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Problem Solving in Mathematics

  • Math Tutorials
  • Pre Algebra & Algebra
  • Exponential Decay
  • Worksheets By Grade

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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  • Study Tips for Math Homework and Math Tests
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5 Teaching Mathematics Through Problem Solving

Janet Stramel

Problem Solving

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.

Needlepoint of cats

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

Back of a needlepoint

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

Teacher teaching a math lesson

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?

problem solving of mathematics

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

Light bulb

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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Mathematical Problem Solving

Current Themes, Trends, and Research

  • Peter Liljedahl 0 ,
  • Manuel Santos-Trigo 1

Faculty of Education, Simon Fraser University, Burnaby, Canada

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Department of Mathematics Education, Cinvestav-Instituto Politecnico Nacional, Mexico City, Mexico

Brings together some of the latest research on problem solving

Offers international perspectives on current themes, trends, and research on problem solving

Presents multiple frameworks and views on problem solving

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

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Table of contents (15 chapters)

Front matter, problem solving heuristics, “looking back” to solve differently: familiarity, fluency, and flexibility.

  • Hartono Tjoe

Future-Oriented Thinking and Activity in Mathematical Problem Solving

  • Wes Maciejewski

Problem Solving and Technology

A model of mathematical problem solving with technology: the case of marco solving-and-expressing two geometry problems.

  • Susana Carreira, Hélia Jacinto

Mathematical Problem Solving and the Use of Digital Technologies

Manuel Santos-Trigo

The Spreadsheet Affordances in Solving Complex Word Problems

  • Nélia Amado, Susana Carreira, Sandra Nobre

Inquiry and Problem Posing in Mathematics Education

Is an inquiry-based approach possible at the elementary school.

  • Magali Hersant, Christine Choquet

How to Stimulate In-Service Teachers’ Didactic Analysis Competence by Means of Problem Posing

  • Uldarico Malaspina, Carlos Torres, Norma Rubio

Assessment of and Through Problem Solving

The impact of various methods in evaluating metacognitive strategies in mathematical problem solving.

  • Mei Yoke Loh, Ngan Hoe Lee

Assessing Inquiry-Based Mathematics Education with Both a Summative and Formative Purpose

  • Maud Chanudet

Beyond the Standardized Assessment of Mathematical Problem Solving Competencies: From Products to Processes

  • Pietro Di Martino, Giulia Signorini

Toward Designing and Developing Likert Items to Assess Mathematical Problem Solving

  • James A. Mendoza Álvarez, Kathryn Rhoads, R. Cavender Campbell

The Problem Solving Environment

Creating and sustaining online problem solving forums: two perspectives.

  • Boris Koichu, Nelly Keller

Conditions for Supporting Problem Solving: Vertical Non-permanent Surfaces

Peter Liljedahl

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  • Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
  • Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
  • Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
  • Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
  • Linear Algebra Matrices Vectors
  • Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
  • Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
  • Physics Mechanics
  • Chemistry Chemical Reactions Chemical Properties
  • Finance Simple Interest Compound Interest Present Value Future Value
  • Economics Point of Diminishing Return
  • Conversions Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time
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  • Linear Algebra
  • Trigonometry
  • Conversions

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Problem Solving

Problem Solving Strategies

Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

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]

George Pólya ca 1973

 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!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

  • What if the picture was different?
  • What if the numbers were simpler?
  • What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (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, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

  • Describe all of the patterns you see in the table.
  • Can you explain and justify any of the patterns you see? How can you be sure they will continue?
  • What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

problem solving of mathematics

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

  • What is the sum of all the numbers on the clock’s face?
  • Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
  • How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

problem solving of mathematics

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

  • Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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10 Strategies for Problem Solving in Math

Created: December 25, 2023

Last updated: January 6, 2024

strategies for problem solving in math

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

Math for Kids

Guess and Check

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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1.5: Problem Solving

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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.

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.

Example \(\PageIndex{1}\)

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:

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found a 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.

Check in question 1:

clipboard_e6c749e22aea062a3b0e3594708210726.png

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

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.

Problem Solving Strategy 2 (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 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.

Example \(\PageIndex{2}\)

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

Last term = 3(200-1) +2

Last term is 599.

Check in question 3:

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

Problem Solving Strategy 3 (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.

problem solving of mathematics

Example \(\PageIndex{3}\)

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 4 (Looking for a Pattern)

Definition: Sequence

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 \(\PageIndex{4}\)

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 \(\PageIndex{5}\)

1, 4, 9, 16 … Find the next 2 numbers.

