I want to do these in order of their occurrence today because it set the tone for me. I was giving an exam in AP Stats (hence no new info about AP Stats) and at the end the exam a learner handed me a note. It was very personal, but it essentially said “please forgive me if I seem out of it this week, a person close to me passed away this week a couple of years ago and I always have a bad week around now.”
I sat there thinking about the struggle this learner has this week, the memory of the passing, and the fact that teaching is not about content but about the relationships. This is a concept that I did not have 8 years ago. I jumped into teaching thinking I could teach math and rock the content like no one else.
Today I know better. Today I know I know content; I am confident in my content; but I KNOW for a fact that all of that knowledge is useless if my learners do not trust me and I don’t trust them. I have not always had that knowledge. I have made mistakes on this issue in the past.
By the way, I thanked that learner for communicating with me. I will watch this person closely to make sure there are no problems. I owe that much for sure.
Then I introduced rational exponents later. I put one question on the board.
“Given that you know what 8^(1/3) means. Given that you know what 8^2 means. What do you think 8^(2/3) means?”
I let them think about it for several minutes.
Eventually, one learner broke it down into two parts, cube root of 8 is two; two squared is 4.
Excellent. We discussed why that works, we discussed what happens if the order is reversed.
Then we rocked some complicated problems I put on the board.
Yes, they made mistakes, but the mistakes made were procedural mistakes. Mistakes about not distributing to all terms, or multiplying fractions wrong, or moving all terms instead of only the term with negative exponent.
I am telling you, this was absolutely successful. I did not approach rational exponents like this last year, but it works. Let them create the meaning.
Never say something a learner can say.