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Mathematics in Everyday Life: Most Vital Discipline

Categories: Mathematics in Everyday Life

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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.

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

Updated on
Dec 22, 2023

Essay on Importance of Mathematics in our Daily Life

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.

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.

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.

Deepansh Gautam

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Essay on Importance of Mathematics in Our Daily Life

Students are often asked to write an essay on Importance of Mathematics in Our Daily Life in their schools and colleges. And if you’re also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic.

Let’s take a look…

100 Words Essay on Importance of Mathematics in Our Daily Life

Introduction.

Mathematics is a crucial part of everyday life. It helps us make sense of the world around us and solve practical problems.

Mathematics in Daily Tasks

From shopping to cooking, we use math. It helps us calculate costs, quantities, and time.

Mathematics in Professions

In professions like engineering, computer science, and finance, math is indispensable.

Mathematics in Decision Making

Math helps us make informed decisions by analyzing data and predicting outcomes.

Thus, math plays a vital role in our daily lives, making it an essential subject to learn.

250 Words Essay on Importance of Mathematics in Our Daily Life

The pervasive presence of mathematics.

Mathematics, often perceived as a complex and abstract discipline, is in fact an integral part of our everyday lives. It forms the foundation for many of the decisions we make and the actions we perform daily, from managing finances to navigating directions.

A Tool for Logical Reasoning

Mathematics fosters logical reasoning and problem-solving skills. It cultivates an analytical mindset, enabling us to break down complex problems into simpler, manageable parts. This approach is not just confined to mathematical problems but extends to various real-life situations, thereby honing our decision-making abilities.

Mathematics in Technological Advancements

The rapid progress in technology, which has become an inseparable part of our lives, is deeply rooted in mathematical principles. Algorithms, which form the basis of computing, are mathematical models. The internet, smartphones, GPS, and even AI owe their existence to mathematical concepts.

Financial Management and Mathematics

Managing personal finances, a critical life skill, is essentially a mathematical exercise. Budgeting, calculating interest, understanding the implications of loans and mortgages, or even evaluating investment options, all require a good grasp of mathematics.

Mathematics and Scientific Understanding

Mathematics is the language of science. It helps us quantitatively understand and describe the physical world around us, from the trajectory of planets to the behavior of subatomic particles.

In conclusion, mathematics is a vital part of our daily lives. It is not just a subject to be studied in school, but a tool for understanding, navigating, and shaping the world around us. Its importance cannot be overstated, as it is the foundation of critical thinking, technological progress, financial management, and scientific understanding.

500 Words Essay on Importance of Mathematics in Our Daily Life

Mathematics, often perceived as a complex and abstract subject, is in fact deeply intertwined with our daily lives. It is the foundation of numerous activities we engage in, from basic tasks such as shopping and cooking to more complex ones like planning finances or solving problems.

The Ubiquity of Mathematics

Mathematics is everywhere. It is used in our everyday activities, often without our conscious realization. When we shop, we use mathematics to calculate prices, discounts, and taxes. When we cook, we use it to measure ingredients. When we travel, we use it to calculate distances, time, and fuel consumption. Even in our leisure activities such as playing games or music, mathematics plays a crucial role in understanding patterns, probabilities, and rhythms.

Mathematics in the Professional Sphere

In the professional world, the significance of mathematics is even more pronounced. Engineers use mathematical principles to design and build infrastructure. Economists use it to predict market trends. Computer scientists use algorithms and data structures, which are fundamentally mathematical in nature, to design efficient software. Even in fields that are traditionally considered non-mathematical, such as literature and arts, mathematical concepts such as symmetry, geometry, and proportion play a key role in creating aesthetic appeal.

Mathematics and Problem-Solving

Mathematics also enhances our problem-solving skills. It teaches us to approach problems logically and systematically. It encourages us to break down complex problems into simpler parts, solve them individually, and combine the solutions to solve the original problem. This skill is not just applicable to mathematical problems but to any problem we encounter in life.

Mathematics and Critical Thinking

Furthermore, mathematics fosters critical thinking. It trains us to question assumptions, identify patterns, and draw conclusions based on evidence. It also teaches us to understand the limitations of our solutions and consider alternative approaches. These are valuable skills that can be applied in various aspects of life, from making informed decisions to evaluating the credibility of information.

Mathematics and the Digital Age

In the digital age, the importance of mathematics has grown exponentially. It is the backbone of modern technologies such as artificial intelligence, machine learning, data analysis, cryptography, and quantum computing. Understanding mathematics is essential to navigate and thrive in this digital world.

In conclusion, mathematics is not just a subject we learn in school. It is a powerful tool that helps us understand and navigate the world around us. It enhances our problem-solving and critical thinking skills, and it opens up a world of opportunities in the professional sphere. Therefore, it is essential that we appreciate the importance of mathematics in our daily lives, and strive to improve our mathematical literacy.

That’s it! I hope the essay helped you.

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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.

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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.

$Essays about Math: Why do many people despise math?$

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 !

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

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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.

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How mathematical practices can improve your writing

Writing is similar to three specific mathematical practices: modelling, problem-solving and proving, writes Caroline Yoon. Here, she gives some tips on how to use these to improve academic writing

Caroline Yoon

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I feel for my students when I hand them their first essay assignment. Many are mathematicians, students and teachers who chose to study mathematics partly to avoid writing. But in my mathematics education courses, and in the discipline more generally, academic writing is part of our routine practice.

Mathematicians face some challenging stereotypes when it comes to writing. Writing is seen as ephemeral, subjective and context-dependent, whereas mathematics is seen as enduring, universal and context-free. Writing reflects self, but mathematics transcends it: they are distinct from each other.

This is a false dichotomy that can discourage mathematicians from writing. It suggests writing is outside the natural skill set of the mathematician, and that one’s mathematics training not only neglects one’s development as a writer but actively prevents it. Rather than capitulate to this false dichotomy, I propose we turn it around to examine how writing is similar to three specific mathematical practices: modelling, problem-solving and proving.

Three mathematical practices that can improve your writing

Mathematical modelling.

Let us consider a hypothetical mathematics education student who has spent weeks thinking, reading and talking about her essay topic, but only starts writing it the night before it is due. She writes one draft only – the one she hands in – and is disappointed with the low grade her essay receives.

She wishes she had started earlier but she was still trying to figure out what she wanted to say up until the moment she started writing. It was only the pressure of the deadline that forced her to start; without it, she would have spent even more time thinking and reading to develop her ideas. After all, she reasons, there is no point writing when you do not know what to write about!

This “think first, write after” approach, sometimes known as the “writing up” model is a dangerous trap many students fall into, and is at odds with the way writing works. The approach allows no room for imperfect drafts that are a necessary part of the writing process. Writing experts trade on the generative power of imperfect writing; they encourage writers to turn off their internal critics and allow themselves to write badly as a way of overcoming writing inertia and discovering new ideas. The “shitty first draft” is an ideal (and achievable) first goal in the writing process. Anyone can produce a sketchy first draft that generates material that can be worked on, improved and eventually rewritten into a more sharable form.

Mathematical modelling offers a compelling metaphor for the generative power of imperfect writing. Like polished writing, polished mathematical models are seldom produced in the first attempt. A modeller typically begins with some understanding of the real situation to be modelled. The modeller considers variables and relationships from his or her understanding of the real situation and writes them into an initial mathematical model.

The model is his or her mathematical description of the situation, written in mathematical notation, and the modeller who publishes a mathematical model has typically created and discarded multiple drafts along the way, just as the writer who publishes a piece of writing has typically written and discarded multiple drafts along the way.

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Problem-solving

Writing an original essay is like trying to solve a mathematics problem. There is no script to follow; it must be created by simultaneously determining one’s goals and figuring out how to achieve them. In both essay writing and mathematical problem-solving, getting stuck is natural and expected. It is even a special kind of thrill.

This observation might come as a surprise to mathematicians who do not think of their problem-solving activity as writing. But doing mathematics, the ordinary everyday act of manipulating mathematical relationships and objects to notice new levels of structure and pattern, involves scratching out symbols and marks, and moving ideas around the page or board.

Why do I care that mathematicians acknowledge their natural language of symbols and signs as writing? Quite frankly because they are good at it. They have spent years honing their ability to use writing to restructure their thoughts, to dissect their ideas, identify new arguments. They possess an analytic discipline that most writers struggle with.

Yet few of my mathematics education students take advantage of this in their academic writing. They want their writing to come out in consecutive, polished sentences and become discouraged when it does not. They do not use their writing to analyse and probe their arguments as they do when they are stuck on mathematical problems. By viewing writing only as a medium for communicating perfectly formed thoughts, they deny themselves their own laboratories, their own thinking tools.

I am not suggesting that one’s success in solving mathematical problems automatically translates into successful essay writing. But the metaphor of writing as problem-solving might encourage a mathematics education student not to give up too easily when she finds herself stuck in her writing.

Our hypothetical student now has a good draft that she is happy with. She is satisfied it represents her knowledge of the subject matter and has read extensively to check the accuracy of its content. A friend reads the draft and remarks that it is difficult to understand. Our student is unperturbed. She puts it down to her friend’s limited knowledge of the subject and is confident her more knowledgeable teacher will understand her essay.

But the essay is not an inert record judged on the number of correct facts it contains. It is also a rhetorical act that seeks to engage the public. It addresses an audience, it tries to persuade, to inspire some response or action.

Mathematical proofs are like expository essays in this regard; they must convince an audience. When undergraduate mathematics students learn to construct proofs of their own, a common piece of advice is to test them on different audiences. The phrase “Convince yourself, convince a friend, convince an enemy” becomes relevant in this respect.

Mathematicians do not have to see themselves as starting from nothing when they engage in academic writing. Rather, they can use mathematical principles they have already honed in their training, but which they might not have formerly recognised as tools for improving their academic writing.

Practical tips for productive writing beliefs and behaviours

Writing can generate ideas. Free writing is a good way to start. Set a timer and write continuously for 10 minutes without editing. These early drafts will be clumsy, but there will also be some gold that can be mined and developed.
Writing can be used to analyse and organise ideas. When stuck, try to restructure your ideas. Identify the main point in each paragraph and play around with organising their flow.
Writing is a dialogue with the public. Seek out readers’ interpretations of your writing and listen to their impressions. Read your writing out loud to yourself: you will hear it differently!

Caroline Yoon is an associate professor of mathematics at the University of Auckland.

This is an edited version of the journal article “The writing mathematician” by Caroline Yoon, published in For the Learning of Mathematics and collected in The Best Writing on Mathematics , edited by Mircea Pitici (Princeton University Press).

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How to conclude an essay | Interactive example

How to Conclude an Essay | Interactive Example

Published on January 24, 2019 by Shona McCombes . Revised on July 23, 2023.

The conclusion is the final paragraph of your essay . A strong conclusion aims to:

Tie together the essay’s main points
Show why your argument matters
Leave the reader with a strong impression

Your conclusion should give a sense of closure and completion to your argument, but also show what new questions or possibilities it has opened up.

This conclusion is taken from our annotated essay example , which discusses the history of the Braille system. Hover over each part to see why it’s effective.

