[Edu-sig] Age groups

[Kirby Urner]

>I think this raises a larger issue, one of the few wholly valid points in
>the recent infamous criticism of CP4E:  you need to define the goals of a
>curriculum.

My view is that programming languages give us more opportunities
to present some of the same content you might find in an older,
pre-computer math notation.  Instead of using a big, funny-looking
greek letter, called 'sigma', with these little i,j subscripts,
we might try communicating some of the same ideas using a do-loop
instead.  Get back to the sigma later.  Lots of kids will get 
into it deeper if we "lose the geeky greek" and just do it in 
something less alien -- like APL <grin>.

The advantage of taking a math notation and re-expressing it in
a programming language is you get something that runs, executes.  
This means you can start automating processes which are too tedious 
to do by hand, and your math curriculum can start taking advantage 
of this ability to do operations "in bulk" (every "add" operation
now only costs you a penny, in terms of student time/energy, instead 
of 10 dollars).  You can really start to take advantage of the 
computer's ability to render spatial forms. "Beyond Flatland" is 
very much a theme of the emerging 21st century math curriculum, 
the way I see it.

BTW, I agree with comments made earlier that bubble sorts, or 
sorts of any kind, aren't particularly exciting.  A lot of texts 
just get bogged down in examples that are dry as bones -- are in 
no way more interesting than the most boring of math books.  And
that's pretty boring.

I think what's more challenging and interesting is to see math as 
Keith Devlin does, as a discipline which "makes the invisible 
visible".  In this sense, a math teacher's job should be about 
deobfuscating, demystifying, helping students to flesh out their 
understanding of systems and infrastructure that runs "behind the 
scenes" in the real world.  In this sense, the teacher is more a 
storyteller, weaving interesting narratives which interlace history, 
world affairs, technological innovations.

But in order to tell these stories from a math-literate point of 
view, you have to impart numeracy and, as we get closer to talking 
about our own time, computer literacy.

So in the course of your storytelling, you'll want to toss out a 
lot of connected "artifacts" which point beyond themselves to how 
the world works (sometimes I call them "cave paintings" -- because 
they're rather sketchy, schematic, idealized).  Patches of notation, 
snippets of working code, flow charts, diagrams, graphical displays...  
always trying to up the comfort level, the confidance level, around 
working with "technical communications" in their many forms.

To this end, I'd like to have sessions where we just play with some 
of the operatators a runtime (REPL) environment gives us.  Play with 
entering expressions, getting back results (that's satisfying in 
itself).  Play with writing short functions. In whatever language.
Part of the fun is reading a short explanation of the syntax, and 
then testing it for real, to see if you're getting it right. 'map' 
and 'reduce' and so on are just more toys, fun to tinker with.

Python is good because it provides in an immediate feedback setting, 
as do Scheme, APL, Logo, Xbase, Smalltalk...  That's a number one 
criterion for a "teaching environment" -- this ability to enter 
an expression and have it be evaluated immediately.

It's also good because you can set up an object structure very 
simply, and start talking about what OOP is all about.  That's 
helpful, because then you can get kids to imagine how much more 
complicated systems of objects might be used to talk about heart 
surgery, transportation, toy manufacturing, banking -- you can 
start to bring the real world into view, in terms of objects.

In sum, I'd like to see a more "Smithsonian Institution" approach 
to math learning, where we take time out to watch documentaries 
and to really appreciate what challenges people faced in the 
past, and how their mathematical abilities often entered into 
whatever solutions they managed to come up with (e.g. radar, 
encryption devices, navigation devices, ways to paint in 
perspective). 

As we approach the present day, we should be looking very 
closely at what people actually do on the job, and how 
mathematical thinking enters into that.  This helps students 
to realistically "look over the shoulders" of people in many 
walks of life -- because if you're a creative teacher, you'll 
find a "math angle" on just about anything people are doing 
(and without making it seem highly contrived -- I'm really 
against all the "faux problems" which fill math texts, 
which have the ring of inauthenticity (kids know these aren't 
real problems, and don't want to waste their time on them -- 
understandably (not talking about abstract problems, which 
don't pretent to be "real world", but the ones which do 
so pretend, and fall way short of being convincing in 
this regard))).

Kirby