Why do we prove things we already know?

Question

As math majors and math educators we take for granted the importance of proofs and being precise. However with I have found that non-math majors are content with anything that looks reasonably quantitative, whether the logic is correct or not. I get two types of responses why it is a waste of time.

• the statement is obviously true and so we don't have to analyze
• the computation is too difficult therefor we should just -- insert superstitious mumbo jumbo here --

In math courses, a typical examples are showing that $-1\times a = -a$ or that $\displaystyle \lim_{n \to \infty} \tfrac{1}{n} = 0$.

Outside of math, words like good or better are usually signs of trouble since they mean a different thing to just about everybody.

When I am trying to discuss a computation, I am trying to motivate the idea there can be several choices when we are trying to be precise. Then we have to pick the best one and sometimes we get it wrong on the first try.

How do I explain the notions of proof and mathematical proof and why it is necessary? Maybe it's not?

Answer 1

You've touched on why it's problematic for educators to only talk about proving in the context of formal proofs. Students need to be accustomed to mathematical reasoning and justification well before they ever see a formal proof.

This is the reason that Common Core State Standards Mathematical Practice 3 exists:

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

You can see important foundations of formal proofs in this mathematical practice. Practices are meant to be developed throughout the mathematics curruculum, across all grade bands (at whatever level is appropriate for the students).

Ways that some of these practices can be addressed are found in many places. For teachers with students before secondary school I suggest looking at Developing Essential Understanding of Mathematical Reasoning for Teaching Mathematics in Grades Pre-K-8. Central ideas of this book are conjecturing and generalizing, along with justifying and refuting, and how they can be a part of the mathematical activity of students in the classroom.

If you're getting the idea that I am not suggesting a quick fix, you've got the right idea. I really think that students should come out of their public school education realizing that mathematics is something other than calculation -- that the mathematical practices themselves are ways of thinking that help you understand and act in the world more effectively.

So, my ideal solution is to be adamant that the mathematical practices are a focus of school mathematics. That way students will have a chance to know why conjecturing, generalizing, justifying, and refuting are important.

Okay, but how do you convince people in "everyday life?"

There is another book in the Essential Understandings series that deals with proof and proving for secondary school age students: Developing Essential Understanding of Proof and Proving for Teaching Mathematics in Grades 9-12. This is not a book on "how to do a proof" but, rather, how and why proofs and proving are a part of mathematics and life. I wouldn't say it addresses your question directly (although, perhaps if you consider everyday people as similar to secondary school students, there is some utility here). Even so, I like these ideas from the book:

Big Idea 4: Not all arguments are proofs. A proof is not an argument based on authority, perception, popular consensus, intuition, probability, or examples.

A good deal of the rest of the book talks about what proof is, but this idea about what it isn't may be especially helpful when explaining to people that proof has a unique place in mathematical reasoning that is not fulfilled by these other sorts of arguments.

Why do we need this aspect of mathematics? The role of proof is discussed in Big Idea 5's subcategories.

• to verify the truth or falsehood of a statement
• to provide insight as to why a statement is true
• as an entry point into the development of a new idea
• as a structure for communicating mathematical knowledge
• as an impetus for the use of precise mathematical language

As an example of how that last one can be demonstrated, I worked on a math ed technology research project that used software to collect a whole classroom's piecewise-defined position vs. time functions and graph them on one Cartesian space. Students created narratives to go along with their functions (the context was an "exciting race" in which all the participants run for the exact same amount of time, have the same start and end position, but all have different ways of completing the race). One student is chosen to read his narrative, and then the whole class tries to pick out his graph from all the other ones.

Generally, it is difficult to chose one such graph out of the 30 or so student functions. And, usually, it is because their narratives lack mathematically precise language. So this becomes an opportunity to show how precise mathematical descriptions communicate much more than a loose narrative, but also how those mathematical statements can be used to justify the conjecture that "Sam's race is the red graph." "Why?" "Because he ran 6 meters/second for 10 seconds, then stopped for three seconds, and that's what the red graph represents." "How do you know the red graph represents that?" Etc. It is much harder to make a convincing argument without the mathematical language and the precision it brings.

So, my advice is to come up with examples that fit into the categories listed above in the role of proof. Cater the examples to what your audience cares about. But if it's students, there are some decent resources. Look specifically to resources that focus on mathematical practices (especially CCSS MP3).

Answer 2

I would say that the important point is to determine, for each public and course, which axioms are relevant and try to convince (often this is very easy) student that the axioms can reasonably be accepted. Then everything that you want to teach and that is not an axiom should be proved, simply because if the proof was not interesting enough for the given curriculum you would have chosen the result as an axiom.

Now with this I have not really answered your question, but I am not so far: with this point of view, you know precisely why you want to show any given proof to your students, so its usually easier to justify them. Also, it makes teachers quite free to admit some things without guilt.

We should not forget that we never prove everything, and it takes a long time even for math majors to be able to prove every basic theorem from the classical axioms (ZFC say), if it happens at all. The difference between proof-oriented courses and math-as-a-tool courses is simply the deepness of the axioms compared to the most sophisticated results of the course. In some case, all results can be axioms, and the only proofs are mere application of assumed theorems. This is ok as long as it fits the goals of the course.

Let me finally consider a couple of examples.

First, take $\lim \frac1n = 0$. If you are teaching biology majors who needs to be able to interpret asymptotic behavior of simple models, then you do not need to prove it; but you want your students to know it, so you will take it as an axiom. If you teach math majors a course one of whose goals is to teach epsilon-delta proofs, then you will rather assume a set of axioms for real numbers, define rigorously limits, and prove that $\lim \frac1n = 0$.

Second example: for a first or second year course in analysis for math majors, I prefer not to develop an integration theory (Riemann's one is cumbersome, and Lebesgue is out of the question; there are other possibilities but that's not my point here). I rather assume as axioms that there exist a theory of integration that takes a piecewise continuous function on a segment and yield a number; and that it is positive, linear, and satisfy the fundamental theorem of analysis. Then I develop the rest of the theory (e.g. integration by part, etc.) from these basic principles. It emphasizes the basic properties, contains some model proofs, and concentrates on what we do daily. For a later course, I think it is important to come back to integration and explain a way to construct integrals from deeper first principles.

As students get more mathematically mature, we should both teach more sophisticated things and go deeper into the foundations. Hopefully, at each step we prove things that do not sound too obvious, or only after students had time to realize how nontrivial foundations of mathematics are.

Answer 3

Addressing the relatively narrow issue of the question itself, the first proof of a theorem, especially an important theorem, is not always the "easiest" or "most insightful." Also different proofs provide different insights. A good example is Euler's Polyhedral Formula, and the associated graph theory version about planar graphs. Euler's attempts to prove

V(vertices) + F(faces) - E(edges) = 2

for (convex) 3-dimensional polyhedra was not correct (but can be fixed). David Eppstein has collected 20 proofs of this result which show different aspects of this remarkable theorem.

http://www.ics.uci.edu/~eppstein/junkyard/euler/

From an educational point of view since the "polyhedral formula" is not true for graphs embedded on the torus, a proof at some point must use some aspect of the space into which the graph is embedded - in the plane this is the Jordan Curve Theorem.