Pedagogical Content Knowledge for Mathematics

Pedagogical Content Knowledge (PCK) is knowledge that transcends understanding for the subject for oneself and reaches an understanding of subject matter for teaching in a way that makes the subject ‘comprehensible to others’ (Shulman 1986: 9). As Hodgen (2011) describes: a mathematical question may examine ‘how do you show the statement is true’; a pedagogical question would enquire ‘how to you enable others to see the statement is true’

(2011: 27).

The impact of Pedagogical Content Knowledge (PCK) on student outcomes, was identified as ‘strong’ by Coe et al (2014) (see Table 2.8 in 2.2.1). Carter (2015) and others (Sadler et al, 2013 and Hill et al, 2005) suggest that teachers who employ PCK - and understand the way students tackle different topics, strive to make the subject accessible and meaningful, understand the thinking behind students’ methods, promote connections, and can identify common misconceptions - are ‘more likely to have a positive impact on pupil outcomes’

(2015: 8). Kahan et al (2003) suggest that it is PCK which sets apart those who are simply good at mathematics and those who are also good at teaching mathematics: although PCK is content-specific, it ranges beyond a knowledge of mathematics and therefore ‘a mathematician may not possess it’ (Kahan et al 2003: 223). Grouws and Schultz (1996) suggest pedagogical content

knowledge includes ‘useful representations, unifying ideas, clarifying examples and counter examples, helpful analogies, important relationships, and

connections’ (1996: 443). An et al (2004) also emphasize connections and define PCK as the ‘knowledge of effective teaching’ which includes three

interconnected components: knowledge of content; knowledge of curriculum;

and knowledge of teaching.

Anthony and Walshaw (2009) also acknowledge the need for students to make mathematical connections and believe teachers need to know ‘how to extend and challenge students’ thinking’ by having ‘substantial pedagogical content knowledge and a grounded understanding of students’ as learners’ (2009: 158).

Liping Ma (1999) considers this as an important quality in Chinese teachers’

professional practice: ‘One thing is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed’ (Liping Ma,1999: 136).

Mason and Spence (1999) describe different types of knowledge: knowing-to and knowing about. Knowing-to describes ‘active knowledge which is present in the moment’; knowledge which is dynamic, current, accessible and useful, enabling a person to ‘act creatively’ in fresh and novel situations ‘rather than merely react’ to rehearsed, trained or habitual positions (Mason and Spence 1999: 135-136). Knowing about is constituted from ‘knowing-that’ (factual),

‘knowing-how’ (techniques and skills), and ‘knowing-why’ (a backstory) (Mason and Spence 1999:135) – and are an attempt to encompass teacher perceptions of the meaning of understanding. These three ‘knowings’ also captured in the components of teacher knowledge identified by Shulman (1987), namely:

subject content knowledge; pedagogical content knowledge; knowledge of related content; knowledge of curriculum; knowledge of learners; knowledge of educational aims; and general pedagogical knowledge.

Shulman’s range of knowledge types is intended to equip the effective

practitioner to act, ‘but knowing-to act when the moment comes requires more

than having accumulated knowledge-about’ (Mason and Spence 1999: 139).

‘Knowing-to’ act is developed by establishing connections between the past, the present and the future, ‘so that in the future, past experience informs (literally) practice in the moment’ (Mason and Spence 1999:148). Rich connections can be made when there are multiple links, or a ‘web of meaning’ (Mason and Spence 1999: 150); links that activate the senses. This ‘knowing-to’ knowledge

‘requires relevant knowledge to come to the fore so it can be acted upon’

(Mason and Spence 1999:139) so, for example, teachers can not only analyse an error but have high levels of fluency and mathematical reasoning to rapidly,

‘often on the fly’, redress the misconception (Ball et al 2008: 397).

Watson and Mason (2007) suggest Schulman’s distinctions between subject content knowledge and pedagogic content knowledge are ‘not necessarily useful for the task of educating mathematics teachers’ (2007: 209) and that a

‘considerably more complex model of teacher-knowledge’ is required. Such a model should be ‘augmented by, among other things, understanding how being knowledgeable about mathematics teaching influences classroom actions’;

‘knowing-to act in the moment through having pertinent possibilities come to mind’ (Watson and Mason, 2007: 208).

Hodgen (2011) suggests the nature of pedagogical content knowledge is itself,

‘something of a contested idea within the education research community’ and there is ‘no clear distinction between subject knowledge and pedagogical content knowledge’ with pedagogical content knowledge possibly a ‘useful metaphor to locate teachers’ knowledge as embedded within the complex and unpredictable practice of teaching’ (2011: 28).

