Chapter 6 - What is Proof?

When students begin a class in geometry, one of the first things that they encounter is the idea of writing a mathematical proof. Unfortunately, many students are confused and dismayed, and some never do like geometry from this moment on. It is truly unfortunate that they feel this way, because it colors the rest of their experience with geometry.

In my geometry classes, I introduce the idea of a proof by first asking the students some questions, to put their thought processes to work. How do you know when you hear someone state a "fact", that it is true? Are there some things that people used to think were true actually turned out to be false? I give them a short assignment: "Think of something that people used to think was true but which turned out to be false." They were somewhat intrigued by this question, as it was not the kind of question they were used to hearing from a math teacher.

They came up with some interesting answers to my question: a student wrote‚"Everyone used to think that the world was flat, but then they realized that it is a sphere" and another did some research and discovered that‚"People used to believe that the size and shape of your head determined your personality and intelligence; this was called Phrenology Theory." Other students mentioned "spontaneous combustion", and "The Ptolemaic Theory, that the Earth was the center of the universe." We discussed these beliefs, and the students began to accept the idea that proof, is part of learning. They even began to feel that it might be of value, as well!

A second project helped them to understand proof; this was also a project that they thought was "fun". They were happy to find that Geometry looked like it might be interesting, after all, if they got to play games! I called this game "Getting Blood from a Stone". But of course, I had to first explain what this saying meant: that you can't get something from someone if they don't have it, for example you can't ask a poor man to give you a million dollars.

So here are the rules for this word game, including a step-by-step analogy to the steps in a mathematical proof:

1) You must start with the word STONE and end up with the word BLOOD by a series of logical steps (just like the logical steps in a mathematical proof).

2) You must change only one letter in each step. Each step must follow directly from the previous step. (in a proof, you must address one concept at time. Each step must follow directly from the step before it - including from the Given, and lead into the next)

3) The word in each step must be a real word, and in the dictionary. (In a proof, each step must be true, a definition, postulate or theorem.) I tell my students that if they use an unusual word, they need to give me the definition.

4) Some people can do this in 10 steps, others might take 15, and both can be correct. There is more than one route, and no one way is necessarily better - some may be longer, but the best one is the one that the student "sees" when trying to do the problem! (This is very true of a proof; there are dozens of proofs of the Pythagorean Theorem, for example, and even in classroom assignments, there is often more than one method.)

5) Some people like to work this backwards, or even from both ends towards the middle. (This works just fine with proofs, too, and I highly recommend it!)

I usually offer this extra credit after we have done an introduction to proofs and they have written a few medium-length proofs on the board. I start by showing them an easy example, and explain all the ideas #1-5 above. For example: "Write the steps you would use to start with the word CAT and end with the word DOG".

Here is one solution:

CAT - BAT - BAG - BOG - DOG

. . . and here is another: CAT - COT - DOT- DOG

Then I make the analogy between this "game" and proofs.

The students thought that it was rather interesting that we would play word games in a math class! Students really enjoy this exercise, and come up with some very creative solutions, as well as creative ways of presenting their work.

In a writing assignment, I asked the question "What is proof? What are some of the ways we have been proving things in Geometry? Is measurement proof? When you construct a figure in geometry software and then drag points around to experiment with the figure, is this proof? Is a "convincing argument" proof? Can you give an example of an apparently convincing argument that seems to "prove" something but is actually a fallacy? Why do we prove things in Geometry?"

In a portion of her response to these questions, a student wrote: "Proof is a way of showing beyond doubt. Proof is using evidence to establish the truth and has to be verified by others. We prove things in geometry for a couple of reasons. One reason is that it is an exercise in logic and discipline which are both necessary in the real world. This discipline is needed in any field of study in school or profession. Constructing proofs helps build your skill in using logic and reason."

Another student, Shelby, answered "Proof is the method by which a statement or idea is shown to be true without a doubt."

