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Is it better to design an exam with fewer questions and relaxed timing or with more questions and a resulting time-pressure?

One the one hand, it seems that students who really know the stuff will be able to complete an exam quickly, and a time-pressure exam detects this, allowing me to sort the students by this measure. On the other hand, perhaps it would be better to have the exam test only for knowledge and not speed. But with fewer or easier questions this would naturally mean that fewer mathematical ideas are tested on the exam.

In practice, my choice boils down to this: should I drop questions from my exam, even though this means fewer topics on the exam, in order that more students will not experience time-pressure during the exam?

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Is it the case that a time-pressured exam won't have deep problems and a non-time-pressured exam will? –  Git Gud Mar 25 at 14:16
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Perhaps it depends on what you want to test. Time pressure will make creative thinking a high-risk endeavor and cause students to opt for mundane, but reliable methods. –  dtldarek Mar 25 at 14:34
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For what it's worth, as a student I lose a lot of time first writing the scratch work and then cleanly presenting the proofs. I could jumble everything together to answer faster without losing any correctness, but I choose not to. Another thing I lose valuable time in is trying to avoid computations (even when they are unavoidable), I hate the fact that I have to do them and I often just stare at the paper in despair. This is my problem, but it also shows that a time-pressured exam might not properly evaluate what I know. –  Git Gud Mar 25 at 14:47
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It is quite difficult to avoid time pressure on term tests in the $50$-minute period tests I am accustomed to. On final exams, time pressure should and can be avoided. –  André Nicolas Mar 25 at 17:58
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I don't believe in time pressure at all, but sometimes I've found it hard to avoid inadvertently creating time pressure. I once gave a brief quiz on which I decided to allow ten full minutes so that there would be no hint of time pressure. I thought the last stragglers might be finishing up after six minutes. Not one student was done after 40 minutes, and that exhausted the available time. –  Michael Hardy Mar 25 at 18:10

13 Answers 13

The time-pressure not only sorts students by knowledge and speed, but also by who is susceptible to math anxiety. Mathematics Educator Jo Boaler comments in a popular media piece (which includes a few links to more formal research) that:

research ... has shown that timed tests are the direct cause of the early onset of math anxiety.

Her comments are mostly about K-12 students, particularly on the younger end, as she also remarks:

It is critical that we take a moment to review the emerging evidence on the impact of timed testing and the ways in which it transforms children’s brains (1) (2), leading to an inevitable path of math anxiety and low math achievement.

And:

Policies in education rarely draw from research knowledge. But I would argue that this particular policy—of giving young children timed math tests—is one of the clearest ways schools damage children, and we now have evidence of the extent of the damage.

I do not think it is a leap to suggest that timed tests could have a similar impact on undergraduate (or even graduate...) students. More precisely, timed tests may not only interfere with students' affect during the test-taking process (note affect is crucial to mathematical success in the classroom; cf. e.g., beliefs and belief systems in Schoenfeld's (1985) "Mathematical Problem Solving") but afterwards as well.

As a closing remark, I leave you with a quotation from G. Perelman's graduate advisor, Y. Burago:

"[Perelman] was not fast. Speed means nothing. Math doesn't depend on speed. It is about deep."

So: My vote goes to fewer questions and relaxed timing.

