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I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how to deal with correctly. Namely, should students be asked to justify or explain their answers to true-false questions?

Some reasons to ask for explanations:

  1. It may inspire them to think harder.
  2. It makes it possible to give partial credit.
  3. It reduces the incentive to guess randomly.
  4. It's easy to give the right answer for a wrong reason, and if I don't ask for an explanation I may never notice. (I find this reason the most compelling.)

Some reasons not to ask for explanations:

  1. Writing an explanation takes a lot more time than just writing "true" or "false", partially negating the efficiency of true-false questions versus "free-response" questions. In particular, a student who has no clue may spend a long time writing rubbish in hopes of partial credit.
  2. For some otherwise nice true-false questions, it's not clear what sort of "explanation" could be given. Sometimes I find that one of "true" and "false" (often the correct one) has a much easier explanation than the other. It's also hard to communicate to the students what level of detail and precision is expected, which makes it hard to grade fairly.
  3. Often students will give an explanation that's basically correct, but also contains inaccuracies or false statements. If I let it slide, then they won't learn to be careful with language; but it's hard to take off something while still giving credit for basic correctness when I usually assign only a couple of points per true-false question. As with #1, this pushes the true-false question towards more of a free-response question.
  4. Not a really good reason, but for completeness, I'll include it: it takes longer to grade.

Is there a good solution? Do you ask your students to give explanations for true-false questions? How do you deal with the issues raised above?

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  • $\begingroup$ [true or false] explorers sailed to the new world during the renaissance $\endgroup$ – user2078 Sep 2 '14 at 23:18
  • $\begingroup$ @jon Welcome to the site. I think OP was looking for your thoughts on how to deal with the issues surrounding explanations of true/false responses. $\endgroup$ – JPBurke Sep 3 '14 at 0:02
  • $\begingroup$ The above two comment were originally an answer and a comment on it. I converted the answer to a comment as it is not an answer as far as I can tell (possibly it is too cryptic for me). $\endgroup$ – quid Sep 3 '14 at 11:29
  • $\begingroup$ A pure True-False question really has 3 possible answers rather than 2, because no answer should be regarded as better than an incorrect answer and worse than a correct answer. So the scoring should have three values, not two (the lowest value can even be negative if one likes). This remedies the problem of guessing to some extent. On quizzes I typically penalize wrong answers about half of what I award correct answers (and zero for nothing). To my mind 5-0-0 is better than 6-0-4 (Right-Nothing-Wrong) on 10 T-F questions. $\endgroup$ – Dan Fox Nov 21 '17 at 20:01
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Your question is in fact three questions.

  1. Is there a good solution?

  2. Do you ask your students to give explanations for true-false questions?

  3. How do you deal with the issues raised above?

My answer for 1 is that I would not include pure true or false questions.

My answer for 2 is yes, I would ask for explanations; however, with regard to pure T/F questions, I think some are better than others, and will include some of my own commentary below.

My answer for 3 is twofold. First, I might ask questions that don't require justification (see below) but not where the choices are T/F. Second, I think a better general approach is to hand out a review sheet with many statements that are either true or false, and to include an "answer key" with the truth value for every (or every other) statement.


Here are a couple of examples of true or false questions and how I would either re-phrase them into a better T/F question (and why I think it is better) or how I would re-phrase it into a question that similarly does not require a full proof or counterexample.

To start with, here is a question that I think is a bad example:

True or false: Every cyclic group with non-composite order is isomorphic to $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$.

If instead you made this a "prove or disprove" question, then - it would still be bad - but at least students who tried to prove the answer is true could earn themselves a fair bit of partial credit. The issue here is that the true or false answer is more of a subtle trick, namely, noting that a trivial group of order 1 is cylic, but that 1 is not prime; so the correct answer is false.

Here is, I think, a pretty good example of a true or false question:

True or false: Every nontrivial cyclic group of prime order is isomorphic to $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$.

Again, this could make for a nice iff proof; however, my sense is that this is a fact that students should know cold, and, while the proof can be made pretty succinct, writing out all the details during a timed examination seems to me like more trouble than it is worth.

Here is another bad example:

True or false: For every finite group $G$, if every subgroup is normal, then $G$ is abelian.

