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.

Teaching How to Read Mathematics

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.
  • Readers should make note of what they don’t understand and ask about it in class.

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:
    1. In your textbook read section ___.__,  pages ____ to ____
    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.


Absolute Value

The majority of students learn about absolute value long before high school. That is, they learn a lot of wrong things about absolute value.

  • They learn that “the absolute value of a number is the number without its sign” or some such nonsense. All numbers, except zero have a sign!  This sort of works with numbers, but becomes a problem when variables appear. True or false | x | = x? True or false | –x | = x? Most kids will say they are both true; in fact, as you know, they are both false.
  • They also learn that “the absolute value of a number is its distance from zero on the number line.” True and works for numbers, but what about variables?
  • They learn that “the absolute value of a number is the larger of the number and its opposite.” True again. How do you use it with variables?
  • They learn \left| x \right|=\sqrt{{{x}^{2}}} which is correct, useful for order-of-operation practice, and useful in other ways later, But they still compute \sqrt{{{\left( -3 \right)}^{2}}}=-3 and  \sqrt{{{x}^{2}}}=x since the square and square “cancel each other out.”

So here is a good vertical team topic. Get to those teachers in elementary and middle school and be sure they are not doing any of the above. They should start with the correct definition in words:

  • The absolute value of a negative number is its opposite.
  • The absolute value of a positive number (or zero) number is the same number.

This works all the time and will continue to work all the time. Teaching anything else will eventually require unlearning what they are using, and unlearning is far more difficult than learning.

When they start using variables and reading symbols translated into English, then the definition becomes their first piecewise define function:

  • \text{ If }x\ge 0,\text{ then }\ \left| x \right|=x;  and if x<0,\text{ then }\left| x \right|=-x
  • \left| x \right|=\left\{ \begin{matrix} x & \text{ if }x\ge 0 \\ -x & \text{ if }x<0 \\ \end{matrix} \right.

When reading this definition be sure to say “the opposite of the number” not “negative x” which in this case is probably a positive number.

Give variations of the two True-False questions above on every quiz and test until everyone gets it right!

When you see absolute value bars and want to be rid of them the first question to ask is, “Is the argument positive or negative? “Any time there is an absolute value situation, this is the way to proceed.

And yes, this does show up on the AP Calculus exams. Consider \int_{0}^{1}{\left| x-1 \right|dx} which appeared as a multiple-choice question a few years ago. Give it a try before reading on.

On the interval of integration, [0,1], \left( x-1 \right)\le 0 so \left| x-1 \right|=-\left( x-1 \right)

\displaystyle \int_{0}^{1}{\left| x-1 \right|dx}=\int_{0}^{1}{-\left( x-1 \right)}dx=\left. -\tfrac{1}{2}{{x}^{2}}+x \right|_{0}^{1}=-\tfrac{1}{2}+1-0=\tfrac{1}{2}

Now try \displaystyle \int_{0}^{1}{\sqrt{{{x}^{2}}-2x+1}\,dx}, or did we do this one already?

A Note on Notation

For quite a while I’ve been writing sin(x), ln(x) and the like with parentheses instead of the usual sin or ln x .

The main reason is that I want to emphasize that sin(x), ln(x), etc. are the same level and type of notation as f(x). The only difference is that sin(x) and ln(x) always represent the same function, while things like f(x) represent different functions from problem to problem. I hope this makes things just a little clearer to the students.

I also favor using (sin(x))² instead of sin²(x), again to make clearer just what is getting squared. Notation can be inconsistent: I don’t think I’ve ever seen ln²(x) or even ²(x).  So this helps in that regard as well.

Of course, when entering functions in calculators or computers you almost always must use the “extra” parentheses in both cases. (Except for the new Casio PRIZM which will understand sin x and ln x, but not sin²(x).)

Now we can use that spot in the notation exclusively for inverse functions, as in {{\sin }^{-1}}\left( x \right) and {{f}^{-1}}\left( x \right). Maybe that will lessen the confusion there.

Another possible inconsistency is trying to write sin′(x)  for the derivative as you do with {f}'\left( x \right)Although, if I saw it I would understand it. (LaTex won’t even parse  sin′(x).)

The First Week

After reading my post on “The First and Second Day of School” Paul A. Foerster,  was nice enough to share  these problems from a recent presentation. They give a taste of derivatives and integrals in the first week of school and get the kids into calculus right off the bat.

The First Week of AP Calculus

Paul who recently retired after 50 (!) years of teaching, is Teacher Emeritus of Mathematics of Alamo High Heights School in San Antonio, Texas. He is the author of several textbooks including Calculus: Concepts and Applications, Second Edition, 2005, published by Kendall Hunt Publishing Company, (Formerly published by Key Curriculum Press).

Many years ago (almost 50) I remember teaching from his fine Trigonometry book 

Thank you, Paul!

The First & Second Day of School

Some suggestions for the first days of school:

There has to be a first day, so make the most of it. Take roll; make sure everyone is in the right place. Give out the textbooks. Explain about calculators and so on.

A word about reviewing at the beginning of the year: Don’t!

If you textbook’s first chapter is  a review of pre-calculus, then assign the class to read this chapter. Tell the class that the next day your will answer any questions they have and, after that, for the rest of the year they should refer back to this chapter when they need more information on these topics.

Start day 2 with your first lesson on limits.

Plan to review material from Kindergarten thru pre-calculus when the topics come up during the year – and they will come up. In some cases, plan for them to come up. For instance, the year usually begins with the study of limits. In connection with limits you will be looking at lots of graphs: this is a perfect time to review the graphs of the parent functions, a lot of the terminology related to graphs, discontinuities, asymptotes, and even the values of the trigonometric functions of the special angles.

Months from now you will teach about the derivatives of inverse functions.  That is when you review inverses. Review inverses now and you will have to do it later anyway.