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The rate of vocabulary memorization of the average student in a foreign language course is given by
d v d t = 40 t + 1 \frac{d v}{d t}=\frac{40}{t+1} d t d v = t + 1 40
where t t t is the number of continuous hours of study, 0 < t ≤ 4 0<t \leq 4 0 < t ≤ 4 , and v v v is the number of words. How many words would the average student memorize in 3 3 3 hours?
Lightfoot Inc., a software development firm, has stock outstanding as follows: 40,000 shares of cumulative preferred 1% stock, $125 par, and 100,000 shares of$150 par common. During its first four years of operations, the following amounts were distributed as dividends: first year, $36,000; second year,$58,000; third year, $75,000; fourth year,$124,000. Calculate the dividends per share on each class of stock for each of the four years
Casey is a real estate agent. She earns 8.15% commission on each sale she makes. After working with a potential buyer for 3 1 2 3 \frac{1}{2} 3 2 1 months, she finally sold a house for $877,000. a. What did Casey earn in commissions for this sale? b. This was her only sale for the entire$3 $\frac{1}{2} month period. What was her average salary per month for this period, to the nearest cent? c. For the next 4 months, Casey is not able to sell a house due to a faltering economy. What is her average monthly salary for the past$7 $$\frac{1}{2}$ months, to the nearest cent?
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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 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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K-5 Math Centers
K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, math problem solving 101.
Have you ever given your students a money word problem where someone buys an item from a store, but your students come up with an answer where the person that bought the item ends up with more money than he or she came in with?
Word problem solving is one of those things that many of our children struggle with. When used effectively, questioning and dramatization can be powerful tools for our students to use when solving these types of problems.
I came up with this approach after co-teaching a lesson with a 3rd grade teachers. Her kids were having extreme difficulty comprehending a word problem she presented. So we devised a lesson that would help students better understand problem solving.
The approach we took included the use of several literacy skills, like reading comprehension and writing. First, we started the lesson with a “think aloud” modeled by the teacher.We read and displayed the problem below but excluded ALL of the numbers. See the images below:
The purpose of reading the problem without the numbers is to get the students to understand what is actually happening in the problem. Typically some students focus solely on keywords when solving word problems, but I do not advise using this approach exclusively. With math problems, the context of the problem and actions in the problem determine how the child should go about solving it.
Read the Problem Without Numbers & Ask Questions:
After reading the problem (without numbers) to the students, I asked the following questions:
- Can you describe what is happening in your own words?
- What is the main idea of the problem?
- How could you act this out?
Make a Plan & Ask Questions:
After the students articulated what was happening in the problem, we made a plan to solve the problem. I used the following guiding questions:
- Sample Answers include- We know that Kai has some goldfish. Kai donated or gave away some of the goldfish.
- Sample Answers include – We need to know how many goldfish Kai has. We also need to know how many he gave anyway. We also need to know how many bowls there are.
- Sample Answers include- We need to find out how many fish belong in each bowl.
The class discussed the answers to the questions above. As we discussed the questions above the responses were written out on a problem solving template.
As part of this process, we clarified student understanding of the problem and determined what we needed to find and do to solve the problem. Next, we walked the students through the process of showing their work using pictures. Lastly, we checked our answers by writing an equation that matched the pictures to finally solve the problem.
Team Work Counts
After going through the process with the class, we decided to split the students into small groups of 3 and 4 to solve a math problem together. The groups were expected to use the same process that we used to solve the problem. It took a while but check out one of the final products below.
Benefits to Using this Process:
- Students understood what the problem is asking them to do
- Students are required to think and communicate as a team
- Students avoid making errors that can come with only using keywords
- Students are required to record their math reasoning using the problem solving template
- After using this process a couple of times, students get used to explaining and justifying their answers
- You become the facilitator of the learning by asking more questions, thereby making students independent thinkers
Things to Consider Include:
- This process in NOT quick. It requires TIME. You should not rush the process and expect to have it completed in 20 – 30 minutes in one day.
- This process is not a one time lesson. Students may not get it the first time. It should be seen a routine that can be used when solving word problems.
