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Math for seven-year-olds (hamkins.org)
262 points by liotier on June 13, 2014 | hide | past | web | favorite | 66 comments

I'm surprised he used the formal terminology here. When I teach graph theory to a younger audience I usually make up (or have them make up) names. I think it gives them a feel for inventing mathematics, but then again it's usually aimed at high school students who have already been convinced as to what math is and isn't, and saying words like "chromatic number" turn them off. I imagine these girls don't suffer from the same afflictions.

Still, this blog post was excellent. I don't understand why all elementary math education isn't in the form of games and activities like this.

Here is TEDx talk, "Why Math Education is Unnecessary", by a math teacher arguing that compulsory math classes should be replaced with voluntary math-based games: https://www.youtube.com/watch?v=xyowJZxrtbg

Here is an essay by a mathematician, A Mathematician's Lament by Paul Lockhart (2002), arguing that math education is completely stupid, like teaching art only through paint-by-number exercises, and should be replaced by play: https://www.maa.org/external_archive/devlin/LockhartsLament....

And here is a short video of a democratic free school, Sudbury Valley School, where kids play freely all day long and learn math through video games, card games, and voluntary study: https://www.youtube.com/watch?v=awOAmTaZ4XI

Music is taught by the equivalent of paint by number, which is reading sheet music. Or play, which is improvisation. There's room for both.

Music as paint-by-number is one of the worst failings of music education. The most robust and life-long music in the world is in cultures that actually use notation only as appropriate if at all and don't think of it as the music. Most of the world's music is not so notation-focused.

I don't think this is the fault of music notation. Prior to widespread availabity of recorded music, playing music from sheet music was a common activity in the West. If anything, I would say the death of lifelong musical performance is more the result of recording -- "this disc here is the music" rather than "this sheet music is the music" -- the sheet music still needs a performer.

I agree, but I think the amount of "paint by number" in mathematics should be minimized, and most of it (learning multiplication tables, isolating variables in simple equations) is already in the form of games. I'd be interested to hear of a topic in math that you would argue can't be turned into a non-superficial game.

This is a little off-topic, I know, but I'd really appreciate some suggestions for good math-based games for my kids, ages 7 and 10.

So does anyone have good math games to suggest?

DragonBox ( http://www.dragonboxapp.com/ ) is fantastic. I have literally seen elementary school kids argue over who gets to be the next one to solve equations playing it.

If you have an iOS or Android tablet, they also have a new game called DragonBox Elements up that teaches geometry. After playing it for a while, I'd see triangles everywhere.

If DragonBox gets them hooked on algebra, another fun algebra game is Algeburst (currently only available for iOS platforms AFAIK), which employs a basic get-three-of-the-same-color-in-a-row mechanic, but requiring you to interpret simple algebraic expressions in order to figure out which tiles are of which color. E.g. if there's a tile called 3X + 2X, you can recognize it as being a 5X tile and give it whatever color 5X tiles happen to have on that particular level. (I was only going to try this game to see whether it was any good, but then ended up losing several hours to it.) They also have a "topics in arithmetic" version which doesn't have algebra but rather focuses on basic arithmetic.

10 might be getting a little old for it, and it teaches the basics of programming not math, but in general I would also recommend the Robot Turtles board game. On the topic of games teaching programming, I've also heard good things about the CodeSpells computer game, in which you're exploring a world and can cast spells by writing Java code, but I haven't tried it myself.

Well, I find Dad Meyer's talk to be helpful in thinking about math school education: https://www.ted.com/talks/dan_meyer_math_curriculum_makeover .

Interesting co-incidence. Just last week, I sent out an email to my friends, saying that I want to teach mathematics to their kids by a math-by-email service, kinda like the chess-by-email of the old days.

The idea is quite simple: I will send out a daily email with a grade appropriate set of math questions and/or games. Your child provides the answers back by email. I check the answers, provide corrections/feedback. And the next day's worksheet is customized to the child's history. If there is an interest, I could follow the child all the way from pre-K to graduate level Math subjects. Think about that -- wouldn't it be awesome if when you are getting your PhD, you could look back over 20 years of daily problems you solved and how you progressed in your conceptual understanding?

