learning in progress ….

A few of us just visited the Aus Maths and Science school in Adelaide; an open plan, highly connected / immersive, ICT rich environment, that we seem to be heading for. Even there, though, integrating maths and science is proving difficult; and i'm not sure IT is fully harnessed, yet, for maths.   

I also read a book recently, what “Video Games have to teach us about Learning and Literacy”. It  takes a reflective view  - a player’s view - on the challenges of learning video games; why do people pay good money for long, difficult learning experiences? how do game designers scaffold the levels so the experience is hard but learnable? what might schools learn from this? I like his thoughts on role and identity – you get to “be” something cool from day one; similar to when schools allow you to work as a scientist, rather than learning 10,000 things “about science” for the “future”. David Perkins of Project Zero had similar ideas of letting students “play the whole game” in school, albeit in a cut down form, not just endless “learning about”. Traditional technology offers the same role based learning, you get to be the cook, or the metal worker. Anyway, I wonder if we can do the same with maths; maybe using computers for modelling.    

(warning - long post!)
seems to me computers are supposedly the latest thing in maths; but its not always obvious, beyond a few useful applications*, what we do with them. But from a “players point of view” I can think back to computers and maths in my schooling . It seems remarkable now, but in 1985, when computers had only been in schools for a few years, the year 12 “applied maths” textbook contained an optional programming module, which assumed that students could cope with the BASIC language, that the first generation of personal computers all seemed to have inbuilt.

.Now, I was not a precocious maths student -  I had struggled for a while with transposition of formulae in year 9. On the IT front, we had a little Commodore Vic 20 at home, and using the well written manual that came with it, I had learnt BASIC. Like a number of my classmates I was quite interested in these early computers, and tinkered a bit, largely independently of school, playing and trying to write simple games etc.   

So when the maths course allowed a programming and modelling option, a few years later, most of us took it. I usually pretty textbook bound in approaching the calculus problems. But something about playing with the functions on the computer opened up more options for me. I recall one task was to calculate area under a curve, numerically and analytically, and compare results. I can still remember that task, and some ways I found to extend and generalise it, while 100% of bookwork exercises have faded from memory.

A pure mathematician might despise the mechanical nature of this. But the at least the programming allowed this student a little more playfulness and extension – not qualities I’d normally claim in maths.

This “algorithmic implementation” (not sure if you’d call it modelling?) might cure, to some extent, the inert (dead) nature of maths in the mind of many students like myself, who were otherwise limited to a dutiful respect for the forms of the subject. (That is, given enough sweat and examples, we could usually chase x through various convoluted problems, but it never felt generative). 

The greater skill one develops when teaching these formal problems can also be a bit deceptive, and limited in similar ways. A colleague, a physics teacher with a engineering background, recently put it in words for me : “most maths teachers are not really mathematicians” – he included himself in the “not  mathematician” category.

He wasn’t talking about capacity to pass or teach a subject, by mastering the various problem types, and recognising mild variations and disguises; he meant the capacity to generate novel or insightful combinations, based on deep understanding. By comparison Richard Feynman mentions watching a student toss a plate  up in the air, in the cafe, and noticing how it spun and wobbled. He tried to work it out mathematically, from first principles; how the spinning was related to the wobble - purely for fun. This piece of reasoning, entertained for no practical application, later fitted right into some of his work on quantum electrodynamics .Against this sort of playful creativity with maths, I think most teachers would concede they are not mathematicians nor physicists. But my point is that modelling or developing algorithms might give all of us – at lower levels - a taste of that creative work, more “creative role” in the game.

(For example in my first year of teaching I taught programming and maths to year 10s (seperate subjects as usual!) and I remembered reading something about prime spirals. The idea is you make a number spiral, and colour the primes. The patterns are interesting – not random, not ordered, but something between –diagonal “prime lodes” emerge;

So I wrote a little routine to do this; to show the students that the Excel programming we were doing could have unusual applications. (Here’s the excel file that made that diag). I have no idea of the formal approaches to this problem, but I could get it to work with a simple program.

I’m in no position to tell anyone how maths should be taught; beyond the fact that for me, as a student and then as a teacher,  programming made some it come more alive; added some power to it. Maybe its a "constructivist" angle - make your own connections.

As for the "industry" angle  - what of the engineer or scientist who supposedly uses all this maths, (and is certainly used to justify the need to teach all the top end formal stuff?)  I'm pretty confident most of these technical scientists and engineers know more "about" maths than they can ever creatively use. If you have good look at these people, most of them are also broadly literate in higher maths, without being deep mathematicians. My first job was in a hi-tech workplace, and there were some very good maths thinkers there - maybe not discovering new maths, but certainly applying it creatively; who might say things like “I’m a bit stuck on my equations with 4 dimensional 'spherical' coordinates” – or “why don’t we run some Fourier transforms on that chart trace you’ve got there”.

But even in a workplace full of scientists and engineers, the few really deep maths people were in the minority; and it was quite acceptable for the rest of us  - from project manager down - to not really follow the fine detail and just grasp, in broad terms, the outline or the result.  For example as a chemist I could test the “curve fitting” routines of the software without knowing much about the algorithms it used  - and it was also ok that the programmer had coded it  without necessarily being able to derive the algorithm from first principles. (We both got the general idea – stitching together a curve, from segments of polynomials of various degree, over a set of points –the algorithm was borrowed from a code library and was tweaked and tested till it worked.**.

I wonder if we shouldn’t aim at the tweaking and modelling approach more often. Take procedures and processes like this and code them up and experiment till they work. I think it might give more ownership of the content for those who are supposedly “ok at maths”, in the textbook sense.

might give more mileage for some students. I think there’s something in this; allowing students to play with parameters, tweak processes etc. Especially given that I think it also works for lots of people in the workplace, as a working skill set (numerical algorithms). 

(i notice some others  have the same idea  - calling for more of a modelling approach ... and this no gamer left behind video  shows how one school has done it)

(we might need to get a more  consistent programming mindset into schools. Since simple BASIC has disappeared as the default approach there is no agreed introductory language (Logo? VB? Java?).

(The idea for all this came from visiting the Maths and Science school in Adelaide – I wondered why they don’t seem to use these approaches – it seems student programming there is still largely self taught, outside the formal classes and curriculum, and they can often exceed their regular teachers in that - and programming is not widely used as a modelling tool, (with a few notable exceptions in robotics etc). Not that much difference from 1985!   (to be round out the picture, though, ICT was a big thing at that school - a lot of students cited constant communication with their study group - home or school- as a key factor in learning. that has certainly changed.)

* I know graphmatica helps to some extent with these approaches- play with parameters and see the result,  and “Maths 300” stuff uses some mini-models on the PC, and I gather Cabri is ok for geometry. I also see that many primary computers seem full of maths games, but I don’t know much about them. But none of them seem to quite get you to develop the model / algorithm yourself (?)