Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics

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18.5 The quest for rigour 175

the teacher to give me the proof why speed were added

this way. He couldn’t gi ve a proof and I couldn’t move for-

ward. For half a year I was stuck and was getting bad

grades until someone realized what my problems and ex-

plained to me that physics i s not mathemati cs and that

physics is based on experiments rather than proofs. Also,

that vector speed addition is an approximation.

After that my grades immediately got back to my usual

A’s and physics became a breeze.

GCS

Age 6, a state primary school in a working class London area. I

always enjoyed playin g with numbers. A teacher tried to tell us

that when you b roke a 12 inch ruler into two pieces that were the

same, each would be 6 inches long.

I went to see her, because I couldn’t see how you could break

the ruler into equal pieces, because of the point at 6; it wouldn’t

know which piece to join.

In adult notation [0, 6] and [6, 12] are not dis joint, but [0, 6),

[6, 12] are not isometric. No matter how I tried to explain the prob-

lem, she didn’t und erstand. It was a valuable lesson, because from

that point on my expectations of schoolteachers were much re-

duced.

Another aspect of mathematical thinking manifested itself in a

very literal frame of mind. I can remember when the head teacher

told us one day to ‘run round the playground’, all the other children

(correctly) interpreted his instruction as an invitation to go outside

and play. I, of course, stuck to th e perimeter and performed laps.

Mihai Putinar

I think my mind was shaped by the generous and beautiful n a-

ture (mountain region) where I grew up, and mainl y b y playing all

my childhood along a river which passed in the proximity of our

house. The ﬂowing water imposes a grammar of thought which I

feel remains deeply rooted inside me.

Why mathematics? At 12–14 years of age I had serious trou-

bles understanding the n atural sciences the way were presented in

school (physics, biology, chemistry). I could not accept the dogmatic

approach, with ad-hoc working deﬁni tions (a la “the energy is the

quantity numerically equal to . . . ”), all melted into a descriptive

story, sparkled with many examples and without a structure.

Mathematics came as the uniqu e salvation: crystal clear, sim-

ple, founded on a few principl es, totall y mental yet s o effective. The

deductive method, well exposed in the ﬁfth-sixth grade by planar

geometry, the implacable power of algebraic identities, the endless

GCS i s male, English , a lecturer of mathematics in a British universi ty.

MP is male, Romanian, a professor of mathematics in an American un i-

versity.

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176 18 Bein g i n C ontrol

mysteries of nu mbers won my heart. Since then I could not s top

doing math with great pleasure and with all my energy.

And Pierre Arnoux

returns to an earlier told story (page 22):

Concerning the ﬁrst story I told you, when I was 10 years old,

I remember very clearly the feeling I had when I ﬁrst learnt the

idea of a variable. The best comparison is that I felt I walked on

very thin ice, whi ch could break at any moment, and I only felt

safe when I arrived at the solution.

Olivier Gerard:

I concur with Pierre Arnoux with a similar image.

When I started solving equations in junior high school, I had

the feeling that the variable was a very precious thing, that could

become ugl y i f badly han d led and following the rules such as

“passin g numbers from one side to the other and inverting signs”

carefully was like walking softly or stepping slowly when mov-

ing a large stacks of things or books in one’s ha nds. If something

was done too roughly the things ended on the ﬂoor, some of them

bruised or broken. As many children experience, the broken glass

or the damaged object ended to be found missing by their parents.

With potential bad consequences.

18.6 Suspicion of easy options

Olivier Gerard raises another issues which I have never seen dis-

cussed in literature on mathematics ed ucation.

In most exercises we had to do, there were miraculous simpliﬁca-

tions, especially with fractions, giving round numbers and “palat-

able” expressions at the end as the solution. This was a good indi-

cation that we succeeded i n f ollowing the rules correctly but then

it was a lso misleading if used too systematically by teachers. I un-

derstood more and more with time that this was in great part a

convenience for teachers when designing exercises: they could eas-

ily grade that way. I remember saying to myself: they must start

from the result and then they go backwards to the question to

make it look complicated. They are so good, they probably can do

that in their head.

