Mathematics under the Microscope

My colleague EHK told me today about a difficulty she experienced in her first encounter with arithmetic, aged 6. She could easily solve “put a number in the box” problems of the type

,

buy counting how many 1’s she had to add to 7 in order to get 12

but struggled with

,

because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults.

A brief look at Peano axioms for formal arithmetic provides some insight in EHK’s difficulties. I quote Wikipedia, with slight changes:

The Peano axioms define the properties of natural numbers, usually represented as a set N or [I skip axioms for equality relation -- AB.]

The [...] axioms define the properties of the natural numbers. The constant 1 is assumed to be a natural number, and the naturals are assumed to be closed under a “successor” function S.

1. 1 is a natural number.
2. For every natural number n, S(n) is a natural number.

Axioms 1 and 2 define a unary representation of the natural numbers: the number  2 is  S(1), and, in general, any natural number n is S^n-1(1). The next two axioms define the properties of this representation.

1. For every natural number n other than 1, S(n) ≠ 1. That is, there is no natural number whose successor is 1.
2. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.

The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers; it is the only second order axiom.

1. If K is a set such that:
   • 1 is in K, and
   • for every natural number n, if n is in K, then S(n) is in K,

then K contains every natural number.

Thus, Peano arithmetic is a formalisation of that very counting by one that little EHK did, and addition is defined in a precisely the same way as EHK learned to do it: by a recursion

Commutativity of addition is a non-trivial (although still accessible to a beginner) theorem. Try to prove it — you will be forced to feel some sympathy to poor little EHK. If it is a trivial task for you, write a recursive rule for multiplication and prove, from Peano axioms, commutativity of multiplication. Then write a recursion for exponentiation and try to explain, why this time commutativity fails, even if the recursive scheme appears to be the same.

In an ideal world (I emphasise, in an IDEAL world), primary school teachers should be taught Peano arithmetic — of course not because they have to teach it to their pupils, but because they have to appreciate intellectual challenges that their pupils have to overcome.

It is a strange concept of British model of teachers training that teachers need to know only the stuff that they pass to pupils. For a successful teaching, at least in mathematics, a teacher has to know much, much, more.

I have to emphasise: I do not propose to introduce Peano arithmetic into teacher training courses. I talk about an IDEAL world.

And this news from BBC is even more depressing:

English teachers who went to school when grammar was not on the curriculum struggle to teach it, research shows.

A review of international studies on the effective teaching of complex writing says there is a need to improve the teachers’ own skills.

The work was done by Exeter University for the Department for Children, Schools and Families in England. [...]

The study concludes: “For English teachers, who themselves attended schools when grammar was not part of the English curriculum, there is a significant issue of lack of assurance in grammatical subject knowledge, leading to difficulties in addressing grammar meaningfully in the writing classroom.

“In particular, effective teaching requires a secure understanding not simply of grammatical terminology, but of applied linguistics and an awareness of the ways in which grammatical constructions are used in different texts for different communicative purposes.” [...]

Another study described a “significant knowledge gap” in terms of teachers’ pedagogical knowledge.

One piece of research on the linguistic subject knowledge that teachers and trainee teachers bring to their teaching of writing found “a persistent theme in teachers’ attitudes to grammar is hostility to anything that makes formal structure the central object of study”.

What can I say? Grammar is an expression of the intrinsic logic of language. Non-teaching or poor teaching of grammar directly affects students’ capacity of logical, and, therefore, mathematical thinking. “Teachers’ [...] hostility to anything that makes formal structure the central object of study” is hostility to logic. It is a seed of future math phobia in students.

From BBC: The shortage of qualified maths teachers in England and Wales is to worsen.

Stephen Jones’ column in The Times Educational Supplement of 9 May 2008 (not placed yet on the newspaper’s website) caught my eye:

“Why do we have to study this?” It’s the question that every teacher must have heard 100 times over. [...] The objection this time round was to the study of poetry as part pf an English course. As the student had actually signed up for an access course in social care, I suppose some might think that she had a point.

