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5 Teaching Mathematics Through Problem Solving

Janet Stramel

Problem Solving

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

Needlepoint of cats

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

Back of a needlepoint

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Teacher teaching a math lesson

Choosing the Right Task

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

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

outline five importance of problem solving approaches mathematics

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

  • Inside Mathematics Number Talks
  • Number Talks Build Numerical Reasoning

Light bulb

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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Module 1: Problem Solving Strategies

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

Pólya’s How to Solve It

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

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

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

outline five importance of problem solving approaches mathematics

Looking back: How would you find the nth term?

outline five importance of problem solving approaches mathematics

Find the 10 th term of the above sequence.

Let L = the tenth term

outline five importance of problem solving approaches 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

Cambridge University Faculty of Mathematics

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
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For younger learners

  • Early Years Foundation Stage

Advanced mathematics

  • Decision Mathematics and Combinatorics
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Published 2018

The Problem-solving Classroom

  • Visualising
  • Working backwards
  • Reasoning logically
  • Conjecturing
  • Working systematically
  • Looking for patterns
  • Trial and improvement.

outline five importance of problem solving approaches mathematics

  • stage of the lesson 
  • level of thinking
  • mathematical skill.
  • The length of student response increases (300-700%)
  • More responses are supported by logical argument.
  • An increased number of speculative responses.
  • The number of questions asked by students increases.
  • Student - student exchanges increase (volleyball).
  • Failures to respond decrease.
  • 'Disciplinary moves' decrease.
  • The variety of students participating increases.  As does the number of unsolicited, but appropriate contributions.
  • Student confidence increases.
  • conceptual understanding
  • procedural fluency
  • strategic competence
  • adaptive reasoning
  • productive disposition

outline five importance of problem solving approaches mathematics

ED

E ducational D esigner

Journal of the international society for design and development in education, introduction, the value of critiquing alternative problem solving strategies., development of the problem solving lessons: the designers’ remit, an example of a problem-solving lesson., sample and data collection, potential uses of “sample student work”, the design and form of sample student work, students needed exposure to a wide range of methods, difficulties in using sample student work in the classroom., discussion of the design issues raised, acknowledgements, about the authors.

Note: If you browse this site it will use temporary 'session cookies' to assist navigation and keep an anonymous record of page visits. More info…

Sheila Evans

Developing students’ strategies for problem solving in mathematics:

The role of pre-designed “sample student work”, sheila evans and malcolm swan centre for research in mathematics education university of nottingham, england.

This paper describes a design strategy that is intended to foster self and peer assessment and develop students’ ability to compare alternative problem solving strategies in mathematics lessons. This involves giving students, after they themselves have tackled a problem, simulated “sample student work” to discuss and critique. We describe the potential uses of this strategy and the issues that have arisen during trials in both US and UK classrooms. We consider how this approach has the potential to develop metacognitive acts in which students reflect on their own decisions and planning actions during mathematical problem solving.

An accompanying paper in this volume ( Swan & Burkhardt 2014 ) outlines the rationale, design and structure of the lesson materials developed in the Mathematics Assessment Project (MAP) [1] . In short, the MAP team has designed and developed over one hundred Formative Assessment Lessons (FALs) to support US Middle and High Schools in implementing the new Common Core State Standards for Mathematics. Each lesson consists of student resources and an extensive teacher guide. About one-third of these lessons involves the tackling of non-routine, problem-solving tasks. The aim of these lessons is to use formative assessment to develop students’ capacity to apply mathematics flexibly to unstructured problems, both from pure mathematics and from the real world. These non-routine lessons are freely available on the web: http://map.mathshell.org.uk

One challenge in designing the FALs was to incorporate aspects of self and peer-assessment, activities that have regularly been associated with significant learning gains ( Black & Wiliam 1998a ). These gains appear to be due to the reflective, self-monitoring or metacognitive habits of mind generated by such activity. As Schoenfeld ( 1983 , 1985, 1987, 1992 ) demonstrated, expert problem solvers frequently engage in metacognitive acts in which they step back and reflect on the approaches they are using. They ask themselves planning and monitoring questions, such as: ‘Is this going anywhere? Is there a helpful way I might represent this problem differently?’ They bring to mind alternative approaches and make selections based on prior experience. In contrast, novice problem solvers are often observed to become fixated on an approach and pursue it relentlessly, however unprofitably. Self and peer assessment appear to allow students to step back in a similar manner and allow ‘ working through tasks’ to be replaced by ‘ working on ideas’ . Our design challenge was therefore to incorporate opportunities into our lessons for students to develop the facility to engage in metacognitive acts in which they consider and evaluate alternative approaches to non-routine problems.

One of the practices from the Common Core State Standards that we sought to specifically address in this way, was: Construct viable arguments and critique the reasoning of others. Part of this standard reads as follows:

Mathematically proficient students are able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. ( NGA & CCSSO 2010 , p. 6)

A possible design strategy was to construct “sample student work” for students to discuss, critique and compare with their own ideas. In this paper we describe the reasons for this approach and the outcomes we have observed when this was used in classroom trials.

