Nice vs. ugly numbers in homework and tests
I think I'm going to be fundamentally disagreeing with a lot of the answers here.
Nice numbers definitely make problems easier, and I make a habit of using them when first introducing a concept; they make the students more comfortable, and let them focus on the key idea that I'm trying to teach. But I never rely on nice numbers for tests or assignments. There's three major reasons here:
- I have had many students over the years who genuinely stop understanding a concept when presented with "ugly" numbers. For example, I've had students who can easily find the average of 2 and 6 but when asked to find the average of 2.3 and 6.7 don't even know how to start. This isn't an issue of getting confused by the calculations; it's that they think about "nice" numbers differently than they think about "ugly" ones. In the case of the average, the issue was probably that the student in question thought of the average as "the number in the middle", not "the sum divided by two", which makes sense when working with integers but not otherwise. The problem is that you can't capture failures of comprehension like this without using ugly numbers at least occasionally.
- Not a grad student and Per Alexandersson pointed out that many students use the "nice number" test to tell whether their answer is right - they tend to trust the answer if they get "2", not so much if they get "2.134". So from the perspective of "be nice to your students", you should use nice numbers; but the thing is that, in literally any application they will have for this material later in life, they won't be working with problems that were carefully curated to produce nice numbers. If you're teaching them something you expect them to use later, it's a disservice to allow them to continue to use the "nice number" test.
- Loosely speaking, there are far fewer "nice" numbers than "ugly" ones. I've had students "solve" problems by assuming the answer will be a whole number and then guess-and-checking their way to success.
That said, if you're using ugly numbers, you do need to make some concessions to make that work. Here's what I do:
- I allow scientific (not graphing) calculators on every assignment and test.
- I allow unsimplified answers, except when the problem is about simplification; so, they're welcome to leave their answer as a complicated mess of radicals if they want to.
- I warn them specifically that the numbers involved in some problems may be messy, and I go through "messy" problems in class.
- I take some class time to teach techniques for judging whether your answer is correct that don't rely on niceness of number; my preferred one is "ballparking", where you use the context of the question to estimate the general size of the answer (is it positive or negative? Bigger than a thousand? Etc.).
- Problems that involve ugly numbers tend to take longer than ones that involve nice numbers - even I find that I go slower when a problem involves weird fractions or decimals. Take that into account when writing tests.
- Problems that involve ugly numbers are more prone to minor error than ones that involve nice numbers; for example, you probably don't want to be counting an answer in a calculus class as "completely wrong" because they typed "2.146" instead of "2.156" into their calculator. I always offer extensive partial credit based on work shown, and do not generally mark off for errors that don't show a lack of comprehension or change the difficulty of the problem. For online tests, to make this work, I allow students to submit their work alongside their answers.
You write "I think I have a method for automatic grading, so that is not a problem.". If you are going to depend on automatic grading, you should use easy, simple numbers.
There are two ways of getting a wrong answer, not getting the method right and making a mistake copying from question to calculator and from calculator to answer sheet. During manual grading you can distinguish those by requiring the students to show their work and grading that. Automatic grading tends to give the same weight to not knowing how to do a calculation and entering one incorrect digit.
Using simple, easily checked numbers reduces the risk of calculator error.
While part of me thinks that learning concepts should be the same for nice numbers and ugly numbers, my personal experience says that there is a difference (perhaps just a small one) between these two.
I'd expect a difference: Ugly numbers get in the way of applying and learning a concept. E.g., average (−1, 0, 1, 2, 3), (½, ⅓, ⅗), and (−1.254, 42.72). The first I can do in my head, simply by applying the concept of averaging, the addition is trivial, the division easy, I'm just thinking about the concept. For the others, I'm not think about the concept, I'm think about fractions and more complex addition/division.
In order to facilitate better learning and better application of course material to real-world problems, should you also include homework with ugly number inputs and answers?
I've just argued that ugly numbers are a barrier to learning, so nice numbers are preferable, imo.
How about tests?
The same. (Plus, do students have calculators?)
Ultimately, it depends what you're trying to teach.