I think this is bad because it feels like it is probably false, but actually dreaming up the counterexample is not so straightforward unless it has been given some thought beforehand.

A better question might be:

A. Classify (without justification) all groups up to isomorphism for orders 1 through 7.

B. Give an example (without justification) of an 8 element nonabelian group $G$ with every subgroup normal.

C. Identify two nontrivial and non-isomorphic proper subgroups of $G$; prove they are not isomorphic, and then prove that each is a normal subgroup of $G$.

In this case, parts A and B are both quick and not too tough to guess, though writing out the details (e.g., of why there are only two groups of order 6) is a bit of a hassle. Part C also allows students to get deeper into the problem in a way that discourages guessing for B.

Lastly, if you are concerned about students guessing, you could always start the test with, say, 5 true or false questions: make them +2 if correct, -3 if incorrect, and 0 if left blank. (This is a general suggestion: Adjust the actual numbers as you see fit.)

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    $\begingroup$ I'm a little confused about what you're advocating. Your first paragraph says that you would ask for proofs or counterexamples on the exam, but that seems to contradict the rest of your post in which you give some examples that don't ask for justification. What general principle are you proposing, and how does it deal with the issues I raised in the question? $\endgroup$ – Mike Shulman Apr 16 '14 at 14:07
  • $\begingroup$ Ok, thanks for the clarification. Although I still don't feel that you really answered the question that I asked. $\endgroup$ – Mike Shulman Apr 17 '14 at 15:19
  • $\begingroup$ @MikeShulman I re-edited; if you still feel the question is not answered, then I hope there is at least something useful! $\endgroup$ – Benjamin Dickman Apr 18 '14 at 2:11
  • $\begingroup$ Maybe I'm missing something obvious here, but how is the cyclic group of order 4 isomorphic to some $\Bbb Z/p\Bbb Z$ for $p$ prime? Did you miss a word in there? $\endgroup$ – user37 Apr 18 '14 at 6:51
  • $\begingroup$ @Mike Thanks! Yes, I omitted "of prime order" from the first two examples; I have edited them in. $\endgroup$ – Benjamin Dickman Apr 18 '14 at 8:53
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I rarely asked true/false questions, but when I did I usually made them worth $2$ or $3$ points each (not $1$ point each), and incorrect answers with a somewhat appropriate and at least a somewhat correct explanation could earn one or two points back. In fact, if it was clear from their explanation that they accidentally circled the wrong "T/F" letter (rare, but it did happen), they would get full credit.

(ADDED NEXT DAY) In reply to your comment about how I would handle the issues you raised, these issues did not really show up for me in the lower level classes where I sometimes asked T/F questions, such as trigonometry (e.g. T/F: $\pi$ does NOT belong to the domain of $\cot{x}).$ In graduate level courses I often asked (on in-class tests) a type of question that is somewhat similar to T/F: "For each of the following statements, give a brief justification or give a counterexample, according as whether the statement is true or false." Maybe you can use this type instead. To take care of your concerns of students not knowing what is appropriate to write, you can try doing what I did -- give lots of examples of appropriate justifications in advance (write up answers to short quiz problems, solve sample test problems of this type when appropriate during lecture, etc.).

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  • $\begingroup$ How did you deal with the issues I mentioned in the question? $\endgroup$ – Mike Shulman Apr 17 '14 at 15:26
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First of all, it depends on the situation. I use true-false questions mainly because of your point 4, as a complement to a standard exam. With 150-200 mathematics major students, it is common practice at our university to complement "real" exam questions with tests, and there true-false questions are great for the reasons you mention.

In out practice, there is a part of the exam consisting only of true-false questions, and those who reach a required point (like 75%) can take part at the second part of the exam, which may be oral or written depending on the course.

