Reflections on teaching senior mathematics
An overview of the 2 Unit course^*

Bill Pender, Sydney Grammar School

My purpose is to discuss a few of the many ways in which the 2 Unit course can be drawn into a unity. Syllabuses and textbooks inevitably suggest a list of disconnected topics, because it can be very confusing if links are made before a new piece of work has been properly understood. Disconnection is absolutely not the case with any of the 2/3/4 Unit courses, but for people new to these courses, the coherence and cleverness may not be immediately obvious. I hope we are all aware that the examination papers in the 1990s have been particularly concerned to emphasize this unity, and I’ll be considering some examples from them.

The MIS/MIP courses, or whatever replaces them have a vocational orientation, consolidating earlier work and teaching reasonably straightforward applications of it. The 2/3/4 Unit courses however, are academic courses, intended to provide intellectual foundation, whether or not the student ever continues with mathematics. The whole point therefore of the 2/3/4 Unit courses is to push a little further into understanding the real nature of mathematics, quite a way in 4 Unit, not very much in 2 Unit.

Let me start by saying that elegance is an excellent characterization of the nature of mathematics. Elegance can roughly be divided into two aspects:

1. Structure: an imaginative appreciation of structure and the interrelatedness of ideas.
2. Language: the careful control of the language and logic by which these structures are enunciated and proven.

Saying it this way means that the subject is equally appealing whether a student’s fundamental motivation lies in the humanities or the sciences, and I intend it to counter any false ideas about some humanities/science divide, which can do so much damage to our thinking. One of the great strengths of the 2/3/4 Unit courses is that they really do allow us to lead students into a deeper understanding of the structure and language of mathematics.

The sums of odd numbers

It’s asking a lot for a teacher to get this sort of thing across. I can demonstrate the kind of complexities involved by looking at a very simple and well-known pattern and then discussing a few mathematical responses to it:

Any sensible Year 7, or even late Primary student with some suitable prompting should be able to conclude that the sum of the first n odd numbers is It’s a fragment of structure, but there’s no context here, and no proof besides the engineer’s argument that it has worked so far.

Now let’s play mathematician and seek a proof and a context for this fragment. The clearest proof is the dot proof, and like all geometric or diagrammatic proofs it seems to give a completely transparent understanding. The approach is generalized routinely in combinatorics, but not in our school courses, so I won’t continue. But it should indicate the connection with area and perimeter in geometry. It also suggests integration, because after a while very thin pieces of area are being added on, gradually adding up to something substantial.

Here is a second proof. When the sequence is run backwards, the terms decrease by 2 instead of increasing by 2:

1 + 3 + 5 + 7 + 9

9 + 7 + 5 + 3 + 1

And we all recognize that this little insight contains the whole theory of arithmetic sequences and their partial sums. This structure, with its formulae and proofs, is developed quite fully in the 2 Unit course. As you know, the examiners love to apply this to all sorts of situations.

Thirdly, someone might object that we’ve only given particular cases, and then left it to the imagination of the reader to conclude that the argument still works whatever the number of terms. If you meet this objection and follow it through to a solution, you’ll arrive at the formalization of proof by induction, but that’s a 3 Unit topic, so I’ll say no more.

Fourthly, here’s another pattern that illuminates why the sum turns out to be a square. This time I’ve started by writing out the sequence of squares, and then underneath I’ve taken their successive differences:

0 1 4 9 16 25 36 49 64 …

1 3 5 7 9 11 13 15 …

and look, it’s the sequence of odd numbers. This can be proven very easily using the difference of squares:

All of this we recognize as the standard 2 Unit problem: ‘Given the partial sums of the series, find the original terms of the series’. It is obviously related to differentiation, and it is obviously the reverse process of what we were doing before.

All this in turn suggests that a more general structure may hold for cubes and higher powers. For example, I’ve written here the sequence of fourth powers, and then taken successive differences several times, until we get a constant row of 24s:

0 1 16 81 256 625 1296 2401 4096 6561 10000 …

1 15 65 175 369 671 1105 1695 2465 3439 …

14 50 110 194 302 434 590 770 974 …

36 60 84 108 132 156 180 204 …

24 24 24 24 24 24 24 …

This is obviously successive differentiation, and we remember that differentiated four times is 4! = 24. But not that even the 4 Unit course makes it possible to deal properly with this pattern, so we’ll abandon that road.

