Karl Menger on teaching
Karl Menger wrote many articles on teaching mathematics which make fascinating reading. We list some of these below with a brief extract from each paper. In almost all cases we quote the part where Menger sees a problem, but not the bulk of the paper in which he proposes his solution. We, therefore, strongly recommend that readers looks at the papers that are of interest to them and read them in their entirety. We present the papers in chronological order.
1. What is Calculus of Variations and What Are Its Applications?
The Scientific Monthly 45 (3) (1937), 250-253.
The calculus of variations belongs to those parts of mathematics whose details it is difficult to explain to a non-mathematician. It is possible, however, to explain its main problems and to sketch its principal methods for everybody. The first human being to solve a problem of calculus of variations seems to have been Queen Dido of Carthage. When she was promised as much land as might lie within the boundaries of a bull's hide, she cut the hide into many thin strips, put them together into one long strip, the ends of which she united, and then she tried to secure as extensive a territory as possible within this boundary. History does not describe the form of the territory she chose, but if she was a good mathematician she covered the territory in the form of a circle, for today we know: Of all surfaces bounded by curves of a given length, the circle is the one of largest area. The branch of mathematics which establishes a rigorous proof of this statement is the calculus of variations. Newton was the first mathematician to publish a result in this field. If a body moves in the air, it meets with a certain resistance, which depends on the shape of the body. The problem Newton studied. was, what shape of body would guarantee the least possible resistance? Applications of this problem are obvious. The rifle bullet is designed in such shape as to meet with a minimum resistance in the air. Newton published a correct answer to a special case of this problem, namely, that the surface of the solid considered is obtained by revolving a curve around an axis. But he did not give the proof or the calculations that had led him to the answer. So Newton's solution had no great effect on the development of mathematics. ... What is common to these problems is this: A number is associated with each curve of a certain family of curves. In the first example (that of Queen Dido) the family consists of all closed curves with a given length, and the associated number is the area of the inclosed surface; in the second example (that of Newton) the number is the resistance which a body somehow associated with the curve meets in the air.
2. What is Dimension?
Amer. Math. Monthly 50 (1) (1943), 2-7.
Strictly speaking, all material objects are 3-dimensional. Yet, only such objects as a metal sphere, a wooden block, or a rock are considered to be typical representatives of 3-dimensional entities (solids). A piece of sheet-iron, paper, and a membrane approach what we mean when we speak of 2-dimensional objects (surfaces). Wire, threads, and streaks of chalk represent our idea of 1-dimensional entities (lines). What is the difference between objects of different dimensions? Originally, mathematicians believed it to be a difference in quantity, in the sense that a surface contains more points than a line and less points than a solid. Now primarily the words "more," "less," and "equally many" are restricted to finite sets while surfaces, as well as lines and solids, contain infinitely many points. But Georg Cantor extended their use to all sets. We say that two sets - finite or infinite - contain equally many elements if we can establish a one-to-one correspondence between their elements. Cantor found that two infinite totalities do not necessarily contain equally many elements. For instance, among geometrical objects a straight line segment contains more points than some dispersed infinite sets, e.g., the set of all points on a straight line whose distances from a certain point are integers. However, a straight line segment, a square, and a cube do contain equally many points. Since these objects are of different dimensions, it follows that dimension is not a quantitative property. Later, geometers thought that the difference between a 1-dimensional and a higher-dimensional object lay in the fact that the former, but not the latter, can be traversed by a continuously moving point. Indeed, lines on a paper or a blackboard are drawn, i.e., traversed by the point of a pencil or chalk. However, Peano found that a continuously moving point can traverse a square surface or a solid cube though nobody would call these objects 1-dimensional. On the other hand, 1-dimensional objects were found which cannot be traversed by a continuously moving point. The fact that an object is the path of a point is interesting in itself, but has no bearing on the question of the dimension of the object.
3. On the Teaching of Differential Equations.
Amer. Math. Monthly 51 (7) (1944), 392-395.
