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

Подождите немного. Документ загружается.

Fractions

Customer: How big is a portion

of half roast chicken?

Does it co me with legs?

Waiter: It comes w ith one leg.

A conversation in a diner.

Adding fractions is a notorious issue in mathematics education ,

and appears to grow in its notoriety.

We start our discussion by quoting a testimony from Simon J.

Shepherd

My main problem while quite young was adding fractions. I was

quite happy mul tiplying them, since you simply multipli ed the two

top numbers to get the answer top number, then multiplied the

two bottom numbers to get the answer bottom number.

But with adding you had to ﬁnd a common denominator b y

cross-multiplying and I remember it took me years to crack this

technique!

Funnil y enough, many of our foundation year students have

exactly the same problem. Perhaps learning the multiplication be-

fore the addition a ctu ally inhibits learning the addition rule?

My personal opinio n is that, ind eed, addition should come before

the multiplication. I will try to explain that in the next section.

8.1 Fractions as “named” numbers

In response to my call for personal stories about difﬁculties in

studying ( early) mathematics Alex Grad

sent me the following

e-mail:

SJS is male, British, a professor of computational mathematics.

AG is male, Romanian, a student of computer science. His other stories

are on Pages 10 and 14.

86 8 Fractions

When I was about 9 years old, I ﬁrst learned at school about frac-

tions, and understood them quite well, but I had difﬁcul ties in

understanding the concept of fractions that were bigger than 1,

because you see we were thought that fractions are part of some-

thing, so I could understand the concept of, for example 1/3 (you

a take a piece of something you divid ed in 3 equal pieces and you

take one), but I couldn’t understa nd what meant 4/3 (how can you

take 4 pieces when there are only 3?). Of course I got it in several

days, but I remember that I was bafﬂed at ﬁrst.

I am surprised how frequently such memories are related to the

subtle interp lay of hidden mathematical structures, like the dance

of shadows in a moonlit garden; these shadows can both fascinate

and scare an imaginative child. As a child, I myself was puzzled by

expressions like 5/ 4; but it appe ars that my worries were resolved

by pedagogical guidance: I was taught to think about fractions as

named numbe rs of a special kind: quarter apples. Fractions like

5/4 are n ot the result of dividing 5 apples between 4 people, since

this ope r ation of division is not yet deﬁned; they come from mak-

ing a sufﬁcient number of material objects of a new kind, “quarter

apples” and then counting ﬁve “quarter apples”.

AG was less fortunate:

The word for fraction in Romania n is fractie, and th at was the ter-

minology used by my teacher an d i n textbooks. The word is used

frequently i n common language to express a part of something big-

ger, like fraction is used in En glish (I think . . . ), so I think that it’s

very likely that my difﬁculty could be of a linguistic nature, I can’t

eliminate also the fact that actually that was how fractions were

introduced to us—pupils (like parts of an object) and only la ter the

notion was extended, and so maybe I had problems accommodat-

ing to the new n otion. Or maybe both reasons . . .

I was luckier: in Russian the word for “fraction” is re se r ved

for arithmetic and is n ot used in e veryday language

. Also, I was

clearly told to think about simple fractions 1/n as “named num-

bers” of a special kind. In effect, when dealing with quarter apples,

we are working in the additive semigroup

N generated by

What happens next is mu ch more interesting and sophisticated:

we have to learn how to add half apples with quarter apples. This

is done, of course, by dividing each half in two quarters, which

amounts to constructing a homomorphism

N −→

Unless you are a hunter and use the same word “drobь” for lead shot

pellets.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

8.2 Inductive l imit 87

Since both

N and

N are canonically isomorphic to N, we, being

adults now, can make a shortcut in notation and write this homo-

morhism simply as

N −→ N, z 7→ 2 × z.

In effect, we have a direct system

k×

−→ N, k = 2, 3, 4, . . .

—or, if you prefer less abstract notation—

−→

Then w e do somethin g outrageous: we take the inductive limit of

this direct system. In primary school, of co urse, taking the inductive

limit is called bringing fractions to a common denominator.

Fig. 8.1. Fraction-of-an-inch Adding Machine, 1952, Patent no. 169941.

