houses, the cutoff temperature for heating can be lower than 65 degrees (sometimes as low as 55 degrees) because of heat generated by light bulbs, appliances, cooking, people, and pets. At temperatures sufficiently below the cutoff, linearity turns out to be a good assumption. Linear regression on the daily usage data (collected as suggested above) ought to find an equation something like U = -.251( T - 65), where T is the average temperature and U is the gas usage. Note that the slope, -.251, is the gas usage per heating-degree-day, and 65 is the cutoff. Note also that the accumulation of heating-degree-days takes a linear equation and turns it into a proportion. There are some important data analysis issues that could be addressed by such an investigation. It is sometimes dangerous, for example, to assume linearity with only a few data points, yet this widely used model essentially assumes linearity from only one data point, the other point having coordinates of 65 degrees, 0 gas usage.

Over what range of temperatures, if any, is this a reasonable assumption? Is the standard method of computing average temperature a good method? If, for example, a day is mostly near 20 degrees but warms up to 50 degrees for a short time in the afternoon, is 35 heating-degree-days a good measure of the heating required that day? Computing averages of functions over time is a standard problem that can be solved with integral calculus. With knowledge of typical and extreme rates of temperature change, this could become a calculus problem or a problem for approximate solution by graphical methods without calculus, providing background experience for some of the important ideas in calculus.

Students could also investigate actual savings after insulating a home in their school district. A customer might typically see 8-10% savings for insulating roofs, although if the house is framed so that the walls act like chimneys, ducting air from the house and the basement into the attic, there might be very little savings. Eliminating significant leaks, on the other hand, can yield savings of as much as 25%.

Some U.S. Department of Energy studies discuss the relationship between heating-degree-days and performance and find the cutoff temperature to be lower in some modern houses. State energy offices also have useful documents.

What is the relationship between heating-degree-days computed using degrees Fahrenheit, as above, and heating-degree-days computed using degrees Celsius? Showing that the proper conversion is a direct proportion and not the standard Fahrenheit-Celsius conversion formula requires some careful and sophisticated mathematical thinking.

Traditionally, vocational mathematics and precollege mathematics have been separate in schools. But the technological world in which today's students will work and live calls for increasing connection between mathematics and its applications. Workplace-based mathematics may be good mathematics for everyone.

High School Mathematics at Work illuminates the interplay between technical and academic mathematics. This collection of thought-provoking essays—by mathematicians, educators, and other experts—is enhanced with illustrative tasks from workplace and everyday contexts that suggest ways to strengthen high school mathematical education.

This important book addresses how to make mathematical education of all students meaningful—how to meet the practical needs of students entering the work force after high school as well as the needs of students going on to postsecondary education.

The short readable essays frame basic issues, provide background, and suggest alternatives to the traditional separation between technical and academic mathematics. They are accompanied by intriguing multipart problems that illustrate how deep mathematics functions in everyday settings—from analysis of ambulance response times to energy utilization, from buying a used car to "rounding off" to simplify problems.

The book addresses the role of standards in mathematics education, discussing issues such as finding common ground between science and mathematics education standards, improving the articulation from school to work, and comparing SAT results across settings.

Experts discuss how to develop curricula so that students learn to solve problems they are likely to encounter in life—while also providing them with approaches to unfamiliar problems. The book also addresses how teachers can help prepare students for postsecondary education.

For teacher education the book explores the changing nature of pedagogy and new approaches to teacher development. What kind of teaching will allow mathematics to be a guide rather than a gatekeeper to many career paths? Essays discuss pedagogical implication in problem-centered teaching, the role of complex mathematical tasks in teacher education, and the idea of making open-ended tasks—and the student work they elicit—central to professional discourse.

High School Mathematics at Work presents thoughtful views from experts. It identifies rich possibilities for teaching mathematics and preparing students for the technological challenges of the future. This book will inform and inspire teachers, teacher educators, curriculum developers, and others involved in improving mathematics education and the capabilities of tomorrow's work force.

READ FREE ONLINE

Welcome to OpenBook!

You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

Do you want to take a quick tour of the OpenBook's features?

Show this book's table of contents , where you can jump to any chapter by name.

...or use these buttons to go back to the previous chapter or skip to the next one.

Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

Switch between the Original Pages , where you can read the report as it appeared in print, and Text Pages for the web version, where you can highlight and search the text.

To search the entire text of this book, type in your search term here and press Enter .

Share a link to this book page on your preferred social network or via email.

View our suggested citation for this chapter.

Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

Get Email Updates

Do you enjoy reading reports from the Academies online for free ? Sign up for email notifications and we'll let you know about new publications in your areas of interest when they're released.

Essays on Mathematics

A lightweight introduction to the foundations of mathematics and physics., what is mathematics.

Table of Contents

For the typical primary school or high school student, the following definition of mathematics would suffice:

(Naive definition) Starting from obviously true axioms, use obviously correct inference rules to derive additional truths.

In this sense, mathematics is all about discovering indisputable truths. For example, the theorems proved in geometry would be literally true statements about the physical space. Someone may argue that there is no such physical object as a geometrical point, or a geometrical line, but this is no issue because we can reply that geometrical objects are nothing more than locations in the physical space, and thus they can happily exist even if nobody can see them materialized. As for the exotic topic of complex numbers, they can be viewed as a man-made tool that sometimes comes in handy for mathematicians in describing reality.

Historically, this looked plausible until the early nineteenth century, when non-Euclidean geometries were invented. It wasn't generally believed any longer that Euclidean geometry was the true description of physical space. Curiously, it turned out to be impossible to decide which geometry, if any, was the true one. But then it raises the question: if mathematics does not necessarily describe reality, what does it describe then? After some meditation, we may adjust our previous definition as follows:

(Modern definition) Starting from obviously clear assumptions, use obviously correct inference rules to derive consequences.

Here we acknowledge that mathematics is all about discovering the logical consequences of given assumptions, where it is not required that the assumptions are actually true. It's comparable to making logical deductions based on the "facts" set out in a detective fiction.

Again, this looks plausible until we find out that different groups of mathematicians don't even agree on which logical rules should be permitted when deriving consequences. Most notably, certain groups don't accept the unconditional application of the law of excluded middle. This stems from their divergent views on the existence of mathematical objects.

What was told so far shows that defining mathematics is far more involved than expected. And we didn't even try to define the scope of mathematics, only its methodology was considered. To move things forward, the attempt to define mathematics must be accompanied by a better understanding of existing approaches to the foundations of mathematics.

One remark before we continue: in the definitions given above, "obviously true", "obviously correct", and "obviously clear" do not mean true, correct, and clear, respectively. Such a wording was used solely to indicate things that, for most people out there with a brain, would or used to appear as obvious. It is this entanglement with the obvious what makes mathematics possess the illusion of indisputability.

Philosophy of mathematics

Philosophy offers reasonable arguments about topics where, given our current level of knowledge, there is no feasible way of testing or verifying any theory. The foundations of mathematics is a philosophical topic of active research. This suggests that today we are still far away from a definitive answer as to what mathematics really is.

Generic framework

(Metaphorical definition) Mathematics is the intellectual discovery of nature's eternal, immutable infrastructure.

By "infrastructure", I mean that the discovery ultimately targets fundamental, ubiquitous ideas that we sense when observing appropriate configurations of things. (I am deliberately imprecise here, in the spirit of the quotation under the title of this essay.) By "intellectual", I mean that the process of discovering happens entirely within one's mind.

All ideas in mathematics are tied to observations. For example, when we look at {apple, apple, apple}, or {orange, orange, orange}, we sense "threeness", denoted by the symbol 3. When we observe the edge of a ruler, we sense a "straight line segment", even in spite of knowing that it would not look smooth through a magnifying glass. Looking at a computer network plan, we sense the idea of a "graph". Imagining a row of natural numbers, starting from 0 and fading away in the distance, can lead us to sense what we'd call the "set of natural numbers", denoted by ℕ. When we throw the dice, we sense "randomness". When we think of all humans ever born, we sense "potential infinity" (the actual set is finite at any given point in time, it's never completed, and may grow indefinitely). When we write down √ -1 , we sense something weird as if a number whose square is -1 existed (it sounded really weird before complex numbers became widely accepted). When we ask if there may be a positive quantity smaller than any positive real number, we sense the "infinitesimally small".

The ideas in the previous paragraph are sensed with varying vagueness. While "threeness" is pretty clear, √ -1 and "infinitesimally small" feel like guessing. Since all the ideas are unclear to some (varying) extent, we need to invent theories in order to describe them and their relationships. (1) (2)

As for the methodology, the starting point of mathematical theories are fundamental, ubiquitous ideas (incl. logic) about which we have accumulated so much and so consistent day-to-day experience that makes it possible to confidently rely on our intuition. We just close our eyes, and in an iterative process, come up with new ideas based on the already available ones, think out or take note of assumptions about the ideas we have, and discover the logical consequences of the assumptions. (3) The only constraint is that the resulting theories must constitute conceivable stories about our world. As a minimum, statements including "clear" ideas (e.g. natural numbers, finite sets) should coincide with our experience, and statements including "unclear" ideas (e.g. √ -1 , infinitesimals, infinite sets) should not lead to known contradictions. (4) (5)

The reason for mathematics transcending cultures and millennia is that humans have always had very similar experiences about the fundamental, ubiquitous ideas from which mathematics emerged. An alien civilization, if any, might develop a mathematics very different from ours, provided they exist in a very different environment and/or have very different sense organs.

The merits of a mathematical theory are assessed based on its beauty, success, and consistency. With regard to the development of mathematics, the first aspect is the most important guiding principle. The reason is, as once somebody put it, that beauty is felt as a result of sensing a deep law of nature. (I am deliberately imprecise here too.) As opposed to other sciences, the aim in mathematics is thus not to pinpoint and test, but rather to reflect laws of nature. That is, mathematics is both a science and an art.