In this setup it is important that your true-false questions are straightforward, do not need deep thinking but the understanding of concepts, definitions or theorems. Then you also do not really need deep explanations. I think of statements like (from an calculus course designed for maths majors):

  • If a function is continuous on $(a,b)$, then it is differentiable.
  • If a function is differentiable on $(a,b)$, then it is continuous.
  • $\int f^2(x)dx = \frac{f^3}{3}+C.$
  • If $f(x)=\sqrt{5-x}$ with $x\leq 5$ and $g(x)=5-x^2$, then $(f \circ g)(x)=x.$
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  • $\begingroup$ So in brief, your answer is "for some purposes, you can choose very simple true-false questions that don't suffer from not asking for explanations"? That makes sense, but I don't think I'm in that situation; I'm only asking a few true-false questions as part of a longer exam given to 30-50 students. $\endgroup$ – Mike Shulman Apr 16 '14 at 14:09
  • $\begingroup$ @MikeShulman: yes. This is the case where I can imagine using this technique: to complement other parts of the exam. When asking more sophisticated questions then some explanations are always good to show where possible mistakes in the reasoning are. $\endgroup$ – András Bátkai Apr 16 '14 at 16:33
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Instead of "true/false" it might be better to have mutiple choice questions. Around here the physics people are fond of them, but ask for explanation of the result. AFAIU, only questions with the correct choice are graded. Not my choice at all, but somebody you might consider.

I prefer to give one question (out of 5 or 6) that has a few very short subquestions, like asking for a definition or a very simple, one line application of an important result. This way, if they even know what the course is all about, they navigate away from the dreaded 0 grade... In any case, I try to get it across that the correct answer with a wrong derivation (or a derivation that can't be deciphered) is wrong. The aim in my field (Computer Science, programming particularly) is to get understood (by yourself fixing the program a month later, by whoever gets to win the lottery of having to fix it when you are long gone). Mathematics (really all science and engineering) should strive for this in the very first place.

About level of detail requested, I make that clear in the question. Students have access to older exams and solutions, that should give them some guidance. As a general rule, I ask for at most one handwritten page (most of the time less), and they know that. "Last will and testament as an answer gets discounted points for each extra sheet" (not really).

One last comment: When there are longer "true of false, and comment on your answer" questions, I tell them explicitly that often there is no "right" answer, that they get full credit for the discussion, not the "correct choice."

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  • $\begingroup$ Note that the question was not "should I ask true/false questions?". $\endgroup$ – Mike Shulman Apr 17 '14 at 15:25
  • $\begingroup$ @MikeShulman, well, my answer is "it is a bad idea, in general." $\endgroup$ – vonbrand Apr 17 '14 at 15:32
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I recall one test I had as an undergraduate for a Linear Algebra course. It was, bar none, the most educational test I'd ever taken. It was a set of 50 questions, of which we only had to answer 30. Every single one was a conjecture, which we had to answer T/F and either formally prove or provide a counterexample for. Naturally, some of these questions were far easier to answer than others. You got 1 point for guessing T/F correctly, and 2 points for a correct explanation (half credit for kind of getting there). He'd grade the first consecutive 30 you made an attempt at, to prevent too much guessing.

Why was do I adore this test so much?

  1. It provides flexibility, and allows you to play to your strengths. Since you only have to answer a subset of questions, you don't have to worry about knowing everything. Still, you have to answer a majority of questions so it doesn't allow you to have too many weaknesses either.

  2. It requires you to identify which questions are even feasible for you to answer. I'm not saying put "T/F P=NP" on your test to gull weaker students into trying to prove things they can't. I'm saying that even being able to recognize trivial counterexamples and do those before they start on the more elaborate proofs (or trickier counterexamples) is a great skill and measures (IMO) a lot about their grasp of the material. I'm not usually a fan of time pressure on tests, but it really works here, and is made up for by the fact that they don't have to answer every question. Heck, this fact alone makes it a good candidate for an open book test.

  3. It encourages simple, concise, elegant proofs. Of course, we all know that theorems don't spring up out of the aether, fully formed, with no rough drafts. Those simple proofs in books were worked and polished to death. However, the time pressure forces students to really consider "what is the easiest way to show this is true/false?" Even if it doesn't get you the best proof possible, it gets them thinking in that direction.

Obviously I'm not saying that this is the one true test. Different tests have different goals. What I do think is that tests like these not only test mastery, but also forces the student to think in new ways and enhances their education in addition to just testing them. It uses time pressure in a constructive way. It uses it to get them to think critically, rather than just staring at them angrily for rederiving a 30/60/90 triangle from an equilateral instead of memorizing it.