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Speaking of university examinations, I rather liked the way they worked at mine. You could answer as many questions as you wanted on each paper. 4 good questions in each 3 hour exam was more than sufficient for a first-class degree but the option to do more (a) helped distinguish the top students, (b) gave people a reason not to leave early. I wouldn't advocate it for young children, though, since I would not expect them all to be capable of truly accepting that it's their choice whether to be relaxed about timing or push themselves to answer additional questions. –  Steve Jessop Mar 25 at 16:21
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I don't have research to cite, but my experience is that time pressure more generally sorts students according to test anxiety. (I've seen plenty of students who are unfazed by the subject of math, but sometimes have anxiety issues in test situations.) –  Mark Meckes Mar 25 at 16:50
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Note that time pressure also wipes out students with ADHD, which, though official accomodations exist, often goes unreported due to stigmatization. Designing tests so that everybody has enough time anyway avoids this issue entirely. –  Alexander Gruber Mar 26 at 18:02
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@AlexanderGruber Forget stigmatization. Competitive students don't want the extra time. I have a handwriting disability - basically, I write more slowly than normal - and I had terrible trouble finishing more-or-less every test in every upper-level math course in college. I have all the documentation that would be necessary to give me extra time, but I never used it because I didn't want any semblance of an unfair advantage against other students. I now realize this was silly, but there it is. –  Brian Gordon Mar 27 at 21:04
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Perelman won the international mathematics olympiad with a perfect score, and several USSR national olympiads at a time when those were harder than the internationals. Which is to say that he was one of the best mathematicians of his generation on high pressure timed tests. There are examples of "slow but deep" mathematicians but Perelman is very far from being one of them. Rather he is exceptionally fast, exceptionally deep, exceptionally perseverant, exceptionally original, exceptionally accurate, (etc). –  zyx Apr 7 at 4:29

A common approach in elementary school math is the use of "mad minutes" wherein a child attempts to madly complete as many arithmetic problems as they can. This is an extreme example of a timed assessment.

In this description of some research, the authors suggest that when students have math anxiety, some of the mental resources they have available are used to deal with the anxiety, rather than work on the mathematics. Hence, if we want all students to be able to utilize their full mental resources for an assessment (which seems like a reasonable goal) we should do what we can to reduce anxiety, and timed tests are one practice that likely results in anxiety.

In this article, the author talks about how her belief that should could not do mathematics started with the practice of mad minutes in elementary school. At 6 years old, she was convinced she could not do mathematics.

This next point comes from my personal experience. The way we assess students, and our emphasis on these assessments, tells students what we value in mathematics class. I have assessed students a variety of different ways, and found that students define "what it means to do mathematics" based on what is valued in class. Hence, if we want students to learn that one view of mathematics is that it involved careful, patient, creative problem solving, we need to demonstrate that with the tasks we ask them to do.

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It's very interesting since I can identify the "mad minute" as the beginning of my excitement about mathematics, which eventually led me to be a member of this community. To be clear, I'm not disagreeing with the answer. –  Chris Cunningham Mar 25 at 21:06
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I think what we've established is that mad minutes are a surprisingly emotional experience! –  Chris Cunningham Mar 25 at 21:29
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"Skill building" is a consequence of feedback. In other words, just getting the questions wrong doesn't help you learn anything, except which problems you need feedback on. Where you learned the actual math facts was on reflecting on your mistakes, looking at the multiplication tables again, or some other feedback mechanism. This all doable without the time limit. –  David Wees Mar 26 at 2:30
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"Mad minute" testing got my brother mis-diagnosed as profoundly math-disabled: ten problems, 60 seconds, and when time ran out, he'd have written down one, maybe two answers. After months of failed treatment of his "math disability", someone thought to give him an oral test, which he aced in ten seconds. For him, the "mad minute" was testing his handwriting skills, not his math skills. –  Mark Mar 26 at 4:35
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So then a related question is, is it fun for all students? –  David Wees Mar 31 at 13:09

Disclaimer: I'm not an educator, so my basis for this answer is quite limited anecdotal evidence.

If you apply time-pressure to mathematics, you switch in some sense from testing what the student can do, to testing what the student can do easily. I would think that most of the time you want to test the former. The difference between "not knowing" and "knowing" is usually far more important than the difference between "knowing" and "really knowing".

I'd speculate that the latter depends to a large extent on something that most of the time is quite uninteresting: inherent aptitude. I'm not saying you don't want to know who in your classes is a mathematical prodigy, but I'm pretty sure you have an inkling of that anyway without needing to destruction-test the ones who aren't. Mathematics is a subject in which it's really quite important to stretch the best students with additional material[*], I think not so much with the same material done quicker.

