[HN Gopher] Thoughts about what worked in math circles
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Thoughts about what worked in math circles
Author : sberens
Score : 285 points
Date : 2023-08-26 20:20 UTC (1 days ago)
(HTM) web link (buttondown.email)
(TXT) w3m dump (buttondown.email)
| thenobsta wrote:
| Huh, my daughter and I taught her first grade class SET last year
| (I did most of the pedagogy...). They seemed both able to learn
| it and engaged. It wasn't in a math circle setting, but a lesson
| to the whole class. Would be cool to investigate the differences!
| rahimnathwani wrote:
| We played Set with my son when he was ~6.5yo, and he
| immediately loved it. After a couple of games, he was spotting
| sets almost as quickly as we were. But his performance and
| interest varied based on distraction, tiredness etc.
|
| If you want a one-player Set game, there's a nice open source
| one here:
|
| https://playset.netlify.app/
|
| The 'AI' looks for solutions randomly. Every 2 seconds (in easy
| mode) it picks two of the visible cards at random, and sees if
| the card needed to make a set is visible.
|
| If your kid is totally new to Set, you probably want to adjust
| the delay to 5 seconds or something.
|
| Set is more fun against human players of course.
| thenobsta wrote:
| ~6 seems to be the right age. I tried with with my kiddo at 5
| and with another kid when they were 5 with no success. A
| bunch of first graders, on the other hand, were quick
| studies.
| hgsgm wrote:
| Another implementation here, with different (non-infringing)
| art http://misterbeebee.github.io/3qual
| rahimnathwani wrote:
| This looks promising. I like the breadth of customization
| options.
|
| I was confused at first because clicking 'Play game' took
| me to a screen that never loaded a set of cards. But then I
| realised you must click 'New game' first, before 'Play
| game'.
| RheingoldRiver wrote:
| Yeah Set has been dear to me since I was about 5 years old, and
| I remember bringing it into class for "favorite board game day"
| in 1st grade (so when I was 6). Really surprised the kids
| didn't like it!
|
| There's a LOT of math you can do with it too, starting with
| some modular arithmetic. and I guess just the idea of
| abstracting the attributes to 0, 1, and 2. Then you can do a
| little bit of group theory and how the game is Z3^4 (iirc this
| is the right notation? been a while), 4 copies of Z3. And there
| are similar card games that represent other games that you can
| print out cards for if they're getting excited about group
| theory, so you can talk about axioms of groups, and how the
| games represent the groups, and why it makes sense.
|
| Also you can introduce a bit of programming too and explain
| like how would you teach a computer how to play this game using
| the invariant property that a definition of a set is that sum
| c_i = 0 for all i 0 -> 3. and you can also easily generalize
| the game to other lengths by adding additional traits
| (background color of the card to add one) or fewer (remove bg
| color of the symbol).
|
| Anyway yeah this game is the best.
| verbify wrote:
| I'm disappointed to hear that ruler and compass constructions
| didn't engage them. I'm currently working on digitising a Hebrew
| translation of Euclid's Elements, and I was hoping that the very
| physical aspect (almost like art class) would be more engaging
| than theoretical stuff, and the fact that it's independent of any
| numbers would make it less scary than usual maths class. I still
| think there's educational value in Euclid's Elements beyond the
| historical importance, but it seems that I might be wrong on it
| being more engaging in an educational context for most kids.
|
| > Trying to figure out who is better at penalty kicks based on
| counts of scores/misses.
|
| If you have kids who care about a particular sport, this is a
| great way to teach linear algebra. There's a book 'Who's #1? -
| The Science of Rating and Ranking' that goes through different
| methods of ratings/rankings in great detail (it was one of the
| required readings for my MSc in Data Science).
| noncovalence wrote:
| I only gained an appreciation for ruler and compass
| constructions early in secondary school, at age 7 I'm not sure
| I could fully understand the beauty of deriving a huge system
| from axioms.
|
| For making Euclid interesting to children, I remember really
| enjoying a game called Euclidea: https://www.euclidea.xyz/
| mbivert wrote:
| In case this is of interest, and unknown, there's an
| aesthetically pleasant digitization of Euclid's elements[0].
| But I'm not sure whether it's best for studying: usually,
| actively engaging with the material is key (e.g. performing the
| constructions, without a model), thought the effective rules
| for kids might be different.
|
| I'm actually happy to see them interested in propositional
| logic, given how foundational it can be to coherent thinking. I
| would have guessed, as the author, that manual activities would
| have been preferred.
