Goals for Mathematics Instruction

Goals for Mathematics InstructionAuthor(s): R. C. BuckSource: The American Mathematical Monthly, Vol. 72, No. 9 (Nov., 1965), pp. 949-956Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2313327 .

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GOALS FOR MATHEMATICS INSTRUCTION R. C. BUCK, University of Wisconsin

In recent years, the mathematical community has been well supplied with articles and brochures which set forth detailed recommendations for curricula and courses, both at the college level and for K-12. More often, the suggestions which follow a title such as the one above have been very specific, saying such things as: "The number line should be done before fractions," or "In linear algebra, one should treat matrices after one has discussed linear transformations," or "The central core of grades 1-5 should be Naive Set Theory and an introduction to formalized Intuitionistic Logic."

In the remarks to follow, I wish to present some goals which are less specific and I hope more universally applicable. They are intended to be more or less independent of courses or content; some may help to supply an answer to the person who asks: "Aside from its technological importance, what are the educa- tional values of mathematics?"

G o AL 1: To provide understanding of the interaction between mathematics and reality.

The point here is one that has been said many times, but I feel it is important enough to repeat the plea, [I].

"Regardless of his ultimate interests and career, a student of mathematics ought to understand something about the way in which mathematics is used in applications, and the complicated interaction between mathematics and the sciences. For many mathematicians, engaged in pure research, contact with other sciences may be infrequent; they are apt to see their subject as one that is largely self-sufficient, breeding its own sub-disciplines and creat- ing its own research problems with only occasional stimulus from outside. Others who are more directly involved in neighboring disciplines may find it difficult to agree with this position, and indeed may lay great stress upon the role of the physical sciences as a source for mathematical ideas and techniques. "

These apparently opposing views are, of course, merely two sides of the same coin, and it is certainly not fair to a student to present only one. Continuing the same quotation:

"What is then the role of mathematics in the sciences? We think that mathematics offers the scientist a vast warehouse full of objects, each available as a model for various aspects of physical reality."

By 'model,' we do not mean the plaster form on the shelf but a mathematical structure which represents one of the possible abstractions from the concrete problem. The model can be a system of differential equations, or a group with its associated representations, or a differentiable manifold. A wide variety of illustrations may be found in the September 1964 issue of Scientific American, devoted to "Mathematics in the Modern World."

949



950 GOALS FOR AMATHEMATICS INSTRUCTION [November

Finally, there is a philosophical question which must be posed. Why is mathematics useful? The physicist E. P. Wigner addressed himself to this in an article [12] with the title: "The unreasonable effectiveness of mathematics in the natural sciences." From our 'model oriented' point of view, a possible answNer can be given [1]:

"The richness and diversity of this supply [of mathematical structures] are central reasons for the importance of mathematics; another is that-along with these objects-mathematics offers a system for using the models to help raise or answer questions about physical reality, as well as techniques for exploring the behavior of the models themselves."

GO AL 2: To convey the fact that mathematics, like everything else, is built upon intuitive understandings and agreed conventions, and that these are not eternally fixed.

Ever since Euclid, students have been reminded that mathematics rests upon a collection of undefined terms and accepted axioms. It is perhaps less often pointed out that we must also agree upon a system of deduction. (Anyone in doubt about the last point should reread what the Tortoise said to Achilles! [2]). At the same time, however, I think we should point out the flexibility of these starting points, and that dogmatism in mathematics is not a virtue. An instance can be found in the notions of function and set.

During the last decades of the 19th century, mathematics went through a period of intensive self-examination and purification in the attempt (ultimately vain) to achieve a complete codification of its foundations. The notion of 'set' was selected as a basic undefined term. The rules of the game then require you to impose a collection of axioms, and then to define all later concepts in terms of the primitive ones. The purpose here is not primarily mathematical but philosophical. As mathematicians, we operate in a world of entities whose existence and properties we understand on the intuitive level. The approach of the formalist is to construct models for these entities within a selected system (say 'set theory') which mimic the platonic ideas in our mind, and then to propose these models as definitions for the concepts themselves, thus identifying the model with the object.

