February 27, 2008

ICT – honour the flipside

Category: proof of concept, maths, learning — rob @ 10:51 pm

I went back to my desk the other day, and found half a dozen large art works had been placed there, to return to some year 8 students.

The glowing creativity, and the teachers effusive praise, made me pause, since I’d just been ruling perpendicular axes on the board, and the same students were learning how to handle y=mx+c, plotting sets of numbers, drawing letters on graph papers, decoding graphical puzzles etc; and even with these variations, it all seemed relatively dry by comparison - not withstanding kids using a bit of interactive whiteboard and graphmatica

Looking at the art, I wondered again, about a topic I've considered in the last couple of years – why do we not use more ICT to explore the world of maths, from the "inside" – that is, get kids to build the simulations, build the models?

Seems that approach should be congenial in Maths, of all places. ICT and maths share some similarities, derive from the same imaginative heritage. There’s also a serious challenge in teaching and learning either well enough to really do the hard stuff.

Maths though, enjoys enough respect that we still take the top end seriously. (Most of the population might wince at the memory of simultaneous equations, and maths teachers might get bad press for being recalcitrant in the face of demands to be more "integrated", "collaborative" or other progressive terms– but overall, it seems that higher maths is justified, by whatever percentage go on to further studies).

And although ICT looks a bit more popular, with its web pages and animations and video conferencing, we seem we seem to have bought the idea that a much lower threshold of difficulty will do.As long as the kids can make something novel out of the glittering array of tools, that their parents didn’t know how to do; there is no need to remember that this flexible software is built on computer science.


(More generously, maybe ICT supports the thinking in other subjects – helps to gather or display data in science, or write essays in English, supports creativity in art with photoshop images or video in media studies etc. All of which is great).

But the art on my desk, got me thinking- i googled "interactive flash art" and recalled these beautiful, dynamc art works


(from www.levitated.net/bones/doubleForm/ - site has author's contextual notes)

(right click forward/rewind to replay) same site for this - and more - here

I know this is built on a synthesis of programming and maths - iteration of functions, rendered with flash's nice vector graphics. The interesting maths is a little hidden - one needs a decompiler to get at the code, to see how it was done - make the parameterised drawing routines visible.

Or take the apparently simple but very popular lineRider game, also coded in Flash, which the students - and a few teachers - seem to love playing. Behind the scenes, all that free flowing sketching and intuitive physics is stacked full of maths; equations for wind resistance etc ( i have had a look at this one- and its full of nice maths we could use).

Its a pity the maths isn’t more visible; not hidden away in the finished product; (although its not hard to decompile with ASV). Recalls Papert – the mathematician – and his brilliant ideas of Logo for children. That fact that "Logo" now sounds like it is relic of a dim era, rather than a high point or seminal approach to educational technology , is part of another story. To use a logo term, we are inclined not to make a serious effort to grapple with the "flip side" of ICT; the detailed control side; which Papert hoped would render geometry and physics and other fields (dynamic systems etc) open to intuitive play - or the play would inform the code - without the excessively formal pathways that are often the way in school.


These days we usually let someone else produce the code - we don't see that as part of the learning; maybe because we think ICT is meant to be simple. Graphic calculators, or Graphmatica - interesting as they are, tend to provide a short cut to the results; and so we are limited to "using the application’. The irony of this is it can kill the promise of building it; of taking the challenge of students being producers as well as consumers of digital simulations, as this visionary school puts it

Anyway given these art and lineRider game examples, I thought, why not use flash, with its slick vector graphics, in maths? - its just as powerful as the old turtle wandering around, but might be more familiar today. Lots of kids already know how to use the drawing tools in Flash, and can animate cartoons – little South Park skits. Its rare for them to tacke the drawing commands, though they are simplicitly itself:

moveTo (x,y) //moves the virtual pen to x,y
lineTo(x,y) //draws with the virtual pen x,y

nothing hard there – in fact almost too simple - but the essence of wandering around the plane, drawing is all there

(the equivalent of the old "pen down" or pen color" commands is :


lineStyle (thickness, color, transparency)

using nothing much more than that, one can make a simple cartesian grapher :

(type numbers into the equation and press draw) (better when viewed larger )

(a minor point for the curious– Flash, like lots of computer graphics, puts 0,0 at the top left corner of the screen – so I’ve used an empty movie clip called ‘origin’ at the mid point of the screen – and by drawing there we inherit its idea of 0,0 at the centre of the screen)

Anyway, the point of this is not that we needed yet another graphing tool.

