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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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A Guide to Problem Solving

When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. How do I become a better problem-solver? First and foremost, the best way to become better at problem-solving is to try solving lots of problems! If you are preparing for STEP, it makes sense that some of these problems should be STEP questions, but to start off with it's worth spending time looking at problems from other sources. This collection of NRICH problems  is designed for younger students, but it's very worthwhile having a go at a few to practise the problem-solving technique in a context where the mathematics should be straightforward to you. Then as you become a more confident problem-solver you can try more past STEP questions. One student who worked with NRICH said: "From personal experience, I was disastrous at STEP to start with. Yet as I persisted with it for a long time it eventually started to click - 'it' referring to being able to solve problems much more easily. This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous 'similar' problems." A Problem-solving Heuristic for STEP Below you will find some questions you can ask yourself while you are solving a problem. The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It". Understanding the problem

  • What area of mathematics is this?
  • What exactly am I being asked to do?
  • What do I know?
  • What do I need to find out?
  • What am I uncertain about?
  • Can I put the problem into my own words?

Devising a plan

  • Work out the first few steps before leaping in!
  • Have I seen something like it before?
  • Is there a diagram I could draw to help?
  • Is there another way of representing?
  • Would it be useful to try some suitable numbers first?
  • Is there some notation that will help?

Carrying out the plan STUCK!

  • Try special cases or a simpler problem
  • Work backwards
  • Guess and check
  • Be systematic
  • Work towards subgoals
  • Imagine your way through the problem
  • Has the plan failed? Know when it's time to abandon the plan and move on.

Looking back

  • Have I answered the question?
  • Sanity check for sense and consistency
  • Check the problem has been fully solved
  • Read through the solution and check the flow of the logic.

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:

  • Am I getting stressed?
  • Is my plan working?
  • Am I spending too long on this?
  • Could I move on to something else and come back to this later?
  • Am I focussing on the problem?
  • Is my work becoming chaotic, do I need to slow down, go back and tidy up?
  • Do I need to STOP, PEN DOWN, THINK?

Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that. As a last resort, read the solution, but not until you have spent a long time just thinking about the problem, making notes, trying things out and looking at resources that can help you. If you do end up reading the solution, then come back to the same problem a few days or weeks later to have another go at it.

Four-Step Math Problem Solving Strategies & Techniques

  • Harlan Bengtson
  • Categories : Help with math homework
  • Tags : Homework help & study guides

Four-Step Math Problem Solving Strategies & Techniques

Four Steps to Success

There are many possible strategies and techniques you can use to solve math problems. A useful starting point is a four step approach to math problem solving. These four steps can be summarized as follows:

  • Carefully read the problem. In this careful reading, you should especially seek to clearly identify the question that is to be answered. Also, a good, general understanding of what the problem means should be sought.
  • Choose a strategy to solve the problem. Some of the possible strategies will be discussed in the rest of this article.
  • Carry out the problem solving strategy. If the first problem solving technique you try doesn’t work, try another.
  • Check the solution. This check should make sure that you have indeed answered the question that was posed and that the answer makes sense.

Step One - Understanding the Problem

As you carefully read the problem, trying to clearly understand the meaning of the problem and the question that you must answer, here are some techniques to help.

Identify given information - Highlighting or underlining facts that are given helps to visualize what is known or given.

Identify information asked for - Highlighting the unknowns in a different color helps to keep the known information visually separate from the unknowns to be determined. Ideally this will lead to a clear identification of the question to be answered.

Look for keywords or clue words - One example of clue words is those that indicate what type of mathematical operation is needed, as follows:

Clue words indicating addition: sum, total, in all, perimeter.

Clue words indicating subtraction: difference, how much more, exceed.

Clue words for multiplication: product, total, area, times.

Clue words for division: share, distribute, quotient, average.

Draw a picture - This might also be considered part of solving the problem, but a good sketch showing given information and unknowns can be very helpful in understanding the problem.

Step Two - Choose the Right Strategy

It step one has been done well, it should ease the job of choosing among the strategies presented here for approaching the problem solving step. Here are some of the many possible math problem solving strategies.

