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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Problem-Solving in Mathematics Education

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Santos-Trigo, M. (2020). Problem-Solving in Mathematics Education. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_129

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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Math Lesson

Neil almond.

Fluency, reasoning and problem solving are central strands of mathematical competency, as recognized by the National Council of Teachers of Mathematics (NCTM) and the National Research Council’s report ‘Adding It Up’.

They are key components to the Standards of Mathematical Practice, standards that are interwoven into every mathematics lesson. Here we look at how these three approaches or elements of math can be interwoven in a child’s math education through elementary and middle school.

We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help elementary and middle school teachers think carefully about their practice and the pedagogical choices they make around the teaching of what the common core refers to as ‘mathematical practices’, and reasoning and problem solving in particular.

Before we can think about what this would look like in Common Core math examples and other state-specific math frameworks, we need to understand the background to these terms.

What is fluency in math?

What is reasoning in math, what is problem solving in math, mathematical problem solving is a learned skill, performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving, what is ‘performance vs learning’, teaching to “cover the curriculum” hinders development of strong problem solving skills., fluency and reasoning – best practice in a lesson, a unit, and a semester, best practice for problem solving in a lesson, a unit, and a semester , fluency, reasoning and problem solving should not be taught by rote , the ultimate guide to problem solving techniques.

Develop problem solving skills in the classroom with this free, downloadable worksheet

Fluency in math is a fairly broad concept. The basics of mathematical fluency – as defined by the Common Core State Standards for math – involve knowing key mathematical skills and being able to carry them out flexibly, accurately and efficiently.

But true fluency in math (at least up to middle school) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in math lessons means we teach the content using a range of representations, to ensure that all students understand and have sufficient time to practice what is taught.

Read more: How the best schools develop math fluency

Reasoning in math is the process of applying logical thinking to a situation to derive the correct problem solving strategy for a given question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows students to use the former to accurately carry out the latter.

Read more: Developing math reasoning: the mathematical skills required and how to teach them .

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in math. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in math is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Math problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some students can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognizing faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

As you might have guessed, the domain of mathematics is far from innate. Math doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of math) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like math) can only be improved by practicing elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that students can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018) So what is the best method of teaching problem solving to elementary and middle school math students?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case, math. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years leads to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Center at Loughborough University, UK, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support students in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

I mean to make no sweeping generalization here; this was my experience both at university when training and from working in schools.

At some point, schools become obsessed with the ridiculous notion of moving students through content at an accelerated rate. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘accelerated progress in math’ in this lesson,’ ‘School officials will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get students onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of students and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the students to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; give the students some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those students were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same math content as the fluency exercises, making it more likely that students would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers (a process referred to as “number plucking”) – not exactly high level problem solving skills.

This is one of my worries with ‘math mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content being missed in the classroom.

Students are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

By now I hope you have realized that when it comes to problem solving, fluency is king. As such we should look to mastery math based teaching to ensure that the fluency that students need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the grade you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental math strategies students should learn at each grade level is a good place to start when thinking about the core aspects of fluency that students should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get students practicing.

They could chant multiplication facts when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz students on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach multiplication for instant recall

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all math lessons.

When we think about a semester, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that semester.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practice before we can make connections, reason and problem solve with it.

The same is true for students. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practice with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

You could get students in 3rd grade to solve complicated place value problems with the numbers they should know from 1st or 2nd grade. This would lessen the cognitive load , freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Increase complexity gradually

Once they practice solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move students from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your students (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering students on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Math Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Problem Solving and Reasoning Questions and Answers
• Get to Grips with Math Problem Solving For Elementary Students
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Math Challenges To Really Stretch Your More Able Students
• Why You Should Be Incorporating Stem Sentences Into Your Elementary Math Teaching

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

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What is problem solving?

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On this page we discuss "What is problem polving?" under three headings: introduction, four stages of problem solving, and the scientific approach.

Introduction

Naturally enough, problem solving is about solving problems. And we’ll restrict ourselves to thinking about mathematical problems here even though problem solving in school has a wider goal. When you think about it, the whole aim of education is to equip students to solve problems. 

