Teach 180: Math ≠ Exercises (Day 139)

This weekend (Friday through Sunday) I am working with a group of AP Statistics teachers.  Although our work day was done at 5 PM, we naturally slid in to talking shop through much of dinner.  There are two Khan Academy representatives with us and of course, we talked about flipping the classroom a bit. 

Several years ago, I flipped some of my lessons and found out that not all of my students enjoyed it.  The main reason was that I was not there beside them for them to ask clarifying questions.  Yes, they watched the video, but they wanted to know more at that moment or were not convinced they they fully understood the concepts.

Now I only flip lessons on snow days or to do some quick review of content and vocabulary at the start of a unit.  Proponents of flipping would say it is better, because the students can "go at their own pace" with the video and you can then use class time to work on problems and help individual students.  That model is fine if your teaching philosophy is "I Do" (video watching) "We Do" (working on problems with a partner) and "You Do" (individual practice) math instruction.

Consider a lesson on arithmetic sequences.  In the flipped lesson, students would watch a video where the speaker would talk about the pattern and derive the formulas for the nth term and the sum of the first n terms.  A few examples would also be shown.  In class, the student works through problems like he or she saw either alone or in groups.  The student demonstrates "understanding" by being able to mimic the examples.  After twenty-five years of teaching, I would say that this student is not doing math. 

Mathematics is making conjectures and discovery.  Mathematical thinking involves reasoning and discussion.  The student watching the videos has no joy of discovery and his or her learning of the content was certainly not a social experience.  At no point did the student have to "look for and make use of structure."  The structure was explained and the video teacher made use of it.  At no point did the student get to "construct viable arguments and critique the reasoning of others."  The student never spoke to anyone.  I am not saying that these last two things couldn't happen post-video.  However, by having the content be taught by video we strip mathematics down to a mindless set of rules and procedures.


What would I propose instead?  List the following sequence on the board.  

Then ask students for the 100th term in the sequence.  You can let them use their calculators.  At some point they will lose track of where they are.  If they get to the 100th term, ask for the 1000th term.  At some point, they will stop using their calculator as a hammer to solve the problem.  They will be forced to "look for and make use of structure".  Given enough time to think, work together and ask questions, they will "construct viable arguments and critique the reasoning" of their peers.  They will find the traditional formulas for themselves AND truly understand why they work.