Mathematical Reasoning

One of the things I try to stress to my students is that the “answer” to a problem is unimportant.  It makes no difference to anyone’s life whether the answer you got to problem 7 was “3” or “-0.8672 repeating.”  What does make a difference is that they learn how to think, reason, justify, and explain.  Today, I was not disappointed.

During my 4th block class, we had been talking about writing rational exponents as roots, and were going through simplifying rational exponent expressions.  Looking at the example 8^{1/3}, I was helping them work through how to rearrange it using inverse operations so it looked like 8=x^3 instead (I think rearranging helps the students to remember that they know the answer to this without using a calculator.  Just like they are quicker to know x=2 when looking at 2x=4 than 2={4/x}) when they suddenly got stuck. Our first step was to rewrite the rational exponent as a cube root.  Next, they decided we needed an exponent to cancel out the radical.  This is where the brakes hit.  They couldn’t decide what exponent to use.  This was almost a non-issue in the other class, but I had crickets in 4th block.  At this point I paused because a couple things happened.  First, I did not want to give the answer away for free so I had to stop to think.  Second, the principal came in to do an observation and that always makes me stop for a moment nervously, even though I love the constructive feedback I get.  When I came  back around, I asked the students to stop and take some private think time to try and figure out the correct exponent.  From there we moved to partner talk, and I fished the numbers out of the crowd. 1, 3, 9.  We went back to partner talk to discuss the numbers they hadn’t come up with themselves, and everyone decided to throw out 9.  A couple people wanted 3, but I had some adamant young men who swore the answer was 1.

From here I cannot explain what happened to my students.  It’s like they ran into the phone booth and came out as mathematicians.  We moved into public discussion and both sides presented an argument.  Both sides were doing an amazing job using definitions, examples and prior knowledge to justify their reasoning and prove their stance.  The “1” team was actually doing a little better at it, which I thought was hilarious because they were wrong, of course, but I did not comment on the correctness of either side.  I eventually sat down with the main “3” advocate and encouraged him since he was having trouble putting his thoughts across (and I needed him to not give up, he had the right answer!).  He proceeded to explain to me in detail why the answer was 3 citing examples from what we did yesterday, so since it was obvious he already knew what he was doing, I helped him figure out how to show his reasoning to the others.  Then I invited both sides to give what would be their final arguments.  It turned into their final arguments, not because the “1” group was convinced, but because they finally were able to explain convincingly what I had suspected earlier… it was time to intervene.  The one group was arguing that the answer was 1 because we had just talked about how with x^{a/b} “a” becomes the exponent of the radicand.  They had misunderstood what we were looking for.  So was the class a waste of time, arguing an answer to the wrong question? HECK NO! The fervor with which the “1” group argued their point only caused the “3” group to work even harder to justify, and BOTH groups ended up successfully justifying a completely correct statement in front of the entire class using mathematical language and examples.  It was around this point that the principal left while giving me a thumbs-up (good sign!) and we debriefed what had happened.  For the rest of the class the students were “on.”  Math had become a class where they don’t just sit and listen but they are encouraged to get up and reason.  There was a noticeable difference in the way the students listened to what I had to say before and after our discussion.  Somewhere in the middle of the chaos of reasoning, a student said “I feel like we’re in court,” referring to the way the two sides where going back and forth building and presenting arguments, and I thought, “YES! Yes, you are! That is what this classroom was made for!”

I think there must be gas leak in my room or something.  This whole week my students have been on.the.ball.



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