Maybe You Can Teach an Old Dog New Tricks

First of all, I need to apologize to any students that I had in my classes from 1992 to about 2010.  You are probably wondering why I feel the need to apologize.  Because when I first began teaching, I somehow used to think that it was about me.  Since my transition into a Math Teacher Educator, I have come to realize that it is about the students and their experience with the content.

Here is an article in the Chronicle documenting that there has been a true shift in instruction at the college-level.  Faculty are moving away from lecture and toward a learning-centered approach. 

http://chronicle.com/article/Professors-Place-in-the/149975/

I can vouch for the fact that it works.  Here is an example this semester from my Geometry for Teachers classroom.

We were discussing similar figures and finding the scale factor for a pair of similar figures.  I then asked the students if the ratio of the area of one of the figures over the area of the other figure would be the same as the scale factor.  One student immediately said, “I don’t think so.”

I told him to go to the board and give an example to illustrate why he doesn’t think the ratio of the areas is the same as the scale factor.

He drew the following picture:

Area of large square = 4 square units

Area of small square = 1 square unit

The ratio of the areas is 4 to 1,  but the scale factor is 2 to 1 .

Then I asked:  “What happens if your figures are triangles?” 

He drew the following picture and said that he was going to draw right isosceles triangle to ease the explanation.

Area of large triangle = 2 square units

Area of small triangle =  1/2 square unit

The ratio of the areas is 4 to 1,  but the scale factor is 2 to 1 .

Then I asked, “Can you see a pattern for the ratio of the areas of two similar figures?”
Then we discussed that since area is a two-dimensional measure that it would make sense that the ratio of the areas is the scale factor squared.

I informed the student that he had just taught a section of the textbook!  No lecture, no PowerPoint—just one good question led the student to explain an important topic.

FYI—I don’t have this great of a success story every day, but when I do, it is very rewarding!