Braille paved the way for dramatic cultural changes in the way blind people were treated and the opportunities available to them. Louis Braille’s innovation was to reimagine existing reading systems from a blind perspective, and the success of this invention required sighted teachers to adapt to their students’ reality instead of the other way around. In this sense, Braille helped drive broader social changes in the status of blindness. New accessibility tools provide practical advantages to those who need them, but they can also change the perspectives and attitudes of those who do not.

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Step 1: return to your thesis, step 2: review your main points, step 3: show why it matters, what shouldn’t go in the conclusion, more examples of essay conclusions, other interesting articles, frequently asked questions about writing an essay conclusion.

To begin your conclusion, signal that the essay is coming to an end by returning to your overall argument.

Don’t just repeat your thesis statement —instead, try to rephrase your argument in a way that shows how it has been developed since the introduction.

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Next, remind the reader of the main points that you used to support your argument.

Avoid simply summarizing each paragraph or repeating each point in order; try to bring your points together in a way that makes the connections between them clear. The conclusion is your final chance to show how all the paragraphs of your essay add up to a coherent whole.

To wrap up your conclusion, zoom out to a broader view of the topic and consider the implications of your argument. For example:

Does it contribute a new understanding of your topic?
Does it raise new questions for future study?
Does it lead to practical suggestions or predictions?
Can it be applied to different contexts?
Can it be connected to a broader debate or theme?

Whatever your essay is about, the conclusion should aim to emphasize the significance of your argument, whether that’s within your academic subject or in the wider world.

Try to end with a strong, decisive sentence, leaving the reader with a lingering sense of interest in your topic.

The easiest way to improve your conclusion is to eliminate these common mistakes.

Don’t include new evidence

Any evidence or analysis that is essential to supporting your thesis statement should appear in the main body of the essay.

The conclusion might include minor pieces of new information—for example, a sentence or two discussing broader implications, or a quotation that nicely summarizes your central point. But it shouldn’t introduce any major new sources or ideas that need further explanation to understand.

Don’t use “concluding phrases”

Avoid using obvious stock phrases to tell the reader what you’re doing:

“In conclusion…”
“To sum up…”

These phrases aren’t forbidden, but they can make your writing sound weak. By returning to your main argument, it will quickly become clear that you are concluding the essay—you shouldn’t have to spell it out.

Don’t undermine your argument

Avoid using apologetic phrases that sound uncertain or confused:

“This is just one approach among many.”
“There are good arguments on both sides of this issue.”
“There is no clear answer to this problem.”

Even if your essay has explored different points of view, your own position should be clear. There may be many possible approaches to the topic, but you want to leave the reader convinced that yours is the best one!

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Literary analysis

This conclusion is taken from an argumentative essay about the internet’s impact on education. It acknowledges the opposing arguments while taking a clear, decisive position.

The internet has had a major positive impact on the world of education; occasional pitfalls aside, its value is evident in numerous applications. The future of teaching lies in the possibilities the internet opens up for communication, research, and interactivity. As the popularity of distance learning shows, students value the flexibility and accessibility offered by digital education, and educators should fully embrace these advantages. The internet’s dangers, real and imaginary, have been documented exhaustively by skeptics, but the internet is here to stay; it is time to focus seriously on its potential for good.

This conclusion is taken from a short expository essay that explains the invention of the printing press and its effects on European society. It focuses on giving a clear, concise overview of what was covered in the essay.

The invention of the printing press was important not only in terms of its immediate cultural and economic effects, but also in terms of its major impact on politics and religion across Europe. In the century following the invention of the printing press, the relatively stationary intellectual atmosphere of the Middle Ages gave way to the social upheavals of the Reformation and the Renaissance. A single technological innovation had contributed to the total reshaping of the continent.

This conclusion is taken from a literary analysis essay about Mary Shelley’s Frankenstein . It summarizes what the essay’s analysis achieved and emphasizes its originality.

By tracing the depiction of Frankenstein through the novel’s three volumes, I have demonstrated how the narrative structure shifts our perception of the character. While the Frankenstein of the first volume is depicted as having innocent intentions, the second and third volumes—first in the creature’s accusatory voice, and then in his own voice—increasingly undermine him, causing him to appear alternately ridiculous and vindictive. Far from the one-dimensional villain he is often taken to be, the character of Frankenstein is compelling because of the dynamic narrative frame in which he is placed. In this frame, Frankenstein’s narrative self-presentation responds to the images of him we see from others’ perspectives. This conclusion sheds new light on the novel, foregrounding Shelley’s unique layering of narrative perspectives and its importance for the depiction of character.

If you want to know more about AI tools , college essays , or fallacies make sure to check out some of our other articles with explanations and examples or go directly to our tools!

Ad hominem fallacy
Post hoc fallacy
Appeal to authority fallacy
False cause fallacy
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Your essay’s conclusion should contain:

A rephrased version of your overall thesis
A brief review of the key points you made in the main body
An indication of why your argument matters

The conclusion may also reflect on the broader implications of your argument, showing how your ideas could applied to other contexts or debates.

For a stronger conclusion paragraph, avoid including:

Important evidence or analysis that wasn’t mentioned in the main body
Generic concluding phrases (e.g. “In conclusion…”)
Weak statements that undermine your argument (e.g. “There are good points on both sides of this issue.”)

Your conclusion should leave the reader with a strong, decisive impression of your work.

The conclusion paragraph of an essay is usually shorter than the introduction . As a rule, it shouldn’t take up more than 10–15% of the text.

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FOR STUDENTS : ALL THE INGREDIENTS OF A GOOD ESSAY

Mathematics essays

Our free mathematics essay examples include popular topics such as algorithms, applied mathematics, calculus, knot theory, linear algebra, and more.

Euler’s identity

Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as “the most beautiful equation.” It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures “our jewel” and “the most remarkable formula in … Read more

The Banach-Tarski paradox

Mathematics is seen by many as a mysterious and often unsettling subject. Answers often hide behind layers and layers of complicated equations, formulas and ciphers, the application of advanced concepts to real life is limited and I often find myself more confused after class than when I first entered. However, the real beauty of Mathematics … Read more

Pascal’s triangle, binomial theorem

What is Pascal’s Triangle? Pascal’s Triangle was named after Blaise Pascal. Pascal’s triangle starts with the number 1 and goes down the scale. When you start with one, add more numbers in a triangular shape, like a pyramid of some sort. All the numbers on the surrounding right and left sides of the triangle are … Read more

My journey of teaching and learning mathematics since embarking on a PGCE Mathematics course

Before I came to study a Post Graduate Certificate (PGCE) Mathematics course at University College London Institute of Education (UCL IOE), I had been working as an Academic Tutor at a behavioural centre, linked to a mainstream secondary school for the past 7 months. Students placed here had either learning difficulties or behaviour issues experienced … Read more

Aircraft – mathematics

Math SL Internal Assessment Lift and Drag Introduction When you look at aircrafts, they look like they shouldn’t be able to leave the ground because of how big they are. I always watched aircrafts, take off and land, over and over. According to Newton’s Third Law, every action has an equal and opposite reaction, lift … Read more

The World of Mathematics

Mathematics is often considered a useless discipline because people think they do not use advanced math in their life. It is a misunderstanding. Not using advanced math does not mean it is not vital. If you trade stocks, you will read many stock analysis reports. These reports use mathematical knowledge. Many students in the United … Read more

The history of algebra

We all use algebra. Even if it’s for the simple stuff, we use some form of algebra in our everyday lives. While reading chapters 1-10, I came across the word algebra and became quite curious about the subject for I have never really understood nor cared for it honestly, I just figured it’s the usual … Read more

Mathematics introduction

Mathematics belongs to the science discourse community. The word science means knowledge and comes from the Latin “Scientia”. In university, science is made up of a lot of discourse communities, such as Mathematics, Physics, and Chemistry. By searching the definition of science in Webster’s New Collegiate Dictionary, science is “knowledge covering general truths of the … Read more

Niels Henrik Abel

For over two centuries, mathematicians had trouble in finding a solution to the quintic equation, that is until Niels Henrik Abel formulated a theory. Abel was a Norwegian mathematician born on August 5, 1802, and his talent and potential in the field of mathematics was already present at a young age, leading him to become … Read more

The narrative of zero

Numbers surround us. They stamp our days, light our evenings, foresee our climate, and keep us on course. They drive business and support human progress. The beginning of the numerals makes disarray between the historical backdrop of mathematics and the historical backdrop of our modern numerals. The narrative of zero alludes to something can be … Read more

Emmy Noether

She was more than a mathematician to the people she met and to the people she inspired. She even has managed to inspire people long after she has passed. Emmy Noether was born on March 23, 1882 in Bavaria Germany. Growing up she wanted to go to college but back then women weren’t allowed to … Read more

Bitopological Approximation Space with Application to Data Reduction in Multivalued Information Systems

Abstract: In this work we generalize Pawlak approximation space to bitopological approximation space. One binary relation can define two subbases of two topological spaces. Membership, equillity and inclusion relations using rough approximations are defined and studied in bitopological aapproximation space. Some new measures that measure the accuracy and the quality of approximations are defined and … Read more

Statistics overview

Statistics is a form of mathematical analysis that uses quantified models, representations and synopses for a given set of experimental data or real life studies. Statistical analysis involves the process of gathering and evaluating data and then summarizing the data into a mathematical form. Statistics is a term used to summarize a process that a … Read more

Numerical Weather Prediction

You turn on the television, and often the first channel that pops up is the weather. It’s going 24/7 with predictions that go from weekly all the way down to hourly, with conditions that go from humidity to temperature. But what goes on behind the scenes is heavily entrenched in mathematics– meteorology’s backbone is a … Read more

The Mathematics of Our Universe

Abstract In this report, we start by defining key aspects of classical Lagrangian mechanics including the principle of least action and how one can use this to derive the Euler-Lagrange equations. Momentum and Conservation laws shall also be introduced, deriving relations between position, momenta and the Lagrangian of a given system. Following this, we develop … Read more

Ideas for your next mathematics essay

Stuck for a title for your next essay? Here are some ideas to inspire you:

The Mathematics of Music: Exploring the Relationship between Mathematics and Music – This essay would examine the connections between music and mathematics, including the use of mathematical concepts in musical composition and the study of the mathematics of sound.
The Golden Ratio: A Mathematical and Aesthetic Marvel – This essay would discuss the concept of the golden ratio and its applications in art, architecture, and design. It would explore the beauty and symmetry of this mathematical principle.
Mathematics in Sports: Analyzing the Numbers Behind Athletic Performance – This essay would explore the use of mathematics in sports, including the use of statistics and analytics to analyze athletic performance and predict outcomes.
Chaos Theory: The Science of Nonlinear Systems – This essay would discuss the concept of chaos theory and its applications in various fields, such as meteorology, physics, and economics. It would explore the idea that small changes in initial conditions can have a significant impact on the final outcome of a system.
The Mathematics of Cryptography: Securing Information in the Digital Age – This essay would examine the use of mathematics in cryptography, including the principles of encryption and decryption, and how these concepts are applied to secure information in the digital age.
Fractals: The Beauty of Infinite Complexity – This essay would explore the concept of fractals and their applications in art, nature, and science. It would discuss the beauty and complexity of these repeating patterns found in nature and how they are used in various fields of study.
Mathematical Models in Biology: Understanding the Complexities of Life – This essay would discuss the use of mathematical models in biology, including the modeling of population growth, the spread of disease, and the behavior of organisms. It would explore how these models help scientists understand the complex systems that make up living organisms.
The Mathematics of Finance: Analyzing Investments and Markets – This essay would examine the use of mathematics in finance, including the principles of financial analysis, investments, and risk management. It would explore how mathematics is used to understand and predict market trends.
Geometry in Art: The Intersection of Math and Creativity – This essay would discuss the use of geometry in art, including the use of shapes, patterns, and symmetry. It would explore how artists use mathematical concepts to create beautiful and compelling works of art.
The History of Mathematics: From Ancient Times to Modern-Day Advances – This essay would trace the history of mathematics, from its origins in ancient civilizations to modern-day advancements in the field. It would explore the contributions of key mathematicians throughout history and the evolution of mathematical concepts and principles over time.