The purpose of this note is the exposition of a well-known fact of physics which might be used with benefit in teaching differential equations, especially, in helping students to visualize them. In solving the equations we shall point out minor shortcomings, from the logical point of view, of the traditional exposition in textbooks, and indicate how they should be corrected. The simplest introduction to the theory of differential equations is offered by nature. Cover a thin bar magnet with a horizontal piece of cardboard, and sprinkle fine iron splinters over the latter. Each splinter after coming to rest will assume a definite direction, depending upon its location. Before us is displayed the direction field associated with a first order differential equation. The splinters fit together along lines, called the lines of force of the magnet in the plane of the cardboard. These lines represent the solutions of the differential equation. If the magnet covered by the horizontal cardboard is in a vertical position touching the cardboard at one pole, then each splinter will point toward the pole. If we choose this pole as the origin of a Cartesian coordinate system, the splinter at the point (x, y) thus has the slope y/x. Hence the differential equation associated with the magnetic direction field reads y'(x) = y(x)/x for x ≠ 0.
4. Methods of Presenting e and π.
Amer. Math. Monthly 52 (1) (1945), 28-33.
Initiating a student into calculus is about what sailing through the straits of Messina used to be: On one side the Charybdis dragging the boat into her whirlpool, on the other side the Scylla waiting for the vessel to shatter on her rock. The whirlpool engulfing so many teachers consists of the false statements concerning infinitely small quantities, the rock on which the beginner goes to pieces is the solid foundation of analysis. The proper initiation into calculus must painstakingly avoid all senseless statements and at the same time avoid unduly rigorous reasoning. If we add that even a first introduction should aim at conveying to the student the understanding of calculus rather than a mere mechanical ability to handle formulae, then we have about described the difficulties confronting the teacher. The solution is to present only statements and arguments which the student can easily visualize and which are capable of rigorous proofs, but to present them without any attempt at rigorously proving them beyond what may come up in answering the questions of intelligent students. As an example of a presentation in this spirit, in what follows I outline methods of introducing e and π which for years I have found useful in teaching.
5. Are Variables Necessary in Calculus.
Amer. Math. Monthly 56 (9) (1949), 609-620.
While variables are not necessary for the presentation of fundamental results of calculus, there remain two questions. To what extent are variables necessary in proving these results? And, are variables not desirable even in formulating the theorems? Since most students learn calculus as a tool, and since books on physics, engineering, statistics, mathematical economics, etc., are written in the classical notation, it is clear that, in initiating students into calculus, we have to use the classical notation. Yet I feel that the possibility of a consistent notation without variables should influence our teaching, namely, in the direction of reducing the use of variables. I further think that, at least in a few cases, we should mention the alternative form and, in particular, make the student aware of the possibility of a consistent notation which dispenses with dummy variables. I even believe that the ability to grasp, say, integration by substitution without variables is a gauge for a student's real understanding of calculus.
6. Tossing a Coin.
Amer. Math. Monthly 61 (9) (1954), 634-636.
A basic fact concerning random experiments treated in practically all books on probability and statistics is stated in one of them in the following words: "If an ordinary coin is rapidly spun several times, and if we take care to keep the conditions of the experiment as uniform as possible in all respects, we shall find that we are unable to predict whether, in a particular instance, the coin will fall 'heads' or 'tails'." The page devoted to a general discussion of this experiment ends with the following remark. "A moment's reflection will show that even extremely small changes in the initial state of the motion must be expected to have a dominating influence on the result. In practice, the initial state will never be exactly known, but only to a certain approximation." Since a quantitative description of the problem can be achieved in terms of the most elementary mathematics and in about as many words as are required by a general discussion, there is no reason why every beginner should not learn the quantitative details of the case.
7. Why Johnny hates math.
The Mathematics Teacher 49 (8) (1956), 578-584.
"Mathematics the most-hated Subject" was the title of an editorial that recently appeared in a large newspaper. ... Discussion has brought forth a great variety of partial explanations. Experts have blamed the teachers and the students. They have criticized the universities, the teachers colleges, the high schools, and the grade schools. They have found fault with the choice of classical topics, their arrangement, and the absence of modern mathematics from the curriculum. They have emphasized that the current textbooks are obsolete, the classrooms overcrowded, and the teachers underpaid. They have, in other words, searched for the roots of the trouble practically everywhere - everywhere except in the procedures of mathematics itself. Mathematics is looked upon with a mixture of awe and gratitude. And indeed everyone has reason to be grateful. Mathematics has been the decisive factor in understanding the universe and the most powerful tool in controlling nature. All this is beyond any doubt. What is on shakier ground, however, is the general belief that the tremendous achievements of mathematics are due to the clarity of its basic procedures and to the precision of its current language. The time has come when it must be frankly admitted that some mathematical processes are successful not because of those presumed qualities, but rather in spite of obscure foundations, ambiguous expressions, and lack of articulateness.