It is this process which led to the statement which has been so fashionable in the last several decades (mostly in very elementary books): "A function is a certain sort of class of ordered pairs." Indeed, if this is done in the proper 19th century spirit, one should also define the ordered pair (a, b) to be the set { {a, b }, { a } }, and it does not take much effort to see that a formalized presen- tation of analysis in this fashion is not very rewarding. A function of two vari- ables becomes a class of pairs ((a, b), c), each of which has the set description:

Ut,} a, { aI} },c, c {I, b}, {a} } } } For example, it is tedious to check that another correct description of ((a, b), c) is:

U U { I a}, { a, b}}}, t {I a}l, { a, b}} , c}}



1965] GOALS FOR MATHEMATICS INSTRUCTION 951

While there are times when it is necessary to work with formal models for 'ordered pair' and 'function,' most mnathematicians find it far easier to leave these unformalized, and to treat both 'set' and 'function' as independent primitive concepts, defined only by their properties; the collection of ordered pairs associated with a function is then the graph of the function, and is regarded as something you construct from the function, not the function itself. In support of this position, Saunders MacLane recently wrote [7]:

"For effectiveness in teaching mathematics, it would be more in order to follow a balanced approach, using the intuition of both 'set' and 'function.' Unfortunately, recent American reformers of elementary mathematics teaching have been over-zealous in propagandizing the notion of set. This has led to elementary books grossly overemphasizing sets (and logic). This over- emphasis should be reversed: for example, one should no longer preach that a function is a certain sort of set of ordered pairs."

But, students should also know that this is not the only possible axiomatiza- tion of the foundations of mathematics. In his 1925 thesis [10], von Neumann proposed an alternate development in which "function" is the primitive con- cept, and "set" is derived. 1M'oreover, it should also be pointed out that recent work in category theory has revived interest in these matters, and that it now appears that another fruitful approach [5] is to adopt "morphism" as the basic primitive. The moral is that there is no single approach, and that coming gener- ations may adopt different views about the axiomatics of mathematics, but that the test is whether they agree with our intuitive picture of mathematics, not whether they are 'right' in some absolute sense.

Go AL 3: To demonstrate that mathematics is a human activity and that its history is marked by inventions, discoveries, guesses, both good and bad, and that the frontier of its growth is covered by interesting unanswered questions.

We speak here of the historicity of mathematics. Examples from the past can be given which make this clear, and which show individuals at work and the nature of their contributions. There is, for instance, Decartes' invention of the exponential notation for integral powers, followed by the emotional need to attach meaning to ax for general choices of x which led to a considerable amount of research work culminating in the general treatment by Euler in 1750 which allowed complex x. Another illustration is to be found in the work of Hamilton who tried for ten years to solve the problem of making appropriate definitions of nmultiplication and addition so that 3-space became a division algebra, and who at last took the leap into 4-space and discovered quaternions [3 1. The effect of Hilbert's 1900 speech on the directions of mathematical activity in the half century which followed is again such an illustration [4]. (In this, it is also interesting to speculate upon the effect of this speech, had Hilbert himself not been one of the major contributors to mathematics during this same period.)

We all recognise that it is difficult to present problems of current interest to an unsophisticated audience. In certain areas, however, this can and has been



952 GOALS FOR MATHEMATICS INSTRUCTION [November

done, and it is important for the future development of mathematics and its status in the world that more attention be given to exposition of this sort.

GOAL 4: To contrast "argument by authority" and "argument by evidence and proof"; to explain the difference between "not proved" and "disproved," and between a constructive proof and a nonconstructive proof.

In many fields of study, questions are settled by citation of authorities, and even by oratory and polemic. In the sciences, one more often finds argument which is based upon evidence, and is inductive or deductive in structure, Throughout all its history, mathematics has made "proof" itself a subject of study, and has a special contribution to make.

Let us examine a special situation. Suppose someone asks: "Is it possible to . . . ," where the blank is to be filled with some specific objective. In many areas of human concern, the answer may have to be: "No, that is impossible; assuming, of course, that our present understanding of the world is correct." (Unfortunately, too often the last qualification is omitted.)

As I have indicated, answers of this sort must be time-bound. Answer for yourself, each of the following:

1845 Can you see through a sheet of metal? 1950 Can argon form chemical compounds? 1965 Can you transmit information at a speed faster than light?

In mathematics, it seems possible for many questions to provide eternally negative answers. In 1965, we do not know how to square a circle; we believe, in fact, that we can prove that no one will ever be able to square the circle. We also believe that we can prove that no one will ever be able to find integers x, y, z such that x3+y3 ==z3, xyz,#O0. Fermat's conjecture, of course, remains "not proved."

