Clocktime and personaltime
clock by Micthev
It took the same incident, in a school in which I was working as a supply teacher, repeated several times to awaken my awareness of how time relates to the significance of events. Such was my belief about people that it took someone else to force me to learn a lesson about how we use time. In response to my requests for guidance, the head of department arranged to meet me one lunch break. He didn’t turn up, and knowing how busy he was I left our meeting place without offence. However, this request and failure to meet occurred twice more, and, when a colleague said, “Perhaps he doesn’t want to meet you”, I protested, ever willing to believe in the good will of the head of department.
clock by Micthev
In thinking about time in mathematics classrooms we need to move beyond the notion of mere clocktime – the ‘ticktock’ of the incessant linear breakdown of lived experience into metered chunks. What other kind is there? How about the more elastic concept that is associated with particular events – time rushes by when you’re having fun, but slows when you’re heading for a crash, they say?
On a week’s holiday I find that the first few days go slowly, say Saturday to Tuesday. Then something happens – suddenly it’s Saturday again and time to drive home. I’m not sure whether this phenomenon is controllable but I recall that Edward de Bono suggested that one way to slow down the perceived speed of a holiday is take along with you the most boring person you can find. Next Saturday would then retreat towards eternity!
clock by Micthev
I will call my two versions of time clocktime and personaltime.
Assuming that personaltime is composed from irrational moments squeezed between the rational ‘ticktocks’ of clocktime (as irrational numbers are squeezed between the rationals), what sort of things might be done in personaltime, and how may we plan to employ it? I suppose some might think it best ignored – life is, after all, hard enough already! But don’t we make rods for our backs by acting as if learning proceeds smoothly, linearly, in clocktime chunks? It’s only teaching that often proceeds like that: learning happens in fits and starts. Perhaps only pure training can be ‘shoehorned’ into clocktime – I imagine WWII navigators learned the mathematics of their craft that way.
In distinguishing between two senses of time – each sense of time capable of being in harmony or in conflict with the other – I link the way that we allocate time to how we value things. For example, I question the response to advocating classroom discussion that goes “Oh, I don’t have time for all that.” If, as I believe, values determine available time, that response is equivalent to “I don’t value classroom discussion.”
The wellresearched finding that mathematics teachers typically leave no more than about three seconds between asking a question and getting (or giving) an answer points to our living as much in personal as in clocktime. Nobody should consider a mere three seconds as a decent allowance of thinking time for a mathematics question that is worth asking.
clock by Micthev
We can relate time allocation to decisionmaking, and note how events are comprised of microdecisions.
Oliver says breathlessly to his father, “I’ve just heard that the bunch I was due to drive to Paris with yesterday were hit by a lorry. It was a bad crash, Tom and Penny were killed.” After a pause, Oliver quietly reflects, “I’m glad I didn’t go.” “But if you had gone”, his father replies, “the accident wouldn’t have happened.” Silence.
This story brings to mind the thought that events are a result of a series of tiny decisions. Can that be stated in your scheme of work?
clock by Micthev
Sometimes a microdecision breaks into clocktime producing an effect way out of proportion to the clocktime spent on making and carrying out the microdecision – it happens in almost zero clocktime and yet it is extremely significant. Such unplanned events figure large in personaltime.
For example, on one occasion I observed a teacher using the space between ‘ticktocks’ tellingly. She was walking past a table at which students were working when she suddenly stopped by one student, pointed at her work, and said, “You’re wrong!” Fierce scribbling and ‘headscratching’ followed. Then the student rushed up to the teacher and exclaimed “No, I’m not, because…!”. Knowing that particular student well, the teacher made a microdecision to intervene in a way that positively resulted in the student thinking hard about what she was doing – on another occasion, with another student, the same intervention might have had a negative effect.
