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Why Johnny Can’t Add Without a Calculator
June 30, 2012 9:09 AM   Subscribe

Slate: Technology is doing to math education what industrial agriculture did to food: making it efficient, monotonous, and low-quality.

A math teacher responds
Response to the response
posted by beisny (70 comments total) 23 users marked this as a favorite

 
Yes. Because learning multiplication tables by rote was not monotonous.
posted by Sys Rq at 9:18 AM on June 30, 2012 [5 favorites]


Sys Rq, I think that's the point: it's monotonous, *and* low quality. At least you learned your times tables.

The second week of college, I ran over my backpack trying to straighten my car out in a parking space. My brand-new graphing calculator, a required item (and expensive at $80 in 1993, especially for someone who had to work while going to school) had a blue screen of death - the LCD leaked. It was unusable, and I couldn't afford a replacement.

So I spent a three-day weekend studying graphs in Excel, so I could visualize things without the calculator. In other words, I learned what I should have learned in high school but was too lazy.

I made it through an engineering degree at a Top 20 university without a graphing calculator. Why are they required, again? (I know, so kids can use the infrared communication port to cheat on tests.)

My mom teaches math at the community college. She doesn't allow calculators until College Algebra (starting from basic math, it's the fifth course in the sequence). Some of the other teachers think she's nuts, but her students do better in successive classes.
posted by notsnot at 9:30 AM on June 30, 2012 [7 favorites]


(I am a math teacher but I'm not your math teacher)

I'm not at all a "Khan Academy will save us all" tech evangelist, but I'm not really on board with this article. He's right that the dichotomy between "learning by rote" and "learning concepts" is a false one, but for the wrong reason. He thinks you can't understand the ideas of mathematics unless you've mastered computational fluency. I think you can understand some ideas without fluency, and you can gain fluency to some extent without learning ideas, but if you don't master both you haven't learned mathematics.

I think it's mathematically crippling not to have memorized the multiplication table. But I also think it's mathematically crippling if your only answer to "why is 5x6 = 6x5" is "because I memorized the table and they're both 30."
posted by escabeche at 9:33 AM on June 30, 2012 [20 favorites]


I am, honestly, not sure why knowing *why* the product of two negative numbers is positive is important. I myself can't come up with a good explanation for this. Does someone want to explain it?

I am a professional software engineer, and as compared to the general population, probably fairly good at math, but I don't know the reasoning behind many of math's simple axioms (why does 2 + 2 = 4? Well, because it just *does*). Similarly, I know that gravity accelerates me toward the center of the earth, but I don't know *why* that happens, and as far as I know, neither does Stephen Hawking, but these things haven't been much of a practical limitation in my own life, so far as I can tell.
posted by tylerkaraszewski at 9:45 AM on June 30, 2012


(There's a link to a proof of the product of 2 negatives being a positive if you want to follow it--but I'm not sure why knowing this proof is desirable to a non-mathematician.)

The real question to ask is: why do we teach mathematics at all? Is it so people can multiply if their calculators break? And why do we need to memorize times tables, for that matter? I'm not asking this because I have an answer. I'm asking because there are a lot of different things people seem to want a mathematics education to do.

For one, it's supposed to allow you to understand the language of mathematics so that you can read (or maybe write) science papers or statistics in the news. For two, it's supposed to allow you to learn different kinds of thinking, such as what is an instantaneous rate of change or what is a binary operation. For three, it's supposed to be an aesthetic experience--that reading the proof that there are an infinite number of primes is like reading a work of literature.

Of those three (and I'm sure there are others I left out) we mainly teach the first, and barely that since people memorize lots of stuff and mostly forget it after their exams. Does technology interfere with that? I think it makes it easier for people to pass their exams and makes it easier to forget stuff afterward. But since education is about getting a degree and a job, this is how we mean it to be. It's a hurdle you need to show you can jump over.

Insofar as education will be anything other than that, I don't think it will be the technology which causes the problem, but is, like the guns which kill people, just something that is part of the implementation.
posted by Obscure Reference at 9:50 AM on June 30, 2012 [5 favorites]


"But the essence of calculus is the proof of its fundamental theorem: that the procedures for determining the area under a curve and a curve's rate of change are in fact opposites in a precisely determined sense. Real mathematical understanding lies not in the idea that this is plausible, but in a rigorous proof of it."

According to this Konstantin Kakaes fellow, Isaac Newton apparently didn't have "real mathematical understanding" of calculus, because he certainly didn't have a rigorous proof of any of it (that came much later with Cauchy in the 19th century. Newton made use of infinitesimals, which wasn't made rigorous until the 20th century!)

Scoff.
posted by nhamann at 9:59 AM on June 30, 2012 [6 favorites]


The real shortfall in math and science education can be solved not by software or gadgets but by better teachers.


The author really should have started out with this line instead of trying to blame technology for causing the downfall of math education. It isn't the fact that calculators and Wolfram Alpha are making kids worse at math, it's just that teachers tend to not be that knowledgeable, especially in elementary and middle schools.

The parts of math that require extensive use of calculators are the most inane and boring parts. Math education, in my opinion, should encourage kids to go above and beyond the level of high school math which involves insane amounts of number crunching for calculus. They should help kids want to learn more, not only about math, but also about how math really works, like why 2 negatives give a positive. It needs to foster the question, "Why?" not the question, "What's the solution?"
posted by astapasta24 at 10:05 AM on June 30, 2012 [2 favorites]


When I went back to college, twenty years after I last even thought about algebra, the first class I took used something called the Hawkes Learning System. It was homework software. In order to pass my homework, I had to get a certain number of problems right, for every concept we covered in class.

When I did a problem, it took me through the steps of how to do the problem, demonstrated what I did wrong if I was incorrect, and gave me another chance to do a different, but similar, problem.

Instant feedback, instead of waiting days (or weeks) to figure out if I understood a concept. Plus the ability to do as many of any type of problem that I wanted to, in case I wanted to make sure I really understood a concept. Available to me 24 hours a day.

I aced that class, and every subsequent math class I took for my degree.

I don't remember who taught that class, but I damned well remember that software package. I hated algebra up to that point, vehemently. Flunked it in high school. That software made it possible for me to pick up the basic concepts that I'd missed, and actually made math enjoyable for me.

The problem with teachers who use old-school methods is that we're relying on every single one of them to be excellent teachers. Not all of them are. And getting a terrible teacher for a single class is enough to handicap a student for the rest of their academic career.

Judicious use of classroom technology can make learning math much easier, and convey the concepts better than the standard blackboard-recitation-homework delayed feedback loop.
posted by MrVisible at 10:08 AM on June 30, 2012 [32 favorites]


Scoff. What deep mathematical understanding Newton had!

