[IAEP] reconstructed maths

[Costello, Rob R]

recent comments on source code proved not very accurate ... 

So the position here - that direct instruction is simply the best way to
do maths, and every other approach is basically a failed fad pushed by
those who refuse to take a hard headed look at the clear evidence - also
seems to me to be a bit overstated (generalising too far from k-3 is
also risky)  

Yet I think its useful to consider claims like this - helps formulate /
clarify a clearer exposition of the middle ground 

I'm not really going to attempt that in any serious way - I'm sure the
second half of Alan's fragment will do it much more deeply than I could 

I will say that while I like direct instruction for K-3 - I have kids
this age - by the time kids reach middle school, where I teach, they
generally get so much 'direct instruction' in various algorithms for
manipulating symbolic expressions, that it kills all joy and interest in
the topic, all passion of mathematics, for any but the most able

 Historically, most "learnt" how to "complete the square" or solve
"simultaneous equations" whether they wanted to or not, - and so we have
the phenomenon where the majority of the populace has a lingering
distaste for these manipulations, a sense of being exiled from that sort
of maths, and maybe some respect for the few for whom it seemed to make
some sense 

(and even the majority of "good maths students" are those whom
examinations certified that with enough study they could reproduce the
manipulations; firstly the specimens lined up in exercises, then the
mild disguises as worded problems, finally the jumbled mix on the exam
- and so there is a level of advanced mimicry - but original thought and
expressiveness in the genre is not really likely. Might not be air
guitar; but not much sense of moving away from the sheet music either) 

it is here that I personally think computer programming and modelling
etc helps ... gives a fluency and power to using variables and
developing models 

my original question, though, was how does this sort of approach map
back onto the traditional curriculum ... 

eg  I have wondered, at times, in what sense does turtle geometry really
work as a child friendly introduction to differential geometry .... 

... I like Paperts explanation that 

Repeat 360 [fd 1 rt 1] 

Is an alternate and possibly more intuitive way for kids to explore a
circle, than the classic analytical description 

(x-a)2 + (y-b)2=r2 

(and when differentiated according to the rules for doing this...  
dy/dx =   --- [an expression too full of indices, brackets and square
roots, 			to be formatted into an email ]

and so what intuitive meaning does a student see in the rate of change
of y with respect to x 

and which version will a child most likely see in school?

The former (logo) approach gives more "feeling" for the differential
changes; and so maybe in later years curl and div all that be likely to
make more sense, if one had played with the 'feeling' of curves like
this ... or at least, maybe one is more primed for some sense of the
mathematical objects ... maybe ... (like paperts 'gears' becoming the
mental tool he spun to appreciate what 20=4x+5y meant)  

But did Papert move from the insight that computers could be used for
exploring vector calculus via incremental changes, to developing
Logo?... or did his mathematical mind propose this rationale for the
interesting software that he had generated ... ? 

And why is the mapping between these domains, across them, so weak in
school maths - with all the effort expended in both ICT and maths ...
why not better allied  ... I can barely find a colleague, apart from a
few via email, who think this is a key 

Yet I know I move off the sheet music when I model or program maths
ideas - and have ever since I was a kid -... though not anywhere near
the level of some here no doubt --- and thus find the maths makes more
music ... and I want the kids to experience all this via modelling and
programming ...  

But is it legit .... are these disciplines really intimately reflecting
each other, is it just an interest of a few   ... 

I think this question will be in the minds of many .... its neat, its
powerful, but is it the real deal, math wise?  

(and for the K-3 study - I'm happy with lots of DI for my son - and it
takes more teacher skill to set up an enquiry and see it go somewhere
.... so I don't blame them if they don't do it ... but don't through out
the DI as well ... and don't do it to death in later years either ..

http://www.contemporarycollegealgebra.org/national_movement/an_urgent_ca
ll.html

and for the contestable nature of all such educational studies ... 
http://math.coe.uga.edu/tme/issues/v14n2/v14n2.Clarke.pdf

... and the somewhat risky link between assessment and learning theory -
the risk of reifing a set of techniques as 'math' 
http://epicpolicy.org/publication/psychometricians-beliefs-about-learnin
g


re the "Singapore maths" curriculum : 

my understanding is that Singapore in spite of its high achieving status
(as per tests) revamped its curriculum to reduce the simple recall of
technique, however successful that was, and promote more creative
thinking ... 

"In 1999, Singapore's Ministry of Education decided to reduce the
content in the curriculum in order to provide room for teachers to
implement key initiatives (namely the infusing of thinking skills and
integrating the use of Information Technology in lessons and the
delivery of the National Education messages).  Curriculum content were
reduced by up to 30% for most subjects."
www.singaporemath.com/Singapore_Math_Story_s/10.htm

so ... not meaning to do write anything here, but I'm too interested in
it to let it go ... 

cheers 

rob 

> -----Original Message-----
> From: its.an.education.project-bounces at lists.lo-res.org
> [mailto:its.an.education.project-bounces at lists.lo-res.org] On Behalf
Of
> Albert Cahalan
> Sent: Friday, 18 July 2008 1:58 AM
> To: Alan Kay
> Cc: its.an.education.project at tema.lo-res.org
> Subject: Re: [IAEP] reconstructed maths
> 
> On Thu, Jul 17, 2008 at 10:55 AM, Alan Kay <alan.nemo at yahoo.com>
wrote:
> 
> > However, where did I say anything about skipping arithmetic?
> 
> Technically you did not. Can I get you to agree that all children
> must memorize traditional arithmetic methods long before getting
> any exposure to vector calculus? Can I get you to agree that
> constructionism does not work for teaching math?
> 
> In case not, please note that you're up against an independently
> reviewed study that would cost about 3.3 billion in 2008 dollars.
> In this real-world test, all 5 constructionist programs failed.
> Personal experience, even 35 years of it, does not compare.
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