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Home > Fischler > Transformations > Vol. 1 (2016) > Iss. 1
The Problem-Solving Process in a Mathematics Classroom
Enrique Ortiz , University of Central Florida
Problem solving provides a working framework to apply mathematics, and well chosen mathematics problems provide students with the opportunity to solidify and extend what they know, and can stimulate students’ mathematics learning (NCTM, 2001). Using this framework, students may utilize ways to learn mathematics concepts and skills that are rich with meaning and connections, and pre- and in-service teachers may implement teaching and assessment procedures to establish teaching and assessment environments.
Ortiz, Enrique (2016) "The Problem-Solving Process in a Mathematics Classroom," Transformations : Vol. 1: Iss. 1, Article 1. Available at: https://nsuworks.nova.edu/transformations/vol1/iss1/1
Since May 20, 2016
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5 Teaching Mathematics Through Problem Solving
In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)
What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.
According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.
There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.
Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.
Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.
Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.
Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):
- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.
Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.
Key features of a good mathematics problem includes:
- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.
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 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.
Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?
What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.
Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.
One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”
You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can
- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.
Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.
any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method
should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning
involves teaching a skill so that a student can later solve a story problem
when we teach students how to problem solve
teaching mathematics content through real contexts, problems, situations, and models
a mathematical activity where everyone in the group can begin and then work on at their own level of engagement
20 seconds to 2 minutes for students to make sense of questions
Mathematics Methods for Early Childhood by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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Theses & Dissertations
How problem-solving tasks work in a mathematics classroom in relation to developing students' mathematical thinking
Basiliana Caroli Mrimi , Aga Khan University, Institute for Educational Development, Karachi
Date of Award
Master of Education (M. Ed.)
Institute for Educational Development, Karachi
The efforts to improve the teaching and learning of mathematics involve varied approaches; the problem-solving approach is one of them. This research study sets out to understand how problem-solving tasks work in a mathematics classroom in relation to developing students' mathematical thinking. The objective was to teach mathematics in a lower-secondary class using problem-solving tasks and to find out how it works in a natural setting, i.e. classroom environment. A qualitative research paradigm was used, the action research in particular, because the study intended to implement the problem-solving approach in teaching mathematics and to find out its impact on the students' mathematical thinking. The findings of this research study show that problem-solving tasks can work in a mathematics classroom in relation to developing the students' mathematical thinking, and that there are processes involved in this regard which include: finding out and using what the students' already know, planning for the problem solving tasks, preparing the conducive classroom environment, and implementing problem solving tasks in a mathematics classroom. The related supporting factors include: the teaching approach, the teacher's role, students' understanding of why learn mathematics, and addressing the students' learning problems as they emerge. However, there are issues that need to be addressed in order to facilitate this kind of classroom practice. The main hindering factors found out include: the right answers myth, the time factor, and the use of the English language. The study concludes by highlighting the lesson learnt as a result of conducting this research study as a teacher-researcher, a mathematics teacher educator and as an agent of change. It provides my overall learning of researching in school, the research implications, and it recommends areas for further research and offers questions for reflection as a guide for needs analysis in the respective contexts to mathematics teachers, mathematics teacher educators, decision makers, and the related educational stake holders. It is expected that the answers to the needs analysis reflective questions would identify specific intentions' for a particular context.
Mrimi, B. (2005). How problem-solving tasks work in a mathematics classroom in relation to developing students' mathematical thinking (Unpublished master's dissertation). Aga Khan University, Karachi, Pakistan.
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Mathematics through problem solving.
What Is A 'Problem-Solving Approach'?
As the emphasis has shifted from teaching problem solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a problem-solving approach to teaching mathematics. The focus is on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterised by the teacher 'helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying' (Lester et al., 1994, p.154). Specific characteristics of a problem-solving approach include:
- interactions between students/students and teacher/students (Van Zoest et al., 1994)
- mathematical dialogue and consensus between students (Van Zoest et al., 1994)
- teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991)
- teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991)
- teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994)
- teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994)
- A further characteristic is that a problem-solving approach can be used to encourage students to make generalisations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).
Schoenfeld (in Olkin and Schoenfeld, 1994, p.43) described the way in which the use of problem solving in his teaching has changed since the 1970s:
My early problem-solving courses focused on problems amenable to solutions by Polya-type heuristics: draw a diagram, examine special cases or analogies, specialize, generalize, and so on. Over the years the courses evolved to the point where they focused less on heuristics per se and more on introducing students to fundamental ideas: the importance of mathematical reasoning and proof..., for example, and of sustained mathematical investigations (where my problems served as starting points for serious explorations, rather than tasks to be completed).
Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations. He described three characteristics of mathematical thinking:
- valuing the processes of mathematization and abstraction and having the predilection to apply them
- developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure - mathematical sense-making (Schoenfeld, 1994, p.60).
- As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just to solve specific problems, but to 'encourage the interiorization and reorganization of the involved schemes as a result of the activity' (p.187). Not only does this approach develop students' confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail.
The Role of Problem Solving in Teaching Mathematics as a Process
Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics listed at the outset of this article: functional, logical and aesthetic. Let us consider how problem solving is a useful medium for each of these.
It has already been pointed out that mathematics is an essential discipline because of its practical role to the individual and society. Through a problem-solving approach, this aspect of mathematics can be developed. Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context. Such motivation gives problem solving special value as a vehicle for learning new concepts and skills or the reinforcement of skills already acquired (Stanic and Kilpatrick, 1989, NCTM, 1989). Approaching mathematics through problem solving can create a context which simulates real life and therefore justifies the mathematics rather than treating it as an end in itself. The National Council of Teachers of Mathematics (NCTM, 1980) recommended that problem solving be the focus of mathematics teaching because, they say, it encompasses skills and functions which are an important part of everyday life. Furthermore it can help people to adapt to changes and unexpected problems in their careers and other aspects of their lives. More recently the Council endorsed this recommendation (NCTM, 1989) with the statement that problem solving should underly all aspects of mathematics teaching in order to give students experience of the power of mathematics in the world around them. They see problem solving as a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others.