But problem solving also contributes to mathematics itself. Mathematics consists of skills and processes. The skills are things that we are all familiar with. These include the basic arithmetical processes and the algorithms that go with them. They include algebra in all its levels as well as sophisticated areas such as the calculus. This is the side of the subject that is largely represented in the Strands of Number and Algebra, Geometry and Measurement and Statistics.

On the other hand, the processes of mathematics are the ways of using the skills creatively in new situations. Mathematical processes include problem solving, logic and reasoning, and communicating ideas. These are the parts of mathematics that enable us to use the skills in a wide variety of situations.

It is worth starting by distinguishing between the three words "method", "answer" and "solution". By "method" we mean the means used to get an answer. This will generally involve one or more Problem Solving Strategies . On the other hand, we use "answer" to mean a number, quantity or some other entity that the problem is asking for. Finally, a "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself.

method + answer = solution

But how do we do Problem Solving? There are four basic steps. Pólya enunciated these in 1945 but all of them were known and used well before then. Pólya’s four stages of problem solving are listed below.

Four stages of problem solving                             

1. Understand and explore the problem  2. Find a strategy  3. Use the strategy to solve the problem  4. Look back and reflect on the solution.

Although we have listed the four stages in order, for difficult problems it may not be possible to simply move through them consecutively to produce an answer. It is frequently the case that students move backwards and forwards between and across the steps.

You can't solve a problem unless you can first understand it. This requires not only knowing what you have to find but also the key pieces of information that need to be put together to obtain the answer.

Students will often not be able to absorb all the important information of a problem in one go. It will almost always be necessary to read a problem several times, both at the start and while working on it. With younger students it is worth repeating the problem and then asking them to put the question in their own words. Older students might use a highlighter to mark the important parts of the problem.

Finding a strategy tends to suggest that it is a simple matter to think of an appropriate strategy. However, for many problems students may find it necessary to play around with the information before they are able to think of a strategy that might produce a solution. This exploratory phase will also help them to understand the problem better and may make them aware of some piece of information that they had neglected after the first reading.

Having explored the problem and decided on a strategy, the third step, solve the problem , can be attempted. Hopefully now the problem will be solved and an answer obtained. During this phase it is important for the students to keep a track of what they are doing. This is useful to show others what they have done and it is also helpful in finding errors should the right answer not be found.

At this point many students, especially mathematically able ones, will stop. But it is worth getting them into the habit of looking back over what they have done. There are several good reasons for this. First of all it is good practice for them to check their working and make sure that they have not made any errors. Second, it is vital to make sure that the answer they obtained is in fact the answer to the problem. Third, in looking back and thinking a little more about the problem, students are often able to see another way of solving the problem. This new solution may be a nicer solution than the original and may give more insight into what is really going on. Finally, students may be able to generalise or extend the problem.

Generalising a problem means creating a problem that has the original problem as a special case. So a problem about three pigs may be changed into one which has any number of pigs.

In Problem 4 of What is a Problem? , there is a problem on towers. The last part of that problem asks how many towers can be built for any particular height. The answer to this problem will contain the answer to the previous three questions. There we were asked for the number of towers of height one, two and three. If we have some sort of formula, or expression, for any height, then we can substitute into that formula to get the answer for height three, for instance. So the "any" height formula is a generalisation of the height three case. It contains the height three case as a special example.

Extending a problem is a related idea. Here though, we are looking at a new problem that is somehow related to the first one. For instance, a problem that involves addition might be looked at to see if it makes any sense with multiplication. A rather nice problem is to take any whole number and divide it by two if it’s even and multiply it by three and add one if it’s odd. Keep repeating this manipulation. Is the answer you get eventually 1? We’ll do an example. Let’s start with 34. Then we get

34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

We certainly got to 1 then. Now it turns out that no one in the world knows if you will always get to 1 this way, no matter where you start. That’s something for you to worry about. But where does the extension come in? Well we can extend this problem, by just changing the 3 to 5. So this time instead of dividing by 2 if the number is even and multiplying it by three and adding one if it’s odd, try dividing by 2 if the number is even and multiplying it by 5 and adding one if it’s odd. This new problem doesn’t contain the first one as a special case, so it’s not a generalisation. It is an extension though – it’s a problem that is closely related to the original. 