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Essay on Importance of Mathematics for Students

Importance of Mathematics

Mathematics is the queen of science. We cannot imagine our life without mathematics. Mathematics makes human life more Organized. The knowledge of numbers and their basic operation is very important to survive in today’s world.

Essay on Importance of Mathematics in Daily Life for Students

Mathematics has its application in daily life such as Time -management, Finance management, Profit and loss in business, Cooking, Purchasing and selling, Consumption of electricity, and patrol. There is hardly any field of science, which does not use mathematics as a tool to arrive at the desired result.

Engineering , Medical , Pharmacy, Geography, Economics, Accountancy and all field related to science uses mathematical methods to study the subjects.

Definition of Mathematics:

How we can define mathematics “Mathematics is the body of knowledge based on numbers and figures”.

Importance of Mathematics in Daily Life:

Math has multiple advantages in day-to-day life. Let’s discuss the importance of Math in day-to-day life.

The most important thing about Maths is that it helps to develop your brain. We know that we can strengthen our muscles by exercise in the same way we can strengthen neurons in our brain by doing Maths. Our brain is made of neuron’s when we do Math neurons in our brain are stimulated.

Research work done by Dr. Tanya Evans of Stanford University proves that the student who performs well in Math can recruit certain brain regions more reliably and have more gray matter in those regions, as compared to those who perform poorly in math.

If you want to succeed in life then you must learn the art of time management math helps you to read the time and manage it. If you understand fractions in math in a better way you can understand time in a better way. Learning to read the analog clock is the first step of time management and it is possible only because of math. In school children are taught to read both times of watching the 12-hour clock and 24-hour clock.

Math helps you with your finances it is said that you should never spend more than your income. Math helps you to calculate your monthly expenditure and helps you keep your monthly expenses as low as possible. Math helps you to calculate what part of your monthly income you are spending on different services and goods. Math helps you to keep your monthly expenditure below your monthly income.

It also helps to calculate your saving according to your future needs. Knowledge of math is also required in investing money in property, bank schemes Mutual funds, or the stock market. This thing required sound knowledge of math and economics.

Math and knowledge of proportion can make you an excellent cook. Let’s take an example if you are making tea. Then to make one cup of tea you will need half a cup of milk, half a cup of water, 1 spoon of tea powder, and 2 spoons of sugar. Now if you have to make two cups of tea you have two take the above ingredients in twice the quantity i.e. 1 cup of milk, 1 cup of water, 2 spoons of tea, and 4 spoons of sugar.

In this way, math helps you to decide the proportion of food ingredients according to their consumers. All this is possible because of math. It is observed that people who are good at math are also having very good problem-solving skills.

Math teaches us to think logically and creatively. Math improves our thinking and improves our Analytical thinking and reasoning ability.

We all face problems in our life our ability to solve problems depends on the way we think. All of us can think but very few think rationally. math develops rational thinking in man and hence improves his problem-solving skills.

Math also helps you to stay fit and healthy. To keep your body in shape you should know which exercise and the number of repetitions you must perform. The counting repetitions during exercise are based on math. You also need a healthy diet to stay fit. We already discussed that knowledge of math plays a vital role in deciding the proportion of ingredients in the diet.

Math helps you to understand sports in a better way. Take the example of cricket knowledge of math helps us to understand cricket scores in a better way. Batting average, bowling average, run rate, strike rate all these cricket terminologies are based on math. Using math sometimes you can also predict the winning team before the match.

Math also helps us while we drive our 2-wheeler or 4-wheeler. It helps us to keep track of the distance covered and the distance to be covered. How much petrol is left inside the tank? What should be the speed of the vehicle to reach the destination in time? All these calculations are based on math.

Conclusion:

Math is the backbone of all science we cannot imagine human life without science. To survive in this world you will need the math you cannot live for a single day without the use of math. The importance of math is like oxygen in the air without which no human progress is possible.

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Definition to Remember:

Thesis + Wisdom + Catchy Last Line = Conclusion

Rules to Remember:

Much as your introduction gives readers a first impression of who you are and what you hope to accomplish, your conclusion is your chance to offer final wisdom. For many readers, your concluding words are what they will remember long after they have finished reading your piece. For that reason, your concluding paragraph is critical.
Always end your essay in a way that reinforces your thesis and your purpose. A conclusion must provide a sense of closure. Readers should recognize your final paragraph as an ending. If you feel compelled to type the words “The End,” you’re not there yet.
Remember to look ahead. Is there future research that you intend or would recommend? Is there something specific you hope your readers will do with the ideas you have shared? Is there a new direction to turn? How can you use your conclusion to keep your readers thinking, even after they have set your essay aside?
Remind your readers of your overall thesis . Do not merely repeat your thesis. If you have added sufficient evidence in your essay to support your claim, your thesis should sound different to your readers than it did in the introduction. As you remind your readers of your purpose, allow your thesis to express the fullness of all of the evidence you have brought to bear.
a related story
a provocative question or series of questions
a hypothetical scenario
a surprising fact or series of facts
an engaging direct quotation
a striking statement
background information or context
an opposing argument
the who, what, where, when, and why of the paper’s focus
a combination of the types listed above
Finish with a catchy last line that is both conclusive-sounding and memorable. Much like a catchy first line, an effective last line should be concise, poetic, persuasive, and provocative. “Writing well may offer little respect, but writing poorly certainly loses it.” David Hartmann, Director of Client Success

Common Errors:

Tacking on a placeholder conclusion. Writers are often fatigued by the time they are ready to write that final paragraph, and, unfortunately, it shows. As with any kind of writing, if you are finding the work tedious, imagine how uninterested your readers will be. Always save time to set your work aside and refresh before writing your conclusion; the added effort will always pay off.
Repeating what has been said already. While many of us were taught in elementary school to use the conclusion as an opportunity to remind your readers of everything you just said, an effective post-elementary school conclusion should aspire for more than merely repetition.

Exercise 13.1

Consider a writing assignment you will need to undertake in the near future. How might you approach a conclusion using each of the following approaches? Be specific as you answer.

A related story:
A provocative question or series of questions:
A hypothetical scenario:
A surprising fact or series of facts:
An engaging direct quotation:
A striking statement:
Background information or context:
An opposing argument:
The who, what, where, when, and why of the paper’s focus:
A combination of the types listed above:

Exercise 13.2

Consider at least five paragraphs you have written in the past week, whether for work, school, or personal use. Write the last line of each on the lines below. If you had been a member of your own audience, would you have found the last line conclusive but memorable? Why or why not? If not, what revisions would you make?

Exercise 13.3

Select one of the examples from Exercise 13.1 and write a conclusion. Once you have finished, ask yourself the following questions:

Which of the suggestions listed in Exercise 13.1 have you used to interest and inspire your readers? Why?
Have you included a repetition of your thesis statement that is a fuller, more complete version of the statement you included in your introduction?
Have you included a catchy last line?
If you were a member of your own audience, would you find the conclusion memorable? Why or why not?
What further revisions do you need to make?

The Simple Math of Writing Well Copyright © 2017 by Dr. Jennie A. Harrop is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Sat / act prep online guides and tips, the complete ib extended essay guide: examples, topics, and ideas.

International Baccalaureate (IB)

IB students around the globe fear writing the Extended Essay, but it doesn't have to be a source of stress! In this article, I'll get you excited about writing your Extended Essay and provide you with the resources you need to get an A on it.

If you're reading this article, I'm going to assume you're an IB student getting ready to write your Extended Essay. If you're looking at this as a potential future IB student, I recommend reading our introductory IB articles first, including our guide to what the IB program is and our full coverage of the IB curriculum .

IB Extended Essay: Why Should You Trust My Advice?

I myself am a recipient of an IB Diploma, and I happened to receive an A on my IB Extended Essay. Don't believe me? The proof is in the IBO pudding:

If you're confused by what this report means, EE is short for Extended Essay , and English A1 is the subject that my Extended Essay topic coordinated with. In layman's terms, my IB Diploma was graded in May 2010, I wrote my Extended Essay in the English A1 category, and I received an A grade on it.

What Is the Extended Essay in the IB Diploma Programme?

The IB Extended Essay, or EE , is a mini-thesis you write under the supervision of an IB advisor (an IB teacher at your school), which counts toward your IB Diploma (learn more about the major IB Diploma requirements in our guide) . I will explain exactly how the EE affects your Diploma later in this article.

For the Extended Essay, you will choose a research question as a topic, conduct the research independently, then write an essay on your findings . The essay itself is a long one—although there's a cap of 4,000 words, most successful essays get very close to this limit.

Keep in mind that the IB requires this essay to be a "formal piece of academic writing," meaning you'll have to do outside research and cite additional sources.

The IB Extended Essay must include the following:

A title page
Contents page
Introduction
Body of the essay
References and bibliography

Additionally, your research topic must fall into one of the six approved DP categories , or IB subject groups, which are as follows:

Group 1: Studies in Language and Literature
Group 2: Language Acquisition
Group 3: Individuals and Societies
Group 4: Sciences
Group 5: Mathematics
Group 6: The Arts

Once you figure out your category and have identified a potential research topic, it's time to pick your advisor, who is normally an IB teacher at your school (though you can also find one online ). This person will help direct your research, and they'll conduct the reflection sessions you'll have to do as part of your Extended Essay.

As of 2018, the IB requires a "reflection process" as part of your EE supervision process. To fulfill this requirement, you have to meet at least three times with your supervisor in what the IB calls "reflection sessions." These meetings are not only mandatory but are also part of the formal assessment of the EE and your research methods.

According to the IB, the purpose of these meetings is to "provide an opportunity for students to reflect on their engagement with the research process." Basically, these meetings give your supervisor the opportunity to offer feedback, push you to think differently, and encourage you to evaluate your research process.

The final reflection session is called the viva voce, and it's a short 10- to 15-minute interview between you and your advisor. This happens at the very end of the EE process, and it's designed to help your advisor write their report, which factors into your EE grade.

Here are the topics covered in your viva voce :

A check on plagiarism and malpractice
Your reflection on your project's successes and difficulties
Your reflection on what you've learned during the EE process

Your completed Extended Essay, along with your supervisor's report, will then be sent to the IB to be graded. We'll cover the assessment criteria in just a moment.

We'll help you learn how to have those "lightbulb" moments...even on test day!

