The Value of Patience

I have mentioned in the past that part of the preparation for the change from a standard lecture-and-mimic classroom to a Harkness classroom was creating the worksheets for the class, since the textbooks we have (and, in fairness, none of the textbooks I have seen) have the types of questions we need, let alone in the order we wanted.  One reason for this is that we chose to not teach the material one chapter at a time, but rather to teach all of the topics for the course concurrently.  So, for the first half of our pre-calculus sequence, this means that pretty much every worksheet has exercises about polynomials or rational functions, and exercises about exponential and logarithmic functions, and exercises about discrete mathematics (sequences, series, binomial theorem, and the like).  Throw in a review question and a "think-outside-the-box" question (the online materials from Exeter are great for this), and you have a good worksheet.

However, one of the struggles we had was determining the pacing of the material.  We needed to create questions that led the students to discover the new material on their own, that challenge the students, that "stretch" them every day, but that don't make such a huge leap that the students can't put the pieces together.  For example, today's worksheet asked the students to evaluate 7 nCr k for k from 0 to 7 inclusive.  Monday's worksheet asks them to expand (a+b)^7.  Layer by layer, worksheet by worksheet, the material contained in the section on the binomial theorem will be covered (and thus, discovered).  However, at the same time, other exercises on the same worksheets are building the layers of solving polynomial and rational inequalities.  Other exercises are building proof by mathematical induction.  You get the idea.  This is a drastic departure from the "normal" way of delivering the material, where all of the material for the binomial theorem is delivered in a lecture on one day, all the material about proof by induction is on a different day, and the material about polynomials was covered weeks ago.  In doing this, one of the huge advantages is that the students are working with all of the topics, at some level, throughout the trimester.  I'm hoping that this will pay off on the trimester exam, since all of the topics will have been seen recently, and none of the topics will have been essentially ignored for several weeks.

On the other hand, there are times when a question on a worksheet seems to come out of nowhere and seems to not be connected to anything we are doing.  A good example of this is the question about 7 nCr k on today's worksheet.  We have not done any combinatorics or factorials this year, and the question is not immediately put into the context of expanding binomials.  It's just sitting there, asking the kids to make the calculation.  Even at this stage of the trimester (we are 8 of the 12 weeks through the term), when they know they need to be patient and let the worksheets gradually reveal the material to them, the students are a bit irritated by the fact that the some of the material seems to move so slowly.  Or at the very least it gets to them that they have to wait for the "big picture".  From my standpoint, several good things come from this:

(1) the kids are working on more than one topic at a time;
(2) the kids get to see and/or use the basic mechanics of a topic over several days, which gives the mechanics more time to sink in; and
(3) there is an almost constant sense of anticipation, waiting for the pieces on which we have been to finally come together.

In the "instant knowledge" world in which these kids have been raised ("I'll just look it up on Google."), this kind of patience is not easy to for the kids to handle.  But, in the "real world", whether it is scientific research or business strategies, this kind of patience is required.  So, in addition to promoting a better understanding of mathematics and having a better grasp of what problem solving is really all about, Harkness promotes and teaches patience.  Sounds good to me.

Discussion vs. Debate

One of the things I have been watching in my classroom during the past week is the dynamics of the different classes.  Two of my classes are very much into discussing the material, making sure every one understands an exercise before moving to the next one, putting different methods of solution to the same exercise on the board for comparison, and in some cases even putting an incomplete solution on the board so the group can finish it together.  For lack of better terms, these two classes "get it", and I couldn't be happier with the progress they are making.  The students in one of my classes are still very hesitant about putting solutions on the board unless they are absolutely certain they have a correct answer.  This has led on many occasions to one person putting multiple exercises on the board, and rather than having a discussion it becomes a student lecture.  Because of this, on Monday a "new rule" will be put in place: no one is allowed to put more than one solution on the board on a given day.  Finally, there is my remaining class, a group that is very much into debating one another, trying to prove "who's right" and, by implication, "who's wrong".  Unlike the other classes, solving the exercises in this particular class has become more of a competition rather than a collaboration.  Though they make a reasonable amount of progress each day, they are nowhere close to as productive as the other periods.  It often takes then a lot longer to complete an exercise, if for no other reason than neither side will "give in".  This really becomes a problem when both of the students are actually correct, but have used such different methods to solve an exercise that the answers they get look as different as their methods, despite the fact that the answers are actually the same.

