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5 Advantages and Disadvantages of Problem-Based Learning [+ Activity Design Steps]

Written by Marcus Guido

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• Teaching Strategies

Advantages of Problem-Based Learning

Disadvantages of problem-based learning, steps to designing problem-based learning activities.

Used since the 1960s, many teachers express concerns about the effectiveness of problem-based learning (PBL) in certain classroom settings.

Whether you introduce the student-centred pedagogy as a one-time activity or mainstay exercise, grouping students together to solve open-ended problems can present pros and cons.

Below are five advantages and disadvantages of problem-based learning to help you determine if it can work in your classroom.

If you decide to introduce an activity, there are also design creation steps and a downloadable guide to keep at your desk for easy reference.

1. Development of Long-Term Knowledge Retention

Students who participate in problem-based learning activities can improve their abilities to retain and recall information, according to a literature review of studies about the pedagogy .

The literature review states “elaboration of knowledge at the time of learning” -- by sharing facts and ideas through discussion and answering questions -- “enhances subsequent retrieval.” This form of elaborating reinforces understanding of subject matter , making it easier to remember.

Small-group discussion can be especially beneficial -- ideally, each student will get chances to participate.

But regardless of group size, problem-based learning promotes long-term knowledge retention by encouraging students to discuss -- and answer questions about -- new concepts as they’re learning them.

2. Use of Diverse Instruction Types

You can use problem-based learning activities to the meet the diverse learning needs and styles of your students, effectively engaging a diverse classroom in the process. In general, grouping students together for problem-based learning will allow them to:

• Address real-life issues that require real-life solutions, appealing to students who struggle to grasp abstract concepts
• Participate in small-group and large-group learning, helping students who don’t excel during solo work grasp new material
• Talk about their ideas and challenge each other in a constructive manner, giving participatory learners an avenue to excel
• Tackle a problem using a range of content you provide -- such as videos, audio recordings, news articles and other applicable material -- allowing the lesson to appeal to distinct learning styles

Since running a problem-based learning scenario will give you a way to use these differentiated instruction approaches , it can be especially worthwhile if your students don’t have similar learning preferences.

3. Continuous Engagement

Providing a problem-based learning challenge can engage students by acting as a break from normal lessons and common exercises.

It’s not hard to see the potential for engagement, as kids collaborate to solve real-world problems that directly affect or heavily interest them.

Although conducted with post-secondary students, a study published by the Association for the Study of Medical Education reported increased student attendance to -- and better attitudes towards -- courses that feature problem-based learning.

These activities may lose some inherent engagement if you repeat them too often, but can certainly inject excitement into class.

4. Development of Transferable Skills

Problem-based learning can help students develop skills they can transfer to real-world scenarios, according to a 2015 book that outlines theories and characteristics of the pedagogy .

The tangible contexts and consequences presented in a problem-based learning activity “allow learning to become more profound and durable.” As you present lessons through these real-life scenarios, students should be able to apply learnings if they eventually face similar issues.

For example, if they work together to address a dispute within the school, they may develop lifelong skills related to negotiation and communicating their thoughts with others.

As long as the problem’s context applies to out-of-class scenarios, students should be able to build skills they can use again.

5. Improvement of Teamwork and Interpersonal Skills

Successful completion of a problem-based learning challenge hinges on interaction and communication, meaning students should also build transferable skills based on teamwork and collaboration . Instead of memorizing facts, they get chances to present their ideas to a group, defending and revising them when needed.

What’s more, this should help them understand a group dynamic. Depending on a given student, this can involve developing listening skills and a sense of responsibility when completing one’s tasks. Such skills and knowledge should serve your students well when they enter higher education levels and, eventually, the working world.

1. Potentially Poorer Performance on Tests

Devoting too much time to problem-based learning can cause issues when students take standardized tests, as they may not have the breadth of knowledge needed to achieve high scores. Whereas problem-based learners develop skills related to collaboration and justifying their reasoning, many tests reward fact-based learning with multiple choice and short answer questions. Despite offering many advantages, you could spot this problem develop if you run problem-based learning activities too regularly.

2. Student Unpreparedness

Problem-based learning exercises can engage many of your kids, but others may feel disengaged as a result of not being ready to handle this type of exercise for a number of reasons. On a class-by-class and activity-by-activity basis, participation may be hindered due to:

• Immaturity  -- Some students may not display enough maturity to effectively work in a group, not fulfilling expectations and distracting other students.
• Unfamiliarity  -- Some kids may struggle to grasp the concept of an open problem, since they can’t rely on you for answers.
• Lack of Prerequisite Knowledge  -- Although the activity should address a relevant and tangible problem, students may require new or abstract information to create an effective solution.

You can partially mitigate these issues by actively monitoring the classroom and distributing helpful resources, such as guiding questions and articles to read. This should keep students focused and help them overcome knowledge gaps. But if you foresee facing these challenges too frequently, you may decide to avoid or seldom introduce problem-based learning exercises.

3. Teacher Unpreparedness

If supervising a problem-based learning activity is a new experience, you may have to prepare to adjust some teaching habits . For example, overtly correcting students who make flawed assumptions or statements can prevent them from thinking through difficult concepts and questions. Similarly, you shouldn’t teach to promote the fast recall of facts. Instead, you should concentrate on:

• Giving hints to help fix improper reasoning
• Questioning student logic and ideas in a constructive manner
• Distributing content for research and to reinforce new concepts
• Asking targeted questions to a group or the class, focusing their attention on a specific aspect of the problem

Depending on your teaching style, it may take time to prepare yourself to successfully run a problem-based learning lesson.

4. Time-Consuming Assessment

If you choose to give marks, assessing a student’s performance throughout a problem-based learning exercise demands constant monitoring and note-taking. You must take factors into account such as:

• Completed tasks
• The quality of those tasks
• The group’s overall work and solution
• Communication among team members
• Anything you outlined on the activity’s rubric

Monitoring these criteria is required for each student, making it time-consuming to give and justify a mark for everyone.

5. Varying Degrees of Relevancy and Applicability

It can be difficult to identify a tangible problem that students can solve with content they’re studying and skills they’re mastering. This introduces two clear issues. First, if it is easy for students to divert from the challenge’s objectives, they may miss pertinent information. Second, you could veer off the problem’s focus and purpose as students run into unanticipated obstacles. Overcoming obstacles has benefits, but may compromise the planning you did. It can also make it hard to get back on track once the activity is complete. Because of the difficulty associated with keeping activities relevant and applicable, you may see problem-based learning as too taxing.