In the following, standpoints of various philosophical schools are presented and commented. They can be used to customize the framework outlined above. It's a matter of personal preference.

Fictionalism

According to fictionalism, mathematics is a collection of useful fictions whose statements are, despite their usefulness, actually all false. In these fictions there are recurring "characters" like numbers, straight lines, graphs and many others, all entirely fictitious. Nevertheless, the fictions are useful because they convey (or rather, reflect) truths about our world. Furthermore, discussing our experiences in terms of carefully chosen, representative fictional characters greatly facilitates communication.

Although I agree that mathematics is a collection of stories, I still think that the ideas (i.e. the characters) in those stories are real, in one way or the other, simply because we do sense them. The assumptions the stories make about the ideas, however, may well (all) be fictional. It's like writing a guide about an existing city without knowing it well.

Also, in my view the ideas exist right here with us (just like the city in the previous analogy), not only in a separate "world of ideas" as platonism would suggest. E.g. in a computer network, there is a graph right there belonging to the network; where else could it be? Putting it another way, I don't think the network is more real than the graph.

Constructivism

Loosely speaking, constructivism means seeing is believing. The principle is that only those ideas and properties exist for which we can exhibit an appropriate configuration in terms of an agreed way of representation. For example, if it's agreed to represent real numbers via Cauchy sequences of rational numbers, then the square root of 2 exists only after one has constructed an appropriate Cauchy sequence. Before that, the square root of 2 does not exist, however esoteric this may sound.

The allowed ways of construction differ in different flavors of constructivism. In most cases though, the set of natural numbers is either assumed to exist or allowed to be constructed, either as actual (completed) infinity or as potential (incomplete) infinity. Moreover, instead of carrying out a construction (e.g. that of a square root), it may be agreed that it suffices just to provide a feasible method for the same.

Truth values have to be constructed too. Here the "appropriate configuration" is the concrete proof, and the "agreed way of representation" is the allowed forms of proof. A statement does not have a truth value until it has either been proved or disproved (or until a feasible method has been provided that would certainly result in a proof of truth or falsity). (6) If there is no such truth value to observe, it simply does not exist for a constructivist, leaving the statement undecided. (7)

To prove "A or B", we need to prove at least one of them. This requirement lies at the very heart of constructivism, and follows from a "seeing is believing" interpretation of logical disjunction. Accordingly, to be able to say that "A or (not A)" is true, we need to either prove A or disprove A. This is different from classical mathematics where "A or (not A)" is always true in itself, since there it is taken for granted that every statement has a truth value, even if nobody can actually observe it. Similar applies to "not (not A) ⇒ A": in constructivism, the left hand side only means we've demonstrated that there is no way of disproving A; but it does not necessarily imply that A is true, or that A has a truth value at all.

Adhering to the principle of constructivism lends constructive mathematics certainty and confidence, and leaves little room for unpleasant surprises like paradoxes or contradictions. Eventually, it's hard to imagine a more obvious and tangible evidence of existence than that of a constructed representation. The price we pay is that proofs tend to become unusually cumbersome. (8)

Strict finitism

Infinity in mathematical theories has always been a major source of controversy. We can only perceive a finite thing in its entirety, and this is true for our imagination too. With respect to intuition, this means we can only guess what a real infinity would look like in terms of our finite perception, and whether there exists infinity of any kind (actual or potential) at all. To eliminate this guesswork, theories in strict finitism are free from infinity.

In classical mathematics, one can state that the formula n 2 -1=(n+1)·(n-1) is true for all natural numbers. In strict finitism, the corresponding statement is that any concrete equation that matches the above formula is true. In other words, we only state that if someone gets hold of a concrete natural number, say, 19, then the resulting concrete equation, 19 2 -1=(19+1)·(19-1), will be true. That is, while in classical mathematics the statement refers to all the elements of a (hypothetically) existing infinite set, in strict finitism it's merely an abstraction of concrete individual occurrences.

Infinite sets of classical mathematics may have counterparts in strict finitism. For example, the "set" of even numbers is basically defined as the property of being even. If a concrete number has this property, we say that the number is an "element of the set". So it's not really a set in the classical sense, but rather a common property that ties concrete occurrences together.

A "sequence" of rational numbers can be defined as a (finite) method that expects a single input, a natural number, and is guaranteed to produce a rational number as output in finitely many steps. Such methods can then be used to represent real numbers by finitary means.

Strict finitism is usually coupled with constructivism (constructivism taken to the next level, so to speak), but nevertheless it's possible to develop non-constructive theories that do not make use of infinity. (9)

Pure vs. applied mathematics

Pure mathematics deals with discovering about the ideas we sense, while applied mathematics means modeling real-world phenomena using a mathematical theory.

As an example, developing (the story of) Euclidean geometry, i.e. intellectually discovering the properties of and relationships between ideas like points, straight lines and planes, is pure mathematics. On the other hand, modeling shapes and trajectories of physical objects by means of Euclidean geometry, with the aim of making measurable predictions about them, is applied mathematics. (10)

Another example of applied mathematics is to model asset prices as continuous quantities, while knowing that real prices have a finite number of decimal places, given e.g. in cents.

In summary, mathematics is the intellectual discovery of nature's infrastructure. It consists of theories about ideas that we sense with varying vagueness.

A theory begins with a number of ideas and assumptions, from which its story unfolds via the derivation of more and more logical consequences. The entire process happens within one's mind, relying fully on one's intuition.

The aim in mathematics is to reflect deep laws of nature; that's where its beauty comes from, and that's how it is connected with arts.

What is still missing is a clarification of the terms "logical consequence" and "fundamental idea", which were both used informally all along. Discussing these in detail will be the topic of another essay.

Related reading

M. Balaguer, E.N. Zalta [ed.] (2011), Fictionalism in the Philosophy of Mathematics E. Bishop, D. Bridges (1985), Constructive Analysis J. Bolyai (1831), F. Kárteszi [ed.] (1987), Appendix: The Theory of Space D. van Dalen [ed.] (1981), Brouwer's Cambridge Lectures on Intuitionism K. Devlin (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory H. Field (2008), Saving Truth from Paradox H. Field (1980), Science Without Numbers: A Defence of Nominalism T. Gowers [ed.] et al. (2008), The Princeton Companion to Mathematics J.L. Heiberg [ed.] (1883-1885), R. Fitzpatrick [trans.] (2008), Euclid's Elements of Geometry D. Hilbert (1926), E. Putnam & G.J. Massey [trans.] (1964), On the infinite D. Hilbert (1898-1899), E.J. Townsend [trans.] (1902), The Foundations of Geometry D.R. Hofstadter (2008), I Am a Strange Loop H.J. Keisler (2011), Foundations of Infinitesimal Calculus E. Nelson (2005), Completed versus Incomplete Infinity in Arithmetic E. Nelson (2010), Confessions of an Apostate Mathematician E. Nelson (1977), Internal Set Theory: A new approach to nonstandard analysis E. Nelson (2000), Mathematics and Faith E. Nelson (1987), Radically Elementary Probability Theory E. Nelson (2002), Syntax and Semantics E. Nelson (2006), Warning Signs of a Possible Collapse of Contemporary Mathematics B. Russel (1919), Introduction to Mathematical Philosophy E. Schechter (2001), Constructivism Is Difficult I. Stewart (2010), Alien mathematics: is Pi universal? P. Suppes (2001), Finitism in geometry P. Suppes (2000), Quantifier-free axioms for constructive affine plane geometry P. Suppes (2002), Representation and Invariance of Scientific Structures P. Suppes (2010), The nature of probability M. Tiles (1989), The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise E.P. Wigner (1959), The Unreasonable Effectiveness of Mathematics in the Natural Sciences F. Ye (2011), Strict Finitism and the Logic of Mathematical Applications

(1) Isn't "threeness" crystal clear? Good question. I don't think it is.

(2) Is √ -1 a ubiquitous idea? No, it's neither fundamental nor ubiquitous, but it bears a close relationship to such ideas, which at the end of the day makes it interesting for mathematical discovery. On a related note, the various kinds of mathematical spaces (vector spaces, topological spaces, etc.) are all about discovering what emerges from the interplay between the fundamental, ubiquitous ideas of set, relation, and operation.

(3) What about discoveries where computers are used? There is a philosophical concept of the "extended mind", which for the mathematical practice means that the mind can be aided by things like pencil and paper, calculator, or computer. Such extensions boost already existing capabilities of the mind, especially the memory and the ability to derive logical consequences.

(4) What if it turns out that a story is not true? No theory is true, there are only circumstances under which it cannot be falsified. The story will continue to have its own life in our mind and can be further developed via logical deductions.

(5) Wouldn't it be more productive to allow experiments in mathematics? Thought experiments are allowed, of course. Others would be of very limited use, since they would either be rendered superfluous by logical proofs (due to the "unreasonable effectiveness" of logic in mathematics), or just couldn't be performed at all (e.g. due to infinity involved).

(6) What does "disproved" mean in constructivism? A statement is disproved after a contradiction has been derived from the assumption that it's true (= proved). This is how the truth value "false" is defined. The following are synonymous: "A is false", "A is disproved", "not A". To paraphrase, a statement is false if and only if it has been proved that the situation described by it cannot occur.

(7) Assuming that the square root of 2 has not yet been constructed, can we say that it does not exist, or only that its existence is undecided? If existence is meant literally, then given the assumption, the statement "the square root of 2 exists" is false. However, if by "exists" we actually meant "can be constructed", then the statement is (currently) undecided.