I wasn't challenged much by mathematics testing before I went to university, and I almost always finished exams early. I don't think it's unusual to see quite a wide range of speed in mathematics (and other subjects). So to me time-pressured exams would have been more interesting and would have showed off what I could do, since some other people would have got much worse marks and I would only have got slightly worse marks due to a somewhat increased rate of errors. But so what? How does that help the class? It's not the only and probably not even the best way to identify the most able students, and anyway one should not design routine examinations for the benefit of the best students.

One summer when I was a child (can't remember how old) I did do some time-pressured tests of multiplication tables. IIRC it worked like:

  • my mother wrote the numbers 2-12 across the top and side of a grid, in different random orders.
  • I filled in the products as quickly as possible.
  • I tried to beat my time the next day.
  • I can't remember for sure the rules on errors, I think that any error was a disqualification.

However the purpose was training by repetition (so that I would know my multiplication tables, not just know them in order). Fairly quickly it became a test of hand-writing under time pressure, to avoiding writing any 0 that looked like a 6. The purpose was not assessment. OK, it's an extreme example of testing purely by time-pressure since the task itself was trivial (to me at that age) absent time pressure. But I think it's clear that it would not be useful as a classroom exam.

[*] "what subject isn't?" chime in all the non-mathematicians. Well, I doubt there's any subject where the best can't benefit from advanced studies, but mathematics seems to have a particular quality that any given material will be frustratingly challenging for one third of the class and boringly easy for another third. At least in English the whole class can study the same novel at different depths :-)

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While I agree in large parts with the other answers posted already, I would like to say a little something in defence of time pressure.

In my experience teaching at the university level, many students fall into the trap of feeling like they understand a concept that they really ought to think about more. I fall into this trap myself sometimes: everything seems nice and clear and straightforward while someone is guiding me through it, and I only realize the gaps in my knowledge later, when I'm forced to do it on my own.

While this phenomenon is not in itself an argument for time-pressured tests, I do believe that such tests can be used quite effectively (and efficiently) to help nudge students into realizing when they've fallen into such a trap. I've found that sprinkling a few shorter tests or quizzes here and there with relatively easy questions but a tight time constraint can help students see for themselves what they really understand and what they just think they understand.

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Hi Adam! Interesting -- maybe you are leaning toward using time-pressure as a sort of baseball doughnut? en.wikipedia.org/wiki/Baseball_doughnut -- Useful for training despite not being part of the final assessment? –  Chris Cunningham Mar 25 at 21:09
    
Hi @ChrisCunningham! Yeah, that sounds about right. The quizzes are generally worth something (for motivation), but not very much. –  Adam Bjorndahl Mar 25 at 23:10

Long ago, I read some fascinating research that I have not been able to find online. The title (or part of it) was in French, though the article was in English. It started with something like y'a'til.

It discussed the disparate impact of standardized testing on women and men. A large calculus course at a university with relatively high cutoffs of SAT scores was looked at. Among women and men who had the same SAT scores, the women were earning A's and the men were earning C's. This was apparently because of the timed nature of the SAT.

Looking now, I find this research article suggesting that timed tests have a disproportionately negative impact on women.

And why do you want to make the test harder? I allow my students to retake tests (using a new version each time), so they can show what they know. If they know what they're supposed to, I want their grade to reflect it.

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Unfortunately likely not relevant to find it again, but just in case: if it was something like y'a'til it presumably was "y a-t-il" which (alone) just would mean "Is there?" ("Il y a" would be "There is"; the word order is changed to signal a question and the t is inserted in between the two vocals just for pronunciation's sake.) –  quid Mar 27 at 11:48

I would like to bring up something for consideration that's really come from the discussion in the comments on Benjamin Dickman's answer.

There is a problem with the dichotomy presented in the question:

my choice boils down to this: should I drop questions from my exam, even though this means fewer topics on the exam, in order that more students will not experience time-pressure during the exam?

