mathematical problem solving as a process

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

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Teaching Thinking through Mathematical Processes: A Review

Prof. Mbuthia Ngunjiri

• Nov 22, 2023

Laikipia University, Nakuru, Kenya

DOI: https://dx.doi.org/10.47772/IJRISS.2023.701161

This paper focuses on seven mathematical processes that describe the action of doing mathematics. It is clear that doing mathematics means engaging in processes. These include: problem solving, reasoning and proving, tools and computational strategies, connecting, representation, and communication. The mathematical processes can be seen as the processes through which students think, acquire, and apply mathematical knowledge and skills in everyday use of mathematics. In mathematics instruction, students are largely unaware of the processes involved in mathematics which requires surface and deep thinking and addressing this issue is important. Therefore, mathematics teachers need to understand how the mathematical processes can be taught to improve student thinking and understanding of the subject. It is also recommended that a study be done on how self-efficacy beliefs of students in mathematics can be influenced by teaching of the mathematical processes.

Keywords : Problem Solving, Reasoning, Proving, Reflecting, Tools, Computational Strategies, Connecting, Representing, Communicating.

INTRODUCTION

Critical thinking plays a significant role in mathematics. When faced with problems to solve, students routinely make reasoned judgments about what, and how to think. While thinking about mathematical concepts, procedures, strategies, tools, representations, and models, decisions are made through the use of criteria and appropriate evidence. To think like a mathematics student is to think critically through the mathematical processes (NCTM, 2005). By promoting, teaching and assessing critical thinking through   the processes, teachers, not only help students to think like a mathematician, they also ensure students think to learn about mathematics. Therefore, it is through mathematical processes that teachers promote thinking. Students must learn how to problem solve, communicate, reason and prove, reflect, represent, connect, and select tools and computational strategies to be able to share ideas, observations, and problem-solving processes in or outside mathematics classrooms (Reys et al.,2001).

REVIEW OF LITERATURE

The mathematical processes that appear to support effective learning   in mathematics are as follows: problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating. In the context of teaching: (i) Mathematical processes develop through different grades and support lifelong learning, (ii) they are taught in ways that address the different needs of different types of learners, and (iii) a variety of groupings and instructional strategies help students improve their mathematical processes (Fernandez – Cezar et al., 2020). The following literature addresses the seven mathematical processes.

The first mathematical process is problem solving which is central to the learning of mathematics (NCTM, 2000, 2005, 2020; Reys et al., 2001; The Ontario Curriculum, 2020). Garfola and Lester (1985) suggested that students are largely unaware of the processes involved in problem solving, and addressing this issue within problem solving instruction may be important. According to Kantowski (1977), to become a good problem solver in mathematics, a student must develop a sound base of domain specific knowledge in mathematics.

In their view, how effective one is in organizing that domain specific knowledge also contributes to successful problem solving. This argument is supported by Schoenfeld and Herrman (1982) who posited that novices appear to attend to surface features of problems, whereas, experts categorize problems on the basis of fundamental principles involved in the problems. According to Schoenfeld (1985), providing explicit instruction and use of heuristics should enhance problem solving performance. In their view, heuristics are kinds of information available to students in making decisions during problem solving that are aids to the generation of a solution. For example, tendency to “write an equation”, to “set sub- goals”, “restate a problem”, and to “draw a figure” are heuristic in nature.

Schoenfeld (1985) and Kantowski (1997) have provided two requirements for successful problem solving. These are: (i) use of heuristics, and (ii) possession of domain specific knowledge. However, Mayer (1987) offered more specific types of knowledge needed for success in mathematical problem solving. In Mayer’s view, students’ performances differ because they possess differing amounts and kinds of knowledge.

First, a student’s needs to be able to translate each statement of a problem into some internal representation. Furthermore, this translation requires linguistic and factual knowledge. Second, a student needs to be able to integrate each of the statements of a problem into a coherent representation. The integration process requires that a student should be able to recognize problem types and also be able to distinguish between information that is relevant for a solution from that is which is not, which is called schematic knowledge. Third, a student needs to devise and monitor solution plan which is called strategic knowledge, and finally, a student should be able to apply the rules of arithmetic (i.e., computational skills) which is called procedural knowledge. In all, according to Mayer (1987), a student may be unable to generate a correct answer due to lack of linguistic and factual knowledge, in particular difficulties in comprehending sentences that express relations among variables.

This implies that students need practice to represent each sentence in a problem. Further, students who are unable to generate the correct answer in a problem may be lacking knowledge of problem types. If a problem does not fit into one of the student’s existing categories, the problem is likely to be misinterpreted. This implies that students need practice in recognizing those problems that go together and those that do not. Moreover, student may fail to solve a problem due to lack of appropriate strategies. This implies that students need instruction and practice in determining which strategies to apply and when to apply. Lastly, a student may fail to generate the correct answer due to lack of appropriate computational skills. This implies that students need practice in solving computational problems.

According to Polya (1965), there is a four-step procedure of solving problems. These are: (i) reading and understanding the problem, (ii) devising a solution plan, (iii) carrying out the plan, and (iv) looking back at the solution process. The first Polya’s step is similar to Mayer’s (1987) argument for students’ ability in linguistic and factual knowledge. The Polya’s second step is similar to Mayer’s argument for the student’s ability in knowledge of problem types to be able to come up with a solution plan. The third Polya’s step is similar to Mayer’s arguments for students’ ability in knowing solution strategies. These strategies include: (i) drawing a picture or a diagram, (ii) looking for a related problem, (iii) restating the problem, (iv) trying to think about other information appropriate to determine the unknown, and trying to look backwards (Polya, 1965).

The second mathematical process in the learning of mathematics is reasoning and proving (NCTM, 2000, 2005, 2020; The Ontario Curriculum, 2020; Rey et al. ,2001). According to the Ontario Curriculum (2005), in mathematics, students will develop and apply reasoning skills (i.e., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make guesses or mathematical conjectures, assess conjectures, justify conclusions, and plan and construct organized mathematical arguments. This process of reasoning and proving involves exploring phenomena, developing ideas, making mathematical conjectures, and justifying results.

