*By William McCallum*

The number line is a seemingly simple object: a straight line with two points marked 0 and 1. Those two points are the seeds of great complexity, however. Whole numbers are located at positions marked off by iterating the interval. Fractions are located at equal subdivisions of the spaces between whole numbers. Flip all those numbers to the other side of 0 and you have negative rational numbers. Then, although the line is completely dense with rational numbers, you find you can sneak between them with infinite decimal expansions to define a whole universe of irrational numbers. Given all of these layers of complexity, when exactly is the right moment to introduce this marvelous object to students?

In IM K–5, we wait until grade 2. One reason is that it’s important for children who are first learning to count to see situations where the things they are counting are arranged in different ways. We want them to learn that the number they arrive at when they finish counting, say, 5 teddy bears, tells them how many teddy bears there are, no matter how they are arranged. This may seem obvious to adults, but adults are not born with this understanding. It is something they learned in early childhood. Researchers have observed the learning: children who have not yet acquired it, when asked “how many teddy bears are there?” after finishing the count, will repeat the count. Later on, they will gesture to the collection of teddy bears and say “5 teddy bears!”. For this concept to take hold it is important that the teddy bears not always be arranged in a line, and to privilege objects over representations at this stage.

Another reason is that the number line is not a very good representation for children who are working hard on the concept that numbers are for counting things. Indeed, it can be positively confusing. The National Research Council report Mathematics Learning in Early Childhood says that “young children have difficulties with the number line representation because they have difficulty seeing the units—they need to see things, so they focus on the numbers instead of on the lengths. So they may count the starting point 0 and then be off by one, or they focus on the spaces and are confused by the location of the numbers at the end of the spaces.”

Later, as children start learning addition, there is a progression of strategies. They might initially calculate 37 + 5 by counting on from 37, a strategy that builds naturally on their conception of numbers as being for counting things. By the time they get to adding 37 + 25, we want them to start thinking in terms of place value. They might adapt their counting on strategy to counting on by tens as well as by ones. For example, they might calculate 37 + 25 by counting on with the 2 tens first and then the 5 ones: 37, 47, 57, 58, 59, 60, 61, 62. They could write this out as

37 + 10 = 47

47 + 10 = 57

57 + 1 + 1 + 1 + 1 + 1 = 62

They could also record this thinking on a line with jumps of ten followed by jumps of one. This line is sometimes called a mental number line or an open number line, and can be useful in early number work. But there is an important way in which the full meaning of the number line is not being used here, namely the aspect of measurement. It doesn’t particularly matter for the purposes of recording the thinking that the jumps accurately represent the actual distances on the number line, and they often don’t in students’ drawings. By the time students get to representing fractions on the number line, however, the measurement aspect is crucial. It is important that when you mark a number line in thirds you understand that the subdivisions are of equal length, and that the number represented by a point on the number line is a length, the length of the interval from 0 to that point.

Although you can see place value on the number line, it doesn’t really shine out. It is true that you can distinguish tens and ones on the number line with major and minor tick marks. However, to go beyond that to hundreds and thousands you need to zoom out, losing the ones and tens. And the bundling of one unit into a higher unit is not transparent on the number line. It is this bundling that we ultimately want students to use in their thinking about 37 + 25 = 62. We want them to see the 37 as 3 tens and 7 ones, and the 25 as 2 tens and 5 ones, and then combine the tens with the tens and the ones with the ones, writing something like

37 + 25 = 30 + 7 + 20 + 5

30 + 7 + 20 + 5 = 30 + 20 + 7 + 5

30 + 20 + 7 + 5 = 50 + 10 + 2 = 62.

The number line is not well suited to the shifting around of tens and ones required for this thinking.

The distinction between open number lines and true number lines here is like the distinction I made in my previous blog post between tables as methods of recording calculations and ratio tables. In each case the two things look more or less the same, but the way they are being used and their implicit properties are very different. Drawing open number lines is a transitional method of recording thinking, to be discarded once students acquire more efficient methods. The true number line is a conceptual tool that unifies counting numbers and fractions, to which students later add negative and irrational numbers, all elements of one coherent number system. It is a representation that stands students in good stead for the rest of their lives.

##### William McCallum

Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.

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