[Numeracy 499] Re: Equality, unary signs, and operations...

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Chip Burkitt chip.burkitt at orderingchaos.com
Wed Aug 18 01:06:27 EDT 2010


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 <http://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/ =

> 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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