So far so good. Stephen Jones starts to giving an excellent answer:

It’s certainly not the subject for those who see life purely in terms of white and black. If shades of grey are too painful to contemplate, then poetry is not for you.

Instead of stopping at that powerful point, Jones then descends into a quasi-poetic rubbish too banal to quote. In my humble opinion, the answer is very simple: learning and analysing poetry calibrates a student’s scale of grey between white and black — and maybe not only shadows of grey, but perhaps also all hues and  colours of rainbow. Poetry is about subtle variations of meaning, emotional charge, colour of a word printed in black on white; an ability to detect these variations is a very essential skill for life.

For a future social worker, ability to read the true meaning of the phrase “I am OK, thank you” is a very essential skill. An instinctive, innate, “emotional literacy” perhaps suffices for a face-to-face interaction (dogs are quite good at that — and without studying poetry). However, detection of the emotional state of a writer of a letter (and even worse, an  e-mail) requires  a certain cultural conditioning.

Unfortunately, poetry suffers at school because it is a classical example of teaching critirea. Interestingly, the same teaching of critirea that Bichenkov talks about in his article (still not translated, sorry).

A talk under this title will be given by Nick Trefethen (University of Oxford) at my School’s Colloquium on 11 June 2008, 2:00 pm, Alan Turing Building, G.205. The abstract says

Numerical algorithms are traditionally applied to functions discretized in space, but one may ask what their analogues would be if we could compute with continuous functions directly. This question is no longer just theoretical, thanks to the development of the chebfun system in object-oriented MATLAB. We will show the system in action and ask whether it can live up to the vision of “computing with the feel of symbolics but the speed of numerics”.

This is a very interesting story with some potential implications for the way we teach calculus (see previous discussions, Donald Knuth: Calculus via O notation and Calculus without limits), even if the project appears to be concerned with a technical enhancement of MATLAB.

This becomes clear from Nick Trefethen grant application, available on Web:

Conventional MATLAB is built on the most advanced algorithms for vectors and matrices; this is a source of its power, since so much of scientific computing in the end comes down to numerical linear algebra. The idea of chebfun computation is to create a MATLAB class that behaves syntactically like the usual MATLAB vectors. The difference is that with chebfuns, the usual vector commands in MATLAB are overloaded with analogues for continuous functions.

For example, the command

>> f = chebfun(’real(airy(10*x))’);

calls the chebfun constructor to produce a chebfun object that will behave, up to the usual IEEE double machine precision of about 16 digits, like the Airy function .

How is all this done? The mathematical basis of the chebfun system is a pair of closely related ideas:

• Polynomial interpolation in Chebyshev points implemented by Salzer’s barycentric formula
• Chebyshev expansions implemented by the Fast Fourier Transform

These are combined together with state-of-the-art numerical algorithms for integration, zero finding and other operations. In particular, two features that make the system fast and accurate are

• Adaptive determination of the number of points needed to represent each function to about 16 digits
• Zero finding via eigenvalues of Chebyshev companion matrices with divide-and-conquer recursion.

The result is a system with, we have found, rather astonishing capabilities. For example, the Airy function above may seem complicated, but its chebfun representation to 16-digit precision is a polynomial of degree just 59. The more irregular function

>> f = chebfun(’sin(3*x)+.5*exp(x).*sin(100*x./(1+3.*x.^2))’)

plotted below only needs a polynomial of degree 237. Even a function represented by a polynomial of degree 100,000 is integrated by sum to full accuracy on our workstation in 0:2 secs.

Some of the mathematics underpinning the extraordinary speed, accuracy, and stability of high order polynomial interpolants is old, and some of it is new, but it is safe to say that until the arrival of the chebfun system few people were aware of these possibilities. The investigations proposed here, despite their classical foundations, will explore poorly charted territory even from an algorithmic point of view. From a software point of view they are completely new.

Basically, a function on a segment is replaced by its appropriate Chebyshef polynomial approximation, but, if it has name, it retains that name.