It looks like each number is a perfect square. \(1^2=1\), \(2^2=4\)

So the next numbers would be

Example \(\PageIndex{6}\)

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

-5 – 3 = -8

Example \(\PageIndex{7}\)

1, 2, 4, 8 …Ffind 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

Problem Solving Strategy 5 (Make a List)

Example \(\PageIndex{8}\)

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. But note that this is just an observation. To answer this question, one would need a mathematically rigorous proof.

Example \(\PageIndex{9}\)

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

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 6 (Process of Elimination)

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

Example \(\PageIndex{10}\)

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.
  • We know it is an odd number between 1 and 100.
  • 21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.
  • 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.

  • References (2)

Fallacies in Common Language

For each of the following statements, name the type of logical fallacy being used.

If you don’t want to drive from Boston to New York, then you will have to take the train.

Every time I go to Dodger Stadium, the Dodgers win. I should go there for every game.

New England Patriots quarterback Tom Brady likes his footballs slightly underinflated. The “Cheatriots” have a history of bending or breaking the rules, so Brady must have told the equipment manager to make sure that the footballs were underinflated.

What you are doing is clearly illegal because it’s against the law.

The county supervisor voted against the new education tax. He must not believe in education.

“Apples a day keeps doctors away.” No one has said apples are bad, so this old saying must be true.

Wine has to be good for your health because… I mean, can you imagine a life without wine?

Studies show that slightly overweight senior citizens live longer than underweight ones. The extra weight must make people live longer.

Whenever our smoke detector beeps, my kids eat cereal for dinner. The loud beeping sound must make them want to eat cereal for some reason.

There is a scientist who works at a really good university, and he says there is no strong evidence for climate change, especially global warming. Some politicians also question climate change. So I don’t really believe it.

My neighbor cheats on his tax returns. I don’t believe anything he says.

A: “Don’t fight over small things. Just let them go.”

B: “What exactly are ‘small things’? How do I know what is small and what isn’t?”

A: “Well, small things are things you really don’t want to fight over.”

Propositions and Logic

List the set of integers that satisfy the following statement: A positive multiple of 5 and not a multiple of 2

List the set of integers that satisfy the following statement: Greater than 12 and less than or equal to 18

List the set of integers that satisfy the following statement: Even number less than 10 or odd number between 12 and 10

You qualify for a special discount if you are either

a full-time student in the state of California or

at least 18 and your income is less than $20,000 a year.

For each person below, determine if the person qualifies for this discount. If more information is needed, indicate that.

A 17-year-old full-time student at a California community college with no job

A 28-year-old man earning $50,000 a year

A 60-year-old grandmother who does not work and does not go to school

A boy in first grade

A mother making $18,000 a year and not enrolled in any college

A 22-year-old full-time student at Arizona State University with no job

A 18-year-old earning exactly $20,000 a year while attending UCLA part-time

Write the negation: Everyone failed the quiz today.

Write the negation: Someone in the car needs to use the restroom.

How can you prove this statement wrong?

“Everyone who ate at that restaurant got sick.”

“There was someone who ate at that restaurant and got sick.”

“There is no baseball player who can excel at both pitching and hitting. Everyone must choose one or the other.”

“Every student must have an ID number before registering for classes.”

Truth Tables

Translate each statement from symbolic notation into English sentences. Let A represent “Elvis is alive” and let G represent “Elvis gained weight.”

A ⋁ G

~( A ⋀ G )

G → ~ A

A ↔ ~ G

A ⋀ ~ G

~( A ⋁ G )

Create a truth table for each statement below.

A ⋀ ~ B

~(~ A ⋁ B )

( A ⋀ B ) → C

( A ⋁ B ) → ~ C

Complete the truth table for ( A ⋁ B ) ⋀ ~( A ⋀ B ).

We have been studying the inclusive or, which allows both A and B to be true. The exclusive or does not allow both to be true; it translates to “either A or B , but not both.” For each situation, decide whether the “or” is most likely exclusive or inclusive.

An entrée at a restaurant includes soup or a salad.

You should bring an umbrella or a raincoat with you.

We can keep driving on I-5 or get on I-405 at the next exit.

Use Gate 1 if you are at least 35 years old, or Gate 2 if you are younger.

You should save this document on your computer or a flash drive.

I am not sure if my pregnant wife is going to have a boy or a girl.

Consider the statement “If you are under age 17, then you cannot attend this movie.”

Write the converse.

Write the inverse.

Write the contrapositive.

Consider the statement “If you have a house in Beverly Hills, you are rich.”

Assume that the statement “If you swear, then you will get your mouth washed out with soap” is true. Which of the following statements must also be true?