I can't speak highly enough of how much I think tests like this do tests correctly from both a teaching and a learning perspective. Though it may be a bit intense for students conditioned to only expect "standard" math exams. Though one downside is that it does disproportionately penalize students particularly susceptible to the sunk cost fallacy that keep wasting their time on one question they just can't answer because DAMMIT I SPENT THIS MUCH TIME ALREADY I NEED TO FIGURE THIS OUT.

(Disclaimer: it did take him like a month and a half to grade this doozy)

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  • $\begingroup$ It surprises me that nobody commented. How long was the test? I like this idea of T/F quiz (except the grading time :/), since I usually make very long tests, to encourage students to be efficient (solving the easy questions first) and to be sure that low level students can show the few they know. $\endgroup$ – Taladris May 29 '14 at 7:38
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I use such questions frequently, but in a limited fashion. Typically my exam contains five or six question, and one of those consists of, say, four true/false questions. The way I do it is that I make four claims, and students are to either give a quick proof or give a counterexample. I design them in a way that either it is easy to construct a counterexample using the body of examples covered in class, or there is a two line argument reducing the claim to a result in the lecture notes. I think explanations are a must.

As I also include trick questions. Then a good explanation for a wrong choice earns partial credit. For example, if a student gives an argument that covers the generic case, but fails in a special case that actually leads to a counterexample, then I will be relatively generous.

For example in first semester calculus I might ask something like:

Fact or fiction?

  • A) If the sequence $(a_n)$ converges, then so does the sequence $(a_n/n)$.
  • B) If the sequence $(a_n/n)$ converges, then so does the sequence $(a_n)$.
  • C) If the sequences $(a_n)$ and $(a_nb_n)$ both converge, then so does the sequence $(b_n)$.

Here a student is expected to A) for example appeal to the result about the product of two converging sequences being convergent itself, B) come up with a counterexample, C) come up with a counterexample, but will get partial credit for realizing that $(b_n)$ should converge as a quotient of two converging sequences, but forgets about the possibility of an attempt to divide by zero (that actually is crucial in finding a counterexample).

In second semester calculus I might ask something like:

Fact or fiction?

  • A) If a function is differentiable on the interval $[0,1]$, then it is also Riemann integrable over that interval.
  • B) If the sequence $(a_n)$ converges, then the power series $\sum_n a_nx^n$ converges for all $x$ with $|x|<1$.
  • C) If the sequence $(a_n)$ converges, then the power series $\sum_n a_nx^n$ converges for all $x$ with $|x|\le1$.

A student should be able to prove A by recalling from first semester that a differentiable function is continuous and combine that with a theorem from second semester telling that a continuous function is integrable over a closed interval. Parts B and C simply test whether they absorbed the basic facts from the chapter of series, power series, and the body of examples covered.

I'm sad to say I often include claims testing whether students understand basic "if..then" clauses at all. The way I grade these comes as a surprise to some, so to soften the blow, I include a sample question like this in the HW set prior to midterms, and explain the philosophy.


In case you were wondering: I teach a section for non-math majors. Some of them major in theoretical physics or some such discipline, and I need to cater to their need to be at least somewhat prepared for more advanced courses. Some of the students shouldn't be in that course, but are there because they did well in high school math, and think they can waltz through this. Yet they wouldn't recognize a basic logical fallacy if it hit them on the nose. Schizophrenic, yes. But somehow we cope.

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You have some really good pros and cons for explanations on a true/false test. If that is the method that you would like to keep using, you could always try having students explain why the find a statement/problem true or false. For example, have the directions read something like: Mark the statement either true or false, if you find the statement to be false, explain your reasoning. Or you could have them explain themselves if they find it true. because I do not know whether or not you give tests that consist of just true/false questions, I would like to suggest that a combination of test structures, i.e. multiple choice, free response, true/false, etc. This would allow your students to have other avenues to express their understanding of the material than just "do you think this is correct or not?" But it is your classroom and you know how much time you have to dedicate to test grading, and what test structure you prefer.

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I would definitely say that there needs to be some part of the question asking justification. The sole purpose of the test is to see if they have sufficiently learnt the material and giving the ability to just plug and chug or even guess isn't doing mathematics. Also if the student is like me they may have made programs on their calculators to do the work for them which ultimately in the long prohibits their ability to come to the conclusion on a step by step basis. Just by the way I answer this as a college freshman, not a teacher. Thought a student's perspective may help.

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