Number lines

[Duane]

Number lines appear in a few places in the Standards and the MD Measurement Progressions discusses their importance at the bottom of page 14, relating to 2.MD.6. The general impression I get is that the number line is seen as a very strict tool, exactly like a ruler: the intervals between consecutive whole numbers must be exactly the same for the entire length of the number line. As the Progressions note, it is a measurement model.

Another way of using number lines is as “open” or “empty”, such as are used most notably in the Netherlands, but this idea also appears in a variety of textbooks. In this case, so long as the relative order of numbers is correct (e.g. 12 is before 16 but after 3) the absolute distance between numbers is irrelevant. There’s a lot of benefit in using number lines in this way for addition and subtraction strategies, as it cuts down on a lot of unnecessary detail.

These two models of number lines seem incompatible and it would seem that the measurement model is clearly intended to be used with the Standards. Is this correct, or is their an intention (or at least scope) for open/empty number lines to be used too? I’m guessing it depends on the particular purpose but I could see that a mix of the two could cause confusion.

[Bill McCallum]

In the examples I have seen of empty number lines, distance still matters. The number line starts out empty, and then a student who is, for example, subtracting 37 from 52 might put down a 37 and a 52, then indicate a jump from 37 to 40, a jump from 40 to 50, and a jump from 50 to 52, and add up all the jumps. They don’t have to get the placement exactly right, but there is still the idea that there is a precise placement. This is not so different from the number line with a 0 and a 1 marked, where a greater or lesser degree of precision will be required, depending on the task at hand. For example, a student might argue that 3/4 > 2/3 by placing each on the number line and recognizing that 3/4 is closer to 1 than 2/3. There are various ways a student could indicate the argument without being exactly right in placing the numbers.

[Duane]

I’d probably see it as more important that the physical distances are exact when reasoning about the closeness of fractions to 1. To my mind, estimating their position implies that students already know which fraction is closer as there’s not much physical difference between 2/3 and 3/4 – an estimate on a number line is not that informative.

However, the subtraction example Bill gave is just what I had in mind for when correct physical distance on the number line is unimportant. 2.MD.6 seems to really emphasize the “equally spaced points” concept of number lines which is not necessary for addition and subtraction – the numerical, not physical, size of the jumps should be the focus. Perhaps page 14 of the relevant Progressions doc could be revisited if this is idea is important.

[Cathy Kessel]

I think that part of the issue is that students need to build their understanding of number line before they get to fractions in later grades. They may be able to function without it earlier if they don’t count 0 as discussed below.

Here are some relevant quotes about from the National Research Council report Mathematics Learning in Early Childhood. This has some discussion of the difference between a “number path” and a “number line” (which CCSS calls a “number line diagram”). This can be read online here: http://www.nap.edu/openbook.php?record_id=12519&page=167

From page 167:

children in kindergarten and Grade 1 are using the number word list (sequence) as a count model: Each number word is taken as a unit to be counted, matched, added, or subtracted. In contrast, a number line is a length model, like a ruler or a bar graph, in which numbers are represented by the length from zero along a line segmented into equal lengths. 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.

From page 168:

The number line is particularly important when one wants to show parts of one whole, such as one-half. In early childhood materials, the term number line or mental number line often really means a number path, such as in the common early childhood games in which numbers are put on squares and children move along a numbered path. Such number paths are count models—each square is an object that can be counted—so these are appropriate for children from age 2 through Grade 1.

• This reply was modified 12 years ago by Cathy Kessel.
• This reply was modified 12 years ago by Cathy Kessel.

[bcohen]

Duane,

Because 2.MD.6 requires “…equally spaced points corresponding to the numbers 0, 1, 2, …” on a number line diagram so that it accurately represents “whole numbers as lengths from 0” we are using ‘closed’ number lines (with all numbers shown) to address this in grade 2.

In grade 3, we begin using open number lines in ways like Bill’s example above.  We also use the strategy to solve problems involving operations with non-base-10 numbers (ex., elapsed time) because it side-steps issues of regrouping non-base-1o numbers.  When we use open number lines (beginning in grade 3), we like the jumps to be ‘relatively’ proportional (i.e., a jump of 100 should take up more space than a jump of 10), but we do not focus on making the jumps exactly proportional.  One of the benefits of the open number line, in my opinion, is that it relies on and fosters number sense and its appropriate use in computation.  While asking students to make the length of jumps ‘relatively proportional’ seems like it supports this sort of number sense, focusing on whether the length of each jump is represented precisely to scale seems like it would undermine the same goal.

For what it’s worth,

Brian

[Author]

Posts