Lastly, here’s a standard geometric interpretation in terms of area that will put us on solid 2 Unit ground again. Obviously the sequence of odd numbers is some sort of linear function. In this diagram, each odd number is interpreted as a trapezium of width 1, and with the vertical sides having a difference of 2. When we put them against each other, their upper boundaries form the straight line y = 2x. So adding them up is now really truly integration, and in this case no theory is needed because the area forms a triangle of base 5 and height 2 C 5:

1 + 3 + 5 + 7 + 9 = C 5 C (2 X 5) =

That was all a bit unsystematic, but I want to draw several conclusions from it all:

1. Adding up a sequence is integration, and taking its successive differences is differentiation.
2. Sensitivity to structure, and the desire for rigorous proof, have led us a long way past pattern recognition in various fruitful directions.3. There’s something unbounded here; the mathematics seems to leak past every explanation we put on it, past every container we construct for the original pattern. Well, life’s like that, and so is mathematics. Any thoughtful group of kids will constantly come up with things we are not ready for, and its our job as teachers of mathematics to stop, to understand the new idea, and to appreciate it whether or not it is part of a formal 2 Unit lesson. Squashing the unsystematic insight will damage any pupil’s confidence and undermine our credibility.4. I hope it illustrates how hard it is to write a coherent syllabus, to build, from the ever-widening implications of even the simplest observation, a unified body of theory that will lie within the capacity of our students, and will also satisfy the mind and give some impression of the nature of mathematical knowledge. This task has, I believe, been achieved in the present 2, 3 and 4 Unit courses.

Differentiation

I’ll turn now to content. Unquestionably, one centre of the course is differentiation, integration, and the relationship between them called the fundamental theorem of calculus. I want to make some remarks about all this, and about the underlying geometric ideas.

Differentiation probably has more interpretations than an election promise, but one major understanding of it is that it is about tangents. Students will probably have already met tangents in Euclidean geometry and coordinate geometry. In Euclidean geometry they’ve learnt specifically about tangents to circles. They know that there is a tangent at every point, and that the tangent is the perpendicular to the radius (Figure 1).

Figure 1	Figure 2
Figure 3	Figure 4

In coordinate geometry, tangents arise in two places:

1. When a quadratic is a perfect square, the parabola is tangent to the x-axis (Figure 2).
2. In non-linear simultaneous equations, when there is a double solution to a line meeting a parabola, the line is a tangent. It’s this idea of a double point which is the basis of the idea of a tangent as the limit of a secant (Figures 3 and 4).

These two mentions of tangents have both moved now from the Core to the Options of the new Years 9&emdash;10 Advanced syllabus, so it’s important when considering a school’s program in these years to make sure they are both taught to all potential 2/3/4 Unit students.

Secondly, the work on graphs in Years 9&emdash;10, particularly the work added in the new syllabus, should have prepared students to understand intuitively what differentiation means. The HSC examiners have made it clear that they want this intuitive understanding maintained, asking questions like Question 7(c) from the 1995 2 Unit paper and Question 7(c) from the 1992 2 Unit paper. In these questions, students were required to interpret rates of change from graphs and to provide brief explanations of their interpretations.

The third point I want to make is that there is a logical reason for teaching sequences and series before differentiation. Using the formula for the sum to n terms of a GP:

From this comes the factorization of the difference of powers:

Then can be differentiated along these lines:

But I’ve indicated earlier that there is also a more intuitive relationship between partial sums of series, and integration and differentiation.

Fourthly, what is the derivative applied to? The derivative is such a fundamental mathematical object that it will soon escape from any constraints put on it, but there are some quite explicit applications of the derivative in our course.

