Posted by: Alexandre Borovik | February 21, 2021

A look from lockdown at horrors of school mathematics

Kit Yates in The Observer: Home schooling: ‘I’m a maths lecturer – and I had to get my children to teach me’  A few quotes:

A senior lecturer in the department of mathematical sciences at the University of Bath, Yates has a PhD in Maths from Oxford and is the author of The Maths of Life and Death. So when he began home schooling his son Will, five, and daughter Emmie, seven, during lockdown, he was pretty confident he already knew everything they would be expected to learn in maths.

He was wrong. “I’d never heard of a ‘bar model’ or a ‘part-whole model’. I had to get my kids to teach me.” He was shocked by how many of these different, “intimidating” methods and models primary school children are expected to use to solve basic maths problems. “I’ve never needed to use them – you don’t need to know all these different mental models to do maths,” he says. […]

But what he really finds frustrating is the lying. The curriculum is forcing teachers to deliberately teach children lies, he says, which then have to be unpicked later. For example, after years of being taught there are no numbers between zero and one, his seven-year-old is suddenly expected to understand that there are such things as fractions.

Posted by: Alexandre Borovik | February 21, 2021

Why is the constructivist theory applicable in teaching mathematics?

My answer to a question on Quora: Why is the constructivist theory applicable in teaching mathematics?

I suggest to modify the question a bit:

Why do some people find the constructivist theory applicable in teaching mathematics?

This wold allow me to express my surprise: indeed, why?

I have lived in the world of professional research mathematics for almost 50 years now, and I wonder why constructivist theory in mathematics education so blatantly ignores the experience accumulated in the mathematics research community. I feel that the  constructivist theory talks about some different kind of mathematics, not the one known to me and my many friends and colleagues from all around the world. But I am Vygotskian by my philosophy upbringing, and I can see how Vygotsky’s sociocultural approach explains the invention of this mock image of mathematics. I will look for an opportunity to explain that- I hope Quora sooner or later will give me chance to do that.

A very important question. As it was already explained in this thread, this is a well-known and quite usual phenomenon (called childhood amnesia), caused by re-wiring of the brain at the critically important stage of development. The timing is slightly different in different people, and, I feel, in respect to different kinds of brain activity — for example, you cannot forget how to swim or ride a bicycle, if you learnt these skills at the age covered by amnesia. Also, it appears that children do not unlearn how to read or do arithmetic — but they can eventually forget how did they learn to read.

In general, I think non-one should worry about their childhood amnesia — these were natural changes in one’s brain, and they were to one’s benefit.

I collected hundreds of testimonies from people about their very first memories of learning mathematics — and discovered, that it seems that majority of people simply do not remember anything at all about their earliest encounters with school mathematics — including, it appears, many teachers of mathematics. Unfortunately, I had a day job to do and had no time to run a proper statistical analysis. But I think, this is something that should be taken into account in professional education of future school teachers of mathematics.

My answer to a question on Quora: What are suggestions for patterns in daily lives that deal with mathematics?

The first thing that comes to mind is the most mundane: numbering, first of all, house numbers on streets in towns and cities. They make quite an expression on a 4 years old child when first explained to her:

  • house numbers are odd on one side and even on another;
  • they grow in one direction;
  • if look in the direction of increase of numbers, then odd numbers are on the left hand side of the street, even are on the right hand side.

It is useful to bring child’s attention to street signs with street names on them, as well as shops’, cafe’s, barbers’, nail salons’ signs: in some older cities there is a custom to include the name of the street in the name of establishment, so Coronation Butchers are likely to be on the Coronation Street. In short: at the very first opportunity, when child just starts to read and count, explain to her the structure of the street.

Please notice that I am talking about structures, not about patterns.

Mathematics is not a science of patterns, as some people claim,

Mathematics is the science of structures hidden behind patterns.

Structures are much richer and more interesting than patterns.

Let us look at another episode with the same child: he and the adult observe an ant on a trunk of a tree in a city park. Adult invites the child to observe that the trunk for the ant looks like a street, and patches of algae and moss are like lawns and bushes. Child: “And branches are side streets”.

A year later, the child is already able to use a standard city map and confidently guide the adult and a little sister through an unknown to them part of the city. Adult: “And where is our next turn?” Child, glancing at the map: “at this T-junction ahead  of us, to the right”. Adult: “And the name of another street?” Child, after quickly checking the map: “Station Road”.