(8) Why are constructive proofs cumbersome? I believe it has to do with our education, namely that we have been trained in classical mathematics from an age of 6 or so. As a result, textbooks on classical mathematics can be written in an informal style where many details that would clutter the main line of thought are omitted. It does not compromise rigor, since it capitalizes on the readers' solid understanding of and intuition about the fundamentals. Alternative mathematical theories don't have this luxury. In their textbooks, theorems and proofs are stated either in great detail at the expense of increased clutter, or with less details risking that the audience gets confused or misunderstands what is written. Either way, the exposition is likely to be difficult to follow.

(9) Does strict finitism deny the existence of infinity? No, not necessarily. It rather says that we don't need to care whether infinity exists or not, because we can still do useful mathematics without it. As of today, the resulting theories seem sufficient for the purpose of modeling the finite world we know about.

(10) Is "applied mathematics" mathematics at all? Yes, because in the models we work with mathematical ideas. However, the application of a model is unlikely to yield deep mathematical discoveries, for the aim is to solve problems of another discipline, not that of mathematics.

You caused me to recall Bertrand Russell’s definition of mathematics: Mathematics is the subject in which we don’t know what we are talking about and whether what we say is true or false, and furthermore we don’t give a damn.

BTW, Zero is not a NATURAL number, but it is a WHOLE number.

Indeed, I tend to have a similar attitude.

Some people consider zero as a natural number and others do not. Where I went to school, zero was a natural number. But I don’t give a damn 🙂

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

25,000+ students realised their study abroad dream with us. Take the first step today

Meet top uk universities from the comfort of your home, here’s your new year gift, one app for all your, study abroad needs, start your journey, track your progress, grow with the community and so much more.

Verification Code

An OTP has been sent to your registered mobile no. Please verify

Thanks for your comment !

Our team will review it before it's shown to our readers.

• School Education /

Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

Related Reads:

Hope you find this information useful. For more information on such informative topics for your school, visit our essay writing and follow Leverage Edu.

Deepansh Gautam

Leave a Reply Cancel reply

Save my name, email, and website in this browser for the next time I comment.

Contact no. *

Connect With Us

25,000+ students realised their study abroad dream with us. Take the first step today.

Resend OTP in

Need help with?

Study abroad.

UK, Canada, US & More

IELTS, GRE, GMAT & More

Scholarship, Loans & Forex

Country Preference

New Zealand

Which English test are you planning to take?

Which academic test are you planning to take.

Not Sure yet

When are you planning to take the exam?

Already booked my exam slot

Within 2 Months

Want to learn about the test

Which Degree do you wish to pursue?

When do you want to start studying abroad.

January 2024

September 2024

What is your budget to study abroad?

How would you describe this article ?

Please rate this article

We would like to hear more.

Have something on your mind?

Make your study abroad dream a reality in January 2022 with

India's Biggest Virtual University Fair

Essex Direct Admission Day

Why attend .

Don't Miss Out

Home — Essay Samples — Science — Mathematics in Everyday Life — Mathematics In Everyday Life: Most Vital Discipline

Mathematics in Everyday Life: Most Vital Discipline

• Categories: Mathematics in Everyday Life

About this sample

Words: 795 |

Published: Mar 14, 2019

Words: 795 | Pages: 2 | 4 min read

Works Cited

• Benacerraf, P. (1991). Mathematics as an object of knowledge. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (pp. 1-13). Cambridge University Press.
• EdReady. (n.d.). Home. Retrieved from https://www.edready.org/
• Puttaswamy, T. K. (2012). Engineering mathematics. Dorling Kindersley (India) Pvt. Ltd.
• Steen, L. A. (Ed.). (2001). Mathematics today: Twelve informal essays. Springer Science & Business Media.
• Suter, B. W. (2012). Mathematics education: A critical introduction. Bloomsbury Academic.
• Tucker, A. W. (2006). Applied combinatorics. John Wiley & Sons.
• Vakil, R. (2017). A mathematical mosaic: Patterns & problem solving. Princeton University Press.
• Wolfram MathWorld. (n.d.). MathWorld--The web's most extensive mathematics resource. Retrieved from http://mathworld.wolfram.com/
• Wu, H. H. (2011). The mis-education of mathematics teachers. Educational Studies in Mathematics, 77(1), 1-20.
• Ziegler, G. M., & Aigner, M. (2012). Proofs from THE BOOK. Springer Science & Business Media.

Cite this Essay

Let us write you an essay from scratch

• 450+ experts on 30 subjects ready to help
• Custom essay delivered in as few as 3 hours

Get high-quality help

Verified writer

• Expert in: Science

+ 120 experts online

By clicking “Check Writers’ Offers”, you agree to our terms of service and privacy policy . We’ll occasionally send you promo and account related email

No need to pay just yet!

Related Essays

1 pages / 649 words

1 pages / 610 words

1 pages / 513 words

1 pages / 470 words

Remember! This is just a sample.

You can get your custom paper by one of our expert writers.

121 writers online

Still can’t find what you need?

Browse our vast selection of original essay samples, each expertly formatted and styled

Related Essays on Mathematics in Everyday Life

Background information on field experience school: Miami Coral Park Senior High School has 2,891 students of which 92% are Hispanic, 2% are Black, 4% White, and 2% Asian. 13% of the school’s students are ELL/LEP. The grades [...]

In this 'Mathematics in Architecture' essay, the researcher delves into the question if it is possible for architecture to stand alone without the help of mathematics? This research paper will also determine how Mathematics is [...]

Algebra, a branch of mathematics that involves symbols, equations, and variables, is often met with mixed feelings among students. Some see it as a challenging and abstract subject, while others recognize its practicality and [...]

Mathematics has thousands of branches, and each branch means something different to every person. Some may know it as a useful tool that is a key to getting civilizations rolling. Others may just see it as bothersome and a tough [...]

Archimedes compared the area enclosed by a circle to a right triangle whose base has the length of the circle’s circumference and whose height equals the circle’s radius. If the area of the circle is not equal to that of the [...]

Golden ration is a common mathematical ratio existing in the nature that is used to construct compositions in design work. The Golden ratio describes the perfectly symmetrical relationship between two proportions. It has been in [...]

Related Topics

By clicking “Send”, you agree to our Terms of service and Privacy statement . We will occasionally send you account related emails.

Where do you want us to send this sample?

By clicking “Continue”, you agree to our terms of service and privacy policy.

Be careful. This essay is not unique

This essay was donated by a student and is likely to have been used and submitted before

Download this Sample

Free samples may contain mistakes and not unique parts

Sorry, we could not paraphrase this essay. Our professional writers can rewrite it and get you a unique paper.

Please check your inbox.

We can write you a custom essay that will follow your exact instructions and meet the deadlines. Let's fix your grades together!

Get Your Personalized Essay in 3 Hours or Less!

We use cookies to personalyze your web-site experience. By continuing we’ll assume you board with our cookie policy .

• Instructions Followed To The Letter
• Deadlines Met At Every Stage
• Unique And Plagiarism Free

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career, collected their thoughts on mathematics (its aesthetic, purposes, methods, etc .) and on the work of a mathematician in written form.

For instance:

• W. Thurston wrote the lovely essay On proof and progress in mathematics in response to an article by Jaffe and Quinn ; some points made there are also presented in an answer given on MathOverflow ( What's a mathematician to do? ).
• More recently, T. Tao shared some personal thoughts and opinions on what makes "good quality mathematics" in What is good mathematics? .
• G. Hardy wrote the famous little book A Mathematician's Apology , which influenced, at least to some extent, several generations of mathematicians.

Personally, I've been greatly inspired by the two writings listed under (1.) -- they are one of the main reasons why I started studying mathematics -- and, considering that one of them appeared on MathOverflow , I'd like to propose here -- if it is appropriate -- to create a " big-list " of the kind of works described in the above blockquote.

I'd suggest (again, if it is appropriate) to give one title (or link) per answer with a short summary.

• A related question, which I've found very interesting, is Good papers/books/essays about the thought process behind mathematical research .
• Only slightly related (but surely interesting): Which mathematicians have influenced you the most?
• A single paper everyone should read? is not quite related, but still somewhat relevant (especially the most up-voted answer).
• reference-request
• soft-question
• 1 $\begingroup$ Hardy's apology is available here: math.ualberta.ca/~mss/misc/A%20Mathematician%27s%20Apology.pdf $\endgroup$ –  Goldstern Oct 5, 2015 at 15:33
• $\begingroup$ This seems a little broad--can you be a bit more specific? I gave one answer, but do you want things like Dyson's "Birds and Frogs" or Gower's "Two cultures"? $\endgroup$ –  Kimball Oct 5, 2015 at 22:29
• $\begingroup$ @Kimball, first of all, thanks for your answer, the book you suggested seems very interesting. Then, yes, I've read both those articles and, although they didn't come to my mind when I asked the question, they are surely two very insightful additions to this list. Thanks again. :) $\endgroup$ –  user81051 Oct 6, 2015 at 18:12

22 Answers 22

There are many snippets that can be found. I like the following bit of the foreword by Thurston to J. H. Hubbard's Teichmüller Theory . I share the remarks because I think you simply can't have enough of Bill Thurston's insights:

"Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then zooming away and leaving us stranded in a blank haze. Why? It is easy to forget that mathematics is primarily a tool for human thought. Mathematical thought is far better defined and far more logical than everyday thought, and people can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality. Logic, formalization, and symbols can be very powerful tools for humans to use, but we are actually very poor at purely formal reasoning; computers are far better at formal computation and formal reasoning, but humans are far better mathematicians. The most important thing about mathematics is how it resides in the human brain. Mathematics is not something we sense directly: it lives in our imagination and we sense it only indirectly. The choices of how it flows in our brains are not standard and automatic, and can be very sensitive to cues and context. Our minds depend on many interconnected special-purpose but powerful modules. We allocate everyday tasks to these various modules instinctively and subconsciously. The term `geometry', for instance, refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics. One illustration of this is the concept of correlation between two measurements on a set, which is formally nearly identical with the concept of cosine of the angle between two vectors. The content is almost the same (for correlation, you first project to a hyperplane before measuring the cosine of the angle), but the human psychology is very different. Each mode of thinking has its own power, and ideally, people harness both modes of thought to work together. However, in formalized expositions, this psychological > difference vanishes. In the same way, any idea in mathematics can be thought about in many different ways, with competing advantages. When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze. I've loved to read all my life. I went to New College of Sarasota, Florida, a small college that was just starting up with a strong emphasis on independent study, so I ended up learning a good deal of mathematics by reading mathematics books. At that time, I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I'd just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments. Even so, when something was ``left to the reader'', I generally left it as well. At the time, I could appreciate that the mathematics was an impressive intellectual edifice, and I could follow the steps of proofs. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited. It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)^n remark)^m style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind. John Hubbard approaches mathematics with his whole mind. If you page through the current book, you will see many intriguing figures. That is a first sign: figures are one of the most important ways to keep our thought processes going in our whole brains, rather than settling down into the linguistic, symbol-handling areas. Of course, the figures in your imagination are even more important. Geometric ideas can be conveyed with words and with symbols, sometimes more effectively than with pictures, but a lack of figures is a good indication of a lack of geometry. Another important part of human thinking is the emotional aspect. In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial; they are not incidental, they shape how we think. Personally, my thinking was shaped by boredom: I develop intense urges to come up with `easy' methods in order to avoid tedious computations that are opaque to me. Hubbard, a principal participant in the mathematics he is discussing, has done an excellent job in conveying the drama."

There are also many very good interviews that can be found, such as this one with Connes , as well as the advice to young mathematicians in the Princeton Companion to Mathematics .

A Mathematician's lament by Paul Lockhart: Reflections on how badly mathematics are taught these days. Imagining how it would be if music was taught the same way.

Indiscrete Thoughts by Gian-Carlo Rota and Discrete Thoughts by Kac, Rota, and Schwartz.

Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos: The sequence of steps through which mathematical ideas can be made to grow in an informal setting is explained through Socratic dialogues between a teacher and students. A beautiful read.

Since you mentioned A Mathematician's Apology : Michael Harris' Mathematics Without Apology .

Here's an excerpt explaining the title:

These attempts at justifications are the 'apologies' of the title. They usually take one of three forms. Pure research in mathematics as in other fields is good because it often leads to useful consequences (Steven Shapin calls this the Golden Goose argument); it is true because it offers a privileged access to certain truths; it is beautiful , an art form. To claim that these virtues are present in mathematics is not wrong, but it sheds little light on what is distinctively mathematical and even less about pure mathematicians' intentions . Intentions lie at the core of this book. I want to give the reader a sense of the mathematical life -- what it feels like to be a mathematician in a society of mathematicians where the first and second lives overlap.

Love and Math: The Heart of Hidden Reality by Edward Frenkel is, in my opinion, a lot better than Lockhart's lament.

The Mathematical Experience by Philip J. Davis and Reuben Hersh is a wonderful collection of essays on mathematics and on the experiences and culture of mathematicians. Written back in the 1980's, it has extremely insightful discussions of many of the same topics that nowadays are discussed on MO. For example, the essay "The Ideal Mathematician," which describes a hypothetical "ideal" mathematician working on the made-up area of "non-Riemannian hypersquares" is absolutely hilarious. Highly recommended!

• 1 $\begingroup$ The "Ideal Mathematician" is, to my mind, a poor mathematician. (It was a caricature, yes, but one which was a little too extreme for me.) $\endgroup$ –  Todd Trimble ♦ Oct 5, 2015 at 16:29
• 1 $\begingroup$ @ToddTrimble, I disliked it too. For myself, the more bearing what I'm working on has on undergraduate or even high-school mathematics, the more excited I am about it. $\endgroup$ –  goblin GONE Aug 23, 2016 at 14:55

Mathematics as Metaphor by Yuri Manin (both the title of the linked book which is a collection of essays, as well as the title of one particular essay in there). At least some of the essays you can find online.

I Want to be a Mathematician , by Paul Halmos.

• $\begingroup$ Indeed I love that book. Thanks for adding it. $\endgroup$ –  user81051 Oct 6, 2015 at 18:13

Eugene Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The statement that the laws of nature are written in the language of mathematics was probably made three hundred years ago [It is attributed to Galileo]. It is now more true than ever before … Surely complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is close to being a necessity in the formulation of the laws of quantum mechanics. It is difficult to avoid the impression that a miracle confronts us here , quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The closest explanation [for this mathematical universe] is Einstein’s statement that “the only physical theories which we are willing to accept are the beautiful ones” … the concepts of mathematics have this quality of beauty.

• 2 $\begingroup$ I have to disagree. Wigner's assertion that "mathematics is the science of skillful operations with concepts and rules invented just for this purpose" is the whole basis of his piece, and it doesn't have much to do with mathematics. The article is quasi-religious speculation based on this false premise. (The example that Wigner opens the article with is a case in point - he marvels at the appearance of $\sqrt{\pi}$ in the pdf for the normal distribution, as if this were magic. But probability theory was developed with very practical applications in mind.) $\endgroup$ –  Paul Levy May 23, 2017 at 9:31

A Drifter of Dadaist Persuasion by Matilde Marcolli, published in Art in the Life of Mathematicians (Edited by Anna Kepes Szemerédi) American Mathematical Society, 2015, pp.210-231

The Psychology of Invention in the Mathematical Field (Jacques Hadamard's 1945 essay)

• $\begingroup$ This book was very influential to me, and made a huge difference in helping me understand m own process of doing mathematics. $\endgroup$ –  Zach H Jul 17, 2017 at 17:13
• $\begingroup$ I love "the Poincare-Hadamard metaphor" described there! It says that our thoughts conscious and unconscious ones and their interactions could be explained via a mechanical model of states of a system of particles(the details inside). Very inspiring and still I haven't found an enough obstruction to the presented point of view there to the modern neuroscience, but I do not know much about it. An expertise needed! :) $\endgroup$ –  P. Grabowski Apr 14, 2020 at 18:42

The Mathematician by John Von Neumannn.

Enigmas of Chance , by Mark Kac.

I would add "Letters to a Young Mathematician" by Ian Stewart

I recommend:

Vladimir Arnold: Yesterday and Long Ago . This is a very enjoyable and highly interesting collection of anecdotes and historical remarks. The latest Russian edition of this book contains some more chapters. Richard Hamming: You and Your Research , transcribed and edited by J F Kaiser, reprinted in Tveito et al: Simula Research Laboratory . This is the text of a lecture of Hamming.

Birth of a Theorem , by French candidate for Parliament Cédric Villani

• 4 $\begingroup$ Now French member of Parliament Cédric Villani. $\endgroup$ –  Michael Lugo Jul 17, 2017 at 15:16

Here are additional mathematicians' thoughts.

S. Ulam, Adventures of a mathematician .A recollection of his life, from Lwow to Los Alamos. I am linking to excerpts. The book is still available for purchase.

Advices to a Young mathematician , a collection of advice and anecdotes by M. Atiyah, B. Bollobas, A. Connes, D. McDuff and P. Sarnak.

A. Borel, Art and science (Math. Intelligencer vol.5 1983, translation from German). A text for a general audience about the relationship between art and mathematics.

R. P. Langlands Is there beauty in mathematical theories? , this text is actually about number theory, old and new.

T. Gowers The two cultures of mathematics , another take on the dichotomy between problem solving and theory building.

A. Connes A view of mathematics , a thorough exposition of A. Connes'philosophical stance about space and physics. Targeted at a scientific audience.

D. Mumford, the dawning of the age of stochasticity , from algebraic geometry to statistics.

Y. Manin, Interrelations between Mathematics and Physics , on the divergence between mathematics and physics in the XXe century.

M. Gromov, ergobrain , one of the most surprising inquiry about life and mathematics.

I end that list with a text from a french mathematician about the future of mathematics: Poincare, l'avenir des mathematiques .

Perhaps a little broader in range/scope than the original question intended — but then again, perhaps not — the essays collected in

Mathématiques, mathematiciens et société. Publications Mathématiques d'Orsay no. 86 74-16 (1974)

I was led to this when someone somewhere posted a link to Vergne's Témoignage d'une mathématicienne , which is one of the essays in this volume, and — I must confess — is the only one I've read, although the other ones do look interesting

In the Princeton Companion to Mathematics , there is a section entitled Advice to a Young Mathematician (pdf), containing essays by Atiyah, Bollobás, Connes, McDuff and Sarnak.

A Mathematician's Miscellany (reprinted, with additional material, as Littlewood's Miscellany by CUP in 1986) is worthwhile reading.

Clifford Truesdell published a series of essays as An Idiot's Fugitive Essays on Science Methods, Criticism, Training, Circumstances (Springer, 1984), which sets out in a forthright manner the author's views on mathematics and science.

A really nice article by Andrei Toom about mathematical education, especially in the US, got recently mentioned in a comment to this question.

• Featured on Meta
• Our Partnership with OpenAI
• Imgur image URL migration: Coming soon to a Stack Exchange site near you!

• Mathematics

What is Mathematics?

Mathematics  is the science and study of quality, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions". Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Through abstraction and logical reasoning mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, in fitful bursts, until the Renaissance, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.