  • Look for a pattern - This might be part of understanding the problem or it might be the first part of solving the problem.
  • Make an organized list - This is another means of organizing the information as part of understanding it or beginning the solution.
  • Make a table - In some cases the problem information may be more suitable for putting in a table rather than in a list.
  • Try to remember if you’ve done a similar problem before - If you have done a similar problem before, try to use the same approach that worked in the past for the solution.
  • Guess the answer - This may seem like a haphazard approach, but if you then check whether your guess was correct, and repeat as many times as necessary until you find the right answer, it works very well. Often information from checking on whether the answer was correct helps lead you to a good next guess.
  • Work backwards - Sometimes making the calculations in the reverse order works better.

Steps Three and Four - Solving the Problem and Checking the Solution

If the first two steps have been done well, then the last two steps should be easy. If the selected problem solving strategy doesn’t seem to work when you actually try it, go back to the list and try something else. Your check on the solution should show that you have actually answered the question that was asked in the problem, and to the extent possible, you should check on whether the answer makes common sense.

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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?

steps in 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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Breadcrumbs

How to Ace Math Problem Solving

steps in problem solving of mathematics

When your kids struggle with their math, it’s time to take a step back and take a deep breath. They need to slow down and take their time. Here’s a step by step guide that will help your kids get through those tough math problems.

We’ll use a grade 3 addition word problem as an example to clarify:

Pinky the Pig bought 36 apples while Danny the Duck bought 73 apples and 14 bananas. How many apples do they have altogether?

Read the problem

Carefully read through the problem to make sure you understand what is being asked.

Pinky the pig and Danny the duck bought apples and bananas. The question is how many apples they have together.

Re-read the problem

Read through the problem again and as you read through it, make notes.

Pinky the pig –36 apples. Danny the duck –73 apples and 14 bananas. How many apples together?

What is the problem asking

In your own words, say or write down exactly what the question is asking you to solve.

The question is asking how many apples the pig and the duck bought together.

Write it down in detail

Go through the problem and write out the information in an organized fashion. A diagram or table might help.

Turn it into math

Math problem solving

Figure out what math operation(s) or formula(s) you need to use in order to solve this problem.

The problem wants us to add the number of apples Pinky the Pig and Danny the Duck have together. That means we need to make use of addition to add the apples.

Find an example

Are you still struggling? Sometimes it’s hard to work out the solution, especially if the math problem involves several steps. It’s time to present the problem in an easier way. As teachers and parents we can often help our kids simplify the problem from our own math knowledge. If the problem is a bit harder, there are lots of resources online that you can look up for similar problems that have been worked out on paper or a video tutorial to watch.

In our example, let’s say the double-digit numbers are intimidating our student, so we’re going to simplify the equation for the sake of helping our student understand the operation needed.

Let’s say Pinky the Pig bought 3 apples and Danny the Duck 7 apples and 1 banana. Now, how many apples have they bought together? With 3 apples and 7 apples bought, the total number of apples is 10.

Work out the problem

Now that we have got to the bottom of what is being asked and know what operation to use, it’s time to work out the problem.

Pinky the Pig bought 36 apples. Danny the Duck bought 73 apples. (The 14 bananas do not matter) We need to add up the apples. 36 + 73 = 109

Check and review your answer

Check that your answer is correct. Always ask: does this answer make sense?  You can use estimation using mental math, for example.

Let’s round the numbers: 30 + 70 = 100. That is close to the exact number so it’s in the correct range.

The beauty of the basic operations is that addition and subtraction can be used to check answers too.

If we use the sum and take away one of the numbers, it should equal the other number.

109 – 73 = 36 109 – 36 = 73

If our student did not work out the sum correctly, we would not come to these sums.

(By the way, the same can be done with multiplication and division.)

Finally, go back and review the problem one last time. By going over the concepts, operations and formulas, it will help your kids to internalize the process and help them tackle harder math problems in the future.

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steps in problem solving of mathematics

steps in problem solving of mathematics

4 Best Steps To Problem Solving in Math That Lead to Results

Eastern Shore Math Teacher

Eastern Shore Math Teacher

What does problem solving in math mean, and how to develop these skills in students?  Problem solving involves tasks that are challenging and make students think.  In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. Therefore, teachers need to provide safe learning spaces that foster a growth mindset in math in order for students to take risks to solve problems.   In addition, providing students with problem solving steps in math builds success in solving problems.

A teacher working on problem solving in math.

By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics.  Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students. 

