[Edu-sig] True or False? - A translation is a function.
kirby urner
kirby.urner at gmail.com
Fri Nov 28 20:30:21 CET 2008
My preferred way of talking about translation is in conjunction with
rotation and scaling, as like the three major things we might do with
shapes that are angle-preserving (not just surface angles, but also
central angles).
My Python polyhedron class will implement these (rotation, translation
and scaling) as three primitive functions, with subclasses holding the
data for actual polyhedra, following the design pattern of "abstract
class" hold the "meta methods" common to all polys.
But what's a Polyhedron then? By this time we have a primitive Vector
concept, except I keep those originating from the origin i.e. (0,0,0)
in 3D XYZ. My Edge class is the generic "between any two points" line
segment concept, and so Polyhedra tend to be encoded as faces (going
clockwise or counter, as seen from without), from which Edges are
distilled, each defined by a pair of Vectors (using whatever
addressing scheme).
That's all one curriculum segment, whereas another would be the
standard trignonmetric A*cos(B*x) + C, other trig functions
substituting for cos. This goes with the Fourier piece, talked about
in this archive quite a bit in the early days, what with our 'Who is
Fourier?' by LEX Institute & company.
I really dislike all the focus on parabolas minus any attention to
"dish" as a concept. The reason that's called a "focus" is because
you can get your heat vectors or radio vectors all concentrating to
that point, lots of calcs as to what's a useful threshold, below which
we don't meet the relevance test.
Modern K-12 mathematics textbooks barely hint at the engineering
aspects, being completely watered down to keep out any "impurities" (
= healthy proteins).
Most people mired in the current math tracks rarely get outdoors, have
no concept of tool use, can't do anything practical with those skills
they get tested on. We waste zero energy apologizing for such
nonsense, assure parents and students alike that we're not that poor
quality, would never retreat to that level, having experienced the
disaserterous consequences of overspecialization first hand.
Kirby
2008/11/28 michel paul <mpaul213 at gmail.com>:
> We were talking about translations in math class, and I was going to show
> the students how the translation notation in their text could TRIVIALLY be
> turned into Python code, and it occurred to me that maybe I should first
> find out if they understood that translations already WERE already functions
> or if functions and translations were two different kinds of things.
>
> The results were very interesting. Lots of hesitation and disagreement
> among the students. Some had no problem at all saying that yes, a
> translation is a function. Others got stuck on the idea of a vertical
> translation somehow violating the 'vertical line test', so they thought
> vertical translations would not be functions. That was what I was
> suspecting - our curriculum creates a really boxed in sound-byte
> understanding of what functions are all about. The kids get trained to
> respond "is a function" when they see a 'U' shaped parabola and to respond
> "is not a function" when they see a 'C' shaped parabola. However - the
> important issue is not a vertical line test - the important issue is clearly
> designating our input and output. A horizontally opening parabola is simply
> a function in the form x = f(y). They also tend to develop the boxed in
> notion that functions are things that operate on and produce single scalar
> values. That's the effect the current curriculum has. The idea that a
> function can take in ordered pairs or lists seems strange to most students
> who haven't been exposed to programming.
>
> I found that a useful discussion to have, and I thought it might be fun to
> see what the teachers in the math department believed. So, I went around
> and asked. I just wanted to see what their honest in the moment response
> would be.
>
> Amazing - I got the same responses as from the students. Some had no
> problem with the idea. Others did get it right, but with much hesitance.
> And, several did not get it right. The reasoning on the part of the
> teachers who were either unsure or incorrect was the same as the students'.
>
> It really does seem that our curriculum fosters among both students and
> teachers an understanding of functions as things that operate on and produce
> single numbers and as graphs that pass the 'vertical line test'. All the
> more argument for weaving in a computational thread in the curriculum.
>
> - Michel
>
> P.S. A couple of weeks ago I gave my first presentation at a math
> conference. I had a small group in my session, but they really seemed to
> get it. The title of my presentation was Fractions are Objects, not
> Unfinished Division Problems. It was the first time any of them had seen
> Python. They loved it.
>
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