From the Origin provides a forum for lively discussion of issues of importance to the mathematical community. The Michigan Section-MAA Newsletter solicits opinion pieces for publication in this column from anyone in the Michigan mathematical community. In addition, comments on pieces published in earlier issues are welcomed.
Items for From the Origin should be submitted to the editor by the beginning of October to be considered for inclusion in the December issue and by the beginning of February for the April issue. Main opinion pieces should be at most 1800 words long, and responses at most 400. The editors reserve the right to shorten responses, if necessary, in order to fit as many as possible within the available space.
Most of us are aware of the growing proportion of college students taking courses at a level below calculus. I am talking about precalculus courses and the courses leading up to them, not courses that are taught at the same level as a first year calculus course that do not use calculus. There may be both good and bad aspects of this, but it is surely troubling that many such students took calculus courses in high school. While this is a difficult and multifaceted problem, there is one aspect that we can address with a realistic hope of making some improvement.
Some students wind up taking precalculus in college because they took a "calculus" course in high school having skipped over the standard precalculus topics. These students are not doing this because that is what their high school teachers want them to do. Conscientious high school teachers often feel pressure from parents, administrators, and school board members to teach courses that are not the ones they would professionally consider the most appropriate courses for their students. This is nothing new, but the increased pressures of "accountability" on just about everyone in K-12 education only intensify this problem. I speak as a second-term school board member.
Some high school teachers have suggested that it would be helpful for them to be able to point to position statements from an organization such as ours to buttress them when they need to defend themselves from such pressures. The statement below was drawn up in response to those suggestions. We have discussed this statement and similar ones within the Michigan Section's Executive Committee and at the Building Bridges Workshop at the last annual meeting of the Section.
I would like to move these discussions to the general membership level. The general question I propose is, Should the Michigan Section issue a position statement along these lines? In doing so, I want to point out several considerations to bear in mind.
Both AP (advanced placement) and IB (international baccalaureate) calculus are rigorous courses with proven records. Many high schools offer these courses to the benefit of their students. They have been singled out because they are the two nationally recognized such programs. It is quite possible that other, similar programs will come along. The statement is not intended to confer a monopoly on these particular programs.
Another consideration is that this is a very small issue. Many would argue, and I would be hard put to dispute, that there are more important considerations such as whether to introduce alternative courses in both high school and first year college. I believe that we will do the most if we focus on this small issue and leave the bigger issues for a later discussion.
In contemplating an official statement, it is essential to recognize that such things need to be done carefully. Advice from outsiders is not always welcome and may in fact be counter-productive when it is not tendered in a constructive manner. Relations between high school and college teachers have generally been good in my experience, but they have been far from ideal in many instances.
Please read this statement as a proposal to think about and discuss, not as a completed document.
Statement on Preparation for College Mathematics Work
It is vitally important for students to receive adequate preparation to succeed in college level mathematics courses. Such preparation includes the development of skills and capabilities as well as the acquisition of a particular body of knowledge.
For students who will eventually take calculus, the skills include the manipulation of algebraic expressions, or mathematical statements. It is also extremely important for students to develop the capability to convert ordinary statements into mathematical statements. This requires extensive practice. In addition to mastering algebraic techniques, students should have a knowledge of geometry, the rectangular co-ordinate system, and transcendental functions, as well as basic trigonometry.
Experience has shown that it take four years of systematic course work for most students to develop these prerequisites. In practice, this means that students will generally need to have developed substantial algebraic skills prior to entering ninth grade if they are to benefit from a calculus course before they complete high school. Students who have not mastered a significant amount of this material before entering ninth grade and who aspire to take calculus should be encouraged to take mathematics courses in high school that will prepare them to take calculus in college.
Recommendation
Calculus courses are appropriate for high school students when both of the following conditions are met.
Many students will be better served by a course that develops the fundamental skills, capabilities, and knowledge needed to master calculus than by a calculus course that does not lead to college credit. By making such preparations, they will stand a much better chance of success in their college level mathematics courses.
- The students are adequately prepared. This means that they have both mastered the prerequisite material and developed the capabilities and skills.
- The course is assessed by a widely recognized organization and will lead to college credit. Examples of such courses include the AP and IB courses.
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