# Pedagogical Content Knowledge Within “Mathematical Knowledge for Teaching”

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_123-5

## Definition

**Pedagogic content knowledge**, in Shulman’s (1986, p. 7) terms, refers to: “the most powerful analogies, illustrations, examples, explanations, and demonstrations — […] the most useful ways of representing and formulating the subject that make it comprehensible to others.”

## Characteristics

Intense focus on the notion of “pedagogical content knowledge” (PCK) within teacher education is attributed to Lee Shulman’s 1985 AERA Presidential address (Shulman 1986) in which he referred to PCK as the “special amalgam of content and pedagogy” central to the teaching of subject matter. His widely cited follow-up paper (Shulman 1987) elaborated PCK as follows:

the most powerful analogies, illustrations, examples, explanations, and demonstrations — […] the most useful ways of representing and formulating the subject that make it comprehensible to others…. Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the...

## Keywords

Pedagogic content knowledge Mathematical knowledge for teaching Lee Shulman Deborah Ball COACTIV## References

- Askew M (2018) Mediating primary mathematics: measuring the extent of teaching for connections and generality in the context of whole number arithmetic. ZDM. https://doi.org/10.1007/s11858-018-1010-9CrossRefGoogle Scholar
- Askew M, Brown M, Rhodes V, Johnson DC, Wiliam D (1997) Effective teachers of numeracy. Report of a study carried out for the Teacher Training Agency 1995–96 by the School of Education, King’s College London. Teacher Training Agency, LondonGoogle Scholar
- Ball D, Hill H, Bass H (2005) Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide? Am Educ, Fall 2005Google Scholar
- Ball D, Thames MH, Phelps G (2008) Content knowledge for teaching: What makes it special? J Teach Educ 59(5):389–407CrossRefGoogle Scholar
- Baumert J, Kunter M, Blum W, Brunner M, Voss T, Jordan A, … Tsai Y (2010) Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. Am Educ Res J 47(1):133–180CrossRefGoogle Scholar
- Blömeke S, Gustafsson J-E, Shavelson R (2015) Beyond dichotomies – competence viewed as a continuum. Z Psychol 223(1):3–13Google Scholar
- Clark-Wilson A, Robutti O, Sinclair N (eds) (2014) The mathematics teacher in the digital era. An international perspective on technology focused professional development. Springer, DordrechtGoogle Scholar
- Fennema E, Franke ML (1992) Teachers’ knowledge and its impact. In: Grouws DA (ed) Handbook of research on mathematics teaching and learning. Macmillan, New York, pp 147–164Google Scholar
- Hill H, Rowan B, Ball D (2005) Effects of teachers’ mathematical knowledge for teaching on student achievement. Am Educ Res J 42(2):371–406CrossRefGoogle Scholar
- Kinach BM (2002) A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: toward a model of effective practice. Teach Teach Educ 18:51–71CrossRefGoogle Scholar
- Lo ML, Pong WY (2005) Catering for individual differences: building on variation. In: Lo ML, Pong WY, Pakey CPM (eds) For each and everyone: catering for individual differences through learning studies. Hong Kong University Press, Hong Kong, pp 9–26Google Scholar
- Ma L (1999) Knowing and teaching elementary mathematics: teachers’ understandings of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates, MahwahGoogle Scholar
- Petrou M, Goulding M (2011) Conceptualising teachers’ mathematical knowledge in teaching. In: Rowland T, Ruthven K (eds) Mathematical knowledge in teaching. Springer, Dordrecht, pp 9–25CrossRefGoogle Scholar
- Rowland T, Huckstep P, Thwaites A (2003) The knowledge quartet. Paper presented at the British Society for research into learning mathematicsGoogle Scholar
- Shulman LS (1986) Those who understand: knowledge growth in teaching. Educ Res 15(2):4–14CrossRefGoogle Scholar
- Shulman LS (1987) Knowledge and teaching: foundations of the new reform. Harv Educ Rev 57(1):1–22CrossRefGoogle Scholar
- Silverman J, Thompson PW (2008) Toward a framework for the development of mathematical knowledge for teaching. J Math Teach Educ 11:499–511CrossRefGoogle Scholar
- Turner F, Rowland T (2011) The knowledge quartet as an organising framework for developing and deepening teachers’ mathematical knowledge. In: Rowland T, Ruthven K (eds) Mathematical knowledge in teaching. Springer, Dordrecht, pp 195–212CrossRefGoogle Scholar