Be sure to let me know how this process works in your classroom in the comments below.
- Read more about: K-5 Math Ideas
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Teaching Through Problem-solving
- TTP in Action
What is Teaching Through Problem-Solving?
In Teaching Through Problem-solving (TTP), students learn new mathematics by solving problems. Students grapple with a novel problem, present and discuss solution strategies, and together build the next concept or procedure in the mathematics curriculum.
Teaching Through Problem-solving is widespread in Japan, where students solve problems before a solution method or procedure is taught. In contrast, U.S. students spend most of their time doing exercises– completing problems for which a solution method has already been taught.
Why Teaching Through Problem-Solving?
As students build their mathematical knowledge, they also:
- Learn to reason mathematically, using prior knowledge to build new ideas
- See the power of their explanations and carefully written work to spark insights for themselves and their classmates
- Expect mathematics to make sense
- Enjoy solving unfamiliar problems
- Experience mathematical discoveries that naturally deepen their perseverance
Phases of a TTP Lesson
Teaching Through Problem-solving flows through four phases as students 1. Grasp the problem, 2. Try to solve the problem independently, 3. Present and discuss their work (selected strategies), and 4. Summarize and reflect.
Click on the arrows below to find out what students and teachers do during each phase and to see video examples.
- 1. Grasp the Problem
- 2. Try to Solve
- 3. Present & Discuss
- 4. Summarize & Reflect
- New Learning
WHAT STUDENTS DO
- Understand the problem and develop interest in solving it.
- Consider what they know that might help them solve the problem.
WHAT TEACHERS DO
- Show several student journal reflections from the prior lesson.
- Pose a problem that students do not yet know how to solve.
- Interest students in the problem and in thinking about their own related knowledge.
- Independently try to solve the problem.
- Do not simply following the teacher’s solution example.
- Allow classmates to provide input after some independent thinking time.
- Circulate, using seating chart to note each student’s solution approach.
- Identify work to be presented and discussed at board.
- Ask individual questions to spark more thinking if some students finish quickly or don’t get started.
- Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)
- Actively make sense of the presented work and draw out key mathematical points. (All students)
- Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)
- Monitor student discussion: Are all students noticing the important mathematical ideas?
- Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.
- Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.
- Write journal reflection on their own learning from the lesson.
- Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.
- Ask students to write in their journals about what they learned during the lesson.
How Do Teachers Support Problem-solving?
Although students do much of the talking and questioning in a TTP lesson, teachers play a crucial role. The widely-known 5 Practices for Orchestrating Mathematical Discussions were based in part on TTP . Teachers study the curriculum, anticipate student thinking, and select and sequence the student presentations that allow the class to build the new mathematics. Classroom routines for presentation and discussion of student work, board organization, and reflective mathematics journals work together to allow students to do the mathematical heavy lifting. To learn more about journals, board work, and discussion in TTP, as well as see other TTP resources and examples of TTP in action, click on the respective tabs near the top of this page.
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Problem Solving: An Approach to Understanding and Critiquing Our World
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Problem Solving: An Approach to Understanding and Critiquing Our World July 2022
Problem-solving and critical-thinking skills prepare students to understand and critique their world, preparing them to be contributing leaders in society. They are skills that all individuals need in decision-making, in the workforce, in their communities, and in daily life. “All children have remarkable abilities to learn substantial mathematics when provided mathematics learning opportunities that emphasize sense making and problem solving” (NCTM 2020a, p. 28).
Many aspects of problem solving are valuable. In PK–12 classrooms these include but are not limited to tasks and problems that have multiple entry points and varied solution strategies and that consider the role of context and multiple representations along with cultural and community connections. A classroom environment that promotes problem solving provides opportunities for creativity, supports developing a positive mathematics identity for all learners, and provides a pathway for developing a deep understanding of mathematics.
Hypatia of Alexandria, Egypt, known as one of world’s first female mathematicians, often traveled with her father in his work. When she had a wondering, he would regularly respond that the answer was math, and thus began her investigations. As a result, she fell in love with mathematics. Goris (2017) wrote about Hypatia in a series of STEM-focused books. My favorite quote is, “Hypatia learned to ask big questions and ponder big answers.” We want all our students to ask those big questions and ponder those big answers as they engage in problem solving.