|
| [0]: https://www.c82.net/euclid/#books
| Mathnerd314 wrote:
| I remember learning about hyperbolic and spherical geometry in
| middle school, and that was cool. Not really because of the
| axiomatic aspect, but more of the "how many lines can you draw
| through two points? Up to you!" sort of questioning of
| mathematical assumptions and the funny diagrams. The Fano
| projective plane model was interesting, the Poincare disk model
| was interesting. I remember some animations and some
| interactive software you could play with. But yeah, after about
| a week I got bored of it. I would say one 2-hour session
| devoted to different geometries and constructions would
| probably be about right.
|
| The Euclid's Elements approach of axiomatic geometry is
| interesting, and suitable for maybe a high school course.
| Before students learn algebra they don't really have an
| appreciation of deriving equations or proofs from a small
| starting point. And coordinate geometry is much more practical
| (some things are simply unconstructible with ruler and
| compass).
| elesiuta wrote:
| I don't know the setup used for the ruler and compass
| constructions, but there's a game called Euclidea [1] which may
| be of interest. I found the level progression, scoring system,
| and solving proofs to unlock new "tools" was done very well.
|
| [1]
| https://play.google.com/store/apps/details?id=com.hil_hk.euc...
| [deleted]
| gilleain wrote:
| Well from the way the linked article describes it, there is no
| 'story' to maths like Euclidian geometry.
|
| After all, what were the original motivations for compass and
| straightedge constructions? Ideas about perfection, symmetry,
| and constructability are all very abstract.
|
| I found https://www.euclidea.xyz/ to be fun, but then I find
| drawing Girih patterns fun, and I'm also not 8 years old :)
| godelski wrote:
| I want to add to this list for something that I teach to kids of
| similar ages: counting to a thousand on your hands. This is
| natural for any computer programmer, but there's some key aspects
| to it that help with math education.
|
| This is quite fun for them as your question of "Do you know how
| to count to a thousand on your hands?" appears like magic or a
| superpower to them. So I demonstrate the beginning counting to 20
| or so (quickly moving through 4 and 6). Then I start to ask them
| to predict certain unseen configurations (i.e. zero shot
| generalization). Re-demonstrate when failure to predict. Once the
| pattern is successfully learned, then I present a quiz/puzzle,
| and ask how many fingers "this many" is (all fingers unfurled).
| Always stumped, I provide the hint "if I had an additional finger
| and that finger were open and all others were closed, how many
| would that be? Can you figure out the other number from here?" It
| takes time, but they almost always get it.
|
| The beauty of this is that we have a low barrier to entry, as the
| kid just needs to know how to multiply by 2 and know the names
| for numbers up to 1024. It surprisingly has many avenues of
| thinking that can help a kid better generalize concepts of math
| while still being entertaining (similar to concepts in this
| article). First, we teach the kids that there are multiple
| representations of things, and that we need to formulate things
| to match our goals. That we can break away from the common and
| expected thinking that most people have to gain "super powers"
| (i.e. not count like most people). Another important aspect is
| the above puzzle, where we specifically teach them that there are
| often better ways to go about solving a problem if we can find
| patterns. Rather than brute-forcing your way through this
| (summing each finger) you can exploit the iteration pattern to
| know that hinted at position is only one away from the desired.
| Frame of reference is such a crucial concept to mathematics and
| is at the root of solutions to many famous problems. Obvious post
| hoc, but inconceivable a priori.
|
| We can even go quite deep and talk about proofs and how to design
| algorithms! I'll explain the algorithm identical to how we would
| perform a proof by induction (this is not how I teach kids, at
| the beginning):
|
| k0th step: starting palms facing user, and an initialized
| position where all fingers are closed (thumb is a finger and at
| left most and right most positions). Starting from the right,
| increment the right most finger (thumb)
|
| knth step: start from the right most position. If finger is
| closed, then unfurl. If finger is unfurled, close it and attempt
| to increment the next right most finger recursively following
| this condition.
|
| There's more that you can build off of this one concept and
| similarly that with the topics in the article. What I've found is
| that which ever "game" the kid likes best is the one you should
| focus on and formulate your basis around. When they have
| difficulties with one game you use a different game that they are
| successful at to teach the difficult one. After all, math is a
| language and so many things can simply be rephrased.
|
| I find that one of the difficulties many have with math is that
| the internalize it as quite strict. That it is often taught "this
| is the way," with no other methods accepted and thus people gain
| quite low generalizability of the concepts. Something that "word
| problems" are intended to resolve, but this approach is quite
| brute forced and more akin to how one might teach a machine
| rather than a human. This is coupled with the fact that so many
| are at a young age taught by people void of passion for the
| subject. This dispassion only passes from teacher to student (I'm
| sure many people can remember the breath of fresh air if they
| were lucky enough to find a teacher who loved math and encouraged
| the creative side of it. Honestly, that's how I came the love the
| subject and prior to that Junior in High School class, I hated
| the subject despite being good at it and in advanced classes).
