"Grey Area" Mathematics

[Cognitive Gear]

I recently discovered a document published by the Department of Public Instruction in Washington written by Jerry Johnson. The title of this document is "Teaching and Learning Mathematics: Using Research to Shift From the Yesterday Mind to the Tomorrow Mind. It's a long read, but the gist of it is as follows:

The US's falling position in mathematical proficiency is caused by "dysfunctional beliefs". The examples of "dysfunctional beliefs" given are:

“The goal of mathematical activity is to provide the correct answer to given problems, which always are well defined and have predetermined, exact solutions."

“The nature of mathematical knowledge is that everything (facts, concepts, and procedures) is either right or wrong with no allowance for a gray area.”

Now, while I do believe that students of advanced mathematics should be given partial credit for problems that were solved in the correct manner with a slight addition or subtraction error, this isn't what they are talking about. The author is speaking of elementary mathematics. In other words, 2x2 is no longer equal to 4.

I wonder how the author would feel about walking into a building engineered with "grey area" mathematics.

[Slater]

2 + 2 is .......................

Ten.



In Base-4, I'm FINE!

[Mr. SmartyPants]

2 + 2 is .......................

Ten.



In Base-4, I'm FINE!

I was totally going to say this! =(

[Azier the Swordsman]

I'm going to sue my old school for giving me a B on that math quiz I took in first grade.

5 x 5 = 32 was a grey area!

[Radical Dreamer]

This is why I prefer English and grammar to math.

"I before E except after C or when sounded like 'A' as in neighbor and weigh," even when that statement is proven untrue (weird, etc.), is seriously MUCH less complex for me to understand than any algebra, ever. XD

But seriously, what is this guy talking about? XD Relativity in math? "Well, if 4+4=12 is true for you, then who am I to judge?"

Humanity has reached an all-time low. XDD

[Warrior4Christ]

Statistics is a rather grey area... *is currently procrastinating from that*

I'm not quite sure what the author is trying to get at yet...

But seriously, what is this guy talking about? XD Relativity in math? "Well, if 4+4=12 is true for you, then who am I to judge?"

Humanity has reached an all-time low. XDD

'Zactly...

[Technomancer]

Now, while I do believe that students of advanced mathematics should be given partial credit for problems that were solved in the correct manner with a slight addition or subtraction error, this isn't what they are talking about. The author is speaking of elementary mathematics. In other words, 2x2 is no longer equal to 4.

Coud you point to where in the report this opinion is discussed? I've skimmed through it and found at least a few things that I do agree with (and others that require more thinking), but I have not encountered this sentiment. However, I should point out that most of the focus of the report seems to be upper level elementary years (when basic algebra is introduced) and high school.

As an aside I recently came across this article:
http://scienceblogs.com/cortex/2008/03/algebra_education_and_john_dew.php

[Kaligraphic]

Interestingly, 2 plus 2 can equal 5. If you are showing one significant digit, you don't know what's beyond the decimal. Traditional rounding rules on a .50 would round to the nearest even number, so 2.50 would round to 2. 2.5 + 2.5 = 5.0, of course, so 2 + 2 = exactly 5. Even using the nontraditional ".50 always rounds up" rule that many people now use, 2.4 + 2.4 = 4.8, which would round up to 5.

Mathematics is an exact science only to the degree that its precision is known, and no further. Precision beyond what is truly known (e.g. assuming 2 means 2.0000000...) will get you into trouble in the real world. Concepts, even in mathematics, may still approximate, rather than define reality. If you look at the statements in the context of the broader world of mathematics, they're quite sensible. Sometimes procedures must be modified. Sometimes new procedures must be developed. Occasionally, one may find that one is mistaken about a particular concept, and must therefore adopt a slightly different understanding of that concept.

The statements quoted, read in context, appear to refer to the robotic assumption that mathematics consists solely of rote operations which may be carried out without need to achieve a deeper understanding of how things work. Read as such, It is quite accurate to say that these beliefs are incorrect.

[Cognitive Gear]

Coud you point to where in the report this opinion is discussed? I've skimmed through it and found at least a few things that I do agree with (and others that require more thinking), but I have not encountered this sentiment. However, I should point out that most of the focus of the report seems to be upper level elementary years (when basic algebra is introduced) and high school.

Note at the beginning, in the "About this Document" area, it notes that it is to apply to Mathematics for K-12. That would indicate to me that it would apply all across the board.

This document was published in 1989. We can see the results of this document by looking at textbooks currently used in Washington State schools.

