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Math for eight-year-olds: graph theory for kids (hamkins.org)
350 points by AllTalk on April 10, 2015 | hide | past | favorite | 45 comments

I'm very surprised that we don't teach basic graph theory in <= high school education. It has such a nice, gradual slope from puzzles and pictures to proofs, algorithms, and all sorts of science. I've done activities/lectures on the topic to all levels of high school (some with notes [1]), and it is routine to see amazement and engagement. And Hamkins shows it can go all the way down to eight-year-olds. It seems like a win-win to include it in every math teacher's repertoire. It can go as deep as the students want it to go.

[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...

I found your site through HN a while back, and since then I've sent a link to your site to pretty much every math/STEM teacher I meet (and I encounter quite a lot of them). Many of them have specifically mentioned to me what a great resource you've created for teachers and how you got their kids excited about math. I hope you know that your work is having a tremendous impact on math education and that impact will grow exponentially.

Wow, I really appreciate that, thank you.

I just wish I had more opportunities (and time, money) to try more things in classrooms.

Your write up on EC and tensors, I enjoyed it a lot. Thanks for the excellent effort!

Wow, what a nice testimonial!

Agree, This applies to pretty much every topic. If you make it relevant, interesting and create a safe environment where mistakes are ok a lot can be taught to kids.

I was fascinated as a kid from various types of sailor knots. I was in the air modelling, then computer club, and our neighbours were the young sailors, but I've never got into it.

But I know kids around me were fascinated by them too (it could be that we were after all living in a fishing/sea town with lots of big ships, etc.).

But then even for girls (knitting, or other activities) - there is something about graph theory to be found there. Just wondering...

Or the graphs of social relations.

>I'm very surprised that we don't teach basic graph theory in <= high school education

Not US, but I learned graph theory in HS, so it is done in some countries (CS class though, not maths)

How did it look like?

I faintly remember that we did `use` graph theory in our HS assignments, but haven't actually touched hypothesis or proofs, that I now associate with it.

I remember we wrote programs to higlight a graphs skelet, find shortest path, search for components, e.t.c.

Discussions on correctness and efficiency were informal.

> And Hamkins shows it can go all the way down to eight-year-olds.

At the bottom he mentions his previous visit - Math for seven-year-olds: graph coloring, chromatic numbers, and Eulerian paths and circuits http://jdh.hamkins.org/math-for-seven-year-olds-graph-colori...

I had like one grade where it was taught, but I mean, it would've been so much cooler to talk of things in Matrix-algebra land in terms of graphs.

I think Bourbakian formalism should be kept where it belongs : > upper undergrad. Math should be about pictures and things. Also see, http://pauli.uni-muenster.de/~munsteg/arnold.html

This is very interesting. I was recently looking over Leslie Lamport's writings[0] and noticed that his first publication was in high school, and was actually about graph theory. Similar to another comment on here he was interested in braids, a branch of graph theory.

Obviously he is a prodigy compared to the general populace. But perhaps an earlier introduction to some of these mathematical concepts led to the man we know today? If so, one would hope that a similar interest would motivate young students to take a different path, one where mathematics isn't quite so scary.

[0] http://research.microsoft.com/en-us/um/people/lamport/pubs/p...

When I was in high school and early on in college--I didn't enjoy math. I always thought of math as the drone of memorizing formulas and plug+chug.

I later took a class that used a textbook, Laboratories in Mathematical Experimentation. The book and that class were the first time math became play for me. A big part of that class was digging into graph theory.

Really fun stuff.

That's a great book. It's got a lot of number theory and dynamical systems as well, and the topics covered fit really well with experimentation using a computer.

This is very cool. I actually learned something from this even though I'm much older than the kids being taught :) Are there any resources for learning graph theory that present it in a fashion like this?

I apologize for the self promotion, but some people told me they immediately grasped the Havel-Hakimi Algorithm by solving this puzzle I made: http://jacquerie.github.io/hh/

No need to apologize! I haven't heard of the Havel-Hakimi Algorithm before, and I don't have the time to actually try and solve your puzzle right now, but I appreciate the link and will check it out later.

Although just reading the description, I do have one question: What is a "degree sequence"? I going to go ahead and assume that it's https://en.wikipedia.org/wiki/Degree_(graph_theory)#Degree_s.... If that's correct, I'd encourage you to add a hyperlink to the term "degree sequence" with that URL, for people like me with zero background in graph theory.

Yes, that's what I meant with "degree sequence". Thanks for the suggestion! That description is more succinct than it needs to be.

Whenever a sequence is graphical, to construct a graph pick any arbitrary node, and connect it to the n other nodes with the largest current numbers (reducing their numbers by one, as the puzzle here does), where n is the number of the node in question. Then pick another arbitrary node, and do the same, iterating until you’ve gone through every node.

As always in these situations, finding an algorithm that works in every case you've tried is fairly straight-forward. Convincing yourself that it's going to work is not that hard.

That challenge is always to prove for definite that it will always work. The proof is in, well, the proof.

That was fun!

I wonder if instead of V-E+R it would be more intuitive to use V+R-E so there are no intermediate negative numbers in the calculation.

The V-E+R follows the pattern of zero-dimensional, one-dimensional, two-dimensional, and this alternating sum formulation continues for the Euler characteristic in higher dimensions. So that is one reason to keep V-E+R.