If you don’t swear, then you won’t get your mouth washed out with soap.

If you don’t get your mouth washed out with soap, then you didn’t swear.

If you get your mouth washed out with soap, then you swore.

Write the negation: If Luke faces Vader, then Obi-Wan cannot interfere.

Write the negation: If you look both ways before crossing the street, then you will not get hit by a car.

Write the negation: If you weren’t talking, then you wouldn’t have missed the instructions.

Write the negation: If you score a goal now, we will win.

Assume that the biconditional statement “You will play in the game if and only if you attend all practices this week” is true. Which of the following situations could NOT happen?

You attended all practices this week and didn’t play in the game.

You didn’t attend all practices this week and played in the game.

You didn’t attend all practices this week and didn’t play in the game.

Use De Morgan’s Laws to rewrite the disjunction as a conjunction: It is not true that Tina likes Sprite or 7-Up.

Use De Morgan’s Laws to rewrite the disjunction as a conjunction: It is not true that the father or the mother of that child will be required to testify.

Use De Morgan’s Laws to rewrite the conjunction as a disjunction: It is not the case that both the House and the Senate passed the bill.

Use De Morgan’s Laws to rewrite the conjunction as a disjunction: It is not the case that you need a dated receipt and your credit card to return this item.

Analyzing Arguments

Determine whether each of the following is an inductive or deductive argument:

The new medicine works. We tried it on 100 patients, and all of them were cured.

Every student has an ID number. Sandra is a student, so she has an ID number.

Every angle of a rectangle is 90 degrees. A soccer field (pitch) is rectangular, so every corner is 90 degrees.

Everything that goes up comes down. I throw a ball up. It must come down.

Every time it rains, my grass grows fast. Rain speeds up the growth of grass.

Sports makes a person strong. My daughter plays basketball. She will be strong.

Analyze the validity of the argument.

Everyone who gets a degree in science will get a good job. I got a science degree. Therefore, I will get a good job.

If someone turns off the switch, the lights will not be on. The lights are off. Therefore, someone must have turned off the switch.

Suppose the statement: “If today is Dec. 25, then the store is closed.”

“Today is Dec. 25. Thus, the store is closed.” Which property was used? Is this valid?

“The store is not closed. So today is not Dec. 25.” Which property was used? Is this valid?

“The store is closed. Therefore, today must be Dec. 25.” Which property was used? Is this valid?

“Today is not Dec. 25. Thus, the store is open.” Which property was used? Is this valid?

For the following questions, use a Venn diagram or a truth table to determine the validity.

If a person is on this reality show, they must be self-absorbed. Laura is not self-absorbed. Therefore, Laura cannot be on this reality show.

If you are a triathlete, then you have outstanding endurance. LeBron James is not a triathlete. Therefore, LeBron does not have outstanding endurance.

Jamie must scrub the toilets or hose down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie will hose down the garbage cans.

Some of these kids are rude. Jimmy is one of these kids. Therefore, Jimmy is rude!

Every student brought a pencil or a pen. Marcie brought a pencil. Therefore, Marcie did not bring a pen.

If a creature is a chimpanzee, then it is a primate. If a creature is a primate, then it is a mammal. Bobo is a mammal. Therefore, Bobo is a chimpanzee.

Every cripsee is a domwow. Mekep is not a domwow. Therefore, Mekep is not a cripsee. (This sentence has a lot of made-up words, but it is still possible to check for the validity of the argument. This is a good practice for abstract thinking.)

Whoever dephels a kipoc will be bopied. I did not dephel any kipoc. Therefore, I will not be bopied. (Again, you do not need to know the meaning of each word to do this exercise.)

Problem Solving

For the following exercises, apply any problem-solving strategies and your critical-thinking skills to solve various types of problems. There is single formula or procedure to follow. Be flexible and consider all possibilities.

There are 13 postage stamps on the table. Some are 20-cent stamps while others are 45-cent stamps. The total postage value of these stamps is $4.10.

If they were all 20-cent stamps, would 13 of them add up to $4.10?

If there were five 20-cent stamps, would these stamps add up to $4.10?

How about ten 20-cent stamps? OK, you probably got some idea now.

How many 20-cent stamps are there?

Can you think of another way to solve the problem?

What would you say to a friend of yours who tries to help you out by writing a system of two equations to solve this problem?

Find the next two terms of each of the following sequences and explain why.

9, 7, 5, 3, …

0, 1, 4, 9, 16, 25, …

3, 6, 12, 24, …

0, 1, 3, 6, 10, 15, …

5, 7, 5, 5, 7, 5, 5, 7, 5, 5, …

1, 1, 2, 3, 5, 8, 13, 21, … (Fibonacci Sequence) We will study it later.