| sieste wrote:
| > quickly moving through 4 and 6
|
| Indeed, beware of playing this game with kids! Mine started
| shouting "645!" at each other.
| hgsgm wrote:
| Why 5?
| sieste wrote:
| 645 in binary is 1010000101 which results in an amusing
| 2-hand gesture if you're 6 years old.
| opportune wrote:
| Wow, all I can think is how exceptionally lucky these kids are to
| be taught these concepts/exposed to these problems at such a
| young age by a willing and enthusiastic teacher. Most people
| professionally employed as K-12 math teachers wouldn't even be
| able to teach this kind of curriculum well. I looked at the
| author's blog history to find out if they were doing this for
| some huge amount of money and found a blog that suggests not only
| were they doing it for free (skimmed it, but didn't see any
| reference to pay) but they had to try to convince other parents
| to let their kids participate!
| https://buttondown.email/j2kun/archive/a-foray-into-math-cir...
|
| In my opinion, there is a relatively unknown (to those outside
| mathematics) huge "privilege" gap in mathematics education that
| makes it so those that only follow a cookie standard or
| accelerated curriculum are relatively unprepared for careers in
| mathematics compared to those tutored (or taught in special
| magnet programs, or by their mathematician parents) in these
| kinds of non-standard-curriculum concepts from a young age.
| Mostly, the problem is that the standard curriculum is almost
| purely rote-computational until you become a college ~Sophomore
| and it abruptly changes to being open ended and proof-based
| (which is the world most pro mathematicians live in) requiring
| skills in creatively applying logic. So students with this kind
| of exposure from a young age have a much easier transition to
| that while also scooping up all the math-career-builders like
| early papers and contest wins on the way.
|
| Those other parents probably don't know this but OP is providing
| an immensely valuable service that is hard to find in some areas
| and which some parents would pay a huge amount of money for.
| pro-kythera wrote:
| > those that only follow a cookie standard or accelerated
| curriculum are relatively unprepared for careers in mathematics
|
| Culture-dependent? I recall the story from France of the
| second-grader who, asked what 2x3 equals, replied "3x2",
| knowing only that multiplication was commutative.
| dfdz wrote:
| > the story from France of the second-grader who, asked what
| 2x3 equals, replied "3x2", knowing only that multiplication
| was commutative.
|
| This is a classic joke making fun of the issues with French
| education based on the Bourbaki [1] school of mathematics,
| see [2] for more discussion. Different issues than the USA,
| but also bad in my opinion.
|
| [1] https://en.m.wikipedia.org/wiki/Nicolas_Bourbaki
|
| [2] https://web.archive.org/web/20230315185224/https://www.un
| i-m...
| zarzavat wrote:
| [2] is excellent satire because I honestly can't tell if it
| is a parody of physicists with a contempt for mathematics
| (e.g. Feynman), or if the author truly believes it.
| svilen_dobrev wrote:
| [2] is interesting read. Detaching any science (or other
| knowledge-gathering-activity) from reality may well turn
| it into teology :/
|
| And This observation:
|
| > "genuine mathematicians do not gang up, but the weak
| need gangs in order to survive."
|
| applies to programmers as well, and may be just about any
| profession/activity..
| ironSkillet wrote:
| Feynman had contempt for mathematics? Never heard that
| before.
| panda-giddiness wrote:
| "Physics is to math what sex is to masturbation" -Feynman
|
| That said, I'm not sure that he actually held contempt
| for math. It's fairly essential for theoretical physics.
| Tainnor wrote:
| It's not satirical. It's a very well known opinion piece
| by one of the most famous former Soviet mathematicians.
|
| FWIW, I understand where he's coming from but I
| fundamentally disagree (not in the least because for me,
| computer science applications of mathematics are much
| more interesting than physics ones, and these can be
| incredibly abstract).
| alexdowad wrote:
| Did you get that phrase backwards? It's hard for me to
| see how [2] could be interpreted as being about
| "physicists with a contempt for mathematics"; it's
| actually about mathematicians with a contempt for
| physics.
|
| Vladimir Arnold was a well-known pure mathematician with
| a deep interest in physics. If you read his math books
| (many of them are good), he constantly uses examples from
| physics to explain math concepts.
