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

[jing]

My only supporting evidence that this idea should be considered is that almost every university lecturer I know thinks that the current state of affairs is appalling, and here's one way a lot of us would like to see tried.

There are not many teachers I know that do not bemoan the teaching their students have had up to the time they receive them.

Interestingly when I was a postgraduate 30 years ago university lectures were saying similar things about the quality of undergraduates and in those days students 14-16 were taught formal Euclidean Geometry with proofs and theorems. (Here we will probably agree, I think the logic of Euclidean Geometry was a tremenedous help in understanding maths at university)

 

[jing]

If they're taught properly, they know that -1 is the unique number that adds to 1 to give zero, and the rest follows. They can then have a simple derivation that they should be able to reproduce any time they start to wonder again. Pique their interest in the abstract.

In George Bernard Shaw's play Man and Superman a character,John Tanner is the author of "The Revolutionist's Handbook and Pocket Companion" in this book is said (or as close as I remember)

Do not do unto others as you would be done by. They may have different tastes.

To paraphrase

Do no teach from the point of view of what interests you the students may not have the same interests.

Often you need to find their understanding first and use that to bring them to a point where they do find the abstract interesting.

 

[Garth]

You could ask them to analyse this and find out where it goes wrong:

1 = 1
1 = +\sqrt{+1}
1 = +\sqrt{(-1)*(-1)}
1 = (+\sqrt{-1})*(+\sqrt{-1})
1 = (+\sqrt{-1})^2
1 = -1 Hint: see posts #8 & #19

Garth

 

[matt grime]

Often you need to find their understanding first and use that to bring them to a point where they do find the abstract interesting.

And I haven't said that you should not do this. However, I would not use 'word problems' that have supposed bearing on real life to do this.

As it is I would rather teach maths to someone who can do cryptic crosswords than someone who has got an A* at GCSE. Numeracy, which is almost all that high school can be said to impart these days, is not the same as mathematical skill. As it is I lament my own education for being too 'lowest common denominator', so it is not mere nostalgia for a not so long ago era.

So, shall we ask for a 'mathematics for the mathematician' option at GCSE?

 

[kesh]

Applications of maths in "real-life" can just about always be found, and they happen to be irrelevant and obscuring to the actual understanding of maths.

not necessarily, and I'm from a very pure maths background.

one stumbling block in the teaching of new ideas is just simple acceptance. there is a kind of stubborness, a mental block, against abstraction that melts away when "real world" examples, however clumsy and inaccurate, are given. and the example isn't latched onto to prevent further understanding, it's just a way in.

 

[arildno]

not necessarily, and I'm from a very pure maths background.

one stumbling block in the teaching of new ideas is just simple acceptance. there is a kind of stubborness, a mental block, against abstraction that melts away when "real world" examples, however clumsy and inaccurate, are given. and the example isn't latched onto to prevent further understanding, it's just a way in.

Well, I have nothing against VISUALIZATION, whenever that does not warp the issues at hand. The best example of a good visualization is to portray the set of real numbers as a number line. Far too little use is made of this visual tool in elementary maths.

Also, even earlier, we should do our best to keep maths on a level where the TACTILE perception of the child will aid them to understand the LOGICAL issues in maths.

In no case, however, should these approaches neglect the really fundamental issue: Namely, that maths is LOGIC, and LOGICAL, anf that a study of maths is first and foremost one way to develop our logical faculty.

 

[kesh]

Well, I have nothing against VISUALIZATION, whenever that does not warp the issues at hand. The best example of a good visualization is to portray the set of real numbers as a number line. Far too little use is made of this visual tool in elementary maths.

Also, even earlier, we should do our best to keep maths on a level where the TACTILE perception of the child will aid them to understand the LOGICAL issues in maths.

In no case, however, should these approaches neglect the really fundamental issue: Namely, that maths is LOGIC, and LOGICAL, anf that a study of maths is first and foremost one way to develop our logical faculty.

i wasn't particularly talking about visualisation, just mundane acceptance of a concepts utility or even its tenuous relationship with the student's notion of "reality".

but seeing as you mention it there is a strong relationship between visual-spatial and mathematical ability. the part of the brain that makes a "map" from our visual (and tactile) perception of the world is the same part of the brain that makes a map out of mathematical ideas. i would hate to stifle a child's intuitive and pictoral exploration of mathematics and the strengthening of this mental muscle by pressing dry logic upon them too soon or too much

 

[jing]

The best example of a good visualization is to portray the set of real numbers as a number line. Far too little use is made of this visual tool in elementary maths.

Yes the use of a number line is useful but can also confuse students unless thought through when you teach it.

Consider 3.4 + 2.2

Do 3.4 and 2.2 represent distances? What do 3.4 and 2.2 represent on the number line?

Do 3.4 and 2.2 represent a position on the line? If so how do you add positions?

Does 3.4 represent a position and +2.2 represent a translation?

Do 3.4 and 2.2 both represent translations? If so what do 3.4 and 2.2 represent on the number line itself?

We may understand that there is a LOGIC behind any of these representations but for many students these ideas are ILLOGICAL and trying to insist on the LOGIC we understand is not always helpful.

 

[arildno]

Well, the simplest way of representing this, is to regard "summation" between numbers as putting two numbers, each represented as an arrow, after each other.

The "answer" is then the arrow of equal length as the combined lengths of the two vectors, but with only one arrowhead (at the end).
This is not difficult to grasp; you should start adding such arrows together in a physical and tactile manner.

 

[jing]

Ok but now we have moved away from the real number line to a vector representation.

Having defined 'summation' do we now define 'difference' or 'inverse'.

