MPAC 6 Communicating

foxtrot-2Saving 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

Students can:

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:

Teaching How to Read Mathematics

Writing on the AP Calculus Exams

The Opposite of Negative

What’s a Mean Old Average Anyway?

Others

foxtrot-1


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MPAC 4: Multiple-representations

We used to call it the “Rule of Four.” Maybe that’s why its MPAC 4.

MPAC 4: Connecting multiple representations

Students can:

a. associate tables, graphs, and symbolic representations of functions;

b. develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology;

c. identify how mathematical characteristics of functions are related in different representations;

d. extract and interpret mathematical content from any presentation of a function (e.g., utilize information from a table of values);

e. construct one representational form from another (e.g., a table from a graph or a graph from given information); and

f. consider multiple representations (graphical, numerical, analytical, and verbal) .of a function to select or construct a useful representation for solving a problem.

AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.

tools_four-actions-frameworkThis is another concept not just of use in the calculus. Students should be using symbols, geometric representations (not just graphs), and numerical ideas along with reading and writing about mathematics from their first days in school.

The symbolic (analytic) aspect of the Rule of Four is perhaps a bit more important in doing mathematics. Things have to be proved analytically. The proper use of symbols in mathematics is the subject of MPAC 5.

The verbal part of the Rule of Four also includes writing and explaining mathematics. This is the subject of MPAC 6.

Technology makes using graphs and table of values very easy. Back in ancient times (that is, in BC – before calculators) when I was in high school getting a graph or a table of values required a lot of work. Now these things are easy and quick when using a calculator; now we can spend our time on what the graphs and numbers mean and what they tell us about the situation we’re investigating.

How/where can you make sure students use these ideas in your classes.

The Rule of Four is definitely not restricted to calculus. Using and relating the parts of the Rule of Four should start way back to the students’ earliest work in mathematics long before Algebra 1. “Graphically” should be expanded to “geometrically;” students should be using drawings and pictures and the like before they learn graphing; and continue to use non-graph representations where appropriate after they learn graphing.

While symbolic or analytic work (working with equations, matrices, etc.) is still where you go when you want to be sure something is true (i.e. to prove things), the others have their place in investigations, in helping to form conjectures, and helping to understanding meaning. By the time they get to the calculus, students should be familiar with looking at functions and other mathematical objects from all four perspectives.

Many problems lend themselves to working with only one or two of the Four. This is natural. While you do not have to force all four aspects into every problem, always consider the others. It is not unusual that one of the other might make things clearer. Students who are required to explain verbally or in writing what they are doing (MPAC 6) will benefit even if that is not strictly required. 

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 1/4 of the multiple-choice and about ½ of the free-response questions on both AB and BC exam reference MPAC 4.


PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.


 

 

 

 

 

 

MPAC 3 Computing

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. computing-1

MPAC 3: Implementing algebraic/computational processes

Students can:

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.

Step-by-step solving with a CAS calculator. The fourth line is an intentional mistake. The user not seeing what he expects on the right recovers nicely in the next line. The last two lines are the check.

Step-by-step solving with a CAS calculator. The fourth line is an intentional mistake. The user not seeing what he expects on the right recovers nicely in the next line. 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 3Speaking 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.

Well, not really. A photo from a schoolroom in Russia, taken on my vacatin this summer.

Three out of four – could be better.  A photo of a poster in a math schoolroom in Russia, taken on my vacation this summer.

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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MPAC 1 Reasoning

My series on calculator/technology use will continue during the year. Meanwhile, today I am starting a short series about Mathematical Practices for AP Calculus or MPACs as they are called.

Earlier this year I did some work verifying the alignment of several textbooks to the Essential Knowledge (EK), Learning Objectives (LO), and MPACs of the new Course and Exam Description for AP Calculus. The publishers provided a reference (page or exercise number) for each LO and EK in their books and a separate reference for each MPAC. The books I looked at all hit the LOs and EKs pretty well at over 95%. But the MPACs, not so much.

The MPACs are intended to get students thinking and working like mathematicians. The more I look at them, the more I think they can do that.

Each MPAC starts with the statement “Students should ….” It’s easy for authors to do a good job explaining the calculus in the LOs and EKs; that’s what authors do. The writers do what’s listed in the MPACs because they are good mathematical practices and the authors are good mathematicians, but the authors do not often point out what they are doing in this regard.

The MPACs are for students to do. Textbooks need to provide opportunities for students to do the them. This pretty much has to be in the exercises. Some of the exercises provide the opportunity to do some of the things listed in the MPACs, but this is often more accidental than intended.

Little or no opportunity is intentionally provided to learn and practice the MPACs. It is up to the teachers to provide these opportunities.

While they are called Mathematical Practices for AP Calculus, in fact they really apply to all of mathematics. The calculus examples in the MPACs may easily be changed to apply to mathematics teaching and learning earlier in the curriculum. Properly applied they should have an impact on the entire curriculum. If the goal is to help students learn to think and work like mathematicians, then starting in AP Calculus is way too late.

I will use this and the next few post to discuss the MPACs in detail and provide some suggestions as to where and how teachers can help their students to think and work like mathematicians.

MPAC 1: Reasoning with definitions and theorems

Students can:

a. use definitions and theorems to build arguments, to justify conclusions or answers, and to prove results;

b. confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem;

c. apply definitions and theorems in the process of solving a problem;

d. interpret quantifiers in definitions and theorems (e.g., “for all,” “there exists”);

e. develop conjectures based on exploration with technology; and

f. produce examples and counterexamples to clarify understanding of definitions, to investigate whether converses of theorems are true or false, or to test conjectures.

AP® Calculus AB and AP® Calculus BC Course and Exam Description Effective Fall 2016, The College Board, New York © 2016. Full text is here.

While the word logic does not appear here, these six items (with the possible logic-spockexception of e) are the tools of logic and the basis of mathematical reasoning. The word prove has appeared very rarely on the AP Calculus exams, students have been asked to justify their answers, apply a definition or theorem to a particular function, and show that they know what ideas can be used in a situation and show that they can use them. Conjecturing, producing examples and counterexamples are the basis of mathematical reasoning.

Some suggestions about how and where you can make sure students work with these ideas in your classes.

These items reflect the structure of mathematics. None of the points are specific to the calculus; they can and should be used and developed in all the classes leading up to calculus. Definitions and theorems come into students’ mathematical education before the first year of algebra. The form and structure of axioms, definitions, and theorems, in addition to their meaning, should be made clear to students. So this is something that should start long before calculus and be included every year.

One way you can help students learn how the items in MPAC 1 work is to use True or False (TF) questions; better yet are Always, Sometimes or Never (ASN) questions. These are similar to TF questions except that the students have a middle choice. These questions are an excellent place to hone ones’ skills using the fine points of theorems and definitions. With either TF or ASN questions students should not just answer with a word, but rather be required to explain how they know their answer is correct. They can do this by citing some theorem or definition or producing an example or counterexample. Students can also be asked to discuss, defend, and compare and contrast their answers with other students.

We all know that AP exam questions often require students to “Justify your answer” or “Explain your reasoning.” Here, too, is a good place to practice with the skills of MPAC 1, since justifications and explanations are based on the theorems and definitions

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam that is in the new format contains very detailed solutions for both the multiple-choice and free-response questions that include these references. About 1/3 of the multiple-choice and all six free-response questions on both AB and BC exam reference MPAC 1.

Here are some previous posts on these subjects:

I have discussed definitions and theorems in previous posts. Here are links to some of them:


PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.