According to Resnick (1987) a problem-solving approach contributes to the practical use of mathematics by helping people to develop the facility to be adaptable when, for instance, technology breaks down. It can thus also help people to transfer into new work environments at this time when most are likely to be faced with several career changes during a working lifetime (NCTM, 1989). Resnick expressed the belief that 'school should focus its efforts on preparing people to be good adaptive learners, so that they can perform effectively when situations are unpredictable and task demands change' (p.18). Cockcroft (1982) also advocated problem solving as a means of developing mathematical thinking as a tool for daily living, saying that problem-solving ability lies 'at the heart of mathematics' (p.73) because it is the means by which mathematics can be applied to a variety of unfamiliar situations.
Problem solving is, however, more than a vehicle for teaching and reinforcing mathematical knowledge and helping to meet everyday challenges. It is also a skill which can enhance logical reasoning. Individuals can no longer function optimally in society by just knowing the rules to follow to obtain a correct answer. They also need to be able to decide through a process of logical deduction what algorithm, if any, a situation requires, and sometimes need to be able to develop their own rules in a situation where an algorithm cannot be directly applied. For these reasons problem solving can be developed as a valuable skill in itself, a way of thinking (NCTM, 1989), rather than just as the means to an end of finding the correct answer.
Many writers have emphasised the importance of problem solving as a means of developing the logical thinking aspect of mathematics. 'If education fails to contribute to the development of the intelligence, it is obviously incomplete. Yet intelligence is essentially the ability to solve problems: everyday problems, personal problems ... '(Polya, 1980, p.1). Modern definitions of intelligence (Gardner, 1985) talk about practical intelligence which enables 'the individual to resolve genuine problems or difficulties that he or she encounters' (p.60) and also encourages the individual to find or create problems 'thereby laying the groundwork for the acquisition of new knowledge' (p.85). As was pointed out earlier, standard mathematics, with the emphasis on the acquisition of knowledge, does not necessarily cater for these needs. Resnick (1987) described the discrepancies which exist between the algorithmic approaches taught in schools and the 'invented' strategies which most people use in the workforce in order to solve practical problems which do not always fit neatly into a taught algorithm. As she says, most people have developed 'rules of thumb' for calculating, for example, quantities, discounts or the amount of change they should give, and these rarely involve standard algorithms. Training in problem-solving techniques equips people more readily with the ability to adapt to such situations.
A further reason why a problem-solving approach is valuable is as an aesthetic form. Problem solving allows the student to experience a range of emotions associated with various stages in the solution process. Mathematicians who successfully solve problems say that the experience of having done so contributes to an appreciation for the 'power and beauty of mathematics' (NCTM, 1989, p.77), the "joy of banging your head against a mathematical wall, and then discovering that there might be ways of either going around or over that wall" (Olkin and Schoenfeld, 1994, p.43). They also speak of the willingness or even desire to engage with a task for a length of time which causes the task to cease being a 'puzzle' and allows it to become a problem. However, although it is this engagement which initially motivates the solver to pursue a problem, it is still necessary for certain techniques to be available for the involvement to continue successfully. Hence more needs to be understood about what these techniques are and how they can best be made available.
In the past decade it has been suggested that problem-solving techniques can be made available most effectively through making problem solving the focus of the mathematics curriculum. Although mathematical problems have traditionally been a part of the mathematics curriculum, it has been only comparatively recently that problem solving has come to be regarded as an important medium for teaching and learning mathematics (Stanic and Kilpatrick, 1989). In the past problem solving had a place in the mathematics classroom, but it was usually used in a token way as a starting point to obtain a single correct answer, usually by following a single 'correct' procedure. More recently, however, professional organisations such as the National Council of Teachers of Mathematics (NCTM, 1980 and 1989) have recommended that the mathematics curriculum should be organized around problem solving, focusing on:
- developing skills and the ability to apply these skills to unfamiliar situations
- gathering, organising, interpreting and communicating information
- formulating key questions, analyzing and conceptualizing problems, defining problems and goals, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations
- developing curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).
One of the aims of teaching through problem solving is to encourage students to refine and build onto their own processes over a period of time as their experiences allow them to discard some ideas and become aware of further possibilities (Carpenter, 1989). As well as developing knowledge, the students are also developing an understanding of when it is appropriate to use particular strategies. Through using this approach the emphasis is on making the students more responsible for their own learning rather than letting them feel that the algorithms they use are the inventions of some external and unknown 'expert'. There is considerable importance placed on exploratory activities, observation and discovery, and trial and error. Students need to develop their own theories, test them, test the theories of others, discard them if they are not consistent, and try something else (NCTM, 1989). Students can become even more involved in problem solving by formulating and solving their own problems, or by rewriting problems in their own words in order to facilitate understanding. It is of particular importance to note that they are encouraged to discuss the processes which they are undertaking, in order to improve understanding, gain new insights into the problem and communicate their ideas (Thompson, 1985, Stacey and Groves, 1985).
It has been suggested in this chapter that there are many reasons why a problem-solving approach can contribute significantly to the outcomes of a mathematics education. Not only is it a vehicle for developing logical thinking, it can provide students with a context for learning mathematical knowledge, it can enhance transfer of skills to unfamiliar situations and it is an aesthetic form in itself. A problem-solving approach can provide a vehicle for students to construct their own ideas about mathematics and to take responsibility for their own learning. There is little doubt that the mathematics program can be enhanced by the establishment of an environment in which students are exposed to teaching via problem solving, as opposed to more traditional models of teaching about problem solving. The challenge for teachers, at all levels, is to develop the process of mathematical thinking alongside the knowledge and to seek opportunities to present even routine mathematics tasks in problem-solving contexts.
Carpenter, T. P. (1989). 'Teaching as problem solving'. In R.I.Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.187-202). USA: National Council of Teachers of Mathematics.
Clarke, D. and McDonough, A. (1989). 'The problems of the problem solving classroom', The Australian Mathematics Teacher, 45, 3, 20-24.
Cobb, P., Wood, T. and Yackel, E. (1991). 'A constructivist approach to second grade mathematics'. In von Glaserfield, E. (Ed.), Radical Constructivism in Mathematics Education, pp. 157-176. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Cockcroft, W.H. (Ed.) (1982). Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's Stationery Office.
