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See the attached image:

a photo of five true-or-false blanks with ambiguous "T" and "F" letters in each

I am having a difficult time grading this paper as I am not sure if the student intentionally wrote the answers in such a way that each answer looks like both "T" (or "t") and "F" (or "f") or if it is just his normal style of writing. He did a terrible job in answering other questions, if this information is even relevant.

Questions:

  1. How should I grade this student's answer? (The correct answers are "F", "T", "T", "T", and "F".) If the student intentionally wrote ambiguously, should I give him zero for all these five questions? If yes, how to prove that he did this intentionally?

  2. How should I design my questions to prevent this from happening in the future?

Edit 2022-03-04. Mystery solved! I have talked to the student and he admits that he wrote his answers in frustration. His intended answers are TTTFT. Many thanks to everyone for your input.

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    $\begingroup$ Please, if you have an answer, try to post it as an answer instead of as a comment. Mini-answers in comments lead to strange sub-comment chains that discourage others from posting an answer and then end up deleted anyway, causing the answer to be lost permanently. If you have an answer to the question, even if it is a frame challenge, post it as an answer so it can get upvotes, comments, be edited, and be accepted. $\endgroup$
    – Chris Cunningham
    Mar 2, 2022 at 2:07
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    $\begingroup$ I've deleted yet another set of mini-answers in the comments. Some of them were good, too! Post answers as answers, even if they are short, so they survive long-term. $\endgroup$
    – Chris Cunningham
    Mar 2, 2022 at 14:49
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    $\begingroup$ @MadeleineBirchfield Ok. Obligatory image $\endgroup$
    – DeepSpace
    Mar 2, 2022 at 15:01
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    $\begingroup$ Sorry if I am too cynical, but IMHO this student is trying to con you. In assessments it is the student's responsibility to make their answer clear. If you cannot read it, it is wrong. If there is some good reason why they cannot write clearly, e.g., disability, then your administration should inform you about this in advance so that you can provide an alternative (maybe tickboxes for T and F instead of having to write a letter). $\endgroup$
    – David
    Mar 4, 2022 at 2:32
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    $\begingroup$ You're missing one key fact here. How old is the student? If they're in second grade and still haven't really mastered using a pen, it's forgivable. In that case you need to think about how you run your tests. If they're older than 7-8 though, this is intentional and you can treat it appropriately. $\endgroup$
    – Graham
    Mar 4, 2022 at 16:44

12 Answers 12

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I read them as TTTFT. But the fourth is very hard to tell, the first somewhat hard to tell, and all show poor writing, perhaps indicating a lack of engagement. You could tell him that such terrible penmanship will get him zeroes in the future (not even the 50/50 chance he has going now).

You could also print the T and F and have the kids circle one. I actually find this quicker and easier to scan when grading. (But don't feel like you have to change.) E.g.

Circle one answer, T true or F false:

Choice Question
T F y=x^4 is the equation of a parabola.
T F x/5 + y/3.1415 =0 is the equation of a line.

FWIW, I wouldn't be so worried about static from the kid, unless you've had squabbles before. He's going to get crushed on the test, overall, and will probably just take it on the chin.

Edit: I disagree with the comment that you should only have essay style (long answer) problems. You should do a mix. T/F is effective for checking tricky things, like the (very mildly) tricky questions I mentioned. On the other hand, you should probably have more standard questions (sans tricks) for the long answer, worked problems. In other words, have a blend of problem types.

Edit: I disagree with the suggestion to orally quiz the kid. You don't have the time for that, especially if you have to initiate it yourself, versus the kid coming to you. Plus he could have reviewed the materials, now. If he actually meant something different, tough...this is a lesson to write more precisely in the future. He will learn that harsh lesson much better from losing the points, than from pleas to write better. And, we need to be a little sanguine about the possibility that at some point a student gets misgraded. School is a practical process, not a theorem of Euclid.