Naturally, you want to balance the gaming aspect with the rigorous aspects. You can start learning Graph Theory with diagram filling, but as you get more serious, there is no substitute to solving several hard problems to get a deeper/intuitive understanding of the concepts. There is no question that people learn different ways, some visually, others through games, and others through mental modeling. I am convinced that if we could tailor math teaching to each kid, we could get rid of the stigma that "Math is hard", or, worse, "Girls can't do Math".

Math, as I say, is a contact sport, not a spectator sport. You have to grab a pencil and a piece of paper to work on 20-30-40 harder and harder problems to master each concept. But to learn new concepts, you also have to cross the significant hurdle of climbing the first few rungs of each concept, so to speak. So let's learn by balancing games and theory.

or like math in the old days [1].

Israel Gelfand's stuff is model for enrichment at an upper level. Alexander Zvonkin's "Math from 3 to 7" book published by MSRI could be a model for younger kids.

I think the "new math"materials for the '60s and '70s and the Eastern European materials from a similar period provide a model for education. Unfortunately, they require mathy folks to present and interpret the material.

PS. I'm not criticizing public school teachers. I also feel inadequate. In order to teach DS7, I feel like I should probably work through Herstein's Abstract Algebra. I'll put it on the list... Currently I'm working through Euclid... Next Fall our homeschool will have a geometric focus... after that... who knows.

[1] See http://gcpm.rutgers.edu/books.html ... this was a pale imitation of what Soviet era math circles would offer but is still way beyond anything extant.

This reminded me of a book that is, I think, an excellent way to teach math to kids: Mathematical Circles: Russian Experience, by Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg (http://www.amazon.com/Mathematical-Circles-Russian-Experienc...)

Your URL is broken -- has extra dots at the end.


EDIT 4: The solution: use a dash. 😤


EDIT 3: Okay, I give up.


EDIT 2: Wow, okay, so my ZERO WIDTH SPACE was automatically escaped instead of treated as a separator. Anyway, I’m pretty sure the comment system accepts Markdown, so you should be able to use Markdown’s URL syntax to clearly mark which part is the URL without using a space to do the same:

[1] See [http://gcpm.rutgers.edu/books.html](http://gcpm.rutgers.edu/... this was a pale imitation of what Soviet era math circles would offer but is still way beyond anything extant.


EDIT 1: Scratch that! Looks like as long as there’s no space, it’s still considered a part of the URL regardless! Perhaps using U+200B ZERO WIDTH SPACE would still be considered a valid separator and could be used in place of U+0020 SPACE:

[1] See http://gcpm.rutgers.edu/books.html​… this was a pale imitation of what Soviet era math circles would offer but is still way beyond anything extant.


Just a tip—instead of adding a space after your three periods, you could have used U+2026 HORIZONTAL ELLIPSIS instead, like so:

[1] See http://gcpm.rutgers.edu/books.html… this was a pale imitation of what Soviet era math circles would offer but is still way beyond anything extant.

The HORIZONTAL ELLIPSIS isn’t be interpreted as part of the URL since '…' isn’t a legal character to use in URLs (unless you write it in escaped format (UTF-8 encoding), which would be '%E2%80%A6').

  > Anyway, I’m pretty sure the comment system
  > accepts Markdown, ...
I suspect that's where you're wrong - I suspect the comment system is hand-rolled.

I wish I was your friend, and on that email list. Your idea of "math by mail" sounds great, I am interested to be included on your list, please contact me if it is acceptable. Email in profile, thank you in advance!

I may be of assistance. Send me a PM

Fantastic! I'm going to print this out for my 9-year-old daughter to play with, and to teach to her friends when they come over (she loves to play school teacher). This is the kind of intuitive math games that I've been looking for to challenge her a bit without scaring her off with dry, boring stuff.

There's also the game of Sprout (or Sprouts) that is easy for kids to learn and has interesting mathematical implications.


I introduced graph theory to my son when he was about 8 or 9 years old. By Nature, he's a networker (like to connect to people) so "friends of a friend" was very intuitive to him.

I have also tried to introduce the concepts to teenagers. In Denmark, we have an annual, national science weeks in primary and secondary schools. I have given http://www.slideshare.net/geisshirt/naturvidenskabsfestival-... as a talk/lecture at 4-5 schools and most 13-15 years old children get the ideas quickly - Facebook and other social medias are a big help :-)

I appreciate the polish of making this into a small book. When educating kids, the 'take away' of having things in a format to take home and play with / show parents makes a big difference (as it does for adult learners).

