Mathematics under the Microscope

muriel and Scott Carter brought to my attention to a recent paper in Science, The Advantage of Abstract Examples in Learning Math, by Jennifer A. Kaminski, Vladimir M. Sloutsky and Andrew F. Heckler. It appears to make quite a splash. From abstract:

Undergraduate students may benefit more from learning mathematics through a single abstract, symbolic representation than from learning multiple concrete examples.

The conclusion could be hardly characterised as surprising, but the redeeming quality of the paper is its experimental confirmation. Here I have some difficulty. The experiment was concerned with symbolic and concrete representation for cyclic group of order 3:

Unfortunately, the “concrete” representation, by measuring cups of liquids, looks unnecessary complicated and therefore methodologically flawed: it is much more natural to represent the identity element by the empty cup. BTW, why the empty cup is not present in the scheme? In the bottom row, the most natural “remaining” is the empty cup. Maybe this is the reason why Concrete A representation on the right is harder than the Generic one on the left? Concrete B and Concrete C examples were formulated in terms of slices of pizza or tennis balls in a container, rather than portions of a measuring cup of liquid. Why not in terms of a switch which could be rotated through angles ?

Basically, the paper proves that a symbolic representation void of real-world connotations is better than bad and overloaded with unnecessary details “real world” representation. Not much to prove.

Linear independence over of numbers

is equivalent to uniqueness of decomposition of integers into a product of prime numbers. This observation, apparently, belongs to Harad Bohr.

An enrerteining paper in the NY Times:

SCIENCE | April 22, 2008
When Language Can Hold the Answer
By CHRISTINE KENNEALLY
Does language shape what we perceive or are our perceptions pure sensory impressions?

An excerpt:

Elizabet Spaepen, a doctoral student at the University of Chicago, examined the ability of home-signing adults in Nicaragua to use numbers. Ms. Spaepen emphasized that although the subjects had never been taught a formal sign language, including counting, they were fully integrated in society. They have jobs and they are paid as much as hearing or signing adults.

Ms. Spaepen asked the home-signers to match an array of objects laid out before them. For example, she placed plastic discs on a table and encouraged the subjects to lay out the same number of discs. If the number was small, as in one, two or three, the home-signers got it right all the time.

If the number was larger, the home-signers got it right just approximately. If Ms. Spaepen laid out four discs, the subjects might lay out five or six. Although they were never quite right, they were never completely wrong. The home-signers would not lay out one or 15 discs in response to four.

Scientists have shown that the understanding of small, specific numbers is a trait with long evolutionary history. Monkeys and other animals can compute the exact number of a small set of objects at a glance without explicitly counting. The ability is called subitization.

Ms. Spaepen suggests that when home-signers correctly use small numbers, they are relying on this innate trait. The count list we learn with most languages (some languages do not have a count list or words for specific numbers greater than three) has enabled humans to build on this heritage, taking the specific and uniform gap between “one” and “two” and “two” and “three,” and extending it out through four and higher, theoretically to infinity.

M.C. Escher was not the first who invented patterns of interlocked fish and birds.

The British Museum’ description of this object:

Length: 30.500 cm, Width: 24.300 cm, Height: 6.200 cm
Gift of Dr Bernhard Landan
Asia JA 1946.10-12.1.a-c

Lacquer document box, From Japan, Edo period, 19th century AD

The interlocking design of black crows and white egrets is very unusual, and has no real precedent in Japanese art.

However, it does have something in common with the painting of the Rimpa school, and can be placed in the tradition of depicting flocks of birds on Japanese screens, common from the seventeenth century onwards.

The crows’ eyes, inlaid with gold lacquer and mother-of pearl, and the conspicuous pinkish bills of the egrets both help us to pick out the birds from the puzzling design.

L. Smith, V. Harris and T. Clark, Japanese art: masterpieces in (London, The British Museum Press, 1990)

Date and time: May 15th at 4.30 for a 5pm start!

Venue: University of Manchester, Simon Building,
Brunswick Street (off Oxford Road), second floor seminar room (2.57)

Christian Greiffenhagen will present an aspect of his recent research:

Video analysis of scientific practice? An attempt to study a ‘thinking’ science.

