> I agree that PhD level math is not what we should > attempt to teach people in K-12 education. However, > "general numeracy skills" isn't what most people want > for their children either.
Really? I'd think they'd want that for their kids.
> The problem with math is that there is more of a > disconnect between what mathematicians do and the > math we teach than exists in the other things we > teach.
It's maybe only a problem if we package our curriculum with misleading labeling.
> And the reason for that is because unlike most other > fields math lives always in the service of "more > important" subjects.
I think the model should be: all subjects may need to import assets from other subjects to get the job done. A scientist might need a strong illustrator. A sculptor might use molecules for inspiration. An historian might study complex numbers in terms of timelines and personages (e.g. Big Bird).
> Well, ultimately it is a matter of what "we" really > want. The problem is that "we" are for the most part > uninitiated. And yet, only we can be trusted to > evaluate what is important.
Yeah, "we" are somewhat uncoordinated and moronic a lot of the time.
I *still* can't come up with a reasonable hypothesis as to why there's such resistance to sharing even this *basic* set of math facts with children:
tetrahedron : cube : octahedron : rhombic dodecahedron == 1 : 3 : 4 : 6, where all are defined in the dense packed context of the CCP.
With this set of blocks, you can get into all kinds of interesting subjects, from virology to architecture to crystallography to American history (Alexander Graham Bell was a big CCP buff, like Kepler, though in AGB's case it was the corresponding *spaceframe* he studied, what Bucky Fuller later named the IVM).
> On a personal level -- in terms of what I want my > kids to learn -- I don't disparage programming as a > skill and a vocation. Like many "mathematicians", I > do a lot of programming. However, there is something > out there. Most people go through life completely > oblivious to it. It is ancient and timeless. It is > a lot bigger than our transient technological > interests. Like most parents, I very idealistically > want my children to catch a glimpse of that. There > will be plenty of time to pick a vocation. But, they > will likely never get an education as busy adults off > pursuing a career. And, that is why now is the time > for these impractical ideals that lead to no > particular vocation.
I promise we won't completely forsake the dizzying heights when rubbing their noses in supermarket SQL.
As you noted about my curriculum elsewhere, it rubs up against a lot of real math ideas, including the Golden Proportion (big in Pentagon Math, as in "fee *phi* foe fum" -- although some, like Vic, say *fee* for phi).
> Well, certainly group theory is a lot closer to "real > math" than highschool trig. But, just because you do > clock arithmetic, that doesn't mean you're doing > group theory or number theory.
We're doing more than just clock arithmetic. We go over the properties you need for Group, Ring and Field, just like some Algebra 2 texts would do, but without much in the way of an interactive demo. Computers change that. Vegetable Group Soup is only the beginning. Just wait'll some real artists get on the payroll!
> I do think that some of the selection of topics is to > your credit. It is a lot closer to the core of what > has made math great. However, in my opinion, it is > not important that you teach set theory or > combinatorics or group theory or number theory. > These subjects are just associated with "higher > math". What makes the difference is the *math > proof*. You don't have to hold someone to the
For me, it's more about nailing the concepts. More focus on proofs down the road.
Kids have a lot of trouble with dividing fractions. But if they've played games with groups, and know how the p/q symbol might be used as shorthand for p*q.inverse(), then we're ahead in the game of providing a tightly woven framework of positively self-reinforcing ideas.
Get all this infrastructure nailed down *before* you get too carried away with the proving. First you need to learn the *language*. No need to skip ahead too quickly, unless you prove up to it.
> standard most graduate students are held to in order > to start training them to prove theorems. But, that > is in fact what we really only do -- we really only > start teaching people how to formally justify their > assertions as graduate math majors. We really could
S'OK. There's a lot of basic numeracy skills required for other jobs, other walks of life. *Those skills* are the proper focus of K-12, not how to win in a PhD math program (like on 'Survivor').
> start on that project at the very first moment of > abstraction. In other words, instead of teaching the > generalized arithmetic we teach as "algebra" we could > start up with the theorem proving.
