Making Connections

Well, we are nearing the end of the second trimester.  Exams are next Thursday and Friday, and while the students have made an incredible amount of progress in their problem-solving skills (most especially in their perseverance), there is still one area in particular in which the students are still struggling.  In writing the worksheets for the course, the way we handled the more difficult topics was to spread out what would be considered one exercise over several worksheets.  For example, during the past week we covered rotation of conics and polar equations of conics.  So, on successive worksheets, I had the students do the following:

In the exercises of these same worksheets, we were building the general polar equation for a conic in a similar way, so that exercise #7 on worksheet #29 should have brought everything together.  However, almost without exception, the students looked at this exercise as brand new, and tried to complete the square on it.  Little to no connection was made with the exercises from the worksheets that immediately preceded it.  And this is not the first time something like this has happened. 

Now, in all fairness, they have successfully made similar connections in the past.  But for me, that just adds to my confusion: having made similar connections between worksheets in the past, why are the students not actively looking to make these kinds of connections all the time?  As mathematics educators we constantly stress the fact that from day to day and from year to year, the material in our classes builds upon itself to the point that it is necessary to essentially remember everything.  However, do we really hold the students accountable to that standard?  Or do we instead only teach one section at a time or one chapter at a time, and thereby inadvertently teach the students that each topic in mathematics stands in isolation?  It has been a real struggle for me this year to understand just how little the students even go looking for these connections.  These are the honors students, they are in pre-calculus, and yet somewhere along the way during the past 11 years of school, they haven’t realized (or haven’t been shown) that all of this math stuff is interrelated.  No wonder when they get to calculus they struggle, not so much with the calculus itself, but with the algebra from the courses that preceded it.  

More disturbing to me is this: why am I only now realizing that the students are struggling with this?  How have I taught this course for so long (10+ years now) and not seen that the students are not making these connections?  In many ways I believe it is because prior to this year, I fell into the trap of treating the topics in isolation, working through a textbook section by section and chapter by chapter, focusing only on the current material.  Another possibility is that I made the connections for the students during the course of my lectures, so they didn’t have to put the pieces together the way they are required to this year. 

Whatever the case may be, the only possible answer to why I am only now seeing this is that this year I’m running a Harkness classroom.  I concluded a long time ago that I have a better feel for each of the students as individuals this year than I ever have in the past.  Day by day, I have a better feel for where they currently are with the material, a better grasp of where they are struggling, and the reason for this is the individual accountability that Harkness brings to the classroom.  There is nowhere for the students to hide during the discussions, and I am more actively involved with them during the discussions.  The amount and level of informal assessment that happens every day in my classroom has increased more than I can possibly relate to you, and perhaps this is why this year I am more aware of the struggle students have with making connections than I have been in the past.  Regardless, as we wrap up this trimester, and more importantly as we begin the next, I am now aware that I need to specifically focus the students on making connection, not only from worksheet to worksheet, but from unit to unit and with the courses that preceded honors pre-calculus.  Of course, all I will do is persistently mention that they should be looking for the connections; I won’t actually make the connections for them.  My hope is that by consistently asking them to be on the lookout for the connections, they will discover them as they have discovered so much of the material we have worked through this year.  And who knows, maybe they will make some connections I’ve never noticed before.