Map is a mathematical object, and is in a mathematical relation with real streets. etc. in the city; this relation is called scaling (or, in more geometric terms, similarity or homothety). But the exactly the same concept of scaling can be introduced to a child using an ant on a tree trunk as an example.


if you want to see mathematical structures in the world around you, try to see the world through the eyes of a child.


Posted by: Alexandre Borovik | February 15, 2021

Some good answers are already given in this thread, I wish only to hint at the whole class of metaphors which can be used as a quick and cheap answer:

The difference between mathematics and mathematics education is the same as

  • between religion and religious education
  • between ***** and ***** education (you may wish to continue the list using this pattern)

Answering this question, it is very easy to switch into cynicism — one of the responses in this thread, from a former student, is already highly critical of mathematics education. I see myself as working professionally in mathematics education, and it is painful for me to see popular sayings as this one, from American popular culture:

Those who can, do. Those who can’t, teach. Those who can’t teach, teach teachers.

— but I have to live with it. We have to accept that this is a popular attitude to our profession.

My answer to a question on Quora: What are some unusual ways you’ve applied the math you learned in high school to your life?

I once was asked to act as a reviewer of a paper submitted for publication in an academic journal on mathematics education. It was a double blind review: the draft paper sent to me contained no names of authors or their affiliation.

The paper described how the authors set up a website and run online questionnaire among staff at mathematics departments of two British universities on the following issue: what kind of examinations, closed book, or open book, better discriminates between different levels of students’ attainment, and what kind is preferred by the respondents? Three pieces of data were given by the authors:

  • Closed book examinations were selected as the most discriminating or second most discriminating of the assessment methods by 79% of the participants.
  • Closed book examination was selected by 86% of the respondents as their most preferred of the assessment methods.
  • The response rate of the questionnaire was 15%,

What surprised me is that the total number of responses to the on-line questionnaire has not been given in the paper, although omitting the size of the sample from statistical data was unacceptable in published academic research.

However, I calculated the number of responses, and explained in my report to the editors how I did that essentially by mental arithmetic.

This is a cute arithmetic problem; one more general piece of information is needed for solution, but it is something commonsense. Try to think for yourself, it is easy. A solution is given below these warning signs:

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Solution. Indeed, 79% and 86% rates of positive answers to particular questions suggest that 86% – 79% = 7% corresponded to an integer number of people (those who answered positively to one question but not to the other). If 7% consists of 1 person, the number of respondents is 14 or 15. If 7% consist of 2 persons, then the number of respondents is between 28 and 30, but in this case, since the response rate was 15%, the two departments have about 200 mathematics lecturers, which was unlikely in UK universities (here the common sense is used). Hence there were 14 or 15 respondents.

Very conveniently, 11/14 rounds up to 0.79 and 12/14 to 0.86 (here I used a calculator – previous steps had been done by mental arithmetic) 15 respondents would produce not so good rounding of percentages.

I recommended to reject the paper — in my opinion, the paper contained no representative data; a chat in a staff lounge during coffee break, or, even better, on in a pub after a seminar was likely to yield a more representative sample. However, the editors accepted the paper for publication, but asked the authors to reveal the number of respondents – indeed, it was 14.

Posted by: Alexandre Borovik | February 15, 2021

How can one remain a mathematician?

My answer to question on Quora: How can one remain a mathematician?

It is next to impossible to answer your question without knowing your circumstances.

Being a mathematician is a way of life.

The way of life could change for a variety a reasons: for example,

  • external pressures (say, money problems)
  • illness
  • marriage
  • just because life became too boring
  • drug addiction
  • gambling addiction
  • taking certain types of medication
  • and so on …

The list can be expanded, and every situation calls for a different answer. I do not want to take your time and say only that drug addiction is incompatible with mathematics — to remain a mathematician, stop doing drugs. Also taking, for extended periods of time, medication about which you are warned: “when taking this medication, do not make important decisions, do not drive or operate machinery”. If you were given this warning, speak to your doctor and ask for an alternative treatment; explain to your doctor, that mathematics is all about making serious decisions; it is also a mental equivalent of operating heavy machinery.

My answer is based on  many years of my experience as a personal academic advisor to mathematics students.

Posted by: Alexandre Borovik | February 14, 2021

What are some of your favorite basic math hacks?

My answer to a question on Quora:  What are some of your favorite basic math hacks?

I have to apologise: I do not have favorite math hacks. I have never used hacks. The essence of mathematics is in universal methods for solving all problems in a particular class of problems. Use of “hacks” is a replacement of mathematics by cheap surrogates. Hacks can work at a certain level, but frequently obstruct students’ progress at the next level of learning of mathematics. Hacks are frequently favoured by teachers who always taught students at a certain level but never at higher levels. These teachers tend not to care much about their students’ progress beyond their class.