I'd rather start with data structures, sequences of figurate and polyhedral numbers, branching to graph theory, with a good look at the city bus system, metro system, if we're in a city that has them.
Lots of focus on networks, switching, the need to connect (container shipping, trucks, railways, pack switching ala 'Warriors of the Net'). Katrina Math.
We also study what goes on in casinos (why so many losers), and in pyramid schemes (scams to watch out for -- a big part of K-12, is it alert kids about predators, a big focus at the police station I worked at).
> I think that that's what most people, without > realizing it, want. They just want some math, and > that's it. It could be accomplished with a little > good old fashioned Euclidean Geometry -- they could > even just memorize porpositions and their proofs > (although that probably is not the best pedagogical > approach). They will then go on to be a car mechanic > or whatever. They will find little application for > whatever actual theorems they learn. But, acquiring > the skill and mental discipline of formally > justifying their assertions will somehow make them > "smarter". Or at least that's what most people think > -- that's what is so special about an "education".
I think that's an oversold bill of goods. We could do far more for future car mechanics by focusing on the bigger picture, adding more real world grist to the mill.
Let's talk about gears and timing belts. Lots of ma... er "computer science" in those.
> philosophically vague distinction. However, it is a > lot more about what than it is about how. And, that > is why it falls into the mathematicians camp not the > education camp. Education folks should be ex-math > people (if they are math-ed -- ex-classics if they > are latin-ed or whatever) rather than sort of > travelling in their own circles for the most part > barely rubbing shoulders with actual math people.
Or we should just drop math as a subject.
> Our public schools probably shouldn't teach college > math, but at the same time, what they do teach > should be descended from that -- not sort of made out > of a completely separate thing. At least, inasmuch > as we want to teach math, it should.
Given how superspecialized is the math at the top of your totem pole, I don't see how to derive the wealth of topics we need to cover (e.g. databases and SQL) by simply inheriting from it.
> The problem is that most people think that math is > math and that is what we're teaching. And so, I > don't think they do know that they are making the > choice that they are -- to teach two periods of > science rather than one period of math and one of > science.
I still think we can give the flavor of pure math by going through some proofs, as you see. Like of the V + F = E + 2 theorem, which is front and center in our spatial geometry curriculum, along with Descartes' Deficit and stuff about Hexapents (HP4E).
> Well, ultimately, I don't think that we should have > to agree on what to teach, personally. I should be > able to teach what I want to and you to teach what > you want to. So, I guess in that sense, it's sort of > beside the point for me. The question is "Is that > math?" And, yes, I do have a much more specific > notion of what math is than you guys seem to. What a > theoretical physicist does, for instance, is not > math. Theoretical Physicists *use* math -- what they > *do* is theoretical physics. But, if someone gets a > degree in math and does the same or a similar thing, > you would be inclined to say they are "a > mathematician" doing "math". And, perhaps it does > look all the same to you, but I would call them a > physicist if they are really doing the same thing. > If it is similar but maybe not quite at the level of > contributing to the field of physics, then I guess > I wouldn't call them a physicist. But, I wouldn't > call them "a mathematician" either unless they were > really doing what a mathematician does, namely > contributing to the field of mathematics.
I get your viewpoint.
> Just solving someone's math problems is generally not > contributing to the field of mathematics. You have > to do something that advances the field -- of > interest to other mathematicians both present or in > the future. So, that's why not just anything > mathematical that isn't clearly and formally > something else is automatically "mathematics". > Mathematics is its own field. If you are doing > something mathematical -- that could just as well > be advancing the field of engineering as math. (In > fact, that is probably more likely than not.) Simply > solving a mathematical problem -- the result -- > normally does belong in some other field. The proof > of that result is what belongs in the field of > mathematics. > > Going back to the tired old field of Calculus. > Newton's Calculus is really physics. The calculus > of Weierstrass, Cauchy, et al is math.
Best wishes recruiting for your discipline. I'm hoping K-12 doesn't *turn off* would be mathematicians. But we can't afford to close too many doors too early, so this exclusive focus on proving will just have to wait until later in a student's career.