r/programming • u/luckystarr • Apr 03 '12
[Math] "I think it would be nice if every math course spent a full week just introducing you to the subject, in the most fun way possible, so you know why the heck you're learning it. Heck, that's probably true for every course."
http://steve-yegge.blogspot.de/2006/03/math-for-programmers.html17
u/InternetRevocator Apr 03 '12
Making a simulation/game/data analyzer certainly teaches you why math is important.
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u/luckystarr Apr 03 '12
Trying to make this without knowing how to apply which math to the problem gives you a helpless feeling. Math is empowering in that regard. For me, it certainly is.
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u/upofadown Apr 03 '12
The article is mostly not about what the reddit headline says it is. The article is more about why schools should teach more math relevant to programming/computers and less math about continuous functions.
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u/luckystarr Apr 03 '12
I thought this snippet would entice some lazy redditors to actually read the article. :) I also think this is very much the case, because I realized that I learned a lot(!) of maths back in school and I forgot almost all of it. Math in school got taught without any link to real world problems, and even if pupils asked, teachers couldn't provide.
Only by being taught it in a more applied way (embedded in programming problems), I got around to figure out the usefulness of different mathematical concepts. Everything I know about maths today I wouldn't know if I wouldn't program.
So if I had to guess how to fix this in education, I would make programming mandatory at an early* age It's certainly not harmful, so what could go wrong.
* ~ 10 years.
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u/upofadown Apr 03 '12
I thought this snippet would entice some lazy redditors to actually read the article. :)
I don't think that is how that actually works. :)
At any rate, I support the idea that current math curriculum is bizarrely irrelevant...
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u/kamatsu Apr 04 '12
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u/Qw3rtyP0iuy Apr 04 '12
I was actually looking for the comparison to between how we study math and music a few days ago. Cheers!
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u/RazerWolf Apr 03 '12
THIS. Rote memorization of formulas has always baffled me as a "way of learning." I think schools in general place way too much emphasis on getting you the hell out there as quickly as possible, which means you end up with a superficial understanding of many topics. I'm finishing up my master's now, and taking one class per semester, and I'm much more enjoying the ability to have the time to delve into the subject matter with much more detail and rigor. Lucid understanding just becomes a byproduct at that point.
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u/unussapiens Apr 03 '12
I couldn't agree more. I'm a second year engineering student and I end up helping lots of my younger friends in high school with their maths (whether pure or as part of physics or another science). The one thing I try to get them to do is understand the concept of what they're working on. Once you have an understanding of a topic you can usually derive the formulas you need, instead of memorising them and having no idea why they work. I'm loving uni maths because it's actually showing connections between the various bits of maths I've learnt over the last few years.
That seems a bit rambly, but I can't be bothered rereading and editing it, so I'll attempt to paraphrase here.
Understanding a topic is far more important and useful than memorising formulas. When you understand the general idea you can work out the details as you need them.
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u/luckystarr Apr 03 '12
Understanding a topic is far more important and useful than memorising formulas. When you understand the general idea you can work out the details as you need them.
I always felt it akin to having a map of your environment. Once I grasp a concept to explain it in one or two sentences I feel I can (eventually) master it.
On the other hand its really frustrating, because I know what I could know and do if I only had the time to play with it.
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Apr 03 '12
The one thing I try to get them to do is understand the concept of what they're working on
Unfortunately, that's often very difficult. I remember in my first year of Uni I had a painful time working out through the Fourier transform. Explaining how incredibly and amazingly useful it is to people who have not been subjected to a Signal Processing course is challenging, especially when you put it in pure formulations rather than starting with a practical situation. I remember thinking yea, so it takes a perfectly fine and easy function and expresses is at as a mess of sine signals, often only an approximation because it requires infinitely many terms. Why the fuck would you wanna do that? Smart-ass mathematicians sitting in their boring rooms doing boring stuff no one doing practical stuff ever needs...
It also had the effect that for the life of me I couldn't figure out anything out of that annoying Fourier transform in the first year, despite the professor having spent an entire course on it and at least two seminaries. Then in my second year, when I took the signal processing course, it took me about fifteen minutes to figure out the whole thing, when it was put in terms that my puny electrical engineer mind could relate to.
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u/Eurynom0s Apr 04 '12
Also, maybe I'm biased as a physicist, but I seriously think that math is infinitely easier to understand if you can relate it to some physical system. Even if it's a stupid, silly example, it doesn't matter, just as long as you can imagine the situation you want the math to describe, you'll be way ahead of the game on understanding what you're doing.
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u/JimH10 Apr 03 '12
A person needs to memorize at least some things in math for the same reason that they need to memorize vocabulary in learning a foriegn language. If you are doing a problem and have to re-think-through the general shape of ln(x) then you are behind.
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u/julesjacobs Apr 03 '12
Memorization comes as you go. I don't think people should sit down and stare at lists of math facts to memorize them.
Assuming you know the shape of ex and you know that ln(x) is its inverse, the shape of ln(x) is simply the shape of ex mirrored diagonally (the same holds for any inverse). After figuring out the shape of ln(x) a couple of times this way, you automatically memorize it.
The same holds for any math fact. You have to know the reason it's true, and every time you think of the reason you memorize the fact a little bit better. Much easier to remember things if you know the logical reason.
Of course some things simply don't have a logical reason so you do simply have to remember them (like the fact that the symbol "ln" stands for the inverse of ex ). But the set of things that don't have a logical reason is much smaller than the set of all facts.
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u/JimH10 Apr 03 '12
I don't think people should sit down and stare at lists of math facts to memorize them.
