by Steve Reifman, Monthly Columnist
When many of us were elementary school students, we learned that math was primarily about memorizing isolated facts and applying procedures. Math was something created by someone else that we either “got” or didn’t “get.” If we were fortunate to understand how to apply these procedures, we did well. If we weren’t, then math became a foreign language, and the skills just didn’t click. This need to understand and use other people’s strategies often led to lifelong math difficulties and, in some cases, even phobias.
Nowadays, thanks to wonderful approaches such as Cognitively Guided Instruction (CGI), the focus is on student understanding of mathematical concepts and principles, not memorization. Children are encouraged to create their own strategies or employ others that their classmates have shared that make sense. One of the main goals of approaches like CGI is to help students develop strong number sense, the greatest asset students have when learning math.
As teachers, it is critical that we emphasize the development of long-term number sense rather than short-term success using memorized procedures. Perhaps my favorite example of this emphasis involves multiplying a one-digit number times a three-digit number. Children tend to love using the traditional algorithm when solving this type of problem because it is fast. The problem is that most students tend not to understand why the algorithm works, and as a result, they can make errors without even realizing it. If we forget to carry the one or forget to add the one that we carried in the previous step, for example, we can wind up with an answer that our number sense tells us could never be correct. The problem is we don’t discover these errors because our number sense is turned off when we are merely applying a series of memorized steps.
In recent years my students and I have come up with and shared numerous ways to multiply a one-digit number times a three-digit number, and because these methods all make sense to the kids, their number sense is improving as they use these methods, not being stifled as it is when algorithms are employed.
Last year, the most popular method involved the distributive property of multiplication. For example, imagine we need to multiply 5 x 125. Using the distributive property, we separate 125 into its component parts, 100, 20, and 5. Separating the numbers in this manner makes sense to kids and reinforces their understanding of place value. Then, we multiply 5 times each of these parts and add the three products together.
5(100) + 5(20) + 5(5) =
500 + 100 +25 =
With testing season on the horizon, it becomes extremely tempting to seek short-cuts, such as algorithms, to help children learn the skills that we know will be on the test. My recommendation is always to emphasize long-term understanding and number sense. Fortunately, this emphasis will also help improve performance on the tests, but more important it will lead to a lifelong enjoyment of and proficiency with math.