If the advantages outweigh the disadvantages -- or you just want to give problem-based learning a shot -- follow these steps:

1. Identify an Applicable Real-Life Problem

Find a tangible problem that’s relevant to your students, allowing them to easily contextualize it and hopefully apply it to future challenges. To identify an appropriate real-world problem, look at issues related to your:

• Students’ shared interests

You must also ensure that students understand the problem and the information around it. So, not all problems are appropriate for all grade levels.

2. Determine the Overarching Purpose of the Activity

Depending on the problem you choose, determine what you want to accomplish by running the challenge. For example, you may intend to help your students improve skills related to:

• Collaboration
• Problem-solving
• Curriculum-aligned topics
• Processing diverse content

A more precise example, you may prioritize collaboration skills by assigning specific tasks to pairs of students within each team. In doing so, students will continuously develop communication and collaboration abilities by working as a couple and part of a small group. By defining a clear purpose, you’ll also have an easier time following the next step.

3. Create and Distribute Helpful Material

Handouts and other content not only act as a set of resources, but help students stay focused on the activity and its purpose. For example, if you want them to improve a certain math skill , you should make material that highlights the mathematical aspects of the problem. You may decide to provide items such as:

• Data that helps quantify and add context to the problem
• Videos, presentations and other audio-visual material
• A list of preliminary questions to investigate

Providing a range of resources can be especially important for elementary students and struggling students in higher grades, who may not have self-direction skills to work without them.

4. Set Goals and Expectations for Your Students

Along with the aforementioned materials, give students a guide or rubric that details goals and expectations. It will allow you to further highlight the purpose of the problem-based learning exercise, as you can explain what you’re looking for in terms of collaboration, the final product and anything else. It should also help students stay on track by acting as a reference throughout the activity.

5. Participate

Although explicitly correcting students may be discouraged, you can still help them and ask questions to dig into their thought processes. When you see an opportunity, consider if it’s worthwhile to:

• Fill gaps in knowledge
• Provide hints, not answers
• Question a student’s conclusion or logic regarding a certain point, helping them think through tough spots

By participating in these ways, you can provide insight when students need it most, encouraging them to effectively analyze the problem.

6. Have Students Present Ideas and Findings

If you divided them into small groups, requiring students to present their thoughts and results in front the class adds a large-group learning component to the lesson. Encourage other students to ask questions, allowing the presenting group to elaborate and provide evidence for their thoughts. This wraps up the activity and gives your class a final chance to find solutions to the problem.

Wrapping Up

The effectiveness of problem-based learning may differ between classrooms and individual students, depending on how significant specific advantages and disadvantages are to you. Evaluative research consistently shows value in giving students a question and letting them take control of their learning. But the extent of this value can depend on the difficulties you face.It may be wise to try a problem-based learning activity, and go forward based on results.

Create or log into your teacher account on Prodigy -- an adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s used by more than 350,000 teachers and 10 million students. It may be wise to try a problem-based learning activity, and go forward based on results.

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

5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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Multiple Methods

• Posted April 13, 2015
• By Mary Tamer

Do you remember watching your math teacher solve a problem on the blackboard and then diligently trying to copy her technique to solve the other problems on your worksheet? That’s the way many of us learned math. The problem is, we absorbed some counterproductive messages in the process. As it turns out, there isn’t always one best way to solve a given problem.

In his research, Associate Professor Jon Star is pushing hard to craft some new messages, by showing students how important it is to use multiple strategies when solving math problems.

“Math problems can be approached in many different ways,” says Star, an educational psychologist and former math teacher. “When a teacher insists that there is only one way, or only one best way, to solve a problem, students are missing out. There is great value in allowing them to explore and contrast many different ways to solve problems.”

Star and colleague Bethany Rittle-Johnson of Vanderbilt University have conducted a number of studies over the past decade that demonstrate the benefits of comparing a variety of problem-solving approaches for learning math, especially algebra. And their work has paid off: the US Department of Education’s Institute of Education Sciences echoed their findings in two recent publications by the What Works Clearinghouse : a new problem-solving guide for grades 4-8 and a new algebra practice guide for middle and high school students.

Building on this work, Star, Rittle-Johnson, and colleague  Kristie Newton of Temple University developed a set of curriculum materials  designed to be used in middle and high school algebra classrooms. The goal is to expose students to multiple problem-solving strategies and to build deep and flexible mathematical knowledge.

“In math class, you should have opportunities to talk about different approaches, and comparison helps us to think not only about what works in mathematics, but also about how and why things work,” says Star. “Our materials are designed to be used by algebra teachers to supplement their regular curriculum, to provide a stronger focus on the learning of multiple strategies.“ The curriculum materials were developed with middle and high schoolers in mind, but there are some applications for elementary schoolers as well. Educators can access the curriculum online at no cost.

In several recent studies, Star and his colleagues have studied the impact of teachers’ use of these materials on their students’ learning. He calls the results quite promising.

“Our research suggests that using our curriculum materials was not especially difficult for teachers, and that students enjoyed and benefited from the emphasis on multiple strategies,” says Star. “Many teachers already include multiple strategies for certain topics that they teach; our materials are designed to expand this focus across all topics in algebra.”

Additional Resources

• Read " Developing Flexibility in Math Problem Solving ."
• Visit the  Contrasting Cases website .
• Use the  Contrasting Cases curriculum .
• Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

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Using a problem solving approach to teaching and learning maths is of value to all students and especially to those who are high achieving. Some of the reasons for using problem solving are summarised below.

• Problem solving places the focus on the student making sense of mathematical ideas. When solving problems students are exploring the mathematics within a problem context rather than as an abstract.
• Problem solving encourages students to believe in their ability to think mathematically. They will see that they can apply the maths that they are learning to find the solution to a problem.
• Problem solving provides ongoing assessment information that can help teachers make instructional decisions. The discussions and recording involved in problem solving provide a rich source of information about students' mathematical knowledge and understanding.
• Good problem solving activities provide an entry point that allows all students to be working on the same problem. The open-ended nature of problem solving allows high achieving students to extend the ideas involved to challenge their greater knowledge and understanding.
• Problem solving develops mathematical power. It gives students the tools to apply their mathematical knowledge to solve hypothetical and real world problems.
• Problem solving is enjoyable. It allows students to work at their own pace and make decisions about the way they explore the problem. Because the focus is not limited to a specific answer students at different ability levels can experience both challenges and successes on the same problem.
• Problem solving better represents the nature of mathematics. Research mathematicians apply this exact approach in their work on a daily basis.
• Once students understand a problem solving approach to maths, a single well framed mathematical problem provides the potential for an extended period of exploration.