According to Nyein and Thein (2018) there are four important points about mathematics reasoning: (1) reasoning is about making generalizations, (2) reasoning leads to a web of generalizations, (3) reasoning leads to mathematical memory built on relationships, and (4) learning through reasoning requires making mistakes and learning from them. In their view, students should be encouraged to reason from the evidence they find in their exploitations and investigations or from what they already know to be true, and to recognize the characteristics of an acceptable argument in the mathematics classroom. Therefore, the reasoning process supports a deeper understanding of mathematics by enabling students to make sense of the mathematics they are learning.

The third mathematical process is reflecting (The Ontario Curriculum, 2005, 2020). In their view, good problem solvers regularly and consciously reflect on and monitor thought processes. By doing so, they are able to recognize when the technique they are using is not fruitful, and to make conscious decision to switch to a different strategy, rethink the problem, and search for related content knowledge that may be helpful.

According to the Ontario curriculum (2005), one of the most valuable processes for students is to reflect in their learning with others. One way of doing this is to bring students together following an investigation to share out strategies and solutions. In the process, students are expected to share their thinking, defend and justify the strategies they used and solutions they reached, and talk about any challenges that faced. Exposing them to this information from a variety of students allows them to compare their own thinking and process to those of others, and evaluate and deepen their own understanding of the mathematical concept (The Ontario Curriculum, 2005).

The fourth mathematical process is selecting tools and computational strategies (Brumbaugh & Rock,2006: The Ontario Curriculum, 2005, 2020). Brumbaugh and Rock (2006) posits that students will always select a variety of concrete, visual, and electronic tools, and appropriate computational strategies to investigate mathematical ideas and to solve problems. For example, the mathematics teacher can encourage students to use technology to solve problems when the focus is on problem solving rather than on paper –and- paper skills (e.g., calculators, spreadsheets, and geometer’s sketch pad). According to the Ontario Curriculum (2005), students need to develop the ability to select the appropriate electronic tools, manipulatives, and computational strategies to perform mathematical tasks, to investigate mathematical ideas, and to solve problems.

The fifth mathematical process is connecting. Successful mathematical thinking means noticing how ideas are related. Costa and Kallick (2000) posits that it is making higher level connections that allows the students to draw forth a mathematical event and apply it to a new context in a way that connects familiar ideas with new concepts or skills. Furthermore, making good connections means seeing how mathematical concepts are connected to others and to the real world.

The Ontario Curriculum (2005), states that students need to see the connections and the relationships between mathematical concepts and skills from one topic of mathematics to another. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and they can use their learning in one area of mathematics to understand another. Moreover, seeing connections among procedures and concepts also helps to deepen students’ mathematical understanding.

The sixth mathematical process is representation (NCTM, 2000, 2005, 2020; Reys, et al., 2001; The Ontario Curriculum, 2020). According to the Ontario Curriculum (2005), students represent mathematical ideas and relationships and model situations using concrete materials, pictures, diagrams, graphs, tables, numbers, words and symbols. For example, a teacher can introduce new concepts using concrete materials. The various forms of representation help students to make connections and develop flexibility in their thinking about mathematics.

Reys et al. (2001) gives three major goals for representation as a process in mathematics. (1) Creating and using representations to organize, record, and communicate mathematical ideas, (2) selecting, applying, and translating among representations to solve problems, and (3) using representations to model and interpret physical, social, and mathematical phenomena.

The seventh mathematical process is communication (NCTM, 2000, 2005, 2020; The Ontario Curriculum, 2020; Rey et al., 2001). A student who is poor at communicating cannot explain his/her thinking which means there is no ability to justify with examples and does not see feedback as important. Students who are successful at mathematical communication, however, seek clarification (Reys et al., 2001). In their view, communication allows interaction and enable students to question, criticize, and clarify.

According to the Ontario Curriculum (2020), a student who is successful in mathematical communication is able to: (i) explain his/her thinking clearly and concisely, (ii) seeks clarification, (iii) realizes that it is normal to make mistakes, and (iv) when others come up with new ideas, he/she asks them to explain those ideas or tries to figure out why that makes sense.

Mathematics plays a key role in shaping how individuals deals with the aspects of social, scientific and technological life, but today, as in the past, many students struggle with mathematics and become discontented with the subject as they progress in their schooling. Therefore, it is important for mathematics teachers to understand how mathematical processes can be taught effectively to improve students’ knowledge and understanding of concepts and skills. The seven mathematical processes highlighted in this paper (i.e., problem solving, reasoning and proving, reflecting, use of tools and computational strategies, connecting, representing, and communicating) can support the acquisition and the use of mathematical knowledge and skills, and should form the base of practice in mathematics classrooms.

• Brumbaugh, D.K. & Rock, D. (2006). Teaching secondary mathematics (3 rd ). Lawrence Erlbaum Associates.
• Costa, A. L., & Kallick, B. (2000). Getting in the habit of reflection. Educational Leadership, 57, 60-62.
• Fernandez- Cezar, R., Nunez, R.P, & Suarez, C.A.H. (2020). Mathematical Processes and pedagogical practice: Characterization of the teachers in basic and middle education. Espacious, 41(8), 8
• Garfola, J, & Lester, F.K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal of Research in Mathematics Education, 16,163-176.
• Nyein, H. A. & Thein, N. N. (2018). An investigation into the mathematics process skills of the middle school students. J. Myanmar. Acad. Arts. Sci, Vol.XVI. No. 9A
• Kantowski, M. G. (1977). Processes involved in mathematical problem solving. Journal for Research in Mathematics Education, 8(3), 163-180
• Mayer, R. E (1987). Educational psychology: A cognitive approach. Little Brown and company.
• NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
• NCTM (2005). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
• NCTM (2020). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
• Polya, G. (1965). Mathematical discovery. Wiley and Sons.
• Reys, R. E., Lindquist, M. M, Lambdin, D. V., Smith, N. L & Suydam, M. N. (2001). Helping children learn mathematics. John Wiley and sons.
• Schoenfeld, A.H. (1985). Mathematical problem solving. Academic press
• Schoenfeld, A.H., & Herrman, D. (1982). Problem perception and knowledge structure in expert and novice problem solvers. Journal of Experimental Psychology: Learning, Memory and Cognition, 8,484-494.
• The Ontario Curriculum (2005). The Ontario curriculum, grades 1-8, mathematics. Ontario Ministry of Education.
• The Ontario Curriculum (2020). The Ontario curriculum, grades 1-8, mathematics. Ontario Ministry of Education.