Students who feel successful in math class are happier and more engaged in learning.  Check out  The Bonus Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys for students to use in your classroom to cultivate a positive classroom community in mathematics.    You can also sign up for other freebies from me Here at Easternshoremathteacher.com .

Have you ever given students a word problem or rich task, and they froze?  They have no idea how to tackle the problem, even if it is a concept they are successful with.   This is because they need problem solving strategies.  I started to incorporate more problem solving tasks into my teaching in addition to making the 4 steps for problem solving a school-wide initiative and saw results.  

Bonus Growth Mindset Classroom resources to use to cultivate a growth mindset classroom.

What is Problem Solving in Math?

When educators use the term problem solving , they are referring to mathematical tasks that are challenging and require students to think.   Such tasks or problems can promote students’ conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interests and curiosity (Hiebert & Wearne, 1993; Marcus & Fey, 2003; NCTM, 1991; van de Walle, 2003).

When educators use the term problem solving, they are referring to mathematical tasks that are challenging and require students to think.

How Should Problem Solving For Math Be Taught?

Problem solving should not be done in isolation.  In the past, we would teach the concepts and procedures and then assign one-step “story” problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as “draw a picture” or “guess and check.”  Eventually, students would be given problems to apply the skills and strategies.  Instead, we need to make problem solving an integral part of mathematics learning. 

In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. As students solve problems, they can use any strategy. Then, they justify their solutions with their classmates and learn new ways to solve problems. 

Students do not need every task to involve problem solving.  Sometimes the goal is to just learn a skill or strategy.   

List of Criteria for Problem Solving in Math

Criteria for Problem Solving Math 

Lappan and Phillips (1998) developed a set of criteria for a good problem that they used to develop their middle school mathematics curriculum (Connected Mathematics). The problem:

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

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

The real value of these criteria is that they provide teachers with guidelines for making decisions about how to make problem solving a central aspect of their instruction.  Read more at NCTM .

Resources to Use for Problem Solving Steps in Math.

Problem Solving Teaching Methods

Teaching students these 4 steps for solving problems allows them to have a process for unpacking difficult problems.  

As you teach, model the process of using these 4 steps to solve problems.   Then, encourage students to use these steps as they solve problems.   Click here for Posters, Bookmarks, and Labels to use in your classroom to promote the use of the problem solving steps in math.  

How Problem Solving Skills Develop

Problem solving skills are developed over time and are improved with effective teaching practices.  In addition, teachers need to select rich tasks that focus on the math concepts the teacher wants their students to explore. 

Problem Solving 4 Steps

Understand the problem.

 Read & Think

  • Circle the needed information and underline the question. 
  • Write an answer STEM sentence.  There are_____ pages left to read. 

Plan Out How to Solve the Problem

Make a Plan

  • Use a strategy.  (Draw a Picture, Work Backwards, Look for a Pattern, Create a Table, Bar Model)
  • Use math tools.

Do the Problem

Solve the Problem

  • Show your work to solve the problem.  This could include an equation. 

Check Your Work on the Problem

Answer & Check

  • Write the answer into the answer stem.
  • Does your answer make sense?
  • Check your work using a different strategy.

Check out these Printables for Problem Solving Steps in Math .

Problem Solving steps for Math poster.

Teaching Problem Solving Strategies

A problem solving strategy is a plan used to find a solution.  Understanding how a variety of problem solving strategies work is important because different problems require you to approach them in different ways to find the best solution. By mastering several problem-solving strategies, you can select the right plan for solving a problem.  Here are a few strategies to use with students:

  • Draw a Picture
  • Work Backwards
  • Look for a Pattern
  • Create a Table 

Why is Using Problem Solving Steps For Math Important?

Problem solving allows students to develop an understanding of concepts rather than just memorizing a set of procedures to solve a problem.  In addition, it fosters collaboration and communication when students explain the processes they used to arrive at a solution. Through problem-solving, students develop a deeper understanding of mathematical concepts, become more engaged, and see the importance of mathematics in their lives. 

Girl Problem Solving.

NCTM Process Standards

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.  With these process standards, the focus became more on mathematics through problem solving.   Students could no longer just develop procedural fluency, they needed to develop conceptual understanding in order to solve new problems and make connections between mathematical ideas. 