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Open Access

Peer-reviewed

Research Article

The effectiveness of Realistic Mathematics Education approach: The role of mathematical representation as mediator between mathematical belief and problem solving

Contributed equally to this work with: Putri Yuanita, Effandi Zakaria

Roles Conceptualization, Data curation, Funding acquisition, Investigation, Project administration, Resources, Writing – review & editing

Affiliation Department of Mathematics Education, Faculty of Teacher Training and Education, University of Riau, Pekanbaru, Indonesia

Roles Formal analysis, Methodology, Writing – original draft

* E-mail: [email protected]

Affiliation Department of Mathematics and Science Education, Faculty of Education, University of Malaya, Kuala Lumpur, Malaysia

Roles Supervision

Affiliation Department of Innovation in Teaching and Learning, Faculty of Education, National University of Malaysia, Bangi, Selangor, Malaysia

• Putri Yuanita, 
• Hutkemri Zulnaidi, 
• Effandi Zakaria

• Published: September 27, 2018
• https://doi.org/10.1371/journal.pone.0204847
• Reader Comments

This study aims to identify the role of mathematical representation as a mediator between mathematical belief and problem solving. A quasi-experimental design was developed that included 426 Form 1 secondary school students. Respondents comprised 209 and 217 students in the treatment and control groups, respectively. SPSS 23.0, ANATES 4 and Amos 18 were used for data analysis. Findings indicated that mathematical representation plays a significant role as mediator between mathematical belief and arithmetic problem solving. The Realistic Mathematics Education (RME) approach successfully increased the arithmetic problem-solving ability of students.

Citation: Yuanita P, Zulnaidi H, Zakaria E (2018) The effectiveness of Realistic Mathematics Education approach: The role of mathematical representation as mediator between mathematical belief and problem solving. PLoS ONE 13(9): e0204847. https://doi.org/10.1371/journal.pone.0204847

Editor: Christine E. King, University of California Irvine, UNITED STATES

Received: March 2, 2018; Accepted: September 14, 2018; Published: September 27, 2018

Copyright: © 2018 Yuanita et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting Information files. Data availability also from author contacted at [email protected] .

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Education equips younger generations with important skills and knowledge. Effective learning enables students to learn through creative teaching methods and acquire knowledge in class; the latter becomes an exciting activity through the effort of teachers [ 1 ]. Mathematics education motivates students to become critical and innovative and to cultivate sound reasoning in problem solving. Mathematics education is an active, dynamic and continuous process; activities in mathematics education help students develop their reasoning, think logically, systematically, critically and thoroughly and adopt an objective and open attitude when dealing with problems [ 2 ]. Teaching and learning consist of three main components, namely, teachers, students and content. Students must be equipped with knowledge and high-level skills and teachers must possess knowledge and professionalism. Problem-solving skills enable students to think creatively and critically by using progressive and challenging thought processes; creative and critical thinking will help develop a nation and address its needs [ 3 ]. Teaching and learning processes in the classroom serve as a study ground for researchers. A future educator can determine effective teaching methods through this process. Teachers and students in Indonesia acknowledge the need to improve the current status of teaching and learning mathematics. Since 1970, Indonesia has applied a modern approach towards teaching mathematics. However, this approach has created problematic situations in various schools.

Mathematics learning in Indonesia remains below average compared with developing countries in Asia, such as China, Singapore and Malaysia [ 4 ]. In the past, China surpassed other western countries in internationally scaled mathematics achievement, such as in PISA and International Mathematical Olympiads (IMO) [ 5 ]. One of the challenges faced by mathematics teaching is the constantly changing curriculum. Traditional mathematics teaching persists in secondary schools. If the paradigm is to be changed, then teachers must find a teaching and learning approach that is consistent with the constructivist approach. One of these teaching and learning approaches is Realistic Mathematics Education (RME), which was introduced in 2001 in Indonesia by the Realistic Mathematical Education of Indonesia (known as Pendidikan Matematik Realistik Indonesia or PMRI). The goal of PMRI is to revolutionise and improve mathematics education [ 6 ].

The RME approach was first developed by the Freudenthal Institute in the Netherlands in 1971. The RME approach for mathematics is widely known as the best and most detailed approach, which was expanded from the problem-based approach for mathematics education [ 7 ]. Teaching and learning RME have five main criteria, namely, students’ experience in daily life; changing reality to a model and changing the model through a mathematical vertical process before turning it into a formal system; use of students’ active style; use of discussions and question and answer methods to cultivate the mathematics skills of students and formation of a connection between concepts and topics until learning becomes holistic and complete [ 8 ]. Since 2001, many teachers in Indonesia have been trained to use the RME approach. RME has been implemented in 13 of 33 provinces. On the basis of this finding, a study is conducted to develop a teaching module that uses RME and to examine the effects of teaching and learning using the mathematics learning module for secondary schools in Indonesia. Teaching and learning via RME aim to solve the problems faced by teachers and students.

The purpose of RME is to transform mathematics learning into a fun and meaningful experience for students by introducing problems within contexts. RME starts with choosing problems relevant to student experiences and knowledge [ 4 ]. The teacher then acts as a facilitator to help students solve contextual issues. This contextual problem-solving activity brings positive impact to the mathematical representation of students, which is related to their problem solving skills [ 9 , 10 ]. The best way to teach mathematics is to provide students with meaningful experiences by solving the issues they face every day or by dealing with contextual problems. Realistic mathematics education enables the alteration of the mathematical material concept and its relationship. Realistic mathematics education changes the culture towards a dynamic one, but still in the corridor of the educational process. Therefore, realistic mathematics education is an innovative learning approach that emphasises mathematics as a human activity that must be associated with real life using real world context as the starting point of learning [ 11 ].

Mathematical belief is the key idea in the application of mathematical teaching approaches [ 12 ]. The mathematical belief of a student is formed from his or her attitude towards his or her mathematical knowledge, thereby enhancing one’s mathematical value. This view is supported by Anderson, Roger and Klinger [ 13 ], who found that positive mathematical belief influences the performance of secondary school students in Canada. According to The National College of Teachers of Mathematics (NCTM) [ 14 ], this belief influences the ability of students to evaluate their skills, desire to perform mathematical tasks and mathematical disposition. Knowledge of these steps is not enough in performing mathematical tasks because students must also believe in the truth of concepts and procedures. The mathematical belief of students consists of three main factors, namely, students’ belief in their ability, in the mathematical discipline and towards mathematical teaching and learning.