8. What are Variables and Constants?
Science, New Series 123 (3196) (1956), 547-548.
The notion variable has not in the literature attained the degree of clarity that would justify the almost universal use of that term without explanations. Recent investigations have resolved it into an extensive spectrum of meanings, some pertaining to reality as investigated in science, others belonging to the realm of symbols studied in logic - two altogether different worlds. As a corollary, these distinctions yield a clarification of the notion constant. ... No clear distinction has heretofore been made between numerical and fluent variables. Moreover, in the literature, numerical variables and variable quantities, notwithstanding the profound differences between them, are indiscriminately referred to as "variables"; and the two concepts have actually been confused.
9. What are x and y?
Mathematical Gazette 40 (334) (1956), 246-255.
An ordered pair whose second member (or value) is a number, while its first member (or object) may be anything, will be referred to as a quantity. By a consistent class of quantities - briefly, c.c.q. - we mean a class of quantities that does not contain two quantities with equal objects and unequal values. Reviving Newton's term, we will refer to c.c.q.'s also as fluents. The class of all objects (of all values) of the quantities belonging to a fluent is called the domain (the range) of the fluent. ... Some mathematicians (if hardly any scientists) call all fluents "functions". But only fluents whose domains consist of numbers or systems of numbers have the power to connect c.c.q.'s. The logarithm connects tihe radius r with log r, which is the class of all pairs (K, log r(K)) for any circle K; similarly, the area a with log a; the exponential function with the identity function; the function cos with log cos; etc. The radius r lacks the power to connect two fluents. There is no radius of the area nor a radius of the cosine. Nor can fluents other than those of the type of log be differentiated or integrated: dr/dK and ∫ r dK are meaningless. Hence a special name for those fluents whose domains consist of numbers or systems of numbers is practically indispensable, and no name for them is more appropriate than "functions". Many scientists and some mathematicians call fluents "variable quantities" - often simply "variables". The latter usage is quite unfortunate. It certainly has not been conducive to the maintenance of a clear and sharp distinction between what above have been called fluents (specific classes !) and variables (replaceable symbols !). Yet this distinction is of the utmost importance in pure and applied mathematics if terms and symbols are to be used according to articulate rules.
10. New Approach to Teaching Intermediate Mathematics.
Science, New Series 127 (3310) (1958), 1320-1323.
[The new approach] is based on a resolution of the spectra of meanings of the letter x and the term 'variable'.
Nothing is more distasteful to an active mathematician or scientist than discussions of symbolism and notation, and that dislike is perfectly understandable. After having overcome in his youth whatever difficulties the formal expression of ideas presents, the mathematician finds that certain ways of writing have become his second nature and regards any suggestion of a change, even if he recognizes its merits, as nothing but a trivial nuisance. There are, however, situations in which a thorough discussion of such matters on the highest level is inevitable. They occur when, at turning points in the history of culture, it becomes imperative to make certain techniques and ideas of mathematics available to wider strata of the population. In the large groups to be initiated, many persons lack the ability to overcome the difficulties that the specialist overcame in his youth. Moreover, an immense collective benefit results if even persons with that ability are spared unnecessary complications.
11. Gulliver in the Land without One, Two, Three.
Mathematical Gazette 43 (346) (1959), 241-250.
About the year 1700, in the course of heretofore unrecorded travels, Gulliver met islanders who used a rather odd arithmetical vocabulary. They counted: stix, stixpair, stixtrip, four stix, five stix, six stix, and so on; and they wrote
(1) |, | p, | t, 4|, 5|, 6|, and so on.
The symbol | and the word stix also denoted kauri sticks - the insular currency and principal object of applied mathematics. The symbols |, | p, | t and the corresponding words for the first three numbers had, as Gulliver learned, always been in use. But beyond stix, stixpair, and stixtrip, the islanders originally counted: four, five, six, and so on (and they wrote: 4, 5, 6, and so on). The combination of these two types of original symbols, however, created difficulties. For instance, while one could write
(2) | + | p = |t and 4 + 5 = 9,
the symbols | + 4 and | p + 5 as well as the words stix plus four and stixpair plus five appeared to be incongruous. As a remedy, mathematicians proposed a uniform symbolism; and they achieved uniformity by assimilating the higher numerals to the lowest three. After this reform, (1) became the official sequence of insular numerals. Ever since, sums were indicated by
(3) | p + | t, | + 4|, | - 5|, 4| + 5|, ...