In a traditional classroom, a task is often used by the teacher to introduce a new technique, then students practice the technique using similar tasks. This is what some refer to as ‘Triple X’ teaching: ‘exposition, examples, exercises.’ There is no need for the teacher to connect or compare alternative approaches as it is predetermined that all students will solve each task using the same method. Any student difficulties are unlikely to surprise the teacher. This is not the case in a classroom where students employ different approaches to solve the same non-routine task; the teacher’s role is more demanding. Students may use unanticipated solution-methods and unforeseen difficulties may arise.

The benefits of learning mathematics by understanding, critiquing, comparing and discussing multiple approaches to a problem are well-known ( Pierce, et al. 2011 ; Silver, et al. 2005 ). Two approaches are commonly used: inviting students to solve each problem in more than one way, and allowing multiple methods to arise naturally within the classroom then having these discussed by the class. Both methods are difficult for teachers.

Instructional interventions intended to encourage students to produce alternative solutions have proved largely unsuccessful ( Silver, et al. 2005 ). It has been found that not only do students lack motivation to solve a problem in more than one way, but teachers are similarly reluctant to encourage them to do so ( Leikin & Levav-Waynberg 2007 ).

The second, perhaps more natural, approach is for students to share strategies within a whole class discussion. In Japanese classrooms, for example, lessons are often structured with four key components: Hatsumon (the teacher gives the class a problem to initiate discussion); Kikan-shido (the students tackle the problem in groups or individually); Neriage (a whole class discussion in which alternative strategies are compared and contrasted and through which consensus is sought) and finally the Matome , or summary ( Fernandez & Yoshida 2004 ; Shimizu 1999 ). Among these, the Neriage stage is considered to be the most crucial. This term, in Japanese refers to kneading or polishing in pottery, where different colours of clay are blended together. This serves as a metaphor for the considering and blending of students’ own approaches to solving a mathematics problem. It involves great skill on the part of the teacher, as she must select student work carefully during the Kikan-shido phase and sequence the work in a way that will elicit the most profitable discussions. In the Matome stage of the lesson, the Japanese teachers will tend to make a careful final comment on the mathematical sophistication of the approaches used. The process is described by Shimizu:

Based on the teacher’s observations during Kikan-shido, he or she carefully calls on students to present their solution methods on the chalkboard, selecting the students in a particular order. The order is quite important both for encouraging those students who found naive methods and for showing students’ ideas in relation to the mathematical connections among them. In some cases, even an incorrect method or error may be presented if the teacher thinks this would be beneficial to the class. Once students’ ideas are presented on the chalkboard, they are compared and contrasted orally. The teacher’s role is not to point out the best solution but to guide the discussion toward an integrated idea. ( Shimizu 1999 , p110)

In part, perhaps, influenced by the Japanese approaches, other researchers have also adopted similar models for structuring classroom activity. They too emphasize the importance of: anticipating student responses to cognitively demanding tasks; careful monitoring of student work; discerning the mathematical value of alternative approaches in order to scaffold learning; purposefully selecting solution-methods for whole class discussion; orchestrating this discussion to build on the collective sense-making of students by intentionally ordering the work to be shared; helping students make connections between and among different approaches and looking for generalizations; and recognizing and valuing students’ constructed solutions by comparing this with existing valued knowledge, so that they may be transformed into reusable knowledge ( Brousseau 1997 ; Chazan & Ball 1999 ; Lampert 2001 ; Stein, et al. 2008 ). However, this is demanding on teachers. The teachers’ concern that students participate in these discussions by sharing ideas with the whole class often becomes the main goal of the activity. Often researchers observe teachers sticking to a ‘show and tell’ approach rather than discussing the ideas behind the solutions in any depth. Student talk is often prioritized over peer learning ( Stein, et al. 2008 ). Merely accepting answers, without attempting to critique and synthesize individual contributions does guarantee participation, is less demanding on the teacher, but can constrain the development of mathematical thinking ( Mercer 1995 )

In our work prior to the Mathematics Assessment Project (MAP) project, however, we have found that approaches which rely on teachers selecting and discussing students’ own work are problematic when the mathematical problems are both non-routine and involve substantial chains of reasoning. Teachers have only limited time to spend with each group during the course of a lesson. They find it extremely difficult to monitor and interpret extended student reasoning as this can be poorly articulated or expressed. Most of the ‘problems’ discussed in the research literature are short and contain only a few steps, so the selection of student work is relatively straightforward. We have attempted to tackle this issue by suggesting teachers allow students time to work on the problems individually in advance of the lesson, and then collect in these early ideas and attempt to interpret the approaches before the formative assessment lesson itself. This time gap does allow teachers an opportunity to anticipate student responses in the lesson and prepare formative feedback in the form of written and oral questions. In addition, we have suggested that group work is undertaken using shared resources and is presented on posters so that student reasoning becomes more visible to the teacher as he or she is monitoring work. The selection and presentation of student approaches remains difficult however, partly because the responses are so complex that other students have difficulty understanding them. We often witness ‘show and tell’ events where the students present their approach only to be greeted with a silent incomprehension from their peers.

