Klaus Hoechsmann, Aug. 22, 1995

Higher math for elementary school teachers.

The following is a brief description of the way Math335 was understood and taught by the person who was in charge of it for all but one of its seven years of existence. It is not meant to be normative but illustrative and perhaps suggestive.

Rationale. Math335 at UBC is a low-level, high-intensity introduction to mathematics for students intending to enrol in the “elementary” programme of the Faculty of Education. They could have satisfied the faculty’s minimal mathematics requirement by taking any normal introductory math course, but those who queue up for Math335 have usually waited till the very last moment for their plunge into the subject which is, after all, the only science commonly associated with the word “anxiety”.

Mathemaphobia, popularly known as math anxiety, is not a purely academic concern. It not only obstructs access to mathematics itself but to a whole range of activities, from econometrics to engineering, seen as “too technical”. Thus it deprives our increasingly technical societies of human resources, both in creative energy and informed political judgment, while undermining its victims’ self-confidence and contributing to their sense of alienation. In some cases, it may be blocking a person’s one major talent.

Like most phobias, it does not always show its component of fear, but often masquerades as disinterest or disdain. While school may not be the sole cause of such attitudes, it is clearly a fertile breeding ground. A modest step toward their elimination would be to ensure that teachers themselves are largely free of them. This cannot be accomplished through good-will campaigns or intensive rote training. Teachers, especially in elementary school, need to move about their subjects with a certain measure of ease and spontaneity: they must be seen as people who can live what they teach. A few brave slogans and formulas, grafted on a stem of indifference and superstition, will not convince the children that mathematics is worth learning. They need to see its human face.

There seems to be no great problem in the early grades: coloured rods, geoboards, and the like help to make math a favorite subject among the youngest children and their teachers. Leaving aside the insoluble conundrum of secondary school, our remedial efforts will probably have their most beneficial effect on the crucial second half of the elementary cycle. There, in the context of fractional quantities, teachers usually come to the end of their tether^{1}, and are reduced to enforcing the laws proclaimed by the text-book.

As mathematicians, we are grateful for the opportunity of some input into the school system. Some of us may bemoan the depth of “innumeracy” we find in these students and be tempted to refuse to deal with them on their terms. However, we had better recognize that they are normal young people in all other respects. Some of them, though mathematically well below university standards, are actually quite bright, and — like it or not — most of them will probably wind up in schools as teachers of our offspring. We have a small (3 months) window of opportunity to influence their attitude toward mathematics and, beyond that, other things technical.

1 Anyone who finds this is exaggerated should make the following experiment. Ask a well-educated layman how much one US dollar is worth in terms of the Canadian dollar, when the latter is pegged at 74 US cents. If he knows “the answer”, ask him to defend it.

Syllabus. Math335 is supposed to be neither a course in teaching arithmetic, nor a course about mathematics, but an excursion into the subject itself. In designing such a course, it is important to strike a balance between puzzles and problems on the one hand, and coherent theory on the other. For relieving math anxiety, games, puzzles, and explorations are clearly indispensible. However, a steady diet of them would not only lead to indigestion, but also be perceived as unreal and condescending. People need to feel that something significant is being developed, and that they are beginning to master it.

Good cases can be made for various exotic topics (groups, graphs, etc.) as being natural entry-points for neophytes, since they can be started from scratch and quickly lead to interesting problems. However, Math335 has opted for an introduction to the classical school curriculum. After all, prospective teachers might as well deal with questions likely to arise in their future environment (e.g., middle school): about high school math^{2}, about some standard calculator functions, etc. Moreover, the traditional curriculum has a certain historical^{3} and practical^{4} legitimacy. Finally, our departmental committee had prescribed in 1988 that the course should have three components: geometry, arithmetic, and combinatorics (with probability).

