I just started using GeoGebra, the open source dynamic geometry software, to create an exploration activity for my Intro to Proofs class. The activity itself was an extension of a discussion in class about the Fundamental Theorem of Algebra. It can also be used in a precalc class which stresses concepts.
(Click here to see in a larger window.)
I was surprised by how easy it was to create - no need to learn Java and the user interface for GeoGebra was very intuitive. I plan to do more with this software, given the speed with which I can make some very interesting acivities.
Tuesday, October 27, 2009
Wednesday, October 7, 2009
Not so Elementary Mathematics
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.
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.
Saturday, October 3, 2009
GeoGebra
My latest tool for interactive math is GeoGebra. Just started to explore it. A great resource outlining many possibilities is this wiki page by Dr. Linda Fahlberg-Stojanovska.
I'm looking forward to using GeoGebra in my math for elementary teachers course as well as my graduate course for high school math teachers.
I'm looking forward to using GeoGebra in my math for elementary teachers course as well as my graduate course for high school math teachers.
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