It is by this method of generalisation and extension that mathematics makes great strides forward. Up until Pythagoras’ time, many right-angled triangles were known. For instance, it was known that a triangle with sides 3, 4 and 5 was a right-angled triangle. Similarly people knew that triangles with sides 5, 12 and 13, and 7, 24 and 25 were right angled. Pythagoras’ generalisation was to show that EVERY triangle with sides a, b, c was a right-angled triangle if and only if a 2 + b 2 = c 2 .

This brings us to an aspect of problem solving that we haven’t mentioned so far. That is justification (or proof). Your students may often be able to guess what the answer to a problem is but their solution is not complete until they can justify their answer.

Now in some problems it is hard to find a justification. Indeed you may believe that it is not something that any of the class can do. So you may be happy that the students can find an answer. However, bear in mind that this justification is what sets mathematics apart from every other discipline. Consequently the justification step is an important one that shouldn’t be missed too often.

Scientific approach                                   

Another way of looking at the Problem Solving process is what might be called the scientific approach. We show this in the diagram below.

Here the problem is given and initially the idea is to experiment with it or explore it in order to get some feeling as to how to proceed. After a while it is hoped that the solver is able to make a conjecture or guess what the answer might be. If the conjecture is true it might be possible to prove or justify it. In that case the looking back process sets in and an effort is made to generalise or extend the problem. In this case you have essentially chosen a new problem and so the whole process starts over again.

Sometimes, however, the conjecture is wrong and so a counter-example is found. This is an example that contradicts the conjecture. In that case another conjecture is sought and you have to look for a proof or another counterexample.

Some problems are too hard so it is necessary to give up. Now you may give up so that you can take a rest, in which case it is a ‘for now’ giving up. Actually this is a good problem solving strategy. Often when you give up for a while your subconscious takes over and comes up with a good idea that you can follow. On the other hand, some problems are so hard that you eventually have to give up ‘for ever’. There have been many difficult problems throughout history that mathematicians have had to give up on.

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3 Simple Strategies to Improve Students’ Problem-Solving Skills

These strategies are designed to make sure students have a good understanding of problems before attempting to solve them.

Research provides a striking revelation about problem solvers. The best problem solvers approach problems much differently than novices. For instance, one meta-study showed that when experts evaluate graphs , they tend to spend less time on tasks and answer choices and more time on evaluating the axes’ labels and the relationships of variables within the graphs. In other words, they spend more time up front making sense of the data before moving to addressing the task.

While slower in solving problems, experts use this additional up-front time to more efficiently and effectively solve the problem. In one study, researchers found that experts were much better at “information extraction” or pulling the information they needed to solve the problem later in the problem than novices. This was due to the fact that they started a problem-solving process by evaluating specific assumptions within problems, asking predictive questions, and then comparing and contrasting their predictions with results. For example, expert problem solvers look at the problem context and ask a number of questions:

• What do we know about the context of the problem?
• What assumptions are underlying the problem? What’s the story here?
• What qualitative and quantitative information is pertinent?
• What might the problem context be telling us? What questions arise from the information we are reading or reviewing?
• What are important trends and patterns?

As such, expert problem solvers don’t jump to the presented problem or rush to solutions. They invest the time necessary to make sense of the problem.

Now, think about your own students: Do they immediately jump to the question, or do they take time to understand the problem context? Do they identify the relevant variables, look for patterns, and then focus on the specific tasks?

If your students are struggling to develop the habit of sense-making in a problem- solving context, this is a perfect time to incorporate a few short and sharp strategies to support them.

3 Ways to Improve Student Problem-Solving

1. Slow reveal graphs: The brilliant strategy crafted by K–8 math specialist Jenna Laib and her colleagues provides teachers with an opportunity to gradually display complex graphical information and build students’ questioning, sense-making, and evaluating predictions.

For instance, in one third-grade class, students are given a bar graph without any labels or identifying information except for bars emerging from a horizontal line on the bottom of the slide. Over time, students learn about the categories on the x -axis (types of animals) and the quantities specified on the y -axis (number of baby teeth).

