[Edu-sig] More re "Python Notation"

[Kirby Urner]

I meant to shut up for a week and concentrate exlusively
on writing Xbase for money, but this edu-sig is just too=20
interesting -- can't resist...

Math notations assume human "parcers" -- increasingly=20
numerate readers as the training progresses.  A lot of=20
work goes into getting those humans "programmed" to read=20
without stumbling.

Software engineers typically have two "clients" for their=20
output, a stupid but extremely fast and efficient machine
on the one hand, and the "naive user" on the other. =20

The progammer is the mediator, writing all sorts of error=20
checking into the code, trying to anticipate what that
"naive user" might do (or a malicious hacker) and trying=20
to protect the integrity of the application (keep it from=20
crashing and/or becoming hopelessly confused).

Some of what I'm doing with Python flies in the face of=20
all this "reflex conditioning" on the part of programmers.
I'm not trying to make robust code that runs on expensive=20
computers, where crashes mean big bucks, angry clients,=20
loss of jobs.

Rather, I'm going back to the "trained human readers" paradigm=20
(not naive).  They're reading Python as an alternative way=20
to express some ideas, such as:=20

  * how to generate Pascal's Triangle using expressions=20
    with factorial in them=20
  * how to spit out the number of spheres in a triangular,=20
    square, cubic, half-octahedral or cuboctahedral packing =20

I've posted some code for all of the above in earlier posts,=20
except for this last, which I'll give here:

  def cubocta(L):
    # L is number of layers in a cuboctahedral packing
    # Returns cumulative number of spheres
    return (10.0/3.0)*L**3 + 5*L**2 + (11.0/3.0)*F + 1

In action:

>>> cubocta(1)
13.0
>>> cubocta(0)
1.0
>>> cubocta(2)
55.0
>>> cubocta(100)
3383701.0

Notice:  I don't check for negative numbers (which in this
case would actually work -- but have no geometric interpretation=20
in this context -- i.e. are "meaningless").  L is supposed to=20
be "the number of layers around a nuclear sphere" and is a whole=20
number.  In my classroom context, it may be assumed that students=20
simply know this.  We aren't going to clutter the function with=20
a lot of error trapping.

Here's a web page with an animated GIF showing a growing=20
cuboctahedron, plus other animations showing how this relates
to crystallography.  I did these GIFs using Java + Povray,=20
but could have done them using Python + Povray (I just=20
hadn't written oop7.html yet):

    http://www.inetarena.com/~pdx4d/ocn/xtals101.html

Although math books include the concept of "domain of a function"=20
(as one subscriber pointed out by private email), the economy=20
of expressions like n!/k!(n-k)! depend on our _not_ cluttering=20
the notation with a lot of redundant "error checking" to make=20
sure the archetypal "naive reader" doesn't substitute -10,=20
Root(2), or a character string for n or k. =20

In a "for humans only" text, you've codified the "domain" of=20
factorial elsewhere, probably in a different book, at a different=20
grade level. Now you just assume all this "context sensitivity"=20
on the part of your readers -- something software engineers=20
have rarely felt their workaday world allowed them to assume=20
(machines don't do context, and the naive users need to have=20
their errors trapped).

So, as a classroom-focused math teacher, my bias is different=20
from the software engineer's.  The Python I write is for human=20
readers to scan and comprehend, much as they learn to scan=20
and comprehend those "crypto-compressed" math notations we=20
already expect them to learn (a lot of geeky greek letters,=20
like SIGMA, PHI, PI, THETA). =20

Plus Python has the added benefit of being machine-interpreted=20
out the back -- so you can set up these interactive situations=20
with a computer that offer immediate feedback and reinforcement. =20
Every time you test an expression or short function definition,=20
you get rewarded with a response (kind of like having a text=20
book with "answers in the back" -- you can immediately confirm=20
your understanding, or lack thereof).

