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Teaching and Learning Theorems

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Theorems are carefully worded statements about mathematical facts that have been proved to be true. Important (and some not so important) ideas in calculus and all of mathematics are summarized as theorems. When you come across a theorem you need to understand it; the author of your textbook would not have included it and the AP Exams would not test it if it were not.  This post discusses some things about theorems in general. Students often do not realize these things; understanding them will help students understand a new theorem when it is presented to them.

Theorems have the form of IF one or more things are true (called the hypothesis), THEN some other thing is true (called the conclusion).  This is abbreviated p\to q where p represents the hypothesis and q represents the conclusion. This is read as “if p, then q” or “p implies q.” Theorems are also known as conditional statements.

In a certain instance, once you are sure all the conditions of the hypothesis are true, then you may be absolutely certain the conclusion is also true. When trying to determine if something is true about some function, check to see if the conditions of the hypothesis are all true.

People using a theorem need to know the hypotheses as well as the conclusion.

When teaching about a particular theorem, one thing that is often helpful to students is to “play” with the hypothesis and see how it affects the conclusion. For instance, if the hypothesis requires a function to be continuous, see what happens if the function is not continuous. If there are several parts to the hypothesis, see what happens if one or the other is changed. Hint: A change in part of the hypothesis will make some difference – good theorems do not have extra, unneeded, or superfluous conditions.

To make them read better, some theorems are not stated in if …, then… form. If there is any confusion, restate the theorem in if…, then… form.

  • The theorem often stated as, “Differentiability implies continuity,” really means: IF a function is differentiable at a point, THEN it is continuous at that point.
  • The geometry theorem, “The diagonals of a rhombus are perpendicular,” really means: IF a quadrilateral is a rhombus, THEN its diagonals are perpendicular.

The Contrapositive

Since if p is true, q must be true, what happens if q is false? The answer is that p must also be false. This is a related conditional statement (theorem) called the contrapositive of the theorem. The contrapositive is abbreviated IF not q, THEN not p, or IF q is false, THEN p is false, or \tilde{\ }q\to \tilde{\ }p.  The contrapositive of a theorem is also true, always.

There are several theorems in the calculus where the contrapositive seems to be used more often than the theorem itself.

  • Differentiability implies continuity. The contrapositive of this theorem is: IF a function is not continuous at a point, THEN it is not differentiable at that point. This is a quick way to tell if a function is not differentiable.
  • IF an infinite series converges, THEN the limit as n goes to infinity of its nth term is zero. Here the contrapositive is IF the limit as n goes to infinity of its nth term of an infinite series is not zero, THEN the series does not converge. This is called the nth term test for divergence.
  • IF the diagonals of a quadrilateral are not perpendicular, THEN the quadrilateral is not a rhombus.

The converse

The converse of a theorem is a statement formed by switching the hypothesis and conclusion of the theorem: IF q, THEN p (or q\to p). The converse is not necessarily true. It may be true, in which case it need to be proved as a separate theorem. Students (among others) often assume that the converse is true – this is called the fallacy of the converse.

  • IF a function is continuous at a point, THEN it is differentiable there, is the converse of our previous example. It is false: a simple counterexample is f(x) = |x|. This function is continuous at the origin but not differentiable there.
  • Another theorem states that IF the derivative of a function is positive on an interval, THEN the function is increasing on the interval. The converse is, IF a function is increasing on an interval, THEN its derivative is positive on the interval. The converse is false: for example, f(x) = x3 is increasing everywhere, yet its derivative at the origin is zero. (See Going up?)
  • IF the diagonals of a quadrilateral are perpendicular, Then the quadrilateral is a rhombus, if false. The quadrilateral may have perpendicular diagonals, but unless they intersect at their midpoints the figure is not a rhombus (it is a kite shape).

The inverse

The inverse is the contrapositive of the converse or IF not p, THEN not q (or\tilde{\ }p\to \tilde{\ }q ). The inverse will be true if the converse is true, and false if the converse is false.

  • The inverse of our fist example is,IF a function is not differentiable on an interval, THEN it is not continuous there. This is false, since the function my fail to be differentiable even though it is continuous. An example is f(x) = |x|. again.
  • IF a quadrilateral is not a rhombus, THEN he diagonals are not perpendicular (false – the kite again).

Biconditional statements

A biconditional theorem is a theorem whose converse is also true (and therefore its inverse and contrapositive are true). These are written in the form p if, and only if, q or p\leftrightarrow q (or for that matter q\leftrightarrow p). It is equivalent to p\to q\text{ and }q\to p.

  • An example from Geometry: IF two sides of a triangle are congruent, THEN the angles opposite them are congruent. The converse of this theorem is, if two angles of a triangle are congruent,THEN the sides opposite them are congruent. Since both the theorem and its converse (and its inverse, and its contrapositive) are true, you may write, “Two sides of a triangle are congruent if, and only if, the angles opposite them are congruent.”

