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

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18.1 Leo Harrin gton: Who is in control? 165

(1) Wh en I was 5, I was once playing with toys early in the

morning whi le all the rest were sleeping. When my mother saw

me alone, she th ought I would like some company and asked my

elder brother to go to me. He came and immediately started mess-

ing around soldiers. I shouted at him and asked to put everything

were it used to be. He put them back but I said h e did not. He

asked why but I could not ﬁnd any explanation for the feeling that

the probability of that is 1. [. . . ]

Or to SC:

As a very young lad walking home from school in the Euclidean

grid of streets that are the suburbs of Chicago, I thought about

avoiding sidewalk cracks: “Step on a crack, break you back.” Some-

how I knew, or had been told by an ol d er brother that lines were

inﬁnite. I reasoned that I didn’t know if sidewalk cracks were per-

fectly regula r, and the cracks running north to south from a block

to the east might extend to my path. It was Chicago and from the

point of view of a child, virtually inﬁnite, so there clearly was no

way to avoid sidewalk cracks. I missed an opportunity to become

obsessive compulsive.

Yes, SC missed a chance. In their extreme cases, mathematical

obsessions could becom e clinical:

“One of our ﬁrst patients, just 1 7 years old, was brought to us

in a wheelchair,” says Professor Christer Lindq uist, a pioneer in

the use of a brain surgery technique for people with Obsessive

Compulsive Disorder (OCD), known as Gamma Knife. “This boy

would set himself maths problems, which he had to solve before he

could eat. His OCD had become so severe, and the maths p roblems

he set himself so complex, that he couldn’t solve them a ny more,

so he couldn ’t eat.” [238]

Computer programming could be a n atural outlet for a child’s

drive to control the world by mean s of mathem atics. Here is a tes-

timony from IH

As a child, I wanted to write a computer program which draws a

chessboard (this was on an APPLE II). I wanted to do it usin g two

nested loops both going from 1 to 8 and then having s ome condition

which decides, depending on x and y, whether the corresponding

square s hould be black or white. I knew a lready how to check for

the parity of an integer, but it did not know how to use th is to get

the correct condition on x and y. (I could have treated the cases

when x is even and when x is odd separately and then check the

IH is male, German, has a PhD in mathematics, teaches mathematics

at an university; the episode told took pla ce when he was about 9 years

old.

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166 18 Bein g i n C ontrol

parity of y. But I had the f eeling that there should be a simpler

solution.) So I tried to test for the parity of x · y. Somehow, th is

seemed like the most natura l thin g to do. (I guess that I though t

that adding is a too simple operation to yield interesting results

like chessboards.)

When I noticed that this did not work, I probably started think-

ing a bit more, trying by hand to check which operation would do

the right thing. At some point, I must have found out that after all,

x+y works, but I guess that before, I tried several other non-linear

polynomials in x and y.

And another story, from Antonio Jose Di Scala

I do not remember any concrete stories about numbers or math-

ematics in the period of my life before I was 15. From that pe-

riod what I do remember is that I had probl ems to understand the

mathematics from school.

When I was 15 (in 198 7) I met my ﬁrst computer (Commodore

64) and my life changed f orever.

When I told these stories to my wife Anna, she instantly re -

sponded by telling me how she, aged 9, was using the Russian word

priuiqitь, “to tame”, to describe acco mmodation of new concepts

that she learnt at school: the concept had to become tame, obed ient

like a we ll trained dog. The word was her secret, she nev er men-

tioned it to parents or teachers.

Anna was not alone in her invention; here is a story from Ya

mur Denizhan:

Although I obviously knew th e word before, my real encounter

with and comprehension of the concept of “taming” is connected

with my reading The Little Prince. As far as I can ﬁgure out I

must h ave been nearly 12 years old. Saint-Exupéry offered me a

good framework for my potential critiques in face of the world of

grown-ups that I was going to enter.

I also must h ave embraced the concept “taming” so readily that

it became part of my inner language. Some years ago a friend of

mine told me of a scene from our university years:

One day when he entered the canteen he saw me sitting at a

table with notebooks spread in front of me but seemingly doing

nothing. He asked me what I was doing and I said (though I do not

remember having said it, it sounds very much like me) “I am tam-

ing the formulae”. (Having heard thi s story I can recall the feeling.

Most probably I must have been studying quantum physics.)

So, let us look at what being in control means.

AJDS is male, Argentinean, has a PhD in mathematics, h ol ds a research

position in an Ital ian university.

