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In this commentary, I focus on the notion of competence and issues related to the distinction between knowing how mathematical problems are solved versus knowing how to teach mathematics. Although definitions of competence may necessarily be affected by value judgements and thus less amenable to factual answers, providing a defensible definition is important because it affects eligibility for intervention and treatment. One way to tackle this issue is to focus on the identification of prerequisite skills and concepts needed for particular domains of mathematics. Recent work on fraction and algebra has shown that long held assumptions may need to be re-examined. On knowledge versus application, some cautionary notes are made on the importance of not losing sight of translating our knowledge of processes involved in mathematical problem solving into better pedagogical practices. [Commentary on: Alcock, L., Ansari, D., Batchelor, S., Bisson, M.-J., De Smedt, B., Gilmore, C., . . . Weber, K. (2016). Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41. doi:10.5964/jnc.v2i1.10]

Laying out a research agenda is seldom an easy task; the more so when there are multiple investigators and when one is aiming at developing an agenda for the whole field. I applaud the authors for putting together a thoughtful and useful set of questions. My comments are focused largely on the notion of competence and the distinction between knowing how mathematical problems are solved versus knowing how to teach mathematics.

In the section on mapping predictors and processes of competence development (

The issue of criterion- versus norm- referenced definition is pertinent to Questions 5 to 9. Questions 6, for example, asks “What are the key mathematical concepts and skills that children should have in place prior to the start of compulsory education?” Ostensibly, this question is concerned with a criterion-based assessment of competence. It may have attracted a criterion approach because of the sense that early mathematical concepts, developed prior to formal schooling, are of such a fundamental nature that children either develop these skills to a satisfactory level or they do not. Questions 7 to 9, on relations between early numeracy and other aspects of mathematics, seem to focus more on a norm-referenced view of competence and are focused on identifying factors that contribute to individual differences in competence.

Given the hierarchical nature of mathematics, the question of the kind of mathematical concepts and skills that should be in place prior to the introduction of higher order concepts can and should be posed for each domain of mathematics in the K-12 curriculum. In evaluating this suggestion, it should be noted that notions of hierarchy in mathematical concepts is not uncontroversial. With respect to fraction knowledge, there are well established difficulties associated with the over-generalisation of whole number concepts to fractions (

A similar debate concerns algebra. Many curricula are arranged based on the belief that a foundation in arithmetic is required before the introduction of algebra. However, there is now a growing literature that shows that even children in primary school exhibit algebraic thoughts (

These observations suggest that amongst typically developing children at least, the relation between early acquired and later acquired, more complex mathematical skills may be weaker and less hierarchical than previously thought. However, whether strict hierarchy or necessary prerequisites exist for certain domains of mathematics seem to be questions that can be answered empirically and thus are not entirely dependent on value judgements. In contrast, questions regarding timing of acquisition of specific skills, though possibly related to periods of sensitivity (as found in the acquisition of language), seem more dependent on societal expectations regarding the kind of problems children of particular ages can be expected to solve.

The question of necessary skills and concepts is of particular importance for children with developmental difficulties. At present, diagnoses are often made on the bases of mathematical performance that is poorer than would be expected from the child’s general intellectual abilities. However, even here, judgement of competence will likely be influenced by ideological beliefs regarding curricular orientation. As

With reference to atypical development,

To help organise my thoughts regarding the overall structure of the questions, I took the liberty of putting them on a concept map (see

Domain areas and questions identified in “Challenges in mathematical cognition: A collaboratively-derived research agenda”. Dotted lines denote potential interconnections between questions and domain areas.

With respect to promoting both procedural fluency and conceptual understanding,

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