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    $\begingroup$ I guess you assumed 4 is F because it has two horizontal lines? $\endgroup$
    – Malady
    Mar 2, 2022 at 3:24
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    $\begingroup$ TTTFT was also my best guess. $\endgroup$
    – Andreas Blass
    Mar 2, 2022 at 4:10
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    $\begingroup$ imo these answers are unambiguous, although the rest of this answer is very good. Other than 4, they all have a single horizontal line, so cannot be F. 4 has two horizontal lines, so cannot be T $\endgroup$
    – Tristan
    Mar 2, 2022 at 10:35
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    $\begingroup$ 4 could be a lowercase T $\endgroup$
    – Alice Ryhl
    Mar 2, 2022 at 15:36
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    $\begingroup$ @GlenYates, or maybe all the others are lowercase t's, since the vertical lines extend above the horizontal one, which doesn't happen in an uppercase T. $\endgroup$
    – ilkkachu
    Mar 2, 2022 at 17:57
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This is an answer to the second question.

If you are having problems distinguishing the written single-letter answers of this student, there are better ways to design the true/false questions such that you won't have the problems in the future. You could require the students to write the entire word "true" or "false", to increase the chances that two answers are always distinguishable. If handwriting is still a problem, you could treat a true/false question as a multiple choice question with only 2 choices and have students circle the correct answer.

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There seem to be two questions here. I am going to answer them separately:

How should I mark ambiguous answers?

Students often write things which are ambiguous or hard to understand. For example, does

enter image description here

mean $\mathrm{e}^x$ or $\mathrm{e}x$? is this multiplication or exponentiation? Arguably, the intended meaning is entirely clear from context, but there is still a problem in that one must rely on that context to mark the answer. Ambiguous responses like these represent a problem in the student's communication. Depending on the objectives of your class or your personal philosophy of instruction, you might handle them differently:

  • I explicitly include learning objectives related to communication in my syllabi:

    [students will be able to] demonstrate an ability to clearly and effectively communicate mathematical ideas, both in writing and speech.

    Because I have identified clear communication as a learning objective for the class, I consider ambiguous answers to be wrong—they fail to meet that objective.

    In practice, when I see an ambiguous answer on an assessment, I will mark it as wrong. However, I also make it clear to students that they can appeal—if they believe their answer to be correct, they can discuss it with me, and I will generally regrade the assessment item. In particular, if a student can explain the meaning of their notation verbally, they have (at least partially) met the communication objective, and the more mathematical objectives are more clearly met. Most of the time, students end up with something like 2.999/3 points (the minuscule point deduction is a finger waggle in the direction of the student).

    The chief difficulty with this approach is one of scalability—it does not work well in a 600 student lecture. However, I am at an institution where class sizes are small, so I don't have that problem.

  • Even if "effective communication" is not something which has been explicitly spelled out as a learning objective, I do not think that it is unreasonable to mark ambiguous answers with "0 points; see me". Particularly in a large lecture class, but also in smaller classes, your time is a resource. Is it more effectively spent trying to parse ambiguous answers, or engaging with a student after the fact about their ambiguous answer?

    In my opinion, it is better for both you and the student to grade quickly, to err on the side of deducting points, and to be generous when regrading work when a student asks you to take a second look. It is better for you because the reality of the world is that most students are not going to ask for regrading (they should, but they won't); and it is better for the students because the ones that do ask for regrading are going to have to engage in a conversation with you, and are likely to learn something from that interaction.

    In either case, it is unlikely that a few ambiguous answers are going to tank a student's grade, and you can take the "default harshness" into account when determining your grading scale.

  • You could work very hard to interpret the student's ambiguous answer and grade it "as correctly" as possible. For example, as many other people have noted, the answers to 1, 2, 3, and 5 all include only a single horizontal line, while the answer to 4 includes two horizontal lines. It is nigh certain that the student intended TTTFT. Grade accordingly.

  • Alternatively, as noted above, it is unlikely that one or two questions are going to have much of an impact on the student's overall grade. If you see an ambiguous answer which could be correct, just give the student the benefit of the doubt and move on.

    Personally, I don't think that this is the best approach, as the student might miss genuine opportunities to learn. On the other hand, experience seems to indicate that most students won't bother to look for feedback, anyway, so there are only minimal opportunities to learn from mistakes, anyway. Since a few ambiguous answers are not make-or-break, and since your time is valuable, assume the student meant whatever the right answer is, mark it as such, and go to the next exam.

How do I write assessments to avoid ambiguous answers?