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18.3 The quest for understanding 167

18.2 The quest for logic

Solomon Garfu nkel

This one may be too far aﬁeld, as it relates to my ﬁrst memory

of constructing a logical argument. But as my research ﬁeld was

mathematical logic, you may allow it. I was probably no more than

5 or 6. I came to the conclusion that adults were insane. M y rea-

soning went l ike this—ice cream was the most wonderful food ever

invented. Any sane person wh o could would eat ice cream all day

every day. Adults could do whatever they wanted (as opposed to us

kids). They could in fact eat ice cream all of the time. They d idn’t,

ergo they were insane. By the way, I still feel that this argument

holds up.

18.3 The quest for understanding

At 11 I had just entered a Comprehensive ‘Middle’ School where,

due to our Education system in Kent, I had to remain before pro-

gressing to the Grammar School at 13. The standard of teaching

was very poor and a lady, who didn’t seem to know too much math-

ematics, attempted to teach us how manipula te equations. I was

bemused by her instructions (only verbal, I seem to remember) to,

for example, ‘take the 6 over to the other sid e a nd turn the plus to a

minus’. My maternal grandmother, an intelligent but not parti cu-

larly well educated lady, happened to be visiting. When I explained

my problem s he thought about it for a moment and explained that

an equation was like scales for baking, if it bal ances and you want

it to stay that way, if you ta ke 6 ounces off one side then you must

take si x ounces off the other. After that I got the idea an d school

algebra was never difﬁcult. But the experience made me realise

how much a clear explanation as to WHY one is following a math-

ematical procedure gives one conﬁdence in doing the subject at an

early age.

The story is echoed in Vadim Tropashko’s accoun t of how the

“balancing the scales” explanation was missed in his ﬁrst encounter

with equations:

The introduction of variable x in the ﬁrst grade! (Public school in

Minsk). The story went like this:

SG is male, American, a well-known expert in mathematical education.

VG is female, Briti sh, has a PhD in mathematics, teaches in a British

univeristy.

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168 18 Bein g i n C ontrol

Teacher: “And now I’ll show you something that you’ll carry

over to the 10th grade (at least)—the mysterious thing called x”.

Then, she carried over how to solve equations, e.g. x + 3 = 5. Her

method was algorithmic: if there is something added to x then the

solution is subtracting i t f rom R HS, whereas if it was subtracted. . .

Memorizing th is simple rules seems to be easy, but not for me. I

vividly remember struggling with this until much later when I was

shown that one can ad d/subtract the same value on both sides of

an equation!

Ulrich Kortenkamp

I was four years old and sitting on our sofa in the living room,

moving cushions from one place to the other. When my mother

asked me what I was doing (apparently creating a mess in the

living room), I proudly told her that I ﬁnally understood the reason

why 6+7 equals 13. I do remember that I found out that I can split

the 7 cushions into 3 and 4, and that I could use the 4 to make 1 0

out of 6. I still fall back to this visualization all the time, 36 years

later, and I cannot think of 6+7 without splitting the 7 in 3 and 4.

Natasha A lechina

When I was, I guess 8 or 9, I remember reading a book ab out school

children where the hero struggled with mathematics, and a fri end

was explaining how to solve a problem to him, which I remember

I appreciated as a good explanation, but i t was not a problem I

struggled with myself. I think it was something like “a boy and

a girl were collecting hazelnu ts, and together they collected 30,

the boy collected twice more than the girl, how many did each of

them collect”? The friend explained it with pictures, the boy had

two pockets and the girl one, each pocket held the same number of

hazelnuts, etc. The book was called Vitya Maleev at school and at

home, don’t remember the author,

and this is also the only thing

I remember from the book. . .

Nicola Arcozzi

I do not whether the short story I am going to tell you ﬁts the re-

quirements of a story about math education, since it takes place

in the fami ly. I was about ei ght-nine years old (Italian third-fourth

grade) and I was learnin g about conti nents. “Is Australia a conti-

nent or an island?” I asked my father. He answered it was both

a continent a nd an island ; an answer I found deeply unsa tisfac-

tory. I thought for a while about islands and what makes them

UK is male, German, professor of mathematics and education.

NA is female, Russian, holds a P hD, teaches computer science in a

British university.

Nikolay Nosov, a wonderful writer for children.—AVB

NA is male, Italian, has a PhD i n mathematics, teaches at an university.

