Ian Stewart’s (2005) book Game, Set and Math includes 12 puzzles linked to the 12 days of Christmas. A favourite is Five Gold Rings. The five gold rings are arranged to maximise the overlaps. At every overlap there is a diamond. You have diamonds of 1, 2, 3, 4, 5, 6, ... carats. Arrange the diamonds so the total number of carats on each ring is the same.

Try starting small—just two rings, two overlaps and each ring has a total of three carats (1 + 2). Now three rings, there are six overlaps, arrange diamonds of 1, 2, 3, 4, 5 and 6 carats so each ring has the same total.

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Seasonal Values
Each of the symbols in the grid has a value. From the totals given for each row and each column can you work out what they are?

The eight symbols have values 1 to 8 and each has the same value wherever it appears.

An excerpt from the freely-available https://atm.org.uk/write/MediaUploads/Journals/MT298/08.pdf

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The Association for Mathematics in Education launched on 1 December 2025 https://amie.org.uk/

These articles are also available to non-members

AMET and AMiE https://atm.org.uk/write/MediaUploads/Journals/MT298/11.pdf
Sally Bamber and Ashley Compton, with current AMET committee members reflect on AMET’s legacy and potential contribution to AMiE.

ATM and AMiE https://atm.org.uk/write/MediaUploads/Journals/MT298/12.pdf
Helen Madeley on behalf of General Council reflects on ATM’s legacy and the potential contribution to AMiE.

From the MA to AMiE: celebrating a legacy, forging a stronger future https://atm.org.uk/write/MediaUploads/Journals/MT298/13.pdf
Andy Kemp and current trustees of the MA reflect on the MA’s legacy and potential contribution to AMiE.

NAMA and AMiE https://atm.org.uk/write/MediaUploads/Journals/MT298/14.pdf
Ruth Trundley and the current executive committee of NAMA reflect on NAMA’s legacy and potential contribution to AMiE. Ruth Trundley

NANAMIC and AMiE https://atm.org.uk/write/MediaUploads/Journals/MT298/15.pdf
John Barton and current trustees of NANAMIC reflect on NANAMIC’s legacy and potential contribution to AMiE.

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@SmartmanApps
Thanks for you insights.

I think the threading on my server means it's easier to view the whole thread directly at https://dotnet.social/@SmartmanApps/115524576264052792

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@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
A puzzle from Seymour Papert's Mindstorms:

"A monkey and a rock are attached to opposite ends of a rope that is hung over a pulley. The monkey and the rock are of equal weight and balance one another. The monkey begins to climb the rope. What happens to the rock?"

What is your reasoning and what were your assumptions? If you know the answer, what was your first answer?

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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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