Rational Thinking Required

A large part of any Maths teacher’s job involves dealing with kids struggling with Maths. We often, as teachers, take things for granted when talking about familiar concepts with them and in trying to explain terms and definitions for familiar things. My biggest gripe with this comes down to fractions. We all know what a fraction is, right? However the language used in these everyday objects can cause a huge amount of confusion and frustration for students. We will start at the beginning.

We introduce the language of fractions from a young age. A fraction is a number that has a numerator and a denominator. At age 8 or 9 we begin the process of explaining fractions using the pizza analogy. Any search for fractions on the internet will throw up hundreds of images of Pies, Pizzas and such in various segments.

The idea is to familiarise students with the idea of a fraction being part of a whole, and to introduce similar fractions. Below is a perfect example of this.




These types of fractions, where we are dealing with part of a unit (a half a pizza) are called proper fractions.

We are then introduced to the idea of a mixed fraction, or a mixed number. These are numbers like 1 \frac{2}{3} . Here we have a whole number and and fraction slapped together to describe things like how much pizza each person would get if we were to equally divide 5 pizzas amongst 3 people.

We then bring the children along to another type of fraction, an improper fraction or top-heavy fraction. These are fractions where the numerator is greater than the denominator. For example, \frac{7}{3} is an improper fraction. Students are always encouraged to simplify improper fractions to mixed fractions. So, an answer of \frac{7}{3} would be marked incomplete and would need to be simplified to 2 \frac{1}{3} to get the required correct answer. How, in any sense of the word, can we call such a fraction improper?

They learn how to add and subtract and multiply and divide these “different” types of fractions together. By the time they get to second level these ideas are deeply ingrained. The language is fixed. We all use these terms in everyday life.

Teaching fractions in this manner is completely counter-productive for actual Mathematics. By the time students come to me they are confident in their understanding of these fairly simple rules. Then I have to teach them about formal number systems and sets and formalise all they have learned in a new framework, that of Algebra.

We introduce concepts like Natural numbers, which is fine, and Integers, which is fine, and then we get to The Rational Numbers\mathbb{Q}.

These are clearly defined as \mathbb{Q} =\lbrace \frac{a}{b} ; a, b \in \mathbb{Z} , b\neq 0 \rbrace but universally understood as “the set of all fractions” as this is the language students have been using for years at this stage. They feel on solid ground. Now the confusion starts.

First of all, there is no special distinction between \frac{7}{3} and \frac{3}{7}. They are both “proper” rational numbers. I explain that  \frac{7}{3} is simply the number I get when I divide 7 by 3. Similarly,  \frac{3}{7} is the number I get when I divide 3 by 7.

I explain the rules of rational numbers, how to add, subtract, multiply and divide them. “OK. So, rational numbers are fractions”, they think, and who could blame them? Then I introduce the set of Irrational Numbers like  \sqrt{2} and I say something like  “\sqrt{2} is irrational because it cannot be written as a rational number.” Everything is still ok. This translates to them as, “\sqrt{2} cannot be written as a fraction”.

Then, a few weeks later I am introducing Trigonometry and telling them and writing on the board  \cos{30} =\frac{\sqrt{3}}{2}, i.e. an irrational fraction!!! Now they are all confused and afraid to ask what is going on. How can we say that  \sqrt{3} is irrational and yet we can write it in a fraction? Then they question the whole thing. Everything gets confused. They know irrational as “not a fraction” so now they think “not a proper fraction” so now it’s an improper fraction to them and the whole idea of irrationality is lost on them. The language and methods we use to teach them are actually, literally, irrational.

Eventually they don’t know the difference between a fraction, a proper fraction, an improper fraction, a rational number, an irrational number, an irrational fraction, and we have a mess of confusion.

Then we try to do really counter-productive things like rationalising a surd. Here we say that if we see  \frac{1}{\sqrt{3}} we should rationalise this to  \frac{\sqrt{3}}{3}, an irrational fraction!!

The language is all over the place and counter to what Maths should be about. It is self-contradictory, nonsensical, and not at all formal . No wonder they hate me.

Then we start on algebraic fractions and continue on to rational functions.

We need to stop this madness and start again, in primary school, by doing away with all the rubbish that sticks and teaching them one way. Is there any need at all for the distinction between proper and improper fractions? It is counter-productive, however intuitive it may be. I would also propose that mixed fractions be an afterthought rather than the method of teaching. Could we not all agree that  \frac{7}{3} is simply the number we get when we divide 7 by 3. To get it’s numeric value, write it in decimal form. I know it would be a huge change but I cannot see the benefit of them.

To be honest, the language of mixed fractions is also problematic. Saying two-and-a-half gives the word “and” the operation of addition. Its use in this context permeates through our understanding. One and One is Two. Four and Seven is Eleven. Now the word “and” is intrinsically linked with addition, when in most formal maths the word “and” implies multiplication. The number 2.5 would be more apt and avoids imparting preconceptions on daily words like “and” that have very specific meanings in Maths.

When we see these problems from the struggling student’s perspective, we can better understand their confusion and feelings of frustration. The problem really isn’t them, it is us, the system that teaches them incoherently. In fact, the way we teach them fractions is irrational.

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