MPAC 6 Communicating

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

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:

Writing on the AP Calculus Exams

The Opposite of Negative

What’s a Mean Old Average Anyway?

Others

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The Math Book

Ants can count.

I did not know that.

I found this interesting fact in a book I bought recently called The Math Book by Clifford A. Pickover. The book is similar to a coffee table book in that it is nicely made with high quality paper and illustrations, although its size is about the same as a regular book. It consists of 250 entries one page in length with a color illustration on the facing page.

The topics cover the whole range of mathematics and mathematicians.

The entries are dated by year from c.150 million B.C. (Ant Odometer) to 2007. Most of the years are from the current era of course. Each entry discusses the important mathematics discovered or developed in that year or a mathematician active in that time.

Some entries are devoted to curiosities such as Aristotle’s Wheel Paradox (c.320 B.C.) or the Birthday Paradox (1939). Numerous mathematicians and their work are discussed. Games with mathematical features such as Go (548 BC), Mastermind (1970) and Awari (2002) make interesting entries. Unsolved and recently solved problems like the Riemann Hypothesis (1859) and the Four-Color Theorem (1852) are included. Mathematical inventions like the slide rule, (1621) and the Curta Calculator (1948) among others are discussed.

Every page seems to have new ideas many of which I have never heard of (which doesn’t prove a whole lot). Do you know about Johnson’s Theorem (1916) or Voderberg Tilings (1936)?

As the author points out, one page is not enough space to go deeply into each topic, but that is not the purpose of this book. Each essay includes reference to other related topics in the book.  For those who want to know more or want their students to dig a little deeper, there are notes, references and further readings for every topic the end of the book.

Each entry is fascinating, informative and fun.

How am I going to use this in my classes? I thought that each day I would have a student pick a date and then read the entry closest to that date. My goal will be simply to give the class a hint of the breadth of mathematics, the people who made mathematics, and the wide range of things mathematical.

The Math Book, by Clifford A. Pickover, © 2009, Sterling Publishing, New York, NY. ISBN 978-1-4027-5796-9 (hardcover), ISBN 978-1-4027-8829-1 (Paperback)

The Opposite of Negative

Next year, for the first time in 15 years, I am going to be teaching high school full-time. While I have enjoyed writing and working primarily with teachers for the last 15 years, I’m looking forward to “going back to the classroom” as they say. It looks like I’ll be teaching BC calculus and Algebra 1 – two of my favorite classes. I’m very positive about that.

With that in mind I have been thinking of some of the things I want to be sure I get right in the Algebra 1 classes to get the kids off to a good start. So a few of my blogs in the coming year may be on Algebra 1 topics with the view of having students do things right from the start and not having to relearn things when they get to calculus.

So here is the first thing I want to be sure to work on: the m-dash also known as the minus sign.

According to Wikipedia:

The minus sign () has three main uses in mathematics:

1. The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.
2. Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5.
3. unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2.

Using the same symbol understandably can confuse beginning math students. I am not going to invent new symbols so I will just have to be careful with what I say and let the kids say. And I have to say it right , if I expect them to.

When used between two numbers or two expressions with variables the symbol means subtraction. That’s pretty easy to spot and understand in context.  But when used alone in front of something the minus sign means different things.

The m-dash may always be read “opposite.” So “–a” is read “the opposite of a” and not “negative a.”  Likewise, –5 is read “the opposite of five.”

There is only one instance where the m-dash may be read “negative.” When it is used in front of a number it indicates a negative number so      “–5” is also correctly read “negative five.” This is the only time the m-dash should be read “negative.”  Things like “–a” should always be read “the opposite of a” and never read “negative a.”

There was a time when the folks who write math books tried to make the distinction by using a slightly raised dash to indicate negative number so negative 3 was written  “3.” This has carried over into calculators where the key marked “(–)” is used for “negative” and “opposite.” and is printed on the screen as a shorter and slightly raised dash. The subtraction key is only used for subtraction.

Oh, if it were only that simple. What do you do with –(–5)? Not really a problem the “opposite of the opposite of 5” and the “opposite of negative 5” are both 5.

I’ll know I’ve succeeded when everyone can get 100% on this little True-False quiz:

1. The opposite of a number is a negative.
2. x < 0
3. x > 0
4. | x | = x
5. |– x |= x

Answers are in my next post.

Revised 10-27-2018

Theorems and Axioms

Continuing with some thoughts on helping students read math books, we will now look at the main things we find in them in addition to definitions which we discussed previously: theorems and axioms.

An implication is a sentence in the form IF (one or more things are true), THEN (something else is true). The IF part gives a list of requirements, so to speak, and when the requirements are all met we can be sure the THEN part is true. The fancy name for the IF part is hypothesis; the THEN part is called the conclusion.

Implications are sometimes referred to as conditional statements – the conclusion is true based on the conditions in the hypothesis.

An example from calculus: If a function is differentiable at a point, then it is continuous at that point. The hypothesis is “a function is differentiable at a point”, the conclusion is “the function is continuous at that point.”