Here is a link to a video which demonstrates how they are currently teaching mathematics.

http://www.youtube.com/watch?v=Tr1qee-bTZI

It's a long video, but towards the end there's a bit that I found more issue with than the way they are teaching math. (It's not wrong, and it may help students grasp a few advanced concepts later, but it also is much harder for a new student to follow). What I found to be more problematic is that a large portion of the mathematics book is used as a "World Tour" atlas. Apparently mathematics isn't worth class time anymore. :/

[chibiphonebooth]

[ChristianKitsune]

you know, believe it or not....but I've often wondered what defined Mathematics in the first place?

you guys are gonna give me strange stares and wierd looks XD

but for a while now I've just been kinda like this:

"if 2+2= 4 who said that value even exists?"

Are we just taught to do math by people who have been taught themselves? Why are these old things still true today?

XD that sort of thing....

yeah.. XD I'm wierd.

GLAD I PASSED COLLEGE ALGEBRA AND I R DUN WITH MATH CLASSES FOREVERRRR

[blkmage]

So, a Dinosaur Comic came to mind when I saw this thread.

Math is built on apriori assumptions: you start with a few things that you assume are true and build stuff on them! Stuff like "1+2 is the same as 2+1". Then you build up a mathematical system around that, but that doesn't mean math is true. It's just consistent!

...

When our mathematical assumptions APPEAR to match up with reality, the results may also seem to apply - but math isn't any more true because of that! It's still based on unproven/unprovable assumptions. And you can actually build up systems where you decide that 1+2 does not equal 2+1, and they're just as consistent!

So does 2+2=4? Under, say, Peano arithmetic, it's possible to mechanically prove that it works, and so the system is consistent. But, we had to work under the assumption that the Peano axioms held.

And I guess, kind of related is the fact that there's a theorem (Godel's first incompleteness theorem) that basically says that there is essentially no system that can be both consistent and complete (completely true).

[Cognitive Gear]

[chibiphonebooth]

NO.

NO IT IS NOT. IT NEVER IS. *meltss*

[Nate]

So, a Dinosaur Comic came to mind when I saw this thread.

patrick stewart is checking his watch and sighing dramatically

In response to Savannah,

[Angel Tifa]

patrick stewart is checking his watch and sighing dramatically

In response to Savannah,

Lol I like that picture XD. I always just prefered English and grammar to anything math or anything that has to do with math.

I'm just a right-brained oriented person and I know that has a lot to do with how good one can be at math. Math is a left-brained subject with is something I'm pretty terrible at XD.

[termyt]

And don't even get me started on math involving 0. It took thousands of years to come up with 0 just to make the math you learn in elementary school work.

Where would we be without the zero concept?

Is zero even a number?

A number implies a quantity, but how many is zero? I can tell you all of the tings I have one of, but I can’t even begin to think of all the things I have zero of.

English is not much better. Look at this thread. We are in a grey area? What's a grey? Perhaps the Technomancer can explain?

[mathgrant]

WINWINWIN. XD

. . . I don't understand what the original post is about. D= But 8x3=4! That much, I know. =D

[mitsuki lover]

I flunked the entire semester we had the New Math in Junior High.I think that must have been a rather black instead of grey area.Seriously though
I think the point is that we think of math as being concrete too much of the time when most of the time,especially when referring to higher mathematics it's really abstract as the article that Technomancer linked puts it.
I also think Dewey had a good point in trying to show how math and science could be taught practically.

[Warrior4Christ]

WINWINWIN. XD

. . . I don't understand what the original post is about. D= But 8x3=4! That much, I know. =D

In mod 5 you'd be okay.
[/nerd]

[Angel Tifa]

WINWINWIN. XD

. . . I don't understand what the original post is about. D= But 8x3=4! That much, I know. =D

From what I interpreted, the original post of the thread was to discuss this weird article saying "there's proof that 2 + 2 does not equal 4!" I was just like "um okay right :shake:" @___@.

[Technomancer]

And don't even get me started on math involving 0. It took thousands of years to come up with 0 just to make the math you learn in elementary school work.

Where would we be without the zero concept?

Is zero even a number?

A number implies a quantity, but how many is zero? I can tell you all of the tings I have one of, but I can’t even begin to think of all the things I have zero of.

But you still have zero of them. Zero is indeed a number, and in this case it has a clear relationship to our quite concrete notions of quantity. As you said you can have zero items of something. However, numbers need not depend on such physical things; they exist quite independent of them in a much more abstract realm. All of our concrete realizations of them are derivable from much more idealized and abstract principles.

This is not a new argument by the way. The Pythagoreans had considerable trouble accepting the existence of irrational numbers. When they were first discovered, people also had trouble accepting the reality of negative numbers, and of course "imaginary" numbers. There is by the way, a very good discussion of these ideas in Ch. 1 and 3 of Roger Penrose's recent book "The Road to Reality".