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Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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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]

Mathematical Problem-Solving: Techniques and Strategies

by Ali | Mar 8, 2023 | Blog Post , Blogs | 0 comments

Introduction to Mathematical Problem-Solving

Mathematical problem-solving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem. It is an essential skill that is required in a wide range of academic and professional fields, including science, technology, engineering, and mathematics (STEM).

Importance of Mathematical Problem-Solving Skills

Mathematical problem-solving skills are critical for success in many areas of life, including education, career, and daily life. It helps students to develop analytical and critical thinking skills, enhances their ability to reason logically, and encourages them to persevere when faced with challenges.

The Process of Mathematical Problem-Solving

The process of mathematical problem-solving involves several steps that include identifying the problem, understanding the problem, making a plan, carrying out the plan, and checking the answer.

Techniques and Strategies for Mathematical Problem-Solving

1.      identify the problem.

The first step in problem-solving is to identify the problem. It involves reading the problem carefully and determining what the problem is asking.

2.      Understand the problem

The next step is to understand the problem by breaking it down into smaller parts, identifying any relevant information, and determining what needs to be solved.

3.      Make a plan

After understanding the problem, the next step is to develop a plan to solve it. This may involve identifying a formula or method to use, drawing a diagram or chart, or making a list of steps to follow.

4.      Carry out the plan

Once a plan is developed, the next step is to carry out the plan by solving the problem using the chosen method. It is important to show all steps and work neatly to avoid making mistakes.

5.      Check the answer

Finally, it is essential to check the answer to ensure it is correct. This can be done by re-reading the problem, checking the solution for accuracy, and verifying that it makes sense.

Know About: HOW TO FIND PERFECT MATH TUTOR 

Importance of using online calculators while learning math.

Utilizing online calculators can prove to be a beneficial resource for learning mathematics. There are numerous reasons why incorporating them into your studies is a wise choice.

Firstly, online calculators offer the convenience of being easily accessible at any time and from anywhere. No longer do you need to carry a physical calculator with you; you can use them on any device that has internet connectivity.

In addition, online calculators excel in accuracy and can efficiently handle complex calculations that may be difficult to do manually. They can perform arithmetic at a faster speed, saving you time and increasing productivity.

Another advantage is that some online calculators include built-in visualizations such as graphs and charts, which can help students grasp mathematical concepts better.

Furthermore, feedback can be provided by certain online calculators, assisting students in identifying and rectifying errors in their calculations. This feature can be especially useful for students who are new to learning mathematics .

Online calculators have a versatile range of functions beyond basic arithmetic, including algebraic equations, trigonometry, and calculus . This makes them useful for students at all levels of math education.

Overall, online calculators are an invaluable tool for students learning math. They are convenient, accurate, efficient, and versatile, and aid in the understanding of mathematical concepts, making them an essential component of modern-day education.

Common Errors in Mathematical Problem-Solving

There are several common errors that can occur in mathematical problem-solving, including misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and double-check the answer for accuracy.

Improving Mathematical Problem-Solving Skills

There are several ways to improve mathematical problem-solving skills, including practicing regularly, working with others, seeking help from a teacher or tutor, and reviewing past problems. It is also helpful to develop a positive attitude towards problem-solving, persevere through challenges, and learn from mistakes.

Must Know: WHICH IS THE BEST WAY OF LEARNING ONLINE TUTORING OR TRADITIONAL TUTORING

Mathematical problem-solving is a crucial skill that is required for success in many academic and professional fields. By following the process of problem-solving and using the techniques and strategies outlined in this article, individuals can improve their problem-solving skills and achieve success in their academic and professional endeavors.

Frequently Asked Questions

What is mathematical problem-solving.

Mathematical problem-solving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem.

Why are mathematical problem-solving skills important?

What are the steps involved in the process of mathematical problem-solving, how can online calculators aid in learning mathematics.

Online calculators can aid in learning mathematics by providing convenience, accuracy, and efficiency. They can also help students grasp mathematical concepts better through built-in visualizations and provide feedback to identify and rectify errors in their calculations.

What are common errors to avoid in mathematical problem-solving?

Common errors to avoid in mathematical problem-solving include misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and double-check the answer for accuracy.

We are committed to help students by one on one online private tutoring to maximize their e-learning potential and achieve the best results they can.

For this, we offer a free of cost trial class so that we can satisfy you. There is a free trial class for first-time students.

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1.6: Problem Solving Strategies

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• Michelle Manes
• University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

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

How to Solve It

George Pólya was a great champion in the field of teaching  effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, How to Solve it . Pólya died at the age 98 in 1985. [1]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

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

• First, you have to understand the problem.
• After understanding, then make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

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

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

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

A Problem Solving Strategy: Try Something!

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

Note that being "good at mathematics" is not about doing things right the first time. It is about figuring things out. Practice being okay with having done something incorrectly. Try to avoid using an eraser and just lightly cross out incorrect work (do not black out the entire thing). This way if it turns out that you did something useful, you still have that work to reference! If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what was left after paying Brianna. Finally, Alex saw David and gave him 1/2 of the remaining money. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

A Problem Solving Strategy: Draw a Picture

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

A Problem Solving Strategy: Make Up Numbers

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

Try this: Assume (that is, pretend) Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person.

Or try working backward: suppose Alex has some specific amount left at the end, say $10. Since he gave David half of what he had before seeing David, that means he had $20 before running into David. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

Most people want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. Instead of asking the teacher, “Is this right?”, you should be ready to justify it and say, “Here’s my answer, and here is how I got it.”

A Problem Solving Strategy: Try a Simpler Problem

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said, “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

The ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

A Problem Solving Strategy: Work Systematically

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

A Problem Solving Strategy: Use Manipulatives to Help You Investigate

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

A Problem Solving Strategy: Look for and Explain Patterns

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table. If possible, actually describe these to a friend.
• Explain and justify any of the patterns you see (if possible, actually do this with a friend). If you don't have a partner to work with, imagine they asked you, "How can you be sure the patterns will continue?"
• Expand this to find what calculation(s) you would perform to find the total number of squares on a 100 × 100 chess board.

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What progress have you made?

A Problem Solving Strategy: Find the Math, Remove the Context

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

A Problem Solving Strategy: Check Your Assumptions

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

Common Core State Standards Initiative

• Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up : adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Standards in this domain:

Ccss.math.practice.mp1 make sense of problems and persevere in solving them..

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. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize —to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize , to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

CCSS.Math.Practice.MP4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

CCSS.Math.Practice.MP6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

CCSS.Math.Practice.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x 2 + 9 x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3( x - y ) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y .

CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y - 2)/( x - 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x - 1)( x + 1), ( x - 1)( x 2 + x + 1), and ( x - 1)( x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

• How to read the grade level standards
• Introduction
• Counting & Cardinality
• Operations & Algebraic Thinking
• Number & Operations in Base Ten
• Measurement & Data
• Number & Operations—Fractions¹
• Number & Operations in Base Ten¹
• Number & Operations—Fractions
• Ratios & Proportional Relationships
• The Number System
• Expressions & Equations
• Statistics & Probability
• The Real Number System
• Quantities*
• The Complex Number System
• Vector & Matrix Quantities
• Seeing Structure in Expressions
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations*
• Reasoning with Equations & Inequalities
• Interpreting Functions
• Building Functions
• Linear, Quadratic, & Exponential Models*
• Trigonometric Functions
• High School: Modeling
• Similarity, Right Triangles, & Trigonometry
• Expressing Geometric Properties with Equations
• Geometric Measurement & Dimension
• Modeling with Geometry
• Interpreting Categorical & Quantitative Data
• Making Inferences & Justifying Conclusions
• Conditional Probability & the Rules of Probability
• Using Probability to Make Decisions
• Courses & Transitions
• Mathematics Glossary
• Mathematics Appendix A

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Improving mathematical reasoning with process supervision.

Illustration: Ruby Chen

We've trained a model to achieve a new state-of-the-art in mathematical problem solving by rewarding each correct step of reasoning (“process supervision”) instead of simply rewarding the correct final answer (“outcome supervision”). In addition to boosting performance relative to outcome supervision, process supervision also has an important alignment benefit: it directly trains the model to produce a chain-of-thought that is endorsed by humans.

More resources

Introduction.

In recent years, large language models have greatly improved in their ability to perform complex multi-step reasoning. However, even state-of-the-art models still produce logical mistakes, often called hallucinations . Mitigating hallucinations is a critical step towards building aligned AGI.

We can train reward models to detect hallucinations using either outcome supervision , which provides feedback based on a final result, or process supervision , which provides feedback for each individual step in a chain-of-thought. Building on previous work [^reference-1] , we conduct a detailed comparison of these two methods using the MATH dataset [^reference-2] as our testbed. We find that process supervision leads to significantly better performance, even when judged by outcomes. To encourage related research, we release our full dataset of process supervision.

Alignment impact

Process supervision has several alignment advantages over outcome supervision. It directly rewards the model for following an aligned chain-of-thought, since each step in the process receives precise supervision. Process supervision is also more likely to produce interpretable reasoning, since it encourages the model to follow a human-approved process. In contrast, outcome supervision may reward an unaligned process, and it is generally harder to scrutinize.

In some cases, safer methods for AI systems can lead to reduced performance [^reference-3] , a cost which is known as an alignment tax . In general, any alignment tax may hinder the adoption of alignment methods, due to pressure to deploy the most capable model. Our results below show that process supervision in fact incurs a negative alignment tax, at least in the math domain. This could increase the adoption of process supervision, which we believe would have positive alignment side-effects.

Solving MATH problems

We evaluate our process-supervised and outcome-supervised reward models using problems from the MATH test set. We generate many solutions for each problem and then pick the solution ranked the highest by each reward model. The graph shows the percentage of chosen solutions that reach the correct final answer, as a function of the number of solutions considered. Not only does the process-supervised reward model perform better across the board, but the performance gap widens as we consider more solutions per problem. This shows us that the process-supervised reward model is much more reliable.

We showcase 10 problems and solutions below, along with commentary about the reward model’s strengths and weaknesses.

Explore examples in 3 categories:

• True positives
• True negatives
• False positives

Model attempt

This challenging trigonometry problem requires applying several identities in a not-at-all obvious succession. Most solution attempts fail, because it is hard to choose which identities are actually helpful. Although GPT-4 usually can’t solve this problem (only . 1 % .1\% .1% of solution attempts reach the correct answer), the reward model correctly recognizes that this solution is valid.

It is unknown how broadly these results will generalize beyond the domain of math, and we consider it important for future work to explore the impact of process supervision in other domains. If these results generalize, we may find that process supervision gives us the best of both worlds – a method that is both more performant and more aligned than outcome supervision.

• Hunter Lightman
• Vineet Kosaraju
• Harri Edwards
• Ilya Sutskever

Acknowledgments

Contributors.

Bowen Baker, Teddy Lee, John Schulman, Greg Brockman, Kendra Rimbach, Hannah Wong, Thomas Degry

Math Interventions

• Introduction
• Subitizing Interventions
• Counting Interventions: Whole Numbers Less Than 30
• Counting Interventions: Whole Numbers Greater Than 30 (Place Value)
• Counting Interventions: Fractions
• Counting Interventions: Decimals
• Composing and Decomposing Numbers Interventions
• Rounding Interventions
• Number Sense Lesson Plans
• Addition and Subtraction Facts
• Multiplication and Division Facts
• Computational Fluency Lesson Plans
• Understanding the Problem Interventions
• Planning and Executing a Solution Interventions
• Monitoring Progress & Reflecting on a Solution Interventions
• Problem-Solving Process Interventions

Problem-Solving Process

Response to error: using the problem-solving process, feedback during the lesson, strategies to try after the lesson.

• Problem-Solving Lesson Plans
• Identifying Essential Variables Interventions
• Direct Models Interventions
• Counting On/Back Interventions
• Deriving Interventions
• Interpreting the Results Interventions
• Mathematical Modeling Lesson Plans
• Math Rules and Concepts Interventions
• Math Rules and Concepts Lesson Plans

A student who has difficulty understanding the problem, planning and executing a solution , self-monitoring progress toward a goal, and evaluating a solution will benefit from intervention around the problem-solving process. The following interventions  support  students  in internalizing this process from start to finish. This page includes intervention strategies that you can use to support your students in this area. Remember, if you're teaching a full process from start to finish, you probably want to use the Self-Regulated Strategy Development approach, which spreads explicit instruction of a full process across a series of intervention lessons.  As you read, consider which of these interventions best aligns with your student's strengths and needs in the whole-learner domains.