Evan, R. and Lappin, G. (1994). 'Constructing meaningful understanding of mathematics content', in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 128-143. Reston, Virginia: NCTM.
Gardner, Howard (1985). Frames of Mind. N.Y: Basic Books.
Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon, V.M. and Raymond, A.M. (1994). 'Learning how to teach via problem solving'. in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 152-166. Reston, Virginia: NCTM.
National Council of Teachers of Mathematics (NCTM) (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s, Reston, Virginia: NCTM.
National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and Evaluation Standards for School Mathematics, Reston, Virginia: NCTM.
Olkin, I. & Schoenfeld, A. (1994). A discussion of Bruce Reznick's chapter. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 39-51). Hillsdale, NJ: Lawrence Erlbaum Associates.
Polya, G. (1980). 'On solving mathematical problems in high school'. In S. Krulik (Ed). Problem Solving in School Mathematics, (pp.1-2). Reston, Virginia: NCTM.
Resnick, L. B. (1987). 'Learning in school and out', Educational Researcher, 16, 13-20..
Romberg, T. (1994). Classroom instruction that fosters mathematical thinking and problem solving: connections between theory and practice. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum Associates.
Schifter, D. and Fosnot, C. (1993). Reconstructing Mathematics Education. NY: Teachers College Press.
Schoenfeld, A. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 53-69). Hillsdale, NJ: Lawrence Erlbaum Associates.
Stacey, K. and Groves, S. (1985). Strategies for Problem Solving, Melbourne, Victoria: VICTRACC.
Stanic, G. and Kilpatrick, J. (1989). 'Historical perspectives on problem solving in the mathematics curriculum'. In R.I. Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.1-22). USA: National Council of Teachers of Mathematics.
Swafford, J.O. (1995). 'Teacher preparation'. in Carl, I.M. (Ed.) Prospects for School Mathematics , pp. 157-174. Reston, Virginia: NCTM.
Thompson, P. W. (1985). 'Experience, problem solving, and learning mathematics: considerations in developing mathematics curricula'. In E.A. Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, (pp.189-236). Hillsdale, N.J: Lawrence Erlbaum.
Van Zoest, L., Jones, G. and Thornton, C. (1994). 'Beliefs about mathematics teaching held by pre-service teachers involved in a first grade mentorship program'. Mathematics Education Research Journal. 6(1): 37-55.
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Mathematics as a Complex Problem-Solving Activity
By jacob klerlein and sheena hervey, generation ready.
By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: they respond to their environment and the reactions of others. This making sense of experience is an ongoing, recursive process. We have known for a long time that reading is a complex problem-solving activity. More recently, teachers have come to understand that becoming mathematically literate is also a complex problem-solving activity that increases in power and flexibility when practiced more often. A problem in mathematics is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem—it is a straightforward application.
Mathematicians have always understood that problem-solving is central to their discipline because without a problem there is no mathematics. Problem-solving has played a central role in the thinking of educational theorists ever since the publication of Pólya’s book “How to Solve It,” in 1945. The National Council of Teachers of Mathematics (NCTM) has been consistently advocating for problem-solving for nearly 40 years, while international trends in mathematics teaching have shown an increased focus on problem-solving and mathematical modeling beginning in the early 1990s. As educators internationally became increasingly aware that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways (Wu and Zhang 2006) little changed at the school level in the United States.
“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”
(NCTM, 2000, p. 52)
In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. For many teachers of mathematics this was the first time they had been expected to incorporate student collaboration and discourse with problem-solving. This practice requires teaching in profoundly different ways as schools moved from a teacher-directed to a more dialogic approach to teaching and learning. The challenge for teachers is to teach students not only to solve problems but also to learn about mathematics through problem-solving. While many students may develop procedural fluency, they often lack the deep conceptual understanding necessary to solve new problems or make connections between mathematical ideas.
“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.”
(Burns, 2000, p. 29)
Learning to problem solve
To understand how students become problem solvers we need to look at the theories that underpin learning in mathematics. These include recognition of the developmental aspects of learning and the essential fact that students actively engage in learning mathematics through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning” (Copley, 2000, p. 29). The concept of co-construction of learning is the basis for the theory. Moreover, we know that each student is on their unique path of development.
Beliefs underpinning effective teaching of mathematics
- Every student’s identity, language, and culture need to be respected and valued.
- Every student has the right to access effective mathematics education.
- Every student can become a successful learner of mathematics.
Children arrive at school with intuitive mathematical understandings. A teacher needs to connect with and build on those understandings through experiences that allow students to explore mathematics and to communicate their ideas in a meaningful dialogue with the teacher and their peers.
Learning takes place within social settings (Vygotsky, 1978). Students construct understandings through engagement with problems and interaction with others in these activities. Through these social interactions, students feel that they can take risks, try new strategies, and give and receive feedback. They learn cooperatively as they share a range of points of view or discuss ways of solving a problem. It is through talking about problems and discussing their ideas that children construct knowledge and acquire the language to make sense of experiences.
Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something directly (Hiebert1997). The teacher’s role is to construct problems and present situations that provide a forum in which problem-solving can occur.
Why is problem-solving important?
Our students live in an information and technology-based society where they need to be able to think critically about complex issues, and “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998). Mathematics education is important not only because of the “gatekeeping role that mathematics plays in students’ access to educational and economic opportunities,” but also because the problem-solving processes and the acquisition of problem-solving strategies equips students for life beyond school (Cobb, & Hodge, 2002).
The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures. It is through problem-solving that students develop a deeper understanding of mathematical concepts, become more engaged, and appreciate the relevance and usefulness of mathematics (Wu and Zhang 2006). Problem-solving in mathematics supports the development of:
- The ability to think creatively, critically, and logically
- The ability to structure and organize
- The ability to process information
- Enjoyment of an intellectual challenge
- The skills to solve problems that help them to investigate and understand the world
Problem-solving should underlie all aspects of mathematics teaching in order to give students the experience of the power of mathematics in the world around them. This method allows students to see problem-solving as a vehicle to construct, evaluate, and refine their theories about mathematics and the theories of others.