To ensure the theoretical cohesion of these three components, each of them was organized around a couple of simple themes, which would serve as anchors while still allowing a lot of exploratory movement around them. In geometry, the central themes are the Pythagorean theorem and the yoga of scaling^{5} (which happen to be the main ingredients of trigonometry). Arithmetic includes an informal introduction to approximations, but has one focus in the relation between fractions and periodic decimals, the other in the computation and applications of (fractional) exponentials. The third component does the inevitable minimum on multiplicative counting and then settles around binomial probabilities and (by extrapolation) the bell curve. A more detailed syllabus is given in the appendix.

Notation. Modern mathematical notation is a double-edged tool: for some (fortunately including most engineers) it provides a fast track into the heart of the Newtonian world, but for the vast majority it looms as a frightening obstacle. Its slick syntax can effortlessly handle subtleties (say, about negatives) which formerly befuddled great minds. However, by shifting the onus of thought toward formal operations, it has created a two-fold trap for the learner: the temptation to skip conceptualization and, at the other extreme, the futile pursuit of elusive meanings^{6}. Not only mental sluggards thus fall by the wayside, but also those who were simply too critical or inquisitive at the wrong time.

For the lay public, however well-educated, the medium has become the message: math and its notation are one — formulas forever. In academic debates, this may be an acceptable epistemological stance (cf. Hilbert’s Programme), but it is certainly not a valid psychological or historical one, as it would have to deny the existence of any real mathematical activity prior to the seventeenth century. The first book making systematic use of formulas was published by John Wallis in 1685. By then, the entire content of the high school curriculum was well established — most of it very old, with only probability (by accident) relatively new.

2 On her first job, one former Math335 student wound up teaching Math11 at a senior high school.

3 Except for its unfortunate downgrading of geometry, it gives a fair overview of pre-Cartesian math.

4 Apart from balancing their cheque-books, most users will wish to do unit conversions, interest calculations, and simple exercises in indirect measurement (lengths, areas, and volumes).

5 Scaling was not part of the original plan but seems to be almost everyone’s Achilles heel.

6 What exactly is one and a half “divided by” three point seven six? Explain by example!

Ideally it should therefore be possible to give a course for prospective teachers without ever writing a formula. Math335 does not strain to practice such total abstinence, but it does try hard to avoid “black boxes” of any kind. Thus, calculator functions (even sin and log) are not legal until their results were repeatedly approximated by hand; formulas are admitted only if they are transparent abbreviations for familiar procedures; laws and rules are banned altogether, or rather, left implicit. For instance, no “law of cosines” is invoked to compute a length BC from an angle BAC and the lengths of its legs. Instead, an auxiliary triangle BAD is introduced, with the same angle at A and a right angle at D; its side-lengths DA and DB can be inferred from the tabulated dimensions of right angle scale models with unit hypotenuse^{7}; the rest is Pythagoras. In other words, the computation re-enacts the derivation of the hidden “law”, with little extra effort but a great gain in conceptual control.

Implementation. Math335 meets five hours a week. According to the UBC calendar, two of these make up a “required tutorial”, but experience shows that this distinction cannot be upheld too rigidly. What happens in reality is that, once or twice a week, there is a kind of lab hour: in groups of four or five, students are engaged in some collaborative project. For instance, they might be approximating the area of the unit circle by that of a regular polygon with 6, 12, 24, …, 6144 sides. Each student acts like a line of computer code; accepting input from the neighbour on the left, processing it, and handing the output to the neighbour on the right, they go round and round the loop^{8}. At the 6144-gon, their area is 3.141592 . . ..

Not every activity is so tightly organized. Sometimes they just jointly work their way through a problem set — which usually includes a couple of items on the frontier of their current understanding. In any case, the scheduling must be kept flexible: it often happens that an unforeseen burning question postpones a lab hour to the next day.

The team structure of the class is maintained throughout the course. In almost every hour, lab or not, there comes a point where a few minutes of joint work on a problem or conceptual difficulty are required to set things straight. The amazing clarifying power of even the most muddled discussion, so familiar to professionals, also works for these students — and incidentally gives them valuable first hand experience about teaching and learning.

