[Winstonm]

I was much younger then, and still quite trusting. There was no way to know then the pain I would later have to endure because of that misguided trust.

They were good. Really good. Well-trained. Disciplined. They didn't tip their hand at first. It started slowly, with lessons like 1+1. We moved through a progression of possibilites, and all made perfect sense. A little later, we were introduced to the fascination of multiplication and division, and were shown how these activities were nothing more than shortcuts for addition and subtraction, that 2x2 was really like saying 2+2, and that 2x3 was really the same as 2+2+2.

I was enthralled, enraptured, entertained, and ensconced by the beauty of it all. And all the while they smiled, put stars on our papers, and smiled.

But then, in twinkling of an eye, in the page turn of a calender, in the move from one grade to the next they changed, as did my view on life and trust.

They introduced negative numbers.

Still, they were good at masking intentions, of hiding their motives until we had almost totally bought in - some even did, and I shudder to think what became of them in later life. Mathematicians, perhaps, though no one really thought anyone could spend a life doing math and live. New rules were put in place, seemingly as understandable as before: 1+1 was still 2. 1+-1=0. -1+-1=-2. O.K. That seemed to make sense.

Until they sprung their trap: multiplication!

All of a sudden the world turned upside down. We had been taught that multiplication was the same as addition, but now it wasn't?

It was on the test. (-2)x(-3). Simple, that was the same as saying (-2)+(-2)+(-2). I proudly wrote down my answer: -6.

Later, SHE showed up - standing beside the desk, handing back tests.

C+. And a big red checkmark beside question #8: (-2)x(-3). And a handwritten notation: a negative times a negative equals a positive.

That's when I knew. My eyes opened. My heart sank. It was a conspiracy. No way would we ever be able to figure out mathematics by common sense - we would have to learn rules! Rules! That meant study.

But how can one study when The Love Boat comes on at 7:00 and Karen Valentine is the special guest star?

It was most definately a conspiracy.

I feel lucky, now, having been awakened at such a young age to the possibilities of nefarious motives. That knowledge has helped me spot other conspiracies, like Roswell, the moon landing, 9-11, and the Kennedys' assassinations, just to name a few.

Now, if they will only let me out of this cell for a bit, I can show them all where they are wrong. Wrong! Wrong, I tell you! It's minus 6. See? Minus 6!

No! Stay away! Help! (rattle, rattle) Let me out of here! (rattle, rattle) H-E-L-P!

Math. It's good food for the brain.

[kenberg]

If you delete (aka take away) 2 debts of 3 dollars each you would be 6 dollars richer, would you not? So (-2)X(-3)=+6.If you take on 2 more debts of 3 dollars each you will be 6 dollars poorer. So (2)X(-3)=-6.

Or: Since 2X3=6 it would not make sense for (-2)X3 to also be 6, that answer is already spoken for. And 8 seems like an unlikely choice. So -6 it is. Now repeat: Since (-2)X3=-6 it would not make sense for (-2)X(-3) to also be -6. Maybe it's pi. But probably it is 6.

Also: When I was a child my mother explained: If you say that you don't want none of that then apparently what you do want is some of that. Same thing.

If all else fails: It's in the book. Learn it and shut up.

[babalu1997]

kenberg, on 2011-February-06, 13:14, said:

If you delete (aka take away) 2 debts of 3 dollars each you would be 6 dollars richer, would you not? So (-2)X(-3)=+6.If you take on 2 more debts of 3 dollars each you will be 6 dollars poorer. So (2)X(-3)=-6.

Or: Since 2X3=6 it would not make sense for (-2)X3 to also be 6, that answer is already spoken for. And 8 seems like an unlikely choice. So -6 it is. Now repeat: Since (-2)X3=-6 it would not make sense for (-2)X(-3) to also be -6. Maybe it's pi. But probably it is 6.

Also: When I was a child my mother explained: If you say that you don't want none of that then apparently what you do want is some of that. Same thing.

If all else fails: It's in the book. Learn it and shut up.

no one can succeed in life till they understand euler`s equation.

[Winstonm]

Quote

If you delete (aka take away) 2 debts of 3 dollars each you would be 6 dollars richer, would you not? So (-2)X(-3)=+6

I don't know. Just sounds like a perspective variance to me.

If I started in a hole 6 feet deep, and two guys came by and each said, oh, by the way, you don't have to pay me back that shovel full of dirt you owe, well, I'm still not closer to the top, am I?

[Gerben42]

What do you get when you multiply six by nine?

42 (base 13)

[jschafer]

I have to agree with the sentiment that the multiplication of two negative numbers is much harder to conceptualise than two positive ones or one of each. For a positive and a negative one you can always use the +ve number as the number of times you multiply: ie. -3x2 = -3-3 and 2x-3 = -3-3 (something twice or twice something has to be the same). The debt debt analogy seems to be more along the lines of -(-2 x 3) rather than -2 x -3 (which although the same, still means you need to learn the rules on brackets and such).

[manudude03]

When I first saw this topic, I assumed it was going to be another discussion about zero or infinity. One of my teachers explained it as a number line and any time there was a change of sign, there was a change of direction (and zero is signless). So with (-2)x(-3), it would be written rather neatly as 0-(-2)-(-2)-(-2) which is +6.

The other way of thinking of it is do the times tables backwards:
-2x3 = -6
-2x2 = -4
-2x1 = -2
-2x0 = 0
And since we're in fact adding 2 each time:
-2 x -1 = 2
-2 x -2 = 4
-2 x -3 = 6

And yes, I was tripped up all the time by it too when I was in school . Between that and thinking Of as in 1/4 of 24 = 6. Since the answer is smaller than the number you took the fraction of, you're obviously dividing.


ps. I have no qualifications of comedy or joke-telling. Sorry if I ruined the fun.

[Winstonm]

manudude03, on 2011-February-06, 16:49, said:

When I first saw this topic, I assumed it was going to be another discussion about zero or infinity. One of my teachers explained it as a number line and any time there was a change of sign, there was a change of direction (and zero is signless). So with (-2)x(-3), it would be written rather neatly as -(-2)-(-2)-(-2) which is +6.

The other way of thinking of it is do the times tables backwards:
-2x3 = -6
-2x2 = -4
-2x1 = -2
-2x0 = 0
And since we're in fact adding 2 each time:
-2 x -1 = 2
-2 x -2 = 4
-2 x -3 = 6

And yes, I was tripped up all the time by it too when I was in school . Between that and thinking Of as in 1/4 of 24 = 6. Since the answer is smaller than the number you took the fraction of, you're obviously dividing.


ps. I have no qualifications of comedy or joke-telling. Sorry if I ruined the fun.

Don't worry about the jokes - you have admirably supported my theory that mathematics is a conspiracy in order to give out bad grades.

[manudude03]

Of course mathematics is a conspiracy. Some more basic examples.
2>1 Multiplying by a constant doesn't affect the equation so 4>2, 6>3 etc
Therefore since -1 is a constant, -2>-1.

Since 2x0=1x0, 2=1.

In mathematics we are taught that 2 is greater than 1, how do we know it is greater? How do we know the "signs" that are 2 and 1 respectively have different values? The same can be said for the alphabet, who decided it started a-b-c etc and not another random collection of characters in a vacuum.

[Winstonm]

manudude03, on 2011-February-06, 17:20, said:

Of course mathematics is a conspiracy. Some more basic examples.
2>1 Multiplying by a constant doesn't affect the equation so 4>2, 6>3 etc
Therefore since -1 is a constant, -2>-1.

Since 2x0=1x0, 2=1.

In mathematics we are taught that 2 is greater than 1, how do we know it is greater? How do we know the "signs" that are 2 and 1 respectively have different values? The same can be said for the alphabet, who decided it started a-b-c etc and not another random collection of characters in a vacuum.

Aha! So that's how they got the pictures that made the Earth look smaller than the moon. I knew it!

What do you know about Roswell? And calculus?

[kenberg]

I am not sure how seriously to take this all. But if a person really wants to think through how it works, mathematics is probably the area of study that is the most likely to reward careful thinking.

Think of mathematics as a machine. Of course any machine can be misused. If you don't read the instructions or make any attempt to find out what the various levers and buttons are for, the results are not so good. But on the other hand, there are reasons why mathematics has wormed its way into almost every productive human activity. If you are happy with 2X(-3)=-6 as expressing the fact that 3 debts of 3 dollars each gives a debt of 6 dollars, it is not a big stretch to move on to (-2)X(-3)=6. 2X(-3) expresses a debt, so (-2)X(-3) undoes that debt and expresses a gain. But of course there are many interpretations. Mathematics is a flexible tool.

If I sound a little authoritarian here, that's right. A true story that I may have told before: I was teaching advanced calculus and after a test a student was taking me to task for the grade I had given him. He insisted that what he said was the same as what I said, just worded differently. I repeated that it was not the same and his way was not correct. After ten minutes or so I suggested "Look, we could go on like this for the rest of the day, you saying that it's the same and me saying that it's different. Alternatively, you could accept, at least provisionally, that I have this job because I know what I am talking about and it is your job to understand what I am saying." He thought that over, decided to give it a try, and became a very good student.

An alternative approach is to simply set out axioms for manipulations. This has a lot of merit, but I think when fundamental issues of interpretation are raised, the axiomatic response is not all that satisfying.

[mike777]

kenberg, on 2011-February-06, 19:47, said:

I am not sure how seriously to take this all. But if a person really wants to think through how it works, mathematics is probably the area of study that is the most likely to reward careful thinking.

Think of mathematics as a machine. Of course any machine can be misused. If you don't read the instructions or make any attempt to find out what the various levers and buttons are for, the results are not so good. But on the other hand, there are reasons why mathematics has wormed its way into almost every productive human activity. If you are happy with 2X(-3)=-6 as expressing the fact that 3 debts of 3 dollars each gives a debt of 6 dollars, it is not a big stretch to move on to (-2)X(-3)=6. 2X(-3) expresses a debt, so (-2)X(-3) undoes that debt and expresses a gain. But of course there are many interpretations. Mathematics is a flexible tool.

If I sound a little authoritarian here, that's right. A true story that I may have told before: I was teaching advanced calculus and after a test a student was taking me to task for the grade I had given him. He insisted that what he said was the same as what I said, just worded differently. I repeated that it was not the same and his way was not correct. After ten minutes or so I suggested "Look, we could go on like this for the rest of the day, you saying that it's the same and me saying that it's different. Alternatively, you could accept, at least provisionally, that I have this job because I know what I am talking about and it is your job to understand what I am saying." He thought that over, decided to give it a try, and became a very good student.

An alternative approach is to simply set out axioms for manipulations. This has a lot of merit, but I think when fundamental issues of interpretation are raised, the axiomatic response is not all that satisfying.

Great post I love it.


Granted as a nonmath major I dont understand all of it but I do understand enough to see true wisdom and discipline.

I took beginner calculus as a 15 year old and knew I had hit my math limit.

With that said I do enjoy in my old age reading math stuff for nonmath majors..
--


btw as a student not a teacher I do think students learn in different ways, this is not just some PC thing.....I found out, as an adult, I do.

As an adult I think my math teachers as well as most other teachers make one huge common mistake. They think or expect their students to know/remember the basics of the subject, they dont.

I think this is really true for bridge.

[Winstonm]

Quote

Granted as a nonmath major I dont understand all of it but I do understand enough to see true wisdom and discipline.

Strange, all I saw in math was an authoritary saying, Because I say so, that's why, never a satisfactory answer. I didn't want someone to hand me a fish sandwich; I wanted someone to teach me how to catch fish.

Unfortunately, my high school math teachers all moonlighted at McDonald's. I didn't learn much math but I did learn to love tarter sauce.

Quote

As an adult I think my math teachers as well as most other teachers make one huge common mistake. They think or expect their students to know/remember the basics of the subject, they dont.

Really good and valid point, Mike. I found out later in life when I went back to school and changed career paths that it has always been the foundational language that I had been so resistant to learning because I do not learn languages easily or well.

To this day I doubt I could define much more that the word "sum" as far as language of math is concerned.

In my career change, I was motivated to learn new terminology and so did it, but it still was the hardest part for me - once I knew the terms, the analaytical parts of learning were pretty simple.

[kenberg]

The authoritarianism is not, imo, that you must accept this as true because I say so. Rather:

You say that math, as you think about it, seems to make no sense even at a fairly elementary level. I say that to me it does make sense, and I suggest how I look at it. Then, and this is the authoritarian part, I suggest that instead of insisting on looking at it your way and asking someone to show you how you can make sense of it your way, I suggest you try thinking of it my way. Put yourself in my hands for a while. I am not a hypnotist, I won't get you to perform any indecent acts, I'll just suggest ways of thinking with mathematics that reflect the views of a practicing mathematician. The alternative is to say "Mathematics makes no sense as I look at it, and I refuse to look at it in any other way". Not surprisingly, this is a dead end.

When my younger daughter was a high school freshman she was having trouble with algebra. I am sort of the negation of a home schooler. Like my parents, I am perfectly content to let the teachers do the teaching. But she needed help, that was clear, so I announced I would be helping her. Her comment was "It won't do any good, I'll understand it when you say it but I won't remember it". I didn't challenge this I just said that we were going to do it anyway. At first she was right, no change. But then a big change, from C and D to A, skipping the in between, as she got the point of it. Pretty much that's the way of it with math. Once you think it through so that you get the point, it's easy. Until then, it's a mess. Very little space in between.

[PassedOut]

kenberg, on 2011-February-06, 22:21, said:

Pretty much that's the way of it with math. Once you think it through so that you get the point, it's easy. Until then, it's a mess. Very little space in between.

That happened to me as a freshman in college. I had gone to three different high schools (long story), and as a senior I had no math because I had already taken everything available (my last high school was a lot of fun socially, but not very good academically).

The next year I found myself in an Ivy League math class that started very fast, and I got totally lost. After three weeks, I decided that I had to do whatever it took to get a handle on it or I was doomed. I bought some extra notebooks, started from page 1 in the math text, and wrote down in my own words exactly what was being expressed so compactly, paragraph by paragraph. Drew diagrams to help me visualize. Did every exercise. After a few days of that, the math did click in and I understood what was going on from then on. From then on I rewrote in my own words everything I was learning, and found that to be a technique worked for me.

[BunnyGo]

For those who are not complaining about their difficulty with mathematics but instead how they were taught: it's been said many times before. This article is a very well written description of everything wrong with primary school mathematics (written by a mathematician turned primary school teacher).

http://www.maa.org/d...hartsLament.pdf

[kenberg]

Yes and no about the Lockhart article. It shows, perhaps, that if you can hire a Ph.D. from Columbia with research interests in automorphic forms and Diophantine geometry, have him teach in a private school that accepts only the top students, and give him free rein, the results might be pretty good. You think?

So yes, there is something to be learned there but we need to think about what can be replicated on a broad scale. The fundamental thing about mathematics is that it actually does make sense, and moreover just about anyone can find the sense of it with a little guidance and a little work. Like Lockhart, I definitely favor presenting mathematics in terms of problems that can be solved, as a challenge where the memorization of rules plays a minor role, and as an aesthetic enterprise. I think that this can be done by ordinary people, although probably not to the extent that Lockhart is capable of.

OTOH:
A very creative friend wrote an article "In Defense of "Mindless Rote", where he presents a different view http://www.nychold.c...kin-rote01.html

He begins with a quote from Alfred North Whitehead (co-author with Bertrand Russell of Principia Mathematica, a revered work that I once checked out of the library but, like everyone else I know, quickly abandoned):

Quote

It is a profoundly erroneous truism repeated by all copybooks, and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations which we can perform without thinking about them. Operations of thought are like cavalry charges in battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.
- Alfred North Whitehead, Introduction to Mathematics

Note: The above quote probably suffices to get the drift of Ethan's essay.


So mathematics is a large animal with many heads. But honest, there are some fundamentals that everyone (almost everyone) can grasp with a bit of effort. These fundamentals can be understood rather than memorized, and once they are understood it is much easier to recall them and, when necessary, adapt the techniques they support to deal with the problem at hand.

[Winstonm]

Well, if you try to figure it out for yourself and write it down, better make sure your thinking is right. To wit:

Ken,

Quote

If you delete (aka take away) 2 debts of 3 dollars each you would be 6 dollars richer, would you not?

Therefore, (-2)x(-3)=6

Me,

If we were using actual -1 dollar bills, then 2(-1 dollar bills) x 3(-1 dollar bills) would give me a pile of six -1 dollar bills for a total of -6. If you wanted to make me richer, you would still have to give me another six +1 dollar bills.

So (-2)x(-3)=(-6)

Me again,

I have a debt (a hole) of 6 feet of dirt. If someone deletes 2 debts of 3 feet each, I have only filled in the hole, got back to ground level, and I don't have a 6-foot-tall pile of positive dirt left over.

