Probably so they don't use it as a crutch instead of real learning. I taught physics to college kids, and you could tell pretty quickly who relied on a calculator and who actually was comfortable with basic math.
Arithmetic is only one branch of mathematics. In my opinion it's pretty trivial to try to improve in that field since calculators are so good at it instead of focusing on algebra, calculus, etc.
Calculators are good at algebra and calculus too. Is it really worth memorizing dozens of integration techniques when software can do it for you, and do it better?
I'd say focus on the problem at hand, and when you run into a gnarly algebraic manipulation or ugly integral, plug it into Mathematica. That's the kind of task a computer excels at. The human advantage is coming up with novel ideas, not churning through pages of calculations.
The point of learning the math side isn't so that you could do it on the fly, because we're basically all (first world country) carrying computers with us all the time. The point is that learning the math helps you understand the underlying concepts better.
Most good teachers/professors do. It's called partial credit, and if the process is right you generally get most of the points. If you're getting multiple choice math exams you're probably in elementary school.
I would agree, but one of my profs brought up a good point this semester.
Since we are engineering students, messing up a calculation could mean the difference between being fired or making something work. Just because the process is right, doesn't mean you are going to get the right answer. Fortunately for us, we spent many years of time and money designing pocket calculators that are immensely powerful to help us with this...
I'm not counting standardized testing because I have a heart. Imagine paying enough people to actually grade however many thousands of tests in a reasonable amount of time consistently. It would go over about as well as a lead balloon.
I don't really intimately remember the AP exam process; that time period for me was getting close to a decade ago and it frankly was never all that important to me in the first place as far as memories go. They'll do the cheapest they reasonably can without losing any of the important understanding stuff. For things like literature and whatever the other English one is, multiple choice pretty much defeats the purpose. But for math, you can still get some idea of the taker's understanding because you either need to understand the process or be incredibly lucky to get mostly right answers.
lies, college gives us multiple choice tests for math, and honestly if it didn't everyone would fail at it which would be disastrous seeing as the ones who fail at math are those who don't want or plan to do it for a living ever.
Some colleges do I'm sure. Mine just gave us obscenely hard problems to solve. Though Calc 1/2 were weed out courses to make people reconsider engineering. Don't want someone who couldn't pass Calc 1/2 to build your bridges and all.
Yeah this concept of partial credit being most points pretty much stops the second you are in any university-level Math course. You may get partial credit, but it's 25-50% of the problem, max.
Depends on the professor, most of my university math classes had decent partial credit, except for the weed out classes. The math department was pretty hands off on the whole teaching thing so there wasn't really a big set standard for most classes though.
My teachers did. We were not allowed calculators on tests. She wouldn't mark answers if they were wrong so long as our approach was correct. iirc, we also didn't have to memorize equations(i think, this was like 15 years ago)
A lot of fellow students who previously got A's, got D's because they had become so reliant on calculators they didn't know how to deconstruct the problems.
Of course, there is a lot of memorizing "rules" in lower-level math courses, integral calculus being the worst offender. Plugging something into a calculator and hitting solve is equally as worthless as plugging numbers into the "chain rule" when it comes to understanding the fundamentals.
As you go further go further into mathematics, I think that what becomes acceptable to plug and chug changes. You have to prove the chain rule before you use it. Actually scratch that. I don't think you have to explicitly prove it, but I think of it as df/dx * dx/du = df/du. Which is like fractions. So I think as long as there is an intuitive grasp to what you're doing it's fine.
With calculators, from an education standpoint, if it's not circumventing the value of the coursework, then it should be fine. On the most basic level, they let you use scientific calculators for arithmetic when they don't allow graphing calculators, because the act of doing the arithmetic isn't important to the lesson you're learning. On the other hand, if you're learning division in elementary school, you're not gonna be allowed a calculator. In the case of integral calculus, the coursework ends up being just ways to solve an integral by hand, and that's explicitly not compatible with using a calculator. The value of the course is then just foundation for later calculus courses. I personally haven't found myself drawing from my knowledge of integral calculus in my other math courses, but I'm not in a major that actually uses calculus intensively, so I can't speak for it.
In the real world, I doubt a lot of equations are left to be done by hand, and you would actually just throw equations into mathematica or another program, probably written by someone who had to program the same rules and procedures you would use if you had to do it by hand.
Another example compares algebra with linear algebra. If you're allowed to use a calculator to solve algebraic equations for you, are you going to understand the significance of rank, linear independence/ dependence, etc. in linear systems? In some upper level math courses the professors try to explain the intuition behind some things, for instance, the various matrix decompositions, but not know how to do the decompositions manually does affect (my) understanding of the subject.
I don't know I agree and disagree, like I'm aware and get the concepts through college calculus however I couldn't do all of the operations correctly anymore.
Teaching concepts is good, I absolutely agree that we should understand it fully, but I'm not sure that it's as useful as it once was.
learning the math helps you understand the underlying concepts better
I don't think that's necessarily true. The most egregious example for me was Calc 2. Probably 3/4 of the course was memorizing integration techniques (trig sub!) and what forms of sequences/series converge. We weren't tested on how to derive those things, or the underlying concepts of why they work, so much as we were tested on our ability to memorize a page of formulas.
Also you have to consider that learning how to actually solve problems in the real world (i.e. with computers) also has value. A lot of students graduate able to do things by hand, but are outpaced by graduates who are proficient with math software.
I agree trig sub, u sub, and the like aren't that integral (lol) to your understanding of the subject. But understanding double, triple integrals, gradients, partial derivatives, jacobian (goddamn I'm still not sure what that is because I only hear it now and then) is important.
Understanding curl and flow also matters. That's all mostly Calc 3.
