
Math for seven-year-olds - liotier
http://jdh.hamkins.org/math-for-seven-year-olds-graph-coloring-chromatic-numbers-eulerian-paths/
======
j2kun
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.

~~~
kirsebaer
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](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....](https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)

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](https://www.youtube.com/watch?v=awOAmTaZ4XI)

~~~
sgustard
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.

~~~
quadrangle
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.

~~~
rquantz
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.

------
lifeofanalysis
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.

~~~
oldbuzzard
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](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.

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

~~~
oldbuzzard
Fixed

~~~
arm
_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/...](http://gcpm.rutgers.edu/books.html\]\(http://gcpm.rutgers.edu/books.html\)…)
this was a pale imitation of what Soviet era math circles would offer but is
still way beyond anything extant.

―――――――――

 _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​…](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…](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').

~~~
ColinWright

      > 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.

------
blisterpeanuts
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.

[http://en.wikipedia.org/wiki/Sprouts_(game)](http://en.wikipedia.org/wiki/Sprouts_\(game\))

------
kneth
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-...](http://www.slideshare.net/geisshirt/naturvidenskabsfestival-2012)
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 :-)

------
peteretep
All of this looks awesome: [http://jdh.hamkins.org/category/math-for-
kids/](http://jdh.hamkins.org/category/math-for-kids/)

------
zhte415
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).

------
vanderZwan
> _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!

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

~~~
crasshopper
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.](http://arxiv.org/list/math/recent.)) I
see graphs as just a common object that relates to a variety of mathematical
areas.

~~~
ColinWright

      > 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.

~~~
oldbuzzard
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.

------
mollerhoj
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.

~~~
mkesper
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.

------
aidos
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.

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

~~~
j2kun
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...](http://jeremykun.com/2011/07/14/graph-coloring-or-proof-by-crayon/)
[2]:
[http://en.wikipedia.org/wiki/Toroidal_graph](http://en.wikipedia.org/wiki/Toroidal_graph)
[3]:
[http://en.wikipedia.org/wiki/Euler_characteristic#Examples](http://en.wikipedia.org/wiki/Euler_characteristic#Examples)

------
pbhjpbhj
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.

~~~
j2kun
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...](http://jeremykun.com/2011/06/26/teaching-mathematics-
graph-theory/)

~~~
pbhjpbhj
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.

~~~
j2kun
I think a good answer is going to depend heavily on your goals (applied or
theoretical). Maybe you could elaborate on that?

~~~
pbhjpbhj
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.

~~~
j2kun
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_...](http://www.maths.lse.ac.uk/Personal/jozef/LTCC/Graph_Theory_Bondy_Murty.pdf)

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

------
oldbuzzard
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/](http://http://www.moebiusnoodles.com/)
[2]
[http://www.valerieslivinglibrary.com/math.htm](http://www.valerieslivinglibrary.com/math.htm)

------
slackpad
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.

~~~
kaitai
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.nsf.gov/statistics/seind12/c1/c1s3.htm)
,
[http://www.math.vcu.edu/g1/journal/Journal7/Part%20I/Sterlin...](http://www.math.vcu.edu/g1/journal/Journal7/Part%20I/Sterling.html)

[2] [http://education.ohio.gov/Topics/Teaching/Educator-
Evaluatio...](http://education.ohio.gov/Topics/Teaching/Educator-Evaluation-
System/Ohio-s-Teacher-Evaluation-System/Student-Growth-Measures/Value-Added-
Student-Growth-Measure)

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

~~~
j2kun
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.

~~~
kaitai
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.

~~~
j2kun
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.

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

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

------
shirro
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.

------
Pxtl
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.

------
arctangent1759
Using CHILDREN to solve NP-Complete problems? What have we become? Monsters, I
tell you.

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

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keppy
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!

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

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mathattack
This is great - thanks for sharing!

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jestinjoy1
Graph Theory for Maths :D

