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Henry Story (Jul 17 2021 at 10:39):I have twins age 5 and a daughter soon to be 7. I am playing Go with them, which is a great game to play with children as one can always set it up so that they win 50% of the time. But I think it should soon be time to introduce some basics of maths.
Has anyone put together a curriculum on how to teach children maths starting from Cats?
That occurred to me just recently
One could start by teaching Category Theory where the fundamental Lemma - explained clearly by @math3ma here https://www.math3ma.com/blog/the-yoneda-perspective and discovered by Nobuyo Yoneda, a Japanese Mathematician https://en.wikipedia.org/wiki/Nobuo_Yoneda puts relations at the heart of mathematics. https://twitter.com/mathematichole_/status/1414653170232532999
- The 🐟 BabelFish (@bblfish) Amar Hadzihasanovic (Jul 17 2021 at 10:49):@Bob Coecke & collaborators have been doing some proper experimental work on teaching string diagrams to children, so probably a good person to ask...
Henry Story (Jul 17 2021 at 10:51):yes, I linked to that work in the Tweet :-) but I have not yet read the papers, and I guess it would help me and other time constrained parents to know how to go about it... Where to start? Are there videos? things to do and not to do...
Amar Hadzihasanovic (Jul 17 2021 at 10:52):Oh, sorry, didn't see the tweet.
Henry Story (Jul 17 2021 at 10:55):(Ah yes, I see the tweet gets hidden in the UI).
I think the topic deserves a full channel.
When I was 4 perhaps as I was in Washington, I was shown ovals with arrows from elements in one to elements in the other, and I remembered that a few years later in primary school in France, doing a similar exercise, which made me realise that mathematics was at least one thing that was the same across such distances.
Timothy Porter (Jul 17 2021 at 11:44):Years ago, Ronnie Brown and I used knot theory in Masterclasses and also in our Maths and Knots exhbition. There is no explicitl category Theory but we did have activities that used comparing knots using Reidemeister moves, colouring knot diagrams, etc. (The exhibition is still available online here and is still worth looking through although as it is some years old, it could be redone and augmented in ways that were impossible when it was built.)
Another source for related material is in Heather McLeay's
[Mathematics in School
Vol. 20, No. 1 (Jan., 1991), pp. 28-31 (4 pages)
Published By: The Mathematical Association]
(https://www.jstor.org/stable/30214750)
Jon Awbrey (Jul 17 2021 at 13:06):Cf: Inquiry Driven Systems • Discussion 2
Way back in the Summer of Love I met a girl who had just graduated in Chemistry and was thinking about grad school in Education, the hot new field of Instructional Media, we got to talking and dreamed up a vision of using media, at first just shapes in motion, to teach people math from scratch. Long time passing, we got married, she did a dissertation — The effect of the hausdorff-besicovitch dimension of figure boundary complexity on hemispheric functioning — studying the effects of fractal figure complexity on cognitive processing, Mandelbrot gave her permission to use a series of his figures and ranked them by fractal dimension for her, and I pursued an array of parallel lives in math, stat, computing, philosophy, and psych.
Here is one of our later collaborations aimed toward integrating inquiry learning and information technology into education.
Cheers,
Jon
Eduardo Ochs (Jul 18 2021 at 13:39):I did not start from categories and I didn't have access to young children, but take a look at the introduction of this paper - "Planar Heyting Algebras for Children" - and at these and these slides. The last link has some slides in Portuguese - just skip them.
Spencer Breiner (Jul 19 2021 at 16:42):Thanks for bringing this up @Henry Story! I have been talking about this a bit with @Brendan Fong and some others at Topos.
Spencer Breiner (Jul 19 2021 at 16:43):I think it's important to distinguish between teaching categories, versus using categories (possibly without mentioning it) to teach other subjects. Coecke-Kissinger is a great example here.
Spencer Breiner (Jul 19 2021 at 16:44):I also love this paper about vector calculus:
https://arxiv.org/abs/1911.00892
I didn't realize until this year that Penrose also introduced a notation for the derivative!
Spencer Breiner (Jul 19 2021 at 16:47):Personally, I've come around to the view that middle/high school education is the place to start, because it has the broadest impact. As I understand it, Algebra I is where we lose the most people.
I've been (half-)joking about how ridiculous it is that we use letters from the beginning and end of the alphabet (a,b,c,.. vs. x,y,z) to indicate this fundamental distinction in algebra between constants and variables. This is in contrast to string diagrams, where you can see the difference right in the picture.
Christian Williams (Jul 19 2021 at 17:55):This is important to me. I'm surprised we don't already have a stream for it --- I'll make one now.
Christian Williams (Jul 19 2021 at 17:59):Hm, it's complaining about subscribing a large number of people.
Christian Williams (Jul 19 2021 at 18:00): Mike Shulman (Jul 19 2021 at 18:36):I'm surprised no one in this thread has yet mentioned Lawvere-Schanuel Conceptual mathematics. Not itself something I would give to a 5-year-old, but there may be some helpful ideas in there.
