In the last blog post I described the origins of pedagogical content knowledge—‘that special amalgam of content and pedagogy that is uniquely the province of teachers’—by Shulman and colleagues. PCK, as it is commonly known, is an important acknowledgment of the professional skills of teachers in combining knowledge of subject-matter and of teaching. But what does it look like in practice?

In another classic paper, ‘*Content Knowledge for Teaching: What Makes It Special?*’, Deborah Loewenberg Ball, Mark Thames and Geoffrey Phelps reviewed the literature published in the 20 years after PCK was first introduced. Among their findings, they noted that PCK was typically couched only in the broadest sense, such as the description I gave earlier. In response, they set out to refine the concept of PCK by identifying precisely what teachers need to know and how this knowledge is used in teaching effectively. They did this through an empirical approach, including documenting and analysing mathematics instruction and associated work (planning, assessing, parent-teacher conferences, staff meetings etc) both episodically and longitudinally.

**Domains of mathematical knowledge for teaching**

In analysing the mathematical knowledge needed to carry out the work of teaching mathematics — what they termed ‘*mathematical knowledge for teaching*’ — Ball’s research team devised the following map of the domain of knowledge for teaching. The map makes refinements to Shulman’s initial categories of subject matter knowledge and pedagogical content knowledge with new subdomains that are empirically detectable, that is, evident in the work of teachers.

To help illustrate the distinctions between the different categories, we draw on an example of calculating 1 3/4 ÷ 1/2. (This is one of three contexts used by Ball (1990) in probing the pre-service teachers’ understanding of division particularly in the context of teaching.)

### Unpacking the subdomains of subject matter knowledge

The left half of the diagram concerns **subject matter knowledge **which is, in a sense, independent of knowledge of the particular learners in the room or of teaching. Subject matter knowledge is far from straightforward mastery of mathematical content. The authors make a helpful distinction between *compressed *and *decompressed *mathematical knowledge. People with mastery of mathematical concepts (for example, mathematicians) work with mathematical ideas in compressed formats — definitions, factual knowledge, routinised procedures. However, teachers need to decompress the content to make it accessible to learners, and that requires additional understanding of content that ‘goes beyond the kind of tacit understanding … needed by most people’, as we will see by elaborating the three subdomains.

**Common content knowledge** is mathematical knowledge that is used in variety of settings, and is not unique to teaching. In our example, it is the knowledge used to correctly compute 1 3/4 ÷ 1/2 = 3 1/2. For a teacher, it is basic competence with the content, including terminology, notation, and pronunciation. It is the ability to recognise if a student — or a resource — gives an incorrect solution or explanation.

In contrast to common content knowledge that is commonly understood by many, not just teachers, is **specialised content knowledge** — mathematical knowledge and skill unique to teaching. Specialised content knowledge is where teachers need to unpack mathematical knowledge to ‘make features of particular content visible to and learnable by students’.

In our example, it includes knowing and being able to justify why fraction division can be computed by inverting the second fraction and multiplying, as shown below:

or judging whether a non-standard method a student might invent will work in general:

Specialised content knowledge includes choosing an appropriate representation to illustrate a concept (e.g. recognising advantages and disadvantages of using rectangles or circles when comparing fractions), selecting which story problem best illustrates a mathematical point, and appraising and adapting resources and tasks.

The third category, **horizon content knowledge, **relates to the landscape of the content — looking forwards, backwards and across. It involves understanding the mathematical ideas that a concept builds on, and connections to later concepts. In our example, it may be understanding the progression of fractions to rational numbers to rational expressions, and that the four operations can be used on all three types of mathematical objects.

### Unpacking the subdomains of pedagogical content knowledge

The right half of the diagram concerns **pedagogical content knowledge, **and brings in knowledge of particular learners and of effective teaching strategies. Again, it has been divided by Ball et al into three distinct subdomains.

**Knowledge of content and students** requires an interaction between specific mathematical understanding and familiarity with students and their mathematical thinking. It involves anticipating what students are likely to think, what they might find easy or hard, and what they might find confusing. In our example, it might involve addressing the following: What common misconceptions will students have (e.g. division makes a number smaller)? How will students’ prior experiences with division of whole numbers support their understanding of division fractions? How might it confuse them? What difficulties do students typically have interpreting the answer to a division of fractions problem (e.g. dividing by 2 not by 1/2)? What are common errors that students might make?

**Knowledge of content and teaching** brings together specific mathematical understanding and an understanding of pedagogical issues that affect student learning. It involves sequencing instruction, choosing which examples to start with, and selecting appropriate representations at a given point in the learning progression. In our example, it might be choosing the sequence to develop a conceptual understanding of division of fractions (e.g. whole number multiplied by whole number, whole number divided by whole number, fraction divided by whole number, whole number divided by fraction, fraction divided by fraction) and the choice of examples within (2 x 3, 6 ÷ 2, 3 ÷ 1/2, and so on).

