From The Origin

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.

A Comparison of High School and University Math Performance

by Richard O. Hill (MSU)

Over the last decade, it slowly became apparent that there are things in the background of some of our incoming freshmen that we did not understand. For example, if you look at the students who place into technical calculus vs. those who place into precalculus, there are small differences in ACT scores, small differences in high school GPAs, but very large differences in what they can handle and how hard they can be pushed. As a related issue, there has been considerable controversy over the effectiveness of reform mathematics curricula. In order to get some insight into these issues so that we might better understand our students and hopefully serve them better, about two years ago I organized a study aimed at comparing the high school and university mathematics performance of students. The study involves some 3000 students who graduated from mid-Michigan high schools and came to MSU in the years 1996–1999. The high schools provided us with what math courses, if any, each student took their senior year together with the grades in those courses. We matched these with ACT scores, MSU math placement scores, what math courses they took here, and grades in those courses

For me, one of the most pleasing aspects of this project has been my interaction with high schools. I have had good conversations with teachers, principals, and administrators. The success of this project depended on their cooperation in providing data. In all, 34 high schools participated in the study. In return, we provided two things to the high schools: first complete confidentiality in their data and second an individualized report on their students. Most schools were delighted to have this feedback about their mathematics program, information otherwise extremely difficult for them to get.

The rigorous analysis of the data has just begun. Thus please keep in mind that the conclusions given below come from the rough overview we have from compiling the individualized reports, unless otherwise indicated. This article is just to give an idea of what we are doing; papers with detailed results and the statistical analysis will be forthcoming. Here are some of the things we have observed.

  1. Schools vary tremendously as to the number of courses, kinds of course, and effectiveness. Our data agree with the TIMSS results that how well a student does in college significantly depends upon which high school the student attended.A Another difference for schools, which is particularly troublesome, is that the percentage of students who come to MSU having taken no math their senior year varies from 0 to 38%.
  2. AP Calculus works well and at some schools it is spectacular. By this we mean that students who do well in an AP calculus course do well in whatever math course they end up in at MSU. If they do less well in the AP course, they generally do less well at MSU. The first thing that you can ask of a high school is that their grades reflect how much the students learn; this seems to happen in AP calculus. However, initial statistical analysis seems to indicate anomalies (possibly caused by some students who take AP Calculus but are not serious about it, overestimating what they have learned, not working at college, and then doing badly in the college courses).
  3. About 25% of incoming freshmen eventually take and pass technical calculus 1. We will try to see what we can say about the success of all students.
  4. Precalculus varies tremendously. Some schools’ precalculus courses (which have various names) prepare students well for beginning university-level mathematics. On the other hand, there are others that produce almost no students who go on to pass MSU’s first technical calculus course.
  5. “Value added” happens at some schools during the senior year; thus ACT scores (which are from the end of a student’s junior year) should not be used for math placement. Some schools have a significant number students score in the mid 20s on the ACT math test (which would not place them into calculus) but score high on MSU’s math placement test, take calculus, and do well there. On the other hand there are schools where this does not happen. This is another of the differences among schools, and it is one reason why it is crucial for colleges and universities to have math placement exams.
  6. AP Statistics seems problematic and hence this requires further study. A high percentage of students taking this course perform just slightly better than students who take no math their senior year. Two high school teachers teaching AP Stat explained to me that only about half of Algebra 1 is required for this course. I do think that statistics is important, but instead of a whole course it should be at most a part of a course that is algebraically demanding if the students are going on to college. This is born out by our data on probability/statistics courses in general.B
  7. Who does poorly at MSU? Roughly 25% of incoming freshmen place into remedial math at MSU. I have heard some criticism of MSU for this. However, roughly 35% of these students took no math their senior year and another 45% either took a low-level course (e.g., Informal algebra, business math, etc.) or did poorly in what course they took. Roughly 80% of the students who took no math their senior year placed into remedial math or college algebra (a high percentage doing poorly) or took no math at MSU. We hope to end up with a description of which students place into these two lowest courses and which place into the rest (the rest satisfy MSU’s math graduation requirement).
  8. Are high school grades too generous? The answer depends upon both school and course, mostly not in upper level courses. However, in some low-level courses a really bad grade is a B+. This is one reason why high school overall GPAs do not mean too much in terms of comparing students; you need to know the courses.

