Multiplication

Why know multiplication and division over addition and subtraction?

Addition and subtraction definitely have their advantages over multiplication and division. Working with smaller numbers, it is not quite as important to know the fact families and have them ready to go at all times. However, once the numbers start growing, fact families become much more important. Students who do not know their fact families are at a disadvantage. Without knowing these groups, it takes much longer to determine the answer to larger problems. Ann H. Wallace and Susan P. Gurganus' article, "Teaching for Mastery of Multiplication," breaks down this area of math and goes over several different categories of teaching multiplication.

Types of Multiplication

Reading through this article, I had no idea there were so many categories of multiplication. I am sure I learned this information years ago, when multiplication was new. However, everything is engrained, now, so it is just something I now know.

1. Repeated Addition- the groups exist simultaneously. One factor describes the number of items in each group, while the other factor describes the number of groups.
2. Scalar Model- the scalar multiple expresses a relationship between the original quantity and the product, but the scalar multiple is not a visible quantity. I.E.: Marcus has eight marbles. His brother has three times as many marbles. How many marbles does Marcus’s brother have?
3. Rate Model- the product is the total value or distance associated with all the units, usually represented in a number line.
4. Cartesian Product Model- two disjoint sets exist and the size of each set is known. The sets are paired and the product is represented by the number of pairings.
5. Area- a region is defined in terms of units along its length and width. The product is the number of square units in the region.

Repeated Addition Model

Scaler Model

Rate Model

Cartesian Model

How Should We Teach Multiplication?

In the past, teachers have taught by following the book.That is likely how most of us were taught: in whatever order the book went. Actually, it wasn't until college that I had teachers go out of order of the texts, because it made more sense that way sometimes. Recent studies have shown that the most effective course of instruction should be:

1. Introducing the concepts through problem situations and linking new concepts to prior knowledge
2. Providing concrete experiences and semiconcrete representations prior to purely symbolic notations
3. Teaching rules explicitly
4. Providing mixed practice

Instruction should be incorporated with realistic problems with hands on materials. For me, hands on activities is the way to go. I learn and understand most things kinesthetically over audio or visual. Allowing students to learn fact families by their own drawing is also beneficial. This not only lets them start understanding fact families, but lets them practice it by also skip counting.