Its to see how easy it is to make a zoomable linear grapher, in a few dozen lines – about as much as a page of working - final file is only 5k.

A little more extension, and it would be able to plot any function – exponential, sine, whatever - might be a nice task for some programmer kid somewhere. So if an ordinary teacher – who is certainly no maths genius - can do it, and could of on the old Commodore 64 - what of our "digital natives"?Why do we not see them there? Students seem to teach themselves, or watch siblings and friends, use Flash from a design point of view; but might need help to get started on the coding side; but I argue its no harder than the maths they’re being asked to do – and empowers it; both for learning and attractive application


if ever a rationale was needed, try this –

(a) in the space it takes to wrestle through a few problems, you can write your own graphing tool

(b) getting kids to do this forces them to grapple with the concepts and modelling involved

(c) can be extended into art or games.

(d) gives an inner perspective on wrestling with functions and space

(e) If this is the information age, modelling simulations should surely be a useful tool, and valid in maths

(f) taps the natural engagement many minds have with a computer

(g) can be extended and modified. (eg first version of this didn’t have zoom –which made it easier to start off - and then extend; unlike a pen and paper problem which stays fixed)

(i) Maybe there’s an inter-disciplinary approach sitting dormant - computer art & maths - here – but would take more than a semester to unlock, given where we are starting from.

So just a little proof of concept.

(I’m not, though, about to reinvent the year 8 course on these 'lines" :) , for a few good reaons:

- in the current curriculum, with its boundaries (this is Maths, this ICT) there is not enough time or access to layer the programming mindset into maths

- its not as easy trying a new approach ; needs some support to make it work - eg a few people to bounce it around with - since canonical maths approaches, and content, both get modified in the process - and notwithstanding the promise of taking control at this level, in the world of "using applications" ("i just want it use it") it looks too slow and laborious (and while i'm suggesting the payoff would be worth the effort, it doesn't seem a common angle for some reason).

This content requires Adobe Flash Player.


anyway code is above if anyone wants to extend it; eg to calculate values for other functions etc (download here)

might be of interest - rob

February 16, 2008

maths wars : speciality or generalist approach

Category: maths, learning, math wars — rob @ 6:05 pm


I wrote originally wrote this post ages ago on the local maths network blog ; but think its worth repeating here. (At the time i saw this as "KLA specialists versus middle years approaches" because that was the tension i saw in the local system - and the way maths teachers spoke of it. I have since realised thats just the local expression of a larger tension about maths - the so called "maths wars").

I think i'll write something else on the "maths wars" thing - but first here's my original post :

There is a philosophical divide in many schools; do we emphasise “middle years” (more generalist teachers, less segmentation of learning into discrete subjects, more emphasis on engagement, social and emotional competencies, individual approaches, generic thinking skills), or do we for go for subject discipline (emphasis on specialisation, structured content, exams) as soon as possible?

On of the purposes of (the maths network) is to have some debate about this (a lot of maths teachers are on the latter side, and they feel they can’t air their views, since it feels unpopular in the current environment, against VELS etc). And some others would say, we can do both; we can engage them early on with generalist strategies and build healthy foundations – good thinking skills etc - for later specialisation. But for the sake of argument, lets “pretend” many middle years proponents and KLA specialists don’t agree with each other.

