Mastodawn

#mathematics #number #askfedi

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)

The incredible palindromic hat-trick

#mathematics #geometry #askfedi

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)

Erich's Packing Center

#mathematics #puzzle #puzzles #askfedi

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.)

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

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)

https://github.com/constructive-symmetry/constructive-symmetry

@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.

@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.

#mathematics #programming #Logo #TurtleGraphics #ITeachMath #MathematicsEducation #MathEd #MathsEd #SeymourPapert #Mindstorms

@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

Mindstorms: what did Papert argue and what does it mean for learning and education?

I turned 37 years old today. People like to point out that I don’t really look my age, and so I leverage this to act younger and feel…

Medium

#mathematics #ITeachMath #MathematicsEducation #MathEd #MathsEd #SeymourPapert #Mindstorms

@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)

#puzzle #MathEd #MathsEd #iTeachMath #physics #mechanics

@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?

The rock goes up.

64.7%

The rock goes stays at the same position.

29.4%

The rock goes down.

5.9%

Poll ended at Nov 10 at 2:30pm.

@leon_p_smith
I thought the rock would go up but I couldn't explain why in a convincing way 🤭

@foldworks I think there's an ambiguity here: which rope is the monkey climbing? If he climbs the rock's rope, the rock obviously goes down, which isn't very interesting.

If the monkey climbs his own rope, yeah now you have to consider the inertia and dynamics of the system. And I suspect that the friction between the pulley and its bearing is a key factor to consider.

My physical intuition is that on a frictionless pulley, the rock will momentarily move upward when the monkey first starts climbing, but then starts falling due to the monkey's upward inertia. But I'm not terribly confident about it.

@leon_p_smith
A diagram for the question (with the rock swapped for a bunch of bananas)

foldworks Nov 8

@leon_p_smith
No votes for down, so is there a good argument for why the rock doesn't go downwards?

Developing Stacy

Nov 8

@foldworks @leon_p_smith

The way I see it the relative heights of the monkey and rock don't matter as regardless of their position, they apply the same force. If the monkey climbs a foot of its rope, it is applying tension on the rope and with a frictionless bearing the rock and monkey will both rise half a foot as they have the same inertia so the force will affect each equally.

foldworks Nov 8

@The4thCircle @leon_p_smith
How could you persuade someone else that the fulcrum/beam model isn't relevant/suitable?

Developing Stacy

Nov 8

@foldworks @leon_p_smith

I'm not sure. I guess its to do with the angles of the force relevant to the angle of the beam, with the upper load pulling toward the fulcrum whereas the lower load pulls away and therefore has less angular momentum force on the beam itself.

Whereas the pulley model they both pull straight down at all times.

foldworks Nov 10

@leon_p_smith
The consensus is that the rock goes up. How can you convince others that would happen?

foldworks Nov 10

"I have presented this problem to several hundred MIT students, all of whom had successfully passed rigorous and comprehensive introductory physics courses. Over three quarters of those who had not seen the problem before gave incorrect answers or were unable to decide how to go about solving it. Some thought the position of the rock would not be affected by the monkey's climbing because the monkey's mass is the same whether he is climbing or not; some thought that the rock would descend either because of a conservation of energy or because of an analogy with levers; some guessed it would go up, but did not know why. The problem is clearly "hard." But this does not mean that it is "complex." I suggest that its difficulty is explicable by the lack of something quite simple. When they approach the problem, students ask themselves:"Is this a 'conservation-of-energy' problem?" "Is this a 'lever-arm' problem?" and so on. They do not ask themselves:"Is this a 'law-of-motion' problem?" They do not think in terms of such a category. In the mental worlds of most students, the concepts of conservation, energy, lever-arm, and so on, have become tools to think with. They are powerful ideas that organize thinking and problem solving."

foldworks Nov 10

For a student who has had experience in a "laws-of-motion" microworld this is true of "law of motion." Thus this student will not be blocked from asking the right question about the monkey problem. It is a law-of-motion problem, but a student who sees laws of motion only in terms of algebraic formulas will not even ask the question. For those who pose the question, the answer comes easily. And once one thinks of the monkey and the rock as linked objects, similar to the ones we worked with in the Turtle microworld, it is obvious that they must both undergo the same changes in state. Since they start with the same velocity, namely zero, they must therefore always have the same velocity. Thus, if one goes up, the other goes up at the same speed.