• When the horizontal axis is time, the derivative is the rate of change, and there are plenty of questions on this.
• As a special application of rates, motion is a beautiful way to visualize the derivative because the second derivative is acceleration, and we can actually feel it in our muscles through the forces associated with it.
• Maximization and minimization are obviously important applications. They involve examining points where the tangent is momentarily horizontal. This has always been a big concern of the examiners.
• Curve sketching means making a formula visible in a sketch of its graph. It’s a place where many skills come together, but analysis of the stationary points using the first derivative, and of inflections using the second derivative, is a routine application of the course, and will occur in virtually every examination paper.
• The geometry of tangents and normals. This topic moves from the calculus back to the geometry that provided its initial impulse, and uses the derivative as a tool to investigate the particular geometric properties of important curves. In the 2 Unit course, this means particularly parabolas and cubics, and it’s only in the 3 Unit and 4 Unit courses that this is turned into a systematic exploration of the geometry of parabolas, ellipses and hyperbolas.

Integration and the fundamental theorem of calculus

Integration must have a life of its own. It cannot be introduced and defined as the reverse process of differentiation. Substitution into the primitive to evaluate a definite integral is the result of the fundamental theorem, fundamental to calculus and to our course, and must never be confused with the meaning of integration.

The definite integral is defined as an area, with qualifications allowing negative area. As with differentiation, the basic idea of integration is geometric, but it is developed through the theory of functions and their graphs in the coordinate plane. The examiners have made it quite clear that they want this original definition of the definite integral as a signed area to remain in the student’s understanding as in Question 5(b) from the 2 Unit 1993 HSC paper where any approach through primitives would be quite inappropriate.

This means that all the work in Years 7&emdash;10 on area now gets absorbed into integration theory - the formulae for areas of things like triangles, trapezia and circles become special cases of the definite integral.

The central problem of integration is finding areas of regions bounded by curves, rather than lines. Area is defined by length times breadth, the formula for the area of a rectangle, so areas of curved regions can only be established by infinite dissection. Here is the way the area of a circle was established, probably in Years 7 or 8, by an infinite dissection of a circle, and this should probably be revisited now so that it can be explained that they’ve already seen integration (Figure 5).

Figure 5

Once this understanding is there, the fundamental theorem can take the appropriate centre stage in our teaching. It is the cornerstone of our course, and the proof is important, but unfortunately, it is difficult, and as always must be tailored to the ability of the class. Even the simplest work on the theorem and its proof will help students to understand that evaluating a definite integral using a primitive is a rather spectacular theorem, not a definition.

The four groups of functions

Now let’s turn to the functions to which the calculus is applied. One of the great changes from Year 10 to Year 11 is that functions and their graphs become the centre of attention. A large number of 2 Unit students find the change bewildering; they don’t like it at all, and they say so. They rebel against tables of values, they don’t want to draw graphs, they just want to cling onto the numbers and equations that got them through Years 9&emdash;10. We all have our methods of dealing with this. For example, at our school we spend an intensive period at the start of Year 11 working on curve sketching and applying it to equations and inequations. In that way we begin to set in motion the characteristic way in which the 2/3/4 Unit courses run arguments backwards and forwards between the graph and the algebra. There’s a great deal to do before the calculus begins.

The functions our impressive methods of calculus and curve sketching are applied to are not very numerous. There are linear and non-linear functions, and I’ve placed the non-linear functions into three groups .

1. Linear functions.

2. Non-linear functions:

(a) Quadratic functions, principally

(b) (i) The trigonometric functions, principally and

(ii) The circle, principally

(c) (i) The exponential and logarithmic functions, principally and

(ii) The rectangular hyperbolas, principally

 

Linear functions

Linear functions are called linear because their graphs are straight lines. Although they are well covered in Years 9&emdash;10, it’s good to remember that the understanding of their graphs requires quite a lot of geometric and trigonometric insight. For a start, implicit in the idea of angle of inclination are corresponding angles, the supplementary adjacent angles in a straight angle, vertically opposite angles, angle sum of a triangle, and the complementary angle with the y-axis (Figure 6).

Figure 6

Figure 7

Linear functions express proportionality, which is firmly grounded in similarity and in the intercept theorems, which are the only new geometry in the 2 Unit course. But even gradient requires similarity, because the uniqueness of gradient requires that whatever two points are taken on the line, the ratio of rise over run remains constant (Figure 7). This gradient ratio is just the tangent of the angle of inclination, which means trigonometry is essential even for linear functions.

