How do I Introduce Converse Theorems in Class?

Question

I'm teaching geometry to grade 8 and 9 students, and I find that they often mistake theorems with their converses. For example, when I give them a 3, 4, 5 triangle and ask them to decide whether or not it is a right triangle (using the converse of the Pythagorean Theorem), many of them begin incorrectly with $a^2+b^2=c^2$, thereby assuming what they're trying to prove.

I'd like to clarify converses for my students, but I have little time to spare. I'd rather not take my grade 8's into propositional logic, though I know that's an option.

How can I meaningfully teach my students converses without going too deeply into propositional logic?

As far as I can see, there are at least 3 possible "depths" for teaching the idea of a converse. Ideally I'd like to teach my students at a level somewhere between 2 and 3 below.

1) Avoid using the word converse altogether and just explain all theorems separately and carefully.

2) Explain that converse means sort of the same thing as an opposite, like in the same way I used the word opposite to help my students understand that $+$ and $-$ are inverse operations.

3) Teach a bit of propositional logic, then explain that the converse of $A \implies B$ is $B \implies A$.

Finally, here's a list of some of the theorems with which my students make mistakes related to the converse.

The Pythagorean Theorem $\iff$ converse of Pythagoras

Parallel lines $\iff$ Angle relationships formed by transversals

Type of triangle/quadrilateral $\iff$ properties of triangle/quadrilateral

Segment connecting midpoints of a triangle $\iff$ consequences of the midpoint theorem

Answer 1

Teaching converses at the level of secondary Geometry is complicated by the fact that the vast majority of the theorems you want your students to learn are in fact biconditional, so that for each theorem they know, the odds are that its converse is a theorem, too, and therefore the distinction you want to make seems (to students) like hair-splitting.

The distinction between a theorem and its converse only seems meaningful when one of them is true and the other one is false. Here are two examples:

1. If quadrilateral $ABCD$ is a rectangle, then it has congruent diagonals.
2. If quadrilateral $ABCD$ is a rhombus, then it has perpendicular diagonals.

One approach to teaching about converses is to begin with a proposition that is true in only one direction, illustrate it with examples, and then pose the converse question. Odds are a lot of students in the class will think it is true. Challenge the class to find a counterexample. Assuming one is found, use that discovery -- the fact that "If $A$ then $B$" is true, but "IF $B$ then $A$" is false -- as an opportunity to introduce the word "converse". You can also use the same opportunity to ask them what they think about "If $B$ is false, then $A$ is false" (contrapositive) and "If $A$ is false, then $B$ is false" (inverse).

Again I think it's important to have this conversation in a setting where the proposition under consideration is true only in one direction. But you can mix it up: Give them a proposition that is false and ask them if its converse / inverse / contrapositive is true. Keep stressing that there is no automatic connection between the two directions of a proposition: they might both be true, or both be false, or either one could be true and the other false.

Then move into some examples that are biconditional, such as the ones you listed in your question: The Pythagorean Theorem and its converse, the various parallel lines/angles properties, etc. The fact that these are biconditional seems much more significant if you already know that isn't always the case.

Answer 2

Converses may be clearer with quantifiers than with propositional logic.

• The converse of "all As are B" is "all Bs are A".

• The converse of "all basketball players are over 5' tall" is "all people over 5' tall play basketball".

• The converse of "All right triangles have $a^2+b^2=c^2$" is "all triangles with $a^2+b^2=c^2$ are right."

Analyzing this with implication may not be necessary. Euclid had the converse to the Pythagorean theorem without any propositional logic.

Answer 3

As you will be unsurprised to learn, this is a common error for students learning about proofs. Two authors who write about this topic are A and J Selden; here is a citation and excerpt (pdf 472/598):

Selden, A., & Selden, J. (1987, July). Errors and misconceptions in college level theorem proving. In Proceedings of the second international seminar on misconceptions and educational strategies in science and mathematics (Vol. 3, pp. 457-470). Link.

(Side-note: It seems that "if it rains" is a common example! I see AndrewC also includes the statement: If it's raining, there are clouds in the sky. See also my example regarding the statement I'll go unless it rains as posted on MESE here.)

I have heard, in the past, instructors for proof-writing courses emphasize the last point about mathematical definitions using the if convention when they really mean if and only if. Moreover, this if vs. iff problem is mentioned by mweiss, who notes that many of the "if, then" propositions in eighth grade geometry are, in fact, biconditional. So: We have here one suggested answer, which is that you emphasize when the word "if" is used to mean "if, then" and when it is used to mean "if and only if"; perhaps you give examples in which you ask students more often to prove a statement, and then to figure out whether or not it is biconditional.

Toscho's comment gives a nice example: If the polygon $S$ is a square, then the polygon $S$ is a rectangle. Prove the statement. Is the converse true, too? Why or why not?

Looking at your three points, I have now gone beyond 1 (by using the word "converse").

You mention that you would like to teach somewhere between 2 and 3. I would actually object to the idea of 2, insofar as the word "opposite" can have different meanings. In this particular context, you give the example of $+$ and $-$ as being "opposites" because they are "inverses." Of course, in propositional logic, the implication $P \implies Q$ has both a "converse" (the subject of your question here) and an "inverse." Which one is the "opposite"?

There are other subtle issues with figuring out the converse of a proposition. For an example from group theory, see:

Hazzan, O., & Leron, U. (1996). Students' use and misuse of mathematical theorems: The case of Lagrange's theorem. For the Learning of Mathematics, 16, 23-26. Link.

in which the authors write:

The phenomenon of experts vs. novices relating to mathematical statements/problems at different levels (i.e., on a structural vs. superficial level, respectively) is recurrent. For another example of this and related citations, I refer you to my MESE response here.