Others have already covered much of this, but I have a few additional comments to offer (which are largely redundant, but I don't see an answer which captures all of what has been said in the comments):

  • Avoid questions which require an answer which is likely to be ambiguous. One letter answers are a potential source of ambiguity. Why accept T or F when you could insist that they write out the entire word? or, better yet, have them circle the correct answer or fill in a "bubble" corresponding to the correct answer?

    For example, you might try one of the following:

    $\bigcirc$ True / $\bigcirc$ False : Completely fill in the circle which corresponds to the correct answer.

    T / F : Circle the correct answer.

    T ___ / F ___ : Mark the correct answer with an x.

    ________ : The correct answer is either true or false. Write the entire word on the line.

    All of these are potentially open to abuse (i.e. a student could draw a large circle which captures both answers, or shade both bubbles, then erase them both in such a way that it is impossible to tell which answer is meant), but a student who is confident in their answer is much less likely to give an ambiguous answer to any of the above.

  • Don't ask true/false questions at all! Or, better yet, ask true/false questions, but require justification. Instructions for such a question might read:

    Determine if the following statement is true or false and justify your answer. For example, if the statement is true, prove it; if the statement is false, give a counterexample: ...

    The specific instructions or forms of justification will vary by level (e.g. I don't think that you would generally ask a middle schooler to prove a statement), but it should be possible to ask this kind of question at any level.

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    $\begingroup$ The circle-something design leaves less room for unintentional ambiguity. Intentional ambiguity like circling both T and F with no attempt to erase either, or half-way between, should be treated as a non-answer. The student did not pick one or the other. $\endgroup$
    – Peter Cordes
    Mar 4, 2022 at 1:45
  • $\begingroup$ Pablo's comment on @guest's answer points out a plausible well-intentioned way to get ambiguity (erasing to change one's mind), but if this is a problem you should probably tell your students to make sure their answers are unambiguous. If they need to cross out both after repeated circling / erasing and write in a "true" or "false" next to it, as the only thing not crossed out, or ask for a new copy of the page after making a mess, they can; it's up to them to make sure they've clearly indicated an answer. $\endgroup$
    – Peter Cordes
    Mar 4, 2022 at 1:46
  • $\begingroup$ @PeterCordes Indeed, that was essentially what I meant by "but a student who is confident in their answer is much less likely to give an ambiguous answer to any of the above". A student who knows the answer will circle one, or fill in one bubble, or whatever---unintentional ambiguity is less likely than when you have to rely on a student (perhaps with dyspraxia) writing one clear glyph. A student who is uncertain or trying to get away with something can potentially give an ambiguous answer, but if that happens, reread the first section. $\endgroup$
    – Xander Henderson ♦
    Mar 4, 2022 at 2:18
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It is perfectly fine to ask students to write answers in as you have asked them on the exam, as simple T or F. But it is also acceptable to require clarity in letter formation. In fact it is good to require clarity in student's penmanship as penmanship is an aspect of literacy and an expectation of all educational institutions. That is despite idiosyncrasies in one's letter formation, and so any student who retorts "that is my style of writing" can simply be informed that official documents have to be written in non-idiosyncratic styles, atleast as far as clarity is concerned. Any letters written unclearly are thus marked wrong because of a lack of clear communication.

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Does it matter in terms of passing or failing?

If it does you could mix those answers up and just ask him which letter is which.

You can also make polls in future classes ;)

An essential skill of an academic is making unambiguous statements. At least he clearly failed at that. If he's not a first-year student just giving him 0 points seems reasonable to me.

You could provide boxes for them to check:

T [_] F [_] How are you doing today?