Problems that are “Problematic”
The teacher’s expectations of the students are essential. Students only learn to handle complex problems by being exposed to them. Students need to have opportunities to work on complex tasks rather than a series of simple tasks devolved from a complex task. This is important for stimulating the students’ mathematical reasoning and building durable mathematical knowledge (Anthony and Walshaw, 2007). The challenge for teachers is ensuring the problems they set are designed to support mathematics learning and are appropriate and challenging for all students. The problems need to be difficult enough to provide a challenge but not so difficult that students can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed. Problems need to be within the students’ “Zone of Proximal Development” (Vygotsky 1968). These types of complex problems will provide opportunities for discussion and learning.
Students will have opportunities to explain their ideas, respond to the ideas of others, and challenge their thinking. Those students who think math is all about the “correct” answer will need support and encouragement to take risks. Tolerance of difficulty is essential in a problem-solving disposition because being “stuck” is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially. Effective problems:
- Are accessible and extendable
- Allow individuals to make decisions
- Promote discussion and communication
- Encourage originality and invention
- Encourage “what if?” and “what if not?” questions
- Contain an element of surprise (Adapted from Ahmed, 1987)
“Students learn to problem solve in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”
(Copley, 2000, p. 29)
“…as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas.…This enables learners to become clearer and more confident about what they know and understand.”
(Fosnot, 2005, p. 10)
Research shows that ‘classrooms where the orientation consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners’ (Anthony and Walshaw, 2007, page 122).
Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Students often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the problem without telling them exactly what to do. In 1945 Pólya published the following four principles of problem-solving to support teachers with helping their students.
- Understand and explore the problem
- Find a strategy
- Use the strategy to solve the problem
- Look back and reflect on the solution
Problem-solving is not linear but rather a complex, interactive process. Students move backward and forward between and across Pólya’s phases. The Common Core State Standards describe the process as follows:
“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary”. (New York State Next Generation Mathematics Learning Standards 2017).
Pólya’s Principals of Problem-Solving
Students move forward and backward as they move through the problem-solving process.
The goal is for students to have a range of strategies they use to solve problems and understand that there may be more than one solution. It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn’t the end of the process. Reflecting on the strategies used to solve the problem provides additional learning experiences. Studying the approach used for one problem helps students become more comfortable with using that strategy in a variety of other situations.
When making sense of ideas, students need opportunities to work both independently and collaboratively. There will be times when students need to be able to work independently and other times when they will need to be able to work in small groups so that they can share ideas and learn with and from others.
Effective teachers of mathematics create purposeful learning experiences for students through solving problems in relevant and meaningful contexts. While word problems are a way of putting mathematics into contexts, it doesn’t automatically make them real. The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it. Realistic does not mean that problems necessarily involve real contexts, but rather they make students think in “real” ways.
Planning for talk
By planning for and promoting discourse, teachers can actively engage students in mathematical thinking. In discourse-rich mathematics classes, students explain and discuss the strategies and processes they use in solving mathematical problems, thereby connecting their everyday language with the specialized vocabulary of mathematics.
Students need to understand how to communicate mathematically, give sound mathematical explanations, and justify their solutions. Effective teachers encourage their students to communicate their ideas orally, in writing, and by using a variety of representations. Through listening to students, teachers can better understand what their students know and misconceptions they may have. It is the misconceptions that provide a window into the students’ learning process. Effective teachers view thinking as “the process of understanding,” they can use their students’ thinking as a resource for further learning. Such teachers are responsive both to their students and to the discipline of mathematics.
“Mathematics today requires not only computational skills but also the ability to think and reason mathematically in order to solve the new problems and learn the new ideas that students will face in the future. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, are encouraged to make mathematical conjectures and test them, and are helped to develop their reasoning skills.”
(John Van De Walle)
“Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.”
How teachers organize classroom instruction is very much dependent on what they know and believe about mathematics and on what they understand about mathematics teaching and learning. Teachers need to recognize that problem-solving processes develop over time and are significantly improved by effective teaching practices. The teacher’s role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants their students to explore. A problem-solving approach is not only a way for developing students’ thinking, but it also provides a context for learning mathematical concepts. Problem-solving allows students to transfer what they have already learned to unfamiliar situations. A problem-solving approach provides a way for students to actively construct their ideas about mathematics and to take responsibility for their learning. The challenge for mathematics teachers is to develop the students’ mathematical thinking process alongside the knowledge and to create opportunities to present even routine mathematics tasks in problem-solving contexts.
Given the efforts to date to include problem-solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable professional learning, resources, and more time are essential steps, it is possible that problem-solving in mathematics will only become valued when high-stakes assessment reflects the importance of students’ solving of complex problems.
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Making Math Meaningful for Young Children
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Children are natural mathematicians. They push and pull toys, stack blocks, and fill and empty cups of water in the bathtub. All of these activities allow young children to experience math concepts as they experiment with spatial awareness, measurement, and problem solving (ETFO 2010; NAEYC 2010). Young children easily learn as they describe, explain, and consider the ideas from their immediate environment. Am I as tall as Yancey? How can I find out? I know! We can both stand next to each other in front of the mirror.
Early math is not about the rote learning of discrete facts like how much 5 + 7 equals. Rather, it’s about children actively making sense of the world around them. Unlike drills or worksheets with one correct answer, open-ended, playful exploration encourages children to solve problems in real situations. Because the situations are meaningful, children can gain a deeper understanding of number, quantity, size, patterning, and data management (Grossman 2012). For example, it is easier to understand what six means when applied to a real-life task such as finding six beads to string on a necklace or placing one cracker on each of six plates.
Creating a math-rich classroom
Research suggests that preschool classrooms can be the ideal environment for learning about math (ETFO 2010). Children sort materials into corresponding bins at cleanup time, explore patterns and shapes while creating at the art table, tell time while using the visual schedule to predict which activities come next, and measure when they squeeze their bodies through the climber on the playground (ETFO 2010).
