> I am going by what you say, yourself, that you are > "covering it" empirically and heuristically. > Sketching a proof every now and then is not treating > the subject the way it exists in reality.
Which subject is that? Mathematics? I'm not teaching it, according to you.
But we *do* look at some proofs. I've got that obvious visual proof that every square number is the sum of two consecutive triangulars, for example.
And we go over why N(N+1)/2 generates the triangulars in the first place, using a story about the young Gauss being forced to add 1...100 as busy work.
This *might* be construed as a proof, though as I mentioned earlier, sometimes we call it an *explanation* of an algorithm, a more "how things work" mentality (why does a clock work? How do we *prove* that it works?).
> Or, do you think you are grooming them to prove > theorems?
I'm in the business of keeping a lot of doors open. If they want to switch to a pure mathematics track at some point, or pursue one in parallel, who am I to stop them?
> If you say you are, then that's one thing, but you > said you weren't. At any rate, I looked around > your websites. I am not going to spend days and days > reading through every single thing you say. If I > have a mistaken impression about what you do, then > now is the time for you to just come out and retract > your use of heuristic and empirical methods to teach > math and to say that you are primarily focussed on > proving theorems -- that you do group theory the way
No, no, on the contrary, I'm into building alphanumeracy skills across the board. We generate sequences (like the triangular and tetrahedral numbers) to motivate early programming, as well as to connect the graphical to the lexical. 1, 4, 10, 20... this is a growing tetrahedron [cartoon].
But including proofs is not verbotten, or at least I haven't heard any boots at my door yet.
> an algebraist would. Otherwise, stop acting like you > "cover all these subjects" when you specifically > don't -- you just have your students learn the > theorems without the proofs.
We use *some* theorems without proofs, yes. Not all of them though. Euler's V + F = E + 2 is easy to prove. We also prove Descartes' Deficit in some segments. I have a lot of mathematicians able and willing to advise.
> If by "walking the walk" you mean practicing what I > preach, then I am by homeschooling my children. If > you want actual examples of secondary school > textbooks that I endorse, then I have already > mentioned a few. Gelfand's Algebra, for instance, > has a number of good proofs built into the exercises > and moves in a fashion that someone with a teacher > and who knows only arithmetic with numbers could > follow. This guy, Solomonovich, wrote a textbook on > Euclidean Geometry that I like. For that matter, > simply picking propositions from the first book of > the elements to try to construct a proof of with a > straight edge and compass isn't all that bad of a > math class either. You would just cover some and > leave others to the student to try. S I Gelfand > wrote Sequences, Combinations, Limits which I also > like. The Art of Problem Solving looks like it might > have some good textbooks, as well, but I haven't > actually put my hands on one.
OK, so you've decided to go it alone, as a homeschooling parent. I respect that decision.
But I remain hopeful that those concerned to teach a purist brand of mathematics, such as you've advertised wanting to teach, will organize and do so, right down to the kindergarten level, in order to have full control of the sequence right up through grad school.
That isn't my agenda *for myself*, other than to help see to it that such a motivated cadre of pure mathematicians maintains full access to world wide publishing, video distribution channels, all the same tools *we* use, to push *our* stuff (which is a very different can of worms, not at all what you tout).
I'd like to *compete* with you, if you get it together, but it sounds like you're just going to retreat into a private shell. I understand. Sometimes that's the best one can do.
> > Sounds good, go for it. > > What do you mean by this? "Go change the world"? >
It's what I do. What else is there to do, besides changing the world? Kicking a stone changes the world. Might as well make it be for the better, while you're at it, one kick at a time. Make sense?
> > > > > proving theorems as much and as quickly as > > > possible because that is the business of > > > mathematics. > > > > > So we've heard. > > Then why are you asking me for the meaning of "well > educated"?
Well, the devil is in the details. Exactly what theorems in what sequence using what proofs? I know you've referred me to some books that you like, so I suppose that should do. I could add them to my collection.
I like 'Polyhedra' by Cromwell, and did I mention 'In Code'? I frequently cite 'The Book of Numbers' as an inspiration and source of ideas, by Conway and Guy. But then I have all these philosophical works to draw on as well, like 'Philosophical Investigations' (Wittgenstein) and 'Synergetics' (Fuller). I can't say all my sources are mathematicians (as you define them), even if some of them are.
In the meantime, I continue looking for ways to distill essential content to its essence. I've got the SQL Supermarket for data management, the Warring Castles for ballistics, domain and range, inverse functions, and Energy Cost Accounting, linking solar calories to food calories (we have this focus on food labeling, to get kids in the habit of caring).
I go around yakking with parents and public school teachers about how they might update their courses to better reflect the needs of employers in the region, how they might open more doors. In K-12, that's what it's about, broadening horizons, not so much narrowing them. I've even taught demo classes in the public system, both for free and for pay.
So far, I'm feeling zero.zero competition from the pure mathematicians. They might as well not exist. My advisers are more like that David Feinstein I was mentioning, but you said he wasn't really a mathematician, because he wasn't just like your teachers in Ohio, who think they have a corner on what "math" *really* means (which was news to me, an Oregonian known for my "gnu math" curriculum).
