Posted by: Alexandre Borovik | August 18, 2008


In response to my call for personal stories about difficulties in studying (early) mathematics AG sent me the following e-mail:

When I was about 9 years old, I’ve first learned at school about fractions, and understood them quite well, but I had difficulties in understanding the concept of fractions that were bigger than 1, because you see we were thought that fractions are part of something, so I could understand the concept of , for example 1/3 (you a take a piece of something you divided in 3 equal pieces and you take one), but I couldn’t understand the what meant 4/3 (how can you take 4 pieces when there are only 3? ). 🙂 Of course I get it in several days, but I remember that I was baffled at first.

I am a boy, the language of my mathematical instruction is Romanian, which is also my mother tongue. Currently I am a student at Computer Science.

I am surprised to see how frequently such memories are related to subtle play of hidden mathematical structures, like 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 appears that my worries were resolved by pedagogical guidance: I was taught to think about fractions as named numbers of special kind: quarter apples. Fractions like 5/4 are not result of dividing 5 apples between 4 people, since this operation of division is not yet defined; they come from making sufficient number of material objects of new kind, “quarter apples” and then counting five “quarter apples”.

In effect, we are working in the additive group \frac{1}{4}\mathbb{Z} generated by \frac{1}{4}.

What happens next is much 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

\frac{1}{2}\mathbb{Z} \longrightarrow \frac{1}{4}\mathbb{Z}.

Since both \frac{1}{2}\mathbb{Z} and \frac{1}{4}\mathbb{Z} are canonically isomorphic to $\mathbb{Z}$, we, being adults now, can make a shortcut in notation and write this homomorhism simply as

\mathbb{Z} \longrightarrow \mathbb{Z}, \quad z \mapsto 2 \times z.

In effect, we have a direct system

\mathbb{Z} \stackrel{k\times}{\longrightarrow}  \mathbb{Z}, \quad k =2,3,4,\dots

— or, if you prefer less abstract notation —

\frac{1}{n}\mathbb{Z} \stackrel{{\rm Id}}{\longrightarrow} \frac{1}{kn} \mathbb{Z}.

Then we do something outrageous: we take its inductive limit. In the primary school, of course, taking the inductive limit is called bringing fractions to a common denominator.

The result, of course, is the additive group of rational numbers \mathbb{Q}. Only then we define multiplication on \mathbb{Q} — I leave a category theoretical construction of multiplication as an exercise to the reader.

I wish to reiterate my principal thesis:

Pedagogically motivated intermediate steps in introduction of mathematical concepts to children very frequently reflect the presence and behaviour of hidden underlying structures of mathematics.

Please, send me more stories like AG’s.

Please do not forget to mention the age when you had your particular experience, your gender, language of mathematical instruction (and whether it was your mother tongue), and the level of mathematical education you have eventually received.



  1. Typo: The map in the inductive system should be multiplication by 1. The displayed map is an isomorphism.

  2. Thanks — corrected.

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