‘If you want something done’, the saying goes, ‘ask a busy person’. It’s not that they know how to work harder, just that they know how to work with time not just in time.
clock by Micthev
Here are some observations about pedagogical events that are large in personaltime  owing to the significance of their effects  yet brief in clocktime. They are presented in the style of Wittgenstein’s Philosophical Investigations because they are intended to provoke reflection – a little effort may be needed to interpret them.
1.0 Manipulation both of algebraic and arithmetical expressions is partly structured by the ways we put things on paper and partly by associated mental action – ‘in our heads’. On a bad day, for me, they don’t cooperate, and I can be surprised by what I write down. Fleeting thoughts trigger microdecisions that may elucidate or may lead to mistakes.
1.1 Recently, for a reason I have not been told, in school textbooks the distinction between the binary operation of subtraction, and the additive inverse element, which is indicated by a unary operator, has been removed. So where we once wrote, for example,
^{}7 – ^{}7, we now write –7 – (–7)
I presume this reversion to a much earlier practice is in the interest of fluency, i.e. unconsciously efficient symbol manipulation. In which case the brackets which hint at one of the ‘–’ signs being used as additive inverse are superfluous, as the aim is to react to mathematical signs not interpret them^{1}. Perhaps we would do well, when say going through on the whiteboard a rehearsed routine of teaching signed numbers, to insert between the ticktocks of the current formulation a lighthanded exploration using '^{}' and '–'. Such a light “bytheway …” might be significantly illuminating, thus retrospectively looming large in students’ personaltime. We don’t want to make everything a matter of explicit rules.
1.11 Years ago it was the fashion to teach about mathematical objects called sets, as in ‘sets, relations and functions’. As part of clocktime textbook exercises about ‘sets’ students had to do trivial things that made little sense. What if, instead, the language of sets – union, intersection, complement, empty set, … had been part of the nonformal communication of mathematics, along with gesticulating and pointing? Wouldn’t that have been a good way of relating clocktime communication to personaltime? Mischievous forays into set theory might include for example, ‘is a shed that is empty of cows the same as a shed that is empty of pigs?’
2.0 The National Strategies go beyond arithmetic as mere computation:
The laws of arithmetic
Primary children need to understand how the laws of arithmetic work in practice if they are to multiply and divide successfully (but they do not need to know the names of the laws, or see them expressed algebraically).
 commutative law of multiplication: a × b = b × a
 associative law of multiplication: (a × b) × c = a × (b × c)
 distributive law of multiplication over addition: (a + b) × c = (a × c) + (b × c)
 distributive law of multiplication over subtraction: (a − b) × c = (a × c) − (b × c).
Written formally like this the laws can look daunting, but anyone who does a multiplication calculation probably uses them unconsciously.
I wonder if many teachers give classtime to these laws. Perhaps their proper place is as hinted generalisations that slip almost unnoticed into spaces between clocktime practice of arithmetic.
3.0 Efficiency isn’t in every way good. Sometimes we need to slow things up to allow students to employ personaltime. For example, a teacher writes on the board:
When a polynomial, f(x) is divided by (x – a), the remainder is f(a).
Then the teacher sets exercises. I can imagine circumstances that would justify for the teacher that decision, but wouldn’t it more often be better to enable student exploration of quotients and remainders of polynomial division in order to allow students reflection in their personaltime?
It might be argued that students should be shown a proof, perhaps along the lines of
f(x) = (x – a).g(x) + R; when x = a
f(a) = R
Student belief might be thereby coerced, but how would such swift brevity chime with their existing knowledge of polynomials and their factors?
3.1 Similarly, rather than presenting ‘completing the square’ as a prepackaged formula – with or without proof – might not the formulation emerge naturally from trying to control coefficients b, c in quadratics of the form x^{2} + bx + c by experimenting with examples of the form (x + p)^{2} + q?
THEOREM 1.0
Discussion promotes use of personaltime.
clock by Micthev
ticktock!
^{1} Language and Mathematics, pub ATM, ISBN 0 90095 31 8
Image Credits
Page header  clock face  photograph by H Grobe some rights reserved
Clock diagrams by Micthev some rights reserved