One of the greatest strengths of recent HSC papers has been the constant linking of topics. The relationship between Euclidean geometry, coordinate geometry and trigonometry is often explored; for example, below is Question 10(c) of the 2 Unit 1997 HSC, where Euclidean geometry, coordinate geometry and trigonometry are combined. There’s more to ask here. Once you start, you can see rates of change, circular motion, maximization of the area of D OPQ. There’s an attitude of mind here that simply knows the topics are linked, and proceeds.

Students seem to have great difficulty doing Euclidean geometry on the coordinate plane. It’s the same even in 4 Unit - give them a rhombus and try getting them to use the fact that the diagonals are perpendicular - they’ll go back to the modulus&emdash;argument form and rationalizing denominators every time. All our 4 Unit students did in our recent Trial. But the examiners are insisting. The final two questions in the 1997 4 Unit paper should be sufficient evidence of that for those who have read them.

Question 10(c) 2 Unit, 1997

In the above diagram, Q is the point (&emdash;1, 0), R is the point (1, 0), and P is another point on the circle with centre O and radius 1. Let PQR = a and PQR = b , and let tan b  = m.

Non-linear functions

(a) Quadratic functions

Quadratic functions arise naturally in two ways. They are the area of a linear figure, and they are the integral of a linear function. I hope it is now obvious that these two things are exactly the same.

This is how our colleagues in physics prove the formula They draw the graph of then calculate the area under the curve by simple mensuration (Figure 8).

What I’m meaning to show by this example is that the study of linear phenomena absolutely requires the study of quadratic phenomena. This is the reason why quadratics occur everywhere, and why so much time is given to the discriminant and the vertex and completing the square and so forth.

Figure 8

Factorizing of quadratics is quick when it works, but it only works in exceptional cases. The systematic study of quadratics requires them to be reduced to completed square form:

From this completed square comes the vertex, the formula for the roots, and discriminant theory.

Conversely, the geometric questions about the parabola can be reinterpreted as algebraic questions about quadratics.

One can make the same remark about cubics being volume and being the double integral of a linear function. If you continue this, you have a theory of polynomials, but that is 3 Unit work. Note that in Question 9(b) of the 2 Unit 1990 HSC paper, cubics, similarity, maximization, and the mensuration of cylinders and cones all come together.

(b) The special functions

Figure 9	Figure 10

But as I said, it’s very hard to keep mathematics in a box. The area of the sector POQ is a linear function of the angle q at the centre, and a quadratic function of the radius (Figure 9). The area of the isosceles D POQ, however, is a sine function of the central angle q , and the sine function can’t be defined algebraically (Figure 10). We have to pass to what are called ‘special functions’.

This course essentially introduces only two of them - and sin x or cos x. Every mathematics teacher should know a little secret here, that and sin x, are really the same thing, because of the formula:

However, that’s not even in the 4 Unit course, so we can keep it our secret, until the students start reading about it and asking questions, in which case for our own credibility we had better know.

Calculus only really gets going with and sin x, and one very satisfactory thing about our 2 Unit course is that it does give a reasonable first treatment of them - only just, but there’s no more time, and it all just fits in.

These functions are desperately important if anyone is going to do anything with mathematics:

If you put them together, for example as you can describe the dying vibrations of a plucked guitar string, or the oscillations of a Kingswood after it has gone over a bump. In statistics, the formulae for the normal and Poisson distributions use and the limit of the binomial involves In fact most things in the natural world start off being described by one or both of these functions.

This is an intensely practical course for anyone who is going to need any reasonable mathematical understanding later on, and my intention is to contrast these remarks with my previous idealistic comments about structure and language and the poetry of things. We do indeed have a great subject to teach.

There is a table below comparing the two special functions. Precisely because they are really the same function, everything matches up quite neatly between them (Table 1).

First, they are both deeply associated with special numbers, the exponential function with e, and sin x and cos x with . These are irrational numbers; more than that they are transcendental - not the solution of any polynomial equation - and they are of course the two most important numbers in calculus. Although the relationships between them as real numbers are very obscure, the relationship through complex numbers is very simple, just but unfortunately we can’t really tell them that, even at 4 Unit level.

