@SmartmanApps
Skemp was writing about England in the 1970s.

However, even today, are all learners in all the world’s schools taught the reasons for all the rules, and use constructivist learning?

I’m not sure in fact that instrumental cf. relational understanding are direct analogies of rote cf. constructivist learning.

Perhaps a relevant analogy is atomised cf. connectionist learning, e.g. ‘Alternatives to atomisation’ by Colin Foster (2025), Mathematics Teaching 295 https://www.foster77.co.uk/Foster,%20Mathematics%20Teaching,%20Alternatives%20to%20atomisation.pdf and Teaching to Big Ideas https://www.youcubed.org/resource/teaching-to-big-ideas/

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@SmartmanApps
It's worth read Skemp's articles as he describes the advantages of Instrumental Understanding as well as Relational Understanding.

Here's a link to the version published in The Arithmetic Teacher, 1978 https://teamone.msuurbanstem.org/wp-content/uploads/2014/07/Skemp-Relational-Instrumental-clean-copy-AT-1978.pdf

This work has been built on by others, including Jo Boaler.

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@leon_p_smith
‘Mindstorms’ is full of quotable text, here’s just one:

“Imagine that children were forced to spend an hour a day drawing dance steps on squared paper and had to pass tests in these ‘dance facts’ before they were allowed to dance physically. Would we not expect the world to be full of ‘dancophobes’?Would we say that those who made it to the dance floor and music had the greatest ‘aptitude for dance’? In my view, it is no more appropriate to draw conclusions about mathematical aptitude from children’s unwillingness to spend many hundreds of hours doing sums.” — Seymour Papert (p. 43)

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@leon_p_smith
Yes, I think programming can be a good way to learn mathematics (and more).

Seymour Papert wrote about this in Mindstorms (1980).

I was fortunate to learn Logo and turtle programming on a modest home computer in my early teens. I don't know which was cause and which was effect, but I still like geometry and programming today.

For those not familiar with Mindstorms, a good summary is at
https://medium.com/bits-and-behavior/mindstorms-what-did-papert-argue-and-what-does-it-mean-for-learning-and-education-c8324b58aca4

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The first two episodes of MT Talk discuss articles from Mathematics Teaching 296 https://atm.org.uk/MT-Talk

Episode 1
Elizabeth Bridgett, Lisa Coe and Jay Timotheus compare and contrast Mike Askew's 1999 article, ‘Teaching numeracy: will we ever learn?’ from MT168 with Jane Hawkins' much more recent, 2025 article, ‘Why purposeful talk in mathematics lessons matters’ in MT296.

Also mentioned in this episode is the 1999 paper ‘Arbitrary and Necessary Part 1: A Way of Viewing the Mathematics Curriculum’, Dave Hewitt's first in a three-part series in For the Learning of Mathematics, as well as Mike Askew’s book, ‘Transforming Primary Mathematics’, published by Routledge in 2011.

Episode 2
Lisa Coe, Fin McLaughlin and Jay Timotheus talk about John Mason's 1987 article, ‘Only awareness is educable’ from MT120, alongside Tom Francome's 2025 article, ‘Everyone can think mathematically’ from MT296.

Fin recommended two books: ‘Thinking Mathematically’, by John Mason, Leone Burton and Kaye Stacey (Pearson, 1982, and 2010) and ‘Researching Your Own Practice: The Discipline of Noticing’, by John Mason (Routledge, 2001).

https://podcasts.apple.com/us/podcast/mt-talk/id1838379708
https://open.spotify.com/show/3cJF2EXKyIe1FTNrRO8xm5

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