Teaching for Meaning
Tteaching mathematics in a way that makes it meaningful for students is imperative. Recognizing that meaning is a criterion for long-term storage, teachers at all grade levels should purposefully plan for meaning in their lessons. Closure is a valuable strategy for helping students attach meaning to their new learning. Here are two basic ideas, with examples, about how to teach arithmetic for meaning using models (Dehaene, 1997) and closure (Sousa, 2006).
Use multiple models. Arithmetic and mathematical knowledge should be based first on concrete situations rather than abstract concepts. Numerical representations help students develop mental models of arithmetic that connect to their intuitive number sense. For instance, a simple subtraction problem such as 8 − 3 = 5 can be presented in different ways using concrete situations. It can be shown using a set of objects model: A box has eight toys.
Take away three, and there are five toys left. It can also be applied to a temperature model: If it is only 8 degrees outside and the temperature drops 3 degrees, then it will be 5 degrees. A distance model is another option: In a board game, a chip moving from space 3 to space 8 requires 5 moves. While these examples may seem the same to an adult, they are new for a young student who must discover that subtraction is the arithmetic process applied to them all.
The use of various models is important because relying on just one model may not be sufficient. Suppose you introduce negative numbers, for example, and you ask the class to compute 3 − 8. A student who relies solely on the set of objects model will say that this operation is illogical and impossible because you cannot take eight toys away from three. But this problem would be logical using the temperature model because most young students can comprehend the concept of negative degrees.
Select the correct model. Children encounter fractions in real life long before they meet them in school. They have a few concrete examples, such as portions of pie or cake. When first confronted with the problem of adding the fractions 1/2 and 1/3, they can relate these numbers to their intuitive notions of sections of a pie. They may soon realize that these two portions will add up to just less than 1. However, children who have no intuitive understanding of fractions are very likely to simply add the numerators and denominators and get the incorrect result, 1/2 + 1/3 = 2/5.
This result is not as far-fetched as it seems because it does have a concrete representation in the real world. If a baseball player gets one hit out of two times at bat, his average is 1/2. In his next game, if he gets one hit out of three times at bat, his average is 1/3. For both games, his total performance is 2 hits for 5 times at bat, or 2/5. Here is a situation where 1/2 “plus” 1/3 equals 2/5. How do you explain this seeming conflict? When teaching fractions, it is important to make clear to students that they should have the “portion of pie” model in their head, not the “scoring average” model.
Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.
Sousa, D. A. (2006). How the brain learns (3rd ed.). Thousand Oaks, CA: Corwin Press.