In Norway we seem to have done our best to do away with time pressure. Our exams are 4 hours long and are not (in the lower level courses) designed to take anywhere near that length of time. Nevertheless, it is rare that students leave before the time is up and it is still common to see "I ran out of time" written on exam scripts. So simply making the exam shorter does not remove the time pressure. It also has some perhaps undesirable side effects. Since it is now feasible for a reasonable student to complete all the tasks, a good student is expected to do so and to do so well. This makes the exam even more sensitive to "computation error" than before since a good student cannot compensate for making a few such errors by completing more questions. Sometimes this is appropriate, but not always. So shortening the exam not only does not completely alleviate time pressure but also increases the accuracy pressure.

An alternative is that which is (or was - my experience is now approaching 20 years ago) practised by Oxford University. The exams were designed so that it was impossible to complete all the questions. Even just writing out an answer to a question without having to think took about half an hour, so in a 3hr exam the most even the best students could do was about six questions. There were nine on each exam.

To emphasise the point: in my year, the best student's best papers consisted of 6 good questions (given that the best of these consisted of Representation Theory and $C^*$-algebras, it's a bit confusing why he's gone in to Differential Topology, but there's no accounting for taste).

This had the effect of replacing time pressure by choice pressure. You now had a choice as to which questions to answer, and thus could, to a limited extent, compensate for lack of ability in one area by excelling in another (of course, the questions were designed so that you couldn't get away with being completely ignorant of the core). This was exaggerated by the fact that the final grade was cumulative across all exams, not just one; but I recognise that that is not possible in the modular system practised by most universities.

The emphasis on choosing questions to do well on was further strengthened by the exotic mark scheme that Oxford used (sadly, I hear no longer) which put far greater emphasis on completed questions than on partial questions. So one complete question was worth four half completed questions.

Overall, the purpose of this type of exam is to give the good students sufficient space to show that they are good, whilst giving the excellent students room to show off, and giving the less able students every opportunity to show that they have learnt something.

In a system where you don't have complete freedom to revise the overall grading system, this could still be implemented in a course simply by revising the grading scheme for that exam. The idea is to make sure that it is not out of 100 and to not scale the points. So an A is not 90% but is some number of points (actually, I wouldn't go for a point scheme at all, but that's a different issue), and a B is not 80% but is some slightly smaller number of points.

In the Oxford system, in my year, then one of the criteria for a first class degree was (roughly) to get at least 20 good questions and 20 fairly good questions. That's spread over 13 papers, so that's just under 2 good and 2 okay questions per paper. That's from a total of 9 questions per paper, so there's plenty of room there for a candidate to choose their questions wisely, and plenty of time for a good candidate to achieve that level.

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I like that concept. What is clear or even communicated to the students that is was impossible to complete the questions? –  Markus Klein Mar 27 at 11:19
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@MarkusKlein I don't remember ever being told that, though the "Sum Square" was a thing of legend whispered about on dark nights in student bars. Our end-of-first-year exams were the same style so I guess we got the hint from those (and those exams did not count towards our final degree classification). –  Loop Space Mar 28 at 8:21

To start off, I'm a mere highschool student. There are a lot of things wrong with how education is done nowadays, but I'll keep focus on this issue at hand.

Time is quite a big deal. For the last three years I've been competing in higher level mathematics with students around the state of Florida, and I've come to realize that time isn't a very good test of who knows math and who doesn't. Each test is comprised of 30 questions, and an hour, making each question about 2 minutes long. I can say with confidence that on a given day, I can complete every single question on the test, just, maybe not in the allotted hour. On the last few tests, I've come to review my answers after the exam with my friends, as we're allowed to take our test with us after the exam, and more often than not do I notice that I made a simple mistake, like adding wrong. (Mind you this is calculus, so evaluating a definit integral of a higher degree polynomial is a lot of tedious arithmetic). Often do I sit in the testing room on one or two of the harder questions, and after 5 or 10 minutes, figure them out. It feels good, I'll say. There is no extra reward for knowing how to do any of the harder questions, and all of the grading is done by machine. What sets me apart from the students who win, is the fact that as much as I practice, I will still write 2*4cos(72) as my answer, and select the answer choice with 4cos(72). This rush for time throws out the smart students, and only rewards students for being able to be accurate, not clever, or intuitive. Quite a shame if I may say.

