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Background/rant:

I am in charge of teaching our single quarter course on vector calculus (don't ask me why the department head thinks the area can be covered in half a semester). The two biggest groups of students in that course are A) math/statistics majors, B) physics majors. There are smaller groups, but those two form the bulk.

The course is designed to be taken roughly by the sophomores. During their respective freshman years the math/statistics majors have seen a variant of 1st analysis that concentrates more on concepts and proofs as opposed to learning methods of calculus. The physics majors have seen a variant of the same, but with emphasis on the methods, but also a touch of epsilons, upper and lower sums when defining integrals and such. Furthermore, the physics majors have taken a year long "methods" course. I am definitely concentrating on giving the students insight/intuition on how to apply the tools of calculus/elementary analysis in several dimensions.

In the past the students, math majors more than the others, have come forward and asked, whether they can use the fancy calculators they bought for their high school work in the finals. The general department policy is to only allow "calculators from the 70s-80s" (guess what decade the senior staff, yours truly included, did time in high school!): no graphical output, nothing symbolic... The thinking has been that the students need to learn to do the integrals by hand, just in case their survival on a desolate island depends on... Yeah, right. So to make allowances for all this I have had to dedicate a bit of time to extra exercises (outside of lecture hours) just to make sure that the math majors also know how to evaluate "cheat sheet" integrals like $\int_0^{\pi/2}\cos^mx\sin^nx\,dx$. Otherwise they would be helpless when calculating integrals in spherical coordinates and such. This is also time that would be better spent learning something more essential. Such as figuring out how to set up the boundaries of an iterated integral – something that is essential when doing several variables.

The idea.

I would be happy to accommodate for this sorry state of affairs, and offer the students their choice of a final exam. Either their fancy symbolic/graphical calculator or the standard edition. Making the two variants and their grading absolutely comparable/fair is not easy, but I think I can manage.

Is it a bad idea to offer variants of a final exam based on the type of allowed calculators? Have you ever tried this? What pitfalls did you meet? Do you foresee any?

Remarks, sporadic thoughts:

  • For a long time I have been rewarding the students for performing "reality checks" on their answer. Say, if I ask them to calculate the volume of some object that (judging from the plot I gave) is on top of the unit disk, with height varying between $1$ and $3$, they should expect an answer in the ball park of $2\pi$. I have been gradually making such mental checks obligatory parts of the answer, and can use this to compensate.
  • On the other hand, I do suffer from pangs of guilt. Am I just sweeping things under the rug instead of making sure the students know how to calculate basic integrals? Irrespective of what was covered in their freshman courses.
  • I asked the colleague in charge of organizing the exams, whether he can arrange the proctoring of such "dual" exams. He did not foresee any problems. We have several exams simultaneously in the same auditorium, and it is trivial to arrange the seating in such a way that the students sit on distinct rows according to the type of allowed aids, making rule enforcement feasible.
  • I have no idea yet, whether the students would actually buy this idea :-)
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    $\begingroup$ "don't ask me why the department head thinks the area can be covered in half a semester" --- If the course involves covering something like Div Grad Curl and All That by H. M. Schey, then half a semester is probably sufficient time. However, if the course involves covering something like Vector Calculus by Jerrold E. Marsden and Anthony J. Tromba, then two semesters is probably too little time! $\endgroup$ – Dave L Renfro Aug 7 at 18:04
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    $\begingroup$ Why not just make nearly all of the problems involve simply setting up appropriate definite integrals? Due to time constraints, that's how I handled a lot of the "applications of definite integrals" problems on in-class tests when I taught (U.S.) 2nd semester calculus (volume of revolution, work, fluid pressure, center of mass, moments of inertia, etc.). Indeed, even when students had to carry out the computations, I still graded mostly on their set-up work (including supporting hand-sketches, etc. for partial credit purposes when they made various minor oversights or major conceptual errors). $\endgroup$ – Dave L Renfro Aug 7 at 18:23
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    $\begingroup$ It is really really sad that departments still think calculators should be allowed/required for exams. There is no justification whatsoever for creating exam problems which either require or are facilitated via calculator use. Call me a grumpy old man (2 out of 3 aint' bad), but I remain adamant that nobody learns math by way of a calculator -- not to mention that nobody post-school uses a handheld anyway. Either it's pure math, done with pencil or with Mathematica / MacSyma, or it's complicated enough to require R/NumPy/Julia/MATLAB. $\endgroup$ – Carl Witthoft Aug 8 at 14:28
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    $\begingroup$ I've had students ask about calculators because they says it is literally impossible to get a calculator that meets the requirements of some exams - they need to have trig functions, ie a scientific calculator, but those now sold can also do a fair amount of calculus and solving equations, which is disallowed. $\endgroup$ – Jessica B Aug 11 at 15:41
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    $\begingroup$ "The thinking has been that the students need to learn to do the integrals by hand, just in case their survival on a desolate island depends on..." Personally, I think the real thought, which not many people have heard explicitly, is that if you reply on calculators for the lower-level stuff, you don't have the mental capacity to actually follow a longer argument. The rote calculations have to be automatic and subconscious so that the concentration can be spent on the overarching logic. THAT'S why elementary school students should learn the multiplication table by heart, for instance. $\endgroup$ – Arthur Aug 11 at 19:45
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Let's start by saying that I strongly advice against such a dual-exam. Even if you and everyone involved in the planning think it is fair, students might think differently. In this way, you open up the floodgates for grade complaining. They might not succeed, but even the fact that some might try will cost you a lot of time and possibly reputation.

