[Numeracy 496] Re: What does equality mean?
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Tue Aug 17 13:28:47 EDT 2010
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Michael,
You are probably right. There is undoubtedly a consistent definition we
can apply. I was not accurate in saying that certain transformations
change the equality. What may change, however, is the amount of
information present in the expressions. For example, squaring an
expression loses information about whether it is less than or greater
than 0. Conversely, taking the square root of an expression introduces
ambiguity about the sign (unless there are contextual clues to eliminate
one of the possibilities). Likewise, in the second example, transforming
the equation from /y//(/x/ − 1) = 3 to /y/ = 3/x/ − 3, loses the
information that /x/ ≠ 1 unless we specifically preserve it. The nature
of the equality is unchanged.
The point I intended to make was that the balance metaphor serves as an
excellent touchstone for students learning about what the equal sign
means. For basic operations (+, −, ×, ÷), it is flawless as far as I can
tell. If anyone has an alternative or wishes to critique the balance
metaphor, please reply.
Equality is reflexive, symmetric, and transitive. That is, for any real
numbers /a, b, c/:
/a/ = /a/
if /a/ = /b,/ then /b/ = /a/
if /a/ = /b/ and /b/ = /c/, then /a/ = /c/
I think that these three properties are sufficient to define equality
for real expressions, but I am not sure. Perhaps a real mathematician
could weigh in on this.
Chip Burkitt
On 8/17/2010 12:34 AM, Michael Gyori wrote:
> Chip,
> I fail to understand why the definition of equality is subject to
> variation - say in relation to the level of numeracy of our students.
> I have always understood equality in math to signify equality in
> value. If you perform the same operation on identical values with
> different expressions, of course doing so might result in inequality,
> as you appear to state in your post below. My question is, why can we
> not apply a consistent definition to equality when teaching math at
> whatever level we may be doing so?
> Michael
>
> Michael A. Gyori
>
> Maui International Language School
>
> www.mauilanguage.com <http://www.mauilanguage.com/>
>
>
>
> ------------------------------------------------------------------------
> *From:* Chip Burkitt <chip.burkitt at orderingchaos.com>
> *To:* numeracy at nifl.gov
> *Sent:* Sat, August 14, 2010 6:59:24 AM
> *Subject:* [Numeracy 493] Re: What does equality mean?
>
> I think the difficulty is that mathematics requires rigorous
> definitions and logic, especially as one advances in it. However, for
> ABE or GED students, it is usually enough to know that the equal sign
> is like a balance scale. In order for the sides to be in balance, the
> expressions on both sides must have the same value. If you add
> something to one side, you must add it to the other side as well to
> maintain the balance. If you subtract from one side, you must subtract
> it from the other side as well. When students get into algebra, they
> need to know that some transformations of an expression can change the
> character of the equality. For example (−/a/)^2 = /a/^2, but it does
> NOT follow by taking the square root of both sides that −/a/ = /a/.
> Likewise, /y//(/x/ − 1) = 3 needs to be qualified by /x/ ≠ 1, even
> though the equation can be transformed to /y/ = 3/x/ − 3, which has a
> solution for /x/ = 1 at /y/ = 0. For most purposes in ABE or GED
> classes, the balance analogy works well without getting into abstract
> discussions about various kinds of equivalence relations and the
> transformations that change the relation or leave it unchanged. If
> anyone has a better explanation of the equal sign for ABE and GED
> students, I would like to hear it.
>
> Chip Burkitt
>
> On 8/14/2010 1:04 AM, Michael Gyori wrote:
>> Greetings all,
>> After all this discussion about what the equal sign (or equality)
>> means, I find myself somewhat in a maze. A discussion of equality
>> takes us into a potentially esoteric realm from the perspective of
>> our students.
>> Might it be time to attempt to more clearly (and simply!) define
>> terms among those who teach math?
>> Michael
>>
>> Michael A. Gyori
>>
>> Maui International Language School
>>
>> www.mauilanguage.com <http://www.mauilanguage.com/>
>>
>
>
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