[Numeracy 495] Re: What does equality mean?
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Tue Aug 17 11:28:37 EDT 2010
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I think we can; I think he did (each side represents expressions of
equal value).
On a tangential note, though, one "big idea" I bring up early and often
is the idea that while it seems to the student that we keep changing the
rules in math (you can't subtract 4 from 1... oh, yes, you can... 78 is
bigger than 4 -- but not when that's your fraction denominator...), what
we're asking is for the student to gradually understand things at a
deeper, more complicated level -- and that humans do this all the time.
I usually talk about how little kids learn what a dog is, and call every
hair four legged thing a dog... then learn better.
I try to show how even when it seems like the rules are changing, the
principles haven't; that 4 + 3 is still 7, even if 4 + -3 is 1. One
rather huge challenge is when a student looks at numbers and the
emotions go into SURVIVE mode, wherein abstract thinking is unlikely.
Susan Jones
Academic Development Specialist
Center for Academic Success
Parkland College
Champaign, IL 61821
217-353-2056
sujones at parkland.edu
Webmastress,
http://www.resourceroom.net
http://www.bicycleuc.wordpress.com
>>> Michael Gyori <michael_gyori at yahoo.com> 8/17/2010 12:34 AM >>>
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
________________________________
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 = 3x − 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
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