One possible solution we explore in the rest of this paper, is the use of pre-prepared “sample student work”. This is carefully designed, handwritten material that simulates how students may respond to a problem. The handwritten nature conveys to students that this work may contain errors and may be incomplete. The task for students is to critique each piece and compare the approaches used, with each other and with their own, before returning to improve their own work on the problem.

Here, we explore the use of sample student work in the classroom. We first describe how the sample student work fits into the design of a problem solving FALs; then consider its potential uses, its design and form and then the difficulties that have been observed as it has been used within the classroom. We conclude by discussing the design issues raised and possible directions for future research.

The design of the MAP lessons has been explained elsewhere in this volume ( Swan & Burkhardt 2014 ), so we refrain from repeating that here. The process was based on design research principles, involving theory-driven iterative cycles of design, enactment, analysis and redesign ( Barab & Squire 2004 ; Bereiter 2002 ; Cobb, et al. 2003 ; DBRC 2003 , p. 5; Kelly 2003 ; van den Akker, et al. 2006 ). Each lesson was developed, through two iterative design cycles, with each lesson being trialed in three or four US classrooms between each revision. Revisions were based on structured, detailed feedback from experienced observers of the materials in use in classrooms. The intention was to develop robust designs that may be used more widely by teachers, without further support.

Figure 1

The remit for the designers was to create lessons that had clarity of purpose and would maximize opportunities for students to make their reasoning visible to each other and their teacher. This was intended to ensure the alignment of teacher and student learning goals, to enable teachers to adapt and respond to student learning needs in the classroom, and to enable teachers to follow-up lessons appropriately ( Black & Wiliam 1998a , 1998b; Leahy, et al. 2005 ; Swan 2006 ). The lessons were designed to draw on a range of important mathematical content, be engaging and feature high-level cognitive challenges. They were intended to be accessible, allowing multiple entry points and solution strategies. This allowed students to approach the task in different ways based on their prior knowledge. The lessons were also designed to encourage decision-making, leading to a sense of student ownership. Opportunities for students to conjecture, review and make connections were embedded. Finally, the lessons were designed to provide opportunities for students to compare and critique multiple solution-methods ( Figure 1 Figure 1 ).

Research indicates that it is not sufficient for teachers to be simply handed non-routine tasks. Lessons such as these can proceed in unexpected ways and, without teacher guidance, can often result in teachers reducing the cognitive demands of the task and the corresponding learning opportunities ( Stein, et al. 1996 ). In order to support teachers in developing skills to successfully work with these lessons, detailed guides were written. The guides outline the structure of each lesson, clearly stating the designers’ intentions, suggestions for formative assessment, examples of issues students may face and offering detailed pedagogical guidance for the teacher.

Figure 1 (1/4)

In Figure 2 we offer one example of a problem-solving task [2] , and below outline a typical lesson structure:

  • An unscaffolded problem is tackled individually by students Students are given about 20 minutes to tackle the problem without help, and their initial attempts are collected in by the teacher.
  • Teachers assess a sample of the work The teacher reviews the sample and identifies the main issues that need addressing in the lesson. We describe the common issues ( Figure 3 ) that arise and suggest questions for the teacher to use to move students’ thinking forward. (In Having Kittens , these included: not developing a suitable representation, working unsystematically, not making assumptions explicit and so on).
  • Groups work on the problem The teacher asks students to work together, sharing their initial ideas and attempt to arrive at a joint, group solution, that they can present on a poster. The pre-prepared strategic questions are posed to students that seem to be struggling.
  • Students share different approaches Students visit each other’s posters and groups explain their approach. Alternatively a few group solutions may be displayed and discussed. This may help for example, to begin discussions on the assumptions made, and so on.

Figure 4

  • Students discuss sample student work Students are given a range of sample student work that illustrate a range of possible approaches ( Figure 4 ). They are asked to complete, correct and/or compare these. In the Kittens example, students are asked to comment on the correct aspects of each piece, the assumptions made, and how the work may be improved. The teacher’s guide contains a detailed commentary on each piece. For example, for Wayne’s solution, the guide says: Wayne has assumed that the mother has six kittens after 6 months, and has considered succeeding generations. He has, however, forgotten that each cat may have more than one litter. He has shown the timeline clearly. Wayne doesn’t explain where the 6-month gaps have come from.
  • Students improve their own solutions Students are given a further opportunity to act on what they have learned from each other and the sample student work.
  • Whole class discussion to review learning points in the lesson The teacher holds a class discussion focusing on some aspects of the learning. For example, he or she may focus on the role of assumptions, the representations used, and the mathematical structure of the problem. This may also involve further references to the sample student work.
  • Students complete a personal review questionnaire This simply invites students to reflect on how their understanding of the problem has evolved over the lesson andwhat they have learned from it.

