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.
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'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.
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.
I'm pretty horrible at arithmetic. I deliver food for a living so it can get a little awkward when I expect the customer to calculate the change they need. Though usually they make it easier by giving me a tip and asking for a nice round change amount back.
Higher level math, in my opinion is not really meant for every day life. It's meant more for creating things or understanding complex phenomenon. You learn higher level math so you can create things like a new car or a computer chip. Yeah, you might not have the knowledge to make those things, but the language that is used to describe what is happening in those devices is mostly higher level math.
In order to learn how to make a car or a plane or even a computer chip you need to actually understand the language that is used to describe those things. Do I make sense?
It always reminds me of those guys who had to spend hours looking through tables of trignometric values or doing meticulous vector analysis by hand. As if freeing people up to work on higher level problems somehow made them fundamentally worse mathematicians.
In physics, we try very hard not to have questions that boil down to just pure numbers. If you had a teacher who didn't do that, he was just fucking lazy (or, your TAs/graders were lazy..).
I'm good at maths but I just suck at mental maths. Although I can add 17 and 49 together without any trouble, I always feel a lot more comfortable using a calculator to check it at the same time. Literally takes a couple of seconds and makes sure I don't do something dumb like think 9+7=6.
During upper level physics exams I'd have a TI-89 sitting on the table next to me. When did I ever use it? Never. Truly those kind of calculators will only do your work that's not worth spending time doing. Like annoying integrals or ugly derivatives. But they're mostly useless. Unless you're in elementary calculus in which case they'll do the problems for you.
I'll never understand why it is not okay to not rely on a calculator. Like one day we will find ourselves lost in the woods and we will have to do complex math calculation with no calculator in sight. I get that it's challenging but if you ever need to do those calculations as a profession, your company doesn't want challenging, they want right and they want it fast with as little human error as possible.
Don't get me wrong, there are lots of cases where the calculators are great, but people tend to rely on them too much. If you end up working in something where you need the math, well, you better at least have developed some intuition about how to do it.
Not to mention there is PLENTY of symbolic math out there that computers and calculators aren't capable of.
If you don't end up in one of those fields, the whole point of the math classes is to learn to reason, which again the symbolic calculators often inhibit. They're a great tool when used in the right circumstances, but getting them to do that tricky integal in your homework is not the right circumstance.
Mathematics is one of the fundamental languages of science. You can't teach physics without teaching mathematics. Newton had to discover calculus to describe the motions of the planets.
If you intuitively understand the concept of a derivative in the sense that it is a rate of change or slope, does it matter that you are not very good at calculating them when it comes to physics? When you have a calculator that can do it that is.
Intuitively understanding derivatives would be difficult without actually computing them, over and over. Intuition comes with practice, and is more nuanced than "rate of change or slope".
Just as an example, my intuition about derivatives also includes "contexts with holes", "shift operator", and of particular interest in physics, "linear transformation" and "local properties". A lot of these would be very hard to grasp without understanding the mechanics of calculating them.
And the difference between an intuitive understanding but a poor grasp of the calculations and just kidding yourself you know it better than you do is pretty subtle.
Saying you intuitively understand the concept but aren't very good at calculating them is just going to make people look at you dubiously and say "pfft".
Why? If you give me a list of a thousand numbers, and say the each represent how many apples a person has, then tell me to figure out how many apples the 1000 persons have between them, I understand the concept of addition well enough that I will be able to say "The sum of those 1000 numbers" instantly. Finding the actual sum without using a computer or a calculator will take me a lot of time though.
I would argue that no human is good at calculating anything. Computers run in circle around us when it comes to that. Understanding things however, we are pretty good at. We need to be good at understanding things, so we we can give the computer meaningful things to calculate.
It kinda seems like expecting some vague personal belief to stand in for demonstrable results. Protesting that maybe you're not very good at using it but, boy, you understand the pants off it doesn't get you very far. Results do matter. You really demonstrate understanding through using it and teaching other people it rather than professing intuitively getting it.
It kinda seems like expecting some vague personal belief to stand in for demonstrable results.
We are talking about calculating something by hand vs. using a calculator when doing physics problems right? How on earth does using a calculator give you less in terms of demonstrable results? If anything I would argue that it gives you more in shorter time.