Self-Regulated Strategy Development 

Self-Regulated Strategy Development (or SRSD) is one way to teach the problem-solving process. The SRSD model "requires teachers to explicitly teach students the use of the strategy, to model the strategy, to cue students to use the strategy, and to scaffold instruction to gradually allow the student to become an independent strategy user." (Reid, Leinemann, & Hagaman, 2013). The steps of teaching SRSD are slightly different from the steps of explicit instruction because, in SRSD, each step must be mastered before the next one is started. For example, you might spend an entire lesson on Developing Background Knowledge before moving on to Discuss It (see below). The longterm goal of SRSD is for students to be able carry out the strategy independently, and so time is dedicated to teaching each step of the strategy in such a manner as enables students to internalize the material. 

Teaching SRSD model requires six steps:

• Develop Background Knowledge. Define the key ideas that students need to know in order to apply the strategy.
• Discuss It. Tell the student what the strategy is called, and describe each step.
• Model It. Use a think-aloud to demonstrate the strategy.
• Memorize It . Internalize strategy.
• Support It. Gradually release responsibility to students.
• Independent performance. Give students opportunities to practice strategy without support.

SRSD Explicit Instruction Six-Step Model: 

To support your students' ability to apply SRSD, you should start by explicitly teaching the six-step model. Keep in mind that this type of explicit instruction may take place over a number of days. 

Step 1: Set the Context for Student Learning and Develop Background Knowledge.  

• Introduce Word Problem Mnemonics, and discuss the use of the mnemonic: "Today you will be learning a new trick to help you solve problems. This strategy is called CUBES." (Teacher gets out chart paper and markers and writes down C, U , B, E, and S vertically.) "CUBES is a self-regulated strategy, which means that you will learn to memorize the strategy and use it without my support. Let's go through each step of CUBES and see how it will help you go through the problem-solving process. First, C-Circle the Numbers" (Teacher write this next to C.) "U - Underline important words." (Teacher writes next to U.)  B- Box the question " (Teache r writes next to B). E- Eliminate unnecessary information. S - Solve and Check. (Teacher writes these terms next to E and S). "Now, what do we need to know when we are doing CUBES?  We need to know which words are important. We also need to eliminate unnecessary information" (Teacher goes on to define these terms.)

Step 2: Discuss It. 

• Discuss the significance and benefits of using CUBES. Discuss and determine goals for using the strategy. At this point, students can examine their past work to set an individual goal: "So, how is a self-regulated strategy going to help us? Well, it gives us an easy way to remember the five steps to solving the problem. How else does it help us?" (Teacher elicits student responses.) "When we are using a SRSD, we ask ourselves questions to make sure we are following the steps. We call these self-statements.  My self-statements are 'What's my first step?' and 'What am I supposed to do now?' I ask myself self-statements so I can make sure that I am using each step of the strategy, and that I don't miss any steps." (Teacher and students discuss benefits of self-statements.)  "Now let's take some time to set goals for using this strategy...." (Teacher and students set goals, such as "students will each have two self-statements they use when employing the CUBE strategy.")

Step 3: Model It.

• The teacher models the strategy using think alouds and self-statements: "Watch as I show you what CUBES looks like when I use it. See if you can notice my self-statements. What am I supposed to do? I'm supposed to to follow the five steps to solve a problem. What is my first step? C. That's right, C. I need to circle the numbers. I'll do that now, and then check that off my CUBE S  list. (Teacher circles numbers). Okay, I'm going to check my CUBES list again. I've already completed C. Now, on to U. I have to Underline important words. (Teacher continues to model the entire CUBES process with 1- 3 problems. The session ends. Teacher starts Model It with new problems on Day 2.)

Step 4: Memorize It . 

• Students memorize the mnemonic and each of the steps of CUBES. The idea is that the students will not be able to implement the strategy if they cannot recall the steps. "Next, we are are going to take some time to memorize each step. What is C?" "Circle the numbers!"What is U?" (Teacher completes this process for all the letters. At this time, students also write the mnemonic down so they can use it as a reference. If they need to, they can come up with a beat or a chant to remember the mnemonic.)

Step 5: Support It.

• In step 5, the teacher gradually releases responsibility to the students. This is the most important stage, especially for struggling readers. In order for students to be able to implement this strategy on their own, they must be supported as needed. Graham, Harris, Mason, and Friedlander (2008), SRSD experts and authors, often tell their teachers, "Please Don't P.E.E. in the Classroom - P ost, E xplain, E xpect. Success with SRSD depends on using all the stages for students who have difficulty with [reading]." SRSD instruction and implementation are only successful when students are given multiple opportunities to practice using their strategy with teacher support before trying it on their own.  "Let's read the next problem and do CUBES together this time..." Teacher follows the steps of gradual release to transfer responsibility to students. The teacher first engages students with guided support. She might read the problem and allow students to complete different parts of the strategy. Then, students might do CUBES in groups. This part of the strategy might take multiple days, until students are effectively completing the strategy by using self-statements. 

Step 6: Independent Practice

• In the final step, students practice using the strategy independently. "Now, you are ready to use CUBES on your own! Remember to use your self-statements, like What do I do next? and What am I supposed to do now? and I'll look at my CUBES sheet to see what I do next. as you employ this strategy!" Teacher circulates and provides support for students who are not yet ready to work independently.  

Activity A: Word Problem Mnemonics

One way to support your student's problem-solving ability is to teach her a mnemonic for a series of steps to take whenever she encounters a story problem. The following brief, developed by the Evidence Based Intervention Network at the University of Missouri, describes this strategy. As you read, consider how each mnemonic breaks down the problem-solving process.

Click here  to read the brief. 

Word Problem Mnemonics in Action

In the video below, Emily Art explicitly models how to use the word mnemonic, CUBES, to teach the problem solving process.

As you watch, consider: How do mnemonics support a student's ability to independently carry out the problem solving process?

Another strategy to use to teach your student the problem-solving process is called Self-Organizing Questions. Gifford (2005) advocates for teaching students a series of questions to ask themselves that will guide them through the problem-solving process. Read through each prompt below and consider its purpose. 

• Getting to Grips:  What are we trying to do?
• Connecting to Prior Knowledge:  Have we done anything like this before?
• Planning:  What do we need?
• Considering Alternative Methods:  Is there another way?
• Monitoring Progress:  How does it look so far?
• Evaluating Solutions:  Does it work?   How can we check? Can we make it better?

  Self-Organizing Questions in Action 

Give the student a problem. Then, go through the six self-organizing questions to guide the student through the problem-solving process. This example refers to the problem below. 

