Elaborating mathematical knowledge for teaching

Unpacking or decompressing of mathematical ideas, which is an aspect of specialised content knowledge (SCK – discussed below) has been identified as an important element of the knowledge- in- action (mathematical practices) that mathematics teachers need to enact as they do their work (Ball & Bass, 2000; Ball, et al., 2004). Elements of unpacking are identified to include designing mathematically accurate explanations that are comprehensible and useful for learners; interpreting and making mathematical and pedagogical judgements about learners’ questions, solutions, problems, and insights which are both predictable and unusual. The authors further argue that content that is in the curriculum should be part of the mathematical knowledge that teachers need. Ma (1999) also holds the same view and argues that a teacher’s subject matter knowledge which develops in the context of teaching and learning is relevant to teaching and is likely to be used in teaching.

This understanding contradicts Krauss et al.’s (2008) findings as discussed in Chapter 1 (Section 1.2) in that they found no positive correlation between teachers’ knowledge base for teaching and their years of teaching experience. What this suggests, and Ball et al. (2008)

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also argue such, is that this is inevitable as teacher education is crucial. However, both Ball et al. and Ma (1999) argue about the importance of developing teachers’ knowledge base for teaching through deliberate study of practice. For Ma, this requires experience in practice while Ball et al. argues for the importance of teacher education. This is interesting because Krauss et al. in their Coactive project are working at the secondary school level and Ball et al. and Ma are working at the elementary school level, and are arguing contrary positions pertaining to sites for developing teachers’ knowledge base for teaching. Ma is saying teachers’ knowledge base for teaching is learned in the context of teaching and Krauss et al. and Ball et al. are saying what is significant is what you have learned before entering the school.

Apart from being able to solve mathematical problems, teachers should also enable learners to access the content; interpret learners’ questions and productions; generate contexts in which the content arises; explain and represent the content in multiple ways (Ma, 1999; Thames, et al., 2008). In addition, all these tasks that the teacher has to engage with require mathematical knowledge and reasoning. My study thus explores the kind of mathematical knowledge and reasoning entailed by student-teachers as they engage in a discourse of and about LMT.

Ball et al. (2008), building on Shulman (1986), and from their study of practice, categorise mathematical knowledge for teaching (MKT) as subject matter knowledge (SMK) comprising of common content knowledge (CCK), specialised content knowledge (SCK), and Horizon content knowledge. They further categorise pedagogical content knowledge (PCK) as comprising knowledge of content and students (KCS), knowledge of content and teaching (KCT) and knowledge of content and curriculum (Thames, et al., 2008).

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Figure 1: Domains of Mathematical Knowledge for Teaching (MKT) by Ball et al.

(2008)

Figure 1 is an illustration of this proposed model of MKT whose knowledge domains are analytically distinct. CCK has been described as the ability by the teacher to solve mathematics problems in a way any other person would manage, for example, the mathematician or other users of mathematics. SCK describes the mathematical knowledge required for teachers to do their work, for example, looking for patterns in learners’ errors, difficulties and misconceptions. SCK is also about unpacking or decompressing of mathematical ideas as earlier described. CCK is likely to resonate with what Shulman means by SMK (concerned with the substantive and syntax of the discipline of mathematics and is discussed in Section 3.2) while SCK is new in terms of its conceptualization but are both mathematical knowledge (Hill et al. 2008). For Hill et al. CCK and SCK do not entail any specific knowledge of learners or instruction.

To exemplify these sub-domains, Ball et al. (2008) states that manipulation of a mathematical problem and knowing that the manipulation is faulty is CCK in that everyone who has been enculturated in the discipline of mathematics is able to do. However, carrying out an analysis of why the error has been made and how it can be corrected by using a variety of representations requires a specialised way of knowing mathematics (SCK) – an indication that teachers work with mathematics in its decompressed or unpacked form while learners strive to develop fluency with compressed mathematical knowledge (Ball & Bass, 2000; Ball,

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et al., 2008). Furthermore, the complexity of SCK in this instance is that if the teacher is aware of the errors or difficulties their learners experience, then they will require KCS. In this instance, for their work, Ball et al. assert that making distinctions between categories of knowledge was not easy in that it affected precision. They further define horizon content knowledge as awareness of how mathematical topics are related over the spam of mathematics included in the curriculum.

KCS focuses on knowledge about students and knowledge about mathematics, for example, learners’ anticipated thoughts and likely confusion; or interesting, motivating, anticipated easiness or hardness of a task by learners. A specific example Hill et al. (2008) provide is that of teachers’ knowledge and awareness that given two fractions to add, learners who are unable to conceptualize the multiplicative nature of fractions would possibly add numerators and denominators. This suggests that the teacher has to attend to both the specific content of adding fractions and the likely learner mistakes or misconceptions that might arise. Such knowledge would then guide the teacher to design appropriate instruction to address the error prior to it occurring or when it occurs. This is based on the argument that:

“Logically, teachers must be able to examine and interpret the mathematics behind students’ errors prior to invoking knowledge of how students went astray” (Hill, et al., 2008, p. 390).

To help clarify what KCS is, Hill et al. (2008) state that it is not about knowledge of best teaching strategies to build on learners’ mathematical thinking including their errors. It is also not about knowledge of curriculum materials. This suggests that KCS resonates with the first two of the three stages for carrying out error analysis, which are identifying the possible errors learners make, and explaining reasons for the error (interpret and evaluate) as conceptualized by Peng & Luo (2009) and Jacobs, Lamb & Philipp (2010). In this case KCS is not about stage three which entails suggesting appropriate remediating strategies. The three stages of carrying out error analysis are discussed in detail in Section 3.3 of this chapter. Therefore, Hill et al. (2008) argue that there is a distinction between teachers’ SMK and their KCS. This implies that one can have strong SMK without necessarily understanding how learners learn particular content, and vice versa.

