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Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my derivative? No. Is it going to give me my integral? No. It can sure give me the answer to my integral, but will it give me the calculations? and steps? No.

In linear algebra, will the calculator give me the matrix reduced form? No. Will it give me the answer to a set of equations? Maybe, but without work obviously.

Teachers mark the "work", and award part marks for each correct piece of work you show. That work cannot be done by a calculator.

Something that frustrates me is that I am in post-secondary, and that my level of math right now is not basic. So, like many, I tend to forget the basics of math, like adding fractions, reducing fractions to lowest form fast, subtracting big numbers. Sure, I can do all those, but I would be more likely to make a mistake doing these and loose marks then if I were to just use a calculator.

Teachers want to make sure we "understand" the concept, but I ask, does "understanding the concept" really have to do with our ability to "add, multiply"? Are they testing whether we can do basic arithmetic that we have so long been used to performing with a calculator?

When your working with integration and derivatives, you will get ugly fractions, and adding them, finding denominators, is really a waste of time and effort for me, when the calculator can just do that for me. In the real world I will have a calculator, so why not here? As I said before, what are they testing? Are they testing how I can do basic math, or how I can compute the limit, evaluate the integral, calculate the deriviative?

Why?

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This question came from our site for people studying math at any level and professionals in related fields.

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    $\begingroup$ What makes you think that doing the arithmetic on a calculator will be any less error-prone? If you don’t have at least an estimate of what the answer should be, how will you know that you didn’t press the wrong key at some point or make some other mistake in entering the computation? $\endgroup$ – amd Feb 13 '17 at 20:40
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    $\begingroup$ Note that some universities are more lenient than others on computational issues. So for example, in my university, if your final answer is $\frac{1}{3} + \frac{3}{19} + \frac{1}{17} + \frac{1}{13}$ you need not simplify this to $\frac{7898}{12597}$. A "calculator-ready" expression suffices. $\endgroup$ – MathematicsStudent1122 Feb 13 '17 at 21:13
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    $\begingroup$ With a cell phone and a subscription to wolfram alpha, you can get step by step worked solutions to many exercises that might appear on a calculus or algebra exam. In general calculators are more sophisticated than the question supposes, and with access to a decent phone the sky is the limit. What saves the day is that students who need to cheat on exams are also often incompetent cheaters. $\endgroup$ – Dan Fox Feb 15 '17 at 21:33
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    $\begingroup$ Even a TI83 is able to show work, if you give it a bit of help: zoommath.com I've never used it, but I know its "discovery" has been passed around my department. $\endgroup$ – pjs36 Feb 16 '17 at 0:21
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    $\begingroup$ You lost me at the title. Who says calculators are not allowed in these courses? Are you claiming it's some kind of universal rule? $\endgroup$ – Ben Crowell Feb 18 '17 at 3:37
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As a professor/teacher I have some insight. You just answered your own question:

"my level of math right now is not basic. So, like many, I tend to forget the basics of math, like adding fractions, reducing fractions to lowest form fast, subtracting big numbers"

By avoiding using a calculator you'll strengthen the basics.

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In addition to @BobaFret's answer I'd like like to point out following:

You said:

Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my derivative? No. Is it going to give me my integral? No. It can sure give me the answer to my integral, but will it give me the calculations? and steps? No.

This is not true, it really depends on the calculator, and there is a whole range of them, from those that cannot even find square roots to programmable calculators with a full CAS. So it would be unfair to give some people an advantage just because they use a "better" calculator. And just restricting certain types of calculators is also very difficult as some might still have some more advanced functions than others. Another solution would be forcing everyone to use the exact same calculator which is also a bad idea.

Instead it is just a lot easier to disallow all forms of calculators. This makes you really strengthen the basic skills (look at it as a positive thing, since you already mentioned that you forgot how to do some of them) and it also makes you think of elegant ways to solve problems instead of memorizing "brute force" attacks.

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One thing that really freaks me out about students on this site is an apparent inability, at least an unwillingness, to draw ordinary graphs on graph paper, maybe $y = x^3 - 3 x + 2$ or the like. If I comment about the desirability of doing this, I generally add a link from which graph paper can be downloaded (as a pdf) and printed, https://www.printablepaper.net/category/graph

From what I can see, no MSE user has ever drawn a graph by hand owing to a suggestion of mine. When a reason is given sometimes it is "I'm not allowed a calculator in exams," as though the only possibility for getting a graph is a machine.

The end of the world is nigh.

enter image description here

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I'm going to add onto flawr's answer, which added onto BobaFret's answer. :-)

I'm sympathetic, because I know plenty of mathematicians who are not particularly good at arithmetic. However, I think that if a test is not intended to test someone's ability at arithmetic in the first place, and is written well in the second place, it should not unduly tax your arithmetic skills. Expressions should "clean up nicely" in the mathematics exam world. When I write an exam, I tend to try to make it so that as little arithmetic computation as possible is required. That being said, some level of arithmetic is needed. That's unavoidable, but probably not to the level of requiring a calculator.

What's more, as amd's comment implies, there is more to mathematics than understanding the concepts, as important as that understanding is. One should learn not just to apply concepts correctly, but to understand when they are being applied incorrectly, whether it was you who made the mistake or someone else. In "real life" (assuming facts not in evidence), I might be asked to look over someone else's work, either as a colleague or as a supervisor. The ability to perform a quick sanity check to see whether it makes sense that the area of a circle $5$ cm in diameter is $79$ cm$^2$, and to identify the likely cause of any error, is invaluable. (You shouldn't need a calculator to figure out what likely happened in this case.)

