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Alcock et al (2016, this issue) have set out and discussed a potential research agenda for mathematical cognition. It is timely that research topics, along with knowledge uncovered to date, should be incorporated into a coordinated agenda for further research. This commentary focuses on the perspectives that learning difficulties, and dyscalculia, reveal. These perspectives potentially add much to that research agenda. [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]

The time for generating a coherent agenda for research on mathematical cognition is long overdue. Problems with learning maths are international and often subject to deeply entrenched beliefs, for example, that all children can rote learn all times table facts and cultures, for example, that mental arithmetic should be executed rapidly, that handicap the learning of maths for dyscalculic students and many others much higher on the learning spectrum.

As this problem is worldwide, it is surprising to find that there are no contributors to this Challenge from, for example, India, Asia, Israel, Australia or New Zealand. However, I am sure that expertise in dyscalculia and maths learning difficulties will be comprehensively represented among the researchers. I firmly believe there is much to be learned from the ‘outliers’, the students who perform at the extremes of the normal distribution (echoing here

I shall base my commentary and observations on the research and wisdom of the 63 authors who, between them, wrote 31 chapters for ‘The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties’. I compiled and edited the Handbook which was published in 2015.

For the Introduction in the Handbook I identified and discussed 16 key questions pertinent to learning difficulties in maths, relating each question to the contributions from the Handbook authors. The learning difficulties spectrum can provide new perspectives on old problems. For example, two of the research questions from the 26 listed by

There are many observations on early learning in the Handbook, ranging from

Amongst the observations about early learning experiences from the Handbook, I hope that those involved in early education take note of the gentle advice on using story telling (

At the base of early learning in terms of developmental trajectories is an understanding of place value (

Section E of the

A key influence in my early days in the field was

Finally, I often revisit

The ability for logical thought in the sphere of quantitative and spatial relationships, number and letter symbols; the ability to think in mathematical symbols.

The ability for rapid and broad generalisation of mathematical objects, relations and operations.

Flexibility of mental processes in mathematical activity.

Striving for clarity, simplicity, economy and rationality of solutions.

The ability for rapid and free reconstruction of the direction of a mental process, switching from a direct to a reverse train of thought.

Mathematical memory (generalised memory for mathematical relationships), and for methods of problem solving and principles of approach.

These components are closely interrelated, influencing one another and forming in their aggregate a single integral syndrome of mathematical giftedness. (pp. 87-88)

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