Restructuring Definitions for ProofsLesson Plans > Mathematics > Geometry > Logic
Restructuring Definitions for Proofs
I recently worked through a short proof with a geometry student - it was the first proof provided in his textbook - the Midpoint Theorem. It states that if X is the midpoint of segment AB, then segments AX and XB are congruent. It is a short and simple proof, and it relies on just two definitions - the definition of midpoint, and the definition of congruent segments.
When we finished working through the short proof, my student stared at me in astonishment and said, "I get it!"
"Well, yeah," I said, "Why wouldn't you?"
Then it came out that this was his second go-round with geometry, and in his first attempt, he couldn't make any sense of proofs at all. "It's like we were just randomly sticking statements and reasons on paper without rhyme or reason. But it's actually all logical!"
I have a theory why this student struggled through Geometry without really getting what it was all about, and then suddenly it clicked for him. I think there's a significant issue with many Geometry curricula which, if remedied, will make proof-writing much more understandable to students. And it's all about the definitions. I'll explain what I mean by that, but first I need to review a little bit of logic.
In our Geometry curriculum, we spend a unit talking about logic, and especially the Law of Detachment and the Law of Syllogism.
The Law of Detachment states that if p -> q is a true statement, and p is a true statement, then q is a true statement. (written symbolically as [(p -> q) p] -> q)
The Law of Syllogism states that if p -> q is a true statement and q -> r is a true statement, then p -> r is true statement (written symbolically as [(p -> q) (q -> r)] -> (p -> r)
These two laws form the backbone of every proof. We begin every proof with a conditional statement, and we treat the hypothesis of that conditional statement as given (true). Then through repeated use of the Law of Detachment, we build a series of one conditional statement after another, and when we are done, we have formed a sort of "daisy chain" of statements, and the Law of Syllogism tells us that if the first hypothesis implies the last conclusion.
So why is it so hard to understand proofs? In part, I think it is because many geometry textbooks don't structure their definitions in if-then form, so the connection between proof and the laws of logic are not obvious.
For example, to use the example above (the Midpoint theorem) we need two definitions. Our textbook defines midpoint and congruence as follows:
Midpoint: The midpoint of a segment is the point halfway between the endpoints of a segment. If X is the midpoint of segment AB then AX = XB.
Congruence (segments): Segments that have the same measure are congruent.
In the first case, the second sentence is a conditional statement, which is great, but it's not strong enough; this should actually be a bi-conditional statement: X is the midpoint of segment AB if and only if AX = XB and X is between A and B.
In the second case, the definition is not written in if-then form at all. It should be rewritten: Two segments are congruent if and only if they have the same measure.
If our definitions are not in conditional form, it is not at all obvious that we are doing Law of Detachment in our proof. But if we have these definitions in the proper form, the proof is a more simple application of these laws.
Furthermore, since the first proofs we do rely heavily on definitions, and since definitions are rarely written in the proper form, students' first introduction to proof-writing works entirely outside the logic framework they've just learned. So now let's consider the proof of the Midpoint Theorem, as I discuss it with my students, once we've rewritten the definitions.
The first statement of the proof is the given hypothesis "X is the midpoint of AB". So I say to my students, "Can you find anywhere in your notes a conditional statement in which the hypothesis matches the statement "X is the midpoint of AB"? "Yes," they say. "The definition of Midpoint."
"Okay, so if the hypothesis of that conditional is true, what does the Law of Detachment allow us to conclude?" They reply (readily!), "AX = XB."
"Right. Since the hypothesis is true," I say, "the Law of Detachment lets us conclude that the conclusion is true." So I write statement 2: "AX = XB", and give "Definition of Midpoint" as my reason.
Now I ask, "Can you find anywhere in your notes a true conditional statement which has something about segments with equal measures in the hypothesis?"
Here is where things get really interesting, because students may skip right by the definition of congruent segments. Why? Because they forget that it is a bi-conditional statement. I remind them that an if-and-only-if means that both the statements serve as hypothesis and conclusion. Thus, the definition of congruent segments is this pair of conditionals:
- If two segments are congruent, then they have the same measure.
- If two segments have the same measure, then they are congruent.
Clearly, statement #2 in the proof matches the hypothesis in the second conditional statement.
Therefore, I write statement #3: "Segment AX is congruent to segment XB." My reason is, "Definition of congruent segments"
It's really important to note that if we had not rewritten the definition of congruent segments in its conditional form, it would have been unreasonable to expect my students to recognize that it fit into the chain of logic.
So what do we do about this? Every time we come to a definition in the textbooks, I force my students to rewrite it in conditional (usually bi-conditional) form. We work out the details together, and they write the new version of the definition in a section of their notebook set aside for definitions. Similarly, if we happen to come across a postulate that is not written in conditional form, we rewrite it as well.
It's a time consuming process, and part of me wishes I didn't have to take so much class time working through this process, but the reality is, if it helps students actually understand what they're doing when they're writing proofs, it's worth it.