NCTM has advocated for problem solving for decades. From 1980 in the Agenda for Action , which included as its first recommendation that mathematics curriculum have a focus on problem solving; to the release of NCTM’s Curriculum and Evaluation Standards (1989), reminding us that problem solving is a process in which developing students as problem solvers is essential if students are to become productive citizens, develop confidence in using mathematics meaningfully, and be able to model real-world problem situations. Problem solving was further emphasized in NCTM’s Principles and Standards for School Mathematics (2000), focusing on building new mathematical knowledge, solving problems that arise in mathematics and other contexts, and applying and adapting appropriate strategies to solve problems. Later in NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014) and the Catalyzing Change series , (2018, 2020a, 2020b), a focus on problems solving was seen as an essential part of effective equitable instruction.
And yet, even with this focus for decades on problem solving and acknowledging its importance and value, we still have been challenged to fully embrace problem solving at times, often finding a continued focus on learning procedures, a lowering of expectations for some students, and limited access to instructional resources to support problem solving. Furthermore, because of the impact of high-stakes assessment in states/provinces and districts, skills and facts are often emphasized rather than problem solving, critical thinking, and reasoning (NCTM 2014). We must address these challenges and barriers to ensure equitable access to mathematical problem solving for each and every student.
Why should we include a problem-solving approach to learning and exploring mathematics? A few reasons include that problem solving—
- supports making connections across disciplines;
- prepares students for future professional opportunities;
- develops students’ positive mathematical identity;
- is a matter of equity and access;
- builds students’ confidence, persistence, flexibility, creativity, perseverance, and curiosity;
- gives students voice and promotes discussion;
- shifts math authority to students; and
- has a positive effect on learning.
(NCTM 2000; 2014, 2018, 2020a, 2020b)
Both mathematical thinking and creativity are key parts of problem solving because “children experience joy when they are encouraged to be creative and are provided with mathematical choices, such as generating their own approaches and strategies for solving mathematical problems” (NCTM 2020a, p. 21). A problem-solving approach to instruction provides a learning environment that promotes thinking mathematically and provides a space for nurturing creativity and innovation. It is essential that we offer students the resources and space for being creative through multiple opportunities for students to examine problems and issues of interest to them, that have meaning in their lives, and that help them makes sense of their world. (NCTM 2018, 2020a, 2020b). We as teachers can and must intentionally facilitate this kind of environment, providing needed resources and supports to each and every student to nurture creativity and mathematical thinking.
Recently Guadalupe Siclia, a grade 3 teacher at Gardens Elementary School in Pasadena, Texas, shared with me, “It's important to me that my students engage in problem solving and problem-solving-related tasks because I want to help build a generation of mathematicians that are able to think, discuss, and persevere when working through problems that can be challenging.” Powerfully stated! A classroom environment and culture of problem solving supports all students in their learning as thinkers and doers of mathematics.
Approaching learning mathematics that includes a focus on problem solving is exciting, rewarding, and essential. What is your plan for preparing students to understand and critique their world through problem solving? Let’s look for opportunities for supporting all PK–12 students on their journey in problem solving!
Trena Wilkerson NCTM President @TrenaWilkerson
References
Goris, Brittany. 2017. Hypatia: Explorer of Geometry. Seattle, WA: Girls Rock Math.
National Council of Teachers of Mathematics (NCTM). 1980. An Agenda for Action: Recommendations for School Mathematics in the 1980s . Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics . Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All . Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2018. Catalyzing Change in High School Mathematics : Initiating Critical Conversations. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2020a. Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations . Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2020b. Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations . Reston, VA: NCTM.
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Mathematics as a Complex Problem-Solving Activity
By jacob klerlein and sheena hervey, generation ready.
By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: they respond to their environment and the reactions of others. This making sense of experience is an ongoing, recursive process. We have known for a long time that reading is a complex problem-solving activity. More recently, teachers have come to understand that becoming mathematically literate is also a complex problem-solving activity that increases in power and flexibility when practiced more often. A problem in mathematics is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem—it is a straightforward application.