In my opinion, teachers have to be assessed not by their students’ marks in their classes, but by their students’ success at the next stages of education. In that environment, any desire to teach “hacks” somehow disappears.

I apologise to be so firm in my opinion on that matter, but for at least a decade I taught mathematics at a Foundation Studies programme in a very big university: up to 400 students in a lecture theatre who failed mathematics at secondary school and had to be brought up to the level where they could start studies in (relatively) mathematically intensive degree programmes at the university level. Hacks were the last thing they needed.

Posted by: Alexandre Borovik | February 14, 2021

Why is math in school boring? Is it the curriculum’s blind spot?

My answer to a question on Quora: Why is math in school boring? Is it the curriculum’s blind spot?

My answer:

  • Math in school is boring because mathematics education lost its purpose and meaning.

Lockdowns during this pandemic laid bare the true social and economic role of schools:

  • Schools are storage rooms for children — to free their parents’ time for paid employment.
  • Mathematics in schools is boring because teaching mathematics is reduced to coaching children to pass exams.
  • Exams dominate mathematics education because they are loved by employers: exam grades provide very cheep, for employers, way to select future employees, compliant and obedient. And mathematics exams could be made hard.
  • From the point of view of an employer, a good grade in exciting and very interesting subject is of less value than the same grade in an excruciatingly boring subject — because the latter requires much more effort, I would even say, strong will, to get.
  • You know, office work is painfully boring; it is advisable to be preparing for that from childhood. Being boring, school mathematics helps to achieve that.

As simple as that.

Posted by: Alexandre Borovik | February 14, 2021

Intuitively, what is a finite simple group?

My answer to a question on Quora: Intuitively, what is a finite simple group?

There are two ways to describe an object: how it is made and what it is doing. For example, a knife can be desribed as “an elongated flat piece of metal, sharpenenned on one edge, with a handle attached” (it is how it is made), or “a thing to cut bread” (how it is used).

I have not red every answer, but the discussion of groups in this thread so far appears to be restricted to the viewpoint of “how they are made”. But what do finite groups do?

They act. They act on sets of various nature; this sets are made of elements. The same group may have many different actions. For example, the group of symmetries of the cube can be seen as acting on

  1. the set of 8 vertices of the cube
  2. the set of 6 faces,
  3. the set of 8 edges seen as non-oriented segments
  4. the set of 16 oriented edges,
  5. the set of 4 non-oriented “main diagonals”,
  6. the set of 3 non-oriented axes passing through the centers of opposite faces.

Indeed, every symmetry of the cube is moving elements of each of this sets within that set (perhaps actually fixing some of them or even all of them).

Elements of the group, for the purpose of this discussion, can be called actors (I invented that name specifically for this post on Quora). What follows is a description, not a rigorous definition.

Each actor moves elements in the set in some way (and this could be an identity move, when nothing is actually changed in the set) . What follows are properties of actions of a group:

  • There is an actor which does nothing.
  • For every actor there is another actor, which reverses its moves.
  • For any two actors, there is an actor who is doing the combination of movements of the first two actors.

An action of a group is called trivial if every actor does not move anything.

An action of a group is called faithful if different actors do different movements. In the example with the group of symmetries of the cube, actions 1 to 4 are faithful, 5 and 6 are not faithful.

The key point: if an action of a group is not faithful, the same movements of elements can be achieved by an action on the same set of another group with smaller number of actors.

Definition: A finite group is simple if all its actions are trivial or faithful.

In short, a simple finite group cannot be replaced, in its non-trivial action, by a smaller group.

This is why finite simple groups are atoms of finite group theory, and why classification of finite simple groups has tremendous importance for combinatorics.

As we can see, the group of symmetries of the cube is not simple. Moreover, its action 6 above contain only movements of 3 elements which can be done but the symmetric group \(Sym_3\) on three letters, or, which is the same, by the group of symmetries of an equilateral triangle (this triangle can be easily seen within the cube). By contrast, the groups of rotations (symmetries which do not change orientation) of the equilateral triangle or the regular pentagon are simple.

By the way, “how it is made” and “what it is doing” are called, in Hegelian dialectics, essence and phenomenon. Questions “intuitively, what is …” refer to phenomena. Intuitively, a knife is a thing to put butter on bread. Intuitively, a group is a mathematical object that acts, or which can be used to describe action. There are other mathematical objects that also can act, in their own way: rings and algebras, for example. On the other hand, you can use a spoon to put butter on bread.

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