Adam wrote an excellent essay when asked the question "Why do we prove things?" His answer was: "We prove things in geometry for a couple of reasons. We don't do it because we will use it in our jobs when we grow up, although it may be useful depending on what profession you go into; I think lawyers need to use logic to prove things. We prove things in geometry because it is an exercise of logic and discipline which are both necessary in the real world no matter what profession you go into. Geometry is challenging and can be frustrating at times; it requires a strong discipline of mind to stick with frustrating problems, work through them logically, not give up, and write out many-stepped proofs which can be tedious at times. This discipline is needed in any field of study in school or profession. Geometrical proofs are also logical and require the student to think along logical lines and follow a train of thought to derive the proof in a clear, concise manner that others may follow and understand. Logic and reason are definitely important skills in life. They allow us to see when propaganda is being used, when incorrect interpretations of experiments are used, when there is a fallacy in someone's argument, etc. It is necessary in all facets of life, not just math. Logic and reason are tools that are very useful in understanding the world around you and having educated views. Constructing proofs helps build your skill in using logic and reason to forge a chain of thought that can bridge the gap between two sides of a river and help you get across."

A proof is a mathematical way of showing that a statement is true. But more importantly, in proving things, we learn more about why they are true. For example we can prove the theorem "The base angles of an isosceles triangle are congruent" by drawing the altitude to the base, and we find that the altitude creates 2 triangles, which are congruent by HL. In writing this proof we have also discovered that the isosceles triangle is perfectly symmetrical along the line of the altitude, and that the altitude bisects the vertex angle and the base.

Kyle wrote: "The symmetry of the isosceles triangle also can lead us to further conjectures, proofs, and elucidation of the properties of other figures. For example, If we think of a parallelogram as a pair of isosceles triangles, one rotated 180 degrees about the midpoint of a leg of the other, we can conclude that opposite angles of a parallelogram are congruent."

When we experiment with a figure using geometry software, we can make conjectures about figures and observe how they behave. We can even measure segments and angles, and may be convinced that a certain property is true. But this is not considered mathematical proof. There may be a fraction of error in the measurement, or there may be an exception to our conjecture that is not revealed in the construction. Mathematicians believe that a proof, if written correctly, not only proves the concept but illuminates it as well.

The computer can be a tool for demonstration, and with carefully designed activities, can be a "math lab" for discovery. When students become proficient with geometry software, they can create their own demonstrations, activities and projects. We learn the most when we teach others, and giving students that opportunity, they will amaze you, and perhaps even themselves! My vision is to have each student's experience not only the excitement of discovery, but the feeling of accomplishment that comes from helping other students; it is what the student becomes the teacher that he or she learns the most.

But one does not need special software to create a climate of experimentation and discovery in the mathematics classroom. Technology is not the only answer to every problem. It is not a replacement for the teacher, for cardboard models, rubber bands, sticks or clay, but it is one more tool. String, metal contraptions, plastic models, oranges, soup cans and even a potato chip. A "Pringle" potato chip is a perfect model for geometry on a curved surface - a "hyperbolic paraboloid", a saddle-shaped surface, as shown below:

Imagine the enthusiasm generated when you pass a bag of potato chips around the classroom, so that each student has their own model of a hyperbolic paraboloid. They will be very interested in "internalizing" the information!

Pringles

Now, a teacher would not necessarily expect students to write proofs in hyperbolic geometry, but it is vitally important that students, when studying geometry, understand the history and importance of Euclidean Geometry, and proof within that framework, and there are other geometries that are equally valid in a different context. While Euclidean Geometry is the best model for much of the material world, we have learned that other models are needed to describe other situations.

The importance of these other geometries is explained as follows: "The development of non-Euclidean geometries proved very important to physics in the 20th century. Given the limitation of the speed of light, velocity additions necessitate the use of hyperbolic geometry. Einstein's theory of relativity describes space as generally flat (i.e., Euclidean), but elliptically curved (i.e., non-Euclidean) in regions near where matter is present. Because the universe expands, the space where no matter exists could be described by using a hyperbolic model. This kind of geometry, where the curvature changes from point to point, is called Riemannian geometry."

(From http://en.wikipedia.org/wiki/Non-Euclidean_geometry#Importance)

If students are uncomfortable with the notion of other, "curved" geometries, remind them that although we spend our days walking on flat surfaces, be they the floors of a classroom or the grass of our front yard, the Earth itself is a sphere, and not a flat surface. It may be hard to imagine, but at any given moment, when viewed from outer space, we may standing sideways or even upside down, depending on where we live! And you might point out to the class that besides Euclidian and Hyperbolic ("Pringle") Geometry, there is also "Spherical Geometry" as well!

Students who are interesting in learning more can find the information at http://en.wikipedia.org/wiki/Spherical_geometry

"The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word."

Galileo Galilei (1564 - 1642)