Is it still worthwhile to teach platitudes?

A teaching platitude is a truism arising from the hackneying of beliefs and of values central to what is commonly understood to be the ideal attitudes governing learning. And there are many we are familiar, for instance, in failure. Failure is part of success, as they say. But if you have never failed, would you be considered to be successful?

It’s probably not the wisest to suggest that a school normalise failing, even though teachers use certain means to replicate the effect of failing. So to say, the likes of grade deflation, a poorly crafted exam problem (not least plagued by the worst standards of English), and so on. Junior Colleges are most guilty of these.

A textualist take on the platitude is that a student would not have succeeded if he had never failed, but countervailing attitudes would have it so that a student never fails, in the most prevailing sense of the word, through supplementary classes, tuitions, and Asian magic. And so it would occur to me that failing has lost its textual meaning in today’s education landscape. Okay I suppose it is a stretch to put it in this way, but failing is indeed more elusive, both in occurrence and in attitudes. Certainly it is almost heresy to suggest a Singaporean student should fail his exam. Nonetheless, it does not follow ipso facto that if a student has failed, then he would have succeeded.

I did a fair bit of math when I was still in school (before I was forced to take a 2 year break) and it was probably the classes where I listened to the least. I should probably qualify this. If you imagine the 2D cartesian plane, with the horizontal axes being “pureness” and the vertical axes being “time spent attentive in class”, math would be at the most extreme of the fourth quadrant. The value of this plot is in essence a metric of “ideas”. The further a class is from the origin, the more important “ideas” is in that class or subject. I would argue that math is wholly made of ideas - that’s why math is thought of as concepts and not individual and mutually exclusive bullet points in a syllabus (although it could certainly be taught as such).

That is gestalt, a school of thought on its own that represents the belief as the whole is something besides its parts. I should discern this from the oft-miscited adage that the whole is greater than the sum of its parts: Aristotle, if capable of addition, would not have said that. It is not that math as a system acts as an ultra vires system alien to Aristotle’s Metaphysics, but that math is definably as the union as all of its concepts, both discovered and not. Nonetheless, the correct quote is not just useful in psychology, but also in inspiring learning outcomes.

When in Junior College, I studied the full suite of mathematics available under (what I consider) a very optimistic lecturer. His reputation (and notoriety) precedes him but for good reasons. One of his platitudes that he set in action was that tests should be a learning journey, whatever that meant. My first time hearing this was probably early in the year before any tests occurred, but I brushed it off as just another overzealous educator scaring us into preparing better for exams. After all, what could possibly appear in the exams outside of what was taught in class?

The moral of this tale is that I was gladly proven wrong because I was bored out of my mind in secondary school listening to teachers saying words to similar effect, but not feeling the least bit challenged in school. Perhaps my peers were, though I had not felt the cautionary effects of the tale until my first test. And till today I could probably recite the full problem and the idea behind that problem. As well as every other problem that challenged me during one of his “learning journeys”. In fact I can recite it here:

Let $A$ be a nilpotent matrix and let $V_k=ker(A^k)$. Prove that the subspaces $V_k$ form a flag of subspaces. The key idea is to prove that each element in one subspace is contained in the successive subspace.

This is obvious in hindsight, or with sufficient mathematical maturity, but certainly not apparent in exam condition when this is only one part out of a larger problem, in a 1.5-hour exam comprising of four such problems.

I vividly remember this idea because I failed to find the idea during the test. Generalising, this is what failing means to me - not least because I failed the test, but because I failed to find the idea that I could have found if given enough time. Fortunately a quick convene after the test convinced me that I was just having a mental block and I quickly wrote up the idea when I went home.

It is neither sufficient nor necessary that one should fail an exam before a student could be said to have failed. This failure is not in learning, nor in outcome, nor in any fault of the student - it is plain, unadulterated failure. If the student was capable of cognition, this is in fact a success on my lecturer’s (or exam setter) part because he managed to construct a problem that left students pondering in some shape or form even after the exam.

This style of examining is perhaps not to the liking of many, or even most Singaporean students who need their grades as some form of justification. Understandable. Though, I’d argue that this is not to the student’s detriment: a well-constructed test, if a teacher is capable, would stretch a wide range of students. Using the problem above for example: a faltering student would not be expected to solve it nor even put his mind to attempting it; a passing student could reasonably put his mind to it and solve it given sufficient time, at the expense of marks elsewhere; an achieving student could do the same, at lesser expense of marks throughout the test. This means that a challenging question, if well-placed, would be the asterisk in a test (although it is not Singapore’s practice to have students avail themselves to the difficulty of a problem).

A more correct qualification of the platitude would be: a student who has failed and have learnt, will eventually succeed, for large values of “eventually”.