# Take-Home Examination on Ordinary Differential Equations?

I am planning to give my students a take-home examination on ODE. The main topic that I would like to cover is Linear Differential Equations of Order Greater than One. For example, I will give my students question such as

Solve the following Differential Equation: $$y''+y'=x^2+2x.$$

But my concern is, if the students are allowed to answer such questions in a take-home examination, there is almost no reasonable way that I can prevent them from using some online ODE solvers, such as WolframAlpha or this one.

Since some solvers not only give the solutions but also the details, it will defeat the purpose of this take-home examination. I can make "no Differential Equation Solvers" as a rule for this take-home examination, but I am not sure how to monitor my students and thus enforcing the rule.

Surely, having an in-class exam solves the problem. But if it is not possible, is there a way to reasonably test ODE's in the format of a take-home examination?

Edit: I gave them the take-home exam asking them to sign to promise that they have not used any unauthorised resources when finishing the exam. From my observation, all of them seem to have finished the exam honestly. By the way, I work in a small religious college where most students have very high moral values.

• You could have two components to the assignment. A take-home portion could be assigned knowing that students have access to technology. The questions could be conceptual in nature, asking students to explain an idea; or they could involve heavy computations that you wouldn't ask them to do during a timed, paper exam. Meanwhile, an in-class portion could ask them to solve simple examples by hand, like the one you mentioned. Dividing it in this way makes it clearer to students what they'll be expected to do. Oct 16 '19 at 18:55
• @BrendanW.Sullivan, thanks for the comment! But even for the heavy computations they can let a computer do the job for them. For conceptual ones, google will probably tell them the answer. Oct 16 '19 at 18:58
• @Namaste, I can do it but I am running behind my schedule of this course and would love to save one class period to cover some materials if possible. Oct 16 '19 at 20:51
• I'm glad you found you have honest students. In our experience running online calculus, typically half the students try to cheat using an online wolfram-alpha type site. There are a lot of websites which show steps these days. If accreditation bodies were doing their due diligence there ought soon be an expectation that we proctor out of class tests by some reasonable method. Too many students do not have honor for your solution to scale. Doctors have to have licenses, I have to insure my car etc. it ought to be an expectation that universities police cheating severely. Feb 27 '20 at 23:42

This is not really a math problem, it's a social problem. Some schools, such as West Point and Cal Tech, have their own honor code systems for this sort of thing. From what I understand, they work very well. However, most schools do not have any such system. Social scientists and psychologists have studied what factors promote or prevent cheating. Cheating is promoted by the following factors:

• a perception that everyone does it
• having a single, high-stakes exam, or a small number of such exams, rather than systematic, ongoing assessment
• a lack of self-efficacy.

At a school that doesn't have an honor code, it's going to be hard to avoid the issue that everyone will think they have to cheat in order to keep up with others who are cheating.

You don't say in your question how many exams you plan to give this term. For example, I'm giving 8 short exams over the course of 15 weeks in one of my classes this semester. If I was having the problem, as you describe, of running low on time, I would just cut one exam and do 7 exams. If you're doing only a couple of high-stakes exams, such as a midterm and a final, then you have one of the risk factors in place, and you're asking for trouble if you make the midterm a take-home.

I think you're kidding yourself if you think that the only issue is that students will use software to solve problems. They can also get help from human beings online. For example, there is a service called Chegg that many students on my campus use to cheat in STEM courses. They sign up for a recurring credit card charge, and in return they get full solutions to textbook problems and can also get help on problems they post. We have even had students use Chegg on their phones during a test to get live help, when the teacher wasn't vigilant enough about preventing access to phones. I believe they were both cooperating with each other and getting help from people off campus. You can't get out of this problem by constructing problems that are AI-proof.

If your problem is lack of time, one good option is not to give in to the assumption that students will never read a textbook and must be spoon-fed every fact and method. Tell them what chapters of the book they're responsible for, and focus class time on helping them over any hard parts.

• Meanwhile I listened to a national academic institutional coach yesterday come to my campus and announce that we have to admit that none of our students will ever read textbooks no matter what we say <grumble grumble grumble> Feb 29 '20 at 1:55

Comment-answer, but too long for a comment:

Tests are some of the MOST valuable hours in a course. They are high stakes performances (like in music or sports). Preparation for them drives a lot of learning. Then the actual execution and subsequent feedback is often much more valuable training than routine drill problems, because of caring about the result. You should maximize test utility versus another hour of lecture (which probably has less learning impact even than an hour of them working drill problems).

