Shad’s Blog for Week #2
If I started to think like a conventional Mathematician in order to teach Math(s), my students would suffer from a common ailment known as VTL or Very Twisted Learning. VTL is associated as you know, with VTP (Very Twisted People).
Imagine explaining the practical application of Number Properties in these terms to Grade 6 students:
“The general idea of numbers comes from the properties we ascribe to the way they develop naturally, in a known sequence, like 1, 2, 3. okay. All mathematical reasoning is derived from our implicit faith in these properties. Here is the first practical application of these properties. Observe closely. You will notice that 7 + 2 = 2 + 7. Ah-ha! What is the general rule here? Right! That the sum does not depend on the order of its terms. Which means that addition is a commutative operation…..etc. etc”
“What…..are …..you……thinking??!!” is how 10-12 year olds will likely respond to such raving. Sheer blinking bewilderment.I can just see them roll their eyes inside their heads and wonder: “Why is s/he making such a big deal about 7 + 2 = 2 + 7, and what is so practical about these properties?”.
The teacher attempting to sound like a mathematician might end up taking his/her students anywhere but to the gates of Math(s) learning. Before long, s/he may start filtering out the “smarter” ones from among the students, while punishing (psychologically, of course) the “dumb ones”. The latter may appear too stupid to understand the essential Math(s) principles, the lofty theories on which they are based, the underlying philosophy that guides mathematics learning… blah-blah-blah. God alone knows how hard s/he tries to simplify them, to reduce those ideas as much as possible to help these kids get it!
The mathematician and the Math(s) teacher are two different beings, both on the outside and from the inside. They live in two different universes, come from very different sources of knowledge and have very different goals. They work in contrasting environments and are guided by different motives, which in turn impacts on their reasoning processes. There is yet another difference that often separates them. The proverbial absent-minded mathematicians are invariably authentic versions of their true selves. Not so those obsessive, my-way-or-the-highway minded Math(s) teachers with correction-impulse genes grafted into their DNA’s. Many of them, in all seriousness and good faith, wear the mathematician’s mantle in order to portray themselves as true believers in the faith they espouse.
To mathematicians, “practical applications” that appear rather petty to the ordinary mortal, carry extremely significant thought-applications in a mathematically ordered universe. Here, numbers and their relationships have a well defined mathematical existence. In this universe, even intuition attains a level of abstraction. The mind intuits ambiguity like a hunter senses danger in a forest. It also intuits exploratory pathways of logic leading one out of the beguiling uncertainties of ambiguity into the lush comfort of Mathematical certainty. For the mathematician, mathematics is a language very unlike the language of words that possess content. The purpose of mathematical content is not to convey meaning (as human words do) but to “construct pure forms of thought”. ( In some ways it does beckon something in us that strives to reach beyond the senses, beyond the boundaries of human words and their ascribed meanings). The mathematician is mystified by how the human mind can keep splitting the tiniest of imaginable fragments, repeatedly, into tinier and tinier fragments, forever. Or how we can keep increasing the same given quantity by the same amount over and over again (5, 10, 15, 20…….) until it disappears into the vague and distant mist of infinity. Why do we conduct such infinite thought-experiments so naturally?
The world of numbers provide the perfect medium to micro-manage such questions. Most importantly, mathematicians get a rise out of a truth they stumble upon over and over again, namely: (a) certain concrete, tangible phenomena can be distilled into abstract forms via numerical representations of the relationships that connect them (b) once these are encapsulated in their abstract forms, the forms grow strikingly vulnerable to human (well, mathematical) manipulation and (c) certain manipulations lead to concrete results that alter the real world in breath-taking, mind-boggling ways.
In sharp contrast, the Math(s)-teacher’s world is often a classroom full of noisy kids bursting at their seams with crazy hormones. This world of unpredictable learners can be far more complex (than the world of numbers!) in terms of how students associate together or singly. How they learn, needs to be considered. As also why. Ideas they find most compelling need to be understood. To add to these headaches, we also need to be informed of their status as socialized intellectual beings.
Finally, there is, for the Math(s) teacher, the inevitable world of mathematics itself. But what a different world it is from the mathematician’s! As a Math(s) teacher, I can only speak for myself. What is my world of Math(s) like? Well, for one thing, it is an overwhelming universe all its own that I am compelled to organize into some kind of an architectural structure. I have to visually expose the inherent composition of its many interconnected parts for learners to see. My visual distillations of Math concepts must be simple and elegant in design. And the design should express both intent and content. This has taken years and years of work and it ( surfacing as CLSO-MATH ) never ceases. I have to extract meaning out of Math(s) at a deep personal level, and knowing about the mathematician’s world helps, but never as much as knowing about how the human mind learns number sense, numeracy, and numerical reasoning. Unlike the mathematician, I have to learn not to speak Math(s) but to explain it, communicate what I understand. Of course, I enjoy the ability of Math(s) to embellish human intelligence. Sometimes a mathematical concept generates a train of abstract thinking that appears to be converging upon some “essence”, and yet, that essence eludes definition, like the answer to the question pitched at moody adolescents: What do you want to be when you grow up?. I like to explore my experience of Math(s) as an interested individual seeking glimpses behind the veil of outward reality not with the eyes of a mathematician, but that of a teacher.
Finally, I have to hypothesise cautiously about how the learners’ minds react to Math(s): how they make meaning out of numbers and number patterns; what puzzles them, astonishes them, excites them or makes them wonder; how they associate Math(s) with brains. How much of what they think, resonates with me? How do I invest their gift of resonance into my teaching practice?
Often I wrestle with questions around teacher-training: what does it take to become an effective Math(s) teacher?
If there is a mathematician’s mind lurking within us somewhere, it could, I think, be transformed into something powerful and compelling as an approach, to channel a learner’s natural curiosity towards mathematics. But bringing the original cast of the “Mathematician-teacher” into Math(s)-teaching is to regress into medieval times. The medieval-classical approach towards Math(s) education began a few centuries ago with all Math(s)-teaching materials and approaches developed and authorised by pure mathematicians. They decided, quite arbitrarily, how Math should be taught to children and teen-agers, without the foggiest notion of how minds develop number sense and learn Math(s).
The shadow of such a tradition falls darkly across many Math(s)-classrooms throughout the contemporary world. We see plenty of evidence of that in today’s frustrated majority of Math(s) underachievers.
And so… teacher-training…? Well, that could be a topic for another blog. 
Shad