Therefore, (-2)x(-3)=0

And there are some who would dare say I was wrong.

Edit: Actually, I'm only partly kidding, as my brain tells me that eliminating debt means getting rid of it, erasing it, subtracting it, and as a debt is a negative already it should be written as: -(-2)X -(-3)=6.

Truthfully, not even that is right. What my brain is really telling me is that -(-2)X -(-3)= 0, back to even, out of debt, and now I'll need to earn $6 dollars in order to be +6 dollars.

I think the whole problem is no one along the way ever seriously took time to explain the equal sign and what it really means. Serious.

[Echognome]

Trying to get your head around a particular mathematical subject can be difficult no matter how far along you are in the chain. I do think that until you get into real analysis, most of the focus is on doing rather than understanding. You learn rules, not proofs. Most of calculus is focused on learning rules. I do think it would be useful to learn more of the whys rather than the hows. I just don't think it's all that practical.

As a continuation of the questions above, seeing the proof for why 1 > 0 blew my mind. I can't recall the exact proof, but I do remember we had been discussing Dedekind cuts. Think of it this way, if you are trying to prove 1 > 0, then what are you starting with as your assumptions?

The real number line also blew my mind. We learned that rational numbers are dense in the real number line. An implication of this is that between any two real numbers lies a rational number. In my mind, there are "more" irrational numbers than rational ones. If we just take any rational number and multiply it by the square root of 2, we have an irrational number (zero being the exception). But then we can also multiply the rational number by the square root of 3 or the square root of 2 divided by 2. In addition, we can multiply by the transcendental numbers, such as pi or e. It just seemed "clear" to me that there were many more irrational numbers than rational numbers. But now I'm being told that you can take any two real numbers (rational or irrational) and you'll find a rational number sitting right there in between. What?

Once I had down some knowledge of real analysis, complex analysis wasn't so bad. Yeah we had to use imaginary numbers, but since they were imaginary, I didn't have to come to grips with them in the same way. Their properties certainly seemed reasonable and the extension from real analysis to complex analysis flowed naturally.

The next blowing of my mind came with abstract algebra. I thought Fermat's Little Theorem was elegant. Where I struggled was with field extensions. Why are there no general formulas for the roots of fifth (or higher) degree polynomials?

All this just taught me that somewhere along the mathematical chain of learning you are going to struggle. I'm sure that PhD mathematicians struggle when trying to understand a theory that is yet to be proven. The difficulty in math is that there is often a chain of understanding that you must grasp in order to understand the next concept. Sometimes the chain is linear, sometimes it is not. I always thought of econometrics as a field where the chain is very "broad" (by that I mean you have to have studied calculus, linear algebra, and statistics). What happens when you struggle with one link of the chain early on? Maybe your parents split up? Maybe you had a bad teacher one year? Maybe you got into a bad crowd? If you break that chain of understanding, it is very difficult to catch up. I don't think other subjects in school are the same way, not relying nearly as much on knowledge learned the year before. That is a challenge for teaching math.

Anyway, I've rambled enough, but it's certainly a subject I find interesting and worth discussing.

[WellSpyder]

Echognome, on 2011-February-08, 01:12, said:

All this just taught me that somewhere along the mathematical chain of learning you are going to struggle. I'm sure that PhD mathematicians struggle when trying to understand a theory that is yet to be proven.

Reminded me of some advice I heard once from a Professor of Mathematics (http://www.maths.qmu...w/inaugtext.pdf)

Quote

In case any of you get confused by what I am about to say, let me reassure you
with this quote from John von Neumann, a famous mathematician who was one
of the pioneers of the computing age in the 1940s.

"In mathematics you don’t understand things, you just get used to
them."

And although this sounds like a flippant remark, it contains a lot of truth. I
think many people make the mistake, in trying to learn mathematics, of trying to
understand something which cannot in fact be understood. (It’s a bit like trying
to understand God, I suppose.) I remember as a student going to a course on
Quantum Mechanics, and getting into quite a state of frustration because I could
not understand where Schr¨odinger’s Equation came from. It was years before I
understood that it didn’t come from anywhere—it was just an assumption. And
years later still I realised that actually nobody understands quantum mechanics.
But quite a lot of people have got used to it and can use it effectively.