How I learned to divide by four
by Willie Wong
mathematicsarithmetic class one day in fourth grade, we (the students) were asked to convert the fraction 1/4 into a decimal representation. Other students put pencils to their scratch papers and started applying the rules of long division: 4 doesn’t go into 1, so we add a decimal point and find that 4 goes into 10 twice, etc. My hand was already up in the air before they finished drawing the long-division sign.
“Zero point two five,” I answered when the teacher acknowledged me.
“Oh that’s nice and fast,” he said, “and how did you get the answer?”
A look of confusion came over his face as I related how I did the computation without the rote-rules that he spent the class teaching us: “Well, four quarters make up a dollar, and each quarter is worth twenty-five pennies. So a quarter is 25 cents, or zero-point-two-five dollars.”
I was just shown Peter Gray’s article in Psychology Today with the preposterous premise that teaching less maths in school may actually improve student’s mathematical abilities. My knee-jerk response was to rise to the defence of the status quo. But after actually looking at what he wrote, and considering my own mathematical upbringing, I feel strongly compelled to agree with him (with some minor caveats).
A cursory look at my education may suggest that I am the product of the rote method of instruction. My early years were spent in Taiwan (with an exception for one year in fourth grade which I spent in the States, lest the earlier anecdote confuse my readers), where for the most part recitation and repetition is key for education. Actual successes and failures aside, the long hours in school and the mountain of homework, coupled with the frequent testing and broad range of subjects, certainly presents a facade of imparting erudition. During high school in the United States, I often credit my ability to out-perform fellow students to the comprehensive education I received when I was younger; at the same time I was also appreciative of the American system of encouraging self-expression and creativity. But I’ve also held the belief that imagination let run amok is pointless without a firm foundation, and leads to things like rap-so-called-music, time-traveling-Viking romance novels, and Frank Gehry.
Recent introspection led me to question those impressions (except about Frank Gehry). How much of my being is shaped by my education? Here are some points that I definitely picked up from my childhood instruction:
- An appreciation for intellectual merit. The Taiwanese school systems are really good at moralistic indoctrinations.
- A fascination with facts and trivia. The history, geography, civics, and science classes all focused on memorisation of facts. Some notion of the scientific method was taught starting around fifth grade, and actual physics and chemistry around seventh grade. I can’t imagine how geography can be taught any differently. History and civics never involved any class participation/discussion when I was in school in Taiwan, but allegedly there are some discourse in high schools.
- A love of language. There is a different sense of aesthetics in the Chinese language compared to English.
- An ability to cope under pressure. With a ton of homework and every test being high-stakes, it is a survival trait.
I was also involved in the G-and-T programme, which taught me
- To think. The G-and-T teacher 李華隆 (Andrew Li), who now works at Da Hu public school in Taipei, has always been a pioneer in pedagogy. He championed the Socratic method when he taught me, was a firm believer of the hands-on experience, and then went back to school and obtained a master’s degree on integration of technology in public education. His success has led to a paradigm shift in primary education in Taiwan. My parents sown the seed of my intellectual curiosity, and allowed it to germinate. Mr. Li tended it, watered it, and fertilized it.
What I learned during those formative years is not the curricula mandated by the school system, nor skills obtained from the numerous and frequent drills. What I learned is a value system which makes me want to learn and investigate the unknown. Over the years, I’ve probably forgotten more facts and tidbits of information than I now retain: what’s important to me is that I once sought out those nuggets of knowledge and treasured them. I am attracted to the natural sciences at first, and then to mathematics, for the simple reason that I want to know, I want to know why, and I want to know how.
Coming back on topic, I learned relatively little mathematics in school. I am certainly not a child prodigy, but I did have a knack of discovering mathematics for myself a bit before their formal introduction in school. My parents taught me the numbers. After school in kindergarten, my mom would sometimes take me (and my sisters) to meet my father at work. Sitting there and waiting, I soon got bored of the counting game (you know, the one where little kids try to count as high as they can). The growth rate of the numbers is too slow! I want to use those fancy words I learned for large numbers (in English it would be Thousand, Million, Billion, etc. In Chinese they were 千,萬,億,兆,京). But the rate of appearance of large-number-names is logarithmic. So I discovered that I can reach those numbers faster if instead of counting up one-by-one, I just added my current number to itself. To make things interesting, I will often start with different numbers. So I would count, “three, six, twelve, twenty-four, forty-eight, ninety-six …” and was pleased that the rate of appearance of large-number-names becomes linear.