Of course, all that group work must be balanced by the appropriate amount of individual concentration, i.e., homework. There is only one assignment, but a massive one (worth 50% of the course grade): to keep a detailed up-to-date chronicle of their explorations, insights, and difficulties — as articulate, complete, and original as possible — written in the manner of a personal mathematical diary for serious future reference. Such a “journal” may seem strange in a math course, but it would be difficult to devise another, equally challenging, task for a class of about 40 mathemaphobia victims with diverse abilities and backgrounds^{9}. It is important for the students to know that a perfunctory job, too obviously produced just “for school”, will not win approval. The best journal I have ever seen was handed in by a woman in her thirties, with no prior penchant for math, who had spent about 20 hours a week on it. She also said:“ I feel that this is what I came back to school for.”

The final exam, too, has an expository component. It contains no mere exercises, but its “problems” are neither too novel nor too difficult.

7 These are, of course, (home-made) trig tables, but that terminology is avoided at first, in order to prevent panic and mechanical responses.

8 This uses only Pythagoras. As a by-product, they obtain the trig tables mentioned above.

9 Those who believe themselves saddled with a “math block” (almost always imaginary) can be encouraged to concentrate on obtaining and recording first-hand insight into learning disabilities.

Appendix: A Sample Syllabus.

The minor variations in the syllabus mainly concern the geometric component.

1. From Meno to Pythagoras. The first math lesson on record: Socrates shows how to double a square. Extending the method, we find the theorem of Pythagoras: how to add two squares. Quadrature of rectangles and general polygons by cutting and pasting. No numbers yet; lengths and areas remain unquantified.

2. Diagonals and Heights. If you can find square roots, Pythagoras allows indirect measurement. We compute square roots by repeatedly averaging the sides of rectangles (Newton). Lengths related to regular polygons and certain three-dimensional objects. Scale and proportion.

3. Areas and Volumes. Stacking strips or slabs, we determine areas and volumes of triangles and pyramids, hence of (regular) polygons and polyhedra. We approximate the area of the unit circle by repeatedly doubling the number of vertices of a polygon. From the same data, we make a trigonometric table.

4. Cavalieri’s Principle. We get the area under a parabola by reinterpreting slices of pyramids. Comparing a hemisphere (slice-by-slice) with a cratered cylinder, we find the volume of a sphere. Seeing the latter as a composite of slender cones, we also deduce its surface area.

5. Rational vs. Irrational Numbers. We show that periodic decimals represent rational numbers (cf. Achilles and tortoise), and prove that many square roots (e.g. \sqrt{2}) are irrational.

6. The Lore of Large Numbers. We find spatio-temporal representations for several very large quantities, like the national debt. Scientific notation and orders of magnitude. The technique of “counting factors” gives easy estimates for products and quotients.

7. Fractional Powers: Logarithms. Using square roots, we make a table of 10^{r}, where r = k/32 with 0 < k < 32. The resulting graph gives a way of writing any positive number as 10^{a}, with obvious benefits for multiplicative calculations.

8. Growth and Decay. Many quantities evolve according to a scheme of the form Q(t) = Q_{0}a^{t}, where a > 1 (growth) or a < 1 (decay). We look at a typical crop of such problems. Pitches in the well-tempered chromatic scale follow such a pattern with a^{12} = 2; irrational versus rational (“Pythagorean”) musical intervals.

9. Multiplicative Counting. Multiplying the number of choices available at each stage of a composite task. Counting with or without replacement. Permutations. Probabilities obtained by comparing counts.

10. Binomial Coefficients. If k elements become interchangeable, the total count is divided by k!. Reshuffling “words” with repeated letters. Binary words. More probabilities.

11. From Pascal to Gauss. Histograms derived from different lines of Pascal’s triangle show a strong resemblence. We draw one for line N = 1600 on graph paper. Probabilistic interpretation. Extrapolating the limiting case. Counting squares on our approximate “bell curve”, we make a table of normal probabilities.

12. Testing and Sampling. The normal curve shows how the values of many (but certainly not all) random variables tend to cluster. It is also useful for computing binomial probabilities, especially when the number of “trials” is large. Standard deviation versus sample size. The mysterious reappearance of square roots (inferred only from numerical evidence).