Saving the best, or perhaps the most important, until last, MPAC 6 is the verbal part of the Rule of Four. Problems and real-life situations are “translated” from ideas or words into symbols, equations, graphs, and tables where they are examined and manipulated to find solutions. Once the solutions are found, they must be communicated along with the reasoning involved. The aspects of good mathematical communications are those listed in this MPAC.
MPAC 6: Communicating
a. clearly present methods, reasoning, justifications, and conclusions;
b. use accurate and precise language and notation;
c. explain the meaning of expressions, notation, and results in terms of a context (including units);
d. explain the connections among concepts;
e. critically interpret and accurately report information provided by technology; and
f. analyze, evaluate, and compare the reasoning of others.
— AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.
Justifying answers and explaining reasoning in words has long been required on AP calculus exams. The exams have also required students to explain the meaning of expression involving definite integrals and the value of a derivative in the context of the questions.
How/where can you make sure students use these ideas in your classes.
Since to write mathematics well textbook authors do the things listed under this MPAC, but they rarely require students to write about or explain mathematics. They do not show students how to write good explanations of their work and solutions nor, do they provide exercises requiring explanations. Therefore, teachers must do it.
When you get to the end of the year and start working on old AP calculus exams for review you find many questions requiring students to communicate their methods and reasoning, the meanings of their work and results, the connections among different concepts, interpreting what their technology has shown them.
But waiting until the end of the year is way too late. This kind of work should be included in students’ mathematical work from the beginning, before Algebra 1. It can and should be done at every level. By the time they get to calculus, students should not be at all surprised at being asked to explain verbally and in writing what they are doing and why they chose to do it that way.
Find or provide opportunities for students to consider the reasoning of others (MPAC 6f) as well as explain their reasoning to each other. This can be accomplished with group projects, study groups, checking each other’s work, etc. You can also provide templates hits and tips for writing well.The Course and Exam Description includes an entire section on “Representative Instructional Strategies” (pp. 33 – 37). Among the suggestions are various ways to have students work together and separately on improving their communication skills. The following section (pp. 37 – 38) discusses what a “quality response will include:
- a logical sequence of steps
- an argument that explains why those steps are appropriate, and
- an accurate interpretation of the solution (with units) in the context of the situation”
Provide less than perfect answers for students to critique and improve. (Hint: Use the sample student responses that are released each year along with the exams to show good and not-so-good answers and reasoning.
When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam given out at summer institutes this summer is in the new format and contains very detailed solutions for both the multiple-choice and free-response questions that include these references. (This version is not available online as far as I know.) None of the multiple-choice question, but all six free-response questions on both AB and BC exam reference MPAC 6 (although see 2014 AB 18 for an idea of how MPAC 6f may be tested).
Here are some previous posts on these subjects:
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