In first grade, not long after learning about subtraction from my father, I discovered the negative numbers. In second grade I was taught multiplication. Not just the multiplication tables by rote. My introduction to the topic was when my dad showed me a fast way of counting things. I’ve already known that to count a lot of small things quickly, I can group them into groups of five, and then pair up groups of five, and count the groups of ten first. (I did it in two steps because it is easier to visually group things into five than to ten. The human brain’s normal threshold is somewhere between 6 and 8 [I think there was some psychological research about this]. To make sure each collection has ten objects requires actually counting the collection; for five objects I can just look and it “obviously is”. Try it sometime.) My dad showed me (using go stones) that it can be faster to, instead of making even groups of five, just make even groups of some number: by arranging the objects in a grid, you don’t need to care particularly how big each group is, because the brain can easily process whether a visual grid pattern is regular. So we just shuffled things around until a squarish-rectangular grid of objects lay in front of us. And at this point we count the width and the height of the grid, and with some multiplication, voila! And at the same time I learned that multiplication is useful for (a) counting things in groups and (b) finding areas.
Divisions, fractions, decimals, and prime numbers came soon after, the first three all taught by my dad. Prime numbers I discovered by myself in my quest to divide everything. Algebra I invented in fifth grade, except I didn’t have the notion of symbolic manipulation: I had phrasal manipulation. It was for a complicated story problem for homework. One had to parse the verbal descriptions into a string of arithmetic relations on numbers. But I found that it was hard to keep track which number represents what, and which sentences I have already used in the calculation. So I first re-wrote the problem into something like
The age of the monkey’s mother in years = the length of the banana in centimeters + the weight of the stone in kilogrammes.
and solved it via symbolic manipulation of phrases. I liked it because it was quicker and less error-prone for computations, and started using it on all my homework problems. I taught it to a good friend of mine in the G-and-T class, who came back to me with the suggestion the next day that, instead of writing “the age of the monkey’s mother in years” on every line, we can just use a short hand (say, a five-pointed-star) for it. And thus symbolic manipulation was born. The two of us were rather unsuccessful in trying to teach it to other classmates.
Anecdotes aside, my point is that I am fond of mathematics now because, when I was younger, I discovered much of it (sometimes with adult help) myself. In middle school a few friends and I kept pace working through the maths workbook about three months ahead of what is taught in class: so instead of using the workbook for its intended purpose of “reinforcement through repetition”, we took it as a challenge to figure out the necessary tricks involved in doing those problems. By learning in this fashion, I developed an appreciation for the art of problem solving.
And I suspect my story is not that unique among mathematicians.
Returning to Peter Gray’s article, I find that it makes a lot of sense. A successful and efficient education experience should be a partnership between the teacher and the student. When one party professes without regard to the other party’s ability to absorb, we end up with a system where only marginal improvements are had despite large additional time spent on a subject. It is rather strange that, for professional mathematicians, we expect our peers to provide ample motivation in the writing of academic treatises. We want to be told why the problem being considered is interesting, and what uses the solution has. Yet as grown adults we expect youngsters to just trust our judgment that “these subjects will be essential” for a successful life.
Just piling on more and more hours into our education system exposing our children to more and more facts will not make them smarter or more competitive in the labour market. In this modern day and age, mathematics should not be about figuring sums. Nor should it be a collection of obscure facts that not even professional mathematicians care about. A class in mathematics should be a class imparting the joy and the art of sound reasoning.
