Note: This resource guide also includes a summary of lessons that deal with proof.
Proofs are fundamental to all of mathematics, yet they are often thought to exist only in high school geometry, where theorems are deduced from axioms. This is an overly narrow view of proof, both for school mathematics and for mathematics in general. A proof is any explanation that uses logically valid reasoning to confirm the truth of a given claim. For example, the observation that 2 = 2/1 is supporting evidence for the claim that every integer is a rational number. A proof, however, asserts that, by definition, a rational number is a quotient of integers with a nonzero denominator and that any integer is equal to itself divided by 1.
In a Japanese grade 8 mathematics text, proof is defined as “a proper explanation of why a certain fact is true, without relying on experiments or intuition but rather on previously established properties” (Kodaira, 1992). The level of detail in a proof depends on the nature of the audience for the proof. As a result, proofs often appear to contain gaps. For example, stating simply that every integer is rational because every integer equals itself divided by 1 might shorten the proof above. For someone who is unsure what a rational number is, this is not sufficient as a proof, but for someone who knows the above definition of a rational number, this is a proof. Teachers need to be aware of how much detail students require when discussing a mathematical argument in class.
How much proof school mathematics should include is a complicated matter. First, the question of proof arises in something as simple as solving a linear equation, say 2x + 4 = x + 6, which is the first equation discussed in HK4. A student puts a solution on the board and concludes that x = 2. The teacher asks what x = 2 means and then summarizes with “It means when x = 2, left hand side equals right hand side. Let’s check it.” The checking process leads to a proof that x = 2 is a solution. When the student reduces the equation to x = 2, what has been done is to show that if there is a solution, then it must be 2. The teacher does not mention this, but to help students understand what the solution x = 2 means, he not only has students evaluate both sides of the original equation when x = 2, to see that they are equal, but also has them evaluate both sides when x = 3 so they can see that the two sides are different in this case.
S4 addresses similar equations. One equation is 2x - 7 = -19. One student adds 7 to both sides to get 2x = -12, and another student says to multiply by 1/2 to get x = -6. Earlier the teacher had said that they were to use the theorems of equivalence, which he restated as one could add a number to both sides, add an expression to both sides, and multiply both sides by the same non-zero number. If students understand that the theorems of equivalence mean that an equation and a second equation obtained by using one of these three rules have the same solutions, then the above argument is a proof that x = -6. If the students do not understand the theorems of equivalence in this way, then the student’s solution is an incomplete proof; one can complete it (as in the HK4 lesson) by showing that when x = -6, both sides of 2x - 7 = -19 are the same number.
Some proofs contain gaps that would involve mathematics that is too advanced for the audience. For example, in grade 5 or 6, one can derive a formula for the area of a rectangle by defining the area of a square with the side of one unit to be one square unit, and then building up rectangles with sides whose lengths are positive integers times the unit, to see that the area of a rectangle with sides of length b and h is bh square units. In grade 6 or 7, one could extend the formula for the area of a rectangle to the case when b and h are positive rational numbers. When b and/or h is not rational, then bh can serve as the definiton of the area of a rectangle with sides of lengths b and h . It is possible to prove this formula, but the mathematics involved is too advanced for high school.
To develop students’ abilities in constructing proofs, teachers need first to help them learn to reason mathematically and logically (Epp, 1999). (For a discussion of high school examples, see Epp, 1998, pp. 711–712.) Such reasoning requires a school mathematics analogue of definitions, axioms, and rules of inference, but none of these definitions needs to be stated formally. Students can learn to refine their understanding of everyday language in ways that allow them to make correct inferences about mathematical statements (Epp, 1998). Epp further unpacks some of the subtleties of understanding and validly determining the truth of statements such as “the square of any rational number is rational” (Epp, 2003).
Deductive reasoning plays a central role both in proof and in the process of mathematical discovery itself. Success in the discovery phase normally requires mastery of the very same logical principles required for the proof phase. The main difference between the uses of logic in the two phases of the problem-solving process is that proof employs logic more systematically than does discovery. In a formal proof, logical principles are put in the service of developing a coherent, essentially complete line of reasoning from a hypothesis to a conclusion. In discovery, one often makes assumptions that leave big gaps in arguments, may jump from one point to another in the problem, and sometimes reasons backward from the conclusion to deduce what conditions would make it true. Throughout the discovery process, however, one needs to have an automatic instinct for various aspects of logical reasoning.
In summary, it is important to reassess our understanding of the use proof in K-12 mathematics. One must recognize that (1) formal proof is not suitable at all levels because the mathematics is beyond them, (2) what constitutes proof resides in part in the receiver (what he or she knows or doesn't know), and (3) there is a difference between evidence or definition and mathematical proof.
Epp, S. (1998). A unified framework for proof and disproof. Mathematics Teacher 91(8), 708–713. (Available from Susanna Epp at sepp@condor.depaul.edu.)
Epp, S. (1999). The language of quantification in mathematics instruction. In Developing mathematical reasoning in grades K-12. Stiff, L.V. & Curcio, F.R. Eds. Reston, VA: NCTM Publications, pp. 188–197. (Available from Susanna Epp at sepp@condor.depaul.edu.)
Epp, S. (2003). The role of logic in teaching proof. American Mathematical Monthly, 110, 886–899.
Kodaira, K. (Ed.). (1992). Japanese Grade 8 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP), p. 121. Available from UCSMP.
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