Photo by Windell H. Oskay. Source: http://www.evilmadscientist.

com/article.php/inchadder.

 2009 Evil Mad Scientist Laboratories,

used with permission.

8.2 Inductive limit

Here are the f ormal deﬁnitions o f a direct set, direct system, direct

(inductive) limit taken from Wikipedia.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

88 8 Fractions

A directed set is a nonempty set A together with a reﬂexive and

transitive binary relation 6 (that is, a preorder), with the addi-

tional property that every pair of elements has an upper bound.

Notice that the directed set in our deﬁnition is the set N or-

dered by the divisibility relation. It is not linearly ordered and has

a pretty sophisticated structure by itself!

Let (I, 6) be a directed set. Let {A

| i ∈ I} b e a family of ob jects

indexed by I a nd suppose we have a family of homomorphisms

: A

→ A

for all i 6 j with the following properties: f

the identity in A

, f

= f

◦ f

for all i 6 j 6 k. Then the pair

({A

}, {f

}) is called a d irect system over I.

In our case, the directed set is the set N (or rather the set

{1/n | n ∈ N}) ordered by divisibility; the direct system is formed

by semigroups

N for k ∈ N with natural embeddings

N →

N if n divides m.

The upper bound of n and m (or of

N and

N) is mn (or

N).

Of course, taking the upper bound in the directed set



| n ∈ N



is nothing more but brin ging the fractions to a common d enomina-

tor. It is not a simple and straightforward operation. In words of

SJS

My main problem while quite young was adding fractions. I was

quite happy mul tiplying them, since you simply multipli ed the two

top numbers to get the answer top number, then multiplied the

two bottom numbers to get the answer bottom number.

But with adding you had to ﬁnd a common denominator b y

cross-multiplying and I remember it took me years to crack this

technique!!!

But let us return to construction of an inductive limit.

The underlying set of the direct (inductive) limit, A, of the d irect

system ({A

}, {f

}) is deﬁned as the disjoint union of the A

’s

modulo a certain equivalence relation ∼:

A =

/ ∼

Here, if x

is in A

and x

is in A

we set x

∼ x

if there is some k

in I such that f

) = f

The sum of x

∈ A

and x

∈ A

is deﬁned as the equivalence

class of f

) + f

) for some k s uch that i 6 k and j 6 k.

SJS is male, British, a professor of computational mathematics.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

8.2 Inductive l imit 89

The inductive limit of the direct system

N →

N if n divides m.

is, of course, the additive semigroup of positive rational numbers

. Fo r a child, this may look like a magic, as a story from Tony D.

Gilbert

conﬁrms:

I greatly l iked arithmetic at primary school. In my 3rd year (age

9/10, now more than 50 years ago), I had an inspirational teacher

(across all subjects) by the name of Miss Jesse (I think that’s the

correct spelling, b ut her name was pronounces ‘Jessie’). Among

other things she taught us fractions (it seems amazing now that

she taught us this, very successfully as far as I was concerned,

whereas I regularly see ﬁrst year stu d ents of Science who are quite

unable to cope with these). My recollection is that she used di-

viding up a cake as her model for fractions (just as I have done

with many a student since!). Having established fractions in low-

est terms, she then went on to deal with canceling down, multi-

plication and division, and also addition and subtraction of frac-

tions with the same denominator. . . but addition and subtrac-

tion of fractions with distinct denominators were a mystery that

awaited us. Then we ha d a detour into highest common factors

(HCF) and lowest common multiples (LCM). Finally, she at last

explained how if you wanted, say, to do the addition 1/2 + 1/3, the

trick was to put each fraction over the LCM of the two denomi-

nators and then like magic, the problem had been reduced to an

earlier one, the problem of addition or subtraction of fractions with

the same denominator.

The point of this story was my reaction to that ﬁnal explana-

tion. I can still remember my puzzlement before the explanation

as to how to one would deal with the problem of distinct denomi-

nators, and my really wanting to have the problem resolved. Then

once we had patiently (and in my case enjoyably) gone through the

detour of HCFs and LCMs, my pleasure and appreciation of the

cleverness (or so it seemed to me) of the resolution of the problem,

once the explanation was given.

