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[LordBucket]

You were explicitly attacking the basic method of solving such problems in favour of shortcut methods.

I disagree with your characterization. I would have said that I was disapproving of "cookie cutter one method for all cases" solutions in favor of understanding the various tools, and choosing the one most appropriate for the task at hand. Yes, when going from point A to B it's useful to know how to go to google maps and print out an satellite map of your trip complete with step by step instructions of every turn along the way. But it's silly to go to all that effort if point A is your front door and point B is your neighbor's.

[da_nang]

You were explicitly attacking the basic method of solving such problems in favour of shortcut methods.

I disagree with your characterization. I would have said that I was disapproving of "cookie cutter one method for all cases" solutions in favor of understanding the various tools, and choosing the one most appropriate for the task at hand. Yes, when going from point A to B it's useful to know how to go to google maps and print out an satellite map of your trip complete with step by step instructions of every turn along the way. But it's silly to go to all that effort if point A is your front door and point B is your neighbor's.

Ok, now I feel like I need to pop my head into this. On the matter of exactness vs approximation:

Both are useful and both are needed by each other. For someone who is in the engineering field, this is quite apparent. Exactness is what builds the theorems, the building blocks. However, the tools and the work come from the approximations. Even your calculators are approximate since they work with floats. If your calculator has π in it, multiply it by say a million or a thousand, remove the integer parts and repeat. Eventually, you'll end up with a bunch zeroes possibly preceded by some other digits. Graphing calculators are also approximate. Finding that zero of a function? Newton's method or secant method. Integrating a function? Trapezoidal rule or Simpson's rule possibly followed by Richardson "extrapolation".

Are these approximate answers less correct? Maybe.

Why is exactness useful? Because it removes any shroud of doubt, it enables wondrous mathematical constructions not possible in the real world. Why is approximation useful? Because it gives us a larger picture of the problem and usually increases efficiency.

Are these approximate answers less correct? Maybe.

Why are both needed by each other?
What can approximation give exactness? Intuition, guidance, coherence and correspondence in the context of truth theory. Dividing two large numbers? An approximate answers gives you an idea of where the exact number should lie.
What can exactness give approximations? Reliability, efficiency and new ways of thinking. Without exactness, you can never know how precise and accurate the approximate answer is. The approximate answers will contain errors. An approximate answer without an error threshold, even if it's implicit due to simplicity of the problem, is useless. Every calculation involving approximations propagates error. Read up on propagation of errors. Loss of significant digits, divisions of numbers of different orders of scale, all these creates errors. But how do you measure the error? With theorems built up with exactness.

Are these approximate answers less correct? Maybe.

[LordBucket]

Ok, now I feel like I need to pop my head into this. On the matter of exactness vs approximation:

Who were you intending to respond to? You quoted me...but at no point have I disputed the usefulness of exactness. The methods discussed in the opening post, as well as my second post were all precise methods. Approximation only even came up in response to a challenge by palsch on page 5 that he already admitted was deliberately made to be difficult to work with, and in his response he seemed to take no particular issue with it.

This precision vs approximation thing came up on the following page, with this exchange between completely other people:

We really shouldn't teach math in schools to such an exact level unless people are planning to go into math. Approximation is an important skill in life

I detest approximation; it just bothers me.

...so...I realize that in lengthy thread like this one it's sometimes easy to read posts by several different people and mentally blur them together. Yes, I gave an example that involved rounding off numbers, but I'm not suggesting that we stop teaching precision.

I'm basically just trying to say that it's good to understand things, and that when you do understand how something works, it's not always necessary to go through all the complications that somebody who doesn't understand might have to  in order to get the same result. If you really understand multiplication and division, it's not necessary to write out the same full formal 5 line process that might be appropriate for a complicated problem if you're only doing  a simple problem. If you're driving to New York, yes by all means print out a google map. But if all you're doing in walking next door it's silly to go to all that effort.

And finally I'm making the observation that the methods that are apparently being used to teach math very often don't result in this sort of understanding. Rather, they result in people who don't understand and therefore apparently need to go through all sorts of silly and unnecessary complicated stuff to do very simple things...because all they know to do is follow a generic recipe.

And I think that's not so good.

[da_nang]

...so...I realize that in lengthy thread like this one it's sometimes easy to read posts by several different people and mentally blur them together.

...That is what happens when I hastily try to write a post in the morning.

And finally I'm making the observation that the methods that are apparently being used to teach math very often don't result in this sort of understanding. Rather, they result in people who don't understand and therefore apparently need to go through all sorts of silly and unnecessary complicated stuff to do very simple things...because all they know to do is follow a generic recipe.

And I think that's not so good.

This is about the US math education system, right? Since YMMV, I can't really comment on it as I was not taught there. What I can say after having read Lockhart's paper, however, is that at some point you need to start teaching mathematical formalism. By all means, you can write proofs elegantly or inelegantly, all that matters is that the assumptions and arguments are true. However, it must be written in a way that prevents confusion. Natural languages have this problem due to semantics. That is why we have formalism. This doesn't exclude the presence of natural languages. Sure enough, you can fill the gaps with whatever amount of fluff you want as long as the actual proof remains formal. A purely formal proof just happen to be concise, to the point and easily understood by one's peers and can stand the test of time regardless of changes in the natural languages.

