Sunday, March 14, 2010

M1N3 f - Mastering the basic addition and subtraction

M1N3. Students will add and subtract numbers less than 100, as well as understand and use the inverse relationship between addition and subtraction.
f. Know the single-digit addition facts to 18 and corresponding subtraction facts with understanding and fluency. (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)

Many of today's elementary school mathematics textbooks discuss a variety of thinking strategies children can use to figure out the basic addition facts. Some textbooks even organize their addition units according to those strategies: add 1/2, doubles, doubles plus/minus 1, make 10, etc.. Young children often "invent" these strategies. In fact, these strategies are the results of children's developing number sense. Enriching children's number sense, for example, composing and decomposing numbers (MKN2b, M1N3c), is a major emphasis in primary grades. Thus, this particular standards has two purposes: helping children master basic facts and helping them further their number sense.

Clearly, we want children to be able to recall the basic addition facts quickly, and some people may wonder why we need to bother with these different strategies. There are many reasons to include students' invented strategies in primary grades mathematics instruction, but to me the following three are the major reasons. First, these strategies are natural for children. If we take the idea of "starting with where children are," then we should think about how to take advantage of children's natural thinking processes. Another reason is that these invented strategies are the results of and promote further development of children's number and operation senses. I believe that the ability to see numbers and calculations flexibly is a powerful mathematical tool. If that is the case, it seems to make little sense to squash children's natural ability to think and force them to memorize the basic facts first then try to teach these flexible ways of thinking later. Such an approach seems to be rather inefficient. Finally, I believe that a major reason we teach mathematics in elementary schools is to help students become better thinkers. Thus, we should be always encouraging students to think. Quick recall is a goal, but if we want students to continue developing their thinking ability, we must dedicate some time in mathematics classrooms that focuses on children's thinking.

Anyway, although these strategies should be discussed as children naturally "invent" them, there is one particular strategy that should be treated intentionally. That strategy is the make-10 strategy. For example, 9 + 6 can be thought of as (9 + 1) + 5 = 10 + 5 = 15. For subtraction, like 13 - 8, children can think 13 - 8 = (10 - 8) + 3 = 2 + 3 = 5, or 13 - 8 = (13 - 3) - 5 = 10 - 5 = 5. 10 is such an important number in our numeration system. Thus, developing the ability to think with 10 systematically must be a major goal of mathematics teaching. For some of the invented strategies, I don't think it is necessary for all children to be able to use them. However, the make-10 strategy is mathematically so significant that all children should understand and be able to use it effectively. This way of thinking also helps students to go beyond the counting-by-one approach. If we consider older students counting on their fingers a problem, we have to offer them an alternative that can be just as effective and perhaps more efficient. The make-10 strategy is one such strategy.