This is the (slightly modified) content for a submission to the MathCountS column in InRoads, the bulletin of the ACM SIGCSE group. This is the Association for Compuing Machinery Special Intrest Group in Computer Science Education — acronyms, like stereotypes, exist for a reason!
Though theories of learning differ on the details, all indicate that abstract reasoning develops from some form of concrete experience. As children we learn arithmetic of integers and fractions, and then we generalize the rules when we study school algebra. In college we learn “arithmetic” of polynomials and matrices, perhaps even some modular arithmetic, and then we generalize the rules when we study abstract algebra. When we teach mathematical proof to beginning college students, the abstraction forms the primary barrier to understanding for many. Hence, we ask the question, “What are the concrete experiences from which we should start?”
We add a caveat about our goal in this paper, because there are certainly benefits of approaching “mathematical proof” as a mechanical process that applies formal logic via axioms and deduction schema. Our premise is that these are simply other benefits, and therefore not at the heart of the matter. Specifically, when a proposition is later prove in an upper level course, the proof will have a certain form, recognizable to instructors as The Way It Is Done, but it will not remotely look like formal predicate logic. Our goal is to provide beginning students with concrete exercises and experiences so that mathematical proof seems natural to the student.
The approaches and activities described here are used in a freshman-level Discrete Mathematics course at Shippensburg University. The course is required for both mathematics majors and computer science majors, and it is a popular elective among mathematics minors. We use a textbook that we wrote specifically to expose our first year students to the philosophy described here. The book and website references are given below.
WHO IS READING YOUR PROOFS
Writing instructors emphasize establishing a “narrative voice” for all forms of writing from creative fiction to persuasive prose. There is no reason that we cannot accomplish the same thing when writing mathematical proofs. A big first step in establishing a voice as a proof writer is to identify the point of view of the reader of the proof. To accomplish this, we begin our proof unit with an activity that leads the student to adopt this point of view through a sort of role playing. The activity consists of a series of statements purported to be “universal truths” about the natural numbers. We instruct the students to decide which statements represent true properties of the natural numbers. We encourage them to work with partners so they can verbalize their decisions, and we warn them that after the activity we will ask them to explain the process they have used to come to their conclusions.
For each of the following statements, decide if you believe it is true or false. If you believe the statement is false, give an example that supports your answer.
- If n is even, then n2 + n is divisible by 3.
- If n is divisible by 3, then n(n+1)(n+2) is divisible by 4.
- If n2 – 1 is not divisible by 3, then n is divisible by 3.
- If n is even or m is even, then n + m is even.
- If n2 – 1 is divisible by 5, then n is divisible by 2 or 3.
- If n is divisible by 2 or 3, then n(n+1) is divisible by 6.
- If n ends with the digit “2”, then n is divisible by 2.
- If n ends with the digit “3”, then n is divisible by 3.
Assessment of this activity produces some interesting results. A seasoned instructor will not be surprised to learn that in a survey of all freshmen taking our discrete mathematics course, the most frequently missed problem was an assessment by over 75% of the students that statement #5 is true. The complexity of the hypothesis (antecedent) in this statement might account for this difficulty, but it is surprising to learn that an almost equal number of students errantly analyzed statement #2.
The subsequent course discussion brings out many issues worthy of discussion including the meaning of “or” and the formal definition of divisibility, and even and odd. However, there is a much more important outcome. Without instruction or prompting, the students decide for themselves that a universally quantified “if, then” statement is false precisely when there is an example that makes the hypothesis true and the conclusion false. Moreover, the students establish a uniform method for analyzing the statements: They choose an example that makes the hypothesis true and then they check the conclusion for that same example. The stubborn seeker of a counterexample is exactly the persona of the Reader that we want to establish, and this activity allows the student to play that role for a while.
In order to extend the experience beyond the twenty minutes in class, we have created a simple applet that randomly selects similar questions and allows a student to consider the existence of a counterexample and receive instant feedback on his or her conclusion. An important feature of these problems is the significant proportion (about 30%) of statements that are actually true. This not only maintains a strong requirement of critical thinking from the student, but it also allows us to segue from finding counterexamples of false statements to writing proofs of true statements.
The CounterExample activity is very easy to revisit with other structures in the course and indeed in other courses. In a superficial way, the students’ familiarity with this activity helps them make connections and transfer knowledge from one topic or course to another.
THE VOICE OF THE AUTHOR
After all the preliminary work in building the students’ experience with “if,then” statements, we are ready to state the following relationship between the proof Author and the proof Reader: The proof of a claim is a narrative written for a stubborn reader who is searching for a counterexample to the statement.