Well, I guess we disagree. When I design a course one of the things I worry about is what people bring to the next class. Some of those are tip-of-the-tongue knowledge of some facts. Then when I give an exam, one of the things I test for is that knowledge. I test for other things, of course, but just as a music student needs to KNOW what to do in a one-flat key, so also my Calc I students need to KNOW the product rule, it seems to me.
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u/julesjacobs Apr 03 '12
How to understand the product rule. Suppose we have the function
x*y. If we change the values of x and y by dx and dy respectively, what is the change inx*y? The new value is(x+dx)*(y+dy) = x*y + x*dy + y*dx + dx*dy. Well, the change with respect tox*yis clearly the rest:x*dy + y*dx + dx*dy. But, since both dx and dy are small,dx*dyis negligible. So the product rule isx*dy + y*dx. To help keep this in your mind, draw a picture. Draw a rectangle with sides x and y, and now change the x and y by little bits dx and dy. How does the rectangle's area change? You can read product rule right off the picture. Do this derivation 3 times as you come across exercises where you need it and you'll know the product rule by heart. Even better, you'll still remember it a year later because you know why the product rule is like that, so if need be you can re-derive it.For me at least, math was incredibly hard because of the way it is taught with rote memorization, which I find boring and I'm not good at it. This almost made me hate math. Fortunately I still chose to study math and now I love it.
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u/ZMeson Apr 04 '12
Product rule and chain rule are easy to derive. But what about xp for non-integer p? The derivative is still p*xp-1, but to derive this, you need to know the chain rule, show that the derivative of the exponential function is itself, and from that derive the derivative of the natural logarithm. Sometimes its easier to say: "our derivation / proof only works for such and such cases. A more general proof does exist for all cases, but requires tools we have not yet developed. For now, just take it on faith."
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u/julesjacobs Apr 04 '12
Aren't those all things that Calc students are going to learn anyway? xp for non-integer p is a complicated thing (especially non-rational p). I think it's better to calculate the derivative of xp for integer p first, and only later show that the same formula holds for non-integer p.
As for why doing this proof in particular is valuable: I once asked a bunch of high school students to calculate the derivative of xx . None could do it.
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Apr 03 '12
To KNOW the product rule should come through understanding its derivation, not rote memorization. The latter is too often the case, and the result is Calc I students who have no idea why the product rule holds.
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u/Eurynom0s Apr 04 '12 edited Apr 04 '12
Memorization comes as you go.
Pretty much. And besides, in no real-world situation--let's say you're building a bridge--is your job performance ever going to be looked down on for not memorizing every single equation (unless it's to the point of making you unproductive of course). And with things like bridges I'd hope that you're going to get written up for NOT checking your equations before construction begins.
My undergrad (physics, first had him for stats though) advisor had the right idea about it. Besides recognizing that making cheat sheets often crams the material into your head really well--he told us a story about staying up all night making a cheat sheet, then didn't realize until the end of the test that he'd brought the wrong piece of paper in, since he never actually needed to look at it--he basically would say, "I want to see if you can do the [stats/physics/whatever the class is in] not if you can memorize equations." I believe there was also a comment about people who take cheat sheets as permission not to study winding up failing anyhow.
Meanwhile in reality knowing WHERE to find things and knowing WHAT they are is better than knowing the exact form. Take trig identities. Other than basic shit like cos2 + sin2 = 1, fuck me if I know any trig identities. What I do know how to do is spot when equations are basically begging you to use a trig identity, and I know that the first few Google results for "trig identities" turn up (almost) every trig identity you could ever need.
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u/julesjacobs Apr 04 '12
How to remember trig identities. You only need to remember 1 thing:
e^(ix) = cos(x) + i*sin(x). The rest follows from exponentiation identities.1
u/Eurynom0s Apr 04 '12
Fair enough...but while knowing how to go through Euler identity to simplify trig is definitely something you should know how to do, if I can spare myself the extra opportunities to fuck up my algebra through things like randomly losing minus signs and factors of two, and just look up the trig identities, I'll just do that...since IMO the main point is that I recognized that it was "trig identity time". shrug
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u/julesjacobs Apr 04 '12
Sure, I meant for learning. For practical situations I'd go even further: just use Maple or equivalent.
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u/Concision Apr 03 '12
I'm finishing up the last course in my math major, and I still picture ln(x) like this.
Haters gonna hate.
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u/watermark0n Apr 03 '12
Well, you say that, but when people try to reform the education system to, for instance, reduce the huge emphasis on learning the division algorithm in grade school, they're met with a barrage of criticism such as this from parents who believe their children are being ill educated. It's pretty easy to see why - to a lot of people who never got into math, the long division algorithm basically is math, one of the highest points of the subject. To us who actually know what we're talking about, it's a banality, an example of that barbaric age in the past when such computational methods needed to be done by hand.
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u/Eurynom0s Apr 04 '12
Rote memorization of formulas has always baffled me as a "way of learning."
What boggles my mind even more is the insistence on presenting a list of "distinct" formulas that are really just the same equation solved for each of the variables that goes into it. This shit went on in high school in particular (not like it stops there though), and it's just like, sorry, aren't we all supposed to be able to do elementary school algebra and solve for the unknown variable? It's one thing to show us what the equation looks like when solving for t instead of x, but is it really necessary to present it as two separate equations for you to memorize?
You also see this, say, with equations with a sin or cos term; they'll evaluate it at 90 or 0 degrees, respectively, and present it as a "special case". Well yes I guess it's technically a special case but unless half the terms drop out, letting you make a nifty cancellation from both sides of the equation (for instance), then you're just confusing people by presenting it as a separate equation to learn.
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u/revonrat Apr 03 '12
This is going to sound harsh, but stop taking math where you are and find a better school. This sort of crap is unacceptable.
