## Key Ideas > [!abstract] Core Concepts > > - Pedagogical content knowledge blends content understanding with pedagogical skill (Shulman, 1987) > - The most distinguishing feature separating expert teachers from subject matter specialists > - Includes knowledge of representations, student thinking, and common misconceptions > - Mathematical knowledge for teaching differs from knowledge needed in other professions (Ball et al., 2008) ## Definition 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. ## Three categories of content knowledge Shulman (1986) identified three categories essential for teaching: **Subject matter content knowledge**: 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**: Understanding the subject matter for teaching purposes. This 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**: Knowledge of how to teach content at specific levels, familiarity with available instructional materials, and understanding when, why, and how to use different approaches. This 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. **Subject matter knowledge** includes: **Common content knowledge**: Mathematical knowledge used in many settings, knowledge any educated adult might use. Example: ability to calculate correctly. **Specialised content knowledge**: Mathematical knowledge unique to teaching, not typically needed by adults in other professions. Examples include 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**: Awareness of how mathematical topics are related across curriculum, understanding where mathematics is heading, knowledge of advanced mathematics that informs current teaching. **Pedagogical content knowledge** includes: **Knowledge of content and students**: Combining knowledge of students and mathematics. Knowing what students find interesting or challenging, common student conceptions and misconceptions, student thinking patterns in mathematics. **Knowledge of content and teaching**: Combining knowledge of teaching and mathematics. This includes sequence of instruction, choosing examples, advantages and disadvantages of representations, and likely impact of instructional choices. **Knowledge of content and curriculum**: Familiarity with mathematics curriculum, understanding curriculum materials and their uses, knowledge of standards and assessments. ## Specialised content knowledge examples Specialised content knowledge distinguishes teachers from other mathematically literate adults: **Evaluating non-standard methods**: When a student uses incorrect but interesting approach, teachers must identify mathematical validity. This requires seeing mathematics from learner's perspective. **Choosing representations**: Multiple ways exist to represent multiplication. Each has advantages and limitations. Teachers must know which representation fits which purpose. **Explaining why procedures work**: Not just how to calculate, but why it works. Making mathematical structure explicit. Connecting procedures to concepts. **Responding to student questions**: Questions like "Why can't we divide by zero?" require clear, mathematically accurate explanation. Must be accessible to students. Research shows teachers with stronger mathematical knowledge for teaching produce greater student gains (Ball et al., 2008). Effect is independent of general intelligence or other teacher qualities. Specialised content knowledge matters more than advanced coursework alone. ## Model of pedagogical reasoning and action Shulman (1987) proposed a model showing how teachers transform content knowledge into instruction: **Comprehension**: Understanding purposes, subject matter structures, ideas within and outside discipline. **Transformation**: This includes preparation (critical interpretation and analysis of texts), representation (use of analogies, metaphors, examples), selection (choosing from instructional repertoire), and adaptation (tailoring to student characteristics). **Instruction**: Observable performance of teaching acts, management, presentations, interactions, questioning. **Evaluation**: Checking for student understanding, testing student learning, evaluating own performance. **Reflection**: Reviewing, reconstructing, re-enacting, critical analysis of performance, grounding explanations in evidence. **New comprehension**: Enhanced understanding of purposes, subject matter, students, teaching. This cyclical process shows that comprehension leads to action which leads to new comprehension. Teaching is not simply applying existing knowledge but developing new understanding through practice. ## Three forms of teacher knowledge Shulman (1986) identified three forms in which knowledge exists for teachers: **Propositional knowledge** (evidence-based): This includes principles (theoretical claims from empirical research), maxims (practical claims from experience), and norms (ideological or philosophical commitments). **Case knowledge** (specific documented events): This includes prototypes (exemplify theoretical principles), precedents (capture principles of practice), and parables (convey norms or values). **Strategic knowledge**: Knowing what to do when principles conflict. Requires metacognitive awareness for professional judgment. Applied when no simple solution exists. All three forms prove necessary for effective teaching. Propositional knowledge without case knowledge remains abstract. Case knowledge without principles lacks generalisation. Both without strategic knowledge cannot handle complexity. ## Development and assessment Teacher education curricula must address all three categories of content knowledge (Shulman, 1986). Training programmes should include all three forms of knowledge. 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. Reform efforts must address what teachers know and how they think. Balance is needed between subject matter knowledge and pedagogical training. > [!tip] Implications for teaching > > - **Deep subject knowledge is necessary but insufficient**: teachers need pedagogical content knowledge specific to teaching > - **Know common student difficulties** and misconceptions for each topic taught > - **Develop repertoire of representations** for key concepts, not relying on single explanation > - **Understand why content is easy or difficult** for students at different stages > - **Specialised knowledge differs from general knowledge**: knowing how to solve problems differs from knowing how to teach problem-solving > - **Teacher preparation must combine** subject matter expertise with pedagogical training > - **Assessment of teaching must consider content** being taught, not just generic behaviours > - **Multiple representations of concepts** are necessary for effective instruction ## 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.