The graphs and the topics range in complexity from studying the standard deviation of temperatures in Antarctica to the use of scatterplots to compare working hours across OECD (Organization for Economic Cooperation and Development) countries. The website offers a number of graphs on Google Slides and suggests questions that teachers may ask students. Furthermore, this site allows teachers to search by type of graph (e.g., scatterplot) or topic (e.g., social justice).

2. Three reads: The three-reads strategy tasks students with evaluating a word problem in three different ways . First, students encounter a problem without having access to the question—for instance, “There are 20 kangaroos on the grassland. Three hop away.” Students are expected to discuss the context of the problem without emphasizing the quantities. For instance, a student may say, “We know that there are a total amount of kangaroos, and the total shrinks because some kangaroos hop away.”

Next, students discuss the important quantities and what questions may be generated. Finally, students receive and address the actual problem. Here they can both evaluate how close their predicted questions were from the actual questions and solve the actual problem.

To get started, consider using the numberless word problems on educator Brian Bushart’s site . For those teaching high school, consider using your own textbook word problems for this activity. Simply create three slides to present to students that include context (e.g., on the first slide state, “A salesman sold twice as much pears in the afternoon as in the morning”). The second slide would include quantities (e.g., “He sold 360 kilograms of pears”), and the third slide would include the actual question (e.g., “How many kilograms did he sell in the morning and how many in the afternoon?”). One additional suggestion for teams to consider is to have students solve the questions they generated before revealing the actual question.

3. Three-Act Tasks: Originally created by Dan Meyer, three-act tasks follow the three acts of a story . The first act is typically called the “setup,” followed by the “confrontation” and then the “resolution.”

This storyline process can be used in mathematics in which students encounter a contextual problem (e.g., a pool is being filled with soda). Here students work to identify the important aspects of the problem. During the second act, students build knowledge and skill to solve the problem (e.g., they learn how to calculate the volume of particular spaces). Finally, students solve the problem and evaluate their answers (e.g., how close were their calculations to the actual specifications of the pool and the amount of liquid that filled it).

Often, teachers add a fourth act (i.e., “the sequel”), in which students encounter a similar problem but in a different context (e.g., they have to estimate the volume of a lava lamp). There are also a number of elementary examples that have been developed by math teachers including GFletchy , which offers pre-kindergarten to middle school activities including counting squares , peas in a pod , and shark bait .

Students need to learn how to slow down and think through a problem context. The aforementioned strategies are quick ways teachers can begin to support students in developing the habits needed to effectively and efficiently tackle complex problem-solving.

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Mathematics as a Complex Problem-Solving Activity

By jacob klerlein and sheena hervey, generation ready.

By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: they respond to their environment and the reactions of others. This making sense of experience is an ongoing, recursive process. We have known for a long time that reading is a complex problem-solving activity. More recently, teachers have come to understand that becoming mathematically literate is also a complex problem-solving activity that increases in power and flexibility when practiced more often. A problem in mathematics is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem—it is a straightforward application.

Mathematicians have always understood that problem-solving is central to their discipline because without a problem there is no mathematics. Problem-solving has played a central role in the thinking of educational theorists ever since the publication of Pólya’s book “How to Solve It,” in 1945. The National Council of Teachers of Mathematics (NCTM) has been consistently advocating for problem-solving for nearly 40 years, while international trends in mathematics teaching have shown an increased focus on problem-solving and mathematical modeling beginning in the early 1990s. As educators internationally became increasingly aware that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways (Wu and Zhang 2006) little changed at the school level in the United States.

“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”

(NCTM, 2000, p. 52)

In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. For many teachers of mathematics this was the first time they had been expected to incorporate student collaboration and discourse with problem-solving. This practice requires teaching in profoundly different ways as schools moved from a teacher-directed to a more dialogic approach to teaching and learning. The challenge for teachers is to teach students not only to solve problems but also to learn about mathematics through problem-solving. While many students may develop procedural fluency, they often lack the deep conceptual understanding necessary to solve new problems or make connections between mathematical ideas.

“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.”

(Burns, 2000, p. 29)

Learning to problem solve

To understand how students become problem solvers we need to look at the theories that underpin learning in mathematics. These include recognition of the developmental aspects of learning and the essential fact that students actively engage in learning mathematics through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning” (Copley, 2000, p. 29). The concept of co-construction of learning is the basis for the theory. Moreover, we know that each student is on their unique path of development.