This paper will report on my ongoing project of developing a sociology of mathematical practice. Within the sociology of science, there is a clear lack of studies of ‘thinking’ (rather than ‘experimental’) sciences such as mathematics and theoretical physics. There have been a number of historical case studies (most notably Lakatos’s “Proofs and Refutations”), but very few researchers have tried to adapt the ethnographic ‘laboratory approach’ for studying experimental sciences to studying mathematics.

I will briefly review two previous observations-based studies of ‘thinking’ sciences, Livingston (1999) and Merz & Knorr-Cetina (1997), before reporting from my own study of graduate lectures in mathematical logic, focussing on the issue of how a lecturer tries to make the mathematical reasoning of a proof visible to students.

The paper Mathematics, mathematicians, and mathematics education by Hyman Bass [Bull. Amer. Math. Soc. 42 no. 4 (2005) 417–430] contains a remarkably compact formulation of what makes mathematics education so special among other disciplines:

Upon his retirement in 1990 as president of the ICMI (International Commission on Mathematical Instruction), Jean-Pierre Kahane spoke perceptively of the intimate connection between mathematics and mathematics education in the following terms:

• In no other living science is the part of presentation, of the transformation of disciplinary knowledge to knowledge as it is to be taught (transformation didactique) so important at a research level.
• In no other discipline, however, is the distance between the taught and the new so large.
• In no other science has teaching and learning such social importance.
• In no other science is there such an old tradition of scientists’ commitment to educational questions.

The recent discussion of Calculus in O notation gave a wonderful proof how close the concept of didactic transformation is to the hearts of mathematicians/computer scientists—even if the the words themselves are not in common use. I thank all contributors for taking part in the discussion.

Later in April I will give a talk on Didactic Transformation at a curious meeting of British HE educationalists, titled The Teaching - Research Interface: Implications for Practice in HE and FE. I am setting myself an unrealistic aim to try to find arguments in support of a simple thesis: teaching of mathematics is very different from teaching any other discipline and for that reason mathematics should be treated differently from other university disciplines.

The concept of didactic transformation is the principal stumbling block; I know from experience how difficult it is to sell to educationalists the idea that didactic transformation of mathematical material is, first of and above all, a mathematical problem.

A brief historic note: the concept of transformation didactique can be traced back to Auguste Comte.

This is from Auguste Comte’s preface to Catéchisme positiviste (1852):

A discourse, then, which is in the full sense didactic, ought to differ essentially from one simply logical, in which the thinker freely follow his own course, paying no attention to the natural conditions of all communication. [...]

On the other hand, this transformation for the purposes of teaching is only practicable where the doctrines are sufficiently worked out for us to be able to distinctly compare the different methods of expanding them as a whole and to easily foresee the objections which they will naturally elicit.

This was a question which intrigued me when I was a student: is there a meaningful mathematical statement about finite fields which is proven by induction on the characteristic of a field?

Finally and many years later I got a partial answer: a meaningful statement about prime numbers proven by induction on the size of prime number in question.

This is a note A Lemma on Divisibility by Peter Walker in a recent issue of The American Mathematical Monthly 115 no. 4 (2008 ) p. 338:

(*) For all primes , implies or .

His proof uses division with remainder but not Euclidean algorithm for finding the greatest common divisor of two integers.

We say that a prime is genuine if it satisfies (*) for all integers and . We shall prove by induction on that all primes are genuine.

Basis of induction. is a genuine prime number. Indeed if a product of two integers is even then at least one of the multiplicands is even.

Inductive step. Let be the least non-genuine prime number so that but does not divide and does not divide .

Using division with remainder, we can write and where . Of course, . If either or then divides or as required. If not, then both and are at least and can be factorised into primes: and (existence of factorisation can be proved earlier). Now and for some integer we have wher all , are less than and are therefore genuine prime numbers. Since none of , can divide , they divide and can be cancelled out one by one from the equation, leaving an obviously contradictory equality .

Continuing the theme of alternative approaches to teaching calculus, I take the liberty of posting a letter sent by Donald Knuth to to the Notices of the American Mathematical Society in March, 1998 (TeX file).

Professor Anthony W. Knapp
P O Box 333
East Setauket, NY 11733

Dear editor,

I am pleased to see so much serious attention being given to improvements in the way calculus has traditionally been taught, but I’m surprised that nobody has been discussing the kinds of changes that I personally believe would be most valuable. If I were responsible for teaching calculus to college undergraduates and advanced high school students today, and if I had the opportunity to deviate from the existing textbooks, I would certainly make major changes by emphasizing several notational improvements that advanced mathematicians have been using for more than a hundred years.