Hwang, Chen, Dung and Yang [ 15 ] defined representation as the process of turning a concrete model in the real world into an abstract concept or symbol. In mathematical psychology, representation refers to the relationship between objects and symbols. The five outer levels used by representation in mathematics education are real-world objects, multiple representation, arithmetic symbol representation, oral representation and picture or graphic representation. The last three representations are abstract and are considered high-level representation in solving arithmetic problems. Ratio with the aid of arithmetic symbol representation involves translating mathematical problems into arithmetic formulas. Language ability representation involves interpreting characters and relationships in mathematical problems into verbal or vocal forms. Picture or graphic representation involves interpreting mathematical problems into pictures or graphics. In this study, the mathematical representations applied by students consist of picture representations, graphic representations, tabular representations, symbolic representations, mathematical notes, written text representations, words and language.

Problem solving is one of the higher-order thinking skills that require students to think critically and creatively [ 16 ]. Ibrahim [ 17 ] claimed that the ability to solve problems involves the use of learned principles to solve problems to achieve certain meanings. In the present study, problem solving skills refer to the ability to solve problems given in the learning context using the RME approach. The problems are based on daily routines and real situations that students were previously aware of. Problem solving skills in this study refer to the ability of students to solve related concepts and procedures in arithmetic problems.

Problem statement

Varying teaching styles increases the difficulty of learning and understanding mathematics. Moreover, students are afraid of mathematics [ 6 ]. The research object in mathematics is abstract and traditional teaching approaches are ill suited for such matters. The unsatisfactory understanding of mathematics and performances of students are attributed to several factors. Firstly, teachers dominate the learning process of a classroom by applying unidirectional and traditional teaching methods. According to Roberg [ 18 ], traditional learning focuses on skill and concept acquisition. Thus, this approach is unsuitable for improving problem solving skills. Secondly, teachers merely present theories and definitions. For example, a theorem is explained through examples and students are assessed through a series of exercises and questions. Teaching is the process of obtaining facts from definitions, attributes and formulas in the mathematics textbook of students. Teachers simply follow the steps given in textbooks without considering whether the process is correct or not. Thus, the learning process becomes mechanical, wherein teachers simply set formulas and solutions for students [ 19 ]. Findings on the application of modern mathematics show that mathematical learning is a low-value learning process [ 6 ].

Mathematical literacy in Indonesia cannot improve with the way mathematics is taught in schools. The current teaching approach does not focus on logical, analytical, systematic, critical and creative thinking among students; rather, teachers simply depend on textbooks [ 20 ]. This approach requires students to memorise the correct steps for answering questions. However, students encounter difficulty when they are given questions that cannot be solved using such steps. The students learn passively and memorise formulas without understanding what the questions actually mean. Thus, they do not benefit from what they are learning and often make mistakes. Zainal [ 21 ] stated that students prefer to memorise the formulas and steps provided by their teachers without comprehending the actual concept. Thus, students only know how to calculate, but they cannot solve everyday problems that involve a mathematical concept or skill. Many students perceive that mathematics is difficult to learn and requires a long time to gain understanding. Students are considered to have learned successfully when they can remember and restate facts or use them to answer questions in examinations. Thus, students have low understanding and mastery of mathematical concepts.

According to Taat, Abdullah and Talip [ 22 ], teachers must use an approach that deeply influences the understanding of students. Sabandar [ 23 ] pointed out the need for challenging settings and problems to encourage students to learn more than they used to. Mathematics is mainly problem solving-oriented. Thus, teachers have to connect mathematics with everyday problems. To improve the problem-solving skills of students, mathematics teachers must provide open, realistic problems with multiple probable answers [ 24 ]. In realistic mathematical learning that uses open problems, students use their problem solving methods and understand the methods used by others. This ability is important because mathematics is used in almost every aspect of life.

Few studies show the relationship between mathematical representations and solving mathematical problems. Hwang, Chen, Dung and Yang [ 15 ] mentioned that good problem-solving skills are the key to obtaining the exact solution to a problem. Gagatsis and Elia [ 25 ] studied the role of four-way representations, namely, verbal, decorative picture, informal picture and counting line representations, in solving mathematical problems. Students generally achieve better problem-solving skills when the four representation models are used than when the single-representation learning model is applied. Ling and Ghazali [ 26 ] found that symbols of numeric and arithmetic representations are the most frequently used models by students in solving problems; these symbols include answer verification from a whole set of questions. This study must be expanded to measure samples until the findings can be generalised. Representation assessment and problem solving strategies are needed to create a specific rubric. Hwang, Chen, Dung and Yang [ 15 ] studied the influence of the ability and creativity of various representations in mathematical problem solving using a multimedia whiteboard system. They found that the representation ability of various students is key to effectively solving mathematical problems. The study should be expanded from the aspect of research subjects until the findings can be generalised because the focus was not on the direct influence of representation and creativity on real-life problem-solving skills.

Mathematical belief is one of the components of the affective domain, which plays a critical role in mathematical learning. The affective aspect determines student success in learning mathematics and includes attitude, interest, self-concept and belief [ 27 ]. The NCTM revealed the roles of cognitive and affective aspects in mathematical learning [ 28 ]. Both aspects are influential in the mathematical performance of students. Student belief in mathematics can influence the view towards mathematical discipline, which is related to mathematical teaching and learning [ 3 ]. According to Kloosterman [ 29 ], many students have strong mathematical belief. Mathematical belief attracted the attention of many educational mathematics researchers, particularly in other countries. However, only a few studies were conducted in Indonesia on the mathematical belief of students. The mathematical belief of students can be improved through the teaching method applied by teachers. Lee, Zeleke and Mavrotheris [ 30 ] studied the development of student belief, which can be expanded to the influence of the students’ condition and setting. Greer, Verschaffel and de Corte [ 31 ] believed that the mathematical belief of students is influenced by teachers, textbooks, learning strategy and the use of problems that exist in their surroundings during learning activities. Interrelated factors influence changes in students’ mathematical belief. Therefore, all related factors should be considered to increase the mathematical belief of students.

