Multiplication and Division:
Models and Strategies
Multiplication: A "Digit Approach"
The digit approach to Multiplication is an algorithm approach. Students who multiply using the algorithm are carrying numbers and writing a zero to hold the ones place below to solve two digit by two digit have no idea as to why they are doing that unless it is explained to them or they have great number sense. If they continue to use these strategies throughout their upper years it is going to become more difficult for them to understand harder concepts. It also talks about students learning facts by memorizing and tricks. But if students have a good sense of magnitude and relationships they wont have to learn those tricks. For example if they know 3 X 5 = 15 and understand that 3X 7 is just two more groups of 3. Then they would have to just add 6 to get 21.
The Ratio Table and the Array
Learning multiplication has two visual ways of learning. You can learn the ratio table way with proportions or use the array way. The ratio table way will give them a better understanding when students move into slopes, ratio of change, etc. But learning the array way is a very basic understanding of multiplication.
5 is half of Ten.
Double and Half.
In addition to the division algorithm, two other strategies work really well - a ratio table and the distributive property of multiplication as well as using a constant ratio. For example, 300/4 could be done by simplifying 300 and 4 by 4 to give you 25 to 1.
Implications for Higher Mathematics
Multiplication and division number sense allow students to reason and justify thinking. The partitive meaning of division and the ability to distribute are just two areas that have the potential to cause major problems in higher level math if not understood on a basic level early on. Merely teaching alogorithms will not provide enough understanding to ensure correct reasoning later on. There are geometry/calculus implications as well. When students learn to create area formulas based on the rectangle, they have a better foundation for geometry and calculus than students who simply memorize formulas. Building a strong foundation of multiplication and division based on number sense gives students a much greater chance for success in higher level mathematics.