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.

It's About the Questions

As teachers, we have been somewhat caught during the last year or two, trying to prepare for the changes that are coming with the implementation of the Common Core standards and the resulting mandatory end-of-course testing that will follow.  Specifically, we have been looking at the Common Core standards and, other than the placement of some of the topics (for example, the amount of statistics written into the algebra 1 and algebra 2 standards), the actual content isn't really changing.  As a result, our initial reaction to the Common Core was, quite honestly, "So what?"  However, now that we are (finally) getting to see some sample questions from the end-of-course tests, it's clear that questions on the new tests are going to be significantly more difficult than anything on the current standardized tests. How is this possible?

Simple: the tests are going to include "performance tasks", extended questions that require the students to put the content together in ways they may not have seen before.  They will require the students to really understand the material, to the point that they will be able to get a bit creative with it.  This is a far cry from the current testing (at least it is here in Ohio) that requires the students to repeat one more time pretty much what they have done in class.  With enough practice and enough rote memory, the current test is reasonable.

It is also unreasonable, in that it does not indicate whether or not the students actually understand the material.  I've said for a while now that the traditional way of teaching does not teach the kids to solve problems; it teaches them to mimic the way we solve problems.  The students we produce tend to be really good at memorizing processes and following algorithms, but not at using the content in ways other than those they have seen in class.  The textbooks are built according to this model as well: here are a few examples, now do a bunch of problems just like it with new numbers.

So what's the difference?  It's all in the questions.  On a surface level, the content is, in fact, the same.  But if you're asking the kids to actually understand that content rather than just memorize a process, then the same content can make for a very different result when it comes to the grade on a test.  Unfortunately, there are a lot of teachers who are stuck looking at the content and, upon seeing the same content in the Common Core standards, think they are going to be able to continue teaching the way they always have and that their students are going to be successful on the new tests as they have been on the old ones.

Nothing could be further from the truth.  Even with the same content, we need to fundamentally change the way we teach the kids.  We need to be asking them to truly understand the material, and the only way to do this is to have them put the content together themselves, to discover it themselves, to make sense of it for themselves.  We need to get them to actually solve problems and not just imitate us or the textbook, and the only way to get them to do this is to ask them to do this on a daily basis.

Enter Harkness.  The more I look at what the Common Core and the subsequent testing will demand of the students, the more I become convinced that the Harkness model is perfectly suited to meet those demands.  However, to implement Harkness requires the teacher to write the questions to be used in the class, because the textbook questions by and large simply don't cut it.  It requires the teachers to get creative with the questions they ask, and to do that means that they actually have to understand the content of the course rather than just having it memorized themselves.  It requires the teachers to leave the old methods behind, because, quite simply, the old methods won't produce students who have the kind of understanding and creativity being demanded of them, not only on the new tests, but in college and in life.

Nothing about the process of implementing Harkness has been easy.  But the central exercise in the entire process for me as a teacher was the writing of the worksheets and the tests we are using this year.  It has been in the writing of the questions, in breaking free from the textbooks and truly designing the course that I have grown as an educator.  In short, it has been in putting the pieces together for myself, and not just following along with the textbook, that I have come to truly understand a course that I have taught for over a decade now.  And to be clear, it's not because the content of the course changed this year.  What has changed is the questions.

Teaching Creativity

It's been a relatively quiet week.  The part A kids are back to having consistenly good discussions, the part B kids are headed into the final set of worksheets, and I'm feeling more and more comfortable with the "basics" of running a Harkness classroom.  In short, the new has become the norm, but there is still enough of an edge to the new that seems to be keeping complacency at bay.

The only glitch to the week was the test in part B, where most of the grades were some sort of B. After revisiting the test itself and then looking at the solutions the students gave, I came to two conclusions:

(1) The students are much better now than they were at the beginning of the year in terms of persistence.  They are far more willing to dig in to an exercise and at least try something, as opposed to simply skipping an exercise and waiting for someone to show them where to begin.  The tests in my class are 6 questions long, and each of the questions is essentially an open-ended problem solving activity. The overwhelming majority of the students gave at least a partial answer to all of the exercises on the test...something that did not happen at the beginning of the year.

(2) The students are still struggling with creativity.  Some of the exercises on the test required the students to put together pieces from both this trimester and last, pieces which individually I am certain they are able to do with ease...I've seen it in action in class and on previous tests.  However, when it came to the most recent test, the creativity required to put those pieces together was lacking.  Now I am certain that when I go through the test with the class the students will respond with phrases along the lines of, "Why didn't I think of that?"  But that leads me to ask the question, "Why didn't they think of that?"  

That leads to the next obstacle for me in the classroom: how do I teach creativity?  The persistence in problem solving the students have gained throughout the course has been acquired by asking the students to problem solve, a lot.  But the problem solving has been within certain "boundaries" until this test.  Specifically, the material on the tests has been essentially contained to the skills required to solve the exercises on the worksheets.  Some of these skills were from previous units or even previous courses, but the test questions did not stray far from these skills regardless of whether or not the skills were new or review.  This test, however, required the kids to make use of some of the skills they already knew (for instance, one of the questions required factoring the likes of which they have seen a lot, both earlier this year and in algebra 2), but which they were not specifically required to use as we went through the worksheets.  

Fortunately, the final set of worksheets has exercises that require the use of essentially all the skills with which we have worked this year.  Those skills that have been ignored recently will be forced back to working memory, and as we do so I will be making a point to emphasize the fact that rapid recall of these skills will be necessary for success in calculus next year.  My hope is that the students, as they have done all year, will take the up the challenge and commit themselves to finding a way to rapidly recall any and all of these basic skills as required.  And honestly, I have no doubt that they will.

Slowing Down

It is a simple fact that every worksheet must have a some number of exercises, be it 5 or 50.  On the worksheets that the other honors pre-calculus teacher and I wrote to implement Harkness this year, the number of exercises per worksheet is 8.

It is also a fact that regardless of the number of exercises on a worksheet, the students will take completing the worksheet as the "goal for the day".  This has been the prevailing attitude in my classes, and it occurred to me recently that this attitude has been getting in the way.  You see, in the race to the end of the worksheet,  the students were rushing through some of the exercises, at the expense of having the good, in-depth discussions.  The reality is that even the "easy" exercises should be discussed thoroughly to make sure that the understanding the students think they have actually goes beyond the superficial and mechanical to a true, concept-driven understanding.  However, if the goal is to finish the discussion of the current worksheet by the end of the class, then the easy stuff gets pushed aside and if the difficult stuff starts taking too long, it gets shorted as well.

This prevailing attitude came to an end late last week, when I explicitly reminded the students that having the in-depth discussion of the exercises was the goal, and that getting to the end of the worksheet was not.  That day, we did not finish the worksheet and I did not freak out about it (I'm fairly certain the students were expecting me to panic at least a little bit).  In planning this trimester, we made 30 worksheets for the 60 days...clearly, even with review days and test days figured in, we were expecting the worksheets to take longer than one day apiece.  Somewhere along the way both the students and I lost sight of this, and the discussions were suffering as a result.  Since the reminder and the subsequent refocus by the students, the discussions have been going much better...and actually, that's an understatement.  So many of the students mentioned that the more deliberate pace and the lack of pressure to get to the end of the worksheet has improved the discussions and the understanding on the current worksheets...I honestly lost count how many students said this and thanked me for it.

That being said, I also realize that there is a balancing act here.  We still have a certain amount of material we need to cover, so the students and I need to find the fine line between too slow and too fast, get to that line quickly every day, and do everything we can to not stray from it.  Hmmm...life lesson?  I think so.