> > Sounds good. They'll get unobstructed access to > > bandwidth, and will therefore be well positioned, > > on a level playing field, to compete with other > > disciplines, many of which feature similar content, > > but only "impurely" (in alloy) by design. > > > > That isn't a level playing field as I've already > shown. Nevertheless, the sort of programs I would
Why not? Because the mean old government has singled out pure math teachers for persecution? Was that your thesis?
> pick for "pure math" for school aged children already > exist in a format like that -- the Gelfand > Correspondence Program, for instance. Is that what > you are asking for in terms of an example? (Again, I > have already said this before.) And it has certainly > had a following. The Math Olympiads are another good > one. > > But, these are viewed as special -- only for the most > gifted students. It isn't treated as mainstream as > it ought to be because the public thinks that the > typical thing encountered is more or less good math > for the average high school student. What I am > talking about is math reform and I am saying that we > would be doing that sort of thing a whole lot more if > the actual math teachers took courses that were > closely aligned with what mathematicians do rather > than what engineers and physicists want for their > students.
And what I'm saying is we engineers, physicists, computer scientists, anthropologists, psychologists, historians, linguists and so on, may not *want* to force our kids through anything even remotely like what you're branding and selling as the "pure thing". So we're busy setting up and staffing our numeracy and literacy tracks, making sure we've got our ducks in a row.
But we don't want to be pigs about it. You math purists should have equal opportunity to step up to the plate and bat your balls around. We should make sure you've got access to the same tools. Beyond that, it's up to the pure mathematicians to self organize. I'm not going to waste my time trying to manage it all for them. That's not in my job description.
> > Yes, we agree. Maybe more than just temporary. > > > > No -- it would be just temporary because at the > professional level, there is a lot of demand for > theorem proving. And, it is really that demand that > drives the support for math departments. Things like > having calculus in your department, in the long run, > doesn't buy the department nearly as much as people > often imagine.
OK, sounds reassuring. Just remember you can't stake a monopoly over theorem proving. Anyone with the proper insights can stake out a proof. They don't have to be mathematicians to do it. But of course it takes a lifetime of study to reach "the frontier" in some specialties (a safe and secure niche).
Here's an example of something a physicist I know proved, Robert Gray: any tetrahedron defined by four noncoplanar vertices in the CCP has a whole number volume relative to the 4-ball tetrahedron of unit volume. That's something from my web site. We were studying these things called Waterman Polyhedra (which I named after Steve Waterman), now a part of the geometry literature.
Coxeter, a geometer, was another who celebrated the contributions of non-mathematicians to his field. He liked those geodesic domes and thought Fuller deserved to crow about his 10*f*f + 2 for generating cuboctahedral numbers, new information at the time. Sloane gives Fuller a lot of credit in the appropriate OEIS entry: http://www.research.att.com/~njas/sequences/A005901
People like Coxeter and Sloane give pure math a good name, by being friendly to folks outside their discipline. They're good diplomats, poster children for your trade.
> > But I'm *very* clear I'm not one of your pure > > mathematicians, with my students, with myself. > > That'd be a huge step down for me, this late in > > my career. > > > > Kirby > > So you really feel like most 8th graders graduating > from your group theory class could, say, prove that > every group of prime order is cyclic or that a cyclic > group is abelian? Of course, I don't mean give an
No way. About all I'm getting out of group theory for them at first is some familiarity with the concepts, plus a feel for what "inverse" means (multiplicative, additive), so that when we get to rational numbers again (aka fractions), they'll understand in what sense - and / (subtraction and division) are "syntactic sugar". I've explained this before.
We might also write a little Python program to convert a permutation into cyclic notation (something J has a primitive operator for, but in Python we'd need to write it).
Here's some code we might play with, just by way of example:
> example of an abeliean cyclic group or something like > that. Can they put together a coherent and > sufficiently formal argument that proves these > assertions or an assertion like it? If that is > indeed what you are doing, then perhaps there is more > to it than you at first let on. If not, then I think
Not what I'm doing. They learn about CAIN and Abelian, a fun Biblical allusion I didn't make up.
For teachers, I supply more background at my Vegetable Group Soup (cited earlier).
> we are back to teaching them some particular fact > without teaching them the real justification for that > fact. Instead what you are doing is giving them a > rationale for why someone might think the fact is > true, not the actual justification for the fact that > *proves* that it is true. Something like that is > little more than a mnemonic device to remember the > fact by.
Actually, I'm helping them learn Python (a computer language) and use group theory as grist for the mill, an interesting topic.
We'll let the pure mathematicians teach it the formal way. I hope they'll keep up with that gig. Not my job to see that they do though. Not my problem.
> It's really much deeper even than math education -- a > fact without justification is not knowledge. What do > your students *know* when they are done? Have they > acquired any knowledge? Or, do they just have vague > rationales for facts that they believe without > justification? I would rather my children have a > little bit of actual knowledge than a lot of facts > without justification. And, I would almost rather > they go completely ignorant than to have a lot of > fallaciously justified beliefs.
What they maybe get are a lot of skills. They're probably not so hung up on "justification" as you are. Nor are they completely ignorant of the role of proofs in some circles.
I'll leave it to the purists to justify their lifestyles, their teachings. I'm not paid to do their jobs for them.
Criticize all you want. You're not my director, and the people you know as pure math teachers don't consider themselves my boss.
Your only way to keep your vision alive, is to work with your colleagues to make sure you pass the torch effectively. Every discipline has this challenge.