Whereas in maths and science work the preferred structures are often fragile and linear. So that if something goes wrong the whole thing collapses and we notice the error

So the dominant culture of maths / science (fragile linear) is at odds with our normal ways of learning something well (robust, cross linked)

In real life, our minds must always tolerate beliefs that later turn out to be wrong. It's also bad the way we let teachers shape our children's mathematics into slender, shaky chains instead of robust, cross-connected webs. A chain can break at any link, a tower can topple at the slightest shove. And that's what happens in a mathematics class to a child's mind whose attention turns just for a moment to watch a pretty cloudThis is why we need Papert's turtle or Resnick's cat

- Marvin Minsky, Society of Mind, 18.8 Mathematics Made Hard, p 193

## 5 comments:

Wow, thanks. I forgot I wrote that almost poetic paragraph—and so is yours. And your comment suggests a good next step: maybe math can be made more robust if teachers would ask for not one, but two or more different proofs for a theorem. For example, I've seen literally dozens of geometric proofs that in a right triangle,

a-squared + b-squared = c-squared. So then the students could discuss

-- why there are so many different proofs

---what they have in common

---how does a person think of a proof

etc.

I recently saw a 7th grade teacher do just that, and I admired it, but until your comment I did not clearly see how this could in fact make a a fragile concept more robust.

(Also, this might lead the student to have something intriguing to daydream about, even though it is still mathematics.

Or, perhaps we could call it "psychological metamathematics" until we can think of a better name.

Bill,

I like the idea that we build mental models and that in the case of maths and science, they are often runnable simulations. We interrogate these models by running them in our mind(http://tonyforster.blogspot.com/2008/02/problem-solving-creating-runnable.html)

and when their output differs from reality we modify them.

Thus they become detailed and robust through testing them in as many ways as possible and adjusting them when they break down.

Here is an example, a student's question just made my models of Snell's law of refraction and division by zero more robust and cross linked. He pointed out that for normal incidence sini/sinr =0/0 or undefined. That is because in this case, the refractive index is undefined, material of any refractive index will give the same experimental result if the angle of incidence is zero. That's what a refractive index of 0/0 means, any refractive index will do.

Students should be given the opportunity to interrogate their mental models in as many ways as possible. That includes as many representations of the material from as many points of view as possible. This includes negative numbers, limits at zero and infinity, on different planets, opposite hemispheres etc.

Tony

that is neat Tony, when things like that happen.

i can think of a handful of examples where that has happened when thinking about maths; a better mathematician might have hundreds i guess

i think it happens rather a lot for lots of maths/science teachers (and maybe in other disciplines as well) in their first couple of years of teaching - at least it did for me - all the stuff one had previously "learnt" seemed to have more connections than previously seen or felt, even if had been used to pass exams - "ah thats how it connects; what it means"

i read somewhere (i think in a thesis on Alan Kay) that the most powerful ideas are the most "networked" - extend into the most areas.

maths does tend to be inert, non generative, for most people

- my hope is that modelling with computers might enliven it, give more sense of application and network to start with. one of the few examples from my own schooling that made it "live" was developing commputer models of numerical vs integral solutions to the "area under the curve" problem

hence my interest in "scratch" as a possible starter for this thinking again with today's students.

much as scratch is great, i wonder why we don't have other tools aimed at this mix of maths and programming/modelling in school.. help develop some resonance and timbre for those for whom maths otherwise does not sing

rob:

>much as scratch is great, i wonder why we don't have other tools aimed at this mix of maths and programming/modelling in school

1) quite a lot of good stuff came out of papert's lab - logo, LEGO logo and the ISDP project are examples - it has been a tortuous evolution over 30 years, with ebbs and flows, as documented by the John Maxwell thesis - commercialisation and much of the WWW has been a distraction perhaps

It's one thing to develop an innovation - another to develop an innovation based on sound learning principles that the mass of teachers understand and implement - that is really hard!

2) have you seen Maths300 , the modern equivalent of "MCTP (Mathematics Curriculum and Teaching Program) is arguably the most successful professional development program" - not computer based but the old MCTP had excellent activities IMO - developed in Australia in situ as far as I am aware

I left this comment on tony's blog, problem solving: creating runnable mental models :

"great post tony

I've been having another look at idit harel's work on teaching kids the deep meaning of fractions using an instructional design approach.

One thing she found was that some kids initially had absurdly rigid ideas of what fractions were, eg. "a fraction is just the shaded semi circular part of a whole circle, the unshaded part is not a fraction, it is nothing"

Then after a month (sic) of discussing and building fractions with logo, the same student started to see fractions everywhere in the world - and built a logo screen showing fractions in a house, cars and the moon.

So the factors here would be play with the right tools in the right environment using a relaxed time frame - pretty much the opposite to how maths is taught in school most of the time.

The notion of situated knowledge rather than generalised higher order thinking is also covered by Papert and Harel in a later summary of this work (this in response to the Pea criticism). Although the popularisation of the word "situated" was done by others (Suchman, Lave, Brown) Papert / Harel argue that the idea of situatedness is an important theme in the development of logo based constructionism - along with the idea of fluency (another important word here)

Software Design as a Learning Environment by Harel and Papert, 1990, page 22 ( I just looked on line, it is referenced but doesn't seem to be available)

Visual thinking, situated knowledge and fluency are all connected concepts, I think

Another thought I had is that competitive chess is a great way to teach runnable mental models, with the enforcement of the touch and move rule. Chess players are continually testing alternative mental models, some win and some lose, so the penalty of getting it wrong is high. The debugging has to be done mentally before committing the "code", not after."

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