Arithmetic is one of the mathematical learning topics applied in daily life. Students experience difficulty in understanding arithmetic-related problems. The concept acquired by students is not formed by the students solely. Hence, students fail to retain the concept in their memory. Once students learn a new concept, they forget the old one. Many students do not solve problems by understanding the concept and rely instead on intuition or memorisation. Many everyday problems can be solved using comparison to facilitate the selection of contextual problems as a first step of the learning process. This step enables students to form their concepts, principles and mathematical procedures related to the topic. In accordance with the objective of mathematical learning, which is to prepare students to use mathematics and its way of thinking in daily life, we attempt to develop an arithmetic module that fits the RME approach. According to Sunismi [ 32 ], the learning approach and increased cognitive development showed the presence of interaction in the understanding of mathematical concepts in solving problems for Form 2 secondary school students. Haji [ 33 ] mentioned the lack of significant interaction between the approach and ability of students to solve problems.

Other studies revealed the RME function in mathematics learning. The study unveiled the relationship among mathematical representation and belief and problem-solving skills. Warsito, Darhim and Herman [ 9 ] examined the effect of RME on improving mathematical representation ability. Meika, Suryadi and Darhim [ 10 ] applied RME in students’ errors in solving combinatoric problems. Yuanita and Zakaria [ 34 ] investigated the differences in the mathematical belief of students based on their abilities in RME and students enrolled in regular classes. The results of the previous study showed that RME can be effectively used to predict the mathematical representation, belief and problem-solving skills of students. A previous study suggested a highly effective learning approach in RME; this approach includes designing instructional materials in accordance with real-life contexts that train student thinking skills. Mathematical learning should be delivered in a form that gives students an opportunity to reinvent ideas and mathematical concepts along with teacher guidance through exploration of various contextual issues and the effects of RME on students’ attitude, problem-solving ability, learning interest or other variables related to mathematics learning.

Radzali, Meerah and Zakaria [ 35 ] examined the relationship between mathematical belief and representation with mathematical problem solving. Results show that mathematical belief and representation contributed to the problem solving of students. The findings of this study are important because no other study has examined the factors mentioned. A previous study focused on examining each separately stated factor. However, studies that incorporate all three factors into inside or outside of the country are lacking. Therefore, the current study attempts to investigate these three factors simultaneously to identify the effect of mathematical representation as a mediator between mathematical belief and problem solving.

The significance of this study is its emphasis on mathematical representation, mathematical belief and problem-solving skills, which are vital to building mathematical discipline. Mathematical representation and belief and problem-solving skills are often misconceived. Therefore, the use of RME in the classroom can provide examples for students based on their daily activities. This approach could assist them in mathematical representation and belief and improve their problem-solving skills. Thus, this study investigates the difference in mathematical representation and belief and problem-solving skills of students who learned with RME and students who were engaged in conventional learning. This study also investigated the effect of mathematical representation as a mediator between mathematical belief and problem solving.

Fig 1 shows that this study was performed to identify the effectiveness of the RME approach in mathematical belief and representation and problem solving. In addition, this study identified the role of mathematical representation as a mediator between mathematical belief and problem solving. This study was conducted to answer the following research questions:

• Does the use of the RME approach have any significant effect on mathematical belief, mathematical representation and problem solving?
• Is mathematical representation a significant mediator between mathematical belief and problem solving?

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https://doi.org/10.1371/journal.pone.0204847.g001

Methodology

Participants.

The study involved 426 Form 1 secondary school students, who were divided into control and treatment groups. RME and traditional approaches were used by 209 and 217 students, respectively. The treatment group had 95 male and 114 female students. Fifty-six students had low ability, 96 had average ability and 57 students had high ability. The control group had 103 male and 114 female students. Sixty of them had low ability, 96 had average ability and 61 students had high ability. The mathematics ability of students was based on the results of their mathematics achievement in the past semester. The results were then categorised using Anates software into low, moderate and high [ 36 ]. The demographic profile is shown in Table 1 .

https://doi.org/10.1371/journal.pone.0204847.t001

Research design

The study used the quasi-experimental design with non-equivalent pre- and post-test control groups. The control group was created for comparison with the experimental group [ 37 , 38 ]. The quasi-experimental design refers to an experiment that consisted of units with treatment. This approach was utilised because the study used the existing class [ 39 ], which indicated that the research subjects were not selected randomly [ 40 ]. The quasi-experimental design was used to determine the effectiveness of the RME approach in improving problem solving skills, mathematical representation and belief of students. The research design is shown in Table 2 .

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Pre- and post-tests were conducted in both groups. The pre-test ensured similarity between groups and statistical control by comparing the mean of mathematical belief, representation, and problem solving with significant value of more than 0.05. The treatment group was given a task using the RME approach in teaching, whereas the traditional method was used as control group. Students in both groups were taught during 10 two-hour sessions in their respective classrooms. The post-test was given to both groups after they were taught social arithmetic to determine the effectiveness of the RME approach. The test questions for pre- and post-tests were similar. The researcher observed each session for both groups throughout the discussion. Observations were conducted for 5 weeks in 10 sessions for both groups. A post-test was given to the two groups after social arithmetic and ratio were taught.

Internal and external validities were determined with reference to Johnso and Christensen [ 40 ]. Internal validity is a controlled variable set by the researcher that aims to identify the actual effect on the treatment variable. External validity sees how far the findings can be applied to individuals and settings other than the ones in the study. Issues, such as selection of research and lost subjects (mortality), emotional maturity, intellectual and physical well-being, testing, research instrument and validity of research objects, can arise from the quasi-experimental design of pre- and post-tests. These issues refer to factors related to the study and the attitude and emotion of students.

Experimental group

The experimental group was taught using the RME approach. Teachers followed three main phases to teach this approach. In the first phase, teachers introduced realistic problems to students and helped them understand the problem setting. Teachers revised previous concepts and connected them with the experience of students. In the second phase, students worked in groups. Each student had a book that contained contextual questions and constructed situational problems, shared ideas, analysed patterns, made guesses and expanded problem-solving strategies based on knowledge or formal experience. The third phase of assessment showed the progress of students in problem solving. They discussed their problems and discovered useful strategies. Teachers guided and instructed students throughout the discussion on how to solve problems efficiently and effectively.

Traditional group

Students in the control group were taught using a marker and whiteboard. They participated in the exercises given by the teachers. The exercises are based on reference books provided by the school. Each school uses different reference books. Teachers narrated and jotted down information on the whiteboard. The enhanced educational curriculum unit requires every teaching method to be contextual. Thus, all teachings conducted in low secondary schools are traditionally contextual teaching.

Training for teachers

Six teachers were involved in the RME approach. They were selected based on the criteria of the RME approach training organised by the Ministry of Education in Indonesia. The teachers underwent training for one month to ensure the success of the study and consistency with the design plan. The study objectives, RME and traditional approaches, planning and execution process and assessment methods were introduced to the teachers. The same teachers were assigned to treatment and control groups. The study was conducted after they understood the entire concept. The researcher observed throughout the study to determine whether the teachers were using the RME approach. Observation began from the start until the end of class for every session. The teachers were given feedback about their teaching. The researcher observed the traditional class to ensure that the teachers were not using the RME approach or any other teaching method.

