Some suggestions for curious and interesting ‘Numbers’ based on personal knowledge (or ignorance). What would you add?

Sum of three palindromes (Javier Cilleruelo, Florian Luca and Lewis Baxter; try it at https://somethingorotherwhatever.com/sum-of-3-palindromes h/t @christianp)

Proof of Fermat's Last Theorem (Andrew Wiles)

Twin primes conjecture is still open, but there are infinitely many pairs of primes that differ by 246 (Yitang Zhang, James Maynard and Terence Tao @tao)

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Some suggestions for curious and interesting ‘Geometry’, more in my comfort zone but still plenty of ignorance 😊

What would you add?

Aperiodic monotile (Smith, Myers, Kaplan & Goodman-Strauss)

Noperthedron (Steininger & Yurkevich)

Progress in various packing problems (https://erich-friedman.github.io/packing/)

15 types of convex pentagonal tilings (Rao and others)

Solving cubics with origami (Beloch & Lill; see http://origametry.net/papers/amer.math.monthly.118.04.307-hull.pdf)

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Now for curious and interesting ‘Puzzles’.

What would you add?

Sudoku

Thinking / puzzle video games like Sokoban

Hmm, I like puzzles but my knowledge is a bit thin (and we no longer have a galvanising figure like Martin Gardner.)

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Now suggestions for curious and interesting ‘Mathematics’. This means strange facts, anecdotes, paradoxes, portraits of eccentric mathematicians, mathematical philosophy, quotes and mathematics education.

“The only way to learn mathematics is to do mathematics.” – Paul R. Halmos

“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.” – William Paul Thurston

“The most exciting phrase to hear in science, the one that heralds new discoveries, is not ‘Eureka!’ but ‘That’s funny…’” — Isaac Asimov

“I used to think that maths teachers were all teaching the same subject, some doing it better than others. I now believe that there are *two effectively different subjects being taught under the same name, ‘mathematics’.*’

...‘relational understanding’ [means]… knowing both what to do and why. Instrumental understanding I would until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’, without realising that for many pupils *and their teachers* the possession of such a rule, and ability to use it, was what they meant by ‘understanding’.” Richard Skemp (https://atm.org.uk/write/MediaUploads/Journals/MT077/Relational_Understanding_and_Instrumental_Understanding_–_Richard_R._Skemp.pdf)

#mathematics #MathEd #MathsEd #iTeachMath #iTeachMaths #education #AskFedi #books #quote #quotes

@foldworks
"rules without reasons" - him not knowing the reasons doesn't mean there aren't any...

borrowing - is easy to prove why we have that rule

Invert and multiply - ditto

Change sides and sign - trivially easy to demonstrate the reason

He then makes a wrong assumption that teachers don't cover anything that isn't in the textbook - which is only an aid for the teacher - the SYLLABUS (not the textbook) says what is actually taught, and includes REASONING, so just a big strawman then 🙄

@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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@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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