Lamont had 14 pumpkin seeds. He also had 32 apple seeds. He planted 41 of the seeds. How many seeds did Lamont have left?

Teacher: We are going to use the self-organizing questions to solve this problem. Frank, what are we trying to do?

Frank: We are trying to figure out how many seeds Lamont has left, after he plants the pumpkin and apple seeds.

Teacher: Let's think about similar problems we've had in the past. Have we done anything like this before?

Frank: Yes, yesterday, we solved a problem about how many baseball and soccer balls Jamie had. 

Teacher: So, what do we need to do to plan to solve this problem?

Frank: We need to add up the total number of seeds, and then subtract how many he planted.

Teacher: Is there another way to solve this problem?

Frank: We could probably draw it, or use manipulatives to help us. 

Teacher: Okay, go ahead and execute it! How does it look so far?

Frank: It's working for me. I added the types of seeds together, which gave me 46. Then, I subtracted the 41 seeds he planted. That gave me 5 seeds leftover, which seems about right. 

Teacher: How can we check our answer?

Frank: I'll see if I can add it back up. My solution was 5, so I'll add that to 41, which gives me 46. Then, I'll add the number of seeds he had total, which gives me 46! So, it matches!

Activity C: Solve It

If your student has particular struggles with understanding the problem, use Solve It, which is an explicit approach to teaching the problem-solving process, with an emphasis on understanding what the problem is about. The following brief, developed by the Evidence Based Intervention Network at the University of Missouri, describes this strategy. As you read, consider how this approach supports student understanding of problems.

Click  here  to read the brief. 

Solve It in Action Read the sample lesson plan (Montague, 2006) below to see what Solve It looks like in action. For your reference, click here to access a  self-regulation script  for students.

SolveItLesson.pdf

Gifford, S. (2005). Teaching mathematics 3-5: Developing learning in the foundation stage. Berkshire:  McGraw-Hill Education. Graham, S., & Harris, K.R. (2005).  Writing better: Effective strategies for teaching students with learning difficulties.  Baltimore, Maryland: Paul H. Brookes Publishing Co. Hughes, E.M. (2011). Intervention Name: Solve It! Columbia, Mo: The Evidence Based Intervention Network, The University of Missouri. Retrieved from https://education.missouri.edu/ebi/math-acquisition/ Hughes, E.M. & Powell, S. (2011). Intervention Name: Word-Problem Mnemonics. Columbia, Mo: The Evidence Based Intervention Network, The University of Missouri. Retrieved from https://education.missouri.edu/ebi/math-acquisition/ Montague, Marjorie. (2006). Self-regulation strategies for better math performance in middle school. In M. Montague and A. Jistendra (Eds.), Teaching mathematics to middle school students with learning disabilities. New York: The Guilford Press.   Reid, R., Lienemann, T. O., & Hagaman, J. L. (2013). Strategy instruction for students with learning disabilities. New York: The Guilford Press.

Think about the following scenario, which takes place after a teacher has explicitly taught a student to use the problem-solving process. The following example refers to the problem below. 

Lamont had 14 pumpkin seeds. He also had 32 apple seeds. He planted 41 of the seeds. How many seeds did Lamont have left?      Teacher: "Now that you understand the problem, what are you doing to do next?"      Student: "Solve it! 41-32 = 9. He had nine seeds left." 

In such a case, what might you do? 

When you are planning your lessons, you should anticipate that your student will make errors throughout. Here are a series of prompts that you can use to respond to errors. Keep in mind that all students are different, and that students might respond better to some types of feedback than to others.

If your student struggles to meet your objective, there are various techniques that you might try in order to adjust the activity so as best to meet your student's needs. 

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• Our Mission

Problem-Based Instruction in Middle and High School Math

PBI allows students to investigate real-world mathematical questions, increasing engagement with and understanding of course material.

Coach , facilitator , and guide on the side are phrases we have heard being used to describe the teacher’s role in PBI (problem-based instruction). Our idea of PBI is that students are exploring, inquiring, and crafting their own knowledge instead of being spoon-fed information by their teacher. In PBI the teacher moves from being the main disseminator of knowledge to a tool students use to help them guide their own exploration. The teacher must be well prepared and well versed in the content to be able to guide students to appropriate resources, answer questions, and ensure students remain on the correct trajectory with their inquiry. It is the role of the teacher as the content expert to come alongside the students to share resources, encourage, ask probing questions, and ensure students have a supportive environment in which to explore, inquire, and craft their understanding.

The preparations for a teacher in a PBI setting come largely before a PBI lesson is launched in the classroom. Teachers need to prepare resources, craft the driving question or problem scenario, and ensure all project aspects are planned and clear. If connections are being made to community entities or entities outside of the classroom, those arrangements must be secured by the teacher before initiating the PBI so that student experiences are well crafted and flow smoothly. While planning the PBI experience, teachers must also be intimately aware of their students’ learning needs.

Students learn at different rates and will seek different levels of content exploration. Scaffolding a lesson to accommodate student learning needs is a necessity in PBI. Teachers must know how to meet individual learning needs and what accommodations will have to be made, and then seek ways to provide support and structure within this framework. Teachers should also be familiar with the instructional learning goals targeted by the PBI lesson to ensure accuracy and adherence to these learning goals throughout the lesson.

Perhaps the most crucial trait of a teacher in a PBI lesson is flexibility. While teachers can plan, plan, and plan, it is almost guaranteed that something will not go as planned. Sometimes students stretch beyond the planned learning target and go deeper with their inquiry than expected. Other times students will hit a roadblock and will require extra support and encouragement. Sometimes schedules change, unexpected events occur, and the pacing for the lesson becomes offset. Teacher flexibility and fluidity will help encourage students to remain focused on their inquiry while knowing their knowledge journey is supported by their teacher.

PBI FROM THE STUDENT’S PERSPECTIVE

“When am I ever going to use this?” “Can you just tell me the steps needed to do this?” If you are or ever have been a math teacher, these are questions you have probably heard from students repeatedly and probably have become frustrated by. However, instead of getting frustrated, we should ask ourselves why these questions continue to permeate our mathematics classrooms. The answer? Students are not engaged in authentic mathematics while they are learning, but rather they are following prescribed steps in a rote memorized fashion to reach an answer. True learning is not regurgitating steps but rather seeing the connectedness of content and understanding the practical usability of different solution strategies.