Exams in other quantitative subjects, like physics or chemistry, may well allow calculators, because the practical aspects of problem-solving are weighted more heavily. I'm not convinced that the same applies to a mathematics course.

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    $\begingroup$ I don't know any mathematicians that are not excellent at arithmetic unless one understands "arithmetic" the sense of Serre to include quadratic reciprocity, zeta functions, and modular forms. $\endgroup$ – Dan Fox Aug 28 '18 at 13:22
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One major concern is cheating. People can enter information into their calculators that may give them an unfair advantage on the exam, in essence using the calculator as a "cheat sheet". Having a professor verify that a couple hundred students don't have any information stored in their calculators isn't feasible.

There is also an economic argument: a calculator, especially a graphing calculator, is expensive enough that not all students may be able or willing to buy one. If calculators are useful then they should therefore be banned so that wealthier students don't have an unfair advantage, and if they aren't useful then why allow them?

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    $\begingroup$ What about allowing 4 function calculators only? These could be easily obtained inexpensively and can't store information. As an elementary school teacher, for certain topics I distributed such calculators to my students. They thought that the tests would be easier but of course they were harder because there was more thinking! $\endgroup$ – Amy B Feb 17 '17 at 11:38
  • $\begingroup$ This could be quite easily addressed by having the school provide the same calculators to all test-takers, and structuring the exam around that $\endgroup$ – yoniLavi Feb 17 '17 at 17:56
  • $\begingroup$ @AmyB: Do such calculators even exist anymore? Does anyone own them? $\endgroup$ – Dan Fox Aug 28 '18 at 13:22
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    $\begingroup$ "Basic office calculators" do still exist, e.g.: at Amazon. However, the logistics would be monumental. I estimate in my department we might need 2,000 or more to supply all classes testing at once = $10K, plus resources to store, transport, distribute, collect, track, replace, deal with "inventory shrinkage", etc., etc. By analogy, my mother works at a school that supplies laptops to all students, and they need a delivery every single day of new, replacement, to-be-replaced laptops. $\endgroup$ – Daniel R. Collins Aug 28 '18 at 13:48
  • $\begingroup$ @DanFox Yes such calculators exist and I had 25 in my classroom that had been donated by a parent. You can google 4-function calculators to see more information. They generally also have memory, a square root function, and a percent button, BUT nothing else. You can get 10 for $37.50 a reasonable price. $\endgroup$ – Amy B Aug 29 '18 at 6:49
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When I taught, we permitted students to have a symbolic calculator at all times. The implication for students of this is that the questions in the exam take this fact into account. When such a decision is taken, the questions in an exam become more theoric and/or demand more imagination.

For example, in an exam on optimisation, the focus would be on setting up the right function, not on deriving it or solving $f'(x)=0$. Once the possible optimums were found, the reason for accepting or rejecting them was the important part. The presentation of the solution was also important. We did sometimes have exams not using the calculator, for example on derivating or integrating functions.

Another example: I used to have a 2-hour exam which consisted in 1) the class helping me to find the formula for the $n$-th derivative of $x\cdot e^x$ and proving the formula by induction (about 20-30 minutes). Then 2) the students had to find and prove a formula for the $n$-th derivative of $x^2\cdot e^x$ working in teams of 2.

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I very much agree with already posted great answers (I do share most of the same opinions), but I'd like to add something else. Talking about math instruction we're not talking about math only — there's a psychological dimension to it too. And from this perspective, from a pedagogical point of view, I see a lot of harm done by calculators, or rather by thoughtlessly relying on them. In schools where calculator usage is rampant, students are lead to believe that mastering math is equivalent to memorizing which buttons to press on a calculator for each task. So discouraging calculator use is intended to break this misconception and force our students to actually learn math. The sad truth is that for some of them it will be the first time they would actually be forced and encouraged to think with their own heads in a math class! So for pedagogical reasons, we have to do something about it, and forbidding calculators on tests is actually a reasonable solution.

This may sound like too strong an opinion, but I see this kind of thing on a daily basis. Often when I see a student in a Calculus class who doesn't have any conceptual knowledge of pretty much any Precalculus topic, and when I talk to such a student to find out more about their background, I would find out that he/she transferred from some other "calculator-reliant" college or school, where they "did everything on a calculator" (this is an actual quote).

Truth be told, I don't walk my talk. We allow non-graphing calculators on tests in our classes. (Long story...) And inevitably I see students wasting half of their time incessantly punching buttons even to compute things like "$-3+3$" and even "$0*5$" — and I'm not making this up! (Let's not even start on things like "$\sin\frac{\pi}{2}$", which is kinda funny, in a sad way, when their calculator is in degree mode.)

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In a perfect world, after students learn some new technique they would program themselves how to do it into a computer and then they would be allowed to that computer program to solve problems. The problem isn't using a calculator, it's using a calculator that was programmed by someone else. That's not testing the students' knowledge, that's testing the knowledge of the person who programmed the calculator.

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Instructors are looking for M & M : memorization and manipulation. We're training brains, not button pushers. Thus, no calculator on math tests. For engineering/applied courses, a scientific is allowed. Your graphing calculator required in high school? Sell it on eBay. When high schools stop teaching button pushing, education will take a turn for the better.

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    $\begingroup$ "Instructors are looking for memorization and manipulation"? Can you expand on that, please? While memorization and manipulation are certainly aspects of what I expect from my students, that only gets a student up to a passing grade. What I really want is for students to analyze and synthesize. I typically encourage students to use calculators (and computers; I really like Desmos and GeoGebra for teaching) in order to build intuition. I remove these tools from the exam setting only to create a level playing field (and I adjust exam questions accordingly). $\endgroup$ – Xander Henderson Aug 28 '18 at 21:31

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