The difficulty for me is that, despite the extra time and the difference in "styles", the discussions being had by the debate class are still really good.  The students are getting into the depths of the exercises and as such I don't really want to stop the debates, if for no other reason than debating really does seem to fit the personality of the class.  They come in each day, ready to state their case (or if they are unsure of their answer, to take sides between two people who are sure of theirs), they work through the exercises, and leave at the end of class still liking each other.  Who am I to complain about that?

On the other hand, the students have gotten used to completing one worksheet every day, and there are times when a debate that occurs makes that impossible.  And there is the matter of making sure this group makes close to the same amount of progress during each class period as the other classes.  Several of the students are very uncomfortable with falling behind (and told me so in their weekly journals), so Monday I will need to try to strike the balance between making sure the discussions go just as deep, but that they are doing so in a more efficient manner.

Despite it all, I am seeing the benefits of Harkness.  I am seeing how much better, how much more permanent the students are learning and understanding the material.  And because of that, I'm more than willing to continue to work through the difficulties, and work with the students to make the class everything I know it can be.

Summary and Review

Over the course of the last few weeks, one piece of the "puzzle" has been lost in the discussions, and that is the importance of review.  By this, I don't mean reviewing the material at the end of a unit immediately before a test, because we have been doing that.  I mean reviewing every day, for about five minutes at the end of the period, summarizing the stuff we covered that day, and I'm as guilty of forgetting to make sure this happens as the students are.  My recommendation to them from the beginning has been to summarize each problem once it has been completed, writing a clean set of notes in a separate notebook that lists the important skills, formulas, etc., that were needed to solve the exercise.  Lately, however, this piece is being left out, due to time constraints (shortened bells) or just getting so deeply involved in the discussion that  the end of the class arrives long before any of us are expecting it.  And while I love it when those discussions happen (and they have been happening more frequently), the price to be paid is the students not having a good set of notes from which to study when the test rolls around.  So, the sacrifice of leaving some questions unanswered until the next day is going to have to be made.

I also noticed this week a particular quality of the more effective groups: they all have the student at the board explaining their exercise as they are writing, rather than writing everything on the board and then explaining what they wrote.  The group is more actively engaged in the problem from the beginning, the mistakes get corrected earlier in the process, and students at the table are not just scribbling down what's on the board, but instead are asking questions about what is being written and including the explanation of the process in their notes.  So, I mentioned to the students on Friday (the beginning of the new unit) that this seems to be more effective, and so far...OK, today...the groups were all working in this way, and in that the discussions were more effective.

Finally, I mentioned above that the good, deep discussions are happening more frequently.  One of the reason I believe this is happening is that the material on the worksheets is taking more than just recall, more than just brushing off the dust and getting the algebra 2 material back up to speed.  The "new material" from the beginning of the year is now being used, and the "newer material" is being built on top of it.  Because of this, the discussions are not only covering the introduction of the new material, but are also solidifying the "older new material".  And in that, none of the material is being forgotten...at least not for long.  My hope is that this will pay off at the end of the trimester as we review for the exam.  I guess we'll see.

A Normal Day...

The link below is not intended to show anything other than what a normal 10 minutes of class looks like.  The video was taken this morning.  I didn't "hand pick" this because it shows something over-the-top spectacular.  Instead, I picked it because it is a good example of what normal looks like.  The three conversations, me floating from group to group answering a few questions, the kids presenting and explaining the material...in other words, the new normal.  This is the first day of a new unit, and the problem they are working through is a review exercise from algebra 2.

Class Video - 5 October