• Our Mission

6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

Photo of elementary school teacher with students

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

Logo for FHSU Digital Press

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

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 in school 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.

Share This Book

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

Module 1: Problem Solving Strategies

  • Last updated
  • Save as PDF
  • Page ID 10352

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

Pólya’s How to Solve It

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

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

Screen Shot 2018-08-30 at 4.43.05 PM.png

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

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

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

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

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

Problem Solving Strategy 1 (Guess and Test)

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

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

Step 1: Understanding the problem

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

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

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

Step 2: Devise a plan

Going to use Guess and test along with making a tab

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

Make a table and look for a pattern:

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

Step 3: Carry out the plan:

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

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

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

Videos to watch:

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

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

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

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

Check in question 1:

clipboard_e6298bbd7c7f66d9eb9affcd33892ef0d.png

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

Check in question 2:

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

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

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

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

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

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

Gauss's strategy for sequences.

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

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

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

Last term = 3(200-1) +2

Last term is 599.

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

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

Sum = 60,100

Check in question 3: (10 points)

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

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

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

Videos to watch demonstrating of “Working Backwards”

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

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

1. We start with 11 and work backwards.

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

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

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

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

Problem Solving Strategy 5 (Looking for a Pattern)

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

We first need to find a pattern.

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

Example 1: 1, 4, 7, 10, 13…

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

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

So the next number would be

25 + 11 = 36

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

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

-5 – 3 = -8

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

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

So each number is being multiplied by 2.

16 x 2 = 32

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

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

Problem Solving Strategy 6 (Make a List)

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

List all the squares of the numbers 1 to 20.

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

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

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

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

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

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

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

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

Let’s set up a table:

Term Number what did we do

problem solving in school mathematics

Looking back: How would you find the nth term?

problem solving in school mathematics

Find the 10 th term of the above sequence.

Let L = the tenth term

problem solving in school mathematics

Problem Solving Strategy 8 (Process of Elimination)

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

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

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

b. It is greater than 20 but less than 35

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

c. The sum of the digits is 7

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

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

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

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

Book cover

Mathematics & Mathematics Education: Searching for Common Ground pp 113–135 Cite as

Reflections on Problem-Solving

Problem Solving in Mathematics and in Mathematics Education

  • Boris Koichu 5  

2594 Accesses

1 Citations

Part of the Advances in Mathematics Education book series (AME)

The chapter includes four contributions on different aspects of the relationship between problem solving in mathematics and in mathematics education. Gerald Goldin points out that besides the importance of teaching students how to solve certain classes of problems, problem solving is a means of achieving some more general purposes pertaining to mathematics learning. Israel Weinzweig develops the claim that certain sequences of mathematical questions can provide students with problem-solving experiences similar to those of research mathematicians, and that such experiences are beneficial for promoting students’ conceptual understanding. Shlomo Vinner discusses the role of schemata and creativity in mathematical problem solving, and argues that the notions “problem solving in mathematics” and “problem solving in exam-oriented mathematics instruction” are incompatible. Roza Leikin presents a study aimed at identifying unique cognitive traits of intellectually gifted students who have the potential to become research mathematicians in the future. The chapter concludes with a reflective summary, in which the points made by the contributors are considered as parts of a longer-term debate on the relationships between problem solving in mathematics and in mathematics education, a conversation that has developed over the years according to a certain spiral pattern.

  • Conceptual understanding
  • Dimensions of mathematical giftedness: creativity
  • Neuro-cognition
  • Insight-based and routine problems
  • Evolution of problem solving within mathematics education
  • Problem-solving expertise
  • Problem solving by mathematicians
  • Teaching for problem solving
  • Teaching through problem solving

With contributions by

Gerald A. Goldin, Rutgers University, Piscataway, NJ, USA

A. Israel Weinzweig, University of Illinois at Chicago, Chicago, IL, USA

Shlomo Vinner, Achva College of Education, Hebrew University, Ben-Gurion University of Negev, Jerusalem, Israel

Roza Leikin, University of Haifa, Haifa, Israel

This is a preview of subscription content, log in via an institution .

Buying options

  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
  • Available as EPUB and PDF
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
  • Durable hardcover edition

Tax calculation will be finalised at checkout

Purchases are for personal use only

Teachers, prospective teachers and middle school students.

The team includes Roza Leikin with responsibility for the mathematical content of the study and research on creativity and giftedness; Mark Leikin who is responsible for cognitive and neuro-cognitive research dimensions; Shelly Shaul who is an ERP-research specialist at the Faculty of Education. The team of researchers collaborates in supervision of a group of Ph.D. students in the design of a multidimensional research puzzle: Ilana Waisman, Nurit Paz and Miri Lev.

Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: an emergent multidimensional problem-solving framework. Educational Studies in Mathematics , 58 , 45–75.

Article   Google Scholar  

Confrey, J. (1995). Student voice in examining “splitting” as an approach to ratio, proportions and fractions. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th international conference for the psychology of mathematics education (Vol. 1, pp. 3–29). Recife: Universidade Federal de Pernambuco.

Google Scholar  

DeBellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: a representational perspective. Educational Studies in Mathematics , 63 , 131–147.

Dreyfus, T., & Eisenberg, T. (1986). On the aesthetics of mathematical thoughts. For the Learning of Mathematics , 6 (1), 2–10.

Dudley, U. (2010). What is mathematics education for? Notices of the American Mathematical Society , 57 , 608–613.

Eisenberg, T. A. (1975). Behaviorism: the bane of school mathematics. International Journal of Mathematical Education in Science and Technology , 6 , 163–171.

Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.), MAA notes series: Vol.   19 . Visualization in teaching and learning mathematics (pp. 25–37). Washington: Math. Assoc. of America.

Eisenberg, T., & Fried, M. N. (2009). Dialogue on mathematics education: two points of view on the state of the art. ZDM. The International Journal on Mathematics Education , 41 , 143–150.

Fischbein, E. (1987). Intuition in science and mathematics—an educational approach . Dordrecht: Reidel.

Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. The Journal of Mathematical Behavior , 17 , 137–165.

Goldin, G. A. (2000). Affective pathways and representation in mathematical problem solving. Mathematical Thinking and Learning , 2 , 209–219.

Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: a hidden variable in mathematics education? (pp. 59–72). Dordrecht: Kluwer Academic.

Goldin, G. A., & Kaput, J. J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 397–430). Hillsdale: Erlbaum.

Goldin, G. A. & McClintock, C. E. (Eds.) (1984). Task variables in mathematical problem solving . Philadelphia: Franklin Institute Press.

Goldin, G. A., Epstein, Y. M., Schorr, R. Y., & Warner, L. B. (2011). Beliefs and engagement structures: behind the affective dimension of mathematical learning. ZDM. The International Journal on Mathematics Education , 43 , 547–560.

Greeno, J. G. (1980). Trends in the theory of knowledge for problem solving. In D. T. Tuma & F. Reif (Eds.), Problem solving and education: issues in teaching and research , (pp. 9–23). Hillsdale: Erlbaum (cited in Heller and Hungate 1985).

Hadamard, J. (1945/1996). The mathematician’s mind: the psychology of invention in the mathematical field . Princeton: Princeton University Press.

Harel, G. (2013). Intellectual need. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 119–151). New York: Springer.

Chapter   Google Scholar  

Heller, J., & Hungate, H. (1985). Implications for mathematics instruction of research on scientific problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: multiple research perspectives (pp. 83–112). Hillsdale: Erlbaum.

Hembree, R. (1992). Experiments and relational studies in problem solving: a meta-analysis. Journal for Research in Mathematics Education , 23 , 242–273.

Hmelo-Silver, E., Duncan, R., & Chinn, C. (2007). Scaffolding and achievement in problem-based and inquiry learning: a response to Kirschner, Sweller, and Clark. Educational Psychologist , 42 , 99–107.

Isoda, M., & Katagiri, S. (2012). Mathematical thinking: how to develop it in the classroom . Singapore: World Scientific.

Book   Google Scholar  

Jeeves, M. A., & Greer, B. (1983). Analysis of structural learning . London: Academic Press.

Kahneman, D. (2011). Cognitive limitations and the psychology of science. A public lecture given on December 29, 2011, at the center for the study of rationality at the Hebrew University of Jerusalem, on the occasion of its 20-th anniversary.

Kirschner, P., Sweller, J., & Clark, R. (2006). Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist , 41 , 75–86.

Koichu, B., Berman, A., & Moore, M. (2006). Patterns of middle school students’ heuristic behaviors in solving seemingly familiar problems. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th international conference for the psychology of mathematics education (Vol. 3, pp. 457–464). Prague: Charles University.

Koichu, B., Berman, A., & Moore, M. (2007). The effect of promoting heuristic literacy on the mathematic aptitude of middle-school students. International Journal of Mathematical Education in Science and Technology , 38 , 1–17.

Krulik, S. (Ed.) (1980). Problem solving in school mathematics (1980 NCTM yearbook) , Reston: NCTM.

Larkin, J., McDermott, J., Simon, D. P., & Simon, H. A. (1980). Expert and novice performance in solving physics problems. Science , 208 (4450), 1335–1342.

Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Rotterdam: Sense Publishers.

Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference? ZDM. The International Journal on Mathematics Education , 45 , 183–197.

Leikin, R., Waisman, I., Shaul, S., & Leikin, M. (2012). An ERP study with gifted and excelling male adolescents: solving short insight-based problems. In T. Y. Tso (Ed.), Proceedings of the 36th international conference for the psychology of mathematics education , Taiwan, Taipei (Vol. 3, pp. 83–90).

Leikin, M., Paz-Baruch, N., & Leikin, R. (2013). Memory abilities in generally gifted and excelling-in-mathematics adolescents. Intelligence , 41 , 566–578.

Lester, F. K. Jr., & Garofalo, J. (Eds.) (1982). Mathematical problem solving: issues in research . Philadelphia: Franklin Institute Press.

Lubinski, D., & Benbow, C. P. (2006). Study of mathematically precocious youth after 35 years: uncovering antecedents for the development of math-science expertise. Perspectives on Psychological Science , 1 , 316–345.

Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior , 24 , 385–401.

Mason, J., & Pimm, D. (1984). Generic examples: seeing the general in the particular. Educational Studies in Mathematics , 15 , 277–289.

Mason, J., Burton, L., & Stacey, K. (2009). Thinking mathematically (2nd ed.). London: Pearson Education.

NCTM (1980). An agenda for action: recommendations for school mathematics of the 1980s . Reston: National Council of Teachers of Mathematics. http://www.nctm.org/standards/content.aspx?id=17278 . Accessed 20 June 2012.

Newell, A., & Simon, H. A. (1972). Human problem solving . Englewood Cliffs: Prentice Hall.

Niss, M. (2011). Reflection on the state of and the trends in research on mathematics teaching and learning—from here to utopia. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 1293–1312). Reston: National Council of Teachers of Mathematics.

Poincaré, H. (1952). Science and hypothesis . New York: Dover.

Pólya, G. (1945/1957). How to solve it . Princeton: Princeton University Press.

Pólya, G. (1962/1965). Mathematical discovery: on understanding, learning, and teaching problem solving (Vols. I and II). New York: Wiley.

Pólya, G. (late 1960s). A lecture on teaching mathematics in the primary schools. http://cmc-math.org/members/infinity/polya.html . Assessed 27 Dec 2012.

Schoenfeld, A. (1979). Explicit heuristic training as a variable in problem-solving performance. Journal for Research in Mathematics Education , 10 , 173–187.

Schoenfeld, A. (1983). Problem solving in the mathematics curriculum: a report, recommendations, and an annotated bibliography . Washington: Math. Assoc. of America.

Schoenfeld, A. (1985). Mathematical problem solving . New York: Academic Press.

Schoenfeld, A. (1992). Learning to think mathematically: problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334–370). New York: Macmillan Co.

Schoenfeld, A. (2007). Problem solving in the United States, 1970–2008: research and theory, practice and politics. ZDM. The International Journal on Mathematics Education , 39 , 537–551.

Schroeder, T. L., Lester, F. K. Jr. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New directions for elementary school mathematics. Yearbook of the national council of teachers of mathematics (pp. 31–42). Reston: National Council of Teachers of Mathematics.

Shaul, S., Leikin, M., Waisman, I., & Leikin, R. (2012). Visual processing in algebra and geometry in mathematically excelling students: an ERP study. In The electronic proceedings of the 12th international Congress on mathematics education (Topic study group-16: visualization in mathematics education) , Coex, Seoul, Korea (pp. 1460–1469).

Silver, E. A. (Ed.) (1985). Teaching and learning mathematical problem solving: multiple research perspectives . Hillsdale: Erlbaum.

Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching , 77 , 20–26.

Sriraman, B. (2005). Are giftedness and creativity synonyms in mathematics? An analysis of constructs within the professional and school realms. The Journal of Secondary Gifted Education , 17 , 20–36.

Stanovich, K. E. (1999). Who is rational? Mahwah: Erlbaum.

Vinner, S. (1997a). From intuition to inhibition—mathematics, education and other endangered species. In E. Pehkonen (Ed.), Proceedings of the 21st international conference for the psychology of mathematics education , Lahti, Finland (Vol. 1, pp. 63–79).

Vinner, S. (1997b). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics , 34 , 97–129.

Vogeli, B. R. (1997). Special secondary schools for the mathematically and scientifically talented. An international panorama . New York: Columbia University Press.

Vygotsky, L. S. (1930/1984). Imagination and creativity in adolescent. In D. B. Elkonin (Ed.), Child psychology: Vol.   4 . The collected works of L.S. Vygotsky (pp. 199–219). Moscow: Pedagogika (in Russian).

Waisman, I., Shaul, S., Leikin, M., & Leikin, R. (2012). General ability vs. expertise in mathematics: an ERP study with male adolescents who answer geometry questions. In The electronic proceedings of the 12th international Congress on mathematics education (Topic study group-3: activities and programs for gifted students) , Coex, Seoul, Korea (pp. 3107–3116).

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education , 27 (4), 458–477.

Download references

Author information

Authors and affiliations.

Technion—Israel Institute of Technology, Technion City, Haifa, 32000, Israel

Boris Koichu

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Boris Koichu .

Editor information

Editors and affiliations.

Ben Gurion University of the Negev, Beer Sheva, Southern, Israel

Michael N. Fried

Tel Aviv University, Ramat Aviv, Tel Aviv, Israel

Tommy Dreyfus

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter.

Koichu, B. (2014). Reflections on Problem-Solving. In: Fried, M., Dreyfus, T. (eds) Mathematics & Mathematics Education: Searching for Common Ground. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7473-5_8

Download citation

DOI : https://doi.org/10.1007/978-94-007-7473-5_8

Publisher Name : Springer, Dordrecht

Print ISBN : 978-94-007-7472-8

Online ISBN : 978-94-007-7473-5

eBook Packages : Humanities, Social Sciences and Law Education (R0)

Share this chapter

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

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

Provided by the Springer Nature SharedIt content-sharing initiative

  • Publish with us

Policies and ethics

  • Find a journal
  • Track your research
  • Open access
  • Published: 19 December 2019

Problematizing teaching and learning mathematics as “given” in STEM education

  • Yeping Li 1 &
  • Alan H. Schoenfeld 2  

International Journal of STEM Education volume  6 , Article number:  44 ( 2019 ) Cite this article

104k Accesses

76 Citations

13 Altmetric

Metrics details

Mathematics is fundamental for many professions, especially science, technology, and engineering. Yet, mathematics is often perceived as difficult and many students leave disciplines in science, technology, engineering, and mathematics (STEM) as a result, closing doors to scientific, engineering, and technological careers. In this editorial, we argue that how mathematics is traditionally viewed as “given” or “fixed” for students’ expected acquisition alienates many students and needs to be problematized. We propose an alternative approach to changes in mathematics education and show how the alternative also applies to STEM education.

Introduction

Mathematics is commonly perceived to be difficult (e.g., Fritz et al. 2019 ). Moreover, many believe “it is ok—not everyone can be good at math” (Rattan et al. 2012 ). With such perceptions, many students stop studying mathematics soon after it is no longer required of them. Giving up learning mathematics may seem acceptable to those who see mathematics as “optional,” but it is deeply problematic for society as a whole. Mathematics is a gateway to many scientific and technological fields. Leaving it limits students’ opportunities to learn a range of important subjects, thus limiting their future job opportunities and depriving society of a potential pool of quantitatively literate citizens. This situation needs to be changed, especially as we prepare students for the continuously increasing demand for quantitative and computational literacy over the twenty-first century (e.g., Committee on STEM Education 2018 ).

The goal of this editorial is to re-frame issues of change in mathematics education, with connections to science, technology, engineering, and mathematics (STEM) education. We are hardly the first to call for such changes; the history of mathematics and philosophy has seen ongoing changes in conceptualization of the discipline, and there have been numerous changes in the past century alone (Schoenfeld 2001 ). Yet changes in practice of how mathematics is viewed, taught, and learned have fallen far short of espoused aspirations. While there has been an increased focus on the processes and practices of mathematics (e.g., problem solving) over the past half century, the vast majority of the emphasis is still on what content should be presented to students. It is thus not surprising that significant progress has not been made.

We propose a two-fold reframing. The first shift is to re-emphasize the nature of mathematics—indeed, all of STEM—as a sense-making activity. Mathematics is typically conceptualized and presented as a body of content to be learned and processes to be engaged in, which can be seen in both the NCTM Standards volumes and the Common Core Standards. Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way. Indeed, much of the inductive part of mathematics has been lost, and the deductive part is often presented as rote procedures rather than a form of sense making. If we arrange for students to have the right experiences, the formal mathematics can serve to organize and systematize those experiences.