One argument that is often heard against the middle years goes like this : its all very well to aim at engagement, and to explore your own individual learning plans etc, but sooner or later, they have to do leave that stuff and do Real Exams. You know, Real VCE or Real University exams. And here the middle years proponents might say – well, our approaches prepare them better for this than just pushing the content earlier down the school (and reteaching it every year). And the specialists reply, they come to us with gaping holes, and our subject has a definite body of knowledge and skills that are not being taught properly, if at all.

And so it goes …

So, lets call this drive to prepare for Real Exams, the push to specialise earlier, the “top down” approach. Top Down says that Education in general, and Maths is particular, is tough at the top end, so lets prepare for that as early as possible, and limit the generalist, middle years approaches as soon as possible.

Notice how this works: Uni >> VCE >> 7-10 >> Primary

The logic of this top down approach is that the sooner specialised maths teaching kicks in the better, preferably before the end of primary school.

OK, we’re all familiar with this landscape. But what would happen if the top changed its mind? If the tertiary sector starts looking for less segmentation of learning into discrete subjects, and started talking about pedagogy and engagement. How would we validate that chain of events, the need to prepare for the Really Big Exam, now? heres a quote from Melb Uni’s vice chancellor, Glyn Davis, on their new approach to undergraduate courses:

As specialisation becomes more and more the way the world we live in is, there’s a tendency to drive specialisation very early on in degrees. [a] Year 11 student … will be choosing subjects for Year 11 and Year 12 long before they get to university, with an eye to the enter score, and an eye to the course they want to do to university.
That‘s not a particularly good way to run your education system. It‘s better that at school, you can explore your potential, and that when you come to university for the first three years, you’re in the broad area that interests you and you want to develop in, but you’re using the time to experience a broader arrange of offerings that you might and to make your mind up pretty clearly about what you want to do. One of the key messages we’re been getting from employers - and we ‘ve been talking in great detail with them - in, say, the accounting firms and the law firms likewise are saying,

“The students you ‘re sending us are well-trained, they ‘re everything we want, except they’re not necessarily motivated because they made this choice at 17 and 18". By the time they ‘ve done five years or four years of university, and they come to us, they ‘re still asking themselves, “Is this the career choice I want?”"

 A student has spent three years in a broad undergraduate degree getting to know themselves and the possibilities and the subject areas, we think is more likely to choose precisely and carefully and with great motivation about what they want to do in graduate school.

Now that seems a bit of tertiary cat among the secondary pigeons. This was on LateLine (transcript and video here) (italics added) I did discuss this with a maths teacher, with strong academic connections, who sees Melb Uni’s new approach as economically motivated - a way to get students to study longer and spend more. I don’t know the truth of this, except to note that this was discussed in the same interview :

(INTERVIEWER) of course, some of your rivals see this move by you as a way of boosting
Melbourne’s private revenue stream. I understand in fact, you ‘ve got agreement from the minister that, in fact, your graduate students can in part be funded by a transfer of HECS-funded placements, is that right?

GLYN DAVIS: Yes. We proposed to the Minister late last year and early this year, and she has publicly endorsed a suggestion that a significant number of places move from undergraduate to graduate. That means they remain HECS-based places, which means the cost for students in undertaking the new model is unchanged. It is no more expensive.

And their professed view is different:

This is drawn from our point of view from pedagogic incentives. This is about a better education model. That’s what’s driving our changes. One of the problems with just saying, “Let’s have undergraduate degrees and then graduate degrees,” is that there has been significant fragmentation in the curriculum. There isn’t a lot of coherence necessarily in the undergraduate curriculum, and undergraduate degrees don’t necessarily articulate strongly into graduate school.

These fine motives might not be the case, of course, but just noting the arguments. Not sure what it does for the mythology of the Big Exam Coming Up, thats used to pass the message down the chain (VCE, 7-10, P-6) that they better stop playing with middle years approaches and do more Formal Mathematics, and that of course advanced maths is useful for all, if for no other reason than its looming on the final exam.