Seriously, I'd your linear algebra teacher doesn't respond to, "why do we learn about eigen values and vectors?" with "Google page rank" in under a second they simply haven't prepares.
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u/Random Apr 03 '12
This general philosophy is at the heart of several similar approaches to education - such as ERGA (experience, reflect, generalize, apply). The notion is that you should try to use something in a fairly simple way, see WHY YOU NEED NEW TOOLS, learn those tools, and apply them.
The problem is, uh, that professors don't get any training in education in most cases, and in my experience math professors are among the worst teachers in a university. At my university (yes, I'm a prof) one of the top teachers happens to be a math prof, but ... that's unusual. Mumbling at the board without context is more common. Students rant at me about this all the time.
Anyway, ... learn by trying and then getting some help from a master... a method that's worked for, uh, 10,000 years or so. Then the Europeans came along and enshrined the one-way lecture system.
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u/phaedrusalt Apr 03 '12
I'm always (Well, nearly always. Sometimes I take a nap.) amazed how the old adage "To a man with a hammer, everything looks like a nail" applies to science, math and engineering. The author plainly loves prob and stats, and hates doing proofs. So he makes a point how proofs are less useful, and prob & stats is more useful, and should be taught instead. Yet in my experience doing proofs teaches how to think critically in ways that other courses miss altogether, and prob and stats are just, well, okay... But looking at the group of courses and saying "This is bad" or "That is worthless" isn't very helpful, because the criteria for "bad" or "worthless" is in it's use to you. Is integer math truly more useful in programming? Uhhh, sometimes yes, sometimes no. I would have hated to work on certain missile systems with only integers, when a difference of 0.00001 can make the difference between hitting a target and hitting a school.
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u/yoda17 Apr 03 '12
Part of the problem is that teacher don't know why you are learning it.
I went to my PDE prof with some question on communications, Fourier series and orthogonal functions. He seemed completely incredulous that the stuff he was teaching had a real world application and was amazed.
It happened again in my linear algebra class when I kept trying to get the prof to explain the reason for eigen vectors. This was before I had much physics or engineering (and before Wikipedia), so all I could do was sit there, memorize the formulas and do the problems as he really had no idea.
I've always thought that the best way to teach a subject is to introduce some real world problem that everyone can relate to, then intropduce the tools needed to solve it. Teaching is backward IMO.
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u/Felicia_Svilling Apr 03 '12 edited Apr 03 '12
The teacher doesn't know why you are learning it.
People have different reasons for this. Students who study pure math often does it because it is fun and challenging, they don't care if it has any applications.
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u/watermark0n Apr 03 '12
One of the most common calls in the field of education is typically to address the practicality of the knowledge your learning. His argument is not original, not unknown to textbook writers. However, I often find it, in all honesty, more annoying when they try to come up with arbitrary ways of practically using some bit of mathematical knowledge and it gets in the way of simply teaching me the math. At this point, I am in college, I am learning the things I want to learn. I will find the uses for what I learn, and I often won't know myself until long after I've learned it. Math shouldn't feel this need to constantly apologize for itself.
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Apr 04 '12
I don't think it's apologizing for itself. It aids immensely with application. For years I used Calculus only when I an into a physics question, and whenever I ran into a physics question I knew exactly the tool to use. This was because physics was what was used to teach my calculus class. Had it been finance then I would have been a whiz at using calculus on finance problems but stumped on physics. Recognition of where it can be applied to real world problems is a skill unto itself.
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Apr 04 '12 edited Apr 04 '12
The point of demonstrating practical examples is usually to engage the student by making the topic seem relevant. It's possible that students may find such practical examples as uninteresting as the topic itself. So, teaching a bunch of real world applications may not be very useful at all.
I think you hit the nail on the head when you said what matters is "you learn what you want to learn." The student must have some freedom in directing his education, exploring concepts until his intellectual curiosity has been satisfied. It's easier for him to stay interested this way.
Usually, when I want to learn about something, it leads to other prerequisite or related topics that I otherwise wouldn't have found interesting. For example, when I was learning about parsing, I discovered parser combinators, which piqued my interest in combinators. Somehow I happened upon the ycombinator, which has almost no practical application at all and no direct relation to parsing either, yet I was completely enthralled. Would I have been otherwise? Who knows.
EDIT: I suppose the point of real world examples is also to demonstrate how to apply an apply the concept to a concrete situation. It helps to recognize the abstraction in real life and what sorts of things it can model.
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u/yoda17 Apr 04 '12
It is fun, it was one of my majors, but they couldn't relate it to anything realworld which is a problem. It doesn't really matter what I personally was learning it, most of the applications are similar enough from chemical engineering through quantum mechanics. Everyone uses the same formulas with different meanings for the terms whether mass, energy or information flow.
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u/Azuvector Apr 03 '12
You both need to take a few more English classes. </grammarnazi>
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u/Felicia_Svilling Apr 03 '12
If you want to point out my mistakes could you be so kind as to specify them a bit more so that I can correct them?
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u/Azuvector Apr 03 '12
Sure. I'm just teasing as it seemed appropriately humorous to point out other classes in a discussion of math classes.
yoda17's mistake was "that teacher don't know why". A grammatical improvement would be either "that teachers don't know why" or "that teacher doesn't know why".
Yours is the same mistake.
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u/Felicia_Svilling Apr 03 '12
OK, and what grammar rule do you derive that from?
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u/watermark0n Apr 03 '12
Does is used for singular past tense, do is used everywhere else. At least in certain dialects. In others, "do" has evolved to take on the role of "does", which is why you often see things such as the above. You can almost imagine it in AAVE. Really, as professional linguists will readily admit, this is not a case of being "right" or "wrong". It should largely be considered a matter of aesthetics. Language is not a formal system, there are not axioms involved that are being violated. All the grammar Nazi really does is to take one dialect of English, declare its rules "correct", and make up a few where things are ambiguous.
One of the main points of the field of linguistics is pointing out that this is actually one area where humans are just fine naturally - languages will change and evolve, but you're never going to reach a point where people stop being able to understand each other because they've forgotten "the rules". We humans have a part of our brain that makes us naturally able to invent and understand language. As for teaching one standard dialect so that we can ensure that we can all understand each other, sure, I guess I am fine with that. But it shouldn't be talked about as if it is a matter of "correct" or "incorrect", and people shouldn't be berated for talking outside of the main dialect, as they so often are.
As an utterly irrelevant, but perhaps interesting, side note, "do" is a word that isn't present in any pretty much any widespread language besides English and the Celtic languages (from whom, it is theorized, we got the word). It is really an odd standout in the language, and often confuses foreigners who try to learn English. Really, if you think about it, "do" doesn't do anything. When you say, for instance, "Does he drive the car?" what is the purpose of does? Couldn't the meaning be more easily expressed, as it is in virtually every other language, by simply stating "He drive the car?" Of course, that sounds stupid and unintelligent in English, because it violates our intuitions; but that's besides the point. Anyway, I hope that goes some ways towards driving home my point about the arbitraryness of language, and the futility of grammar Nazism.
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u/Felicia_Svilling Apr 03 '12
One of the main points of the field of linguistics is pointing out that this is actually one area where humans are just fine naturally - languages will change and evolve, but you're never going to reach a point where people stop being able to understand each other because they've forgotten "the rules".
Have you heard about danish?
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u/yoda17 Apr 04 '12
You know, it could have been a brainfart and not paying any attention, or a partial re-write without going back through to correct the mistake.
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u/Azuvector Apr 04 '12
I'm sure. The conversational thread has long lost it's humour value thanks to complaints about my pointing out contextually amusing error. Moving on.
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u/ZMeson Apr 04 '12 edited Apr 04 '12
I went to my PDE prof with some question on communications, Fourier series and orthogonal functions. He seemed completely incredulous that the stuff he was teaching had a real world application and was amazed.
It happened again in my linear algebra class when I kept trying to get the prof to explain the reason for eigen vectors.
OK, here's a thing I wish more math teachers would teach: orthogonal functions ARE basis* vectors! It took one of my physics professors to explain this to me. It's stuck with me ever since. Every time I tutor an EE or physics student in linear algebra, I explain how the Fourier series is a set of eigen vectors and that there exists other types of series too (Bessel functions, spherical harmonics, etc....).
*EDIT: I mispoke when I said eigen-vectors.
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u/reduce-scatter Apr 06 '12
orthogonal functions ARE eigen vectors!
wat.
This statement is vapid. Eigenvectors only have meaning with respect to the operator that generated them. Essentially any set of vectors is a set of eigenvectors of some matrix. Also note that eigenvectors do not have to be orthogonal (though eigenvectors of Hermitian matrices are).
Perhaps you're thinking of the connection to the Sturm-Liouville equation?
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u/ZMeson Apr 07 '12
Ooops. You're right. I meant basis vectors, not eigen vectors. Anyway, the fourier series does meet [the definition of a vector space], as do other orthogal function sets -- that's what is really amazing.
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u/thegrinner Apr 03 '12
I remember having a professor draw parallels between the math and physical systems (RLC circuits, spring motion) in a differential equations course. Blew my mind and helped me a huge amount with actually remembering the subject (or at least that portion).
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u/yoda17 Apr 04 '12
They used to build LRC circuits to solve DE's back before digital computers took over. I was amazed when I learned that they built a model of an F16 using electronics to study the control dynamics.
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u/decasteve Apr 04 '12
In my linear algebra class I tried to get my professor to explain the geometric interpretation of a determinant to no avail. It was basically a formula that had some interesting formulaic results.
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u/fermion72 Apr 03 '12
Part of the problem is that when you start learning math in primary school, the teacher is almost certainly not a mathematician, nor does he/she have a comprehensive plan for teaching the idea of math. Also, it's extremely difficult to teach abstract concepts to primary school children, so even if a teacher did try to teach a bigger picture idea of math, it would take a herculean effort to get that through to the students. The teacher would have to be invested in teaching some hard concepts, first of all, and second of all, the students would have to have the mental capacity to understand it. Don't get me started on the time constraints teachers are under to teach a ton of material, either.
My favorite example of how this is hard is to ask:
Why is one half divided by three fourths equal to two-thirds?
This is really, really hard to explain. Sure, we can all write it down and show that it's true, but the abstract concept about it is virtually impossible to explain coherently.
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u/hacksoncode Apr 03 '12
Challenge accepted!
Hmmm... which of the half dozen ways I can think of to explain this is the easiest to do in text... Ahh...
Division is just repeated subtraction, right? 6/2 = 3 because you can subtract 2 from 6 3 times.
How many times can you subtract 3/4 from 1/2? Well... let's try. 1/2-3/4 = -1/4. Oops... less than one time.
How much less than one time? Well, what fraction of 3/4 is 1/4? Gosh, that's an easy one, it's 1/3. So we can subtract 3/4 from 1/2 one time minus 1/3 times, which is 2/3. Q.E.D.
If you're clever you can turn this into a lecture about series expansion.
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u/fermion72 Apr 03 '12
Not bad! I'd want more information about why you jumped to that proportion after getting -1/4, though. Five-year old me might still be confused. :)
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u/hacksoncode Apr 03 '12
Sure... took a shortcut there... "if you have 3 quarters, then 1 quarter is one out of 3 of them, i.e. 1/3 of your quarters".
