Making Sense of Decimal Multiplication

In this article by Margaret M. Rathouz, teachers are reminded of how students first developed their understanding of numbers and operations and are encouraged to extend that knowledge to include multiplication of decimal fractions using a similar means: meaningful representation.

Students first come into an understanding of multiplication using the 'equal groups' analogy. They have learned that three times four is synonymous with three groups of four whole objects. Multiplication of decimals might seem a leap of faith. Students don't think in terms of 3.3 students with 1.5 gum balls each. It is necessary, therefore, to move students from counting multiples of whole numbers to the decimal fraction-replete world of measurement. The author notes that distance, time, weight and volume are all measures commonly associated with the use of decimal fractions. Story problems using such quantities make the act of multiplication of decimal fractions more natural event.
In addition to using measurement, the author recommends the use of visual representation that extends the students understanding of multiplication to include decimal fractions. In one figure, she depicts how one can calculate the weight of  3.2 meters of wood if it weighs 0.46 kg per meter. She uses both a number line and a three-dimensional drawing to emphasize the relationship between length and weight. In two other diagrams, she uses grid diagrams to illustrate the multiplication of fractions. Finally, to help students to understand algorithms involving decimals really work, she emphasizes that it is not the decimal that moves, but the digits. She also encourages teachers to have students to explain why their answer makes sense.

This article brought me right back into ETE 107. I am thankful that my training is so up to date with what is being recommended in the literature. I fully expect to use no end of visual representations in my classroom. In fact, I used some just this evening as my daughter was struggling with adding and subtracting decimals. (I have my own manipulatives, didn't you know.) As science is my concentration, I expect it will be quite natural to me to move into using measurement as a means of illustrating the practical use of decimal fractions. I particularly liked how the author explained how to teach students the rationale behind what happens to the decimal when working through a decimal fraction multiplication algorithm. Changing unit names and scaling dimensions up and down are very 'science-like.' Renaming decimals as fractions will bridge the gap between decimals and decimal fractions for those students that resist change. The important message of this article is the importance of deepening student's understanding such that they don't blindly implement methodology without comprehension. I had to agree.