With hindsight these round number solutions can have per-

verse consequences.

Inside math teaching, th ere is the fact that pupils can infer

incorrect informal rules for themselves about what the s teps a nd

PA is male, French, a professor of mathematics in a French university.

Another story from him i s on pages 22 and 67.

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18.6 Susp icion of easy options 177

results should be, based on the very special subsets of applica-

tions and n umbers they see doing exercises. Some develop a kind

of mysticism of what the value of x or f (x) should l ook like across

exercises. They are not ready nor willing to start with a bla nk slate

about them with every problem. When interviewing some young

pupils that had a lot of difﬁculty with mathematics in high school,

I discovered that some of them actually thought that they could

learn more about x or f each time they were doing another exer-

cise, and the harder they tried, the b etter memory they had, the

more confused they got.

It is worth n oticing that, with beneﬁt of hin dsight, Olivier Ger-

ard sees danger in artiﬁcial easy options open to an inquisitive

child—but he himself did not succumb to te mptation of adopting

a grossly simpliﬁed approach to mathematics.

As a child of 10 or 11, I could easily guess answers to exercises

in the arithme tic textbook that required e valuation of com plex ex-

pressions involving brackets and all four arithmetic operations. I

quickly realized the artiﬁcial nature of expressions—all numbers

were integers and all divisions were exact, without remainders.

This r evealed too much, and crude estimates of dividends and d ivi-

sors frequently produc ed almost instant result. I remember devel-

oping a bit more sophisticated tricks for more complex exercises,

but I cannot recall them now.

In any case, ev en as a child, I was uneasy about the pedagogi-

cal wisdom immortalized in the famous Ru ssian painting, see Fig-

ure 18.1, and I still cann ot d ecide whether it is good or bad.

But I agree with Olivier Gerard ’s fur ther comments:

In relation with physics, where “numerical applications” a re

messier or are intended as approximations, that should be trun-

cated as appropriate, it will give or prepare a false intuition of

simplicity or correctness, more based on numerology than relative

dimensions and units.

In relation with a little more advanced mathematics, when you

discover some of the increasing complexity say of solutions to poly-

nomials equation like the 4th degree (and of course the i mpossibil-

ity of solving it generally for degree greater than 4), the fact that

many important constants such as π, e, are both very simple but

not that much if you do not know integrals and inﬁni te series, the

difﬁculty of constructing exact polygons of odd order, the fact that

triplexes do not exist, etc. you can have a deception if you antici-

pated mathematics to be always cutting through complexity with

very clean and minimal closed formula e.

I know of some of my former classmates who felt deceived in

that way and decided to turn more ful ly to other disciplines such as

biology, chemistry, history,. . . Perhaps to be further disappointed in

their expectations. I cannot say.

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178 18 Bein g i n C ontrol

Fig. 18.1. Mental arithmetic. Nikolai Petrovich Bogdanov-Belskii, 1896.

Tretyakov Gallery, Moscow. Public domain.

The teacher is Sergei Aleksandrovich Rachinskii (1833–1902), biology pro-

fessor and a philanthropist; Bogdanov-Belskii (1868– 1945) was his pupil

and protege.

An exercise for the reader: solve the example by mental arithmetic.

And, last bu t not least, child’s suspicion that teachers cheat and

build exercises from ready answers is already an im portant m ath-

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18.7 “Everything had to be proven” 179

ematical discovery. Olivier Gerard’s concludes his reminiscences

with a charming episode:

I remember that when 11 I wondered how you could make a

labyrinth puzzle like the one I liked to solve on activity and puzzle

books. So I took squared paper and tried to design one myself in

order to ask my sister to solve it. I had a very hard time trying to

do it without success, I was always walling myself into it. It was

also because I wanted to d es ign the most difﬁcult l abyrinth puzzle

possible, that only me could solve. So I did not try to do ﬁ rst some-

thing simple. Then one year later while going home from school,

I thought “I have ﬁrst to draw the path I want, then I build the

labyrinth around it”. I started doing it that way, tracing a solution

with lea d pencil in order to erase it and beginning walls with bl ack

ink. But I soon stopped. I was conﬁdent I had found the way to do

it, it was not i nterested anymore, I was not motivated an ymore ei-

ther by challenging others to solve the resu lting puzzle. I switched

to play with something else.

