A “structuring precalculus” tangle
A month ago I asked for comments on what might belong in a precalc course. I thank my readers for a nice group of responses, running through a thoughtful range of views. I will return to some of them, but later. First, I need to explain who this course is for. Think, for a second. It makes sense to design a course not just in the abstract, but to meet the needs of the students who will take it.
My school is just 9 years old. And we have tinkered several times with our math sequences. And we are in year 3 (the interesting year) of phasing in one which I believe we will keep.
Philosophy, first. We go slowly. Very bright kids, good at math (they all can add fractions), but we move more slowly, with more depth, than at similar schools. Choosing between acceleration (really just jumping to the next course) or moving from knowledge to mastery (staying in a course where the student already has some, but not complete, knowledge) we choose the latter.
We offer two placements for incoming freshmen – Algebra (honors), full year, or 2nd term Algebra (honors), starting Geometry in the Spring.
(Aside. Previously we started the “advanced” group in Geometry. We found that when the work was easy, we got immature 9th grade behavior that made the class hard to teach. When the work got hard – first definitions in a statement/reason system – we got immature frustrated 9th grade behavior. Our school gets kids from different middle schools with very different levels of preparation – and we could get kids acting out from the work being ‘too easy’ and others acting out in frustration, at the same point, in the same lesson.
Further, we found that kids who had “seen” everything in Algebra I in middle school usually raced through it. They might have excellent scores on the Regents, but the exam is only 50% Algebra, and that part is quite shallow. Our freshmen, we explain to them, have taken “tourist algebra” – seen all the sights, but not stayed long enough to appreciate them. So kids with good Regents Algebra benefit from 1 term of more challenging algebra where we might not need to dwell on the basics. They also get one term of hard high school under their belts, with some maturation that leads to more success as we start proof-based geometry).
And we added what I think is a wonderful twist: A student in our regular level who does very well for the first two-and-a-half years, can in the second term of junior year take the second term of Algebra II (this would be normal), and the second term of precalculus (this is the twist), in preparation for joining the more advanced students in AP Calculus AB the following year. We’ve made “moving up” easy.
The trick – what should that precalculus class look like?
Our three sequences:
|regular||regular||regular (adv)||regular (adv)||advanced||advanced|
|9th Grade||Algebra (1)||Algebra (2)||Algebra (1)||Algebra (2)||Algebra (2)||Geometry (1)|
|10th Grade||Geometry (1)||Geometry (2)||Geometry (1)||Geometry (2)||Geometry (2)||Algebra II|
|11th Grade||Algebra II||Trigonometry||Algebra II||Trig + Prec (2)||Trigonometry||Precalc (2)|
|12th Grade||Precalc (1)||Precalc (2)||AP Calc AB (1)||AP Calc AB (2)||AP Calc AB (1)||AP Calc AB (2)|
So we’ve got an interesting structural precalc issue.
- The course serves as a full year course for students who will not take calculus in high school, but most of whom will take calculus in college.
- The second term alone provides a one-term bridge for advanced students from Algebra II/Trig (taught, as we are in New York State, with assorted regents content, ie, a little standard deviation and normal curve, a little calculator regression, just to add breadth and steal depth) to Calculus.
- And there will be a number of students who take the 2nd term of precalc simultaneous with the second term of Algebra II/Trigonometry, and that will lead them to AP Calculus.
Next post, I’ll discuss the issues relating to choosing which topics we chose to include, which we decided not to use, and how we are sequencing the course. And in a final post I will discuss my experience with giving only quizzes (no tests), an interesting, yet standard-free, experience.