[Numeracy 500] Re: Equality, unary signs, and operations...
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Wed Aug 18 02:32:49 EDT 2010
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Chip,
My only reservation is that a unit of measurement of temperature is entirely
abstract, while being in the red is not. When we *owe* money, our balance is
negative, even if the money we owe is positive...
Michael
Michael A. Gyori
Maui International Language School
www.mauilanguage.com
________________________________
From: Chip Burkitt <chip.burkitt at orderingchaos.com>
To: Michael Gyori <michael_gyori at yahoo.com>
Cc: The Math and Numeracy Discussion List <numeracy at nifl.gov>; Ladnor Geissinger
<ladnor at email.unc.edu>
Sent: Tue, August 17, 2010 7:06:27 PM
Subject: Re: Equality, unary signs, and operations...
Michael,
I agree. A balance scale makes an excellent metaphor for equality.
I remember a bit about the unary minus sign and your contention that it has no
physical representation. Of course, the same could be said for the concept of
number. Counting is actually a fairly sophisticated exercise. It requires
knowledge of a number system, understanding of one-to-one correspondence, and
confidence that the quantity does not change if items are counted in a different
order. Even tallying, which does not require a number system, still requires
fairly powerful abstract reasoning. For example, if I say, "I have fourteen
chickens," the number fourteen is an abstraction. It cannot be derived from the
chickens; it has to be brought to the chickens. So fourteen (or any number, for
that matter) also has no physical representation. It could be argued that
fourteen is what all groups of fourteen items have in common, but that reasoning
is circular and neglects other properties that groups of physical items have in
common (e. g., occupies space, color, mass, duration).
Nevertheless, numbers certainly have more common applications in the lives of
students than unary minus signs do. When I introduce negative numbers, I give
examples likely to be common to everyone: below zero temperatures (especially
here in Minnesota), overdrawn bank account, and below sea level elevations. It
is especially easy to understand operations on temperatures (e. g., Most
students know that warming up 10° from −17° is −7°).
Chip Burkitt
On 8/17/2010 5:42 PM, Michael Gyori wrote:
Hello Chip,
>
>I wonder whether Ladnor might be willing to weigh in further at this poin with
>respect to the nature of equality. I don't know what a "real" mathemetician
>is, but if there is such a person, I would consider him to be one... :)
>
>As for equality, the reason I like the metaphor of a scale in discussing
>equality is that it connects numbers with physical physical phenomenon and
>shared background knowledge (by and large). Scales don't compare what's on
>either side, they just tell you that weights are the same when they're balanced.
>
>On another note, we had some very interesting discussions about unary signs a
>while ago. I contended and still contend that they had no physical
>existence. I'd be really interested in further discussion about unary signs
>vs. operations.
>
>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: Tue, August 17, 2010 7:28:47 AM
>Subject: [Numeracy 496] Re: What does equality mean?
>
>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 = 3x − 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
>>
>>
>>
>>
>>
>>
>>
________________________________
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
>>
>> ---------------------------------------------------- National Institute for
>>Literacy Math & Numeracy discussion list Numeracy at nifl.gov To unsubscribe or
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>>chip.burkitt at orderingchaos.com
>>
>
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