Secondly, the use of these special numbers allows the derivatives to take a very simple form without any constant multipliers, and both their graphs consequently have gradient exactly 1 at the x-intercepts and
y-intercepts. This simple form of the derivative, and the gradient 1 at the intercepts, is the reason why in calculus, radians are used to measure angles for the trigonometric functions and the base e is used for the exponential function.

Thirdly, these functions are common in the natural world, and a mathematician would explain this by saying that they are solutions of the very simplest first and second order differential equations, which can’t be solved by purely algebraic functions.

Notice that both these equations in their more general form involve proportionality, which is a linear relationship, because has a derivative proportional to itself, and has a second derivative proportional to itself. This brings us back to linear functions and again emphasizes the primacy of proportionality and of linear functions. Of course the symmetry of the two differential equations is lost in 2 Unit because simple harmonic motion is too hard for the course.

Fourthly, their study requires knowledge of various associated functions, the inverse function log x, the various other trig functions, and in 3 Unit, the inverse trig functions. Because the 3 Unit course goes a lot further in the study of these other functions than the 2 Unit course, for some 3 Unit students this can actually cloud the fact that it’s the exponential function and the wave function that our courses are mostly about. All the other work on logarithms, inverse trigonometry, secant and cosecant, compound angles and so forth is there to support and deepen that study. Let me qualify that immediately by saying that this material goes all over the place if you let it. Even in 4 Unit, all sorts of other things are happening.

Fifthly, these special functions give us primitives of the purely algebraic functions , and . These are algebraic functions whose primitives are not algebraic functions, and we have to pass over to these special functions to find them. Again the 2 Unit course misses the analogy here because these integrals and inverse trig functions are too complicated.

Figure 11

And sixthly, the associated conic. The trigonometric functions are associated with the circle, and the logarithmic function is the primitive of the reciprocal function. But the most dramatic way to see the analogy is this. The number p is the area of the unit circle. The number e is the bar you put in here on the rectangular hyperbola so that the area between the curve and the asymptote is exactly 1. So we come back to these three special conics at the very foundation of the 2/3/4 Unit course - the rectangular hyperbola for , the parabola for and the circle for . All three are just different cuts through a single cone. Note how the rational number 3 is associated with the parabola using these regions (Figure 11).

Trigonometric functions

Any study of the trigonometric functions should begin with Pythagoras’ theorem:

which is as much a theorem about circles as about triangles. Circles and triangles are of course very closely related, as the Years 9&emdash;10 geometry should have made clear.

If you hold r constant, this is the equation of a circle. But you can do more. If r is regarded as a third variable, then Pythagoras’ theorem is the equation of a cone in 3D space. Slicing this cone by planes in different directions is 4 Unit work. It gives different curves which are all called conics because they can be sliced from a cone:

1. the circle, and hence trigonometry;
2. the rectangular hyperbola, and hence
3. the parabola, and hence

The sine function predates coordinate axes, and was usually defined as the length of the semichord subtended by an angle of size q at the centre of a unit circle. Then cos q is the length of the altitude from the centre to the chord (Figure 12).

Figure 12

Note that circle geometry with chords and radii is reasonable 2 Unit work because it is just an application of isosceles triangles and congruence. It is angles at the circumference and the cyclic quadrilateral which are 3 Unit work.

Our definition of the trigonometric functions is really identical to the semichord idea, as a comparison of the two diagrams shows. It is best expressed using ratios in a circle of radius r:

because then it is clear that the sine and cosine are ratios rather than lengths (Figure 13).

This also ties in with the ratio definition of the size of an angle in radians:

Figure 13

The foundation of the calculus of the trigonometric functions is the proof that the derivative of is . At 3 Unit, everything is carefully prepared for the task, but 2 Unit has no compound angles, and the fundamental limit is usually too hard and abstract, which leaves us in a tricky situation. Nevertheless the mensuration of sectors and segments is part of the 2 Unit course, as the examiners have made quite clear in recent years.

Whatever solution you adopt, the really important thing is to differentiate the sine function graphically by having it drawn on graph paper, drawing tangents, measuring their gradients, and plotting these gradients on a new graph. It’s obvious from the graph that the derivative of sin x is cos x, but it always surprises me how few HSC pupils seem to have seen this (Figure 14).