This is often shortened to, “Differentiability implies continuity.” Many implications are shortened to make them easier to remember or just to make the English flow better. When students get a new idea in a shortened form, they should be sure to restate it so that the IF part and the THEN part are clear to them. Don’t let them skip this.

Related to any implication are three other implications. The 4 related implications are:

1. The original implication: if p, then q.
2. The converse is formed by interchanging the hypothesis and the conclusion of the original implication: if q, then p. Even if the implication is true, the converse may be either true or false. For example, the converse of the example above, if a function is continuous then it is differentiable, is false.
3. The inverse is formed by negating both the hypothesis and the conclusion: if  p is false, then q is false. For our example: if a function is not differentiable, then it is not continuous. As with the converse, the inverse may be either true or false. The example is false.
4. Finally, the contrapositive is formed by negating both the original hypothesis and conclusion and interchanging them, if q is false, then p is false. For our example the contrapositive is “If a function is not continuous at a point, then it is it is not differentiable there.” This is true, and it turns out a useful. One of the quickest ways of determining that a function is not differentiable is to show that it is not continuous. Another example is a theorem that say if an infinite series, an, converges, then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=0$. This is most often used in the contrapositive form when we find a series for which  $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\ne 0$; we immediately know that it does not converge (called the nth-term test for divergence).

The original statement and its contrapositive are both true or both false. Likewise, the converse and the inverse are both true or both false.

Any of the 4 types of statements could be taken as the original and the others renamed accordingly. For example, the original implication is the converse of the converse; the contrapositive of the inverse is the converse, and so on.

Definitions are implications for which the statement and its converse are both true. This is the real meaning of the reversibility of definitions. For this reason, definitions are sometimes called bi-conditional statements.

Axioms and Theorems

There are two kinds of if …, then… statements, axioms (also called assumptions or postulates) and theorems. Theorems can be proved to be true; axioms are assumed to be true without proof. A proof is a chain of reasoning starting from axioms, definitions, and/or previously proved theorems that convince us that the theorem is true. (More on proof in a future post.)

It would be great if everything could be proved, but how can you prove the first few theorems? Thus, mathematical reasoning starts with (a few carefully chosen) axioms and accepts them as true without proof. Everything else should be proved. If you can prove it, it should not be an axiom.

Theorems abound. All of the important ideas, concepts, “laws” and formulas of calculus are theorems.  You will probably see few, if any, axioms in a calculus book, since they came long before in the study of algebra and geometry.

Learning Theorems

When teaching students and helping them read and understand their textbook, it is important that they understand what a theorem is and how it works. They should understand what the hypothesis and conclusion are and how they relate to each other. They should understand how to check that the parts of the hypothesis are all true about the function or situation under consideration, before they can be sure the conclusion is true.

For the AP teachers this kind of thing is tested on the exams. See 2005 AB-5/BC-5 part d, or 2007 AB-3 parts a and b (which literally almost no one got correct). These questions can be used as models for making up your own questions of other theorems.

Definitions 2

In helping students read and understand mathematics knowing about definitions, axioms (aka assumptions, postulates) and theorems. By this I mean knowing the parts of a definition or theorem and how they relate to each other should increase the students’ understanding. Today I’ll discuss definitions; theorems and assumptions will be discussed in a future post.

A definition names some mathematical “thing.” A good definition (in mathematics or elsewhere) names the thing defined in a sentence. The sentence may contain symbols, which are really just shorthand for words. A definition has 4 characteristics:

1. It should put the thing defined into the nearest group of similar things.
2. It should give the characteristics that distinguish it from the other things in the group.
3. It should use simpler terms (previously defined terms).
4. It should be reversible.

I will discuss each of these with an example first from geometry and then from calculus. First however, a word or two about “reversible.” Definitions are what are known technically as bi-conditional statement, meaning that the statement and its converse are true. More on this in the next post.

An example from geometry:

Definition: An equilateral triangle is a triangle with three congruent sides.

The term defined is “equilateral triangle.”

1. Nearest group of similar things: triangles
2. Distinguishing characteristic: 3 congruent sides. We all know that an equilateral triangle also has 3 congruent angles, and that all the angles have a measure of $\displaystyle \tfrac{\pi }{3}$, and all the angles add up to a straight angle, and lots of other great things, but for the definition we only mention the feature that distinguishes equilateral triangles from other triangles. It would be possible to use instead the 3 congruent angles or the fact that all three angles measure are $\displaystyle \tfrac{\pi }{3}$, as the distinguishing characteristic, but whoever wrote the definition choose the sides. (We could not use the fact that the angles add to a straight angle, because that is true for all triangles and therefore doesn’t distinguish equilateral triangles from the others.) Definitions do not list all the things that may be true, only those that make it different.
3. Simpler terms: triangle, sides (of a triangle) and congruent. We assume that these key terms are already known to the student. Of course there were no previously defined terms for the very first things (points, lines and planes) but by now we are past that and have lots of previously defined terms to work with.
4. Reversible: If we know that this object is an equilateral triangle, then without looking further we know it has 3 congruent sides AND if we run across a triangle with 3 congruent sides, we know it must be an equilateral triangle.

