A thought provoking article on mathematics for elementary school teachers appeared in the recent issue of the American Educator. It was written by Dr. Hung-Hsi Wu, Professor Emeritus of Mathematics at UC- Berkeley. He writes that the basic operations of addition, subtraction, multiplication and division involve more conceptual processes than most people realize, and suggested that there should be separate fourth and fifth grade teachers in math.
Of note in his article is the very basic notion that mathematical thinking involves breaking up a complex task into several easier components. To add 15+16, it is conceptually easier to break up the numbers into 10’s and 1’s. Even if the rote algorithm is taught at some point, teachers definitely need to understand the ideas behind place value that are inherent in the standard algorithms for arithmetic. I mentioned to my students, in my graduate level math course for high school teachers, that 79*85 can be rewritten as 79(80+5), which leads to the distributive property used in algebra.
Many had never thought it about that way. They quickly pointed out that students are less likely to make errors when calculating 79(80+5) as opposed to the standard way of multiplying. Both methods require the same number of arithmetic operations. The partial products method simply requires a little more space. At any rate, it does require some organization of thought for the student, which is another aspect of mathematical thinking.
I should note that the same issue of American Educator also has an excellent article on the “science wars” and sheds some light why one needs both content and reasoning in science. That is, scientific reasoning cannot exist without content. Likewise, understanding mathematical concepts cannot exist without sound mathematical content.