FTFY.
posted by metaplectic at 10:17 AM on June 30, 2012


I as impressed by the article's author:
One problem asked me to calculate the width of a doorframe, given the frame’s height and a diagonal measurement of the door. After 30 seconds’ work with pen and paper, I submitted my answer: 93.7cm.
I think it would have taken me more than 30 seconds to work out the square root of 8784 with pen, paper and an understanding of the Newton-Raphson method.
posted by fredludd at 10:24 AM on June 30, 2012 [7 favorites]


Let me clarify: I was impressed.
posted by fredludd at 10:25 AM on June 30, 2012


I'm generally anti-calculator. I suspect because once you stop using one, they become a pain in the butt.

However, I disagree with the article's vilification of graphing calculators on the grounds that the author doesn't seem to actually know what students use them for. As a TA, I get to see a lot of under-prepared students taking their first college math class. Yes, they overwhelmingly are reliant on their calculators. This calculator reliance has tended to lead their ability to work with fractions and radicals to atrophy. They have a distressing belief than if something isn't a decimal it isn't 'precise' and therefore can't be right.

However, they overwhelmingly don't have calculators more sophisticated than the TI-84 (which is nothing but a re-branded, more expensive TI-83Plus with a USB port, which was the TI-83 with some added financial math ability no high school student uses, if I recall correctly). That my TI-89 can factor polynomials blows their minds. (Less so now that they know Wolfram Alpha exists, which took a while after its introduction.) You can't blame their sometimes limited ability to factor polynomials or remember that the quadratic equation exists on their calculators. Yes, if you can graph, you can find the roots, but for the most part, students who can't factor confidently haven't discovered that feature of their calculators.* I also don't think you can blame weak problem solving skills on a calculator--you have to be able to figure out what to stick into the calculator first.

My department splits on whether to allow calculators. Some people do, on the grounds that they're pretty useless in a calculus course. Other people ban them on the same grounds. I'm in the second camp, partly to convince the students they don't need a calculator and partly so I don't have to grade quizzes with 10 decimal places copied down as the answer. It should be noted, though, that the textbooks assume access to a calculator. They write problems with ugly decimals for no reason. Then someone asks about one of those in class. So we go through it on the board and then I have to stand at the front of the room and fiddle with a calculator for 30 seconds to get a number, which is hardly a good use of time. But saying 'It's a giant mess and I can't be bothered.' isn't a great option either.

*Coincidentally, this seems to happen in all courses of mostly first year students, regardless of the level of the course. Once one student starts being timid about factoring, they all suddenly lose confidence in their ability to do it.
posted by hoyland at 10:39 AM on June 30, 2012 [1 favorite]


tylerkaraszewski: "I am, honestly, not sure why knowing *why* the product of two negative numbers is positive is important. I myself can't come up with a good explanation for this. Does someone want to explain it?"

You have 2 businesses losing money every month. But don't worry, they're only losing two dollars monthly, and you're certain the right person could turn the whole thing around.

So you sell both businesses (-2). As a result, how much money are you saving every month?

-2 * -2 = 4 dollars saved every month.
posted by pwnguin at 10:57 AM on June 30, 2012 [3 favorites]


So you sell both businesses (-2). As a result, how much money are you saving every month?

-2 * -2 = 4 dollars saved every month.


That can't be right. If it were, then if you had three businesses losing two dollars every month, selling them all would lead to your losing eight dollars a month! Since (-2)*(-2)*(-2) = -8. What you should be doing is adding those various losses (or savings).
posted by Jonathan Livengood at 11:11 AM on June 30, 2012


It's amazing how picking the wrong numbers for an example can obscure the point so easily.

JL, those two -2's don't represent one store each. The first -2 is "change in dollars per business" and the second one is "change is number of businesses".
posted by benito.strauss at 11:14 AM on June 30, 2012 [4 favorites]


@tylerkaraszewski: A complex number interpretation would have -1 = exp(i*pi). So -1 * -1 = exp(i*pi) * exp(i*pi) = exp(i*pi + i*pi) = exp(i*2pi). One of the properties of exp(phi) is that exp(phi + i*2pi) = exp(phi), so exp(i*2pi) = exp(0) = 1.
posted by DetriusXii at 11:14 AM on June 30, 2012


No, if you sell both businesses, you're modifying the number of businesses you own by -3, and each business is making you -2 dollars a month, so the total gain to you is (-3)*(-2), or 6.
posted by escabeche at 11:14 AM on June 30, 2012


I would like to see some real statistical analysis with respect to the article's questions. I mean, we're getting an anecdote about one good math teacher who doesn't like the new technology. Okay, how many good math teachers don't like the new technology -- as a proportion of (good) math teachers? How do good (bad) math teachers who use technology compare to good (bad) math teachers who do not use technology in terms of outcomes for their students?
posted by Jonathan Livengood at 11:16 AM on June 30, 2012


Ah, okay. My mistake.
posted by Jonathan Livengood at 11:17 AM on June 30, 2012


The most "practical" application of math I actually learned in general chemistry. Our instructor spent most of the year drilling it into our brains to keep track of units because otherwise it didn't matter how correct your calculations were, if one didn't keep track of units then you wouldn't know what your numbers meant and what is the point of that? She had this "unit cancellation" method for stoich that caused me no end of trouble until it finally clicked and then IT WAS AWESOME.

I work with weavers and there is a lot of arithmetic and simple algebra used in weaving design and the people who have problems with it aren't bad at computation necessarily, but they do not keep track of units and they'll say to me, "I need 24" and I'll say "24 what?" because I don't know and then they realize they don't know either, or they think they know but like their calculation represents an output in inches and we are trying to figure out how many threads they need or whatever.

But I emphasize because for so long I sat in gen chem and I watched my prof cross out moles from the numerator and moles from the denominator and why is the answer in liters? I thought we were calculating moles? No we started with moles and grams, we want liters what? It's a practice thing that doesn't get practiced.

I guess see the discussion directly above.
posted by newg at 11:32 AM on June 30, 2012 [6 favorites]


I just want to testify that a couple of years ago I saw a student in a community college technical program use a calculator to multiply a two digit number by one hundred. I'm curious to know what he would have done if required to multiply by ten.
posted by BigSky at 11:35 AM on June 30, 2012


The problem with teachers who use old-school methods is that we're relying on every single one of them to be excellent teachers. Not all of them are. And getting a terrible teacher for a single class is enough to handicap a student for the rest of their academic career.

I just feel the need to echo this. I did fine with basic math through high school but didn't like it. In order to do what I wanted to do, in undergrad I attempted calculus. I was a good student in that I was able and willing to work hard, but I am not a natural with math, and my teacher was so terrible that despite going to every office hour with him (I would go to him with a question and explain, for example, where the book had skipped a step that I didn't understand, and his response was always 'well let's look at the book' with no further input) I only barely passed the class.

So I gave up on it. I just kind of decided to avoid it for as long as possible. But I had to really learn it for graduate school, and so now at an entirely different university I took it again with no small amount of dread. My teacher was amazing though. I can't even remember anything he did that was particularly special, but he was clear and patient and willing to take the time to discuss every step with you and frame it different ways if necessary. For the duration of that class I honestly loved calculus and had a good amount of 'eureka' moments. If I hadn't had that teacher I don't think I ever would have understood calculus.