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18.3 The quest for understanding 169

different from continents, until—weeks later—I reached the con-

clusion that, by stretching and contracting, Eurasia coul d be a n

islan d of the Oceans as well as an island of the Como lake ( “all its

shores are on the Como lake”). A sunny day right after rain I was

walking with my mother, I p oi nted to a p uddle and I said: “we are

on the island of that puddle”. She shrugged and replied “why do

you always say such stupid things”. (Only many years af terwards

I learned that was part of something called topology).

Teresa Patten

When I was a girl of about 7, being instructed in English (my na-

tive language), I remember asking my teacher to explain, “What

is two plus one?” She told me the answer is three, and explain ed

that if I have two oranges and my friend gives me one orange, I

will then have three oranges. “Yes,” I s aid, “but what is it?”

What I was really trying to as k was “What is the nature of

number?” I wanted to know how this abstract concept can apply

universally to any unit we determine to be a unit, and how this

correlates to our sensory experience of things as individual items.

In particular, I wanted to know which is more ‘real’, the abstract

concept or the thing itself? But of course as a 7-year-old I did not

have the mastery of l anguage to express this, and even if I could

have done so I sincerely doubt my teacher would have understood

the question. So instead of pursuing the idea she concluded that

I could not do simple addition and put me into the lowest math

group (this was California in the 197 0s, and at that place and time

children were taught in groups determined by aptitude), whi ch is

where I remained unti l I was about 11.

I think the saddest part of thi s is that until my late teens I be-

lieved I had no ability to do math whatsoever. I simply assumed

that my classmates all understood the nature of number and other

theoretical questions that seemed so difﬁcult to me. It never oc-

curred to me that the other children probabl y never even thought

to ask them.

Jens Høyrup

I remember being puzzled about certain physical questions at ages

3 to 5, including one pertaining to geometrical optics: at age 5, I

believe, before schooling, I had the impression that things when

becoming more distant did not d iminish as much as they “s hould”

(that is, proportionally to distance, as predicted by a model with

linear visual rays), and tried to solve the apparent p uzzle by draw-

ing a model where visual rays were conic in shape; I had a feeling

that my model did not work sati sfactorily, but neither made exper-

iments checking my initial erroneous intuition nor th e model, nor

did I really see my mistake; so I gave up.

TP is female, has an undergraduate degree in mathematics.

JH is male, Danish, has a PhD in ph ysics, a historian of mathematics.

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Johan Swanljung

I do remember one story, and it wasn’t in school at all—it was sim-

ply something that I found so perplexing th at it stayed with me.

I was 11 or 12 and I was passionate about Dungeons and Drag-

ons, which meant I had a lot of dice. If you roll a large number of

dice, you don’t expect them all to come up the same. If you roll,

say 10 six-sided dice and they all come up si xes, you’re very likely

to sus p ect trickery. This seemed obviously true, but there was a

logical problem. If the dice were fair, s houldn’t every possible com-

binati on be equally likely? In th at case, why is getting 1 0 sixes

strange? L ogically I could convince myself that there was nothing

special about that outcome, but something still felt deeply wrong

about that conclusion. After all, rolling a pair of sixes isn’t more

likely than rolling a 3 and a 4, is it? So why would that be different

when you roll ten sixes? It wasn’t until several years later that it

became clear to me that it had to do with how we count outcomes

(and that yes, two sixes is half as likely as a 3 and a 4 if you roll

two dice). There are many ways to get 5 sixes out of 10, but only

one way to get 10 si xes. The outcomes are only equally likely if we

keep track of which d ie is which. I don’t remember the moment of

understanding, unfortunately. Perhaps it was in school. I have no

memory of studyin g probability in school, although I’m certain I

must have.

18.4 The quest for power

Joseph Lauri

My childhood memories of mathematics are almost nightmares.

We had a very competitive 11+ exam to pass from primary to sec-

ondary (our school system is basically the same as the English

one). The maths exam u sed to have problems like: Tom, Dick and

Harry have so much marbles. Tom h as three more marbles tha n

Dick and Harry together bu t if they give him so much each he

will have twice as much. Find how many marbles. . . I still have a

vague recollection of a similar type of problem involving three jugs

of liquid an d if one pours so much into the next one then. . .

At that age these problems were simpl y beyond me. They had

some way of teaching us how to solve them using "shares" or some-

thing.

JS is male, Swedish, a teacher of mathematics. He says about this

episode: Language of mathematical instruction: English. I am bili ngual

(Swedish/English) and at that age English was my stronger la nguage.