I spent my last summer studying mathematics at the local university, since I am quite passionate about it, and my highschool really isn't. All of the professors I had curved their tests. So nearly every test, when there was a problem I couldn't do, or was too lazy to do (I know, I know.) I just wouldn't do it. And the curve would boost me up to an A+ every single time. The system was broken, and it was proving that without minimal effort, you could succeed. And an aside, the 95% of the class hated mathematics, and had an irrational fear of working with numbers. I suppose the stress brought on from schools did this. Very unfortunate.

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I see that your experience with these kind of tests is relevant to the question, but I don't see how the last paragraph relates to the initial question. –  Roland Mar 28 at 21:26

For a number of reasons, I would say time pressure is not generally good for math exams.

  1. It forces to students to be more concerned with getting done quickly than with checking their work. In any real-world job that uses math, being correct is much more important than being blazingly fast. I'm much more concerned with teaching an engineer to check his work thoroughly than teaching him to do calculations very, very quickly, for instance. The same goes for accountants and, really, almost other profession that makes significant use of mathematics skills. Being really, really fast at solving math problems is simply not a particularly valuable skill, especially today when we have computers that can do it much, much faster than you no matter how fast you are. We don't hire people to sit there and perform computations very quickly. We use computers for that. He hire people to use mathematics to create new solutions to practical or theoretical problems, a skill which doesn't strongly depend on one's ability to quickly perform computations.

  2. It measures things completely unrelated to mathematics, including how quickly the student can read the questions (which can be limited by eyesight and, at least judging by my anecdotal experience, is not strongly correlated to students' mathematical abilities.)

  3. As others have mentioned, it further disadvantages students who have test or math anxiety, which is more likely to discourage them than help them in any way.

So, in short, no, you shouldn't design your exams to have time pressure.

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Disclaimer: Most answers say in a nutshell "No, you should not have time pressure due to the following (comprehensible) reason(s)". I agree with most of the arguments there, but want to add some arguments that time-pressure is not such a bad things, at least on university level.

  • Assuming a fixed time constraint for the exam, you have three posibilities: Either you risk to have time pressure, or you only ask about a minor part of the course, or you avoid any deeper questions and ask questions about (almost) every part of the course but without going in to deep.
    • The second possibility is bad for those who lack in that particular part of the course and at the same time maybe some students are lucky since they only learned that part. In my opinion, such an exam is unfair (exept maybe, there is one particular main topic which is obvious, or you have announced that particular topic as being relevant for the exam). Moreover, if someone has to build up on your course and needs some other topics, there is a possibility that students might have no clue of that topics.
    • The third possibility is maybe a way between, but if you do not have any questions with deeper understanding, this is also bad for the following courses as well as you cannot distinguish between good and excellent students. This argument was also elaborated in some answers in the question What to do when your students are all getting A's.
    • If you want to have a broad knowledge and some sort of deep understanding, you might run out of time.
  • Most of the students will be working in industry and economy after their study, where the hiring is mostly based on their final grad in their study. Since it is not only important that someone is able to things right, but also to do things quickly, it is not unfair to also test how quick a student is in solving problems. Also, if two students are able to solve the same level of problems, but one of them is significant quicker, he/she should get a better grade.
  • Even for mathematicans in academia, (a certain amount of) speed is to some way important. Of course, if you are a genius and can work on his own and solve big problems, you don't need speed. But most of the students will at some point write a thesis and have to talk to their advisor. At sessions with the advisor, you may do a brainstorming or discuss possible solutions for a problem where a student can in general not say: "Let me think about it for a few hours and then I will come back", but a student who is capable of telling quickly if a solution might lead into a good direction has a lot of advantages here. The same holds for discussions with colleagues at conferences. The ability for a quick scan if ideas make sense or not cannot be trained within a few days or weeks and exams where time is also important can help to train this.
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While I agree in general, I'd contend that the ability to check if something makes sense after a superficial scan isn't something learnt through time-pressured exams. –  vonbrand May 4 at 16:09

It depends on the type of math being tested and the students who are being tested.

Firstly, ALL exams are timed. However, some exams are designed to be easily finished in the time allowed.

The concept of a timed exam is that it would be impossible or extremely challenging to complete the exam in the time allowed.

The purpose of this should be to measure a persons skill or ability, not to measure their knowledge.

Since most elementary schools measure knowledge, not skills or ability, then a timed exam would not be appropriate for them. Also, if the test is only a knowledge test, then again a timed exam is not appropriate.

A timed exam is appropriate when you are measuring a person's mastery of skills and their ability to apply knowledge to complex problems.

EDIT After rereading your question, I think the true question is at the end - the true question is "Should I drop topics to allow students time to finish." In fact, whether you drop topics or not, the students will not "finish" the exam, either due to lack of time or lack of questions!

I think the best solution is to create two exams if one exam cannot adequately cover all the topics, or arrange for a special double-long testing session.

Further, the distribution of your topics should be reconsidered if a certain period's teaching cannot be tested in one hour. If it cannot be tested in the allotted time, it probably cannot be learned in the alloted time either. If that is the case, then the student is doomed regardless of the testing format.

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Giant disclaimer: I am a future math undergrad with experience in completely separate fields. So far I've just done high school and "hobby" math, i.e Numberphile and the like. This is also my very first post.

There is the option of doing two exams, one for "pass" and one for showing greater depth. The first one being a greater number of easier questions with a larger questions/time quota, while the latter is just a few different problems to battle with that require greater insight. I know this is used in a few engineering programs in Sweden, but I don't know how successful it's been.

Another option is giving homework that gives credit in the upcoming exam. I'm also familiar with the x-amount of questions answered in A/B/C-difficulty version, where the student is not required (nor expected, in some cases) to answer everything, but at least gets the opportunity to show what level they're at. This is where I usually (as a student) get to show that I'm terrible at grinding out questions, but not too shabby in understanding difficult concepts. More than once I've answered the difficult once but failed the "easy" ones.

I guess they all have their pros and cons, but I think it's quite clear that the traditional time pressure test is not the only one, nor necessarily the best.

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What about taking a page from the SAT? The exam is designed such that it's not necessarily expected that the student completes every problem within the time allotted (although there are certainly students who do). Instead of starting at 100% and losing points for every incomplete or incorrect answer, the student starts at 0%, gains points for every correct answer, and loses a fraction of that for every incorrect answer¹ – incomplete answers are ignored.

This style is probably superior for gauging where a student is at over a wide range of topics, possibly to help direct further teaching energy, as opposed to testing full comprehension of a single topic.


¹ According to Wikipedia, the SAT will stop deducting points for incorrect answers in 2016.

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Do you feel the SAT, along with its ongoing changes over the decades to lower the bar, is an ideal test? –  JoeTaxpayer Mar 25 at 19:37
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@JoeTaxpayer, The test scoring was set up to make the median score close to 500, and the scoring was 'recentered' in 1995 in response to declining median scores. This was a change to the scoring system, not the test content. (Scores from 1995-2001 are reported with an 'R' to reflect the scoring change.) The test's actual difficulty was increased in 2005 in response to a rising number of perfect scores. There have been other changes to the test over the years which might affect perceived difficulty, but I don't know how much they actually affected scores in practice. –  Brian S Mar 25 at 19:49
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As far as "ideal test," I don't think such a statement can be made in the abstract. I do think the SAT's scoring format is a good one for discovering a student's aptitude across a range of topics, however. (I don't think the content of the SAT itself is relevant to this question.) –  Brian S Mar 25 at 19:50
    
TIL that the SAT is going to remove the penalty on incorrect answers. I always considered that a staple of that exam. –  Joe Z. May 28 at 21:11

The reason I went to university was to gain experience dealing with time-pressure.

Exam questions rarely require too much thinking---they are there to test familiarity with concepts and the deftness with which students can apply techniques.

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Interesting -- but which of the choices in the question would be better? –  Matt F. May 4 at 13:08

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