[IAEP] reconstructed maths
alan.nemo at yahoo.com
Mon Jul 14 16:08:58 CEST 2008
A few thoughts about reconstructing mathematics ...
When considering a subject, we need to set some thresholds as well as try to define it. In tennis, there is a definite notion of a threshold called "intermediate", where lots of real tennis is happening at increasing levels of fluency. Tim Gallwey, the great tennis teacher pointed out that most of the activity in beginning tennis is "chase the ball" and because the most time is spent doing that, the beginners are actually getting more fluent in chasing than playing. Gallwey therefore tried to come up with an introduction to tennis that, using various tricks and scaffoldings, got the novice immediately involved in intermediate actions. We thought that this was such a good insight that we incorporated it into the new GUIs we were designing at PARC.
Similarly, from the POV of a former guitar teacher and player, "playing guitar" has a threshold that excludes "air guitar", "Guitar Hero", certain kinds of noise, and "not enough fluency to make music yet". We can use "air guitar" as a metaphor for (a) taking such a small subset of an activity that only some form and essentially no important content is being done, and (b) for using form over content to fool ourselves that we are "players" and "part of the club".
And, similarly, in teaching guitar, the careful teacher tries to find ways to get the novice started off as a low intermediate rather than a "beginner playing the equivalent of 'chase the ball' ", and to get into the start of the "real deal".
The above-threshold real-deal stuff could be called "hard fun" -- and most who get beyond the thresholds will say that "hard fun" is also "more fun".
Sports and music are usually optional choices made by learners. But
most societies make obligatory the learning of certain human knowledge
and thought patterns, and mathematics is usually one of these. Two problems that arise is that not everyone is well-motivated or well-setup to put in the hard fun, and there are many temptations for all parties to find "air maths" as a way of pretending the real learning and understanding is happening. A few decades of this and most of all ages no longer understand what the real deal might and should be. This makes it much easier to continue the "air math" process.
John von Neumann defined math as "relationships about relationships", and Bertrand Russell quipped that it is "just p implies q". It is not an error to categorize "really careful thinking" as "doing math". And, mathematics is plural for good reason so it can admit a wide variety of "careful thinkings" from the past and allow new ones to be invented and included.
So computing certainly does qualify as "a valid math". But the place(s) we have to be really careful about is (are) the threshold(s) that need to be defined to avoid "programming as 'air math' and 'Computing Hero math' " which can be difficult distracters to learning the real above threshold stuff. So we need to find out just what constitutes "hard fun" with our new mathematical materials.
To continue with the "materials" metaphor, we can notice that not a lot about architecture, structural engineering, etc., is learned merely by laying bricks or making a doghouse from scrap materials. One of the above threshold ideas that moved the ad hoc math special cases math of the Babylonians and Egyptians to a much simpler and more powerful scheme of thinking was the invention by the Greeks of various kinds of abstractions -- and ways to abstract -- and normalizations -- and ways to normalize -- large numbers of special cases into a few compact expressions of general and powerful ideas.
Much of the problem with "school math" is that it is much more about brick-laying than architecture and design -- there's a lot of math in arithmetic, but essentially none of it is shown to the kids, and they are dragooned into bricklaying and special case rules rather than powerful principles.
Similarly, a very large amount of programming today is more like brick laying and doghouse making with lots of special cases and weak to non-existent abstractions -- and this doubtless is one of the largest factors in the almost lack of correlation between getting skilled at programming and getting fluent with math. This is a real pity, since programming can be used and taught in much stronger ways to both make programming better and to build real mathematical abilities.
Two of Seymour Papert's most important insights about above threshold math-with-computers for children were to (a) find and use the real mathematical thinking that children could do at each stage of development, and (b) to both pick from the large body of existing mathematics and to invent new mathematics that embody the most "powerful ideas" that humans have come up with. One of many such examples is how to use the children's ability to add (and to think additively) and to physically move their bodies to make for them a powerful and valid version of Gauss' Differential Geometry which covers some of the most important parts of vector calculus in a way easily learnable by children.
An example of new and powerful mathematics -- that is not particularly tractable without computers but works really well with them -- is using feedback in dynamic systems for both stability and making progress with incomplete information.
Another important idea here is
----- Original Message ----
From: "Costello, Rob R" <Costello.Rob.R at edumail.vic.gov.au>
To: Bill Kerr <billkerr at gmail.com>
Cc: dfarning at sugarlabs.org; its.an.education.project at tema.lo-res.org
Sent: Saturday, July 12, 2008 1:03:19 AM
Subject: Re: [IAEP] reconstructed maths
Love that paper
But it does tell me that all this is still
pretty speculative and early days
not much in the education system that
builds this thinking it seems
hopefully sugar / olpc might give a boost
to this, be a catalyst for these approaches
From:Bill Kerr [mailto:billkerr at gmail.com]
Sent: Saturday, 12 July 2008 5:28
To: Costello, Rob R
Cc: dfarning at sugarlabs.org;
its.an.education.project at tema.lo-res.org
Subject: reconstructed maths
On Sat, Jul 12, 2008 at
11:51 AM, Costello, Rob R <Costello.Rob.R at edumail.vic.gov.au>
> what should the ""reconstructed mathematics" look like?
