M2N1 - Number lines

M2N1. Students will use multiple representation of numbers to connect symbols to quantities.

When I visit primary grade classrooms, I often see a large number line posted above the whiteboard in the front of the room. Sometimes I also see number lines taped on students' desks. A variety of experts, including the National Math Panel, state that number lines are powerful representation tools and mathematics instruction should develop students' proficiency with number lines. Singapore elementary mathematics textbooks are famous, in part, because of their use of "tape diagrams" to help students deal with complicated mathematical problems. As I have discussed previously, double number lines can be powerful thinking tools to support students' comprehension of multiplication and division of rational numbers. So, on the surface, the display of number lines in the primary grades (K-2) seems to be a sound teaching practice. But, is it?

When you examine Japanese elementary mathematics textbooks, the formal term, "number line" does not appear until Grade 3. However, that does not mean number lines are not used in Grades 1 and 2 (there is no Kindergarten in Japanese elementary schools). As usual, Japanese textbooks carefully and gradually develop number line representations. Thus, students' first encounter with something like number line is simply placing number cards 1 through 10 in order going from left to right. They will be asked to fill in the missing number in a sequence like 3 - 4 - [ ], or 7 - [ ] - 9. A little later on, once the range of numbers has been extended up to 20, there is a question which asks how far a space alien character hopped along a number line, starting at 0. Missing number problems may also involve number cards sequenced in backward (from large to small). When students are studying numbers up to 100, students are asked to locate given numbers on a number line, and similar questions are asked in Grade 2 when the range of numbers is extended to 1000.

What is conspicuously absent in the Japanese primary mathematics textbooks is the use of number lines to deal with addition and subtraction. Rather, number lines are used to represent visually relative sizes of numbers. I recently heard that some people distinguish number paths and number lines. Number paths, as I understand it, simply string together numbers, 1, 2, 3, ... On a number path, numbers are represented more by their positions (or orders) whereas on a number line, a number is represented by the distance of the tick mark from the origin, i.e., 0. So, the way the Japanese textbooks introduce and use number lines are much more along the line of number paths.

So, why don't Japanese textbooks use number lines to represent addition and subtraction, as is often done in some US textbooks? There are at least a couple of reasons. The idea that a number is represented by the distance of the tick mark from the origin is a difficult one for students in primary grades. This is difficult, in part, because those students are still learning about measuring length. So, they really don't have the prerequisite knowledge to interpret number lines in that manner. What they tend to do is to simply count the tick marks. However, when students count, they start with "1," and this is another reason number lines are complicated for young children. I have yet to meet a child who started his/her counting by saying, "zero... one, two, three, ..." For many young children the role of 0 (the origin) on a number line is mysterious. So, when they have to use number line to solve 5+3, they will start with the tick mark labeled "5," some will point to "5" and say, "one." Most, if not all, teachers of primary grades have seen young children line up their rulers starting at "1." It's the same problem.

Some people suggest that number lines are inappropriate for primary students, and we should not use number lines. However, I do think it is important that number lines are introduced in primary grades. However, we should be careful about how we use them. We can use them to think about relative sizes of numbers. However, it is probably a good idea to wait to use number lines as a tool for arithmetic. We can use something like tape diagram for that purpose. But, developing the idea that numbers can be represented on number lines is an idea that should start in primary grades, and we should guide students to understand how numbers are represented (as distance from the origin) on number lines, perhaps connecting to the study of linear measurement.