Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Report of the Content Panel for Mathematics (2002) / Chapter Skim
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3. Analysis of AP and IB Curricula and Assessments
3. Analysis of AP and IB Curricula and Assessments
Pages 16-31
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mathematics programs for advanced high school students, the panel considered the degree to which the curricula and assessments of each emphasize four areas listed below. These areas represent the kind of mathematical learning the panel believes advanced mathematics students should experience in their high school calculus course: · Technical skill, including flexibility with symbolic manipulation, graphical representation, numerical approximation, and function notation · Conceptual understanding, including, for example, the derivative as a rate of change · Theory, including precise definitions, carefully reasoned mathematical arguments, and mathematical rigor · Applications and modeling, including situations in which students must set up the model (for example, choose a formula that represents a real situation)
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In this sense, the panel sees the AP calculus programs as minimal rather than maximal courses for advanced study. Teachers should be encouraged to challenge their students with problems that may go beyond the scope of the syllabi, teaching a course that is more demanding than the tests.
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Finding: 17oeAP curricula are largely sound. 77' e recently revised syllabi with more emphasis on conceptual understanding havesignificantly improved
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Since the types of problems that are on the end-of-course assessments largely determine the structure and content of classroom and homework problems, evaluation of the effectiveness of the programs must include an evaluation of the problems on the examinations. AP Program On the basis of the four areas of emphasis described above, the panel believes the AP examinations should include questions that emphasize the following: ..~ Complex problems that may require significant technical skills · Problems that probe students' conceptual understanding and take into account what research tells us about common misconceptions · Problems that require mathematical reasoning, clear exposition, and the ability to write precise mathematical statements · Problems in which students must construct a mathematical model from a verbal description The panel reviewed 5 years of publicly released AP Calculus AB and AP Calculus BC examinations, related course descriptions, teachers' guides, and scoring rubrics (College Entrance Examination Board ICEEB]
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The College Board's analysis of grades earned on the 1998 AB examination indicated that fewer than 15 percent of the students who earned a 3 obtained a score above 23 out of 45 on the multiple-choice questions.20 Most of these multiple-choice questions tested very basic calculus knowledge. The panel speculates that the underlying reason students could not answer these questions was poor algebra skills rather than a lack of understanding of the necessary calculus.
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and by placing some items in realworld contexts. However, there are missed opportunities to include questions that address common student misconceptions identified by educational research (for example, in understanding the derivative, slope fields, and functions)
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Theory anc' Proof Theoretical questions do not appear explicitly on AP examinations; there is little emphasis on proof in the AP syllabi. In the last decade, however, the emphasis on conceptual understanding has increased in the examinations.
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The panel notes that the portfolio component of the IB program does require students to perform mathematical investigations, extended closedproblem solving, and mathematical modeling. The panel is concerned that the need to standardize AP has led to a course with the rough edges smoothed out as much as possible.
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Me examination does not include enough problems thatfocus on conceptual understanding. Moreproblems are needed that involve multiple steps, test technical skills in the context of applied problems, ask for interpretation and explanation of results, include substantial realistic applications of calculus, and test reasoning or theoretical understanding.
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The panel is concerned that the procedural aspects of the IB examinations could lead to superficial instruction in the mathematics. The mathematical investigations, extended closed-problem solving, mathematical modeling, and mathematical research options for the internal portfolio assignments are a large step in the right direction to mitigate this problem.
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The calculators must be of a preapproved type, which include graphing calculators, but not those with a computer algebra system. Although the IB examinations require substantial skill with symbol manipulations, they do not require much graphical and numerical skill.
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However, there is such a broad range of topics on the examinations that it would be difficult for students to do well without understanding mathematical concepts. The IB examinations test conceptual understanding indirectly.
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Applications and Modeling The questions on the 1999 IB examinations for Mathematical Methods SL and Mathematics HL make almost no attempt to connect calculus with the real world (although the graph theory and statistics questions do feature quasirealistic settings)
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Although the panel was unable to obtain precise figures, the director of the North American IBO office estimated that more than 90 percent of IB students take the examination. In contrast, College Board data suggest that nearly 40 percent of the students who enroll in AP calculus courses do not take the exam.
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However, the IB examinations reflect even less of an attempt than the AP examinations to connect to the real world. · The AP calculus examinations are still fairly predictable, even after the recent revisions to the syllabi.
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MATHEMATICS Finding: Both UP and IB examinations lack good applications and connections to the real-world uses of mathematics. The IB examinations are weaker than the AP examinations in this regard.
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