Problem-solving Models and Strategies

12 Key excerpts on "Problem-solving Models and Strategies"

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  Teaching and Learning Mathematics

  A Teacher's Guide to Recent Research and Its Application

  • Marilyn Nickson(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Chapter 6 Problem-solving Introduction Problem-solving is a somewhat global term and, taken in the context of mathematics, it leads to the consideration of problem-solving processes from a variety of perspec-tives, e.g. the meta-perspective of Polya (1957) or the more focused perspective of a single area of mathematics, e.g. algebra (Kaput 1987) or geometry (Chinnapan 1998a). Research considered in this chapter will approach problem-solving in the context of specific areas of mathematics as well as in the context of word problems. Because much of recent research in problem-solving in mathematics education is approached through modelling, we shall also briefly examine the role of modelling and the way in which it informs some of the studies carried out in the field. Before considering research into this area, however, we should also note here that there are those in mathematics education who consider that the focus should be on problem-posing as opposed to solving (Brown 2001, Goldenberg and Walter, in press). Part of the argument for this call for a change in emphasis is the way in which mathematical modelling used in problem-solving ignores the imaginative and creative aspects of engaging in mathematics. Another part of the argument is that while supposedly engaging learners of mathematics in * real-life' problems, modelling procedures adopted actually exclude many aspects of * real life* in doing so. This is an innovative approach to considering problems from a mathematical perspective and is one that stresses that problems need not specifically be concerned with application to be 'real' but can also arise from within mathematics itself. The 'reality' arises from the learners direct involvement in posing the original problem, the lack of which has been seen as a drawback by many researchers (e.g. Davis 1988, Douady 1997). However, there is some evidence that problem-posing is becoming a focus for studies by researchers (e.g. Crespo 2003).

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  Mathematics for Elementary Teachers

  A Contemporary Approach

  • Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Although the problems that people encounter may be very diverse, there are common elements and an underlying structure that can help to facilitate problem solving. Because of the universal importance of problem solving, the main professional group in mathematics educa- tion, the National Council of Teachers of Mathematics (NCTM) recommended in its 1980 Agenda for Actions that “problem solving be the focus of school mathematics in the 1980s.” The NCTM’s 1989 Curriculum and Evaluation Standards for School Mathematics called for increased attention to the teaching of problem solving in K-8 mathemat- ics. Areas of emphasis include word problems, applications, patterns and relationships, open-ended problems, and problem situations represented verbally, numerically, graphically, geometrically, and symbolically. The NCTM’s 2000 Principles and Standards for School Mathematics identified problem solving as one of the processes by which all mathematics should be taught. This chapter introduces a problem-solving process together with six strategies that will aid you in solving problems. Key Concepts from the NCTM Principles and Standards for School Mathematics PRE-K-12–PROBLEM SOLVING Build new mathematical knowledge through problem solving. Solve problems that arise in mathematics and in other contexts. Apply and adapt a variety of appropriate strategies to solve problems. Monitor and reflect on the process of mathematical problem solving. Key Concepts from the NCTM Curriculum Focal Points KINDERGARTEN: Choose, combine, and apply effective strategies for answering quantitative questions. GRADE 1: Develop an understanding of the meanings of addition and subtraction and strategies to solve such arithmetic problems. Solve problems involving the relative sizes of whole numbers. GRADE 3: Apply increasingly sophisticated strategies … to solve multiplication and division problems. GRADE 4 AND 5: Select appropriate units, strategies, and tools for solving problems.

