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Old 01-04-2019, 04:32 AM
 
Location: Pacific 🌉 N, 🌄W
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Quote:
Originally Posted by rodentraiser View Post
I think some of the other things that we can get rid of in math are things that I don't see a connection to. If I'm wrong, correct me.
I can't think of a worse situation than someone wanting to annihilate a valuable tool...such as math because they don't get it?! You are very mistaken to proclaim that math should be gotten rid of simply because you don't understand it.

That's how societies become dumbed down.
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Old 01-04-2019, 05:25 PM
 
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Originally Posted by tnff View Post



Been doing this for nearly 40 years now and f(x) still drives me bats*&t crazy.
Huh?


You do realize that it's important to understand which independent variable your function is a function of, don't you?


If you have a three dimensional function, for example, you have a rate of change of z wrt x, and a different rate of change of z wrt y. Or in other words, dz/dx vs. dz/dy. And of course what you really have (let's assume orthogonal functions here) is that dz/dx is itself a function of y. So you could represent that as dz/dx(y). Or you could figure d2z/dydx.


This kind of clarity with respect to exactly what the dependent variable is varying with, is very important when you analyze complex incomplete data sets (and all data sets met with in the real world are incomplete). This is why Leibnitz's notation for derivatives is the one that's most generally used, because it makes clear exactly what is being differentiated with respect to what.


Real world example: volumetric efficiency of a piston type gas compressor is a function of the piston-to-wall clearance; of the dead/re-expansion volume; of the friction losses in the suction passage and the discharge passage. Plus a bunch of other things.


Re-expansion volume is a function of the discharge hole size (if reed valves are used), but friction loss in the discharge passage is also a function of the discharge hole size. Smaller the hole, less the re-expansion volume (good), but more the friction loss (bad). You need to know the slopes of the relevant curves with respect to the relevant independent variables to be able to make a reasonable choice for the size of this hole.


This one item - how large should the discharge hole in the valve plate be? - is a tiny fraction of the hundreds or thousands of decisions that will have to be made in order to design a new or modified compressor. You will make some of those decisions based on past product knowledge, some on rules of thumb, and some you will have to model the thing, perform calculations, analyze the results, and then make prototypes, collect messy incomplete data, and make final design decisions based on those messy incomplete data.


How will you know whether you need to fit the piston more closely to the cylinder ($$$) or change the re-expansion volume ($$$), unless you can understand the concept that the resulting quantity (let's focus on volumetric efficiency, but there are many) varies according to many different inputs and that its response to inputs can be wildly different between different inputs?
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Old 01-04-2019, 07:41 PM
 
Location: Washington state
4,904 posts, read 2,456,767 times
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Quote:
Originally Posted by Matadora View Post
I can't think of a worse situation than someone wanting to annihilate a valuable tool...such as math because they don't get it?! You are very mistaken to proclaim that math should be gotten rid of simply because you don't understand it.

That's how societies become dumbed down.
I'm not saying get rid of it because I don't understand it. I understood it fine. But if it's not relevant to anything else you're going to learn in the course, why keep it?

Take vectors, for instance. They're like a lesson entirely by itself. Vectors have absolutely no relation to the physics lesson that comes after vectors, which is always acceleration. You learn vectors and then you forget about them because they're not used again.

Speaking of being dumbed down, when was the last time you manually figured out what the square root of 5 was without using a calculator? I bet you don't feel dumb if you don't know how to do that manually or for using a calculator to find out the answer. It's just silly to keep reinventing the wheel every time we want to go somewhere. Saving the time by doing the "grunt work" on a calculator makes us smarter, not dumber.

Quote:
Originally Posted by turf3 View Post
Huh?


You do realize that it's important to understand which independent variable your function is a function of, don't you?


If you have a three dimensional function, for example, you have a rate of change of z wrt x, and a different rate of change of z wrt y. Or in other words, dz/dx vs. dz/dy. And of course what you really have (let's assume orthogonal functions here) is that dz/dx is itself a function of y. So you could represent that as dz/dx(y). Or you could figure d2z/dydx.
We both realize that. We're simply objecting to using f(x) when we could be using y. That's all. I understand completely about needing to realize what variable the function is a function of, but when you're learning algebra, is that really needed at that time?

People coming into algebra aren't always math whizzes and you need to see things the way they see them in order to teach them. People have a hard time with numbers and algebra and making it more difficult isn't helping anyone.

Algebra is moving numbers around. You take 2x = 3y and change that to 2x - 3y = 0. You have to see the numbers. You have to make that mental jump from 2 x 3 = 6 to 2x = 6. Believe it or not, that's not easy for a lot of people. It's a big jump to be able to mentally understand the 2x can be broken up into 2 times x = 6. So then when you see an equation like f(x) = 2x, it becomes confusing.

