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

[matt grime]

As a teacher I would be providing a demonstration that would lead them to the required result rather than let them founder in a range of possible solutions and then provide them will the acceptable one.

But the problem with all pattern recognition type arguments is that they are silly. One man's pattern is another man's chaos. For many years fatally flawed IQ tests have used the same arguments. If you wish to lead your students to the answer as some natural progression that does what it should then teach it to them properly. The proof is elementary and has been given in this thread, I am sure. If you want to aid their process of discovery then point them in the right direction for the right reasons, not for this fallacious argument.

 

[jing]

Of course seeing a pattern is not the same as a proof but seeing a pattern may be what leads to a hypothesis which in mathematics may be one that is interesting to see if there is a proof for.

As students develop in mathematics they need some understanding of the pattern before they consider the need for a proof. Thus at AS level (for our non UK contributors AS exams are taken at 17 as part of a pre-university course) for instance calculus is not proved rigourously but may be demonstrated sufficiently for students to grasp the technique. Those students going on to university may then follow a rigourous path to the concepts of differentiation and integration. Different levels of understanding are required at different stages.

The proofs earlier in the thread are indeed elementary to me and you but for my students giving them a such a proof is way over their heads. They are students who having not obtained a C at GSCE (for our Non UK contributors -an GCSE is an examination taken at 16 where a grade C in Maths GCSE is required to enter university) and are retaking it at college. They are at the stage of needing 'hooks' that will help them add, subtract, multiply and divide integers correctly.

 

[arildno]

There is a pattern - right up to 0, when the pattern stops... ignoring that I see that when I multiply by -1 I put a minus sign in front of the digit. So (-1)*(-1) must therefore be --1. And (-1)*(--1)=---1, etc...

The pattern I see is:
(-1)*(-1)=-(-1) whatever that is, in the minds of the young.

I agree with matt grime that arbitrary pattern arguments are not that useful.
However, I would like to say that when we analyze and break up some expression, we often do so because the partial expressions each is recognizable as something we know of beforehand, and are therefore transformable according to some FUNDAMENTAL pattern we know of.

Thus, to take a trivial example, a student ought to recognize simple formulae so that they might make any of the substitutions a/(-b)=(-a)/b=-(a/
b) for whatever expressions a and b might stand for.


That they ALSO ought to understand WHY these formulae hold, i.e understand the proof of them is not only desirable, but IMO, mandatory.

It is PRECISELY because the proofs of such simple identities are SIMPLE PROOFS that they are ideal as an introduction to learn how to prove something in maths.

 

[Werg22]

-1*-1 doesn't mean anything in real life. It's just a "way" of doing algebra which gives the "right" awnsers. Why is -1*-1 defined as = 1 can be easily explained. If you have the sum of a sequence of numbers... (a+b+c...) and it's opposite: -(a+b+c...) and you add them (a+b+c...) - (a+b+c...) = 0. It only follows that,

-(a+b+c... )= -a-b-c... Now if |a| > a, then, since a - a = 0, -a = |a|. This way, it cancels out.

 

[jing]

Thus, to take a trivial example, a student ought to recognize simple formulae so that they might make any of the substitutions a/(-b)=(-a)/b=-(a/b) for whatever expressions a and b might stand for.

However my point is that for some students this is not a trivial example it is incomprehensible. These are students who remember something about two negatives making a positive and so conclude -3 - 4 = 7 as there are two negatives. This is not because they have not been taught correctly or are lazy but because a sequence of symbols of this sort has no meaning for them however often it is repeated with them. These are clever students in other academic areas but maths remains a mystery.

 

[arildno]

However my point is that for some students this is not a trivial example it is incomprehensible. These are students who remember something about two negatives making a positive and so conclude -3 - 4 = 7 as there are two negatives. This is not because they have not been taught correctly or are lazy but because a sequence of symbols of this sort has no meaning for them however often it is repeated with them. These are clever students in other academic areas but maths remains a mystery.

