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Home > Features > Mathematics at the School-to-College Transition
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Mathematics at the School-to-College Transition
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by Bernard L. , Madison
University of Arkansas
Fayetteville, Arkansas
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| | Too Much of the Same Thing
The transition from school to college mathematics is one of the most troublesome in all of U.S. education. More students report difficulty succeeding in college mathematics than in any other discipline, and more students are dissatisfied with their college mathematics than with any other subject. Why is this?
Part of the problem can be lack of articulation, that is, school mathematics and college mathematics not fitting together well. Lack of fit can be caused by differences in content or in teaching methods. For example, most high schools use calculators or computers, but colleges and college faculty differ widely in use of technology; teaching methods in colleges are generally different from those in high schools; students are asked to assume more responsibility in colleges; and college courses are faster paced.
On the other hand, the topics covered in the major high school sequence -- geometry, algebra, and trigonometry -- fit very well with the topics in the major college sequence -- algebra, trigonometry, and calculus. The problem is not the lack of agreement in topics covered; it is more too much agreement and too much overlap. Add to that the fact that this sequence consists mostly of material preparing students for future study rather than being an end in itself, and much of the difficulty and dissatisfaction is accounted for.
It is difficult to tell from lists of course topics where school mathematics ends and college mathematics begins. In fact, some 80 to 90 percent of the enrollments in college mathematics courses are in courses whose content is taught in high schools, and the fastest growing part of high school mathematics is in courses for college credit. Many students with three to four years of high school mathematics find themselves in a college course with content much the same as their high school courses, very often heavy on algebraic skills that they have failed to demonstrate on a placement test. This can lead to discouragement and disinterest, and trouble ensues. The abstract and technical ideas and skills of geometry, algebra, and trigonometry are best learned the first time through. During a second or third effort over the same material, bad habits and misconceptions are difficult to change, and the motivation to work hard is weighted down by the disappointment of repeating study and lack of progress. Of course, the extension of study of high school topics can lead to greater understanding, increased facility, and stronger interest.
The big overlap between college and high school mathematics has fueled the recent growth of dual-credit courses in high schools. A recent national survey1 estimated that half (approximately 3.5 million) of all the juniors and seniors in U.S. high schools are enrolled in courses that carry credit both for high school graduation and college degrees. Some of these courses are in the examination-based programs of Advanced Placement (AP) and International Baccalaureate (IB), where college credit is dependent on the score on a national or international examination and not the high school grade. But, according to the data in this report2, most (57 percent) dual-credit enrollees are in courses that, unlike AP and IB, have no uniform examination. Because the recent growth of dual credit has been so large, there are not good data on how students with this credit will fare in college. However, if nothing significant has changed except the awarding of college credit, then whatever deficiencies in the students' knowledge and skills that existed before will still exist but will be complicated with the credit issue.
The GATC Sequence
The geometry, algebra, trigonometry, and calculus sequence (GATC) dominates mathematics at the school-to-college transition. Much of the mathematics taught in GATC is in preparation for later mathematics, either in GATC or in mathematics after GATC. The rest of the content is basically determined by the needs of students in the physical sciences and engineering. For many students, this for-later-use nature obscures the utility of the mathematics, and, unfortunately, many students never get to the point where they actually use the mathematics in GATC to solve problems in their everyday lives or even in other school or college courses. Courses whose content is for later use in other mathematics courses are not usually appropriate as the last mathematics course in a student's education. Experiencing these courses as last courses is like reading only the early chapters of a Tom Clancy novel and never finding out how all those seemingly unrelated story threads come together at the end.
The best use of the GATC sequence by students is to learn the mathematics well as they proceed through the sequence and to follow the sequence to the end. Learning well through calculus is important, even if a student does not choose a college major that uses calculus. The ideas of calculus are important intellectual ideas -- the manipulative techniques are much less important intellectually. Knowing how to find derivatives mechanically may teach discipline and organization but is of little value to the student who will not continue the practice of calculus in other courses or in a chosen profession. However, understanding the ideas of approximation, rates of change, and optimization has intellectual application in many pursuits.
Students and teachers in the GATC sequence are often overwhelmed by the rapid pace of new ideas and new techniques. There seems to be little time to stop and use the ideas and techniques in real-world contexts to gain fuller understanding, utility, and facility. Consequently, too often the emphasis is on gaining enough facility to survive and move on rather than understanding. This makes it doubly important for students to continue in the sequence so that they can patch up the lack of understanding in later study after the techniques learned by cramming for survival have disappeared from their skills list.
