Thinking Mathematically Problem Solving and Critical Thinking

Inductive Reasoning “Inductive reasoning” is the process of arriving at a general conclusion based on observations of specific examples. Example You purchased textbooks for 4 classes. Each book cost more than $ Conclusion: College textbooks cost more than $50.00.

Inductive Reasoning There is no guarantee that the conclusions reached by “inductive reasoning” are correct with no exceptions. Even though all your textbooks this term cost more than $50.00 it is still possible (and even probable) that a textbook for some course will cost less than $50.00.

Inductive Reasoning The conclusion is really a conjecture or hypothesis and a case where the conclusion is not true is a counterexample. It takes one counterexample to show a hypothesis is false.

Inductive Reasoning In mathematics “inductive reasoning” is often used to find patterns. Find the pattern in the following sequence of 6 numbers and use that pattern to decide what the next number should be. 1,4,16,64,256,1024,? Each number is obtained from the previous one by multiplying by 4. The next number is 4096

Deductive Reasoning “Deductive reasoning” is the process of proving a specific conclusion from one or more general statements. A conclusion that is proved true by deductive reasoning is called a theorem. For example: The catalog states that all entering freshmen must take a mathematics placement test. Conclusion: You will have to take a mathematics placement test. You are an entering freshman.

Deductive Reasoning Examples from Mathematics Suppose 3x = 12. We conclude x = 4. The length of a rectangle is 6 and its width is 5. We conclude its area is 30.

Select a number. Multiply it times 10. Subtract 6 from the result. Divide what you have by 2. Add 3 to this. Repeat a few times with other numbers and make a conclusion [hypothesis]. What type of reasoning did you use. Show why this works. Relate deduction to logic and proof.

Can you think of examples of inductive and deductive reasoning in your areas of interest? In social sciences, humanities, arts, and sciences?