JM is male, Mal tese, Professor of Mathematics at the Universi ty of

Malta. His initial mathematics education was in Engl ish, with Maltese

being used in the classroom as a meta-language.

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18.5 The quest for rigour 171

I did not even sit for the 11+ exam but managed to enter a

secondary school which had an easy maths qualifyin g test. In the

ﬁrst few months we we ta ught algebra. I discovered that I could do

the 11+ exam using al gebra in about h alf the time allotted. It was

then that I felt the sense of power that mathematics gives you,

and I was hooked.

Now I have a son and a daughter in the last years of primary

school. The pendulum has swung the other way. They try to avoid

anything which is intrinsica lly difﬁ cult. The difﬁculties are cre-

ated by th e modern insistence of avoiding algorithms or any ef-

ﬁcient way to work out a probl em. Constructivism and all that

leading to the situation that kids ﬁnish primary without knowing

how to add 1/2 plus 1/3 (it is ofﬁcially so in our N ational Cur-

riculum) and they never actually get round to doing it becau se in

the ﬁrst year of secondary they start using calculators and spread-

sheet (even this is in the curricu lum if you read between the lines).

Now I am not for one moment suggesting going back to my

nightmare days. But today, children are not shown how powerful

maths is, so they cannot feel the kick and the buzz I felt when I

was 11.

18.5 The quest for rigour

We are usu ally convinced more easily

by reasons we have found ourselves than

by those which have occurred to others.

Blaise Pascal, Pensées (1670)

Here is one thing which I remember well when p laying with a

compass as a child: a f ter having drawn a circle with the compass,

I started at a particular point P on th e circle a nd - not changing

the radius of the compass—tried to create successively points on

the original circle by getting to the next point clockwise with the

help of the compass by using the previous point as centre. After

doing this six times, I always seemed to get back almost to the

original point P , but never precisely. At that time, I wasn’t sure

whether this was based on my drawing not being precis e (all my

points and l ines had some width) or whether it really was some-

thing which was only almost tru e. I cannot precisely say at what

age I encoun tered this puzzle. I am certain that no foreign lan-

guage was involved at that time.

I wrote to NP asking him:

What would be your reaction if, instea d of a compass, you were

using a software pa ckage like GEOGEBRA or CINDERELLA which

NP is male, German, has a PhD in mathematics, teaches in a British

university.

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172 18 Bein g i n C ontrol

automatically adjus ts the picture and helps the user to achieve

formal mathematical perfection of the picture?

His response:

Hypothetically assuming I had this tool availab le at the time of my

compass experiments, I might have found out that this fact hold s

precisely and it might have encouraged me to ﬁnd some theoretical

reason for this to be true.

But I can only guess that this would have happ ened.

I was about 12 years old , at secondary school. We had been taught

about ruler and compasses constructions, including angle bisec-

tions, and h ad spent time practicing various constructions using

the geometrical instruments which I enjoyed. The teacher made a

remark in class about it being impossible to trisect an a ngle. Hmm

I though to myself—it was very easy to bisect an angle, why not

trisect? I spent a long time working on ways to trisect an angle

with ruler and compasses later at home. But, I remember feeling

that I could never be sure whether I had done it and my execution

of the ins tru ments was not accura te, hence explaining the error I

detected by measurin g with the protractor, or whether I had not

done it and the small error measured did indicate the fact of non-

trisection.

PS I was delighted to ﬁnd that angles in origa mi-geometry can

be trisected!

The one memory that is still strong, for whatever reason, was of

an interchange with my father when I was 6. (He was a lawyer,

but he liked the math he knew and had been head of the math

team, he told me, at Stuyvesant High School in NYC—he grad-

uated in 1920, I beli eve. I don’t think h e ever learned calculus.)

One afternoon I had been doing a page of drill on ad ding and sub-

tracting one-digit integers and I noticed that if you added 5 or

subtracted 5 from the same number, the results were th e same in

the units digit. I was quite delighted with this observation, and

told him about it when he came home from work. His response

was “Oh yes, and that’s because . . . ”, givi ng a reason. I remember

thinking—who cares why? it’s just cool, here was something that

had made me happy to discover. So you might say that at that time

I was challenged in the analytical part of mathematical curiosity.

MR is female, English, has a PhD, an university lecturer and a re-

searcher in mathematical education.

RL is male, American, and professor of mathematics in an American

university.

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18.5 The quest for rigour 173

But the deepest meaning of this memory is no doubt of the emotion

and family dynamic.

