learning in progress ….

now, what cutting edge curriculum is this from?

and what would most teachers say - we don't have enough access? thats all very well for the laptop school down the road?

this from my 1985 year 12 text book, at a time when the computer ratio was perhaps 1:20 in the school

like most of my friends though, we tinkered a bit at home, typing in game code etc ...

(i learnt the DO and IF statement for example, on a Vic 20 that took 20 minutes to load 3k of memory from tape drive ... and it seemed unremarkable that the beautifully written manual would teach you how to write a binary search routine etc, or by plugging in a 16k system module, design your own characters using binary addition on an 8x8 grid .. and i was not the computer geek in the class, just a bit curious about these things

one page in to this maths 'option' :

anybody game to try even that introductory example with kids today? ( i know Bill grappled with that one .. but its unusual .. )

i was no maths genius, but something about tinkering here was a positive thing for me ...it stayed with me over the years; and i found myself wanting to open that possibility to kids i taught

but i think few would disagree that this looks like it has faded from the curriculum ... even from the IT curriculum, and certainly from the maths

for example it seemed quite possible in my first year of teaching to ask the year 10 kids to find and plot every prime number up to 20,000 ... and when that proved a little ambitious, to show them how .. i'd seen a Ulam spiral somewhere and it looked quite do-able and interesting

So while my maths and programming were rightly judged as unexceptional at the time - what surprises me since then is its quite rare to see it attempted or used as a teaching approach

here's the rest of the module - its overly loaded with questions and content; not saying its the best way it should be done ... but at least it was done

it seems to me that the maths faculty should be leading edge of these technical skills... unless i'm missing something?

so repeating some diagrams ...

this omission and exclusion seems most striking in the maths faculty ...

i guess the questions that remain for me are
(a) in terms of my research - can i justify all these diagrams ... maybe i need some hard data to back up my impressions
(b) technology in maths can soak up some hackwork... but does this mean letting a black box of a program - one we would no longer presume to try to write - do the thinking for you?
(c) should technology change content as well as make some of the work more efficient... (should it open new ways of exploration - like the humble Ulam spiral example)

(anyone following this might notice the maths wars posts 3,4  have disappeared. I'm trying to hone in one a research topic, and trying out a few ideas by blogging them ... but decided there was little mileage in those mid posts

I began to realise that was a typical expression of a bigger, and more widespread, debate about how school mathematics should be taught. This year, the debate came up again in another context so i had the opportunity to contribute to a broader discussion article that examined the issue in more detail, reviewing the issue ..

i notice this sort of discussion is current at the Research in Practice bloginteresting timing, since while i had come up with a continuum of sorts ...

and discussed the claims and counter claims under each 'side', i'm also trying to work out what to do with a masters ... and deciding which research methodology suits my style and interest has been tricky. One that appeals to me goes by the rather overblown name of 'phenomenography' - which basically looks at how people perceive things! It uses the idea that simpler views of something can often be considered to be contained within more complex ones

Chris Cope, introduced me to it ... and one his articles says

the research has found that the different ways of understanding a phenomenon are related in a hierarchy of depth of understanding based on logical inclusiveness. Deeper levels of understanding of a phenomenon are inclusive of shallower levels of understanding. An example to demonstrate these findings is a study of the different ways of understanding mathematics as a course area in ... first year undergraduate students ... A hierarchy of five different levels of understanding of mathematics was identified. The lowest level of understanding represented a fragmented view of mathematics as numbers, rules and formulae. The deeper levels described a more complex view of mathematics as a means of solving complex problems and developing new insights used for understanding the world.

The deeper understandings were found to be inclusive of the lower levels of understanding. Students who experienced mathematics as a means of solving complex problems were aware that numbers, rules and formulae were the tools used to solve problems. Students who only experienced the study of mathematics as learning rules and formulae had a view of mathematics that would limit them in terms of solving complex problems they had not experienced before.

the idea being that limited views can be contained within deeper and more encompassing ones, rather than being opposing points on a continuum, suggests this sort of thing to me:

so the view of maths as predominately skills and formulae etc is not necessarily opposed to a more open ended problem solving approach, but can be contained within it

don't know that fully resolves how its taught but it might suggest we don't just focus on skills for a purpose that is always defined as 'you'll need it later' (as in, years later)

i asked Alan Schoenfeld a similar question when writing that discussion article

Dear Alan,
Rereading your 1994 'What do we know about Mathematics Curricula' article

You mention at one point that

Things about which I am confident: (1) most mathematics can be taught in the style of my problem solving courses; (2) large amounts of mathematics can be learned as sensible answers to sensible questions - i.e., as part of
mathematical sense-making, rather than by "mastery" of bits and pieces of knowledge; (3) many basic skills can be picked up in the context of meaningful mathematical work. Things about which I am not confident: (1) how much mastery of some basics is required for competent, flexible performance on more demanding tasks; (2) what the best ways of mastering some of those basics might be; (3) how best to think about organizing a curriculum in a way that does justice to what is important in the traditional content, while engaging students meaningfully with the mathematics.