For many people, a more interesting example is to be found in the so-called Arrow Paradox. (No relation to Zeno.) The problem posed is to obtain a completely fair method for arriving at a group consensus ranking of a set of alterna- tives when one has been given a collection of individual rankings of these alter- natives. (Each judge has ranked the beauty queen contestants from 1 to 10; what should the final ranking be?) Being objective about this, we list a set of criteria which any admissible system should satisfy. (For example, if each judge ranks contestant C higher than contestant D, then the group ranking should also place C higher than D, although the interval need not be the same.) Having completed this, we seek those group ranking schemes which satisfy the criteria. The paradox is that, in spite of the apparent utter reasonableness of the criteria, there is no admissible solution! Here, again, we prove that no one will ever arrive at a completely 'fair' system for deciding such matters [6].

I also feel that the typical non-constructive proof of existence is peculiar to mathematical thought, and that it is therefore important for students to be made aware of this consciously at some time. En passant, it should be remarked




that the recent argument which was used in nuclear physics to deduce the existence of a particle by the absence of interactions sounds a lot like this mode of proof !

GOAL 5: To demonstrate that the question "Why?" is important to ask, and that in mathematics, an answer is not always supplied by merely giving a detailed proof.

Suppose we have just finished proving an interesting and perhaps somewhat surprising result. Is it meaningful to ask why it is true? In many cases, I think it is, and that a correct answer may not be to repeat the proof just given. What I would hope for is an insight which illuminates the entire matter, which goes to the heart and lets the student see why the whole result came out as it did. Such insights are seldom unique, for one may look at a situation from many viewpoints.

Perhaps several illustrations will explain what I mean. In number theory, Euler's generalization of Fermat's theorem asserts that if c is relatively prime to m, then

C+(m)1 = (mod m).

For me, the clarifying insight is that the integers between 1 and m which are prime to m form a multiplicative group. Since the order of this group is k(m), the Euler relation becomes an instance of Lagrange's theorem on groups.

As another illustration, suppose we have just shown by the usual process that the real and imaginary parts of a holomorphic function obey the Cauchy- Riemann equations. Is there a viewpoint which makes this look sensible? For me, the following has been helpful. Suppose the function is given by f: z->w, with z = (x, y) and w -(u, v). The differential of this mapping is represented by the matrix

[Ou au]

df= Lx 9y9.

_8(x y_

The Cauchy-Riemann equations assert: au/ax = av/ay and au/ly = o-v/iax. Looking at df, this means that the matrix has the form

A B1 df=[ d= _B A_

The illuminating insight is that matrices of this form are precisely those which arise in the representation of the complex field as an algebra of real 2-by-2 matrices. (The actual connection is made by re-examining the nature of the differentiation process for complex valued functions.)

Finally, the change of variable formula for multiple integrals in elementary calculus is made more transparent for me by the following observations. Sup-



954 GOALS FOR MATHEMATICS INSTRUCTION [November

pose that T is an orientation preserving homeomorphism of class C', defined in a region R in n-space, and let D be a compact quadrable set in R, and D* its image under T. Let f be a continuous function defined on D*. We want to understand why it is true that

fX *. L* * f ... fDf(T(x))J(x) dxdX2 . . . dx

where J(x) is the Jacobian of T at x. First, introduce a set function F for suitable sets SCR by

F(S) = f TT J T(S)

(Note that we want to calculate the value of F(D)). As a general rule (the fundamental theorem of calculus), set functions are the integrals of their de- rivatives. If g is the point function which is the derivative of the set function F, i.e.

g(xo) = lrn F(S) Vol (S)

where the limit is taken as the set S closes down on xo, then we would have for any set S,

F(S) = **fg.

(Looking at the answer, we want to show g(x) =f(T(x))J(x).) If T(xo) =yo and S is closing down on xo, then T(S) is closing down on yo. When T(S) is small enough, f is essentially constantly f(yo) on T(S), so that F(S) is approximately f (yo) vol(T(S)). This gives us

vol (T(S)) g(xo) = f (yo) lurn ol(S

Now, the differential of T, dT, is a linear transformation which gives a local approximation to T. Linear transformations alter the volumes of sets by a multiplicative factor, namely their determinant. But, the determinant of dT is J, so we arrive at

g(xo) -f(yo) lim J(x-) Vol (S) Vol (5)

- f(T(xo))J(xo).

(In a simplified form, and with the details supplied, this approach is to be found in [1], 300.)




G o AL 6: To show that complex things are sometimes simple, and simple things are sometimes complex; and that, in mathematics as well as in other fields, it pays to subject a familiar thing to detailed study, and to study something which seems hopelessly intricate.

