04/15/18

# If a quantity is not assigned a concrete numerical value, it does not exist for students

I ses this example in my lectures when I explain the difference between arithmetic and harmonic means:

A car traveled from A to B with speed 40 miles per hour, and back from B to A with speed 60 miles per hour. What was the average speed of the car on the round trip?

Anatoly Vorobey and Vladimir Kramchatkin made a useful comment on Facebook on this quite standard and well-known problem:

“The answer is obviously 48 [miles per hour]. 95% can not solve this problem the first time. But if they are told in advance that the distance between A and B is 120 [miles], 95% of schoolchildren will easily solve this problem.”

A concrete number, 120 km, serves as a strong hint that students are expected to do something with this number. But, for majority of students, if a magnitude or a quantity is not assigned a concrete numerical value, it does not exist. This is one of the flaws of mathematics education at schools: no-one tells students that they have to be able to see hidden parameters in arithmetic problems.

But this is not the only flaw: students are also not told how to check solutions. Checking answers frequently benefits from seeing a problem in a wider context and varying the data. The standard answer that students give to the problem with the car is 50 miles per hour, the arithmetic mean of the two speeds. But this solution immediately collapses if we slightly change the problem: what would happen if the speed of the car on its way back from B to A was 0 miles per hour?

04/14/18

# Unreasonable ineffectiveness of mathematics in biology

This post appeared first 2006 in a now-defunct blog, reposted in 2011 and  is now reposted again as a source of a quote from Israel Gelfand which appeared in Wikipedia.

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

I heard that from Israel Gelfand in a private conversation. Because of Gelfand’s peculiar style of work, I was often present during his conversations with his biologist co-authors about structure of proteins. The quote was included by me in an earlier version of the book ‘Mathematics Under the Microscope‘, but is not present in the published version (originally I planned to insert it in Section 11.6).

Besides being one of the most influential mathematicians (and mathematical physicists) of 20th century, Gelfand also had 50 years of experience of research in molecular biology and biomathematics, and his remark deserves some attention.

Indeed biology, and especially molecular biology, is not a natural science in the same sense as physics. Indeed, it does not study the relatively simple laws of the world. Instead, it has to deal with molecular algorithms (such as, say, the transcription of RNA and synthesis of proteins which ensures the correct spatial shape of the protein molecule) which were developed in the course of evolution as a way of adapting living organisms to the changing world. If they solve a particular problem in an optimal way, they should allow some external description in terms of the structure of the problem. Indeed, this is the principal paradigm of physics; it is an experimental fact that the behavior of physical systems is governed by various minimality / maximality principles, and the optimal points have, as a rule, especially nice mathematical properties.

But why should a biological system to be globally optimal? Evolution is blind, and there is no reason to assume that the optimal solution is reached. The implemented solution could be one of myriads of local optima, sufficiently good to ensure survival. Lucky strikes could be so rare that the huge search space and billions of years of evolution produced just one survivable algorithm, which, as a result, dominates the living world, and is perceived by us as something special. But it might happen that there is absolutely no external characterization which allows us to distinguish it from other possible solutions, and that its evolutionary phylogeny is its only explanation.

However, I am not a philosopher and cannot claim that my solution of Gelfand’s paradox is correct. What I claim is that philosophers ask wrong questions. The classical conundrum of relations between mathematics and physical world starts to look very different — and much more exciting — as soon as we include biology into consideration. I will try to continue this discussion.

04/14/18

# Perfectionism: Type A and Type B

There are two very different types of perfectionism.

Type A: interiorised perfectionism driven by personal, internal criteria. As an apocryphal story goes, one of the presidents of Harvard University was once asked what was so special in teaching at Harvard to justify their extortionate fees. His answer was of just three words: “We teach criteria”. Cambridge appears to be the only university in Britain which teaches criteria. I know — I myself was lucky to get my own education at a boarding school and an university which taught criteria. Among my mathematician colleagues (and co-authors) I know a number of Type A perfectionists. Some of my friends on Facebook teach or spread criteria — by means of art classes, or mathematics circles, or lectures on history of mathematics, or poetry evenings …

Type B: external perfectionism, a Pavlovian dog reflex to meet crude criteria, at the level of metrics, rankings, “likes” on social  media — all of them imposed from outside. In modern world, most perfectionists belong to Type B. IMHO, the best inoculation against the soul-destroying Type B perfectionism is development of a system of deeply interiorised personal criteria. In principle, this is what education should give to every child. It fails. Moreover, the vast majority of schools and universities spread the disease.

04/11/18

# Education is a zero sum game, and it is rigged

In a market economy, education is a zero sum game. And it is rigged.

