Teachers are like personal trainers

    Over the past couple of weeks I have been posting about the quadratic unit that we are covering in my freshmen math class.  Today I am going to deviate from that a bit to talk about a lesson that occurred today.  The first part of the next unit is using exponent rules to simplify expressions.  There are a lot of mechanics involved, and the practice problems can get quite tedious.  
    Lately I have been telling my freshmen that teachers are like personal trainers.  I gave them this scenario: imagine you go to the gym with your personal trainer.  For an hour you watch your trainer run 5 miles, then afterwards you watch them lift weights.  At the end, you leave the gym and say that you "worked out."  Ridiculous right?  The personal trainer should show you some exercises and guide you to best practices, but at some point you need to perform those exercises and routines yourself, and go through the sweat and tears.  Teaching and learning is much like that.  The teacher can guide and show certain things, but the students must perform the exercises and go through the sweat and tears much the same way as in the gym.
   I thought to myself: 'How can I guide my students through these exponent rules while giving them the opportunity to go through the sweat and tears necessary to deepen their learning?'

​​First, I created examples for them and handed them out.  Above is one of the examples that I gave the students.  You can download the full worksheets here.  As you can see, the worksheet isn't anything special - it is something that I've more or less handed out every year.  I then organized the class in to six groups using the Team Shake app.  You can see the organization of the teams on the right.  After the class re-organized themselves in teams, I said the following:
"Okay, all of you are in your teams.  Team 1 is responsible for Example 1 in the packet I gave you, Team 2 is responsible for Example 2, and so forth.  Your team needs to think about the problems in your example and talk together to determine how to do them.  Perhaps some of you have seen this before and you can start the group in that direction, or maybe you want to go to www.wolframalpha.com, type in your question, see the answer, and then work backwards.  As always, you can ask me well thought out questions.  In addition, each team gets one "spoon feed" - I will work out ONE example for you if your group requests it.  I won't tell you what I am doing, though, it will just be the work and process written on a small white board.  Okay - enjoy!"

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​I left the team arrangements showing on the screen. The students went off in their teams and began their problem set.  Here are the things I observed:

• Some of the teams would call me over for their one spoon-feed problem, but others in the team would say, "wait, don't use it yet, I can explain it to you!"
• Many students would ask me 'is this right?'  I would turn it back on them and ask them to type it into www.wolframalpha.com.  It turns out, though, that wolfram alpha didn't always have the answers in the perfect form.  Below is one example.  On the left is the original problem; on the lower right is Wolfram alpha's simplification of the problem.  It turns out that it is correct, but not quite as simplified as one would wish.  This led us to a discussion of helpful math websites.

• ​We talked about using www.symboab.com, www.mathpapa.com, and desmos.com.  It turns out that symboab.com and mathpapa.com didn't simplify the original problem down very nicely either.  We used desmos.com to help us.  We entered the original expression as well as the proposed solution into desmos (see the picture at right).  It turns out that desmos wouldn't process it because it got confused over the x and y being in the same expression.  So, we changed the x and y to a and b, and then added sliders.  As you can see, the answer boxes are the same for both expressions, even if you change the 'a' and the 'b.'  The bottom line is that this gave the students and I the opportunity to discuss different ways to check answers using online tools.  
• As the groups finished their assigned problem, they began heading to other groups to get help with the next problem set.  At one point, a student was walking towards me and I thought he was going to ask me a question, and he walked right by me to head to another group so he could ask them a question!
• I commended the students after about a half hour because of their effort and for asking each other so many thoughtful questions.  I opened it up for about 5 minutes for them to ask me anything about any of the problems or rules that they encountered.  There were a few questions, and I was able to offer some 'helpful tips' that I have used after years of experience which they appreciated. 
• It was interesting that not one of the teams ended up using their 'free spoon feed' ticket that I offered them at the beginning of class.  Many teams didn't want to 'waste it', but by the end of class they didn't use it anyway!
• I reflected on the cultural forces that were leveraged during this class:

1. Expectations - my expectations for the students on this day was for them to be less dependent on a teacher's answer key, and more dependent on the math tools that exist online such as wolframalpha.com or symbolab.com.
2. Time - giving the students time to struggle through exponent rules and exercises in class.  It would have been very easy for me to demonstrate all the rules at the front of the classroom with lack of time as an excuse for doing so.  
3. Routine - one routine that is in place in my classroom is putting students in groups my number, assigning a problem to that group, and then having that group be 'responsible' for that problem.  Students know that other students in the class will be coming to them for help at some point during the block.
4. Interactions - the interactions between the students on this day were invaluable as they leaned on each other. I believe that students in math should 'answer questions' and should 'question answers' and there was a multitude of this going on between students.

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