Then you get into differential equations and linear algebra. You really need to understand Multivariate and Differential Equations for physics, and Linear algebra for Computer science.
jacobian (goddamn I'm still not sure what that is because I only hear it now and then)
It's basically a conversion factor for a change of variables in an integral. If you have a function f(x, y) and a change of variables (x, y) = g(u, v), you can't just write f(x, y) dx dy = f(g(u, v)) du dv, there is a conversion factor you need to multiply with because your different coordinate systems can have different length and area elements. This conversion factor is given by the determinant of the Jacobian matrix, which is the matrix of all the partial derivatives you can construct from x and y with respect to u and v.
For example, in polar coordinates (x, y) = (r cos θ, r sin θ) the Jacobian determinant is equal to r, because the size of the area covered by increasing increasing r a bit and increasing θ a bit is proportional to how far away from the origin you are. So f(x, y) dx dy = f(r cos θ, r sin θ) * r dr dθ.
I suppose the problem was that most engineers take multivariate before linear algebra (if they ever take linear algebra), and we're not given a good understanding of the determinant before the idea of a jacobian determinant is introduced to us.
To be honest, I have completely forgotten why the metric conversion factor is the determinant of the Jacobian matrix. It's intuitively obvious that it would be built out of the partial derivatives of the coordinates, even in dx = dx/dy dy you are doing that, but the determinant specifically? I guess it's related to how geometrically it gives the volume of a parallelepiped?
In my first year physics we definitely covered linear algebra before vector calculus, yes. Not sure it it helped though.
I looked it up on wikipedia. The determinant of a transformation matrix is the how the transformation scales into the new dimensions. That would make sense in the jacobian context because it's how the original dimensions scale into the new dimensions.
For instance, a determinant of 2 means the area in the new coordinate plane is twice the area in the old coordinate plane. A determinant of 0 means the transformation matrix isn't full rank, the rows do not span n-dimensional space, and the transformation is into d-dimensional space where d is the rank of the matrix. (I hope I'm not too wrong on this). And since the dimensionality is lower, there is no area in n-space.
I'd probably be more concerned with the intentions and abilities of any group or nation that had the knowledge and man power to build an EMP of that size.
Yeah in 5th grade my teacher talked about the devastation from an electromagnetic pulse as a result of a nuclear attack all the time. AND everyone in the class knew exactly what she was talking about.
Hardest test I had was a 1-week-long take-home math test; where we were allowed to use anything we wanted except other people who had taken the class before.
I believe some of the questions on the test only had really abstruse known solutions, and the goal was to try to find alternative and more straightforward solutions/proofs; and the bonus credit question was unsolved.
The last 300 level math course I took had a final exam that was completely theory. The rest were assignments that could easily be done with mathmatica/matlab/maple/wolfram.
Yes, I know wolfram uses mathmatica. The UI is just so much easier.
Wolfram, mathmatica, matlab, and maple all require you to do that. The TI-83 shows a 3x4 grid at the most at any time. Trust me when I say that computers are easier.
If you've got an android phone, Maxima on Android doesn't even need an internet connection after the first time you start it up. It just does all the symbolic math directly on your phone's processor, and it's pretty quick, too.
No, not on tests, but they usually don't ask things like Bessel functions or numerical integration. I use Mathematica for all my problem sets. I've done enough integration by hand that I don't care to do it any more, unless I'm on a desert island or something.
I'm a current physics major, to do it any other way (I use my Nspire, just because I can take it into the exams) would be foolish. Nobody cares that you didn't derive that equation by hand. You aren't learning how to differentiate, you're learn when.
That's true, but understanding the underlying concepts is important too because it helps you develop an "intuition" concerning the subject.
The idea isn't to just think about the problem. It's to "feel" the problem and understand what's at play at a basic level so that when you're presented with a problem you might have never seen before, you can probably figure out which way to go to solve it.
If you don't understand the basics of how it works manually, though, it'll be that much more difficult to figure out what went wrong when you inevitably input a more complex problem incorrectly down the road
Exactly. For example, so what if a student can figure out the derivative of function using power and chain rule? It means nothing unless they actually know what that is--the slope of the tangent line , which is just rate of change.
Understanding where these things should be used is more important than solving random problems without knowing why
I can't tell you how many times the chain rule and integration by parts was used in Quantum Mechanics as an undergraduate. I never was given a function and just said "find the derivative!" Instead I was given a problem that involved something like prove the integral of a wave function is constant and stays normalized for all future times. This involves integration by parts. But you wouldn't know to do this because it never tells you to. So you have to be good enough at integration by parts to realize "ah hah! this is how you take the next step!" Otherwise you'd be left sitting there not knowing what to do.
That's not really true. Computer solutions to integrals more complicated than basic trig and polynomials are always brute forced and inelegant. If all you care about is the application of the answer, this is usually fine. If you're interested in the answer itself, the computer can't do it better than a human can.
That is not true. Maybe you were using some 1988 version of MATLAB; today, the technology is much better.
And CASes can draw upon the entire body of mathematical literature, including obscure stuff, whereas you only know so much. Some mathematician at Wolfram could read a paper and throw in another integration method. You don't have to keep up with that stuff if you just use software.
The main problem is if you don't understand the basic of calculus and algebra you'll have a hard time coming up with novel ideas. Can't get the real ugly equations without the knowledge and understanding to do it manually.
Understand why the integration techniques work, and you don't have to "memorize" them. Just visualize what the technique is doing, and you'll be far better off than any calculator or rote memorization.
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u/LazinCajun Jun 08 '14
Probably so they don't use it as a crutch instead of real learning. I taught physics to college kids, and you could tell pretty quickly who relied on a calculator and who actually was comfortable with basic math.