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Nov 8

@foldworks @leon_p_smith
"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” - and yet, he drew this conclusion without asking any Maths teachers/tutors about what the root issues are. In my experience, students who don't like Maths are poor at mental arithmetic, and once you get them up to speed they don't hate Maths anymore, and become confident with it. It's that simple.

@foldworks I had a commodore 64 at home, but never had a Logo for it that I can recall.

I did have some minimal exposure to Logo on the Apple IIe at school, but we never did anything with it that was more programming oriented, and I didn't have enough time with the software to really get into it very deeply.

BASIC, then Pascal, then SML and Haskell, then Scheme, was the progression of programming languages I spent significant time with.

I definitely was the most attracted to abstract algebra, number theory, combinatorics, discrete math, and logic as an undergrad. I probably should have leaned more into geometry.

@leon_p_smith
In the UK BBC Basic was a good structured Basic that meant Gosub and Goto were deprecated.

I used a C64 but mostly as a games machine 😆

After that, C/C++, SQL and Visual Basic for work. Courses studied included Smalltalk, Java, Lisp and a tiny amount of assembler. I did some Python on my own.

The visual side of things and geometry were the attraction and the motivation 😀

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Nov 7

@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 🙄

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Nov 7

@foldworks
"I used to think that maths teachers were all teaching the same subject" - they are. All teachers HAVE TO teach what is in the syllabus - tests are based on what's in the syllabus (questions which use the word "explain" are getting the student to show their reasoning, not just follow the rule). Another strawman from him

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Nov 7

@foldworks
P.S. it never ceases to amaze me that there are people who go to all that effort to write a long diatribe, which is based on a wrong assumption, when they could've just checked their #facts first and saved themselves all that trouble... AND avoided spreading #misinformation

#mathematics #ITeachMath #MathematicsEducation #MathEd #MathsEd

foldworks Nov 8

@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.

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Nov 8

@foldworks
"Instrumental Understanding as well as Relational Understanding" - teachers already know the difference - it's part of a degree in teaching - and we call it ROTE learning vs. Constructivist learning. Again the issue is he made a wrong assumption, based on what was in textbooks, about teachers only ever using ROTE learning, instead of looking in a syllabus or actually asking them about it. We absolutely do teach the reasons for all the rules he claimed that we don't. It's a strawman.

#mathematics #education #iTeachMath #MathematicsEducation #MathEd #MathsEd

foldworks Nov 9

@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/

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Nov 10

@foldworks
"in the 1970s" - I know that. The reasons were being taught then. This isn't a change in modern education

"are all learners in all the world’s schools taught the reasons for all the rules" - easy enough to look in the syllabus. Guaranteed you'll find the word "understanding" many times

"I’m not sure in fact that instrumental cf. relational understanding are direct analogies of rote cf. constructivist learning" - not teaching the reasons vs. teaching the reasons. Not complicated!

#geometry #mathematics #puzzle

OK, a bit of a cheat as this could be geometry, but it's tough when you want to avoid sophisticated trigonometry: the Tantalus problem https://www.cambridge.org/core/books/abs/art-of-mathematics-take-two/tantalus-problem-from-the-washington-post/E4EBC04AAB2DFD6B0436F1E69F04347F

https://www.chegg.com/homework-help/questions-and-answers/referred-tantalus-problem-isosceles-triangle-72-0-420-240-need-findangles--q227796 has a cleaner diagram

The Tantalus Problem – from The Washington Post (Chapter 15) - The Art of Mathematics – Take Two

The Art of Mathematics – Take Two - June 2022

Cambridge Core

https://www.mscroggs.co.uk/blog/64

A geometry entry that could have been published at the time is the plastic ratio https://en.wikipedia.org/wiki/Plastic_ratio

But the Harriss spiral is newer https://chalkdustmagazine.com/regulars/on-the-cover/on-the-cover-harriss-spiral/

#geometry #spiral #PlasticRatio #GoldenRatio #MathArt

Plastic ratio - Wikipedia

Something newer: an elegant "almost orthogonal polyhedron: a polyhedron whose adjacent faces are orthogonal to each other, except on one edge." https://www.gathering4gardner.org/g4g15gift/ExchangeArchive-AlmostOrthogonalPolyhedra-G15-093-1.pdf

h/t @robinhouston

#geometry #polyhedron

bit101 Nov 8

@foldworks I bought up a bunch of used copies of his books from ebay and amazon some years ago. I think I have most of these. Good stuff to peruse from time to time.