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!


My Quora answers


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.


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?


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.



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.

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“If on a Winter’s Night a Traveller”

The title of Italo Calvino‘s book is great. When I first time saw the book on a shop shelf, I bought it on impulse because on its title. To the Russian year, it has immediate connotations with the famous passage from Chekhov’s “Ionych”:

Then they all sat down in the drawing-room with very serious faces, and Vera Iosifovna read her novel. It began like this: “The frost was intense… .” The windows were wide open; from the kitchen came the clatter of knives and the smell of fried onions… . It was comfortable in the soft deep arm-chair; the lights had such a friendly twinkle in the twilight of the drawing-room, and at the moment on a summer evening when sounds of voices and laughter floated in from the street and whiffs of lilac from the yard, it was difficult to grasp that the frost was intense, and that the setting sun was lighting with its chilly rays a solitary wayfarer on the snowy plain. Vera Iosifovna read how a beautiful young countess founded a school, a hospital, a library, in her village, and fell in love with a wandering artist; she read of what never happens in real life, and yet it was pleasant to listen — it was comfortable, and such agreeable, serene thoughts kept coming into the mind, one had no desire to get up.

In Russia in “the period of stagnation”, the expression “The frost was intense” (“мороз крепчал”) became proverbial and was transformed into a less politically correct, but more politically charged, derivative.

And I was delighted to discover that my instinctive choice was correct and that, indeed, Calvino’s book “did exactly what it said on the tin”!


Evaluating students’ evaluations of professors

This paper contains some bizarre observations:
Michela Braga, Marco Paccagnella, Michele Pellizzari, Evaluating students’ evaluations of professors. Economics of Education Review 41 (214) 71-88.
Abstract: This paper contrasts measures of teacher effectiveness with the students’ evaluations for the same teachers using administrative data from Bocconi University. The effectiveness measures are estimated by comparing the performance in follow-on coursework of students who are randomly assigned to teachers. We find that teacher quality matters substantially and that our measure of effectiveness is negatively correlated with the students’ evaluations of professors. A simple theory rationalizes this result under the
assumption that students evaluate professors based on their realized utility, an assumption that is supported by additional evidence that the evaluations respond to meteorological conditions.

Meta-analysis of faculty’s teaching effectiveness: Student evaluation of teaching ratings and student learning are not related

An interesting paper:

Bob Uttl, Carmela A.White, Daniela Wong Gonzalez, Meta-analysis of faculty’s teaching effectiveness:  Student evaluation of teaching ratings and student learning are not related. Studies in Educational Evaluation, Volume 54, September 2017, Pages 22-42.

Abstract: Student evaluation of teaching (SET) ratings are used to evaluate faculty’s teaching effectiveness based on a widespread belief that students learn more from highly rated professors. The key evidence cited in support of this belief are meta-analyses of multisection studies showing small-to-moderate correlations between SET ratings and student achievement (e.g., Cohen, 1980, 1981; Feldman, 1989). We re-analyzed previously published meta-analyses of the multisection studies and found that their findings were an artifact of small sample sized studies and publication bias. Whereas the small sample sized studies showed large and moderate correlation, the large sample sized studies showed no or only minimal correlation between SET ratings and learning. Our up-to-date meta-analysis of all multisection studies revealed no significant correlations between the SET ratings and learning. These findings suggest that institutions focused on student learning and career success may want to abandon SET ratings as a measure of faculty’s teaching effectiveness.

The epigraph is great:

For every complex problem there is an answer that is clear, simple, and wrong.” H. L. Mencken

title = "Meta-analysis of faculty's teaching effectiveness: Student evaluation of teaching ratings and student learning are not related",
journal = "Studies in Educational Evaluation",
volume = "54",
number = "",
pages = "22 - 42",
year = "2017",
note = "Evaluation of teaching: Challenges and promises",
issn = "0191-491X",
doi = "http://dx.doi.org/10.1016/j.stueduc.2016.08.007",
url = "http://www.sciencedirect.com/science/article/pii/S0191491X16300323",
author = "Bob Uttl and Carmela A. White and Daniela Wong Gonzalez",
keywords = "Meta-analysis of student evaluation of teaching",
keywords = "Multisection studies",
keywords = "Validity",
keywords = "Teaching effectiveness",
keywords = "Evaluation of faculty",
keywords = "SET and learning correlations"