18.7 “Everything had to b e proven”

A child’s quest for rigor in mathematics can be supported by his

or her intellectual env ironment. The following story, from Azadeh

Neman

, comes from her childhood in the political hotspot of Iran:

I was talking to a colleague here and he mentioned something

which I thought might be useful for your book. It was about the

idea of proof and how hard it is for students to grasp what i s a

rigorous proof.

Since I assume most people in the ﬁeld of mathematics to be

a lot more clever than I am, I found it interesting that I did not

experienced such a problem. After considering the matter a bit I

realized the problem comes from the late introduction to the idea

of rigorous proofs. Since I was brought up in an environment of

constant political argument and everything one claimed had to be

proven there, it was not so strange to see the same principl e ap-

plied, more coherently and clearly of course, to two-dimensional

geometry. Adding one sheer good luck in my ﬁrst year in High -

school I ended up with a very methodical teacher who would give

us marks for clear statement (and deduce for the absence) of the

hypothesis and the goal. Her method kind of stayed in mind ever

since.

AN is female, Iranian, educated in Iran and Britain, holds a PhD i n

mathematics from a French University.

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180 18 Bein g i n C ontrol

And now we move to another ho tspo t, Turkey. Ali Nesin

tells

about his childhood and about his father Aziz Nesin, a famous

writer and political activist:

I spent my early childhood in drawing, dreaming a nd reading. I

loved to be left alone and be immersed in my thoughts. I remem-

ber looking at oil paintings for hours. I believe what made me a

mathematician is this love of loneliness.

Also, I loved to listen to the intense debates of intellectuals.

Most often about politics a nd literature. Indeed every other night

our h ome was full of writers, poets, painters, actors. . . I was trying

to understand whi ch one was the winner of the debate!

Most people at these “raki tables” were h ighly educated and

clever and excellent speakers. Bu t none of them could equal my

father. His ﬂawless logic, the examples h e picked, the analogies he

found, strong images he offered, the eloquent and the soft way he

formulated his ideas, and a witty but warm humor on top of all

that was a real delight; not only to me but to everyone.

I believe these conversations also played an immense role in

my becoming a mathemati cian.

18.8 Raw emotions

In our discussion so far, the issue of control was m ostly mathe mati-

cal. When classroom politics and relations with teachers and other

students are involved, the stories told are sometimes bitter, some-

times good humored, but they are almost always quite emotional.

Here is a testimony from Charles Leedham-Gre en

The ﬁrst time I went to school I was three years and seven

months old. There were two other pupils at the school, girls rather

older than me, so there was no streaming. The teacher was prob-

ably in her teens. We had a mathematics lesson. We were each

given a piece of cardboard that had the outli ne of a one (1) with all

the trimmings, thus

____

/__ |

| |

__| |_

|______|

AN i s male, Turkish, professor of mathematics, an inﬂuential teacher,

popularizer and promoter of mathematics.

CLG is male, English, professor of mathematics in a British university.

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18.8 Raw emotions 181

We were required to colour this in red, and after some effort we

more or less achieved this task, and the best effort, far far better

than mine, was held up for general admiration, a nd we were asked

what this might be. The two girls answered ‘Th e ﬁgure one’ in

unison. I had the impression that they had been coached to d o this.

However, I was not so sure and remained dumb. I had a suspicion

that as even this high quality art work was inclined to go over the

lines, and leave some parts of the interior un coloured, it might not

be quite a ﬁgure one.

I am still not certain of whether I was right to be cautious, but

I went right to the bottom of the class [of three].

Later of course I went to a big school, with some twelve pupils

and two teachers. It was an excellent school; I thi nk the teachers,

who were sisters, were in their seventies; perhaps a little younger;

unmarried because of the slaughter of the men in the ﬁrst world

war. I was at this school till the age of six or seven, and was tau ght

a fair amount of French, Latin, History, Norse mythology, spelling,

art and mathematics. We were taught arithmetic up to the extrac-

tion of square roots and the calculation of hcf’s and lcm’s by fac-

torisation. I was reasonably competent, except perhaps for the art,

but was inclined to think about things (or gaze out of the window)

rather than getting on wi th my work, a tendency that the teacher

regarded with some un d erstanding.

