# As an evaluator in mathematics, how should I grade responses that answer the question asked, but in less detail than desired?

I want to know whether there is a need for responding to students who ask for points/marks by arguing that they only answered the question that was asked., a.k.a "literalism". So, I am giving a simple hypothetical scenario. Note: This is a simple scenario and the questions can be different.

Consider the following simple question and the three classes of responses by three hundred different students.

Question

Define "finite set." (5 points)

A set with a finite number of elements is called a finite set.

A set with a finite number of elements is called a finite set.

Example: {1, 2, 3}

A set with a finite number of elements is called a finite set. All the finite sets are countable. The sets which are not finite are infinite sets. Any given set is either finite or infinite and neither both nor otherwise.

Example: {1, 2, 3}

And the set of natural numbers is countable but not finite.

As an evaluator, suppose I decided to give three points to answer #1, four points to answer #2, and full points (5) to the last answer. Is it not recommended to do it that way?

The reason for such awarding is to encourage the students who understood the concept well and are presenting the answers that show their understanding.

But the students who answer with #1 might argue that their answer is worth all five points, and it's unfair to take off points since the question was "Define 'finite set'" and not "Define 'finite set' and 'infinite set' with examples and their relationship with countable sets."

Although I have complete authority in awarding grades, I am not sure how to respond to those students. Is it recommended to simply ignore them by saying it is a subjective evaluation based on the presentation and clarity of the answer? Or should all answers presented here earn all five points?

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– cag51
Apr 30 at 7:32
• Answer 1 should obviously get full credit May 1 at 19:55

I don't think this is an issue with "literalism", in the sense of students taking some farcical literal interpretation of the question. For example, a prompt to "Write a definition of finite set" and the student responds "a definition of finite set".

Rather, this is a poorly worded question that does not make the expectations clear if your expectations were for a particular level of detail in the definition. If the third answer is the answer the instructor(s) wanted, they need to make this clear in the question as asked.

I question the value of asking this question at all, as I don't see what knowledge it demonstrates beyond an ability to regurgitate something from memory. Something more meaningful might be to ask to distinguish between finite and countable sets.

Since it's too late for this particular exam, I would assign full points for any correct answers, even if they are brief and not as comprehensive as was desired, and change the wording for future versions of the exam.

• (+1) for I question the value of asking this question at all, as I don't see what knowledge it demonstrates beyond an ability to regurgitate something from memory. This was my thought also. If the word "finite" is allowed to be used in the answer, then I don't see what purpose this serves, as this is like asking students what a novel is in an introductory literature class -- if they don't know this term, how can they follow anything in the course? Maybe for a more advanced class one could ask for a definition that doesn't make use of the word "finite", but that's not the case here. Apr 28 at 20:44
• I question whether the third answer reflects superior knowledge. It sounds like a student who is flailing around, grasping for the answer, and trying out all the answers that might work, without being able to judge a credible answer. I give it a lower score than the first answer. On appeal I would ask the student to write brief statement, based on the text and lectures only, why their answer is at least as good as, or better than, the basic information needed, eg, as shown in answer 1. Apr 28 at 22:17
• Making sure students understand a rigorous definition of finite and infinite with respect to cardinality is valuable, in my personal opinion. Of course the answers listed as acceptable by the asker are almost entirely without rigor. Apr 29 at 4:00
• Regurgitating something from memory is a useful skill. It is also much easier to memorise definitions and proofs if you understand them so it’s also a useful test. Apr 30 at 8:01
• @Dave L. Renfro Agree with you on using the question's main term in the answer without explanation. But this was just an example question to show what the issue is - it is not a real math exam question. Strange how many respondents here missed that. Apr 30 at 14:47

As an evaluator, suppose I decided to give three points to answer #1, four points to answer #2, and full points (5) to the last answer. Is it not recommended to do it that way?