> Next, we considered Eulerian paths and circuits, where one traces through all the edges of a graph without lifting one’s pencil and without retracing any edge more than once. We started off with some easy examples, but then considered more difficult cases.

Wait, there's math devoted to that? I used to do that as a kid for fun!

Ha! There's a whole branch of math dedicated to variations on it (http://en.wikipedia.org/wiki/Graph_theory).

For graph colouring I think the right WP link is Ramsey theory.

Also not sure graphs are a "branch" of mathematics, that would be like saying polygons are a "branch" of mathematics. (Branches would be like CA, AT, DG, ... as in http://arxiv.org/list/math/recent.) I see graphs as just a common object that relates to a variety of mathematical areas.

  > For graph colouring I think the right WP link
  > is Ramsey theory.
That turns out not to be the case: Ramsey theory is an area of Graph Theory that overlaps with, but neither subsumes, nor is subsumed by, questions related to coloring graphs.

  > Also not sure graphs are a "branch" of mathematics, ...
It may be the case that you are unsure, but my PhD is in Graph Theory, specifically, and I can assume you that it is a recognized area of mathematics.

  > I see graphs as just a common object that relates
  > to a variety of mathematical areas.
Similarly groups, topological spaces, sets, etc. Math is built on abstraction - the idea is to find commonality, extract it, then study it in its own right. Then anything you prove there applies to everything it came from. Graph Theory is like that, and it is its own subject within math.

Hmmm... Sets, Groups, Topology, Algebraic Structures... Sounds alot like the Bourbaki "Mother Structure's".

Sets and Algebra, I can teach as a homeschool dad. Group theory, topology, and abstract algebra are to difficult for me.

I don't think this is intrinsic. Lots of Rosen is hard for me.I suspect Smaullyan's math logic book coming out this Summer will be transparent(Euler diagrams alone make it clearer). Likewise, lots of Stewarts Calc is obscure. Yet, strangely Spivak's Calc is clear and simple. I suspect there are similar resources for abstract algebra and topology.. I assume that I will have worked through baby Rudin before DS finishes highschool... however, there must be a more direct, less abstract path.

It also underpins the foundations of my field (GIS). It's used to describe network routing (e.g. streets, streams, utility systems) and topology (for vector datasets--points, lines, polygons).

Yes, but it is somewhat dull math because the problem is relatively simple. See http://en.wikipedia.org/wiki/Eulerian_graph, and compare it with http://en.wikipedia.org/wiki/Hamiltonian_path and http://en.wikipedia.org/wiki/Hamiltonian_path_problem

Not dull. Fascinating, but small and quickly used up. For a novice it is a wonderfully accessible and inspiring intro on the field. It shows the magic of math, who tiny patterns underlying whole classes of seemingly disparate individual cases.

This! Graph theory should be taught very early to all children. It is as fundamental and important as simple algebra, and yields great insights without requiring tough computations.

When I was a child, we had set theory in elementary school. They dropped it shortly afterwards, though I still think it's very elementary.

Great project. Strangely enough I was thinking about going through something similar just yesterday (to teach my work mate about graphs).

My daughter isn't even 3 yet so this would still be a little beyond her. Not by much though, given how approachable it has been made.

I've downloaded the kit for later in life.

Amazing project! Does anyone know why a (2D?) map will need at most four colors while avoiding neighboring areas with the same color?

This is a monumental theorem called the four-color theorem, which specifically states that ever planar graph is 4-colorable. It actually took mathematicians many years and many pages (and computer programs!) to prove. There is a simpler proof that every planar graph needs at most 5 colors. [1]

Also, 2D is not quite specific enough. It turns out that there are non-planar graphs that can be drawn on a different kind of 2D-surface (the surface of a torus) that need more than four colors. [2] It turns out that for tori, the max number of colors is seven. And you can keep going up, culminating in this cool thing called the Euler characteristic. [3]

[1]: http://jeremykun.com/2011/07/14/graph-coloring-or-proof-by-c... [2]: http://en.wikipedia.org/wiki/Toroidal_graph [3]: http://en.wikipedia.org/wiki/Euler_characteristic#Examples

"The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer.":


What I'd like is a companion booklet that looks with a bit more rigor and formalism at the topics - but not too much! - that would suit a Y13 (17-18yo) or an introductory lecture at undergrad level. I've never really had a proper introduction to graph theory.

