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.
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