# Using terminology for the different concepts of rational number

In elementary maths education literature, they distinguish multiple concepts that rational numbers are used to represent: fractions, quotients, ratios, rates, and possibly more. These words seem to be used to refer to the concept, rather than to the number itself per se. For example, when they say "fractions" they are referring to part-whole relationships, and the numbers themselves are rational numbers.

Most high-school and university teachers don't distinguish between all of these though. When we say "fractions", we normally mean simply numbers produced via division.

I myself am not completely sure on the definitions and usages of these words in an elementary education context. For example, I don't know if it would be considered bad to say "Find the rate of change of volume with time and express your answer as a fraction."

My question is twofold:

1. In the elementary maths education context, what are the definitions of these words, and how are they used?

2. Is it important or useful to distinguish these words in this way for maths learners, or are they simply words we use to describe the concepts to ourselves in papers etc? - for example, should the word fraction always have a part-whole concept attached to it even when we use it with learners?

You can find a discussion of these considerations in:

Chapin, S., & Johnson, A. (2006). Math matters: Grade K-8 understanding the math you teach. Sausalito: Math Solution Publications.

Specifically, pp. 99 - 131, Chapter 5, Fractions.

Let me just quote from Teaching Fractions on p. 131:

Fractional numbers are a rich part of mathematics. However, many students find them difficult to understand. To help students learn about and use fractions, it is important to introduce the multiple meanings of fraction and to emphasize sense making in all mathematical activities. Instruction in the early grades should focus on the part-whole interpretation of fraction but include all other interpretations as well.

To give you an idea of the multiple meanings, I will also quote from the start of the chapter on pp. 99-100:

Interestingly, fractions have multiple meanings and interpretations. Educators generally agree that there are five main interpretations: fractions as parts of wholes or parts of sets; fractions as the result of dividing two numbers; fractions as the ratio of two quantities; fractions as operators; and fractions as measures (Behr, Harel, Post, and Lesh 1992; Kieren 1988; Lamon 1999).

The book then goes through each of these five meanings.

Alternatively: The National Council of Teachers of Mathematics (NCTM) has released a relevant book:

Barnett-Clarke, C., Fisher, W., Marks, R., & Ross, S. (2010). Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5. Reston, VA: National Council of Teachers of Mathematics.

The entire book would be a good read if you are interested in these subjects. I will just quote the "Essential Understanding" definition given on p. 13:

The rational numbers are a set of numbers that include the whole numbers and integers as well as numbers that can be written as the quotient of two integers, $a \div b$, where $b$ is not zero.

But, of course, this single definition does not do the book justice, as it goes on to provide multiple interpretations and explore the ramifications of each.

1. In the context of elementary school mathematics, there are many definitions; see either of the sources provided above as well as the works they cite.

2. To re-quote from the first excerpt: "it is important to introduce the multiple meanings of fraction and to emphasize sense making in all mathematical activities."

I cannot imagine that there would ever be a sufficient consensus on the meanings of the various words. And even if many teachers agree on the "correct" meaning and usage of every fraction-related word, students of such teachers should be able to cope with teachers and students with different views — including the chance of no view at all. And there is, of course, the additional problem that the verbiage does not have exact counterparts in other languages. This is to say that if such distinction is to be made, I doubt it can be meaningfully made beyond very elementary math education. Everyone will eventually be exposed to sloppy language, and I see no reason why this should not start in school.

But there is a point to making some distinctions, though. For example, pupils should understand that cutting a pie in six equal pieces and eating two of them is a different action than cutting it in three equal pieces and eating one, although $\frac26=\frac13$. (I see this so that the fraction stands for a number, not the process behind it.) I would find it beneficial that pupils are given suitable words to express this, although I'm not sure if this is too confusing to too many. The gain from this distinction is not a huge one: after all, both action lead to the same amount of pie eaten, which is reflected by the equality of the two fractions. Instead of making the distinction by giving exact definitions and sticking to them, it might be better (also for the sake of developing one's skill in explaining and arguing) to discuss this issue in free words.