I am not a teacher of mathematics. I am also neither a philosopher of mathematics nor a designer of math questions. I am not even good at math.
But after reading this: http://www.channelnewsasia.com/stories/singaporelocalnews/view/1010464/1/.html , it made me--out of the spirit of (mathematical) objectivity motivated even by a greater curiosity--to imagine what is mathematics education.
After reading this story in an objective spirit, I state what mathematics education is NOT:
(1) Mathematics education is not coming home from an exam, albeit an important one, and gesturing to your mom that your throat has recently been slit.
(2) Mathematics education is not coming home from an exam, albeit an important one, and getting very emotional before crying your heart out while trying to tell your mom amid tears and mucus that your hopes of an A* (i.e., read A-Star: the equivalent of a North American A+. For the simple at heart, it means either a perfect grade or somewhere a iota south of that--one can go no further academically or intellectually, which are of course, untrue) are dashed.
(3) Mathematics education is not about using a calculator, so that 'computational errors can be reduced'. If that were the case, the rocket that NASA just crashed on the Moon would have created a bigger plume of ash. As a matter of fact, use of calculators under examination pressures tends to entail 'errors of dexterity'. This means for a simple calculator, instead of keying in '8', you either key in '7', '5' or '9'--or all four at once.
(4) Mathematics education is not about big numbers so that inadvertently, you need the assistance of a calculator; which truly for the initiated, an abacus would have been much more efficient for elementary mathematics.
(5) Thus clearly, mathematics education at least for this level is not about calculations requiring calculators at all, unless of course, calculators are nice bling for examinations but we know that bling has nothing to do with math. QED on calculators.
(6) Mathematics education is not about 'breaking down in tears right after the exam'. As far as I know there are two kinds of tears associated with math. One, when you discover something like the incomplete theorem before you lose your mind; and two, when you win the Fields Medal for it. Or by an act of mathematical randomness, you win a Nobel Peace Prize for applying the incomplete theorem, which has happened by induction at least once in human history.
(7) And mathematics education is not about this kind of question:
"Jim bought some chocolates and gave half of it to Ken. Ken bought some sweets and gave half of it to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. The ratio of Jim's sweets to chocolates became 1:7 and the ratio of Ken's sweets to chocolates became 1:4. How many sweets did Ken buy?"
Why? (we should truly ask more 'why' questions for mathematics education).
It took my unpracticed hand and mind nervously scrawling on two pieces of paper before I finally worked out the answer. Honestly, I would have done no better than the average kid on this type of questions. In fact, I would have flung if you locked me in the room with them and gave me the same time as everyone else.
But in solving this problem, I revisited how to do mathematical sums long-hand (without a calculator); remembered how to call concrete things X and Y again; relearned how to perform elementary functions with ratios and fractions; once again toyed with simultaneous equations; discovering where I made errors (yes, I was working backwards from the answer after a few 'computational' errors: the advantage of adulthood) and in it all, remembered that mathematics should be less frustrating and more fun, which concisely just summed up about all Polya said on this subject.
But this question--if this was in fact a representative of all questions for Paper 2 in the PSLE this year--sums up just about everything that is frustrating and obnoxious in mathematics education.
First, a mathematics word-question and its heuristics should aid in clarification and not in exacerbating confusion. About two minutes into re-reading and expressing the algebraic relationships to this question, I forgot who was Jim and what Ken bought. Perhaps a 12 year old brain has greater clarity. Nonetheless, in about another two minutes down this same track, I had forgotten who bought chocolates and who bought sweets. The 50-50 heuristic created by the question designer--doubtlessly trying to make things easy for that poor student--ended up mirroring and hence, worsening the identity problem in this math problem.
Second, a mathematical problem should never constrain possible solutions through the phrasing (or design) of the word problem itself. The problem asked, "How many sweets did Ken buy?". Why, I am also interested in finding out how many pieces of chocolates Jim bought! Because if I do so (which I did but more later), I can by that route find out how many sweets Ken bought using the same mathematical concepts and methods. As a matter of mathematical principle, there is no priority attached to either Ken's or Jim's purchase. So why prejudiced the math question from the angle of Ken? Clearly, the question-designer has not consulted some of the important literatures pertaining to psychological bias in the way we asked questions.
Third, if I did go by Jim's route, guess how many pieces of chocolates Jim bought? 308!!! (*correct me if I am wrong; the urge to write might have compromised my ability to do simultaneous equation).
Now, this is troubling. Because if I were 12 once more (I rather not be 17 again), and looking around even in prosperous Singapore, to have a kid, or a chocolate loving adult like myself, buying 308 pieces of chocolates at one go would have seriously revoked common sense. I might then look around the examination hall, and asked myself if anyone in the world would go out to a gourmet chocolate stand (that's where they sell pieces of chocolates and not 'some chocolates') and buy 308 pieces of chocolates. Because if I, the 12 year old boy with a precocious appetite for upmarket chocolates, can count, 308 pieces of chocolates would have cost me $924 at $3 a piece without 7% Goods and Service Tax. That's serious cash. If I am just as precocious in rudimentary economics, I would think $924 is cash one commits to a mortgage or rent, not to chocolates. In fact, it is downright irrational (i.e., unbelievable). That said, personally I find it comforting that the question designer has glimpsed a world where a boy might part with 154 pieces of chocolates for his friend, or $494.34 for the candy utility of his friend.
And because of this, I, as the 12 year old again (this is getting fun), would be inclined to tell myself that this is an utterly unbelievable answer. Although I am standing on firm ground, my psychology would be telling me otherwise. Should I believe in my psychological intuitions (believe me this counts)? Or should I believe unwaveringly in my mathematical capabilities (believe me this counts relatively little in the crisis scenario of an examination)? Assuming that a 12 year old may not know how to differentiate between his psychological intuitions and his mathematical certainties, then we have a recipe for disaster. Guess we know now why even A-Star students could not complete this examination--if this was in fact the careless way questions were designed and phrased.
Lastly, I am no fan of large numbers whether as a 34 year old adult-still-in-school, or a 12 year old schoolkid. In solving this problem, I had to work out calculations that went beyond my ability to calculate mentally. Maybe I am a notch below average on mental computation--I have been told that quite a few times. Nonetheless, does this impress fans of big numbers? Maybe. But mathematically, big numbers do nothing as far as mastery of the relevant mathematical concepts is concerned. By the highest common denominator of '4', Jim might have bought 77 pieces of chocolates and Ken 17 sweets and this problem could have been still the same. But wait, this would mean half a piece of chocolate and half a piece of sweet by the 50-50 heuristics right? Indeed. So in fact on our hands we have a compound problem--the 50-50 heuristics has complicated even the good intention to reduce these numbers. Thus if designers want to keep the heuristics, change the numbers; but if designers want to keep to a small number in the same ratio, change the heuristics. Mathematical thinking tells me this much, no? To force a choice on these preferences shows some guts in concept prioritization in mathematics; to waver shows incomplete understanding or worse, the absence of responsibility.
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In sum, no child should be put through this sort of un-mathematical mental anguish, because this is not a test of proficiency for mathematics but something like a test of luck.
I behoove designers of such questions to take more responsibility so that students do not feel that their throats have been slit, or doomsday has arrived, or suffering from any of these un-mathematical fates. Most important of all, think like a child and feel the common sense of that child once more. After all, is this not the essence of all great mathematics?
