Frameworks combining mathematical content knowledge and mathematical thinking

It is well documented that pedagogical content knowledge is subject and topic specific (Lehrer & Franke, 1992; Hadfield, Littleton, Steiner & Woods, 1998; Hashwe, 2005). This study, however, argues that it is also mathematical thinking processes specific; teachers of mathematics need to understand both mathematical concepts, processes, and should have pedagogical knowledge of how to teach mathematical concepts through mathematical processes. Thus, the pedagogical content knowledge needed for teaching problem-solving, and reasoning and proof will be different to the pedagogical content needed for teaching mathematical facts and procedures. There is therefore, a need for teachers to hold specific kinds of knowledge to teach for the development of mathematical problem-solving as both processes and mathematical thinking. I draw on the following case studies to support this argument.

Mathematical knowledge for teaching mathematical problem-solving

Foster et al.’s (2014) case study stemmed from questions such as, what would a learner’s development in mathematical communication in a problem-solving context comprise and what

30

pedagogical knowledge of problem-solving is needed by teachers in order to support leaner engagement in mathematical communication. In response to this question, Foster et al. (2014), proposed, “(a) robust conceptualisation of mathematical process knowledge (MPK) and pedagogical process knowledge (PPK). Foster’s (2014) study was underpinned by an adapted version of Ball et al.’s, (2008), MKT - model, in which each occurrence of the word “content” in Ball’s framework was replaced with the words “concepts and processes”, as illustrated in Figure 8, adapted from Foster et al. (2014, p. 98).

CONCEPT AND PROCESS KNOWLEDGE PEDAGOGICAL CONCEPT AND PROCESS KNOWLEDGE

Common content concept and process

Knowledge Specialized

concept and process knowledge Knowledge of concepts and processes and students Knowledge of concepts and processes and curriculum Horizon concept and

process knowledge

Knowledge of concepts processes and teaching

Figure 8: Foster et al.'s (2014) adapted version of Ball et al.'s (2008) MKT model (Source: Foster et al., 2014, p. 98)

In their case study, Foster et al. (2014) investigated secondary school (age 11 -18) in-service teachers’ lesson study informed by the Japanese model of lesson planning, with an added focus on problem-solving. The authors were particularly interested in how the three sub-domains of pedagogical concept and process knowledge (PPK), i.e. knowledge of concepts and processes and students (KPS); knowledge of concepts and processes and teaching (KPT), and knowledge of concepts and processes and curriculum (KPC) can be supported by a carefully designed lesson study programme.

31

In discussing, what the authors meant by each of the PPK sub-domains, they drew parallels with Ball et al.’s (2008) pedagogical content knowledge sub-domains. Thus, parallel to Ball’s ‘knowledge of content and students’, KPS, is referred to as the blending of knowledge of processes and “common ways in which students think about processes, what context motivate them to learn the processes and what difficulties they have” (Foster et al., 2014, pg. 102). Knowledge of processes and teaching refers to “knowing and being able to use effective strategies for teaching problem-solving processes” (ibid). Teachers’ ability to select and sequence appropriate mathematical tasks to assist in a “coherent development in students’ process skills” is known as knowledge of processes and the curriculum” (ibid).

Findings regarding ‘knowledge and processes and students’, suggested learners in the study often misinterpreted requests for mathematical communication from the teacher and tasks to show their ‘working out instead of seeing communication as a “reasoned mathematical argument” (Foster et al., 2014, pg. 102). With reference to, the teachers’ knowledge of processes and teaching, findings suggested this knowledge domain was underdeveloped. However, the teachers demonstrated a specific kind of knowledge of processes and the curriculum by being able to design a sequence of lessons to develop a single process such as reasoned argumentation through problem-solving. The authors explained that their empirical findings are limited to this particular case study and that it is not possible for them to report on the development of a clear approach of the learning of problem-solving over time as it is beyond the scope of their study (Foster et al., 2014). These authors concluded that they anticipated that the empirical conceptualisation of the mathematics knowledge for teaching has not to date identified knowledge of mathematical thinking processes as essential to everyday classroom practice. They conclude that problem-solving does not receive the required attention in the day- to day teaching of mathematical concepts. Furthermore, they conclude that teachers’ understanding of process skills and how to assess learning through process skills are currently largely underdeveloped.