What Should You Write About in Your IB Extended Essay?

You can technically write about anything, so long as it falls within one of the approved categories listed above.

It's best to choose a topic that matches one of the IB courses , (such as Theatre, Film, Spanish, French, Math, Biology, etc.), which shouldn't be difficult because there are so many class subjects.

Here is a range of sample topics with the attached extended essay:

Biology: The Effect of Age and Gender on the Photoreceptor Cells in the Human Retina
Chemistry: How Does Reflux Time Affect the Yield and Purity of Ethyl Aminobenzoate (Benzocaine), and How Effective is Recrystallisation as a Purification Technique for This Compound?
English: An Exploration of Jane Austen's Use of the Outdoors in Emma
Geography: The Effect of Location on the Educational Attainment of Indigenous Secondary Students in Queensland, Australia
Math: Alhazen's Billiard Problem
Visual Arts: Can Luc Tuymans Be Classified as a Political Painter?

You can see from how varied the topics are that you have a lot of freedom when it comes to picking a topic . So how do you pick when the options are limitless?

How to Write a Stellar IB Extended Essay: 6 Essential Tips

Below are six key tips to keep in mind as you work on your Extended Essay for the IB DP. Follow these and you're sure to get an A!

#1: Write About Something You Enjoy

You can't expect to write a compelling essay if you're not a fan of the topic on which you're writing. For example, I just love British theatre and ended up writing my Extended Essay on a revolution in post-WWII British theatre. (Yes, I'm definitely a #TheatreNerd.)

I really encourage anyone who pursues an IB Diploma to take the Extended Essay seriously. I was fortunate enough to receive a full-tuition merit scholarship to USC's School of Dramatic Arts program. In my interview for the scholarship, I spoke passionately about my Extended Essay; thus, I genuinely think my Extended Essay helped me get my scholarship.

But how do you find a topic you're passionate about? Start by thinking about which classes you enjoy the most and why . Do you like math classes because you like to solve problems? Or do you enjoy English because you like to analyze literary texts?

Keep in mind that there's no right or wrong answer when it comes to choosing your Extended Essay topic. You're not more likely to get high marks because you're writing about science, just like you're not doomed to failure because you've chosen to tackle the social sciences. The quality of what you produce—not the field you choose to research within—will determine your grade.

Once you've figured out your category, you should brainstorm more specific topics by putting pen to paper . What was your favorite chapter you learned in that class? Was it astrophysics or mechanics? What did you like about that specific chapter? Is there something you want to learn more about? I recommend spending a few hours on this type of brainstorming.

One last note: if you're truly stumped on what to research, pick a topic that will help you in your future major or career . That way you can use your Extended Essay as a talking point in your college essays (and it will prepare you for your studies to come too!).

#2: Select a Topic That Is Neither Too Broad nor Too Narrow

There's a fine line between broad and narrow. You need to write about something specific, but not so specific that you can't write 4,000 words on it.

You can't write about WWII because that would be a book's worth of material. You also don't want to write about what type of soup prisoners of war received behind enemy lines, because you probably won’t be able to come up with 4,000 words of material about it. However, you could possibly write about how the conditions in German POW camps—and the rations provided—were directly affected by the Nazis' successes and failures on the front, including the use of captured factories and prison labor in Eastern Europe to increase production. WWII military history might be a little overdone, but you get my point.

If you're really stuck trying to pinpoint a not-too-broad-or-too-narrow topic, I suggest trying to brainstorm a topic that uses a comparison. Once you begin looking through the list of sample essays below, you'll notice that many use comparisons to formulate their main arguments.

I also used a comparison in my EE, contrasting Harold Pinter's Party Time with John Osborne's Look Back in Anger in order to show a transition in British theatre. Topics with comparisons of two to three plays, books, and so on tend to be the sweet spot. You can analyze each item and then compare them with one another after doing some in-depth analysis of each individually. The ways these items compare and contrast will end up forming the thesis of your essay!

When choosing a comparative topic, the key is that the comparison should be significant. I compared two plays to illustrate the transition in British theatre, but you could compare the ways different regional dialects affect people's job prospects or how different temperatures may or may not affect the mating patterns of lightning bugs. The point here is that comparisons not only help you limit your topic, but they also help you build your argument.

Comparisons are not the only way to get a grade-A EE, though. If after brainstorming, you pick a non-comparison-based topic and are still unsure whether your topic is too broad or narrow, spend about 30 minutes doing some basic research and see how much material is out there.

If there are more than 1,000 books, articles, or documentaries out there on that exact topic, it may be too broad. But if there are only two books that have any connection to your topic, it may be too narrow. If you're still unsure, ask your advisor—it's what they're there for! Speaking of advisors...

Don't get stuck with a narrow topic!

#3: Choose an Advisor Who Is Familiar With Your Topic

If you're not certain of who you would like to be your advisor, create a list of your top three choices. Next, write down the pros and cons of each possibility (I know this sounds tedious, but it really helps!).

For example, Mr. Green is my favorite teacher and we get along really well, but he teaches English. For my EE, I want to conduct an experiment that compares the efficiency of American electric cars with foreign electric cars.

I had Ms. White a year ago. She teaches physics and enjoyed having me in her class. Unlike Mr. Green, Ms. White could help me design my experiment.

Based on my topic and what I need from my advisor, Ms. White would be a better fit for me than would Mr. Green (even though I like him a lot).

The moral of my story is this: do not just ask your favorite teacher to be your advisor . They might be a hindrance to you if they teach another subject. For example, I would not recommend asking your biology teacher to guide you in writing an English literature-based EE.

There can, of course, be exceptions to this rule. If you have a teacher who's passionate and knowledgeable about your topic (as my English teacher was about my theatre topic), you could ask that instructor. Consider all your options before you do this. There was no theatre teacher at my high school, so I couldn't find a theatre-specific advisor, but I chose the next best thing.

Before you approach a teacher to serve as your advisor, check with your high school to see what requirements they have for this process. Some IB high schools require your IB Extended Essay advisor to sign an Agreement Form , for instance.

Make sure that you ask your IB coordinator whether there is any required paperwork to fill out. If your school needs a specific form signed, bring it with you when you ask your teacher to be your EE advisor.

#4: Pick an Advisor Who Will Push You to Be Your Best

Some teachers might just take on students because they have to and aren't very passionate about reading drafts, only giving you minimal feedback. Choose a teacher who will take the time to read several drafts of your essay and give you extensive notes. I would not have gotten my A without being pushed to make my Extended Essay draft better.

Ask a teacher that you have experience with through class or an extracurricular activity. Do not ask a teacher that you have absolutely no connection to. If a teacher already knows you, that means they already know your strengths and weaknesses, so they know what to look for, where you need to improve, and how to encourage your best work.

Also, don't forget that your supervisor's assessment is part of your overall EE score . If you're meeting with someone who pushes you to do better—and you actually take their advice—they'll have more impressive things to say about you than a supervisor who doesn't know you well and isn't heavily involved in your research process.

Be aware that the IB only allows advisors to make suggestions and give constructive criticism. Your teacher cannot actually help you write your EE. The IB recommends that the supervisor spends approximately two to three hours in total with the candidate discussing the EE.

#5: Make Sure Your Essay Has a Clear Structure and Flow

The IB likes structure. Your EE needs a clear introduction (which should be one to two double-spaced pages), research question/focus (i.e., what you're investigating), a body, and a conclusion (about one double-spaced page). An essay with unclear organization will be graded poorly.

The body of your EE should make up the bulk of the essay. It should be about eight to 18 pages long (again, depending on your topic). Your body can be split into multiple parts. For example, if you were doing a comparison, you might have one third of your body as Novel A Analysis, another third as Novel B Analysis, and the final third as your comparison of Novels A and B.

If you're conducting an experiment or analyzing data, such as in this EE , your EE body should have a clear structure that aligns with the scientific method ; you should state the research question, discuss your method, present the data, analyze the data, explain any uncertainties, and draw a conclusion and/or evaluate the success of the experiment.

#6: Start Writing Sooner Rather Than Later!

You will not be able to crank out a 4,000-word essay in just a week and get an A on it. You'll be reading many, many articles (and, depending on your topic, possibly books and plays as well!). As such, it's imperative that you start your research as soon as possible.

Each school has a slightly different deadline for the Extended Essay. Some schools want them as soon as November of your senior year; others will take them as late as February. Your school will tell you what your deadline is. If they haven't mentioned it by February of your junior year, ask your IB coordinator about it.

Some high schools will provide you with a timeline of when you need to come up with a topic, when you need to meet with your advisor, and when certain drafts are due. Not all schools do this. Ask your IB coordinator if you are unsure whether you are on a specific timeline.

Below is my recommended EE timeline. While it's earlier than most schools, it'll save you a ton of heartache (trust me, I remember how hard this process was!):

January/February of Junior Year: Come up with your final research topic (or at least your top three options).
February of Junior Year: Approach a teacher about being your EE advisor. If they decline, keep asking others until you find one. See my notes above on how to pick an EE advisor.
April/May of Junior Year: Submit an outline of your EE and a bibliography of potential research sources (I recommend at least seven to 10) to your EE advisor. Meet with your EE advisor to discuss your outline.
Summer Between Junior and Senior Year: Complete your first full draft over the summer between your junior and senior year. I know, I know—no one wants to work during the summer, but trust me—this will save you so much stress come fall when you are busy with college applications and other internal assessments for your IB classes. You will want to have this first full draft done because you will want to complete a couple of draft cycles as you likely won't be able to get everything you want to say into 4,000 articulate words on the first attempt. Try to get this first draft into the best possible shape so you don't have to work on too many revisions during the school year on top of your homework, college applications, and extracurriculars.
August/September of Senior Year: Turn in your first draft of your EE to your advisor and receive feedback. Work on incorporating their feedback into your essay. If they have a lot of suggestions for improvement, ask if they will read one more draft before the final draft.
September/October of Senior Year: Submit the second draft of your EE to your advisor (if necessary) and look at their feedback. Work on creating the best possible final draft.
November-February of Senior Year: Schedule your viva voce. Submit two copies of your final draft to your school to be sent off to the IB. You likely will not get your grade until after you graduate.

Remember that in the middle of these milestones, you'll need to schedule two other reflection sessions with your advisor . (Your teachers will actually take notes on these sessions on a form like this one , which then gets submitted to the IB.)

I recommend doing them when you get feedback on your drafts, but these meetings will ultimately be up to your supervisor. Just don't forget to do them!

The early bird DOES get the worm!

How Is the IB Extended Essay Graded?

Extended Essays are graded by examiners appointed by the IB on a scale of 0 to 34 . You'll be graded on five criteria, each with its own set of points. You can learn more about how EE scoring works by reading the IB guide to extended essays .

Criterion A: Focus and Method (6 points maximum)
Criterion B: Knowledge and Understanding (6 points maximum)
Criterion C: Critical Thinking (12 points maximum)
Criterion D: Presentation (4 points maximum)
Criterion E: Engagement (6 points maximum)

How well you do on each of these criteria will determine the final letter grade you get for your EE. You must earn at least a D to be eligible to receive your IB Diploma.