Preschool classrooms also celebrate curiosity and risk-taking as children engage in inquiry-based exploration at various learning centers and outdoors. Interesting items in the environment encourage children to find answers to their questions and solve problems across all curricular domains. Children measure as they clap out the beats to music. They repeat rhythmic patterns as they dance. They describe, sort, and count objects in the discovering science center and look for patterns while on a nature walk. They count the rungs while climbing up the ladder to the loft. Many familiar children’s songs, stories, and poems contain mathematical messages that help familiarize children with counting, measuring, and patterning. For example, children can count along with “One, two, buckle my shoe” and “Ten little monkeys jumping on the bed.”
In addition to offering blocks, buttons, and other loose materials to touch and explore, teachers can ask open-ended questions that promote problem solving and probe and challenge children’s mathematical thinking and reasoning (Ontario Ministry of Education 2010). Such questions are not meant to elicit correct answers but rather to engage children in open-ended conversations that promote high-level thinking, such as What do you notice about these objects? How might we sort the toys? One of the foundations of play-based learning is that the teacher is active in the play, asking questions and adding knowledge and insight. The teacher learns together with children throughout the inquiry process.
Every preschool classroom needs to be rich with materials that encourage math exploration and learning. A well-stocked math and manipulatives center includes found objects such as shells, stones, bread tags, and sticks, as well as purchased materials. The center can include photos of completed geoboard creations or of children sorting coins in the dramatic play center. There might be narratives of children’s learning, such as transcripts of children’s comments and conversations, and artwork featuring pattern or shape exploration. Teachers can post documentation of math learning as a way of encouraging children to reflect on past experiences and motivate them to plan and revise future ones. These visuals can inspire even deeper, more connected learning. They help children maintain their focus on a particular topic, refine and expand their ideas, communicate their learning to others, and reflect on their experiences before making new plans.
Encourage children to play mathematically
Young children need to see themselves as capable mathematicians. Child-guided and child-focused explorations and teacher-guided math activities help children practice and consolidate their learning. This helps them feel confident about what they know and can do. Although many preschoolers learn some math concepts on their own, it’s important for teachers to include math in authentic experiences, resulting in a deeper understanding by children (ETFO 2010).
In addition to creating a rich math and manipulatives learning center, teachers can encourage children to use math tools and strategies in all areas of the classroom. Children might use a set of plastic links to measure their buildings in the block center, use play money to pay for a train ticket in the dramatic play center, and use rulers to measure the growth of spring bulbs in the discovering science center. Take a set of scales outdoors so children can figure out who found the heaviest rock. Using math tools for real-life tasks frees both teachers and children to act spontaneously, resulting in richer interactions and a calmer learning environment (Wien 2004).
In addition to the freedom to use materials in authentic ways, children also need freedom of time and space to deeply engage in math. The preschool schedule should include plenty of time for uninterrupted play so children have the time they need to work on sustained tasks of interest. This allows children to explore materials thoroughly, often resulting in more complex and evolved experiences over time. If a child spends all of his time at one learning center, he is not missing out on learning opportunities elsewhere. Instead, his deep connection to the center is often indicative of rich learning. Teachers can model the use of other materials at the center, such as using writing materials to draw plans for a structure to be built, or pose challenges that encourage the child to think beyond her play, such as How tall can you build this tower before it falls?
To support learning, it is important to encourage children to communicate their explorations and findings. Teachers can establish a routine through which children share their experiences at group time. For example, a child might explain how he built a structure with blocks, do a dance with repeating steps, or share a photo of a complex pattern made with colorful buttons. While circulating through the room, a teacher might notice high-quality work and suggest that a child share it with her peers during group time. The child making the presentation grows in confidence and the onlookers may want to try the experience themselves.
Most children enter preschool knowing a lot about math. In a safe and supportive classroom they will feel comfortable taking risks and engaging in self-directed problem solving. Weaving math into all areas of the curriculum will heighten children’s play experiences and allow all learners to experience success. Children will soon see themselves as capable mathematicians who apply their skills in a number of ways. Their growing math skills, confidence, and interests will serve them well in school and life.
Supporting Dual language learners
Children who are DLLs can learn math concepts and skills without being fluent in their second language. Much of the meaning is found in the right materials. If families send to the classroom familiar items from home, the children will know the name and function of the items in their home language. They can use this prior knowledge as a foundation to help them learn math. For example, young children may not understand how to sort plastic shapes, but they already know it is important to sort the baby’s socks and daddy’s socks in separate piles—a math activity that has real-life meaning in any language.
ETFO (Elementary Teachers’ Federation of Ontario). 2010. Thinking It Through: Teaching and Learning in the Kindergarten Classroom . Toronto, ON: ETFO.
Grossman, S. 2012. “The Worksheet Dilemma: Benefits of Play-Based Curricula.” Early Childhood News . www.earlychildhoodnews.com/earlychildhood/article_view.aspx?ArticleID=134 .
NAEYC. 2010. “Early Childhood Mathematics: Promoting Good Beginnings.” A joint position statement of NAEYC and the National Council of Teachers of Mathematics (NCTM). www.naeyc.org/files/naeyc/file/positions/psmath.pdf .
Ontario Ministry of Education. 2010. The Full-Day Early Learning-Kindergarten Program (draft version). Toronto, ON: Queen’s Printer for Ontario. www.edu.gov.on.ca/eng/curriculum/elementary/kindergarten_english_june3.pdf .
Wien, C.A. 2004. “From Policing to Participation: Overturning the Rules and Creating Amiable Classrooms.” Young Children 59 (1): 34–40. www.naeyc.org/files/yc/file/200401/Wien.pdf .
Deanna Pecaski McLennan , PhD, is a passionate early childhood educator, researcher, and writer from Amherstburg, Ontario, Canada, who has spent over 20 years working with young children. As a teacher in Ontario’s full-day kindergarten program and a university instructor, she devotes her research and practice to exploring the potential for rich mathematics learning through playful inquiry and exploration.
Vol. 8, No. 1
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Jul 16, 2021
Metacognitive Strategies in the Math Classroom
By dr. lanette trowery, sr. director of learning at mcgraw hill school and margaret bowman, academic designer at mcgraw hill school.
Metacognition refers to individuals’ knowledge concerning cognitive processes and regulation of these processes in relation to cognitive objectives (Desoete & De Craene, 2019; Flavell, 1976; Jin & Kim, 2018).