Personally, the use of a graphing calculator helped me more than hurt me in that it was easy to plug in an equation and see how it looked graphically. Visually examining the patterns generated by different equations was something I would actually do for fun just because it was so simple to do, and I learned a lot for that. Manually producing those graphs was important too, but I felt like the calculator helped me understand those concepts more quickly than I would have without them, and with no less 'quality' in the end result.
posted by six-or-six-thirty at 11:35 AM on June 30, 2012 [2 favorites]


Judging what I am currently seeing, scoring standardized high school algebra and geometry tests, somewhere around 1% of high school students are ready for high school math classes.

I took math in high school at the very moment the slide rule became obsolete and the portable scientific calculator was released. When I was a junior, I bought a very expensive K&E slide rule. Our math classroom had one of those huge 6 foot long slide rules hanging over the blackboard, so the teacher could demonstrate operations to the whole class. I remember thinking what an enormous pain in the ass it was getting precise values when the teacher required us to calculate like 6 digits of precision when a slide rule was only capable of about 3 at best. You had to run iterative successive approximations. Since this was mostly based on logarithms, I hated doing these repetitive calculations, which were prone to error.

When I was a senior, I bought a hideously expensive HP-35 calculator, I think I recall it cost like $295. With an HP-35, it was easy to get precise answers to simple calculations that would require a ton of effort to calculate with that number of significant digits. I also bought the HP manual of calculations that were beyond the means of a non-programmable calculator. You could look up the calculation method, and step through the instructions, one keypress at a time, following the exact algorithm on the page. Essentially you were manually performing the programmable function of a programmable calculator. I could rarely make it through a complex calculation without an incorrect keypress somewhere, and often the error wouldn't become obvious until you completed the whole algorithm. It was maddening. But I think this sort of model of mathematical thinking prepared me for computer science classes on a higher level than most students. That is sort of a futile accomplishment, everyone knows CompSci people can't do math.

So I think I am kind of the worst of both worlds. When I was a senior, I should have been hammering home the knowledge of direct relationships between the slide rule scales. But instead I was using a calculator a lot of the time. But my teacher wouldn't allow calculators on homework or in class. So I was straddling both methods. Consequently, I have a fairly good understanding of most of the relationships used in calculations, but I have terrible troubles doing manual calculations and keeping track of the mantissa and exponents simultaneously. So I often get the right answer, but it's off by an order of magnitude (or two or more). Or I often get the order of magnitude right, but miscalculate the mantissa, giving me a vaguely correct answer in an entirely different way.

Then it got worse. In college, I took an honors calculus class. The university sprung a surprise on us: an entirely new experimental textbook , Kiesler's Infinitesimal Calculus. It was a disaster. We didn't get a printed, bound book until the second semester. For the first semester, the book was delayed at the publisher, so we worked from xeroxed galley proofs. Every monday, the instructor distributed errata from Kiesler, and we would spend the largest part of the class period making corrections. So that usually cut our 3 hours a week of class time down to 2 hours. Then with our scarce class time, the instructor would give us poorly constructed explanations of the material, since nobody had ever taught elementary calculus this way before. It was nearly impossible to make up for the insufficient instruction time. I would work with tutors and they'd be completely baffled at this method, so they'd teach me the usual ways of doing derivatives, integration, etc. Then I'd mix up the methods, making it even more incomprehensible. When I occasionally used standard methods on homework and tests instead of the infinitesimal methods, I'd get my papers back marked with big red letters, "This isn't the way we taught you to do this in class."

The result was that this honors section, with the most proficient math students in the entire university, many of them math majors, had the lowest grades and lowest pass rate of any calculus class ever taught at the university. The experiment was abandoned immediately after the first year it was attempted. It would have been abandoned after the first semester, but we were already in too deep. I barely managed to pass the class, but now 35 years later, I probably remember more calculus than the average students from any of the other non-infinitesimal methods.

Math instruction generally sucks, and methods to improve it are generally even worse. Math is the hardest subject to teach since it is the foundation for all the exact sciences. Oh well, I don't care. I carry my iPhone and I have PCalc and I can call up Wolfram Alpha any time I need it. I may not remember how to get the right answers, but at least I know how to ask the right questions. IMHO it is more important to know how to frame the questions correctly, than to know how to calculate the answer. With enough higher order thinking, you can always learn the method of calculation. The solution is left as an exercise for the reader.
posted by charlie don't surf at 11:42 AM on June 30, 2012 [2 favorites]


I think it's mathematically crippling not to have memorized the multiplication table. But I also think it's mathematically crippling if your only answer to "why is 5x6 = 6x5" is "because I memorized the table and they're both 30."

Is there a good answer to that question other than "because multiplication is commutative because we say so" that doesn't involve a proof from the Peano postulates?

I'm not sure which side to come down on here, though I lean towards the suspicion that computational facility has something to do with conceptual facility -- it's easier to get a feel for patterns when you've already seen a lot of examples of how they manifest. As an example, I'm pretty good at doing arithmetic in my head, and I think that really helped me in proving results about congruences when I was learning about modular arithmetic.
posted by invitapriore at 11:50 AM on June 30, 2012


The calculator question is a huge red herring, the real issue is the people:

1) who gets to teach mathematics
2) who sticks with teaching mathematics
3) who gets to make decisions about how mathematics is taught.

Right now, the system and culture of pre-college mathematics in the US works to favor people who can switch pedagogy fluidly with the ever changing directives from above. They are more comfortable with the process of teaching than mathematics itself. Fundamentally, the question is whether a teacher is someone who *knows* something vs. a teacher is someone who *teaches*, a sort of professional explainer.

Mind you, math knowledge has little to do with degrees, especially at the primary school level, but aptitude and interest. We would rather invest in new technologies and new methodologies than invest in people. Of course, you could say that about the rest of american society... why should education be any different.

But again, the issue is not about pedagogy really, it's about the politics of education itself.
posted by ennui.bz at 11:54 AM on June 30, 2012


In some cases focusing on how to do the computations can get in the way of really teaching the material. For example, often linear algebra is taught in such a way that the bulk of the time is spent learning how to manually perform various operations on matrices (inversions, determinants, factorizations, finding eigenvalues, etc.), when in practice nobody does any of these operations by hand, and so people get the idea that linear algebra is about doing various apparently-pointless operations on 5x5 matrices. Furthermore, because the focus is on manually doing operations to small matrices, often the more powerful tools (like the singular value decomposition) are glossed over, because they are too complex to compute by hand.
posted by Pyry at 12:02 PM on June 30, 2012 [1 favorite]


Is there a good answer to that question other than "because multiplication is commutative because we say so" that doesn't involve a proof from the Peano postulates?