The physical sciences have made much of their progress by repeating older experiments with greater and greater precision. In a similar way, progress in mathematics has often been made by re-examining some apparently well-worked vein with totally new tools and viewpoint.

Surely, the nature of the real numbers system and the elements of calculus are sterile ground! The classical work of Weierstrass and others succeeded in erasing the embarrassing unrigor of Leibnitz and Cauchy, and we know where we stand. Nevertheless, those with an ear to the ground and an eye on Mathe- matical Reviews know that "infinitesimals" have once more crept out into the open, and with new and impressive credentials! The subject of nonstandard analysis owes its origin to recent work in mathematical logic, model theory, and the study of syntactical descriptions. The nonstandard reals not only have infinitesimals, but also their reciprocals, and we can construct tangents in nonstandard calculus by drawing secants through points which are infinitely close! Moreover, it even seems possible to work within nonstandard analysis and obtain solutions to previously unanswered problems. If significant new results in classical mathematics continue to emerge from beneath the cloak of this heresy, it may become necessary for us to re-examine the nature of the elementary calculus we teach in colleges. (See [8] and [9].)

In another direction, the amorphous collection of all social games must surely seem an unlikely arena in which to employ the tools of mathematics. The simple steps by which von Neumann analysed this, to emerge at the end [11] with the common abstraction of a game in normalized form, about which certain general statements could be made, is a perfect example of the last sentence in Goal 6. Another example on a much simpler level occurs when a student replaces

(2x2+5)2 + 2 2x2 + 5 + 4

+ 4

x+ 4

by

y2 + 4y + 4

in order to obtain the simplification

(2X2 + 2x + 7)2

(X + 1)2

In another sense, the same pattern occurs whenever one divides out the radical of an algebra to get something one may hope to work with more easily, postponing the analysis of the radical and the original algebra until later. In applications, it often pays to isolate certain components of a complex structure



956 A CLASS OF INTEGRAL EQUATIONS [November

and study their interaction, before attempting to investigate the internal structure of each of the components.

I hope by these remarks that I have succeeded in showing that mathematics need be neither austere nor remote, that it has relevance for almost all human activities, and that it can be of value to any sincerely interested person.

References 1. R. C. Buck and E. F. Buck, Advanced Calculus, 2nd ed., McGraw-Hill, New York, 1965. 2. Lewis Carroll, What the Tortoise said to Achilles, The World of Mathematics, Simon &

Shuster, New York, 1956, pp 2402-2404. 3. W. R. Hamilton, Lectures on Quaternions, Dublin, 1853. 4. David Hilbert, Mathematical problems, Bull. Amer. Math. Soc., (1902) 437-479. 5. F. W. Lawvere, An elementary theory of the category of sets, Proc. Nat. Acad. Sci., 52

(1964) 1506-1510. 6. R. Duncan Luce and Howard Raiffa, Games and Decisions, Wiley, New York, 1957

(Chap. 14). 7. S. MacLane, in Proc. Prelim. Meeting on College Level Math. Educ., US-Japan Prog.

Sci. Coop., Katada, 1964, pp. 68, 69. 8. A. Robinson, Non-standard analysis, Nederl. Akad. Wetensch., 23 (1961) 432-440. 9. A. Robinson, Introd. to Model Theory and the Metamathematics of Algebra, North Hol-

land, Amsterdam, 1963 (Chap. X). 10. John von Neumann, Die Axiomatisierung der Mengenlehre, Math. Zeit., 27 (1928)

669-752. 11. J. von Neumann and 0. Morgenstern, The Theory of Games and Economic Behavior,

Princeton University Press, Princeton, 1947. 12. E. P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences,

Comm. Pure Appl. Math., 13 (1960) 1-14.

A CLASS OF INTEGRAL EQUATIONS A. G. MACKIE, Victoria University of Wel-lington, New Zealand

In a recent paper Buschman [1 ] solved the integral equation

(1) 1 PX (I) 1 g(xo) dxo = f(x),

where x> 1 and P. is a Legendre polynomial. In a later note Erdelyi [2] gave a more direct proof of the inversion formula with fewer restrictions on the function f(x). These papers prompted the author to examine whether it would be possible to solve the integral equation (1) when the limits in the integral are from 0 to x instead of 1 to x. Because of the singularity at x0 = 0 there will need to be certain restrictions on the f and g functions to ensure convergence of the integral but once these are met it is in fact possible to solve for g(xo) explicitly. Moreover, the solution is valid for any real value of n and not merely for positive integral values.



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Goals for Mathematics Instruction