Stix plus four stix (in contrast to stix plus four) sounded unobjectionable and so did stixpair plus five stix.
12. Gulliver's Return to the Land Without One, Two, Three.
Amer. Math. Monthly 67 (7) (1960), 641-648.
Gulliver's interest in advanced mathematics can be traced to adventures not mentioned by Swift and only recently recorded. They took place on an island which, because of the current sophisticated symbols for numbers, the traveller called the Land without One, Two, Three. Upon his return to Europe, Gulliver studied, both in England and on the Continent, what then (that is, in the early 1700's) was the most modern topic of investigation: the functions x, x2,x3,√x, cos x, log x, The more the traveller penetrated into analysis, the more he admired the ingenious theory. Yet he was puzzled by several features of those sophisticated symbols for functions, which somehow reminded him of the numerals in the Land without One, Two, Three - features of analysis that, incidentally, are still apparent today.
(a) Each symbol for a function includes the letter x and is equivocal in that it also designates the value assumed by that function for x. For instance, the function log x assumes the value log x for x.
(b) Power functions are denoted by symbols that differ in structure from those in which x is preceded by signs such as √, cos or log. In each symbol for a higher power, x is followed by a raised small numeral. The first power, also called identity function, is denoted simply by x.
(c) Symbols for the results of substitutions into a function, say into log x, include merely the x-free part of its symbol. For instance, mathematicians write log cos x and not log x(cos x) when substituting cos x into the function log x. Similarly, the x-free part of a function symbol is all that is used in applying the function to what Newton called fluents, e.g., the time t and the gas pressure p; that is to say, mathematicians write, for instance, t2 and log p, and not x2(t)or log x(p).
(d) The symbol x, because of its equivocal use, is unfit to express in a simple way some of the most important properties of the identity function. For instance, the law that substitution of the function x into the function x yields the function x cannot be expressed by writing xx = x or x(x) = x.
13. The Geometry Relevant to Modern Education.
Educational Studies in Mathematics 4 (1) (1971), 1-17.
For some time now, the shortcomings of Euclid's Elements have been recognized in precollege teaching; and the age that identified the study of geometry with reading Euclid's book is definitely past. But it has also become clear that a presentation of the whole of Euclidean geometry with modern rigour would require more time than secondary schools can apportion to this task. As a result of this situation, many educators feel that they are forced to adopt one of two courses: either to spend too much time on the teaching of geometry or to present the subject without meeting the requirements of modern rigour - a task in which they are supported by textbooks some of which actually have much greater shortcomings than Euclid's Elements. Two ways out of this dilemma have been proposed. The first is to give up the teaching of geometry altogether - a step welcomed by those students who doubt that geometry is, to use a term in vogue, relevant. But young people who are not taught geometry miss the great experience of seeing a deductive theory developed before their eyes - a theory, moreover, that is accompanied by mental pictures and drawings on paper which motivate its assumptions and illustrate its conclusions. The study of rings and fields in algebra cannot compensate for this loss. The second proposed way out of the dilemma, initiated by G D Birkhoff, is the deduction of geometry from postulates involving arithmetical ideas, in particular the profound concept of real number. This amounts to a synthetic presentation of something akin to analytic geometry. A related method is the embedding of geometry into linear algebra and its presentation as a theory of vector spaces over the real numbers. But although these ideas can be presented in the form of deductive systems they deprive the student of the experience of developing such a theory from assumptions about really simple and purely geometric concepts such as points, lines, and incidence. And just such a treatment is of particular importance because of certain applications. I propose a third way out of the dilemma. It is based on the conviction that geometry should be included in the curriculum, that it ought to be taught with perfect rigour, and that it should be presented within a reasonable span of time. The way to achieve these aims is to abandon the idea of teaching Euclidean geometry in its entirety and to present only a part or an aspect of Euclidean geometry - but that part or aspect with absolute rigour - as well as some simple related theories which, from the points of view of various students, are "relevant".