In which case they don't understand fundamental concepts like "what is a term", "what is a factor" "what property has the negative of a number" and so on. They have, indeed, been taught INCORRECTLY.

It is equally silly to formulate the sentence "-3-4=7" as writing the name "James" as "John".
One of the primary deficits with the teaching of elementary maths, is that the teaching of the math LANGUAGE and what the symbols stand for is thoroughly neglected, in favour of so-called "applied maths".

The pupil must be taught to focus on what is actually written on the paper, rather than encouraged to make a hasty calculation that might, or might not, yield the correct "answer". (Whatever is meant by the silly word "answer", that is often very unclearly stated what should be)


While you are, indeed, right in saying that what appears as a mystery will soon be forgotten again, it does not at all follow that a presentation of the matter in such a manner that it no longer will seem mysterious to them is impossible.
Since we know that what is perceived as LOGICAL is what appears as LEAST mysterious, it follows that we should teach maths in a LOGICAL MANNER, i.e with proofs, rather than "intuitively" or with pictorial thinking.

 

[arildno]

-1*-1 doesn't mean anything in real life.

Eeh, if I always move with a speed of 1 m/s to the left, and ask where was I 1 second before someone started a clock when I whizzed past him, isn't it meaningful to say that the answer to my question is that 1 second before I made contact with the clock-holder, I was in a position 1 m to the right of him??


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.

 

[matt grime]

I'm absolutely behind arildno here. If they do indeed from hazy memory decide that -3-4=7, then they have indeed not been taught correctly. There are simple rules. Learn them as you would learn the rules of any other language, or subject. Maths suffers in its teaching precisely because we refuse to teach mathematics like we would teach any other subject.

Try reading the VSI to Mathematics by Tim Gowers, or Polya's writings, or even Gowers's writings in the stlye of Polya, for some clear expositions of these points that anyone can understand.

 

[Werg22]

Eeh, if I always move with a speed of 1 m/s to the left, and ask where was I 1 second before someone started a clock when I whizzed past him, isn't it meaningful to say that the answer to my question is that 1 second before I made contact with the clock-holder, I was in a position 1 m to the right of him??


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.

It is somehow "meaningful" to say that -1*-1 = 1, but from a purely algebraic point of view, I think it's more of a definition. In fact multiplication by -1 simply means "the inverse". The inverse of a negative is obviously a positive. Like I said, with [a+b+c...], an inverse result is [-a-b-c...], and their sum is 0. So when you apply those "rules" in physics, you are truly dealing with the "inverse" and the mathematical definition allows you to access the needed results. And don't blame schools, I am self-taught.

 

[jing]

we should teach maths in a LOGICAL MANNER, i.e with proofs, rather than "intuitively" or with pictorial thinking.

Presuming you mean we should do this because doing so would improve students understanding of maths what evidence do you have that this statement is true?

 

[matt grime]

Presuming you mean we should do this because doing so would improve students understanding of maths what evidence do you have that this statement is true?

How about - the current method of teaching doesn't emphasize formal logic and reasoning, just hand wavy nonsense, and the output of this is a generation of mathematically under-educated people? Just ask anyone who teaches maths at university, or people in industry. Today's high-school graduates in the US and the UK are mathematically far behind where they should be. So let's try teaching maths as if it were maths. Your method doesn't work, so give ours a try, and then you are in a position to evaulate the relative merits. Get a student and guide them with hints that are mathematically sound, rather than just guessing from patterns *without thinking*.

 

[Werg22]

The problem is that mathematical maturity is obtained very much like what we call "wisdom", meaning, a certain part of it is very similar to philosophy. There is no "rules" in mathematics, there is simply the true and the false. Most students try to approach a problem with a set of rules, as if they were playing a game of chess and their strategy is dictated by the allowed movement for each piece. In mathematics, rules are only there to remind of the truth and should not be at the core of any kind of real understanding.

 

[jing]

There are simple rules. Learn them as you would learn the rules of any other language, or subject.