| mangodrunk wrote:
| Satirical song by Tom Lehrer regarding New Math sounds
| relevant: https://youtu.be/p0LDOAYcXAY?si=FCgSLx_PRlVL1Hc0
|
| https://en.wikipedia.org/wiki/New_Math_(song)
| mycologos wrote:
| +1, I didn't think of my upbringing as "underprivileged" in any
| way until I got to college and later did a math-adjacent PhD
| and was increasingly surrounded by people who had been doing
| math circle-type enrichment throughout their childhoods. I was
| OK at math but not especially precocious. I represented my
| middle school at a couple of local math competitions and didn't
| do very well, but looking back, it's kind of weird that I
| didn't have any help preparing at all.
|
| Thinking about this more in the last year or two has led me to
| shift a lot of my charitable giving to math circle-type
| programs, even though I know they're less verifiable than a lot
| of the (pre-longtermist) effective altruism causes -- I think
| that kind of mathematical thinking is a very valuable tool that
| is not so easy to come by without these kinds of programs.
| rahimnathwani wrote:
| If this has piqued your interest, there are some (but not lots)
| of paid options for things like this, e.g.
|
| - (San Francisco) afterschool math team for 7-11yo kids:
| https://www.meetup.com/chess-games-inner-fire/events/2956372...
|
| - (San Francisco) afterschool math circles at a couple of SFUSD
| schools: https://www.sfmathcircle.org/jose-ortega
| https://www.sfmathcircle.org/starr-king
|
| - (Remote) 501(c)(3) nonprofit running weekly online competition
| math sessions for elementary and middle school students:
| https://mrmathonline.com/
|
| If you're looking for something in between classes and circles,
| then perhaps:
|
| - Engaging Math Circles (https://emc.school)
|
| - Russian School of Math (RSM)
| avital wrote:
| For those who are interested in getting involved with online math
| circles (as parents or potential instructors), check out
| https://theglobalmathcircle.org (Jeremy, the author of this post
| graduated from our training program)
| jacknobody wrote:
| This is great! Home school teachers - take notice!
| hgsgm wrote:
| I think the specific topics and activities are hit and miss based
| on the specific skill level, interests, and daily mood of the
| group. Presentation style is a common theme though.
| dan-robertson wrote:
| If you're interested in more about 'math circles':
| https://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf
| rahimnathwani wrote:
| Yes, in a previous blog post the author says he read Zwonkin's
| book in preparation for running his circles.
| msla wrote:
| https://web.archive.org/web/20141231040717/https://www.msri....
| dools wrote:
| That link gave me a 404
| dan-robertson wrote:
| Ah it was working when I posted it I think. Here it is in
| book form: https://www.amazon.com/Math-Three-Seven-
| Mathematical-Prescho...
|
| I suspect you can find an online pdf if you search for the
| title.
| rahimnathwani wrote:
| That URL used to contain a PDF of this book:
|
| https://bookstore.ams.org/mcl-5
| benlivengood wrote:
| This is probably the most important recent scientific article
| about the (possible) future of STEM in the U.S.
| [deleted]
| amai wrote:
| On a side note: I can recommend the book Mammoth math for your
| kids:
|
| https://www.penguinrandomhouse.com/books/704500/mammoth-math...
| akermany wrote:
| I am wondering if this has been tried with children on the Autism
| Spectrum. I suspect a very different pattern of interests.
| BoppreH wrote:
| Different age group, but I had success in engaging with high-
| school students with "Bitcoin mining". This was a ~15 minute
| exercise after a lecture on blockchains, during a cryptography
| seminar.
|
| The original Bitcoin proof-of-work algorithm is to tweak the
| middle input of a hash so that the result starts with many binary
| zeroes (find x such that `sha256(sha256(a || x || b)) < H`). We
| simplified it down to `x^2 % N < 10^H` (calculators and computers
| allowed). You can freely tweak N and H.
|
| The students had a blast, and I believe it was a lucky
| combination:
|
| - It's more topical than ancient puzzles.
|
| - The students were racing against each other.
|
| - Rewards were semi-random (faster/smarter groups still had an
| advantage).
|
| - The rewards were "physical bitcoins" (chocolate coins).
|
| - Winning was more or less guaranteed by brute force, but there
| were plenty of shortcuts to find.
| [deleted]
| gnicholas wrote:
| > _The Function Machine game (guess a function given the ability
| to query it as a black-box)_
|
| My kid has loved this one since I read about it on HN when she
| was 6. As she learned more advanced mathematical operations, we
| added them to the toolkit. It's great! I can tell she's mastered
| a concept when we can swap roles and she can accurately answer my
| queries.
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