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Calculus Reform--For the Millions comm-mumford.qxp 3/14/97 10:02 AM Page 559 Calculus Reform— For the Millions David Mumford About twenty years ago I was part of a group of bank, and it is running a special promotion with professors from many fields who met once a a new account which pays imaginary interest at month for dinner and an after-dinner talk given the rate of 100%. The audience immediately sees by one of the members. It was entertaining to that you get imaginary interest building up and hear glimpses of legal issues, historical problems, that the interest on the interest is decreasing discussions of what freedom meant in different your total of real dollars. You run them through cultures. But then I had to give one of these a few more numbers, and they see that their real talks! I was working on algebraic geometry then, funds will go to 0, while their imaginary funds and I tried to figure out what I could say that build up to about 1.i; then they go into real would hold my colleagues’ attention while di- debt, next imaginary debt, and finally get their gesting a large meal. In the end I decided to stick real $1 back, which they immediately withdraw! to a rather anecdotal level but to inject one bit A little picture in the complex plane convinces of real math. I thought I would try to explain to them that this will take 2π years, while after π them the first mathematical formula that I had years they were in debt $1 : voilà, eiπ = 1! seen in school which totally bewildered me. This What is the point of this struggle to commu-− was eiπ = 1. It seemed to me that here was a nicate some tiny bit of math? For me, the lesson nugget of −real math and maybe it could be ex- was that I think my audience got a bit of hon- plained. Here is how I tried. est math from this and that what they struggled I got e into the picture by discussing a sav- to learn consisted of some numerical fiddling, ings bank which pays 100% interest, and con- some geometry (the circle showing the evolution vinced them that in a year they would get more of your balance in the complex plane), and some than $2 for each $1 invested. It was not hard to thinking about the rules which underlie arith- convince them they would get between $2.50 metic and exponentiation. I do not think I said and $3 per dollar invested, and we could define anything mathematically dishonest, yet I cer- e to be their balance after a year. Then I needed tainly gave no proof of anything. I think this is i. Most people have heard of i, and I just de- the same approach as that taken by many cal- scribed it as part of a game invented by math- culus reform texts, and it is exactly the philos- ematicians to get enough numbers so every equa- ophy of the Gleason-Hallett Calculus Consor- tion has solutions: starting with the one new tium. rule that multiplying i by i you get 1, you get The problem of communicating comes up in all the numbers a + ib and their arithmetic.− Next, many situations other than after-dinner imagine you go to the neighborhood savings speeches. For nearly fifteen years I have been doing applied mathematics, and I have to talk David Mumford is University Professor, Division of Ap- especially to biologists, psychologists, and en- plied Mathematics at Brown University. His e-mail ad- gineers. The same rules seem to apply. If I men- dress is [email protected]. tion Lp, they have me pegged as “one of them”. MAY 1997 NOTICES OF THE AMS 559 comm-mumford.qxp 3/14/97 10:02 AM Page 560 Of course, the tolerance level varies. Some en- tegration through Archimedes’ great achieve- gineers have been rather thoroughly math- ment, calculating the volume of a sphere. He ematicized: in control theory everything is done does this in fifty pages! Was he successful? Well, in multiple Banach spaces. But I know a psy- the book went through four editions over more chologist who hates math yet understands ab- than thirty years, apparently being read by lit- solutely correctly the meaning of robust statis- erally millions. How did he do this? Here is how tics. For pure mathematicians and statisticians, he introduces the derivative: robust statistics refers to statistics that work If the points p and q in the course when the variables being measured are not nor- (of a cyclist) are very close together, mally distributed and still give you good esti- … the curved line joining them is dif- mates of things like the mean and variance of ficult to distinguish from a straight the variable. In real life I think it is fair to say line, and the pointer of the that nothing is ever normally distributed be- speedometer will not shift apprecia- cause there are always “outliers”, exceptional bly during the interval representing cases which are off the scale. My psychologist the difference between the x co-or- friend spent much of his time measuring reac- dinates of q and p. When p and q are tion times and would average over his subjects very close together, so that we can- to get a mean. When I did the same and got ter- not distinguish them, the line pass- rible results, he said, “Maybe you forgot to throw ing through them becomes the tan- out the slowest 3 percent?” The point is that gent at the point p = q , and the about 3 percent of the time, his subjects’ minds gradient of this line corresponds with wandered, and he got absurdly slow responses. the speedometer-reading at the in- Is this math? In fact, yes: there is a very sub- stant represented by the x co-ordi- stantial body of theory on what “α-trimmed nate of p. means” do for you with unknown distributions. My friend has an excellent intuitive grasp of this .......... without knowing any of these theorems. Often a picture is what facilitates communi- The tangent method is equivalent to cation. When you were in grade school, you taking two points with x co-ordinates might have been puzzled, as I was, when asked xp and (xp + x) and y co-ordinates 1 to accept the formula 1 = a. Of course, modern yp and (yp + y) so close together a that x and ∆y are too small to mea- books “prove” this, more or less, manipulating y sure. The gradient∆ is = tan a. the axioms in the usual way. But does it not be- x When∆ y and∆ x are immeasurably come just as clear from a picture which com- ∆ small, we write the ratio (pronounced∆ pares: dy as dee-wy-by-dee-eks)∆ ∆ dx. (pp. 521–522, 3rd edition) ((••)(••)(••)) Hogben was a genius at putting things in How many pairs is 6? As a formula: plain English. He used , but no or δ. He was Answer: 36/2=3also very clear about the need to explain calcu- lus in down-to-earth terms.∆ He railed against Newton himself: The intellectual leaders in the New- tonian period did not realize that every intellectual advance raises a constructive problem in education. How many quarter As a formula: Newton himself devoted much of his pies in a whole pie? 1/(1/4) = 4 energy to devising long-winded Answer: 4 demonstrations in Euclidean geome- try instead of trying to make his own An example, a picture, an explanation is pre- methods intelligible to his contem- sented. Would it be better to present a proof? poraries. One result of this was that I learned calculus during high school when I conspicuous progress in Newtonian stumbled across a great classic of the pedagog- mechanics did not take place in his ical literature: Lancelot Hogben’s Mathematics for own country during the century the Million. Hogben explains the essence of cal- which followed the publication of the culus, including differentiation, integration (both Principia. with many examples up through trig functions), the fundamental theorem, and multivariable in- (op. cit., p. 567) 560 NOTICES OF THE AMS VOLUME 44, NUMBER 5 comm-mumford.qxp 3/14/97 10:02 AM Page 561 A striking contrast for me was the curriculum We drill students in conjugating French verbs, which my oldest son encountered learning Eu- so why not in algebra? clidean geometry in a Paris high school in 1976. But everyone agrees that this approach of Unfortunately, I no longer have the textbook, but memorizing and game playing has real limits. the following is close to the definition presented Learning to correctly convert among fractions, there of a “Euclidean line”: a Euclidean line is an decimals, and percents was such a case for some ordered pair X, . Its first member is a set X of my children. I do not think this can be learned { } whose elements will be called “points”. Its sec- either by memorization or by pretending it is a ond member is a Φset of bijections between X bizarre game required to pass tests. It is also an and the real line R which satisfies two axioms. essential skill in later life in thinking about bud- First, for all φ, Φ , the composition gets, inflation, savings, and using recipes. What ∈ f = φ 1 is a map from R to itself of the form it seems to require is an understanding of what ◦ − f (x)= x + a for some realΦ number a.
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