Now that presents a problem. I am extremely capable of learning the rules, grammar and syntax of mathematics. When it comes to learning French no matter how well the rules, syntax and grammar were taught and how much I understood them at the time of explanation for some reason I was not able to incorporate them into my knowledge and learn French.

So it is with some students. As one maths student said to me "I can understand the rules as you state them and I can follow the rules when the questions just relate to the rules you have just stated but tomorrow everything will get mixed up and I will not know which rules to apply to what"

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

 

[Werg22]

If you were to verify the orthography of each of your words when you write an essay, it would take a decade. I think learning how to write well is a matter of reading allot and, to certain extent, intuition.

 

[jing]

Your method doesn't work, so give ours a try, and then you are in a position to evaulate the relative merits.

As a maths teacher of thirty years I have tried and evaluated a variety of methods to try to help my students to come to an understanding of maths. For very many students the methods you are suggesting just do not work.

You are asking me to take you hypothesis and test it for you. I am asking you what evidence you have for making the statement you did or is it something you just believe with no evidence for its truth?

 

[Werg22]

I feel out of place, this is between teachers, not a simple high school student like me! I will now retreat.

 

[matt grime]

Have you tried teaching maths as a university lecturer (not in the American sense of Calc 101) understands the term, not a high school teacher, from year 1, from the very beginning? Did you check the information I gave about Gowers (Fields Medal)? Or Polya? I teach/taught 7 years of incoming undergraduates, and they are not equipped to start learning proper mathematics. They become befuddled at the notion of a group. So start teaching them maths properly earlier, at the earliest possible stage. And if they struggle there is nothing wrong with streaming them and letting those who can't cope just do arithmetic. It is a bottoms up revolution I'm suggesting here. Some one should evaluate it as a serious idea, not dismiss it because it has not been evaluated. 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. Forget teaching them boring stuff, give them logic problems, let them try to figure out how to solve the old 'prisoner in the sand' problem, or something that bears some resemblance to logic and mathematics.

 

[homology]

Thanks all for the renewed interest in this topic. I agree with all of you to some extent. Or perhaps I understand all of you. I don't think students should have to swallow content. They should at least find it relevant. Its our obligation as teachers to make the material either relevant or interesting or perhaps even belivable.

Taking the "suck up and learn it" attitude doesn't do much for the learning of mathematics. I'm not easy on my students. I expect a lot out of them, but one of the things I don't expect is that they find any of this stuff interesting or relevant because it isn't to them. So I'm just trying to find ways to change this perspective.

I really like arildno's post and will work with that. Jing I downloaded the webpages but couldn't get them to work on my computer. Any ideas?

Thanks again,

Kevin

 

[d_leet]

It is somehow "meaningful" to say that -1*-1 = 1, but from a purely algebraic point of view, I think it's more of a definition.

Even from a purely algebraic point of view it isn't really a definition because it follows from the axioms of a commutative ring that (-a)(-b)=ab. So with the set of real numbers and te operations of addition and subtraction defined as usual we have a commutative ring because the real numbers under these operations satisfy the axioms of a commutative ring, and it can be proved that as a consequence of these axioms we must have (-a)(-b)=ab ((-1)*(-1)=1 is a special case of this). If I am incorrect in any of this please correct me as I haven't had that much experience with modern/abstract algebra .

 

[matt grime]

Taking the "suck up and learn it" attitude doesn't do much for the learning of mathematics.

I don't think anyone said that was what should be done. Mathematics should be taught as an interesting and intellectually stimulating subject within its own right. That is after all what it is. Doing integrals, say, is dull, so let's give them something more interesting to do. Formulating completely unrealistic word problems isn't exactly fascinating. And they should ban calculators too, but that's another story.

If a student asks 'why is (-1)*(-1) equal to plus one, why invent silly examples like 'well, if you're owed one dollar, and some one converts all your debts to profits, you have one dollar'. No, ask them what they think they mean by -1. 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.

Teach them about concrete groups, matrix groups - there's plenty of scope for real life application there. Ask them to work out an algorithm to sort an unordered list, then ask them if they can think of a better one, once they've figured out what 'better' means'.

 

[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.