Mathematicians have always understood that problem-solving is central to their discipline because without a problem there is no mathematics. Problem-solving has played a central role in the thinking of educational theorists ever since the publication of Pólya’s book “How to Solve It,” in 1945. The National Council of Teachers of Mathematics (NCTM) has been consistently advocating for problem-solving for nearly 40 years, while international trends in mathematics teaching have shown an increased focus on problem-solving and mathematical modeling beginning in the early 1990s. As educators internationally became increasingly aware that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways (Wu and Zhang 2006) little changed at the school level in the United States.
“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”
(NCTM, 2000, p. 52)
In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. For many teachers of mathematics this was the first time they had been expected to incorporate student collaboration and discourse with problem-solving. This practice requires teaching in profoundly different ways as schools moved from a teacher-directed to a more dialogic approach to teaching and learning. The challenge for teachers is to teach students not only to solve problems but also to learn about mathematics through problem-solving. While many students may develop procedural fluency, they often lack the deep conceptual understanding necessary to solve new problems or make connections between mathematical ideas.
“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.”
(Burns, 2000, p. 29)
Learning to problem solve
To understand how students become problem solvers we need to look at the theories that underpin learning in mathematics. These include recognition of the developmental aspects of learning and the essential fact that students actively engage in learning mathematics through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning” (Copley, 2000, p. 29). The concept of co-construction of learning is the basis for the theory. Moreover, we know that each student is on their unique path of development.
Beliefs underpinning effective teaching of mathematics
- Every student’s identity, language, and culture need to be respected and valued.
- Every student has the right to access effective mathematics education.
- Every student can become a successful learner of mathematics.
Children arrive at school with intuitive mathematical understandings. A teacher needs to connect with and build on those understandings through experiences that allow students to explore mathematics and to communicate their ideas in a meaningful dialogue with the teacher and their peers.
Learning takes place within social settings (Vygotsky, 1978). Students construct understandings through engagement with problems and interaction with others in these activities. Through these social interactions, students feel that they can take risks, try new strategies, and give and receive feedback. They learn cooperatively as they share a range of points of view or discuss ways of solving a problem. It is through talking about problems and discussing their ideas that children construct knowledge and acquire the language to make sense of experiences.
Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something directly (Hiebert1997). The teacher’s role is to construct problems and present situations that provide a forum in which problem-solving can occur.
Why is problem-solving important?
Our students live in an information and technology-based society where they need to be able to think critically about complex issues, and “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998). Mathematics education is important not only because of the “gatekeeping role that mathematics plays in students’ access to educational and economic opportunities,” but also because the problem-solving processes and the acquisition of problem-solving strategies equips students for life beyond school (Cobb, & Hodge, 2002).
The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures. It is through problem-solving that students develop a deeper understanding of mathematical concepts, become more engaged, and appreciate the relevance and usefulness of mathematics (Wu and Zhang 2006). Problem-solving in mathematics supports the development of:
- The ability to think creatively, critically, and logically
- The ability to structure and organize
- The ability to process information
- Enjoyment of an intellectual challenge
- The skills to solve problems that help them to investigate and understand the world
Problem-solving should underlie all aspects of mathematics teaching in order to give students the experience of the power of mathematics in the world around them. This method allows students to see problem-solving as a vehicle to construct, evaluate, and refine their theories about mathematics and the theories of others.
Problems that are “Problematic”
The teacher’s expectations of the students are essential. Students only learn to handle complex problems by being exposed to them. Students need to have opportunities to work on complex tasks rather than a series of simple tasks devolved from a complex task. This is important for stimulating the students’ mathematical reasoning and building durable mathematical knowledge (Anthony and Walshaw, 2007). The challenge for teachers is ensuring the problems they set are designed to support mathematics learning and are appropriate and challenging for all students. The problems need to be difficult enough to provide a challenge but not so difficult that students can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed. Problems need to be within the students’ “Zone of Proximal Development” (Vygotsky 1968). These types of complex problems will provide opportunities for discussion and learning.