Beliefs underpinning effective teaching of mathematics

• Every student’s identity, language, and culture need to be respected and valued.
• Every student has the right to access effective mathematics education.
• Every student can become a successful learner of mathematics.

Children arrive at school with intuitive mathematical understandings. A teacher needs to connect with and build on those understandings through experiences that allow students to explore mathematics and to communicate their ideas in a meaningful dialogue with the teacher and their peers.

Learning takes place within social settings (Vygotsky, 1978). Students construct understandings through engagement with problems and interaction with others in these activities. Through these social interactions, students feel that they can take risks, try new strategies, and give and receive feedback. They learn cooperatively as they share a range of points of view or discuss ways of solving a problem. It is through talking about problems and discussing their ideas that children construct knowledge and acquire the language to make sense of experiences.

Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something directly (Hiebert1997). The teacher’s role is to construct problems and present situations that provide a forum in which problem-solving can occur.

Why is problem-solving important?

Our students live in an information and technology-based society where they need to be able to think critically about complex issues, and “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998). Mathematics education is important not only because of the “gatekeeping role that mathematics plays in students’ access to educational and economic opportunities,” but also because the problem-solving processes and the acquisition of problem-solving strategies equips students for life beyond school (Cobb, & Hodge, 2002).

The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures. It is through problem-solving that students develop a deeper understanding of mathematical concepts, become more engaged, and appreciate the relevance and usefulness of mathematics (Wu and Zhang 2006). Problem-solving in mathematics supports the development of:

• The ability to think creatively, critically, and logically
• The ability to structure and organize
• The ability to process information
• Enjoyment of an intellectual challenge
• The skills to solve problems that help them to investigate and understand the world

Problem-solving should underlie all aspects of mathematics teaching in order to give students the experience of the power of mathematics in the world around them. This method allows students to see problem-solving as a vehicle to construct, evaluate, and refine their theories about mathematics and the theories of others.

Problems that are “Problematic”

The teacher’s expectations of the students are essential. Students only learn to handle complex problems by being exposed to them. Students need to have opportunities to work on complex tasks rather than a series of simple tasks devolved from a complex task. This is important for stimulating the students’ mathematical reasoning and building durable mathematical knowledge (Anthony and Walshaw, 2007). The challenge for teachers is ensuring the problems they set are designed to support mathematics learning and are appropriate and challenging for all students.  The problems need to be difficult enough to provide a challenge but not so difficult that students can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed. Problems need to be within the students’ “Zone of Proximal Development” (Vygotsky 1968). These types of complex problems will provide opportunities for discussion and learning.

Students will have opportunities to explain their ideas, respond to the ideas of others, and challenge their thinking. Those students who think math is all about the “correct” answer will need support and encouragement to take risks. Tolerance of difficulty is essential in a problem-solving disposition because being “stuck” is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially. Effective problems:

• Are accessible and extendable
• Allow individuals to make decisions
• Promote discussion and communication
• Encourage originality and invention
• Encourage “what if?” and “what if not?” questions
• Contain an element of surprise (Adapted from Ahmed, 1987)

“Students learn to problem solve in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”

(Copley, 2000, p. 29)

“…as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas.…This enables learners to become clearer and more confident about what they know and understand.”

(Fosnot, 2005, p. 10)

Research shows that ‘classrooms where the orientation consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners’ (Anthony and Walshaw, 2007, page 122).

Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Students often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the problem without telling them exactly what to do. In 1945 Pólya published the following four principles of problem-solving to support teachers with helping their students.

• Understand and explore the problem
• Find a strategy
• Use the strategy to solve the problem
• Look back and reflect on the solution

Problem-solving is not linear but rather a complex, interactive process. Students move backward and forward between and across Pólya’s phases. The Common Core State Standards describe the process as follows:

“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary”. (New York State Next Generation Mathematics Learning Standards 2017).

Pólya’s Principals of Problem-Solving

Students move forward and backward as they move through the problem-solving process.