When doing calculations, if you're the one who's only ever going to read it then by all means leave out obvious steps. I do that all the time. If you're showing it off to someone else, however, some of those steps may need to be written out to avoid confusion.
I mean if I'm solving the integral of cos(x)² at university for an assignment, this is an answer I can give:

∫cos(x)²dx = ½∫[2cos(x)² - 1 + 1]dx = ½∫[cos(2x)+1]dx = (1/4)∫2cos(2x)dx+x/2 = x/2 + sin(2x)/4 + C = ½(x+sin(x)cos(x)) + C

I do not need to write out all the identities, they're implicitly understood at this level of education. At best I might point out that e.g. 2cos(x)² - 1 happens to be cos(2x). Sometimes I might even get away with leaving out the third step.

If I was writing a proof on the other hand, then the standards would obviously be higher especially if it's for proving a thesis or theorem.

Now I don't know much about the US education system, but I'm pretty sure High School is a secondary education level. At that point, the students are at least 14-15 years old, they've developed the basic mathematical knowledge. If they're not supposed to learn mathematical formalism here, then boy will they have a surprise waiting for them in tertiary education levels.

[palsch]

Yes, when going from point A to B it's useful to know how to go to google maps and print out an satellite map of your trip complete with step by step instructions of every turn along the way. But it's silly to go to all that effort if point A is your front door and point B is your neighbor's.

Except that when you are teaching people to navigate from A to B as a learning exercise, you kinda want them to go through all the steps.

I'll agree that teaching using trivial examples can be a problem, which is why I bothered creating a non-trivial example. Going back to my original post;

Learning the most general methods for classes of problems is the only thing that makes sense, even when shortcuts exist. Those shortcuts usually only apply to a subset of problems, and relying on them too much is setting you up for failure when they aren't around. I'd almost say that teaching the problems where the shortcuts exist is worse than just diving into the deep end, because it creates the illusion that the full methods are overly complex and torturous for those students who recognise the quicker methods.

As it is, I'd let people get away with using the quicker methods when available, but it does hurt learning the methods that most need practice and drilling to actually fix in your mind. There are whole classes of problem I'm sketchy about solving because I didn't run through enough example problems that didn't have neat and cute solutions that let me skip the whole handle turning exercise. The result is I'm not confident enough to turn that handle without constant outside reference.

As for trying to teach people to recognise and take the shortcuts, sure. Except I'm not convinced there is a universal and reliable method for teaching that. So many such shortcuts and tricks involve having well trained intuition at applying more basic mathematical techniques and pattern recognition. Those are things that require drilling and a desire (or at least a willingness) to learn and explore at an early stage. Expecting students who didn't have that to learn those methods is likely futile, especially in any classroom or mandated lesson situation (yes, even one-on-one).



As far as the whole approximation thing, physicist, remember? Pi = 3.

Quick approximation methods and Fermi calculations are my bread and butter. In day to day life I'll use them constantly. But the level of mathematical (and physical) formalism I have had to internalise in order to use them that reliably is significant. And in any serious attempt to solve an actual problem they serve only as a sanity check and calibration to make sure my actual calculations are in the right ballpark.

[Vector]

I think I should clarify about my availability for math questioning, since there's been a few instances of confusion.  Please, no one feel bad over this, okay?  I don't think I was clear enough.

I am more than happy to help out with a few small points here and there and point people towards resources.  However, I am not willing to write a complete course for you or just generically "teach math" unless you are willing to send me the $6,000-$10,000 that this is worth.  I'm not going to explain the entirety of differential calculus when there are many, many guides online.  That wouldn't make sense, would it?  I'm off doing stuff like getting myself shocked with electricity to the tune of $20 per hour to pay for schooling; I really don't have the ability to take non-paying students.

Google is free, and it doesn't get you in debt with Jabba the Hutt.

Seriously, folks, you should ask me for resources!  I'm a second-semester senior at Berkeley and now know all the stuff that I wish I'd known.  I know all the best undergrad textbooks, I know which sorts of professors to look out for and the different "schools" of education from various countries, I know how to solve a lot of cool problems and the intricacies of many methods.  I know the arcane social rules of the math department inside and out.  I know the best ways to learn various subjects, so if you've got a problem like "I can't understand calculus for X, Y, and Z reasons--which resources should I use?" I'll be more than happy to help.  Please ask me specific questions about the fine points of any of these things!  I honestly look forward to it.

But let's use our resources well, all right?  If you don't understand Euclidean geometry but are motivated in that direction, you're much better off finding someone at your own level to work things through with, and then asking someone who knows the subject well when you're stuck at a particular point.  The more heavy lifting you do on your own, the stronger you'll be.  So let's all concentrate on increasing our powers, and building the buds on our math-plants.

[Skyrunner]

Incidentally, I found that math is pretty darned fun outside of a highschool context :/

[Remuthra]

Lockhart's Lament was quite inspirational to me. It made my math classes much more interesting. All you have to do for a good laugh is listen to the teacher telling someone how horrible they must be at math because they don't remember what an asymptote is.
In collusion with the essay, I began working on my own number system, which arbitrarily used fractions as whole numbers and switched the uses of Greek and English letters.

