{X:X is full of baloney}

Turning our attention to beginning algebra courses. First of all, the text I’ve been using most recently is called Intermediate Algebra; “intermediate between pre-algebra and actual (university-credit earning) algebra” is the most charitable spin I can put on that. Anyhow, here again, set theory is used so clumsily that it’s hard not to attribute malice to somebody along the line (Hanlon’s Razor notwithstanding).

Consider, then, an abomination like
\{ x | x is a natural number less than 3\}.

If we’re going to go around calling a perfectly inoffensive set like \{1, 2\} out of its name in order to make some point about our notations, we’d be much better off to actually pretend we believed these very notations were actually good for something and write instead
\{x | x \in N, x \, \langle\, 3\}.

Or how about
\{x| x is a real number and x is not a rational number \}? Doesn’t it just make you want to, I don’t know, hurl the chalk at something? Actually, I have to admit that I’ve copied this display on several blackboards in my time … but only to illustrate a point (namely, that it was created by enemies of mathematics and that one of course really means
\{x| x\in R, x\not\in Q\}).

The symbols in question (N, R, Q, \in, \not\in) appear in this very section and are used in the exercises.  Though not, of course, in the exercises about “set-builder” notation — no, these have all been carefully contrived to reinforce the reader’s impression that our goal in presenting this material is to make easy things hard by way of the whole ignore-the-point word & symbol mishmosh I’ve just been complaining of.

But then, that brings us to the saddest part of the whole sorry business. We don’t actually need (still less want) these symbols — or the set-builder notation itself! — for whatever follows in the whole rest of the book! And pretty much every 9th-grade-algebra-for-college-students text that’s come out in several (admittedly very short) generations does things in exactly the same way!

I have what feels like a pretty coherent theory of how things got this way (though essentially no idea as to “what, then, are we to do?”).  But I promised myself I’d keep it brief, so I’ll just wave my hands in the direction of The Muddle Machine.

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  1. I could forgive them if they used those examples just to say “look at how awful it would be to write Math without good notation!” or “We can misuse notation to make simple things look complex!”.

    You’re right, 9th-grade-algebra-for-college-students does not need so much notation. If you introduce the machinery when it actually makes the students lives easier, students tend to be more receptive. That’s my experience anyway.

    By the way, I enjoyed your links. Thanks!

  2. in faithfulness to my source here
    i followed the convention that
    N = {1, 2, 3, …}.
    this is quite a commonplace convention.
    i’ve come to strongly prefer the *other* convention…
    the one that holds that
    0\in\Bbb N
    (“zero is a natural number”)
    and so
    N = {0, 1, 2, … }.

    part of the point of mentioning this now
    is another complaint about the set i was
    complaining about… these “conventions”
    are *both* quite common and so one
    needs a *context* (and one more
    precise than “any old math book” or
    “… math blog”) for this set even to
    qualify as well-defined.

  3. this “yahoo answers” page displays an abuse
    far worse than those i ranted about here.
    {x: x is all real numbers}.

    i’ve seen this on a blackboard written by a teacher.
    i’m pretty sure i blogged about it too but now you
    don’t have to take my word for it.




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