The most important of these changes would be to introduce the notation and related ideas at an early stage. This notation, first used by Bachmann in 1894 and later popularized by Landau, has the great virtue that it makes calculations simpler, so it simplifies many parts of the subject, yet it is highly intuitive and easily learned. The key idea is to be able to deal with quantities that are only partly specified, and to use them in the midst of formulas.

I would begin my ideal calculus course by introducing a simpler “ notation,” which means “absolutely at most.” For example, stands for a quantity whose absolute value is less than or equal to . This notation has a natural connection with decimal numbers: Saying that is approximately is equivalent to saying that . Students will easily discover how to calculatewith :

I would of course explain that the equality sign is not symmetric with respect to such notations; we have and but not , nor can we say that . We can, however, say that . As de Bruijn points out in [1, 1.2], mathematicians customarily use the sign as they use the word “is” in English: Aristotle is a man, but a man isn’t necessarily Aristotle.

The notation applies to variable quantities as well as to constant ones. For example,

if and

if

Once students have caught on to the idea of notation, they are ready for notation, which is even less specific. In its simplest form, stands for something that is for some constant , but we don’t say what is. We also define side conditions on the variables that appear in the formulas. For example, if is a positive integer we can say that any quadratic polynomial in is . If is sufficiently large, we can deduce that

I would define the derivative by first defining what might be called a “strong derivative”: The function has a strong derivative at point if

whenever is sufficiently small. The vast majority of all functions that arise in practical work have strong derivatives, so I believe this definition best captures the intuition I want students to have about derivatives. We see immediately, for example, that if we have

so the derivative of is . And if the derivative of is , we have

hence the derivative of is and we find by induction that

Similarly if and have strong derivatives and , we readily find

and this gives the strong derivative of the product. The chain rule

also follows when has a strong derivative at point and has a strong derivative at .

Once it is known that integration is the inverse of differentiation and related to the area under a curve, we can observe, for example, that if and both have strong derivatives at , then

I’m sure it would be a pleasure for both students and teacher if calculus were taught in this way. The extra time needed to introduce notation is amply repaid by the simplifications that occur later. In fact, there probably will be time to introduce the “ notation,” which is equivalent to the taking of limits, and to give the general definition of a not-necessarily-strong derivative:

The function is continuous at if

and so on. But I would not mind leaving a full exploration of such things to a more advanced course, when it will easily be picked up by anyone who has learned the basics with alone. Indeed, I have not needed to use “” in 2200 pages of The Art of Computer Programming, although many techniques of advanced calculus are applied throughout those books to a great variety of problems.

Students will be motivated to use notation for two important reasons. First, it significantly simplifies calculations because it allows us to be sloppy — but in a satisfactorily controlled way. Second, it appears in the power series calculations of symbolic algebra systems like Maple and Mathematica, which today’s students will surely be using.

For more than 20 years I have dreamed of writing a calculus text entitled O Calculus, in which the subject would be taught along the lines sketched above. More pressing projects, such as the development of the TeX system, have made that impossible, although I did try to write a good introduction to notation for post-calculus students in [2, Chapter 9].

Perhaps my ideas are preposterous, but I’m hoping that this letter will catch the attention of people who are much more capable than I of writing calculus texts for the new millennium. And I hope that some of these
now-classical ideas will prove to be at least half as fruitful for students of the next generation as they have been for me.

Sincerely,

Donald E. Knuth

Professor

[1] N. G. de Bruijn, Asymptotic Methods in Analysis (Amsterdam: North-Holland, 1958).

[2] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics (Reading, Mass.: Addison-Wesley, 1989).

An extreme Platonist manifesto which reached me via Samizdat: Max Tegmark (MIT), Shut up and calculate, arXiv:0709.4024v1 [physics.pop-ph] 25 Sep 2007. One quote:

I argue that our universe is not just described by mathematics — it is mathematics.

A new (paper) journal, Confluentes Mathematici,  will specialize in transversal articles (touching at least two distinct mathematical disciplines) and survey papers. It is scheduled to be launched in March 2009.

A very good idea. A growth in highly specilaised journals is disturbing.