Overcorrecting

Overcorrecting is something I do on a regular basis, so I get it.  And I’ve always had students who do it, but not so many in the same exact way as now.

One common misconception that students seem to arrive with is that “=” means “and here’s the next step”.  So you get bizarre false statements like

5x-2=13=5x=15=x=3 ,

and the worst part of it is that students don’t realize that what they have written claims that 13=15=5 .

And so we have conversations in my class about the meaning of symbols, and how it is acceptable to write = as a random connector symbol in your history notes, but in math class = is used to mean actual equality and nothing else.  I stress the importance of clear communication, and give examples of basic mathematical communication standards — like how

\displaystyle \lim_{h \to 0} = 2x-5

might be fine as a note to yourself, but would be poor communication.

All this I did the same as always.  It usually results in an ongoing conversation, but at the end of the semester, they have improved a lot.  And the result this semester?  I had fourteen students (out of 88) that did not write a single = sign on the last test.  Equality is problematic, so it’s best to avoid it altogether?  Yeesh.  Overcorrection.

So I think I need to write some explicit standards of mathematical communication.  I think there will need to be separate standards for explanations in words, sequences of statements, and possibly graphs/visual stuff (yeah, my own communication might need some work here).

Brainstorming here:

  • When writing a sequence of mathematical statements, each statement must have a “verb”, which will usually be an =.
  • A line can start with an =, which means it is equal to the line above it.
  • Someone with a strong mathematical background who does not attend our class must be able to follow your work
  • Use standard mathematical notation

I know there needs to be more.  Thoughts?

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2 thoughts on “Overcorrecting

  1. Ethan Bolker says:

    In your (common) example the student wants the third and fifth ‘=’ to mean “is the same equation as” or “is logically equivalent to”. So I ask/require them to write

    5x-2=13
    so/therefore/which tells me/any other reasonable English logical connecting phrase
    5x=15
    so
    x=3 ,

    I also remind them that paper is cheap, so they should use separate lines.

    Shameless plug: if you need to teach quantitative reasoning (not algebra) you might enjoy
    http://www.cs.umb.edu/~eb/qrbook/

    • I do like the idea of requiring them to connect equations with an actual word if they’re written on the same line.

      Your book looks interesting at first glance– after the semester’s over I’ll find time to take a closer look.

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