Today, mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.   

• For Prospective Students
• Alumni and Careers
• Undergraduate Program
• Fast Track Program
• B.S. in Math Goals and Reports
• Graduate Program
• Degree Requirements
• Application Instructions
• M.S. in Math Goals and Reports
• Master's Degrees Awarded
• Course Descriptions and Syllabi
• Scholarship Information
• Frequently Asked Questions
• Resources - Calculus Readiness Tools

Why Study Mathematics?

Because it's fun and can prepare you for a variety of excellent careers! If you like solving puzzles and figuring things out, then a mathematics major may interest you. In addition, applications of mathematics are everywhere and a strong background in mathematics can help you in many different careers.

The sections below provide information about careers in mathematics and the opportunities available to our mathematics majors.

Careers The following links are to pages that provide information about the careers available to students of mathematics.

American Mathematical Society

American Statistical Association

This is Statistics

Mathematical Association of America

Society of Industrial and Applied Mathematics (SIAM)

Society of Actuaries

Undergraduate Research If you are considering graduate school in mathematics, you should think about participating in some research as an undergraduate. There are opportunities to do so with professors in our department or at other institutions during the summer at an REU (Research Experience for Undergraduates). REUs are usually four to eight weeks long and typically pay the student a stipend.

Bold. Fearless. Confident. Kind. Resilient.

Visit us to see what sets us apart.

Quick Links

• Tech at a Glance
• Majors & Concentrations
• Colleges & Schools
• Student Life
• Research at Tech
• Tech Express
• Current Students
• Faculty & Staff
• Mission and Vision
• Facts about Tech
• University Rankings
• Accreditation & Memberships
• Maps & Directions
• Board of Trustees
• Office of the President
• Strategic Plan
• History of Tech
• Parents & Family
• International
• Military & Veteran Affairs
• Tuition & Fees
• Financial Aid
• Visit Campus
• Scholarships
• Dual Enrollment
• Request Information
• Office of the Provost
• Academic Calendar
• Undergraduate Catalog
• Graduate Catalog
• Volpe Library
• Student Success Centers
• Honors Program
• Study Abroad
• Living On Campus
• Health & Wellness
• Get Involved
• Student Organizations
• Safety & Security
• Services for Students
• Upcoming Events
• Diversity Resources
• Student Affairs
• Featured Researchers
• Research Centers
• ttusports.com
• Social Media
• Student Resources
• Faculty & Staff Resources
• Bookstore/Dining/Parking
• Pay Online - Eagle Pay
• IT Help Desk
• Strategic Planning
• Office of IARE
• Student Complaints

• Math Careers

Search form

• MAA Centennial
• Spotlight: Archives of American Mathematics
• MAA Officers
• MAA to the Power of New
• Council and Committees
• MAA Code of Conduct
• Policy on Conflict of Interest
• Statement about Conflict of Interest
• Recording or Broadcasting of MAA Events
• Policy for Establishing Endowments and Funds
• Avoiding Implicit Bias
• Copyright Agreement
• Principal Investigator's Manual
• Planned Giving
• The Icosahedron Society
• Our Partners
• Advertise with MAA
• Employment Opportunities
• Staff Directory
• 2022 Impact Report
• In Memoriam
• Membership Categories
• Become a Member
• Membership Renewal
• MERCER Insurance
• MAA Member Directories
• New Member Benefits
• The American Mathematical Monthly
• Mathematics Magazine
• The College Mathematics Journal
• How to Cite
• Communications in Visual Mathematics
• About Convergence
• What's in Convergence?
• Convergence Articles
• Mathematical Treasures
• Portrait Gallery
• Paul R. Halmos Photograph Collection
• Other Images
• Critics Corner
• Problems from Another Time
• Conference Calendar
• Guidelines for Convergence Authors
• Math Horizons
• Submissions to MAA Periodicals
• Guide for Referees
• Scatterplot
• Math Values
• MAA Book Series
• MAA Press (an imprint of the AMS)
• MAA Library Recommendations
• Additional Sources for Math Book Reviews
• About MAA Reviews
• Mathematical Communication
• Information for Libraries
• Author Resources
• MAA MathFest
• Proposal and Abstract Deadlines
• MAA Policies
• Invited Paper Session Proposals
• Contributed Paper Session Proposals
• Panel, Poster, Town Hall, and Workshop Proposals
• Minicourse Proposals
• MAA Section Meetings
• Virtual Programming
• Joint Mathematics Meetings
• Calendar of Events
• MathFest Programs Archive
• MathFest Abstract Archive
• Historical Speakers
• Information for School Administrators
• Information for Students and Parents
• Registration
• Getting Started with the AMC
• AMC Policies
• AMC Administration Policies
• Important AMC Dates
• Competition Locations
• Invitational Competitions
• Putnam Competition Archive
• AMC International
• Curriculum Inspirations
• Sliffe Award
• MAA K-12 Benefits
• Mailing List Requests
• Statistics & Awards
• Submit an NSF Proposal with MAA
• MAA Distinguished Lecture Series
• Common Vision
• CUPM Curriculum Guide
• Instructional Practices Guide
• Möbius MAA Placement Test Suite
• META Math Webinar May 2020
• Progress through Calculus
• Survey and Reports
• "Camp" of Mathematical Queeries
• DMEG Awardees
• National Research Experience for Undergraduates Program (NREUP)
• Neff Outreach Fund Awardees
• Tensor SUMMA Grants
• Tensor Women & Mathematics Grants
• Grantee Highlight Stories
• "Best Practices" Statements
• CoMInDS Summer Workshop 2023
• MAA Travel Grants for Project ACCCESS
• 2024 Summer Workshops
• Minority Serving Institutions Leadership Summit
• Previous Workshops
• Frequently Asked Questions
• Course Resources
• Industrial Math Case Studies
• Participating Faculty
• 2020 PIC Math Student Showcase
• Previous PIC Math Workshops on Data Science
• Dates and Locations
• Past Programs
• Leadership Team
• Support Project NExT
• Section NExT
• Section Officers Meeting History
• Preparations for Section Meetings
• Bylaws Template
• Editor Lectures Program
• MAA Section Lecturer Series
• Officer Election Support
• Section Awards
• Section Liaison Programs
• Section Visitors Program
• Expense Reimbursement
• Guidelines for Bylaw Revisions
• Guidelines for Local Arrangement Chair and/or Committee
• Guidelines for Section Webmasters
• MAA Logo Guidelines
• MAA Section Email Policy
• Section Newsletter Guidelines
• Statement on Federal Tax ID and 501(c)3 Status
• Communication Support
• Guidelines for the Section Secretary and Treasurer
• Legal & Liability Support for Section Officers
• Section Marketing Services
• Section in a Box
• Subventions and Section Finances
• Web Services
• Joining a SIGMAA
• Forming a SIGMAA
• History of SIGMAA
• SIGMAA Officer Handbook
• MAA Connect
• Meetings and Conferences for Students
• Opportunities to Present
• Information and Resources
• MAA Undergraduate Student Poster Session
• Undergraduate Research Resources
• MathFest Student Paper Sessions
• Research Experiences for Undergraduates
• Student Poster Session FAQs
• High School
• A Graduate School Primer
• Reading List
• Student Chapters
• Awards Booklets
• Carl B. Allendoerfer Awards
• Regulations Governing the Association's Award of The Chauvenet Prize
• Trevor Evans Awards
• Paul R. Halmos - Lester R. Ford Awards
• Merten M. Hasse Prize
• George Pólya Awards
• David P. Robbins Prize
• Beckenbach Book Prize
• Euler Book Prize
• Daniel Solow Author’s Award
• Henry L. Alder Award
• Deborah and Franklin Tepper Haimo Award
• Certificate of Merit
• Gung and Hu Distinguished Service
• JPBM Communications Award
• Meritorious Service
• MAA Award for Inclusivity
• T. Christine Stevens Award
• Dolciani Award Guidelines
• Morgan Prize Information
• Selden Award Eligibility and Guidelines for Nomination
• Selden Award Nomination Form
• AMS-MAA-SIAM Gerald and Judith Porter Public Lecture
• Etta Zuber Falconer
• Hedrick Lectures
• James R. C. Leitzel Lecture
• Pólya Lecturer Information
• Putnam Competition Individual and Team Winners
• D. E. Shaw Group AMC 8 Awards & Certificates
• Maryam Mirzakhani AMC 10 A Awards & Certificates
• Two Sigma AMC 10 B Awards & Certificates
• Jane Street AMC 12 A Awards & Certificates
• Akamai AMC 12 B Awards & Certificates
• High School Teachers
• MAA Social Media

You are here

What is mathematics, really.

• From Amazon
• From the MAA Bookstore

temp hardcoded node--book.tpl.php

Reuben Hersh

• Table of Contents

This is an interesting, important, ambitious, and infuriating book, one that deserves both attention and response from the mathematics community. It has many good things to say, and it has the ambition to reshape the debate on the philosophy of mathematics.

There was a time, early in the twentieth century, in which mathematicians were passionately interested in the philosophy of mathematics. People were deeply concerned about what mathematics is, what sort of existence mathematical objects have, and their opinions on these questions actually influenced the mathematics they did.

There were three major points of view in the debate about the nature of mathematics. The formalists argued (roughly: the short summaries that follow are really caricatures) that mathematics was really simply the formal manipulation of symbols based on arbitrarily-chosen axioms. The Platonists saw mathematics as almost an experimental science, studying objects that really exist (in some sense), though they clearly don't exist in a physical or material sense. The intuitionists had the most radical point of view; essentially, they saw all mathematics as a human creation and therefore as essentially finite. The intuitionists refused to have any dealings with completed infinite sets, rejected the "law of the excluded middle" (i.e., the claim that a mathematical statement is always either true or false), and were willing to give up large tracts of classical analysis that didn't fit this point of view.

