Math for seven-year-olds
jdh.hamkins.org
jdh.hamkins.org
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 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
So does anyone have good math games to suggest?
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.
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.
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.
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EDIT 3: Okay, I give up.
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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.
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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.
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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, ...
There's also the game of Sprout (or Sprouts) that is easy for kids to learn and has interesting mathematical implications.
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 :-)
Wait, there's math devoted to that? I used to do that as a kid for fun!
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.
> Also not sure graphs are a "branch" of mathematics, ...
> I see graphs as just a common object that relates
> to a variety of mathematical areas.
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.
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.
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
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'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.
[1]: http://www.maths.lse.ac.uk/Personal/jozef/LTCC/Graph_Theory_...
[1] http://http://www.moebiusnoodles.com/ [2] http://www.valerieslivinglibrary.com/math.htm
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...
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.
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 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.
(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... )
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.
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.