Thursday, May 21, 2009

M3N5 - Simple cases of fraction addition/subtraction

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators.
Addition and subtraction of fractions are discussed in three different grades (M3N5e; M4N6b; M5N4g). Both this current standard and M4N6b involve fractions with like denominators. For M4N6b, there is a note stating that denominators should not exceed 12. So, what is the difference between M3N5e and M4N6b? If one of the reasons for developing the GPS was to minimize repetitions, why is this topic repeated in Grade 4?

One of the differences is that in Grade 3, the sum or the minuend must be less than or equal to 1 as students will not be studying improper fractions and mixed numbers until Grade 4. Thus, 2/5 + 1/5 is appropriate in Grade 3 but not 4/5 + 2/5. However, the most important reason for discussing simple addition and subtraction in Grade 3 is to help students understand fractions as numbers, just like whole numbers.

Fractions are often introduced as parts of a whole. Although this way of looking at fractions is relatively easy for students to grasp, research also shows that this is a very limiting view of fractions. In other words, if students can consider fractions only as parts of a whole, they will have difficulty dealing with fraction arithmetic. Part of a whole is a relationship, and we cannot perform arithmetic operations on relationships. We can only add, subtract, multiply, and divide numbers. Thus, students must understand fractions as numbers in order to make sense of fraction arithmetic. So, how do we help students to see fractions as numbers? Well, one way is to help students experience situations where fractions are added or subtracted. From those experiences, students can realize that fractions are numbers because they can be added or subtracted. It sounds like a circular argument, and it probably is. However, I would like to think this relationship more of reflexive, i.e., neither one is a prerequisite for the other, and an understanding of one can actually promote and deepen the understanding of the other.

What is important, though, is that experiences students will encounter are something that they can determine as addition/subtraction situations. For example, we can ask students what is the total length of a tape if a 2/5-meter segment and 1/5-meter segment are put together end to end. They can see that this situation is an addition situation - you would use addition if the lengths of the segments were 2 meters and 1 meter, respectively.

Another key idea is the unitary view of fractions. In other words, students should understand 2/5-meters as made up of 2 1/5-meter segments. Then, 2/5 + 1/5 is really 2 1/5-units and 1 1/5-unit put together, or 2+1 1/5-units. By recognizing that fractions may be added or subtracted, and having a way to reason through to find the answers, students can develop the understanding of fractions as numbers. With this knowledge as the starting point, students in Grade 4 can explore fraction addition and subtraction more formally.

Wednesday, May 6, 2009

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.