Pilot study

The present study was reviewed and approved by the Ministry of Education Pekanbaru City, Riau, Indonesia. A pilot study was conducted with 100 students to determine the validity and reliability of the research instrument. The validity of the research instrument was verified by four experts; two experts for content and two for language. According to the experts, the instrument language is suitable for measuring mathematical belief, representation and problem solving. The data from the pilot study were analysed using SPSS 23.0 and ANATES 4. Findings showed that the reliability of the mathematical belief instrument, problem solving and mathematical representation are 0.93, 0.87 and 0.80, respectively. The discriminant and difficulty index for the mathematical belief test and the mathematical problem solving test are at good and an average levels, respectively. [ 36 ] stated that the difficulty index value is at its best when used at the average level. The discriminant index should be at good and very good levels. The pilot study results indicated that the developed items are solid and strong for the actual study.

Mathematical belief instrument.

The instrument of mathematical belief was adapted from the Mathematical Problem Solving Beliefs Instrument [ 41 ] and students’ mathematics-related beliefs questionnaire [ 42 ]. The latter measures three factors of students’ mathematical belief, which are related to students in terms of mathematics students, mathematical discipline, mathematical teaching and learning. Sixty statements in the mathematical belief scale were used. Each statement could be answered with five responses of strongly agree (SA), agree (A), slightly disagree (SD), disagree (D) and strongly disagree (SLD).

Mathematical representation instrument.

The instruments for mathematical representation consisted of a written test set with four questions on the topic of arithmetic. The instrument was constructed by the researcher to collect information about a representation problem solved by the students and their success in solving mathematical problems. This instrument had four problem statements with an open-question format. These mathematical problems required students to apply comprehension, analysis and interpretation in the context of daily life. The full score for each item was 4 and 0 was the lowest score.

Problem solving instrument.

The Mathematical Problem Solving Beliefs Instrument is used to collect information about the method and the success of how the students solve mathematical problems. This instrument has five problem statements with an open-question format and requires students to comprehend, analyse and interpret these problems in the context of daily life. The full score for each item is 4 and 0 is the lowest score. The problem solving instrument is measured using marking schemes. The full score for each item is 4 and 0 is the lowest score. The total score of the students is changed to a scale of 0 to 100. The marking scheme for each item is shown in Table 3 .

https://doi.org/10.1371/journal.pone.0204847.t003

The marking scheme used for levels of mathematical representation and problem solving is the same as that used by [ 43 ], which was adapted to the arrangement outlined by the government.

Data analysis

The analysis for the actual study was performed using SPSS 23.0 and Amos 18. Analysis of covariance (ANCOVA) was performed to identify the difference in mathematical belief, representation and problem solving between the treatment and the control groups where the pre-test is a covariate. This step was followed in the structural equation modelling (SEM) test to identify the role of mathematical representation as a significant mediator in the relationship between mathematical belief and problem solving.

Research findings

Difference in mathematical belief gain score between treatment and control groups.

Univariate Analysis of Variance (UNIANOVA) was performed to identify the gain scores of the mathematical belief of the treatment and the control groups. Certain requirements for the test needed to be met prior to UNIANOVA. These requirements include normality and homogeneity of variance between groups. The normality test showed the skewness and kurtosis values for the mathematical belief gain score for the treatment and the control groups are (0.07, -0.82) and (-0.36, 0.32), respectively. This result shows that normality requirement was met and data were considered normal if the skewness and kurtosis value ranged from -1.96 to +1.96 [ 44 ]. Therefore, one-way UNIANOVA can be performed to identify the differences in the mathematical belief gain score of the treatment and the control groups, as shown in Table 4 .

https://doi.org/10.1371/journal.pone.0204847.t004

The UNIANOVA test result in Table 4 shows a significant difference in the mathematical belief gain score between the treatment and the control groups [F = 39.963, sig = 0.000 (p < 0.05)]. Students in the treatment group (mean = 0.606, std. error = 0.07) have a higher mathematical belief than students in the control group (mean = -0.027, std. error = 0.07). This finding means that the RME approach has better effect on the increase in the mathematical belief of students than the use of the traditional method. This differential effect size is medium (Cohen’s d = 0.61) [ 45 ]. Inspection of the 95% confidence intervals around each mean indicated that a significant increase in mathematical belief for participants in the treatment group and no increase in mathematical belief for participants in the control group, as shown in Table 5 .

https://doi.org/10.1371/journal.pone.0204847.t005

Fig 2 shows the pre- and post-test means for a two-group design. In the treatment group, post-test results (mean = 3.90) had higher mathematical belief than pre-test results (mean = 3.29). However, in the control group, pre-test results (mean = 3.23) had higher mathematical belief than post-test results (mean = 3.21).

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Difference in mathematical representation gain score of treatment and control groups

UNIANCOVA was performed to identify the difference between the mathematical representation gain score of the treatment and the control groups. The normality test showed the skewness and kurtosis values for mathematical representation pre-test for the treatment (0.09, -0.57) and the control (-0.05, -0.78) groups. These results indicated that the normality requirement was met. Levene’s test obtained F = 1.525, sig = 0.434 (p > 0.05), which showed that the data had similar variances between groups. Thus, UNIANCOVA can be performed to identify the difference in mathematical representation gain scores between the treatment and the control groups.