As for level 3: I think this is the best choice. I am quite confident that you can intersperse real-life examples among geometric examples and still keep everything digestable for eighth grade students. (Incidentally, when I took geometry as a ninth grader, we began by covering propositional logic: implications, negations, and/or, truth tables, etc.)

Summary: Teach students that "if $P$, then $Q$" statements have as their converse "if $Q$, then $P$". Oftentimes, a true "if, then" statement run into by students will have its converse be true as well; however, this is not always the case. To help students understand this point, introduce the word converse and repeatedly ask about "if, then" statements both what their converse would be, and whether the converse is true or false.

Finally: An oft-incorporated technique in proof-writing classes is to show students examples of incorrect proofs. (See, e.g., Brendan Sullivan's MESE response here.) If you are observing a particular error frequently (e.g., students using a proposition's converse erroneously) then try writing up sample "proofs" with this mistake and ask students to critique them.

Answer 4

I don't think they're confusing statements with their converses, I think that, like my own students, they don't initially know that to prove B you can't start with B, but must instead conclude with B.

This is different from the converse - they didn't attempt to use Pythagoras to show that the triangle had sides 3,4,5, they just used Pythagoras because it was mentioned in the question.

Their directionless past

It happens partly as a result of the directionless world they come from - before being introduced to proof, there are just things you write on a page - they factorise stuff, they multiply stuff out, they square stuff, they square root stuff. The rules of mathematics don't have a great deal of direction to them before they meet proof. So far, they think of some maths and they write it down, and they get ticks unless they made a mistake.

You asked them to get to a destination from a start, and they write the destination down first - this is because they didn't see that as cheating since all their questions so far only gave them stuff they were allowed to use whenever they like in their answer. Any feedback they've had so far on what order to write things in has seemed presentational rather than fundamental.

A sense of direction takes time to acquire

This takes time to absorb and learn, so in initial lessons I give non-maths examples and then follow up strictly for about a year - "That's cheating - you can't start with what you're trying to show!". (For fans of football (soccer) or rugby, you can refer to it as offside or a forward pass, but sadly I don't know enough to make a US analogy.)

I don't think there's a short route to this any more than there is to understanding that you can't distribute anything over everything. It takes practice.

Examples I've used to explain

Innocent until proven guilty

I pick on the most innocent, pleasant and helpful student that I get on with in the class (suppose she's called Lauren), and say

One of my erasers went missing! Lauren stole it! Let me prove it to you: Lauren was the one who took it, but she's clever, so she wouldn't have done it when there was anyone else in the room, so clearly she'd have been in the room on her own, and that was when the eraser went missing. Now let's look at the facts we've established: Lauren was in the room, on her own, at the time when the eraser went missing. Before she was in the room on her own, the eraser was there, and after that it wasn't, and no-one else was there in between. We can only come to one conclusion from this evidence: it was Lauren!

I explain that the court would reject my evidence since it was based on Lauren's assumed guilt. We can only say Lauren's guilty after we've used evidence to back that up.

The trigonometrical identity 5 = 9

5 = 9, so subtracting 7 we get
-2 = 2, then squaring both sides we get
4 = 4. Tick! True!

(I use that in response to "proofs" of trigonometrical identities that conclude $3 \sec^2 x = 3 \sec^2 x$, true.)

They pick up the rules of the game eventually through practice and being corrected.

When I am explaining converses

If it's raining, there are clouds in the sky.
Contrapositive: If there are no clouds in the sky, it isn't raining.
Converse: If there are clouds in the sky it is raining.

Answer 5

One of my colleagues was fond of using this kind of proposition to make students understand that the converse is not always true: "If you live in Manhattan, then you live in New York." It is easily translated in an inclusion diagram. It is also easy to see that the inverse ("If you don't live in New York, then you don't live in New York") is true.

Answer 6

Perhaps the best way is to understand any (well-formed) statement as being either true or false. Each statement may be about certain objects or an absolute statement independent of any object. Then it is quite clear that some statements are true and some are false, with no necessary link between them. A theorem of the form (If P, then Q) can be understood to mean that we somehow know that whenever the statement P is true, the statement Q is true too. Examples are important, and can even lay the foundation of first-order logic without ever talking about propositions and logical symbols, not to say "converses".

Example

Ask your students to take out 3 pencils each and place them on the table. Write the following three statements on the board:

(P) The first and second are parallel

(Q) The second and third are parallel

(R) The first and third are parallel

Ask them if there are any situations where:

(0) None of them are true

(1) Exactly one of them is true

(2) Exactly two of them are true

(3) All of them are true

After they experiment, ask them to tell you the answers and the constructions for the possible scenarios. They won't be able to arrange the pencils to make (2) true. Ask them if perhaps they did not try hard enough. When they are convinced that it is impossible, ask them why? Probably some of them will realize that when P and Q are both true, R is also true (which we know as $P \wedge Q \Rightarrow R$). But since (1) can happen, it is possible that R is true while P and Q are both false. Also, notice that this very fact is the same as saying that (2) implies (3). But certainly (3) implies (2), and hence we have (2) and (3) being equivalent.

Emphasize that mathematics is about figuring out what is true based on what we know to be true already. In the above example, if for a certain arrangement of 3 pencils we know that ( P and Q ) is true (someone we trust told us so), we can figure out that R is also true without having to look at the pencils. But if we only know that R is true, we cannot claim that ( P and Q ) is true, because we don't have enough information to rule out the possibility that the first and third pencils are parallel but the second is slanted to both of them.

Note

This same example will also be useful to explain equivalence relations and equivalence classes in the near future. It is also related to the fact that you need at least $n$ equations to determine the values of $n$ unknown real numbers, which in this case are rotations with respect to the first pencil. A lot can be discovered with pencils!