On 10/13/2010 at 12:12 pm, I wrote:
> Dear All,
>
> The following article "Precision of Speech" by Ralph
> A. Raimi gives
> another excellent and strong reason for why
> mathematics education
> is failing our students:
>
> http://www.math.rochester.edu/people/faculty/rarm/prec
> ision.html.
>
> Though he mentions mathematics education, I believe
> the reason
> he mentions here, which is the flood of imprecise and
> meaningless
> speech in the media and in everyday life, language
> that sounds
> impressive but does not say much of anything, carries
> over to
> education in general.
>
> Raimi does say that he does not claim that this
> reason is the
> reason for mathematical education's failures, and I
> would not
> make any such claim either. The reasons are many and
> complex
> and so cannot be reduced to a simple list that is
> also an
> exhaustive one. But imprecision in everyday speech
> and in
> the media certainly contributes much to the problem
> of our
> students' educational failures.
>
> I may have additional comments later.
>
>
>
> Jonathan Groves
Dear All,
Here are my further comments on Ralph Raimi's essay after I have thought
about it for some time after I had posted this message: To be more
accurate about the imprecision of everyday language and the language
in the media as playing their roles in making math education more
difficult for students, I should say that math classes often fail
to acknowledge this fact, that math teachers often seem to forget
that mathematical statements written in everyday English cannot
necessarily be interpreted in the same way as we may intepret them
in English. One good example are conditional statements: In
everyday English, we sometimes state the conditional while also
implying that the inverse is also true. A father may say to his
son, "If you don't eat your veggies, I'll send you to bed without
ice cream." It is almost certain that the father also means the
inverse as well. However, in mathematics, we can never assume the
inverse unless we are told explicitly that we can do so. Another
good example are disjunctive statements: The word "or" in
mathematics is always taken to be the inclusive "or"--that is,
A or B is true when either one or both statements are true.
If we mean the exclusive "or," then we must say so explicitly
unless it is obvious that both A and B cannot both be true.
However, in everyday English, whichever meaning of "or" is
intended is not always explictly stated and often means the
exclusive "or" though not always.
Another problem is that much of K-12 curricular mathematical
language is imprecise, which Clyde Greeno has mentioned
recently. Though such language originated with SMSG not
thinking through such matters carefully, I wonder that
curricular language has continued to remain imprecise
because we are used to imprecise language being all around us.
We can get used to it enough that sometimes--perhaps often--
it can be difficult to tell, especially without serious
thought, that the language is not really all that precise.
Another problem is that much everyday language often wrongfully
assumes that certain key terms are known to the audience and
known to the audience in the same way that the speaker or
writer uses--especially when the key term either has no definition
that everyone agrees on or has never been precisely defined.
Recent discussions here on Math-Teach have shown me that
the term "discovery learning" does not mean the same thing
to everyone. I wonder if we get used to this difficulty
in everyday language and then often fail to recognize it.
That can happen in mathematics education, especially when
students transfer to new schools or programs whose teachers
use alternate terminology not familiar to the student and
the teacher not being aware of that or when the teacher
falsely assumes that students have mastered the terminology
he or she is using simply because the students had seen
it before. The same goes for notation.
Other problems are not only not acknowledging such facts about
the sloppiness of language in the media and everyday life but
also not helping students to understand that mathematical
language is precise and is supposed to give us more information
than everyday, sloppy versions of mathematical statements
and that everyday, sloppy versions of mathematical statements
might actually not really tell us as much as we think they do.
Another problem is many math teachers often allow students to use
mathematical language sloppily. And then many students end up
not realizing that mathematical language often does not mean
what they think it means, that mathematical language is more
precise and explicit than they realize. And many of them end up
believing that there is nothing wrong with being sloppy with the
language as long as "you know what I mean."
In short, the ways we think about everyday language and using
everyday language do not work well in mathematics, yet
mathematics instruction at the lower levels often fails to
recognize this and often fails to do much of anything about it.
We cannot change how everyday speech is used, but everyday
speech does not necessarily need to be an obstruction to
mathematics education if we realize its limitations and the
problems we run into if we try to force the kind of thinking
about everyday language onto mathematical language.
Jonathan Groves
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