Although each criterion has a point value, the IB explicitly states that graders are not converting point totals into grades; instead, they're using qualitative grade descriptors to determine the final grade of your Extended Essay . Grade descriptors are on pages 102-103 of this document .

Here's a rough estimate of how these different point values translate to letter grades based on previous scoring methods for the EE. This is just an estimate —you should read and understand the grade descriptors so you know exactly what the scorers are looking for.

Here is the breakdown of EE scores (from the May 2021 bulletin):

How Does the Extended Essay Grade Affect Your IB Diploma?

The Extended Essay grade is combined with your TOK (Theory of Knowledge) grade to determine how many points you get toward your IB Diploma.

To learn about Theory of Knowledge or how many points you need to receive an IB Diploma, read our complete guide to the IB program and our guide to the IB Diploma requirements .

This diagram shows how the two scores are combined to determine how many points you receive for your IB diploma (3 being the most, 0 being the least). In order to get your IB Diploma, you have to earn 24 points across both categories (the TOK and EE). The highest score anyone can earn is 45 points.

Let's say you get an A on your EE and a B on TOK. You will get 3 points toward your Diploma. As of 2014, a student who scores an E on either the extended essay or TOK essay will not be eligible to receive an IB Diploma .

Prior to the class of 2010, a Diploma candidate could receive a failing grade in either the Extended Essay or Theory of Knowledge and still be awarded a Diploma, but this is no longer true.

Figuring out how you're assessed can be a little tricky. Luckily, the IB breaks everything down here in this document . (The assessment information begins on page 219.)

40+ Sample Extended Essays for the IB Diploma Programme

In case you want a little more guidance on how to get an A on your EE, here are over 40 excellent (grade A) sample extended essays for your reading pleasure. Essays are grouped by IB subject.

Business Management 1
Chemistry 1
Chemistry 2
Chemistry 3
Chemistry 4
Chemistry 5
Chemistry 6
Chemistry 7
Computer Science 1
Economics 1
Design Technology 1
Design Technology 2
Environmental Systems and Societies 1
Geography 1
Geography 2
Geography 3
Geography 4
Geography 5
Geography 6
Literature and Performance 1
Mathematics 1
Mathematics 2
Mathematics 3
Mathematics 4
Mathematics 5
Philosophy 1
Philosophy 2
Philosophy 3
Philosophy 4
Philosophy 5
Psychology 1
Psychology 2
Psychology 3
Psychology 4
Psychology 5
Social and Cultural Anthropology 1
Social and Cultural Anthropology 2
Social and Cultural Anthropology 3
Sports, Exercise and Health Science 1
Sports, Exercise and Health Science 2
Visual Arts 1
Visual Arts 2
Visual Arts 3
Visual Arts 4
Visual Arts 5
World Religion 1
World Religion 2
World Religion 3

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IB Maths EE examples

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Vălenii de Munte, Romania. Photo by Adriana Duduleanu/Getty

How to make a map of smell

We can split light by a prism, sounds by tones, but surely the world of odour is too complex and personal strangely, no.

by Jason Castro + BIO

What is the distance between the scent of a rose and the odour of camphor? Are floral smells perpendicular to smoky ones? Is the geometry of ‘odour space’ Euclidean, following the rules about lines, shapes and angles that decorate countless high-school chalkboards? To many, these will seem like either unserious questions or, less charitably, meaningless ones. Geometry is logic made visible, after all; the business of drawing unassailable conclusions from clearly stated axioms. And odour is, let’s be honest, a bit too vague and vaporous for any of that. The folksy idea of smell as the blunted and structureless sense is at least as old as Plato, and I have to confess that, even as an olfactory researcher, I sometimes feel like I’m studying the Pluto of the sensory systems – a shadowy, out-there iceball on a weird orbit.

In recent years, however, things have changed dramatically, and understanding what one might call ‘the geometry of smell’ is a field that now enlists task forces of neuroscientists working together with mathematically trained theorists and artificial intelligence (AI) experts. While we’re notoriously bad at intuiting how our minds organise phenomena like colours and smells, machines offer a potential route for outsourcing introspection, and doing it with rigour. They can be trained to mimic human performance on perceptual tasks, and they make available the internal representations they use to do this – the abstract spaces and coordinate frames in which the ineffable stuff of thought lives.

The recent publication of an unprecedentedly comprehensive and accurate ‘odour map’ in the journal Science is a declaration of this new paradigm for smell. In the same way that a map of the United States tells you that Buffalo is a bit closer to Detroit than to Boston, the odour map can tell you that the smell of lily is closer to grape than it is to cabbage. That much may seem obvious, but the real magic comes from the fact that any arbitrary chemical’s precise location on the odour map can be calculated. From having only a few facts on hand about a chemical, we can compute that it smells, say, 13 per cent closer to lily than to grape. By analogy, it would be something like having a formula that takes in information about an unknown city’s population size and soil composition, and spits out, correctly, the exact longitude and latitude of Philadelphia.

A map like this isn’t just an accurate, laboriously assembled catalogue of relative locations and perceptual similarities. It’s something much more powerful: a set of derived rules for calculating which odour goes where. Knowing these rules, you can apply them not just to a small handful of chemicals, but to the entire world of odorous chemicals. You can see where the most densely populated areas are, and where the ‘state lines’ are in the world of smells. This is a prospect that dazzles the world’s perfumers and gourmands, and anyone else interested in the notoriously difficult and fickle task of predicting how something will smell from its chemical properties.

But, even more than this, it also raises intriguing philosophical questions about what our noses even think chemicals are , and what it means to measure their similarity. What is it about the world that our noses are ‘mapping’, in other words, when they put lily close to grape? Are they latching on to some single molecular property like a chemical’s weight or size? Are they calculating some kind of average fingerprint across a variety of such properties? Or are they doing something different altogether, like locating molecules in a space of common metabolic reactions?

Interestingly, the last of these seems to be more the case. The perceptual yardstick our brains use to measure, organise and compare smells may ultimately have less to do with what a chemist could discover from running a sample, and more to do with our deep relational histories with the world. Our noses may turn out to be geometers not of the world’s fixed and invariant properties , but of its evolved and Earthly processes .

There is something poetic in the idea that, in order to crack the ‘ancient sense’, the crude, most scientifically incorrigible sense, we’ve had to wait for machine intelligence. This is in contrast with the other sensory modalities, which began to share their secrets in the 17th century to wizard-like seers bearing prisms and tuning forks.

T he basic investigative template for ‘geometrising’ the senses was developed by Isaac Newton in the late 1600s. In his iconic experiments in optics, performed in his Cambridge parlour, he uncovered a relationship between the colour of light and its refrangibility – the degree to which it was bent by a simple prism. The mere description of this fact would have ranked among the most important scientific discoveries ever, but Newton went a step further, and fit his observations to a geometric model. Wrapping the seven primary colours of the visible spectrum along the circumference of a circle (see figure below), he produced the first ‘chromaticity diagram’ – a forerunner of the colour wheels that we use to organise our thinking about colours and their mixtures.

The circle, for Newton, was not just some poetic flourish, but a commitment to a very particular way of encoding colour’s properties. It was an invitation to pull out our protractors and rulers, and make calculations about how colours relate to one another, and combine into mixtures. The components of a three-part mixture of fully saturated red, yellow and green, for example, would be represented as the three vertices of a triangle, with each vertex pinned on the colour circle’s circumference at the appropriately labelled point. The centre of mass of this triangle is a single point in the circle’s interior, and specifies the hue and saturation of the resultant mixture. In the case of mixing all seven primary colours to an equal degree, the centre of mass of the seven-pointed figure would be at the exact centre of the circle, which Newton designated as white.

A circular diagram like a pie chart divided into regions annotated as sections of colour

There is of course a lot more to colour vision than what Newton described in his Opticks (1704), and even his contemporaries noted flaws and shortcomings in his model. Nevertheless, his achievement still encapsulates the ambition of the classical paradigm for sensory mapping. It seeks a mathematical correspondence between measurable and intrinsic properties of the natural world (like light’s refrangibility, which we now attribute to wavelength), and phenomenological qualities of mind (like colour, pitch and smell). There is something like a Pythagorean, the-world-is-mathematics mysticism to the endeavour.

The basic logic of pitch perception was also cracked similarly, with simple tools like tuning forks and spherical ‘resonators’ used to produce pure tones of a single frequency, from which rules about consonant pitch combination could be derived. Pitch perception as a whole is fantastically complex but, in broad strokes, our entire auditory system – from the tiny coiled-up cochlea of our inner ear, to the auditory portions of our sensory cortex – is built on the basic principle of organising low, middle and high tones like the keys of a piano. Striking neighbouring notes on a piano will also ‘strike’ neighbouring neurons in your brain.

S mell could never be parsed with a tool as fundamental as a tuning fork, and it never got its Newton, but it was not for a lack of people trying to follow his lead as a geometer of the senses. The idea that there might exist a small number of ‘odour primaries’ that, by analogy to the prismatic colours, organised the world of smells has occurred to many, and the search for these continued in earnest well into the 20th century.

An early and influential classification scheme for odours by the famed botanist and taxonomist Carl Linnaeus, in 1756, included seven types: aromatic, fragrant, ambrosial (musky), alliaceous (garlic), hircine (goaty), repulsive, and nauseous. A contemporary of Linnaeus’s, Albrecht von Haller, was a bit stingier with his adjectives, and proposed a more austere scheme of three basic odour types: sweet/ambrosiac, stench, and intermediate. One senses that ‘intermediate’ is doing a lot of work here, but perhaps Haller adopted the idea out of a conviction that all odours could be squeezed onto a line, and organised along a single axis. If these early odour taxonomies sound like they have a rather ad-hoc feel to them, it’s because they were the fruit of introspection rather than careful data collection and measurement. Basically, these guys were winging it.

In fairness though, it wasn’t (and honestly still isn’t) really obvious how to not wing it. With all due deference to one of history’s geniuses, Newton had it easy. He could create essentially any visible colour at will by rotating a few pieces of polished glass in a slit of sunlight. The stimulus just showed up, unasked for, when the sun rose, and in a form that was virtually readymade for scientific interrogation. Odours are far less workable. If Newton had wanted to study odour, he would have had to start by grabbing some plants, maybe some spoiled food, a crust of bread, a swab from his chamber pot if he was feeling audacious and naughty. This doesn’t exactly scream ‘Newtonian’. The critical missing abstraction of ‘the chemical compound’ as the basic property-bearing token of smell was still far off, as were the techniques for synthesising pure chemicals for testing purposes.

Henning proposed an odour prism that organised smells into flowery, fruity, resinous, spicy, burnt, and foul

But even if these impediments could have been, miraculously, dealt with, there are additional, deeper complications that make the problem of smell harder than the problem of colour. These boil down to the fact that chemicals aren’t smoothly graded variations of a single underlying phenomenon, like light is. Rather, they’re collections of the world’s particulate stuff. And, ultimately, there’s just a lot of stuff and kinds of stuff out there, making it highly unlikely that some single chemical property – a molecular analogue of light’s refrangibility – will capture all the meaningful variability in the wild and woolly world of chemicals. If there was a map from chemical features to odour qualities, it would have to involve something more complicated than a circle, with more places – in fact, more dimensions – to distribute the chemicals.