If you want students to give examples of finite sets, you have to ask them to do that. It is incumbent upon us as instructors to give students clear and precise directions about what knowledge we expect them to demonstrate. If you perceive that students are using “literalism” as an excuse to not show some knowledge you want them to show, the solution is simply to make your own instructions more precise and in line with your actual wishes.

Edit: I see a lot of discussion here on the question of whether OP’s perception that answer #1 is a technically correct definition of a finite set is actually correct. This seems like a side issue and tangential to OP’s actual question. My answer starts from the premise that this is a course teaching set theory concepts at an intuitive level, such that “a finite set is a set with a finite number of elements” is actually a correct definition in the context of the course.

Some of the other mathematicians here seem to be of the opinion that one should not teach such a naive definition of a finite set. Well, we could have a debate about that, but as I said, I think it misses the point of what sort of advice OP is actually asking for. So I’ll stay out of that particular debate.

• @Buffy I wonder what definition the book/instructor gave. If "a set with a finite number of elements" then what more is there to say? We don't know if there was a prior definition of "finite number". Perhaps infinite was defined as the existence of a bijection with a proper subset of itself and finite as not infinite. Apr 28 at 19:23
• @EthanBolker, Bijection is a set concept, as is "element". If that path were taken then a definition of a finite set would be one that cannot be put into a bijection with a proper subset of itself (or the empty set, for which proper subset doesn't apply). But the answers given in the question are just circular. Apr 28 at 19:28
• @Buffy I think OP is describing a course teaching set theory concepts at an intuitive level, where “finite number” is assumed to be a known concept. In that case, saying that a finite set is a set with a finite number of elements can be a correct definition and not circular. Of course, for a course in axiomatic set theory one will develop a more robust definition. Apr 28 at 20:44
• A similar set of questions in a different field: "Define Totalitarianism", "Define Totalitarianism and provide at least one example each from the 19th, 20th, and 21st centuries", and "Define Totalitarianism, provide at least one example each from the 19th, 20th, and 21st centuries, compare and contrast it with Fascism, Stalinism, and Mao Zedong Thought, and then answer the following: Is the current government of Ruritania Totalitarian? Explain your answer and give at least five reasons for or against." Apr 29 at 2:02
• @DanielR.Collins the example here is not "using a word in its own definition". It is defining the notion of a finite set in terms of another concept (assumed more primitive in this context) of finite number. If you think this is a problem, do you also object to an open map being defined as a function mapping open sets to open sets? Or to the definition of a compact operator being logically dependent on the concept of a compact set? There's countless other examples of this sort of definition pattern in math. Honestly, this whole side debate seems like a storm in a teapot to me. Apr 30 at 5:46

In the unlikely event that "finite number of elements" was previously defined and can therefore be used in a definition of "finite set", all three answers deserve full credit.

In any event, answers 2 and 3 should not get more credit than answer 1, because the additional information they contain is not part of a definition. (It doesn't even have the form of a definition.)

• It is pretty hard in the modern world to define "finite number of elements" prior to defining finite set. After all, what is an "element"? What is "finite". Apr 28 at 19:24
• @Buffy I didn't say it's easy, but here goes: The "number of elements" in a set X is the smallest (von Neumann) ordinal in bijection with X. An ordinal is "finite" if it is a member of every limit ordinal. Apr 28 at 19:28
• @Buffy: What is "finite" --- Without using the word "finite" in the definition (aside from the term being defined), one could say that a finite set is a set such that there does not exist a bijection to a proper subset of itself, and other possibilities exist. But in light of the answers given in the OP's question, the 2nd or 3rd year undergraduate level (U.S. standard) in doing something like this is not the case here. Apr 28 at 20:47
• How about "a set whose cardinality is a natural number"? Apr 29 at 11:30
• @EthanBolker: That is precisely the presentation in the liberal-arts math appreciation text my department uses (Pirnot, Sec. 2.1). Apr 30 at 5:03

If answer #1 is a correct answer to your question, then I would argue that answer #3 deserves fewer points than answer #1, not more. This is something that I see too often on students' exams: they don't really understand the concept they are asked to define, so they regurgitate every sentence from the course notes that contains the needed words and hope something sticks.