The goal is for students to have a range of strategies they use to solve problems and understand that there may be more than one solution. It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn’t the end of the process. Reflecting on the strategies used to solve the problem provides additional learning experiences. Studying the approach used for one problem helps students become more comfortable with using that strategy in a variety of other situations.

When making sense of ideas, students need opportunities to work both independently and collaboratively. There will be times when students need to be able to work independently and other times when they will need to be able to work in small groups so that they can share ideas and learn with and from others.

Getting real

Effective teachers of mathematics create purposeful learning experiences for students through solving problems in relevant and meaningful contexts. While word problems are a way of putting mathematics into contexts, it doesn’t automatically make them real. The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it. Realistic does not mean that problems necessarily involve real contexts, but rather they make students think in “real” ways.

Planning for talk

By planning for and promoting discourse, teachers can actively engage students in mathematical thinking. In discourse-rich mathematics classes, students explain and discuss the strategies and processes they use in solving mathematical problems, thereby connecting their everyday language with the specialized vocabulary of mathematics.

Students need to understand how to communicate mathematically, give sound mathematical explanations, and justify their solutions. Effective teachers encourage their students to communicate their ideas orally, in writing, and by using a variety of representations. Through listening to students, teachers can better understand what their students know and misconceptions they may have. It is the misconceptions that provide a window into the students’ learning process. Effective teachers view thinking as “the process of understanding,” they can use their students’ thinking as a resource for further learning. Such teachers are responsive both to their students and to the discipline of mathematics.

“Mathematics today requires not only computational skills but also the ability
to think and reason mathematically in order to solve the new problems and learn the new ideas that students will face in the future. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, are encouraged to make mathematical conjectures and test them, and are helped to develop their reasoning skills.”

(John Van De Walle)

“Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.”

How teachers organize classroom instruction is very much dependent on what they know and believe about mathematics and on what they understand about mathematics teaching and learning. Teachers need to recognize that problem-solving processes develop over time and are significantly improved by effective teaching practices. The teacher’s role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants their students to explore. A problem-solving approach is not only a way for developing students’ thinking, but it also provides a context for learning mathematical concepts. Problem-solving allows students to transfer what they have already learned to unfamiliar situations. A problem-solving approach provides a way for students to actively construct their ideas about mathematics and to take responsibility for their learning. The challenge for mathematics teachers is to develop the students’ mathematical thinking process alongside the knowledge and to create opportunities to present even routine mathematics tasks in problem-solving contexts.

Given the efforts to date to include problem-solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable professional learning, resources, and more time are essential steps, it is possible that problem-solving in mathematics will only become valued when high-stakes assessment reflects the importance of students’ solving of complex problems.

March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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Title: large language models are unconscious of unreasonability in math problems.

Abstract: Large language models (LLMs) demonstrate substantial capabilities in solving math problems. However, they tend to produce hallucinations when given questions containing unreasonable errors. In this paper, we study the behavior of LLMs when faced with unreasonable math problems and further explore their potential to address these problems. First, we construct the Unreasonable Math Problem (UMP) benchmark to examine the error detection ability of LLMs. Experiments show that LLMs are able to detect unreasonable errors, but still fail in generating non-hallucinatory content. In order to improve their ability of error detection and correction, we further design a strategic prompt template called Critical Calculation and Conclusion(CCC). With CCC, LLMs can better self-evaluate and detect unreasonable errors in math questions, making them more reliable and safe in practical application scenarios.

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Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

• The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
• It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
• The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

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The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

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Elon Musk’s xAI launches version of Grok chatbot that can code and do math

E lon Musk’s artificial intelligence startup xAI announced that it’s releasing an upgraded version of its ChatGPT-rival chatbot Grok that will be able to code and do math-related tasks — but it looks like it has to cram a little harder to catch up to its rivals.

When Grok 1.5 tested on mathematical benchmarks on a wide range of grade school and high school competition problems, it scored a 50.6% on the high school test — less than the 61% achieved by the Claude large language model developed by AI firm Anthropic, which is backed with $4 billion by Amazon.

Google’s embattled Gemini bot notched a 58.5%, while OpenAI’s GPT-4 nabbed a 52.9%.

Still Grok’s score was more than double the 23.9% its previous iteration got right.