In traditional mathematics classrooms, the teacher stands at the front of the room, demonstrates several step-by-step examples (typically devoid of real-world context) of a new skill, and then releases students to try it independently. As soon as students begin to struggle, the teacher walks them through the problem step-by-step. Students are exposed to word problems and applications at the end of the unit and then only minimally. As a result, students have developed a mathematical identity that defines their role in math class not as learners of mathematics and problem solvers but as performers whose only goal is to get questions right (Boaler, 2022). They disconnect from mathematics because they view it as rote procedures with no interesting or practical application.

PBI also allows students to learn and practice critical 21st-century skills needed to be successful no matter where their path in life and career takes them. Because PBI is collaborative in nature, students are learning to work together in team settings. They are learning to discuss their thoughts and share ideas so that others can understand and engage in dialogue around the shared comments and disseminate their findings/comments/ideas through various verbal, written, or multimedia platforms.

No matter how clearly or repeatedly a teacher explains a mathematical concept or skill, understanding can occur only when students connect new information with previously learned skills. Sure, using traditional methods may support students in memorizing enough steps to allow them to pass their unit assessment or even their end-of-course assessment, but is that truly learning? Rote regurgitation of memorized steps rarely results in long-term learning that translates to solving real-life problems or even to subsequent courses taken during their academic careers. To achieve this level of mathematical understanding, students must be able to engage in authentic mathematical tasks that allow them to collaborate, problem-solve, and problematize. In other words, mathematics is not something students learn by watching; it’s something they learn by doing. One of our students described PBI as a puzzle: “You look for pieces you need when you need them, and then all of a sudden, the whole picture comes together.”

In contrast to student experiences in traditional classrooms, students in a PBI environment feel immersed in their learning. They begin to believe that their voice matters and immediately see the applicability and practicality of what they are learning. Instead of “When am I ever going to use this?” and “Just tell me what the steps are,” students ask questions that prompt exploration, resulting in learning in context. Yes, students are working with manipulatives; yes, sometimes they complete practice worksheets; yes, students are working with their teacher(s) and peer(s), but each activity is carefully crafted toward its purpose relative to the problem/task to be solved/completed. In PBI lessons, there is no longer a feeling that students are learning content because it is in chapter 2 and they just finished chapter 1, so chapter 2 is what comes next . . . instead, the content is explored in context to give meaning and applicability.

Transitioning from a traditional classroom environment to one grounded in PBI can be challenging for students. PBI pushes students to think. PBI pushes students to go beyond what they think they know and to use what they know to “figure out” new concepts. PBI is different from how most students have been learning mathematics for years—it pushes them outside their comfort zone. As a result, students will push back. They will complain.

However, we can tell you from firsthand experience that if teachers remain consistent and support students through this struggle without compromising the foundations that PBI is built upon, students will not only accept this new way of learning mathematics but will thrive because of it. One of our students explained it this way: “This class is different. We don’t just cover content through lectures and you [the teacher] telling us what to do. We explore and discuss ideas, and suddenly I feel like I just know it. I feel like I have learned more in this math class than all of my other math classes combined.”

Reprinted by permission of the Publisher. From Sarah Ferguson and Denise L. Polojac-Chenoweth, Implementing Problem-Based Instruction in Secondary Mathematics Classrooms , New York: Teachers College Press. Copyright © 2024 by Teachers College, Columbia University. All rights reserved.

March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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• 20 March 2024
• Correction 21 March 2024

Mathematician who tamed randomness wins Abel Prize

• Davide Castelvecchi

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Michel Talagrand studies stochastic processes, mathematical models of phenomena that are governed by randomness. Credit: Peter Bagde/Typos1/Abel Prize 2024

A mathematician who developed formulas to make random processes more predictable and helped to solve an iconic model of complex phenomena has won the 2024 Abel Prize, one of the field’s most coveted awards. Michel Talagrand received the prize for his “contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics”, the Norwegian Academy of Science and Letters in Oslo announced on 20 March.

Assaf Naor, a mathematician at Princeton University in New Jersey, says it is difficult to overestimate the impact of Talagrand’s work. “There are papers posted maybe on a daily basis where the punchline is ‘now we use Talagrand’s inequalities’,” he says.

Talagrand’s reaction on hearing the news was incredulity. “There was a total blank in my mind for at least four seconds,” he says. “If I had been told an alien ship had landed in front of the White House, I would not have been more surprised.”

The Abel Prize was modelled after the Nobel Prizes — which do not include mathematics — and was first awarded in 2003. The recipient wins a sum of 7.5 million Norwegian kroner (US$700,000).

‘Like a piece of art’

Talagrand specializes in the theory of probability and stochastic processes, which are mathematical models of phenomena governed by randomness. A typical example is a river’s water level, which is highly variable and is affected by many independent factors, including rain, wind and temperature, Talagrand says. His proudest achievement was his inequalities 1 , a set of formulas that poses limits to the swings in stochastic processes. His formulas express how the contributions of many factors often cancel each other out — making the overall result less variable, not more.

“It’s like a piece of art,” says Abel-committee chair Helge Holden, a mathematician at the Norwegian University of Science and Technology in Trondheim. “The magic here is to find a good estimate, not just a rough estimate.”

Abel Prize: pioneer of ‘smooth’ physics wins top maths award

Thanks to Talagrand’s techniques, “many things that seem complicated and random turn out to be not so random”, says Naor. His estimates are extremely powerful, for example for studying problems such as optimizing the route of a delivery truck. Finding a perfect solution would require an exorbitant amount of computation, so computer scientists can instead calculate the lengths of a limited number of random candidate routes and then take the average — and Talagrand’s inequalities ensure that the result is close to optimal.

Talagrand also completed the solution to a problem posed by theoretical physicist Giorgio Parisi — work that ultimately helped Parisi to earn a Nobel Prize in Physics in 2021. In 1979, Parisi, now at the University of Rome, proposed a complete solution for the structure of a spin glass — an abstracted model of a material in which the magnetization of each atom tends to flip up or down depending on those of its neighbours.

Parisi’s arguments were rooted in his powerful intuition in physics, and followed steps that “mathematicians would consider as sorcery”, Talagrand says, such as taking n copies of a system — with n being a negative number. Many researchers doubted that Parisi’s proof could be made mathematically rigorous. But in the early 2000s, the problem was completely solved in two separate works, one by Talagrand 2 and an earlier one by Francesco Guerra 3 , a mathematical physicist who is also at the University of Rome.

