What knowledge do teachers need to teach? Pedagogical Content Knowledge for Mathematics

What knowledge do mathematics teachers need in order to teach successfully?   In a series of four blog posts I want to summarize some of the research on the topic. The first blog post looks at the concept of pedagogical content knowledge of mathematics.  The second will discuss research attempts to measure teachers’ knowledge and link it to students’ learning outcomes.  In the third one, I write about the implications for teachers’ professional development.  In the final blog post, I will relate the first three blog posts to the South African context.

 Many intuitively feel that a thorough understanding of content is necessary to be a good teacher. However, accurately describing what and how much knowledge a teacher should have to teach successfully, has eluded researchers and policy makers.  Nevertheless, the question matters. Ball et al. (1991) argue that insufficient insight in the knowledge it takes to teach well contributes to low numeracy levels and a lack of interest in maths with many people.  Questions such as “What is the probability that in a class of 25, two people will share a birthday? Or Is a square a rectangle? leave many baffled.

This is due to an excessive focus on procedures, rules of thumb and ‘drill and kill’ kinds of practice, instilled by a dated, but tenacious view of mathematics as a fixed body of knowledge rather than an system of human thought (Ball et al., 1991).  This view is reinforced by textbooks that list ‘hints’ instead of developing conceptual understanding and a wide but shallow curriculum. A similar diagnosis, in my opinion, can be made for other sciences.  Many feel that what makes a good teacher is tacit and consider teaching as an art, based n common sense with little need for professional learning.

Research on the relation between teacher knowledge and student learning took off in the 1960s. Quantitative studies used the number of certificates or mathematics courses taken as proxy variables for teachers’ knowledge.  These studies rejected a straightforward relation between more teacher mathematical knowledge and more student learning.  A (weak) positive relation for undergraduate courses was found, but with non-linear relations and threshold effects complicating matters (Ball et al., 1991).  For more advanced graduate courses the relation was absent and in some cases even negative (e.g. Belge, 1979; Monk, 1994).  This may be due to the increasing compression of knowledge in advanced courses, which can complicate the ‘unpacking’ of content necessary in teaching and with higher exposure to conventional teaching approaches in advanced courses (Ball et al., 1991).

In the 1980s, researchers set upon closer probing of mathematical knowledge, rather than using second-order indicators.   Lee Schulman (1986) conceptualized pedagogical content knowledge (PCK) as a unique domain of teacher knowledge covering various aspects of subject knowledge of mathematics that are relevant for teaching including:

  • Knowledge what concepts students develop in various stages of their development
  • Knowledge of common student misconceptions on mathematical concepts
  • Knowledge of curriculum, threshold concepts and in what order they are best taught.

With PCK Shulman intended to vindicate the central role of subject content knowledge in teaching quality, in addition to generic pedagogical knowledge. It offers a fine-grained conceptualization of the kind of content knowledge a teacher requires for teaching successfully. PCK relates to the ability of not only knowing the content, but the ability to enable others to know it.  A powerful example of PCK is given by Deborah Ball and colleagues on multiplication of decimals, worth quoting in full:

‘The teacher had to know more than how to multiply decimals correctly herself.  She had to understand why the algorithm for multiplying decimals works and what might be confusing about it for students.  She had to understand multiplication as iterated addition and as area, and she had to know representations for multiplication.  She had to be familiar with base-ten blocks and to know how to use them to make such ideas more visible to her students. Place value and the meaning of the places in a number were at play here as well.  She needed to see the connections between multiplication of whole numbers and multiplication of decimals in ways that enabled her to help her students make this extension.  She also needed to recognize where the children’s knowledge of multiplication of whole numbers might interfere with or obscure important aspects of multiplication of decimals.  And she needed to clearly understand and articulate why the rule for placing the decimal point in the answer – that one counts the number of decimals places in the numbers being multiplied and counts over that number of places from the right – works.   In addition, she needed an understanding of linear and area measurement and of how they could be used to model multiplication.  She even needed to anticipate that a fourth-grade students might ask why one does not do this magic when adding or subtracting decimals and to have in mind what she might say.’ (Ball et al., 1991, p.448)

Since Shulman introduced PCK, the concept has been refined.  Krauss et al. (2008) distinguish three dimensions of PCK: knowledge of mathematical tasks as instructional tools, knowledge and interpretation of students’ thinking, and knowledge of multiple representations and explanations of mathematical problems.  Ball et al. (1991) include a component of subject knowledge, called ‘horizon knowledge’, that includes insight in curriculum structure and how concepts are gradually introduced over grades. Hill et al. (2008) use the terms ‘common content knowledge’ and ‘specialised content knowledge’.  The former relates to mathematical knowledge that numerically literate non-teachers are likely to know.  The latter refers to specific knowledge for teaching, such as what kind of mistakes are typically made at what age, or what representations create powerful learning.