Sometimes, when doing a long multiplication, I would set it

out the other way from the way we had been tau ght; so I started

by multiplying by the unit entry of the second number instead of

starting with the most signiﬁcant digit. The teacher took this in

her strid e, and learnedly told me that I had set it out in the Ger-

man style. She was a brilliant teacher, but they retired and closed

the school as I was leaving, raising the fees for th e last year from

£2 per term to £2 10s per term. At my next school, which had 70

pupils in 6 classes, having rather lost track, through lack of prac-

tice, with how to s et out multiplicati ons, I set out a multiplication

in some way unknown to God or man, but got the right answer,

to the confusion of the teacher. I also surpri sed him by knowing

the word ‘colloquy’ at the age of seven. Not infantile precocity, but

spelling lessons at the previous school.

In the ne xt story, from FB

, the most interesting point is the

use of the word “eavesdropping” that gives it a distinctive emo-

tional color.

I was a competent at mental ari thmetic but did it as a l ea rned

trick with out real understanding. I understood multiplica tion / di-

vision eventually by eavesdropping on teacher explaining the con-

FB tells about herself: “I was 9 years old; I am female, from UK, stud -

ied mathematics at university but not very happily.” She tea ches at

a British university and actively promotes the use of information and

communications technologies in teaching and learning.

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182 18 Bein g i n C ontrol

cepts usi ng ( not real) squares of chocolate to a child who was s trug-

gling. This experience was useful to me when I taught secondary

mathematics for a while.

Similarly, the key word in Robin Harte’s story appears to be

“escape”.

I have a curious memory for you, I probably got into mathematics

as a n escape from physics wh ich in turn was an es cape from Latin

and Greek—my father was a school teacher who taught Latin and

Greek and it was important to me to stay as far as possible from

that—I was at the same school where he taught Latin and Greek

which was bad karma. [. . . ]

I did go on to do mathematics in Trinity College Dublin, and a

PhD in Cambridge, and never really to escape the system.

Reuben Hersh:

The best I can do—not what you are looking for, I am afraid—was

when my family moved from The Bronx to Mount Vernon (a nearby

suburb) a nd my mother and I met the principal of my new school.

(I was entering the 10th grade, but I was only 12 or 13 years old,

and had a protective mother.) Anyway, my mother told the princi-

pal, “He loves mathematics,” and the pri ncipal said to me, “You’ll

take geometry.” To which I replied (based on my previous two years

of junior high school) “Bu t I like algebra!” To which the principal

simply replied, “You’ll take geometry.”

Of course, I did OK in geometry. But I was always more in-

clined to algebra.

Alexander Bogom olny

[I was 12.] One day, my math teacher asked me to stay after the

class as she wanted to show me something interesting. After every

one left, she wrote on the b lackboard an exampl e of multiplica-

tion of two-digit numbers by 11. She asked whether I saw why it

worked and whether I liked that. I was utterly embarrassed. Of

course I knew that trick. I do not remember how it happened, but

I was aware of many more, this particular one being probably the

RH is male, American, a professor of mathematics and a prominent au-

thor of books on methodology and philosophy of mathematics. He in-

formed me, in particular: “The ﬁrst chapter of our forthcoming book

Loving and Hating Mathematics [127] has plenty of stories of childh ood

experiences of mathematicians, bu t they are mostly happy and positive

experiences.”

AB is male, his mother tongue is Russian. He has MSc in mathe-

matics from the Moscow State University an d a PhD in ma themat-

ics from the Hebrew University. AB runs a wonderful website http:

//www.cut-the-knot.org/ctk/ErrDisc.shtml. The episode hap-

pened when he was 12 years old.

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18.8 Raw emotions 183

most trivial. I blushed, thanked her and ran out of the room with

a realization that, if I wanted to l ea rn anything, I had to do that

on my own.