Perhaps something like a prism would do? If we take it as a loose geometric metaphor, it seems to have some virtues over the circle, and is reaching in the right direction. The prism has faces and facets on different planes, which could be used for organising molecules according to diverse criteria like atom type or chemical group. Its sharp points suggest areas of aggregation and separation in chemical space that emphasise odour’s discrete categories versus light’s continua.

For the German scholar of smell Hans Henning, this was more than just metaphor. In his book Der Geruch (‘Smell’) (1916), he proposed the idea of an abstract odour prism that organised the world of smells, with its six pointy vertices corresponding to what he considered to be the olfactory primaries: flowery, fruity, resinous, spicy, burnt, and foul. Even though there had been considerable developments that allowed for better quantification of how humans perceive odour, and better physical descriptions of odour stimuli, the field was not ready for a proposal like Henning’s. He was, by all accounts, one of those scientists, described by the US neurobiologist Gordon Shepherd, ‘whose imaginations cannot resist the temptation to put together an underconstrained theory.’ Also, Henning didn’t do himself any favours in forcefully promoting his work, and swiping at influential icons of the field like the Dutch scientist Hendrik Zwaardemaker, who had pioneered the use of olfactometers – steam-punky contraptions of valves and tubing that delivered controlled doses of odour. An early reviewer who otherwise spoke very positively of Henning’s book still felt compelled to refer to him, on the record, as ‘a ruthless – in fact very uncivil – iconoclast.’ Henning had bulldozed into the discussion a strongly geometric conception of odour that was inspired and influential but, ultimately, a house of cards.

A diagram of a prism, each corner annotated with a German adjective

To Henning’s credit, despite his theory’s shakiness, it was specific enough to be testable, and the invitation was taken up by several, including the psychologist Malcolm Macdonald. In a comprehensive and devastating critique from 1922 that ended with a long section entitled ‘Logical and Factual Inadequacies of Henning’s Theory’, Macdonald investigated whether there was really a prism behind things in the world of smells. Using judgments of relative odour similarity as a proxy for distance (where ‘similar smelling’ = ‘close’), he performed critical perceptual reality-checks, like asking whether chemicals taken from opposite ends of the long prism diagonal smelled the least similar. When you say that odour is a prism, you should expect that your colleagues will pull out their calculators and check.

‘Odour maps’ gave a bird’s eye view of how the human nose organises the world of chemicals

What, exactly, did Henning get wrong? The question is a bit tongue-in-cheek, because it’s not clear that there’s anything he got right. Still, if we want to be charitable, we might say he was seduced by a Lego-like view of molecules that organic chemists were developing at the time. They saw a modular system in which organic molecules were assembled from a small library of so-called functional groups, resulting in motifs of a few atoms arranged in stereotyped ways. In addition to haunting the dreams of hopeful premeds everywhere, these functional groups were thought to confer a molecule’s specific properties, and define its basic reactivity. In Henning’s view, it was perfectly sensible that the same functional groups should also confer the primary odours he had identified. Indeed, there is something enticing about the idea that the chemist’s alphabet for organic molecules might also be the nose’s alphabet for smell. Ultimately, however, nature chose not to oblige. Few olfactory neuroscientists would claim that functional group is unimportant for determining odour quality, but it is clearly not the whole story.

In modern machine-learning parlance, we might say that Henning didn’t have a rich enough feature set for representing smells. In committing to functional group as his basic smell alphabet, he implicitly adopted a specific idea of what a molecule fundamentally is, and discarded other potentially useful features that could serve as the grist for odour prediction. A molecule, after all, isn’t just a list of the Lego blocks from which it’s made. It’s also a springy little thing that spins and vibrates, and chemists can ring it like a tiny molecular bell to listen for clues about its structure. It is also a list of descriptive attributes like ‘strongly acidic’, or ‘non-polar’ (having symmetrically distributed charges). And it’s also a lump of just so much stuff, maybe a bit bulkier here, and more stretched-out there.

Instead of doing what Henning did, seizing on one characteristic in advance, the best way forward is obviously an agnostic mindset where the data does the talking. Instead of making fantastically wrong guesses about which chemical features determined odour quality, why not winnow them down from a massive list of all conceivable features?

That’s the approach in studies pioneered by the US scholar Susan Schiffman and others in the 1970s and ’80s. The basic idea was to take a set of a few dozen odorous molecules and create a map summarising their relative perceptual similarities. Similar to how one could create a rough map of the US from a table of all (of the many!) between-city distances, these ‘odour maps’ gave a bird’s eye view of how the human nose organises the world of chemicals. With this perceptual map in hand, the question then turned to chemistry: what is it about molecules that assigns them to some particular portion of the perceptual map? To get at this, Schiffman and others used a range of ‘dimensionality reduction’ techniques to see which chemical features – of the many hundreds potentially available – were most effective at recapitulating the map. These approaches generated significant interest for a time, but they too were unable to solve the problem and went dormant for decades – until the age of AI.

I t was in 2017 that data sets were finally democratised enough for machine learning to help scientists widen the search. An important milestone that year was the DREAM challenge to see who might solve the odour map using AI. Published in Science , the winning models were endorsed by the community as potential inroads – suggesting that handing our model-making to the custody of the machines was the right intellectual move.

The best models, the so-called ‘random forests’, used AI to aggregate a host of models. The result could be baroque and inscrutable systems of rules for performing prediction tasks. They can get the right answer, but it’s often by finding lengthy and complex rubrics along the lines of: ‘If the molecular weight is > X, and the number of carbons is > Y, and the Moreau-Broto autocorrelation of lag 7 is < Z, and…, and…, then the molecule will smell like rose.’

It’s of course possible that odour categorisation is handled by similar ‘brute’ computations in the brain, but one is left with the nagging question: is that really how nature solved it? Not through the economy of Occam’s razor but in the thicket of Occam’s forest? Where’s the deep principle? The basic organising axes? The geometric insight? An important and often-asked question about these kinds of ‘data-driven’ models is whether their success at prediction actually indicates understanding or, at least, the kind of understanding that science has historically prized and glorified through tidy parables of discovery like ‘Newton and the prism’.

One way forward is to give up on the idea of an organised odour space whose curves and contours cleanly track some yet-to-be revealed chemical properties. After all, if ‘stinky feet’ and ‘gourmet cheese’ can be two valid descriptions of the same physical object, perhaps odour qualities are just too labile and individual to really serve as the targets for prediction. Perhaps they reflect more what we’ve learned from living in the messy world than anything intrinsic to it, encoding our idiosyncratic experiences with, and predilections for, feet and cheeses. Perhaps there is even something romantic and worth defending here in the idea of a sense that, decades into the age of mass digitisation, remains stubbornly phantom-like and evanescent, unmeasurable, and fundamentally unavailable for capture by geometric concepts.

Scientists are training, tuning and tweaking million-parameter models that ingest digitised molecule after digitised molecule

Or we go in the opposite direction. Hit the problem with even more data and more computing power. This was the big bet made by Osmo, a startup based in Cambridge, Massachusetts that began several years ago as the digital olfaction group at Google Brain, and which now has several dozen neuroscientists, chemists and computer scientists on its staff.

Osmo is the brainchild of Alex Wiltschko, a Harvard-trained neurobiologist who made his mark in grad school developing pioneering computer-vision systems for analysing animal behaviour. After growing up in small-town Texas, where, he drily notes ‘neither computers nor perfume were popular’, his twin passions for aromas and algorithms eventually landed him at the helm of a company that is ‘giving computers a sense of smell’.

This is about as far as one can get from Newton investigating the senses in his lonely parlour, sketching models on vellum with a quill pen. Instead, these scientists are collaborating to develop dense code repositories, training, tuning and tweaking million-parameter models that ingest digitised molecule after digitised molecule under the mandate This one smells like rose, this one smells like grass, figure out how to make that happen. The chemicals are not given to the model as lists of predetermined molecular properties that are served up to some homunculus chemist in the nose. Instead, they’re represented as skeletal and stripped-down graphs that capture only basic information about atom identities and their connectivity. The model is not trying to find what aspects of known chemistry are important for smell. It’s trying to discover whether chemical principles we haven’t yet thought of may hold the key for smell.

The Osmo model is a type of graphical ‘deep net’ that’s loosely inspired by the successive processing stages of the brain’s sensory systems. The analogy isn’t exact, but it’s similar to how your brain captures raw information from the world and passes it downstream to units that will ultimately have something useful or actionable to say about the inputs: ‘It’s a cat!’ or ‘Smells awful!’ The output units are the doers and the deciders whose performance can be evaluated (‘Nope, it’s actually a dog’ or ‘Yup, that chemical really does smell awful’), but many important insights are found in the intermediate, or ‘hidden’, network layers, too. These can be thought of as a transformational space that squeezes and warps raw sensory inputs into sensory judgments. The connections between units that define these transformations are learned gradually and incrementally by an AI system as it continually iterates and self-adjusts based on how well it mimics human judgments. By peeking at these intermediate layers, we can get the insights of a latterday, AI-supercharged Newton of smell. They tell us how we might think about the space that chemicals live in. Or at least, as our nose sees it.

S o how do we get from the transformations within the Osmo net to a literal geometry? The geometry at hand is not a circle, or a prism, or any kind of simple archetypal shape. Instead, it’s more like a world of craggy chemical continents, each demarcating a conspicuous aspect of human ecology, each seeming to invite a set of actions or appetites. There is, for example, a continent of ‘fermentation’, a ‘green’ continent, a land of the ‘meaty and savoury’. The key notion is that, in this space, two chemicals are rendered as close together and similar-smelling not because they necessarily share intrinsic structural features, but because they share ecological roles and have a close and contingent relationship out in the wild, as it were.

The Newtonian story of colour space is about how human perception latches on to the world’s universal and impersonal attributes (think about light’s wavelength and refrangibility). But the developing story of smell is about how our noses have decoded the world as it manifests locally, relationally and idiosyncratically on our planet. Odour space, in other words, is framed in human-centred coordinates, reflecting our histories as foragers and hunters in a world that blooms and withers, with matter that ripens and decays. It is a geometry that invests matter with its meanings and possibilities for us .

But it’s not just some loosely poetic space. Tapping the Osmo model, one can compute distances and angles here, predict which chemical will smell exactly halfway in-between musk and carvone, examine whether a collection of chemicals should trace out a smooth or squiggly path in odour-perceptual space. Moreover, the Osmo model does this demonstrably better than other attempts, suggesting that the way it’s measuring distances between chemicals may point to a deep principle of odour processing.

To smell something is to understand the neighbourhood it lives in

Interestingly, distances computed on the map correlate strongly with what has been termed ‘metabolic distance’ – roughly, how reachable one chemical is from another through common metabolic pathways. If nature can easily move from chemical A to chemical B through a small number of fermentation reactions, say, chances are your nose will find A and B to smell alike, even if they lack obvious structural similarities. The important corollary is that molecules with striking structural similarities needn’t smell alike (though, of course, they often will). A and B might hypothetically differ by only one double bond, but if it’s a very expensive double bond to form or break, requiring a large number of synthesis steps and chemical pirouettes, then the compounds will smell different to us. What the nose seems to know is not the static world of chemicals, but the movements that nature makes through it.

