Pedagogical Content Knowledge for Mathematics
Core content
Candidates should understand how mathematical ideas develop across grade levels and how to represent them with diagrams, symbols, tables, graphs, and manipulatives. Know common misconceptions, including whole-number bias in fractions, additive reasoning where multiplicative reasoning is needed, and overgeneralising rules. Plan tasks with appropriate cognitive demand, use questioning to elicit reasoning, and orchestrate classroom discourse. Formative assessment, error analysis, scaffolding, differentiation, and alignment between learning objectives and assessment tasks are essential.
Trap watch
Hands-on activity is not automatically minds-on learning. A real-world context can be pseudocontext if the mathematics is forced or unrealistic. Answer-getting tasks may build fluency while leaving conceptual understanding untouched. Showing only one method can obscure connections, and premature rule-telling can replace sense-making. Ignoring prerequisite knowledge creates fragile procedures. Assessment must match the intended objective: a multiple-choice item may not reveal reasoning, and a rubric must not reward neatness over mathematical validity.
Worked example
A student writes 1/2 + 1/3 = 2/5, adding numerators and denominators. Diagnose this as a whole-number and additive misconception. Use an area model: shade one-half and one-third of the same whole, then subdivide into sixths to show 3/6 + 2/6 = 5/6. Ask the student to compare 2/5 with one-half to prompt conflict and rethinking.
Rule of thumb: elicit the student's reasoning, interpret the misconception, then respond with a representation or question that creates cognitive conflict.