The second shift is suggested by the first, with specific attention to classroom instruction. Whether mathematics or STEM, the main focus of most instruction has been on the content and practices of the discipline, and what the teacher should do in order to make it accessible to students. Instead, we urge that the main focus should be on the student’s experience of the discipline – on the affordances the environment provides the student for disciplinary sense making. We will introduce the Teaching for Robust Understanding (TRU) Framework, which can be used to problematize instruction and guide needed reframing. The first dimension of TRU (The Discipline) focuses on the re-framing discussed above: is the content conceptualized as something rich and connected that can be experienced and codified in meaningful ways? The second dimension (Cognitive Demand) examines opportunities students have to do that kind of sense-making and codification. The third (Equitable Access to Content) examines who has such opportunities: is there equitable access to the core ideas? Dimension 4 (Agency, Ownership, and Identity) asks, do students encounter the discipline in ways that enable them to see themselves as sense makers, building both agency and positive disciplinary identities? Finally, dimension 5 (Formative Assessment) asks, does instruction routinely use formative assessment, allowing student thinking to become public so that instruction can be adjusted accordingly?

We begin with a historical background, briefly discussing different views regarding the nature of mathematics. We then problematize traditional approaches to mathematics teaching and learning. Finally, we discuss possible changes in the context of STEM education.

Knowing the background: the development of conceptions about the nature of mathematics

The scholarly understanding of the nature of mathematics has evolved over its long history (e.g., Devlin 2012 ; Dossey 1992 ). Explicit discussions regarding the nature of mathematics took place among Greek mathematicians from 500 BC to 300 AD (see, https://en.wikipedia.org/wiki/Greek_mathematics ). In contrast to the primarily utilitarian approaches that preceded them, the Greeks pioneered the study of mathematics for its own sake and pursued the development and use of generalized mathematical theories and proofs, especially in geometry and measurement (Boyer 1991 ). Different perspectives about the nature of mathematics were gradually developed during that time. Plato perceived the study of mathematics as pursuing the truth that exists in external world beyond people’s mind. Mathematics was treated as a body of knowledge, in the ideal forms, that exists on its own, which human’s mind may or may not sense. Aristotle, Plato’s student, believed that mathematicians constructed mathematical ideas as a result of the idealization of their experience with objects (Dossey 1992 ). In this perspective, Aristotle emphasized logical reasoning and empirical realization of mathematical objects that are accessible to the human senses. The two schools of thought that evolved from Plato’s and Aristotle’s contrasting conceptions of the nature of mathematics have had important implications for the ensuing development of mathematics as a discipline, and for mathematics education.

Several more schools of thought were developed as mathematicians tackled new problems in mathematics (Dossey 1992 ). Davis and Hersh ( 1980 ) provides an entertaining and informative account of these developments. Three major schools of thought in the early 1900s dealt with paradoxes in the real number system and the theory of sets: (1) logicism, as an outgrowth of the Platonic school, accepts the external existence of mathematics and emphasizes the form rather than the interpretation in a specific setting; (2) intuitionism, as influenced by Aristotle’s ideas, only accepts the mathematics to be developed from the natural numbers forward via “valid” patterns of mental reasoning (not empirical realization in Aristotle’s thought); and (3) formalism, also aligned with Aristotle’s ideas, builds mathematics upon the formal axiomatic structures to free mathematics from contradictions. These three schools of thought are similar in that they view the contents of mathematics as products , but they differ in whether products are viewed as pre-existing or created through experience. The development of these three schools of thought illustrates that the view of mathematics as products has its long history in mathematics.

With the gradual development of school mathematics since 1900s (Stanic and Kilpatrick 1992 ), the conception of the nature of mathematics has increasingly received attention from mathematics educators. Which notion of mathematics mathematics education adopts and uses has a direct and strong impact on the way of school mathematics being presented and approached in education. Although the history of school mathematics is relatively short in comparison with mathematics itself, we can find ample examples about the influence of different views of mathematics on curriculum and classroom instruction in the USA and other education systems (e.g., Dossey et al. 2016 ; Li and Lappan 2014 ; Li, Silver, and Li 2014 ; Stanic and Kilpatrick 1992 ). For instance, the “New Math” movement of 1950s and 1960s used the formalism school of thought as the core of reform efforts. The content was presented in a structural format, using the set theoretic language and conceptions. But the result was not a successful progression toward a school mathematics that is best for students and teachers (e.g., Kline 1973 ). Alternatively, Dossey ( 1992 ), in his review of the nature of mathematics, identified and selected scholars’ works and ideas applicable to both professional mathematicians and mathematics educators (e.g., Davis and Hersh 1980 ; Hersh 1986 ; Tymoczko 1986 ). Those scholars' ideas rested on what professional mathematicians do, not what mathematicians think about what mathematics is. Dossey ( 1992 ) specifically cited Hersh ( 1986 ) to emphasize mathematics is about ideas and should be accepted as a human activity, not strictly governed by any one school of thought.

Devlin ( 2000 ) argued that mathematics is not a single entity but has four different faces: (1) computation, formal reasoning, and problem solving; (2) a way of knowing; (3) a creative medium; and (4) applications. Further, he contended school mathematics typically focuses on the first face, makes some reference to the fourth face, but pays almost no attention to the other two faces. His conception of mathematics assembles ideas from the history of mathematics and observes mathematical activities occurring across different settings.

Our brief review shows that the nature of mathematics can be understood as having different faces, rather than being governed by any single school of thought. At the same time, the ideas of Plato and Aristotle continue to influence the ways that mathematicians, mathematics educators, and the general public perceive mathematics. Despite nearly a half century of process-oriented research (see below), let alone Pólya’s work on problem solving, mathematics is still perceived of largely as products —a body of knowledge as highlighted in the three schools (logicist, intuitionist, formalist) of thought, rather than ideas that call for active thinking and creation. The evolving conceptions about the nature of mathematics in history suggests there is room for us to decide how mathematics can be perceived, rather than being bounded by a pre-occupied notion of mathematics as “given” or “fixed.” Each and every learner can experience mathematics through different practices and “own” mathematics as a human activity.

Problematizing what is important for students to learn in and through mathematics

The evolving conceptions about the nature of mathematics suggest that choices exist when deciding what and how to teach and learn mathematics but they do not specify what and how to make the choice. Decisions require articulating options for conceptions of what is important for students to learn in and through mathematics and evaluating the advantages and drawbacks for the students for each option.

According to Stanic and Kilpatrick ( 1992 ), the history of school mathematics curricula presents two important and real changes over the years: one is at the turn of the twentieth century when school mathematics was reformed as a unified and applied curriculum to accommodate dramatically increased student populations from diverse backgrounds, and the other is the “New Math” movement of the 1950s and 1960s, intended to integrate modern mathematics into school curriculum. The perceived failure of the “New Math” movement led to the “Back to Basics” movement in the 1970s, followed by “Problem Solving” in the 1980s, and then the Curriculum Standards movement in the 1990s and after. The history shows school mathematics curricula have emphasized teaching and learning mathematical knowledge and skills, together with problem solving and some applications of mathematics, a picture that is consistent with what Devlin ( 2000 ) refers to as the 1st face and some reference to the 4th face of mathematics.

Therefore, although there have been reforms in mathematics curriculum and instruction, there are hardly real changes in how mathematics is conceptualized and presented in school education in the USA (Stanic and Kilpatrick 1992 ) and other education systems (e.g., Leung and Li 2010 ; Li and Lappan 2014 ). The dominant conception remains mathematics as products , frequently referring to a body of static knowledge and skills that need to be learned and acquired (Fisher 1990 ). This continues to be largely the case in practice, despite advances in conceptualization (see below).

It should be noted that conceptualizing mathematics as “a body of knowledge and skills” is not wrong, especially with such a long history of knowledge creation and accumulation in mathematics, but it is not adequate for school mathematics nowadays. The set of concepts and procedures, after years of development, exceeds what could be covered in any school curricula. Moreover, this body of knowledge and skills keeps growing, as the product of human intelligence and scholarship in mathematics. Devlin ( 2012 ) pointed out that school mathematics mainly covers what was developed in the Greek mathematics, plus just two further advances from the seventh century: calculus and probability theory. It is no wonder if someone questions the value of learning such a small set of knowledge and skills developed more than a thousand years ago. Meanwhile, this body of knowledge and skills are often abstract, static, and “foreign” to many students and teachers who learned to perceive mathematics as an external entity in existence (Plato’s notion) rather than Aristotelian emphasis on experimentation (Cooney 1987 ). It is thus not surprising for so many students and teachers to claim that mathematics is difficult (e.g., Fritz et al. 2019 ) and “it is ok—not everyone can be good at math” (Rattan et al. 2012 ).

What can be made meaningful should be critically important to those who want to (or need to) learn and teach mathematics. In fact, there is significant evidence that students often try to make sense of mathematics that is “presented” or “given” to them, although they made numerous errors that can be decoded to study their thinking (e.g., Ashlock 2010 ). Indeed, misconceptions are best thought of not as errors that need to be “fixed,” but as plausible abstractions on the basis of what students have learned—i.e., attempts at sense-making (Smith et al. 1993 ). Conceiving mathematics as about “ideas,” we can help students to play, own, experience, and think about some key ideas just like what they do in many other activities, such as game play (Gee 2005 ). Definitions of concepts and formal languages and procedures can be postponed until students are ready to consider why and how they are needed. Mathematics should be taken and accepted as a human activity (Dossey 1992 ), and developing students’ mathematical thinking (about ideas) should be emphasized in learning mathematics itself (Devlin 2012 ) and in STEM (Li et al. 2019a ).

Along with the shift from products to ideas in mathematics, scholars have already focused on how people work with ideas in mathematics. Elaborated in detail by Schoenfeld ( in press ), the revolution began with George Pólya (1887–1985) who had a fundamental interest in having students learn and understand content via problem solving. For Pólya, mathematics was about inquiry, sense making, and understanding how and why mathematical ideas (instead of content as products) fit together the way they do. The call for problem solving in the 1980s in the USA was (at least partially) inspired by Pólya’s ideas after a decade of “back to basics” in the 1970s. It has been recognized since that the practices of mathematics (including problem solving) are every bit as important as the content itself, and the two shouldn’t be separated. In the follow-up standards movement, the content and practices have been the “warp and weave” of the fabric doing mathematics, as articulated in Principles and Standards for School Standards (NCTM 2000 ). There were five content standards and five process standards (i.e., problem solving, reasoning, connecting, communicating, representing). It is widely acknowledged, also in the Common Core State Standards in the USA (CCSSI 2010 ), that both content and processes/practices are essential and they form the base for next steps.

Problematizing how mathematics is taught and learned, with connections to STEM education

How the ways that mathematics is often taught cause concerns.

Conceiving mathematics as a body of facts and procedures to be “mastered” has been long-standing in mathematics education practice, and it often results in students’ learning by rote memorization. For example, Schoenfeld ( 1988 ) provided a detailed account of the disasters of a “well-taught” mathematics course, documenting a 10th-grade geometry class taught by a confident and experienced teacher. The teacher taught and managed his class well, and his students also did well on standardized examinations, which focused on content and procedures. At the same time, however, Schoenfeld pointed out that the students developed counterproductive views of mathematics. Although the students developed some level of proficiency in content and procedures, they gained (or were reinforced in) the kinds of beliefs about mathematics as being fragmented and disconnected. Schoenfeld argued that the course led students to develop a robust and counterproductive set of beliefs about the nature of geometry.

Seeking possible origins about students’ counterproductive beliefs about mathematics from mathematics instruction motivated Schoenfeld’s study (Schoenfeld 1988 ). Such an intuitive motivation is also evident in other studies. Keitel ( 2006 ) compared the lessons of two teachers (T1 and T2) in Germany who taught their classes very differently. T1 regularly taught the class emphasizing routine individual practice and memorization of specific algebraic rules. T1 emphasized the importance of such practices for test taking, and the students followed his instruction. Even when T1 one day introduced a non-routine problem that connects algebra and geometry, the overwhelming emphasis on mastering routines and algorithms seemed to overshadow in dealing such a non-routine problem. In contrast, T2’s teaching emphasized students’ initiatives and collaboration, although T2 also used formal routine tasks. At the end, students in T2’s class reported positively about their experience, enjoyed working together, and appreciated the opportunities of thinking mathematically. Studies by Schoenfeld ( 1988 ) and Keitel ( 2006 ) indicate how students’ experience in mathematics classes influences their perceptions of mathematics and also imply the importance of learning about teachers’ perceptions of mathematics that likely guide their instructional practice (Cooney 1987 ).

Rattan et al. ( 2012 ) found that teachers with different perceptions of mathematics teach differently. Specifically, Rattan et al. looked at these teachers holding an entity (fixed) theory of mathematics intelligence (G1) versus incremental theory (G2). Through their studies, Rattan and colleagues found that G1 teachers more readily judged students to have low ability, comforted students for low mathematical ability, and used “kind” strategies (e.g., assigning less homework) unlikely to promote their engagement with the field than G2 teachers. Students who received comfort-oriented feedback perceived their teachers’ entity theory and low expectations and reported lowered motivation and expectations for their own performance. The results suggest how teachers’ inadequate perceptions of mathematics and beliefs about the nature of students’ mathematical intelligence contributed to low achievement, diminished self-esteem and viewed mathematics is only a set of static facts and procedures. Further, the results suggest that how mathematics is taught influences more than students’ proficiency with mathematics content in a class. Sun ( 2018 ) made a similar argument after synthesizing existing literature and analyzing classroom observation data.

Based on the 2012 US national survey of science and mathematics education conducted by Horizon Research, Banilower et al. ( 2013 ) reported that a vast majority of mathematics teachers, from 81% at the high school level to 90% at the elementary level, believe that students should be given definitions of new vocabulary at the beginning of instruction on a mathematical idea. Also, many teachers believe that they should explain an idea to students before having them consider evidence for it and that hands-on activities should be used primarily to reinforce ideas students have already learned. The report suggests many teachers emphasized pedagogical practices of “give” and “present,” perhaps influenced by conceptions of mathematics that are more Platonic than Aristotelian, similar to what was reported about teachers’ practices more than two decades ago (Cooney 1987 ).

How to change?

Given that the evidence demonstrates a compelling case for changing how mathematics is taught, we turn our attention to suggesting how to realize this transformation. Changing how mathematics is taught and learned is not a new endeavor for both mathematics educators and mathematicians (e.g., Li, Silver, and Li 2014 ; Schoenfeld in press ). For example, the “Moore Method,” developed and used by Robert Lee Moore (a famous topologist) in the early twentieth century, shifted instruction from teacher-centered lecturing to student-centered mathematical development (Coppin et al. 2009 ). In its purest form, students were presented with mathematical definitions and asked to develop and/or prove theorems from them after class, without reading mathematics books or using other resources. When students returned to the class, they were asked to prove a theorem. As a result, students developed the mathematics themselves, instead of the instructor presenting the proofs and doing the mathematics for students. The method has had its own success in producing many great mathematicians; however, the high-pressure environment also drowned many students who might have been successful otherwise (Schoenfeld in press ).

Although the “Moore Method” was used primarily in advanced mathematics courses at the post-secondary level, it illustrates how a different conception of mathematics led to a different instructional approach in which students developed mathematics. However, it might be the opposite end of a spectrum, in comparison to approaches that present mathematics to students in accommodating and easy-to-digest ways that can be as much easy as possible. Neither extreme is a good option for K-12 students. Again, it becomes important for us to consider options that can support the value of learning mathematics.

Our discussion in the previous section highlights the importance of taking mathematics as a human activity, ensuring it is meaningful to students, and developing students’ mathematical thinking about ideas, rather than simply absorbing a set of static and disconnected knowledge and skills. We call for a shift in teaching mathematics based on Platonic conceptions to approaches based on more of Aristotelian conceptions. In essence, Plato emphasized ideal forms of mathematical objects, perhaps inaccessible through people’s sense making efforts. As a result, learners lack ownership of the ideal forms of mathematical objects, because mathematical objects cannot and should not be created by human reasoning. In contrast, Aristotle emphasized that mathematical objects are developed through logic reasoning and empirical realization. In other words, mathematical objects exist only when they can be sensed and verified by people's efforts. This differs from Plato’s passive perspective, highlights human ownership of mathematical ideas and encourages people to make mathematics make sense, termed as making sense by McCallum ( 2018 ). Aristotelian conceptions view mathematics as objects that learners can actively develop and structure as mathematically meaningful, which is more in line with what research mathematicians do. McCallum ( 2018 ) argued that both sense-making and making-sense stances are needed for a complete view of mathematics and learning, recognizing that not attending to both stances carries risks. “Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.” (McCallum 2018 ).

In addition, there is the issue of personal identity: if students come to avoid mathematics because they are uncomfortable with it (in fact, mathematics anxiety has become a widespread problem for all ages across the globe, see Luttenberger et al. 2018 ) then mathematics instruction has failed them, regardless of test scores.

In the following, we discuss sense-making and making-sense stances first with specific examples from mathematics. Then, we discuss connections to STEM education.

Sense making is much more than the acquisition of knowledge and skills

Sense making has long been emphasized in mathematics education community. William A. Brownell is a well-known, early 20 th century scholar who advocated the value of sense making in the learning of mathematics. For example, Brownell ( 1945 ) discussed how arithmetic can and should be taught and learned not only as procedures, but also as a meaningful system of thinking. He shared many examples like the following division,

Brownell suggested to ask questions: what does the 5 of 576 mean? Why must 57 be the first partial dividend? Do you actually divide 8 into 57, or into 57…’s? etc., instead of simply letting students memorize how to carry out the procedure. What Brownell advocated has been commonly accepted and emphasized in mathematics education nowadays as sense making (e.g., Schoenfeld 1992 ).

There can be different ways of sense making of the same computation. As an example, the sense making process for the above long division can come out with mental math as: I am looking to see how close I can get to 570 with multiples of 80; 7 multiples of 80 gives me 560, which is close. Of course, given base 10 notation, that’s the same as 8 multiples of 70, which is why the 7 goes over the 57. And when I subtract 560, there are 16 left over, so that’s another 2 8 s. Such a sense-making process also works, as finding the answer (quotient, k ) of 576 ÷ 8 is the same operation as to find k that satisfies 576 = k × 8. In mathematics, division and multiplication are alternate but equivalent ways of doing the same operation.

