The journey of maths…

Roger Martin the former dean of the Rotman School of Management in Toronto Canada has put forward the concept of the Knowledge Funnel and lay out the challenge of exploitation of current knowledge versus exploration of new knowledge. I believe this has implications for children’s learning, and certainly how certain subjects are taught and learned at school. In traditional approaches to educating we offer children the things that can be easily captured, canned and codified, but offer little in the way of chaos, calamity and risk. This has an impact if you are focused on ‘chalk and talk’ ‘drill and kill’ and rote methods, or if you engage in messy, participatory, hands-on, project learning.

At every given moment in time we can choose or our unconscious mind chooses for us, to focus on multiple and competing stimulus coming from within our bodies, physical sensations and data from memory, or coming from the external environment around us, including our media and communication devices, the ambient sounds of the street or forest. For those things we know, love and recognise, we have strategies or courses of action, algorithms through which we deal with them. It could be something like this: a great smell of cooking…, decide if I want to eat at this time, work out if I have enough money, choose among various menu options, and order and eat the food or go home and cook it. There is a particular order or rationale involved rather like when computer programs respond, indeed are mapped out to respond, by system designers. They may follow the form: ‘If yes, then go to x, if no then go to y’. There is linearity, a sequence of input and output. This is why we use random elements like shuffling cards and throwing dice to lend us indeterminacy in games, unpredictability – within limits – that add to the tension and fun.

There is a dependability, a reliability, a feeling of security or even safety in such a response, however there is also a price, namely a lack of flexibility, and more than an air of routine and monotonous. There is a kind of black and white, ‘if y, then x…’ it is never ‘maybe’. 1+1 = 2, but while one banana and one orange may make two fruits, it still makes only one banana and not two. To cut to the quick we can only process and/or associate that which we can recognise and name. This brings it firmly within the boundaries of the social, or at least the socially known and defined.


AO Spencer1

To illustrate this further imagine a series of microscopes set up on a lab bench. A first group of laymen are brought in to describe what they witness as they gaze into the eyeglass. The fumble and use analogy and figurative speech to describe the microbes, and virus and parasites they see. A second group of biologists outside a given area of expertise are able to make more accurate descriptions of what they see without being able to name the pathogens. Only a third group expert in their field accurately describe not only what they see but are able to name the pathogen and describe its impact on our bodies if we were infected. Whereas we can praise the prowess of this third group, they may show their weakness when asked regarding another field or area of knowledge. “Architectural Myopia” is an example of how the ‘trained eye’ can literally make people view the world in a different way. all this is similar to what Michel Foucault understood by ‘discourse’ for instance in The Archaeology of Knowledge (1969) . Discourse  – a formal way of thinking that can be expressed through language – therefore is controlled by objects, what can be spoken of; ritual, where and how one may speak; and the privileged, who may speak.

The point is widely understood. The practising of a skill, or the repetitive usage of a certain set of knowledges leads to expertise, effortless delivery, honed reactions. You can learn to tell the difference between a ‘p’ and a ‘q’ after some exposure and moderation and feedback of your responses. You can learn to read but at the same time hardly understand what is being ‘said by the text’, let alone ‘read between the lines’.  You can perform the maths operations without considering what they may be used for. This is after all, what we have often done at school.

knolwldege funnel

The knowledge funnel is a kind of three-part process where raw, unsorted, non-interpretable, stimulus enters the senses, and becomes less wild and more domesticated. Within the funnel raw experiences or things like the letters of the alphabet are drawn together and grouped in simple arrangements that begin to make more sense, before reaching the level where they become fully formed words.

Letters then, from a preliminary phase where a baby can’t recognise them at all, but sounds and tricks to channel attention, moves through phases before becoming the fodder for combinations and re-combinations which concord with certain rules [cultural, linguistic, social, even artistic].   Through  articulation on a social level they are made to be shown to posses characteristics over others, they become styles [say fact from fiction], more heuristics before finally being widely and commonly recognised as the most efficient solution or way of doing or representing something. This could be the manner in which wikipaedia entries are made – i.e. something of the history, major practitioners and characteristics of a given theme or topic.

They become an ever more limiting palette. The more comprehensive wikipeida becomes the more it becomes a one-stop-shop for knowledge – removing the need for multiple resources which in the past, the older ways of coming to know something often offered up serendipity, and even the promise of more ‘depth’. Each child’s journey into knowledge begins with mystery (how, who, what, why, where and when) to algorithm (stripping away uncertainty, ambiguity, and judgement from almost all processes).

“Mysteries are expensive, time consuming, and risky; they are worth tackling only because of the potential benefits of discovering a path out of the mystery to a revenue-generating heuristic”, “The algorithm generates savings by turning judgment… …into a formula or set of rules that, if followed, will produce a desired solution” and “Computer code – the digital end point of the algorithm stage – is the most efficient expression of an algorithm” “In reliability-driven, analytical-thinking companies, the norm is to see constraints as the enemy”, whereas when validity is the goal “constraints are opportunities” and “they frame the mystery that needs to be solved

The Four Pillars Upon Which the Failure of Math Education Rests (and what to do about them)“, by Matthew Brenner, page 55, highlights this in respect ot maths education. I’m not a math educator but Brenner’s comparison of how kids learn math (not sure if he is referring only to the U.S. or not) struck me as so funny, so tragic, and so true all at the same time.

“Kids are taught math as pets are taught tricks. A dog has no idea why its master wants it to perform. With careful training many dogs can be taught to perform complex sequences of actions in response to various commands and cues. When a dog is taught to perform a trick it has no need or use for any “understanding” beyond which sequence of movements its trainer desires. The dog is taught a sequence of simple physical movements in a specific order to create an overall effect. In the same way, we teach children to perform a sequence of simple computations in a specific order to achieve an overall effect. The dog uses its feet to move about a space and manipulate objects; the student uses a pencil to move about a page and manipulate numbers. In most cases, the student doesn’t know any more than the dog about the effect he creates. Neither has any intrinsic motivation to perform nor any idea why the performance is demanded. Practice, practice, practice, and eventually the dog can perform reliably on command. This is exactly how kids are trained to perform math: do a hundred meaningless practice problems, and then try to do the same trick on the test.”

I think of those high-achieving [on exams at least] Asian children with their dragon mothers breathing down their necks and their teachers bending over backwards to produce algorithmic children that answer question effectively and efficiently and pass exams ever so routinely. What do they really learn in their risk free, smoothed out designer and centrally planned worlds? Oliver Burkeman writing The Guardian had this to say on the matter:

“Most lives are too messy to think in terms of a blank canvas. But thinking in terms of elegantly arranging interlocking items is practical, while leaving space for real creativity. There’s an old joke about designers, which I’d always taken as teasing them for being truculent, but perhaps on reflection it’s more flattering: how many designers does it take to change a lightbulb? “Why does it have to be a lightbulb?”

The Archaeology of Knowledge