The fact that the debate never really got resolved, together with the complicating factor of Gödel's incompleteness theorems, seem to have caused most mathematicians to lose interest. In recent years, most mathematicians seem to have been content with an attitude best described by Jean Dieudonné. In everyday life, we speak as Platonists, treating the objects of our study as real things that exist independently of human thought. If challenged on this, however, we retreat to some sort of formalism, arguing that in fact we are just pushing symbols around without making any metaphysical claims. Most of all, however, we want to do mathematics rather than argue about what it actually is. We're content to leave that to the philosophers.

Reuben Hersh wants to change this. His book is an attempt to get us all involved in the debate about the nature of mathematics. To this end, he does a number of things. First, he argues that most writing on the foundations of mathematics is woefully ignorant of actual mathematical practice. Second, he tries to break the three-way tie by making a new proposal as to what mathematics really is. Third, he runs through the history of the philosophy of mathematics to argue that (a) his position is not really new, but has a distinguished pedigree, and (b) that all the other positions are clearly wrong. Finally, he connects philosophical positions on the nature of mathematics to broader philosophical and political issues.

The foundational debates of the early twentieth century turned on the issue of certainty . Everyone agreed that mathematical statements were true in an absolute sense, that one could be certain that they were true. The issue was to find a philosophy of mathematics that guaranteed that certainty. Hersh's position is that the desire for certainty is simply a mistake. In fact, he argues, regardless of our ideals, mathematics is done by fallible people, and so the traditional philosophies cannot really guarantee certainty. So let's give it up: mathematics is a human endeavor, and mathematical truths are uncertain like any other truths.

But if we locate mathematics as a human construction, we need to account for the very strong feeling that mathematical objects have some sort of independent existence. The number pi (or the number two) is not just something in my head! Hersh agrees, and proposes that mathematical "objects" are really socio-cultural constructs. As such, they really do transcend individual minds even while remaining human creations. Hersh calls this view of mathematics humanism .

So far so good: it's an interesting proposal, and has much in its favor. It makes modest claims for mathematics which actually correspond to our human experience as mathematicians, and it takes seriously the fact that mathematics is learned and taught. (It is also, which Hersh does not observe, completely compatible with a Platonist understanding of mathematical objects; all that one needs to do is to delete the (implicit) "merely" in Hersh's claim that mathematical objects are socio-cultural constructs. Our mathematical concepts can certainly be socio-cultural constructs that attempt to grasp and understand Platonist mathematical objects...) The humanist view allows us to escape from metaphysics and to focus attention on things we can actually observe first-hand: the mathematical community and how it learns, teaches, and develops mathematics.

There are other persuasive things about "humanism." For example, it is deeply aware that mathematics has a history . Both formalism and Platonism often give the impression that they deal with mathematics as a completed product, when in fact mathematics is produced by people working in socio-cultural contexts. A look at the history of mathematics certainly seems to undermine a naive formalism (since "formal proof" is a relatively recent phenomenon). History also asks difficult questions for Platonism; does it really make sense to claim that, say, Galois groups "really exist", and that they existed even in Euclid's time?

On the other hand, the humanist view reduces our feeling that mathematical truths are really true to a social consensus, something we learn. This opens the possibility that Little Green People from Mars, if they exist, have a mathematics that is not only different from, but actually contradictory to ours. This is a position that many mathematicians find extremely hard to take.

One should emphasize that the evidence for some sort of "certainty" in mathematical truths goes beyond our intuitive feelings about them. The history of mathematics also points in this direction. We may no longer accept some of Euclid's arguments as rigorous, but we do think every one of his theorems is still true ! What other science can make such a claim? Or consider Fermat's Last Theorem: is there any other field of human endeavor in which a question posed in 1636 can still make sense, in exactly the original terms, 350 years later?

In arguing for his humanist philosophy of mathematics, Hersh has some very good points to make, but he seems to spend much more time attacking the rival positions of formalism and Platonism. Against formalism, he follows and develops the criticisms of Imre Lakatos: he argues that no one actually writes formal proofs of anything, and that the view that mathematics is simply meaningless symbol-pushing is impossible to believe. Against Platonism, he argues that believing in eternal mathematical objects existing independently of human thought is only possible if one believes that God exists, which, he says, no one does anymore.

While there is something to both arguments, neither of them really takes the opposing philosophical position seriously. Formalism is more solid than Hersh makes it seem, Platonism has been the position of serious philosophers who were not theists, and, after all, many philosophers do believe that God exists. So do many mathematicians. This points up the basic problem: all too often, Hersh is willing to dismiss the opinions of eminent thinkers after only a very shallow interaction with their thought.

As a result, Hersh's historical survey of the philosophy of mathematics is, to my mind, the weakest part of the book. The quick sketches of the thought of various philosophers really do justice to no one. (They remind me of John Rist's description of Bertrand Russell's History of Western Philosophy : "sophistic and simplistic misinterpretations of most Western ethics and metaphysics.") An example is his treatment of George Berkeley's very complex philosophy of mathematics, which gets reduced to "using the deficiency of mathematics to support religion." He then adds "His attack on mathematicians is unique since St. Augustine," which is hard to understand, since in his section on St. Augustine Hersh explains that Augustine's "attack on mathematicians" is really nothing of the sort!

Even less convincing is Hersh's attempt, in the last chapter, to correlate philosophers' positions on mathematics with their political stances, in order to reach the conclusion that "the Platonist view of number is associated with political conservatism, and the humanist view of number with democratic politics." Aside from the fact that he takes as axiomatic that being associated with the latter is better, the whole argument is based on a painfully arbitrary distinction of left versus right. (David Hume as a leftist?!).

The book concludes with an invocation of the story of the blind men and the elephant "as a metaphor for the philosophy of mathematics, with its Wise Men groping at the wondrous beast, Mathematics." All except, it seems, the humanist, who can see the whole elephant and laugh at the fallibility of all the others. Hersh doesn't seem to realize how arrogant this attitude is.

What I think is missing from the book is a realization that the philosophy of mathematics is indeed philosophy, and not science, and that therefore it cannot ignore the overarching philosophical issues that relate to it. It is clear that one's beliefs on metaphysics and epistemology, and particularly one's stance with respect to the notion of truth , are going to have an enormous impact on one's positions on the nature of mathematical objects and mathematical truths.

However, despite the serious limitations of Hersh's treatment of other thinkers, and despite the fact that he does not argue for his "humanist" position as forcefully as he might have, this is still a book that deserves attention. It should be widely read, discussed, and argued with. I hope it stimulates many responses, and most of all that it manages to convince more mathematicians of the importance of the questions with which it deals.

Fernando Gouvêa ( [email protected] ) is chair of the Department of Mathematics and Computer Science at Colby College and editor of MAA Online.

• Log in to post comments

Dummy View - NOT TO BE DELETED

MAA Publications

• Periodicals
• MAA History
• Policies and Procedures
• Support MAA
• Member Discount Programs
• Propose a Session
• MathFest Archive
• Putnam Competition
• AMC Resources
• Curriculum Resources
• Outreach Initiatives
• Professional Development
• Communities

Connect with MAA

Mathematical Association of America P: (800) 331-1622 F: (240) 396-5647 Email: [email protected]

Copyright © 2024

• Terms of Use
• Privacy Policy
• Mobile Version

• Share full article

Advertisement

Supported by

current events conversation

What Students Are Saying About the Value of Math

We asked teenagers: Do you see the point in learning math? The answer from many was “yes.”

By The Learning Network

“Mathematics, I now see, is important because it expands the world,” Alec Wilkinson writes in a recent guest essay . “It is a point of entry into larger concerns. It teaches reverence. It insists one be receptive to wonder. It requires that a person pay close attention.”

In our writing prompt “ Do You See the Point in Learning Math? ” we wanted to know if students agreed. Basic arithmetic, sure, but is there value in learning higher-level math, such as algebra, geometry and calculus? Do we appreciate math enough?

The answer from many students — those who love and those who “detest” the subject alike — was yes. Of course math helps us balance checkbooks and work up budgets, they said, but it also helps us learn how to follow a formula, appreciate music, draw, shoot three-pointers and even skateboard. It gives us different perspectives, helps us organize our chaotic thoughts, makes us more creative, and shows us how to think rationally.

Not all were convinced that young people should have to take higher-level math classes all through high school, but, as one student said, “I can see myself understanding even more how important it is and appreciating it more as I get older.”

Thank you to all the teenagers who joined the conversation on our writing prompts this week, including students from Bentonville West High School in Centerton, Ark, ; Harvard-Westlake School in Los Angeles ; and North High School in North St. Paul, Minn.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

“Math is a valuable tool and function of the world.”

As a musician, math is intrinsically related to my passion. As a sailor, math is intertwined with the workings of my boat. As a human, math is the building block for all that functions. When I was a child, I could very much relate to wanting a reason behind math. I soon learned that math IS the reason behind all of the world’s workings. Besides the benefits that math provides to one’s intellect, it becomes obvious later in life that math is a valuable tool and function of the world. In music for example, “adolescent mathematics” are used to portray functions of audio engineering. For example, phase shifting a sine wave to better project sound or understanding waves emitted by electricity and how they affect audio signals. To better understand music, math is a recurring pattern of intervals between generating pitches that are all mathematically related. The frets on a guitar are measured precisely to provide intervals based on a tuning system surrounding 440Hz, which is the mathematically calculated middle of the pitches humans can perceive and a string can effectively generate. The difference between intervals in making a chord are not all uniform, so guitar frets are placed in a way where all chords can sound equally consonant and not favor any chord. The power of mathematics! I am fascinated by the way that math creeps its way into all that I do, despite my plentiful efforts to keep it at a safe distance …

— Renan, Miami Country Day School

“Math isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life.”