Figure 2

Collect students’ responses to the task. Make some notes on what their work reveals about their current levels of understanding, and their different problem solving approaches. The purpose of doing this is to forewarn you of issues that will arise during the lesson itself, so that you may prepare carefully. We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it will encourage students to compare their scores and will distract their attention from what they can do to improve their mathematics To, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this lesson unit. (extract from the Teacher Guide)

By drawing attention to common issues, the contents of the table can also support teachers to scaffold students learning both during the collaborative activity and whole class discussions.

outline five importance of problem solving approaches mathematics

Altogether, these formative assessment lessons were trialed by over 100 teachers in over 50 US schools. During the third year of the project, many of the problem solving lessons were also taught in the UK by eight secondary school teachers, with first-hand observation by the lesson designers.

Although teachers in all of these trials were invited to teach the lesson as outlined in the guide, we also made it clear that teachers should feel able to adapt the materials to accommodate the needs, interests and previous attainment of students, as well as the teacher’s own preferred ways of working. We recognized that teachers play the central role in transforming the design intentions and, inevitably, that some of these transformations would surprise the designers .

Figure 5

We examined all available observer reports on the problem solving lessons and elicited all references to sample student work. These comments were then categorized under specific themes such as ‘Errors in Sample Student Work’ or ‘Questions for students to answer about sample student work’. Additionally, observers completed a questionnaire ( Figure 5 Figure 5 ) designed specifically to help designers better understand how teachers use the sample student work and the supporting guide, and how this use has evolved over the course of the project. This data forms the basis of the findings from the US lesson trials.

outline five importance of problem solving approaches mathematics

The analysis of the UK data is ongoing. Before and after each FAL teachers were interviewed using a questionnaire ( Figure 6 Figure 6 ) intended to help designers better comprehend key teacher behaviors and understandings, such as how the teacher prepared for the lesson, what she perceived as the ‘big mathematical ideas’ of the lesson, what she had learnt from the lesson. At the end of the one-year project, teachers were interviewed about their experiences. Again the questions asked were shaped by the literature and issues that had arisen over the course of the project. For example, how teachers used the guide and their opinions on the sample student work. At the time of writing, all the final interviews have been analyzed, as have the pre and post lesson responses made by two of the teachers. We have also developed a framework to analyze whole class discussions. Twelve class discussions have been analyzed. This data forms the basis of the findings from the UK lesson trials.

outline five importance of problem solving approaches mathematics

During the refinement of the lessons we have gradually become more aware that the purpose of sharing student approaches needs to be made explicit. By combining purposes inappropriately, we can undermine their effect. For example, if a sample approach is full of errors, the student may become so absorbed in working through the sample work that they fail to make comparisons between different pieces of work.

The following list describes some of the reasons we have designed sample student work:

To encourage a student that is stuck in one line of thinking to consider others If a student has struggled for some time with a particular approach, teachers are often tempted to suggest a specific approach. This can lead to subsequent imitative behavior by students. Alternatively the teacher may ask the student to consider other students’ attempts to solve the problem. This offers fresh insight and help without being directive.

“For students who have had trouble coming up with a solution, having the sample student work has helped them think of a way to organize or get started with the task. Since these students are having trouble getting a solution, they usually look over the various sample student work and pick one with which they feel most comfortable. Having Kittens was one task where students benefitted by seeing how other students organized their thinking”. (Observer comment from questionnaire)

Figure 7

To enable a student to make connections within mathematics Different approaches to a problem can facilitate connections between different elements of knowledge, thereby creating or strengthening networks of related ideas and enabling students to achieve ‘a coherent, comprehensive, flexible and more abstract knowledge structure’ ( Seufert, et al. 2007 ).

“I did not routinely, except perhaps at A level, make connections between topics and now I am trying to incorporate this into my practice at a much lower level. The sample student work highlighted how traditional my approach was and how I followed quite a linear route of mathematical progression” (UK teacher during end-of-project interview)

Figure 7 Figure 7 shows an example of sample solutions provided in the FALs that provide students with opportunities to connect and compare different representations.

