This is the second post in a series addressing the topic, “What to Look for In Your Child’s Math Materials and Classes.” In the first post, I provided some preliminary discussion of the principle, "New concepts in arithmetic should be presented initially as manipulations of quantities and described in ordinary language.”
I suspect that it is relatively easy to think of some of the very elementary concepts of arithmetic, such as basic addition and subtraction, in terms of actions done to a quantity. Simple addition, for example, can be demonstrated by having a student count out a small number of objects, then count out another small number of objects, and then count the total number of objects in the combined groups. However, because the math instruction that most of us received didn’t focus on this sort of physical representation, it may be more difficult for many readers to understand the underlying action on quantity represented by more abstract concepts that we meet at a higher level of arithmetic.
For example, what is the underlying action on quantity involved in simplifying fractions? Here is one good way to explore this action. Count out six red cubes and three yellow cubes. (If you don’t have snap cubes or unifix blocks, you can use any counter, so long as they can be sorted according to readily identifiable characteristics, such as color.) Lay them out in a row as shown in the diagram below:
R R R R R R Y Y Y
What fractional part of the cubes is red if we are thinking about individual cubes only? Well, there are 6 out of 9 cubes that are red, so 6/9 are red. However, the cubes can be arranged into equal, uniformly colored stacks. Rearrange the cubes so that you have the greatest number of cubes in each stack possible if all the stacks have exactly the same number of cubes and each stack is composed of only one color of cube. It turns out that we can rearrange the stacks into 3 stacks of 3 as shown in the following diagram.
R R W
R R W
R R W
If we count stacks or columns now (instead of individual cubes) we can say that 2 out of the 3 stacks or 2/3 of the cubes are red. We have just simplified the fraction 6/9. This sort of task is easy for students to do, and the result of the manipulation, the calculation, is obvious given the arrangement and not something mysterious and arbitrary.
It turns out that all of K-8 mathematics can be presented in this way, and when one does so, the student is rarely confused. There are several reasons that such presentation provides so much clarity. First of all, Children have many experiences grouping, sorting, and counting objects in their lives outside theh classroom, and these experiences give them a substantial amount of math knowledge about basic math concepts. By focusing on the manipulation of things, a teacher allows students to connect new, more formal math ideas with their extensive informal math experiences.
Another reason this form of presentation works so well is that the basic patterns and relationships the student is learning can be recovered relatively easily if she forgets. All learners forget some new information, but if they have some sort of physical relationship or activity to refer to, they always have a ready means for recreating and recovering the concepts they have forgotten.
Finally, this method of presentation makes it easy to visualize key patterns, and visualization provides the foundation to develop more sophisticated and potent mental models as one’s understanding becomes deeper and more abstract. As we saw in the case of simplifying fractions, the physical representation makes the result of the calculation obvious and in some sense necessary, rather than something elusive and mysterious.
Those of you who have been in math classrooms recently or who have looked through education catalogs with math supplies, know that there is an enormous number of physical, manipulable materials in many math classrooms, especially at the primary school level. However, to present new math concepts as a manipulation of quantity requires more than merely having counters and models available. It is also critically important to use ordinary language at first to describe these manipulations. The reasons for doing so should sound familiar. Using ordinary language roots new information and new concepts in something that is familiar to the student, and, moreover, helps them to visualize the new patterns and relationships she is learning about.
Students, of course, do need to learn academic math vocabulary, such as “plus,” “minus,” “times,” “denominator,” “reciprocal,” etc. But they don’t have to learn these terms when they are first learning these concepts. For example, talking about “4 groups of 3” to second graders first learning about multiplication will lead to much quicker learning and retention than talking initially about “4 times 3.” (Indeed, the word “times” is especially diabolical for many students because it is a word they are very familiar with a completely different context.) Precise instruction initially uses familiar and functional words such as “groups of,” "put more on", "take some off" or “bottom number” rather than the more formal mathematical terms.
In my next post, I will talk about the principle, “Math symbolization should be taught as a means of recording actions.”
Until then, Happy Teaching!