Nice vs. ugly numbers in homework and tests

Question

This question revolves around using integers (−1, 0, 1, 2, 3) or simple fractions (½, ⅓, ⅗) vs. real numbers (−1.254, 42.72) in teaching concepts, assigning homework, and preparing tests for math, science, or engineering. For the rest of this question, I will call integer or simple fractions nice and real numbers ugly.

For the sake of simplicity, let’s say that you are teaching a math class, and the first topic is basic addition. The first time that you teach it, I would assume that you would want to teach it using nice numbers. For example, using 2 + 2 = 4 would be preferred over 1.234 + 5.678 = 6.912. Sometimes you can get lost in the weeds of the calculations ("just plug these numbers in here and get the answer out") and completely miss the concepts. While concepts are important, it is important for the students to be able to apply the concepts for more complicated problems. 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.

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? How about tests? During my engineering studies, it seemed like there were lots of problems that had nice inputs and/or answers. Most of the questions did not have really ugly answers. Is this typically done to make it better for the students learning, or is this done to make grading easier? Perhaps calculator use can also influence the type of number being used too. Overall, it would be nice to understand why professors and/or teachers often select nice numbers for assignments.

If it would help to know, the main drive of this question is that I would like to automate some of the homework or maybe even tests for classes. I would like to be able to generate multiple versions of homework or tests so that students can’t simply copy answers from each other. If I am generating homework, it might be tricky to find nice solutions vs ugly solutions. I think I have a method for automatic grading, so that is not a problem. The main thing that I want to maintain is a good learning experience for students.

Note on π and other irrational numbers: For my studies, π was of course in lots of the problems, and this technically makes problems have answers that are irrational. For most problems, it is acceptable to include the symbol π in the answer instead of including the numerical form in calculations. These problems could be still written nicely with implied multiplication like 2π or 3π/5.

Answer 1

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:

1. 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.
2. 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.
3. 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:

1. I allow scientific (not graphing) calculators on every assignment and test.
2. 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.
3. I warn them specifically that the numbers involved in some problems may be messy, and I go through "messy" problems in class.
4. 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.).
5. 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.
6. 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.

Answer 2

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.

Answer 3

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?)

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Ultimately, it depends what you're trying to teach.

Answer 4

Since this is a website about higher education, I will answer in that context.

The only "difficult" thing about "ugly" numbers is doing concrete basic operations with them, like adding them and so on. Everything up to that is typically done algebraically using variables (x, y, z…). University students are already supposed to know how to perform basic arithmetic, even with "ugly" numbers. This is never what you want to teach in higher education. So either let your students use a calculator, or use "pretty" numbers in your data. If you are concerned about real-world applicability, then surely you know that today everyone who has to perform this kind of tasks works with a computer that is hugely more capable of performing mathematical computations than any human.

As for computer-generated homework questions, I have had the unfortunate duty of doing it last Spring, like many of us, I guess. It was not particularly hard to produce "pretty" numbers, even when I needed to produce complicated linear systems to solve for example. Make it so that your question depends on a few parameters (let's say 3-5) and make sure that these parameters are taken to be integers in a reasonable range (for example [-5,5]). Then unless you get crazy with how you derive questions from parameters, you will mainly get "pretty" numbers. And since I assume that you wouldn't dare giving a question to students that you haven't even looked at, when you do a cursory check of the automatically generated questions you would quickly pick up bad edge cases.

Answer 5

I want one question on the test to have ugly numbers, since I want students to learn to trust their calculations, and not use "the answer is a nice number" as a verification method. Ugly numbers are great for teaching you to trust the method and the knowledge. But most of the time, they are simply annoying.

Answer 6

I'm not an educator. I'm simply a math graduate working in related industry but my answer would be a hard no-no on nice numbers. Some answers claim that makes students use intuition to know whether the result is correct. There is absolutely no scenario where intutition is a good verification for the result you get (not saying it's useless for choosing the right method of calculation). You absolutely do not want to teach students to rely on the result being nice. Another answer mentions that using them gets rid of the need to verify their calculations. This is an absolutely crucial step, you should always verify your calculation at least once. It is a process that never has any drawback. Otherwise you might hear news stories like "building collapse kill 50 people because civic engineer made a mistake but the number looked nice so he didn't check again".

EDIT: I actually more often checked my calculations when the result was actually nice. If it wasn't then I assumed that I have used the best method I could think of and the worst that could happen was loosing a point for the wrong result. Luckily I mostly had professors and teachers that graded the method, not the result.

EDIT 2: You want to teach your students to think of a solution, not how to game the system. Those same students might learn to later look for legal loopholes to deliver a half completed and sometimes dangerous product instead of doing the expected (see dieselgate scandal, why work on the solution when you can just game the result).