The first statement that we prove in our discrete mathematics class is the following “universal truth” about natural numbers:
Claim 1. The numbers 3,4 are the only example of a prime number followed by a perfect square.
To be clear, this is the first statement that we prove as an example for the students. The first proofs written by students are much less ambitious, but we feel strongly in beginning with a statement that is not even obviously true in order for the students to feel that this statement really does require proof.
The statement of Claim 1 is not in the “if,then” form for which we developed our framework, so the first order of business is to rewrite the statement in that form. Informed by their own difficulties with unfortunately placed negative statements in the CounterExamples, students see the wisdom in opting for the following form of the proposition: “If n > 4 is a perfect square, then n – 1 is not prime.” As a pleasant side effect, the class discussion leading to this decision has a natural outgrowth about the equivalence of an implication and its contrapositive.
Here is the proof of the Claim 1 as we would develop it in class. It is written in the form of correspondence addressed to the critical reader who is stubbornly looking for a counterexample.
Claim 1. If n > 4 is a perfect square, then n – 1 is not prime.
I know you keep writing down perfect squares and then subtract one and check the result for “primeness.” Consider the last perfect square you chose, and let us both refer to this number as n. Since you wrote it down as a perfect square, I know that your n is equal to some integer squared. If we refer to that integer as m, then when I can write n = m2. When you subtract 1 from n, this is the same thing as m2 – 1. To check this resulting number for “primeness,” I know you are trying to factor it, but I’m sure you remember from high school algebra that m2 – 1 = (m – 1)(m + 1). Since you picked n > 4, I know that m > 2, and so this algebraic factorization shows that n – 1 can be written as a product of two integers each bigger than 1. Hence, n – 1 is definitely not prime, no matter which perfect square n > 4 you pick at the outset. Therefore, your attempts to find a counterexample to this statement are fruitless. I hope you are doing well otherwise.
Hugs and kisses,
The following proof is more in line with the expectations of what the student will write in this first part of the course.
Claim 2. If n is an odd integer, then n2 – 1 is divisible by 4.
Consider an odd number n that you have written. Since the number is odd, that means n has the form
2k + 1 for some integer k. In this case, the calculation n2 – 1 is the same thing as (2k + 1)2 – 1, and from algebra we know that
n2 – 1 = (2k + 1)2 – 1 = 4k 2 + 4k = 4(k 2 + k)
Since k2 + k is an integer, this shows that n2 – 1 is divisible by 4. Hence, any number you choose satisfying the hypothesis cannot be a counterexample to this claim.
Please give my best regards to your family,
The style of proof in the course moves from this informal (some would say “chatty”) flavor to the more traditional form as the students gain understanding about what they are writing. But the logic of the proof is not informal at all. Students later opt for a more formal style of writing in order to reduce the verbosity of the proof, but the logic and structure of the proof remain the same.
OTHER HANDS-ON APPLETS
In addition to the CounterExample applet, we use two other applets to provide the student more concrete interactions with formal statements and proofs. These applets encourage students to read proofs critically. Everyone agrees that students should be able to read an argument and either understand that it is a proof or find a mistake if it is not. However, most courses spend little (if any) time on the development of this skill.
The simpler of the two applets is the ProofScrambler, which requires the user to reorder the lines of a proof to put them in the correct order. Technically, this is simple to implement, which means that it is very versatile for use with other topics that arise later. The more complicated is the ProofReader application that requires a student to respond to each line of a proof as if the line is one in a sequence of instructions. This meshes well with the “critical reader” point of view and works well for getting students to proofread their own attempts at direct proofs. In the ProofReader activity, there are false claims with typical mistakes in the arguments as well as true claims with and without correct proofs. The goal is to have students read and respond to each line of a proof the same way that a researcher might read an unfamiliar proof for the first time.
This article is not so much about our specific examples –these are determined largely by the content goals of the course. The real lesson is the curricular focus on building background knowledge and experience to prepare students for mathematical abstraction. Ideally, this focus should be distributed across as many different courses as possible, but this more ambitious topic is a subject for another day.
Ensley, D., and Crawley, W. Discrete Mathematics: Mathematical Reasoning and Proof through Puzzles, Patterns and Games, John Wiley & Sons, 2006.
Ensley, D., and Crawley, W. Flash activities for discrete mathematics, linked from http://webspace.ship.edu/deensley/DiscreteMath/
Ensley, D. and Kaskosz, B. Tutorials for creating mathematics applets in Flash, at http://www.flashandmath.com/