Teachers should strive to use concrete ideas as much as possible. In high school, many students are not at the Piaget level of abstract thinking, and consequently simply cannot understand abstract concepts. They compensate for this inability by learning what to do, to follow others.

 

Here are some examples of how to make properties of numbers concrete.

 

Commutative is a + b = b + a. Tell two students to switch places. Then tell the class that this is an example of this property. Then tell them to switch back.

 

Distributive. Put down two green markers and one orange marker on the table in a row. Then make a second row with two green and one orange marker. We tell the class that this is equivalent to 2(2 green + orange). We write this on the board.

 

Associative. Bring up three students to the front of the room in a row. Let the first and second hold hands. Now let the hands drop, and let the second and third hold hands. This is an example of (a + b) + c = a + (b +c). We also write this on the board, and then tell the students to be seated.

 

This also has the advantage of making the math fun, and gets them thinking.

 

Later on, if there is any confusion with, say, the associative, you remind them that Shawn and Jerry were holding hands, and then Jerry and Sam were holding hands.

 

Divide by ½. To explain how 2 ÷ ½ = 4, M tore up two sheets of paper in half, and got four sheets.