Turner and Rowland (2006) refer to the various forms of knowledge and experience a teacher needs to draw upon as incorporating a ‘Knowledge Quartet’. This Knowledge Quartet, consisting of ‘foundation knowledge’,

‘transformation knowledge’, ‘connection knowledge’ and ‘contingency

knowledge’ (Turner and Rowland 2006: 2), illustrates multiple representations of mathematical knowledge, each of which can usefully emerge and be employed in differing circumstances. Foundation knowledge can be seen to be the

teacher’s self-knowledge about mathematics together with their beliefs and attitudes towards teaching the subject, perhaps best encapsulated with the practise of procedures, use of textbook questions, correct use of terminology and noticing mistakes and misconceptions. Transformation knowledge is concerned with ‘knowledge-in-action’ (Turner and Rowland 2006: 2) and includes the decisions in terms of the choice of examples and explanations offered, and the use and scope of analogies, illustrations and demonstrations.

Connection knowledge is, as it name suggests, about making connections and about the significance of doing so in terms of conceptual understanding. And contingency knowledge is about being able to respond to students in real-time, to answer their tangential questions and to view these ideas as teaching

opportunities; ‘In other words, it concerns teachers’ readiness to react to situations that are almost impossible to plan for’ (Petrou and Goulding 2011:

19).

Recognising a scarcity of specifics, in terms of what teachers needed to know and do, The Joint Mathematical Council of the United Kingdom (JMC) released recommendations on ‘Developing mathematics-specific pedagogy in Initial Teacher Education’ to provide such guidance. These recommendations,

designed to ‘develop a coherent and rich approach to learning mathematics’

(JMC 2017) are outlined in Table 2.14.

• A focus on developing effective use of a variety of approaches to learning

• A need to recognise the value of both procedural and conceptual learning - and the relationships between them

• The use of alternative methods and representations to be informed by a deep understanding of mathematics and an appreciation of how mathematics is understood by learners

• Knowledge of the ‘big ideas’ in mathematics and the connected nature of the discipline of mathematics (within itself and to other subjects/contexts)

• A rigorous use of language and symbols

• Investigative and problem solving approaches to be explored

• Use of reasoning and proof

• The development of skills in mathematical reasoning through the use of high quality questions on the part of both teachers and learners

• Recognizing the importance of talk; Supporting and developing mathematical talk

• Recognizing and working with errors and common misconceptions

• Understanding the role of manipulatives and diagrams in learning and doing mathematics

• Due consideration given as to when to generalise from physical experience to the symbolic and abstract (also often referred to as moving between CPA)

• Promote experiences which progress student learning and challenge thinking, rather than experiences that are repetitive

• Developing learners’ positive attitudes to learning and their confidence to persevere and ‘have a go’; all have the opportunity to make progress and achieve success

• Opportunities for teachers and learners to develop confidence and competence with a range of tools and resources (including digital resources)

• Assessing learning in the specific context of the mathematics classroom

• Considered planning of lessons: recognizing and using prior learning, (within topics in mathematics as well as across topics); skillfully sequencing and selecting

mathematical tasks and classroom activities

Table 2.14 Recommendations Designed to ‘Develop A Coherent and Rich Approach to Learning Mathematics’ (JMC 2017)

The current Teachers’ Standards framework (GOV.UK 2011b) require teachers to be proactive in updating their knowledge and to be reflective and self-critical

in regard to their levels of expertise. Problems may arise with this modus

operandi, when teachers are unable to identify gaps in their own knowledge. As one teacher remarked: ‘You don’t know what you don’t know’ (Ofsted 2012). In-school-variation (ISV) also poses problems, and has been identified as a dominant factor in determining national outcomes – and one that overwhelms the variation between schools (National College 2011). The National College (2011) report states, that if in every school, if each group of students ‘attained the same standards as the best groups in that school, then national outcomes would be transformed’ (National College 2011: 4). A commitment to the

development of their teachers is one way in which some schools are attempting to narrow this gap. Providing professional development (from within schools) may both help reduce in-school-variation and help teachers to identify gaps in knowledge and expertise.

In summary, ‘Teachers must know the subject they teach’ as there is ‘nothing more foundational to teacher competency’ (Ball et al 2008: 404). The McKinsey report (2007), based on effective educational systems from around the world, makes it clear that whilst this is true, teachers also need to observe and reflect and identify what makes for great instruction in their subject and then have in place support in schools to ensure that teachers can deliver great instruction lesson after lesson; that is, provide an environment that sustains great

instruction. Ideas concerning in-school professional development and support-systems, are discussed in the following sections.

2.3.3 Using Classroom Observations to Develop