On Thu, Jan 2, 2014 at 3:46 PM, Robert Hansen <email@example.com> wrote:
> On Jan 2, 2014, at 1:03 PM, kirby urner <kirby.urner@GMAIL.COM> wrote: > > > I think there's overwhelming evidence that integrating more programming > into math class works, and works well. > > I don?t know how you can make that statement. We are at year 50 of this > experiment. And you are complaining how no one except professional > programmers, can program. >
That's not really my complaint. My complaint is the best, funnest [sic], coolest toys are locked away in the closet, out of reach, because of daddy-knows-best theories that don't hold water.
People complain that too many walk away from STEM but then the same people don't bother to keep their own skills up to date. Adults don't share the best toys because they're ignorant of how to use them themselves and are too afraid to admit it. Junior might realize the curriculum has no clothes. Easier to invent theories about ADHD and pass out the amphetamines.
Why can't junior play around with a full featured ray tracer for math credit? There's your XYZ coordinate system, right there, issues of perspective already handled. Drawing an icosahedron by hand is tedious whereas with a ray tracer what you get is textured, has shadows.
Yes, it's still a workout, but look at the payoff. And to get the coordinates in the first place you needed to know about those three mutually orthogonal phi-rectangles -- or that certainly helped. Lots of math here (no CS at all).
> > And I am complaining that even the professional programmers can?t > program.:) >
> To wrap up, this is my position - > > 1. I do not think higher level classes should be mandatory. >
Oregon requires three years of math to get a high school diploma but I don't know if that has anything to do with "higher level".
> 2. I do think that CS should be offered in high schools, as an elective. >
I don't think math and CS need to be split apart at the high school level. Doing so is an artifact of a subculture: a dumb one.
> 3. I do not see any pedagogical advantage of teaching anything with > programming, except, programming. >
Here I see your lack of imagination (and real world experience) as astounding. You've never taught high school math to high schoolers. I have.
> 4. I think technology has a place in math classes starting in the 7th > grade, but only in context, not as part of the lesson. > > A rather unclear statement. Presumably you think all those textbooks that dwell on how to use a TI-calculator are misbegotten. I think so too in 2014.
> This is after watching the ?experiment? for decades. > > Further explanation of (4) - > > When I say technology in context I don?t mean programming or Mathematica > or CAS. People use spreadsheets. Engineers use spreadsheets. Everyone uses > spreadsheets. They use them today. Spreadsheets should be ushered in during > the 7th or 8th grade to do for the students what they do for the 100 > million other users. Release them from tedium. Not to replace or enhance > any learning tasks at hand. Computers can?t perform algebraic reasoning. > And if the student doesn?t get in deep with the parts of algebra, neither > will they. I am assuming they know arithmetic well by now, and like salt, > they get enough practice in their daily lives without artificially adding > more as we had to in grade school. Numbers are everywhere and now that hey > know about them, they will be adding and multiplying enough in their daily > lives. That is my motive and my only motive for introducing spreadsheets. > It has no more pedagogical value than the pencil sharpener. But it relives > tedium. >
Spreadsheets are fine but they tend to hide the formulas and also add a layer of irrelevance by making the cell addresses important. The whole business of absolute versus relative position... lots of poopka you hafta learn, lots of overhead.
In a computer language, which is a mathematics notation that just happens to also be machine executable, you can write what look very much like ordinary textbook functions:
And then you may compose them. Composition of functions is a standard / everyday math topic:
def h(x): return f(g(x))
def k(x): return g(f(x))
How easy is it to compose functions in Excel? Plus you get to start thinking clearly in terms of types? What type of thing can x be? An integer? A float? A complex number? All of the above? That's the mathematical concept of "domain". We say f(x) where x is a "real number" (complex numbers usually forgotten about). No CS here. Just math.
I agree that relieving tedium is a big part of it, but for some people, like those graphic artists you say aren't mathy, a spreadsheet just doesn't hack it. What about fractals? Don't they help people learn about the complex plane and what it means to multiply two complex numbers? You say "no higher level math required" but I've done fractal programming with 7th and 8th graders.
You seem to have this very lexical / algebraic mindset that relegates the right brainy graphical stuff to the "not mathy" bin, but that only shows you're remarkably unaware of how some intelligent good-at-math types don't happen to be biased the same way you are. Your whole attitude is "what works for me is what works for any bright good-at-math type". Wrong, very wrong.
> The reason they use calculators is that when they started using > calculators, PCs were still sparse and calculators were the next best > choice. > > Clearly a totally lame excuse in 2014, right up there with the dog ate my homework. Any teacher explaining why we're using calculators today when we could be using ubuntu should be laughed out of the room.
But if you've been paying any attention at all, you know that math teachers are excited about Geogebra, designed for middle schoolers. The 3D version is out there in beta. They're also getting more interested in SAGE and similar products. The price is right (free). All that's needed is time to build skills and that's precisely what this subculture does not allow teachers. They're on a treadmill set to fast forward. Taking the time to learn new skills is considered saintly beyond the call of duty (meaning unpaid).
> > The reason we hate calculators is that the damn screens are so small. But > we are middle aged. The screens are not small to kids. I remember my youth > rather well. I could disassemble and reassemble a watch using just my naked > eye. Now I can barely even tell the time without my glasses. >
We don't hate calculators, we hate the lazy lame excuses used to keep them entrenched. It's corrupt. It's about money. In the USA, money trumps a lot of things, because of the ideology of "free market forces".
> > But it is high time they get with the 22nd century and use spreadsheets > like everyone else. >
Using spreadsheets would at least bring them into the 1980s or 1990s. Currently most high school mathematics is mired in the earlier 1900s.
> > In higher classes in later years, when they have sufficiently conquered > algebra, then the door is wide open. > > Your insistence that "conquering algebra" is the be-all-end-all is remarkably shallow. Draw two unit edge squares next to each other and draw the diagonal: 2nd root of 5. Draw the body diagonal of a unit-edge cube: 2nd root of 3. The study of roots is part of math. At this point, their historical significance should be discussed since math without history is like butter without bread (people need *orientation* and *context*). But of course in this lame / backward culture of ours, history is completely divorced from math, a travesty and a tragedy.
> Note: There is nothing wrong to exposing them to Mathematica and tools > like that. But don?t try to use it. Not till much later. > > Bob Hansen > > I'm still going to use a ray tracer and some simple programming to teach about fractals in 8th grade.
I'm also going to teach about a unit-volume tetrahedron that divides evenly into other familiar shapes:
(these two are "duals" -- if the concept of dual is not in the common core, write to your state governor).
Rhombic dodecahedron (same long diagonal): 6 tetrahedrons
(a space-filler, rediscovered and studied by Kepler among others)
Cuboctahedron (same edge): 20 tetrahedrons
(the basis for the CCP, a core concept in STEM)
If you don't know this stuff, in my book you're probably operating at about 50% capacity (i.e. you're a moron compared to what you could be). It's inexcusable to not do this kind of basic spatial geometry, starting around 2nd or 3rd grade. The adults in our current generation are out to lunch. But we're liberals. We let morons teach school and go on to work in "business intelligence" or whatever.