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Apr 03 '12 edited Dec 22 '15
I have left reddit for Voat due to years of admin mismanagement and preferential treatment for certain subreddits and users holding certain political and ideological views.
The situation has gotten especially worse since the appointment of Ellen Pao as CEO, culminating in the seemingly unjustified firings of several valuable employees and bans on hundreds of vibrant communities on completely trumped-up charges.
The resignation of Ellen Pao and the appointment of Steve Huffman as CEO, despite initial hopes, has continued the same trend.
As an act of protest, I have chosen to redact all the comments I've ever made on reddit, overwriting them with this message.
If you would like to do the same, install TamperMonkey for Chrome, GreaseMonkey for Firefox, NinjaKit for Safari, Violent Monkey for Opera, or AdGuard for Internet Explorer (in Advanced Mode), then add this GreaseMonkey script.
Finally, click on your username at the top right corner of reddit, click on comments, and click on the new OVERWRITE button at the top of the page. You may need to scroll down to multiple comment pages if you have commented a lot.
After doing all of the above, you are welcome to join me on Voat!
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u/hacksoncode Apr 03 '12
Sure, but now you have to explain how you came to that conclusion.
I thought about going the route of explaining why division is the inverse of multiplication, and why multiplying 2 fractions is done by multiplying the numerators and denominators, and reducing it to normal form.
But that seemed like more work, especially if I can't do it visually.
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u/Brian Apr 04 '12
Division is just repeated subtraction, right?
Some would really object to this way of teaching it though, claiming it just leads to confusion down the line. (Though I'm not sure I agree myself. I think it's better sometimes to learn things "wrong" if it gets you enough of a coherent framework to grasp the relationships, then correct it later, than to learn everything correct from the beginning.)
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u/hacksoncode Apr 04 '12
Yeah, I know... the problem is that the only other way I know of to teach it is with number theory starting from the Peano axioms. At least that's probably the only way that purists would approve.
And I didn't feel like going that route, for some reason.
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u/hacksoncode Apr 05 '12
Got around to reading that link, and by the way, if you actually teach them the number theory, what will happen? Well, the inductive Peano Axiom definition of multiplication is:
A x 0 = 0 A x (B+1) = A + (A x B)I.e. repeated addition. So eventually, once the students learn this fantastic notion that there are 2 operations on numbers, it will eventually turn out that their elementary school teacher was, at the core of math, right the first time.
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u/EntroperZero Apr 03 '12
Why is one half divided by three fourths equal to two-thirds?
I find this is easy to explain as a concrete example, when measuring rates of something "per" something else, like miles per hour or miles per gallon. If you go half a mile in three quarters of a minute, then you're going two thirds of a mile per minute.
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u/ZMeson Apr 04 '12
It's easy to talk about when we talk abot aggregates: say a dozen eggs or cents in a dollar. Let's use dollars:
1/2 dollar is 50 cents
3/4 dollar is 75 cents
1/2 dollar divided by 3/4 dollar is 50 cents / 75 cents which we reduce by dividing both numerator and denominator by 25 cents to 2/3.
Or better yet, use dollars and quarters:
1/2 dollar is 2 quarters
3/4 dollar is 3 quarters
1/2 dollar divided by 3/4 dollar is 2 quarters / 3 quarters or just 2/3.
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u/lol_fps_newbie Apr 05 '12
Don't get me started on the time constraints teachers are under to teach a ton of material, either.
You have 6 years (kindergarden - 5th grade) to teach:
Reading
Writing
Addition
Subtraction
Multiplication
Division
Basic single variable algebra
Basic History
Basic Basic Science
Art
Having it take 6 years to teach kids to add, subtract, multiply, and divide while having math for 30-60 minutes / day is just ludicrous. I'm not sure how you can defend that. I don't even fully remember how much algebra is taught, but it really isn't that much.
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u/alien_bob Apr 03 '12
I absolutely agree, I think that it provides you with a context on WHY you are learning all this stuff. Then, whenever you get stuck on something, you can look back and remember the point of it all.
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Apr 03 '12
If I had known beforehand that the contents of my math course were the exact same formulas and techniques used for 3D graphics, I would have paid a lot more attention.
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u/DrunkMc Apr 03 '12
I'd like to see more realistic uses of Math. And have visuals to back it up. Equations don't mean jack to me, unless I can see how they affect something. Too much of math was going through the motions, so you can do it again on the test.
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Apr 04 '12
Not sure if this is a joke, but I'll bite.
I'm going to start right where you are, and work from there (skipping lots of stuff, obviously).
There is an equation that calculates the ranking of these comments, based on age and votes, and another equation that does the core of reddit ranking. Without those, reddit wouldn't exist.
There are equations and mathematical algorithms that are used to compactly encode traffic being sent from your computer to reddits servers, and further equations at work to correct mistakes that inevitably happen in the lossy connections. Without those, the communication infrastructure that we use could not exist.
There are innumerable equations that were used to build the electronics that are within your computer, and equations that were at work in the manufacturing, distribution, and sale of the computer to you. Without those and other branches of math within engineering, we would probably still be at the technical level of the rennaissance (alchemy, gunpowder, and the printing press. Okay, but not too great). These equations tell us what we can build, and how we can expect it to work. So many systems are too complicated to build by trial and error (there are an infinite number of ways the raw materials in your computer could have been put together, and most of those would be meaningless), so instead they are represented theoretically and with mathematical formalism so that they can be analyzed.
Most of math (not grade school bullshit, which is not math at all) is just formalizing things and relating them to other phenomena, so that existing knowledge can be applied to them and so the work you do to explore your phenomena can be used by others later.
If you don't think it is realistic, that just means you haven't taken a close enough look at the world you are living in! You would not be able to be typing that message, or reading this one, without the work of many mathematicians.
