Abstraction. With multiplication, it can be a real trap, for it either denies a child entrance into larger order multiplication or it allows a computationally adept student to mask his or her misunderstanding of the underlying logic. Luckily, there is a way around both issues. New mathematical concepts become accessible by using physical manipulatives, in exploration, before any symbolic procedures are applied. Currently, the class is exploring the Commutative Property of Multiplication and the Associative Property of Multiplication. Both rules are building blocks to an understanding of multi-digit multiplication. Symbolically, the Commutative Property of Multiplication can be represented as n x p= p x n. In student language, this is translated as two amounts can be multiplied in a switched order as long as the amounts are not changed nor the total. The Associative Property of Multiplication can be represented as n x (p x o) = (n x p) x 0. In student language: the names, amounts and groupings can change as long as the total in the equation does not. It is this rule that allows ten tens to equal one hundred, for instance, when dealing with place value.

But how to get nine and ten year olds to grasp these abstractions? It is all in how they use the blocks. Initially, the students are taught to look at a multiplication equation in the following way. For example,  8 x 3= 24. The 8 represents the number of sets, the 3 is the amount in each set, leading to a total of 24. This first stage revolves around building a series of equations out of blocks to make sure they understand the concept of set and quantity.

Kelly and Cody show the original set and the 

Commutative Property of Multiplication application.

Next, the students are given an equation. They build that equation and then have to build the correlate Commutative Property of Multiplication equation. For instance, the students had to build the original 8x3= 24. Then, they had to build 3x8=24. Physical spacing and proximity of blocks became very important at this point. For instance, if a student formed three rows of 8 but joined them end to end, the blocks represented 24 x 1. The Commutative Property is what is applied when a student learns that the multiplication fact of 4 x8 will have the same total as 8 x 4.

Once students have grasped this, and feel comfortable with showing their understanding both physically and through the use of the written equations, they are ready to move on the Associative Property of Multiplication. Using the same problem as before, the children explored how 3x8 can actually be restated as 3 x ( 2 x2), or 2 x (2 x3 ), or (2 x2) x 3. Any of these variations will produce set series that look very different from each other. By having students explain why they are building what they are building, assessment of their understanding becomes very clear.

Now the Associative Property is in action.

Interestingly, some children try to bypass the concrete stage and go right into a symbolic representation because they are used to being able to manipulate mathematical ideas easily in their heads. Sometimes that works, sometimes, it doesn't. One example that often crops up is having a child assign one to one correspondence to the numerals in the equation without considering what it actually meant. For instance, some of the children drew  the original 3X 8= 24, but drew a clump of three blocks next to a clump of eight blocks followed by a clump of twenty four in their grided math journals. Regarding the Associative Property, many could get the original two items- say, three sets with two in each set, but then were unsure what the next step would be. They didn't understand that configuration became the new set and they had to replicate that one two more times. This transition from the concrete to the semi-abstract representation of the grid blocks on the page can cause some confusion but with some discussion, the students can master it fairly quickly. 

An example of the transfer from the concrete to the semi-abstract form.

All these permeations in logic would not have become evident if the students only had to write their multiplication fact families or give just give an answer. By exposing children to the mathematical reasoning behind what they are doing, first at a concrete level and then moving into more abstract modes, true understanding is fostered.

This particular method ties in particularly well with the National Teacher's of Mathematics Representation standard which states that students should: 

• create and use representations to organize, record and communicate mathematical ideas;
• select, apply, and translate among mathematical representations to solve problems;
• use representations to model and interpret physical, social, and mathematical phenomena.  

If you are interested in finding out more about the NTCM standards, please click on www.nctm.org 

Last updated June 24, 2003
Maintained, and written by, Bridgette Fincher