Students will have opportunities to explain their ideas, respond to the ideas of others, and challenge their thinking. Those students who think math is all about the “correct” answer will need support and encouragement to take risks. Tolerance of difficulty is essential in a problem-solving disposition because being “stuck” is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially. Effective problems:
- Are accessible and extendable
- Allow individuals to make decisions
- Promote discussion and communication
- Encourage originality and invention
- Encourage “what if?” and “what if not?” questions
- Contain an element of surprise (Adapted from Ahmed, 1987)
“Students learn to problem solve in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”
(Copley, 2000, p. 29)
“…as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas.…This enables learners to become clearer and more confident about what they know and understand.”
(Fosnot, 2005, p. 10)
Research shows that ‘classrooms where the orientation consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners’ (Anthony and Walshaw, 2007, page 122).
Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Students often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the problem without telling them exactly what to do. In 1945 Pólya published the following four principles of problem-solving to support teachers with helping their students.
- Understand and explore the problem
- Find a strategy
- Use the strategy to solve the problem
- Look back and reflect on the solution
Problem-solving is not linear but rather a complex, interactive process. Students move backward and forward between and across Pólya’s phases. The Common Core State Standards describe the process as follows:
“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary”. (New York State Next Generation Mathematics Learning Standards 2017).
Pólya’s Principals of Problem-Solving
Students move forward and backward as they move through the problem-solving process.
The goal is for students to have a range of strategies they use to solve problems and understand that there may be more than one solution. It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn’t the end of the process. Reflecting on the strategies used to solve the problem provides additional learning experiences. Studying the approach used for one problem helps students become more comfortable with using that strategy in a variety of other situations.
When making sense of ideas, students need opportunities to work both independently and collaboratively. There will be times when students need to be able to work independently and other times when they will need to be able to work in small groups so that they can share ideas and learn with and from others.
Getting real
Effective teachers of mathematics create purposeful learning experiences for students through solving problems in relevant and meaningful contexts. While word problems are a way of putting mathematics into contexts, it doesn’t automatically make them real. The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it. Realistic does not mean that problems necessarily involve real contexts, but rather they make students think in “real” ways.
Planning for talk
By planning for and promoting discourse, teachers can actively engage students in mathematical thinking. In discourse-rich mathematics classes, students explain and discuss the strategies and processes they use in solving mathematical problems, thereby connecting their everyday language with the specialized vocabulary of mathematics.
Students need to understand how to communicate mathematically, give sound mathematical explanations, and justify their solutions. Effective teachers encourage their students to communicate their ideas orally, in writing, and by using a variety of representations. Through listening to students, teachers can better understand what their students know and misconceptions they may have. It is the misconceptions that provide a window into the students’ learning process. Effective teachers view thinking as “the process of understanding,” they can use their students’ thinking as a resource for further learning. Such teachers are responsive both to their students and to the discipline of mathematics.
“Mathematics today requires not only computational skills but also the ability to think and reason mathematically in order to solve the new problems and learn the new ideas that students will face in the future. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, are encouraged to make mathematical conjectures and test them, and are helped to develop their reasoning skills.”
(John Van De Walle)
“Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.”
How teachers organize classroom instruction is very much dependent on what they know and believe about mathematics and on what they understand about mathematics teaching and learning. Teachers need to recognize that problem-solving processes develop over time and are significantly improved by effective teaching practices. The teacher’s role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants their students to explore. A problem-solving approach is not only a way for developing students’ thinking, but it also provides a context for learning mathematical concepts. Problem-solving allows students to transfer what they have already learned to unfamiliar situations. A problem-solving approach provides a way for students to actively construct their ideas about mathematics and to take responsibility for their learning. The challenge for mathematics teachers is to develop the students’ mathematical thinking process alongside the knowledge and to create opportunities to present even routine mathematics tasks in problem-solving contexts.
Given the efforts to date to include problem-solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable professional learning, resources, and more time are essential steps, it is possible that problem-solving in mathematics will only become valued when high-stakes assessment reflects the importance of students’ solving of complex problems.
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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.
Reviewed by:
Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Nina Klang, [email protected]
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focus on teaching strategies and conceptual understanding.
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