So, now that I've set up all this context, I will conclude by=20
appending a real life post to the Association of Mathematics=20
Teacher Educators (AMTE) listserv.  I don't mention Python by=20
name, but you'll see how I'm practicing this very philosophy=20
in that context:

------------------------------------------------
Archived at the Math Forum=20
thread =3D http://forum.swarthmore.edu/epigone/amte/belgandjy
------------------------------------------------

Subject:      Re: substitution
Author:       Kirby Urner <pdx4d@teleport.com>
Date:         Sun, 30 Jan 2000 16:36:19 -0800

> =3D Sally
  =3D Kirby

> For another example with perhaps makes is clearer,=20
> f(x)=3Dx^2, g(x)=3Dx+1, find f=B0g(x) and g=B0f(x). (The symbol =B0 is=20
> a degree sign on my keyboard, the closest I can come to the=20
> composition symbol, but it might not be that on others'
> screens.) When this is first introduced at least half of=20
> the students will give me f=B0g(x)=3Dx^2+1 AND g=B0f(x)=3D(x+1)^2.

When doing composition of functions, I might have a student
perform the role of x + 1.  "Whenever someone gives you a number,
your job is to simply add 1 to it and hand it back." =20

Sally takes that job.  We test it out a few times.  Looks like=20
she's got the hang of it.  Sally(x) =3D x + 1

Frank's job is to multiply a number by itself and return the=20
result.  We do a little job training to make sure he's got it=20
down.  Fred(x) =3D x * x

So let's say I start with 3 and hand it to Sally.  I get back 4.
Now I take my 4 and hand it to Frank.  I get back 16.  Suppose
I go in the reverse order.  I start with 3 again, get back 9=20
from Frank and then 10 from Sally.  Different answer.  Order
matters.

Or we could talk about other operations, besides those dealing
with numbers. =20

Sally paints anything she's given the color red.

Frank paints green polka dots on whatever he's given. =20

So if I give this toy horse to Sally first, then Frank, I get=20
back a red horse with green polka dots.  But what happens if=20
I give it to Frank first and then Sally?  Again, order matters.

   Frank(Sally(horse)) --> red horse w/ green dots
   Sally(Frank(horse)) --> horse is completely red (dots=20
                           painted over)

If my class has access to an interactive command line (as per=20
earlier posts re my favored model of a math class), they could=20
explore composition of functions by writing two definitions=20
and taking g(f(x)) vs f(g(x)) -- no need for the little circle=20
symbol I don't think, or we can show them that later (it's=20
somewhat redundant).  I'd expect students to make up their=20
own functions and test the outcomes of composition.

And again (as per my last post), I'd like to *move away*=20
from:=20

 (a) the idea that operations are always about numbers (real=20
     or otherwise) and=20
 (b) the convention of using single letters for method and
     variable names

On the contrary, I want my students to:

 (a) see "operations" as more generic than "number functions"
 (b) get in the habit of using longer strings to name their
     variables and operations

In the example below, I follow my own advice:

>>> 'A'*3  =20
'AAA'

[Note: when used with a characters, the operators * and +=20
mean 'repeat' and 'concatenate' respectively.  This exercise
presumes students are already quite familiar with this=20
behavior.]

>>> def Sally(input):
	return input * 3
=09
>>> def Frank(input):
	return "CA"+input
=09
=09
>>> Frank("T")
'CAT'
>>> Sally("T")
'TTT'
>>> Frank(Sally("T"))
'CATTT'
>>> Sally(Frank("T"))
'CATCATCAT'

In the exercise given:

  f(x)=3Dx^2, g(x)=3Dx+1

  f=B0g(x)=3D ?

I might write an alternative next to it, such as:

temp   =3D g(x)
answer =3D f(temp)

This makes it clearer that the output of the innermost
function is the input of the enclosing function.  Or I might=20
use concentric circles, with g inside of f (I've seen text=20
books that do this).

I'm sure other list members have tried many of these ideas
(plus many more), with varying degrees of success.

Sometimes it's just the notation that's confusing, and switching=20
to some other notation, while keeping all the ideas intact, clears=20
up some conceptual confusions.  Then you can go back to the original=20
notation, with more confidance that the concepts are well-anchored.

Kirby

=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D

None of this means that I'm "against" using Python in other
ways (e.g. to teach computer programming).  It's a great
segue into all kinds of other patterns of usage.  But I=20
just wanted to be clearer about my model, which owes a lot
to Iverson's APL.  APL is the first programming language=20
I ever learned, and it was love at first sight.  And APL,
as some of you know, was originally a chalk board notation. =20
It was designed for sophisticated human readers, _plus_ it=20
had the advantage of being executable in silicon circuits.
Just amazing.

Kirby