Has anyone performed a study on the differences between student interpretations of these words?

Background: When I taught high school geometry and later undergraduate precalculus, I noticed that even when explicitly taught how to read and interpret the words "true" and "false," that students have a hard time circling "false" when a statement is "sometimes true."

For example, in my experience, students struggle with "True/False: Dividing two polynomials results in another polynomial," but have less difficulty with the question "True/False: Dividing two polynomials always results in another polynomial," even though these two statements have equivalent truth value and even when students are explicitly taught to interpret the word "true" as meaning "ALWAYS true" and the word "false" as meaning "SOMETIMES false".

With this in mind, I hypothesize that before students have been expected to master mathematical logic, that they will perform better on "Always True /Sometimes but Not Always True/ Never True" questions, and we would get a more accurate picture of their understanding than if we ask true/false questions. Is there any research supporting (or not!) the use of always/sometimes/never instead of true/false questions, before the more formal mathematical understanding of the meaning of "true" and "false" are expected to be grasped?

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    $\begingroup$ The problem with "True/False: Dividing two polynomials results in another polynomial" is that a quantifier is missing, and the reader is expected to ASSUME that the absence of a quantifier means that a universal quantifier is assumed, which beginning students might not realize and even experienced writers sometimes violate. The simplest solution is to simply say explicitly what you mean, and let the question be a test of mathematics and not of some unspoken convention. $\endgroup$ Commented Nov 2, 2018 at 18:17
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    $\begingroup$ Regarding your colleagues, I wonder how many of them have written questions on tests that begin with something like "A bag contains two red apples and four green apples . . ." Here the meaning is clearly "A certain bag", but the kinds of rules some want students to follow would mean the students would have to interpret this as "Every bag contains ..." Such a student could then argue that the hypothesis is false, and thus anything logically follows, and thus their answer (whatever it might be) is correct. (More likely, they'll just think all these rules are unreliable made-up things.) $\endgroup$ Commented Nov 5, 2018 at 10:08
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    $\begingroup$ I remember the nightmare that there was answer "yes/no" or "always/sometimes but not always/never" to questions in negation form, like "A polynomial with integer coefficients has no complex roots ?" $\endgroup$ Commented Nov 5, 2018 at 13:26
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    $\begingroup$ I agree wholeheartedly with @DaveLRenfro that it is important to clearly state quantification and not to assume that anyone will correctly understand when something is unstated. That being said, I also want to point out that I have witnessed the issues described in OP even when quantification is made explicit. For example, in an "intro to proofs" course, we explicitly defined surjectivity of a function using quantifiers: $\forall y\in B.\; \exists x\in A.\; f(a)=b$. When asked whether $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^2$ is surjective, students said "Yes, except when $y<0$! $\endgroup$ Commented Nov 5, 2018 at 17:20
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    $\begingroup$ Following up on my comment: I suspect that this may be a consequence of "sloppy" usage of concepts like this earlier in their mathematical careers, especially when such quantification was not made explicit. For example, it is very common in calculus to talk about a function being "continuous except at $x=0$". This seems conceptually isomorphic to what my students were saying about surjectivity. $\endgroup$ Commented Nov 5, 2018 at 17:21

2 Answers 2


With extensive anecdotal experience, by now I scrupulously avoid such questions (and analogous ones that exactly hit at the incompatibilities between "ordinary" language and mathematical language), at all levels, exactly because of the ambiguities you mention.

Part of the problem is that "order of quantifiers" even in very careful ordinary English need not match the order of occurrence in a sentence. E.g., `property $P(x)$ holds for all $x$' has the for-all after the assertion.

Use of articles (a, an, the) is highly charged in mathematics, and fairly charged in ordinary English, but in different ways.

And the classic "sequence" versus "series", which are synonyms in English, but (traditionally?) we make a big deal about in math?

Altogether, it seems to me perverse to exactly "test" students (at all levels) on the incompatibility of ordinary language versus mathematical usage, when there are more direct and substantive questions available. And when it is easily possible to spend a few more words to clarify what the mathematical (as opposed to linguistic) question really is.

I've seen this on long-answer calculus problems, long-answer upper-division stuff, graduate qualifying exams, and so on, for 40+ years now... and, yes, some colleagues do assert that "this is part of what math is about".

  • $\begingroup$ The following answer of mine to another question on this site is related to your passing comment about sequence vs. series: matheducators.stackexchange.com/a/927/80 $\endgroup$ Commented Nov 5, 2018 at 21:11
  • $\begingroup$ @BrendanW.Sullivan, ah, yes! But I forgot to rant about the adjective "normal"... :) $\endgroup$ Commented Nov 5, 2018 at 21:13

You say that the questions:

"Dividing two polynomials results in another polynomial,"


"Dividing two polynomials always results in another polynomial ?"

are equivalent. In addition, you say that "false" must be understood as "sometime false".

However, in front of a question "always <...> ?" answer "sometimes false" is not a valid construction. Thus, it seems students are receiving contradictory instructions.

About the other possible set of answers "always/sometimes/never". Is "sometimes" exclusive of "always" ? In mathematics we use "for all" and "exists". If "sometimes" is equivalent to "exists" (and assuming "always" is equivalent to "for all"), then "sometimes" is not exclusive to "always", it should be clarified as "sometimes but not always".

  • $\begingroup$ I will edit my post to say "sometimes but not always," but this nitpick is not the spirit of the question. In logic,for a statement to be true, it must "always be true." This is simple for mathematicians and every time I run across a question like this in my graduate exams it is interpreted in this manner. If I were asking a "Always/Sometimes/Never" question, I would not have "always" in the question statement. The "always" is only there as a crutch to help students interpret "True/False" questions correctly before they have any clue what "for all" and "exists" mean in a math context. $\endgroup$
    – Opal E
    Commented Nov 5, 2018 at 1:01
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    $\begingroup$ @OpalE: sorry, but no, in logic true is not "always true" and false is not "sometimes false". In logic, is the question the one whom contains the quantifier, not the answer. The answer is true or false, without need of quantifier, if the question is correctly stated. $\endgroup$ Commented Nov 5, 2018 at 7:34
  • $\begingroup$ @OpalE: more important, as others persons commented, this kind of test breaks the basic rules: 1) skip exams 2) skip tests exams $\endgroup$ Commented Nov 5, 2018 at 7:36

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