Similarly, one may try to deﬁne multiplication on Q

using the

same inductive limit; I leave this construction of multiplication as

an exe r cise for the reader.

The deﬁnition of div ision o n Q

is much helped by the f act that

endomorphisms of the additive sem igroup Q

are invertible and

form a group ( the multiplicative group of positive rational numbers

) which acts on Q

simply transitively: for every non -zero r, s ∈

there exists a unique q ∈ Q

such that s = q · r.

ADG is male, British, has PhD in mathematics, teaches in a Scottish

university.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

90 8 Fractions

And here is a story from BC

which illustrates this point:

I could not u nderstand the “invert and multi ply” rule for dividing

fractions. I could obey the rule, but why was multiplying by 4/3

the same as dividing by 3/4?

My teachers could not explain, but I was used to that. I couldn’t

work it out for myself either, which was less usual.

Finally I asked my father, who was an accountant. He said: if

you divide everything into halves, you have twice as many things.

Suddenly not just fracti ons but the whole of algebra made sense

for the ﬁrst time.

8.3 Field of fractions of an integral domain

There is, o f c ourse, a can onical c onstruction of the ﬁ eld of fractions

of a commutative integ r al domain; it works equally well for the

integers Z and for polyno mials R[x]. Some my correspondents found

this, at a ﬁrst glance more formal and purely sym bolic approach to

fractions and rational functions easier to g r asp.

Listen to a testimony from RW

I recall very few difﬁculties with mathematics in my formative

years: on the whole it was a case of learning a simple rule and

doing exactl y what you were told. I d o recall being introduced to

fractions at age about 7, and wanting to know how to add them

together. The teacher refused to tel l me, sayin g it was difﬁcult and

we’d come to that later. I think I man aged to get the formula

ad + bc

from an older pupil at the school (it was a very small school with

several classes of different age-groups being taught in the same

room, s o this was not difﬁcult), and I remember being totall y mys-

tiﬁed by it. At least I think it was an algebraic formula, but this

might be a false memory implanted by my later understanding:

it might really have b een an algorithm, or even an example. Pre-

sumably my difﬁculties went away once I was taught it properly.

Victor Maltcev

preferr ed a p urely axiomatic approach:

When I was 8 we learn ed at school (2nd grad e) rational numbers.

I had very big problems in gra sping them th e whole 2nd and 3rd.

Eventually I started feeling I will never get along with them and

every time before classes in Maths I felt depressed that again we

BC was 10 years old. He is male, a non-E nglish Western European.

RW is male, British, a professor of mathematics.

VM is male, Ukranian, a PhD student in a British university.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

8.4 Back to commutativity of multipl ication 91

will do something with rational numbers and I still do n ot under-

stand what are th ey. When I was 10, 5th grade (omitting the 4th),

in one of the classes I realiz ed that in order to understand ratio-

nal numbers we sh ould not understand them!—just impose th e

axioms on symbolic [expressions ] “a over b” and deal with them.

This gave really a big push to studyi ng Maths after school clas ses.

Alexey Muranov:

I also remember that probably in the 5th grade I added up fra c-

tions with different d enominators without being taught or asked

:).

Ah, here is something more interesting: to understand that th e

quantity expressed as the fraction m/n (m pieces each of which is

obtained by dividing 1 into n equal pa rts) is the same as m divided

by n (the result of taking m whole pieces and dividing this into n

equal parts) was not at all obvious, I have dif ﬁculty to explain this

even now.

8.4 B ack to commutativity of multiplication

For the construction of the ﬁeld of fractions of an integ r al dom ain

to be straightforward, we need commutativity of multiplication in

the dom ain. And this is also tricky.

Bernhard Baumgartner

Learning multiplication of entire numbers, like 3 ×4, we were con-

fronted with pictures showing three lines with f our objects and

four lines with three objects. This should demonstrate the commu-

tativity. I had a vague feeling, that this demonstration is not really

a proof, but did not dare to say aloud such a nasty thinking to the

teacher.