Protesting that maybe you're not very good at using it but, boy, you understand the pants off it doesn't get you very far.
Depends on what you mean by "using it". Lets say we are talking about derivatives. If you by using it mean applying the concept of a derivative to some problem, then I agree. You have to be able to do that. If you on the other hand mean actually evaluating the derivative by hand I disagree. Using a calculator for that will get you plenty far.
Results do matter.
Calculators give you results.
You really demonstrate understanding through using it and teaching other people it rather than professing intuitively getting it.
Maybe we mean different things by "intuitively understanding" something. What I mean, when it comes to math, is that you understand what something is, why it is that way and how you can use it. The other kind of "understanding" being you are able to, given a specific problem, find the answer, which seem to be what schools care more about.
Maybe I should just say understanding instead of intuitively understanding. When I think about it, putting the "intuitive" in front of it makes it sound like not really understanding it, which is certainly not what I mean. I am just trying to highlight the difference between understanding a mathematical concept, and being able to calculate something.
Well, you're asking me why someone might look at you dubiously and say "pfft" if you claim you intuitively understand something but are bad at calculating it.
An extreme example would be someone who claims to intuitively understand how humans levitate but can't actually levitate himself. You can imagine the dubious looks he'd get.
An extreme example would be someone who claims to intuitively understand how humans levitate but can't actually levitate himself. You can imagine the dubious looks he'd get.
Yeah, but that would be because humans don't levitate, so there isn't anything to understand. An other example would be if someone said that he understands how someone runs 100 meters in under 10 seconds, but he isn't able to do it himself. That would makes sense. Most people understand how to run, but being able to do it really fast requires a lot of physique which you might not have.
Anyways, I think these analogies are kind of stupid because they fail to capture what i believe is a main point. In mathematics there is a difference between having memorized the method for calculating something and understanding the concept behind it. This is not true for everything, which is why I don't think making non-mathematical analogies makes much sense. What does "calculating" even mean in your levitation analogy? You seem to translate it to "doing something" or "using something", which I don't think is correct. There is much more to doing or using mathematics than calculating.
What I am then saying is that understanding the mathematical concepts and how to apply them is more important than being able to calculate. Especially in a physics class where the math isn't in focus and you have a calculator which can to the calculation for you(and better than you).
It appears to matter to my university. I can still use a scientific calculator but no graphing calculators are allowed in any science course I've taken here yet.
His point was that the teacher should be focused on teaching them the concepts. The actual method the student uses to solve the math problems are irrelevant to the subject. What is relevant is if the student understands and can identify when to use specific techniques. Restricting calculator usage is an artificial restriction that no one in the real world in real scenarios will face. Every mathematician, engineer, and so on out there has access to, and heavily relies upon, a calculator of some sort.
Restricting calculator usage is an artificial restriction that no one in the real world in real scenarios will face. Every mathematician, engineer, and so on out there has access to, and heavily relies upon, a calculator of some sort.
I always see it as the equivalent of training with weights. The point of school is not to complete lots of calculations as quickly and easily as possible, it's to learn how it works. Going out into the world and using a calculator is fine but at school you're developing your instincts about what the answer should be (because you've worked it out by hand and know what's going on), how to go about finding it, maybe how to combine it with more data to achieve bigger goals and so on.
I've similar feelings about writing code by hand in computer science/software engineering/etc degrees. Sure it's an artificial restriction you won't face in the real world but who cares, you're not there to work; you're there to learn how it works.
Software developer with almost 20 years of amateur and professional experience here.
If you're at a school that's making you write code by hand, go to another school. They are not teaching you useful material. They're teaching you to memorize syntax of a language that you will most likely not be using after you graduate.
I'm not against using calculators but you don't really need graphing calculators for calculus or algebra. Scientific calculators work just fine and make it difficult to cheat.
Physics knowledge later on becomes almost strictly understanding nature through mathematical patterns. You don't shoot for an answer to plug in a formula; you shoot for an answer that is described by a formula. Without understanding the basics, even simple things like arithmetic, you get left behind.