Henry Story (Jul 19 2021 at 19:51):Oh, I bought Lawvere's "Conceptual mathematics" in Feb 2012 (just checked). It was the first book on Cats I looked at. I appreciated it, yet at the same time it was too easy for me at least, as at a certain age one has to see how what one learns meshes with what one already knows. I felt it difficult to develop an interest for equalisers... But I think children might be quite ok with this type of thing.
( After that I bought Awodey's "Category Theory" and that felt like rocket science lifting off so fast I felt it hard to hang on. Weird, because I looked at it recently again at it felt like a very nice introduction :thinking: In the mean time I had read introductions by Bartosz Milewski, Bart Jacobs and Spivak, all very helpful)
But yes, "Conceptual Mathematics" feels like something one could teach 10-13 year olds. It might actually do the teacher some good too :-)
Henry Story (Jul 19 2021 at 20:45):Q: Could one re-invent the whole curriculum from age 5 onwards built on new Cats concepts?
Jon Awbrey (Jul 19 2021 at 22:00):Cf: Inquiry Driven Systems • Discussion 3
@Henry Story said:
Q: Could one re-invent the whole curriculum from age 5 onwards built on new Cats concepts?
Henry, Eduardo, …
If I were starting from scratch, and I'm always starting from scratch, I would ease my way up to the pons asinorum of logic and math using the types of logical graphs laid down by Peirce and Spencer Brown. That is because I think it's crucial to firm up propositional logic before taking on quantifiers and to grasp classical logic before intuitionistic.
The climb from “zeroth order logic” to first order logic is a lot more interesting and richer in adventure once you have a truly efficient calculus for propositional logic at the ready. An approach to categories, combinators, heyting, etc. can then be made via the propositions as types analogy. For the kiddies, Smullyan's Mockingbird would be the primer of choice.
Regards.
Jon
Eduardo Ochs (Jul 20 2021 at 04:13):@Jon Awbrey, do you have links on how to teach Logical Graphs to children (and to people like me!) and how to use them as a basis for learning Propositional Calculus and quantifiers?
I am not convinced that "To Mock a Mockingbird" would be a good starting point (disclaimer: I have very little contact with real children)... I remember clearly how I was fascinated by the games and the diagrams in the columns by Martin Gardner and Douglas Hofstadter in Scientific American when I was a kid, and how I felt super-powerful every time that I learned a new way of drawing mathematical ideas. If someone had shown me how to draw λ(x,y)∈{0,1,2,3}×{0,1,2}.(x≤y) as a grid of 0s and 1s that would have certainly blown my mind very very much, and I've sort of been trying to make things like this more accessible to "children"...
Jon Awbrey (Jul 20 2021 at 15:28):Cf: Inquiry Driven Systems • Discussion 4
@Eduardo Ochs said:
Jon Awbrey, do you have links on how to teach Logical Graphs to children (and to people like me!) and how to use them as a basis for learning Propositional Calculus and quantifiers?
I am not convinced that "To Mock a Mockingbird" would be a good starting point (disclaimer: I have very little contact with real children)... I remember clearly how I was fascinated by the games and the diagrams in the columns by Martin Gardner and Douglas Hofstadter in Scientific American when I was a kid, and how I felt super-powerful every time that I learned a new way of drawing mathematical ideas. If someone had shown me how to draw λ(x,y)∈{0,1,2,3}×{0,1,2}.(x≤y) as a grid of 0s and 1s that would have certainly blown my mind very very much, and I've sort of been trying to make things like this more accessible to "children"...
Dear Eduardo, ...
There’s a lot of stuff I’ve put on the web over the last twenty years devoted to CSP and GSB and my own versions of Logical Graphs — I’m still working at organizing all that in a step-by-step tutorial fashion. I’ll be doing more of that over time on a number of local streams and topics, e.g.
You might try sampling my Inquiry blog for the daily rushes and discussions or my OEIS user page and OEIS workspace to see if anything engages your interest.
Cheers,
Jon
Henry Story (Jul 23 2021 at 05:30):If one were to think about maths and children's education one would need to look at the needs of other subjects too. It should be easy for people here to work out how cats ties in with physics and biology - having a maths of open systems could help a lot there. But one would also want to help maths tie in with the humanities. In France children sometime after 13 or so read Voltaire's Candide published 1759, where Voltaire makes fun of Leibniz' idea that we live in the best possible world, by having Candide go around the world and witness all the suffering known at the time. It would be good if the maths department then also gave some introduction to fragments of contemporary modal logic, so that the children could see that the "best possible world" idea is abandoned by contemporary modal logics.
Jon Awbrey (Jul 23 2021 at 12:25):Cf: Inquiry Driven Systems • Discussion 6
@Henry Story said:
If one were to think about maths and children's education one would need to look at the needs of other subjects too. It should be easy for people here to work out how cats ties in with physics and biology - having a maths of open systems could help a lot there. But one would also want to help maths tie in with the humanities. In France children sometime after 13 or so read Voltaire's Candide published 1759, where Voltaire makes fun of Leibniz' idea that we live in the best possible world, by having Candide go around the world and witness all the suffering known at the time. It would be good if the maths department then also gave some introduction to fragments of contemporary modal logic, so that the children could see that the "best possible world" idea is abandoned by contemporary modal logics.