If banks and insurance companies were interested in having numerate customers, we would witness the golden age of school mathematics – fully funded, enjoying cross-party political support, promoted and popularised by the best advertising companies in all forms of mass and social media. But they are not; banks and insurance companies need numerate workforce – and even more so they need innumerate customers. 25 years ago in the West, the benchmark of arithmetic competence at the consumer level was the ability to balance a chequebook. Nowadays, bank customers can instantly get full information about the state of their accounts from an app on a mobile phone – together with timely and tailored to individual circumstances advice on the range of available financial products.

04/11/18

# Why is mathematics useful?

Mathematics is immensely useful but what makes mathematics mathematics is not the same as what makes it useful.

Mathematics has its own intrinsic needs that must be addressed for it to stay alive.

Let us compare mathematics with a cow.

The cow is useful, it gives us milk (whole, low-fat, skim, fat-free, organic …), cream (single, double, soured, clotted …), butter (unsalted, salted, spreadable …), a variety of cheeses — the list can be continued. Applied mathematics can be compared with the cow’s udder — it produces milk. Some branches of pure mathematics are best described as the cow’s immune system — they keep the cow alive. The cow of course has other uses. To make a steak, it suffices to take a piece of cow and gently roast it to taste. What is a piece of cow? Mathematicians. Financial industry, security sector, etc. are connoisseurs of a good steak. NSA advertises itself as the biggest employer of mathematicians in the USA.

Pure mathematicians are sometimes accused on focusing on “useless” problems “they invent for themselves”.

But let us look at geneticists’ obsession with a pretty useless creature: Drosophila melanogaster. An article in Wiki devoted to it says: “The species is known generally as the common fruit fly or vinegar fly. Starting with Charles W. Woodworth’s proposal of the use of this species as a model organism, D. melanogaster continues to be widely used for biological research in studies of genetics, physiology, microbial pathogenesis and life history evolution. It is typically used because it is an animal species that is easy to care for, has four pairs of chromosomes, breed quickly, and lays many eggs”.

Very frequently, “famous” mathematical problems are means of concentrating the effort of generations of mathematicians on development of methods of proof in particular areas of mathematics, they are drosophilas of mathematics. In some cases (and the Riemann Hypothesis is the archetypal case) they have exceptional importance for mathematics as a whole. Дети, любите корову – источник мяса!

12/2/17

11/6/17

# As a mathematician, how would you mentor your child?

This is my response to a question in Quora: As a mathematician, how would you mentor your child and help her to learn, do and live mathematics in her free time as she is growing up?

I write from the  position of a mathematician about what a mathematician can do for her child.

First of all, a mathematician understands and can use the fact of life non-mathamticians are not aware of:

Mathematics is done by the subconscious.

Encourage in your child, and help her, to develop all kinds of intuition, guesswork (with subsequent checking, whenever possible, of the correctness of the guess). Help her to train her vision of the world, see relations in the world, identify mathematical structures present in the world.

What follows are a few random examples, chosen from what I did myself with my (grand)children or had seen my colleagues doing with their (young, pre-school or primary school age) children.

1. Adult spends time with a child, aged between 3 and 4, in a garden, watching insects and ants, and discussing with the child how the world looks from the ant’s viewpoint: that the tree trunk is a like a street, and pathches of algae and of moss on the bark are like lawns and bushes along the street. Child: ”and this branches are like sidestreets”.
2. The same adult uses every opportunity to explain to the child the structure of an actual street in a big city: street signs, house numbers which go in progression and odd on one side of the street and even on another side. A year later, the child is able to confidently guide the adult (and his little sister) across a unknown part of the city using a map. Observing an ant on a tree helped. This is an ecouraging sign of mathematical development.
3. Of course, a child’s ability to read is useful. Street names, all kinds of shop signs provide an excellent material for reading and proof that reading gives information about the world.
4. An adult and a child (aged 5) send to each other, from opposite corners of a sofa, small strips of paper with messages written in a substitution cipher: each letter is substituted by the next one (cyclically, z is substituted by a). Suddenly the child exclaimes: “And I invented my own cipher — each letter is replaced by the previous one!” IMHO, this will help the child understand algebraic notation where nubers are substituted by letters. [It is worth remembering that Vieta, the inventor of algebraic notation, was the frist cryptographer known to us by name. His deciphering of intercepted diplomatic coresspondence directly infuenced Europian politics of his time.] The child is now 7 years old and can handle variables in Scratch.
5. Children are invited to guess weight of every household object they can handle by weighing it in hand and check the result by weighing on scales. The same with temperature of water in hte bath, checked by thermometer.
6. All kinds of estimates with subsequent checking: how many steps are in this staircase? How many steps are to the end of the street? How to estimate the number of cars in the parking lot without counting them all?
7. Playing lego ( with child of 4). An adult encourages a child to pick correct bricks (say, 2 by 3 studs) without looking at them, by touch only. They together follow step-by-step instruction in the manual. Building a symmetric model (say, a plane), the adult builds the left wing, the child builds the right wing by mirroring the adult. Very soon the child starts picking details of correct orientation even before the adult touches his detail.
8. Playing snakes and ladders with two dices. A player can pick one of the values or their sum. The catch is that, for winning the game, 100 has to be hit without overshooting — for otherwise the player gets back to the beginning, the path is circular, 97 + 6 = 3 . A fast, furious, and vicious game which trains tactical thinking.
9. Actually the rules of snakes and ladders can be changed in variety of ways. Adult encourages the child to invent her own rules. Crucially, the new rules need to be agreed and written down before the start of the game.