A man bought an old car for $2000. He fixed it up and sold it for $2,500. But he missed it, so he brought it back for $3,200. Later, he sold it for $4,000. How much did he make in these transactions?

Is it $4,000 - $2,000 = $2,000?

He made $500 on the first sale and then $800 on the second. But he lost $700 in between when he re-purchased it. So is it $800 - $700 = $100?

Is it $500 + $800 = $1,300?

See the pattern below. Can you make a conjecture? Is it true? Can you prove it?

1+3=4 (=22).

1+3+5=9 (=32).

1+3+5+7=16 (=42).

A heart surgeon is about to perform a medical procedure on a boy. The surgeon told the nurses, “I want you to know that this is my son. I am operating on my own child today.” Everyone there knew that the boy was the surgeon’s son. But the surgeon was not the boy’s father. How can this be true?

What can “H.D.” in this little story be?

“H.D. sat on a wall. H.D. had a great fall. All the president’s horses and all the president’s men couldn’t put H.D. together again.”

You have decided to work out at the gym at least twice a week, but never on two consecutive days. If you are to keep the same schedule every week, list all possible days of the week) you can exercise at the gym.

A man must be married to have a mother-in-law but can be single and have a brother-in-law. Explain why.

Your sprinklers are set to water your plants at 6 am every morning during summer, when the daylight saving time is in effect. Once the time goes back to regular time (in November, say from PDT to PST), what time do your sprinklers start? (Hint: it’s either 5 am or 7 am.) What is the best way to explain this to your friend?

You are to visit a friend who lives 300 miles away. You drive to his house early in the morning, averaging 60 mph, but you return in the afternoon, when the freeway is jammed, at an average speed of 40 mph. Is the overall average speed 50 mph? Why or why not?

You have 10 identical pairs of black socks and 9 identical parts of white socks in your drawer, except each pair is not “paired up,” i.e., each sock is in the drawer, separated from all others. The room is completely dark, and you cannot see which socks you are taking out. How many socks do you have to pull out if you want to be sure that you get

A pair of white socks?

A pair of black socks?

Any matching pair?

A divorced 49-year-old man with a 25-year-old son marries a young woman whose mother is a widow. The 25-year-old son marries the widow and have a baby girl.

How is that baby related to the 49-year-old man? (The baby is his son’s daughter.)

How else is that baby related to the 49-year-old man? (The baby is his wife’s mother’s daughter.)

Does that baby have a step-brother? If so, under what conditions?

Who is that baby’s step-sister?

Combine your answers to describe how the 49-year-old man is related to the baby.

A person can be ½ Chinese and ½ Italian. Are the following cases possible?

½ Japanese, ¼ Russian, ¼ Irish

3/8 Scottish, 3/8 Vietnamese, ¼ Spanish

3/8 Mexican, ¼ Turkish, and ½ Norwegian

1/6 French, ½ Canadian, 1/3 Brazilian

½ Armenian, ¼ German, ½ Swedish

John says, “I don’t have any brothers, sisters, step-brothers, or step-sisters. See that tall woman? Her father is my mother’s child.” Who is the tall woman?

Contributors and Attributions

Saburo Matsumoto CC-BY-4.0

Teaching and Learning

Elevating math education through problem-based learning, by lisa matthews     feb 14, 2024.

Elevating Math Education Through Problem-Based Learning

Image Credit: rudall30 / Shutterstock

Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?

The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.

problem solving of mathematics

Knowledge Through Experience

The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions , to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information .

You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.

problem solving of mathematics

So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.

I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.

problem solving of mathematics

Skills and Understanding

Despite the challenges, the trend toward PBL in math education has been growing , driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.

At the heart of Imagine IM lies a commitment to providing students with opportunities for deep, active mathematics practice through problem-based learning. Imagine IM builds upon the problem-based pedagogy and instructional design of the renowned Illustrative Mathematics curriculum, adding a number of exclusive videos, digital interactives, design-enhanced print and hands-on tools.

The value of imagine im's enhancements is evident in the beautifully produced inspire math videos, from which the mountaineer scenario stems. inspire math videos showcase the math for each imagine im unit in a relevant and often unexpected real-world context to help spark curiosity. the videos use contexts from all around the world to make cross-curricular connections and increase engagement..

This article was sponsored by Imagine Learning and produced by the Solutions Studio team.

Imagine Learning

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Solving word problem chart

1. Understand the Problem by Paraphrasing

2. identify key information and variables, 3. translate words into mathematical symbols, 4. break down the problem into manageable parts, 5. draw diagrams or visual representations, 6. use estimation to predict answers, 7. apply logical reasoning for unknown variables, 8. leverage similar problems as templates, 9. check answers in the context of the problem, 10. reflect and learn from mistakes.

Have you ever observed the look of confusion on a student’s face when they encounter a math word problem ? It’s a common sight in classrooms worldwide, underscoring the need for effective strategies for solving math word problems . The main hurdle in solving math word problems is not just the math itself but understanding how to translate the words into mathematical equations that can be solved.

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Generic advice like “read the problem carefully” or “practice more” often falls short in addressing students’ specific difficulties with word problems. Students need targeted math word problem strategies that address the root of their struggles head-on. 

A Guide on Steps to Solving Word Problems: 10 Strategies 

One of the first steps in tackling a math word problem is to make sure your students understand what the problem is asking. Encourage them to paraphrase the problem in their own words. This means they rewrite the problem using simpler language or break it down into more digestible parts. Paraphrasing helps students grasp the concept and focus on the problem’s core elements without getting lost in the complex wording.

Original Problem: “If a farmer has 15 apples and gives away 8, how many does he have left?”

Paraphrased: “A farmer had some apples. He gave some away. Now, how many apples does he have?”

This paraphrasing helps students identify the main action (giving away apples) and what they need to find out (how many apples are left).

Play these subtraction word problem games in the classroom for free:

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Students often get overwhelmed by the details in word problems. Teach them to identify key information and variables essential for solving the problem. This includes numbers , operations ( addition , subtraction , multiplication , division ), and what the question is asking them to find. Highlighting or underlining can be very effective here. This visual differentiation can help students focus on what’s important, ignoring irrelevant details.

  • Encourage students to underline numbers and circle keywords that indicate operations (like ‘total’ for addition and ‘left’ for subtraction).
  • Teach them to write down what they’re solving for, such as “Find: Total apples left.”

Problem: “A classroom has 24 students. If 6 more students joined the class, how many students are there in total?”

Key Information:

  • Original number of students (24)
  • Students joined (6)
  • Looking for the total number of students

Here are some fun addition word problems that your students can play for free:

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The transition from the language of word problems to the language of mathematics is a critical skill. Teach your students to convert words into mathematical symbols and equations. This step is about recognizing keywords and phrases corresponding to mathematical operations and expressions .

Common Translations:

  • “Total,” “sum,” “combined” → Addition (+)
  • “Difference,” “less than,” “remain” → Subtraction (−)
  • “Times,” “product of” → Multiplication (×)
  • “Divided by,” “quotient of” → Division (÷)
  • “Equals” → Equals sign (=)

Problem: “If one book costs $5, how much would 4 books cost?”

Translation: The word “costs” indicates a multiplication operation because we find the total cost of multiple items. Therefore, the equation is 4 × 5 = $20

Complex math word problems can often overwhelm students. Incorporating math strategies for problem solving, such as teaching them to break down the problem into smaller, more manageable parts, is a powerful approach to overcome this challenge. This means looking at the problem step by step rather than simultaneously trying to solve it. Breaking it down helps students focus on one aspect of the problem at a time, making finding the solution more straightforward.

Problem: “John has twice as many apples as Sarah. If Sarah has 5 apples, how many apples do they have together?”

Steps to Break Down the Problem:

Find out how many apples John has: Since John has twice as many apples as Sarah, and Sarah has 5, John has 5 × 2 = 10

Calculate the total number of apples: Add Sarah’s apples to John’s to find the total,  5 + 10 = 15

By splitting the problem into two parts, students can solve it without getting confused by all the details at once.

Explore these fun multiplication word problem games:

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Diagrams and visual representations can be incredibly helpful for students, especially when dealing with spatial or quantity relationships in word problems. Encourage students to draw simple sketches or diagrams to represent the problem visually. This can include drawing bars for comparison, shapes for geometry problems, or even a simple distribution to better understand division or multiplication problems .

Problem: “A garden is 3 times as long as it is wide. If the width is 4 meters, how long is the garden?”

Visual Representation: Draw a rectangle and label the width as 4 meters. Then, sketch the length to represent it as three times the width visually, helping students see that the length is 4 × 3 = 12

Estimation is a valuable skill in solving math word problems, as it allows students to predict the answer’s ballpark figure before solving it precisely. Teaching students to use estimation can help them check their answers for reasonableness and avoid common mistakes.

Problem: “If a book costs $4.95 and you buy 3 books, approximately how much will you spend?”

Estimation Strategy: Round $4.95 to the nearest dollar ($5) and multiply by the number of books (3), so 5 × 3 = 15. Hence, the estimated total cost is about $15.

Estimation helps students understand whether their final answer is plausible, providing a quick way to check their work against a rough calculation.

Check out these fun estimation and prediction word problem worksheets that can be of great help:

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When students encounter problems with unknown variables, it’s crucial to introduce them to logical reasoning. This strategy involves using the information in the problem to deduce the value of unknown variables logically. One of the most effective strategies for solving math word problems is working backward from the desired outcome. This means starting with the result and thinking about the steps leading to that result, which can be particularly useful in algebraic problems.

Problem: “A number added to three times itself equals 32. What is the number?”

Working Backward:

Let the unknown number be x.

The equation based on the problem is  x + 3x = 32

Solve for x by simplifying the equation to 4x=32, then dividing by 4 to find x=8.

By working backward, students can more easily connect the dots between the unknown variable and the information provided.

Practicing problems of similar structure can help students recognize patterns and apply known strategies to new situations. Encourage them to leverage similar problems as templates, analyzing how a solved problem’s strategy can apply to a new one. Creating a personal “problem bank”—a collection of solved problems—can be a valuable reference tool, helping students see the commonalities between different problems and reinforcing the strategies that work.

Suppose students have solved a problem about dividing a set of items among a group of people. In that case, they can use that strategy when encountering a similar problem, even if it’s about dividing money or sharing work equally.

It’s essential for students to learn the habit of checking their answers within the context of the problem to ensure their solutions make sense. This step involves going back to the original problem statement after solving it to verify that the answer fits logically with the given information. Providing a checklist for this process can help students systematically review their answers.

Checklist for Reviewing Answers:

  • Re-read the problem: Ensure the question was understood correctly.
  • Compare with the original problem: Does the answer make sense given the scenario?
  • Use estimation: Does the precise answer align with an earlier estimation?
  • Substitute back: If applicable, plug the answer into the problem to see if it works.

Problem: “If you divide 24 apples among 4 children, how many apples does each child get?”

After solving, students should check that they understood the problem (dividing apples equally).

Their answer (6 apples per child) fits logically with the number of apples and children.

Their estimation aligns with the actual calculation.

Substituting back 4×6=24 confirms the answer is correct.

Teaching students to apply logical reasoning, leverage solved problems as templates, and check their answers in context equips them with a robust toolkit for tackling math word problems efficiently and effectively.

One of the most effective ways for students to improve their problem-solving skills is by reflecting on their errors, especially with math word problems. Using word problem worksheets is one of the most effective strategies for solving word problems, and practicing word problems as it fosters a more thoughtful and reflective approach to problem-solving

These worksheets can provide a variety of problems that challenge students in different ways, allowing them to encounter and work through common pitfalls in a controlled setting. After completing a worksheet, students can review their answers, identify any mistakes, and then reflect on them in their mistake journal. This practice reinforces mathematical concepts and improves their math problem solving strategies over time.

3 Additional Tips for Enhancing Word Problem-Solving Skills

Before we dive into the importance of reflecting on mistakes, here are a few impactful tips to enhance students’ word problem-solving skills further:

1. Utilize Online Word Problem Games

A word problem game

Incorporate online games that focus on math word problems into your teaching. These interactive platforms make learning fun and engaging, allowing students to practice in a dynamic environment. Games can offer instant feedback and adaptive challenges, catering to individual learning speeds and styles.

Here are some word problem games that you can use for free:

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2. Practice Regularly with Diverse Problems

Word problem worksheet

Consistent practice with a wide range of word problems helps students become familiar with different questions and mathematical concepts. This exposure is crucial for building confidence and proficiency.

Start Practicing Word Problems with these Printable Word Problem Worksheets:

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3. Encourage Group Work

Solving word problems in groups allows students to share strategies and learn from each other. A collaborative approach is one of the best strategies for solving math word problems that can unveil multiple methods for tackling the same problem, enriching students’ problem-solving toolkit.

Conclusion 

Mastering math word problems is a journey of small steps. Encourage your students to practice regularly, stay curious, and learn from their mistakes. These strategies for solving math word problems are stepping stones to turning challenges into achievements. Keep it simple, and watch your students grow their confidence and skills, one problem at a time.

Frequently Asked Questions (FAQs)

How can i help my students stay motivated when solving math word problems.

Encourage small victories and use engaging tools like online games to make practice fun and rewarding.

What's the best way to teach beginners word problems?

Begin with simple problems that integrate everyday scenarios to make the connection between math and real-life clear and relatable.

How often should students practice math word problems?

Regular, daily practice with various problems helps build confidence and problem-solving skills over time.

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February 5, 2024

How String Theory Solved Math’s Monstrous Moonshine Problem

A concept from theoretical physics helped confirm the strange connection between two completely different areas of mathematics

By Manon Bischoff

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After the star-studded mystery thriller The Number 23 debuted in cinemas in 2007, many people became convinced that they were seeing the eponymous number everywhere. I was in school at that time, and some of my classmates would shudder whenever the number 23 appeared in any context. Other people became fascinated by this form of numerology because as soon as you pay more attention to a certain thing—including a number— you get the feeling that you see it too often to be purely coincidence.

For a long time, people assumed that the late mathematician John McKay might have fallen victim to this same phenomenon, known as the “frequency illusion,” or the Baader-Meinhof phenomenon . In McKay’s case, the number that captured his imagination was 196,884.

It doesn’t seem too surprising that a two-digit number such as 23 might come up repeatedly. But would a six-digit figure do so? McKay came across this number by chance in 1978 when he was looking through a paper in a mathematical field that was not his specialty. He was working in geometry and was studying the symmetry of figures. That day, however, he was looking at results from number theory , which deals with the properties of integers such as prime numbers . He came across a sequence of numbers that started with the value 196,884.

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This figure sounded familiar to McKay. He had previously worked on a mathematical structure—still hypothetical at the time— known as the monster . This strange algebraic structure was intended to describe the symmetries of a geometric object that lives in 196,883 dimensions (only one fewer than the number 196,884). And because a one-dimensional point fulfills every symmetry anyway, the monster can also describe its symmetrical properties. So McKay found the number 196,884 again in an extraordinary way. He added the first two dimensions in which mathematicians believed the monster’s symmetry applied: 196,883 + 1 = 196,884.

Does that sound far-fetched? Others thought so, too. Experts paid little attention to McKay’s result. After all, a structure such as the monster contains a number of numbers, as does the consequence from number theory that McKay had associated with it. “If you have a whole lot of numbers, then a few of them are going to be roughly the same as each other just by coincidence,” said mathematician Richard Borcherds, who has made major contributions to the field, in an explanatory YouTube video .

But McKay couldn’t shake the feeling that the two extremely different mathematical fields of geometry and number theory could be connected. He even reportedly wore T-shirts with the inscription “196,883 + 1 = 196,884” at conferences.

Complete Madness or a Stroke of Genius?

A short time later, mathematician John Thompson realized that there might be something to McKay’s suspicions after all. He succeeded in linking the next higher dimension, in which an object follows the symmetries of the monster, with the next member of the mysterious sequence of numbers from number theory. The dimension is 21,296,876. The values differ—but if you add up all the monster dimensions as before (1 + 196,883 + 21,296,876), the result is 21,493,760.

That was surprising because, as you may recall, when McKay first spotted 196,884, he was looking at a special sequence in number theory. The second number in that sequence is 21,493,760—Thompson’s result. In other words, it began to seem that there really could be a link between two seemingly unrelated areas of mathematics.

At this point the math community began to get curious. Maybe McKay was right after all—even if that sounded totally absurd. What could this strange structure, which described symmetries of unimaginable objects and had not even been fully constructed, have to do with number theory?

By 1979 evidence was mounting that other numbers and dimensions seemed to follow this unexpected pattern. Mathematicians John Conway and Simon Norton finally published a paper entitled “Monstrous Moonshine,” in which they set out the conjecture of a connection between geometry and number theory. “They called it moonshine because it appeared so far-fetched,” said number theorist Don Zagier of the Max Planck Institute for Mathematics in Bonn, Germany, to Quanta Magazine in 2015.

And indeed, there was likely very little hope of ever proving this moonshine conjecture. Quite apart from the fact that there was no indication that the two distant mathematical areas were connected, it was not even completely clear whether the monster really existed.

The Monster in the Moonlight

The monster was a theoretical prediction of group theory, an area of geometry that deals with the symmetrical properties of objects. In the 1970s mathematicians began to create a kind of periodic table of groups: they wanted to find the “atoms” of finite symmetries. According to this way of thinking, every finite group can be represented by a combination of these atoms. After decades of research, the geometers finally seemed to have reached their goal. Unlike the chemical elements, there are an infinite number of “finite simple groups,” but almost all can be divided into 18 categories, the arrangement of which is reminiscent of the periodic table. In addition, the experts came across a total of 26 outsiders that do not fit into these 18 classes.

Triangles labeled to show varied symmetries.