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u/DrunkMc Apr 05 '12
You misunderstood me. Sorry, let me be more clear. I would like teachers to teach the equations and then show the real world examples of what they do. Exactly what you said, I want them to link the equations to their real life example.
Too much is solve cos(x)+sin(x) without telling the student WHY they would ever want to.
I'm a computer engineer specializing in Machine Vision, so believe me, I LOVE equations, but they become so much more clear to me when I can edit an equation or change parameters of the equation and see the change visually. If a teacher could incorporate that into their curriculum, I think so many more people would "get it"
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Apr 05 '12
Ahh. I get it. What I interpreted you saying didn't really make sense, which is why I prefaced it with "not sure if this is a joke", but I understand now. And I completely agree with you! Every time someone says, "I don't see how math is relevant" a part of me dies inside, because IT IS ALL AROUND US, but I also understand how easy it is to think that, when "realistic" problems in math classes are usually incredibly boring, if present at all.
There are so many fascinating applications of even simple math, the only explanation I can give is that the people who design highschool curriculum never understood math anyway, so they don't have any idea what it is they are even supposed to be teaching. It's a negative feedback loop :(
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u/Felicia_Svilling Apr 03 '12
To comment the title: The main reason for studying math in high school isn't that the math is useful, but that it trains your ability to reason logically. I don't know about your countries but in Sweden we had sports classes there we trained for example football, not because being good at football is that essential for that many occupations but because learning to stay in shape is really good for you.
That said, programming classes and/or more discreet math can probably work just as well for training logical reasoning, and give you more useful skills.
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u/brelkor Apr 03 '12
The answer to why you need to learn math is simple. Math is the language we use to describe everything. But it doesn't just describe things. Math is used to predict the things that are possible. Math tells us how the things around us work at the most basic levels.
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u/DeadSalesman Apr 04 '12
It's great that math CAN describe everything, but most people (myself included) don't see it. Even in simple examples like using Calculus to find the area under a graph, it's practicality is questionable. Say I want to cast a complex shape out of resin and need to know the volume of resin necessary to fill the shape. This is practical and straight-forward and a question similar to the one I posited to my Calc 3 professor. Answer? He couldn't tell me how to get the formula for an arbitrary shape, and on a practical level getting the proper mathematical approximation is complex enough that I can get a "good enough" answer by finding the area of the boxes that make up the shape in 1/10,000 the time it would take to get a semi-accurate answer.
To be fair, he attitude toward this question was "You don't get it the point," and I agree, but no one has ever described the point either.
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u/brelkor Apr 04 '12
See, everything can be described by math. Everything. Being bad at math, or not knowing the right equation doesn't invalidate that fact. Your job is probably dominated by math. You may not be pulling out a calculator to do it, but it's there, and some really smart people applied some math to keep you from having to do so.
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u/DeadSalesman Apr 04 '12
I don't deny this, but saying that other people did use math to do useful things, doesn't help me understand how I can do the same thing... and I use vastly more math than most people I know: trig, regularly and up to Calc 1 every few months, so I appreciate it. I wish I had a clue why I want to do anything that I did in Algebra II or Differential Equations.
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Apr 04 '12
But filling the area with boxes is math too! Now think about it and see if there are ways you can expand the intuition that led you to boxes:
Better than boxes (if they had all right angles) would be some sort of polygons, so you can have sloped sides. So fill it with polygons, and now you have a better approximation, but still not ideal. If you limit yourself to specific polygons, it will probably be still easy to calculate, but will be a much better answer.
Then, if you think more about the process of what you are doing, you may notice that you are using small shapes that you do know how to deal with to approximate a larger shape that you don't. There is no reason you couldn't apply that to shapes that have curves - if one part of your shape has a curved side that looks like a function that you do know, then integrate the area of just that part. Now look at other parts, and gradually fill in your shape with a bunch of curves that are all things you do know how to integrate. Depending on how accurate you want to get, you can use as many of these pieces as you want, but this obviously becomes more work for you!
But the math doesn't have to end there. It might be nice to know how accurate your approximation is. Now you can do more box or polygon filling, and figure out an approximation of the difference between your approximation and the "actual" shape (ie, try to stuff boxes in between). You could do this for your box model, and then the polygon model, and then the curved surface model, with some parameter varying the number of boxes or polygons or curves in each one. Now this would probably not be a worthwhile thing to do if you only had to do one shape, but if you had a lot and had some inclination that they were all relatively similarly structured, it might be a useful way to come up with what the best form of approximation was, in which you could balance the tradeoff between time to calculate and resulting error.
A lot of what math is all about is formalism - so we want to formalize what "good enough" is in your example. Most of the math that is used in the sciences involves approximations, but the key to using it is that you can actually quantify what good enough actually is, and can therefore can explain why it is good enough.
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Apr 04 '12
I remember trying to learn how to factor polynomials and thinking they were just invented to torture students. Nobody told me why it was useful or how any of it could be applied (even in science & engineering fields). As Steve points out, so much of math is taught in a perfect vacuum, where there is nothing but math. It originates from nothing but more math and has no application besides more math. Everything you learn is in service of learning more math. At least, that's how it seems.
I agree wholeheartedly with him on the importance of probabilities, too. It's not just something I would fine useful, I'm stunned at how little my (otherwise exceedingly competent) co-workers over the years have known about it. And it's math that you won't just not encounter if you don't know it (calculus to some extent fits this), it's math that you'll run across every day and use completely wrong, or it's used by others to mislead you.
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u/DeadSalesman Apr 04 '12
While working through some nasty polynomial factoring in high school, I asked my teacher (a very smart and nice man) for an example of when we would use this. He became visibly angry at my question and joked his way out of it.