The problem, however, is not just the subject matter, but also its presentation. Why is it that the same Euclidean geometry can call forth a great mind like Bertrand Russell’s, and bore high school students throughout America? In designing a curricula, the boards of education, made of laypersons poorly advised by experts, driven by self-interests and lobbying efforts, ended up putting the cart before the horse. It is absolutely inconsequential that a person knows the various theorems about congruency between triangles. There’s no point it forcing a student to memorise that whereas “two sides and the angle between them” is enough to determine uniquely a triangle, “two sides and one other angle” is not. The teaching of Euclidean geometry should not be so that the student, at the end of the day, can quote a bunch of theorems and pass a simple multiple choice test. The teaching of Euclidean geometry should be a vehicle introducing sound reasoning via axiomatic, logical thinking.
Take yourselves a tour (if you have not the pleasure of being thus educated) of modern American high school mathematics class. See how they tackle the subject of geometry. What is taught is not mathematics. It is cargo cult geometry. It has all the ostensible shapes and features of geometry: formalized two-column proofs, theorems and propositions and lemmata, and definitions of various things, but it is all just a pretense! For what use is memorising theorems and propositions and lemmata about figures on a piece of paper? What use is knowing that technically a point has no size nor shape, and that a line extends in two directions infinitely, whereas a ray extends only in one? What use is knowing that 12 times 13 is 156? And no mathematician actually write two-column proofs. Geometry should be nothing but a logical game, one asking “what are the consequences if we assume X?” or one asking “why is X not necessarily true given we only know Y and Z?” Theorems, propositions, and lemmata are just intermediate footholds that we can start from, to save some trouble from re-deriving everything from scratch. In video-game terminology, they are “save-files”. They are things we have already reasoned out logically previously, and thus can be used as foundations upon which we figure out more consequences. The actual contents of those save-files are besides the point.
And this is the kind of education that we need! We need politicians who can see the big picture and figure out the long-term impact of their policies, ones who are capable of independent (as in, not completely beholden to the whims of the special interest groups) thought; we need financial analysts who can ask, “can we actually use that equation to model the economy? Are all the prerequisites to using that equation satisfied?”; we need military strategists who can think ahead of the enemy; and we need environmentalists capable of more than knee-jerk reactions of the basest form.
And this is not just in mathematics. There’s an old saying that those who forget history are doomed to repeat it. The caveat is that it is not enough to just know what happened (as is wont in primary and secondary education in the United States). It is necessary to understand the various forces and tides that drove to the climax of the event. It is important to know how the various happenings are interlocked, how individuals influenced each other, and how movements arise.
Beyond the most basic of literacy and human decency (education, like mathematics, has to start somewhere; “turtles all the way down” will lead to a never-ending quest, and at one point we have to accept that certain basic functions of life in a society–those crucial to simple social interactions–must be taught by rote), nothing should be forcibly memorised in school. Thousand or thousands of years ago, our ancestors already decided that stories should be recorded in books: the oral tradition is too fragile and inconvenient compared to written records. Starting about four hundred years ago, civilisations started to compile encyclopaedic collections of known facts. And just about a decade ago, factual information has become always available at our figure tips. A system in which the hallmark of education is mental mastery of abstruse facts is hopelessly outdated. We should instead focus on ability to assemble those facts and draw logical and intelligent conclusions from them. And to do this, we need a completely revamped science and mathematics (and also possibly humanities) curricula.
Primary education should teach a lot less. Emphasis should be placed on basic literacy and social etiquette. Beyond that should be a mathematics class that teaches logical thinking (not arithmetic)–and not in a formal way like predicate calculus, but through games and paradoxes–and a science class that teaches the power of analogies–through activities that encourages kids to draw conclusions about the unknown through personal experience of the known. Formally, the two classes focus on deductive and inductive reasoning. Specific subject areas (and facts) can be introduced for illustration, but not as an end to itself. Only at the secondary education level should facts be considered seriously: but the number of facts considered should be few, while the consideration should be deep (and full of discussions). This is one thing that the American education system seems to do well: group projects. Again, the facts merely serve as a platform leading to discussion, so it is not that important for students to amass a collection of knowledge, but for them to be able to quickly digest knowledge and form opinions. (In some ways, this is the D.E.Shaw philosophy: the financial firm hires a lot of maths and physics PhDs on the assumption that while they know less about economic and finance then a bachelor in those departments, they can pick up information and tools very quickly and think critically about those information and tools.)
Of course, this is just my opinion.