In my desire to create a 5-week summer geometry course that would provide students with the concepts most needed to do well in pre-calc and on the ACTs, I think I underestimated how critical it would be to provide instruction on the principles of logic and deductive reasoning. I saved this for the 4th week of the class; looking back on it now, it should have been the first week. There are so many logical concepts that I take for granted, and I kept running up against them in the previous lessons (oh yeah, you guys don't get what "if and only if" means, or why the converse might not be true, etc.)

So, that being said, I am excited to start teaching deductive reasoning this week. I am using the first chapter from __Geometry__ by Harold Jacobs as my guide. I'm starting to like that book more and more. Each section has only a couple pages of well written text, followed by 3 problem sets. The book doesn't fit the typical standards based text pattern (i.e. formula presented, 3 - 5 different examples of problem types, drill on those problem types). Instead, the problem sets seem to be more of an integral part of the lesson itself, with problems that are continuously moving to the next level.

Here is the sequence for teaching logic:

1) Drawing conclusions (determining if a conclusion is necessarily true, not necessarily true, or false)

2) Conditional statements (understanding the "if a then b" structure, using Euler diagrams)

3) Equivalent statements (converse, inverse, contrapositive; logical equivalence)

4) Definitions (it's got a great definition of "definition" - a conditional statement whose converse is necessarily true)

5) Syllogism (if a -> b, and b -> c, a -> c)

6) Deductive reasoning (putting it all together) and proof

I've never taught this before, so I don't know what to expect. Today I did just the first two parts. Some students seemed to really get into it, but a few looked like they were trying to transfer their thought patterns into the tables by staring really hard. I'm telling them that I consider this to be the single most important thing I am teaching them all summer, that it will help them not only in all math classes, but in all their classes, and beyond school. I would like to be able to have them write arguments that have intended logical fallacies in them to try and persuade the class of something, and then have the class try to identify the inconsistencies.

What experiences have you had teaching logic to students? What activities / techniques / scaffolding do you use?

An Prelude to Unit Circle Trigonometry

3 days ago

## 5 comments:

We used to teach a 3-week unit on logic at the beginning of our calculus courses. If you are looking for a good textbook or print guide to the basics, I'd recommend Steve Roman's book "Logic"; it's a 35-page booklet that covers all the basics, most of what you listed above, along with a neat application chapter on digital logic circuits. Of courses there's lots of free stuff on the web too.

Beware that logic, for most students, is a completely alien concept. They are products of a culture that has basically trained them to think of logic and analytical reasoning as "bad". They will have some innate skill, because logic is supposed to describe how rational human beings think under normal circumstances, but don't underestimate the amount of work it will be to get them to think with their brains rather than their emotions.

I would have them do a lot of examples where they write sentences/situations from their own lives that illustrate the concept at hand. Personalizing what converse means is a lot easier than "just remembering" it. (Also, maybe try to do something really nerdy like make up an example that has something to do with the word--like sneakers for converse, etc. The nerdier my way for remembering things is for my kids, the more they remember. Plus, then they feel like they get to make fun of me a bit which is fine by me as long as they remember what I'm being nerdy about!) :)

So far, I've found the Euler diagrams very helpful for getting across the idea of why the converse is not necessarily true (it is simply a circle within a circle, where the inner circle is the hypothesis and the outer circle is the conclusion). I've had a couple of students resisting the idea, but when we use the diagram, it seems to be clearing it up for them, because they can see the difference between being in both circles, in just the outer, or outside of both. It also helps make clear why the contrapositive is necessarily true.

What's also interesting so far is that one of my students who is failing due to weak numeracy and algebra skills is really taking off with these concepts, and is doing a great job of thinking logically. I wonder how I can help her connect this ability back to her mathematical difficulties.

I've been trying to use examples that they will enjoy, as well as having them write their own conditional statements. May favorites so far are "If you have a lover, you will have drama" and "If you have feet, then you have shoes".

When I taught Geometry, I would spend a day or two using Lewis Carroll puzzles. This site has some great ones....

http://home.earthlink.net/~lfdean/carroll/puzzles/logic.html

Thanks for the link Mrs. Simpson - those problems will be quite challenging for my students, but I think I will give them a try.

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