Engaging Students to Learn in Mathematics Class

Engaging students to learn in math class will help students to love math.  Children develop a dislike of math early on and end up resenting it into adult life.   Even in the real world, students will likely have to do some form of mathematics in their personal or working life.  So how can teachers make math more interesting to engage students in the subject? Read more at 5 Best Strategies for Engaging Students to Learn in Mathematics Class

Puzzles in Math with Answers on a computer screen.

Teachers can promote number sense by providing rich mathematical tasks and encouraging students to make connections to their own experiences and previous learning.

Sign up on my webpage to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.  Providing opportunities to do math puzzles daily is one way to help students develop their number sense.  CLICK Here to sign up for  71 Math Number Puzzles and check out my website.

Promoting a Growth Mindset

Research shows that there is a link between a growth mindset and success. In addition, kids who have a growth mindset about their abilities perform better and are more engaged in the classroom.  Students need to be able to preserve and make mistakes when problem solving.  

Read more … 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset

Here are some Resources to Use to Grow a Growth Mindset

  • Free Mindset Survey
  • Growth Mindset Classroom Display Free
  • Growth Mindset Lessons

Growth Mindset in Math Resources on a computer screen.

Using Word Problems

Story Problems and word problems are one way to promote problem solving.   In addition, they provide great practice in using the 4 steps of solving problems.   Then, students are ready for more challenging problems.  

For Kindergarten

  • Subtraction within 5

For First Grade

  • Word Problems to 20
  • Word Problems of Subtraction

Word Problems of Addition and Subtraction on a computer screen.

For Second Grade

  • Two Step Word Problems with Addition and Subtraction
  • Grade 2 Addition and Subtraction Word Problems
  • Word Problems with Subtraction 

Problem Solving in Math with these addition and subtraction word problems with different problem structures. Can be used digitally or as a worksheet.

For Third Grade

  • Word Problems Division and Multiplication
  • Multiplication Word Problems

Use repeated addition to multiply and find the total number of items. See the connection between repeated addition and multiplication when using arrays.

For Fourth Grade

  • Multiplication Area Model
  • Multiplicative Comparison Word Problems

Solving Multiplicative comparison word problems on a computer screen.

Resources for Problem Solving

  • 3 Act Tasks
  • What’s the Best Proven Way to Teach Word Problems with Two Step Equations?
  • 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset
  • 5 Powerful Ideas to Help Students Develop a Growth Mindset in Mathematics

Problem Solving Steps For Math 

In mathematics, problem solving is one of the most important topics to teach.  Learning to problem solve helps students apply mathematics to real-world situations. In addition, it is used for a deeper understanding of mathematical concepts. 

By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics.  Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students. 

Check out  The Free Ultimate Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys to use to cultivate a growth mindset classroom.

Start by modeling using the problem solving steps in math and allowing opportunities for students to use the steps to solve problems.   As students become more comfortable with using the steps and have some strategies to use,  provide more challenging tasks.  Then, students will begin to see the importance of problem solving in math and connecting their learning to real-world situations. 

Kids solving word problems.

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Problem Analysis in Math: Using the 5-Step Problem-Solving Approach

This blog will explore the ways in which problem analysis of student mathematics difficulties can be applied within a problem-solving approach. We will review the core principles of a multi-tier system of support (MTSS) framework, identify the steps within a problem-solving approach, and explore the ways in which problem analysis helps to inform intervention development within the context of mathematics instruction.

Core Principles of an MTSS Framework

A variety of definitions of an MTSS framework exist within the field of education; however, several common principles are apparent and have helped to shape much of the work within this area. The National Association of State Directors of Special Education (Batsche et al., 2006) define eight core principles that capture some of the most important aspects and core beliefs of an MTSS framework:

  • We can effectively teach all children.
  • Intervene early.
  • Use a multi-tier model of service delivery.
  • Use a problem-solving model to make decisions within a multi-tier model.
  • Use scientific, research-based validated intervention and instruction to the extent available.
  • Monitor student progress to inform instruction.
  • Use data to make decisions.
  • Use assessment for screening, diagnostics, and progress monitoring.

The fourth core principle refers to utilizing a problem-solving model to make decisions. More specifically, educators and administrators should use a clearly defined problem-solving process that guides their team in identifying the problem, analyzing the size and effect of the problem, developing a plan for intervention to address the problem, implementing the plan, and examining the effectiveness of the intervention plan.