Incidentally this has come at the perfect time. I was just discussing with my 8 yo different fields of mathematics and which symbologies he's used and starting him on Boolean set operations. Graph theory was mentioned (by me!) so this will be a good flexi-day if he wants to follow up on it.

A nice accompaniment might be a lightbot like game for exploring Eulerian paths.

I have given lectures to students of this age range (13-18 years), but I don't have a booklet. See this blog post: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...

That's great, an enjoyable and inspiring read. Can you - or another reader - recommend a resource that's perhaps just a little more advanced though, please? What's the next step after formalising the language used, understanding connectivity and path traversal ... perhaps, what would the subheadings of a follow up lesson to this be?

I'm not asking to be spoonfed, really!, there are just so many resources on the web and shooting off I tend to get lost in Wikipedia and drift around too much without getting a reasonably rigorous overview.

I think a good answer is going to depend heavily on your goals (applied or theoretical). Maybe you could elaborate on that?

Um ... bit of both. My goal is personal enjoyment - I've done graduate level maths [way in the past] but graph theory never really featured for some reason. Thanks for your continued consideration.

There are a lot of books out there for really applied and really theoretical versions. I personally prefer the CS perspective, which spends a lot of time on problems about graphs (finding matchings, covers, cliques, flows, traversals satisfying certain properties). A good overview text is Bondy & Murty[1]. This text sort of straddles theory and practice in that you get lots of topological intuition (like Euler characteristic), and lots of algorithms and applications. One thing that's conspicuously missing is a treatment of random graphs, which is a huge topic both in theory and in applications. I do like that the book covers some basic Ramsey theory, as this is one of the most popular topics in combinatorics and it gives you a good flavor of what's going on in modern research there.

[1]: http://www.maths.lse.ac.uk/Personal/jozef/LTCC/Graph_Theory_...

Excellent, thank you for your insight and your patient answers.

I remember reading "Graphs and their uses" by Oystein Ore in high school and liking it.

If this sort of thing interests you, then either the modern Moebius Noodles[1] material or the vintage Young Math[2] materials like "Maps, Tracks, and the Bridges of Konigsberg: A Book about Networks by Michael Holt" would also.

[1] http://http://www.moebiusnoodles.com/ [2] http://www.valerieslivinglibrary.com/math.htm

This is the kind of experience would never be allowed with the Common Core type approach. I can't imagine people spending a classroom day on something like this (even though it's awesome and super valuable), but alas it's not content that will appear on the standardized tests.

Why wouldn't this be allowed with the Common Core? The Common Core is a set of standards for what students should have learned by the end of a year, and for fourth grade, for instance, the Core consists of things like "Use the four operations with whole numbers to solve problems," "Understand decimal notation for fractions, and compare decimal fractions, " and "Generate and analyze patterns."

One can fault the Common Core for not including enough discrete math -- it really focuses on numbers, algebra, and eventually modeling with polynomial and trigonometric functions -- and one can fault America for putting people into the classroom who don't know math [1] and then telling them they'll be awarded tenure or not based on student performance on some test [2].

People are confusing the same old problems we've always had (poor teacher training, stupid cobbled-together curricula like Integrated Math mandates by school boards, teaching to tests with high stakes for teachers and little value for students, etc) with Common Core. Read the standards [3], folks, and decide based on what they actually say what you might actually think.

[1] http://www.nsf.gov/statistics/seind12/c1/c1s3.htm , http://www.math.vcu.edu/g1/journal/Journal7/Part%20I/Sterlin...

[2] http://education.ohio.gov/Topics/Teaching/Educator-Evaluatio...

[3] http://www.corestandards.org/Math/Practice/

At a high level you're right, Common Core philosophy does encourage this sort of lesson. The unfortunate part is that Common Core still is largely a list of facts that students need to know, and when they need to know them. This is why graph theory is not "compatible" with the standards, because in the huge list of topics they pretend these things don't exist. And since the Common Core is really a document for non-teachers, I agree with the original comment that this wouldn't fly. It's just due to opposition by administrators, test writers, and standards-compliance checkers who read the document too thoroughly and, paradoxically, too shallowly. They don't look at this lesson to see whether it fits with the philosophy of learning espoused by the standard; rather, they check off which standards it covers and complain when the answer is zero. I put some fault in the document for this.