In other words, metacognition is the process of thinking about thinking.
Metacognitive strategies , as you can imagine, are teaching and learning practices that encourage students to engage in metacognition, or think about their thinking as they learn new things, explore concepts, and apply knowledge.
How do metacognitive strategies help students learn?
Strategic metacognitive engagement has been shown to aid in performance in the classroom and overall academic achievement.
💡 Research Spotlight: In one study, students’ problem-solving processes were qualitatively shown to be supported by engaging in metacognitive regulation — the active monitoring and controlling of cognitive processes (Jin & Kim, 2018). Students were able to help monitor and adjust each other’s thinking through their conversations. As students said things like, “This makes no sense” or “I don’t understand this,” other students would respond with, “Let’s try to think of this another way.” Desoete and De Craene (2019) noted that metacognitive skills were associated with mathematical accuracy. Reflection is also linked to social-emotional learning, as students can benefit from reflecting on the thoughts, feelings, and emotional aspects of what they have learned.
How can teachers encourage metacognitive strategies in the classroom?
Metacognitive strategies can be integrated into regular classroom instruction through:
- Collaborative activities, such as students working in groups while discussing solutions to a given problem (Jin and Kim 2018)
- Thorough explanation of a topic allows students to reflect on what they know about a topic and connect it to new information they learn (Denton, 2011)
- Formative assessments (Denton, 2011), such as verbal discussions or written evaluations in which students complete a chart to explain how they feel about their learning.
💡 Research Spotlight : In one study, students were guided to use metacognitive questioning as part of the process of solving math problems. These metacognitive questions included comprehension, connection, strategic and reflection questions (Mevarech & Kramarski, 2003).
Metacognitive Strategies in Math
Why is metacognition important in the math classroom.
Metacognition is a critical skill in K-5 math education because engaging in metacognitive strategies can help students build a conceptual understanding of content and foster student agency.
- When studying mathematics, metacognitive strategies can play an important role in knowledge acquisition, retention, and application . At the conceptual development stage, when students are first encountering new ideas and skills, thinking about the relationships between their prior knowledge and new knowledge tends to help students have better conceptual understanding (Mevarech & Kramarski, 2003).
- Giving students the time and space to reflect on their own thinking is critical for fostering student agency . Research shows that agency is time bound, where individuals draw on their patterns, habits, and identity to set goals or outcomes, creating plans or actions toward reaching that goal and evaluating how well the plan and actions are helping meet the goal in the current context or if a new plan is needed (Adie, Willis, & Van der Kleij, 2018; Poon, 2018; Klemencic, 2015). The ability to engage in metacognition allows students to recognize and reflect on how their own thinking helps them reach their goals.
- Metacognitive skill development is critical for all learners, including those with learning disabilities .
💡 Research Spotlight : Desoete and De Craene (2019) found that metacognitive activities can help students with learning disabilities build computational accuracy and mathematical reasoning.
Helpful Metacognitive Strategies in Math
Here are just a few practical methods that students can use to reflect on their learning and engage in overall metacognition:
Verbalizing and writing the steps to solving a problem helps students reflect on, monitor, and evaluate their problem-solving abilities and strategies. This has been shown to increase conceptual understanding and provides students the opportunity to evaluate their learning (Gray, 1991; Martin, Polly, & Kissel, 2017).
Writing about their thinking contributes to their mathematical learning (Martin et al., 2017). For example, students may write math journal entries to think about what they learned and what they might not yet understand.
Answering prompts about concepts and encourages students to collaboratively reflect, justify their reasoning, and elaborate on their thought processes. Choose prompts that frame math as a as a set of problem-solving strategies instead of an end result.
Reflecting through formative assessment allows them to consider how well they understand the lesson content and engage in thinking about their own thinking and how they feel about their learning
These metacognitive strategies allow learners greater metacognitive insight into their own thinking — connecting intuition, modeling, and conceptual representation — and are at the very heart of the mathematical practices that foster deeper mathematical learning (Hattie, 2017, p. 136). Metacognition also empowers students to drive their own learning, building from the support of a teacher’s modeling and moving toward independent practice of skills and concepts. An added benefit to this approach is that when teachers use strong focusing questions, they are also modeling how to ask clarifying questions in a way that will serve students better in later phases of learning, when they ask themselves those clarifying questions.
For more on the importance of metacognition in mathematics, and how metacognition is practiced through specific instructional elements in Reveal Math K-5 , see the full Reveal Math K-5 Research Foundations .
About the Authors
Lanette Trowery , Ph.D. is the Senior Director of the McGraw Hill Learning Research and Strategy Team.
Lanette was in public education for more than 25 years, working as a university professor, site-based mathematics coach, elementary and middle school teacher, mathematics consultant, and a professional learning consultant, before coming to McGraw Hill in 2014. She earned her Master’s and Doctorate from the University of Pennsylvania.
Lanette’s team, Learning Research and Strategy, serves as the center of excellence for teaching and learning best practices. Her team conducts market, effectiveness, and efficacy research into products to provide insights and recommendations to product development. They collaborate across internal teams, external experts, and customers to establish guiding principles and frameworks to move from theory to practice.
Margaret Bowman is an Academic Designer in the Mathematics Department at McGraw Hill.
Margaret earned her Bachelor of Science in Education from Ashland University with a teaching license in Middle Grades Education, and her Master of Education from Tiffin University. She was a middle school Math and Language Arts teacher for six years before joining the middle school team at McGraw Hill in 2012, writing and designing print and digital curriculum.
Margaret is also a Research Associate in the Research Laboratory for Digital Learning at The Ohio State University. She is nearing completion of a PhD in Educational Studies with an emphasis in Learning Technologies. Her past research and journal publications have focused on teachers’ value for using technology in the classroom and technology’s impact on student learning. Her current research examines how students’ use of technology can improve the value they have for mathematics and their expectations that they can succeed.
Adie, L., Willis, J., and Van der Kleij, F. (2018). Diverse perspectives on student agency in classroom assessment. The Australian Educational Researcher. 45. 1–12. https://doi.org/10.1007/s13384-018-0262-2
Denton, D. (2011). Reflection and learning: Characteristics, obstacles, and implications. Educational Philosophy and Theory, 43 (8), 838–852.