To help students build numerical reasoning and make sense of computations, many teachers use number talks in their classrooms for students to practice and share these mental math and computation strategies (e.g., Parrish 2011 ). In fact, new terms are being created and used in mathematics education about sense making, such as number sense (e.g., Sowder 1992 ) and symbol sense (Arcavi, 1994 ). Some instructional programs, such as Cognitively Guided Instruction (see, e.g., Carpenter et al., 1997 , 1998 ), make sense making the core of instructional activities. We argue that such activities should be more widely adopted.

Making sense makes the other side of mathematical practice visible, and values idea development and ownership

The making-sense stance, as termed by McCallum ( 2018 ), is not commonly practiced as it is pertinent to expert mathematician’s practices. Where sense making (as discussed previously) emphasizes the process of making sense of what is being learned, making sense emphasizes the process of making mathematics make sense. Making sense highlights the importance for students to experience mathematics through creating, designing, developing, and connecting mathematical ideas. As an example, for the above division computation, 8 \( \overline{\Big)576\ } \) , students may wonder why the division procedure is performed from left to right, which is different from the other operations (addition, subtraction, and multiplication) that are all performed from right to left. In fact, students can be encouraged to explore if the division can also be performed from right to left (i.e., starting from the one’s place). They may discover, with possible support from the teacher, that the division can be done in this way. However, once the division is moved to the high-value places, it will require the process to go back down to the low-value places for completion. In other words, the division process starting from the low-value place would require repeated processes of returning to the low-value places; as a result, it is inefficient. As mathematical procedure is designed to improve efficiency, the division procedure is thus set to be carried out from the high-value place to low-value place (i.e., from left to right). Students who work this out experience mathematics more deeply than the sense-making described by Brownell ( 1945 ).

There are plenty of making-sense opportunities in classroom instruction. For example, kindergarten children are often given opportunities to play with manipulatives like cube trains and snub cubes, to explore and learn about patterns, numbers, and measurement through various connections. The recording of such activities typically results in numerical expressions or operations of these connections. In addition, such activities can also serve as a context to encourage students to design and create a way of “recording” these connections directly with a drawing line next to the connected train cubes. Such a design activity will help students to develop the concept of a number line that includes the original/starting point, unit, and direction (i.e., making mathematics make sense), instead of introducing the number line to students as a mathematical concept being “given” years later.

Learning how to provide students with opportunities to develop mathematics may occur with experience. Huang et al. ( 2010 ) found that expert and novice teachers in China both valued students’ mastering of mathematical knowledge and skills and their development in mathematical thinking methods and abilities. However, novice teachers were particularly concerned about the effectiveness of their guidance, in contrast to expert teachers who emphasized the development of students’ mathematical thinking and higher-order thinking abilities and properly dealing with important and difficult content points. The results suggest that teachers’ perceptions and pedagogical practices can change and improve over time. However, it may be worth asking if support for teacher development would accelerate the process.

Connecting changes in mathematics and STEM education

Although it is commonly acknowledged that mathematics is foundational to STEM, mathematics is being related to STEM education at a distance in practice and also in scholarship development (English 2016 , see additional notes at the end of this editorial). Holding the conception of mathematics as products does not support integrating mathematics with other STEM disciplines, as mathematics can be perceived simply as a set of tools for these disciplines. At the same time, mathematics and science have often proceeded along parallel tracks, with mathematics focused on “problem solving” while science has focused on “inquiry.” To better connect mathematics and other disciplines in STEM, we should focus on ideas and thinking development in mathematics (Li et al. 2019a ), unifying instruction from the student perspective (the Teaching for Robust Understanding framework, discussed below).

Emphasizing both sense making and making sense in mathematics education opens opportunities for connections with similar practices in other STEM disciplines. For example, sense making is very much emphasized in science education (Hogan 2019 ; Kapon 2017 ; Odden and Russ 2019 ), often combined with reflections in engineering (Kilgore et al. 2013 ; Turns et al. 2014 ), and also in the context of using technology (e.g., Antonietti and Cantoia 2000 ; Dick and Hollebrands 2011 ). Science is fundamentally about discovery and understanding of the natural world. This notion provides a natural link to mathematical modeling (e.g., Burkhardt 1981 ). Beyond that, in science education, sense making places a heavy focus on the construction and evaluation of explanation (Kapon 2017 ), and can even be defined as a process of constructing an explanation to resolve a perceived gap or conflict in knowledge (Odden and Russ 2019 ). Design and making play vital roles in engineering and technology education (Dym et al., 2005 ), as is student reflection on these experiences (e.g., Turns et al. 2014 ). Indeed, STEM disciplines share the same conceptual process of sense making as learners, individually or in a group, actively engage with the natural or man-made world, explore it, and then develop, test, refine, and use ideas together with specific explanation. If mathematics was conceived as an “empirical” discipline, connections with other STEM disciplines would be strengthened. In philosophical terms, Lakatos ( 1976 ) made similar claims Footnote 1 .

Similar to the emphasis on sense making placed in the Mathematics Curriculum Standards (e.g., NCTM, 1989 , 2000 ), Next Generation Science Standards (NGSS Lead States 2013 ) prompted a shift in science education away from simply knowing science content and procedures to practicing and using science, together with engineering, to make sense of the world and create the future. In a review, Fitzgerald and Palincsar ( 2019 ) concluded sense making is a productive lens for investigating and characterizing great teaching across multiple disciplines.

Mathematics has stronger linkages to creation and design than traditionally imagined. Therefore, its connections to engineering and technology could be much stronger. However, the deep-rooted conception of mathematics as products has traditionally discouraged students and teachers from considering and valuing design and design thinking (Li et al. 2019b ). Conceiving mathematics as making sense should help promote conceptual changes in mathematical practice to value idea generation and design activity. Connections generated from such a shift will support teaching and learning not only in individual STEM disciplines, but also in integrated STEM education.

At the same time, although STEM education as a commonly recognized field does not have a long history (Li 2014 , 2018a ), its rapid development can help introduce ideas for exploring how mathematics can be taught and learned. For example, the concept of projects is common in engineering professional practice, and the project-based learning (PjBL) as an instructional approach is a key component in some engineering programs (e.g., Berger 2016 ; de los Ríos et al. 2010 ; Mills and Treagust 2003 ). de los Ríos et al. ( 2010 ) highlighted three main advantages of PjBL: (1) development in technical, personal, and contextual competences; (2) students’ engagement with real problems from professional contexts; and (3) collaborative learning facilitated through the integration of teaching and research. Such advantages are important for students’ learning of mathematics and are aligned well with efforts to develop 21 st century skills, including problem solving, communication, collaboration, and critical thinking.

Design-based learning (DBL) is another instructional approach commonly used in engineering and technology fields. Gómez Puente et al. ( 2013 ) conducted a sampled review and concluded that DBL projects consist of open-ended, hands-on, authentic, and multidisciplinary design tasks. Teachers using DBL facilitate both the process for students to gain domain-specific knowledge and thinking activities to generate innovative solutions. Such features could be adapted for mathematics education, especially integrated STEM education, in concert with design and design thinking. In addition to a few examples discussed above about making sense in mathematics, there is a growing body of publications developed by and for mathematics teachers with specific examples of investigations, design projects, and instructional activities associated with STEM (Li et al. 2019b ).

A framework for helping students to gain important experiences in and through mathematics, as connected to other disciplines in STEM

For observing and evaluating classroom instruction in general and mathematics classroom instruction in specific, there are several widely used frameworks and rubrics available. However, a trial use of selected frameworks with sampled mathematics classroom instruction episodes suggested their disagreements on what counts as high-quality instruction, especially with aspects on disciplinary thinking being valued and relevant classroom practices (Schoenfeld et al. 2018 ). The results suggest the importance of choice making, when we consider a framework in discussing and evaluating teaching practices.

Our discussion above highlights the importance of shifting away from viewing mathematics simply as a set of static knowledge and skills, to focusing on ideas and thinking development in teaching and learning mathematics. Further discussion of several aspects of changes specifies the needs of developing and using practices associated with sense making, making sense, and connecting mathematics and STEM education for changes.

To support effective mathematics instruction, we propose the use of the Teaching for Robust Understanding (TRU) framework to help characterize powerful learning environments. With the conception of mathematics as “empirical” and a focus on students’ experience, then the focus of instruction should also be changed. We argue that shift is from instruction conceived as “what should the teacher do” to instruction conceived as “what mathematical experiences should students have in order for them to develop into powerful thinkers?” It is the shift in the frame of TRU that makes it so powerful and pertinent for all these proposed changes. Moreover, TRU only uses a small number of actionable dimensions after distilling the literature on teaching for robust or powerful understanding. That makes TRU a practical mechanism for problematizing instruction.

Figure 1 presents the TRU Math framework that identifies five key dimensions along which powerful classroom environments can be characterized: the mathematics; cognitive demand; equitable access; agency, ownership, and identity; and formative assessment. These five dimensions were distilled from an extensive literature review, thus capturing what the literature considers to be essential. They were tested against classroom videotapes and data on student performance, and the results indicated that classrooms that did well on the TRU dimensions produced students who did correspondingly well on tests of mathematical knowledge, thinking, and problem solving (e.g., Schoenfeld 2014 , 2019 ). In brief, the argument regarding the importance of the five dimensions of TRU Math is as follows. First, the quality of the mathematics discussed (dimension 1) is critical. What individual students learn is unlikely to be richer than what they experience in the classroom. Whether individual students’ understanding rises to the level of what is discussed/presented in the classroom depends on other factors, which are captured in the remaining four dimensions. For example, you surely have had the experience, at a lecture, of hearing beautiful content presented, and then not being able to do any of the assigned problems! The remaining four dimensions capture aspects needed to support the development of all students with respect to sense making, making sense, ownership, and feedback loop. Dimension 2: Cognitive Demand. Are students engaged in sense making and making sense? Are they engaged in “productive struggle”? Dimension 3: Equitable Access. Are all students fully engaged with the central content and practices of the domain so that every student can profit from it? Dimension 4: Agency, Ownership, and Identity. Do all students have opportunities to develop idea ownership and mathematical agency? Dimension 5: Formative Assessment. Are students encouraged and supported to share their thinking with a meaningful feedback loop for instructional adjustment and improvement?

figure 1

The TRU Mathematics Framework: The five dimensions of powerful mathematics classrooms

The first key point about TRU is that students learn more in classrooms that are powerful along the five TRU dimensions. Second, the shift of attention from the teacher to the environment is fundamentally important. The key question is not “Is the teacher doing particular things to support learning?”; instead, it is, “Are students experiencing instruction so that it is conducive to their growth as mathematical thinkers and learners?” Third, the framework is not prescriptive; it respects teacher autonomy. There are many ways to be an excellent teacher. The question is, Does the learning environment created by the teacher provide each student rich opportunities along the five dimensions of the framework? Specifically, in describing the dimensions of powerful instruction, the framework serves to problematize instruction. Asking “how am I doing along each dimension; how can I improve?” can lead to richer instruction without prescribing or imposing a particular style or particular norms on teachers.

Extending to STEM education

Now, we suggest the following. If you teach biology, chemistry, physics, engineering, or any other STEM field, replace “mathematics” in Fig. 1 with your discipline. The first dimension is about rich content and practices in your field. And the remaining four dimensions are about necessary aspects of your students’ classroom engagement with the discipline. Practices associated with sense making, making sense, and STEM education are all be reflected in these five dimensions, with central attention on students’ experience in such classroom environments. Although the TRU framework was originally developed for characterizing effective mathematics classroom environments, it has been carefully framed in a way that is applicable to many different disciplines (Schoenfeld 2014 ). Our discussion above already specified why sense making, making sense, and specific instructional approaches like PjBL and DBL are shared across disciplines in STEM education. Thus, the TRU framework is applicable to other STEM disciplines. The natural analogue of the TRU framework for any field is given in Fig. 2 .

figure 2

The domain-general version of the TRU framework

Both the San Francisco Unified School District and the Chicago Public Schools adopted the TRU Math framework and found results within mathematics sufficiently promising that they expanded their efforts to all subject areas for professional development and instruction, using the domain-general TRU framework. Work is still in its early stages. Current efforts might be best conceptualized as a laboratory for exploration rather than a promissory note for improvement across all different disciplines. It will take time to accumulate data to show effectiveness. For further information about the domain-general TRU framework and tools for professional development are available at the TRU framework website, https://truframework.org/

Finally, as a framework, TRU is not a set of specific tools or guidelines, although it can be used to guide their development. To help lead our discussion to something more practical, we can use the framework to check and identify aspects that are typically under-emphasized and move them to center stage in order to improve classroom instruction. Specifically, the following is a list of sample under-emphasized norms and practices that can be identified (Schoenfeld in press ).

Establishing a climate of inquiry, in which mathematics is experienced as a discipline of exploration and sense making.

Developing students’ ownership of ideas through the process of developing, sharing, refining, and using ideas; concepts and language can come later.

Focusing on big ideas, and not losing the forest for the trees.

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

Where to start? Begin by problematizing teaching and the nature of learning environments

Here we start by stipulating that STEM disciplines as practiced, are living, breathing fields of inquiry. Knowledge is important; ideas are important; practices are important. The list given above applied to all STEM disciplines, not just mathematics.

The issue, then, is developing teacher capacity to craft environments that have the properties described immediately above. Here we share some thoughts, and the topic itself can well be discussed extensively in another paper. To make changes in teaching, it should start with assessing and changing teaching practice itself (Hiebert and Morris 2012 ). Opening up teachers’ perceptions of teaching practices should not be done by telling teachers what to do!—the same rules of learning apply to teachers as they apply to students. Learning environments for teachers should offer teachers the same opportunities for rich engagement, challenge, equitable access, and ownership as we hope students will experience (Schoenfeld 2015 ). Working together with teachers to study and reflect on their teaching practices in light of the TRU framework, we can help teachers to find out what their students are experiencing and why changes are needed. The framework can also help guide teachers to learn what changes would be needed, and to try out changes to learn how their students’ learning may differ. It is this iterative and concrete process that can hopefully help shift participating teachers’ perceptions of mathematics. Many tools for problematizing teaching are available at the TRU web site (see https://truframework.org/ ). If teachers can work together with a focus on selected lessons like what teachers often do in China, the process would help form a school-based learning community that can contribute to not only participating teachers’ practice change but also their expertise improvement (Huang et al. 2011 ; Li and Huang 2013 ).

As reported before (Li 2018b ), publications in the International Journal of STEM Education show a mix of individual-disciplinary and multidisciplinary education in STEM over the past several years. Although one journal’s publications are limited in its scope of providing a picture about the scholarship development related to mathematics and STEM education, it can allow us to get a sense of related development.

If taking a closer look at the journal’s publications over the past three years from 2016 to 2018, we found that the number of articles published with a clear focus on mathematics is relatively small: three (out of 21) in 2016, six (out of 34) in 2017, and five (out of 56) in 2018. At the same time, we should point out that these publications from 2016 to 2018 seem to reflect a trend, over these three years, of moving toward issues that can go beyond mathematics itself, as what was noted before (Li 2018b ). Specifically, for these three articles published in 2016, they are all about mathematics education at either elementary school (Ding 2016 ; Zhao et al. 2016 ) or university levels (Schoenfeld et al. 2016 ). Out of the six published in 2017, three are on mathematics education (Hagman et al. 2017 ; Keller et al. 2017 ; Ulrich and Wilkins 2017 ) and the other three on either teacher professional development (Borko et al. 2017 ; Jacobs et al. 2017 ) or connection with engineering (Jehopio and Wesonga 2017 ). For the five published in 2018, two are on mathematics education (Beumann and Wegner 2018 ; Wilkins and Norton 2018 ) and the other three have close association with other disciplines in STEM (Blotnicky et al. 2018 ; Hayward and Laursen 2018 ; Nye et al. 2018 ). This trend likely reflects a growing interest, with close connection to mathematics, in both mathematics education community and a broader STEM education community of developing and sharing multidisciplinary and interdisciplinary scholarship.

Availability of data and materials

Not applicable

Interestingly, Lakatos was advised by both Popper and Pólya—his ideas being in some ways a unification of Pólya’s emphasis on mathematics as an empirical discipline and Popper’s reflections on the nature of the scientific endeavor.

Antonietti, A., & Cantoia, M. (2000). To see a painting versus to walk in a painting: An experiment on sense-making through virtual reality. Computers & Education, 34 , 213–223.

Article   Google Scholar  

Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14 (3), 24–35.

Google Scholar  

Ashlock, R. B. (2010). Error patterns in computation (Tenth Edition) . Boston, MA: Allyn & Bacon.

Banilower, E. R., Smith, P. S., Weiss, I. R., Malzahn, K. A., Campbell, K. M., et al. (2013). Report of the 2012 national survey of science and mathematics education. Horizon Research, Chapel Hill, NC. Retrieved from http://www.nnstoy.org/download/stem/2012%20NSSME%20Full%20Report.pdf

Berger, C. (2016). Engineering is perfect for K-5 project-based learning. Engineering is Elementary (EiE) Blog, https://blog.eie.org/engineering-is-perfect-for-k-5-project-based-learning

Beumann, S. & Wegner, S.-A. (2018). An outlook on self-assessment of homework assignments in higher mathematics education. International Journal of STEM Education, 5 :55. https://doi.org/10.1186/s40594-018-0146-z

Blotnicky, K. A., Franz-Odendaal, T., French, F., & Joy, P. (2018). A study of the correlation between STEM career knowledge, mathematics self-efficacy, career interests, and career activities on the likelihood of pursuing a STEM career among middle school students. International Journal of STEM Education, 5 :22. https://doi.org/10.1186/s40594-018-0118-3

Borko, H., Carlson, J., Mangram, C., Anderson, R., Fong, A., Million, S., Mozenter, S., & Villa, A. M. (2017). The role of video-based discussion in model for preparing professional development leaders. International Journal of STEM Education, 4 :29. https://doi.org/10.1186/s40594-017-0090-3

Boyer, C. B. (1991). A history of mathematics (2nd ed.) . New York: Wiley.