Another reason, however, is that although some negative numbers show up, the numbers are all very small, and this could be a good opportunity to introduce some exposure to negative numbers.

So it turns out that this formula generalizes to an alternating sum over many dimensions [1] (although it can be stated more simply than Wikipedia does). From that perspective, it makes more sense to put the minus sign in the middle.

In either case, I don't think it is that big of a conceptual stumbling block to have a minus sign in the middle...

[1]: http://en.wikipedia.org/wiki/Euler_characteristic#Generaliza...

But it is an operational stumbling block when subtraction gives negative results. It may lead to calculation errors otherwise avoided. In a pedagogical setting like this one error sources should be minimized.

But the students are at exactly the stage where calculating with negative numbers has been introduced and needs practice and precision, so this works out very nicely with the curriculum.

Also see, http://csunplugged.org/

From the webpage:

CS Unplugged is a collection of free learning activities that teach Computer Science through engaging games and puzzles that use cards, string, crayons and lots of running around.

The activities introduce students to underlying concepts such as binary numbers, algorithms and data compression, separated from the distractions and technical details we usually see with computers.

CS Unplugged is suitable for people of all ages, from elementary school to seniors, and from many countries and backgrounds. Unplugged has been used around the world for over twenty years, in classrooms, science centers, homes, and even for holiday events in a park!

There was a truly fantastic post a few years ago about how to teach binary notation to eight-year-olds.

Children are very capable of understanding basic or even not-so-basic math; only when they learn that «maths are for nerds» do they decide it's not for them.

I believe it was this article: http://www.garlikov.com/Soc_Meth.html

One of the shining ideas in this lesson is the difference between an object and its representation, a nail which I think the author hits squarely on the head.

Yes, it was! Thanks!

There is much emphasis on teaching kids computer and math related subjects these days. Although it is exciting to see educational initiatives like these, they might actually be motivated by economical and institutional needs rather than taking into account individual interests of children. A shortage of computer programmers emerges and there is instantly an interest to teach children graph theory...

Math is not the only thing that is interesting. Why not have, "critical thinking for kids", or "composition for kids", "cinema for kids", "creativity for kids", or something else? A great deal of other skills and literacies are left out just for the sake of meeting the industrial "demand".

That's a bit cynical. I think at least a large portion of it is that people tend to believe that whatever skill sets they have are important.

So a plumber thinks that basic plumbing knowledge is something everyone should have. Chefs think that everyone should be able to cook the basics. Programmers think that programming is valuable knowledge that everyone should have some familiarity with.

If confronted with this, all of them will be able to rattle out perfectly rational arguments in defense of their positions. They aren't wrong. It is just that what they prioritize or think about most often is influenced by what they know and do. Basic plumbing knowledge is useful for anyone who owns a home (or just uses a toilet, sink, or shower), but how often do you see chefs advocating for that sort of practical education? Being able to cook a nice meal for friends or family is a great skill with a lot of social value, and cooking your own food can be very economical, but how often do plumbers advocate for that?

We see a lot of people advocating for tech education and extolling the virtues of tech knowledge because we surround ourselves with people who are already in this field.

This said, I think there is some truth in what you say. I get pretty suspicious when I see national campaigns for tech education being pushed by the elite. Business leaders or politicians; people who have made something other than programming their profession. Maybe there is a plumbing equivalent of things like code.org, but if so I am not aware of it.

Graph theory is not a subject taught solely for computer science. In fact, graph theory has a rich mathematical history independent of computer science, even through today.

The reason graph theory (and mathematics in general) is nice to teach is that it includes "critical thinking" and "composition" and "creativity," it's arguably the most efficient, direct way to teach these skills, and all you need is the knowledge to teach it and a writing utensil.

Graph Theory is pretty widely applicable. It's been a core course in my undergraduate program for years (I think since the beginning of the program). For example, you can apply it to a number of different domains in physics. It can be used to analyze electrical circuits, pipe flow, and mechanics. It's at the core of a number of packages related to these topics. The circuit simulator SPICE uses graph theory as does Modelica and MapleSim. That's only the physics applications. I've read papers that use graph theory in a lot of domains. There was a good book I used as reference when I looked at using graph theory for CAD (as part of an undergrad project), but I don't recall it as that was years ago.

I very likely would have done a graduate degree under a professor studying graph theory (he was my 4th year project supervisor), but he died over New Year's that year and I had to change supervisors for my last four months. But, he was the reason I decided to go to grad school in the first place.

No, I think math is very important. There are a few subjects I wish I was taught in high school though.

1.The scientific method(I can see young eyes rolling already though) 2. The Placebo effect 3. Basic statistics(with an emphasis on how adults often use statistics to deceive) 4. Basic course in Psychology(I don't recall if my high school had any psychology courses. In College, I had a great Psychology instructor. His course covered every thing I listed. When I look back at college, I can honestly state he changed the way I viewed to world. I remember thinking, I wish I had this information in HS. He was a great teacher.)

And that correlation does not imply causation.

Why are all the kids girls?

Probably a girls' school.

Heroic. This is exactly the kind of thing America needs to get over the bizarre general notion that "math is dumb".

I will try this on my 7-year old. I'm fairly certain that he's going to enjoy this.

looks like it's down already.

Screenshot of the site: http://i.imgur.com/yFfyjN0.png

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