Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools: Report of the Content Panel for Mathematics (2002)

2. To what degree is the factual base of information that is provided by the AP and IB curricula and related laboratory experiences adequate for advanced high school study in your discipline?

3. Based on your evaluation of the materials that you received, to what extent do the AP and IB curricula and assessments balance breadth of coverage with in-depth study of important topics in the subject area? In your opinion, is this balance an appropriate one for advanced high school learners?

4. Are there key concepts (big ideas) of your discipline around which factual information and ideas should be organized to promote conceptual understanding in advanced study courses (e.g., Newton’s Laws in physics)? To what degree are the AP and IB curricula and related laboratory experiences organized around these identified key concepts?

5. To what degree do the AP and IB curricula and related laboratory experiences provide opportunities for students to apply their knowledge to a range of problems and in a variety of contexts?

6. To what extent do the AP and IB curricula and related laboratory experiences encourage students and teachers to make connections among the various disciplines in science and mathematics?

Research and experience indicate that assessments of student learning play a key role in determining what and how teachers teach and what and how students learn.

1. Based on your evaluation of the IB and AP final assessments and accompanying scoring guides and rubrics, evaluate to what degree these assessments measure or emphasize:

   1. students’ mastery of content knowledge;

   2. students’ understanding and application of concepts; and

   3. students’ ability to apply what they have learned to other courses and in other situations.

2. To what degree do the AP and IB final assessments assess student mastery of your disciplinary subject at a level that is consistent with expectations for similar courses that are taught at the college level?

Research and experience indicate that learning is facilitated when teachers use a variety of techniques that are purposefully selected to achieve particular learning goals.

1. How effectively do the AP and IB curricula and assessments encourage teachers to use a variety of teaching techniques (e.g., lecture, discussion, laboratory experience and independent investigation)?

2. What preparation is needed to effectively teach advanced mathematics and science courses such as AP and IB?

The National Science Education Standards and the NCTM Standards 2000 propose that the emphases of science and mathematics education should change in particular ways (see supplemental materials).

1. To what degree do the AP and IB programs reflect the recommendations in these documents?

Advanced study at the high school level is often viewed as preparation for continued study at the college level or as a substitute for introductory-level college courses.

1. To what extent do the AP and IB curricula, assessments, and related laboratory experiences in your discipline serve as adequate and appropriate bases for success in college courses beyond the introductory level?

2. To what degree do the AP and IB programs in your discipline reflect changes in knowledge or approaches that are emerging (or have recently occurred) in your discipline?

3. How might coordination between secondary schools and institutions of higher education be enhanced to optimize student learning and continued interest in the discipline?

This outline of topics is intended to indicate the scope of the course, but it is not necessarily the order in which the topics are to be taught. Teachers may find that topics are best taught in different orders. (See the Teacher’s Guide – AP Calculus for sample syllabi.) Although the examination is based on the topics listed in the topical outline, teachers may wish to enrich their courses with additional topics.

• An intuitive understanding of the limiting process.

• Calculating limits using algebra.

• Estimating limits from graphs or tables of data.

• Understanding asymptotes in terms of graphical behavior.

• Describing asymptotic behavior in terms of limits involving infinity.

• Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

• An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)

• Understanding continuity in terms of limits.

• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

• Derivative presented graphically, numerically, and analytically.

• Derivative interpreted as an instantaneous rate of change.

• Derivative defined as the limit of the difference quotient.

• Relationship between differentiability and continuity.

• Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.

• Tangent line to a curve at a point and local linear approximation.

• Instantaneous rate of change as the limit of average rate of change.

• Approximate rate of change from graphs and tables of values.

• Corresponding characteristics of graphs of ƒ and ƒ’.

• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.

• The Mean Value Theorem and its geometric consequences.

• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’.

• Relationship between the concavity of ƒ and the sign of ƒ’’.

• Points of inflection as places where concavity changes.

• Analysis of curves, including the notions of monotonicity and concavity.

• Optimization, both absolute (global) and relative (local) extrema.

• Modeling rates of change, including related rates problems.

• Use of implicit differentiation to find the derivative of an inverse function.

• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

• Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

• Basic rules for the derivative of sums, products, and quotients of functions.

• Chain rule and implicit differentiation.

• Computation of Riemann sums using left, right, and midpoint evaluation points.

• Definite integral as a limit of Riemann sums over equal subdivisions.

• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

• Basic properties of definite integrals. (Examples include additivity and linearity.)

• Use of the Fundamental Theorem to evaluate definite integrals.

• Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

• Antiderivatives following directly from derivatives of basic functions.

• Antiderivatives by substitution of variables (including change of limits for definite integrals).

• Finding specific antiderivatives using initial conditions, including applications to motion along a line.

• Solving separable differential equations and using them in modeling. In particular, studying the equation y’ = ky and exponential growth.

The topic outline for Calculus BC includes all Calculus AB topics. Additional topics are found in paragraphs that are marked with a plus sign (+) or an asterisk (*). The additional topics can be taught anywhere in the course that the instructor wishes. Some topics will naturally fit immediately after their Calculus AB counterparts. Other topics may fit best after the completion of the Calculus AB topical outline. (See the Teacher’s Guide – AP Calculus for sample syllabi.) Although the examination is based on the topics listed in the topical outline, teachers may wish to enrich their courses with additional topics.

• An intuitive understanding of the limiting process.

• Calculating limits using algebra.

• Estimating limits from graphs or tables of data.

• Understanding asymptotes in terms of graphical behavior.

• Describing asymptotic behavior in terms of limits involving infinity.

• Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

• An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)

• Understanding continuity in terms of limits.

• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

• Derivative presented geometrically, numerically, and analytically.

• Derivative interpreted as an instantaneous rate of change.

• Derivative defined as the limit of the difference quotient.

• Relationship between differentiability and continuity.

• Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.

• Tangent line to a curve at a point and local linear approximation.

• Instantaneous rate of change as the limit of average rate of change.

• Approximate rate of change from graphs and tables of values.

• Corresponding characteristics of graphs of ƒ and ƒ’.

• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.

• The Mean Value Theorem and its geometric consequences.

• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’.

• Relationship between the concavity of ƒ and the sign of ƒ’’.

• Points of inflection as places where concavity changes.

• Analysis of curves, including the notions of monotonicity and concavity.

• Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors.

• Optimization, both absolute (global) and relative (local) extrema.

• Modeling rates of change, including related rates problems.

• Use of implicit differentiation to find the derivative of an inverse function.

• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions.

• Numerical solution of differential equations using Euler’s method.

• L’Hô pital’s Rule and its use in determining convergence of improper integrals and series.