Fractions and semiotics.

Fractions haunt elementary school teachers and occupy a disproportionate (1/2? 2/3? 15/17?) amount of time in math teacher professional development. Many college students arrive with a distaste for fractions; they are generally much happier talking about 3.4 rather than 17/5. There are a few likely reasons: it is much easier to work with decimals on a handheld calculator, and numbers are easier to compare as decimals rather than fractions.

But I like fractions, especially comparing them, and teaching fractions raises some important issues at the heart of mathematics. My two favorite fraction topics are the mediant fraction and kissing fractions. Both topics can be found in L. R. Ford’s article “Fractions” from the American Mathematical Monthly, vol. 45, no. 9 (Nov. 1938), pp. 586–601.

The mediant fraction is what happens every day when a child adds fractions the wrong way. The mediant of 1/3 and 1/2 is 2/5. Add the numerator, add the denominator — it’s a natural thing to try, but it doesn’t give you the sum. It gives you the mediant.

I wish more teachers would tell these children something like “What you’ve done isn’t adding the fractions, it’s something much more interesting. You’ve made the mediant fraction!”

What does this operation do? If you start with fractions with positive denominator, the output fraction is between the fractions you start with. But strangely, to a novice, the operation is really an operation on fractions, not on rational numbers. For example, the mediant of 2/6 and 1/2 is 3/8. But hang on a second — the mediant of 1/3 and 1/2 is 2/5, and 1/3 = 2/6, so how can this be?

This is one sign that the mediant isn’t an operation on rational numbers. It makes sense for fractions, but it depends on how the fraction is written. Further explanation is in the two-page spread below.

Screenshot from Chapter 4 of the Illustrated Theory of Numbers
Click to enlarge a two-page spread from Chapter 4

I think teachers and students have trouble with this, and the obstruction is a poor understanding of semiotics. In other words, teachers and students do not adequately distinguish between numbers and how numbers are written. I made a common decision to use the word “fraction” to mean a written expression like “a/b”, and “rational number” to mean a number which can be described by a fraction. So “1/3” and “2/6” are different fractions, which mean the same rational number.

Why is this so important (besides the fact that the mediant is an operation on fractions)? I think this issue arises again and again. Eventually, I would guess most students think that numbers are decimal expansions. The number \pi is the decimal expansion 3.14... for most students. But this too is incorrect, leading to confusion when students or teachers hear that 0.999\ldots equals 1 for the first time. Real numbers are not decimal expansions, nor are rational numbers fractions.

What about kissing fractions? Given integers a, b, c, d, I say that a/b kisses c/d if ad - bc = 1 or ad - bc = -1. I call this kissing, because two fractions kiss when their Ford circles kiss; kissing circles have been around for a long time, sometimes called tangent (touching) circles or osculating circles. Kissing fractions are the algebraic counterpart of kissing circles. Kissing fractions leads to more kissing, love triangles, and more. You’ll have to read Chapter 4 to find out. In case you think it is all a love-fest, Chapter 4 also contains material on the construction of the 17-gon and Dirichlet’s approximation theorem.

The mediant fraction and kissing fractions forced me to make a few notational decisions. Since the mediant fraction is not the sum, nor even really related to the sum, how should one write it? I use the “vee” symbol, as in

\frac{1}{2} \vee \frac{2}{3} = \frac{3}{5}.

I think the “vee” symbol is good since it is symmetric (like the mediant operation) and it points to where the mediant ends up: right between the fractions you start with (if the denominator is positive).

For kissing, I couldn’t resist using the heart symbol, as in

\frac{1}{2} \heartsuit \frac{2}{3}. since (1)(3) - (2)(2) = -1

Again the heart symbol is symmetric, and kissing is a symmetric relation. I hope I’m not getting carried away.

Now here is an activity for teachers who have made it all the way to the bottom of this blog post; an audience of 20-50 students is ideal. Write fractions on index cards; include all the reduced fractions between 0/1 and 1/1, with denominators up to 7 or 8 or 9. Your cards should include fractions like 2/5 and 3/7 and 1/4 and the like, one fraction per card.

Begin the activity by passing out the cards, and asking everyone with a card to stand up and line up in order of their (now personal) fractions. Quickly 0/1 and 1/1 will stand at opposite sides of the room. The activity usually takes a bit of time, as 2/5 and 3/7 have difficulty sizing each other up.

Next is a good opportunity to share how students compare fractions (common denominators, some weird butterfly technique, etc.).

Finally, have the students sit down, except for 0/1 and 1/1, who should stand at the corners of the front of the room. Ask them to add themselves, the wrong way (yielding 1/2), and ask 1/2 to stand between them. Then ask 0/1 and 1/2 to add themselves the wrong way, and ask the result (1/3) to stand between them. The students can take it from there… and they will remember the mediant fraction.

May the readers always understand signs, what they signify, and the difference between the two