I can continue this list, but, I hope, it already gives some idea. Sorry for typos.

11/4/17

# Harmonic mean

This problem is already in my lecture notes, and next week I will discuus it with my students:

A car traveled from A to B with speed 40 miles per hour, and back from B to A with speed 60 miles per hour. What was the average speed of the car on the round trip?

Anatoly Vorobey and Vladimir Kramchatkin made in hteir post on Facebook a useful comment on this quite standard and well-known problem:

“The answer is obviously 48 [miles per hour].  95% can not solve this problem the first time. But if they are told in advance that the distance between A and B is 120 [miles], 95% of schoolchildren will easily solve this problem.”

A concrete number, 120 km, serves as a strong hint that students are expected to do something with this number. But, for majority of students, if a magnitude or a quantity is not assigned a concrete numerical value, it does not exist. This is one of the flaws of mathematics education at schools: no-one tells students that can, and should see hidden parameters in arithmetic problems. But this is not the only flaw: students are also not told how to check solutions. Checking answers frequently benefits from seeing a problem in a wider context and varying the data. The standard answer that students give to the problem with the car is 50 miles per hour, the arithmetic mean of the two speeds. But this solution immediately collapses if we slightly change the problem: what would happen if the speed of the car on its way back from B to A was 0 miles per hour?

10/22/17

# Comment on “Making Oxbridge entry matter less” by Becky Allen

This is my comment that I left at Rebecca Allen’s blog entry  Making Oxbridge entry matter less. Basically, her suggestion was

Establishing robust and comparable degree classification will help fix the extraordinary stratification of universities in the eyes of employers. Getting into Oxbridge rather than, say, Nottingham undoubtedly gives people an easy ride in the labour market. […]

We could fix all these problems by introducing a common core examination in all degree subjects, set externally by learned societies. All students would sit them, say, two-thirds of the way through their degree, thus allowing specialised final year examinations to continue. Performance in this exam, by subject, would determine the number of first-class, upper-second, lower-second and third-class degrees the department is allowed to award that year. It would not determine the degree-class of the student.

Agreeing a common core of the curriculum would be more controversial in some subjects than in others. We should try this first in subjects where this is not controversial: the sciences, maths, economics, and so on.

My comment follows:

Let us have a look at the table of distribution of UCAS Tariff Scores actually achieved by entrants into Mathematics at Cambridge and at Wolverhampton in 2014 (I had no time to look up fresher data; these were taken from from the official source, https://unistats.direct.gov.uk/Compare-Courses):

Cam Wolverhamton
< 120 0% 10%
120 – 159 0% 5%
160 – 199 0% 25%
200 – 239 0% 25%
240 – 279 0% 15%
280 – 319 0% 5%
320 – 359 0% 10%
360 – 399 0% 5%
400 – 439 1% 0%
440 – 479 2% 5%
480 – 519 4% 0%
520 – 559 12% 0%
560 – 599 13% 0%
600+ 68% 0%

Let us do some aggregation:

Cam Wolverhamton
< 400 0% 95%
400 – 479 3% 5%
480+ 97% 0%

100% of new mathematics students at Wolverhampton were within the range of the lowest 3% at Cambridge and 95% at Wolverhampton were below the Cambridge range entirely. And this did not include STEP, compulsory at Cambridge.

We have to accept that Cambridge U. and U. of Wolverhampton belong to two different nations separated by the deep socio-economic, class and caste schism. This is a “first world / third world” division. Any attempt to objectively measure “the outputs” will only increase the level of controversy.

Later addition:  Mathematics is one of the subjects where Rebecca Allen’s proposal will create outcry – exactly because “a common core examination” will be objective. And it is likely to be a death sentence to many mathematics departmenst.

Disclaimer: The views expressed do not necessarily represent the position of my employer or any other person, organisation, or institution.

10/20/17

# Handwriting and “Cortical homunculus”

Правописание – это то, что в старой России в курсе физиологии развития детей (стандартный курс был в пединститутах – интересно, остался?) называлось “мелкая моторика” (видимо, по английски это dexterity). Не знаю, что сейчас говорит наука, но тогда считалось, что в человеском мозгу центры речи близко связаны с центрами, контролиующими движение рук – в частности, отсюда происходит неконтролиуемая жестикуляция во время разговора.Рука – это громадная часть коры головного мозга, как показано на классической картинке Cortical homunculus.