5-Step Problem-Solving Approach

The problem-solving approach utilized by the FastBridge Learning ® system includes the following five steps:

Problem identification

  • Problem analysis
  • Plan development
  • Plan implementation
  • Plan evaluation

Following this 5-step problem-solving approach helps to guide school teams of educators and administrators in engaging in data-based decision making: a core principle of the MTSS framework.

The first step in this problem-solving approach is problem identification. Christ and Arañas (2014) define a problem as a discrepancy between observed and expected performance. Regularly scheduled universal screening plays an important role in problem identification. The FastBridge Learning ® system offers a variety of screening measures for mathematics, which are summarized in the table below. While the results of regularly scheduled universal screening help to inform whether or not a discrepancy between observed and expected performance exists, problem analysis aims to identify the size and effects of the problem.

steps in problem solving of mathematics

Problem Analysis

After a problem has been identified, the problem must be defined through the method of problem analysis. Only after a problem has been sufficiently analyzed and defined, can the significance of a problem be understood (Brown-Chidsey & Bickford, 2016). Problem analysis can occur at both the individual level and the group level.

Problem Analysis at the Individual Leve l

At the individual level, a discrepancy between observed and expected performance may appear within an individual’s universal screening results. FastBridge Learning ® Individual Skills Reports can provide detailed information to support problem analysis. The Individual Skills Report provides a snapshot of a given student’s risk, relative to benchmark goals, and also provides detailed results on an item-by-item basis. This item-by-item analysis provides insight into the skills the student demonstrates and which skills may require additional instructional support.

If the student’s performance is relatively close to the benchmark goal, the problem is likely to be understood as a minor problem. Item-by-item analysis can help a teacher determine if targeted reteaching of specific content may help the student to reach the goal, or if more intensive intervention may be necessary. An example of a FastBridge Learning ® Individual Skills Report for the earlyMath subtest Numeral Identification (Kindergarten) may be seen below.

steps in problem solving of mathematics

The above report suggests that the sample student is at “some risk” for difficulty with numeral identification. It also reveals that the student was able to identify the given numerals with 91% accuracy. Additionally, the item-by-item analysis indicates that the student struggled to identify the following numerals: 12, 14, and 19. In this case, problem analysis may suggest that this is a minor problem, which may be remedied through targeted reteaching of the numerals 12, 14, and 19. In contrast, if during problem analysis, an Individual Skills Report for Numeral Identification indicated very low accuracy and a substantial number of misidentified numerals, the results would suggest a more significant problem. Significant problems warrant planning for more intensive intervention.

Problem analysis at the group level

At the group level, problem analysis seeks to determine the size and effects of a problem at the class, grade, or school-wide level. FastBridge Learning ® Group Screening Reports and Group Skills Reports help to provide insight about groups of students at risk for learning difficulties. An example of a Group Screening Report for earlyMath may be seen below.

steps in problem solving of mathematics

The Group Screening Report pictured above indicates that approximately 77% of Ms. Horst’s kindergarten class scored at or above the benchmark goal on the earlyMath Composite during the fall benchmark period, while 6% and 18% were identified as being at “some risk” and “high risk” respectively.

All students should demonstrate growth in math achievement throughout the school year. If there are students who met the fall benchmark for mathematics, but not the winter and/or spring benchmarks, problem analysis must occur. In Ms. Horst’s class, we can see by the spring benchmark period, only 12% of her kindergarten class scored at or above the benchmark goal on the earlyMath composite, while 18% and 71% were identified as being at “some risk” and “high risk” respectively.

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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

Close up of lightbulb sparkling with teal color outline on black background

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.

Bubbles with numbers and arrows show Collatz conjecture sequences

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 .

A plot lays out the starting number of the Collatz sequence on the x-axis with the total length of the completed sequence on the y-axis

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.

Cryptopolitan

Photomath: Your Math Tutor in Your Pocket

I n the ever-evolving landscape of education technology , a groundbreaking tool has emerged to revolutionize how students approach math. Photomath, developed by Google, has become a go-to solution for those grappling with mathematical concepts. This innovative smartphone app is a personal math tutor, offering instant solutions, step-by-step explanations, and a deeper understanding of mathematical principles.