Today I would subsume this epistemological problem und er

The Unreasonable Effectiveness of Mathematics . . . (Eugene P.

Wigner)

MW:

I think the last time we spoke, I was employed at XX College in

YY but I now ﬁnd myself back in ZZ teaching mathematics and

statistics to Higher Education students at ZZ College. I’d like to

share an experience I had only last week with one of my private

tuition students rather than my day-to-day students:

My student J is female a nd 27. J is an intriguing ca se as she

has a severe aversion to all things mathematical and, in p articular,

division. In fact, from what I have seen, it is beyond me as to how

BB is male, Austrian, a professor of mathematical physics.

See [ 159]—AVB.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

92 8 Fractions

she managed to be accepted on a Higher Education course in the

ﬁrst place!

Here is how part of our conversation went:

MW: “What is 4 times 5?”

J: (slight pause)“Twenty”.

MW: “Correct. Ok, how many 5s are there in 20?”

J: (longer pause) “Four”

MW: "Correct. Ok, how many 4s are there in 20?”

J: (very long pause, counting on ﬁngers) “Five”.

I found this fascinating. The i dea that J was unable to make the

connection between 4, 5 and 20 and how the fact that 4 × 5 equals

20 determines the answer to the quotients 20/5 and 20/4. I decided

to try again. . .

MW: “What is 2 times 9?”

J: (slight pause)“Eighteen”.

MW: “Ok, how many 9s are there in 18?"

J: (longer pause)“Two”.

MW: “Ok, how many 2s are there in 18?”

J: (very long pause, counting on ﬁngers) “Not sure. . . Eigh t?”

MW: “Not quite”.

J: “Nine?"

MW: “Correct".

Yet again, the multiplication was no real p roblem, the ﬁrst d i-

vision only a minor problem, but the second division caused major

problems.

MW: “What is 4 × 11?”

J: (slight pause)“Forty-four”.

MW: “Ok, how many 11s are there in 44?"

J: (longer pause)“Four”.

MW: “Ok, how many 4s are there in 44?”

J: (very long pause, counting on ﬁngers) “Not sure. . . ”

We did make progress in the end but I must ad mit to being

shocked at this lack of understanding from someone of my own

age!

By contrast, a story f rom Jonathan Kirby :

Here is my favourite [true!] story of how I ca me to love mathemat-

ics. When I was 6, every day I had to do a few homework problems

like: 2 + 7 = . . . ; 4 × 5 = . . .

One day I was sitting at home doing these, and couldn’t re-

member the answer to 5 × 3, so I asked my mother:

J. What is ﬁve times three?

M. 15.

J. No, that’s three times ﬁve.

M. It’s the same. It d oesn ’t matter which way round

they are.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

8.4 Back to commutativity of multipl ication 93

J. Really? Are you sure?

To me, ﬁve times three meant adding ﬁve copies of three, which

was clearly different from adding three copies of ﬁve. I was puz-

zling over why they should be the same for the next few days, then

suddenly I realised that I could see why they were the same. On

squared paper, you could colour in ﬁve rows of three on top of each

other, and the total number of coloured squares was the product,

15. But then you could rotate the paper and have three rows of

ﬁve, and the sa me number of coloured squares. I have loved math-

ematics [almost] ever since.

And Alexey Muranov reminds us about the role of v isualization:

I also vaguely remember that some of the algebraic laws studied

in the second grade, such a s associativity and distributivity, were

posing problems. I do not remember now which exactly, but it could

have been the associative laws for multiplication. I am not sure,

but this could be because th e other laws could be explained on 1-

or 2-d imensional pictures, and in class we only used 2-dimensional

pictures, but the best way to explain the as sociativity for mul-

tiplicati on would be to arra nge objects in a 3-dimensional par-

allelepiped. I did not think in terms of pictures, I somehow was

able to imagine those laws, but probab ly I used geometry subcon-

sciously.

Exercises

Exercise 8.1 Deﬁne multiplication on the inductive limit of the system of

additive semigroups

k×

−→ N, k = 2, 3, 4, . . .

Exercise 8.2 Prove th a t the endomorphism ring of the additive group of

rational numbers is a ﬁeld.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK