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

[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.

 

[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.