I believe learning mathematics is both crucial to the learning and development of 21st century students and yet also not to be imposed upon learners too heavily. Aside from the rise in career opportunity in fields centered around mathematics, the skills gained while learning math are able to be translated to many facets of life after a student’s education. Learning mathematics develops problem solving skills which combine logic and reasoning in students as they grow. The average calculus student may complain of learning how to take derivatives, arguing that they will never have to use this after high school, and in that, they may be right. Many students in these math classes will become writers, musicians, or historians and may never take a derivative in their life after high school, and thus deem the skill to do so useless. However, learning mathematics isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life. A student who excels at calculus may never use it again, but with the skills of creativity and rational thinking presented by this course, learning mathematics will have had a profound effect on their life.

— Cam, Glenbard West

“Just stop and consider your hobbies and pastimes … all of it needs math.”

Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works. I love math — especially algebra and geometry — as it all follows a formula, and if you set it up just right, you can create almost anything you want in at least two different ways. Just stop and consider your hobbies and pastimes. You could be into skateboarding, basketball, or skiing. You could be like me, and sit at home for hours on end grinding out solves on a Rubik’s cube. Or you could be into sketching. Did you know that a proper drawing of the human face places the eyes exactly halfway down from the top of the head? All of it needs math. Author Alec Wilkinson, when sharing his high school doubting view on mathematics, laments “If I had understood how deeply mathematics is embedded in the world …” You can’t draw a face without proportions. You can’t stop with your skis at just any angle. You can’t get three points without shooting at least 22 feet away from the basket, and get this: you can’t even ride a skateboard if you can’t create four congruent wheels to put on it.

— Marshall, Union High School, Vancouver, WA

“Math gives us a different perspective on everyday activities.”

Even though the question “why do we even do math?” is asked all the time, there is a deeper meaning to the values it shares. Math gives us a different perspective on everyday activities, even if those activities in our routine have absolutely nothing to do with mathematical concepts itself. Geometry, for instance, allows us to think on a different level than simply achieving accuracy maintains. It trains our mind to look at something from various viewpoints as well as teaching us to think before acting and organizing chaotic thoughts. The build up of learning math can allow someone to mature beyond the point where if they didn’t learn math and thought through everything. It paves a way where we develop certain characteristics and traits that are favorable when assisting someone with difficult tasks in the future.

— Linden, Harvard-Westlake High School, CA

“Math teaches us how to think.”

As explained in the article, math is all around us. Shapes, numbers, statistics, you can find math in almost anything and everything. But is it important for all students to learn? I would say so. Math in elementary school years is very important because it teaches how to do simple calculations that can be used in your everyday life; however middle and high school math isn’t used as directly. Math teaches us how to think. It’s far different from any other subject in school, and truly understanding it can be very rewarding. There are also many career paths that are based around math, such as engineering, statistics, or computer programming, for example. These careers are all crucial for society to function, and many pay well. Without a solid background in math, these careers wouldn’t be possible. While math is a very important subject, I also feel it should become optional at some point, perhaps part way through high school. Upper level math classes often lose their educational value if the student isn’t genuinely interested in learning it. I would encourage all students to learn math, but not require it.

— Grey, Cary High School

“Math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful.”

Although I loved math as a kid, as I got older it felt more like a chore; all the kids would say “when am I ever going to use this in real life?” and even I, who had loved math, couldn’t figure out how it benefits me either. This was until I started asking my dad for help with my homework. He would go on and on about how he used the math I was learning everyday at work and even started giving me examples of when and where I could use it, which changed my perspective completely. Ultimately, I believe that math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful and where they can use it outside of class.

— Lilly, Union High School

“At the roots of math, it teaches people how to follow a process.”

I do believe that the math outside of arithmetic, percentages, and fractions are the only math skills truly needed for everyone, with all other concepts being only used for certain careers. However, at the same time, I can’t help but want to still learn it. I believe that at the roots of math, it teaches people how to follow a process. All mathematics is about following a formula and then getting the result of it as accurately as possible. It teaches us that in order to get the results needed, all the work must be put and no shortcuts or guesses can be made. Every equation, number, and symbol in math all interconnect with each other, to create formulas that if followed correctly gives us the answer needed. Everything is essential to getting the results needed, and skipping a step will lead to a wrong answer. Although I do understand why many would see no reason to learn math outside of arithmetic, I also see lessons of work ethics and understanding the process that can be applied to many real world scenarios.

— Takuma, Irvine High School

“I see now that math not only works through logic but also creativity.”

A story that will never finish resembling the universe constantly expanding, this is what math is. I detest math, but I love a never-ending tale of mystery and suspense. If we were to see math as an adventure it would make it more enjoyable. I have often had a closed mindset on math, however, viewing it from this perspective, I find it much more appealing. Teachers urge students to try on math and though it seems daunting and useless, once you get to higher math it is still important. I see now that math not only works through logic but also creativity and as the author emphasizes, it is “a fundamental part of the world’s design.” This view on math will help students succeed and have a more open mindset toward math. How is this never-ending story of suspense going to affect YOU?

— Audrey, Vancouver, WA union high school

“In some word problems, I encounter problems that thoroughly interest me.”

I believe math is a crucial thing to learn as you grow up. Math is easily my favorite subject and I wish more people would share my enthusiasm. As Alec Wilkinson writes, “Mathematics, I now see, is important because it expands the world.” I have always enjoyed math, but until the past year, I have not seen a point in higher-level math. In some of the word problems I deal with in these classes, I encounter problems that thoroughly interest me. The problems that I am working on in math involve the speed of a plane being affected by wind. I know this is not riveting to everyone, but I thoroughly wonder about things like this on a daily basis. The type of math used in the plane problems is similar to what Alec is learning — trigonometry. It may not serve the most use to me now, but I believe a thorough understanding of the world is a big part of living a meaningful life.

— Rehan, Cary High School

“Without high school classes, fewer people get that spark of wonder about math.”

I think that math should be required through high school because math is a use-it-or-lose-it subject. If we stop teaching math in high school and just teach it up to middle school, not only will many people lose their ability to do basic math, but we will have fewer and fewer people get that spark of wonder about math that the author had when taking math for a second time; after having that spark myself, I realized that people start getting the spark once they are in harder math classes. At first, I thought that if math stopped being required in high school, and was offered as an elective, then only people with the spark would continue with it, and everything would be okay. After thinking about the consequences of the idea, I realized that technology requires knowing the seemingly unneeded math. There is already a shortage of IT professionals, and stopping math earlier will only worsen that shortage. Math is tricky. If you try your best to understand it, it isn’t too hard. However, the problem is people had bad math teachers when they were younger, which made them hate math. I have learned that the key to learning math is to have an open mind.

— Andrew, Cary High School

“I think math is a waste of my time because I don’t think I will ever get it.”

In the article Mr. Wilkinson writes, “When I thought about mathematics at all as a boy it was to speculate about why I was being made to learn it, since it seemed plainly obvious that there was no need for it in adult life.” His experience as a boy resonates with my experience now. I feel like math is extremely difficult at some points and it is not my strongest subject. Whenever I am having a hard time with something I get a little upset with myself because I feel like I need to get everything perfect. So therefore, I think it is a waste of my time because I don’t think I will ever get it. At the age of 65 Mr. Wilkinson decided to see if he could learn more/relearn algebra, geometry and calculus and I can’t imagine myself doing this but I can see myself understanding even more how important it is and appreciating it more as I get older. When my dad was young he hated history but, as he got older he learned to appreciate it and see how we can learn from our past mistakes and he now loves learning new things about history.

— Kate, Cary High School

“Not all children need to learn higher level math.”

The higher levels of math like calculus, algebra, and geometry have shaped the world we live in today. Just designing a house relates to math. To be in many professions you have to know algebra, geometry, and calculus such as being an economist, engineer, and architect. Although higher-level math isn’t useful to some people. If you want to do something that pertains to math, you should be able to do so and learn those high levels of math. Many things children learn in math they will never use again, so learning those skills isn’t very helpful … Children went through so much stress and anxiety to learn these skills that they will never see again in their lives. In school, children are using their time learning calculus when they could be learning something more meaningful that can prepare them for life.

— Julyssa, Hanover Horton High School

“Once you understand the basics, more math classes should be a choice.”

I believe that once you get to the point where you have a great understanding of the basics of math, you should be able to take more useful classes that will prepare you for the future better, rather than memorizing equations after equations about weird shapes that will be irrelevant to anything in my future. Yes, all math levels can be useful to others’ futures depending on what career path they choose, but for the ones like me who know they are not planning on encountering extremely high level math equations on the daily, we should not have to take math after a certain point.

— Tessa, Glenbard West High School

“Math could shape the world if it were taught differently.”

If we learned how to balance checkbooks and learn about actual life situations, math could be more helpful. Instead of learning about rare situations that probably won’t come up in our lives, we should be learning how to live on a budget and succeed money-wise. Since it is a required class, learning this would save more people from going into debt and overspending. In schools today, we have to take a specific class that doesn’t sound appealing to the average teenager to learn how to save and spend money responsibly. If it was required in math to learn about that instead of how far Sally has to walk then we would be a more successful nation as a whole. Math could shape the world differently but the way it is taught in schools does not have much impact on everyday life.

— Becca, Bentonville West High School

“To be honest, I don’t see the point in learning all of the complicated math.”

