## April 21, 2006

### Inductive and deductive reasoning

I have often been unsure about the difference between inductive reasoning and deductive reasoning. The difference was explained to me by my math prof. and I will try and record it here.

DEDUCTIVE REASONING follows from rules of logic. It is an "If a then b" type of reasoning. Conclusions are not drawn from experiment or measurement. For example, you could not prove that global warming existed using deductive reasoning. But to prove that 2 odd numbers when added produce an even number it is ideal. The formalization of deductive reasoning goes back to the Greeks. They were disdainful of physical measurements. They started with a few basis assumptions about lines and points on a plane. These were called axioms and were considered to be "self evident." (They were, however, not comfortable with one axiom the parallel postulate and with good reason). Using the axioms one can deduce one proposition from another proposition.

INDUCTIVE REASONING is based on empirical data. Most proofs outside mathematics invoke empirical data. The law of gravity is not a theorem. It is a probabilistic statement based on the fact that historically it has always been the case. It is a physical law.

Inductive reasoning is not the same as mathematical induction.

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## March 19, 2006

### Mathematics websites

A colleague (thanks Miriam) sent me this link to square puzzles in mathematics. The page is taken from an online magazine called PLUS.

I decided to post a few more mathematics websites that I use from time to time.

I like mathwords for looking up mathematical terms and definitions.

The Understanding Mathematics website was developed at the University of Utah to offer support to undergraduate mathematics students. Its contents are relevant to mathematics learners at many levels.

This website discusses the story of the number zero, in four dimensions.

The John Handley High School mathematics website has many interesting resources from puzzles, to quotes to lesson plans.

The Vocational Information Center offers a link to many mathematics sites.

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## March 14, 2006

### APOS Theory

The professor on the abstract algebra course that I am taking, Dr. Matt DeLong (a visiting professor from Taylor University)introduced me to the idea of APOS theory today. It is a theory developed by Dr. Ed. Dubinsky about the stages that undergraduate students progress through as they learn mathematics: Action, Process, Object and Schema. There are several references to the theory on the web. Here are links to some of the better ones:

APOS: A constructivist theory of learning

Using a theory of learning in college mathematics courses

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