I have been thinking about the nature of expertise over the last few months. What is it? Who has it? Do I have any and if so, in what?

Earlier today, I became aware of some comments made about me on Twitter by Keith Devlin, a British Mathematician now resident in the U.S. Devlin had remarked on the recent Jo Boaler affair and I reacted, prompting Devlin to block me:

The interesting part is where Devlin states that I have, “…no experience of mathematical research or of using math in real world settings.” I agree with the first part — I have never been a maths researcher.

This reminded me of an objection Marty made over on the Bad Mathematics blog. Marty is more sympathetic toward me than Devlin — we shared a common cause in opposing a draft Australian maths curriculum, even if we had different emphases. However, Marty correctly objected to newspaper reports about this campaign referring to me as a ‘maths expert’. I am not a mathematician, I have never conducted maths research and so I cannot reasonably be described as a maths expert.

So what has this got to do with anything? Well, Keith Devlin has a blog in which he opines on maths education. When I read it, I see extraordinary claims such as:

*“The primary focus of mathematics education today, as determined by the mathematical requirements of today’s world, is to prepare the next generation of citizens with the mathematical thinking skills (as opposed to the procedural execution skills of former times) they will need to lead productive and rewarding lives as members of society.”*

From my perspective, such claims are transparently wrong. Abstract, generic mathematical thinking skills simply do not exist — if you think they do then try precisely describing one. If they did exist, we could teach them and solve the knotty problem of transfer of learning. If Devlin’s use of this term is instead intended to refer to conceptual understanding, then he’s still wrong because, in mathematics, conceptual understanding develops alongside procedural knowledge. You cannot ditch the procedural.

And yet, unlike me and by an reasonable definition, Devlin is a genuine maths expert.

What does this mean for the nature of expertise? First, a little personal news.

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