The third category, **knowledge of content and curriculum**, relates to the landscape of the curriculum — looking forwards, backwards and across. It involves locating a specific concept in the curriculum, identifying what students will have encountered in previous years, and connecting to curriculum-specific features such as the mathematical proficiencies, mathematical processes, general capabilities, cross-curriculum priorities. In our example, it might include knowing in which grade level students are typically taught to divide fractions, how division of fractions relates to division of whole numbers in the school curriculum, and what models for fractions and for division students might be familiar with.

To illustrate the interactions between domains, the authors give an example of error analysis. Directly quoting from page 401:

*In other words, recognizing a wrong answer is common content knowledge (CCK), whereas sizing up the nature of an error, especially an unfamiliar error, typically requires nimbleness in thinking about numbers, attention to patterns, and flexible thinking about meaning in ways that are distinctive of specialized content knowledge (SCK). In contrast, familiarity with common errors and deciding which of several errors students are most likely to make are examples of knowledge of content and students (KCS).*

Ball and colleagues note later in the paper that it is not always easy to discern the boundaries between the categories. They also note that the same situation might manifest different knowledges. For example, one teacher analysing a student error might invoke specialised content knowledge whereas another teacher who has seen other students make the same mistake is drawing on their knowledge of content and students. While the authors level these criticisms (and more) at their own work, I see them more as nuance of interpreting the framework rather than fundamental limitations.

I have done scant justice to the paper in this blog post. There are so many gold nuggets that help us to understand the mathematical demands of teaching that I have not included here. This is a paper that deserves to be read multiple times. I also recommend the transcript of a talk on ‘Knowing and Using Mathematics in Teaching’ that Deborah Ball gave in 2006 and which is accessible here.

### Some final thoughts

When the mathematical knowledge required for teaching is broken down in this way, teaching is both complicated and complex. It is complicated (in the generally understood sense of the word) in that there are many interconnecting parts and elements. And it is complex (in the systems theory sense of the word) in that the parts are richly interrelated and change in unexpected ways as they interact. Another way to view complex systems is that the same starting conditions can produce different outcomes, depending on interactions of the elements in the system^{1} which helps describe why every classroom is unique.

Given that this is a complex environment even for experienced teachers to be working in, how do we support novice teachers in acquiring and deploying these various forms of mathematical knowledge for teaching? KCS (knowledge of content and students) and KCT (knowledge of content and teaching), in particular, only really develop through experiences in the classroom. For example, how can you know the common errors that students might make without having accumulated a range of experiences in seeing lots of students make errors? You can ask more experienced colleagues, review existing work samples, and consult text books on common misconceptions, but the rubber only hits the road once in the classroom.

You also may be wondering whether this is all strictly academic, that is, why does it matter how we define and understand the different types of knowledges required for teaching? There is a familiar refrain whenever Australia’s declining performance in standardised testing (i.e. NAPLAN, TIMMS, PISA) attracts attention — that teachers do not know mathematics well enough to teach, and so need increased content knowledge. Those arguments are also echoed when discussing upskilling out-of-field^{2} teachers and contemplating reforms to initial teacher education. However, these arguments only carry weight if we know the kind of mathematical knowledge and skill required for high-quality teaching, and how teacher education and professional development might be designed in response. And, there is a trap in thinking that more content means better quality teaching. For example, there is a common implication that mathematically qualified mid-career professionals have all the knowledge required, but are put off from a career change ‘by the lengthy process of becoming a qualified teacher’ — as if subject matter expertise was the only knowledge required.

By articulating the different forms of knowledge required to teach mathematics well, perhaps there might be more appreciation of the complex work of ‘teaching maths’ rather than ‘telling maths’.

### References

Ball, Deborah Loewenberg. “Prospective Elementary and Secondary Teachers’ Understanding of Division.” *Journal for Research in Mathematics Education* 21, no. 2 (1990): 132–44.

Loewenberg Ball, Deborah, Mark Hoover Thames, and Geoffrey Phelps. “Content Knowledge for Teaching: What Makes It Special?” *Journal of Teacher Education* 59, no. 5 (November 2008): 389–407.

[1] Source: https://investigationsquality.com/2019/12/05/the-difference-between-complex-and-complicated/

[2] There needs to be a blog post unpacking the various definitions of out-of-field teaching and why it is a problematic term to use without nuance. For more on this, I strongly recommend Hobbs, L., Campbell, C., Delaney, S., Speldewinde, C., Lai, J. (2022). Defining Teaching Out-of-Field: An Imperative for Research, Policy and Practice. In: Hobbs, L., Porsch, R. (eds) Out-of-Field Teaching Across Teaching Disciplines and Contexts. Springer, Singapore. https://doi.org/10.1007/978-981-16-9328-1_2

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