The UNIANCOVA test result in Table 6 showed no significant difference between the mathematical representation gain score of the treatment and the control groups [F = 0.430, sig = 0.512 (p > 0.05)]. The mathematical representation gain score of the students in the treatment group (mean = 1.17) was similar to that of the students in the control group (mean = 1.23). This result indicated that the RME approach and the traditional method had the same effect on the increase in the mathematical representation of students. This differential effect size was small (Cohen’s d = 0.06) [ 45 ].

https://doi.org/10.1371/journal.pone.0204847.t006

Fig 3 shows the pre- and post-test means for a two-group design. In the treatment group, post-test results (mean = 2.90) had higher mathematical representation than pre-test results (mean = 1.73). However, in the control group, post-test results (mean = 2.74) had higher mathematical representation than pre-test results (mean = 1.52).

https://doi.org/10.1371/journal.pone.0204847.g003

Differences in mathematical problem-solving gain scores of treatment and control groups

UNIANCOVA was performed to identify the difference between mathematical problem-solving gain scores of the treatment and the control groups. The normality test showed the skewness and kurtosis values of mathematical problem-solving gain scores for the treatment group (-0.27. -0.81) and the control group (0.38, -0.48). Results showed that the normality requirement was met. Levene’s test obtained a value of F = 1.440, sig = 0.231 (p > 0.05), which indicated that the data had similar variances between groups. Therefore, UNIANCOVA can be performed to identify the differences between mathematical problem-solving gain scores of the treatment and the control groups, as shown in Table 7 .

https://doi.org/10.1371/journal.pone.0204847.t007

The UNIANCOVA test result in Table 7 showed a significant difference in mathematical problem-solving gain scores between the treatment and the control groups [F = 6.716, sig = 0.010 (p < 0.05)]. Students in the treatment group (mean = 2.01) had better mathematical problem solving gain scores than the students in the control group (mean = 1.85). These results prove that the RME approach was better than the traditional method at improving problem solving skills. Such differential effect size was small (Cohen’s d = 0.25) [ 45 ].

Fig 4 shows the pre- and post-test means for a two-group design. In the treatment group, post-test results (mean = 2.70) had higher mathematical problem-solving value than pre-test results (mean = 0.68). However, in the control group, post-test results (mean = 2.39) had higher mathematical problem-solving values than pre-test results (mean = 0.54).

https://doi.org/10.1371/journal.pone.0204847.g004

Role of mathematical representation as a mediator between mathematical belief and problem solving for the treatment group

SEM analysis was performed to identify the role of arithmetic representation as a mediator between belief towards mathematical teaching and learning and mathematical problem solving. The analysis result of the SEM path model in Fig 5 shows the following: chi square/df = 3.06, root mean-square error approximation (RMSEA) = 0.07, goodness of fit index (GFI) = 0.91, Tucker–Lewis fit index (TLI) = 0.90 and comparative fit index (CFI) = 0.92. All assessments indicated that the data in the study had reasonable adjustment for the suggested model [ 46 ]. The result of SEM analysis showed that the suggested regression model was suitable when mathematical teaching belief (β = 0.33, p < 0.05) and mathematical learning belief (β = 0.52, p < 0.05) are significant predictor variables for mathematical problem solving. The SEM result showed that mathematical teaching belief (β = 0.52, p < 0.05) and mathematical learning belief (β = 0.70, p < 0.05) are significant predictor variables for arithmetic representation. Bootstrapping test was performed to determine the effect of mathematical representation as a significant mediator.

https://doi.org/10.1371/journal.pone.0204847.g005

Bootstrapping test was applied to determine the effect of arithmetic representation as a significant mediator between mathematical teaching and learning belief and problem solving. Table 8 shows that arithmetic representation is a significant partial mediator between teaching belief (β = 0.19, p < 0.05) and learning (β = 0.29, p < 0.001) towards problem solving.

https://doi.org/10.1371/journal.pone.0204847.t008

Role of mathematical representation as a mediator between mathematical belief and problem solving for the control group

SEM analysis was performed to identify the role of arithmetic representation as a mediator between the belief towards mathematical teaching and learning in mathematical problem solving. The analysis of the SEM path model in Fig 6 shows the measure of chi square/df = 1.31, RMSEA = 0.07, GFI = 0.91, TLI = 0.90 and CFI = 0.92. The result of SEM analysis indicated that the suggested regression model was suitable when mathematical teaching belief (β = 0.36, p < 0.05) and mathematical learning belief (β = 0.57, p < 0.05) were significant predictor variables for mathematical problem solving. The SEM result showed that mathematical teaching belief (β = 0.57, p < 0.05) and mathematical learning belief (β = 0.74, p < 0.05) were significant predictor variables for arithmetic representation. Bootstrapping test was conducted to determine the effects of mathematical representation as a significant mediator ( Table 9 ).

https://doi.org/10.1371/journal.pone.0204847.g006

https://doi.org/10.1371/journal.pone.0204847.t009

The bootstrapping test was applied to check the effect of arithmetic representation as a significant mediator between mathematical teaching and learning belief and problem solving. Table 9 shows that arithmetic representation was a significant mediator for teaching belief (β = 0.19, p < 0.001) and learning (β = 0.25, p < 0.001) towards problem solving. The SEM result indicated that the treatment and the control groups obtained the same results for the role of mathematical representation as a partial mediator between mathematical belief and problem solving.

Students who were taught using the RME approach had higher mathematical belief than students who were exposed to the traditional method. The use of RME increased the confidence of students in mathematics, especially in arithmetic, as reflected in their active participation in the activities presented with the RME approach. According to Fauzan [ 47 ], active students use the RME approach, which develops creative thinking and lessens uncertainty towards mathematics. However, the use of the traditional method successfully increased the mathematical belief of students, although the RME approach had better effect. Saragih [ 48 ] stated that the advantage of the RME approach is its ability to strengthen students’ interest in mathematics. The findings supported Lee, Zeleke and Mavrotheris [ 30 ] who asserted that the RME approach enables students to learn mathematics actively such that their belief can increase through the effort of teachers. Greer, Verschaffel and de Corte [ 31 ] supported this idea by stating that the mathematical belief of students is influenced by factors, such as teachers, textbooks, learning strategies and use of problems that exist in the surroundings of students for learning activities.

The use of the RME approach did not significantly increase mathematical representation compared with the traditional method. Thus, the RME approach was not suitable for all skills or topics. However, the RME approach still successfully increased the mathematical representation of students. This idea was supported by Arsaythamby and Zubainur [ 49 ] who claimed that not all learning activities of students should be conducted using the RME approach. Teaching with the RME approach provided students with the opportunities to come up with ideas that can enable them to solve mathematical problems easily. The traditional method provided opportunities for students to generate ideas, but these opportunities are fewer than those offered by the RME approach. Neria and Amit [ 50 ] mentioned that questions on mathematical representation are given to students to allow them to present situational problems in the form of mathematical notes, numerals, symbols, graphics, tables and pictures, which they will try to solve later. Therefore, the skills of teachers in using the RME approach must increase the mathematical representation of students to guide their gradual learning according to levels.