Do you say 1-1 represents having a '1 arrow' and then removing it or

do we change definition to say is is not the length of the line that represents the number but the translation along the line so we have now suddenly changes from representing a number by an object to representing it by an action. If we are going to do this then maybe we should have started with defining the number as an action in the first place. However perhaps that's too much to expect some students to grasp in the first place but if we do it by using the simple way with just the length of the line then the students might develop fixed views that make it difficult for them to change their perception.

 

[arildno]

Ok but now we have moved away from the real number line to a vector representation.

Not really. Or rather, we are thinking of numbers as oriented line segments of unequal length. That should be easy to understand.

Having defined 'summation' do we now define 'difference' or 'inverse'.

Do you say 1-1 represents having a '1 arrow' and then removing it or

do we change definition to say is is not the length of the line that represents the number but the translation along the line so we have now suddenly changes from representing a number by an object to representing it by an action.

Eeh??
In this visualization, the best way to visualize the "difference" operation is to flip the arrow representing the minuend 180 degrees, and then put it again foot-by-tail onto the first number. The "answer" is then again, the length of the oriented line segment going from the origin to the tip of the last arrow.

Subtraction is literally: Addition, with a TWIST.

 

[Werg22]

As a high school student, there's one thing that I recommend about vectors. Do not teach them addition/multiplication with "triangle" rules. Simply say a vector is a entity that has both a vertical component and horizontal component. Remind the students that always have to separate each vector into a horizontal and a vertical component.

 

[Hurkyl]

As a high school student, there's one thing that I recommend about vectors. Do not teach them addition/multiplication with "triangle" rules. Simply say a vector is a entity that has both a vertical component and horizontal component. Remind the students that always have to separate each vector into a horizontal and a vertical component.

Actually, that's one of the things we usually want people to unlearn. A vector is an entity unto itself, and many things become much clearer when you treat it that way.

 

[Hurkyl]

If we are going to do this then maybe we should have started with defining the number as an action in the first place. However perhaps that's too much to expect some students to grasp in the first place but if we do it by using the simple way with just the length of the line then the students might develop fixed views that make it difficult for them to change their perception.

How will they learn there are many ways to picture something unless they are taught that there are many ways to picture something?

 

[jing]

In this visualization, the best way to visualize the "difference" operation is to flip the arrow representing the minuend 180 degrees, and then put it again foot-by-tail onto the first number. The "answer" is then again, the length of the oriented line segment going from the origin to the tip of the last arrow.

Ah so your original statement below about summation needs refining to be consistent by adding a condition about placing the tail of the first arrow at the origin and length of the answer being from the origin to the tip of the last arrow.

Well, the simplest way of representing this, is to regard "summation" between numbers as putting two numbers, each represented as an arrow, after each other.

The "answer" is then the arrow of equal length as the combined lengths of the two vectors, but with only one arrowhead (at the end).
This is not difficult to grasp; you should start adding such arrows together in a physical and tactile manner.

 

[jing]

In this visualization, the best way to visualize the "difference" operation is to flip the arrow representing the minuend 180 degrees, and then put it again foot-by-tail onto the first number.
Subtraction is literally: Addition, with a TWIST.

OK but where students understanding is firmly based on the idea of difference being taking away a number of objects from another number of objects the concept that difference now means something utterly different can be just seen as some sort of strange nonsense

 

[jing]

How will they learn there are many ways to picture something unless they are taught that there are many ways to picture something?

True and if you look at my earlier posts you will see I am talking about students who have difficulty picturing aspects of maths in one way let alone many

 

[arildno]

OK but where students understanding is firmly based on the idea of difference being taking away a number of objects from another number of objects the concept that difference now means something utterly different can be just seen as some sort of strange nonsense

Eeh? Honestly, I have no idea what you are up to in this thread.
First, you vigorously oppose any sort of logical teaching of maths to pupils, and then, when you are given a few ways as to how we might visualize maths, and even how to handle maths in a tactile manner, you criticize that different visualizations highlight slightly nuanced properties of arithmetic and call this expanding of ideas as utter nonsense.

It seems to me that what you are after is a single, hand-wavy manner in which to "teach" something that no longer bear any resemblance to either maths and logic. I can wish you a good hunt, even though you won't find what you seek, and nothing you find that seems to fulfill your requirements will be desirable to be taught.

For the record, I would like to say it is precisely the ossification tendency, i.e, to regard math symbols to have one and only one application in "real life" that should be combated by math teachers.

The "true" meaning of math symbols are given as parts of a particular system of LOGIC, whereas their applicability is as wide and varied as the world itself is.
Thus, it is not, as you seem to think, unpedagogical to teach pupils how to THINK, and to think LOGICALLY, along with gradually expanding their concepts of how we might interpret maths in a variety of settings and visualizations.

Those of us who have reached this point of view reached these ideas in our adolescence without any help of our teachers at all; if the teachers had made these points explicit to the other pupils, it is probable that they would have reached the same level of competence in maths and physics as us so-called "math geniuses".

 

[jing]

you criticize and ... call this expanding of ideas as utter nonsense.

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.

 

[jing]

Eeh? Honestly, I have no idea what you are up to in this thread.
First, you vigorously oppose any sort of logical teaching of maths to pupils,

It seems to me that what you are after is a single, hand-wavy manner in which to "teach"

Again not true. I do, however, oppose the idea that the logical maths teaching as discussed in this thread is the sole way that will improve the understanding of maths for all students.

I repeatedly make it clear that I am talking about a particular subset of students.

Perhaps I also need to make it clear that we have about 150 students from this subset who arrive at our college with a failing grade in mathematics at the age of 16 and after one year over 70% of these have a pass grade. We achieve this with a variety of teaching methods.

 

[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.