Instant solutions with comprehensive explanations

Photomath’s core functionality lies in its ability to scan, solve, and explain various math problems. The app instantly deciphers printed or handwritten equations using advanced image recognition technology , identifies mathematical concepts, and provides real-time solutions. 

However, what sets Photomath apart is its commitment to fostering comprehension. Instead of merely presenting answers, the app breaks down the solution process into clear, digestible steps, empowering users to understand the underlying logic behind each calculation.

Empowering learners of all ages

Whether you’re a student struggling with basic arithmetic or an individual tackling complex calculus problems, Photomath caters to learners of all levels. Its intuitive interface and user-friendly design make it accessible to anyone with a smartphone, regardless of mathematical proficiency. 

By offering detailed explanations for each step, Photomath helps users find the correct answer and equips them with the skills to tackle similar problems independently.

How to harness the power of Photomath

Download: Obtain Photomath for free from your smartphone’s app store.

Open: Grant camera access and launch the app.

Point and Shoot: Focus your camera on the math problem.

Capture: Take a clear picture of the equation.

Solve: Photomath analyzes the image and provides instant solutions and step-by-step explanations.

Unlocking additional functionality

While the basic version of Photomath offers powerful problem-solving capabilities, users can upgrade to the premium version for added features. These include access to multiple solution methods and support for complex numbers through scientific notation. By catering to diverse learning styles and preferences, Photomath ensures users have the tools to excel in mathematics.

A game-changer for math enthusiasts

Photomath’s impact extends beyond simply providing answers to math problems. It serves as a catalyst for transforming the way individuals approach math education. Photomath empowers users to overcome challenges and build confidence in their problem-solving abilities by demystifying complex concepts and fostering a deeper understanding of mathematical principles.

Whether used as a supplement to traditional classroom instruction or as a standalone learning tool, Photomath has become an indispensable asset for students and learners of all ages.

Embracing word problems and diverse approaches

One of Photomath’s standout features is its easy handling of word problems. The app guides users through translating word problems into actionable equations by analyzing textual content and identifying mathematical elements. 

Furthermore, Photomath often presents multiple approaches to problem-solving, allowing users to explore different methods and choose the one that resonates best with their learning style.

Photomath: Your Math Tutor in Your Pocket

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March 7, 2024

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Drawings of mathematical problems predict their resolution

by University of Geneva

Drawings of mathematical problems predict their resolution

A team from the University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and University of Burgundy (uB), have analyzed drawings made by children and adults when solving simple problems. The scientists found that, whatever the age of the participant, the most effective calculation strategies were associated with certain drawing typologies.

These results, published in the journal Memory & Cognition , open up new perspectives for the teaching of mathematics.

Learning mathematics often involves small problems, linked to concrete everyday situations. For example, pupils have to add up quantities of flour to make a recipe or subtract sums of money to find out what's left in their wallets after shopping. They are thus led to translate statements into algorithmic procedures to find the solution.

This translation of words into solving strategies involves a stage of mental representation of mathematical information, such as numbers or the arithmetic operation to be performed, and non-mathematical information, such as the context of the problem.

The cardinal or ordinal dimensions of problems

Having a clearer idea of these mental representations would enable a better understanding of the choice of calculation strategies. Scientists from UNIGE, CYU and uB conducted a study with 10-year-old children and adults, asking them to solve simple problems with the instruction to use as few calculation steps as possible.

The participants were then asked to produce a drawing or diagram explaining their problem-solving strategy for each statement. The contexts of some problems called on the cardinal properties of numbers—the quantity of elements in a set—others on their ordinal properties—their position in an ordered list.

The former involved marbles, fishes, or books, for example: "Paul has 8 red marbles. He also has blue marbles. In total, Paul has 11 marbles. Jolene has as many blue marbles as Paul, and some green marbles. She has 2 green marbles less than Paul has red marbles. In total, how many marbles does Jolene have?"

The latter involved lengths or durations, for example: "Sofia traveled for 8 hours. Her trip started during the day. Sofia arrived at 11. Fred leaves at the same time as Sofia. Fred's trip lasted 2 hours less than Sofia's. What time was it when Fred arrived?"

Both of the above problems share the same mathematical structure, and both can be solved by a long strategy in 3 steps: 11–8 = 3; 8–2 = 6; 6 + 3 = 9, but also in a single calculation: 11–2 = 9, using a simple subtraction. However, the mental representations of these problems are very different, and the researchers wanted to determine whether the type of representations could predict the calculation strategy, in 1 or 3 steps, of those who solve them.

