## Tuesday, March 10, 2009

### M5N1 a - Even and odd numbers

M5N1. Students will further develop their understanding of whole numbers.
a. Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
When teachers examine the GPS, there are (at least) a couple of topics they are surprised to see so much later in elementary schools than they are used to. The topic of odd/even numbers is one of those topics. Teachers are surprised that a topic that they used to discuss in the second grade (or even in the first grade) is now delayed until Grade 5. Some may be tempted to include this topic in an earlier grade. So, why does the GPS wait to discuss this topic until Grade 5? Although I cannot speak for the committee who developed the GPS, I can share with you the Japanese perspective.

There is no question that we can teach second grade children to distinguish odd/even numbers. We can connect to skip counting or simply tell students that all numbers in the sequence, "2, 4, 6, 8, 10, ..." are called even numbers and the rest are odd numbers. For larger numbers, they can simply use the rule, "if a number ends with a 0, 2, 4, 6, or 8, it is an even number." However, the point is NOT identifying even/odd numbers. Let's look at the GPS statement:
Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
It is important to note that what we want students to understand is that "counting numbers" (whole numbers?) can be classified into different subsets by paying attention to various distinguishing characteristics. So, what is the distinguishing characteristics for even/odd numbers? It is the divisibility by 2. Even numbers are those numbers that are divisible by 2, while odd numbers are those that cannot be divided (with a whole number quotient) by 2. So, the emphasis is not about identifying even/odd numbers, but understanding ways to sort whole numbers. Even/odd numbers are just an example of one such classification schemes.

To help students focus more on ways to classify whole numbers, teachers may want to engage students with a task that require students to sort whole numbers in a similar way - by focusing on the remainder when divided by a number. You can watch a videotaped lesson from Japan, in which the teacher posed an interesting question involving the idea of classifying numbers by looking at the remainders when they are divided by 4.
http://tiny.cc/rkB1X