Brownell, W. A. (1945). When is arithmetic meaningful? The Journal of Educational Research, 38 (7), 481–498.

Burkhardt, H. (1981). The real world and mathematics . Glasgow: Blackie, reissued Nottingham: Shell Centre Publications.

Carpenter, T., Fennema, E., & Franke, M. (1997). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97 , 3–20.

Carpenter, T., Franke, M., Jacobs, V. R., & Fennema, E. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29 (1), 3–20.

Committee on STEM Education, National Science & Technology Council, the White House (2018). Charting a course for success: America’s strategy for STEM education . Washington, DC. https://www.whitehouse.gov/wp-content/uploads/2018/12/STEM-Education-Strategic-Plan-2018.pdf Retrieved on 18 January, 2019.

Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics . Retrieved from http://www.corestandards.org/Math/Practice

Cooney, T (1987). The issue of reform: What have we learned from yesteryear? In Mathematical Sciences Education Board, The teacher of mathematics: Issues for today and tomorrow (pp. 17-35). Washington, DC: National Academy Press.

Coppin, C. A., Mahavier, W. T., May, E. L., & Parker, E. (2009). The Moore Method . Washington, DC: Mathematical Association of America.

Davis, P., & Hersh, R. (1980). The mathematical experience . Boston: Birkhauser.

de los Ríos, I., Cazorla, A., Díaz-Puente, J. M., & Yagüe, J. L. (2010). Project–based learning in engineering higher education: Two decades of teaching competences in real environments. Procedia Social and Behavioral Sciences, 2 , 1368–1378.

Devlin, K. (2000). The four faces of mathematics. In M. J. Burke & F. R. Curcio (Eds.), Learning Mathematics for a New Century: 2000 Yearbook of the National Council of Teachers of Mathematics (pp. 16–27). Reston, VA: NCTM.

Devlin, K. (2012). Introduction to mathematical thinking. Stanford, CA: The author.

Dick, T. P., & Hollebrands, K. F. (2011). Focus on high school mathematics: Technology to support reasoning and sense making . Reston, VA: NCTM.

Ding, M. (2016). Developing preservice elementary teachers’ specialized content knowledge: The case of associative property. International Journal of STEM Education, 3 , 9 https://doi.org/10.1186/s40594-016-0041-4 .

Dossey, J. A. (1992). The nature of mathematics: Its role and its influence. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 39–48). New York: MacMillan.

Dossey, J. A., McCrone, S. S., & Halvorsen, K. T. (2016). Mathematics education in the United States 2016: A capsule summary fact book . Reston, VA: The National Council of Teachers of Mathematics.

Dym, C. L., Agogino, A. M., Eris, O., Frey, D. D., & Leifer, L. J. (2005). Engineering design thinking, teaching, and learning. Journal of Engineering Education, 94 (1), 103–120.

English, L. D. (2016). STEM education K-12: Perspectives on integration. International Journal of STEM Education, 3:3, https://doi.org/10.1186/s40594-016-0036-1

Fisher, C. (1990). The research agenda project as prologue. Journal for Research in Mathematics Education, 21 , 81–89.

Fitzgerald, M. S., & Palincsar, A. S. (2019). Teaching practices that support student sensemaking across grades and disciplines: A conceptual review. Review of Research in Education, 43 , 227–248.

Fritz, A., Haase, V. G., & Rasanen, P. (Eds.). (2019). International handbook of mathematical learning difficulties . Cham, Switzerland: Springer.

Gee, J. P. (2005). What would a state of the art instructional video game look like? Innovate: Journal of Online Education, 1 (6) Retrieved from https://nsuworks.nova.edu/innovate/vol1/iss6/1 .

Gómez Puente, S. M., van Eijck, M., & Jochems, W. (2013). A sampled literature review of design-based learning approaches: A search for key characteristics. International Journal of Technology and Design Education . https://doi.org/10.1007/s10798-012-9212-x .

Hagman, J. E., Johnson, E., & Fosdick, B. K. (2017). Factors contributing to students and instructors experiencing a lack of time in college calculus. International Journal of STEM Education, 4 , 12 https://doi.org/10.1186/s40594-017-0070-7 .

Hayward, C. N. & Laursen, S. L. (2018). Supporting instructional change in mathematics: Using social network analysis to understand online support processes following professional development workshops. International Journal of STEM Education, 5 :28. https://doi.org/10.1186/s40594-018-0120-9

Hersh, R. (1986). Some proposals for reviving the philosophy of mathematics . In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 9–28). Boston: Birkhauser.

Hiebert, J., & Morris, A. K. (2012). Teaching, rather than teachers, as a path toward improving classroom instruction. Journal of Teacher Education, 63 (2), 92–102.

Hogan, M. (2019). Sense-making is the core of NGSS. In Illuminate education blog, https://www.illuminateed.com/blog/2019/03/sense-making-is-the-core-of-ngss/ Accessed 15 Oct 2019.

Huang, R., Li, Y., & He, X. (2010). What constitutes effective mathematics instruction: A comparison of Chinese expert and novice teachers’ views. Canadian Journal of Science, Mathematics and Technology Education, 10 (4), 293-306. https://doi.org/10.1080/14926156.2010.524965

Huang, R., Li, Y., Zhang, J., & Li, X. (2011). Improving teachers’ expertise in mathematics instruction through exemplary lesson development. ZDM – The International Journal on Mathematics Education, 43 (6-7), 805–817.

Jacobs, J., Seago, N., & Koellner, K. (2017). Preparing facilitators to use and adapt mathematics professional development materials productively. International Journal of STEM Education, 4 , 30 https://doi.org/10.1186/s40594-017-0089-9 .

Jehopio, P. J., & Wesonga, R. (2017). Polytechnic engineering mathematics: assessing its relevance to the productivity of industries in Uganda. International Journal of STEM Education, 4 , 16 https://doi.org/10.1186/s40594-017-0078-z .

Kapon, S. (2017). Unpacking sensemaking. Science Education, 101 (1), 165–198.

Keitel, C. (2006). ‘Setting a task’ in German schools: Different frames for different ambitions. In D. Clarke, C. Keitel, & Y. Shimizu (Eds.), Mathematics classrooms in 12 countries: The insiders’ perspective (pp. 37–58). Rotterdam Netherlands: Sense Publishers.

Keller, R. E., Johnson, E., & DeShong, S. (2017). A structural equation model looking at student’s participatory behavior and their success in Calculus I. International Journal of STEM Education, 4 , 24 https://doi.org/10.1186/s40594-017-0093-0 .

Kilgore, D., Sattler, B., & Turns, J. (2013). From fragmentation to continuity: Engineering students making sense of experience through the development of a professional portfolio. Studies in Higher Education, 38 (6), 807–826.

Kline, M. (1973). Why Johnny can’t add: The failure of new math . New York: St. Martin’s.

Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery . Cambridge, England: Cambridge University Press.

Book   Google Scholar  

Leung, F. K. S., & Li, Y. (Eds.). (2010). Reforms and issues in school mathematics in East Asia – Sharing and understanding mathematics education policies and practices . Rotterdam, Netherlands: Sense Publishers.

Li, Y. (2014). International Journal of STEM Education – A platform to promote STEM education and research worldwide. International Journal of STEM Education, 1 , 1 https://doi.org/10.1186/2196-7822-1-1 .

Li, Y. (2018a). Journal for STEM Education Research – Promoting the development of interdisciplinary research in STEM education. Journal for STEM Education Research, 1 (1-2), 1–6 https://doi.org/10.1007/s41979-018-0009-z .

Li, Y. (2018b). Four years of development as a gathering place for international researcher and readers in STEM education. International Journal of STEM Education, 5 , 54 https://doi.org/10.1186/s40594-018-0153-0 .

Li, Y., & Huang, R. (Eds.). (2013). How Chinese teach mathematics and improve teaching . New York: Routledge.

Li, Y., & Lappan, G. (Eds.). (2014). Mathematics curriculum in school education . Dordrecht: Springer.

Li, Y., Schoenfeld, A. H., diSessa, A. A., Grasser, A. C., Benson, L. C., English, L. D., & Duschl, R. A. (2019a). On thinking and STEM education. Journal for STEM Education Research, 2 (1), 1–13. https://doi.org/10.1007/s41979-019-00014-x .

Li, Y., Schoenfeld, A. H., diSessa, A. A., Grasser, A. C., Benson, L. C., English, L. D., & Duschl, R. A. (2019b). Design and design thinking in STEM education. Journal for STEM Education Research, 2 (2), 93-104. https://doi.org/10.1007/s41979-019-00020-z .

Li, Y., Silver, E. A., & Li, S. (Eds.). (2014). Transforming mathematics instruction: Multiple approaches and practices . Cham, Switzerland: Springer.

Luttenberger, S., Wimmer, S., & Paechter, M. (2018). Spotlight on math anxiety. Psychology Research and Behavior Management, 11 , 311–322.

McCallum, W. (2018). Sense-making and making sense. https://blogs.ams.org/matheducation/2018/12/05/sense-making-and-making-sense/ Retrieved on October 1, 2019.

Mills, J. E. & Treagust, D. F. (2003). Engineering education – Is problem-based or project-based learning the answer? Australasian Journal of Engineering Education , https://www.researchgate.net/profile/Nathan_Scott2/publication/238670687_AUSTRALASIAN_JOURNAL_OF_ENGINEERING_EDUCATION_Co-Editors/links/0deec53a08c7553c37000000.pdf Retrieved on October 15, 2019.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics . Reston, VA: NCTM.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics . Reston, VA: NCTM.

NGSS Lead States. (2013). Next generation science standards: For states, by states . Washington, DC: National Academies Press.

Nye, B., Pavlik Jr., P. I., Windsor, A., Olney, A. M., Hajeer, M., & Hu, X. (2018). SKOPE-IT (Shareable Knowledge Objects as Portable Intelligent Tutors): Overlaying natural language tutoring on an adaptive learning system for mathematics. International Journal of STEM Education, 5 , 12 https://doi.org/10.1186/s40594-018-0109-4 .

Odden, T. O. B., & Russ, R. S. (2019). Defining sensemaking: Bringing clarity to a fragmented theoretical construct. Science Education, 103 , 187–205.

Parrish, S. D. (2011). Number talks build numberical reasoning. Teaching Children Mathematics, 18 (3), 198–206.

Rattan, A., Good, C., & Dweck, C. S. (2012). “It’s ok – Not everyone can be good at math”: Instructors with an entity theory comfort (and demotivate) students. Journal of Experimental Social Psychology . https://doi.org/10.1016/j.jesp.2011.12.012 .

Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23 (2), 145–166.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334–370). New York: MacMillan.

Schoenfeld, A. H. (2001). Mathematics education in the 20th century. In L. Corno (Ed.), Education across a century: The centennial volume (100th Yearbook of the National Society for the Study of Education) (pp. 239–278). Chicago, IL: National Society for the Study of Education.

Schoenfeld, A. H. (2014). What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice, productively interwined. Educational Researcher, 43 (8), 404–412. https://doi.org/10.3102/0013189X1455 .

Schoenfeld, A. H. (2015). Thoughts on scale. ZDM, the International Journal of Mathematics Education, 47 , 161–169. https://doi.org/10.1007/s11858-014-0662-3 .

Schoenfeld, A. H. (2019). Reframing teacher knowledge: A research and development agenda. ZDM – The International Journal on Mathematics Education . https://doi.org/10.1007/s11858-019-01057-5

Schoenfeld, A. H. (in press). Mathematical practices, in theory and practice. ZDM – The International Journal on Mathematics Education .

Schoenfeld, A. H., Floden, R., El Chidiac, F., Gillingham, D., Fink, H., Hu, S., Sayavedra, A., Weltman, A., & Zarkh, A. (2018). On classroom observations. Journal for STEM Educ Res, 1 (1-2), 34–59 https://doi.org/10.1007/s41979-018-0001-7 .

Schoenfeld, A. H., Thomas, M., & Barton, B. (2016). On understanding and improving the teaching of university mathematics. International Journal of STEM Education, 3 , 4 https://doi.org/10.1186/s40594-016-0038-z .

Smith, J., diSessa, A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3 (2), 115–163.

Sowder, J. (1992). Estimation and number sense. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 371–389). New York: MacMillan.

Stanic, G. M. A., & Kilpatrick, J. (1992). Mathematics curriculum reform in the United States: A historical perspective. International Journal of Educational Research, 17 (5), 407–417.

Sun, K. L. (2018). The role of mathematics teaching in fostering student growth mindset. Journal for Research in Mathematics Education, 49 (3), 330–355.

Turns, J. A., Sattler, B., Yasuhara, K., Borgford-Parnell, J. L., & Atman, C. J. (2014). Integrating reflection into engineering education. Proceedings of 2014 American Society of Engineering Education Annual Conference , Paper ID #9230.

Tymoczko, T. (1986). New directions in the philosophy of mathematics . Boston: Birkhauser.

Ulrich, C., & Wilkins, J. L. M. (2017). Using written work to investigate stages in sixth-grade students’ construction and coordination of units. International Journal of STEM Education, 4 , 23 https://doi.org/10.1186/s40594-017-0085-0 .

Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5 , 27 https://doi.org/10.1186/s40594-018-0119-2 .

Zhao, X., Van den Heuvel-Panhuizen, M., & Veldhuis, M. (2016). Teachers’ use of classroom assessment techniques in primary mathematics education – An explorative study with six Chinese teachers. International Journal of STEM Education, 3 , 19 https://doi.org/10.1186/s40594-016-0051-2 .

Download references

Acknowledgments

Many thanks to Jeffrey E. Froyd for his careful review and detailed comments on an earlier version of this editorial that help improve its readability and clarity. Thanks also go to Marius Jung for his valuable feedback.

Author information

Authors and affiliations.

Texas A&M University, College Station, TX, 77843-4232, USA

University of California at Berkeley, Berkeley, CA, USA

Alan H. Schoenfeld

You can also search for this author in PubMed   Google Scholar

Contributions

Both authors contributed ideas to conceptualize this article. YL took the lead in developing and drafting the article, and AHS reviewed drafts and contributed to revisions. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to Yeping Li .

Ethics declarations

Competing interests.

The authors declare that they have no competing interests.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Cite this article.

Li, Y., Schoenfeld, A.H. Problematizing teaching and learning mathematics as “given” in STEM education. IJ STEM Ed 6 , 44 (2019). https://doi.org/10.1186/s40594-019-0197-9

Download citation

Received : 15 November 2019

Accepted : 19 November 2019

Published : 19 December 2019

DOI : https://doi.org/10.1186/s40594-019-0197-9

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 in school 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 in school 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 in school 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

More from EdSurge

How Trauma Impacts the Well-Being of Black Women Educators

Research Commentary

How trauma impacts the well-being of black women educators, by sarah wright.

Research to Impact: Four Steps to Build a Successful Edtech Enterprise

Research to Impact: Four Steps to Build a Successful Edtech Enterprise

By john gamba.

Boys Aren’t Excelling in Schools. Would More Male Role Models in Early Learning Help?

Diversity and Equity

Boys aren’t excelling in schools. would more male role models in early learning help, by daniel mollenkamp.

What If Myths, Metaphors and Riddles Are the Key to Reshaping K-12 Education?

EdSurge Podcast

What if myths, metaphors and riddles are the key to reshaping k-12 education, by jeffrey r. young.

Journalism that ignites your curiosity about education.

EdSurge is an editorially independent project of and

  • Product Index
  • Write for us
  • Advertising

FOLLOW EDSURGE

© 2024 All Rights Reserved

problem solving in school mathematics

  • Business & Money
  • Management & Leadership

Amazon prime logo

Enjoy fast, free delivery, exclusive deals, and award-winning movies & TV shows with Prime Try Prime and start saving today with fast, free delivery

Amazon Prime includes:

Fast, FREE Delivery is available to Prime members. To join, select "Try Amazon Prime and start saving today with Fast, FREE Delivery" below the Add to Cart button.

  • Cardmembers earn 5% Back at Amazon.com with a Prime Credit Card.
  • Unlimited Free Two-Day Delivery
  • Streaming of thousands of movies and TV shows with limited ads on Prime Video.
  • A Kindle book to borrow for free each month - with no due dates
  • Listen to over 2 million songs and hundreds of playlists
  • Unlimited photo storage with anywhere access

Important:  Your credit card will NOT be charged when you start your free trial or if you cancel during the trial period. If you're happy with Amazon Prime, do nothing. At the end of the free trial, your membership will automatically upgrade to a monthly membership.

Buy new: $44.50 $44.50 FREE delivery: Friday, Feb 23 Ships from: Amazon Sold by: RGSellers

  • Free returns are available for the shipping address you chose. You can return the item for any reason in new and unused condition: no shipping charges
  • Learn more about free returns.
  • Go to your orders and start the return
  • Select the return method

Buy used: $27.97

Fulfillment by Amazon (FBA) is a service we offer sellers that lets them store their products in Amazon's fulfillment centers, and we directly pack, ship, and provide customer service for these products. Something we hope you'll especially enjoy: FBA items qualify for FREE Shipping and Amazon Prime.

If you're a seller, Fulfillment by Amazon can help you grow your business. Learn more about the program.

Other Sellers on Amazon

Kindle app logo image

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera - scan the code below and download the Kindle app.