In a realistic point of view, I need to know how to cut a cake or a piece of pie or know how to divide 25,000 dollars into 10 paychecks. On the other hand, I don’t need to know the arc and angle. I need to throw a piece of paper into a trash can. I say this because, in all reality and I know a lot of people say this but it’s true, when are we actually going to need this in our real world lives? Learning complicated math is a waste of precious learning time unless you desire to have a career that requires these studies like becoming an engineer, or a math professor. I think that the fact that schools are still requiring us to learn these types of mathematics is just ignorance from the past generations. I believe that if we have the technology to complete these problems in a few seconds then we should use this technology, but the past generations are salty because they didn’t have these resources so they want to do the same thing they did when they were learning math. So to be honest, I don’t see the point in learning all of the complicated math but I do think it’s necessary to know the basic math.

— Shai, Julia R Masterman, Philadelphia, PA

Learn more about Current Events Conversation here and find all of our posts in this column .

Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

More from CBS News

As a Social Security cut looms, should seniors buy long-term care insurance now?

"Absolutely stunning" rare electric blue lobster caught in England

Thousands of students who crossed border from Mexico to U.S. set to graduate

"Operation Catch a Toe" leads to murder suspect with distinctive foot

Featured Topics

Featured series.

A series of random questions answered by Harvard experts.

Explore the Gazette

Read the latest.

Roger Ware Brockett, 84

Providing community support

A change of mind, heart, and soul

So how do you track spread of disease by the numbers.

Kris Snibbe/Harvard Staff Photographer

Anne J. Manning

Harvard Staff Writer

Ivan Specht decided to employ his love of math during pandemic, which led to contact-tracing app, papers, future path

Part of the commencement 2024 series.

A collection of stories covering Harvard University’s 373rd Commencement.

Ivan Specht started at Harvard on track to study pure mathematics. But when COVID-19 sent everyone home, he began wishing the math he was doing had more relevance to what was happening in the world.

Specht, a New York City native, expanded his coursework, arming himself with statistical modeling classes, and began to “fiddle around” with simulating ways diseases spread through populations. He got hooked. During the pandemic, he became one of only two undergraduates to serve on Harvard’s testing and tracing committee, eventually developing a prototype contact-tracing app called CrimsonShield.

Specht took his curiosity for understanding disease propagation to the lab of computational geneticist Pardis Sabeti , professor in Organismic and Evolutionary Biology at Harvard and member of the Broad Institute, known for her work sequencing the Ebola virus in 2014 . Specht, now a senior, has since co-authored several studies around new statistical methods for analyzing the spread of infectious diseases, with plans to continue that work in graduate school.

“Ivan is absolutely brilliant and a joy to work with, and his research accomplishments already as an undergraduate are simply astounding,” Sabeti said. “He is operating at the level of a seasoned postdoc.”

His senior thesis, “Reconstructing Viral Epidemics: A Random Tree Approach,” described a statistical model aimed at tackling one of the most intractable problems that plague infectious disease researchers: determining who transmitted a given pathogen to whom during a viral outbreak. Specht was co-advised by computer science Professor Michael Mitzenmacher, who guided the statistical and computational sections of his thesis, particularly in deriving genomic frequencies within a host using probabilistic methods.

Specht said the pandemic made clear that testing technology could provide valuable information about who got sick, and even what genetic variant of a pathogen made them sick. But mapping paths of transmission was much more challenging because that process was completely invisible. Such information, however, could provide crucial new details into how and where transmission occurred and be used to test things such as vaccine efficacy or the effects of closing schools. 

Specht’s work exploited the fact that viruses leave clues about their transmission path in their phylogenetic trees, or lines of evolutionary descent from a common ancestor. “It turns out that genome sequences of viruses provide key insight into that underlying network,” said the joint mathematics and statistics concentrator.

Uncovering this transmission network goes to the heart of how single-stranded RNA pathogens survive: Once they infect their host, they mutate, producing variants that are marked by slightly different genetic barcodes. Specht’s statistical model determines how the virus spreads by tracking the frequencies of different viral variants observed within a host.  

As the centerpiece of his thesis, he reconstructed a dataset of about 45,000 SARS-CoV-2 genomes across Massachusetts, providing insights into how outbreaks unfolded across the state.

Specht will take his passion for epidemiological modeling to graduate school at Stanford University, with an eye toward helping both researchers and communities understand and respond to public health crises.

A graphic designer with experience in scientific data visualization, Specht is focused not only on understanding outbreaks, but also creating clear illustrations of them. For example, his thesis contains a creative visual representation of those 45,000 Massachusetts genomes, with colored dots representing cases, positioned nearby other “dots” they are likely to have infected.

Specht’s interest in graphics began in middle school when, as an enthusiast of trains and mass transit, he started designing imagined subway maps for cities that lack actual subways, like Austin, Texas . At Harvard, he designed an interactive “subway map” depicting a viral outbreak.

As a member of the Sabeti lab, Specht taught an infectious disease modeling course to master’s and Ph.D. students at University of Sierra Leone last summer. His outbreak analysis tool is also now being used in an ongoing study of Lassa fever in that region. And he co-authored two chapters of a textbook on outbreak science in collaboration with the Moore Foundation.

Over the past three years, Specht has been lead author of a paper in Scientific Reports and another in Cell Patterns , and co-author on two others, including a cover story in Cell . His first lead-author paper, “The case for altruism in institutional diagnostic testing,” showed that organizations like Harvard should allocate COVID-19 testing capacity to their surrounding communities, rather than monopolize it for themselves. That work was featured in The New York Times .

During his time at Harvard, Specht lived in Quincy House and was design editor of the Harvard Advocate, the University’s undergraduate literary magazine. In his free time he also composes music, and he still considers himself a mass transit enthusiast.

In the acknowledgements section of his thesis, he credited Sabeti with opening his eyes to the “many fascinating problems at the intersection of math, statistics, and computational biology.”

“I could fill this entire thesis with reasons I am grateful for Professor Sabeti, but I think they can be summarized by the sense of wonder and inspiration I feel every time I set foot in her lab.”

Share this article

You might like.

Memorial Minute — Faculty of Arts and Sciences

Harvard Allston Partnership Fund awards grants to 26 Allston-Brighton nonprofits

Choosing Harvard took LyLena Estabine down an uncertain path. The former student co-president has no regrets.

Excited about new diet drug? This procedure seems better choice.

Study finds minimally invasive treatment more cost-effective over time, brings greater weight loss

How far has COVID set back students?

An economist, a policy expert, and a teacher explain why learning losses are worse than many parents realize

Epic science inside a cubic millimeter of brain

Researchers publish largest-ever dataset of neural connections

• Business Today
• India Today
• India Today Gaming
• Cosmopolitan
• Harper's Bazaar
• Brides Today
• Aajtak Campus

• Magazine Cover Story Editor's Note Deep Dive Interview The Buzz
• BT TV Market Today Easynomics Drive Today BT Explainer
• Market Today Trending Stocks Indices Stocks List Stocks News Share Market News IPO Corner
• Tech Today Unbox Today Authen Tech Tech Deck Tech Shorts
• Money Today Tax Investment Insurance Tools & Calculator
• Mutual Funds
• Industry Banking IT Auto Energy Commodities Pharma Real Estate Telecom
• Visual Stories

INDICES ANALYSIS

Mutual funds.

• Cover Story
• Editor's Note
• Market Today
• Drive Today
• BT Explainer
• Trending Stocks
• Stocks List
• Stocks News
• Share Market News
• Unbox Today
• Authen Tech
• Tech Shorts
• Tools & Calculator
• Commodities
• Real Estate
• Election with BT
• Economic Indicators
• BT-TR GCC Listing

'Time zone cheating' in IB exams after several question papers leaked on Reddit, Telegram

Time zone cheating occurs when students present in one time zone, after completing their exams, share what they remember about the questions on social media. This activity takes place before students in other time zones appear for the exam.

• Updated May 08, 2024, 4:12 PM IST

The International Baccalaureate Diploma Programme (IBDP) students have reportedly misused the time zone differences to leak the question papers of mathematics and several other subjects through social media platforms, which offered other students who were still supposed to appear for those exams in other time zones, an unfair advantage over others.

Related Articles

• CBSE Results 2024: Results on May 20? Marks verification schedule out for Class 10 and Class 12, check how and where to check results
• CBSE Class 10, 12 Results 2024: 6-digit DigiLocker access codes out, here's how to enable it

Question papers of several IP papers like business management, global politics, mathematics, physics, computer science, biology and chemistry were leaked on social media platforms of Telegram and Reddit, the South China Morning Post reported.

The incident being referred to as 'time zone cheating' has triggered an investigation by the IB authority, as for the first time in the board's 55-year history such a leak has happened. The source of the leak is currently not disclosed by the authorities; suspicions are that Turkey is the source of origin. 

As per the reports, the leaked materials were downloaded over 45,000 times until Sunday. 

“We have identified the source of this activity and are taking appropriate steps to hold those responsible accountable,” IB said in its statement. However, it didn't specify the source.

The issue has raised concerns and questions on the fairness and authenticity of the IB system.

The board disclosed that very few students were involves in what is being termed as 'time zone cheating.'

"To date, there is no evidence of widespread cheating and we are confident that this activity remains at the fringes of what is otherwise a standard exam session," it said in a release on Sunday.

But what is 'Time Zone Cheating'?

Time zone cheating is banned by IB as a part of their academic integrity policy. 

Online petitions have surfaced urging the IB to cancel this year's exams. Many have even asked the board to ensure justice for students unaffected by the leaks. The petition has received over 3,000 signatures. 

TOP STORIES

• Advertise with us
• Privacy Policy
• Terms and Conditions
• Press Releases

Copyright©2024 Living Media India Limited. For reprint rights: Syndications Today

Add Business Today to Home Screen