Now, in the hope that you are still planning your class and aren't already two weeks before the exam, let's tackle the main problems.

First of all:

In the past the students, math majors more than the others, have come forward and asked, whether they can use the fancy calculators they bought for their high school work in the finals.

Of course they have, and you can't blame them. They learned in high school that life is easy, you just enter the information into the calculator and it returns the desired result. Why should they do it the hard way when they don't have to? Thus, your lecture needs to show them the why. Include examples where the calculators fail or, even worse, make wrong assumptions on the parameters included and thus output wrong information.
If you want to, you can also include the calculators. These are mostly stupid beasts, so it should be possible to have an integral that they can't solve, but after applying some transformations that the students learn in class, they suddenly can solve the simpler integrals constructed.

Furthermore, make sure to keep your numbers easy. If I have to compute $$\int_1^2 \frac{1}{x} + 5x^2 dx,$$ then I feel like this should be feasible by hand. If, on the other hand, the same problem comes with numbers like $$\int_1^2 \frac{7.56893}{x} + 5x^2 - 1.7653\cdot 10^8 dx,$$ even though the steps to take are the exact same, I can fully understand if no one wants to do it by hand.

Furthermore, you can ask for intermediate steps and explanations in the computation, something that a calculator can most likely not provide.

So to make allowances for all this I have had to dedicate a bit of time to extra exercises (outside of lecture hours) just to make sure that the math majors also know how to evaluate "cheat sheet" integrals ...

And why was this outside of lecture hours? I'm sure everyone would profit from learning how to do it. Even if they already knew beforehand, repetition is always good. If you can't do it in a single repetition lecture because the differences are way too big, you need to reevaluate the course outline and the prerequisites. See also the next point about that.

I am in charge of teaching our single quarter course on vector calculus (don't ask me why the department head thinks the area can be covered in half a semester).

You are in charge of teaching that class, so you should definitely know how and why to do it in a single quarter. Push back by demanding more time or reducing the content you are teaching. As long as you don't do anything, people will be happy that the students learn that much in a single quarter and don't bother checking if that is even possible.
It is your responsibility as a teacher to make sure that the students have the right knowledge to take the class and that the goals of the class can be achieved in one quarter. If you think that it not possible, something has to change. Not only is it stressful for you, but imagine how the students feel, taking extra lectures and maybe still failing because it is too much.

I don't know how bad it is in your case, I wish that your only problem are lazy students that got too used to a machine thinking for them. Still, I think that meeting their demands is not the right way to handle things. Take time to rethink your lecture so that there is no need for two exams anymore.

