Models of mathematical thinking as problem-solving

Schoenfeld (2016), explained in his book, Mathematical Problem Solving, that his views were influenced by George Polya’s (1945) book, How to Solve It. Schoenfeld (1985), describes how he had read the book for the first time in 1975 and then became interested in finding out what it means to “think mathematically” and how one can help students to accomplish it. Schoenfeld (1992, pg. 51), recognizes that Polya’s influence in, How to Solve It (1945) “planted the seeds of the problem-solving ‘movement’ that flowered in the 1980’s, calling Polya’s book a ‘revelation’ (Schoenfeld, 2016). Polya’s own work addresses pertinent issues regarding the teaching of mathematical problem- solving, such as the role of the teacher as instructor and as example of mathematical thinking (for her ‘apprentices’), the process of systematic problem- solving, especially the importance of ‘looking back’ and also ‘problem posing’. According to Polya (1957), the role of the teacher is to model how mathematical problems should be solved, by ‘thinking aloud’ when solving problems. The point is that students may well need an auditive example of how to think and learn to do this. They learners then have a model of what verbalised thinking entails.

Polya’s (1945) classic four-stage model (see, Figure 10), has been, and still is, instrumental for both the teaching and assessing of mathematical problem-solving. According to the model, problem solvers first need to understand the problem and what it entails before they can devise a plan. By devising a plan, the solver decides on a suitable solution strategy. Once the solver has ensured that the chosen strategy is appropriate, she can carry out the plan. The final stage is to ‘look back’. By ‘looking back’ the solver is reflecting on the solution strategies to determine if the strategy used was the most suitable.

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Figure 10: Polya’s (1945) four-stage problem-solving framework

Building on Polya’s (1945) work on heuristics, Schoenfeld (1985, 1992) developed a four- phase theoretical framework of mathematical thinking as problem-solving that was empirically derived from a decade-long period of his design experiments, aimed at understanding and improving students’ problem-solving (Schoenfeld, 1985). In these design experiments, Schoenfeld (1985) first modelled to his students how to solve non-routine mathematical problems and then they had to solve problems on their own. Based on the findings of these design-experiments, mathematical thinking was conceptualised as comprising of mathematical processes, mathematical strategies and tacit mathematical understanding. These three components - mathematical processes, mathematical strategies and tacit mathematical understanding - were further refined to form an analytical tool, consisting of four key interrelated components, namely, resources, heuristics, control/metacognition, beliefs (Schoenfeld, 1985), as shown in Table 2. Later Schoenfeld (1992) added another component, namely practices. The (1985), framework represents an attempt to explain a range of behaviours that, arguably, constitute mathematical thinking as problem-solving.

Understand the problem

•E.g. Do you understand what are you asked to find or show. Do you

understand all the words in the problem, etc. Plan can only be devised once you understand the problem.

Devise a Plan

•Deciding on a suitable strategy, i.e. can you restate the problem in your own words, guess & check, making a drawing, working from backwards, etc., may be daunting-the more you solve problems the easier it becomes to choose the correct startegy

Carry out the Plan

•Carrying out the plan is easier once it is accurately devised. You need to ensure you have the necessary strategies. If a strategy does not work the first time be persistent, try again. If it still doesnt work try a new one.

Look Back

•The look back phase is an important phase but often overlooked. During this phase you look back at your final solution - to reflect on what strategies you used, are there possibly an easier or better method to find a solution

41 Table 2:

Knowledge and behaviour necessary for an adequate characterization of mathematical problem-solving performance (Source: Schoenfeld, 1985, p. 15)

Resources Mathematical knowledge possessed by the individual that can be brought to bear on the problem at hand:

Intuitions and informal knowledge regarding the domain Facts

Algorithmic procedures

“Routine” nonalgorithmic procedures

Understandings (propositional knowledge) about the agreed-upon rules for working in the domain

Heuristics Strategies and techniques for making progress on unfamiliar or nonstandard problems; rules of thumb for effective problem-solving, including:

Drawing figures; introducing suitable notation Exploiting related problems

Reformulating problems; working backwards Testing and verification procedures

Control Global decisions regarding the selection and implementation of resources and strategies:

Planning

Monitoring and assessment Decision making

Conscious metacognitive acts

Belief systems One’s “mathematical world view” the set of (not necessarily conscious) determinants of an individual behaviour:

About self

About the environment About the topic About mathematics

Table 2, summarizes the key knowledge and behaviour exhibited by a student who can think mathematically. A student who can think mathematically is resourceful, flexible, and efficient in the ability to deal with ‘unfamiliar’ mathematical problems (Schoenfeld, 1985). In addition, the framework serves as a tool to find out what the cause of failure or success is of a person

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attempting to solve a mathematical problem. The failure or success of the problem-solving attempt is dependent the problem solver’s knowledge base (resources), strategies (heuristics), ability to monitor or self-regulate (control), and beliefs about mathematics as a (belief system) and practices (ibid).

Resources

According to Schoenfeld (1985), the solver’s knowledge base is key, because in an attempt to solve a problem, the solver draws from three broad classes of information - relevant facts known by the solver, algorithmic procedures, and relevant competencies. Not only is it important for the solver to have these resources at her disposal, but also knowing how this knowledge is structured, stored and retrieved from memory (ibid). A very important aspect to take note of is that the solver’s knowledge base does not need to be ‘true’, but what the solver believes to be true is important because this is the foundation of all problem-solving (Schoenfeld, 1985).

Heuristics

Resourceful problem-solvers have a wide range of problem-solving techniques (heuristics) at their disposal. Heuristics are methods, tools and strategies that can be helpful in problem- solving (Bruner, 1960). According to Tiong, Hedberg, and Lioe (2005), heuristics are methods that have been useful in previous problem-solving experiences and which one might want to apply to the problem-solving process at hand. Schoenfeld (1985) summarizes how a problem- solver builds a repertoire of problem-solving strategies as:

Occasionally the person solves a problem using a technique that was successful earlier and something clicks. … If that method succeeds, twice the individual may use it when faced with another similar problem. In that, way a method becomes a strategy. Over a period of years, each individual problem solver comes to rely – quiet possibly unconsciously – upon those methods that have proven useful for himself or herself. That is the individual develops a personal and idiosyncratic collection of problem solving strategies. (Schoenfeld, 1985, p. 70 -71)

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The methods used by problem solvers could be obtained through various ways, such as methods taught by teachers, methods self-discovered, methods observed while other where solving problems and those gained form mathematics textbooks (Tiong et al., 2005).

Heuristics play an important role in solving non-routine problems of unfamiliar topics. Polya (1945) suggested several problem-solving strategies including, working backwards, searching for similarities, making a drawing, etc. Tiong et al., (2005) categorized problem-solving heuristics into four broad categories. These categories are representation heuristics, simplification heuristics, pathway heuristics, and generic heuristics. Representation heuristics such as acting it out, using a diagram/model and equations are heuristics that “guide the representation of problems in different forms” (Tiong at al., 2005, p. 3 – 4). Representation media include but are not limited to a mixture of different semiotics, such as words, mathematical symbols, graphs, diagrams, gestures, sound etc. (Quarfoot, 2015). Most popular semiotics used in mathematics classrooms are words, diagrams, mathematical symbols, and graphs. Over the last two decades, there has been an increase in interest on the role of gestures in teaching and learning of mathematics (Radford, 2003; Arzarello & Edwards, 2005; Roth, 2001). Gestures refer to “extra-linguistic modes of expression” (Arzarello, et al., 2009, p. 97), and includes glancing, pointing, using hands or fingers, etc.

Simplification heuristics are heuristics that guide the solver in deciding which information in the problem is important and necessary in the problem statement but before the solver can make these decisions, she needs to fully understand the problem. Simplification heuristics include but are not limited to the following heuristics, look for patterns, solving only part of the problem, and restate the problem in another way (Tiong et al., 2005).

Pathway heuristics refers to choosing “effective and suitable paths” to solve the problem and they may require other heuristics as well (Tiong et al., 2005). Working backwords and using before-after concepts are examples of pathway heuristics.

Generic heuristics are heuristics used to select heuristics, as they do not suggest an immediate heuristic to solve the problem but guides the solver where to find an appropriate solution strategy (Tiong et al., 2005). Generic heuristics, such as guessing-and-checking, also referred to as trial-and-error, are frequently used heuristics to approach a problem (ibid). Generic heuristics also include heuristics such as think of a similar problem and make systemic lists.