outline five importance of problem solving approaches mathematics

  • To signal to students that mistakes are part of learning In so doing the stigma attached to being wrong may be reduced ( Staples 2007 ).
  • To draw attention to common mathematical misconceptions A sample piece of student work may be chosen or carefully designed to embody a particular mathematical misconception. Students may then be asked to analyse the line of reasoning embedded in the work, and explain its defects.
  • To compare alternative representations of a problem For modelling problems, many different representations are possible during the formulation stage. Typically these include verbal, diagrammatic, graphical, tabular and algebraic representations. Each has its own advantages and disadvantages, and through the comparison of these over a succession of problems, students may become more able to appreciate their power.
  • To compare hidden assumptions It is often helpful to offer students two correct responses to a problem that arrive at very different solutions solely because different modelling assumptions have been made. This draws attention to the sensitivity of the solution to the variables within the problem. An example of this is provided by the sample solutions in Figure 3 .
  • To draw students attention to valued criteria for assessment. Particularly when using tasks that involve problem solving and investigation, students often remain unsure of the educational purpose of the lesson and the criteria the teacher is using to judge the quality of their work ( Bell, et al. 1997 ). If they are asked, for example, to rank-order several pieces of sample student work according to given criteria (such as accuracy, quality of communication, elegance) they become more aware of such criteria. This can contribute significantly to the alignment of student and teacher objectives ( Leahy, et al. 2005 ). Also, engaging in another student’s thinking may strengthen students’ self-assessment skills.

Research suggests that students’ self-assessment capabilities may be enhanced if they are provided with existing solutions to work through and reflect upon. Carroll (1994) , for example, replaced students working through algebra problems with students studying worked examples. This was shown to be particularly effective with low-achievers because it reduced the cognitive load and allowed students to reflect on the processes involved.

In our work we have frequently found it necessary to design the ‘student work’ ourselves, rather than use examples taken straight from the classroom. This is often to ensure that the focus of students’ discussion will remain on those aspects of the work that we intend. For example, the work must be clear and accessible, if other students are to be able to follow the reasoning. If each piece of work is overlong, then students may find it difficult to apprehend the work as a whole, so that comparisons become difficult to make. If our created student work is too far removed (too easy or too difficult) from what the students themselves would or could do, then it loses credibility.

It was felt important to use handwritten work, as this communicates to students that the work is freshly created and has not been polished for publication. It reduces the perceived ‘authority’ of the mathematics presented, increases the likelihood that it may contain errors and introduces a third ‘person’ to the classroom who is unknown to the students. This anonymity can be advantageous; students do not know the mathematical prowess of the author. If it is known that a student with an established reputation for being ‘mathematically able’ has authored a solution then most will assume the solution is valid. Anonymity removes this danger. Making ‘student work’ anonymous also reduces the emotional aspects of peer review. Feedback from our early trials indicated that sometimes students were reserved and over-polite about one another’s work, reluctant to voice comments that could be perceived as negative. When outside work was introduced, they became more critical.

In the US trials, we found that, within a single class, the solution methods used by students were often similar in kind. This may be partly due to the common practice of US teachers to focus exclusively on each topic area for an extended period, thus making it likely that students will draw from that area when solving a problem. Alternatively, students may choose to use a solution method they assume is particularly valued, even when this might be inappropriate. The following observer comment would suggest that a numerical solution would be favored over a geometric one, for example:

Due to the ‘traditional’ approaches generally used here in the States, many teachers believe that ‘geometric’ solutions are NOT showing rigor or intelligence and that number is the best way. Students have internalized this… (Observer report)

In our experience, students are unlikely to draw autonomously on methods they are still unsure of or they have only just learned. The mathematics they choose to use will often relate back to mathematics used in earlier years. They may frequently resort, for example, to safe and inefficient ‘guess and check’ numerical methods, that they know they can rely on, rather than graphical or algebraic methods.

The difficulty of transferring methods from one context to another is a common theme in the research literature. For example, students may know how to figure out the gradient, intercept and the equation of a graph, but still find it challenging to recall and apply these concepts to a ‘real-world’ problem. One reason for the low degree of transfer is that students often recall concepts in a situation-specific manner, focusing mainly on surface features ( Gentner 1989 ; Medin & Ross 1989 ) rather than on the underlying mathematical principles. Our UK study supports these findings. On several occasions teachers taught a concept, in advance of the lesson, that they considered would help students to solve the problem and were subsequently surprised that students decided not to use it! Clearly, successful problem solving is not just about students’ knowledge - it is about how, when and whether they decide to use it ( Schoenfeld, 1992 , p. 44).

In the few cases where students did use a wide range of approaches, these rarely included strategies to match all the learning goals of the lesson. For example, students did not necessarily select different representations of the same concept, or use efficient, elegant or generalizable strategies. The mathematical learning opportunities were therefore limited.

For the above reasons we concluded that some fresh input of methods needed to be introduced into the classroom if students were to have opportunities to discuss alternative representations and powerful methods. This could perhaps come from the teacher, but that would then almost certainly remove the problem from students and result in students imitating the teacher’s method. Sample student work provides an alternative input that, as we have said, carries less authority.

In this section we outline a few of the main difficulties we observed when sample student work was used in US and UK classrooms.

Students were analyzing work in superficial ways

In our first version of the teacher’s guide, we suggested that the teacher could introduce the sample work to the class by writing the following instructions on the board:

Imagine you are the teacher and have to assess this work. Correct the work and write comments on the accuracy and organization of each response. Make some specific suggestions as to how the work may be improved.