Answer 7

What kind of students are you teaching? If you are teaching elementary or high school students, then use the kind of numbers that are appropriate to your curriculum. If you are teaching engineering students, then you should use real world numbers.

You say "During my engineering studies, it seemed like there were lots of problems that had nice inputs and/or answers.". Wow, what kind of engineering did you study. After halfway through my first real engineering class, just about every problem I did had no tidy solution - we used trial and error to solve nearly every problem (on first generation programmable calculators (think HP-25)). The numbers made sense - a heat exchanger might be rated 100,000 BTU/hr, for example, not some oddball number. But the pipes going in to that equipment might be 4 inch schedule 40 (which are 4.026 inch inside diameter - I always had a pipe schedule booklet handy, along with steam tables in my bag). When I used the ideal gas constant R, I always used a version with 5 significant digits (and I could rattle those values of R off in 4 or 5 different unit systems - I studied in Canada in the middle of the transition from Imperial to metric units).

When teaching, you want to use numbers that challenge your student to think and to be unafraid to solve the problems that they will see when they do their senior design project or when they get their first job. There's no sense using numbers with much more precision than they will see in real problems, but you are cheating them if you make everything too "cute" by having problems that feature integers as inputs, and particularly integers as outputs.

If you really want to challenge them (and get them understanding the numbers they are using), get them to buy or borrow a slide rule and give them a "no calculators allowed" test (by the way, if you do that, you probably want to make sure that the problems are reasonably easy to solve using a slide rule - lots of multiplications and divisions and little else).

Answer 8

On a test, you don't want students to be always unsure if they got the right or wrong answer when it comes to algebra, so generally nice numbers are preferred. Also, if you just want to test them on knowing the basic methods and you assume that they can work with more complicated numbers, making the numbers messy is a distraction. At the very least you need to give the students an idea of what to expect. If all but one of the answers have nice answers and the other one has a messy answer, the students who get the messy (but correct) answer will spend all their time double-checking their algebra, when they could be spending their time on other problems.

On HW then I think messy numbers are fine, but I think that writing "Round your answer to the nearest hundredth" would be appropriate.

For lower-level classes, though, using some messy numbers at some point on HW is a good idea. Once during a final exam for precalculus, a student thought that she messed up on finding a vertical asymptote because she got a number that wasn't an integer. Apparently vertical asymptotes can only happen at integer values. Well, she looked again at her work and had a flash of insight when she found her mistake and got that the asymptote was in fact at an integer value.

Answer 9

I’m in the habit of giving mostly nice numbers of assignments and tests. As “mostly nice” number is Sqrt[2], or Log[6] or e^7, for instance. This way students can provide answers in exact form (without floating point) without too much difficulty.

I would stay away from things like Sqrt[1+Sqrt[2]]$ which I consider truly ugly. The students know this so if they get an answer like Sqrt[21/213] they suspect there’s probably some bug in their calculation.

Now I also have in some courses assignments questions which are entirely numerical (v.g. plotting some solutions to some nonlinear differential equations). Even in those cases I will try to find “nice” boundary conditions so that the students can verify if their intuition matches the numerical output.

Answer 10

There is a pedagogic advantage in using ugly numbers: users will try to avoid them and teach themselves algebra in the process by maninpulating symbols instead of specific integers, fractions or decimal expansions. The idea is that you simplify the expression as much as you can before plugging in the actual figures.

So it really depends on what you're trying to teach.

By the way, what you call "real" number is referred to as a "decimals of nonintegers", which is a pretty ugly name, as -1, 0, 1/2 or 1 are real numbers too.

Answer 11

It has been suggested that I should turn my comment into an answer.

• Use 'ugly' numbers for an entire page of assignments.
• Take all the results and add a few more 'ugly' numbers as distractors. List them of the bottom of the worksheet.

That means students will practice to work with 'ugly' numbers. They can easily verify that there was no arithmetic error by finding their result in the list, yet guessing will not get a passing grade. If the number is not in the list, the first step is to check if they mistyped on their calculator or got their sums wrong.

Answer 12

Unless the point of the homework is to test them on their ability to do basic arithmetic, why not give variables rather than numbers, and the answer should be an expression in terms of those variables.

Going from that to a numerical answer is just some easy but tedious arithmetical calculations.

Answer 13

Using nicer numbers allow calculation in tests to be more fluid. I think this is generally a good thing.

However there are sometimes situations where you explicitly want to teach the basics of symbolic calculations, where for example rational numbers like 1/3 ensure that the task cannot be solved numerically without leading precision. Sometimes with trigonometry questions they might rely on the fact that intermediate results are expressed in pi fractions.

One of the examples mentioned that when asking for the average of numbers, that 2.3 and 6.7 would better test the understanding than nice numbers. However I would argue that those are actually nice numbers as they add to a round number and can easily divide by 2, so the result is a clear 4.5 with no need for calculator or risk of rounding error.