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AP Calculus Success
Christine S. Lucas, Whitefish Bay High School, Wisconsin
For the first time in ten years of teaching AP Calculus at Whitefish Bay High School, all of my students (strong and weak alike) returned from taking the 1997 AB Calculus exam claiming "it was so easy." Of course, being familiar with the exam, I expected this response. In previous years, however, only a few students ever felt confident enough to make such a claim. Perhaps they felt it would jinx their scores. So what was the variable in the 1996-1997 school year? I would like to say that my abilities as a teacher had peaked-that all of my students fully comprehended every word of our class discussions and as a result, assignments simply involved the act of completing and mastering rather than struggling. This was not the case. The variable was the adoption of a new calculus textbook. I taught and students learned from Calculus, Single Variable, First Edition by Hughes-Hallett, Gleason, et al., and assignments were anything but easy or rote.
Oddly enough, I had refrained from adopting the CCH book in previous years for fear of tampering with an already good track record of AP scores. My concerns were allayed when my students bounded happily into my classroom after the exam, and also on the day I received their scores. Of twenty who took the AB exam, seventeen earned the score of 5, and three earned 4's. Both students (from the same class as the others) who took the BC exam earned the score of 5. (My course includes some topics from the BC curriculum.)
Changing the Book
My initial concern about changing texts was fear of a mismatch: how would students who learned calculus in a reformed setting fare on a traditional exam? Despite this concern, I adopted the CCH materials because what I saw was good calculus and expert pedagogy. I was teaching calculus after all, not simply preparing my students all year for an exam in May. It seemed logical though, that if they truly knew and understood calculus, as I believed this text would aid them in doing, high performance on the AP exam would be inevitable. Rectilinear motion, tangent line, simple differential equation (via separation of variables), and graphical differentiation problems would come naturally to students who had explored derivatives and integrals via physics examples and the CBL (computer based laboratory), extrapolated using data tables and local linearity ideas, worked with slope fields and Euler's method, and differentiated and integrated functions in a multitude of ways. And they did. My students had become careful readers of problem situations and flexible thinkers. They learned that if one approach fails, or they are unable to carry it through, all hope is not lost. Other approaches—numerical or graphical exploration, intuition, conjecturing and testing, utilization of calculus features on their graphing calculators—were employed. My students learned not only calculus but problem solving too.
Course Changes
Because of increased emphasis on "real" problems—real situations, problems that may have worldly impact, problems that exemplify the power, application, importance and therefore beauty of calculus—the structure of my course had to change. Rather than beginning class with a few homework problems, and then moving to the new material, we now typically begin with a warm-up (occasionally in the form of a short quiz), and then make the transition into the new topic. Through questioning, students are carefully led in groups to investigate, discover and take ownership of calculus concepts. This continues for four to five consecutive (fifty-one minute) class periods at a pace of a half to full unit per day. Each day, readings and selected problems from each unit are assigned for homework.
Following this, approximately two classes are taken to discuss some of the assigned problems. Depending on the nature of the problems and the size of the class, students sign up to present their solutions individually or in groups. Problems and their implications are discussed thoroughly, not only in mathematical and realistic senses, but in a pedagogical one as well. Students are called to reflect on why such a question is being asked, and where it is leading. They are able to see the placement of the problem as crucial in the continuum of their calculus learning. Discussing several sections of problems also allows students to gain a more global perspective and an understanding of the interconnectedness of topics.
Assessment Changes
Since the type of problems and the structure of my course changed, my testing had to change, too. I have never been a fan of one-hour calculus tests—particularly when I was in calculus—but in a reformed course, such tests can restrict student success even more. To give tests that reflect the depth of what students learn, one must ask questions that require students to investigate, explore, and call into play not only problem-solving techniques but knowledge accumulated from their (more or less than) seventeen years on earth. Such questions require adequate thinking time whereas repetitive manipulations and rote exercises do not. To accommodate the need for time and to reduce student stress, I break most tests into two parts, taken at separate times. This also affords me the opportunity to separately test some theory and knowledge of common identities (without a programmable utility) and problem-solving and application (with a graphics programmable calculator). Also, because students have many methods of solution at their disposal, I frequently ask them to outline several. If the problem is open-ended and any method is acceptable, clear and succinct written communication on their part becomes necessary for me to assess their work.
Alumni Changes
I believe my change to using the CCH text was a wise and timely move. Positive pressures on the College Board have caused the AP exam to reflect reforms in the teaching and learning of calculus. AP questions will not only call for an increased use of technology, but for a deeper and more flexible understanding of calculus. Therefore, the CCH materials are becoming an even better match for student success on the AP exam. Of interest to me also is how my students fare in their traditional and reformed university calculus courses. I am happy to report the most popular comment of alumni: "I've been tutoring other kids on my floor in calculus!" We know that understanding calculus is one thing, but explaining it is quite another.
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