Mathematics teachers’ knowledge for teaching problem-solving

In a similar vein, Chapman (2015), highlighted the fact that the categories of mathematical knowledge proposed by Ball et al.’s (2008), model does not provide a complete picture of the knowledge needed by teachers to teach for the development of mathematical problem-solving. Chapman (2015), thus, proposed a mathematical problem-solving knowledge for teaching (MPSKT) framework, which addresses the question of what teachers need to know, particularly

32

to teach mathematical problem- solving proficiency. According to Chapman (2015), teachers need to hold interdependent categories of knowledge, but are not limited to these categories only, to develop learners’ mathematical problem- solving proficiency, as diagrammatically illustrated in Figure 9.

Figure 9: Interrelationships of MPSKT (Source: Chapman 2015, p. 32)

Chapman’s (2015) construct of mathematical problem- solving proficiency is defined as the direct relationship between suggested characteristics for successful problem- solving and mathematical proficiency. Mathematical problem-solving proficiency therefore, encompasses the relationship between components of successful mathematical problem-solving as proposed by Schoenfeld’s (1985), Mayer and Wittrock’s (2006), and Kilpatrick et al., (2001) strands of mathematical proficiency. The relationship between the components of successful problem- solving and the strands of mathematical proficiency is illustrated in Table 1, adapted from Chapman (2015, p. 22). PS proficiency PS content knowledge Pedagogical PS knowledge Affective factors and beliefs

33 Table 1:

Perspective of mathematical problem-solving proficiency

Mayer & Wittrock (2006) Schoenfeld (1985) Kilpatrick et al., (2001) Mathematical PS proficiency Concepts Procedures Appropriate Resources Conceptual understanding and procedural fluency Conceptual understanding of mathematical concepts, operations and relations Strategies Heuristic

Strategies

Strategic competence Understanding of general heuristics and specific strategies and when and how to use them Metacognitive

knowledge

Metacognitive control

Adaptive reasoning Capacity for logical thought and understanding of reflection for awareness, monitoring, controlling and overseeing one’s own cognitive processing during PS Beliefs Appropriate

beliefs

Productive disposition Holding beliefs about mathematics, PS and one’s PS competence that support motivation and confidence

In Table, 1, Chapman shows the possible relationship between each of the five knowledges proposed by Mayer and Wittrock (2006), each of the components for successful problem- solving proposed by Schoenfeld (1985) and each of the interwoven strands of mathematical proficiency. Chapman (2015) further argues that similar to mathematical proficiency; mathematical problem-solving proficiency is not a “one dimensional concept” and cannot be developed by just focusing on one or two of the components (p. 21). Thus, the development of mathematical problem-solving proficiency requires teaching practices that will incorporate all the components as illustrated in Table 1. Furthermore, in order for teaching practices to

34

successfully incorporate, all the components of mathematical problem-solving proficiency teachers are not only required to know how to solve mathematical problems but also what create deep knowledge6.

According to Chapman (2015), development of mathematical problem-solving proficiency includes a proficiency in selecting and designing mathematical problems that are largely influenced by teachers’ views regarding the nature of mathematical problems. Teachers’ views about the nature of problems is crucial for the development of students’ mathematical problem- solving proficiency as it could either limit or enhance how students experience, perceive and learn problem-solving (Chapman, 2015). Consequently, Chapman (2015) proposed that mathematical problem-solving knowledge for teaching should include the view that a problem is a mathematical task for which the solver does not have an obvious way to complete the problem task. Furthermore, teachers should not only understand problem-solving as a process but also in terms of mathematical thinking or problem-solving thinking. To show the relationship between problem-solving thinking and mathematical thinking, Chapman (2015) referred to Mason, et al.’s (1982), notion of mathematical thinking that involves specializing, generalizing, conjecturing and convincing.

Secondly, knowledge of problems also includes how teachers understand the structure and purpose of mathematical problems in order to make sense of students’ solutions. Having an understanding of the structure of mathematical problems include understanding how the syntax of word problems influences the way in which students understand the problems which impacts on the solution strategies used. In addition, it also includes understanding how different structural problems have an effect on students’ solutions.

