Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Defining Reasoning and Proof
|Introduction | Investigating Conjectures | Reasoning and Justification | Additional Methods | Your Journal|
It is helpful for the teacher to be aware of several types of reasoning and proof. Through teacher support, students can have positive initial experiences with proof in many areas of mathematics, including number, geometry, probability, and/or algebra.
At least initially, young students are generally more successful at using inductive reasoning to justify the truth of a statement -- that is, making conjectures from examining cases. For instance, students may create an organized set of data related to particular cases of a conjecture and then examine the patterns. A general statement or rule is then made and defended to verify their conjecture. In Part A, we saw students investigate number patterns and look at diagrams of square numbers. They described a relationship between adjacent square numbers. The students were able to defend and modify their argument that the next successive square will always be "one square wider and one more square down."
A deductive argument is built on established facts and follows a logical chain of reasoning to establish that a conjecture is true in all cases. Extending the Products of Numbers problem from Part A, we can illustrate an example of deductive reasoning.
A group of students made a conjecture that you always add an odd amount to get the next square number after a given square number. They collaborated with the teacher and then presented a rough deductive argument as follows:
"The new square has the same number of square units as the previous square, plus an extra row of square units along the top or bottom. That row is just as long as the original square. A column of the same length is added to one vertical side. But so far, that means you've added the same number of square units twice, which is an even number. We can see that the new corner is empty. By adding one more square [to an even number], the total will be an odd number."
In other words, the students recognized that the number added to obtain the next square number will always be odd, regardless of the square number. The models of squares show that the same number of tiles is added to one horizontal and to one vertical edge of the first square, which is like adding n + n = 2n. By definition, 2n is always even. However, one more square tile is needed to fill in the corner, which makes a square with side length of n + 1 -- so, really, 2n + 1 is being added to each square to make a new square number. And the sum of 2n + 1 is always odd, because 1 more than an even number is an odd number.
Finding a Counter-Example
A conjecture is not true unless it can be proven to always be true. Keep in mind, though, that just one example of being false is all that is needed to prove a conjecture false. For example, students sometimes note that 3, 5, and 7 are all odd numbers and all have exactly two factors (that is, are prime). But this does not prove that all odd numbers are prime. The student who tests 9, 15, or any other composite odd number can successfully disprove this particular conjecture.
Determining a Contradiction
This method involves more sophisticated reasoning. The negation (or opposite) of a conjecture is first assumed to be true, but it is then shown that this results in a contradiction. One of the earliest recorded examples of this type of reasoning appeared in Euclid's works, in an effort to prove that there are an infinite number of prime numbers. The proof assumed the opposite -- that there is one largest prime, say, p. To explore and test this proof, take every prime up to and including p, multiply them together, and then add 1. The resulting number is larger than p, the largest prime. But the resulting number must also be prime because it is just 1 greater than a multiple of every prime up to p. It cannot be a composite number with prime factors because it cannot be divided evenly by any prime less than or equal to p. So -- there is a contradiction.
In the process of experimenting with a variety of types of proof under the teacher's guidance, students are not only introduced to the idea of proof, they also see that mathematical ideas come from people's efforts to make sense of patterns they notice and other observations. They also have many opportunities to deepen their conceptual understanding and problem-solving skills.
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