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  Mathematics Across the Curriculum

  Problem-Solving, Reasoning and Numeracy in Primary Schools

  • Sue Fox, Liz Surtees(Authors)
  • 2010(Publication Date)
  • Continuum

    (Publisher)

  Avail-able at: nationalstrategies.standards.dcsf.gov.uk/downloader/e5f49e2d5663a1412acbc1d96623fe00.rtf (accessed on 31/07/09), 2006b, p.4. Mathematics Across the Curriculum 48 The whole concept of problem-solving challenges elements of a traditional linear approach to teaching and learning. An approach that starts with one-step problems, then moves on to two-step problems and so on appears limited in that it does not help children decide on an appropriate strategy when faced with a new problem. If we believe that mathematicians are problem solvers, and as teachers we want our children to be mathematicians, then we need to encourage our children to solve problems in their own way. Furthermore, we need to develop an environment, where children learn to think creatively and critically within a social framework. It is neither sufficient, nor desirable, to limit children’s experience of problem solving to sets of word problems in order to assess whether they can apply skills learnt in previous lessons. Developing children’s creativity in problem solving and encouraging them to think mathematically requires an appropriate learning environment to be established and nurtured by the teacher. This environment needs to include rich tasks and activities and the freedom for children to explore those activities so that they become confident, competent and creative problem solvers. It should also be a safe environment where children are not afraid to ‘have a go’ and possibly make mistakes. Problem solving includes the skills of identifying and understanding the problem, planning ways to solve a problem, monitoring progress as the problem is tackled and then reviewing a solution to a problem. There could also be the potential for the child thinking about ‘what next?’ Sakshaug et al.

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  Mathematical Cognition

  • James M. Royer(Author)
  • 2001(Publication Date)
  • Information Age Publishing

    (Publisher)

  What does a person need to know to be proficient in mathematics? To answer this question, this chapter examines research on the cognitive pro-cesses involved in mathematical problem solving. I begin with an introduc-tion that includes definitions of key terms and a summary of four cognitive processes in mathematical problem solving—translating, integrating, plan-ning, and executing. Then, for each cognitive process, I provide examples and explore exemplary research. Finally, the chapter ends with a conclu-sion in which I suggest some future directions for research on the cognitive psychology of mathematical problem solving. Definitions Let’s begin by defining some key terms—mathematics problem and mathematical problem solving. A problem exists when you have a goal but do not immediately know how to reach the goal (Mayer, 1992). Thus, a problem consists of three elements: a given state (i.e., the current state of the situation), a goal state (i.e., the desired state of the situation), and obstacles that block you from moving directly from the given state to the goal state. A mathematics problem is a problem that involves mathematical content such as numbers, geometric shapes, or algebraic relations. The defining feature of mathematics problems is that they require mathematical reason-ing—such as reasoning based on the rules of number systems, geometry, or algebra. It is important to note that in this definition, whether a situation is a problem depends on the problem solver. Schoenfeld (1985, p. 74) observed: “Problem solving is relative. The same tasks that call for signifi-cant efforts from some students may well be routine exercises for others…. Thus, being a problem is not a property inherent in a mathematical task. Rather, it is the particular relationship between the individual and the task

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  Making Sense of Number

  Improving Personal Numeracy

  • Annette Hilton, Geoff Hilton(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  As we have seen, Polya’s framework is very broad and provides only a general guide to problem solving. As problems are quite often unique, with very particular facts, it is impossible to give a one-size-fits-all means of devising a plan. Instead, it is important to have a range of strategies that you can use when and if they are relevant and it is at this point that our number sense can help. Sometimes we need a few other strategies to help us apply our number sense. Ultimately, the strategy will depend on the nature of the problem and it is fair to say that whatever you choose to do, you will need to decide on the mathematical steps necessary to solve the problem. In short, you will need to transform your representations of the problem into mathematical symbols so that you can carry out your plan. In the following sections we will look at some useful strategies and then consider some scenarios to illustrate them. Using multiple representations Most problem-solving situations begin with consideration of whether one or more representations might be helpful. For example, would it be useful to • draw a diagram or picture of the situation? • use physical materials to help model the problem? • act out the situation (either individually or with others)? • make a table, graph or chart? • make a list of all the information first? • write an equation or number sentence? Chapter 9 Problem solving 171 Being able to visualise a problem can be a helpful strategy when working out a problem. It can help you to understand what the problem is about and sometimes it can help you visualise a strategy or solution or at least decide what a reasonable strategy or solution might be. This step can help you to decide what to do next (e.g. What operations will you use? Will you need more than one step to solve the problem? What assumptions should you make?).

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  Mathematics Education

  Models and Processes

  • Lyn D. English, Graeme S. Halford(Authors)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  mathematical thinking, which she claimed is usually interpreted in terms of the content of mathematics (e.g., thinking about patterns and relationships in the number system). Thinking mathematically is seen as “the style of processing which supports an enquiry which might ultimately lead to the learning of some mathematics but equally might lead to learning in other subject areas” (p. 58). The processes that Burton claimed constitute mathematical thinking, especially for young children, include strategies such as classifying, ordering, enumerating, testing, conjecturing, and generalizing.

  We return to this important area of mathematical thinking when we consider a number of higher order thinking skills that play a significant role in problem solving/posing and in mathematics learning in general.

  Cognitive Components of Mathematical Problem Solving

  Although different terminology may be used, there nevertheless appears to be general agreement on the broad aspects of cognition that play a critical role in mathematical problem solving. We address these here in terms of problem models (mental models comprising problem representation and problem-solving heuristics), strategic processes (goal-directed operations to facilitate problem solution), metaprocesses (comprising analogical reasoning, higher order thinking skills, and metacognitive processes), and affective models (beliefs, attitudes, and emotions). These components are represented in diagrammatic form in Fig. 8.1 . We review each of these components in turn, and consider ways in which they interact to facilitate problem solution.

  Fig. 8.1.   Cognitive components of mathematical problem solving.

  Problem Models: Problem Representation and Knowledge of Heuristics

  Problem Representation.   We addressed the role of mental models in representing computational problems in chapters 5 to 7 . In this section we focus specifically on their role in formulating and solving novel problems.

  Recall that mental models are representations that are active while solving a particular problem and that provide the workspace for inference and mental operations (Halford, 1993). During problem solving, learners modify or extend their existing mental models by connecting new information to their present knowledge structures and constructing new relationships among those structures (Silver & Marshall, 1990). We elaborate on this process in a subsequent section.

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  The Nature of Problem Solving

  • OECD(Author)
  • 2015(Publication Date)
  • OECD

    (Publisher)

  Cooney and Henderson (1971), looking at mathematics content, divided knowledge into deductive, inductive, classifying and analysing, as a means of assisting students to relate different forms of knowledge and apply it in different situations. In another analysis, Henderson (1967) examined how instruction might link aspects of concepts to aid in understanding or structuring instruction. Others have suggested other instances and models for dividing knowledge into scripts, frames or schemata. All indicate that problem solvers tend to find some organisational form for problem contexts and that these forms help govern their behaviour when they encounter similar experiences in the future. Problem-solving strategies As already discussed, there are a number of models that have been developed by mathematicians and mathematics educators to codify approaches to problems and their solutions. Pólya (1945) suggested broad strategies of analogy, auxiliary elements, decomposing and recombining, induction, specialisation, variation, and working backward. However, these have been hard to implement on a general scale, even though they have held up with more mathematically inclined audiences. Studies have indicated that, given strategic guidance in structuring learning sequences and focused work with the strategies, combined with enough time, teachers can separate themselves from the “sage on the stage” role to that of “consultant and guide.” (Charles and Lester, 1984). This study and others reflect how difficult it is to implement the power of teachers who can model problem-solving strategies as Pólya envisioned, even with excellent problems and outstanding teachers. Self-regulation, or monitoring and control Self-regulation, or monitoring and control, is at the heart of metacognition and, as such, at the heart of mathematical problem solving.

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  Mathematics in Middle and Secondary School

  • Alexander Karp, Nicholas Wasserman(Authors)
  • 2012(Publication Date)
  • Information Age Publishing

    (Publisher)

  PART II This part introduces the reader to the theory of problem solving and prob- lem posing and to its possible applications in class activities. Mathematics in Middle and Secondary School, pages 59–89 Copyright © 2015 by Information Age Publishing All rights of reproduction in any form reserved. 59 CHAPTER 3 WHAT IS PROBLEM SOLVING AND HOW IT IS CONCEPTUALIZED IN MATHEMATICS? WHAT IS A PROBLEM? The Meaning of the Word “Problem” One comes across the word “problem” so often, in mathematics as well as everyday life, that this inevitably leads to misunderstandings. In ordinary The purpose of this chapter is to acquaint the reader with the most important approaches and conclusions of classical works in the theory of problem solving. The chapter discusses: ◾ The concepts of “problem,” “exercise” and “problem solving.” ◾ Different approaches to describing the process of problem solving, especially the four-stage schema of George Polya. ◾ Examples of different heuristic strategies. 60  Mathematics in Middle and Secondary School speech “problems” generally refer to obstacles that hinder the achieve- ment of a particular pursuit. In the classroom “problems” sometimes refer to so-called “word problems”—tasks that appear as a verbal description of some situation, but which then needs to be translated into a mathematical language in order for the solution to be reached. “Problems” are some- times, explicitly or implicitly, linked to the mathematical modeling of real life situations.

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  Problem Solving in Mathematics Education

  • Uldarico Malaspina, Manuel Santos-Trigo, Peter Liljedahl(Authors)
  • 2016(Publication Date)
  • Springer Open

    (Publisher)

  Heuristics such as those by Perkins ( 2000 ) and Mason et al. ( 1982 ) have found a way of combining the intentional and mechanical processes of problem solving by design with the extra-logical processes of creativity, illumination, and the AHA!. Furthermore, they have managed to do so without having to fully comprehend the inner workings of this mysterious process. 1.3 Digital Technologies and Mathematical Problem Solving — Luz Manuel Santos-Trigo Mathematical problem solving is a fi eld of research that focuses on analysing the extent to which problem solving activities play a crucial role in learners ’ under-standing and use of mathematical knowledge. Mathematical problems are central in mathematical practice to develop the discipline and to foster students learning (P ó lya 1945 ; Halmos 1994 ). Mason and Johnston-Wilder ( 2006 ) pointed out that “ The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out ” 1 Survey on the State-of-the-Art 19 (p. 25). Tasks are essential for learners to elicit their ideas and to engage them in mathematical thinking. In a problem solving approach, what matters is the learners ’ goals and ways to interact with the tasks. That is, even routine tasks can be a departure point for learners to extend initial conditions and transform them into some challenging activities. Thus, analysing and characterizing ways in which mathematical problems are formulated (Singer et al. 2015 ) and the process involved in pursuing and solving those problems generate important information to frame and structure learning environments to guide and foster learners ’ construction of mathematical concepts and problem solving competences (Santos-Trigo 2014 ). Furthermore, mathemati-cians or discipline practitioners have often been interested in unveiling and sharing their own experience while developing the discipline.

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  Mathematizing Student Thinking

  Connecting problem solving to everyday life and building capable and confident math learners

  • David Costello(Author)
  • 2022(Publication Date)
  • Pembroke

    (Publisher)

  • Recognize the goal state – Cognitive strategies are utilized by the student to recognize the end goal. • Decide on a strategy that leads them from the initial state to the goal state • Enact the strategy – Self-monitor when enacting the strategy. – Determine if the strategy is leading to a solution. – Overcome stumbling blocks. • Determine if the strategy leads to a meaningful response – If it did, review the work. – If it did not, consider alternative strategies. When we approach problem solving as a thinking process, or as a verb, we turn our attention to the thinking that students are engaged within. We can then recognize and appreciate the many emotional states our students experience as they problem solve. Understanding the Different Ways to Problem Solve 51 Problem solving is multifaceted. Problem solving has various stages. At each stage, we want students to think confidently and apply their mathematical under- standing toward solving problems. The Problem Solving Stages When problem solving students are engaged in a range of thinking processes. The work of George Polya (2004) has been used to identify the key thinking points within the problem-solving process. Polya referred to this as the four prin- ciples of problem solving. Some people who have taken up his work refer to the principles as steps, while others refer to the principles as stages. For the purposes of this book, we will refer to Polya’s four principles as the four problem solving stages. Each stage of problem solving is layered. A single stage is comprised of differ- ent thinking points in the decision-making process. I am providing an overview of the stages but each stage is quite in-depth and can engage students in signifi- cant thinking. There can be productive struggle within each stage and the stu- dent must rely on their thinking to move through the stages. Using the work of Polya (2004), the following are the accepted four stages of problem solving: 1.

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  Helping Children Learn Mathematics

  • Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  For example, children can build new mathematical knowledge by having concepts introduced through problems set in familiar contexts (e.g., sports, games, and everyday activities) or through problems involving a variety of repre- sentations (e.g., hundreds charts, fraction bars, and counting cubes). A strong mathematics program builds on the natural, informal problem-solving strategies that the child has encountered before entering school. As the opening quote by Abraham Maslow suggests, the more problem-solving strate- gies we become familiar with, the more appropriately we can handle unfamiliar problem situations. Many of the best prob- lems for elementary children involve everyday situations— for example, first graders will relate to problems such as “How many more chairs will we need if we’re having five visitors and two children are absent?” and “How many cook- ies will we need if everyone is to have two?” while fifth grad- ers might be more interested in problems such as “Who has the higher batting average, Benny or Marianne?” or “Which of these dice games is a fair game?” No matter what type of problem is involved, children who are effective problem solvers plan ahead when given the problem, ask themselves if what they are doing makes sense, adjust their problem-solving strategies when necessary, and look back afterwards to reflect on the reasonableness of their solution and their approach. INTRODUCTION WHAT IS A PROBLEM AND WHAT IS PROBLEM SOLVING? A problem is something a person needs to figure out, something where the solution is not immediately obvious. Solving problems requires creative effort and higher-level thinking. If a child immediately sees how to get the answer to a problem, then it is not really a problem for that child. Skill in solving problems comes through experiences with solving many problems of many different kinds. Children who have worked on many problems score higher on problem-solving tests than children who have worked on few.

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  Mathematics Methods for Elementary and Middle School Teachers

  • Mary M. Hatfield, Nancy Tanner Edwards, Gary G. Bitter, Jean Morrow(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Too often, students with special needs are denied the opportunities to become problem solvers because ‘‘they need to learn the basics first.’’ Or ‘‘these stu- dents are too easily frustrated and just look for someone else to do the work for them.’’ Yet, it is important to remember that ‘‘ . . . there is abun- dant research evidence that proficient calculation skills and basic facts mastery need not precede conceptual understanding and problem solving’’ (Sutton and Krueger, 2002). Researchers and curriculum developers at the Education Development Center, based in Boston, Massachusetts, have developed a profession devel- opment series called Addressing Accessibility in Mathematics. Some strategies for making mathe- matics more accessible for students with learning disabilities are shown in Figure 5.12. A more com- plete list of strategies can be found at their Web site, www2.edc.org/accessmath. ASSESSMENT Chapter 3 described a variety of assessment stra- tegies, tools, and rubrics. Many of those strategies, tools, and rubrics can be applied to assessment of students’ work in problem solving. Here are a few additional ideas related to assessment and pro- blem solving. Rating Group Problem Solving There will be times when you will decide to assess the class during a collaborative problem-solving activity. With this approach, students may be asked to solve a problem as a collaborative group and to submit one group solution. An analytic scoring rubric based on Polya’s plan for problem solving (Figure 5.13) may be used at this time. In this situation, all the members of the group received the same score or rating. Many teachers recommend including some form of individual accountability at the same time. Thus, students might be asked to submit individual papers describing the solution process, or justifying the solution, or applying the process to a similar problem.

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