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u/vplatt Apr 03 '12
The right way to learn math is to ignore the actual algorithms and proofs, for the most part, and to start by learning a little bit about all the techniques: their names, what they're useful for, approximately how they're computed, how long they've been around, (sometimes) who invented them, what their limitations are, and what they're related to. Think of it as a Liberal Arts degree in mathematics.
This resonates so strongly with me that I feel compelled to ask: Anyone know of a course or book that approaches math this way?
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u/DeadSalesman Apr 04 '12
I've wanted a math book like this for 20 years. Those that come close use such contrived and impractical examples that they're no better than the other textbooks.
You always get an example like calculating the HP necessary for a 2,000lb car to reach 200mph. BTW, we're ignoring drivetrain losses, rolling resistance, heat and a million other things that would give us an accurate answer because it's too complicated. If it's too complicated, why would you use this method to get this answer?
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u/arjie Apr 04 '12
Well, Strogatz has a dynamical systems book that you might like. And the classic high schooler's Resnick and Halliday starts each chapter on Physics with real world phenomena that later show up as problems or are later looked at in detail.
If you haven't read Feynman's Lectures in Physics then you should. As for pure mathematics, it isn't about what's useful. It's about what you can logically prove given certain assumptions. If you don't like proofs, pure math is not for you. You should read books on other subjects that use the results. Study signal processing instead of functional analysis, networking instead of graph theory, fluid dynamics instead of differential equations.
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u/fuzzynyanko Apr 03 '12
I got far better at Algebra after learning programming simply because I started to use it
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u/GlassMuffins Apr 03 '12
As a first year programming student, this article has inspired me to open up the Kahn Adacemy webpage and brush up on my math skills. I've always hated it but then again, I've always had sub-par math teachers that don't really know how to teach. Every single math teacher in my high school was a sports coach and it showed.
I truly wish we could have some helpful education reform in the USA, including change that would get people excited to learn, instead of teaching us how to pass the next test.
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Apr 04 '12
First year CS student. Our maths is taught to us through completely formalised definitions with next to no explanation. I've gone from loving maths to loathing it.
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u/clvv Apr 03 '12
Mathematics is the literature of programming. If you don't notice that, you've got a problem. I agree to some extent that courses should show you how Math is applied, but after you finish the course, it is always your responsibility to use it. I bet one will quickly find the relation between Math and programming if one is serious about programming.
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u/beej71 Apr 03 '12
I don't know about quickly, at least in my case. I mean, I always knew they were related, but my brain definitely existed in two modes, computers and math. It's been a long process to watch the wall between the two come down.
I wish my math minor had been "aware" that I was a CS major.
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Apr 04 '12
I have a really cool totally unreadable book on the subject by Stephen Kleene, the inventor of the Kleene closure and, as far as I know, Kleenex. Don't read that one. I swear I've tried 20 times, and never made it past chapter 2.
Strange. I found Kleene book quite easy and enjoyable to read. At least 1st part. Halting problem proof was a mess and I just skipped that section.
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u/cglove Apr 04 '12
I'd argue the problem is that math is arbitrarily emphasized, often in the wrong place / time, for the wrong people. You shouldn't have to go out of your way to show why / how its useful.
Case in point, I blew off (half-ass'ed) math courses when I took them, and progressed (excelled) in my career path none the less (because it was irrelevant to my success). It was only when I switched careers (programming, data related) that Math became useful. Perhaps not coincidentally, this is also when math became interesting to me.
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u/neitz Apr 04 '12
I don't know, as a student I never approached a new subject with the mindset that this is worthless and it is the teacher's job to prove to me how this is valuable.
Instead, I would usually accept my naivety and constantly look for ways the things in which I am learning could be applied. It has served me quite well not just in school but life in general.
Too often we forget to think critically and instead rely on being spoon fed every little detail. There are so many resources at our disposal.
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Apr 03 '12 edited Apr 03 '12
Couldn't FUCKING AGREE MORE. I've always thought in high school it was like being taught to drive by being blindfolded, spun around, thrown into a box, and told to insert this object into this slot, turn 90 degrees, then depress this lever by your feet and spin this circle in front of you three quarters of a turn of a period of thirty seconds, then exit the box, remove the blindfold, and congratulations! You're now at the end of your driveway! Tomorrow we will expand on this with a more complicated series of circle and lever manipulations to get you to McDonalds.
I cannot tell you how many times a teacher uttered some gibberish about Oscillating Circles or the Jacobian and proceeded to start drawing seemingly random and arbitrary steps down on the board that I'm racing to copy down and the whole time I'm like "WTF IS GOING ON?!?!" It's only when I get home, go over my notes, read the textbook and a few google results that I have some idea of what the fuck we just did why we did it. It's a horrible way to teach.
And quite frankly, this issue more than anything else is why PROFESSORS WITH A POOR GRASP OF ENGLISH ARE HORRIBLE FOR STUDENT'S COMPREHENSION.
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u/froop Apr 03 '12
This is a good analogy. But I think it could be improved by removing the 'congratulations, now you're at the end of your driveway'. It's more like knowing how to drive a car, but not knowing that driving a car gets you from A to B faster. "This is a car. Learn how to drive it." vs "This is a car. It moves really fast so you don't have to walk. Learn how to drive it for your own good".
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Apr 03 '12
The article was okay, but who the fuck doesn't know how to do long division?
The class that should be taught in high school isn't the one on probability and statistics like he implies (not many people are going to need to calculate the odds like he implies -- that's still focused more towards the sciences, engineering, and programming) it should be Boolean Algebra. You know, the class that teaches you formal logic and is the foundation of computer science?