I wonder if you have since gained a fuller sense of 'how much mastery of some basics is required for competent, flexible performance on more demanding tasks' ? / or perhaps published on the topic, in relation to secondary, even elementary, curricula ?

his response is interesting:

The long answer is NCTM's Principles and Standards, which I helped write... I don't have a short answer in print, nor do I think it's a matter of percentages - one thing that can be done is to develop a lot of the mathematics in meaningful contexts, in a problem-oriented if not problem-based way. The trap one should avoid is to think that students have to "master" something before they can use it. Often pieces of something (or the whole thing) can be developed in interesting contexts, or via reasonable problems, at which point the math the kids have done can be codified cleanly. Some degree of automaticity is useful if not necessary - Arthur Rubenstein practiced scales, and nobody would say it harmed his piano playing - but he practiced them with purpose...

italics added - makes sense to me - i often think we learn by immersion into conversations and procedures we don't yet fully understand. The issue is that traditionally math curricula have not been structured like this (immersive problems from which skills and principles can be generalised) and so curriculum often seems to default to lists of content and skills, which perhaps build up to 'application'. Nothing wrong with that, but might be overdone as a pathway.

 Compare this with the rich interactvity of game engines.... the huge and sustained development that makes a world fluid and believable and compelling ... where are the versions tweaked to make maths explicit?

school barely even attempts to go there ...-
(here's a minor example of what i mean)

  Alan Kay thinks the real computer revolution hasn't happened yet... and we're still in the early days, like the first 50 years after the printing press, when it just looked like a better way to make manuscripts....

.. when there is a realm of ideas and approaches that can be tapped - click the pic to see what 6 simple building blocks can do, when we move beyond the computer as reproducing text, to having tools that allow new ways of thinking, new ways of seeing the material

 this is literally, a few building blocks ... click to see ... year 7 and 8 kids got this  at first go...

there are 200,000 other projects shared in this space,  and some rich maths ones...

in spite of the limitations of the tool, Scratch is rich enough to allow

• expression of creative ideas
• tinkering
• experimentation

...attributes often rather lacking in school versions of maths

 I’ve often noticed a few kids in school – often totally under the radar of their teachers- have quite strong skills at programming, largely self taught - even though they might be mediocre at maths or english, as judged by results.

just about no-one in school tries to the leverage the overlap...since education still tends think of 'mathematics' as manipulating symbols according to the same rules ... and computing has become a black box, where we are 'users' and others to the work of developing the simulation and model

Contrast with this

"(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."
http://www.cs.berkeley.edu/~bh/v1ch0/preface.html

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....

for example,
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 (where some of this was drafted) ]

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 approach gives more "feeling" for the differential changes; and so maybe in later years 'curl' and 'div' and all that would 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)

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 know I move into a more creative place when i model or program maths ideas - and have ever since my own schooling - and thus find the maths makes more music ... and I want the kids to experience all this via modelling and programming ...

will take some more permeation of ICT into maths... not as a presentation tool, but as modifying the discipline itself...

Papert and David Perkins describe this as "2nd order" change - not just using the new tool to more efficiently represent traditional paper based content, but allowing it to interact, modify the content and discipline itself ... will sound like sacrilege to many maths educators; less so, i suspect, to pragmatic scientists and engineers, whom we like to think we are preparing..

so ... my point is really, not that we can make demo's like this, though its fun... but that we could, in principle, teach whole courses like this ... a hybrid of maths with computers ... or even this sort of interactive art, -ie  not just apps that demo "key concepts" ... but maths thinking and "IT thinking" -(programming etc) supporting each other ... thats the boundary i think we still haven't crossed in school yet; reconceptualising how maths and ict could relate. 

Nothing much novel here - Papert and Kay were suggesting it 30 years ago; just don't think we've gone there in any significant way -programming feels a bit out of favour,  for various reaons - and so i think we are missing a key aspect of what software really is; limiting kids to being software "users" - not experimenting with the most flexible and expressive symbolism devised ... (i'm certainly no expert at this - just feel that in order to take control in creative ways kids need to be exposed to the art and discipline of programming; its what this ICT stuff is made of after all; they need to be literate here, or least get a chance to be - empowered so "behind the scenes" isn't out of reach - and this world of functions and variables could also be very useful for exploring maths in particular; could also mix into art,  word games,  media stories etc)   

(here's a compelling story on how we got to where we are  .. where the software experience is reduced to "using applications" -  i'm just finding out there was a huge educational vision around the initial explosion of IT ... not just logo ... which had the idea of kids making and exploring their own tools - which has largely gone by the wayside)

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.