QR code to download the Kindle App

Image Unavailable

Creative Problem Solving in School Mathematics

  • To view this video download Flash Player

problem solving in school mathematics

Follow the author

George Lenchner

Creative Problem Solving in School Mathematics 2nd Edition

Purchase options and add-ons.

  • ISBN-10 1882144104
  • ISBN-13 978-1882144105
  • Edition 2nd
  • Publisher Mathematical Olympiads
  • Publication date January 1, 2005
  • Language English
  • Print length 284 pages
  • See all details

problem solving in school mathematics

Frequently bought together

Creative Problem Solving in School Mathematics

Similar items that may ship from close to you

Math Olympiad Contest Problems for Elementary and Middle Schools, Vol. 1

Product details

  • Publisher ‏ : ‎ Mathematical Olympiads; 2nd edition (January 1, 2005)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 284 pages
  • ISBN-10 ‏ : ‎ 1882144104
  • ISBN-13 ‏ : ‎ 978-1882144105
  • Item Weight ‏ : ‎ 1.65 pounds
  • #347 in Decision-Making & Problem Solving
  • #1,562 in Unknown

Important information

To report an issue with this product or seller, click here .

About the author

problem solving in school mathematics

George Lenchner

Discover more of the author’s books, see similar authors, read author blogs and more

Customer reviews

Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them.

To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness.

Reviews with images

Customer Image

  • Sort reviews by Top reviews Most recent Top reviews

Top reviews from the United States

There was a problem filtering reviews right now. please try again later..

problem solving in school mathematics

  • Amazon Newsletter
  • About Amazon
  • Accessibility
  • Sustainability
  • Press Center
  • Investor Relations
  • Amazon Devices
  • Amazon Science
  • Start Selling with Amazon
  • Sell apps on Amazon
  • Supply to Amazon
  • Protect & Build Your Brand
  • Become an Affiliate
  • Become a Delivery Driver
  • Start a Package Delivery Business
  • Advertise Your Products
  • Self-Publish with Us
  • Host an Amazon Hub
  • › See More Ways to Make Money
  • Amazon Visa
  • Amazon Store Card
  • Amazon Secured Card
  • Amazon Business Card
  • Shop with Points
  • Credit Card Marketplace
  • Reload Your Balance
  • Amazon Currency Converter
  • Your Account
  • Your Orders
  • Shipping Rates & Policies
  • Amazon Prime
  • Returns & Replacements
  • Manage Your Content and Devices
  • Recalls and Product Safety Alerts
  • Conditions of Use
  • Privacy Notice
  • Your Ads Privacy Choices

U.S. flag

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

  • Publications
  • Account settings
  • Advanced Search
  • Journal List
  • Front Psychol

Middle school students’ mathematical problem-solving ability and the influencing factors in mainland China

1 School of Education, Shanghai International Studies University, Shanghai, China

2 Faculty of Education, Beijing Normal University, Beijing, China

Associated Data

The datasets presented in this article are not readily available because in view of the data confidentiality agreement, the data used in this study is only shared by members of the project team and cooperative institutes. Requests to access the datasets should be directed to moc.621@aixuhciq .

This study investigated the mathematical problem-solving ability of 42,644 ninth-grade students who participated in regional education quality health monitoring from Z province in East China and the factors which influence their performance of mathematical problem-solving. The results are as follows: (1) ~96% of the students’ mathematics problem-solving ability meets the basic academic requirements of the mathematics curriculum standards; (2) boys and children without siblings performed better, and urban students performed significantly better than county and rural students; (3) ~28% of students’ mathematical problem-solving performance came from inter-school variability; urban and rural backgrounds had a greater impact on mathematical problem-solving than did teaching factors, while teaching self-efficacy had the least impact among the school-level influencing factors. In contrast, the influence of individual non-intelligence factors was higher than that of student background variables, including a greater positive effect of self-efficacy and a greater negative effect of mathematics anxiety.

Introduction

Problems inspire the search for knowledge and learning. As such, Zhang (2012) suggests that personal learning and knowledge acquisition are pursued to solve practical difficulties. Thus, the purpose of mathematics learning is to solve various problems in the mathematical context ( Ma, 2009 ). The role of science is not only to explain the different phenomena in the world, but also to solve real-world problems. Thus, problems drive scientific development. Historically, mathematical science developed from two cultural traditions and two models. Culturally, mathematics is derived from Western abstract deductive mathematics represented by ancient Greek mathematics and algorithmic applied mathematics represented by ancient Chinese mathematics ( Liu, 2005 ). The confluence of these two traditions, neither of which can be solved without mathematical constructs, formed modern mathematics. Polya (1944) argued that one of the main purposes of mathematics education is to develop students’ problem-solving ability and teach students how to think. The indispensable role of the ability to solve problems using mathematics in the process of mathematical exploration, discovery, and innovation has gradually attracted widespread global attention. Mathematical problem-solving ability has been introduced into global curriculum reforms ( National Council of Teachers of Mathematics (NCTM), 1980 ; Ministry of Education of the People’s Republic of China, 2012 ; Wang, 2021 ) and international evaluations ( OECD, 2013 ). In addition, scholars from the East and West have focused on the important factors that influence students’ performance of mathematical problem-solving. They can be broadly summarized as internal factors of the individual learner (e.g., cognitive resources, meta-cognition and non-intellectual factors), external factors (e.g., complexity, familiarity, type, context of the problem), and teaching factors ( Schoenfeld, 1985 ; Mayer, 1992 ). But generally speaking, at present, the academic community have not paid enough attention to the non-intellectual factors and teaching methods ( Wang, 2000 ). Moreover, comparisons reveal that American mathematics education promotes the development of students’ mathematical literacy or other core abilities, wherein the problem-solving process focuses on the application of mathematics knowledge and skills. In contrast, Chinese mathematics education has long advocated double-base teaching , which promotes a process of mathematical problem-solving that focuses on the acquisition of basic knowledge and skills instead of reasoning activities ( Peng et al., 2017 ). While numerous studies suggest that Chinese and East Asian students’ overall math problem-solving skills surpass those of Western students, such as those in the United States, various studies indicate that no significant gap exists between the two in solving complex mathematical problems ( Zhao and Shen, 2003 ). In fact, the higher mathematics achievement of middle school students in mainland China is inextricably linked with the learning process of mathematical problem solving. In particular, China’s compulsory education middle school mathematics curriculum standard also emphasizes that students should cultivate mathematical affections in mathematics learning and actively exert the important promotion of non-intelligence factors ( Ministry of Education of the People’s Republic of China, 2012 ). Thus, under the advocacy of domestic double-base teaching , how do Chinese students develop their mathematical problem-solving ability? Which factors have a greater impact on it? These questions still urgently require an intensive investigation of the overall mathematical problem-solving process of mainland Chinese students, especially to determine the key internal and external factors that influence their mathematical problem-solving performance.

Literature review

Significance and value as goals of mathematics teaching.

The advantage of improving problem-solving ability as a goal of mathematics teaching has long been recognized. Since the 1980s, most countries have regarded improving students’ problem-solving ability as one of the primary goals of mathematics teaching ( Silver and Kilpatrick, 1988 ; Kilpatrick, 2009 ). For instance, in 1980, the National Council of Teachers of Mathematics (NCTM) proposed establishing problem-solving as the core of the mathematics curriculum, thereby introducing a primary goal of American mathematics education [ National Council of Teachers of Mathematics (NCTM), 1980 ]. In 1982, the United Kingdom stated that the core of mathematics education is to cultivate the ability to solve mathematical problems, emphasizing that mathematics is meaningful only when it is applied to various situations [ Department for Education and Science (DES), 1982 ]. Since then, many countries have addressed this issue. In 1989, Japan formally integrated the content of Subject-Studying , based on a mathematics class featuring problem-solving, in its newly revised Curriculum Guidelines ( Fang et al., 1993 ). In 1990, Singapore’s mathematics syllabus listed the development of students’ mathematical problem-solving ability as the basic goal of the mathematics curriculum and, for the first time, proposed the pentagonal model of the mathematics curriculum framework, with mathematical problem-solving positioned as its core ( Fan and Zhu, 2003 ). Currently, most countries regard improving students’ problem-solving ability as an important goal of mathematics education, and problem-solving has become a popular topic in international mathematics curriculum and teaching research ( Stacey, 2005 ; Manfreda, 2021 ).

In contrast, the People’s Republic of China (1949–1957) was influenced by the educational climate of the time and adopted the Soviet mathematics teaching model, which emphasized abstraction, rigor, and application. It was not until the late 1970s that elementary and secondary mathematics syllabi noted that students should learn to apply mathematics knowledge to solve real-world problems. In modern China, the Mathematics Curriculum Standards for Compulsory Education (2011) consider problem-solving as the basic goal of school mathematics ( Ministry of Education of the People’s Republic of China, 2012 ), including the later Ordinary High School Mathematics Curriculum Standards (2002) and General High School Mathematics Curriculum Standards (2017). These curriculum standards emphasize learning to discover and pose problems from the perspective of mathematics, apply mathematics knowledge to solve practical problems, enhance application awareness, and improve practical ability ( Ministry of Education of the People’s Republic of China, 2003 , 2020 ). As such, although China’s research on problem-solving began relatively late, it has developed rapidly and is generally valued by the domestic mathematics education community.

Mathematical problems and problem-solving

American mathematician Halmos (1995) argues that the fundamental element of mathematics is the problem and answer, and the problem is the heart of mathematics. Thus, scholars from various countries have investigated what constitutes the problem. Polya (1965) states that a mathematical problem means to drive learners to find appropriate actions to achieve a visible—but not immediately accessible—goal. Similarly, several Japanese scholars believe that problem situations refer to those that do not yet have a direct solution, thus resulting in a cognitive challenge situation ( Chen, 2007 ). Moreover, according to a renowned Chinese mathematics educator, a mathematics problem is a situation that a person wishes to comprehend, but for which standard solutions cannot be applied ( Zhang, 1991 ). Therefore, mathematical problems refer to problems that learners can only solve through active exploration and thinking using existing mathematical concepts, theories, or methods.

However, consensus has also not yet been achieved regarding the concept of mathematical problem-solving. Perspectives can generally be classified into five categories: (1) mathematical problem-solving refers to facing new situations and issues in daily life and social practice that contradict subjective and objective needs and have no ready-made countermeasures, requiring psychological activity to seek solutions to problems that occur ( Shao, 1983 ; Zhao, 2007 ); (2) mathematical problem-solving is considered to be the process of applying previously learned knowledge to new and unfamiliar situations ( Tan, 2004 ); (3) mathematical problem-solving, as an important part of curriculum theory, is a type of teaching ( Dai, 2012 ); (4) problem-solving is perceived as the purpose of mathematics teaching ( Department for Education and Science (DES), 1982 ; Pasani, 2018 ); and (5) mathematical problem-solving is defined as the ability to apply mathematics to various situations ( Mayer, 1992 ; Stacey, 2005 ). Despite the apparent inconsistency in the formation of problem-solving, the preceding explanations emphasize that mathematical problem-solving is not only an essential skill for all students, but also a process in which they use a variety of intellectual activities to find solutions to problems. In addition, it requires teachers to provide students with an environment and opportunities for discovery and innovation in the classroom. Furthermore, for students, mathematical problem-solving refers to the comprehensive and creative application of mathematical knowledge and methods to solve problems that are not pure exercises, including practical problems and problems derived from mathematics.

Psychological analysis of the process of mathematical problem-solving

Mathematical problem-solving is not only the core of mathematics education but also an important part of mathematics learning psychology. Therefore, research on the psychological mechanism of problem-solving is intriguing. However, various psychological theories maintain different interpretations of problem-solving, and there is no comprehensive view to date. The previous behaviorist theory considered problem-solving to be trial and error, while the Gestalt theory considers it to involve insight ( Kilpatrick, 1978 ; Lumbelli, 2018 ). Actually, in the process of problem-solving, trial and error and insight are not mutually exclusive and often occur alternately. In addition, depending on its nature, a problem can be solved through trial and error or by relying on insight. Moreover, these behaviors are not entirely random but are organized behaviors that gradually search for information, establish connections between information, and adopt certain strategies. Cognitive psychology, a prevalent approach in Western psychology ( Neisser, 1967 ), has largely promoted the theory of mathematics education. The information processing theory developed from cognitive psychology states that problem-solving is a process of finding, receiving, and processing information ( Newell and Simon, 1972 ; Chien et al., 2016 ).

Based on psychological analyses of the process of solving mathematical problems, more researchers began to focus on the steps and procedures of problem-solving, especially observing the process of solving complex mathematical problems ( Duncker, 1945 ; Hunt, 1968 ). The theory of information processing gradually aroused people’s interest in the role of heuristic methods in the problem-solving process. The most influential was Polya’s (1957) four-stage problem-solving process: understanding the problem, devising a plan, implementing the plan, and reviewing and testing. In addition, Mayer et al. (1991) also categorize the problem-solving process into three stages: paraphrasing, integration, and planning. In recent years, an increasing number of related studies on problem-solving steps and procedures, such as heuristic training ( Wang, 2020 ), discovery learning ( Hulukati et al., 2018 ), and other teaching procedures ( Goulet-Lyle et al., 2019 ) have been applied to the teaching field.

Influencing factors of mathematical problem-solving

Factors that affect the solution of mathematical problems are elements that impact the problem-solving process. As problem-solving is a complex psychological process, it requires students to process the conditions, reorganize known concepts and theorems from the understanding of the basic relationship and characteristics of the problem, adjust the relationship between the basic elements in the problem, and explore and guess problem-solving strategies and methods. Based on the extant literature, many factors—such as knowledge, experience, motivation, confidence, thinking ability, and meta-cognition ( Wang, 2017 )—influence mathematical problem-solving. These factors can be classified into three categories: (1) the learner’s individual internal factors, such as personal experience (personal characteristics of the problem solver), cognitive factors (intuition, imagination, abstraction, generalization, reasoning, analysis, and synthesis), meta-cognition, and non-intellectual factors, such as care, desire, motivation, interest, will, and belief ( Ye and Zhang, 2004 ; Tan, 2009 ); (2) external factors related to the mathematical problem, such as complexity, familiarity, type, and context of the problem ( OECD, 2013 ); and (3) teachers’ problem-solving teaching, such as teaching self-efficacy of problem-solving and teaching methods for problem-solving ( Schoenfeld, 1985 ).

Evaluation of mathematical problem solving ability at home and abroad

Although many scholars have conducted in-depth research on the steps, procedures, and open-ended questions of mathematical problem-solving, no unified and clear framework and standard for evaluating the ability of mathematical problem-solving exists. For instance, Mayer et al. (1991) designed 18 arithmetic problems using their original problem-solving procedures based on their previous psychological analysis of the mathematical problem-solving process, and the problems were used to compare the performance of English and Japanese fifth-grade students in mathematical problem-solving. However, mathematical problem-solving is not a single component, but an ability that involves simple calculations and reading comprehension as well as extensive reasoning skills ( Kilpatrick, 1978 ). Various Chinese scholars believe that junior high school students’ mathematical problem-solving abilities involve the four major ability elements of reading comprehension, mathematical modeling, problem-solving expression, and evaluation reflection ( Bai, 2011 ). Thus, the mathematical problem-solving evaluation tools developed by scholars have gradually transitioned from simple to complex, and the problem form has changed from closed to open. As such, early mathematical problem-solving tests usually focused on the preparation of traditional arithmetic problems ( Stinger et al., 1990 ). Later, various researchers began to design high-level cognitive diagnostic tools, such as the QUSAR Cognitive Assessment Instrument (QCAI), which highlights the important role of open-ended questions in mathematical problem-solving ( Lane, 1993 ). On this basis, various studies have applied these open-ended problems related to cognitive diagnostic tools to specific problem-solving evaluations. Cai (1995) used the QCAI as a test tool in a comparative study on the mathematical problem-solving ability of sixth-grade students in China and the United States. Ding et al. (2009) also used the QUSAR QCAI in their study of the relationship between the elementary school mathematics classroom environment and students’ problem-solving ability; they concluded that the dimensions of “happy” and “knowledge-related” in the classroom environment scale had a significant positive predictive effect on students’ problem-solving ability and traditional test scores. All in all, few studies have examined the measurement and evaluation of mathematical problem-solving processes or comprehensively considered the relevant influencing factors of the mathematical problem-solving process. Evaluation design concepts are only incorporated in some representative mathematics curriculum standards and the evaluation framework of international comparison projects ( Xu and Qi, 2018 ).

Analysis framework

On the whole, compared with foreign research on mathematical problem-solving, Chinese mathematics education pays special attention to the learning of mathematical problem-solving strategies and skills, such as in-depth analysis of external factors like the form, background and other elements of mathematical problems, but little attention is paid to the analysis of students’ internal cognitive process of mathematical problem solving. On the other hand, although the domestic mathematics curriculum standards for middle schools also emphasize the role of non-intellectual factors such as mathematical affections in promoting learning, their attention is still obviously insufficient in the actual evaluation ( Wang, 2000 ). In fact, research suggests that personal internal psychological factors, such as motivation, learning interest, and self-efficacy, are more significant in mathematical problem solving performance ( Sun et al., 2016 ). Moreover, the impact of teaching factors on students’ mathematical problem-solving performance also cannot be ignored ( Schoenfeld, 1985 ). Therefore, to systematically evaluate the mathematical problem-solving ability, in addition to considering examining the structural elements of mathematical problem solving, the role of internal non-intellectual factors and teaching variables in the process of problem-solving must be valued.

In addition, the empirical investigation on the influencing factors of mathematical problem solving in the existing research is more just for the perspective of students or only considering the intervention of the teaching environment, so it is rare to combine these two together for comprehensive analysis. Therefore, at the technical level, multilevel models can be used to analyze the predictive effect of influencing factors at different levels (such as student level and school level) on the performance of middle school students’ mathematical problem-solving ability, thus helping to find the key influencing factors in the school education environment, so as to promote the cultivation and improvement of students’ mathematical problem-solving ability ultimately.