A philosopher would say that your nose appears to be an empiricist – it classifies and categorises chemicals on the basis of relationships that must be learned from the world either over evolutionary timescales or over an individual’s lifetime. A mathematician, following up, would say that what is learned is the abstract, high-dimensional manifold that tracks the world’s chemical relationships – its partitioning into the branches, cycles and pathways that shuttle around the world’s carbon. To smell something is to locate it on this manifold, to understand the neighbourhood it lives in.

These are still early days for theorising about the structure of odour space, but several investigators have put forward the idea that the space is non-Euclidean, meaning that it’s a far cry from the ‘intuitive’ geometry of secondary school, where the angles of a triangle always add up to 180 degrees. Instead, odour space may have an intrinsic curvature to it (like a potato chip, according to one theorist), driven by the fact that distances in odour space are defined less like the physical distance between two people, and more like their social distance.

Amazingly, Macdonald, the critic of Henning described above, had an intuition of this back in 1922, when he suggested modifications to the odour prism that amounted to replacing it with a ‘hollow hyper-solid with solid tetrahedrons as its sides’. This is difficult to visualise but, basically, it is a higher-dimensional prism that gives odours more space in which they can distribute themselves. ‘There is no reason why mental continua should occur only under Euclidian [sic] limitations,’ he noted. Perhaps smell has been the last standing sensory mystery because its mathematics has proven to be the most esoteric.

Human rights and justice

My elusive pain

The lives of North Africans in France are shaped by a harrowing struggle to belong, marked by postcolonial trauma

Farah Abdessamad

Psychiatry and psychotherapy

The therapist who hated me

Going to a child psychoanalyst four times a week for three years was bad enough. Reading what she wrote about me was worse

Michael Bacon

Consciousness and altered states

A reader’s guide to microdosing

How to use small doses of psychedelics to lift your mood, enhance your focus, and fire your creativity

Tunde Aideyan

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Family life

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After my marriage failed, I strove to create a new family – one made beautiful by the loving way it’s stitched together

The cell is not a factory

Scientific narratives project social hierarchies onto nature. That’s why we need better metaphors to describe cellular life

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Public health

It’s dirty work

In caring for and bearing with human suffering, hospital staff perform extreme emotional labour. Is there a better way?

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Improving Mathematics Education: Resources for Decision Making (2001)

Chapter: 7 conclusion, 7 conclusion.

The eight documents summarized in this report have been categorized as informing content, learning, teaching, assessment, and program changes. The Committee felt that organizing the recommendations in the same way would enable readers to understand the connections among the documents and illustrate common themes. Tables 7–1–5 highlight those recommendations that seem to cut across the eight publications. While there are diverse opinions in the mathematical community on how to improve mathematics education, the reports contain noticeable overlap in both the spirit and the specifics of the recommendations.

The tables do not contain all of the recommendations summarized in Improving Mathematics Education, let alone the entire set of recommendations across the eight documents. Empty cells in the table should not be construed as lack of support; the publication may actually address the particular issue but not make a direct recommendation. And, although some topics were common to several documents, this does not necessarily imply the recommendations were the same. Note that despite the increased interest in assessment and accountability throughout education, little attention seems to be paid to the topic across these publications.

The Committee identified several cross-cutting themes that emerged from the documents by considering the recommendations in each section and categorizing them across topics that emerged from the sets of recommendations. These common themes can provide a background to help frame the decisions we must make to improve school mathematics in the United States.

Change. The world is changing, society's expectations are changing, and our students are changing. It follows that the mathematics needed, its importance, and methods for teaching it are changing as well. As Before It's Too Late states, “It is abundantly clear from the evidence clearly at hand that we are not doing the job we should.” Five of the documents (see Table 7–5 ) reflect the need to change what we do, how we do it, and how for all students. For example, Adding It Up makes direct statements about

we assess the impact of the changes to provide better instructional programs problems in the current system: “Our experiences, discussions, and review of the literature have convinced us the school mathematics demands substantial change” (p. 407). It also suggests coordination of curriculum, instructional materials, assessment, and professional development; school organization should drive the changes. Every Child Mathematically Proficient offers an action plan for changing mathematics education. The Mathematical Education of Teachers calls for “changing expectations for mathematical knowledge.”

Equity. According to the documents, all students can and should be mathematically proficient. The reports consistently deliver the message that school mathematics can and must empower all students (see Table 7–5 ). For example, five of the documents emphasize high expectations for all students. Every Child Mathematically Proficient calls for ending “dead-end” tracks, and High Stakes calls for the end of tracking that discriminates

against certain students. The content areas emphasized in the documents (see Table 7–1 ) underscore that “mathematics for all” does not mean a watered-down mathematics but an authentic, coherent, and functional mathematics that expects more of every student than in the past. For example, Every Child Mathematically Proficient recommends algebra and geometry for all students, and a majority of the documents emphasize mathematical reasoning.

Content. Mathematical content is the centerpiece for teaching, instructional materials, assessments, teacher preparation, and professional development. The documents make strong statements about the importance of reasoning, problem solving, algebra, and geometry ( Table 7–1 ). For example, reasoning and proof is a content strand in Principles and Standards for School Mathematics, and Adding It Up includes the notion of adaptive reasoning as essential for mathematical proficiency. Six of the documents call for strong content knowledge on the part of teachers ( Table 7–3 ) and emphasize the need to align curriculum instruction and assessment ( Table 7–5 ). In particular, Before It's Too Late states “high-quality teaching requires that teachers have a deep knowledge of subject matter” (p. 22), while The Mathematics Education of Teachers suggests the specific number of content hours that should be part of undergraduate teacher preparation programs. If teachers do not fully understand fundamental mathematics, it is unlikely they will impart that knowledge to their students.

Understanding. While research findings in education are always open to further interpretation, it is increasingly clear that learning is a complex cognitive process that builds on prior knowledge and requires active engagement with new situations. As described in How People Learn, a growing body of knowledge suggests that learning is best accomplished when accompanied by conceptual knowledge and making sense of new concepts, with an emphasis on understanding. Adding It Up makes this explicit in terms of learning mathematics, emphasizing the need for both conceptual understanding and procedural fluency. Four of the documents ( Table 7–2 ) support the use of student thinking to build understanding. The notion of learning as a sense-making process also has import for teaching and teacher preparation with recommendations that teachers be prepared to teach mathematics in ways that enable students to learn. For example, Educating Teachers of Science, Mathematics, and Technology suggests that teachers “regularly update the content knowledge and pedagogical tools needed to teach in ways that enhance student learning and achievement in these subjects” (p. 109).

Research. The use of research to support mathematics education is a key element in majority of the documents ( Table 7–5 ). This ranges from How People Learn, which calls on existing research to substantiate claims made in the document, to High Stakes, which advocates advanced research on

topics for which there are more questions than answers. Educating Teachers of Science, Mathematics, and Technology calls for enhanced links between research and practice, and Every Child Mathematically Proficient suggests such links should be translated into the development of curricular materials and professional development programs for teachers. The prominence of research in these documents suggests a shift in the way changes are made in mathematics education and has implications for how program changes are instituted.

Collaboration. Making change in our complex education system is difficult, and improvement requires collaboration among diverse stakeholders who represent different parts of the system. Systemic change requires new forms of partnerships to make the system more productive and to provide solutions that cut across the system components. Educating Teachers of Science, Mathematics, and Technology and The Mathematics Education of Teachers make strong statements about forming K–12 and university partnerships, involving mathematicians and scientists to improve teacher preparation and development ( Table 7–3 ). Several of the documents suggest that parents as well as teachers must be brought into the process of change in school mathematics, including discussions of curricular goals and how teaching and assessments have changed in mathematics classrooms ( Table 7–5 ).

Professional Development. Every report that addresses changing conditions and expectations eventually arrives at the need for more effective forms of professional development. Six of the reports ( Table 7–3 ) support the need for highly qualified teachers who take part in continuous learning that is part of the system for teachers: “High-quality teaching must be continually reshaped” ( Before It's Too Late, p. 22). Given enhanced opportunities for professional growth and development, teachers will be able to implement the vision of mathematics instruction called for in these reports. Moreover, the professional development must be more than a “patchwork of courses” ( Educating Teachers, p. 31) and must be related to the mathematics teachers are expected to teach and to how to helping students understand this mathematics.

Technology. The documents recognize technology as a major factor driving the need for change, creating a different world with different expectations about the mathematics students will need ( Table 7–1 ). The reports, however, are not unanimous in their technology recommendations. Principles and Standards states that technology is essential to the teaching and learning of mathematics. Every Child Mathematically Proficient cautions that further study is needed to explore how technology should be used to enhance student learning in mathematics, and Adding It Up is also cautious, but suggests that technology offers much promise in the search for ways to improve what happens in classrooms.

Mathematics Specialists. As teachers learn new content, new methods of instruction, and new methods of assessment, they need resources and support. There is also immense pressure on elementary teachers, who have multiple subjects to teach. While math specialists are called for in several of the documents ( Table 7–5 ), the nature of such specialists and the recommendations vary. The Mathematics Education of Teachers and Every Child Mathematically Proficient call for math specialists to teach mathematics content by grade 5. Adding It Up recommends that mathematics specialists provide support for K–8 teachers, and Principles and Standards calls for teacher-leaders throughout the system.

Closing Comments

Considering the overall mathematical landscape and potential directions necessary for improvement, the Mathematical Sciences Education Board recommends that educators use reports such as the ones described in this document to help them make informed decisions. The committee also recommends that the decision making process include participation and input from across the community, with diverse contributions from a wide spectrum of individuals—in much the same spirit as these documents were developed.

The committee recognizes that all the information necessary to make the many decisions facing educators is not available. The research-based findings and the recommendations in these documents, however, can be a rich resource for educators engaged in actions and initiatives intended to make real and positive change in the lives of students and teachers in mathematics classrooms across the nation. These documents can help educators as they struggle to make wise and defensible decisions and strive to meet the challenges of implementing truly excellent programs of mathematics instruction.

Improving Mathematics Education has been designed to help inform stakeholders about the decisions they face, to point to recent research findings, and to provide access to the most recent thinking of experts on issues of national concern in mathematics education. The essence of the report is that information is available to help those charged with improving student achievement in mathematics. The documents cited above can guide those who make decisions about content, learning, teaching, and assessment. The report is organized around five key questions:

What should we teach, given what we know and value about mathematics and its roles?
How should we teach so children learn, given what we know about students, mathematics, and how people learn mathematics?
What preparation and support do teachers need?
How do we know whether what we are doing is working?
What must change?

Each of the five main chapters in this report considers a key area of mathematics education and describes the core messages of current publication(s) in that area. To maintain the integrity of each report's recommendations, we used direct quotes and the terminology defined and used in that report. If the wording or terminology seems to need clarification, the committee refers the reader directly to the original document. Because these areas are interdependent, the documents often offer recommendations related to several different areas. While the individual documents are discussed under only one of the components in Improving Mathematics Education , the reader should recognize that each document may have a broader scope. In general, the references in this report should serve as a starting point for the interested reader, who can refer to the original documents for fuller discussions of the recommendations and, in some cases, suggestions for implementation. Improving Mathematics Education is designed to help educators build a critical knowledge base about mathematics education, recognizing that the future of the nation's students is integrally intertwined with the decisions we make (or fail to make) about the mathematics education they receive.

