## Sunday, May 25, 2014

### 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

I teach mathematics content courses for prospective elementary school teachers. Some of my students are parents of elementary school children. Sometimes I hear them complain about the "Common Core math." One complaint I often hear is that their children are made to practice different strategies - strategies like those discussed in 1.OA.6. However, I believe such teaching practice misses the point of 1.OA.6. The standard expects students to add and subtract within 20 (fluency within 10) using strategies. The standard does not say that each students needs to master or be fluent with each strategy.

Let's think about 8 + 9. There are a variety of strategies students can use. Here are some that I can think of:

* count all
* count on (from 8)
* count on (from 9)
* "I knew 8 + 8 is 16, so 1 more is 17."
* "I knew 9 + 9 is 18, so 1 less is 17."
* "I knew 10 + 10 is 20. Since 10 is 2 more than 8 and 1 more than 9, I took away 3 from 20, and the answer is 17."
* "I took 2 from 9 and 8 + 2 = 10. Then added 7 more to get 17."
* "I took 1 from 8 and 9 + 1 = 10. Then added 7 more to get 17."
* "I knew 5 + 5 = 10, and 3 + 4 = 7. 10 + 7 = 17."

I'm sure there are others. When students figure out 8 + 9 using their own reasoning, what is the point we want to emphasize? Are these strategies equally good? If not, how do we decide which strategy is "better than" others? Better in what sense?

If the focus is getting the correct answer to 8 + 9, perhaps all of these strategies are equally good. However, if our goal is more about helping students to think about ways to calculate, then perhaps these strategies have "good" in different ways. For example, those who recognize counting on from the larger added has begun to focus on the more efficient way to find the sum. Those who use doubles, either 8 + 8 or 9 + 9, realize that the known sums can be used to figure out an unknown sum. They have also figured out that if an added increases (or decreases) by 1, the sum also increases (or decreases) by 1. Figuring out such a relationship is an important mathematical practice - perhaps noticing and making use of mathematical structure. Using 10 + 10 also requires the realization that known facts may be useful to figure out unknown sums. In addition, these students are beginning to focus on 10 as a useful benchmark in our number system. Perhaps we can say the similar thing with those who use 5 + 5. Finally, the other two strategies, 8 + 9 = 8 + 2 + 7 or 8 + 9 = 7 + 1 + 9, are not only focusing on 10 as an important benchmark, they are also taking advantage of the way numbers between 10 and 20 can be thought of as "10 and some more."

So, these strategies have different strengths, and it makes no sense to force all students to be fluent with all of these strategies. Moreover, if the strength of a double strategy is the fact that students are beginning to develop the disposition to seek what they already know to figure out something they don't know yet, giving students several problems to solve using this particular strategy seems to be totally inappropriate.

Of these strategies, only the make-10 strategies are the strategies we want to make sure that all students understand and be able to use. So, perhaps assigning some homework problems where students practice these strategies are appropriate. However, we should remember that whether or not students themselves will realize the usefulness of these strategies really depend on how easily they can compose/decompose numbers to 10 and their understanding that numbers between 10 and 20 can be thought of as 10 and some more. If they don't have that prerequisite understanding, they may not see the point of these strategies. On the other hand, if they have that prerequisite understanding, they may not need much "practice" to master these strategies. It is perfectly appropriate to assign some practice problems, but we should be careful about what we want students to practice.