9^(1/2) isn't Purple - The Value of Wrong Answers

Last week, as I was getting ready to introduce rational exponents to my honors Algebra II classes, for some reason I flashed on a session offered by Ed Burger at last year's NCTM conference.  He spoke to the importance of productive failure - practicing being wrong and recovering.  He talked about asking kids to give him answers that they knew were wrong and how learning would follow.

So, I gave it a try.  The question I posed was, "What does 9^(1/2) mean?"  Their job was to give me either a correct answer and reason why it was correct, or a wrong answer and a reason why it was wrong.

I asked first for the wrong answers and reasons.

Student 1: "It's not purple."
Me:  "Why not?"
Student 1:  "Because the answer is a number."

Back in my early teaching days, I probably would have been irritated by that answer.  Experience, however, allowed me to see that even though there's a wise-crack element to the response, at its heart is real truth -- the answer is a number!  And the student is engaged!

Me:  "Okay, that's good.  So we've narrowed the answer down to a number.  Give me another wrong answer and why it's wrong."
Student 2: "It can't be 70 million."
Me:  "Okay, why not?"
S2: "It's way too big."
Me:  "But why 70 million?"
S2:  "Alright, 30,000.  Or 500.  It just can't be bigger than 9."

Bingo.  Had I asked the question "what is this?" I don't believe the student would have seen this fact.  By asking "what isn't this?" the student found it on his own.

Student 3:  "It has to be bigger than one, because 9^0 is one."

Student 4:  "It can't be 4.5, because it's not multiplication."

This was my favorite moment, because I know that if I had asked them right away what they thought 9^1/2 was, many of them would have said 4.5 for lack of a better answer, even if a little reflection would have told them that it was wrong.  By approaching the question from the negative, it forced them to reflect and it made that wrong answer a correct answer.

The conversation continued as we narrowed down the possibilities -- it probably was closer to 1 than it was to 9 because that's the pattern we saw with higher integer powers.

Eventually, I asked for a correct answer.  Some students knew what it was, but it took a little prompting from me for them to see the why.  I also helped them see that we were really "defining" rather than "figuring out."

Approaching an idea through wrong answers was effective for every student in the room.  The students who had been taught the right answer in another class were challenged with explaining why.  Some of them chose to find wrong answers with an explanation because it was easier.  For student who didn't know what the answer was, the entry bar was low -- there are a lot of wrong answers, yet they ended up making critical contributions to the discussion.  In the end, I believe everyone had a better understanding of the idea than they would have had I just allowed those who knew to tell those who didn't what the answer was.

Plus, it was way more fun this way.