Note how having gradient 1 at the origin means that a right angle needs to be about times a unit on the
y-axis, and is about

Along the same lines, the area of one arch of is exactly 2 - this is a characteristic property of the sine curve. The result can be easily confirmed by counting up the little squares under the curve, again reinforcing the meaning of integration.

If the differentiation process is continued, we get successively and at the fourth derivative sin x once again. Note how the phase shifts back a quarter period each time, so that differentiation of sin x is the same as rotation of 90° (Figure 15).

Figure 15

The second derivative is the opposite of sin x; that of course is the simple harmonic differential equation

This example should demonstrate how important it is that calculus remain a visual, geometrically based subject and not be lost in algebraic calculations. Almost every question should be sketched in some way. I often say to my class that the 2/3/4 Unit courses are picture-book courses, in that the graphs are in the foreground almost all the time.

I’ll digress a little to talk about Question 10 from the 2 Unit 1996 HSC.

(a) (i) On the same set of axes, accurately draw the graphs of and for

(ii) Find the gradient of the tangent at the origin.

(iii) For what values of m does the equation have a solution in the domain

(b)

The diagram shows a circle centre O with points P and Q on the circle. The angle POQ = 2a , the length of the chord PQ is 200 metres, and the length of the arc PQ is 300 metres as shown. (i) Show that

(ii) Use your graphs from part (a) (i) to find an approximate value for a .

(iii) Hence find the size of POQ, and also find the radius of the circle.

(c) A and B are two points 200 metres apart. For what values of is it possible to find a circular arc AB of length metres? Justify your answer. You may use the results from parts (a) and (b).

It’s a classic recent HSC question, involving mensuration of arcs, the graph y = sin x drawn in radians, graphical solution, and an unseen investigation of the forms a diagram can take.

The exponential function

In 3 Unit, I believe it is best to begin with the logarithmic function, starting with the problem of integrating , and arriving at the result:

But this is far too ambitious at 2 Unit level, so the usual approach is to look for an exponential function whose derivative is equal to itself. That base must be the special number e.

With more able pupils, one can proceed as follows: It’s clear that every exponential function has a derivative that is a multiple of itself:

The limit here is the gradient of at its y-intercept. It is reasonably clear from the graphs that has a gradient less than 1 at the y-intercept, and has a gradient more than 1 at the y-intercept.

But again, to convince most 2 Unit people, hand out on graph paper, draw tangents, measure their gradients, and plot the result. This time the derived curve is identical to the first (Figure 17). This must be understood as meaning:

1. ‘the gradient at each point is equal to the height’,
2. is a solution to the differential equation

Figure 17

Students taking 2 Unit generally seem pretty resistant to absorbing this level of sophistication, even though the idea itself may seem very simple to us.

Intuition about exponential functions is easier if the base is 2 rather than e. That’s why we have half-lives of radioactive elements and doubling times of populations. It’s only in the calculus that base e is more convenient. Remember that can be written in terms of

so in fact the function covers every exponential function, to whatever base. And beware, the examiners have shown they are interested in

There’s an important link here with series. I’ve already remarked that an arithmetic series is just a linear function. But every geometric series is really just an exponential function, for example the GP:

6, 12, 24, 48, 96, 192, …

is just the values of for positive integer values of x. For example, depreciation is a GP when the value is considered at the end of each year, but an exponential function when the value is considered at any time t.

I’ve remarked on how the rectangular hyperbola under-lies the exponential function, because The rectangular hyperbola also arises from area in geometry by taking a rectangle of constant area and looking at the relationship between the sides. For example, the diagram (Figure 18) generates the hyperbola

Figure 18

Note that Question 8(b) from the 1992 2 Unit paper explored this relationship a little further.

Probability and statistics

The fundamental connection of the elementary probability of the 2/3/4 Unit course is that is the number of ways of choosing r objects from n objects, and is also the coefficients in the expansion of The 2 Unit course studies neither polynomials nor binomial probability, so I really can’t say much here.