Now, let us look at related issues in methodology of mathematical teaching. Assume that chebfun approach became an industry standard for (semi-)numeric computations, and you have to teach a calculus course for engineering students. Should you pay attention tot he following two metamathematical factors?

• The way they will be used by your students, functions become intensional objects with names, not extensional entities.
• Uniform convergence, not a limit at a point, starts to dominate the field.

Isn’t that exactly the setup for Donald Knuth’s Calculus via O notation and Calculus without limits?

My alma mater, FMSh, a preparatory boarding school of Novosibirsk University, celebrates 45th year of its work. My physics lecturer at the School, Evgenii Bichenkov, republished a short article, Физико-математической школе - треть века, written 10 years ago. I discovered it only now; it is a remarkably interesting document, and I promise to translate it in English as soon as I have free half an hour. To wet the reader’s appetite, one line:

A student should have free time just to think: what on the Earth has he actually been taught?

The document is a manifesto of meritocratic eliticism in education, a recipe for building a highly selective and academically intensive school. [A Google search for "meritocratic eliticism" leads mostly to two types of materials: (a) Educational system of Singapore; (b) Barak Obama. Both have no relevance to what Bichenkov says.]

[...] Что нового в практику школьного образования внесла школа и каковы главные результаты ее деятельности в обучении основам наук на школьном уровне? [...]

Итак, что дал отбор учеников? Я глубоко убежден, что сам факт отбора и создания детского коллектива на его основе благоприятен для ребенка. Попав из своих школ, где все роли и места уже распределились и все устоялось, в новую среду, дети начинают свое внутреннее соревнование за распределение по шкале своей ценностной иерархии. Не делать этого они не могут - такова их природа и таков возраст. Важно, что в этом возрасте им предложены достойные нравственные и человеческие ценности для соревнования и показаны хорошие примеры. Кажется, нам в Новосибирской ФМШ это удалось.

Далее. В какой мере отбор произошел по истинным способностями? Соответствуют ли его результаты провозглашенным целям? Здесь я не могу быть однозначен в выводах. Во многом отбор все еще связан со случайностями. Очевидно влияние на выбор развитие личных устремлений ребенка семьи, учителя, друзей, знакомых, а на результаты олимпиад спортивности характера, настойчивости, уровня взрослости, наконец. И, конечно, при отборе проявляется личность учителя, экзаменатора.

Здесь встает вопрос о выборе учителя для отобранных детей. С самого начала мы выдвинули одно ограничение на отбор учителя - учитель должен быть научным работником СО АН. При всей кажущейся слабости это ограничение оказалось довольно тонким и верным признаком отбора, оставив в стороне отдельных претендентов на место учителя ФМШ, не имевших кроме, страстного желания попасть в штат школы, никаких других объективных данных для работы с одаренными детьми. Оказалось, что требование быть научным работником в условиях Академгородка почти полностью соответствует требованию личностной состоятельности как в профессиональном, так и в человеческом плане. Мы живем своим особым сообществом, знаем друг друга в лицо и по работе и обязаны постоянно считаться с этим. Нам повезло, что от основания Академгородка ученого здесь судят по его делам, и судят требовательно. В наших условиях плохой работник просто не мог стать преподавателем ФМШ, а если такое случалось, то по ошибке администратора и на очень короткое время. Я не знаю, как быть с отбором учителей в иных местах, не в Академгородке. Но из нашего опыта я на первое место выдвину критерий отбора по уровню личных достижений в предыдущей работе: если это инженер - то удачливый, с идеями и достижениями, если учитель - то фантазер и любимец школы и тоже с результатами, если студент - то отличник и выдумщик, и обязательно “хороший парень” среди сокурсников. А штат школы должен быть открытым, с живым обменом людей, с протоком. В нем должны собираться очень разные по своим интересам и личностным особенностям люди. Если угодно, при их подборе должен работать принцип взаимодополняемости. В Академгородке все получилось очень естественно. У нас несколько разных школ физики. И представители всех из них собрались на кафедре физики в ФМШ, обогащая друг друга знаниями и сотрудничая. Сначала это произошло случайно, так как работа в школе ни по оплате, ни по престижу не шла ни в какое сравнение с университетом или любым вузом. [...]

