Jeremy Kilpatrick and colleagues came up with a list of aspects of understanding mathematics

• Conceptual understanding – comprehension and mathematical concepts, operations, and relations;
• Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately;
• Strategic competence – ability to formulate, represent, and solve mathematical problems;
• Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification;
• Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (Kilpatrick et al., 2001, p116)

Edwina Michener’s approach was to delineate various kinds of objects which contribute to understanding” [see also Michener’s Types of examples]

• Examples; results; results (theorems, facts); examples (illustrative); and concepts (including formal and informal definitions and heuristic advice) (Michener, 1978, p362)

Susan Pirie & Tom Kieren sketched out  details of how understanding develops in their onion model which includes the phenomenon of learners sometimes appearing to go backwards in their understanding. They called this folding back as if to re-group and consolidate before making a further advance.



Anne Watson (2002) summarised several views on understanding by addressing the somewhat curious phrase ‘teaching for understanding’.

Probes & Prompts

Taking Action

Case Studies

Research Sources

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