In addition, by using a take home you are trying to get an extra hour of their time. In addition, take homes are very hard to control from assistance (book, roommate, online). Can you really expect them to be rigid about a time limit when taking a test at home? If unlimited, then now you're taking more of their time.

In addition, take home tests often lead to instructors giving non ideal questions (ones that are too much like research projects for instance). This is not the case here--your question is fine. However, your urge to ask some sort of strange, hard to cheat on question may already be driving you away from what would be the normal best question in a controlled setting.

In general, ODE books have too much content for a one quarter or one semester course. It is very normal not to cover it all. So if you're already shaving some lessons off, just shave one more off. And learn the core of the topic well. And good tests are key to good learning. Don't shave this 2nd order ODE with constant coefficients off. It's the key part of the course! Very high gain. But surely there is some other lesson (existence, Wronskians, predator prey, transforms, Sturm Louisville, series, etc.) you can skip to allow use of class time for examination.

And I love all the topics and would love to have you cover everything. But you need to prioritize. You got a lifeboat with 15 spots and 25 people, pick the best 15 to live. 2nd order constant coefficient ODE, with a forcing function, will be seen all over the place in physics, engineering, chemistry applications. So it lives. Some other topics go in the water. And the test time needs to live. It's "high gain".

In theory, you could cover more content by telling the kids to learn a lesson sans lecture. But I get the impression you're not training superstars here. They will be unhappy without the scaffolding of some lecture time and Q&A time on the topic. So, just treat this as an optimization problem and do a little less, but do it well.

[Moderator may cut my other answer.]

• I would argue precisely the opposite of this. There is no point whatever in testing people's ability to grind through the solution to problems like the OP's, because even the most basic computer algebra system can get the solution faster and more reliably. On the other hand, understanding something about Sturm Liouville theory, Laplace (and other) transforms, phase-plane diagrams, existence and uniqueness of solutions, etc, is about insight, which is the one thing that can't be automated. The fact that many undergraduate courses haven't changed for 50 years is relevant, of course. Oct 17 '19 at 10:57
• Most of his students are going to be going into engineering or science. Being able to work diffy Qs, comes up all the time in STE. If you're weak on manipulational skill you'll have a hard time doing homework problems in junior year physics (mechanics, E&M, quantum) or in mechE (fluids, thermo, controls) or in EE (circuits). You'll have a hard time even following derivations. (You could make the same argument that knowing fractions or basic algebra is not important...but if you have to look everything up or run to a CAS, you're not going to follow the subject matter well.) Oct 17 '19 at 11:30
• I also think having a good foundation in analytical solution techniques, first, will make it easier when dealing with the more detailed topics like series solutions or Sturm Louisville or transforms. And it tends to "build" in that you get a new lens to look at familiar topics. Humans are not computers...we have practical limits to our brains, but seeing the same thing a couple different ways, helps makes things make sense and stick. For instance seeing the similarity of parallel/series resistors in circuits and head losses in a fluid system. Oct 17 '19 at 11:39
• Note, once you HAVE experienced things well and done a lot of practical manipulation, it may be absolutely fine to start using a calculator for tedious arithmetic or a CAS for solving equations. But at this point, you are a sophisticated user. This is a very different thing to saying a student should never bother to learn initially. Personally, I don't see how you can sit and read a paper, if you can't follow derivations and run to a computer (and having experience working problems makes it easier to follow work, even if not practiced now...just having done it before.) Oct 17 '19 at 11:45
• The theme of this answer - that tests are hugely important - is absolutely true. Many folks who did not experience that value in earlier forms of university may not realize it and have missed out. I just completed a masters and it was well less valuable than the undergrad and the too-easy and/or too-little-thought-out exams that did not focus on deep problem solving were a big reason. Oct 17 '19 at 20:07

Leave more conceptual questions for take home. They have more time, can think things over. Encourage using technology, ask for a sketch of how they think the solution to something hairy looks like (Asymptotes? Increasing? Maxima?). Explore e.g. the effect of initial values on the plot of the solution.