As such, on the basis of implementing academic requirements in the Chinese Compulsory Education Mathematics Curriculum Standards , this study designed test papers for evaluating students’ mathematical problem-solving ability and questionnaires focusing on non-intellectual internal factors and teaching variables which affect students’ mathematical problem-solving performance. It is hoped that this research can help the academic community to clearly clarify the current performance of middle school students’ mathematical problem solving in mainland China, as well as the learning differences between student groups, schools and regions, and find the key factors that restrict the cultivation of students’ mathematical problem solving ability, so as to provide targeted strategies for improving mathematical problem solving ability. The following research questions were posed:

  • What is the overall proficiency of middle school students’ mathematical problem-solving ability in mainland China?
  • Do middle school students’ mathematical problem-solving ability differ based upon gender and urban–rural environment?
  • What are the key factors that influence middle school students’ mathematical problem-solving performance?

Materials and methods

Participants.

This study utilized 2016 survey data provided by the Regional Education Quality and Health Monitoring team of the China Basic Education Quality Monitoring Collaborative Innovation Center. The Regional Educational Quality and Health Monitoring project is an important regional education investigation and evaluation program in China that is implemented annually. The program aims to conduct health monitoring on the quality of domestic mathematics education through standardized tests and questionnaires based on Chinese mathematics curriculum standards, and it proposes targeted improvements based on data analysis and evaluation. This study adopted a three-stage unequal probability sampling method. The first stage utilized the stratified probability proportionate to size (PPS) sampling method to extract counties (cities and districts). The second stage applied the hierarchical PPS method to extract schools. The third stage used random equidistant sampling to select students. Consequently, the sampling results provided a sample that was representative of the overall province and distribution of different groups, including cities, counties, towns, and rural areas. The study selected 42,968 ninth-grade students, who participated in the 2016 Regional Education Quality and Health Monitoring (used as the main data source), from 762 schools of Z province in East China. In addition, in terms of imputation, since the sample is large enough and the missing rate is only 0.75%, this study used the method of list-wise deletion to obtain 42,644 valid samples, including 22,302 boys (52.3%) and 20,342 girls (47.7%).

Instruments

This study was based on the Regional Education Quality and Health Monitoring project, which included middle school students’ mathematical problem-solving test papers and student and teacher questionnaires on the factors influencing mathematical problem-solving.

Mathematical problem-solving test paper

The middle school mathematical problem-solving assessment of the Regional Education Quality and Health Monitoring project was guided by the Mathematics Curriculum Standards for Compulsory Education (2011), drawing on the experience of large-scale international mathematics assessment, this study designed the test paper for evaluating three dimensions (content, context and cognitive) of mathematical problem-solving process and questionnaires focusing on non-intellectual internal factors and teaching variables that affect students’ problem-solving performance. In this study, mathematical problem solving was defined as an individual’s ability to use cognitive processes to face and solve real, interdisciplinary problems. The mathematical problem-solving test paper consists of 10 items, including numbers and algebra, figures and geometry, and statistics and probability as the content dimensions to examine mathematical problem-solving ability; these items also involve three contexts: personal situation, social situation, and pure mathematical situation. Meanwhile, the cognitive processes involved in problem-solving are divided into three domains: knowing (four items), understanding (four items), and applying (two items), respectively. The test paper contains multiple-choice questions and subjective questions (including open-ended questions), with items including two to three questions. The difficulty of the test paper is about 0.70, the discrimination ranges between 0.40 and 0.80, about 76% of the items’ discrimination is >0.40, and the internal consistency of the test paper is >0.9, which indicates that its reliability is good.

Questionnaires on factors influencing the performance of mathematical problem-solving

Based on the extant literature, a questionnaire was designed to identify factors affecting the performance of middle school students in solving mathematics problems. The significance of related influencing factors was investigated from the perspectives of students and teachers. Two questionnaires were compiled—one for students and another for teachers. The student questionnaire included four subscales: mathematics anxiety, mathematics interest, self-efficacy, and teacher–student relationship. The three subscales of mathematics anxiety, mathematics interest, and self-efficacy were adapted from the Student Questionnaire in PISA (translated into the Chinese version scales by the research team for application). Answers were given on a 5-point Likert scale ranging from 1 ( strongly disagree ) to 5 ( strongly agree ). Higher scores indicated higher degrees of expression. Teacher–student relationship comprised a self-reported subscale rated on a 5-point Likert scale ranging from 1 ( strongly disagree ) to 5 ( strongly agree ). The higher the score, the more harmonious the teacher–student relationship. In addition, to focus on the impact of teaching factors on students’ mathematical problem-solving, the teaching self-efficacy of problem-solving and teaching methods for problem-solving subscales, rated on a 5-point Likert scale, were added to the teacher questionnaires for middle school mathematical problem-solving monitoring. Moreover, the internal consistency coefficients of the overall student questionnaire and teacher questionnaire were 0.91 and 0.89, respectively, and both types of questionnaires had good structural validity (CFI = 0.910, RMSEA = 0.054; CFI = 0.921, RMSEA = 0.070).

Data collection and test procedure

In order to collect test data quickly and efficiently, the China Basic Education Quality Monitoring Collaborative Innovation Center cooperated with the Department of Education in Z Province to jointly launch the project of Regional Education Quality and Health Monitoring . With the assistance of cities and counties (county-level cities and districts) in Z province, sampling tests were successfully organized and implemented in 11 cities and 104 districts and counties in Z Province in October 2016. Among them, 42,968 ninth grade students from 762 junior high schools participated in this test, while 42,644 of them finally filled out the Student Questionnaire . In addition, a total of 3,565 principals and vice-principals in charge of teaching of the participating schools filled out the Principal Questionnaire , 10,599 teachers answered the Teacher Questionnaire , and a total of 76,502 parents of students answered the Parent Questionnaire .

Meanwhile, for the test procedure, the mathematics project team has undergone a series of complete evaluation processes from the beginning of 2016 to November 2016, including framework testing and two-way specification table preparation, item collection and polishing, the first interviews with six participants, the round pre-tests of 30 participants, the second-round pre-tests of 300 participants, and the external reviews of domestic and foreign mathematics experts and assessment experts. Thus, implementing these procedures ultimately ensures the scientific and normative nature of the entire testing process ( Qi et al., 2015 ).

Data processing

After going through the above test procedures, the project team first determined the scoring standards based on the standard answers of the test paper of mathematical problem-solving and the students’ final formal participation in the test and then scored objectively according to the scoring standards. Next, the Rasch model from item response theory was used to analyze students’ original scores to obtain their mathematical problem-solving ability value. Then, the ability value was converted into a standardized score (average 300, standard deviation 50), that is, the scale score that represents students’ mathematical problem-solving performance. Simultaneously, the project team used the Angoff method 1 to calibrate the performance of students’ mathematical problem-solving ability, and it divided the students into four levels (A, B, C, D) according to their mathematical problem-solving performance, where level C represents the benchmark of students’ mathematical problem-solving. In contrast, the project also processed the original data from student and teacher questionnaires into a questionnaire database. In addition, this study first used the descriptive statistical analysis method to further describe the proficiency of students’ mathematical problem-solving; then, it used the hierarchical linear model to analyze the inter-school differences in mathematical problem-solving performance and the predictive role of factors from different educational levels.

Overall proficiency of students’ mathematical problem-solving ability

To distinguish the characteristics of mathematical problems of different difficulty levels and the characteristics of students’ mathematical problem-solving performance, this study divides the performance of all students’ mathematical problem-solving ability into three proficiency levels from high to low, namely A level, B level, and C level, with each level representing the expected range of abilities for a different student group. Among them, students at the A level can comprehensively use basic knowledge in the process of mathematical problem-solving, master mathematical concepts, apply appropriate mathematical methods, or establish appropriate mathematical models to solve unfamiliar or open-ended problems. The group of students at the B level can understand the characteristics of mathematical concepts in the mathematical problem-solving process and apply appropriate mathematical methods or build simple mathematical models to solve relatively unfamiliar or unpracticed problems. Finally, students at the C level can only memorize and identify mathematical concepts in the mathematical problem-solving process and use conventional mathematical methods to solve familiar or practiced problems. In addition, below C level is defined as D level; the students at this level cannot analyze and interpret the answers nor evaluate and categorize problem-solving processes and methods. Table 1 shows that the mathematics problem-solving ability of middle school students in Z province in mainland China is relatively good; the majority of students’ mathematical problem-solving skills are at a moderate to high level, and 48% of them have reached the A level, 35% the B level, and 13% the C level, with only 4% having located in the D level.

The ratio of different proficiency levels of students’ mathematical problem-solving ability.

Background differences in the benchmark of students’ mathematical problem-solving performance

As mentioned above, the C level represents the benchmark for students’ mathematical problem-solving performance 2 , which is the minimum requirement for middle school students’ problem-solving skills in the Mathematics Curriculum Standards for Compulsory Education (2011). In other words, when a student’s mathematical problem-solving proficiency reaches the C level and above, their problem-solving ability meets the curriculum standard’s academic requirements. The survey found that 98% of boys’ mathematics problem-solving ability reached the C level and above, 3 percentage points higher than girls, and the gender difference was significant ( p  < 0.01, φ  = 0.12). Simultaneously, the proportion of only children (97%) reaching the C level and above was also significantly higher than that of non-only children (94%), and we observed a significant difference between the two ( p  < 0.01, φ  = 0.11). In contrast, we found no significant difference between leftover students and non-leftover students in the compliance rate of the benchmark of mathematical problem-solving ability ( p > 0.05, φ  = 0.06), but the proportion of non-leftover students (96%) reaching the C level and above was slightly higher than that of leftover students (95%). In addition, we observed significant urban and rural differences in the performance of middle school students’ mathematical problem-solving ability ( p  < 0.01, φ  = 0.21), and 98% of urban students’ mathematical problem-solving ability reached the C level and above, which was 1 and 3 percentage points higher than that of county students and rural students, respectively (see Table 2 ).

The compliance rate of middle school students’ mathematical problem-solving performance.

Students’ mathematical problem-solving ability and influencing factor model setting

Our hierarchical linear model took students’ mathematical problem-solving ability (the scale score) as the dependent variable; gender, leftover situation, only-child situation, mathematics interest, self-efficacy, teacher–student relationship, and mathematics anxiety as the student-level variables; and urban and rural backgrounds, teaching self-efficacy of problem-solving, and teaching methods for problem-solving as the school-or teacher-level variables.

Hierarchical linear model analysis

Due to the nested structure of the school-and student-level data, this study used the hierarchical linear model 3 to process them. Compared with the traditional regression method, this method can make full use of the data information of each level in the analysis of differences in mathematical problem-solving performance and decompose differences at each relevant level; thus, the source and size of the difference can be estimated more accurately. The analysis process involved two basic models: the null model and the random intercept model. The following analysis shows the regression equation model and the corresponding variance component analysis results after including the student-level variables and the school-level variables, respectively (see Table 3 ).

Students’ mathematical problem-solving performance and influencing factors HLM analysis results.

The variance estimation results are nonstandard residual estimates, both of which are significant when p value is 0.001.

In Model 0, Y ij is the mathematical problem-solving performance of i students in j school; β 0j is the average problem-solving performance of j school; r ij is the random error of individual students, which indicates the difference between the i students in j school and the j school’s average problem-solving performance; and γ 00 is the overall average performance. μ 0j is the school’s random error, which indicates the difference between the average problem-solving performance of the j school and the overall average performance.

Based on Model 0’s student level, we establish Model 1, which adds variables denoting students’ gender (male, female), only-child situation (yes, no), and leftover situation (yes, no), and school-level variables denoting urban versus rural background (urban, county, or rural), which study the influence of background variables on students’ mathematical problem-solving.

In Model 2, the following student-level variables are added: mathematics interest, self-efficacy, teacher–student relationship, and mathematics anxiety, which study the influence of individual non-intellectual variables on students’ mathematical problem-solving. Meanwhile, teaching self-efficacy of problem-solving and teaching methods for problem-solving are added into the school level to study the influence of teaching-related variables on students’ mathematical problem-solving.

Model 0 represents the variance component analysis. By calculating the intraclass correlation coefficient (ICC), this study found that the ICC of the influential factors of ninth-grade students’ mathematical problem-solving performance was about 28%, indicating that 28% of the problem-solving performance differences in middle school students in China’s compulsory education come from inter-school differences. In other words, the model shows significant inter-group differences, and thus, it is necessary to use a hierarchical linear model for the analysis ( Zhang et al., 2005 ).

After incorporating the background variables (Model 1), this study found that the student-level background variables (gender, the only-child situation, and the leftover situation) have little effect on students’ mathematical problem-solving performance. Further observation of the regression coefficients of these student background variables showed that the mathematical problem-solving performance of boys was higher than that of girls, and the mathematical problem-solving performance of only children was higher than that of non-only children. By contrast, the urban or rural background, which belonged to school-level background variables, had a larger impact on the average school achievement (the absolute value of the regression coefficient was larger); specifically, the mathematical problem-solving performance of urban students was significantly higher than that of county and rural students. The above findings also corroborate the results of the previous Chi-squared test.

By observing Model 2, we found that the variance of the student-level residuals reduced more when mathematics interest, learning self-efficacy, teacher–student relationship, and mathematics anxiety were added into the student-level variables. Among these individual non-intelligence factors, the absolute value of the regression coefficient of self-efficacy was the largest, followed by mathematics anxiety and mathematics interest, and the smallest was teacher–student relationship. Simultaneously, the addition of two variables that belonged to school-level, namely teaching self-efficacy of problem-solving and teaching methods for problem-solving, greatly reduced the residuals at the school level. Although they were not as prominent as the effect of urban or rural background on mathematical problem-solving performance, teaching self-efficacy and teaching methods for problem-solving did have a significant impact on students’ mathematical problem-solving, and teaching methods for problem-solving had a relatively larger positive effect.

Overall performance of students’ mathematical problem-solving ability

This study shows that in the four-level distribution of students’ problem-solving ability performance, 96% of middle school students in Z province met the minimum requirements of the curriculum standard, and only 4% of students did not. This result is similar to the average level of problem-solving performance of students from OECD countries and regions that participated in the PISA 2012 test. For example, according to the students’ problem-solving performance in the PISA 2012 survey report, the proportion of students in OECD countries and regions whose problem-solving ability was at level 1 and above was 91.8, and 8.2% of the students were still unable to reach the problem-solving benchmark. However, the difference is that in terms of problem-solving performance at the high level of difficulty, the performance of students from Z province in mainland China is more prominent, with the proportion of students at the A level and above as high as 48%, while the proportion of East Asian students at level 5 and above who participated in the PISA problem-solving test is lower than 20%, of which Hong Kong-China is 19.3%, and Chinese Taipei and Shanghai-China are both 18.3% ( OECD, 2014 ). The above results may be due to Z province being located in East China, where China’s education and economy are relatively developed. In fact, East China has always played an important role in the six administrative regions of mainland China, with its population and GDP accounting for more than 30% of the country. Moreover, in terms of basic education, the government of East China attaches importance to education investment, with well-equipped teachers and infrastructure, and balanced development among schools. Especially in mathematics education, mathematics teachers often have more unique teaching art and teaching strategies. For example, they often create a series of mathematical problem situations to stimulate students’ cognition, so that students can understand the whole process of mathematical problem-solving ( Zang, 2006 ). Thus, the students’ overall mathematics academic level and mathematical problem-solving ability are relatively good, and students are especially able to successfully deal with mathematics problems of medium and high difficulty levels. Nevertheless, it cannot be overlooked that this study mainly relies on paper-and-pencil tests for the monitoring of mathematics problem-solving, and the East Asian middle school students participating in the PISA survey may have faced a more complex problem-solving test environment (the testing process, for instance, relied on computer technology); thus, their problem-solving ability performance may have been easily underestimated.

Differences in the benchmark of middle school students’ mathematical problem-solving ability in mainland China

In this study, significant differences are observed with regard to gender and only-child situation in terms of mathematical problem-solving benchmark among students from different backgrounds, these differences are not practical. Many studies have also pointed out that no statistical difference exists in students’ mathematical ability based on gender ( Fennema and Sherman, 1976 ). However, from the perspective of cultural tradition, men in East Asia tend to have more educational expectations than women, which interferes with academic performance and mathematical ability ( Zhu et al., 2018 ). On the other hand, retrospecting China’s population policy changing, the sex ratio of the domestic population decreased from 107.56 in 1953 to 104.88 in 2021. Moreover, from the development trend, although it has been declining, the total number of men is still higher than that of women, and it is worth noting that the gender ratio of the population in East China is also higher than the national average ( Yuan and Wu, 2022 ). Overall,Chinese boys are more likely than girls to perform at higher levels in problem-solving. As for family structure, according to the resource dilution theory, only children who receive family support are more likely to succeed in academic performance and mathematical ability improvement ( Blake, 1981 ). In contrast, the differences in the performance of students’ mathematical problem-solving abilities caused by different urban and rural backgrounds have more practical significance, this may be due to the significant, long-term urban–rural education gap in mainland China. In reality, although the country vigorously implements the policy of Coordinated Development of Compulsory Education in Urban and Rural Areas , objectively, the situation of urban education resources concentration and urban family education investment surge has not reversed, so the current situation of relatively weak education quality in districts, counties, towns and rural schools cannot be changed in the short term ( Liu, 2006 ; Wei, 2018 ). Moreover, even in East China, where the development of basic education is relatively balanced, the educational differences between urban and rural areas are still significant. But the difference is that the gap between urban and rural education in East China is more about the quality of teachers than the hardware conditions such as infrastructure. For example, urban teachers can often get more high-level education and training opportunities (including the interpretation of mathematics curriculum standards), so they have a more accurate grasp of many teaching contents and more effective teaching methods ( Zang, 2006 ). Therefore, on the whole, the performance of mathematical problem solving ability of urban students in Z Province is better than that of students in counties, towns and rural areas.