The RME approach successfully improved the problem-solving skills of students and was better than the traditional method in this aspect. In the RME approach, teachers checked the answers of students by writing down detailed answers and providing reasons or explanations as to how the answer was obtained. Moreover, students were motivated to stand in front of the class and explain their work. Jones, Thornton and Nisbet [ 51 ] found that the RME approach is suitable for arithmetic learning until the students become more confident in solving problems. This statement supported the findings of Viholainen, Asikainen and Hirvonen [ 52 ], who stated that confidence in mathematics has strong influence on mathematical problem solving and determines how a student chooses the approach, technique and strategy to use. The results of study supported Laurens, Batlolona, Batlolona and Leasa [ 4 ], who claimed students who were taught with RME achieved better results than the students who were involved in conventional learning.

The SEM test showed the same match between the treatment and the control groups, wherein mathematical representation was a significant partial mediator between mathematical belief and problem solving. Findings showed that mathematical belief indirectly affected mathematical problem-solving skills. This study indicated no significant difference in mathematical representation, but the mediator effect of mathematical representation between treatment and control groups was the same. This result suggests that mathematical representation is an indirectly important aspect in students to enhance the relationship between mathematical beliefs and problem solving. The use of different methods did not influence the effect of mathematical representation as the mediator of the relationship between mathematical beliefs and problem solving. The findings supported Hwang, Chen, Dung and Yang [ 15 ] in their claim that mathematical representation contributes to the ability of students to solve mathematical problems. This study supported Ling and Ghazali [ 26 ], who reported that arithmetic is the most frequently used representation model by students in problem solving, including answer verification from all the given questions. Moreover, mathematical belief affects the mathematical representation and problem-solving skills of students. This finding means that if students believe in mathematical teaching and learning, then they will possess mathematical representation and reliable problem-solving skills. This statement is consistent with the findings of [ 3 ], who found that the belief of students towards mathematics can influence their view on mathematical discipline, which is related to mathematical teaching and learning. The SEM results showed higher connection of mathematical belief and mathematical representation in problem solving with the use of the RME approach than with the use of the traditional method. This finding is supported in Muchlis [ 53 ] and in Husna and Saragih [ 54 ].

The study successfully proved that the RME approach had a positive effect on mathematical belief, representation and problem solving among students. Thus, teachers need to adjust their teaching methods using RME and encourage students to participate in activities and engage in discussions. The RME approach provides students with the opportunity to generate knowledge on the topics that they have been taught. Students can convey their ideas until they can form concepts for each learning step. Many students provide solutions that consist of different steps but have the same answer. Students believe in producing results that they obtain by themselves, which is a process that they will later find as an arithmetic concept. School administrators must assist teachers in eliminating the negative perception towards teaching and learning mathematics. The effectiveness of RME offers an opportunity to use the approach continuously to teach other topics for secondary school students as a whole. Future studies can examine the use of RME at various educational levels to obtain detailed information.

The contribution of this study is the identification of various learning methods often used by students in everyday life that can be utilised to improve the quality in learning through the creativity of teachers. In additional, the RME approach is among the most effective approaches in fostering mathematical representation, belief and problem-solving skills that could improve student achievement. Few studies examined the relationship of mathematical representation as a mediator between mathematical belief and problem solving. The present study filled the gap by producing a new form of relationship model through a quasi-experimental design.

The findings and results of this study provided information on the differences in mathematical representation, belief and problem-solving skills of students who learned through RME and conventional learning methods. Mathematics teachers should apply RME in the classroom to make abstract mathematical concepts more understandable. Teachers should be creative and innovative in designing learning with this approach. Teachers should develop additional learning media, strategies or models that are more suitable with learning materials or with the contexts of students. Further, schools should create contextual environments that are rich in information on ways to solve real life problems.

The use of RME can increase mathematical belief, representation and problem solving skills. This approach successfully trains students to formulate their own ideas from real-life situations or experiences. Teachers must be encouraged to use the RME approach in teaching and learning mathematics. Efforts pertaining to mathematical representation should be doubled to increase the mathematical problem solving skills of students. The belief of students is another major factor in increasing mathematical problem solving skills. Cooperation from all sides should be improved to encourage the use of the RME approach in teaching and learning mathematics at all school levels to increase mathematical belief, representation and problem solving. This study seeks to serve as a stepping stone for future studies to expand the use of the RME approach from the national to the international level.

Supporting information

S1 file. data availability..

https://doi.org/10.1371/journal.pone.0204847.s001

Acknowledgments

The authors wish to thank Mrs. Dewi Marianti, Mrs. Suarni, Mrs. Arnidar, Mrs. Nurwahyu, Mrs. Furqonati, Mrs. Gusniwati and Mrs. Yulmaliza for their assistance with data collection.

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

• 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.

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.

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.

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.

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.

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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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T1 - The advantages and disadvantages of problem-solving practice when learning basic addition facts

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N2 - How children learn to retrieve answers to basic (single-digit) addition problems and how teachers can support children's learning of retrieval has captivated my attention as a researcher and teacher educator for the last 25 years. In this chapter, I describe this research and explain how I got started with the help of Professor Mike Lawson. I then present findings from a series of microgenetic studies to illustrate the different effects problem-solving practice has on children's development of retrieval.

AB - How children learn to retrieve answers to basic (single-digit) addition problems and how teachers can support children's learning of retrieval has captivated my attention as a researcher and teacher educator for the last 25 years. In this chapter, I describe this research and explain how I got started with the help of Professor Mike Lawson. I then present findings from a series of microgenetic studies to illustrate the different effects problem-solving practice has on children's development of retrieval.

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Learning to teach mathematics through problem posing: teachers’ beliefs and performance on problem posing

• Published: 07 September 2020
• Volume 105 , pages 325–347, ( 2020 )

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• Xinlian Li 1 ,
• Naiqing Song 1 ,
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• Jinfa Cai 2  

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Any effort to integrate problem-posing instruction in school mathematics must attend to teachers’ beliefs about the advantages of teaching through problem posing and especially their beliefs about the challenges of teaching in this way. This study investigated teachers who were learning how to teach mathematics through problem posing. The primary foci were the teachers’ beliefs about teaching through problem posing and their problem-posing performance. The findings demonstrate that the participating teachers were able to engage successfully in problem-posing tasks and that they held a number of different beliefs about the advantages and challenges of teaching through problem posing. After participating in three workshops, a sample of teachers showed gains in problem-posing performance and in the scope and specificity of their beliefs about teaching through problem posing.

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Li, X., Song, N., Hwang, S. et al. Learning to teach mathematics through problem posing: teachers’ beliefs and performance on problem posing. Educ Stud Math 105 , 325–347 (2020). https://doi.org/10.1007/s10649-020-09981-0

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Published : 07 September 2020

Issue Date : November 2020

DOI : https://doi.org/10.1007/s10649-020-09981-0

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