Key ideas
Core concepts
- Pedagogical content knowledge blends content understanding with pedagogical skill (Shulman, 1987) and is the feature that most distinguishes expert teachers from subject matter specialists. It includes knowledge of representations, student thinking, and common misconceptions.
- Mathematical knowledge for teaching differs from knowledge needed in other professions (Ball et al., 2008).
Pedagogical content knowledge is the specialised understanding that combines subject matter expertise with knowledge of how to teach that content effectively, including the most useful representations, common student difficulties, and effective instructional approaches for making concepts accessible to learners.
Connected to
Teacher Expertise | Experts and Novices Think Differently | Curse of Knowledge | Prior Knowledge | Misconceptions | Knowledge-Based Curriculum
The missing paradigm
Teacher education in the late 20th century emphasised generic pedagogical skills without adequate attention to content knowledge, creating a “missing paradigm” (Shulman, 1986). This shift was justified by process-product research on teacher effectiveness that simplified classroom teaching by ignoring subject matter.
California teacher exams from 1875 tested extensive subject matter knowledge across 20 domains. By 1985, teacher evaluation focused on generic processes (organisation, management, cultural awareness) with subject matter largely absent.
Research on teaching effectiveness treated teaching generically, with content of instruction treated as relatively unimportant. Studies of teacher cognition focused on planning and decision-making, neglecting how content knowledge was organised. Policy and practice separated pedagogy from content.
Shulman’s three categories of content knowledge
Shulman (1986) identified three categories essential for teaching. Subject matter content knowledge is the amount and organisation of knowledge in the teacher’s mind. This goes beyond knowing facts to understanding domain structures and intricacies; teachers require comprehension of why topics are important and how they connect.
Pedagogical content knowledge is understanding the subject matter for teaching purposes. It includes knowledge of what makes topics easy or difficult to learn, awareness of student conceptions and misconceptions, and understanding how to represent concepts for effective learning. Pedagogical content knowledge most distinguishes expert teachers from subject matter experts.
Curricular knowledge covers how to teach content at specific levels, familiarity with available instructional materials, and understanding when, why, and how to use different approaches. It includes lateral curricular knowledge (what is taught simultaneously in other subjects) and vertical curricular knowledge (what has been and will be taught in the same subject).
Mathematical knowledge for teaching
Ball, Thames, and Phelps (2008) refined Shulman’s framework for mathematics, identifying that mathematical knowledge for teaching differs qualitatively from mathematical knowledge needed by other professionals.
On the subject matter side, common content knowledge is mathematical knowledge used in many settings, knowledge any educated adult might use, such as the ability to calculate correctly. Specialised content knowledge is mathematical knowledge unique to teaching, not typically needed by adults in other professions: evaluating unusual solution methods, choosing and evaluating mathematical definitions, explaining why algorithms work, representing mathematical ideas accurately, and understanding multiple solution paths. Horizon content knowledge is awareness of how mathematical topics are related across the curriculum, understanding where mathematics is heading, and knowledge of advanced mathematics that informs current teaching.
On the pedagogical side, knowledge of content and students combines knowledge of students and mathematics: what students find interesting or challenging, common student conceptions and misconceptions, and student thinking patterns in mathematics. Knowledge of content and teaching combines knowledge of teaching and mathematics, including sequence of instruction, choosing examples, advantages and disadvantages of representations, and the likely impact of instructional choices. Knowledge of content and curriculum covers familiarity with the mathematics curriculum, understanding curriculum materials and their uses, and knowledge of standards and assessments.
Specialised content knowledge distinguishes teachers from other mathematically literate adults in everyday classroom work. When a student uses an incorrect but interesting approach, teachers must identify its mathematical validity, which requires seeing mathematics from the learner’s perspective. Multiple ways exist to represent multiplication, each with advantages and limitations, so teachers must know which representation fits which purpose. Teachers explain not just how to calculate but why procedures work, making mathematical structure explicit and connecting procedures to concepts. Questions like “Why can’t we divide by zero?” require a clear, mathematically accurate explanation that is accessible to students.
Teachers with stronger mathematical knowledge for teaching produce greater student gains (Ball et al., 2008). The effect is independent of general intelligence or other teacher qualities, and specialised content knowledge matters more than advanced coursework alone.
Pedagogical reasoning and action
Shulman (1987) proposed a model showing how teachers transform content knowledge into instruction. It begins with comprehension: understanding purposes, subject matter structures, and ideas within and outside the discipline. Transformation follows, through preparation (critical interpretation and analysis of texts), representation (use of analogies, metaphors, examples), selection (choosing from the instructional repertoire), and adaptation (tailoring to student characteristics). Instruction is the observable performance of teaching acts: management, presentations, interactions, questioning. Evaluation involves checking for student understanding, testing student learning, and evaluating one’s own performance. Reflection means reviewing, reconstructing, re-enacting, and critically analysing performance, grounding explanations in evidence. The cycle ends in new comprehension: enhanced understanding of purposes, subject matter, students, and teaching.
Comprehension leads to action, which leads to new comprehension. Teaching is not simply applying existing knowledge but developing new understanding through practice.
Forms of teacher knowledge
Shulman (1986) identified three forms in which knowledge exists for teachers. Propositional knowledge is evidence-based and includes principles (theoretical claims from empirical research), maxims (practical claims from experience), and norms (ideological or philosophical commitments). Case knowledge consists of specific documented events: prototypes that exemplify theoretical principles, precedents that capture principles of practice, and parables that convey norms or values. Strategic knowledge is knowing what to do when principles conflict; it requires metacognitive awareness for professional judgment and is applied when no simple solution exists.
All three forms are necessary for effective teaching. Propositional knowledge without case knowledge remains abstract. Case knowledge without principles lacks generalisation. Both without strategic knowledge cannot handle complexity.
Consequences for teacher development
Teacher education curricula must address all three categories of content knowledge (Shulman, 1986). Training programmes should include all three forms of knowledge, and teacher assessment must reference the content being taught, not just generic teaching behaviours.
“Mere content knowledge is likely to be as useless pedagogically as content-free skill.” The ultimate test of understanding is the ability to transform knowledge into teaching. Subject matter experts without pedagogical knowledge cannot teach effectively: deep subject knowledge is necessary but insufficient, and knowing how to solve problems differs from knowing how to teach problem-solving.
In practice, teachers should know common student difficulties and misconceptions for each topic taught, understand why content is easy or difficult for students at different stages, and develop a repertoire of representations for key concepts rather than relying on a single explanation. Reform efforts must address what teachers know and how they think, balancing subject matter knowledge with pedagogical training.
References
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-23.