Shulman’s conceptualisation of PCK (from Depaepe et al., 2013) and Ball’s refinement

Not everyone finds Shulman’s PCK concept helpful . Margaret Eisenhart (1993) dismisses the distinction between subject and pedagogical content knowledge as fuzzy and prefers to use procedural and conceptual knowledge as components of teacher knowledge.  Cochran et al. (1993) coined the term pedagogical content knowing (PCKg) to stress PCK as a dynamic knowing-to-act’ that is inherently linked to and situated in the act of teaching within a particular context.  Shulman’s concept was a theoretical construct, which proved difficult to confirm with empirical data.  Tim Rowland  (et al., 2005) used grounded theory to develop an empirically based classification, the ‘knowledge quartet’ that distinguishes between foundation, transformation, connection and contingency.

Foundation consists of the teacher’s theoretical knowledge and understanding of mathematics and beliefs about the nature of mathematics, including why and how it should be learned.  It’s called foundation because it determines the potential for the three other categories. These describe how foundation knowledge informs teaching decisions. Transformation refers to the capacity to transform it into powerful pedagogical forms, enabling others to learn.  Connection describes teachers’ ability to convey mathematics’ inherent coherence through well-chosen sequencing of topics, tasks and exercises within and between lessons.  Finally, contingency is the preparedness of the teacher to listen to student responses and readiness to suitably respond and even deviate from the set lesson agenda.

Foundation Awareness of purpose; identifying errors; overt subject knowledge; theoretical underpinning of pedagogy; use of terminology; use of textbook; reliance on procedures.
Transformation Choice of representations and explanations; choice of examples, teacher demonstrations
Connection Making connections between procedures; making connections between concepts; anticipation of complexity; decisions about sequencing; recognition about conceptual appropriateness.
Contingency Responding to children’s ideas; use of opportunities; deviation from agenda; swift and correct analysis of student errors and difficulties

Recent research efforts tend to focus more on the practice of teaching rather than teachers’ knowledge.  Strong content knowledge, or even strong PCK does not always translate into strong teaching, due to both teacher factors as environmental constraints.  In her case study of Ms. Daniels, Eisenhart (1993) splendidly describes the tensions between the focus on conceptual understanding in policy documents and the teacher training courses and elements at the personal level (limited conceptual understanding of students) and the school level (beliefs cooperating teachers, wide curriculum) that push teachers towards more procedural approaches.  Deborah Ball hits the nail on the head:

 ‘The pull toward neat, routinized instruction is very strong.  Teaching measurement by giving out formulas – l x w = some number of square units and l x w x h = some number of cubic units – may seem more efficient than hauling out containers, blocks and rulers and having students explore the different ways to answer questions of ‘how big’ or ‘how much’. With focused, bounded tasks, students get the right answers, and everyone can think they are successful.  The fact that these bounded tasks sometimes results in sixth graders who think that you measure water with rulers may, unfortunately go unnoticed’ (Ball et al., 1991).

In the next post I’ll discuss some of the approaches that have been used to measure teachers’ pedagogical content knowledge.  Comments and suggestions welcome!

Selected references:

Ball, D.L. (1990) ‘The mathematical understandings that prospective teachers bring to teacher education’, The elementary school journal, 90(4), pp. 449–466.

Ball, D.L., Lubienski, S.T. and Mewborn, D.S. (2001) ‘Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge’, 4th ed. In Richardson, V. (ed.), Handbook of research on teaching, Washington, DC, American Educational Research Association, pp. 433–456, [online] Available from: http://www-personal.umich.edu/~dball/chapters/BallLubienskiMewbornChapter.pdf (Accessed 12 September 2013).

Baumert, J., Kunter, M., Blum, W., Brunner, M., et al. (2010) ‘Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress’, American Educational Research Journal, (1), p. 133.

Cochran, K. F., DeRuiter, J. A., & King, R. A. (1993) ‘Pedagogical content knowing: an integrative model for teacher preparation’, Journal of Teacher Education, 44, pp.263-272.

Depaepe, F., Verschaffel, L. and Kelchtermans, G. (2013) ‘Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research’, Teaching and Teacher Education, 34, pp. 12–25.

Eisenhart, M., Borko, H., Underhill, R., Brown, C., et al. (1993) ‘Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding’, Journal for Research in Mathematics Education, 24(1), pp. 8–40.

Hill, H.C., Ball, D.L. and Schilling, S.G. (2008) ‘Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students’, Journal for Research in Mathematics Education, (4), p. 372.

Rowland, T., Huckstep, P. and Thwaites, A. (2005) ‘Elementary Teachers’ Mathematics Subject Knowledge: The Knowledge Quartet and the Case of Naomi’, Journal of Mathematics Teacher Education, 8(3), pp. 255–281.

Shulman, L. S. (1986) ‘Those who understand: Knowledge growth in teaching’, Educational Researcher, 15(2), pp.4- 31.


2 comments on “What knowledge do teachers need to teach? Pedagogical Content Knowledge for Mathematics

  1. […] a previous blog post I discussed the concept of pedagogical content knowledge for mathematics.  In this post I look how […]

  2. […] knowledge do mathematics teachers need in order to teach successfully?   In a first blog post I looked at the concept of pedagogical content knowledge of mathematics.  In the second I discussed […]

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