David G. Wells’ “The Penguin dictionary of curious and interesting…” books are excellent and have been great source of inspiration for me.
They are now are almost 30 years old.
What new entries would you put in these books—either because they’re new after publication, or could have (should have) been included at the time?
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)
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)
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.)
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 I've been thinking about the early childhood math curriculum since, well, I was a early child myself.
My idea is to introduce the Stern-Brocot tree, the Symmetry Group of the Square, Pascal's Triangle, and computer programming to youngsters.
These are all incredibly fertile concepts that set up future lessons about mathematics, in lessons that are likely appropriate for children, others that are more likely to be appropriate for teens, and even others that are more likely to be appropriate for undergraduate or even graduate students.
I am very keen on finding concepts that meaningfully connect to advanced research-level mathematics, but are also approachable by the youngest children.
https://github.com/constructive-symmetry/constructive-symmetry
@foldworks Incidentally, since you are interested in tilings, there are a couple of things you might find particularly interesting here:
1. The Stern-Brocot Tree is isomorphic to SL(2,N), the 2x2 matrices with nonnegative integer entries that are of determinant 1.
2. The Symmetry Group of the Square D4 is isomorphic to the 2x2 matrices with a 0 and a ±1 in each row and column. These eight matrices have determinant ±1. Also, I feel this is an ideal setting to introduce and explore matrix multiplication with children.
3. The general modular group GL(2,Z) is those 2x2 integer matrices of determinant ±1. It turns out every element of GL(2,Z) can be written in exactly 4 ways as an an element in D4 times an element in SL(2,N) time an element in SL(2,N), with the exception that the elements of D4 can be written in 8 different ways. This explains the connection between continued fractions and the various flavors of the modular group.
4. GL(2,Z) is isomorphic to the automorphisms of the additive group ZxZ.
5. Of course, two things naturally associated with GL(2,Z) is the modular tiling, and the moduli space of acute triangles.
foldworks
Leon P Smith