Off-Topic

Over the last week or two, the discussions in my "A" section (the first half of the pre-calculus sequence) have been off task quite a bit, and after repeatedly pointing this out to the students to no avail, I asked them to devote part of their weekly journal entry to discussing why they thought this was happening and to proposing solutions.  Student after student kept mentioning the same thing as far as what they see as being the problem: some of the questions are not challenging.

Currently, part of what we are covering is exponential and logarithmic functions to which the students were introduced last year, so they do have a level of familiarity with some of the material.  (However, the way the worksheets have been structured, only two or so of the eight daily exercises are about exponents and logs.)  Since the "rules" came back to them fairly quickly, the exercises that were intended to help them understand where the "rules" came from became the easy exercises for the day.  I have a feeling that this may come back to haunt them on the test, as the questions there are intended to determine whether or not they truly understand the "rules", and are not just seeing if they can use the "rules". Regardless, the fact that the students saw these exercises as being easy was seen as the culprit for the amount of off-topic time, because the student presenting the problem at the board was being ignored by the other students at the table, believing that they didn't need to pay attention since they already understood the problem.

There were other times during the week, however, when the discussions went very well.  The students mentioned this in their journals as well, commenting that they believed the discussions went better when they encountered a more difficult exercise, since the entire group was truly needed to successfully complete the exercise.  This has led me to take a critical look at the exercises on the worksheets, since I firmly believe that the exponent and log problems are just as difficult as the other exercises.  The conclusion to which I have come is that I am assuming the students will put forth the effort to truly understand the material rather than just relying on what they already know about a topic.  Any math teacher who has had to lead students through exponents and logs is well aware that the first time students encounter them, memorization tends to trump understanding.  The difficulty I'm having is that the memorization came back relatively quickly for the students this trimester, so the struggle of trying to solve the exercises didn't happen...they relied on their memorization, were successful at solving the problems, and let it go at that.  For the record, the memorization did not return as easily last trimester, and therefore the discussions were more fruitful.  So, I'm not convinced that it's the exercises that need to be changed (and by changed, I mean made more challenging).  Perhaps the students actually do understand exponents and logs more that I'm giving them credit for, which would not really be a problem from a math standpoint, but rather from a class management standpoint: what do I need to have prepared when this happens in the future?

The other thing that the students consistently mentioned was that, regardless, not paying attention to the students at the board was disrespectful and rude.  Of course, I agree with this completely, and I intend on allowing the students to take their own admonition to heart and make the necessary adjustments themselves: if they know what's wrong, then they should be able to make the corrections without me.

In the big picture, this is again a strength and an obstacle of running a truly Harkness classroom.  Letting the students be responsible for their own education means more than guiding them through the discovery of the academic content of the course.  It means letting them learn what it takes to have a meaningful discussion and what it takes for a group to succeed.  It means letting them work through the difficulties, having them reflect on and uncover what the real cause of the difficulties is, and make the corrections themselves, rather than having the teacher impose a solution (which I did a couple weeks ago with the popsicle sticks...still deciding whether or not that was a good idea).  It means stepping back and trusting that the students can and will make the necessary corrections, and after 22 years of stepping in instead, it's really difficult to have the patience needed to allow it to happen.  I want every discussion to go well, to be as fruitful as it can be.  When it doesn't happen, my instinct is to take over, get involved, "fix" it...which is not what Harkness is about.  I need to constantly keep in mind that I am there to guide the students, not to do the hard work for them.  I've been successful at doing this with the math.  However, I still have some work to do when it comes to the actually dynamics of the discussions themselves.  Fortunately, the students have provided a good road map for me (courtesy of their journals)...which is much more in line with the Harkness philosophy.

Reviewing

If there is one thing with which I am still struggling when it comes to implementing Harkness in my classroom, it's reviewing for tests.  In a traditional classroom, preparing students for a test means making sure they have a specific set of algorithms memorized so there's nothing on the upcoming test that comes as a surprise.  Over the years, the students have taken great comfort in the fact that they are able to completely prepare for a test, and that the night before the test they can feel feel completely prepared for the test.  This does not fit well in a problem-solving centered classroom, where the focus of the tests is not on how well the students have memorized how to solve a specific set of problems, but on how well they can use a set of concepts and skills on a new set of problems.  

