Explaining Why (-1)(-1)=1 in College Algebra

[D H]

Many of you (arildno, MattGrimes, ...) fail to understand the problem most people have with math. Jing said it best here:

You must understand that that the way we see the world as mathematicians is not everybodies way of thinking.

Most of us who frequent this site have a very good mathematical sense. We cannot grasp how mathematics is an inscrutable concept to most people. Teachers can grasp this problem. Jing and homology are looking for simple and visualizable concepts, like the lining up of arrows. Forget the esoteric and abstract, as that is exactly what turns most people away from math.

You mathematicians have forgotten that mathematicians themselves struggled with abstract concepts like zero and negative numbers. Zero is a very abstract notion; negative numbers are even more so. Most people operate at a lower level mathematical sophistication than that at which ancient mathematicians operated.

The problem Jing and homology are confronting is the exact same problem that led to https://www.physicsforums.com/showthread.php?t=147358".

My suggestion is to tie math to things people know about: money and simple physics. The lining up of arrows fits this nicely and provides a way to visualize the extension of the number line to real numbers. Make the math concrete.

 

[arildno]

Not true. Please read my post carefully. It is not I who find this expanding of ideas utter nonsense I was referring to that fact that for some students changing how they have to view numbers makes no sense to them and hence is seem as utter nonsense by them and so are not able to move forward.

For that student group, it is of course crucial that:

a) They are assured and may confirm to themselves that every single truth they have learned of maths remains true even though we change our viewpoint a bit

but equally important:

b) statements that had no meaning in their previous view can in the new view be given a perfectly good meaning. Thus, their earlier viewpoint can be regarded as valid, but limited.

That is, they need to understand that we are expanding their concepts merely than just changing them for the change's sake.

 

[Hurkyl]

You mathematicians have forgotten that mathematicians themselves struggled with abstract concepts like zero and negative numbers.

Mankind used to struggle with abstract concepts like writing too. That doesn't stop us from expecting our kids to become literate, though.

 

[Doodle Bob]

Not a proof that (-1)x(-1)=1 but a discovery using patterns. And surely pattern discovery is fundamental in maths

Presuming that for n>0 it is accepted that (-1)xn=(-n) (if not a similar pattern to below can be used to discover it.)

(-1)x 12 = -12
(-1) x 11 = -11
(-1) x 10 =-10
(-1) x 9 = -9
(-1) x 8 = -8
(-1) x 7 = -7
(-1) x 6 = -6
(-1) x 5 = -5
(-1) x 4 = -4
(-1) x 3 = -3
(-1) x 2 = -2
(-1) x 1 = -1
(-1) x 0 = 0

Get the students to check out the pattern on the left and right hand sides in the sequence of numbers and so predict what the next one in the sequence will give

many of you will scream at this, but my experience has been that this is the best *heuristic* way to motivate why -1*-1=+1. This is after 16 years of teaching undergrads and 10 years teaching both future and current teachers. This has been the one way to get a group of hostile and confused students (face it, most nonmathematicians think we're just making this stuff up in order to be difficult) to suddenly realize why it's got to be that way.

of course, once i have them so hooked, i then show that it is implied by the axioms of the real numbers.

incidentally, this approach is straight from Polya: 1st chapter of Induction and Mathematical Thinking.

 

[D H]

Mankind used to struggle with abstract concepts like writing too. That doesn't stop us from expecting our kids to become literate, though.

Most of us in the civilized world can read. Unfortunately most people, even a lot of very smart people, are numerically illiterate. Wishing it were otherwise is just wishful thinking. We don't teach the illiterate how to read by foisting James Joyce upon them, do we? We start them on "See Spot Run!". So why should we foist abstract thinking on the numerically illiterate? Think of the concepts being discussed in this thread as a kind of "Mathematical Fun with Dick and Jane".

 

[Hurkyl]

Unfortunately most people, even a lot of very smart people, are numerically illiterate.

Which is a serious problem. (even moreso, due to the fact many people don't think it's a problem)

So why should we foist abstract thinking on the numerically illiterate?

Zero and negative numbers aren't any more abstract than any other number! People don't seem to have any trouble with the specialized language we have for dealing with zero and negative numbers. I honestly can't see why the corresponding numbers should be considered more difficult to comprehend.

 

[arildno]

Besides, it is far more important, and intellectually uplifting to be numerically literate than having read James Joyce's pretentious and worthless novels.

 

[robert Ihnot]

arildno: James Joyce's pretentious and worthless novels.

Murray Gull-Mann never thought that! In fact, here is Murray's own explanation of why it is called the "quark."

In 1963, when I assigned the name "quark" to the fundamental constituents of the nucleon, I had the sound first, without the spelling, which could have been "kwork." Then, in one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word "quark" in the phrase "Three quarks for Muster Mark." Since "quark" (meaning, for one thing, the cry of a gull) was clearly intended to rhyme with "Mark," as well as "bark" and other such words, I had to find an excuse to pronounce it as "kwork." But the book represents the dreams of a publican named Humphrey Chimpden Earwicker. Words in the text are typically drawn from several sources at once, like the "portmanteau words" in Through the Looking Glass. From time to time, phrases occur in the book that are partially determined by calls for drinks at the bar. I argued, therefore, that perhaps one of the multiple sources of the cry "Three quarks for Muster Mark" might be "Three quarts for Mister Mark," in which case the pronunciation "kwork" would not be totally unjustified. In any case, the number three fitted perfectly the way quarks occur in nature.
http://hypertextbook.com/physics/modern/qcd/

 

[arildno]

Irrelevant.
Have you ever tried to read Finnegan's Wake? Pretentious b****it.

 

[robert Ihnot]

For me to do that, I have to first of all join The Finnegans Wake Society in New York City to get their slant on it. http://www.finneganswake.org/aboutfwsociety.shtml

But, I am not presently in New York City.

 

[arildno]

Why?
Do you need a chef to tell you whether a dinner tastes bad or not?

 

[robert Ihnot]

I think we are too far off the subjet.