Thirdly, knowledge of problem-solving also includes knowledge of heuristics, but over and above this, teachers need to have conceptual and procedural knowledge of various problem- solving models, such as Polya’ s, Mayer and Wittrock’s and Schoenfeld’s models of problem- solving to understand the thinking and processes involved in finding a solution for a problem. Knowledge of problem posing is the companion of problem-solving as teachers do not only need to know and exemplify how to solve problems, but they should also know how to generate new problems and reformulate problems worked on during the problem-solving process. The generation of one’s own problems can occur before the problem-solving process, during it, or

6 _{Knowledge includes the following factors, knowledge of problem solving, knowledge of problem posing,}

35

afterwards. Problem posing after the problem- solving process usually includes extending the existing problem that was solved. Teachers need knowledge of how to pose problems so that they are able to support their students in generating their own “useful and meaningful problems to develop their problem-solving proficiency” (Chapman, 2015, p. 26). Teachers also need to know common difficulties encountered by students about problem-solving.

Mathematical problem-solving knowledge for teaching should also include knowledge of the difficulties experienced by students when they attempt to solve problems. These difficulties include, but are not limited to, the following: “lack of knowledge of essential facts, rules and formulas, insufficient mastery of computational skills, inability to read, and lack of a method for attacking the problem” (Chapman, 2015, p. 27). In addition to having knowledge of the common difficulties encountered by students, teachers also need to have knowledge of both the cognitive and affective dispositions of successful problem solvers as well as knowledge of students’ mathematical thinking. More specifically, teachers need to understand students’ mathematical reasoning during the problem-solving process because knowledge about students’ reasoning has been shown to be useful in understanding and supporting students’ problem- solving.

Chapman (2015) argues that it is obvious that MPSKT should include knowledge of problem- solving teaching practices that will support and develop mathematical problem-solving proficiency. Holding knowledge of problem-solving teaching practices requires from teachers to be strategic competent, understand the role of metacognition during the problem-solving process. Furthermore, teachers need to anticipate students’ possible solution strategies as well as know when and how to assist students in order to successfully solve the problem without lowering the cognitive demand. It is expected that teachers may at times find themselves in a position of not knowing the solution to a problem. Teachers, thus need to know how to deal with instances of not knowing.

An associated construct is students’ mathematical problem-solving beliefs. Beliefs have been referred to as a cognitive and affective construct (Goldin, 1998), which can either facilitate or inhibit problem-solving. It is important for teachers to have an understanding of students’ mathematical problem-solving beliefs, because their beliefs influence how they approach problem-solving. Not only should teachers be aware of their students’ beliefs about problem- solving but also about their own mathematical problem solving beliefs, because these may influence how they approach and teach problem solving. In conclusion, Chapman (2015)

36

contends that mathematical problem-solving knowledge for teaching is a “complex network of interdependent knowledge” and requires a deep understanding from teachers about the interdependent nature in order to develop problem-solving proficiency through their teaching practices (p. 19).

I have examined a few pertinent frameworks of teachers’ mathematics knowledge, needed for teaching mathematics, as well as mathematical problem-solving processes and thinking. By implication, these frameworks propose specific forms of mathematical learning, which requires that teachers create specific mathematical experiences for students. As Schoenfeld (1985, p. 185) explained, “if the bulk of students’ experience with particular mathematical ideas occurs in the classroom, the students’ mathematical world views – their abstraction of their experiences with those mathematical ideas – will be based on those experiences.” As an example, student teachers’ perspectives and use of geometrical proofs will be greatly influenced by how proofs have been used in the geometry lectures (Schoenfeld, 1985). Similarly, if mathematics content courses for student teachers are focused mainly on memorization of facts and procedures it is highly unlikely that student teachers will develop an appreciation of mathematics as a discipline nor will they make the necessary sense of the discipline (Schoenfeld, 2016). It is also highly unlikely that student teachers will become mathematical thinkers, because students need to experience mathematics content in its full richness to become thinkers of the discipline (Schoenfeld, 2016).