The Mathematical Practices

This is the second of four posts on the new AP® Calculus AB and AP® Calculus BC Course and Exam Description  (CED). If you do not yet have a copy, click the link above to download the PDF version.

What is really new, important, and nice in the new CED are the Mathematical Practices or MPACs.

I recently did some consulting for a company that was aligning calculus textbooks with the MPACs and Concept Outline. Publishers listed where their textbook dealt with each Learning Objective (LO), Essential Knowledge (EK), and MPAC statement from the CED. All of the books I reviewed met the LOs and the EKs at 95% or better. The MPACs not so much.

Part of the reason was that the publishers were looking for the MPACs in the text material. The text material did a good job of explaining the LOs and EKs and the reasoning behind them, applications, etc. The text is what the authors do. The MPACs are what the students do. At best, the texts occasionally provided exercise that addressed tangentially some of the things the MPACs require, but never in a purposeful manner. (The exception was the CPM text.)

What that means is that as an AP Calculus teachers you have to provide opportunities for your students to practice the MPACs. In preparing the questions for the AP Calculus exams the College Board is aware of the MPACs; all the sample questions (p. 46 – 94), list which MPACs apply to each question. While you probably won’t be teaching individual lessons on each MPAC, a number of them will naturally be used in each lesson. Make your students aware of them as you use them.

The MPACs are designed to “capture important aspects of the work that mathematicians engage in, at the level of competence expected of AP Calculus students…. [They enable] students to establish mathematical lines of reasoning and use them to apply mathematical concepts and tools to solve problems…. [T]hey are highly integrated tools that should be utilized frequently and in diverse contexts.” (p. 8)

There are 6 MPACs, with lists of what the students should be able to do. They each start with “Students can:” Here are summaries of the Mathematical Practices for AP Calculus (paraphrased from p 9 – 10).


MPAC 1: Reasoning with Definitions and Theorems. Students can use theorems and definition in justifying answers, proving results, confirming hypotheses, interpreting quantifiers, developing conjectures (possibly from using technology), and solving problems. They can give examples and counterexamples, understand converses, and test conjectures.

MPAC 2: Connecting Concepts. Students can relate the concept of limit to other parts of the calculus. They can use the connection between concepts (such as rate of change and accumulation, or processes such as differentiation and antidifferentiation) to solve problems. They can connect visual representations of concepts with and without technology, and identify common structures in different contexts.

MPAC 3: Implementing algebraic/computational processes. Students can select appropriate algebraic and computational strategies. They can sequence and complete them correctly. They can apply technology, attend to precision analytically, numerically, graphically and verbally, and specify units of measure. They should be able to connect these processes to the question they are working on.

MPAC 4: Connecting multiple representations. This refers to what is known as the Rule of Four: considering mathematics analytically, verbally, graphically, and numerically. Students can associate the various representations of functions, develop concepts, identify characteristics of function, extract and interpret mathematical content, relate and construct one representation from another, and select or construct a useful representation for solving a problem.

MPAC 5: Building notational fluency. Students know and can use a variety of notations, connect the notations to their definitions and to different representations of functions (Rule of Four). They should be able to interpret notation accurately and understand its meaning in different contexts.

MPAC 6: Communicating. Students can clearly present methods, reasoning, justifications and conclusions in accurate and precise notation and language. They can explain the meaning of results, notation, expressions, and units of measure in context. They can explain connections between concepts, and accurately report and critically interpret information provided by technology. Students can analyze, evaluate, and compare the reasoning of others.


The MPACs are written with reference to learning and using the calculus. The MPACs should, I think, become part of mathematics learning and instructions at all levels K – 12 leading up to calculus and then beyond it into higher levels of mathematics. When the College Board announced that starting in 1995 graphing calculators would be required for use on the AP Calculus exams, graphing calculators and other technology quickly worked their way down through the secondary math curriculum in most high schools. I think that was a good thing. I hope the same thing happens with the MPACs. The MPACs should have a big impact.

The third post in this series will look at some of the other items in the CED to help you organize and teach a better course.

The College Board has produced a series of short videos on the new CED. Click here to see these Course Overview Modules.











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