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  • $\begingroup$ Thanks for sharing your thoughts. A few comments. I can and will make it necessary for the students to think, no problem. No calculator will tell them what the boundaries of the iterated integral of interest is. In my opinion the calculator would be more like a substitute for the well-known formula for a trig integral as in the question body. The students may not see it that way, though :-) $\endgroup$ – Jyrki Lahtonen Aug 7 at 12:42
  • $\begingroup$ About getting more time. That is an uphill battle. The key professors are all sold on the idea that computer science is more important as a minor than physics, so this is the place where they cut a few credit units to squeeze the entire degree program into the framework dictated by the university. Sad. $\endgroup$ – Jyrki Lahtonen Aug 7 at 12:45
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    $\begingroup$ The second of those integrals looks easier. Even without a calculator, it's obviously $-1.7653\times 10^8$, up to the precision of the givens :-) $\endgroup$ – Henning Makholm Aug 8 at 16:00
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I think this is a bad idea, because it invites students to try to game the system: those who own a fancy calculator suddenly have to decide if they’re better off using it and taking an exam they think will be harder, and those who don’t have to decide if it would offer enough of an advantage to be worth getting one and taking the other version.

Worse, you invite students to decide, retroactively, that they made the wrong choice. They won’t make that assessment based on full knowledge of the exams - they’ll hear “oh, I had such and such problem on my version” and decide that they would have been better off taking the other exam.

Issues like this come up a lot - where there are a couple reasonable ways to structure an exam (with or without a calculator, a few long problems or an larger number of shorter problems, etc), and it’s tempting to offload that decision to students in the hopes that they might make the best choice for themselves. I think this rarely works out; the weakest students are often the ones most prone to choosing badly for themselves, and it invites students to fixate on these discrete, superficially high stakes decisions rather than the core content of the course.

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    $\begingroup$ Thanks for sharing. Yes, potential problems here. Whatever experience I have tells that fancier tools make the gap between better and weaker students wider. $\endgroup$ – Jyrki Lahtonen Aug 8 at 2:54
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Wanting your students to learn how to solve problems, instead of having them memorize some formulas and do basic arithmetic by for two hours, is a great goal imo.

The first step towards that goal is making the formulas available to your students, just as they would be in the real world, but preventing them from accessing the actual solution of the problem (or something that calculates the solution). The simple answer to that is a cheat sheet with the formulas. You can create a curated cheat sheet with all necessary formulas (in your case it would be some of the most common integrals, if I understood correctly), distribute it to your students or somehow make it accessible to them, use it during the course, and allow it on the exam. Your students will be able to focus on the problem instead of the formula, and they will learn how to solve unknown problems by reducing them to known problems (the formulas themselves).

The second step is to remove the calculators, to prevent any unfair advantage from having an extremely advanced and expensive calculator, or arguments about whether this or that model is allowed and if not why wasn't it made clear before the student wasted money buying it. You wanted to allow calculators in the first place because that would remove basic arithmetic from the grading (i.e. eliminate incorrect results that show correct reasoning but basic arithmetic errors). You can actually achieve that to a very acceptable degree by using constants that are easy to operate with, such as small integers. Dirk has already made a great point about that, so I won't elaborate further.

If you find it impossible or unfeasible to write exam questions that don't need a calculator, having the department acquire a number of very basic calculators is an option, though a more difficult one. In the long run it could benefit many other courses and help standardize rules throughout the department's courses, so imho it's not a bad idea at all, but you're in a better position to judge that and the feasibility.

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There are already answers suggesting you to make an exam that doesn't require a calculator at all, so I won't elaborate it here.

But if you are concerned about students with different backgrounds and different expectations about what they know after it ends, I see no issue with splitting it along these lines. I have had a few classes in a similar setup in my years (including one which was for both grad and post-grad students at the same time). The post-grad group got some additional questions to raise their bar and it worked fine.

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Ugh.

There are few things worse than having to prepare different versions of an exam for different cohorts. It's almost impossible to make the exams comparable.

A better solution is to make the exams "calculator independent." In other words, it should make no difference if you have a fancy calculator or none at all.

Here's an example (I've used something like this on assignments): given the value of a function at a few points (but not the function itself), approximate the integral.

Or: given the graph of the derivative, find where the local max and min of the function exists.

(I haven't taught vector calculus, so I don't have good examples off the top of my head, but you get the idea: the questions don't have anything you can input into a calculator, so it doesn't matter if you're using one)

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Just make the kids do a test sans calculators. Knowing how to manipulate the integrals helps when you are reading derivations or papers in physics and engineering. Then you only need one version. The no calculator version.

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