# Lies and Collaboration

I actually enjoy the puzzle-like aspect of exponent rules, and simplifying radicals.  For me, there is something satisfying about learning ways to manipulate numbers and letters – probably why I love algebra so much.  But I am tuned-in enough to my students to know that many of them don’t find the same satisfaction from doing this work just for the sake of intellectual exercise.   And since calculators came into vogue, it’s been harder to justify the need for rationalizing the denominator or expressing the square root of 50 as 5 times the square root of 2.  But we are tasked to follow standards that often include these kinds of skills and it has been helpful for me to turn this into exploration or game learning as much as possible.

I did some mining of the MTBoS for ideas to teach rules of exponent arithmetic and came across this post, which includes a nice exploratory worksheet from Andrew Stadel.  He describes a similar issue with contextualizing exponent rules for middle schoolers – one of the really great things about our online community are these moments where we are reminded that we are not alone.  He asks his students to find the mistakes in the equations, to explain where the author went wrong, and to find the correct solutions.  He used a bunch of the common misconceptions found on mathmistakes.org to help students to catch themselves in the common errors.  Very nicely done.  This would have been a good lesson as is.

Then I remembered the Bucket O Lies protocol from Nora Oswald at Simplify With Me.  Nora manages to gamify math like no one else that I’ve seen.  She manages to add entertainment even to potentially dry topics like this one.  I combined Andrew’s worksheet with Nora’s idea to make a bucket-o-exponent lies.  I printed the 3 worksheets, cut them out into individual problems, folded them up, and put them into buckets (or baskets).  Voila! Drama and Motivation.  In pairs or threes, learning happened.

Of course, I hammed it up with the students.  There’s nothing like telling teenagers that someone is trying to get one over on them to motivate them.  This has worked well for me in the past, especially when it came from advertisements.  I riled them up by acting outraged that someone had created this whole set of math material, which was full of mistakes!  (Actually, I blamed Andrew :) ) …Lies I tell you… these baskets are FULL OF LIES!  Let’s find the mistakes so we can write a self-righteous set of corrections back to this author who was deliberately spreading bad math.

They quickly saw through my act, but it was enough.  They were already motivated in spite of themselves.  Andrew’s worksheet was just enough for everyone.  I started by coaching the groups who needed help getting started and moved to pairs who were making mistakes with fractional exponents.  For my honors group, I added a few more examples with rational exponents.

Thanks Andrew!  Thanks Nora!  Our generous community is the Best!

# Revealing Learning Targets

I am not always sure about how explicit to be about learning targets.  I have seen some convincing research, which seems to indicate that letting students know exactly what is expected of them for each lesson helps them to take ownership of their learning, and to make sure that they are getting what we think they are getting during each class session.  I agree with this practice in general, and I believe that it definitely has a positive impact on some students.  My current school, as well as the previous one have required that we post targets each day, and there are many educators who I respect who advocate for this practice.  But sometimes, I feel like a learning target can put a limit on where we can go as a class, and can feel a bit stifling, especially when we want a problem or exploration to feel open-ended.

Lately, I have adopted a practice of “Hidden Targets.”  I do post the learning target, but I often leave it covered up during class.

As part of our end of class routines, students make conjectures about what they think that today’s learning target was.  We reveal the target, assess how well the lesson matched the target, and whether the learning matched what was expected.  Although I think that I am good at starting class off, and generating enthusiasm, I sometimes am not as good at synthesizing and wrapping up.  Being conscious of synthesis and wrapping up class in a richer way has been one of my goals for this year, and this routine has been a good protocol for me and for my students.  It quickly reminds us about what we learned during class, and how this lesson fits in to the bigger picture.  Students have been highly engaged in figuring out the day’s learning goals; I hear students talking throughout the class period about what they think is under the flap for today – and you know that they remind me if I forget to do the “reveal!”

…And it doesn’t hurt that we have created this sweet Appolonian Gasket on which to showcase the day’s targets.  Who doesn’t want to stare at this and contemplate infinity?!