[Leatra]

If somebody asks you to divide two by one half, it's a lot faster and easier to simply multiply two by two than it is to convert two into a fraction, give separate treatment to numerator and denominator, then convert the result back into an integer. There's no reason to go to all that effort.

When I was in high school, I occasionally managed to find a shortcut to the answer of a math problem in a way like this. Whenever I showed it to my teacher, I got responses like "okay, but that solution isn't going to work every time" or "but I want you to find the answer by using the technique I just showed". My teacher even once said "you made a mistake and reached the answer coincidentally" Yeah, because randomy finding the answer out of the infinite world of numbers is sooo possible.

Pretty much all the time maths made no sense for me. One time I asked my teacher "what is f(x) and what does it represent?" The answer I got was pretty much like "it just presents a value, like a number" without any further explanation. If f(x) represents, let's say, the number 7, then why don't call it 7? So I decided to ignore my curiosity and just memorize every shit maths threw at me. As you can guess, it didn't work. Everything seemed incredibly abstract. Mathematics seemed so abstract, I felt like I was learning a new language. No wait, that doesn't even explain it. I felt like I was learning the language and culture of a life form that lives like 3 or 5 galaxies away. It was an unfamiliar concept.

I'm glad I stopped trying to be a computer programmer, really.

[Putnam]

f(x) doesn't represent a value, it represents a function. Now, f(1/7) for the function y=1/x would represent 7, as would f(pi) for the function y=e^(ix)+8, but that's neither here nor there.

[Leatra]

Exactly.

That reminds me, I had a maths teacher who was actually an actor. He was a funny guy but he didn't really teach us anything.

I think he defined f(x) as a function too and when I asked what a "function" is in maths, that was the answer I got. I don't remember. I was in 9. grade.

[Vector]

Eh. . . a function f is a pair of values from the product X x Y (consisting of all ordered pairs (x, y) such that x is in X and y is in Y), with every (x, y) and (x, y') in f having the property that y = y'.  Sometimes these functions can be determined by an equation, but most of the functions out there cannot.

[LordBucket]

When I was in high school, I occasionally managed to find a shortcut to the answer of a math problem in a way like this. Whenever I showed it to my teacher, I got responses like "okay, but that solution isn't going to work every time" or "but I want you to find the answer by using the technique I just showed". My teacher even once said "you made a mistake and reached the answer coincidentally" Yeah, because randomy finding the answer out of the infinite world of numbers is sooo possible.

Very likely he was a Person Who Does Not Understand Math and didn't understand why what you were doing worked.

Anyway, I find the "that method won't work every time" argument to be pretty much silly. It would be like someone telling you that it's a bad a idea to drive to the grocery store without looking it up up google maps. Because "pull out your driveway, turn left, turn right, turn left" isn't a method that will work to get you to every place. And that's true. But if those directions get you to the grocery store, and you're going to the grocery store, and you know this...there's nothing wrong with doing that rather than using the method that "works in every case" of printing out a map and directions.

I'm glad I stopped trying to be a computer programmer, really.

In my case, I learned math by doing programming. That occasionally created friction between me and others in class. For example, I remember once in particular being partnered with someone for an assignment, and I wrote out something like:

a = length * width

My partner look at that in complete confusion and complained that it was "supposed to be x" and "what was a?" Then she looked at the rest and wanted to know what all those other variables were. I thought it was pretty obvious, but she'd apparently been so conditioned to see things a certain way that she looked at "length * width" and interpreted it as (l * e * n * g * t * h) * (w * i * d * t * h).

[Loud Whispers]

a = length * width

My partner look at that in complete confusion and complained that it was "supposed to be x" and "what was a?" Then she looked at the rest and wanted to know what all those other variables were. I thought it was pretty obvious, but she'd apparently been so conditioned to see things a certain way that she looked at "length * width" and interpreted it as (l * e * n * g * t * h) * (w * i * d * t * h).

Wait, she deconstructed it into the individual letters?

...

Strange.

[LordBucket]

Eh. . . a function f is a pair of values from the product X x Y (consisting of all ordered pairs (x, y) such that x is in X and y is in Y), with every (x, y) and (x, y') in f having the property that y = y'.  Sometimes these functions can be determined by an equation, but most of the functions out there cannot.

...see...this is why people have a difficult time understanding math.

A function is a magic box with two holes. When you put something in one hole, the box does magic on it and gives you something out the other hole.

"What magic does the box do to stuff I put inside it?"

Whatever. Doesn't really matter. For example, you might have a magic box that squares whatever you put into it. So if you put in 1 you get 1. If you put in 2 you get 4. If you put in 3 you get 9. Etc. Or you might have a magic box that adds one to whatever you put in. If you put in 1 you get 2. Etc.

Basically, a function is a formal way of saying "do stuff."

Wait, she deconstructed it into the individual letters?

Yep. And asked me to do it "the way we were supposed to" because writing out length times width made it "confusing."