In trying to conceptualize and measure KCS by writing, piloting, and analysing results from multiple – choice items, Hill et al. (2008) found that teachers were able to draw from their

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KCS and/or mathematical reasoning. They were familiar with learners’ mathematical thinking such as common learner errors as one element of knowledge for teaching. From the results, they claim that they could talk of how skilful, insightful, and full of wisdom their teachers were compared to the mathematician. However, they could not make claims on how these rich resources teachers had could translate into learner gains in their teaching of mathematics.

Similarly, Even & Tirosh (2002) report on a study done by Tirosh, Even, and Robinson (1998) and argue that there is a link between teachers’ knowledge and understanding about learner learning, and their instructional practice. This implies a relation between teachers’ awareness of learners’ errors and the strategy for instruction. In light of this, Tirosh, Even and Robinson worked with four teachers (two novices and two experienced) on simplification of open algebraic expressions. Their findings are that the two novices who were not aware of learners’ tendency to conjoin used a method of ‘collecting like terms’ that had implications for the quality of instructions given to learners. One teacher drew on the application of rules while the other drew on the ‘fruit salad’ approach. The other two experienced teachers who were aware of learners’ tendency to conjoin planned comprehensive lessons meant to familiarise the learners with the notion of like and unlike terms prior to teaching simplification of algebraic expressions. One teacher focused on identifying like terms while the other teacher used multiple strategies that included substitution, order of operations, and going backwards.

These findings are important for my study in that as you will observe in the unfolding Chapters and more explicitly in Chapter 8, one of the strategies used by the novice teachers, which is the application of rules formed scenario 2, which I named the conjoining problem. One of the questions I ask my data pertaining to scenario 2 is: In what ways do student- teachers talk about the conjoining problem and how might this be explained in relation to their preparedness for the task of engaging with the discourse of LMT?

Moreover, Thames et al. (2008) refer to KCT as knowledge about teaching and content and focuses on knowing how to design instruction, for example, sequencing particular content for instruction and making instructional decisions related to which learners’ contributions to ignore, pursue or reserve for future reference. They further stated that knowledge of content

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and curriculum was provisionally placed while work was still in progress. Overall, this framework of MKT depicts the kind of mathematical problem solving that teachers engage with as they teach mathematics. In my study then, what kind of mathematical problem solving skills do student-teachers demonstrate as they engaged in a discourse of and about LMT in general and in the context of school algebra? To what extent would the kind of mathematical problem solving include aspects of CCK, SCK, Knowledge on the horizon, KCS, KCT and curriculum knowledge?

From the foregoing discussion, I have demonstrated how Ball et al. (2008) carefully mapped and measured their knowledge domains. In this respect, they argue that Shulman’s notions of teachers’ knowledge base for teaching lack clarity on the boundaries between PCK and other forms of teacher knowledge. This, they assert, has lead to researchers who have engaged with Shulman’s constructs to overlook his initial invitation which was directed towards refinement. Ball et al. have observed that, instead, the researchers have used the constructs as though they were fully developed. In particular, PCK can hardly be distinguished from other forms of teacher knowledge. For example, PCK has been referred to what could be viewed as content knowledge and sometimes referred to something that could be considered, to a larger extent, as pedagogical skills. Ball et al. have further observed that the definitions of PCK as a construct that is an amalgam of content and teaching has been broadened to include any package of teacher knowledge and beliefs. Therefore, Ball et al. argue for precision about the concepts and methods involved, and hence their development.

Adler & Patahuddin (2012) also hold similar views on the lack of precision on how researchers have used Shulman’s constructs of PCK and other forms of teacher knowledge. In referring PCK to something that could be viewed as content knowledge, Adler & Patahuddin draw on Ball et al.’s (2008) and Krauss et al.’s (2008) works. They point to that while Ball et al. consider working with different mathematical representations as SCK; Krauss et al. see such an aspect as focus on PCK. For Ball et al., you have to know different representations to mathematical problems irrespective of who the learners are; while for Krauss et al., you are only bothered by different representations because you are teaching and therefore you need to have different ways of explaining. In referring to the issue of being knowledgeable about multiple representations to mathematical problems, the argument is that:

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“One will rarely be able to move flexibly across representations if one has poor understanding of a representation within itself and, vice versa, a representation within itself is likely to be poorly understood if one is not able to carry the meanings over to other representations” (Arcavi, 1995, p. 155).

This suggests that one cannot work with multiple representations if they do not know what it means to get to the answer given any mathematical problem. Despite the debate around multiple representations, my study whose focus is on student-teachers’ preparedness to participate in the discourse of engaging with LMT falls inside figure 1 in PCK. While there is controversy around what is PCK and what is not PCK, there is no controversy even coming from Shulman that engaging with the discourse of LMT is part of PCK, it is not SCK. Therefore, my study is about PCK. Following Even & Tirosh (2002), my conceptualisation of the notion of engaging with LMT is not only about focusing on learner errors but also about developing in learners both instrumental and relational understanding, and creating an environment where teacher can listen to learners. The detail of this is discussed in Section 3.3. However, QUANTUM project by focusing on MfT is concerned with the relationship between SMK and PCK, and I discuss what aspects inform my study.