Greetings, MVC Teachers and Friends,
CALCULUS IN SPACE: I was notified about a new publication, "Calculus in Space", by one of our group members (Alan Lipp). Here is a link to the product page and a sample chapter.
Finally, here is some content. Hopefully it informs and raises some issues that we can discuss.
THE PATH TO MULTIVARIABLE CALCULUS
For those of you not teaching at a public magnet school or a high powered private school, here is how our oridinary school managed to create enough demand to offer a Multivariable Calculus class. I offer only one person's opinions and I hope that they lead to an interesting discussion. Being a California public school, we needed to find at least 25 qualified students although some schools may be able to find funds to offer it with fewer students. The main issues are finding enough prepared students and developing a curriculum to allow them to complete BC Calculus by the end of their junior year. Of lesser concern is how to get the class approved (you can tell from the last two sentences that I am a teacher and not an administrator).
The path starts with students taking Algebra I in 7th grade. I have been a vocal critic of this hyper-accelation. It came about in California when the state decided that Algebra I would be the standard 8th grade class. The consequence was that most parents who thought that their child was above average wanted them to take Algebra I in 7th grade. Then the parents who thought their students were average, wanted them to take Algebra I in 7th grade for social reasons. The adverse result was many students who could have done excellent work in Geometry through Calculus in high school were entering as ill-prepared Algebra II students. This was either because they took math classes they weren't ready for, or because they had watered down classes (often both). Ten years ago, any 9th grade student in Algebra II earned an A+. Now the 9th grade students were getting from an A+ to a D. We have stemmed the tide a little by articulating with the feeder schools and giving them a list of rigorous topics in Algebra I and Geometry the students need to have mastered before being recommended to take Algebra II in 9th grade.
My personal feeling is that students who reach Multivariable Calculus should also be able to qualify for the AIME exam. The operative word is should as a few of my students don't meet this criteria and we do need to find at least 25 in order to offer a class. Working backwards, this means that a 9th grade Algebra II student should score 90 or higher on the AMC 10. Only a few of my students meet this requirement. However, we have a very active math club at the school and integrate discussions of math contest problems into our curriculum. The result has been about 30 students a year for the last few years taking the AIME. This is from a school that usually only has two or three national merit semi-finalists (the number nationally of AIME qualifiers and national merit semi-finalists are fairly comparable). Hence, whereas Multivariable Calculus should be for the top 1 or 2% of students, you don't need to restrict 9th graders in Algebra II to the top 1 or 2% of your students. With a good high school math program, maybe even the top 10% of 9th grade students belong in Algebra II.
The next question is what do you teach in Algebra II and Pre-Calculus to get the students ready for BC Calculus as juniors? Our Algebra II class is entitled Algebra II - Trigonometry. We review all the Algebra I topics (filling in large gaps for many students), cover all the Algebra II topics, and teach a basic Trigonometry class. The class has a great amount of breadth and little depth. I could never teach it myself, but it serves a valuable purpose. By the way, we also offer a couple of sections of Algebra II for students just trying to complete the requirements for entry into state colleges.
The really interesting class is our Pre-Calculus. It is entitled Analytic Geometry - Calculus Honors. It is just the opposite of Algebra II - Trigonometry; it contains a lot of depth and conceptual work. We use Paul Foerster's Pre-Calculus book and spend the first quarter doing statistics, probability, and data analysis. The students learn Excel and do mathematical modeling. We do this first quarter so that students can double up in AP Statistics without having to have first completed this class. The second quarter consists of vectors and a function approach to trigonometry. The third quarter curriculum is analytic geometry, polar coordinates, parametric equations, and infinite series. In the fourth quarter, we do the first three chapters of Foerster's Calculus book. These too are mainly conceptually. Finally, over the summer, the students do five sections from the old Protter-Morrey Calculus book focusing on algebraic manipulations in Calculus.
The final issue is administrative and probably different at every school. We set up a partnership with a local community college that offers Multivariable Calculus. They offer a section on our campus with me as the teacher. It is open to the public, but we haven't had an outside student yet. We are using the UC Berkeley Math 53 syllabus, textbook, and assignments for the class as many of our students apply there for admission. Between the Math 53 syllabus and what I cover beyond that, we satisfy all of the requirements of the community college and then some. We also happen to have an old approved class on the books at our school entitled Independent Study Advanced Placement Calculus Honors. We use that class to give the students dual credit (high school and college) for Multivariable Calculus. This has enabled us to avoid having to go through the time consuming task of creating and getting approval for a new course at our school. I'm trying to avoid this since hopefully the College Board will be offering an AP Multivariable Calculus class in the near future.
Bob Enenstein
Carlmont High School
Belmont, CA
HOME: gplus@telis.org
SCHOOL: renenste@seq.org
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