I wrote a review of a Papert paper about this in April
(I describe the Papert paper as very interesting but all over the place in
terms of its presentation)
On Sat, Jul 12, 2008 at 11:51 AM, Costello, Rob R <Costello.Rob.R at edumail.vic.gov.au>
[impatient developers worried about too much talking and not enough
doing might want to skip this teacher question]
The relationship of mathematics to programming is of interest
Brian Harvey has made some of his text books available online and he
says, in the preface to one of them
(http://www.cs.berkeley.edu/~bh/v1ch0/preface.html ) that :
"(If you like programming, but you hate mathematics, don't panic. In
that case it's not really mathematics you hate, it's school. The
programming you enjoy is much more like real mathematics than the stuff
you get in most high school math classes.) In these books I try to
encourage this sort of formal thinking by discussing programming in
terms of general rules rather than as a bag of tricks."
Papert of course had strong views on this - that school maths was too
dry, and that playing with the turtle gave even young students access to
ideas like vector calculus, in a more intuitive way, without the
formalism normally associated with these ideas
Similarly Alan Kay, ("The real computer revolution hasn't happened
"One of the realizations we had about computers in the 60s was that they
give rise to new and more powerful forms of arguments about many
important issuses via dynamic simulations. That is, instead of making
the fairly dry claims that can be stated in prose and mathematical
equations, the computer could carry out the implications of the claims
to provide a better sense of whether the claims constituted a worthwhile
model of reality.
And, if the general literacy of the future could include the writing of
these new kinds of claims and not just the consumption (reading) of
them, then we would have something like the next 500 year invention
after the printing press that could very likely change human thought for
these ideas are congenial to me .... tasted something of this in my own
as a teacher I've wondered why we don't make more use of the overlap
between maths and programming .... and have tinkered with such
But .... I'd also like to round this out with a question / reflection
Programming, in itself, with variables and functions, is not quite
maths, is it?
Or ... does not seem to map very directly against traditional curriculum
Is the problem traditional curriculum? Papert (Mindstorms):
Faced with the heritage of school, math education can take two
approaches. The traditional approach accepts school math as a given
entity and struggles to find ways to teach it. Some educators use
computers for this purpose. Thus, paradoxically, the most common
use of the computer in education has become force-feeding indigestible
material left over from the precomputer epoch. In Turtle
geometry the computer has a totally different use. There the computer
is used as a mathematically expressive medium, one that
frees us to design personally meaningful and intellectually coherent
and easily learnable mathematical topics for children. Instead of
posing the educational problem as "how to teach the existing
school math," we pose it as "reconstructing mathematics," or
generally, as reconstructing knowledge in such a way that no great
effort is needed to teach it.
If is so - what should the ""reconstructed mathematics" look
Much more modelling?
What sort / style of programming helps?
What sort of thinking involved in mapping programming / modelling onto
Do we have to convince educational authorities to respect recursive
experiments in Scratch/Logo (which my year 8 students enjoyed) for
example, as what maths thinking "really is" ...
Alan Kay talks of wrestling with creating suitable models that span
teacher and kid skills, allow some learning from both, and get at deep
maths .... j
Assessment systems in the western world are also not very tailored to
this - we don't assess these models - which impedes the take up of the
ideas ... whereas I could legitimately program in the final year of
secondary maths course in 1985, I don't think it would fit in today;
relegated outside the maths curriculum
But how isomorphic are the domains of maths and programming - and how
accessible to most kids... questions I wonder about ...
> -----Original Message-----
> From: its.an.education.project-bounces at lists.lo-res.org
> [mailto:its.an.education.project-bounces at lists.lo-res.org]
> David Farning
> Sent: Saturday, 12 July 2008 9:21 AM
> To: its.an.education.project at tema.lo-res.org
> Subject: [IAEP] Sugar Labs, LOGO and Brian Harvey
> What is the status
of LOGO for sugar? Is it a high priority item?
> As much as LOGO I would like to bring Brian Harvey, the original
> of BL, into the project.
> He has a wealth of personal experience teaching people how to program,
> he has a strong interest in LOGO, and is a good guy.
> Brian's page is at http://www.cs.berkeley.edu/~bh/ .
> ucbLOGO's page is at http://sourceforge.net/projects/ucblogo/ .
> If Sugarizing logo is a priority we could do much worse then point new
> contributors to Brian's group to get their feet wet before diving into
> I know neither the value of bringing LOGO into OLPC nor the cost of
> Sugarizing it to make a valid cost benefit analysis. If some one
> do that analysis and it seems like a good idea it will try to get the
> collaboration started.
> In my role as 'wiki watcher' I see quite a few people register, ask
> they can help, and disappear when no one responds.
> Its.an.education.project mailing list
> Its.an.education.project at lists.lo-res.org
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