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  • $\begingroup$ Thank you for your answer! The student will fail this exam since he did terrible on other parts. So it does not matter much. $\endgroup$
    – Zuriel
    Mar 2, 2022 at 13:11
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    $\begingroup$ I don't understand. How can the question "How are you doing today?" have an answer of "true" or "false"? $\endgroup$
    – JRN
    Mar 2, 2022 at 13:48
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    $\begingroup$ @JoelReyesNoche: Yes. ;) $\endgroup$
    – Ilmari Karonen
    Mar 2, 2022 at 13:55
  • $\begingroup$ Don't get the downvote. These are valid suggestions for both questions. @JoelReyesNoche you can always add a Neither [ ] box. $\endgroup$
    – j2L4e
    Mar 2, 2022 at 15:31
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    $\begingroup$ @j2L4e Heck, I gave you up vote just for the 'How are you doing today?' True/False question! $\endgroup$
    – Glen Yates
    Mar 2, 2022 at 18:35
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It appears that answer 4 is an "F" because there's a second right-tick overlapping the line on the form. The others don't have that, so they are "T" responses. I would mark the paper with these interpretations, circled, next to the student's answers, then grade the paper as such.

You should also encourage the student to improve their handwriting in a positive tone, explaining the problem. Showing a little interest that way could have a very positive effect on their attitude and engagement.

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Answers 1, 2 and 4 are not recognisable English characters so should be marked as incorrect.

Answers 3 and 5 are close approximations of the letter T so if you are feeling generous you could them as such, making 3 correct and 5 incorrect.

Communication is important, personally I would give them 0 marks for this but I like @Xander's idea of being open to an appeal as a chance to have a conversation with them.

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    $\begingroup$ 4 looks like a lower-case t to me, with a serif. It doesn't look much like the F that other people are claiming; although I guess they're looking at it as a capital F with the vertical part below the lowest horizontal very truncated? That's a very mis-shapen F. The other letters are closer to lower-case than upper-case, if anything, though, so it seems weird to assume a capital F. But yeah, if a student doesn't unambiguously write a T or F, they aren't answering the question. Fishing for generous interpretations is a waste of the instructor's / grader's time. $\endgroup$
    – Peter Cordes
    Mar 4, 2022 at 1:57
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You've tagged the question undergraduate-education. By this age it is expected that you are able to communicate clearly. If you have special needs that mean this is not possible, then these should be discussed with the academic staff before the exam. If a student has not provided a clear answer to the question, then they should not receive a mark for that question.

Having said that, the answers in your photo all look like clear "t" characters to my eye. 1, 2, 3 and 5 are a straight vertical line that has been crossed, which is a simple way of writing a t. The closest "f" would have a distinct curve to the right at the top of the vertical line. There is a slight curve and forward lean on answer 1, but nowhere near enough for me to read that as an f.

Answer 4 is different in that it has two clear right pointing horizontal strokes, but one of these is at the very bottom of the character. This is typical for a t, but any horizontal line at the bottom of an f would point to the left.

Therefore I would take the answers to all five questions as being "true".

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Student obviously wrote it hastily and carelessly, so you should read it hastily and carelessly.

TTTFT. Done.

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The student I am tutoring also has true false questions on his calculus test. The answer choice is to mark it true or to give a counterexample if it's false. I believe this option eliminates ambiguity and also challenges the student to show some real understanding.

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"How should I grade..."

It depends much on the kid and the dynamic you have with him/her. My first thought was that this kid obviously did it on purpose probably because he/she didn't know the answers so this is better than nothing or simply to be joking around. In my experience, teacher normally would take this as a wrong answer since my handwriting is not legible, after all you are in school to, among many things, read and write correctly.

Now, it could also be that this kid is not joking around and is being 100% serious, so in this case I'd say just ask her/him to clarify the answer for your.

Now your second questions on how to avoid this happening in the future, I loved "Xander Henderson" suggestion, I see it as an absolute way to solve this.

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Just ask the student to clarify their answers, maybe in office hours or whatever's most convenient for you.

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    $\begingroup$ Sure. But it is possible that he has already discussed the answers with other students and figured out what the correct answers should be. $\endgroup$
    – Zuriel
    Mar 2, 2022 at 16:01
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    $\begingroup$ Maybe scramble the questions to make sure they haven't memorized the order of Ts and Fs, but otherwise if the student can match the question to the answer, I'd say they've learned the material! (maybe they learned it after the assessment, but is that bad?) $\endgroup$
    – TomKern
    Mar 2, 2022 at 16:05
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    $\begingroup$ @TomKern the fact that he knows the answer doesn't mean he understands the material and is also unfair to students who didn't have a chance to go back and correct their answers. $\endgroup$
    – Amy B
    Mar 2, 2022 at 20:42

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