Rush to Calculus
Calculus is generally considered the first full-fledged college mathematics course in the GATC sequence; that is why it was chosen as the AP course in mathematics almost 50 years ago. Calculus has enormous appeal academically and intellectually. It is a powerful and coherent body of knowledge, a major human achievement. Success in calculus carries great weight in student achievement. Consequently, calculus becomes a pinnacle of achievement in high school and a weighty asset for achieving entry into competitive universities. No wonder it becomes the goal of ambitious students and hopeful parents.
Geometry, algebra, and trigonometry should not be cut short just to get to calculus; a student should choose calculus only if it is a natural next step in the student's progress. Those who do not intend to pursue science or engineering in college may be better off in AP Statistics or in a course that emphasizes the contextual use of algebra and geometry, such as the College Board's Pacesetter mathematics. Calculus is a very worthy and challenging goal at the end of high school, but it is not the only such goal.
AP Calculus or Just Calculus
The purpose of Advanced Placement Calculus3 is embedded in its name, to achieve placement beyond the beginning of college calculus. Although advanced placement with college credit is preferable, credit is not the important intellectual and educational issue. Moving on and learning more is what's important. AP Calculus does enrich and improve high school education, but it should also enhance and improve college education by allowing for more advanced study and better learning habits.
For college-bound students, taking a calculus course in high school that does not result in advanced placement in college is of little value and can have a negative effect. Such courses, which are often surveys or technically oriented, take the excitement out of the college course and can lead to unwarranted student complacency. Trouble often ensues. In 1983, the Mathematical Association of America (MAA) Committee on the Undergraduate Program (CUPM) appointed a Calculus Articulation Panel to undertake a three-year study of questions concerning the transition of students from high school calculus to college calculus4. The recommendations of that panel -- three college teachers and four high school teachers -- center on such issues as this and remain relevant in face of the growth of both AP Calculus and non-AP calculus in high schools. One of the members of that panel talks about the effects of a calculus survey course in high school: 'It is like showing a 10-minute highlights film of a baseball game, including the final score, and then forcing the viewer to watch the entire game from the beginning -- with a quiz after every inning.' High school calculus that is not aimed at advanced placement has very limited educational value and can lead to problems in more rigorous courses in college.
The ideal of AP Calculus is to provide a smooth transition from school to college mathematics. The successful student should fit naturally into the calculus sequence in most colleges and universities. The content of AP Calculus matches very well with the content of the first calculus courses in many colleges. The AB course matches with the first semester course, and the BC course matches with the first and second semester courses. The pace is faster in college; for example, the AB course in high school is covered over an entire school year meeting five days per week, while in college the same material is covered in a semester (half a year) meeting three to four times per week. Nevertheless, students who complete AP Calculus and achieve grades of 3, 4, or 5 on an AP Calculus Examination are usually prepared to succeed in the second or third semester of college calculus. Students with grades of 3 (roughly equivalent to a grade of C), say on the AB examination, may encounter difficulty in the second semester course in college calculus; however, the college student who passes first semester calculus with a grade of C faces a similar challenge. Short-cutting the content of an AP Calculus course or not successfully completing an AP Calculus Examination is likely to lead to difficulty in the transition to college calculus.
Recommendations to Students for Smoother Transitions to College
- Learn the ideas and skills of algebra, trigonometry, and geometry well in high school. You will likely need them later, and you do not want to have to repeat the courses in college.
- Find out about the placement tests that are used by the college that you plan to attend. Refresh your knowledge and skills to be tested before you sit for the test.
- Take AP Calculus in high school only after you have completed the mathematics courses that are normally required, and then take the AP Calculus Examination. If you do not expect to continue in calculus in college, consider alternatives to AP Calculus in high school. Such alternatives might be AP Statistics or a course that applies the mathematics you have learned in algebra, trigonometry, and geometry.
Footnotes
1Clark, Richard W. "Dual Credit: A Report of Programs and Policies that Offer High School Students College Credits." Pew Charitable Trusts, 2001.
2The report gives the number of students in AP as 1.2 million. This is the number of examinations. The number of students is closer to 800,000. The estimate of 300,000 U.S. students in IB also seems too large, so the percent of students in courses that do not have national examinations is probably higher than the 57 percent cited.
3See Access to Excellence, Report of the Commission on the Future of the Advanced Placement Program (College Board, New York, 2001) for more recommendations on appropriate uses of AP.
4"Report of the CUPM Panel on Calculus Articulation: Problems in the Transition from High School Calculus to College Calculus," The American Mathematical Monthly 94 (1987): 776-785. |
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