Dear Henry,
I've never found much use for modal logic in mathematics proper since mathematics is all about possible existence, in the sense of what is not inconsistent with a given set of premisses. Of course, one can entertain modal logic as an endeavor to construct mathematical models of natural language intuitions about possibility, contingency, necessity, etc. but that is an application of mathematics to an empirical domain.
As far as best possibilities go we certainly do a lot of work on optimization in math and its applications to the special sciences and engineering. For instance, a lot of physics begins with skiers on snowy slopes and their contemplation of gradients. That very sort of thinking by Leibniz led to his personal discovery of differential calculus.
Regards,
Jon
Henry Story (Jul 23 2021 at 13:01):Dear @Jon Awbrey ,
I place Logic within Mathematics and modal logic is a field of Logic, and so of mathematics.
You will find that modal logics comes up a lot working with machines, programs and all state based systems. This thesis is defended in Modal Logics are Coalgebraic and many other papers over the past 20 years. There is also a view of modal logics as arising out of geometric morphisms between infinite topoi - which I hope to one day understand. See the Book "Modal HoTT" by David Corefield for an introduction.
Of course if you narrow your definitions of mathematics so that these issues don't come up, then you can define that reality away. But I think it is better teaching children to be open to a larger view of mathematics that allows them to see the relation of it to what they study in Language (grammar) or Literature and fiction, as in David Lewis' article Truth in Fiction.
Regards,
Henry, The Babelfish
Jon Awbrey (Jul 23 2021 at 13:18):Cf: Inquiry Driven Systems • Discussion 7
@Henry Story said:
Dear Jon Awbrey ,
I place Logic within Mathematics and modal logic is a field of Logic, and so of mathematics.
You will find that modal logics comes up a lot working with machines, programs and all state based systems. This thesis is defended in Modal Logics are Coalgebraic and many other papers over the past 20 years. There is also a view of modal logics as arising out of geometric morphisms between infinite topoi - which I hope to one day understand. See the Book "Modal HoTT" by David Corefield for an introduction.Of course if you narrow your definitions of mathematics so that these issues don't come up, then you can define that reality away. But I think it is better teaching children to be open to a larger view of mathematics that allows them to see the relation of it to what they study in Language (grammar) or Literature and fiction, as in David Lewis' article Truth in Fiction.
Regards,
Henry, The Babelfish
Dear Henry,
Just by way of personal orientation, I tend to follow Peirce and assorted classical sources in viewing logic as a normative science whereas mathematics is a hypothetical descriptive science. That gives a picture of their relationship like the one I drew in the following post.
The way I see it, then, logic is more an application of mathematics than a subfield of it.
Regards,
Jon
Leon Conrad (Jul 25 2021 at 07:47):Hi all -
Thanks to Jon for inviting me to join this discussion.
I teach classical syllogistic logic to (bright) kids aged 9+ using a method derived from George Spencer-Brown's work in Laws of Form.
In this paper, I explain how GSB's approach was intuitively spot on, once the confusion between negation and distribution is eliminated:
https://www.academia.edu/40136796/Laws_of_Form_Laws_of_Logic
It would be very interesting to see how this approach might be developed to help students tackle what are essentially logical gate diagram design problems such as the one found in question 2 in Eton's 2014 General 1 King's Scholarship paper attached. 1-KS-G1-2014-FINAL.pdf
building perhaps on GSB's work on NOR Gates - recently published here: https://www.emerald.com/insight/publication/doi/10.1108/9781839826108.
Leon
Jon Awbrey (Jul 25 2021 at 12:28):Cf: Inquiry Driven Systems • Discussion 5
@Leon Conrad said:
Hi all -
Thanks to Jon for inviting me to join this discussion.
I teach classical syllogistic logic to (bright) kids aged 9+ using a method derived from George Spencer-Brown's work in Laws of Form.
In this paper, I explain how GSB's approach was intuitively spot on, once the confusion between negation and distribution is eliminated:
https://www.academia.edu/40136796/Laws_of_Form_Laws_of_Logic
It would be very interesting to see how this approach might be developed to help students tackle what are essentially logical gate diagram design problems such as the one found in question 2 in Eton's 2014 General 1 King's Scholarship paper attached. 1-KS-G1-2014-FINAL.pdf
building perhaps on GSB's work on NOR Gates - recently published here: https://www.emerald.com/insight/publication/doi/10.1108/9781839826108.
Leon
Ahoy Leon! Welcome aboard, a-synchronicity being what it is,
it may be September before I get both my hands back on this
deck myself as I've got a bunch of long-procrastinated home
and garden and auto and health-related matters to deal with.
If you're an old time web surfer like we all used to be way
back when I could leave you with a link or two to follow up
on your own recognizance -- I know, I know, these days it's
more like you can link a horse to whatev but you can't make
em click it.
I will try to write something more coherent later today but
failing that here's a link to an omnibus Survey page for my
blog, where you can find what's been occupying my trains of
thought for the past half-century. The last-numbered links
under each topic include and update all the earlier entries.
Best regards,
Jon