I think a good answer is "if you make a big box with 5 columns and 6 rows, there are 30 things in it, and if you turn the box on its side, it still has 30 things in it." I do math for a living and I don't think any more rigor than that is necessary (or even necessarily appropriate) at the elementary school level.
posted by escabeche at 12:22 PM on June 30, 2012 [17 favorites]


Personally, the use of a graphing calculator helped me more than hurt me in that it was easy to plug in an equation and see how it looked graphically. Visually examining the patterns generated by different equations was something I would actually do for fun just because it was so simple to do

When you get trained as a college teacher in graduate school, one of the things they impress upon you is that some people like to learn things in terms of formal languages and operations, and other people find it easier to grasp things visually. So we're taught to use both modes throughout every part of the course, so that neither population gets totally alienated and lost.
posted by escabeche at 12:24 PM on June 30, 2012 [1 favorite]


The experiment was abandoned immediately after the first year it was attempted. It would have been abandoned after the first semester, but we were already in too deep.

But it's not bad that the experiment was done! As far as I know, experimentation is the only way to figure out whether novel ways of teaching math are better than the old. Certainly it's not a priori obvious that the approach via infinitesimals wouldn't be a good idea. That's how the subject was originally understood, after all, and in many ways it's closer to physical intuition! Indeed, when Cauchy tried to replace the traditional treatment via infinitesimals with his experimental approach (aka the way we teach calculus now) the students revolted and the dean had to place monitors in the classroom to make sure he was teaching the engineers infinitesimals as he was supposed to. Heck, Don Knuth still thinks we should teach calculus via an approach much closer to infinitesimals than the standard presentation we use now. And when it comes to clear exposition of mathematical ideas I have to concede that Don Knuth's opinion carries a lot more weight than most people's.
posted by escabeche at 12:31 PM on June 30, 2012 [2 favorites]


So we're taught to use both modes throughout every part of the course, so that neither population gets totally alienated and lost.

or both populations are alienated and lost.

I think everyone is better off if the teacher focuses on what makes sense to them, rather than what the research might say is good for some average student.

But it's not bad that the experiment was done!

Yes it was. It was bad for the students in the class, that's the whole point of the anecdote... Bad teaching absolutely has long-term effects on students. My father was in honors math as a freshman at U. Mich. last cetnury, where his 1st year math class was taught by a very young post-doc: Raoul Bott. Bott, for reasons which are probably clear, decided to teach calculus in n-dimensions starting with set theory. For some of the students this was undoubtably great, for my father it led to him permanently dropping out of college.

Are my two points contradictory? In some ways they are, but there are important differences between university education and what comes before it. It also illustrates that some fo the problems with university math education in the US are long-standing. The long desire to delegate teaching calculus to junior or visiting faculty has led to a system where the required calculus textbook (and now included required calculator) prevents young mathematicians from doing what Bott did.
posted by ennui.bz at 12:40 PM on June 30, 2012


tylerkaraszewski: "I am, honestly, not sure why knowing *why* the product of two negative numbers is positive is important. "

I can explain it as like taking directions: So you can think of "-2 × -3" as being the same as "Turn around, then go twice as fast, then turn around, then go three times as fast." And because the order in which you do these things isn't important, that's the same as saying "Turn around, then turn around, then go twice as fast, then go three times as fast." Well, anyone can tell you that if you turn around twice you'll end up in the same direction, so you can just ignore that part. So the real directions are "Go twice as fast, then go three times as fast."

The ability to use math is one of our defining traits as a species. Other animals can do math, but we can use it. We can use math to explain things we'd assumed were inexplicable. We can use math to invent new types of math. We can use it to try to create models to connect different parts of our world, and even when we're wrong we learn things along the way.

The analogy above is imperfect and incomplete, certainly. But it doesn't need to be those things. If I share that analogy with someone, that's a way for me to share my world with them. It's the excitement of creating new languages all the time, and each one is as expressive as you can imagine, but without normal languages' burden of ambiguity.*

I know you asked why understanding the principle is important, rather than why that particular principle is actually true. Well, that analogy is my answer; it's because understanding mathematical principles is the bridge between math and reality.

But if you don't understand the principles, then the only thing you can share is "press the '−' button, then press the '2' button, then press the '*' button, then press the '−' button, then press the '3' button, then press the 'Enter' button. Hooray, your screen shows the same thing as my screen. We'll be best friends.

* ...but just for the record, ambiguity is one of the best things about normal languages. It lets you create a new idea that combines your expression with someone else's interpretation, and you both get to benefit from it. It's no more or less vital to human communication than the clarity we get from mathematics.
posted by Riki tiki at 12:44 PM on June 30, 2012 [8 favorites]


I think a good answer is "if you make a big box with 5 columns and 6 rows, there are 30 things in it, and if you turn the box on its side, it still has 30 things in it."

I had a student in a college algebra class who couldn't multiply two one digit numbers by hand (most of my students are very dismayed that I won't let them use calculators in class). After I told him 5 times 6 was 30, I showed him a rectangle of circles, five circles by six circles, and he was amazed that you could do multiplication like that.
posted by Elementary Penguin at 12:57 PM on June 30, 2012


I can't even remember anything he did that was particularly special, but he was clear and patient and willing to take the time to discuss every step with you and frame it different ways if necessary.

I think why some teachers are better than others comes from insight into the many ways a student can misunderstand something. Often this arises from the teacher's own problems learning the material when he was in school. I particular had (and still have) trouble if there was any way to possibly misunderstand what was being talked about. Of course, it helps if the teacher knows the material very well so he can rephrase it if necessary; being able to receive the same information expressed in different ways is immensely valuable to a student.
posted by JHarris at 12:58 PM on June 30, 2012


But it's not bad that the experiment was done!

Yes, as already noted by ennui.bz, it was bad for the students. Speaking on behalf of the guinea pig students, it was very very bad for us. My grade in that class (which was exactly average for that disastrous course) forced me out of Honors status, which had a whole cascade of bad consequences for me, including me dropping out of college.

Heck, Don Knuth still thinks we should teach calculus via an approach much closer to infinitesimals than the standard presentation we use now.

Yeah, well Knuth still thinks we should learn MIX to understand his work, and of course we should all learn a completely idiosyncratic, hypothetical computer language in order to understand the work of a single person. Sheesh.

Kiesler's book was a predictable disaster. Almost everyone predicted it. Instructors persisted in the face of obvious evidence it was a disaster in progress. That is not how you do an experiment.
posted by charlie don't surf at 1:08 PM on June 30, 2012 [1 favorite]



I dunno. I won't say I've always found math easy - hell, when I was doing my engineering degree, I failed a math class for chrissakes.

But, I did a couple semesters as a pre-calc and first semester calculus tutor, and one thing really stood out among those I would see. They would just decide that they weren't good at math, and then give up - because it's OK to be "bad at math".

Which is sort of crap. I'm not good at woodworking; nobody is ever going to hire me to build some cabinets. But I can hang a shelf, or make a fence. I don't need to be Tom Silva - just knowing which end of the drill to hold onto is OK.