I've never seen f(x) as anything but f times x the first time I look at it because of how fast it came up in algebra and I'd wager a lot of other people do the same. It takes a couple of seconds for me to unsee f times X and mentally translate that into y, which is really all it is. f(x) has its place in math, but let's use it when we come to that place, which would be in calculus.

edited to add: if you're a person who never had trouble in math, you won't understand what I'm saying here. But as an adult doing math (for the first time, really), I can remember exactly what I did and didn't understand when I finally got into classes again. What everyone forgets is that the people who can do math will always be able to do math. Those kids who understand the numbers coming into algebra, trig, precalc, and calculus aren't the problem. It's the kids who don't have a firm foundation in math who are the problem and believe me, the weeding out starts early.

If we want kids today to like and understand how numbers work, then we have to quit looking at math classes as a sink or swim situation. Some kids do fine on minimum instruction or they understand perfectly the instructions in the book. But most kids don't. And if you want those kids to understand the math, then you need to step out of your ivory tower and understand exactly what and why they don't understand. Otherwise, you don't have a hope in hell of trying to get through to them. And if you want test scores to come up, then you have to get through to them.

You could say the same thing about reading skills as well.

Last edited by rodentraiser; 01-04-2019 at 08:03 PM..
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Old 01-04-2019, 07:48 PM
 
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Quote:
Originally Posted by rodentraiser View Post

Take vectors, for instance. They're like a lesson entirely by itself. That have absolutely no relation to the physics lesson that comes after vectors, which is always acceleration. You learn vectors and then you forget about them because they're not used again.


No, vectors have everything to do with acceleration, because acceleration has both a magnitude and a direction - in other words, it's a vector quantity, not a scalar quantity.


This is how the planets orbit in circles (OK, ellipses) - their gravitational acceleration is perpendicular to their velocity (which has a magnitude and a direction, too, thus is also represented by a vector).


Basically, two ways to mathematically represent the quantities studied in mechanics have been developed: quaternions, and vectors. Using vectors allows you to properly understand and calculate the motions of bodies in motion, which can't be done without them (unless you use quaternions which are considerably more complication - I remember that in freshman physics we were assigned a couple of problems to do using quaternions just to show what they were, and the arithmetic was considerably more onerous). Unless, that is, you live in a one-dimensional world where all motion is in a straight line and nothing ever rotates about an axis.
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Old 01-06-2019, 07:47 PM
 
Location: Washington state
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Don't tell me, tell the books. I have yet to see a physics book that incorporates vectors with acceleration. Maybe that's part of the problem.

I watched the old Mechanical Universe shows on TV. I wasn't even in a calculus class at the time, but even I could understand the equations that had to do with orbiting planets. I don't recall them using vectors at all in those episodes.

Sometimes I think things have gotten too complicated. I remember working my way through a small physics book (circa 1940s) that presented the same problems many current college physics books do. Yet the book solved all the problems through nothing more than regular mathematics. No calculus was used. I understand why many aspects of calculus are used. But I also think that maybe they're not necessary in a lot of cases.

I see this in a lot of things things today. People who just want to train a family pet going to a dog training class and being trained to the standards of an obedience class (more work, more frustration - "All I want the dog to do is heel!"). People wanting to learn to draw just for fun and having to face the whole first semester of a beginning drawing class learning the history of drawing. As I said, if we all had to take apart engines and and learn about the mechanics of a car before getting a license, how many of us would even learn to drive? Sure, maybe we should all be doing that before we get a license. But what I'm asking is why someone who just wants to get from Point A to Point B needs to doing all that when there's so many more important things to learn about taking a car out on the road?

There are some things we can skip along the way.
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Old 01-06-2019, 11:03 PM
 
3,952 posts, read 1,724,573 times
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Quote:
Originally Posted by rodentraiser View Post
Don't tell me, tell the books. I have yet to see a physics book that incorporates vectors with acceleration. Maybe that's part of the problem.



Sometimes I think things have gotten too complicated. I remember working my way through a small physics book (circa 1940s) that presented the same problems many current college physics books do. Yet the book solved all the problems through nothing more than regular mathematics. No calculus was used. I understand why many aspects of calculus are used. But I also think that maybe they're not necessary in a lot of cases.

There are some things we can skip along the way.
Well:


My high school physics course, for some unexplained reason, was scheduled the year before we had calculus. So the teacher had to tie himself up in knots to explain things like s = 1/2 at^2 - where does the 1/2 come from? Of course, if you have had a few weeks of calculus it's pretty easy to explain the antidifferentiation and where the 1/2 comes from. No mention of vectors at all in high school.


Now why they had physics, which is deeply integrated with calculus throughout, the year before calculus, and chemistry, which basically requires no math beyond the simplest algebra, run concurrently with calculus, I never understood.