Feedback from the US trials indicated that these instructions were inadequate. Teachers and students were not clear on the purposes of the activity, and student responses were superficial. For example, observers reported US teachers asking:

What is the math we want to have a conversation about? Do we want students to explain the method? Do we want each piece to stand-alone or should students compare and contrast strategies?

Observers reported that students were not digging deeply enough into the mathematics of each sample and, unless asked a direct question by the teacher, they often worked in silence, looking for errors without evaluating the overall solution strategy. Some students mimicked the feedback they often received from their teacher, providing comments such as ‘Awesome’, ‘Good answer’ or ‘Show a little more work’. A clear message came from the observers; the prompts in the guide needed to be more explicit and focus on the mathematics of the problem; scaffolding was required. The decision was therefore made to include more specific questions, such as:

What piece of information has Danny forgotten to use? What is the purpose of Lydia’s graph? What is the point of figuring out the slope and intercept?

Such questions appeared to make the purpose more discernable to teachers. Feedback from the US observers to these changes was encouraging:

I think the questions or prompts about each piece of student work really focus the students on the thinking, bring out the key mathematics and are a great improvement to the original lesson…Last year students just made judgment statements, but this year the comments were focused on the mathematics.

Not all teachers shared this view, however. In the UK, one teacher commented:

Students are being forced along a certain path as a way to engage with the sample student work. Rather, they [the questions] should be more open and students are then able to comment in any way they like. …. I think sometimes they feel themselves kind of shoehorning in certain types of answer.

This teacher preferred to simply ask students to explain the approach; describe what the student had done well and suggest possible improvements. This practice did encourage engagement, and students’ assessment criteria were made visible to the teacher, but at times the learning goals of the lesson were only superficially attended to.

In both the US and UK, many students focused on the appearance of the work, rather than on its content, with comments on the neatness of diagrams and handwriting. Many commented that the sample work was poorly explained, but did not go on to say clearly how it should be improved. Sample comments were: ‘she needs to explain it better’; ‘the diagrams should not be all over the place’. We attempted to remedy this by suggesting that, rather than just making suggestions for improvement, students should actually make improvements. One teacher commented that this focus on effective mathematical communication had resulted in her students writing fuller explanations when solving problems for themselves.

Students were focused on correcting errors, while ignoring holistic issues

The feedback from observers on the use of errors in sample student work presented us with a more complex issue. Observers commented that when understandings were fragile, the errors often made ‘the most complicated ideas more complicated’. It also became apparent from US feedback that when errors were found in sample student work, some students dismissed the solutions as undeserving of further analysis. Similarly, in UK classrooms students and teachers often assumed the only goal of the activity was to locate and correct errors. One UK teacher commented that when the student work was error-free, students were more inclined to make holistic comparisons of strategic approach.

This led us to look carefully at our purposes in using errors. We had originally included two different kinds of errors: procedural and conceptual. Procedural errors are common arithmetic or algebraic mistakes. Conceptual errors are symptoms of incorrect reasoning and are often more structural in nature. In response to feedback from observers, we removed many of the procedural errors. In many cases, however, the design decision was taken to retain conceptual errors that encourage students to understand the solution-method and its purpose.

For example, Figure 10 shows a problem solving task and Figure 11 shows three samples of student work. Each sample contains a conceptual error. Included in the guide is an explanation of these errors:

Figure 10

Ella draws a sample space in the form of an organized table. Although Ella clearly presents her work, she makes the mistake of including the diagonals. This means the same ball is selected twice. This is not possible, as the balls are not replaced. (Teacher’s guide) Anna assumes that there are only two outcomes (that the two balls are the same color or that they are different colors), so that the probabilities are equal. Anna does not take into account the changes in probabilities when a ball is removed from the bag and not replaced. Jordan does not take into account that the first ball is not replaced. When selecting the second ball there are only 5 balls in the bag, so these probability fractions should all have a denominator of 5.

outline five importance of problem solving approaches mathematics

In some lessons we decide that, rather than including errors, we invited students to complete unfinished responses. For example, in the Testing a New Product task ( Figure 12 ) students’ were asked to complete the tables in Penny and Aran’s work and the final column in Harry’s graph ( Figure 13 ).

They were then asked to describe the advantages and disadvantages of each approach to the problem. Most students in a UK trial of the lesson were able to complete the work, they understood the processes, and were able to work out the correct answers. They did however encounter difficulties interpreting the resulting figures in the context of the real-world situation. This struggle prompted students to consider how far each approach is fit for purpose: how well it each one tackles the problem of working with the four variables of packaging, fragrance, gender and preference, and how far useful conclusions may be reached using each approach.

outline five importance of problem solving approaches mathematics

Students were not given time to consider a sufficient range of sample student work

Initial feedback from observers indicated the lessons were taking longer than had been anticipated; teachers were giving out all pieces of sample student work, but there was often insufficient time for students to successfully evaluate and compare the different approaches. In response to this, designers included the following generic text to all lessons guides:

There may not be time, and it is not essential, for all groups to look at all sample responses. If this is the case, be selective about what you hand out. For example, groups that have successfully completed the task using one method will benefit from looking at different approaches. Other groups that have struggled with a particular approach may benefit from seeing another student’s work that uses the same strategy.