And this has actual repercussions. The local rag ran a story one time about a property tax increase. The gist of the story was that the average home owner would see their taxes go up by 300 dollars per year. However, because of the wide disparity in property values, the median home owner would only realize an increase of about 70 dollars per year - but the average sounded scarier, so they ran with that. And they get away with it because math is hard.

You don't have to be Newton Feynman Einstein to understand the difference between the average and the median. It's not fucking rocket science.

But people don't. And won't. Because being good at math is something eggheads do, and being "not good at math" is cool. And easy.
posted by Pogo_Fuzzybutt at 1:16 PM on June 30, 2012 [13 favorites]


I think a good answer is "if you make a big box with 5 columns and 6 rows, there are 30 things in it, and if you turn the box on its side, it still has 30 things in it."

That's not a proof, so much as relying on the assumption that the sizes of objects don't change when you rotate them. It's intuitive, and correct for this problem, but intuition can just as easily lead to wrong answers, too.
posted by Blazecock Pileon at 2:05 PM on June 30, 2012


I'm with Pyry. I try not to make students in my classes do anything by hand that I would not do by hand. I try to reinforce the idea that there is technology out there, let's use it. For example, in Calc III, I have students use Wolframalpha to solve the equations they get from setting the partial derivatives equal to zero when finding extrema of functions of two variables. The problems are long enough as it is- by doing this, I hope I can prevent them from getting lost in the algebra of these problems.
posted by wittgenstein at 3:00 PM on June 30, 2012


A couple calculator anecdotes:
I was doing duty in the chemistry help center (something us grad students had to do) and a student came in with some P-chem quantum mechanics problems. One was: which of the following functions would be acceptable as wave functions? So I went over the rules with her - piece-wise integrable? doesn't tend to infinity at infinity? To which she replied, having plugged the function into her calculator, "How can I tell, there's no infinity button?".
By now, I'm sure that calculators have infinity buttons, I'm too scared to check.
On the other hand, I remember my father talking about designing a piece of (military) equipment that had a spiral with an increasing pitch. To correctly dimension the part, they needed values for the hyperbolic sine function to 6 or so decimal places. They ended up going to a university which had a book of tables, hand written (by some poor grad student) and hand calculated. Now you just hit the "hyp" 2nd function and "sin" and instantly have 8 decimal places.
posted by 445supermag at 3:30 PM on June 30, 2012 [2 favorites]


Yes, as already noted by ennui.bz, it was bad for the students. Speaking on behalf of the guinea pig students, it was very very bad for us.

Well, to be precise, he didn't show that the method was bad, just that it was implemented poorly. Trying to teach off a mess of photocopied pages needing constant revision is no way to teach any class. Also, failing to address the problems in the class and their affects on the students was a critical error for the Departmental and University administration. None of that says that the system would have been better or worse for the students than any other.

I tell my students flat out on the first day of class that I will never let them off the hook for their errors of judgement, but the flip side of that is that I will never let them suffer for my failures of judgement.If I royally screw up an assignment, I refigure the grades into something that is fair. Of course, a lot of Math departments have convinced themselves that failing 30-50% of their students is normal since "math is so hard" rather than a sign of failed pedagogy, so perhaps they didn't even realize what was going on.
posted by GenjiandProust at 3:38 PM on June 30, 2012 [2 favorites]


I think everyone is better off if the teacher focuses on what makes sense to them, rather than what the research might say is good for some average student.

From experience as an instructor, I can say that's an ass-backward approach. A good teacher has the ability to teach the material in whatever format the students understand best, and the perception to tell when a given methodology is or isn't working.
posted by happyroach at 3:50 PM on June 30, 2012 [1 favorite]


so, failing to address the problems in the class and their affects on the students was a critical error for the Departmental and University administration. "

I've been in a class like that too. The F***ing university should give us our money back. Did we ask? No.
posted by sneebler at 5:07 PM on June 30, 2012 [1 favorite]


"I was impressed by the article's author:

One problem asked me to calculate the width of a doorframe, given the frame’s height and a diagonal measurement of the door. After 30 seconds’ work with pen and paper, I submitted my answer: 93.7cm.

I think it would have taken me more than 30 seconds to work out the square root of 8784 with pen, paper and an understanding of the Newton-Raphson method.
"
posted by fredludd

I think he used sines and cosines, as sin theta = o/h, both given, then 1-sin to get cos theta, and cos theta (.78 say) = o/h and o is given. Possibly.
posted by marienbad at 5:20 PM on June 30, 2012 [2 favorites]


I've been in a class like that too. The F***ing university should give us our money back. Did we ask? No.

Most students lack the agency and the conviction to do this. It's frustrating, since most universities have systems to address this, but students just go along most of the time, so finding out what is going wrong is more difficult than you might think. I mean, some of this is just going to happen - there are good instructors and bad instructors. But an experimental program should be prepared to take extreme measures to make sure that the students get the course they signed up for....
posted by GenjiandProust at 5:48 PM on June 30, 2012


That's not a proof, so much as relying on the assumption that the sizes of objects don't change when you rotate them.

I didn't say it was a proof, I said it was a good answer to the question "why is this the case?" In my view there's a lot of airspace between the two. If I ask "why is the area of an isosceles right triangle half the product of the legs" and my student says "because you can put two of them together to make a square whose side length is the leg of the triangle" I'm going to say "Great job" and if another kid then pipes up to object with "but what about Banach-Tarski huh" there is going to be some serious eye-rolling on my part, you betcha.
posted by escabeche at 6:55 PM on June 30, 2012 [6 favorites]


What a poorly reasoned article - Kakaes complains about the use of technology, apparently only because the money could be better spent hiring or training better teachers ("[t]hough serious empirical research fails to show any beneficial effects of technology, it also doesn’t demonstrate any harm."). But he then complains about how the CCSM "fetishize" data analysis, which takes the focus off the "rigorous" approach to mathematics education - which has nothing to do with the quality of teachers or an improper focus on technology. In the end, Kakaes reveals himself as a simple Luddite: "The widespread use of computer technology is inimical to the exercise of intelligence. I fear this is no more than shouting into the wind, but resist it while you can...." In his "response to the response," Kakaes completely fails to address the main criticism in the responding paper, that is that he never questions "what of the traditional pencil-and-paper mathematics is worth teaching?"

At any rate, his analysis is fairly weak. He essentially complains that there might be problems with the data in studies on the effectiveness of technology, and concludes that the results of the studies must therefore be wrong. The only "data" he marshals in support of his position about the harms of technology are a few anecdotes. He apparently does not know what a systematic review or a meta-analysis is, but is fully prepared to criticize one (again, without providing any better data or study). And he takes it as established that "computational fluency is the path to conceptual understanding. There is no way around it." I am confident that many a mathematician (and maybe even a teacher or two) would disagree with that assessment - indeed, very often the causal direction is likely precisely the opposite. And there is something akin to intellectual dishonesty in his claim that the Kamii paper "argues that teaching children to add and subtract is harmful." It does not - it compares different *methods* of teaching addition and subtraction and concludes that a particular algorithmic method is less effective than the other methods used in that particular school. Regardless of the validity of the study or its findings, Kakaes's description is simply wrong.

Technology provides tools, which can be used well or poorly. Simple internet access, for example, now allows thousands of students to enjoy a bit of mathematical exploration provided by the wonderful YouTube videos of Vi Hart, which is a very, very good thing. It is not technology that has destroyed math education - math education was devoid of interest and quality long before anyone used a graphing calculator. The problem is much, much deeper than the expanding use of modern bad analogies to replace those other, outdated bad analogies we all were taught. The problem is the lack of *mathematics* in math education - which has been an issue for a rather long time. I think one of the most eloquent expressions of this problem I have read is Paul Lockhart's famous "Lament" (see also follow-up commentary ).
posted by dilettanti at 8:05 PM on June 30, 2012 [2 favorites]


Most students lack the agency and the conviction to do this. It's frustrating, since most universities have systems to address this, but students just go along most of the time, so finding out what is going wrong is more difficult than you might think.

I have actually tried to do this. Universities have administrators in place specifically to make it impossible to do this.
posted by charlie don't surf at 8:05 PM on June 30, 2012


Then it got worse. In college, I took an honors calculus class. The university sprung a surprise on us: an entirely new experimental textbook , Kiesler's Infinitesimal Calculus. It was a disaster.

And to completely contradict any conclusions you might draw from this story, my freshman honors calculus 1 section used Kiesler's Infinitesimal Calculus too. But in this case, the book was complete and the errors removed. (This was long after you took it.) The professor (Dr. Peter Loeb) knew the material and how to teach it. I don't know how our grades were compared to other classes, but all the people I knew from that section went on to do very well in math classes. I remember feeling that I had learned a secret magic which reduced the epsilons and deltas from high school into mere algebra!
posted by Harvey Kilobit at 2:46 AM on July 1, 2012 [1 favorite]


I have actually tried to do this. Universities have administrators in place specifically to make it impossible to do this.

You really need the class to organize and complain together. Or, at least, in such numbers and at enough levels that it becomes clear what is going on. I've never seen a school where the administration was specifically designed to nullify student complaints, although I have certainly seen systems that have evolved to have that effect, at least some of the time.
posted by GenjiandProust at 3:26 AM on July 1, 2012


Harvey Kilobit: (This was long after you took it.)

It's possible it wasn't really the same book. According to the wikipedia entry (and I was surprised a textbook was controversial enough to get a wikipedia entry) if you used the 2nd edition (1986) the core of the first semester was removed and demoted to an epilog.

GenjiandProust: I've never seen a school where the administration was specifically designed to nullify student complaints, although I have certainly seen systems that have evolved to have that effect, at least some of the time.

It doesn't seem like you even looked at this. I don't want to derail here, but I encountered an administrator who was responsible for arranging accommodations under the Americans with Disabilities Act, who spent all her time helping the departments determine how to avoid giving accommodations. This is standard operating procedure, it is commonplace Regulatory Capture.
posted by charlie don't surf at 8:20 AM on July 1, 2012


I'm on the calculator side.

My last important application of math was writing a rigid body physics engine for a game. My TI-83 sits next to my keyboard and it's heavily used. I also used Wolfram Alpha for some algebra and calculus. They are incredibly useful tools.

I'm of the believe that calculators simply shift the work to a more strategic level. When are you going to be doing hard-core math without a calculator (or cell phone with a calculator)? I could do a problem in my head, but I am likely to make a mistake. So I could do it on paper, and check my work. I'm much less likely to have made a mistake, but in that time, I could have run the problem 10 times on a calculator. There is no comparison.

It will be up to the teachers to adjust their curricula to account for calculators and Wolfram Alpha, and take best advantage of the tools. Many teachers are luddites, though, and it won't always happen.
posted by colinshark at 9:10 AM on July 1, 2012


It doesn't seem like you even looked at this. I don't want to derail here, but I encountered an administrator who was responsible for arranging accommodations under the Americans with Disabilities Act, who spent all her time helping the departments determine how to avoid giving accommodations. This is standard operating procedure, it is commonplace Regulatory Capture.

Did I look into your exact situation? No, of course not. I would have to have a lot more information and probably a time machine to do that.

I do, however, work closely with university administration and with Student Affairs (including the accommodations/disability office), and I can say with considerable confidence that, at all levels, we are really concerned with student success -- it does not benefit the university for students to ever fail -- repeating classes reduces retention and completion rates and takes up seats that could be occupied by incoming students. Of course, if every student passes, that suggests weak evaluations, but more students passing is generally better than fewer. Likewise, we take disability accommodation very seriously. This may be a factor of time (ie accommodation has become more serious in the past few decades); I am reasonably sure that it it's not because my institution is amazingly student focused and dedicated to doing the right thing (I mean, we try our best, but so do most institutions in my experience).

Are there asshole advisers, clueless professors, disengaged administrators? Absolutely. Do many institutions have major processes specifically designed to disempower students? Probably not. There may be systems in place that have that effect, but that's not their goal. I do think STEM disciplines have a tendency to try and hide poor pedagogy behind "this is hard," but that is more disciplinary arrogance than a specific desire to screw students.

Much the same way that whether calculators (or open books or access to the internet or whatever) is largely driven by professorial (or maybe departmental) instinct rather than a close reading of relevant literature. I used to give an exam that was "open internet," since they had to do some online searching. It was timed so that a student who needed to look up every answer would not have time to finish, and most of the students understood that well enough to deploy the tool when necessary and use other tools when they were more appropriate.
posted by GenjiandProust at 11:03 AM on July 1, 2012


Reading this just made me so thankful for the competent teachers I had at the primary and secondary level. I had plenty of teachers who weren't great, but every year of my young life with maybe one or two exceptions, I had a teacher who taught me math in ways that advanced my understanding of mathematics nontrivially, and for that I am grateful.

I remember that as a high school student I used my graphing calculator to check my answers - I'd get my numbers by hand and then actually graph the equation and follow the line on the calculator to make sure I hadn't made an arithmetic error. At the time I kind of felt like I was cheating, since we were supposed to get our answers and be confident they were right without graphing them - but looking back, that was probably the best way I could have used my calculator.
posted by town of cats at 3:53 PM on July 1, 2012


It will be up to the teachers to adjust their curricula to account for calculators and Wolfram Alpha, and take best advantage of the tools.

WolframAlpha is crazy, though. I'm not sure how to even start thinking about teaching classes to take useful advantage of WolframAlpha.

The example I encountered last semester is the following. Ask WolframAlpha "are the vectors {blah}, {blah}, {blah} linearly independent?". Not only will it tell you if they are, but if they're not, it will give you a linear dependency, and show you (if they're vectors in R^3) the subspace spanned by the set. And possibly more information, I forget.

So, suppose I want to find out if students know what it means to say three vectors are linearly independent? Or if I want to determine if they know how to find the subspace spanned by the vectors? It's ok to have access when doing HW, I guess, but still, that's way more information than I'd like students to have access to all the time.

On the other hand, you shouldn't have to demonstrate to me more than once that you can row-reduce a matrix. After that, it's just wasting everyone's time.

So I'm really conflicted. On the one hand, using technology eliminates a lot of stupid calculations that do nothing but waste time. And certainly, the engineers will use technology in the future to do stuff, so it's good for me to teach them using the technology. On the other hand, if you never do the calculations, you don't understand what's going on. And worse, I can't guarantee that all my students have access to any technology at all , much less the same technology. That's one of the reasons I was so excited about WolframAlpha; everyone's got access to a web browser, even if it's by going to the computer lab in the library.

But I don't want WolframAlpha to do my students' thinking for them.

Many teachers are luddites, though, and it won't always happen.
This really isn't helpful.
posted by leahwrenn at 4:42 PM on July 1, 2012


That's one of the reasons I was so excited about WolframAlpha; everyone's got access to a web browser, even if it's by going to the computer lab in the library

The scuttlebutt I hear from my friends working at WR is that Alpha has really tanked their profit margins, afterall, why buy the Mathematica cow when you can get the milk for free? So it will be interesting to see if it stays free.

Interesting Mathematica note, once for this diffraction class I had to calculate great circle distances between major earth cities. The book totally didn't explain this, but fortunately mathematica just does for it for you. That's really the main problem with Mathematica and its child, Alpha, just knowing that it can do to make your life easier.
posted by Chekhovian at 6:34 PM on July 1, 2012


The reason kids don't learn math well is because they don't need too.
One problem asked me to calculate the width of a doorframe, given the frame’s height and a diagonal measurement of the door. After 30 seconds’ work with pen and paper, I submitted my answer: 93.7cm.
But why would you ever need to do this in the real world? No one would ever rely on pencil/paper calculations in real engineering anymore. So what's the point? You only have a finite amount of time to teach kids math, it's better to spend that time teaching concepts that the kids might actually use.
@tylerkaraszewski: A complex number interpretation would have -1 = exp(i*pi). So -1 * -1 = exp(i*pi) * exp(i*pi) = exp(i*pi + i*pi) = exp(i*2pi). One of the properties of exp(phi) is that exp(phi + i*2pi) = exp(phi), so exp(i*2pi) = exp(0) = 1.
You can think of it visually imaging the unit circle on the imaginary plane. Multiplying by an imaginary number is the same as rotating it around zero. If you multiply by i, you rotate it 1 quarter of the circle. i * i = -1, so multiplying a number by -1 rotates it 180 degrees. If you start out in the negative direction, you rotate 180 degrees and get back to positive. (I think)

Anyway, the real reason (if you're not talking about numbers) is that it's just how the sets are defined. If you wanted, you could create an algebraic ring where a negative times a negative was a negative, while keeping the other two things the same, in which case -1 would define an ideal of that ring (just like 2 defines an ideal of the ring of natural numbers - just swap 'even' for 'negative'. I think.)
Is there a good answer to that question other than "because multiplication is commutative because we say so" that doesn't involve a proof from the Peano postulates?
Multiplication is communicative in the field of complex numbers, real numbers, etc. But there are other algebraic structures where multiplication is not communicative. An obvious example is matrices. A*B != B*A if both are matrices, but A+B == B+A.

Real numbers just happen to be a useful set of numbers for real world applications.


-----

Also, I find Wolfram|Alpha pretty annoying. It would be great if there were a standard query language, but instead it tries to be 'smart' and figure out what you're asking. But sometimes it's like -- you ask "quantity A" and it tells you, you ask "quantity B" and it tells you, but you ask "quantity A * quantity B" and it flips out and has no idea what you're asking. Sooo.... you need to copy and paste results in and then they make it difficult to actually do that for some reason. Bleh.
posted by delmoi at 2:53 AM on July 2, 2012


I have no problem with calculators used to free up time for higher order operations. Long division takes a lot of time and at some point you are spinning your wheels, but I feel a bit of mouth-vomit every time I see someone use a calculator for things that should be obvious. Such as..

* Calculating a tip. Unless you think the tip should be 16.334%, this should be a mental exercise.
* Making change.
* Calculating a percent discount or markup.
posted by dgran at 8:49 AM on July 2, 2012


But why would you ever need to do this in the real world? No one would ever rely on pencil/paper calculations in real engineering anymore. So what's the point? You only have a finite amount of time to teach kids math, it's better to spend that time teaching concepts that the kids might actually use.

Except you've said this to disparage an example of precisely the sort of computation one might do while doing, say, DIY. Or you might be a carpenter and do it professionally. Faffing about with a calculator there is certainly not worth it, at least until the numbers get ugly. How you'd end up with the height and diagonal measurement of the door and not the width, I don't know. But the point still stands.

No one really cares if engineers outsource integrals to Mathematica or whatever (that said, I had some engineering students in denial that round-off error could possibly matter). It's the people who don't know what an integral is that we're worried have traded basic skills for (oftentimes lousy) calculator usage.
posted by hoyland at 10:52 AM on July 2, 2012


in which case -1 would define an ideal of that ring

-1 is a unit, so the ideal it generates is the entire ring.
posted by escabeche at 8:25 PM on July 2, 2012 [1 favorite]


Except you've said this to disparage an example of precisely the sort of computation one might do while doing, say, DIY. Or you might be a carpenter and do it professionally.
First of all, it seems extremely unlikely that you would know the height of a door and the diagonal length but not the width. You could just measure the width directly.

But beyond that, how is it more convenient to spend 30 seconds with a pencil and paper then use a calculator? If you're in the middle of a project, you need a pencil, paper, and a flat writing surface, and if it takes 30 seconds that's 30 seconds you can't spend working - over and over again. With a calculator, you just hold it in one hand, and use it when you need it.

And again, why would a professional anything use pencil and paper arithmetic, which is error prone over a calculator? If money was on the line it would be completely irresponsible not to verify with a calculator anyway.
posted by delmoi at 8:47 PM on July 2, 2012


-1 is a unit, so the ideal it generates is the entire ring.
I'm talking about a hypothetical ring which where two negatives multiplied together stay negative, while a positive times a negative would be negative. in which case the set generated -1 would be an ideal, since that's essentially the definition of an ideal.

It would be something like 1,-1, 2, -2, 3, -3... and so on. Just switching even for negative. 1 would still be the unit, and 1+1 = -1 because 1+1 = 1*-1
posted by delmoi at 8:52 PM on July 2, 2012


As this thread has progressed, I have been having this same conversation with my coworkers, as we score standardized algebra tests for high school kids. He's a high school math teacher, he does this job during summer breaks.

A few days ago, he despaired at the obvious disinterest kids had in even attempting the test, and asked me how in the hell could he get kids to take this stuff seriously. I thought about it, and the next day I told him, people basically respond to only two things, seeking pleasure or avoiding pain. The students who find pleasure in mathematics are already motivated to study it. That leaves pain as your only tool. I said you have to make it more pain for them to not study the math, than to study it. He asked me how he was supposed to do that. I said, how the hell would I know? Both of us found pleasure in math, we have no idea what will reach these kids, but we could probably think of ways to make them miserable. I am quite certain that it is best for everybody that I am not a math teacher, because I could think of plenty of ways to make them miserable.

And there is the issue. Day after day I see miserable work from miserable math students. There are questions asking what is the angle of a corner of a specific triangle, and I see answers like 54 dollars, or 857 degrees. That is probably where you could see the most improvement, at the low end. I would just be happy if most of these kids showed any minimal sign of math ability, like doing fractions. The hell with Wolfram Alpha, these kids do not know what to do with a 4 function calculator.
posted by charlie don't surf at 9:12 PM on July 2, 2012


I'm talking about a hypothetical ring which where two negatives multiplied together stay negative

I'm not clear on what you mean by "negative." What's for sure is that (-1) x (-1) = 1. Doesn't matter what the ring is, it follows directly from the axioms. Because

(-1) x (-1) + (-1) x 1 = (-1) x 0 = 0

and

(-1) x 1 = -1

so

(-1) x (-1) + -1 = 0

and adding 1 to both sides get

(-1) x (-1) = 1.

So whatever class of elements you're calling "negative" had better include 1, which makes me feel "negative" is not a very good word for it.
posted by escabeche at 10:50 PM on July 2, 2012 [1 favorite]


Delmoi, not every ring structure allows a splitting into {positives} ∪ {0} ∪ {additive inverses of positives}. And, as escabeche details the consequences of, -1 is not an arbitrary element; it's the additive inverse of the multiplicative identity, which puts serious limits on its behaviour.

(The thing I like about algebra is seeing how few assumptions you can make and still force strict consequences.)
posted by benito.strauss at 11:17 PM on July 2, 2012


I'm not clear on what you mean by "negative." What's for sure is that (-1) x (-1) = 1. Doesn't matter what the ring is, it follows directly from the axioms. Because
Well, the point I was trying to make is that the labels are kind of arbitrary. You can go from set theory to the natural numbers, and then extend the natural numbers to the set of integers. Looking at how it's constructed the integer -1 is 'composed' of the set of all pairs of natural numbers (a,b) such that a - b = -1. So (0,1), (1,2), (2,3) and so on. (of course -1 isn't the set of natural numbers so the construction actually uses an equivalence class such that (a,b) ~ (c,d) if a+d = b + c to define each negative number

So negative numbers can be expressed in the form (k+n,k) for positive number n, and (k, k+n) for -n (where k represents 'every possible natural number). Then you can come up with a multiplication rule that operates on those expressions and gets you what you expect where negative * negative = positive.

But you could create a different ring, give the elements different labels, and get a different result. If you wanted you could call them a, -a, b, -b, c, -c and so on.
I'm not clear on what you mean by "negative."
In the hypothetical ring I'm talking about -1 * -1 = -2 and you could say it was "twice as negative"

As I said in the earlier comment, just imagine the set of natural numbers but switch 'even' with 'negative' i.e: where each even number n was switched with -(n/2) and each odd number were switched with ceil(n/2), (in that case -1 * -1 = -2 because -(2/2) + -(2/2) = -(4/2))

This ring would probably not be useful for anything, I'm only trying to point out how these things work based on the way they're defined. You can define different algebraic structures where different things happen.
posted by delmoi at 11:55 PM on July 2, 2012


But "-1" is not "just a label". It is the additive inverse of the multiplicative identity. And escabeche showed that -1 × -1 equals 1, and he only used rules from the definition of a ring.

If you don't need your structure to be a ring then you can go ahead, but "Ring" implies "-1 × -1 = 1".
posted by benito.strauss at 12:10 AM on July 3, 2012


delmoi, what you're doing is taking the usual integers and giving them new names. Now you're welcome to do this -- the number I'm used to calling "2" you can call "frog" or "Barry Bonds" or "your highness" or what have you. You seem to want to call it "-1" which I would say is a bad choice, because there's already a different number by that name; namely, the additive inverse of 1. YOUR -1 is not the same as this one -- for you, 1 + (-1) = 2.

I would say that the number you call -1 is not negative, except in the very modest sense that you've chosen to express it as a string of characters starting with a "-".

I think your example is analogous to the following: if there were a language in which the word for the number "three" were pronounced as a string of phonemes which matched the English "negative five," but all other number names were pronounced as in English, you could say "in that language, a negative times a negative can be a positive, because "negative five times negative five is nine" is a phonetic string in their language that asserts a true mathematical statement.

Or if I said "the sum of two integers is an integer" and you said, "not necessarily, because my definition of the word "integer" is that an integer is a cream soda, and cream sodas can't be added at all."
posted by escabeche at 5:08 AM on July 3, 2012 [2 favorites]


delmoi, what you're doing is taking the usual integers and giving them new names.
No, just the natural numbers.
I would say that the number you call -1 is not negative, except in the very modest sense that you've chosen to express it as a string of characters starting with a "-".
Right... the point I was making is that you can do that, but then the numbers would behave differently then you would expect. Someone was asking why -1 * -1 = 1, and rather then just saying "because that's how it is" you actually can change it, do something else, and see what would happen. In this particular example you get weird results like -1 + 1 = 2 and so on. It wouldn't be useful for everyday mathematics. But ultimately the integers are just one particular algebraic structure that was chosen from the various ones that can be constructed from natural numbers, which in turn can be derived from set theory.
posted by delmoi at 3:29 PM on July 4, 2012


(Er, actually that probably doesn't meet the strict definition of a ring, maybe if you made it finite so you could get back to zero.)
posted by delmoi at 3:37 PM on July 4, 2012


> (Er, actually that probably doesn't meet the strict definition of a ring, maybe if you made it finite so you could get back to zero.)

"Ring" implies "-1 × -1 = 1".

/I think I'm in a killfile somewhere.

For what it's worth, finite makes it very difficult to put an order on a ring, at least one that is compatible with the ring structure. Finiteness implies that adding one repeatedly eventually gets you back to zero, so you would have:

0 < 1 < 1+1 < 1+1+1 < 1+1+1+1 < ...... eventually ..... < 1+1+.......+ 1 = 0

which is a contradiction.
posted by benito.strauss at 4:27 PM on July 4, 2012


(Er, actually that probably doesn't meet the strict definition of a ring, maybe if you made it finite so you could get back to zero.)

Yeah... as far as I can tell you're circling around a definition of a finite field (having characteristic other than zero) and claiming you're doing something else.
posted by hoyland at 11:06 AM on July 6, 2012


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