Solomon Garfunkel:

One experience immediately comes to mind . I was in third grade,

so likely about 8 years old. My teacher asked the class how far the

moon was from the sun. I raced to th e board and drew a picture

with the earth at the bottom, the moon directly overhead in a cir-

cular orbit an d the sun directly above both. I reasoned that since

the earth was 93 million miles from the sun and the moon was

250 thousand miles from the earth, all we had to do was to sub-

tract and voila the moon was 92.75 mill ion miles from the sun. The

teacher was effusive in her praise, lauding my reasoning ability.

When I got home an d began to tell my father a bout the prob-

lem (and the praise) I quickly realized how foolish I had been

to oversimplify the model as I did and not take into account the

movement of the bodies involved. I also realized that , even as-

suming circular orbits, I didn’t have the technical knowledge to

actually solve the problem and that there were many (I couldn’t

have thought inﬁnite)different solutions. I also realized that my

teacher thought I was correct, which led to a healthy skepticism of

teachers ever after.

Lawrence Braden

When I was twelve years old, Mr. P, my math (maths) teacher, told

the class that π equalled 22/7. Also that π equalled 3.1416. He was

speaking to me in Ojai, California, in a language we mistakenly

called English but whi ch was roughly equi valent to it under most

circumstances.

Excited, I went home with the grandiose notion of ﬁnding π

to a hundred decimal places by the process of long division, and

wondered if anyone had ever done th at sort of thing before. Well,

of course, I kept getting the wrong answer! Not 3.1416 at all! I d id

it over four or ﬁve times, and was really di sturbed. The book said

that π equaled 3.1416 and the teacher s aid that π equaled 3.1416

so I logically came to the conclusion tha t I did not know how to

do long division! A truly disturbing notion; I thought I was pretty

good at it and here I couldn’t even do this simple problem!

“Oh”, Mr. P said the next day. “I didn’t mean that π was exactly

equal to 22/7." It was at that poin t that I learned not to take ev-

erything a maths teacher (or any s ort of teacher) said a s hewn i n

stone. Years later I came upon Niven’s truly beautiful and elemen-

tary proof of the irrationality of π, and Lindemann’s proof of its

transcendence.

Teachers should be careful what they s ay. They will be believed

by their children. I myself taught at least two generations of stu -

LB is male, American, a recently-retired math teacher of 4 0 years expe-

rience.

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dents that parabolas came in different shapes before learning dif-

ferently. But that was out of ignorance. Mr. P knew better.

I became unpopular with my class mates and some teachers be-

cause sometimes, maths ques tions appeared ambiguous to me.

When a question was hidden in the text and you had to think about

how to translate it into a mathematical problem, I frequently came

up with at least one solution different from what the teacher ex-

pected, referring to different ways of understanding what was

written. I used to be very stubborn when a teacher would try to

“sell me” that only one unique answer was correct. Of course I

was wrong sometimes and just misunderstood what was said, but

sometimes I was right and some teachers made me respect them a

lot by discussing my views in an open way (independent of whether

I was right or wrong).

Robin Harte

I went to physics classes (and was told that I had to do extra maths

if I wanted to do ph ysics) and at one point our physics teacher was

explaining something later called ﬂux, which he described as “th e

number of lines of force”.

What I want to recall is a subconscious awareness tha t some-

thing was out of kilter: nothing was said, but this quantity s eemed

not necessarily to be a whole number and that seemed to bother

me.

So in the sixth form at school I believe I had some kind of math-

ematical sense.

I can also report small hairs standing up on the back of my

neck the ﬁrst time I met differentiation.

As Tanya Khovanova’s

story conﬁrms, it could be easier to

accept that it is physics that is non-rigorous, thus protecting the

sacred status of mathematical rigour:

I had problems with physics rather than mathematics. I

remember when I was in ﬁ f th grade (I lived in Moscow at

that time) my f ather told me about the relativity theory

and the fact that the speed of light is constant.

When I started studying mechanics in my 8th grade

I was completely confused. The vector addition of speeds

contradicted to all I new about mechanics. I was bugging

RW is female, German, a professor of mathematics in a German univer-

sity.

RH is male, British. He says about himself: I did go on to do mathemat-

ics in Trinity College Dublin, and a P hD in Cambridge, and never really

to escape the system.

TK is female, Russian, has a PhD in Mathematics, holds a research

position in a n American university. A popularizer of mathematics, see

her website http://tanyakhovanova.com.

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