Definitions are always biconditional statements. They are always true and do not need to be proved; in fact they cannot be proved.

  • The definition of continuous at a point is, “A function is continuous at a point \left( {a,f\left( a \right)} \right) if, and only if, \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=f\left( a \right) and both the limit and value are finite.”
  • A rectangle is defined as a quadrilateral with four right angles or A quadrilateral is a rectangle if, and only if, it has four right angles.

Which is which?

The theorem, its contrapositive, converse, and inverse are all theorems. Any of them could be taken as “the theorem” and the others would rearrange their names accordingly. (Which is good practice for your students.)

One other thing

What if the hypothesis of a theorem is false: can the conclusion still be true? The answer is, yes! The hypothesis of a theorem tells us that if true, the conclusion must be true. But the conclusion may be true anyway.

Consider the Mean Vale Theorem (MVT): IF a function, f, is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), THEN there exists at least one number c in the open interval (a, b) such that \displaystyle {f}'\left( c \right)=\frac{{f\left( b \right)-f\left( a \right)}}{{b-a}}.

  • But consider the function f\left( x \right)=\left\{ {\begin{array}{*{20}{c}} {{{x}^{2}}} & {-2\le x\le 1} \\ \text{4} & {1<x\le 2} \end{array}} \right.
  • This function has a jump discontinuity at x = 1 and therefore is neither continuous nor differentiable on the interval [-2, 2]; the MVT does not apply. Yet \displaystyle {f}'\left( 0 \right)=\frac{{f\left( 2 \right)-f\left( {-2} \right)}}{{2-\left( {-2} \right)}}=\frac{{4-4}}{4}=0. (In fact, c could be 0 or any number between 1 and 2). In this example and not every example, the conclusion of the MVT is true even though the hypothesis is false.
  • On the 2017 International exam AB 15 makes use of the idea that even though the theorem about limits that seems to apply doesn’t because the conditions are not met, but, nevertheless, the conclusion is true.

Proof

The proofs of all the important theorems are given in any good textbook. You study proofs for two reasons: (1) to see why a theorem is true, and (2) to learn how to write a proof of your own. If neither of these reasons concern you or your students (and they may not), then you still need to learn the hypothesis and conclusion, and how to apply the theorem. This cannot be avoided.

Some proofs are rather tricky. That is, the key step is not obvious. A beginning calculus student should not expect to know how to prove most of the theorems; they should, however, be able to follow the proofs in the textbook. The AP Calculus exams never ask for proof, per se, although they may ask you to justify a conclusion you make. The justification should show that the hypotheses are all true and state the name of the theorem that implies your conclusion.

I can recall only one time many years ago where students were asked to “prove” something on an AP Calculus Exam. The usual instruction is “Justify your answer” or “Explain your reasoning.” This means that students are supposed to cite the appropriate theorem and show that the hypotheses are met in the given situation. So, not quite “prove” but close. It’s not “prove” an original theorem, but rather determine which (unnamed) theorem applies (or does not apply) in a particular situation and verify that the conditions are (or are not) met.

As always, look at a number of past exams and see just what is asked and how it is asked.

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Getting Ready

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Today we continue to look at some previous posts that I hope will help you and your students throughout the year. We begin with some posts on graphing calculator use and then a few general things in three posts on beginning the year, followed by some mathematics I hope students know before they start studying the calculus.

Graphing Calculators

There are four things that students may, and are required to know how to do, for the AP Exams. But graphing calculators are not required just to answer a few questions on the exams. They are to encourage investigations and experimentation in all math classes. And not just graphing calculator use but all kinds of appropriate technology. So, don’t restrict yourself and your students to only those operations required on the exam. That said, here are previous posts on exam calculator use; as the year goes on there will be other posts on the use of graphing calculators and other technology in your class.

Starting School

A little late perhaps …

These posts discuss basic ideas that I always hoped students knew about mathematics before starting calculus

 

 

 

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Proof

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When math books present a theorem, they almost always immediately present its proof. I tend to skip the proofs. I assume they are correct. I want to get on with the ideas in the text. Later I may come back and read through them. Is this a good thing to advise students to do? I don’t know.

There are reasons to read proofs. One reason is to help understand why a theorem is true, by seeing the reasoning that leads to the result. Another is to check the reasoning yourself. A third is to learn how to do proofs.

Learning to write original proofs is not usually one of the goals of a beginning calculus course. That comes later in a course with “analysis” in its title. There are many theorems that involve some one-off that rarely will be used again. I’m thinking of a proof like that of the sum of the limits is equal to the limit of the sums, where you add and subtract the same expression and this more complicated form allows you to group and factor the terms of the numerator and arrive at the result. Another example is in the Mean Value Theorem where you consider a new function that gives the vertical distance between a function and its secant line. These always bring the question, “How did you know to do that?”

