Plenty — That ’ s Why Elementary Schools Need Math Teachers

ion. When asked, what is a fraction?, we say it is just something concrete, like a slice of pizza. And when this doesn’t work, we continue to skirt the question by offering more metaphors and more analogies: What about a fraction as “part of a whole”? As another way to write division problems? As an “expression” of the form m/n for whole numbers m and n (n > 0)? As another way to write ratios? These analogies and metaphors simply don’t cut it. Fractions have to be numbers because we will add, subtract, multiply, and divide them. What does work well for showing students what fractions really are? The number line. In the same way that fingers serve as a natural reference point for whole numbers, the number line serves as a natural reference point for fractions. The use of the number line has the immediate advantage of conferring coherence on the study of numbers in school mathematics: a number is now defined unambiguously to be a point on the number line. In particular, regardless of whether a number is a whole number, a fraction, a rational number, or an irrational number, it takes up its natural place on this line. (For the definition of fractions, including how to find them on the number line, see the sidebar on page 12.) Now, let’s describe the collection of numbers called fractions. Divide a line segment from 0 to 1 into, let’s say, 3 segments of equal length; do the same to all the segments between any two consecutive whole numbers. These division points together with the whole numbers then form a sequence of equal-spaced points. These are the fractions with denominators equal to 3: the first division point to the right of 0 is what is called 1⁄3, and the succeeding points of the sequence are then 2⁄3, 3⁄3, 4⁄3, etc. The same is true for 1⁄n, 2⁄n, 3⁄n, etc., for any nonzero whole number n. Thus, whole numbers clearly fall within the collection of numbers called fractions. If we reflect the fractions to the left of 0 on the number line, the mirror image of the fraction m/n is by definition the negative fraction – m/n. Therefore, positive and negative fractions are now just points on the number line. Most students would find marking off a point 1⁄2 of a unit to the left of 0 to be much less confusing than contemplating a negative 1⁄2 piece of pie. The number line is especially helpful in teaching students about the theorem on equivalent fractions, the single most important fact in the subject. To state it formally, for all whole numbers k, m, and n (where k ≠ 0 and n ≠ 0), m/n = m/kn. In other words, m/n and m/kn represent the same point on the number line. Let us consider an example to get a better idea: suppose m = 4, n = 3, and k = 5. Then the theorem asserts that 4 = 5 × 4 3 5 × 3 and, of course, 5 × 4 = 20 . 5 × 3 15 The number line makes the equality clear. To see how 4⁄3 equals 20⁄15, draw a number line and divide the space between 0 and 1, as well as between 1 and 2, into three equal parts. Count up to the 4th point on the sequence of thirds—that’s 4⁄3. Then take each of the thirds and divide them into 5 equal parts (an easy way to make 15ths). Count up until you get to the 20th point on the sequence of 15ths—that’s 20⁄15, and it’s in the same spot as 4⁄3. 0 1 | | | | | | | | | | | | | | | | | | | | | 1⁄3 2⁄3 3⁄3 4⁄3 5⁄15 10⁄15 15⁄15 20⁄15 The use of the number line has another advantage. Having whole numbers displayed as part of fractions allows us to see more clearly that the arithmetic of fractions is entirely analogous to the arithmetic of whole numbers. For example, in terms of the number line, 4 + 6 is just the total length of the concatenation (i.e., linking) of a segment of length 4 and a segment of length 6. | | | 4 6 Then in the same way, we define 1⁄6 + 1⁄4 to be the total length of the concatenation of a segment of length 1⁄6 and a segment of length 1⁄4 (not shown in proportion with respect to the preceding number line). | | | 1∕6 1∕4 We arrive at 1⁄6 + 1⁄4 = 10⁄24 as we would if we were adding whole numbers, as follows. Using the theorem on equivalent fractions, we can express 1⁄6 and 1⁄4 as fractions with the same denominator: 1⁄6 = 4⁄24 and 1⁄4 = 6⁄24. The segment of length 1⁄6 is therefore the concatenation of 4 segments each of length 1⁄24, and the segment of length 1⁄4 is the concatenation of 6 segments each of length 1⁄24. The preceding concatenated segment is therefore the concatenation of (4 + 6) segments each of length 1⁄24, i.e., 10⁄24.** In this way, Because fractions are students’ first serious excursion into abstraction, understanding fractions is the most critical step in preparing for algebra. *Very large numbers are already an abstraction to children, but children tend not to be systematically exposed to such numbers the way they are to fractions. Rational numbers consist of fractions and negative fractions, which of course include whole numbers. See, for example, page 4-40 of the National Mathematics Advisory Panel’s “Report of the Task Group on Learning Processes,” www.ed.gov/about/bdscomm/list/math panel/report/learning-processes.pdf. We exclude complex numbers from this discussion, as they are not appropriate for the elementary grades. **Naturally, the theorem on equivalent fractions implies that 10/24 = 5/12, as 10/24 = (2 × 5)/(2 × 12), but contrary to common belief, the simplification is of no great importance. Notice in particular that there was never any mention of the “least common denominator.” (Continued on page 10)