To me, this is analogous to answering, "What is the first sentence of the most famous soliloquy in Hamlet," with the complete works of Shakespeare, rather than "To be or not to be, that is the question." This is not an acceptable answer, and it demonstrates a worse knowledge than a student who answers straight to the point.

As for answer #2, I'm on the fence. I always try to explain to students the difference between a definition, an example, a theorem, and a proof. This is, after all, the most basic thing about logical reasoning that you can teach; but you would be surprised at how few get it quickly and how long it takes to settle in most students' minds. (This may be a failure of me as a teacher, though.) Giving full points to someone who answers a query for a definition with an example flies in the face of that. But the answer does contain the definition, and the example is clearly demarcated, so this is perhaps acceptable.

In any case, I would not give points to students who answer a question I haven't asked. All the others would have an outcry for unfairness, and I wouldn't be able to do anything but sympathize.

• I'm not sure if this was your intent, but "To be or not" isn't the first line in an Act, and isn't even from MacBeth (it's from Hamlet). Apr 29 at 22:47
• Agreed with the point here. Related question: Test answer rejected for saying more than asked for Apr 30 at 5:08
• @OwenReynolds It turns out I'm an idiot. Thanks for the catch!
– N.I.
Apr 30 at 23:02

I agree with the other answers that this is not an issue of literalism at all, but a lack of clarity in questioning.

Let's consider another field, and three possible exam questions you might ask:

Define Totalitarianism.

Define Totalitarianism and provide at one example each from the 19th, 20th, and 21st centuries.

Define Totalitarianism, provide at one example each from the 19th, 20th, and 21st centuries, compare and contrast it with Fascism, Stalinism, and Mao Zedong Thought, and then answer the following: Is the current government of Ruritania Totalitarian? Explain your answer and give at least five reasons for or against.

The more detail you can provide as to what characteristics make up a top-notch answer, the more your students will be able to focus on providing just that.

One might argue that a student should just provide everything they know, but that is problematic too - your students are not going to know what "additional" details you want or how far they ought to go unless you tell them.

Now, this instruction does not necessarily need to be expressed on the test itself if it is otherwise provided in the course. You could make a note in your course syllabus providing,

Exams for this course will include questions asking you to "define" or "explain" some concept. To receive full credit for a "define" question, you must phrase your definition with respect to another concept covered in this course, as well as provide at least three examples of the defined concept. For "explain" questions, you must "define" the concept as well as compare and contrast it with another concept covered in this course. The following would be an acceptable "define" answer:

The definition of crancorianism is when pre-reticular osmosis occurs within a transfinite non-orientable matrix with n<3 and m=plus or minus 0.5. Examples of this include Smith's Decomposable Spline, Marvin's Ghostly Gradient in eight-space, and the application of the Wossamotta U Framework of Advanced Best Practices in Pre-K Remedial Pedagogy in the Juvenile Justice Context to NP-hard graph traversals.

Of course, we can also come up with true "literalism" in student responses:

Hey, your test said "Choose one of the following", it didn't say "Answer one of the following"! I chose #2, so I should get full credit even though I didn't answer it.

The question said "Find x", so I circled "x". What more do you expect? If you wanted me to "compute" it, you should have said so!

The instructions said to answer the question, not to answer the question "correctly". How was I supposed to know that you wanted a correct answer and not any answer?

Of course I copied my answers off a classmate, there wasn't a "No Copying Answers" warning at the top of the test!

If one of your students provides one of these comments, you can dismiss them as failure to understand the social context of university study rather than a real issue with your test.

• No offense, but this is not good advice. Specifically, 1. It is completely unrealistic to expect students to read and understand complicated explanations from the syllabus of what you mean by “define” and “explain”. Those are English words with a standard meaning. Can you imagine if every course instructor had their own particular, very specific notion of “define” and “explain” that they required their students to grasp in all its nuances? 2. Your third version of the “define totalitarianism” question is much too long to ask in a single paragraph. I’d break it up into multiple subquestions. Apr 29 at 18:20
• Your answer is applicable to fields other than mathematics where a definition is (largely) semantic. In mathematics, definition is a creative act, creating a new concept for study. The closest non-mathematical analog I can easily think of is the term Bunburyist, defined in the play "The Importance of Being Earnest" by Oscan Wilde. The character Algernon created a new concept and named it, rather than just applying a name to something known. The "namer" of totalitarianism didn't invent the concept. It was semantic only, not creative. Apr 29 at 20:02

Consider that you need to know what a full-credit answer looks like ahead of time, since that's how you decide how many total points each Q is worth.

That Q was 5 points. Let's say that's not much, which means you've decided ahead-of-time that a short simple answer is worth full credit.

On the other hand, suppose 5 is a medium or large amount of points. That means you've considered shorter answers and ruled them out. Let's say you decided the answer needs a contrast with finite vs. infinite sets, plus examples for the full 5 points. Obviously, the Q as written doesn't require that -- it will take a "literal" answer. That needs to be fixed, ahead of time (maybe the Q becomes "what are the 2 major cardinalities of sets, give an example of each"). If a Q slips by where you expected a long answer but a simple one works, that's clearly a mistake on your part, and not a call to grade the Q on a curve.

Answer #1 to the question is correct. Full marks.

Both answers #2 and #3 are not as good as answer #1, if you plan to grade differently, they should get less points.

Students must learn to understand a question and its scope. The scope was to say what a finite set is, and not what an example of a finite set is, or what an infinite set is, or what is the added value tax in Belgium (usually 21% if anyone is interested).

Answering too much to a question can be catastrophic in some cases.

If you ask a question that's worth 5 points:

• Every single answer that correctly answers what is asked should be graded with 5 points no matter the wording or the length
• Every single answer that partially or not completely answers what is asked should be graded with 1-4 points, depending on how much of the question is answered

Grading an exam should always be around what is being asked. If "how the answer is written" is part of the criteria, then it should be stated like that. The way you say you assign points leaves your question open both heavily for interpretation, and for some sort of "unexplainable group-knowledge" i.e. "You'll get full points as long as no other student gives a better answer than yours".

Every student is different, and just because someone has better writing skills doesn't mean they understood better the concept. Not only that; in some cases, beign able to use less words to explain something actually means better understanding of the concept.

Now, if you want to encourage students to use examples and extended explanations, award extra points to those who do so, either by giving 5+1 to them, or by changing the question to 3 for those who "just" answer and 5 for those who answer and give examples; but in both cases you should clearly establish that as the rule for grading the question.

• I disagree that any length should get the full grade. If the answer is that long or cumbersome to read that it is hard to extract the relevant part, one may subtract points. Answers who provide several different solutions should also not get the full points. But providing an additional example after answering the question should not be a problem. And sometimes it helps to see the reasoning to give some of the points for incorrect answers which mostly use correct reasoning and only have a minor mistakes that leads to the incorrect conclusion.
– allo
May 1 at 22:42

Actually, I think you were being overly generous with the first two answers and even the third. The first is simply a tautology, not a definition. Zero points. ("'Finite' is, you know, man, FINITE! Know what I mean, bro?"). The second shows a bit of comprehension, maybe 2 points to distinguish it from the first.

Even the third lacks the key element that a finite set can be put into 1-1 correspondence with a bounded subset of the natural numbers. So, maybe 4 points. That answer says that an infinite set is not finite, but that is also a tautology. A finite set is not infinite: not not-finite.

Not to be too harsh, but 3 is quite literally I can't define it, but I know it when I see it.

So, for the complainers, I'd just tell them to go back and study harder and that you were already a bit permissive with the grading. They lack insight that they need to obtain if they are to advance in math.

Caveat, I don't know the level of the course, but for any college level course, I'd expect more.

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– cag51
Apr 30 at 7:37