Reform Mathematics. I would now like to turn to our attempt to find out something about the effectiveness of reform mathematics, at least for students coming to MSU. Reform movements, and often-associated controversy, have been in U.S. education over 150 years. Diane Ravitch in Left BackC gives a scholarly discussion and is critical of educators who draw conclusions with insufficient hard evidence. The current wave of reform math curricula, Core-Plus being one, is based on recommendations of the 1989 StandardsD,E. Unfortunately it is easy to misconstrue the intentions of those recommendations, and some reform projects way overemphasized problem solving and calculator usage at the expense of learning mathematical skills and relationships.6 Fortunately, the 2000 StandardsG made significant steps in the right direction.H,I

Core-Plus is one of the most popular high school math reformed curriculums in Michigan. It has been the subject of controversy, even math wars, at some places. You can find out a lot about it from the developer’s viewpoint from their web site, http://www.wmich.edu/cpmp. Some high schools have tried it and love it; they feel their students are really benefiting from it, staying in math longer. Other high schools have tried it and dropped it, often saying it is algebraically weak. Rhetoric abounds. We really wanted to know what the facts are, so we invited several Core-Plus schools to participate in the study.

Schools use Core-Plus in roughly two ways. Some schools use it as the main sequence; others use a more traditional series for the main sequence, while using Core-Plus as an option for weaker students. Several teachers using it the second way told me how much they like it and how much better the students understand things and are liking math more. From our study’s perspective so far, these students performed roughly the same at MSU as students with similar ACT scores from traditional courses. If a student enjoys math more this is a positive, but we are not sure how to measure this.

We approached six schools that use Core-Plus as their primary curriculum. Unfortunately, five of them turned us down,10 so we are proceeding with the data that we do have. The results are interesting, but difficult to summarize in a fair way in just a few sentences. They will be published in due course.

Conclusion. The purpose of this study is to find relationships between students’ math performance in high school and college. Although we are getting some insight, not surprisingly, we found many areas where more study is needed; i.e., we ended up with more questions than answers. I look forward to the response when the data are analyzed and published.

Post comment. This study required me to seek the collaboration of high school teachers, principals, and administrators. This interaction has been very enjoyable and the results interesting. There are other examples in Michigan of constructive interactions between college and university math faculty and those involved in K–12 math education. The sessions organized at this year's Annual Meeting by Roger Verhey (UM–Dearborn) and Tim Husband (Sienna C) is one example. The collaboration of Deborah Ball and Hyman Bass (UM–Ann Arbor) is another. There are others. However, it is important for more mathematicians to get involved in K–12 education, both to work on the plethora of problems and to be at the table for discussions. I hope others are encouraged to do so.

(Footnotes)

  1. William H. Schmidt, et al., Why Schools Matter, Josse-Boss, Inc., San Francisco, 2001.
  2. Please look at Joel Best, Damned Lies & Statistics: Untangling Numbers from the Media, Politicians, and Activists , UCPress, 2001, about $14 from Amazon.com. This would be a wonderful book to use as a statistical supplement in a course.
  3. ISBN 0-684-84417-6.
  4. Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics, March, 1989.
  5. The 1989 Standards were in part a response to calls for reform in A Nation at Risk: The Imperative for Educational Reform, National Commission of Excellence in Education, Washington, D.C., U.S. Government Printing Office, 1983.
  6. For example, one principal at a high school insisted that, because of technology, it is no longer necessary for students to learn to factor simple quadratic expressions.
  7. Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, 2000.
  8. For example, see p. 24 for their statement about calculator usage, and p. 35, about needing a balance between conceptual understanding and computational proficiency.
  9. Prof. Manuel Berriozabal, UT San Antonio, discussed this and related issues in a MAA Invited Address, Reforms in mathematics education: Best practices and malpractices, given at the AMS/MAA 2002 Joint Meetings in San Diego. A greatly expanded version of that talk can be found at http://www.math.utsa.edu/~prep; click on January 9, 2002 presentation.
  10. Interestingly, the one that participated looks good. Their top course is AP Calculus, its teacher said students coming into it are more confident, more take and pass the AP exam, and “gateway exams” are used to ensure manipulative skills. Their students do well when they come to MSU.



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