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Author Topic: "People Who Understand Math" and "How Math is Taught"  (Read 8438 times)

Skyrunner

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #75 on: March 21, 2013, 09:27:30 pm »

Why the 'when will we do this in real life' argument? D:
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #76 on: March 21, 2013, 09:30:21 pm »

Why the 'when will we do this in real life' argument? D:

Because... honestly? 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, and it's the second best thing to a calculator.

mainiac

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #77 on: March 21, 2013, 09:39:40 pm »

Because... honestly? 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, and it's the second best thing to a calculator.

Real math is the foundation of nearly all important work done in the world today.  It's the logical processes of figuring stuff out once you've turned that stuff into quantifiable data.  Calculators aren't going to be a replacement for that until we invent A.I.  If it's something that a calculator can do it's not real math.  The calculator just does the filler and the mental exercise for you.  A calculator can no more do math then a tractor can farm.

And approximation isn't a second best to a calculator, it's a far more essential skill.
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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #78 on: March 21, 2013, 09:56:52 pm »

Why the 'when will we do this in real life' argument? D:

Because... honestly? 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, and it's the second best thing to a calculator.

I detest approximation; it just bothers me. I want to know, dammit! Understanding math should be an end in and of itself. To understand math is to understand logic, to gaze in wonder at the very building blocks of reality. Language and history may delve into the human condition, but math delves into the universal condition.
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #79 on: March 21, 2013, 10:00:16 pm »

Yes, but in the real world, approximation is an invaluable tool. Of course you're not going to approximate when you're doing math.

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #80 on: March 21, 2013, 10:05:13 pm »

If it's something that a calculator can do it's not real math.
That kind of attitude? Yeah, that's not gonna help draw regular people into math. That's what gets mathematicians labeled as elitist fucks who are disconnected from the real world. Does your version of "real math" even have numbers in it, oh enlightened one?
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PanH

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #81 on: March 21, 2013, 10:05:14 pm »

To understand math is to understand logic
I wish we were taught logic and then mathematics. It would make more sense IMO. Mathematics is a kind of logic.
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Frumple

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #82 on: March 21, 2013, 10:07:02 pm »

Language and history may delve into the human condition, but math delves into the universal condition.
... not really. Math delves into math. Universal condition is more physics, if anything (if it's communicable at all). Math is a construct, completely artificial and divorced from reality, based on its own axioms that are only coincidentally related to what is in any universal sense. We just happen to use it to give approximations of real life stuff occasionally :P It's one of our best tools for doing so with anything approaching precision, which is why it's so damn useful.

And that's okay. Logic is like that too, by and large. That's a kind of a beautiful thing. Concept systems, the things arising from axiomatic foundations, are beautiful. It's basically all we do as a species.

At most (in that direction, anyway) it's a filter, a means of communicating the nature of real things, or things we consider real, or things that are considered real for the purpose of a particular scenario. Just like any other language.
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Zrk2

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #83 on: March 21, 2013, 10:09:31 pm »

If it's something that a calculator can do it's not real math.
That kind of attitude? Yeah, that's not gonna help draw regular people into math. That's what gets mathematicians labeled as elitist fucks who are disconnected from the real world. Does your version of "real math" even have numbers in it, oh enlightened one?

I believe he's getting at the difference between math and arithmetic.

To understand math is to understand logic
I wish we were taught logic and then mathematics. It would make more sense IMO. Mathematics is a kind of logic.

I've always been fond of saying math is merely quantified logic.

Language and history may delve into the human condition, but math delves into the universal condition.
... not really. Math delves into math. Universal condition is more physics, if anything (if it's communicable at all). Math is a construct, completely artificial and divorced from reality, based on its own axioms that are only coincidentally related to what is in any universal sense. We just happen to use it to give approximations of real life stuff occasionally :P It's one of our best tools for doing so with anything approaching precision, which is why it's so damn useful.

If something exactly describes how something works I would say it is enough to be said to actually describe that thing. Building on my previous point one reaches the conclusion that physics is simply math applied to reality.
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Frumple

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #84 on: March 21, 2013, 10:21:49 pm »

If that's all it takes to be something that considers the universal condition, every spoken language under the sun counts :P

Insofar as I'm aware, you can translate any mathematical concept you please into a spoken language without even making reference to numerical concepts. "That thing and that other thing and that other other thing" is just a fairly clumsy way of saying "three" and if you can approximate a number you're pretty much done, the rest is just the working of it. Clumsy as hell and takes forever, but... again. Mathematics is a language. It can be translated. It doesn't do anything particularly special insofar as that's concerned. It's just a particular method of arranging communication in a particular way.

Math and reality are just... separate. Math, like all languages, is a human construct. Physics is just an attempt to communicate the nature of reality that happens to use mathematics as a primary means of communication. But it's not really what pops out when you apply mathematics to reality. S'just not how it works, y'know?
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #85 on: March 21, 2013, 10:25:40 pm »

If it's something that a calculator can do it's not real math.
That kind of attitude? Yeah, that's not gonna help draw regular people into math. That's what gets mathematicians labeled as elitist fucks who are disconnected from the real world. Does your version of "real math" even have numbers in it, oh enlightened one?

oh man how did I miss that

calculators can do lots of real cool stuff, dude. Yes, head-math is cool, but a calculator is all that is needed to get into math. Just get your hands on a TI graphing calculator and play around with the functions. So much can be figured out as long as you have documentation (which can be found on the internet).