The first of these outliers was the “monster,” which mathematicians Bernd Fischer and Robert Griess predicted in 1973. The name comes from the sheer size of this group: it contains more than 8 x 10 53 symmetries. For comparison, the symmetry group of a 20-sided “D20” die ( an icosahedron ) contains 60 symmetries, meaning 60 possible transformations (rotations or reflections) can be carried out without changing the orientation of the D20.

Colorful circles connected by lines represent groups of symmetries.

Because of its sheer size, the monster presented mathematicians with massive challenges. “Most people thought it was going to be hopeless to construct it since much, much, much smaller groups required computer constructions at that time,” explained Borcherds in his YouTube video. Meanwhile even powerful computers struggle with a structure consisting of 8 x 10 53 elements.

Yet this pessimistic forecast ultimately proved wrong. In 1980 Griess constructed the monster and thus proved its existence —without the help of computers.

A Sine Function on Steroids

Number theory is mostly about integers, which seems quite simple at first glance. But to investigate the relationships between them, experts resort to complicated concepts, such as so-called modular forms. These are functions f ( z ) that are extremely symmetrical. As with the sine function, you only need to know a specific section of a modular form to know what it looks like everywhere else.

“Modular forms are something like trigonometric functions, but on steroids,” mathematician Ken Ono told Quanta Magazine .

Colorful arcs represent moduli space

Nevertheless, they play an extremely important role in mathematics. Andrew Wiles of the University of Oxford used them, for example, to prove Fermat’s theorem, and Maryna Viazovska of the Swiss Federal Institute of Technology in Lausanne used them to find the densest sphere-packing arrangement in eight spatial dimensions . Because modular forms are so complicated, however, they are often approximated by an infinitely long polynomial, such as:

f (q ) = ( 1 ⁄ q ) + 744 + 19,688 q + 21,493,760 q 2 + 864,299,970 q 3 + …

The prefactors in front of the variable q form a number sequence with interesting properties from a number-theoretical perspective. McKay associated this sequence of numbers with the monster.

A Surprising Link

Borcherds first heard about the moonshine conjecture in the 1980s. “I was just completely blown away by this,” he recalled in an interview with YouTuber Curt Jaimungal . Borcherds was sitting in one of Conway’s lectures at the time and learned that number theory and group theory could be mysteriously connected. The subject never let go of him. He began to search for the suspected connection until he found it. In 1992 he published his groundbreaking result , for which he received a Fields Medal, one of the highest awards in mathematics, six years later. His conclusion: a highly speculative area of physics, string theory, could provide the missing piece of the puzzle between the monster and the sequence of numbers.

String theory attempts to unite the four fundamental forces of physics (electromagnetism, strong and weak nuclear forces and gravity). Instead of relying on particles or waves to make up the basic building blocks of the universe, as in conventional theories, string theory involves one-dimensional structures: tiny threads vibrate like the strings of an instrument and thus generate the familiar particles and interactions that we perceive in the universe.

Borcherds knew that string theory was based on many mathematical principles related to symmetries. As it turns out, moduli also play a role. When the tiny threads are closed and move through spacetime in a wobbly manner, their track forms a two-dimensional tube. This structure has the same symmetry as modular shapes—regardless of how the thread oscillates.

The type of string theory that Borcherds investigated can only be mathematically formulated in 25 spatial dimensions. Because our world consists of only three visible spatial dimensions, however, string theorists assume that the remaining 22 dimensions are rolled up into tiny spheres or doughnut-shaped tori. But the physics depends on their exact shape: a string theory in which the dimensions are rolled up as cylinders provides different predictions than one in which they form a sphere. In order to describe the particles and their interactions in a way that fits our world, physicists have to find the right “compactification” in their calculations.

Borcherds rolled up 24 dimensions into a 24-dimensional doughnut surface and discovered that the associated string theory had the symmetry of the monster. The fact that only one free spatial dimension remained did not bother him. After all, he was interested in the mathematical properties of the model and not in a physical theory that describes our world.

In this constructed world, the threads swing along the 24-dimensional doughnut. The dimensions of the monster count all the ways in which a thread can vibrate at a certain energy. So at the lowest energy, it only vibrates in one way; at the next highest energy, there are already 196,883 different possibilities. And the trace that the thread leaves behind has the symmetry of a modular shape.

Borcherds had thus proven the connection between the monster group and a modular form. And it was not to remain the only such case: in the meantime, mathematicians have been able to connect other finite groups with other modular forms —and there, too, string theory provides the link. So even if it turns out that the speculative theory is not suitable for describing our universe, it can still help us discover completely new mathematical worlds.

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

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