A couple related thoughts: I'm coming to this largely because of how many of my students in any class (even sophomore statistics) get helplessly tangled up over something as simple as an if/then statement. Or a subset relationship (e.g., normal curves are bell-shaped, but bell-shaped is not the same thing as normal). Or an "and" statement (the z-interval procedure requires a simple random sample, known population standard deviation, and a normal sampling distribution of the mean... the last of which can be established by either a normal population or a large sample size).

I'm reminded of my first programming book in the 6th grade which introduced "and" and "or" operators and just said, "the meaning of these should be obvious", with an example of each. It may not be

*a priori*obvious to everyone, but it really shouldn't take very long, and could pay off enormous benefits later.

Coincidentally, I just came across a delightful blog post by John Barnes on the same subject titled, "The Hobo Queen of the Sciences". Here are a few terrific highlight quotes:

And also:And then I got Ms. Pounding Shouter... She thumped the podium, she pointed at people and accused them of not understanding her, she ordered them to believe what she told them to... "I was totally logical. I pointed things out real loud and told people they were dumb if they didn't believe it, and I yelled so they'd get the point."

Last and far from least, in a related course where I used to teach listening for logic as a way of improving listening comprehension and retention, one student asked me at the end of the class, "Why wasn't I taught this in fourth grade?"

Of course, to his credit John goes on to explain the vested interests that don't want fourth-graders -- or jury members -- knowing the basics of logic and reasoning.

Read it here.

Aha! You've just supplied me with part of my argument for never letting my daughter into a math book. (She's currently 2.75 years old.)

ReplyDeleteI believe her math education should start with learning to read instructions (DVD players, IKEA furniture, etc.) and then continue into good logical practices. After which, if she begs, she can be allowed a "math" book. This will then be consumed with vigor as she'll be able to follow everything. Because she learned logic first.

Thanks so much!

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ReplyDeleteFrom Reuben Hersh's book "What is Mathematics Really".

ReplyDeleteAny proof has a starting point. So a mathematician must start with some

undefined terms, and some unproved statements. These are "assumptions" or

"axioms." In geometry we have undefined terms "point" and "line" and the

axiom "Through any two distinct points passes exactly one straight line." The

formalist points out that the logical import of this statement doesn't depend on

the mental picture we associate with it. Nothing keeps us from using other

words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations

to the terms bleep, ook, and bloop, or the terms point, pass, and line, the

axioms may become true or false. To pure mathematics, any such interpretation

is irrelevant. It's concerned only with logical deductions from them.

Results deduced in this way are called theorems. You can't say a theorem is

true, any more than you can say an axiom is true. As a statement in pure mathematics,

it's neither true nor false, since it talks about undefined terms. All mathematics,

can say is whether the theorem follows logically from the axioms.

Mathematical theorems have no content; they're not about anything. On the

other hand, they're absolutely free of doubt or error, because a rigorous proof

has no gaps or loopholes.

In math, it is quantification (not logic) the students are bored and often disgust with. Add human dimension to a number -lecture will rock.

ReplyDeleteMath students have no problem with logical thinking. It is the quantification they are bored and often disgust with. They have a problem of logic applied to quantities, numbers only. Logic applied to music will be ok for many of them.The point with math is to convince someone to take direction of quantitative thinking. Logic will follow, as it follows ordinary thinking.

And, of course, there is no reason not to introduce the concept of limit right after kids learn rational numbers.

ReplyDeleteInteresting thoughts! mathfour, it seems all my math classes devolve into detailed-reading classes, so your daughter should be ahead of that curve. And Bill, I ADORE the idea of teaching limts early -- just tonight I was trying to grasp for the idea, a bit frustrating not to have it (e.g., central limit theorem)

ReplyDeleteI totally disagree with the statement that "(m)ath students have no problem with logical thinking," a fortiori from my observation that seemingly ALL students have problems with logical thinking. I teach logic, and the difficulty with seemingly-obvious things like the distinction between "necessary" and "sufficient" baffles me. STEM or humanities, none of them seem to get it easily, and that's not the only example I could cite. On the contrary, it's usually only once I start showing them with specific examples that they understand, for instance, why material implication works the way it does.

ReplyDelete