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However, knowing how to use various heuristics cannot replace knowing the relevant subject matter knowledge, because heuristics are only scaffolds that help to solve problems (Tiong et al., 2005). Schoenfeld (1992) pointed out that an extensive number of studies have been conducted to find out what heuristics students employ when they attempt to solve a problem. Ultimately, findings of these studies were unable to provide mathematics education with distinct directions with regard to students’ problem-solving strategies. However, there is enough evidence, to suggest that problem-solving strategies are both student- and problem dependent (Schoenfeld, 1992).

Control and metacognition

Furthermore, having a range of heuristics and mathematical knowledge, the solver also needs to be competent in monitoring progress as well as making the right decisions. Monitoring refers to being able to assess one’s progress towards finding a solution and to determine whether you are able to adapt your approach if you are not on the right solution path. Referring to Polya’s ‘devising a plan’ stage, this is not a once off decision.

Beliefs

Belief systems are also referred to as students’ worldview (Schoenfeld, 1985). Students’ belief systems have an impact on how they use their resources, heuristics and how they control and monitor the problem-solving process. Their beliefs about the nature of mathematics and their self-efficacy beliefs are key determinants of good mathematical problems solvers. Beliefs are one of the affective components as classified by McLeod (1992). McLeod, who classified affective components as beliefs, emotions, and attitudes. DeBellis and Goldin (1997) later added a fourth affective component, namely, values. Hannula (2004) pointed out that the four components do not encapsulate the complete affective field and added motivation. Hannula (2004) pointed out affective components such as beliefs, motivation, emotions and attitudes influence students’ problem-solving processes. He further described the relationship between mathematical thinking and affective domain by stating that:

In mathematical thinking, the motivational aspect determinates goals in a situation. […] Emotions are an evaluation of the subjective progress towards goals and obstacles on the way. […] Cognition is a non-evaluative information process that interprets the

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situation, explores possible actions, estimates expected consequences, and control actions. (Hannula, 2004, p. 55)

Motivation, beliefs, emotions and values are interrelated components that could either hinder or promote problem-solving processes. They also separate good problem solvers form poor ones.

In addition to Polya’s (1945) classical problem-solving framework and Schoenfeld’s (1985), mathematical problem-solving framework, there are other mathematical problem-solving models, e.g. Garafalo and Lester’s (1985), framework, consisting of four distinctive metacognitive activities - orientation, organization, verification and execution. There is also the work of Carlson and Bloom (2005), who argued that mathematical problem-solving processes are cyclic in nature and not linear as depicted in Polya’s framework. They offered a multidimensional framework for investigating and analysing mathematical problem-solving behaviour. As different scholars proposed, various models to analyse mathematical problem- solving behaviour there are also various lenses through which mathematical thinking is viewed. Over the past three decades, mathematics education has seen a range of linguistic terms describing and expounding Schoenfeld’s (1985) ideas. Taking the lead from Schoenfeld (1985), some elaborated on “problem-solving by expanding on the importance of affect in problem-solving, like McLeod (1992). Cuoco, Goldenberg and Mark (1996) developed the theoretical construct of mathematical “habits of mind”, while Mason, Burton and Stacey (1982/2000) examined “thinking mathematically”, which is also the title of their book. Also influenced by Schoenfeld’s (1985) work, the NCTM (1998/2000) proposed “mathematical processes” and later the Common Core State Standards for Mathematics (2010), in which core “mathematical practices” are discussed. The range of linguistic terms of mathematical thinking mentioned here signifies mathematical thinking as processes, activities, practices, behaviours and habits of mind of mathematicians. In addition, it also illustrates that there are various lenses from which to view mathematical thinking, such as mathematical thinking as ‘reasoning and proof’ (Tall, 2013, 2014; Stylianides, 2005; Harel & Sowder, 2007); mathematical thinking as ‘abstraction’ (Devlin, 2000); mathematical thinking as ‘problem-solving’ and mathematical thinking as ‘modelling’ (Zawojewski & Lesh, 2003; Lesh & Harel, 2003). My study views

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mathematical thinking as problem-solving through the ‘lens’ of Polya (1945) and Schoenfeld’s (1992) frameworks.