I used to think...

I recently finished Making Thinking Visible (a book I recommend with moderate enthusiasm). One strategy they recommend is giving students the prompt "I used to think... Now I think..."

Today I read the responses from my PreCalc class last week and I've decided I like this strategy. The warm up that day was a problem from the previous class. Two students had found the inverse of an exponential function using different methods and ended up with log equations that looked different. I asked students to show that they were equivalent. 

"I used to think solving for y would be the easiest way to see if two log equations were equal. Now I think that manipulating the logs is easier."

"I used to think logarithmic and exponential functions were unrelated."

Comic relief:

"I used to think logs were really complicated and only MIT people used them. Now I think I wish I could keep thinking I'd never see a log function unless I went to MIT."

The next part of class we looked at average rate of change over certain intervals for exponential, logarithmic and conic graphs. To compare the average rates of change we used approximate decimal values.

"I used to think slope was only looked at in fraction form."

“I used to think x^2+y^2=4 created two parabolas. Now I think it creates a circle.” 

Cool starting theory he’d never shared!

Mixed in the pile were plenty of "I used to think ___ was hard. Now I think it's easy." and "I used to think ___ was hard. Now I think it's still hard." Less interesting responses but still good information. If I have a good memory I will use this prompt again at the end of the next unit, and I'll share some of these responses with students along with why I like them.