'"Our hypothesis was that cardinal problems—such as the one involving marbles—would inspire cardinal drawings , i.e., diagrams with identical individual elements, such as crosses or circles, or with overlaps of elements in sets or subsets.

"Similarly, we assumed that ordinal problems—such as the one mentioning travel times —would lead to ordinal representations, i.e., diagrams with axes, graduations or intervals—and that these ordinal drawings would reflect participants' representations and indicate that they would be more successful in identifying the one-step solution strategy," explains Hippolyte Gros, former post-doctoral fellow at UNIGE's Faculty of Psychology and Educational Sciences, associate professor at CYU, and first author of the study.

Identifying mental representations through drawings

These hypotheses were validated by analyzing the drawings of 52 adults and 59 children. "We have shown that, irrespective of their experience—since the same results were obtained in both children and adults—the use of strategies by the participants depends on their representation of the problem, and that this is influenced by the non-mathematical information contained in the problem statement, as revealed by their drawings," says Emmanuel Sander, full professor at the UNIGE's Faculty of Psychology and Educational Sciences.

"Our study also shows that, even after years of experience in solving addition and subtraction, the difference between cardinal and ordinal problems remains very marked. The majority of participants were only able to solve problems of the second type in a single step."

Improving mathematical learning through drawing analysis

The team also noted that drawings showing ordinal representations were more frequently associated with a one-step solution, even if the problem was cardinal. In other words, drawing with a scale or an axis is linked to the choice of the fastest calculation.

"From a pedagogical point of view, this suggests that the presence of specific features in a student's drawing may or may not indicate that his or her representation of the problem is the most efficient one for meeting the instructions—in this case, solving with the fewest calculations possible," observes Jean-Pierre Thibaut, full professor at the uB Laboratory for Research on Learning and Development.

"Thus, when it comes to subtracting individual elements, a representation via an axis—rather than via subsets—is more effective in finding the fastest method. Analysis of students' drawings in arithmetic can therefore enable targeted intervention to help them translate problems into more optimal representations. One way of doing this is to work on the graphical representation of statements in class, to help students understand the most direct strategies," concludes Gros.

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MathScale sets itself apart from other models, including LLaMA-2 7B, LLaMA-2 13B, and Mistral 7B, on the MWPBENCH dataset. It not only achieves a micro average accuracy of 35.0% and a macro average accuracy of 37.5% but also surpasses counterparts of equivalent size by 42.9% and 43.7%, respectively. Even on out-of-domain test sets like GaokaoBench-Math and AGIEval-SAT-MATH, MathScale-7B significantly outperforms other open-source models. MathScale-Mistral demonstrates performance parity with GPT-3.5-Turbo on both micro and macro averages, further underscoring its superiority.

In conclusion, researchers from The Chinese University of Hong Kong, Microsoft Research, and Shenzhen Research Institute of Big Data present MathScale, which introduces a straightforward and scalable approach for producing top-notch mathematical reasoning data using cutting-edge LLMs. Also, MWPBENCH provides a comprehensive benchmark for math word problems across various difficulty levels. MathScale-7B exhibits state-of-the-art performance on MWPBENCH, outperforming equivalent-sized peers by significant margins. This contribution advances mathematical reasoning by facilitating fair and consistent model evaluations in academic settings.

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Muhammad Athar Ganaie

Muhammad Athar Ganaie, a consulting intern at MarktechPost, is a proponet of Efficient Deep Learning, with a focus on Sparse Training. Pursuing an M.Sc. in Electrical Engineering, specializing in Software Engineering, he blends advanced technical knowledge with practical applications. His current endeavor is his thesis on "Improving Efficiency in Deep Reinforcement Learning," showcasing his commitment to enhancing AI's capabilities. Athar's work stands at the intersection "Sparse Training in DNN's" and "Deep Reinforcemnt Learning".

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2.3.1: George Polya's Four Step Problem Solving Process

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Step 1: Understand the Problem

  • Do you understand all the words?
  • Can you restate the problem in your own words?
  • Do you know what is given?
  • Do you know what the goal is?
  • Is there enough information?
  • Is there extraneous information?
  • Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)

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  4. What Is Problem-Solving? Steps, Processes, Exercises to do it Right

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  5. What IS Problem-Solving?