Alan Hutchinson:

I asked the head of maths at school what more maths there is to

learn. He laughed at me.

Gwen Fisher:

I think my biggest general challenge with math as a kid was won-

dering what it was useful for. So many of my teachers taught stuff

so abstractly, it was just moving symbols around on paper. But I

was good at that, so I didn’t fuss about it much. One very memo-

rable occasion for me was when I asked my junior high school math

teacher why matrix multiplica tion was u seful. He admonished me

harshly for my question, and never answered it. The effect of this

incident on me was that I never asked “Why do we do this?” in a

math class again. . . until I was in graduate school! I was learning

about ﬁnite ﬁeld s and asked my teacher and then recoiled in an-

ticipation of the tongue lashi ng that I thought would result from

my question. Ala s, he answered me, so I told him this story about

matrix multiplication. And so he told me an application for matrix

multiplica tion that an 8th grader could understand. That was a

very important moment for me as a math teacher.

Theresia Eisenkölbl:

I know that this is not exactly what you are searching for with

your demand for stories, but I have to comment on the “intrinsic

competitiveness” of mathematics.

In mathematics olympiads and in p rofessional mathematics,

competitiveness is ritualized and contained by ethical rules. In my

olympiad trainin g session, you were expected to explain things to

newer participants even though they might well be better than you

at the competition. In olympiads and real life, I have found that

this kind of ethics almost always corresponded to mathematical

capability.

So here is my anecdote on competitiveness:

In primary school (age 6–10), I had about half the running

speed of any other child. On the other hand, I understood every-

thing in maths immediately and was very quick with mental cal-

culations. In sports, we had weekly running competitions of groups

by four, where my group never had a chance at all. In maths, we

had weekly speed calcu lating competitions where each pair of stu-

dents were posed a question a nd the quicker one would remain

standing until one student was left and got a little prize. I won

every time and was only allowed to participate every other week.

The lesson that good performance in math is bad for your social

standing with your peers could not have been more evident.

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184 18 Bein g i n C ontrol

And I would like to round up this section with a bit of very happy

memories. Here is a story from Anthony O’Farrell

I suppose you are referring to kindergarten and primary school?

I never had any difﬁculty with any mathematical ideas presented

to me at this level. Everything made i mmediate sense. However,

most of it was rather un surprising. I remember one or two things

that took my fa ncy, such as the fast way to work out the squares

of numbers such as 45 . For the most part, school was prison, a

place you had to go, where they told you stuff you knew, and made

you memorise stuff you might otherwise ignore. But I went every

day, with out fail, and occasionally something nice happened. The

one day that stands out in my memory was in 6th class (aged 11),

when Brother Skehan explained how to complete the square and

derive the formula for the solution of the q uadratic

+ bx + c = 0.

This was a revelation—the whole way of thinking was like noth-

ing in my experience up to that point. I never forgot that elegant

little argument. About 5 years ago, I met the brother again, and

mentioned the incident to him. He remembered it, too. He told me

that he was not required to do it, it was not on the curriculum, but

it was a slow day and he just felt like doing something dif ferent.

He observed that most of the class looked on in blank incompre-

hension, but that I sat up and drank it in, with shining eyes.

18.9 David Epstein: Give students problems that

interest them

David Epstein

I was about 9 years old when I was given a s a present a book by

H. E. Dudeney of puzzles. (See Wikipedia for an article about Du-

deney. ) I tried to do all the questions, which covered a wi d e vari ety

of different kinds of problems. One p articular type of problem “Al-

ice was twice as old as Henry when she was as old as Henry is now,

etc, etc” intrigued me but I couldn’t solve them. I somehow got the

idea that “algebra” was the answer. I asked my teacher for a more

advanced textbook, but she refused, so one lunch break, when ev-

eryone was outside playing, I took the book from the cupb oa rd.

The teacher denounced me to the class as a thief, and demanded

the book’s return. But I didn’t return it. Instead, I studied it with

great interest, and with a lot of difﬁculty in coming to terms with

what it all meant. After a while I could solve these problems in

AO’F is male, Irish, a professor of mathematics.

DE is male, British, learnt mathematics in English, a professor of math-

ematics.

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