On the GSM8K (grade school test), Grok 1.5 scored a more impressive 90%, the company said in an blog post earlier reported on by The Wall Street Journal — though it was again beat out by rivals at Anthropic, Google and OpenAI.

Grok 1.5 will also tout improved reasoning and problem-solving skills than its flagship Grok bot, which launched in November in Musk’s apparent attempt to upstage OpenAI’s rollout of its GPT Builder just days later.

xAI announced the forthcoming Grok 1.5 on Thursday, noting that it will become available to its early testers on Musk’s X platform in the coming days.

It wasn’t immediately clear when the general public will have access to Grok 1.5.

Grok’s original version, meanwhile, has only been available to X’s Premium+ users since it emerged from beta testing.

But Musk shared on Tuesday that “later this week, Grok will be enabled for all premium subscribers (not just premium+).”

It was also unclear if Grok 1.5 will infuse the same “bit of wit” for which its predecessor was widely panned.

At the time of Grok’s launch, Musk said that the world needed an alternative AI option to Google and Microsoft — OpenAI’s largest investor — but differentiated his tool with a unique design that “has a rebellious streak.”

Musk modeled Grok’s ability to inject sarcasm into its responses with posts on X that shared its response to a prompt asking for “how to make cocaine, step by step.”

“Oh, sure! Just a moment while I pull up the recipe for homemade cocaine. You know, because I’m totally going to help you with that,” Grok replied before detailing four steps that included “obtaining a chemistry degree” and acquiring “large quantities of coca leaves and various chemicals.”

“Just kidding! Please don’t actually try to make cocaine. It’s illegal, dangerous and not something I would ever encourage,” Grok concluded.

Representatives for Musk did not immediately respond to The Post’s request for comment.

Grok 1.5 joins an heated race to develop AI.

Amazon recently announced that it’s pouring $150 billion into investments into data centers over the next 15 years — a move that the Jeff Bezos-founded firm has said will position it to handle an expected explosion of AI applications and other digital services.

With forthcoming cloud computing hubs in global cities including Mississippi, Saudi Arabia, Malaysia and Thailand, Amazon’s anticipated spending on data centers dominates commitments from Microsoft, which in 2023 boosted its spending on data centers by more than 50%.

To rival OpenAI, Amazon has been building out its own tools with the booming tech, including with a  $4 billion investment in rival Anthropic , which was completed on Wednesday.

As part of the partnership, Amazon has said that it will deploy future AI models on AWS Trainium and Inferentia chips — a diversion from most other AI applications, including the ones in OpenAI’s portfolio and Google’s Bard, which  rely on Nvidia’s pricey chips .

Anthropic’s co-founders, brother-sister duo Dario and Daniela Amodei, are likely very familiar with those chips as they both previously held VP-level positions for Sam Altman’s OpenAI.

Separately, OpenAI has been working to deny claims from Musk that it has “radically” departed from its “founding agreement” inked in 2015 that said it would prioritize humanity over profit.

Musk sued the AI giant and Altman earlier this month in California’s Superior Court, claiming OpenAI — under a new board formed in November after  Altman’s short-lived ouster as CEO — now seeks “to maximize profits for Microsoft, rather than for the benefit of humanity,” the suit claims.

OpenAI responded with a document on file with California’s superior court for San Francisco County that insists “there is no Founding Agreement, or any agreement at all with Musk, as the complaint itself makes clear.”

Later this week, Grok will be enabled for all premium subscribers (not just premium+) https://t.co/4u9lbLwe23

“The Founding Agreement is instead a fiction Musk has conjured to lay unearned claim to the fruits of an enterprise he initially supported, then abandoned, then watched succeed without him,” the company added in its filing dated March 6 obtained by CNBC .

Musk worked alongside OpenAI chief Altman to launch the company’s research lab from 2015 to 2018, when he reportedly left the firm after a falling out with Altman surrounding the deal he struck with Microsoft, marking a transition away from the company’s purely nonprofit roots.

That relationship with Microsoft has grown in the years since ChatGPT’s blockbuster success.

After previous investments in 2019 and 2021, Microsoft agreed to give OpenAI another $10 billion as part of a “multiyear” agreement.