Talking about fractions, decimals, and numbers

When students first learn about fractions, we want them to learn that they are just numbers; new numbers, but numbers nonetheless, that fit into the same system as the whole numbers they are familiar with. The number line can help with this, with whole numbers and fractions sitting together, and located in essentially the same way; choose a unit (1, 1/3, 1/10) and then count off a number of those units. It also helps students understand that equivalent fractions are just different ways of writing the same number. When (finite) decimals come along, they get added to the list of representations.

The Common Core emphasizes this unity by treating decimals as just a different way of writing fractions, e.g. in 4.NF.C: “Understand decimal notation for fractions, and compare decimal fractions.” In this view, 0.3 is not a new sort of number, just a different way of writing the number 3/10.

This leads to some difficulties in the use of language, because at some points in the curriculum you do want to distinguish between decimals and fractions, for example when you ask a student to write 4/5 as a decimal or to write 0.125 as a fraction. (“You told me it’s already a fraction!” the smart student might reply.)

The IM curriculum writing team was talking about these difficulties the other day and Cathy Kessel had a useful comment:

There’s a developmental issue. When fractions are introduced, the distinction between number represented and representation is blurred, and similarly for decimals (finite, then repeating). But, when the two types of representations are seen as representing the same thing, then the thing and its representations start to separate more.

Because we want students to develop a conception of the number behind the representation, we start out saying decimals are also fractions. Later we build a negative addition to the number line and add the opposites of fractions. Once we have a robust conception of the number line, inhabited by rational numbers, we want to talk about different ways of expressing those numbers: fractions, decimals, infinite decimals, expressions involving square root symbols and exponents. So we start to distinguish between fractions and decimals, not as numbers, but as forms for expressing numbers. We initially suppress their role as forms in order to gain a robust conception of number; once they are firmly attached to that conception we can distinguish between them.

They only way to do this without giving multiple meanings to the same words would be to invent new words and be consistent in their use. This harks back to the distinction between “numeral” and “number” in the New Math, which didn’t take hold.

3 thoughts on “Talking about fractions, decimals, and numbers

  1. Recently I posed the following problems to my daughter:

    1/8 = 0.1 + ?
    1/7 = 0.14 + ?
    1/3 = 0.3 + ?

    At her stage of learning, these problems are thought-provoking; they are not mechanical exercises. The problems led to some good conversations. In particular, on the topic of your post today, it was clear from talking to my daughter that she has a sturdy conception of an “amount,” independent of the form in which that amount is expressed. (Her conception of ‘amount’ seems to be something like what Pat Thompson calls magnitude.)

    The presence of addition/subtraction in the above problems also reminds me of something that I once read in a paper by (I think) Guershon Harel. The idea as I recall was along the following lines. One way in which new kinds of numbers get integrated into a student’s evolving conception of number is when the student calculates with the new kinds of numbers. In the case of fractions, for example, calculating with fractions tends to ‘make fractions numbers’ simply because, to some extent, ‘numbers’ just means ‘the things one does mathematics with.’ (This was the idea behind the task “Are Fractions Numbers?”

    • Hi Jason,

      In our high school program, we have a motto “different forms for different purposes.” This certainly applies to algebraic expressions, but it also applies to rational numbers. Decimal notation is useful when you are concerned with \emph{value}—it helps you compare the size of a number or to determine how close two numbers are on a number line. Fractions are less good for this. But fraction representation is very useful when you care about the \emph{form} of a number, for example, when you want to investigate patterns in certain calculations. In a way, the distinction is between the analytic and algebraic faces of rational numbers. The representations come together, of course—the key to the length of the period for the decimal expansion of a rational number lies in its representation as a fraction.

      Al (I have no idea why this lists me as “Budapest Education”)

      • Oh, I’ve been wondering who Budapest Education was, thanks for identifying yourself, Al! I will see if I can fix this.

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