Problem-solving for mathematics education

For the purpose of studies in mathematics education, one has to derive some definition of what it means to ‘solve’ a ‘problem’. Mathematical problem-solving cannot be defined with a simple definition because there are multiple notions and meanings of what mathematical problem- solving is. Wilson, Fernandez and Hadaway (1994) noted, “when two people talk about mathematics problem-solving, they may not be talking about the same thing” (p.1). The meaning of problem- solving may range from ‘solving’ (completing) rote textbook exercises across a spectrum to doing mathematics as a mathematician (Schoenfeld, 1992). Polya (1945, 1957), described problem-solving as ‘the art of discovery’. He also regarded the type of problems a student must solve as key to doing mathematics, because these problems should challenge the student’s “curiosity and evoke his creative abilities” (Polya, 1978, p. The National Council of Teachers of Mathematics (NCTM, 1982, 2000) alludes to this view by saying that problem-solving refers to how students engage in a task without knowing which known solution method will lead to the solution. According to Schoenfeld (1985), when a student already knows a (reliable) solution path, there is no problem to solve, but only an exercise to practice a process or to strengthen memory of both factual and procedural knowledge. Schoenfeld (1985) further argues that a problem is not determined or classified as a problem, based on an inherent property (like difficulty) but on the relationship between the solver’s knowledge/ability and the problem. For example, when one poses the following problem, “Find the sum of all the numbers between 1 and 99” to a group of first year student teachers the students will follow different paths. For some it will be a problem because they have not encountered this type of task before and they are likely to embark on finding a solution path – perhaps by adding all the numbers. Others may have solved this problem or similar problems in the past and therefore would recall form experience which solution path to follow. For the latter type of student, “finding the sum of all numbers between two given numbers” and similar tasks are exercises to practice and reinforce facts or processes and are not problems for them because they know what to do. The complexity of the ‘problem’ is thus not dependent on the function of the task but on the student’s exposure and experience to this type of task, and of the knowledge and belief system of the solver (Schoenfeld, 1992; Carlson & Bloom, 2005).

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Finding solution paths requires identifying what the problem/question is, and then reflecting on what path exists toward a solution. I argue that for this to happen certain knowledge has to be in place. For example, a question requires that multiple digit numbers must be added, a pathway has to come to mind about how to proceed. Students cannot perform the task if they do not know of different ways to perform this activity. They are thus not only faced with a mathematical problem, but also with a mental executive decision, a metacognitive decision (Flavell, 1987), utilizing working memory abilities (Dehaene & Changeux, 2011, Cockroft, 2015). Much of mathematics requires learning, and while problem-solving is also a form of learning, it is at the same time a form of applying knowledge and using knowledge and skills to identify a problem and searching for a way to address it and to solve it (Figure 11).

Figure 11: Components of mathematical problem-solving

The ability to see the problem and to act upon it in a strategic way, deliberating on possible solution paths, requires to work from knowledge that is retrieved from memory and applying it in an integrated way to first ‘see’ and then to ‘solve’ the problem. Duncker (1945) describes the generic activity of problem-solving, emphasizing the goal of the solver and the importance of having “recourse to thinking”. I would argue that this recourse is the existing knowledge that a solver uses to ‘see’ and to ‘solve’ the problem:

A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation

Abstract maths knowledge: concepts, facts and skills Identifying the 'problem by using the knowledge Examining the problem and considering pathways to solve it

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simply by action, then there is recourse to thinking…Such thinking has the task of devising some action, which may mediate between the existing and the desired situation. (Duncker, 1945, pg.1)

Although I have started this section by proposing that there is no clear definition for what problem-solving, I would argue that there are at least three distinct features of problem-solving in mathematics learning.

(1) The solver does not have a solution path for solving the task at hand (Lithner, 2008). (2) The solver needs to put some effort to find a solution and the effort is a thinking or

cognitive effort (Lesh& Zawojerski, 2007).

(3) The problem should be worth the while of solving the problem (Johanassen, 2000). An important overarching goal is that the problem-solving episode should serve as a learning experience that can reinforce conceptual understanding and procedural knowledge, utilizing facts and principles that are retrieved from memory.

2.5 MATHEMATICAL PROBLEM-SOLVING AS A CURRICULAR GOAL FOR LEARNING MATHEMATICS

The rationale for learning mathematics, globally, has been informed by the worldwide drive for competencies and skills to succeed in the twenty first century and the 4th Industrial Revolution. Students need to construct knowledge and develop skills in a fast-changing world, where 21st century skills are interchangeably referred to as higher-order thinking skills, deeper learning outcomes and complex skills in curriculums and amongst scholars (Scott, 2015). Several sources7 identified competencies needed for the 21st century. For instance, the Assessment and Teaching of 21st Century skills Project, a worldwide multi-stakeholder partnership categorized 10 competencies into four categories; (i) ways of thinking; (ii) ways of

working; (iii) tools for working and (iv) skills for living in the world (Griffin, McGaw & Care,

2012), see Table 3.

7_{The Delores Report, produced by the International Commission on Education for the Twenty-first Century; The}

Assessment and Teaching of 21st _{Century skills Project ,(ATC21S) & The US-based Apollo Education Group} (cited in Scott, 2015)

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21st Century Skills (Source: Griffin et al., 2012)

Ways of thinking Ways of working Tools for working Living in the world Creativity and innovation Communication Information literacy Citizenship – local and

global Critical thinking, problem-

solving, decision making

Collaboration (teamwork) ICT literacy Life and career

Learning to learn, metacognition

Personal and social responsibility – including cultural awareness and competence

While various sources identified 21st century competencies, there is no single prescribed approach in how to educate students for the twenty-first century (Dede, 2010). However, mathematics has been identified as a pertinent vehicle to develop these skills, which are expressed as mathematical thinking as evidenced in mathematics curriculum frameworks across the globe. 21st century skills included in mathematics curricula, but not limited to these are; problem-solving, logical reasoning, curiosity, creative and ICT proficiency. Learners from countries that have adopted a mathematical thinking approach are Singapore, where the curriculum centres on mathematical problem-solving, and the Netherlands’, which adopted a realistic mathematics approach, plus Japan, China and Finland with their focus on mathematical thinking approaches in the curriculum. There is a move, generally, to revise mathematics curriculum frameworks to address the need to develop mathematical thinking practices needed for the 21st (e.g. South African Curriculum and Assessment Draft Policy for Mathematics, 2018, the US Common Core State Standards, 2010). I give a brief account of mathematics curriculum frameworks with a strong mathematical thinking approach.

2.5.1 From mathematical process standards to common core state standards (CCSS) in