These instructions encourage students to critique and reflect on unfamiliar approaches, to explicate a process and to compare their own work with a similar approach; this, in turn could serve as a catalyst to review and revise their own work. Differentiating the allocation of sample student work in this way may however create problems in the whole class discussion, as not all of the students will have worked on the piece of work under discussion. This instruction places pedagogical demands on teachers, however. They have to again make rapid decisions on which piece of work to allocate to each group. In US trials, however, the suggested approach was not followed:

We have some teachers who give all the sample student work and let students choose the order and the amount they do. This might be less common. Others are very controlling and hand out certain pieces to each group. Others like a certain method to solve problems and like to use that one to model. I think this is a function of the teacher’s comfort level with control and students expectations. (Observer report)

It turned out that very few students were allowed sufficient time to work on all the pieces of sample student work or time to evaluate unfamiliar methods.

These issues were also a concern for the UK teachers. At the start of the project some were reluctant to issue all of the sample student work at the same time, for fear that students would be overwhelmed. As one teacher commented:

At the beginning (of the project) it was too much for pupils to take on all the different methods at once. Even towards the end I didn’t always give them all to them. I believed they became unsettled because the task felt too great. I felt they needed to get used to just looking at one piece first. I also picked out pieces of work that I felt within their ability they could access. (Teacher report)

Students were not using the sample student work to improve their own solutions

Although the teachers clearly recognized that a prime purpose of sample student work was to serve as a catalyst for students to ultimately improve their own solutions, there was little evidence of students subsequently changing their work apart from when they noticed numerical errors. While most students acknowledged that their work needed improving, many did not take the next step and improve it. Only students that were stuck were likely to adapt or use a strategy from the sample student work.

The problem solving lessons were designed to involve cycles of refinement of students’ solutions. They attempted the task individually, before the lesson, then in groups, then considered the sample work and then again were urged to improve their work a third time. For teachers that were used to students working through a problem once, then moving on, this was a substantial new demand.

It is clear that communicating complex pedagogic intentions is not easy. It is made easier by having some common framework with reference points. A strategic goal of these lessons was to build this infrastructure in teachers’ minds

Students were often not invited to make comparisons between the sample approaches.

As mentioned earlier in the paper, the design intention is for students to compare alternative problem solving approaches. As such, all lessons include whole-class discussion instructions of the following kind:

Ask students to compare the different methods: Which method did you like best? Why? Which method did you find most difficult to understand? Why? How could the student improve his/her answer? Did anyone come up with a method different from these?

Feedback from both the US and UK classrooms indicate that teachers rarely encouraged students to make such comparisons. There appear to be multiple reasons for this.

Time pressure was a frequently raised issue. Students need sufficient time to identify and reflect on the similarities and differences between methods and connect these to the constraints and affordances of each method in terms of the context of the problem. The whole class discussion was held towards the end of the lesson. These discussions were often brief or non-existent, possibly reflecting how teachers value the activity. A common assumption was that the important learning had already happened, in the collaborative activity.

Another factor may be lack of adequate support in the guide. Research indicates it is not enough to simply suggest that sample student work should be compared, there need to be instructional prompts that draw students’ attention to the similarities and differences of methods ( Chazan & Ball 1999 ). Teachers and students need criteria for comparison to frame the discussion ( Gentner, et al. 2003 ; Rittle-Johnson & Star 2009 ). Furthermore, these prompts should occur prior to the whole-class discussion. Students need time to develop their own ideas before sharing them with the class.

Rather than compare the different pieces of sample student work, UK students were consistently given the opportunity to compare one piece with their own. Students often used the sample to figure out errors either in their own or in the sample itself. One UK teacher noted that when groups were given the sample student work that most closely reflected their own solution-method, their comments appeared to be more thoughtful, whereas with unfamiliar solution-methods students often focused on the correctness of the result or the neatness of the drawing and did not perceive it as a solution-method they would use.

Most of the teachers involved in the trials had never before attempted to ask students to critique work in the ways described above. They reported that ‘getting inside another person’s head’ proved challenging and students learned to do this only gradually.

I think it has taken most of the year to get the kids to actually be able to look at a piece of work and follow it through to see what that person has done …..

One of the profound difficulties for designers is in trying to increase the possibilities for reflective activity in classrooms. The etymology of the word curriculum is from the Latin word for a race or a racecourse, which in turn is derived from the verb currere meaning to run. Perhaps unfortunately, that is precisely what it feels like for most students. The introduction of problem solving in general, and of analyzing sample student work in particular are seen by many as time-consuming activities that detract from the primary goal of improving procedural fluency or ‘learning more stuff’.