I'd disagree that it's a list of facts; the high school modeling standards are skills. I've read some good discussions of whether the earlier-year standards are useful for their age groups and am not sure (I think they're pretty reasonable). The high school modeling and geometry components are great.

But what does Common Core have to do with graph theory not being taught in elementary school? It never has been, and not because of new standards. It's not taught because too many teachers don't know it, and because too many "administrators, test writers, and standards-compliance checkers" have their noses where they shouldn't! It is terrible that in the US we have this attitude that teachers must be told exactly what to teach and when. We have it because we don't trust teachers, and swapping out NCTM standards for Common Core standards doesn't change that.

The modeling standard is, I feel, the most honest part of the standards. I'm arguing that if someone tried to teach graph theory as part of, say, modeling (which it is by all means), then they would see blowback from the people that make sure teachers are working toward meeting the standards.

There are so many hours in the day.

Either you argue that graph theory is more important than the other topics (which?) or you acknowledge that graph theory is an enrichment topic for some students who learn the core material faster or devote more time to math study.

It is true that Common Core lacks "enrichment" -- by defining a minimum standard, it doesn't specify what to do with student with more capacity to learn. But that's left to individual students, teachers, and schools, not incompatible.

If a teacher can teach their students 7G math standards in 6 months of the year, the teacher has plenty of class time for other topics.

I don't want to argue that graph theory is more important (though I could list which topics I think it's more important than). Rather I want to argue that, partially because primary and secondary school educators don't have so many years of preconceptions about the right way to teach the subject, it can do a better job building the underlying skills we want to teach students in teaching them math. We don't teach math because we want students to know facts (the correct answer to "What's the derivative of arccsc(x^2)?" is almost always "who cares?").

I certainly don't think that basic graph theory should be reserved for "advanced students." This second grade experiment shows that it's accessible to everyone.

That's a fair point - his content is certainly compatible with the Common Core's standard. I was really venting about your point [2] and shouldn't have implicated the standard for that. Thanks for the references.

There is so much to talk about re: point [2]...!

(Evaluating teachers using student test scores has been a push of the Bill & Melinda Gates Foundation, but it's not well-supported by evidence. One more link: http://www.latimes.com/opinion/editorials/la-ed-teacher-eval... )

That's an incorrect slag on Common Core.

It is true that standardized tests would put pressure to ignore this sort of "enrichment" material, but that's a (Pearson) test problem and a (unfairly constrainted) teacher problem, not a Common Core problem.

Put another way:

If a student is exceeding the test standards, there is time to branch out to enrichment topics.

If a student is not passing the basic arithmetic standards of the standardized tests, it's open for debate how to reach a passing level, or if it is appropriate to engage in a major alternative curriculum.

> If a student

I think the problem is that in a vast majority of schools the classes are taught to the lowest common denominator. Sure, one , or two, or ten students could be exceeding the standards but if one or two other students are not passing the standards, I think teachers are very unlikely to introduce these enrichment topics.

Wow, pretty sure I learnt this stuff after high school...kinda awesome/scary.

Yeah, math education is ordinary schools is terrible. I still haven't learned this, although I am 20...

Thanks, this was great. I got through the coloring with my 6yo this morning. I printed out two copies and we shared a desk and did it together. He enjoyed it and got mostly optimal answers. He got bogged down in the maps because coloring big areas frustrates him. So he is drawing robots now. I would suggest not trying to do this in one session with easily distracted boys.

I remember when my wife took a course on set theory I was fascinated with the graph theory stuff. I wonder if there are other academic fields of math that you could customize for elementary schoolers - I know my dad taught me Boolean algebra and logic gates in Grade 4, so there's one.

Using CHILDREN to solve NP-Complete problems? What have we become? Monsters, I tell you.

The coloring of vertices is a great idea. I'm going to try that with my daughter.

Great material! Can we get a follow up article for seventy year olds? Not a lot out there for the elderly. Math for seventy-year-olds Thanks!

The Google Drive download link isn't working for me. Are there any mirrors?

This is great - thanks for sharing!

Graph Theory for Maths :D

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