We want students to understand math, not complete problems by simply memorizing processes that make no sense to them. Manipulatives are a great tool and research suggests that they often too quickly abandoned. It does make sense to eventually represent processes graphically. Unfortunately, there’s sometimes a fine line, and other times a chasm, between the use of graphics and true understanding of processes. I have two examples in mind, one related to decimals and one to division. Today’s post deals with decimals.
When students are first asked to understand decimals, we typically use an area model. The grid above represents .28 and most kids “catch on” to the way tenths and hundredths are represented with this model. However, don’t assume that all kids recognize the relationship between tenths and hundredths, even if they can color them in perfectly. Another problem with the area model often lurks out of sight. Students may not understand that this grid represents a part of a whole. Decimals are fractions but some kids never see the relationship between fractions and decimals. (Sadly, some kids don’t truly learn what fractions represent when all they see are those pie-shaped pieces.) Anyway, by the time kids get to decimal grids, they have used hundreds boards for several years. Hundreds boards represent whole numbers and look a lot like the decimal grid.Using an area model to multiply decimals compounds the confusion for special needs kids. See for yourself. Here is a representation of multiplying decimals, in this case showing that 0.3 x 0.4 equals 0.12. The answer is purple, a combination of the red and blue.
For some kids, this representation is worse than simply memorizing an algorithm. I’ve had kids ask why they should color one part vertically and the other horizontally. After coloring in so many blocks, my kids usually think this is 8 tenths or sometimes 58 hundredths. For students with no understanding of the relationship between tenths and hundredths, this particular area model is meaningless. It doesn’t explain why the product is so much smaller than the factors.
For kids who get stumped by this area model, I may focus on multiplication of the decimal numbers as fractions: 3/10 x 4/10. Another graphic solution is the use of a number line divided into hundredths. (I’d obviously start with common fractions/decimals, such as 5 tenths or one-half, but the idea is the same: I am finding three-tenths of a “group” or line of four-tenths.) Overall, I think using a number line to teach decimals is far more effective than an area model. A number line also conveys the relationship between simple fractions and decimals.
I’d love to hear your thoughts!