MaximumZero

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #86 on: March 21, 2013, 10:33:05 pm »

If it's something that a calculator can do it's not real math.
That kind of attitude? Yeah, that's not gonna help draw regular people into math. That's what gets mathematicians labeled as elitist fucks who are disconnected from the real world. Does your version of "real math" even have numbers in it, oh enlightened one?

I believe he's getting at the difference between math and arithmetic.
The problem I have is twofold. A) Isn't arithmetic real math? I mean, you aren't going to play barre chords without learning how to play single notes. Playing single notes is just as much real guitar as barre chords. Just because it's not difficult to someone who understands the higher echelon concepts of music theory doesn't mean it isn't valid. Furthermore, B) the people who are just learning, just getting interested, or are struggling with the basics don't deserve scorn, and the attitude doesn't help them learn/get interested/fix the problem.
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palsch

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #87 on: March 21, 2013, 10:38:05 pm »

I don't see anyone suggesting that long forms not be taught. The topic is general non-comprehension of math. In my own personal life, I run into far more people facing problems like "how many third-cups of sugar do I need" and who are unable to do them than I run into cases like yours that are resistant to non-longform methods.
See, I did;
In this case, however, I believe the method being taught by the education system is this horribly complicated mess:
You were explicitly attacking the basic method of solving such problems in favour of shortcut methods.

If you want my real problem with this, it's that not everyone is going to have the basic aptitude for such shortcuts. They rest heavily on intuition and spotting patterns. I've never seen a reliable way to train such skills, and I've seen people have day after day of intense, one-on-one training, at various levels and using different methods. Even more than general maths training it also depends heavily on the commitment of the student and their desire to want to see things that way.

On the other hand, learning the full toolkit works for everyone. I'd actually say it's essential for everyone. Especially the people with strong intuition who don't seem to need it. Those are the people most likely to use mathematics in more advanced situations in the future, so most likely to encounter situations where they genuinely need to apply the full methods. Absolutely everyone's intuition and pattern recognition fails at some point, and you need to have something to fall back on when it does.

The last significant time I had to divide fractions by fractions was calculating probabilities of states in quantum systems; normalising the sum of the square of multiple fractions to equal 1, where the fractions involved were complex numbers involving surds. At this level it's not even a significant step in the mathematics - pretty much just done by observation - but that's because the full method has been solidly internalised and can be applied without really considering it. If I had learned to rely on approximations or computations of such problems I would have had a hell of a harder time of it.


To pull up a completely different example, I never had any basic lessons in using matrices. I kinda had them thrown at me as part of physical systems and had to just read up on the various applicable operations. Most physical systems have significant symmetries or fall into one of a few basic patterns which allowed me to get away without having to remember the full mechanisms you are supposed to learn. I did great on all the problems we had assigned and came out of the course happy, having avoided a pile of extra work by finding shortcuts that had done the job perfectly.

Except that a year or so later I needed to use the same mathematics without the crutch of simple symmetries. And I couldn't, off the top of my head, confidently carry out these basic functions. I may as well have never used matrices in the past for all the good that first term's use did me. I ended up having to go back over the exact same ground I'd covered a year before, only now in addition to a far heavier workload and trying to apply these new skills to far harder problems. At which point there was no chance of me going back and actually doing the sort of basic practice problems that would help fix those methods into my memory permanently.

That's far from the only place where relying on shortcuts and simple patterns rather than trying to learn and understand the basic concepts has cost me understanding, but it's one I still feel today. I've never developed the full intuition for matrices I have for, say, trig functions or calc, always feeling slightly worried I'm misremembering something and going to mess up if I don't check I'm using the right methods constantly.
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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #88 on: March 21, 2013, 11:46:37 pm »

If it's something that a calculator can do it's not real math.
That kind of attitude? Yeah, that's not gonna help draw regular people into math. That's what gets mathematicians labeled as elitist fucks who are disconnected from the real world. Does your version of "real math" even have numbers in it, oh enlightened one?
A calculator can't do math the same way a microscope can't do science. It's a tool, nothing more.

* Descan bops MZ.

Also, yeah, it doesn't have real numbers in it. That's the point. You don't -need- the numbers, math is just fiddling with constructs and seeing what happens. Pattern recognition, going "What if I do this?" The numbers just allow you to communicate it, and to be a bit more precise.
« Last Edit: March 21, 2013, 11:51:24 pm by Descan »
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #89 on: March 22, 2013, 12:07:12 am »

Yes, exactly. Calculators are a collection of tools that help you do the math that's truly beautiful.

Just a few days ago, I discovered that e^(iT) on the complex plane makes a circle! That was amazing! Then I did the same for i^x and found... the same thing! But the whole thing is squished because the equation is multiplied by π/2. Which is cool. I was taught in school that i^x=(i^x)%4 (in the way of i^1=i, i^2=-1, i^3=-i, i^4=1, i^5 = i and so on), but I never really thought of that until... well, I did that. I never realized that the graph of i^x is literally a circle even though they taught me that in school. Then I told my math teacher and... he didn't know either. God dammit.
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