Finding motivation

Talagrand’s journey to becoming a top researcher was unconventional. Born in Béziers, France, in 1952, he lost vision in his right eye at age five because of a genetic predisposition to detachment of the retina. Although while growing up in Lyon he was a voracious reader of popular science magazines, he struggled at school, particularly with the complex rules of French spelling. “I never really made peace with orthography,” he told an interviewer in 2019 .

His turning point came at age 15, when he received emergency treatment for another retinal detachment, this time in his left eye. He had to miss almost an entire year of school. The terrifying experience of nearly losing his sight — and his father’s efforts to keep his mind busy while his eyes were bandaged — gave Talagrand a renewed focus. He became a highly motivated student after his recovery, and began to excel in national maths competitions.

Just 5 women have won a top maths prize in the past 90 years

Still, Talagrand did not follow the typical path of gifted French students, which includes two years of preparatory school followed by a national admission competition for highly selective grandes écoles such as the École Normale Supérieure in Paris. Instead, he studied at the University of Lyon, France, and then went on to work as a full-time researcher at the national research agency CNRS, first in Lyon and later in Paris, where he spent more than a decade in an entry-level job. Apart from a brief stint in Canada, followed by a trip to the United States where he met his wife, he worked at the CNRS until his retirement.

Talagrand loves to challenge other mathematicians to solve problems that he has come up with — offering cash to those who do — and he keeps a list of those problems on his website. Some have been solved, leading to publications in major maths journals . The prizes come with some conditions: “I will award the prizes below as long as I am not too senile to understand the proofs I receive. If I can’t understand them, I will not pay.”

Nature 627 , 714-715 (2024)

doi: https://doi.org/10.1038/d41586-024-00839-6

Updates & Corrections

Correction 21 March 2024 : An earlier version of this article stated that Giorgio Parisi won the Nobel Prize in Physics in 2001. He in fact won in 2021.

Talagrand, M. Publ. Math. IHES 81 , 73–205, (1995).

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Talagrand, M. Ann. Math. 163 , 221–263 (2006).

Guerra, F. Commun. Math. Phys. 233 , 1–12 (2003).

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Title: incorporating graph attention mechanism into geometric problem solving based on deep reinforcement learning.

Abstract: In the context of online education, designing an automatic solver for geometric problems has been considered a crucial step towards general math Artificial Intelligence (AI), empowered by natural language understanding and traditional logical inference. In most instances, problems are addressed by adding auxiliary components such as lines or points. However, adding auxiliary components automatically is challenging due to the complexity in selecting suitable auxiliary components especially when pivotal decisions have to be made. The state-of-the-art performance has been achieved by exhausting all possible strategies from the category library to identify the one with the maximum likelihood. However, an extensive strategy search have to be applied to trade accuracy for ef-ficiency. To add auxiliary components automatically and efficiently, we present deep reinforcement learning framework based on the language model, such as BERT. We firstly apply the graph attention mechanism to reduce the strategy searching space, called AttnStrategy, which only focus on the conclusion-related components. Meanwhile, a novel algorithm, named Automatically Adding Auxiliary Components using Reinforcement Learning framework (A3C-RL), is proposed by forcing an agent to select top strategies, which incorporates the AttnStrategy and BERT as the memory components. Results from extensive experiments show that the proposed A3C-RL algorithm can substantially enhance the average precision by 32.7% compared to the traditional MCTS. In addition, the A3C-RL algorithm outperforms humans on the geometric questions from the annual University Entrance Mathematical Examination of China.

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Can you solve NASA's Pi Day 2024 challenge?

Hungry for Pi? Check out NASA's Pi Day challenge and put your wits to the test solving problems just like NASA scientists and engineers.

Happy Pi Day 2024!

Have you ever wondered what it would be like to solve problems for NASA to help with the exploration of other planets in the solar system ? 

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So, let's get solving! You can find each of the problems with an accompanying worksheet you can do all your work online and the answers will be posted by NASA so you can check your work! 

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Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

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This saved my life

I wasn't paying attention in class so I forgot what to do so I panicked so I started searching for an app that can help me. I found a few and installed but they kept telling me it was a word problem and that they can't solve it so I searched and searched until this app came I'm like this is like those apps but let's give it a try and then it said can solve every subject so I'm ok that's a good sign and then I finished my homework so try this app today. Trust me you won't regret installing this life saving homework app

I had no problems with it at first. The first few questions were free and I didn’t realize there was a limit on how many questions you could ask it. I ran out of questions quickly. It is absolutely disgusting how so many of these Ai apps expect large sums of money when all you want to do is study and learn. ESPECIALLY the amount they ask for. Some ask for 10-20$ a month just to talk to a BOT that may not even give you accurate/quality answers. Worst part is they expect you to make an account using apple so you can’t even make a new account or get anymore free questions. I deleted the app almost instantly. This app is no different than any other greedy AI app. Until it’s free or at least gives you more free questions I will be steering clear. Good bye.

The best math helper

This is the best thing I’ve ever seen the other things like Gauth, AI and Chat AI and chat to BT they don’t show me how to do this properly. Whenever I take a picture it shows me the wrong answer and I’ve told them multiple multiple times and it still doesn’t give me the right answer but this actually gives you the right answer. I thought it would be one of those scams, we give you the wrong thing, but it gives you the actual best answer you can get

App Privacy

The developer, Studdy, LLC , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

Data Used to Track You

The following data may be used to track you across apps and websites owned by other companies:

• Identifiers

Data Linked to You

The following data may be collected and linked to your identity:

Data Not Linked to You

The following data may be collected but it is not linked to your identity:

• User Content
• Diagnostics

Privacy practices may vary, for example, based on the features you use or your age. Learn More

Information

English, Arabic, Bengali, French, German, Greek, Hindi, Indonesian, Italian, Japanese, Korean, Polish, Portuguese, Russian, Simplified Chinese, Spanish, Swahili, Thai, Traditional Chinese, Turkish, Ukrainian, Urdu

• Studdy Unlimited $19.99
• Studdy Unlimited $143.99
• Studdy Unlimited $179.99
• Power Monthly $9.99
• Studdy Lite $9.99
• Power Monthly $19.99
• Developer Website
• App Support
• Privacy Policy

Family Sharing

Some in‑app purchases, including subscriptions, may be shareable with your family group when family sharing is enabled., you might also like.

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