Maybe binomial probability can be added to 2 Unit - it’s quite straightforward to prove the binomial theorem without any reference to polynomials, I actually think it would be clearer this way in 3 Unit, and it’s absolutely essential to any understanding of statistics. But in any case, the 2 Unit probability is quite essential preparation for anyone continuing with mathematics, even in the absence of

Apart from probability and the binomial distribution, statistics is hard to fit into our course. The next mathematical object is the normal distribution, and this is just too hard at 2 Unit level. The function involved is which is far too tricky for 2 Unit, and the central limit theorem is the motivation for the normal distribution, and that is surely over the top. Otherwise it’s descriptive statistics, which is not appropriate for an academic mathematics course, and is better handled directly in courses that use it. A really tough problem is what you leave out. As I think I’ve shown, the present course achieves a great deal in its combined geometric-algebraic development of the calculus through to and but it only just makes it, and omissions would bring down the structure.

Geometry and arithmetic

I have tried to show how each topic in calculus grows out of some geometric concept. To put it very roughly:

• Tangents and areas lead to differentiation and integration.
• Parallels, angle theory and similarity underlie linear functions.
• Parabolas and areas are the geometric expression of quadratics.
• Circles, triangles and similarity lead to the trigonometric functions.
• The rectangular hyperbola and the rectangle of constant area underlie the log and exponential functions.

The geometric ideas need to be taken seriously because here, as in all of mathematics, ancient and modern, geometric ideas seem to be the key to a really fundamental understanding of the structure of the material and its interrelationships. The Greeks set each of our modern disciplines on its proper foundations. They developed geometry and arithmetic as the two pillars on which their mathematics rested. These two sets of intuitions, one about number and one about shape, should be understood separately as well as being combined.

Trigonometry already begins to draw the two together, but the coordinate plane is the traditional way in which a union of the two is begun. It is the place in which our calculus is done, and calculus draws its intuitions from both. We turn to geometry for the derivative and the integral, and to some extent for the classification of functions. We turn to arithmetic for the equations of functions, for tables of values, for numerical solutions, and for algorithms for handling rational numbers, surds, logarithms and trigonometric expressions. The real numbers we take from geometry - they are the points on a line. Much of the strength of our course lies in the combining of geometric and arithmetic intuition into the graphical work, and it is important that students bring these intuitions into calculus already well developed from earlier days.

The diagram (Figure 19), which is so central to the trigonometry of our course, is a symbol for what I’m saying. It represents so much about trigonometry and about our course. At the centre is the Greek circle, existing in its own right, a symbol of the perfection of God. We can imagine Descartes, the great rationalist, bringing in his bag of numbers and tying down the circle at the top, the bottom, the left and the right, so that it is brought under human control. There are many buildings in the Renaissance with precisely this structure at the top and sides of their arches. It certainly is a classic representation of humanism. And yet we know that algebraic geometry later became just as much about using geometry to interpret and understand algebraic ideas as about using algebraic ideas to interpret and understand geometric ideas. The harmony of the two is expressed in 4 Unit in the fundamental theorem of algebra and much later in the definition of the real numbers themselves, but that is all quite beyond 2 Unit.

Figure 19

Conclusion

So I return to my opening remarks, that elegance is an excellent characterization of mathematics, and I have spoken about the two contrasting aspects of elegance - first, structure, an imaginative appreciation of structure and the interrelatedness of ideas, and secondly, language, the careful control of the language and logic by which these structures are enunciated and proven.

The excellence of our 2/3/4 Unit courses, very much including the 2 Unit course alone, is that they allow us to teach these aspects of our subject. The material in the courses has been chosen with a great deal of wisdom in order that we can present a unified and coherent course in which everything relates to everything else, and in which these most satisfying structures of introductory calculus can be developed. As I’ve said elsewhere, the present 2/3/4 Unit syllabuses have managed to create a small, self-contained but very lively microcosm to mimic the unified structure of mathematics as a whole, and to give students a genuine experience of the methods and structures of the subject.

Like all coherent courses in mathematics, this unity is simplistic, and I began by showing how even the simplest structures have implications and suggestions that lead far beyond our courses. I think the skill in teaching each of the three courses is to keep the eye on the main structures of the course, emphasizing constantly the many interrelationships and links among the various topics. However, occasionally, and without confusing the class, one can allow certain problems or parts of the theory to suggest further realms of mathematics. Thus our subject does not become closed and complete, but every solved problem has the capacity to throw up further problems and further structures yet to be developed, and occasionally even problems whose solutions are not yet known.