An example from the calculus:

Definition: A function, f, is increasing on an interval if, and only if, for all pairs of numbers a and b in the interval, if a < b, then f (a) < f (b).

This is a little more complicated. The term being defined is increasing on an interval. This becomes important and can lead to confusion because sometimes we are tempted to think functions are increasing at a point. There is no definition for the latter: functions increase only on intervals.

1. Nearest group of similar things: functions
2. Distinguishing characteristic: for all pairs of numbers a and b in the interval, if a < b, then f (a) < f (b).
1. The if …, then … construction indicates a conditional statement (discussed in the next post) inside of the definition. This is not uncommon. It means that if can establish that this is true, then we can say then function is increasing on the interval.
2. The phrase “for all” is also common in mathematics. It means the same thing as “for any” and “for every.” When you come across one of these it is a very good idea to rephrase the sentence with each of them: “for all numbers a and b in the interval…”, “for any pair of numbers a and b in the interval …” and “for every two numbers a and b in the interval…” This greatly helps understanding definitions.
3. Simpler terms: function, interval (could be open, closed or half-open), less than (<), the meaning of symbols like f (a).
4. Reversible:
1. the phrase “if, and only if” indicates that what goes before and what comes after it, each imply the other. This phrase is implicit in all (any, every) definitions although English usage often omits it. The first definition could be written “A triangle is equilateral if, and only if, it has 3 congruent sides” but is a little more user-friendly the way it is stated above.
2. If you can establish that “for all pairs of numbers a and b in the interval, if a < b, then f (a) < f (b)”, then you can be sure the function increases on the interval. AND if you are told f is increasing on the interval, then without checking further you can be sure that “for all (any, every) pairs of numbers a and b in the interval, if a < b, then f (a) < f (b).”

Now that’s a fairly detailed discussion (definition?) of a definition. But it is worth going through any new definition for your students to help them learn what the definition really means. First identify the four features for you students and then as new definitions come along have them identify the parts. Encourage them to pull definitions apart this way. It is worth the little extra time spent.

At this time of year many teachers are picking the calculus book for their class to use next year. At the same time, you will find teachers complaining, quite correctly, that their students don’t read their math textbooks. Authors, editors and their focus groups try their very best to make books “readable,” to no avail, since students won’t read them anyway.

Maybe this is because students have never learned to read math books, because no one has ever taught them how. Have you? Here are some thoughts and suggestions gathered from several sources that may help.

First, some obvious (to us) comments, which, alas, probably won’t make much of a change in students’ ways:

• It takes time to read a math book. Unlike a novel or a non-fiction book, a few pages of mathematics will take longer than reading a story or essay.
• Readers should stop every few lines and make sure they understand what they’ve read
• Readers should have a pencil and paper handy both to take notes, to draw graphs, to work through some of the examples.
• Math books contain examples to help the reader understand what’s going on; so readers should work (i.e. with paper and pencil) through the examples.

Here are some things you can do to help your students at all levels learn to read a mathematics textbook. The sooner students learn to read mathematics the better. Work with your pre-Algebra and Algebra 1 teachers (or earlier) to get them started. The sooner the better, but if they have not done it before they get to your calculus class do it then.

1. Start with short reading assignments and spend some time before and after discussing both what they read and how they read it. Do not do this forever, rather
2. Don’t reread the text to them or follow the text exactly in you class discussions; make them responsible for understanding what you’ve assigned them to read. Of course, you should answer questions on anything they didn’t understand, but expect them (eventually) to learn from what they read.
3. A brief but structured reading organizer can be a help. Have them make notes on what they read in a form like this:
2. What is this section about? What is the main idea?
3. There are ____ new definitions (or vocabulary words) in this section. For each, express the definition in your own words, include a drawing if appropriate.
4. There are ____ new theorems (rules, laws, formulas) in this section. For each, write its hypothesis and conclusion and explain what it means in your own words, include a sketch if appropriate.
5. Which application or example was most interesting or instructive for you? Why?
6. Is there anything you find confusing or do not understand in the reading?

The next day in class meet in groups of 3 or 4 and compare answers: Does everyone in the group agree on the new vocabulary? Which paraphrase is better? Which example/application was the most interesting? Why? What questions do you still have?

Hold the students responsible for doing this work by not repeating what they have just read as a formal lesson on the same material. Approach the material from a different way; probe their understanding with questions.

Instead of a lecture on the material they read, just have a discussion on it. Let the students lead the way explaining what they think the text means, why the examples were chosen, and what they are still unsure of.

In my next two posts I intend to discuss definitions and theorems in more detail – their structure and how to help students use the structure to increase their understanding.

The Electronic Discussion Group or EDG run by the College Board for AP Calculus which has now become the AP Calculus Community is an excellent source of help and information. Some of the ideas here are taken from an EDG discussion on helping students read mathematics textbooks. I’ve also used and expanded ideas from Dixie Ross, Stephanie Sains, Jon Stark, and David Wang that appeared on the EDG. Thanks to them all.