by James Hershberger
As the associate chair of a university mathematics department and as an unabashed proponent of the need to reform mathematics curricula and instruction in both school and university mathematics education, I take the stance that all reform efforts successfully address some of the intended problems, create new problems, and occasionally result in surprises (both positive and negative) in unintended areas. This essay addresses issues that the NSF-sponsored integrated curricula create for university mathematics departments.
The first issue is that many chairs do not know what integrated mathematics is in the NSF sense and that they are only peripherally aware of, or involved with, NCTM's Principles and Standards for School Mathematics (Reston, Va.: NCTM, 2000). As a consequence, their first interaction with integrated mathematics often comes after a call from the admissions office, where someone wants to know whether the university should accept these courses as college-preparatory mathematics. The course titles suggest to many in admissions that integrated mathematics is not college-preparatory mathematics. Admissions personnel are accustomed to high schools and states that create courses with such titles as "Intuitive Geometry" and "Applied Algebra" for non–college-bound students. Persistent parents often force the issue to the mathematics department, in the hope that a more informed decision can be made.
Now the chair has to answer the question: is integrated mathematics appropriate preparation for university work? Available data indicate that it is, even when the question is framed in different, legitimate ways. For example, one question that high school faculty have had to address is whether integrated mathematics prepares students to take Advanced Placement calculus or Advanced Placement statistics. One can argue that an answer of yes is an indicator that students are prepared for university work. The Core-Plus Mathematics Project (CPMP) and the Interactive Mathematics Program (IMP) both have evaluation data that support the claims that students who successfully take three years of integrated mathematics are well prepared to take Advanced Placement statistics and that students who successfully take either three years of integrated mathematics followed by precalculus or four years of integrated mathematics are well prepared to take Advanced Placement calculus.
A second way to address the readiness issue is by asking how well students who take integrated mathematics fare on traditional examinations or tests. Again, evaluations of the integrated projects support the claim that integrated mathematics is appropriate preparation for college work. Analyses of SAT and ACT data consistently show results at least as good as those for similar students who take more traditional coursework.
A third, and possibly most appropriate, way to frame the readiness question is to ask how well students who have taken integrated mathematics have done in university courses. Now we are in an area in which universities have seldom operated. The data that are desired are virtually nonexistent and are difficult to obtain even when one knows what is needed. People tend to operate from anecdotal information. The best study to date has been done with students from Bloomfield Hills, Michigan, who have attended the University of Michigan. Data indicate that students from the high school that uses CPMP have performed better in a variety of university mathematics classes than students from that high school before CPMP was used, and they have also performed better than their peers from the system high school that uses a more traditional approach.
I do not pretend that the preceding paragraphs will persuade someone that NSF-sponsored integrated mathematics curricula are appropriate preparation for college. I think that an integrated approach is appropriate and that it can offer better preparation than traditional coursework. Even so, issues remain: placement, for example. Suppose that students complete an integrated high school curriculum with A's and B's. Those students, as well as students with similar grades in traditional programs, should be ready but still must be placed in the appropriate university course. As in the preceding example concerning how well students do in university coursework, universities have little experimental data to evaluate placement processes. For example, suppose that course A is a prerequisite for course B. Do we ever administer the placement examination to the successful students in course A? What if we did, and significant numbers were still not deemed "ready" for course B? Is that a statement about the placement examination or a statement about course A?
My own preference would be to create articulation arrangements between the university and high schools so that students could seamlessly enroll without a placement examination at all. However, my university's precalculus classes are fairly traditional. Where would someone start who has taken integrated coursework? Even though the programs have as a goal that students will be able to perform symbolically as well as those in traditional curricula, do they? All the reform curricula—whether integrated by title or not—use technology to a greater degree than traditional programs. Are those students inadequately placed if they take paper-and-pencil placement examinations? Do they have an advantage if they are allowed to use graphing calculators on the examinations?
The one great advantage that university mathematics department chairs have in dealing with these and other issues is that they have usually already begun discussing them in light of their own curriculum. The arguments among university faculty concerning reform calculus are virtually identical to issues involving integrated curricula. For example, will the heavy technology use in reform calculus hurt students as they later take what have traditionally been—and are likely to remain—abstract courses, such as modern algebra? What has been left out as students do all the applications, projects, and cooperative activities? What do students know or not know, as contrasted with past students?
The mathematics and mathematics education communities are struggling with the previously mentioned issues and many others at all levels of instruction. I do not know anyone who thinks that all elements of traditional curricula and instruction are undesirable. Conversely, even avid proponents of reform curricula and pedagogy must admit that their efforts are only beginnings. I hope that we will soon be able to get past what I see as essentially political arguments and get on with the task of working to improve the performance and achievement of all students at all levels of mathematics education.
|Jim Hershberger, professor of mathematical sciences at Indiana University—Purdue University Fort Wayne and the Friends of the University Teacher of the Year in 1989, was a member of the Indiana University Faculty Colloquium on Excellence in Teaching in 1992. In 1989, he was the director of the Indiana Governor's Scholars Academy. He is currently a codirector of the Indiana Mathematics Initiative, a National Science Foundation grant in which faculty from Indiana and Purdue University support teachers from 10 urban Indiana districts as they implement reform mathematics curricula in the middle grades and high school.