This morning I observed one of my student teachers teaching a lesson on remainders from the Bridges 2 Math Program. As the student teacher struggled with her understanding of the mathematics involved it became incredibly clear to me why the program is so good. The activities are so rich with mathematical opportunity for what Bob Wright called the mathematization of children's minds.

The third grade activity I was observing involved rolling two die and multiplying the subsequent numbers to get a product. Then, using a neat worksheet, the students had to divide the product by 2,3,4,5, and 6 to see what happens. The demonstration roll the student teacher used was 4 x 4 for 16. Dividing by 2 yielded 8, but 8 what? The children were using tiles to make 2 x 8 arrays vertically. Here the 2 referred to the number of columns and the 8 to the number of rows. So the 8 was 8 rows. She could have shown 8 groups of 2 or 2 groups of 8, demonstrating the commutative property of multiplication. This is only true abstractly as the actual arrangement looks quite different. This is known as psychological non-commutativity.

She then went on to divide 16 by 3 getting 5 R1, then divided it by 4 getting 4 (proving 16 is a square number), then by 5 getting 3R1 (showing the commutative property), and finally by 6 getting 2 R4 .

The third grade activity I was observing involved rolling two die and multiplying the subsequent numbers to get a product. Then, using a neat worksheet, the students had to divide the product by 2,3,4,5, and 6 to see what happens. The demonstration roll the student teacher used was 4 x 4 for 16. Dividing by 2 yielded 8, but 8 what? The children were using tiles to make 2 x 8 arrays vertically. Here the 2 referred to the number of columns and the 8 to the number of rows. So the 8 was 8 rows. She could have shown 8 groups of 2 or 2 groups of 8, demonstrating the commutative property of multiplication. This is only true abstractly as the actual arrangement looks quite different. This is known as psychological non-commutativity.

She then went on to divide 16 by 3 getting 5 R1, then divided it by 4 getting 4 (proving 16 is a square number), then by 5 getting 3R1 (showing the commutative property), and finally by 6 getting 2 R4 .

The interesting thing about the commutative property in division is if you then give the remainder as a fraction. 16 ÷ 3 would give 5 1/3. 16 ÷ 5 would give 3 1/5. So the question is; what do the fractions represent? 1/3 of what and 1/5 of what? A fraction has no value unless you know the size of the one to which it refers. It depends, of course, on what the referents for the 3 and the 5 are. If you had 16 cookies divided between 3 people, each person would get 5 and 1/3 cookies. If you had 16 cookies and divided them into groups of 5, 3 1/5 people could get a group of cookies; which makes absolutely no sense at all.

The neat thing about the Bridges Program 2 activities is that they are so

**rich with mathematical possibilities**.
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