Integrated Math, Part 1 – Don’t Grant the Premise

As I’ve mentioned before, a couple years ago we made a decision as a department and school to replace our Algebra 1 through Precalculus courses with a sequence of integrated mathematics. Since September, a colleague and I have had a one-course teaching reduction so that we could develop these courses. In addition to developing the curriculum itself, we’ve been working on administrative details, including a clear, concise justification of why we’re making this curricular move. We feel pretty good about the justification at this point; in one concise, well-written page (for which I deserve none of the credit), it describes the reasons for moving to integrated math in a way that we can easily convey to prospective students, their parents, and other interested constituencies.

What we don’t have is a similarly strong and concise justification for colleagues in the math department, teaching candidates, and math teachers from other schools who might be interested in what we’re doing. This hasn’t been a pressing concern, but it is something I think we need. We don’t yet have complete buy-in from our department, which isn’t surprising since we’re talking about something that hasn’t been taught before and is still in development. As we continue to talk about it, it’s time to put something in writing that will serve our needs now and for the next few years. I was having trouble figuring out how to structure the argument until recently, when the  College Board’s AP Calculus listserve took care of it for me.

A few weeks ago, there was a question on the listserve about course sequencing. The question, which is frequently posed in math departments around the country, is which is a better sequence of high school mathematics courses: Algebra 1 – Geometry – Algebra 2 – Precalculus, or Algebra 1 – Algebra 2 – Geometry – Precalculus. There are benefits and drawbacks of each option, the most signficant of which revolve around when students forget the algebra they’ve learned and how much of that algebra will have to be reviewed, or retaught, later.

The debate over when in the sequence to teach geometry makes an implicit assumption: high school mathematics should be taught using these four courses to lead to calculus. While mathematicians are taught to look for fallacies in arguments, we seem to miss this one quite often. To argue over when to teach geometry is to grant the premise that these four courses are the best way to prepare students for calculus (and, more broadly, to be mathematically literate adults). If we refuse to grant this premise, the answers to the question suddenly become very different.

So let’s not grant the premise, but instead start with the questions of how do we best prepare students to study calculus, understand what it means to think mathematically, be mathematically literate, and see that there is much more to mathematics than solving equations. These were the questions that we started with and have kept in mind. In addition, we  ignored “the way things have always been done,” what textbooks are available, or what we’re most comfortable with (but, because we’re a college prep school,  we did pay attention to the SAT, ACT, and the college process). By not granting the premise, we came up with what we believe is a strong sequence of courses that pursues these goals. Over the next few weeks I’ll start to share some details of how we’re building these courses, what content we’re including (or excluding), and why we made the decisions we made. I hope people will tell us that they think and (especially) what hidden assumptions exist in our own work that we might have missed.

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The Day My Students (Almost) Invented Calculus

One of my favorite things to do in Precalculus Honors is ask my students, “How do polynomials behave?” They started working on this deceptively simple question in early November and are still plugging away at it. In the process they’ve had to come up with a definition of polynomial, decide what we mean by “behave,” figure out a way to tackle the question, and then look for patterns from which they can develop conjectures that they can make convincing arguments for. In other words, they’re being mathematicians, not just doing math. Some people just stare at me in disbelief when I tell them that my honors class is in its seventh week of working with polynomials, but if didn’t spend this much time on it or do it this way, then we wouldn’t get classes like the one we had today.

A couple of groups in the class had been working on the patterns they were seeing for turning points on a graph, which are things like those labeled “A” and “B” in the graph below:

Cubic1

They had looked at a lot of graphs and were pretty convinced that for a polynomial of degree n, there should be n-1 turning points. (The truth is that there will be at most n-1 turning points, but I wanted them to explore their conjecture themselves, not just wait for the right answer from me.)

While this conjecture made sense to them, it led to some challenges in addressing curves like this:

Cubic2

This curve has no apparent turning points, but they knew that it needed to have some in order for their conjecture to hold, so they spent a lot of time trying to figure out where the turning points were. They come up with something like this:

Cubic3

When they looked at points “A” and “B” they knew that they were not maximum or minimum y-values, which they had figured out happened at turning points, but they felt like there was a distinct change in the shape of the graph at these points and so these must be the turning points.

As they were presenting their work to the class, there was a lot of discussion about these non-extreme-value turning points. Everyone was having trouble figuring out exactly where they were or how to define them. One member of the group said that she wanted to think of them as the places where the rate of change changes, but she knew that wasn’t correct because she knew the rate of change (slope) was continuously changing. This led a classmate to suggest that maybe they could pick a couple of points that were close together, find the slope between those points, and use that to get a sense of the rate of change. Prompted by another classmate, he went on to say that by doing that in a couple different places they’d be able to compare the rates of change, and by keeping the points really close together they should get at least a decent idea of what the rate of change was.

In other words, he was suggesting approximating the rate of change by calculating the slope of the secant line between points that were close together.

In other words, he was about 20 minutes from developing an intuitive definition of the derivative.

I do a lot of student-driven inquiry work with my classes because to me it’s like a road trip – we have a general sense of where we’re headed, but sometimes we take a detour because something interesting catches our eye. This was one of those times. What I really wanted to do was let three or four students starting playing around with this question of approximating the rate of change. To do so would have been an amazing, fun experience that would have let them continue to develop as mathematicians. Unfortunately, it also would have meant that I wouldn’t have gotten to something like sequences and series, plus I would have stolen the thunder of the AP Calculus teacher. Still, it does make me wonder – how flexible could our courses be so that we can follow a line of inquiry without worrying about going past what’s on our syllabus?