This year, using a map of Alberta, I had students discover the formulas for themselves. Using two cities in our province, Calgary and Edmonton, students created their own two points. In collaborative groups, students were required to calculate the slope, midpoint (which is our city, Red Deer), and the distance between the two cities.
At first, students didn't know how to start. I heard comments such as "We know the midpoint is Red Deer, but how do we show that?", "Let’s try measuring the line with a ruler", and my favourite "If we measure with a ruler, that won't be the exact distance."
Most groups started working on slope first and calculated it correctly; from there they realized a right angle triangle can be drawn. After 10 minutes, all groups had calculated all three parts correctly. I then posted on the board two general points (x1, y1) and (x2, y2). The groups were then instructed to work with general points. They didn't realize this, but they were forming the formulas.
At the end of the lesson, students did not only know what the formulas were, but also the mathematical representation and application of them. In recent years, I have taught students how to memorize mathematical ideas and concepts, and when it came to application, just regurgitate the information down. This year, I believe my students truly understood each formula and how to apply them.
David,
ReplyDeleteI remember having a moment of clarity in my own teaching when discussing trig with another math teacher, he talked about the ratios being in the back of an old textbook, and how they used to look up the ratios in a table rather than using a calculator. I looked at the tables and realized, for the first time in my life, that the ratio table and the relationship between them expressed a simple idea: That as one angle got bigger, the other got smaller.
DUUUUHHHHHHHHHHHHHHHHH
Of course it does, but I had never been able to SEE it happening as I did when I saw those tables. Through my education in Math in school and then in the years I taught Trig to Gr. 9 students never once had I taken the time to look at it critically.
Trig is a great example of an opportunity for our students to discover a relationship, too bad I never had that opportunity.
Great post, makes me miss being in a Math classroom!