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  6. The 5 Steps of Problem Solving

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COMMENTS

  1. Module 1: Problem Solving Strategies

    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.

  2. 1.5: Problem Solving

    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 ...

  3. The easy 4 step problem-solving process (+ examples)

    Consider the problem-solving steps applied in the following example. I know that I want to say "I don't eat eggs" to my Mexican waiter. That's the problem. I don't know how to say that, but last night I told my date "No bebo alcohol" ("I don't drink alcohol"). I also know the infinitive for "eat" in Spanish (comer).

  4. 1.3: Problem Solving Strategies

    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.

  5. Step-by-Step Calculator

    To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.

  6. Step-by-Step Math Problem Solver

    QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and ...

  7. Problem Solving in Mathematics

    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.

  8. A Guide to Problem Solving

    A Guide to Problem Solving. When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice.

  9. Microsoft Math Solver

    Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

  10. Four-Step Math Problem Solving Strategies & Techniques

    Solving a math problem involves first gaining a clear understanding of the problem, then choosing from among problem solving techniques or strategies, followed by actually carrying out the solution, and finally checking the solution. See this article for more information about this four-step math problem solving procedure, with several problem solving techniques presented and discussed for ...

  11. Teaching Mathematics Through Problem Solving

    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 ...

  12. PDF Problem solving in mathematics

    Therefore, high-quality assessment of problem solving in public tests and assessments1 is essential in order to ensure the effective learning and teaching of problem solving throughout primary and secondary education. Although the focus here is on the assessment of problem solving in mathematics, many of the ideas will be directly transferable ...

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  14. Step by step guide to solving math problems

    109 - 73 = 36109 - 36 = 73. If our student did not work out the sum correctly, we would not come to these sums. (By the way, the same can be done with multiplication and division.) Finally, go back and review the problem one last time. By going over the concepts, operations and formulas, it will help your kids to internalize the process and ...

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  16. 4 Best Steps To Problem Solving in Math That Lead to Results

    In the past, we would teach the concepts and procedures and then assign one-step "story" problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as "draw a picture" or "guess and check.". Eventually, students would be given problems to apply the skills and strategies.

  17. 10.1: George Polya's Four Step Problem Solving Process

    Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1. Guess and test.

  18. Problem Analysis in Math: Using the 5-Step Problem-Solving Approach

    This blog will explore the ways in which problem analysis of student mathematics difficulties can be applied within a problem-solving approach. We will review the core principles of a multi-tier system of support (MTSS) framework, identify the steps within a problem-solving approach, and explore the ways in which problem analysis helps to ...

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  21. Solve

    Differentiation. dxd (x − 5)(3x2 − 2) Integration. ∫ 01 xe−x2dx. Limits. x→−3lim x2 + 2x − 3x2 − 9. Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.

  22. The Simplest Math Problem Could Be Unsolvable

    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.

  23. 1.1: Introduction to Problem Solving

    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.

  24. Photomath: Your Math Tutor in Your Pocket

    Point and Shoot: Focus your camera on the math problem. Capture: Take a clear picture of the equation. Solve: Photomath analyzes the image and provides instant solutions and step-by-step explanations.

  25. Drawings of mathematical problems predict their resolution

    Both of the above problems share the same mathematical structure, and both can be solved by a long strategy in 3 steps: 11-8 = 3; 8-2 = 6; 6 + 3 = 9, but also in a single calculation: 11-2 ...

  26. This AI Paper from China Presents MathScale: A Scalable Machine

    Large language models (LLMs) excel in various problem-solving tasks but need help with complex mathematical reasoning, possibly due to the need for multi-step reasoning. Instruction Tuning effectively enhances LLM capabilities. However, its effectiveness is hindered by the scarcity of datasets for mathematical reasoning. This limitation highlights the need for more extensive datasets to fully ...

  27. ‎Studdy

    Simply take a photo to break down a complex problem into a solution with step-by-step explanations. Learn foundational skills and check your understanding by asking the Studdy clarifying questions for personalized instruction. Studdy is the most accurate app for solving word problems, math problems, and college-level science problems.

  28. 2.3.1: George Polya's Four Step Problem Solving Process

    Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1. Guess and test. 11. Solve an equivalent problem. 2.