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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

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NPR in Turmoil After It Is Accused of Liberal Bias

An essay from an editor at the broadcaster has generated a firestorm of criticism about the network on social media, especially among conservatives.

Uri Berliner, wearing a dark zipped sweater over a white T-shirt, sits in a darkened room, a big plant and a yellow sofa behind him.

By Benjamin Mullin and Katie Robertson

NPR is facing both internal tumult and a fusillade of attacks by prominent conservatives this week after a senior editor publicly claimed the broadcaster had allowed liberal bias to affect its coverage, risking its trust with audiences.

Uri Berliner, a senior business editor who has worked at NPR for 25 years, wrote in an essay published Tuesday by The Free Press, a popular Substack publication, that “people at every level of NPR have comfortably coalesced around the progressive worldview.”

Mr. Berliner, a Peabody Award-winning journalist, castigated NPR for what he said was a litany of journalistic missteps around coverage of several major news events, including the origins of Covid-19 and the war in Gaza. He also said the internal culture at NPR had placed race and identity as “paramount in nearly every aspect of the workplace.”

Mr. Berliner’s essay has ignited a firestorm of criticism of NPR on social media, especially among conservatives who have long accused the network of political bias in its reporting. Former President Donald J. Trump took to his social media platform, Truth Social, to argue that NPR’s government funding should be rescinded, an argument he has made in the past.

NPR has forcefully pushed back on Mr. Berliner’s accusations and the criticism.

“We’re proud to stand behind the exceptional work that our desks and shows do to cover a wide range of challenging stories,” Edith Chapin, the organization’s editor in chief, said in an email to staff on Tuesday. “We believe that inclusion — among our staff, with our sourcing, and in our overall coverage — is critical to telling the nuanced stories of this country and our world.” Some other NPR journalists also criticized the essay publicly, including Eric Deggans, its TV critic, who faulted Mr. Berliner for not giving NPR an opportunity to comment on the piece.

In an interview on Thursday, Mr. Berliner expressed no regrets about publishing the essay, saying he loved NPR and hoped to make it better by airing criticisms that have gone unheeded by leaders for years. He called NPR a “national trust” that people rely on for fair reporting and superb storytelling.

“I decided to go out and publish it in hopes that something would change, and that we get a broader conversation going about how the news is covered,” Mr. Berliner said.

He said he had not been disciplined by managers, though he said he had received a note from his supervisor reminding him that NPR requires employees to clear speaking appearances and media requests with standards and media relations. He said he didn’t run his remarks to The New York Times by network spokespeople.

When the hosts of NPR’s biggest shows, including “Morning Edition” and “All Things Considered,” convened on Wednesday afternoon for a long-scheduled meet-and-greet with the network’s new chief executive, Katherine Maher , conversation soon turned to Mr. Berliner’s essay, according to two people with knowledge of the meeting. During the lunch, Ms. Chapin told the hosts that she didn’t want Mr. Berliner to become a “martyr,” the people said.

Mr. Berliner’s essay also sent critical Slack messages whizzing through some of the same employee affinity groups focused on racial and sexual identity that he cited in his essay. In one group, several staff members disputed Mr. Berliner’s points about a lack of ideological diversity and said efforts to recruit more people of color would make NPR’s journalism better.

On Wednesday, staff members from “Morning Edition” convened to discuss the fallout from Mr. Berliner’s essay. During the meeting, an NPR producer took issue with Mr. Berliner’s argument for why NPR’s listenership has fallen off, describing a variety of factors that have contributed to the change.

Mr. Berliner’s remarks prompted vehement pushback from several news executives. Tony Cavin, NPR’s managing editor of standards and practices, said in an interview that he rejected all of Mr. Berliner’s claims of unfairness, adding that his remarks would probably make it harder for NPR journalists to do their jobs.

“The next time one of our people calls up a Republican congressman or something and tries to get an answer from them, they may well say, ‘Oh, I read these stories, you guys aren’t fair, so I’m not going to talk to you,’” Mr. Cavin said.

Some journalists have defended Mr. Berliner’s essay. Jeffrey A. Dvorkin, NPR’s former ombudsman, said Mr. Berliner was “not wrong” on social media. Chuck Holmes, a former managing editor at NPR, called Mr. Berliner’s essay “brave” on Facebook.

Mr. Berliner’s criticism was the latest salvo within NPR, which is no stranger to internal division. In October, Mr. Berliner took part in a lengthy debate over whether NPR should defer to language proposed by the Arab and Middle Eastern Journalists Association while covering the conflict in Gaza.

“We don’t need to rely on an advocacy group’s guidance,” Mr. Berliner wrote, according to a copy of the email exchange viewed by The Times. “Our job is to seek out the facts and report them.” The debate didn’t change NPR’s language guidance, which is made by editors who weren’t part of the discussion. And in a statement on Thursday, the Arab and Middle Eastern Journalists Association said it is a professional association for journalists, not a political advocacy group.

Mr. Berliner’s public criticism has highlighted broader concerns within NPR about the public broadcaster’s mission amid continued financial struggles. Last year, NPR cut 10 percent of its staff and canceled four podcasts, including the popular “Invisibilia,” as it tried to make up for a $30 million budget shortfall. Listeners have drifted away from traditional radio to podcasts, and the advertising market has been unsteady.

In his essay, Mr. Berliner laid some of the blame at the feet of NPR’s former chief executive, John Lansing, who said he was retiring at the end of last year after four years in the role. He was replaced by Ms. Maher, who started on March 25.

During a meeting with employees in her first week, Ms. Maher was asked what she thought about decisions to give a platform to political figures like Ronna McDaniel, the former Republican Party chair whose position as a political analyst at NBC News became untenable after an on-air revolt from hosts who criticized her efforts to undermine the 2020 election.

“I think that this conversation has been one that does not have an easy answer,” Ms. Maher responded.

Benjamin Mullin reports on the major companies behind news and entertainment. Contact Ben securely on Signal at +1 530-961-3223 or email at [email protected] . More about Benjamin Mullin

Katie Robertson covers the media industry for The Times. Email: [email protected] More about Katie Robertson

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Amy Ettinger, who inspired readers with her life-affirming essays on dying, succumbs to cancer at 49

( JTA ) — Amy Ettinger, an author and creative writing instructor who chronicled the last months of her life in articles for the Washington Post , died March 20 from cancer at her home in Santa Cruz, California. She was 49.

Ettinger’s essays focused on the things she was able to do and cherish despite her diagnosis with a rare, incurable cancer called leiomyosarcoma : seeing a live performance of “Mamma Mia!” with her 14-year-old daughter, Julianna; eating her favorite pastry from a San Francisco bakery.

“ I’ve learned that life is all about a series of moments, and I plan to spend as much remaining time as I can savoring each one, surrounded by the beauty of nature and my family and friends,”she wrote.

Ettinger was an occasional contributor to Kveller, the Jewish family website that is a Jewish Telegraphic Agency partner. There she wrote about her mother’s kugel recipe (“light brown on its crispy top, and the color of milky coffee in the middle”) , and how she, as a “non-observant Jew,” marked Yom Kippur — which in 2013 happened to fall on her 10th wedding anniversary .

“Like Yom Kippur, a wedding anniversary is a time to take a step back from your daily life — to weigh the good and bad, to contemplate your triumphs and missteps, to make a vow to do better individually and as a couple,” she wrote.

Ettinger was born in Rochester, New York, and grew up in Cupertino, California. She discovered her calling as a journalist in high school. She majored in American literature at UC Santa Cruz and earned a master’s degree in journalism from Northwestern University in 1999.

Her writing appeared in the New York Times, the Washington Post, New York Magazine, Salon, CNN and Newsweek. In a 2021 article for AARP, she wrote how her mother’s death inspired her to learn Sheila Ettinger’s favorite game: mahjong. She taught writing classes at Stanford Continuing Studies.

In 2017, Penguin Random House published her memoir-cum-travelogue “Sweet Spot: An Ice Cream Binge Across America.” In it she wrote how she keeps “between fifteen and thirty dollars’ worth of ice cream in my freezer at all times” — not to eat, but as an “emergency backup system” in case one of her favorite shops or stores runs out.

Her follow-up story to her Washington Post article, titled “I Have Little Time Left. I Hope My Goodbye Inspires You,” appeared on the newspaper’s homepage less than two weeks before she died.

“I am choosing to focus my limited time and energy on doing the things I love with the people I care most about. It’s a formula that works, I think, no matter where you are in your life,” she wrote.

In an article written after she died , her husband, the writer Dan White, wrote that she had dictated her last essay to him from a reading room at UC Santa Cruz with a view of a redwood forest. He said she had gotten hundreds of personal responses: A handful “unwelcome, including missives from ultrareligious people wanting my proudly Jewish wife to get saved to spare herself from hellfire,” but the vast majority saying Ettinger had inspired them to make the most of their lives no matter the cards they’d been dealt.

“Amy had no way of predicting that the lines she composed on the spot would be calls to action for readers from all over the United States, as well as Canada, Poland, France and Greece,” White wrote.

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Emerald Cove Student Among Florida League Of Mayors Essay Winners

The Florida League of Mayors (FLM), an organization for Florida’s mayors, founded and developed by Florida’s mayors, recently announced the 2023-24 “If I Were Elected Mayor” essay contest winners.

The contest, hosted annually, is sponsored by FLM and the Florida League of Cities. Among the winners was Marialucia Hernandez of Emerald Cove Middle School in Wellington, who took second place.

Other winners were Andrew Johnson of Gulf Coast Classical Academy in Clearwater and Aayushi Kamdar of Coral Springs Middle School, who both took first place, as well as Maya Morrison of Coral Springs Middle School, who took third place.

The contest required participants to write a 250-word essay describing how they would make a difference in their community if they were elected mayor, and how home rule, the ability of local governments to address local problems with local solutions, helps make their community a great place to live. All middle school students in the state were invited to participate. More than 1,000 essays were submitted.

“I would prioritize public transportation,” Hernandez wrote in her essay. “Public transportation can aid many citizens, especially elderly people who can’t drive, and minors under the age of 16, who can’t get a driver’s license yet. Public transportation can additionally help better the local economy, since one dollar invested in public transportation is estimated to generate five economic dollars back. Public transportation would be advantageous for essentially everyone.”

Hernandez added that she would also concentrate on affordable housing.

“To conclude, if I were to be elected mayor, I would focus on improving the quality of life, fostering community growth and promoting economic development. I vow to lead the community with integrity and utilize Home Rule to meet civilian needs,” she wrote.

Johnson’s winning essay explained that, as mayor, his goals would be to address the city’s population increase and lower the crime rate by employing more police officers. Kamdar also wrote that his focus would be on policing in the community and placed an emphasis on sustainability.

All four contest winners will be recognized by their mayors during meetings of their local governments. They will be presented with a joint resolution from the Florida League of Cities and the Florida League of Mayors, and their essays will be published on the Florida League of Mayors web site and recognized by FLC’s Quality Cities magazine.

Visit www.floridamayors.org to learn more about the Florida League of Mayors.

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