Я высказал свои суждения о двух фундаментальнейших вопросах для специализированной школы: “Кого учить?” и “Кому учить?”. Остался третий: “Чему учить?”. Обсужу его на примере физики, хотя рискну сделать несколько общих выводов. В практике нашей учебной деятельности мы выработали несколько “граничных условий”, определяющих во многом построение наших учебных курсов. В формальных временных рамках так называемого учебного плана основными из них оказались следующие принципы:

Короткий срок обучения: один или два года. Наши попытки работать в условиях интерната в течение трех лет следует признать неудачными.

Короткие семестры. Занятия осенью идут примерно до 10 декабря, затем две недели зачетов и экзаменов и три недели каникул (детям обязательно надо отдохнуть от общежития). Второй семестр: с 20 января по 20 мая, опять экзаменационная сессия и летние каникулы. Кроме того, бывает несколько нерабочих дней в ноябре и мае.

Короткие недели. ФМШ при всей напряженности занятий работает по пятидневной неделе.

Короткие лекционные курсы. Ни один лекционный курс не может занимать более двух часов в неделю. Общее число обязательных занятий в настоящее время не может превышать 32 часа в неделю.

К этим ограничениям мы пришли далеко не сразу и совсем не прямым путем. Начало нашим поискам положил опять же М.А. Лаврентьев, который высказал несколько афористическое требование: “Ученик должен иметь свободное время, чтобы подумать, чему же его учат!”

Содержание учебных курсов по физике в физматшколе сформировалось в результате деятельности большого количества очень разных преподавателей. Они были из разных институтов, профессионально работали в различных областях физики, сильно отличались по возрасту. Поставленные в жесткие временные рамки и стремясь отразить свои личные научные пристрастия, эти люди могли пойти по пути упрощения в изложении научных знаний и придти к примитививной популярщине науки, от которой пострадали все стандартные школьные учебные курсы. Другая опасность была в глубоком изложении лишь нескольких тем. Поплавав между этими крайностями, мы провели выбор лишь самого важного и самого существенного в современных научных знаниях. В результате наши обязательные учебные курсы содержат лишь фундаментальные знания. И оказалось, что этих знаний очень немного, логика их использования почти очевидна, а прозрачность и глубина внутренних связей поразительна. Как самую высокую оценку успеха нашей программы обучения приведу слова одного из бывших учеников ФМШ, которому уже исполнилось сорок и научная судьба которого сложилась очень успешно. Он сказал: “На физфаке НГУ я изучал детали физики. Все основное, ее стержень и внутреннюю логику я уловил в ФМШ”.

A strange story of David Massey, Professor of Mathematics at Northeastern University, USA, suspended for reasons that have not been disclosed. His case as explained at http://supportmassey.com/.

Peter McBurney sent me a link to works by Danish artist Peter Callesen. He added:

I wonder if anyone has developed a mathematical theory of this work, to determine, for example, which cut-out shapes are possible or impossible. Presumably, there would be a link from such a theory to the math. theory of origami.

My old friend Owl sent me the following quote :

I realized that reading a piano piece is much like solving a math problem. Each note has a beat which is a fraction of a whole bar, and the sum of the notes’ beats in a bar should equal the numerator in the so called “time signature”, the fraction indicated on the left most side of the staff. (A staff is a series of bars.) The denominator in the time signature specifies the kind of note which would receive ONE beat. Thus, a time signature of 2/4 means that a quarter note (1/4) would have one beat, and each bar would have two beats; an eighth note (1/ would receive a half-beat, and the bar may have as many as four eighth notes, or a combination of one quarter note and two eighth notes, and so on and so forth. Whew! Beethoven must have been a good mathematician!

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.