Desoete, A. & De Craene, B. (2019.) Metacognition and mathematics education: an overview. ZDM Mathematics Education, 51 (4), 565.
Flavell, J. (1976). Metacognitive aspects of problem-solving. In L. B. Resnick (Ed.), The nature of intelligence . (pp. 231–236). Hillsdale, NJ: Erlbaum.
Gray, S. (1991). Ideas in practice: Metacognition and mathematical problem solving. Journal of Developmental Education, 14 (3), 24–26, 28.
Hattie, J. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning . Thousand Oaks, CA: Corwin Mathematics.
Jin, Q., & Kim, M. (2018). Metacognitive regulation during elementary students’ collaborative group work. Interchange, 49 (2), 263–281.
Klemencic, M. (2015). What is student agency? An ontological exploration in the context of research on student engagement. In Klemencic, Bergan, and Primozic (eds.) Student engagement in Europe: society, higher education and student governance (pp. 11–29) . Council of Europe Higher Education Series №20. Strasbourg: Council of Europe Publishing.
Martin, C., Polly, D., Kissel, B. (2017). Exploring the impact of written reflections on learning in the elementary mathematics classroom. The Journal of Educational Research, 110 (5), 538–553.
Mevarech, Z. & Kramarski, B. (2003). The effects of metacognitive training versus worked-out examples on students’ mathematical reasoning. British Journal of Educational Psychology, 73 , 449–471.
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The Playful Approach to Math
Math doesn’t have to be serious. One teacher believes play is the key to math comprehension for all ages.
The concept of play is often limited to younger students and less academic endeavors, but play can be a useful strategy in an unlikely discipline: math.
Mathematics is known as cold, logical, and rigorous, but the subject doesn’t get enough credit for its true mischievous spirit, which is well-hidden from the world. The K–12 mathematics curricula can involve time and space to play with concepts and ideas, even in the rush of required topics.
In my research for a 2017 talk I gave at TEDxKitchenerEd called “ Math Is Play ,” I found few materials on the joy of math in upper grades. Much of the literature on play as a learning approach is based on the early years, particularly kindergarten, where it is an accepted pedagogical mode.
Young children at play often reach a state that psychologist Mihaly Csikszentmihalyi calls “flow,” an elusive state of mind where time seems to disappear as they focus deeply on what they’re doing. Getting to this ideal state in the classroom requires more than the freedom to play—teachers must also react to students’ ideas and guide them through concepts like counting and numbers. This type of guided play requires decision-making about how and when to give direct instruction. Creating freedom while at the same time offering direction allows for productive play that opens students’ minds to better understand difficult mathematical concepts.
The amount of play in “serious” academic topics like mathematics is inversely proportional, it seems, to the age of students, but this does not have to be the case. A playful pedagogy of mathematics can be codified and made real, rigorous, and authentic.
Embrace and Incorporate Play
In my book Teaching Mathematics Through Problem-Solving in K–12 Classrooms , I wrote about the need to accept that humans are born to play. Play is irrepressibly human, and we can play to learn. Playing and thinking are not at odds with each other. Though play is usually associated with turning off thinking and giving oneself to a pleasurable activity, working on interesting problems can be a trigger for flow. There is a sweet spot, often about 30 minutes into working on an interesting problem, where ideas start to become solutions.
Create a culture where mathematical ideas are not just formulas on a page but instead are concepts to be discussed and reasoned through. Play moves math instruction beyond rote memorization to a more expansive understanding of mathematics. Encourage students to talk, think, reason, and wonder as they move through problems. Creating a sense of curiosity, even for simple concepts, engages students in a playful way.
Simple strategies like turn-and-talk can create opportunities for collaborative, playful learning. Using prompts as part of the daily classroom routine can make mathematical concepts fun. Sites like Visual Patterns , Fraction Talks , or Estimation180 offer easy, quick ways to make mathematical concepts entertaining.
Lean Into the Unknown
Math is full of surprises that can be interesting and fun. There is no single path or strategy to the solution in many problems. Be receptive to surprises in how your students think about and solve problems. An openness to the unexpected can foster a culture of playful curiosity in the classroom. A playful mathematics learner is hopeful and optimistic—elements of a mindset that helps students improve their understanding of complex concepts.
Embrace the mess of the problem-solving process. Thinking is messy. We don’t always get things right the first time. Course corrections, revisions, and even total transformations of the work are necessary.
Observe your students as they work. Where are the roadblocks? How are they adapting to specific challenges? Listen to your own self-talk while you work, and use your challenges to think through ways your students might be challenged. Solutions are important, but so is the process along the way. Listening and talking to your students while they work allows you to give good feedback and receive assessment data.
Play creates open spaces for thinking where teachers can guide students to engage with big and interesting ideas of mathematics.
Use Concrete Methods
Physical or digital manipulatives like snap cubes, pattern blocks, and relational rods are all tools that can help students bring mathematics into being—a process called representation. Teachers can use decks of cards, dice, or counting objects to help students practice their basic skills.
For example, younger students can practice multiplication facts up to 6 times 6 by rolling two dice and multiplying the results. Older students can use decks of cards to practice integer operations, where red suits are negative and black suits are positive. For young students learning basic number skills, designate one day a week for purposeful practice using games.
Visual presentation of mathematical ideas can be playful. For example, give students a can lid or other circular object, some string, and a measuring tape to try and find the relationship between circumference and diameter.
Using physical or digital elements creates space for students to play with more abstract concepts and ideas. The freedom of a playful environment creates opportunities for deeper engagement. As I said in my talk, “Let them play. Let them talk and think and conjecture and wonder. Let them play.”
Problem Solving Activities: 7 Strategies
- Critical Thinking
Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.
In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough.
I was so excited!
We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies.
It was a proud moment for me!
Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy.
After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name.
What a crushing blow! Just when I thought I did something special, I find out I did it all wrong.
I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom.
When I Finally Saw the Light
To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students.
Problem Solving Activities
Here are seven ways to strategically reinforce problem solving skills in your classroom.
Seasonal Problem Solving
Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!