We are encouraged, however to see that the new Common Core State Standards place explicit value on the development of problem solving, mathematical practices and, in particular, on students being able to critique reasoning. Most students, we suspect, are not aware of this new agenda. Some years ago, we conducted an experiment to see whether students could identify the purposes of a number of different kinds of mathematics lesson. It became clear that students’ and teachers’ perceptions of the purposes of the lessons were only aligned for procedural mathematics. The mismatch between teacher and student perceptions was more pronounced as lessons became progressively more practices-oriented ( Swan, et al. 2000 ). There was some empirical evidence, however, that by introducing metacognitive activities into the classroom that this mismatch could be reduced. These included such activities as discussing key conceptual obstacles and common errors, explaining errors in sample student work – and orally reviewing the purpose of each lesson.

In this paper, we have seen that, left to themselves, students are unlikely to produce a wide range of qualitatively different solutions for comparison, and therefore it may be helpful to create samples of work to stimulate such reflective discussion. We have, however also noted that we have found it necessary to:

  • discourage superficial analysis, by stating explicitly the purpose of the sample student work, and by asking specific questions that relate to this purpose;
  • encourage holistic comparisons by making the sample student work short, accessible and clear, and by not including arithmetic and other low-level errors that distract the students’ attention away from the identified purpose;
  • make the distribution of the sample student work more effective, by perhaps sequencing it so that successive pairwise comparisons of approaches can be made;
  • offer students explicit opportunities to incorporate what they have learned from the sample work into their own solutions;
  • offer the teachers support for the whole class discussion so that they can identify and draw out criteria for the comparison of alternative approaches.

From a designers’ perspective, it is natural to focus on the challenges in creating a design that may be used effectively by the target audiences. We may thus have given the impression that the lessons have been unsuccessful in achieving their goals. This, however, is far from the truth. These lessons are proving extremely popular with teachers and are currently being used as professional development tools across the US. They are also forming the basis for ‘lesson studies’ in both the US and the UK. In the lesson studies, they are viewed as ‘research proposals’ rather than ‘lesson plans’.

Teachers and observers have described on many occasions the learning they have gained from comparing student work in these lessons; teacher comments include:

I now think pupils can learn more from working with many different solutions to one problem rather than solving many different problems, each in only one way.
It moves away from students chasing the answer.
I can now see how much easier it is for a student to recognize that, say a trial and improvement method is inefficient, when it is compared to a sleek geometrical method rather than when simply looking at the solution on its’ own.

To our knowledge, there are no major studies that focus on how teachers work with a range of pre-written solution-methods for a range of non-routine problems. This study raises many issues and in so doing acts as a launch pad for further more detailed studies. More exploration is required into how the use of sample student work affects pupils’ capacity to solve problems. One might expect to see, for example, that students increase their repertoire of available methods when solving problems. So far, however, we have no evidence of this. We do, however, have some early indications that students are beginning to write clearer and fuller explanations as a result of critiquing sample student work.

We would like to acknowledge the support for the study, the Bill and Melinda Gates Foundation, our co-researchers at the University of Berkeley, California and the observer team.

[1] The Maths Assessment Project, based at UC Berkeley, was directed by Alan Schoenfeld, Hugh Burkhardt, Daniel Pead, Phil Daro and Malcolm Swan, who led the lesson design team which included at various stages Nichola Clarke, Rita Crust, Clare Dawson, Sheila Evans, Colin Foster and Marie Joubert. The work was supported by the Bill & Melinda Gates Foundation; their program officer was Jamie McKee. The US observers who provided the feedback from US classrooms were led by David Foster, Mary Bouck and Diane Schaefer, working with Sally Keyes, Linda Fisher, Joe Liberato and Judy Keeley.

[2] The Having Kittens task used in this lesson was originally designed by Acumina Ltd. ( http://www.acumina.co.uk/ ) for Bowland Maths ( http://www.bowlandmaths.org.uk ) and appears courtesy of the Bowland Charitable Trust.

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Sheila Evans is a member of the Mathematics Assessment Project team in the Centre for Research in Mathematical Education at the University of Nottingham. For the last four years she has worked designing, observing, teaching, revising and providing professional development for the MAP Formative Assessment Lessons. She is currently working on a doctorate using teaching resources that have been shaped by this project. Before that, she taught for fifteen years in secondary schools in the UK and Africa, and wrote a textbook Access to Maths aimed at students without traditional qualifications who wished to study at University .

Malcolm Swan is Director of the Centre for Research in Mathematical Education at the University of Nottingham, which incorporates the Shell Centre for Mathematical Education team. He has led the design teams in a sequence of research and development projects. He led the diagnostic teaching research program that established many of the design principles set out in this paper. In 2008 he was awarded the first ISDDE Prize for educational design, for The Language of Functions and Graphs.

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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

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  • 1 Department of Education, Uppsala University, Uppsala, Sweden
  • 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
  • 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
  • 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

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FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

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TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

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TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

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TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

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.

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

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