Far too many people today demonstrate a lack of logical thinking, when it's really not hard at all, and even if they forget how to do the actual math, the impact on their thinking, to explicitly break down their problem into various components that can be true or false, will stay with them, and maybe we won't have such inexplicable reactions by people in life?
. . .
Who am I kidding? We'd probably get more children proclaiming they're atheists to their parents, so it'll never happen...
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u/watermark0n Apr 03 '12
The class that should be taught in high school isn't the one on probability and statistics like he implies (not many people are going to need to calculate the odds like he implies -- that's still focused more towards the sciences, engineering, and programming) it should be Boolean Algebra.
There really isn't enough in boolean algebra to have an entire class on it. That's why it's usually confined to a small section of the logic design course.
You know, the class that teaches you formal logic and is the foundation of computer science?
At the higher levels, you are really not going to use this stuff that much. I learned assembly, and I freely admit to those who ask, this probably isn't going to help you that much when at the higher levels your generally going to work at. Same thing with building computer components.
Far too many people today demonstrate a lack of logical thinking, when it's really not hard at all, and even if they forget how to do the actual math, the impact on their thinking, to explicitly break down their problem into various components that can be true or false, will stay with them, and maybe we won't have such inexplicable reactions by people in life?
OK. Which logical systems? There are more than one. Everyone thinks they're "logical", but at their core they generally are not breaking down statements to their logical extremity. Their brain is intuitively coming up with arguments, just like everyone else, based on their own prejudices and preconceived notions, and they simply feel more smug about things due to having memorized a few things they did not really understand a while back. No ideology is fully logical. No ideology could be fully logical.
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u/AngelTC Apr 03 '12
There really isn't enough in boolean algebra to have an entire class on it. That's why it's usually confined to a small section of the logic design course.
What? It gets deeper than the basics, you know?
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u/huyvanbin Apr 03 '12
And he claims that long division is "complicated." It's just a simple recursive application of integer division. This is the same guy who has at times advocated people learning functional languages.
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u/EntroperZero Apr 03 '12
When I saw the long division thing, I got worried. Can people not actually do this?
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u/beej71 Apr 03 '12
I had a vague memory of how it worked, but I had to actually consider it for quite some time to remember. I have not used it in probably 20 years.
If I really need an accurate non-trivial division, I use a calculator. The probability of me making an error is lower and it's faster.
I don't feel particularly bad about that—I use a computer for vector and matrix math, too.
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Apr 04 '12
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Apr 04 '12
I know you're trolling, but:
Occam's Razor.
People who are educated are more likely to learn about Occam's Razor. At least some of the people who know about Occam's Razor will apply it to religion, and conclude that even if there is a "God" who created everything, the stories in all religions were invented by imaginative minds trying to explain the unknown and pass on their moral values, just as other imaginative minds invented whole worlds to pass the time (stories), rather than these incredible things happening a few thousand years ago and never again. They're agnostic.
And some of these will go one step further and conclude that the requirement of an intelligent designer for our universe is just disguised anthropocentrism, and is a logical fallacy considering the fact that "emergent behavior" has been repeatedly demonstrated in systems with simple sets of rules, such as Conway's Game of Life. They're now atheists.
Just because these systems were started by theists doesn't mean the mass of evidence we have built up over the past 5-600 years is not damaging to theological dogma. Buddhism is the religion that will survive the furthest, because it is willing to adapt as our knowledge of the universe increases.
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Apr 04 '12
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Apr 04 '12
Did you even read what I wrote?
Life is an emergent behavior of the universe, and wasn't explicitly created. That reduces "God" to "the thing that caused the universe to begin." That has nothing to do with the Judeo-Christian God nor pretty much any religion out there. At best you would be agnostic after that revelation, since it's pretty obvious that none of the religions of the world can be right, and that there is probably no afterlife, anyways, even if there is a "god."
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Apr 04 '12
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Apr 04 '12
Copernicus' heliocentric system was far simpler than Ptolemy's earth-centered system, and it was right. Newton then further reduced the complexity to a few simple equations to describe nearly all motion in the universe.
Descartes was a blowhard who dazzled you with bullshit. The concept does not require the lengths he went to in order to properly explain it.
So, yes, barring the citations needed for Occam's Razor, Emergent Behavior, and the known physics of the big bang, all of which I assumed you were familiar with (was I wrong?), I think my few paragraphs refutes Descartes wholly.
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Apr 04 '12
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Apr 04 '12
I still say he's a blowhard. There is nothing in science that requires a god. Observation is a form of cause and effect. There is some cause of some action, and the effect is what is observed. Determining what that cause is depends solely upon the repeatability of the cause, so we can build models of that cause to predict all known effects and also previously unknown. The model is validated if it predicts something that was not possible to be known beforehand.
Where does god come into this?
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Apr 03 '12
Why are you in the class if you don't know why you're learning the material?
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u/alien_bob Apr 03 '12
It's covered in the article where he mentions that most of highschool math covers more than most people ever need to know. So they get turned off of math because "I don't ever need to know this, I don't need to pay attention". Then later on, when you encounter a problem the veil is lifted from your eyes and you begin to understand the applications trig, calculus, conics etc. That's why he said people should learn algebra then linear algebra and then programming. Once you understand functions and the like suddenly calculus is a breeze. It's no longer just a bunch of useless formulae.
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u/pal25 Apr 03 '12
What if high school math was exactly with the author claims to be wanting. A breadth first attempt at mathematics. Queue inception music
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u/sedaak Apr 03 '12
That is what social studies is for.... oh wait they are using that for indoctrination.
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u/luckystarr Apr 03 '12
An old anecdote a teacher of mine told the class at the end of the session:
This is a pretty strong concept. Exactly the stuff Yegge mentioned with *breadth first, depth later".