The predictive effect of student-level and school-level factors on mathematical problem-solving ability

On the whole, this study points out that 28% of the difference in mathematical problem-solving performance among middle school students in East China comes from inter-school variation, which shows that the imbalance of problem-solving between schools in compulsory education in mainland China still requires attention. According to the analysis results of the inter-school differences in PISA 2012, the percentage of the average variation in mathematical problem-solving performance among OECD members, accounting for school characteristics, is 38%. Simultaneously, the percentage of Shanghai samples who participated in the test on behalf of mainland China reaches 42%, while the mathematical problem-solving performance of students in countries such as Finland and Sweden is relatively balanced, with an average variation in problem-solving results across schools lower than 20% ( OECD, 2014 ). The above results fully indicate that there is still space for improvement in the inter-school differences in the mathematical problem-solving of students in compulsory education in mainland China. As far as the current education situation in East China is concerned, the overall development level of basic education is relatively balanced, so the inter school differences in students’ mathematical performance are not particularly prominent, which is mainly due to the positive measures taken in this region, such as paying attention to education layout planning and increasing support for weak schools ( Zang, 2006 ). However, due to the long-term existence of urban–rural dual economic and social development structure, local weak rural schools have always been at a disadvantage in solving problems, and their school running quality and education investment are obviously insufficient ( Liu, 2006 ). In addition, under the influence of social class differentiation, the average socioeconomic status of schools composed of students with different family socioeconomic statuses further exacerbates the Matthew effect of inter-school differences in mathematical problem-solving ( Dumay and Dupriez, 2008 ).

In addition to the significant difference in the performance of middle school students’ mathematical problem-solving ability caused by the gap between urban and rural backgrounds, the study also found that students’ individual non-intelligence factors (e.g., mathematics interest, self-efficacy, teacher–student relationship, and mathematics anxiety) explained the difference in mathematical problem-solving more than students’ background variables (e.g., gender, only-child situation, and leftover situation) did. The development of individual characteristics is always accompanied by the psychological maturity of students, which, when compared to individual background, may better predict problem-solving ( Lu, 2011 ; Alibali et al., 2019 ). In addition, some studies have found that increasing middle school students’ mathematics interest and self-efficacy can effectively improve their mathematical problem-solving, while excessive mathematics anxiety can hinder it ( Xu and Qi, 2018 ). Similarly, this study demonstrated that mathematics interest, self-efficacy, and teacher–student relationship positively influenced students’ problem-solving, while mathematics anxiety negatively affected it. This is because positive learning attitudes and persistence can promote mathematical thinking, while poor learning attitudes and habits can hinder mathematics learning and thinking ( Huang, 2006 ). In view of this, in the future, mathematics teaching of secondary schools in various countries should pay more attention to the regulating role of non-intellectual factors like mathematical affections in the process of problem solving, such as actively creating mathematical problem situations to promote their interest in mathematics learning, increasing the opportunities for students about problem posing, and alleviating the anxiety of mathematical problem solving.

Furthermore, this study remarks that teaching factors are important for students’ mathematical problem-solving ability; in particular, the teaching methods of mathematical problem-solving have played an important role in nurturing this ability because real teaching scenarios can provide students with step-by-step decomposition and reasoning analysis of the problem-solving process ( Pasani, 2018 ). Therefore, for the cultivation of mathematical problem-solving ability in middle schools, the primary task of future mathematical classroom teaching is to improve the teaching strategy of problem-solving to activate students’ mathematical cognition, such as appropriately transforming some open-ended problems with complex problem situations to help students gradually develop their mathematical thinking in the process of exploring the procedures of problem solving.

Limitations

This research has some limitations. First, although the Regional Educational Quality and Health Monitoring project adopted a relatively scientific PPS sampling method and included school students from different districts and counties and urban and rural backgrounds, such as administrative divisions, the main source was a sample of students from the upper levels of education and economy in East China. Therefore, the main findings of this study can provide appropriate reference for mathematics education in the developed regions of other countries, but at the same time, some conclusions still cannot be extended to the other regions of mainland China. For example, there may be differences between leftover and non-leftover students in China’s underdeveloped provinces in mathematical problem-solving performance. In view of this, the follow-up research can further enrich the survey samples, such as expanding to the whole country. Second, from the type of math problem solving in test paper, the authors mainly use the two forms of multiple-choice questions and subjective questions commonly used in math tests in mainland China, which have less reference for the problem solving test questions in mathematics textbooks for secondary schools from other countries. Therefore, future research can consider from the perspective of textbook analysis to further enrich the form of math problem presentation in the current test paper, so as to facilitate subsequent international comparisons. Third, due to the limitation of the variables in the database, as this study did not choose SES and the school average SES as the optimal control variables in different levels but instead replaced them with the leftover situation and urban or rural background situation, the estimated results present deviations to a certain extent. Finally, limited by the volume of the questionnaire survey, the factors affecting the mathematical problem-solving ability selected in this study only involved students’ background and internal non-intellectual factors, with less consideration of factors such as meta-cognition, including learning strategies, which may lead to limitations in the process of impact mechanism analysis. Thus, follow-up supplementary research could consider increasing the content of the student questionnaire on the influencing factors of mathematical problem-solving ability.

This study focused on the systematic monitoring and investigation of mathematical problem-solving ability of middle school students in the compulsory education stage in mainland China. It addressed the overall proficiency, background differences in the benchmark of ability, and the predictive effect of student-level and school-level factors on mathematical problem-solving performance, drawing meaningful conclusions.

First, the mathematics problem-solving ability of middle school students in Z province in mainland China is relatively good, and 96% of the students’ mathematics problem-solving ability meets the basic academic requirements of the curriculum standards.

Second, in the difference analysis of the benchmark for middle school students’ mathematical problem-solving ability performance, we found that the proportion of boys reaching the C level and above was significantly higher than that of girls, and the proportion of only children reaching the C level and above was also significantly higher than that of non-only children. In contrast, the proportion of non-leftover students reaching the C level and above was higher than that of leftover students, but no significant difference was observed between the two.

Finally, in terms of school-level variables, urban and rural backgrounds had a larger impact on mathematical problem-solving than teaching factors. Among the teaching factors, the teaching method of problem-solving had a relatively greater positive impact on problem-solving than the teaching self-efficacy. For student-level variables, the influence of individual non-intellectual factors on mathematical problem-solving was higher than that of student background variables, including a greater positive effect of self-efficacy and a higher negative effect of mathematics anxiety. Moreover, among the effects of student background on mathematical problem-solving, gender had the largest negative effect, followed by the effect of the only-child situation, while the positive impact of the leftover situation was not significant. In particular, only children and boys performed better.

Author’s note

The samples used in this study were provided by the Regional Educational Quality and Health Monitoring project. The schools and education bureaus in the participating areas signed cooperative research agreements with the project team.

Data availability statement

Ethics statement.

The studies involving human participants were reviewed and approved by the Academic Committee of the Faculty of Education in Beijing Normal University. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author contributions

ZX wrote this manuscript. CQ data provided and theoretical guidance. All authors contributed to the article and approved the submitted version.

This study was funded by General Program of NSFC: “Research on Cognition and Brain Mechanism of Students’ Mathematical Creative Thinking in Complex Situations (no: 62277003)” and the project of The China Basic Education Quality Monitoring Collaborative Innovation Center: “Middle School Mathematics Literacy Evaluation and Diagnostic Improvement (no: 110105006)”.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

1 φ represents the effect size of the Chi-squared test.

2 The Angoff method is one of the most commonly used in standard setting procedures and could be also used to determine the academic benchmark. Specifically, two or more split points were used in large-scale assessments to classify students’ academic performance into multiple levels to determine classification criteria for different proficiency.

3 In this study, the performance of the influencing factors is represented by the average value of multiple items that affect students’ mathematical problem-solving ability.

  • Alibali M. W., Brown S. A., Menendez D. (2019). Understanding strategy change: contextual, individual, and metacognitive factors . Adv. Child Dev. Behav. 56 , 22–256. doi: 10.1016/bs.acdb.2018.11.004, PMID: [ PubMed ] [ CrossRef ] [ Google Scholar ]
  • Bai Y. (2011). Study on mathematical problem solving ability of junior middle school students [Master’s Thesis]. Shenyang Normal University. [ Google Scholar ]
  • Blake J. (1981). Family size and the quality of children . Demography 18 , 421–442. [ PubMed ] [ Google Scholar ]
  • Cai J. (1995). A cognitive analysis of U. S. and Chinese students' mathematical performance on tasks involving computation, simple problem solving, and complex problem solving . J. Res. Math. Educ. 7 , 1–151. [ Google Scholar ]
  • Chen T. (2007). High school students’ mathematics problem solving strategy and its training [Master’s Thesis]. Changchun, China: Northeast Normal University. [ Google Scholar ]
  • Chien T. K., Lin H. Y., Ma H. Y. (2016). “A systematic information problem-solving process,” in International Conference on E-commerce in Developing Countries: with Focus on E-tourism . IEEE.
  • Dai S. (2012). Research on middle school mathematics teaching based on problem solving Master’s thesis [Master’s Thesis]. Wuhan, China: Central China Normal University. [ Google Scholar ]
  • Department for Education and Science (DES) . (1982). “ Mathematics counts. Report of the committee of inquiry into the teaching of mathematics in schools ,” in Cockcroft (the Cockcroft Report) . ed. W. H. Cockcroft (London: HMSO; ). [ Google Scholar ]
  • Ding R., Wong N., Ma Y. (2009). Effects of primary school mathematics classroom environment and students’ problem solving ability . Educ. Sci. Res. 12 , 39–42. [ Google Scholar ]
  • Dumay X., Dupriez V. (2008). Does the school composition effect matter? Evidence from Belgian data . Br. J. Educ. Stud. 56 , 440–477. doi: 10.1111/j.1467-8527.2008.00418.x [ CrossRef ] [ Google Scholar ]
  • Duncker K. (1945). On problem solving . Psychol. Monogr. 58 , i–113. doi: 10.1037/h0093599 [ CrossRef ] [ Google Scholar ]
  • Fan L., Zhu Y. (2003). “ A school to think, a country to learn: Singapore’s mathematics curriculum , “in International Vision of Mathematics Curriculum Development. ed. X. Sun (Beijing, China: Higher Education Press; ), 305–357. [ Google Scholar ]
  • Fang X., Zeng G., Gao Z. (1993). An experimental record of Japanese “subject-studying” . Shu Xue Jiao Xue 2 :221. [ Google Scholar ]
  • Fennema E., Sherman J. A. (1976). Fennema-Sherman mathematics attitudes scales: instruments designed to measure attitudes toward the learning of mathematics by females and males . J. Res. Math. Educ. 7 , 324–326. doi: 10.1007/s11858-019-01098-w [ CrossRef ] [ Google Scholar ]
  • Goulet-Lyle M. P., Voyer D., Verschaffel L. (2019). How does imposing a step-by-step solution method impact students' approach to mathematical word problem solving? ZDM - Int. J. Math. Educ. 52 , 139–149. doi: 10.2307/748467 [ CrossRef ] [ Google Scholar ]
  • Halmos P. R. (1995). To count or to think, that is the question . Nieuw Archief voor Wiskunde , 4 :10. [ Google Scholar ]
  • Huang L. (2006). On the cultivation of non-intelligence factors in mathematics teaching . China Educ. Guide 9 , 67–68. [ Google Scholar ]
  • Hulukati E., Zakiyah S., Rustam A. (2018). The effect of guided discovery learning model with superitem test on students' problem-solving ability in mathematics . J. Social Sci. Stud. 5 , 210–219. doi: 10.5296/jsss.v5i2.13406 [ CrossRef ] [ Google Scholar ]
  • Hunt E. (1968). Computer simulation: artificial intelligence studies and their relevance to psychology . Annu. Rev. Psychol. 19 , 135–168. doi: 10.1146/annurev.ps.19.020168.001031 [ PubMed ] [ CrossRef ] [ Google Scholar ]
  • Kilpatrick J. (1978). Research on problem solving in mathematics . Sch. Sci. Math. 78 , 189–192. doi: 10.1111/j.1949-8594.1978.tb09345.x [ CrossRef ] [ Google Scholar ]
  • Kilpatrick J. (2009). The mathematics teacher and curriculum change . PNA 3 , 107–121. [ Google Scholar ]
  • Lane S. (1993). The conceptual framework for the development of a mathematics assessment for QUASAR . Educ. Meas. Issues Pract. 12 , 16–23. [ Google Scholar ]
  • Liu C. (2005). A comparative study of the views of mathematics between ancient China and ancient Greece . J. Harbin Univ. 26 :4. doi: 10.3969/j.issn.1004-5856.2005.08.005 [ CrossRef ] [ Google Scholar ]
  • Liu S. (2006). On the balanced development of urban and rural compulsory education in China [Doctoral Dissertation]. Beijing, China: Beijing Normal University. [ Google Scholar ]
  • Lu L. (2011). Prediction of the academic achievement of primary school students by individual and environmental factors [Doctoral Dissertation]. University of Chinese Academy of Sciences. [ Google Scholar ]
  • Lumbelli L. (2018). Productive thinking in place of problem-solving . Gestalt Theory 40 , 131–148. [ Google Scholar ]
  • Ma R. (2009). The key to solving mathematical problems lies in the insight of the problem situation . Green Apple 1 :2. [ Google Scholar ]
  • Manfreda V. (2021). Mathematical literacy from the perspective of solving contextual problems . Eur. J. Educ. Res. 10 , 467–483. doi: 10.12973/eu-jer.10.1.467 [ CrossRef ] [ Google Scholar ]
  • Mayer R. E. (1992). Thinking, Problem Solving, Cognition . 2nd Edn. New York: Freeman. [ Google Scholar ]
  • Mayer R. E., Tajika H., Stanley C. (1991). Mathematical problem solving in Japan and the United States: a controlled comparison . J. Educ. Psychol. 83 , 69–72. doi: 10.1037/0022-0663.83.1.69 [ CrossRef ] [ Google Scholar ]
  • Ministry of Education of the People’s Republic of China (2003). Ordinary High School Mathematics Curriculum Standards. People’s Education Press. [ Google Scholar ]
  • Ministry of Education of the People’s Republic of China (2012). Mathematics Curriculum Standards for Compulsory Education. Beijing, China: Beijing Normal University Publishing Group. [ Google Scholar ]
  • Ministry of Education of the People’s Republic of China (2020). General High School Mathematics Curriculum Standards. Beijing, China: People’s Education Press. [ Google Scholar ]
  • National Council of Teachers of Mathematics (NCTM) (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA: National Council of Teachers of Mathematics. [ Google Scholar ]
  • Neisser U. (1967). Cognitive Psychology. New York: Appleton-Century-Crofts. [ Google Scholar ]
  • Newell A., Simon A. (1972). Human Problem Solving. New Jersey: Prentice Hall. [ Google Scholar ]
  • OECD (2013). PISA 2012 Assessment and Analytical Framework. Paris: OECD Publishing. [ Google Scholar ]
  • OECD (2014). PISA 2012 Results: Creative Problem Solving: Students’ Skills in Tackling Real-Life Problems. Vol. 5. Paris: OECD Publishing. [ Google Scholar ]
  • Pasani C. F. (2018). The use of problem-solving as a method in the teaching of mathematics and its influence on students’ creativity . Int. J. Eng. Res. Technol. 11 , 451–479. [ Google Scholar ]
  • Peng A., Jing L., Nie B., Li Y. (2017). Characteristics of Teaching Mathematical Problem Solving in China. Netherlands: Sense Publishers. [ Google Scholar ]
  • Polya G. (1944). How to Solve it: A New Aspect of Mathematical Method Princeton, NJ: Princeton University Press. [ Google Scholar ]
  • Polya G. (1957). How to Solve it . 2nd Edn. New York: Doubleday Publishing. [ Google Scholar ]
  • Polya G. (1965). Mathematical discovery. On understanding, learning and teaching problem solving . NJ: John Wiley & Sons. [ Google Scholar ]
  • Qi C., Zhang X., Wang R. (2015). Research on the present situation and influential factors of the mathematics academic level of the eighth graders . J. Educ. Stud. 11 , 87–92. doi: 10.14082/j.cnki.1673-1298.2015.02.0011 [ CrossRef ] [ Google Scholar ]
  • Schoenfeld A. H. (1985). Mathematical Problem Solving. New York: Academic Press. [ Google Scholar ]
  • Shao R. (1983). Educational Psychology: Principles of Learning and Teaching. Shanghai, China: Shanghai Education Publishing House. [ Google Scholar ]
  • Silver E. A., Kilpatrick J. (1988). “ Testing mathematical problem solving ,” in Research Agenda for Mathematics Education: The Teaching and Assessing Mathematical Problem Solving . eds. Charles R., Silver E. (Reston, VA: National Council of Teachers of Mathematics; ), 178–186. [ Google Scholar ]
  • Stacey K. (2005). The place of problem solving in contemporary mathematics curriculum documents . J. Math. Behav. 24 , 341–350. doi: 10.1016/j.jmathb.2005.09.004 [ CrossRef ] [ Google Scholar ]
  • Stinger J. W., Lee S., Stevenson H. W. (1990). Mathematical Knowledge of Japanese, Chinese, and American Elementary School Children. Reston, VA: NCTM. [ Google Scholar ]
  • Sun Z., Cao J., Yao J. (2016). The influence of mathematics self-efficacy and learning motivation on the performance of mathematics problem solving . Math. Teach. Res. 35 , 40–45. doi: 10.3969/j.issn.1671-0452.2016.01.010 [ CrossRef ] [ Google Scholar ]
  • Tan S. (2004). Research on cognitive psychological process in mathematical problem solving [Master’s Thesis]. Hohhot, Chin: Inner Mongolia Normal University. [ Google Scholar ]
  • Tan W. (2009). “Problem solving” and problem solving in mathematics teaching. Education research . Forum 1 :18. [ Google Scholar ]
  • Wang X. (2000). Review of the study on mathematical problem-solving . J. Guangxi Normal Univ. 2 , 188–190. [ Google Scholar ]
  • Wang X. (2017). Research on influencing factors of middle school students’ mathematical problem solving ability [Master’s Thesis]. Yanbian, China: Yanbian University. [ Google Scholar ]
  • Wang P. (2020). The application of heuristic teaching in modern teaching . Global Market. 6 :215. [ Google Scholar ]
  • Wang K. (2021). On problem solving and mathematics curriculum reform . Educ. Rev. 7 , 141–147. [ Google Scholar ]
  • Wei Z. (2018). Research on the status and development countermeasures of China’s compulsory education . New West 457 , 149–150. [ Google Scholar ]
  • Xu Z., Qi C. (2018). Investigation and research on junior middle school students’ math problem solving ability and influencing factors . Educ. Meas. Eval. 7 , 41–46. doi: 10.16518/j.cnki.emae.2018.07.007 [ CrossRef ] [ Google Scholar ]
  • Ye Q., Zhang J. (2004). Several psychological factors affecting the ability of “problem solving” in primary school mathematics . Popular Psychol. 4 :40. [ Google Scholar ]
  • Yuan X., Wu J. (2022). The unbalanced situation of China's population sex ratio: problems and solutions . Population Health 4 :5. [ Google Scholar ]
  • Zang Y. (2006). On the characteristics and thinking of the development of basic education in East China . Zhong Guo Nong Cun Jiao Yu 5 , 20–22. [ Google Scholar ]
  • Zhang D. (1991). Mathematics Education . Nanchang, China: Jiangxi Education Press. [ Google Scholar ]
  • Zhang C. (2012). Apply what you have learned in mathematics learning . Modern Educ. Sci: Teach. Res. 11 :22. [ Google Scholar ]
  • Zhang L., Lei L., Guo B. (2005). Application on hierarchical linear model.. Beijing, China: Educational Science Publishing House. [ Google Scholar ]
  • Zhao F. (2007). How to embody the idea of “problem solving” in middle school mathematics teaching . Chin. J. Educ. Dev. Res. 4 , 66–67. [ Google Scholar ]
  • Zhao X., Shen L. (2003). A comparison of the differences in solving mathematics problems between Chinese and American students . J. Educ. Stud. 11 , 27–30. [ Google Scholar ]
  • Zhu Y., Kaiser G., Cai J. (2018). Sex equity in mathematical achievement: the case of China . Educ. Stud. Math. 99 , 245–260. doi: 10.1007/s10649-018-9846-z [ CrossRef ] [ Google Scholar ]

