Continuing our look at the Mathematical Practices today we consider computations. We require students to do computations so that they will learn how to do computations; the answer and the check are just the last steps.
MPAC 3: Implementing algebraic/computational processes
a. select appropriate mathematical strategies;
b. sequence algebraic/computational procedures logically;
c. complete algebraic/computational processes correctly;
d. apply technology strategically to solve problems;
e. attend to precision graphically, numerically, analytically, and verbally and specify units of measure; and
f. connect the results of algebraic/computational processes to the question asked.
— AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.
Pretty much all calculus involves computations. This MPAC says that students should be able to plan and carry out the computations necessary to solve problems. This includes selecting the right processes to use and using them correctly. There may be more than one way to do a problem. It includes the use of technology when appropriate as well as the Rule of Four (MPAC 3e). The results should apply to the question asked.
How/where can you make sure students use these ideas in your classes.
Of course you are going to have you students solve problems and investigate mathematical situations, so in some ways this MPAC is “boiler plate.” Students are supposed to learn what to do, in what order to do it, do it correctly, and check or apply their results in the context of the problem.
This applies to the calculus, but starts much earlier. Teachers should be sure that students do this from before day one of Algebra 1. For the teacher it also means checking their work not just for the correct answer, but for the correct thinking and best procedure.
Even many multiple-choice questions involve do a computation. In your classroom exams and quizzes it is a good idea to have students show their work and reasoning on multiple-choice questions. I regularly gave partial credit for good work on multiple-choice questions that required a computation, even if the answer was correct.
CAS calculators and computer programs are great at doing computations, but they still have to be told what to do and in what order to do it. Problems with long or tricky computations are a place to use this technology. For this reason, choosing what to do is, I think, more important than the actual doing it. Still students need to know how to do basic algebra and trigonometry.
CAS calculators can be used to teach basic computation. If a student enters a linear equation and types the operation to solve the equation (such as -4x, or +2) the CAS will perform the operation on both sides of the equation and give the resulting equation. If a student chooses the wrong operation, the CAS does it anyway and presents the result; the student will not see what he or she expected to see and know he or she made a mistake.See the figure in which the fourth line shows a “mistake” followed by a recovery; the last two lines are the check.
Aside 1: I once had a student in a pre-algebra course who did division by subtracting the divisor from the dividend until he got down to zero. Then he counted the times he subtracted and presented this as the quotient. After all, division is just repeated subtraction. Correct procedure? Yes. Good way to divide? No. His previous teachers were not checking what he did; they loved his correct answers. Alas, I was unable to break him of the habit, and he was not able to go much farther in mathematics.
Aside 2: When scoring the AP exam, every year we see students finding the area of a region by integrating the difference of the upper function subtracted from the lower function and taking the absolute value when they came up with a negative answer. Correct algorithm? Yes. Good way to do the problem? I think not. (They earn full credit for this, if done correctly.)
Aside 3: Speaking of computing, I recently learned that my youngest son, who just turned 31 never learned his multiplication tables! Yet, he never had any trouble and could do multiplication as quickly as anyone. So I asked him how he did it. He explained that he worked off the perfect squares. If he had to multiply seven times eight, he thought: seven squared is 49 plus another 7 is 56. I suspect his teacher never asked him to explain how he multiplied. On the other hand, if I were his teacher would I consider this a good way or would I make him memorize the tables? I don’t know; what would you have done?
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.) About 2/3 of the multiple-choice and all six free-response questions on both AB and BC exam reference MPAC 3.
Here is a previous post on this subjects:
While many posts include computations, I do not seem to have any posts on just the idea of doing computations. I offer my euphonious theorem as an example of choosing an unusual computational path through a problem (and leaving the actual computations to the CAS).
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