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Reasoning and ProofSession 04 Overviewtab atab bTab ctab dtab eReference
Part C

Defining Reasoning and Proof
  Introduction | Deductive Reasoning | Proof by Contradiction | Inductive Reasoning | Proof by Mathematical Induction | The Teachers' Role | Your Journal

 
 

In contrast to deduction, inductive reasoning depends on working with cases, and developing a conjecture by examining instances and testing an idea about these cases. It is frequently used in mathematics and is a key aspect of scientific reasoning, where collecting and analyzing data is the norm. Consider the following example of student work in a geometry class:


Here is a problem: What is the maximum number of regions made by 10 chords within a circle? To solve this problem, make a conjecture about the relationship between the number of chords in a circle and the maximum number of regions defined by those chords. Keep track of the reasoning behind your conjecture.


Here is what one student presented to his teacher:


Chords Within a Circle

I drew these pictures and made this table:


"Every time I added a chord, I doubled the number of regions. So I know that if I add a third chord, there will be 8 regions. All of the numbers of regions are powers of 2. Since 1 chord made 21, or 2 regions and 2 chords made 22, or 4 regions, then 3 chords will make 23, or 8 regions, and 10 chords will make 210, or 1024 regions."



The student has worked inductively here. He has tried several cases, and now formulated a conjecture by observing a pattern that seems to hold.


Here is some dialogue between the teacher and the student.


Mr. Desjardins: Tell me about your thinking. How did you come up with your doubling conjecture?
Ian: Because each time I tried it, it doubled. So I reasoned that it would continue to double all the way out to 10 or past that.


Mr. Desjardins: How could you know that this would hold?
Ian: Well, I checked the numbers.


Mr. Desjardins: Which numbers did you check?
Ian: On my calculator, I multiplied 2 by 2 itself 10 times and got 1024.


Mr. Desjardins: Okay, so are you saying "if it's true that 2 to the 10th = 1024" that your conjecture about the chords problem is true?
Ian: Yes.


Mr. Desjardins: Could you draw a circle with 10 chords? What if you checked the next case past 2, a circle with three chords?
Ian: (makes this drawing)


Ian's Circle


Ian: I can't see how to make more than six or seven. But my table said I should have 8 and then 16 next. I guess it falls apart.


Mr. Desjardins: It's always good to check cases. This time your idea worked fine for the first 2 cases, but didn't hold always. But what about your 1024? How does that fit in?
Ian: I don't know about that, seems like that had to be true but it isn't.


Mr. Desjardins: Well, it is true that 2 to the 10th in 1024, but how does that relate to your doubling idea?
Ian: Oh, I get it. If my doubling conjecture had been true, then we would have gotten the 1024. That comes after, but I was saying it was reason for it to be true. It was just the answer if what I had said was right. Not a reason that it was right on its own.


Let's think about some of the issues raised by this dialogue. The student is reasoning inductively but he has worked on only two cases, and has leapt to a hypothesis that he is eager to check on a calculator. What's more he considers the calculator's result justification, when in fact it is irrelevant to the validity or invalidity of the proposition.


What are the teaching points here? We must foster good "investigative" instincts in our students for them to rely on when examining cases. For instance, it is generally useful to check data beyond the first three cases, typically 0, 1, 2. When they are working inductively they should also keep in mind to examine what may happen for very large or very small values of x. This can be particularly relevant when thinking about the overall shape of a curve. Sometimes the part of a curve you can see in graphing calculator on in a software program does not provide all the data necessary to form valid inductive insights into a problem. A third good practice is to observe first and draw conclusions second. In this case, the student may have been framing the task as finding the powers of two, rather than investigating the geometric figure that was the real point of the exercise. Students should begin to get a feel for when to move to the "rule finding" or conjectural stage of reasoning. This should happen only after a reasonable amount of data has been found and that data make a strong case for the argument. Students must get a feel for the uses and limitations of inductive methods.

Next  Proof by mathematical induction

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