If a student can accept things like that, then the proof is usually easy enough to follow. But I would never spend a lot of time making every student fight his or her way through each and every proof.

On this other hand, I would never just present a theorem and not give some explanation as to why it is true (and why it is important enough to mention). Unfortunately, I have seen teachers write the Fundamental Theorem of Calculus on the board and proceed to show how to use it to evaluate definite integrals, with no hint of why this important theorem is true. Sure kids can memorize it and use it, but it seems to me they should also have a hint as to why it is true.

Some theorems are easy to understand if explained in ways other than giving a proof. For an example of this, see my post of October 1, 2012 on the Mean Value Theorem. Almost every book will bail out on the Intermediate Value Theorem by claiming (quite rightly) that, “the proof is beyond the scope of this book,” or they give the proof in an appendix. But a simple drawing will convince you that it is true.

So my feeling is that you do not need to labor over a proof for every theorem, BUT, big BUT, you should provide a good explanation of why it is true.

This is important for all students and especially for young women. Jo Boaler writes

“As I interviewed more and more boys and girls, I noticed that the desire to know why was something that separated the girls from the boys. The girls were able to accept the method that were shown them and practice them, but they wanted to know why they worked, where they came from, and how they connected with other methods…. When they could not get access to the depth of understanding they wanted, the girls started to turn away from the subject…. Classes in which students discuss concepts, giving them access to a deep and connected understanding of math, are good for boys and girls. Boys may be willing to work in isolation on abstract rules, but such approaches do not give many students, girls or boys, access to the understanding they need. In addition, high-level work in mathematics, science and engineering is not about isolated, abstract rule following, but about collaboration and connection making.”

[Jo Boaler, What’s Math Got to Do with It? Helping Children Learn to Love Their Most Hated Subject – And Why It’s Important for America, © 2008 Penguin Group, New York. From Chapter 6]

Theorems and Axioms

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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

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Definitions are similar to theorems, but are true in both directions; technically, this means that the statement and its converse are both true (p\leftrightarrow q). The double arrow is read “if, and only if.” Both parts are either true or both parts are false. Definitions usually name some thing or some property.  Definitions are not proved.

The definition of continuity is a good example: A function f is continuous at x = a if, and only if, these three things are true

(1)  f\left( a \right) exist (i.e. is a finite number)

(2)  \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) exist (i.e. is a finite number)

(3) \underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)  (“The limit equals the value.”)

“Play” with it: consider cases where only 2 of the 3 requirements are true – is the function still continuous? What would happen if you removed the requirements about finite numbers?

To use a theorem, one must be sure all the hypotheses are true. To use a definition, one may say that either part is true once you have established that the other part is true. So, if you know a function is continuous at a point, then the three statements are true; or if you can show the three statements are true, you may say the function is continuous.

Here’s an example: A typical AP problem might give a piecewise defined function and ask if it is continuous at the place where the domain is divided.

To get credit for justifying an answer of “yes”, students must show that all the requirements of the definition are met. Specifically, they must show that the limit as x approaches that point must equal the value of  the function at that point (and both are finite).  In turn, to show that this limit exist the student must show that the hypotheses of the theorem that says if the two one-sided limits are equal to the same number, then that number is the limit.

To get credit for an answer of “no”, the student must show that (only) one of the hypotheses is false.

Finally, as with theorems, express definitions in words. With your students, “play” with the theorem or definition by making changes to the hypotheses and seeing how that affects the conclusion. Look at the graphs. Don’t just state the definition and expect students to understand it, remember it and use it correctly.

Theorems

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Theorems are statements that summarize the results that are true in mathematics. Theorems are statements that have been proved true; but the emphasis in AP Calculus is not on proof. Rather, it is on what the theorems mean and how to use them.

Theorems have two parts: the “if …” clause called the hypothesis and the “then …” clause called the conclusion. Students need to know both parts. In many theorems the conclusion is some sort of formula. The students need to know this, but also need to know when they can use it (the hypothesis tells them that).

An early important theorem is the Intermediate Value Theorem (IVT). Take some time with this theorem. “Play” with it. The hypothesis requires that the function be continuous on a closed interval. Use graphs (sketches, no equation needed) to show cases where the conclusion is both true and false when the function is not continuous. Can the function take on values not between f(a) and f(b)? Can you find a case where the hypothesis is met, but the conclusion is false? (Let’s hope not!)

Consider the theorem (p\to q), its converse (q\to p), its inverse (\sim p\to \sim q) and its contrapositive (\sim q\to \sim p) by looking at graphs of each case. (For the IVT the converse and inverse are false. The contrapositive of any true theorem is also true.)

Finally, for this and for all the important theorems that you use this year, express them in words, “play” with them by making change to the hypothesis, and look at graphs. Don’t just state the theorem and expect students to understand it, remember it and use it correctly.

The next post will be about definitions, which are similar to theorems in lots of ways.