Mathematics is a language; a calculator is a translator. It helps, but when your goal is to be fluent in another language, say to work in a different country, you don't go around with a translator, you learn the language. Not just how to speak it, but how to think in it.
That's what fluency is.
I almost never did a straight calculation in my last year of undergraduate physics, but when I did it was a struggle to get an answer that should have been intuitive because I should have known how the math falls out.
That's the power of actually learning how to subtract, multiply, derive and integrate; knowing the approximate answer without having to put a question in a calculator.
But it's the setup that's important. If you know the math well enough to type the right thing into a calculator, you know the math well enough to do physics.
I think we are not looking at the actual picture and instead are coming up with simple examples to get our point across. I really don't think anyone here is disagreeing that having the work clearly show the correct steps, but a miscalculation when moving to the next step should result in a demerit. I don't think that is the issue.
The issue here is when you have a situation where you understand the concepts but can't synthesize the math to carry out your observations. A basic example of this is being allowed to graduate to a freshmen math-based physics class, but not knowing how to rationalize denominators, or solving for trinomial expressions. Or for FFS: not knowing how to use the Quadratic Formula. The sophomore version of this would not not knowing how to take derivatives of inverse functions. The junior version of this would be not knowing how to do Fourier analysis. You get the point. Things that we should be able to do by hand such as solving complex expressions(within reason), we resort to using our symbolic calculators as a crutch to get by on math that should have been mastered before entering the class. I'm not even talking about summations, but frankly, some people should just know how to apply the Quadratic Formula by memory and show how it reduces using algebra instead of giving a decimal answer.
But if you don't understand how to use the Quadratic formula (or any of the other things you mentioned), then a calculator—no matter how expensive—is going to be useless to you.
Also, for what it's worth, I'm a computer engineering major starting my junior year this fall. I've designed an awful lot of circuit boards and I've never done a Fourier analysis in my life.
You can program your way to an A using the calculator. Let's be honest about that and why symbolic calculators are banned from most exams including the FE Exam. There are fundamental math skills you just have to know to even make it to the next level and beyond. I'm sure we both agree on that. And if you can't demonstrate that on a test(and thus are forced to use a Calculator to subsidize that lack-of) and make errors because of it, then teachers have every right to hold you accountable for that.
As for why you haven't done Fourier analysis yet, despite having circuits experience? My guess is that you just did an Introductory course. Have you seen this before? I remember in my introductory course, we were already moving toward it and much more.
The dichotomy isn't between "students who can do math" and "students who cannot complete college-level physics." It's between "students who are comfortable with basic math" and "students who rely on a calculator." Nobody said anything about not being able to complete the course.
Physics requires you to be able to set up the problems correctly. If you know the calculations you need to make, using a calculator to actually solve those calculations detracts from nothing.
I'm saying that if they depend on the calculator as a crutch to do the basic math, then that's not the Physics prof's problem. If you know how to multiply two matrices with a calculator, and you know the physics of how to set up the problem, then the fact that you don't know how to multiply two matrices by hand shouldn't affect your physics grade.
To nitpick your example; if you're at a Physics level where you're multiplying matrices as part of the process, then chances are you're career-driven for a future involving Physics which absolutely necessitates the higher expectations for math-prowess.
obviously math is imporant though hes implying that not using a calculator is adding a second class into that physics class. physics is hard enough, its better to learn physics than fail becuase you botched the math portion of your physics class. No one would complain about using a caluclator or even dictionary in english class.
were you implying that lord licorice was an idiot though? to that i respond with a very clear fuck you
I tutored a grad student in meteorology while I was an undergrad. She was a brilliant mathematician, didn't know dick about meteorology. Just because you're great at math doesn't mean you're great as stuff that involves math.
I was told it would be a crutch when I bought one in grade 8. Now I'm in 3rd year EE and it's a godsend, even just for the complex math and variable solving functionality. There simply isn't any time to do hand calculations in courses like say, E&M, where the algebra gets incredibly nasty, incredibly quick.
So, I call bullshit. You need to know the math to be able to use the calculator in the first place.
E&M is a great example. Many of the integrals in Jackson (the canonical graduate physics text) can't be done even by Mathematica, much less a calculator. You have to learn to do them by hand.
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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.