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?