Cooperative Problem Solving Tasks
Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.
Notice and Wonder
Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it.
Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.
Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here !
Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.
Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!
Three-Act Math Tasks
Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons .
Getting the Most from Each of the Problem Solving Activities
When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next.
Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below.
Which of the problem solving activities will you try first? Respond in the comments below.
Shametria Routt Banks
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This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.
Thank you, Scott! Best wishes to you and your pre-service teachers this year!
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The Role of the Teacher Changes in a Problem-Solving Classroom
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How can teachers help students develop problem-solving skills when they themselves, even though confronted with an array of problems every day, may need to become better problem solvers? Our experience leads us to conclude that there is an expertise in a certain kind of problem-solving that teachers possess but that broader problem-solving skills are sometimes wanting.There are a few reasons why this happens. One reason may be that teacher preparation programs remain focused on how to teach subjects and behavior management techniques. Another reason may be that professional development opportunities offered in schools are focused elsewhere. And, another reason could be that leaders still often fail to engage their faculties in solving substantive problems within the school community.
A recent issue of Education Leadership was dedicated to the topic, “Unleashing Problem Solvers”. One theme that ran through several of the articles was the changing role of the teacher. In a positive but traditional classroom, information is shared by the teacher and the students are asked to demonstrate application of that information. A problem-solving classroom is different. A problem-solving classroom requires extraordinary planning on the part of the teacher. For problems to have relevance, students are engaged in the identification of the problem. Teachers have to become experts at creating questions that require students to reach back to information and skills already attained, while figuring out what they need to learn next in order to solve the problem. Some of us are really good at asking these kinds of questions. Others are not.
Students have to become experts at reflecting on these questions as guides resulting in a gathering of new information and skills, and answers. Teachers have to be prepared to offer lessons that bridge the gaps between the skills and information already attained and those the performance of the students demonstrate remain needed. Often it involves teams of students and they are simultaneously learning collaboration and communication skills.
Problem-Based Classrooms Require Letting Go
Opportunities for teachers to work with each other, to learn from experts, to receive feedback from observers of their work, all allow for skill development. But at the same time, there is a more challenging effort required of the teacher. Problem-based classrooms require teachers to dare to let go of control of the learning and to take hold of the role of questioner, coach, supporter, and diagnostician. In addition to the lack of training teachers have in these skills, the leaders in charge of evaluating their work also have to know what problem-solving classrooms look like and how to capture that environment in an observation, how to give feedback on the teachers’ efforts. Of course, if problem- solving is a collaborative school community process, how does that change the leader’s role? Are leaders, themselves, ready to become facilitators of the process rather than the sole problem solver? Many talk about wanting that but most get rewarded for being the problem solver.
Questions are Essential
There is a place to begin and that place is the shared understanding of what problem-based learning actually is. Because teachers traditionally plan for a time for Q and A within classes, they and their leaders may think of questions as having a correct answer. In moving into a problem-based learning design, the questions also have to be more overarching, create cognitive dissonance, and provoke the learner to search for answers. Here is why it is important to come to an understanding about the types of questions to be asked and shifting the teaching and learning practices to be one of expecting more from the learner.
Students Need Problem-Solving Skills
Problem-based learning skills are skills that prepare for a changing environment in all fields. Current educators cannot imagine some of the careers our students will have over their lifetimes. We do know that change will be part of everyone’s work. Flexibility and problem-solving are key skills. Problem- solving involves collaboration, communication, critical thinking, empathy, and integrity. If we listen to the business world, we will hear that design thinking is the way of the future.
Tim Brown, CEO of IDEO says,
Design thinking is a human-centered approach to innovation that draws from the designer’s toolkit to integrate the needs of people, the possibilities of technology, and the requirements for business success.
The only way for educators to develop these skills in students is to build lessons and units that are interdisciplinary and demand these skills. If we begin from the earliest of grades and expect more as they ascend through the grades, students will have mastered not only their subjects, but the skills that will prepare them for the world of work. How do we best prepare our students? We think problem solving is key.
A nn Myers and Jill Berkowicz are the authors of The STEM Shift (2015, Corwin) a book about leading the shift into 21st century schools. Ann and Jill welcome connecting through Twitter & Email .
Photo courtesy of Pixabay
The opinions expressed in Leadership 360 are strictly those of the author(s) and do not reflect the opinions or endorsement of Editorial Projects in Education, or any of its publications.
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- Published: 07 March 2023
Exploring the effects of role scripts and goal-orientation scripts in collaborative problem-solving learning
- Ke-Ru Li 1 ,
- Zhuo Sun 1 ,
- Ning Ma ORCID: orcid.org/0000-0002-1941-724X 1 &
- Yi-Fan Sun 1
Education and Information Technologies ( 2023 ) Cite this article
Collaborative problem-solving (CPS) learning is increasingly valued for its role in promoting higher-order thinking of learners. Despite the widespread application of role scripts in CPS, little is known about the mechanisms by which roles influence learners' cognition and the impact of goal orientation on roles. In this study, we designed role scripts and goal-orientation scripts to facilitate CPS. Then, a total of 32 postgraduate students participated in CPS and they were divided into 8 groups, among which two roles of analyst and commenter were assigned respectively. Through qualitative and quantitative analysis, this study explored the differences between the two roles in terms of discourse space rotation, types of cognitive activities and epistemic network structure, and the function played by goal orientation. Results showed that there was a general structure in CPS, that analysts and commenters have different functional biases, and that goal orientation influences the function of the roles. This study clarified the cognitive contribution of different roles, and the respective strength of different goal orientation. The findings may provide instructors with implications for designing scripts and organizing CPS in the classroom context.
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Yao Lu, Ke-Ru Li, Zhuo Sun, Ning Ma & Yi-Fan Sun
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Lu, Y., Li, KR., Sun, Z. et al. Exploring the effects of role scripts and goal-orientation scripts in collaborative problem-solving learning. Educ Inf Technol (2023). https://doi.org/10.1007/s10639-023-11674-z
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DOI : https://doi.org/10.1007/s10639-023-11674-z
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- Collaborative problem-solving learning
- Role scripts
- Goal orientation
- Types of cognitive activities
- Epistemic network structure
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