share this!

February 15, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

trusted source

Studies recommend increased research into achievement, engagement to raise student math scores

by La Trobe University

math homework

A new study into classroom practices, led by Dr. Steve Murphy, has found extensive research fails to uncover how teachers can remedy poor student engagement and perform well in math.

More than 3,000 research papers were reviewed over the course of the study, but only 26 contained detailed steps for teachers to improve both student engagement and results in math. The review is published in the journal Teaching and Teacher Education .

Dr. Murphy said the scarcity of research involving young children was concerning.

"Children's engagement in math begins to decline from the beginning of primary school while their mathematical identity begins to solidify," Dr. Murphy said.

"We need more research that investigates achievement and engagement together to give teachers good advice on how to engage students in mathematics and perform well.

"La Trobe has developed a model for research that can achieve this."

While teachers play an important role in making decisions that impact the learning environment , Dr. Murphy said parents are also highly influential in children's math education journeys.

"We often hear parents say, 'It's OK, I was never good at math,' but they'd never say that to their child about reading or writing," Dr. Murphy said.

La Trobe's School of Education is determined to improve mathematical outcomes for students, arguing it's an important school subject that is highly applicable in today's technologically rich society.

Previous research led by Dr. Murphy published in Educational Studies in Mathematics found many parents were unfamiliar with the modern ways of teaching math and lacked self-confidence to independently assist their children learning math during the COVID-19 pandemic.

"The implication for parents is that you don't need to be a great mathematician to support your children in math, you just need to be willing to learn a little about how schools teach math today," Dr. Murphy said.

"It's not all bad news for educators and parents. Parents don't need to teach math; they just need to support what their children's teacher is doing.

"Keeping positive, being encouraging and interested in their children's math learning goes a long way."

Steve Murphy et al, Parents' experiences of mathematics learning at home during the COVID-19 pandemic: a typology of parental engagement in mathematics education, Educational Studies in Mathematics (2023). DOI: 10.1007/s10649-023-10224-1

Provided by La Trobe University

Explore further

Feedback to editors

problem solving in school mathematics

Examining viruses that can help 'dial up' carbon capture in the sea

10 hours ago

problem solving in school mathematics

New research helps create new antibiotic that evades bacterial resistance

12 hours ago

problem solving in school mathematics

From crop to cup: A new genetic map could make your morning coffee more climate resilient

14 hours ago

problem solving in school mathematics

Saturday Citations: Einstein revisited (again); Atlantic geological predictions; how the brain handles echoes

17 hours ago

problem solving in school mathematics

CERN researchers measure speed of sound in the quark–gluon plasma more precisely than ever before

Feb 16, 2024

problem solving in school mathematics

NASA's final tally shows spacecraft returned double the amount of asteroid rubble

problem solving in school mathematics

Harnessing light with hemispherical shells for improved photovoltaics

problem solving in school mathematics

New species of pirate spiders discovered on South Atlantic island

problem solving in school mathematics

Bacteria in the Arctic seabed are active all year round, researchers find

problem solving in school mathematics

Martians wanted: Apply here now for NASA's simulated yearlong Mars mission

Relevant physicsforums posts, higher dimensional spheres viz. cubes.

5 hours ago

Negative radius convention equivalent but not equal?

Feb 14, 2024

Are these two optimization problems equivalent?

Tennis probabilities challenge, degrees of freedom in lagrangian mechanics for a fractal path.

Feb 13, 2024

Requesting constructive criticism for my paper

Feb 12, 2024

More from General Math

Related Stories

problem solving in school mathematics

'Math anxiety' causes students to disengage, says study

Nov 22, 2023

problem solving in school mathematics

Top-rated educational math apps may not be best for children's learning

May 30, 2022

problem solving in school mathematics

Former math teacher explains why some students are 'good' at math and others lag behind

Nov 3, 2022

problem solving in school mathematics

AI can teach math teachers how to improve student skills

Dec 8, 2023

problem solving in school mathematics

How to improve math skills among American children

Jan 10, 2023

problem solving in school mathematics

Why the way you talk to your child about math matters

Sep 7, 2022

Recommended for you

problem solving in school mathematics

Reading on screens instead of paper is a less effective way to absorb and retain information, suggests research

Feb 6, 2024

problem solving in school mathematics

Mathematical model connects innovation and obsolescence to unify insights across diverse fields

Feb 5, 2024

problem solving in school mathematics

Swarming cicadas, stock traders, and the wisdom of the crowd

Feb 1, 2024

problem solving in school mathematics

Researchers use simulations to tackle finite sphere-packing problem and 'sausage catastrophe'

Jan 31, 2024

problem solving in school mathematics

A manifold fitting approach for high-dimensional data reduction beyond Euclidean space

Jan 29, 2024

problem solving in school mathematics

Certain personality traits linked to college students' sense of belonging

Jan 17, 2024

Let us know if there is a problem with our content

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

  • Math for Kids
  • Parenting Resources
  • ELA for Kids
  • Teaching Resources

SplashLearn Blog

10 Best Strategies for Solving Math Word Problems

5 Easy Tips & Tricks to Learn the 13 Time Table for Kids

How to Teach Number Sense to Kids: Step-by-Step Guide

How to Teach Decimals: A Step-by-Step Guide

How to Teach Fraction to Kids – 11 Best Activities

How to Choose Best School For Your Kid: 12 Best Tips

Why Kids Get Bored at School: 10 Tips to Keep Them Interested

11 Best Writing Apps for Kids

Homeschool vs Public School: 12 Tips on How to Choose One

15 Essential Life Skills Activities for Kids: Beyond ABCs

12 Animals That Start With ‘E’: From Elephants to Eels

12 animal that starts with k.

60 Best Essay Topics for Kids: Nurturing Young Minds

How to Teach Sentence Structure to Kids: The Ultimate Guide

72 Best G words for Kids in 2024

11 Best Search Engine for Kids: Protecting Young Minds Online

How to Prevent Teacher’s Burnout – 12 Top Ways

46 Best Teaching Tools for Teachers in 2024

How to Teach 3rd Grade Kids: 25 Tips for a Successful Year

12 Best Online Tutoring Websites for All Grades

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.

SplashLearn: Most Comprehensive Learning Program for PreK-5

Product logo

SplashLearn inspires lifelong curiosity with its game-based PreK-5 learning program loved by over 40 million children. With over 4,000 fun games and activities, it’s the perfect balance of learning and play for your little one.

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:

Card Image

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:

Card Image

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:

Card Image

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:

Card Image

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:

Card Image

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:

Card Image

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.

problem solving in school mathematics

Most Popular

A working mom and her daughter in the bedroom, Mom is working while daughter is playing with her toys.

101 Best Riddles for Kids (With Explanation)

problem solving in school mathematics

15 Best Report Card Comments Samples

Good vibes quotes by SplashLearn

40 Best Good Vibes Quotes to Brighten Your Day

Recent posts.

Collage of animals that start with “E”

Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Watch your kids fall in love with math & reading through our scientifically designed curriculum.

Parents, try for free Teachers, use for free

Banner Image

  • Games for Kids
  • Worksheets for Kids
  • Math Worksheets
  • ELA Worksheets
  • Math Vocabulary
  • Number Games
  • Addition Games
  • Subtraction Games
  • Multiplication Games
  • Division Games
  • Addition Worksheets
  • Subtraction Worksheets
  • Multiplication Worksheets
  • Division Worksheets
  • Times Tables Worksheets
  • Reading Games
  • Writing Games
  • Phonics Games
  • Sight Words Games
  • Letter Tracing Games
  • Reading Worksheets
  • Writing Worksheets
  • Phonics Worksheets
  • Sight Words Worksheets
  • Letter Tracing Worksheets
  • Prime Number
  • Order of Operations
  • Long multiplication
  • Place value
  • Parallelogram
  • SplashLearn Success Stories
  • SplashLearn Apps
  • [email protected]

© Copyright - SplashLearn

IMAGES

  1. EBOOK_HARCOVER LIBRARY Creative Problem Solving in School Mathematics

    problem solving in school mathematics

  2. Primary Problem-Solving in Mathematics 97818465418 |SchoolDepot.co.uk

    problem solving in school mathematics

  3. Mathematics Problem-solving Challenges For Secondary School Students

    problem solving in school mathematics

  4. Math Problem Solving 101

    problem solving in school mathematics

  5. Primary Problem Solving Poster

    problem solving in school mathematics

  6. Teaching Problem Solving in Math: 5 Strategies For Becoming a Better

    problem solving in school mathematics

COMMENTS

  1. 6 Tips for Teaching Math Problem-Solving Skills

    1. Link problem-solving to reading When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem.

  2. Teaching Mathematics Through Problem Solving

    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.

  3. Problem Solving in Mathematics Education

    Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students' development of mathematical knowledge and problem solving competencies.

  4. PDF Developing mathematical problem-solving skills in primary school by

    referred to as a framework to underline different -solving; phases of problem mathematics is more than just filling in the textbook, it could be understood as an activity. Devising a plan and choosing the most appropriate heuristic were supported by visual tools called Problem-solving Keys, which are introduced in Chapter 3.2.

  5. Problem solving in the mathematics curriculum: From domain‐general

    Problem solving is widely regarded as a fundamental feature within the school mathematics curriculum. However, there is considerable disagreement over what exactly problem solving is, and if and how it can be taught.

  6. Problem Solving

    Brief Problem solving plays an important role in mathematics and should have a prominent role in the mathematics education of K-12 students. However, knowing how to incorporate problem solving meaningfully into the mathematics curriculum is not necessarily obvious to mathematics teachers.

  7. PDF Problem solving in mathematics

    Problem solving in mathematics: realising the vision through better assessment June 2016 Introduction Problem solving is an important component of mathematics across all phases of education. In the modern world, young people need to be able to engage with and interpret data and information.

  8. Problem solving in the mathematics curriculum: From domain‐general

    Problem solving is widely regarded as a fundamental feature within the school mathematics curriculum. However, there is considerable disagreement over what exactly problem solving is, and if and how it can be taught. ... of domain-general problem-solving strategies and has the potential to enable all students to benefit from a powerful problem ...

  9. Mathematics Through Problem Solving

    Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics listed at the outset of this article: functional, logical and aesthetic. Let us consider how problem solving is a useful medium for each of these.

  10. PDF Cognitive flexibility: exploring students' problem-solving in

    The vision of mathematics education in Indonesia is to understand mathematical concepts and ideas that are then applied to routine and non-routine problem-solving through the development of reasoning, communication, and connections inside and outside mathematics (Simamora et al., 2018).

  11. Teaching of Problem Solving in School Mathematics Classrooms

    As a result of the publication of Polya's book about solving mathematics problems in 1954, National Council of Teachers of Mathematics and the worldwide educational reforms in school mathematics have recommended the study of problem solving at all levels of the mathematics curriculum.

  12. Problem Solving in Mathematics Education

    Mathematical problem solving is a research and practice domain in mathematics education that fosters an inquisitive approach to develop and comprehend mathematical knowledge Santos-Trigo 2007. As a research domain, the problem-solving agenda includes analyzing cognitive, social, and affective components that influence and shape the learners ...

  13. 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.

  14. PDF On Teaching Problem Solving in School Mathematics

    ards for School Mathematics (NCTM, 2000), problem solving is mentioned as a teaching method with which one can improve the quality of mathematics teaching in school. The key ideas of problem solving seem to have spread around the world, as we can see in the published overview papers. In the last ten years, a number of

  15. PDF Fostering Problem Solving and Critical Thinking in Mathematics Through

    The role of problem solving in Mathematics is undisputed: especially when applications are involved, the ... From digital mate training experience to alternating school work activities. Mondo Digitale, 15(64), pp. 63-82. Camiller, P. and Popper, K., 1999. All Life is Problem Solving. Routledge, London, DOI: 10.4324/9780203431900

  16. Reflections on Problem-Solving

    However, problem solving in school mathematics is a totally different activity than the research mathematician's activity. Because of the currently prevailing educational policy, passing the mathematical exams has become the top priority of teachers and students. The main goal of teaching is, therefore, to prepare the students to pass the exams.

  17. Problematizing teaching and learning mathematics as "given" in STEM

    Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way.

  18. ERIC

    The twenty-two essays on problem solving included in this publication give ideas and examples for use in the classroom at all levels of instruction. Topics discussed in the essays include but are not limited to: problem posing, clues from research, story problems, calculators, textbook problems, and measuring problem solving. Polya's heuristics as well as an annotated bibliography are also ...

  19. Problem Solving in School Mathematics

    Problem Solving in School Mathematics, Volume 42 Yearbook (National Council of Teachers of Mathematics) Contributor: National Council of Teachers of Mathematics: Publisher: National Council of Teachers of Mathematics, 1980: Original from: the University of California: Digitized: Jun 12, 2008: ISBN: 0873531620, 9780873531627: Length: 241 pages ...

  20. PDF A Problem Solving Approach to Mathematics for Elementary School

    An often-used strategy in elementary school mathematics is making a table A table can be used to look for patterns that emerge in the problem, which in turn can lead to a solution. ... A Problem Solving Approach to Mathematics for Elementary School Teachers, Thirteenth Edition, Chapter 1, An Introduction to Problem Solving

  21. On Teaching Problem Solving in School Mathematics

    Problem Solving On Teaching Problem Solving in School Mathematics Center for Educational Policy Studies Journal License CC BY 4.0 Authors: erkki pehkonen liisa näveri Anu Laine University of...

  22. Elevating Math Education Through Problem-Based Learning

    The Traditional Approach. Problem-based learning has a rich history in American education, with John Dewey laying the theoretical groundwork in 1916 and McMaster University pioneering the PBL program for medical education in 1969. More recently, the National Council of Teachers of Mathematics published Principles and Standards for School Mathematics in 2000, setting forth a vision that ...

  23. Problem Solving in School Mathematics

    Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint) The goals of this chapter are (1) to outline and substantiate a broad conceptualization of what it means to think mathematically, (2) to summarize the literature relevant to understanding….

  24. Creative Problem Solving in School Mathematics

    Amazon.com: Creative Problem Solving in School Mathematics: 9781882144105: George Lenchner Stores › Education › Higher Education Enjoy fast, FREE delivery, exclusive deals and award-winning movies & TV shows with Prime Try Prime and start saving today with Fast, FREE Delivery Buy new: $44.95 FREE Returns FREE delivery Wednesday, November 22

  25. Some Texas schools try new way to teach math to students

    Third grade teacher Eran McGowan works through math problems with his students at the Eddie Bernice Johnson STEM Academy in Dallas on Feb. 5. Credit: Azul Sordo for The Texas Tribune

  26. Middle school students' mathematical problem-solving ability and the

    Table 1 shows that the mathematics problem-solving ability of middle school students in Z province in mainland China is relatively good; the majority of students' mathematical problem-solving skills are at a moderate to high level, and 48% of them have reached the A level, 35% the B level, and 13% the C level, with only 4% having located in ...

  27. Studies recommend increased research into achievement, engagement to

    A new study into classroom practices, led by Dr. Steve Murphy, has found extensive research fails to uncover how teachers can remedy poor student engagement and perform well in math.

  28. 10 Best strategies for solving math word problems in 2024

    2. Identify Key Information and Variables. 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.