But as soon as we got to freshman physics addresses these things correctly. I happen to still have the book, "Physics" by Paul A. Tipler. Chapter 2 is "motion in one dimension" - all of 20 pages. Chapter 3 is "Motion in two and three dimensions". Displacement vector is introduced on the second page of this chapter; the velocity vector on the 7th page; the acceleration vector on the 10th page. So there are no more than 20 pages of this almost 1000 page textbook, between the introduction of acceleration as the second derivative of position, and the introduction of vectors to represent acceleration. This is a freshman physics book, mind you, not an advanced text. Surely freshman physics students can remember the basics of vector arithmetic for 20 pages?


Tipler, Paul A. Physics. Worth Publishers, Inc., New York, 1976.


So it certainly is just a tiny bit disingenuous to say you have "yet to see a physics book that incorporates vectors and acceleration". Maybe you have not seen such a textbook, but everyone that attended the same freshman physics lectures I attended, saw it.


As far as whether calculus is necessary to study physics (mechanics) I would say, only if you want to understand what you are doing, rather than simply plug numbers into canned equations. Given that the whole reason calculus was invented was to aid the analysis of problems in mechanics, which could not be properly understood before that, studying physics without calculus results in, as I noted before, having to tie yourself into knots to explain things (like 1/2 at^2) that are dead simple to understand with the aid of the calculus.
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Old 01-07-2019, 01:58 PM
 
Location: Washington state
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Believe me, I looked at calculus and physics books. I checked them out of the library and looked at them in bookstores to see if any of them could explain some things I didn't understand. All of them followed the same basic pattern. I didn't see a great deal of difference in any of them.

I looked at The Mechanical Universe, the only physics book I own right now (it's not your average physics book because it was meant to go along with the TV series). The first chapter is the Introduction of Physics, chapter 2 is The Law of Falling Bodies, chapter 3 is Derivatives, chapter 4 is Inertia, and chapter 5 is Vectors.

This is a book I picked up years ago. It was never a book I used in any calculus or physics class.

I see vectors the way I see Simpson's Rule in calculus. There were two pages devoted to Simpson's Rule in my calc book and nothing else. We discussed Simpson's Rule for maaayybee half an hour in calc class. The teacher and the book never mentioned it again. In retrospect, I think it was one of those things that's important to know. But was it incorporated into all the other info in the book? No. Could it have been left out with no detriment to understanding calculus and the rest of the book? Yes.

I can see the point of calculus. One of our first physics problems had to do with dropping a ball from 40 feet and getting the time till it hit the ground versus dropping it 80 feet. Our teacher went over the homework and several students simply got the time for 40 feet and doubled it, which isn't right since the time accumulates over distance as the ball falls. Using calculus makes understanding that much easier.

edited to add: I can't speak to any math or physics classes in high school. I think I took my last math class in 8th or 9th grade. That was basic algebra and I flunked that completely. I never took another math class before I graduated by passing a math equivalency test. How I passed that, I'll never know. As I said, when I went back to community college at the age of 25 or 26, I started with a basic arithmetic class and found I had no idea how to do anything beyond basic adding, subtracting, multiplying, or division.

Last edited by rodentraiser; 01-07-2019 at 02:11 PM..
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Old 01-07-2019, 03:21 PM
 
253 posts, read 59,318 times
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For me the problem with swapping rigor for "fun" is that important details tend to get lost and information incorrectly communicated. I can't count the number of times I've seen scientific reporting show glaring errors, all in the name of entertaining the public.
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Old 01-09-2019, 08:42 PM
 
Location: Washington state
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Originally Posted by rjshae View Post
For me the problem with swapping rigor for "fun" is that important details tend to get lost and information incorrectly communicated. I can't count the number of times I've seen scientific reporting show glaring errors, all in the name of entertaining the public.
True, that. But a lot of that is because reporters and people today don't educate themselves on issues, either. I can't count how many times I've read a story, researched it on the internet, then had to make numerous corrections to people who only took their facts from the story. The raid on the FLDS in Texas was one such example. People only go for the soundbites these days.

At the same time, you could look on it as learning to ride. Learning centered riding and dressage techniques will come after you learn to stay on the horse, not before you get up on it.
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Old 01-11-2019, 05:21 PM
 
253 posts, read 59,318 times
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Quote:
Originally Posted by rodentraiser View Post
edited to add: I can't speak to any math or physics classes in high school. I think I took my last math class in 8th or 9th grade. That was basic algebra and I flunked that completely. I never took another math class before I graduated by passing a math equivalency test. How I passed that, I'll never know. As I said, when I went back to community college at the age of 25 or 26, I started with a basic arithmetic class and found I had no idea how to do anything beyond basic adding, subtracting, multiplying, or division.
I happen to think that teaching probability and statistics in grade school would be much more beneficial to most people compared to algebra or geometry (and probably a whole lot more fun too). I understand the importance of algebra for STEM careers, for example, but otherwise you're much more likely to encounter statistical data in the news than abstract math. Critical thinking skills about statistical studies on health topics, for example, can be vital.
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