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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