With this shift in focus, the review that happens in the classroom the day or two before a test is, by necessity, also very different.  In a traditional classroom, working through a few more exercises on the board for the students to see, performing a few more examples for them to memorize, or giving them one more set of questions to see how well they have the required material memorized is pretty typical.  This doesn't work in a problem-solving centered classroom, because no matter how many examples the students see, the problems on the test are going to be different.  Yes, the test questions cover the same material, and yes, the students have all of the skills required to successfully complete the exercises, but the specific applications and problems on the test are not going to be the same as the ones the students have seen.  This makes the night before a test unsettling for the students, since their working definition of "completely prepared" is "there will be no surprises" - essentially an impossibility if the goal is, in a way, to see how well they can handle reasonable surprises.  

So the question becomes - and it's the thing with which I find myself having difficulty - how do you help the students review for a test?  Based on their previous experience, students have come to expect that any extra practice problems are, in fact, of the "type" that will be on the test.  While I certainly give the students extra practice problems, the students have needed to adjust to the fact that these extra problems are, instead, at the same level of difficulty as the test questions, but are not necessarily of the exact same type.  And in that, the in-class review is of a different character as well, since it's no longer a matter of showing the kids "just one more example" for the sake of helping the memorization.  In short, the focus of the review, both at home and in class, is on the concepts.  I have come to realize that determining whether or not the students have a solid understanding of the concepts before the test is far more difficult that determining whether or not they have the required algorithms memorized, and the students are experiencing the same thing.  I need to work on this.

On the positive side, the majority of the students have the necessary mechanics on board long before the test review arrives, which means that what used to be the focus of the review is now what the students already understand and are, in some ways, taking for granted.  In terms of accomplishing the goal of improving the students' problem-solving skills, this is a huge step forward.

Uncomfortable

It seems to be human nature that when things get difficult or uncomfortable, we look for ways to regain a sense of familiarity.  This was on display in my classroom this week as the students returned from the break.  It actually started before the break, as the kids had fallen into a pattern of having whichever students at the table completely understood how to do one of the homework problems go to the board and present their solution to the rest of the group.  This led to two (negative) things:

1. they would skip exercises about which no one was absolutely confident, returning to them later but having an incomplete discussion, the lack of absolute confidence getting in the way of having a deep discussion; yes, someone would finally go to the board, but the good discussion simply wasn't happening; or
2. one of the students who did understand the exercise would go to the board and essentially deliver a mini lecture about the problem...no discussion, few questions, just passively listening to the person at the board; almost invariably this ended up being the same students mentioned above going to the board, so we ended up with a small set of students going to the board over and over again.

My hope before the break was that this was a phase that would pass, that the lead-up to the break and the excitement were getting in the way, that after the break things would return to the way they had been.  That hope was dashed on Monday, so on Tuesday, for the first time ever in my teaching career, I numbered a set of popsicle sticks and randomly assigned students to present the homework exercises at the board.  Some of the students mentioned in the journals that they wholeheartedly approved of the random assignment.  Others, of course, were not thrilled with this, at least not initially, but the complaints in the journals actually appeared in two different forms:

1. there were the students who had gotten comfortable with passively listening to the mini lecture who were now forced to go to the board and actively participate, working through a problem with which they were not necessarily comfortable at the board; and
2. there were students who were used to going to the board a lot (and liked it that way) who now were sitting at the table while the another student who did not necessarily understand the exercises was at the board.

Both of these scenarios led to better discussions.  The entire table was involved in the presentation of the exercises, with those who understood the problem actively discussing the exercise with the person at the board, and those at the table who did not understand the problem actively asking questions, both to get the information they needed and to let the person at the board know they weren't the only one who needed help.  Even the students who were not happy about the change admitted that the discussions were more thorough, more detailed, deeper...in short, better.

Since one of the big points of Harkness is to have the students take responsibility for their education, I plan on giving the students a few opportunities to fix the problem themselves, but the popsicle sticks will always be at the ready, just in case.

As a side note: it began to hit me just how many of the "best practices" I've heard over the years, and tricks to make them happen, are coming into play this year.  I've heard of other teachers using some sort of random assignment for presenting homework exercises at the board, but the time constraints that exist in a classroom where lectures are the predominant method of information delivery make this less than desirable (you need to leave enough time for the lecture, so wasting time sending a student who did not or was not able to do a homework problem to the board - essentially by themselves - is not a good idea).  This year, however (or at least this week), the random assignment made sense and promoted better discussions. Of course, the students at the board were not alone, and the lecture-induced time constraint is not there, so the popsicle sticks made sense.  There have been several of these revelations this year, pieces from different methods, parts of best practices, that all seem to be coming together in a seamless whole.  Put simply, Harkness seems to be the perfect blend of all of the best practices I've heard over the years.  Hopefully, it will continue to reveal itself as such.  And honestly, I have no doubt that it will.