For simple expressions with easily derived canonical forms (eg polynomials and simple rational expressions), is there a way to leverage existing tools to verify that two expressions are equal when those expressions are expressed in programming form (ie, 2x^2+3x+1 and (4x^2+6x+2)/2) in two separate cells of a spreadsheet?

I'm drawn toward Google Sheets and Gapps script because I could leverage java libraries, but this might not be a programming question if there is a simpler approach.

To make the connection to math education more clear, the purpose of this question is to allow a teacher to get math responses in a free response quiz (eg, via google forms), then grade the quiz automatically so much as is possible, largely cutting down on the time required to grade.

This is possible via multiple choice responses, but this format is less accurate for assessing student understanding. This question is in the educator's forum because it's possible that someone has accomplished this task already or by using a different approach. Otherwise, this is a programming question (which is asked in the appropriate forum as well).

  • $\begingroup$ Does everything need to fit into two separate cells, or could one, two, or three columns be used? $\endgroup$ – Rory Daulton Sep 27 '18 at 0:10
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    $\begingroup$ Any number of cells could be used. @TommiBrander correct, this is about answer validation. $\endgroup$ – Zediiiii Sep 27 '18 at 13:50
  • $\begingroup$ This appears to be a duplicate of your prior question, both of which are off topic on this site. $\endgroup$ – Bill Dubuque Sep 27 '18 at 23:24
  • $\begingroup$ This is off-topic because there isn't a pedagogical answer, or because using computerized grading tools for math education isn't something math educators concern themselves with? I'm not looking for programming answers here, I'm looking for pedagogical answers, such as the great answer by Daniel Collins. $\endgroup$ – Zediiiii Sep 29 '18 at 8:34

Generally speaking, this issue is exactly why the global direction to "simplify your answer" is given. Properly defined for different structures, it provides a way of generating a canonical single answer that can be more easily checked (among two students, student-and-book, student-and-instructor, etc.)

And I don't think that's merely a pedagogical or ease-of-grading issue. In practice and industry, there has to be some language in which two people communicate and express answers and check whether they agree or not. That is precisely the point of the simplification process (again, defined for a variety of different objects). I would recommend you use that; include the direction "simplify your answer" (and define it in class!) and then you'll just have one form of each answer to check.

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  • $\begingroup$ I agree with your conclusion, but without some really pedantic instructions (which don't work well in less advanced situations) telling students to "simplify" will still create problems. For instance, (x+3)(x-2)^2=(x-2)(x-2)(x+3)=(x-2)^2(x+3). I could introduce ordering, make a database full of possibly correct solutions, etc, but this isn't ideal. Do you have any suggestions for ambiguous ordering cases? $\endgroup$ – Zediiiii Sep 29 '18 at 8:22
  • $\begingroup$ @Zediiiii: Well, "simplifying" generally means removing all parentheses (among other things; e.g., that's what my slide on the subject in elementary algebra says). But obviously your example involves "factoring fully" -- specifying "use exponents for repeated factors" is commonly done. I agree that the order of the factors is more troublesome, but in your example there would be only two cases to check, really. $\endgroup$ – Daniel R. Collins Sep 29 '18 at 13:28
  • $\begingroup$ Also a good point. Building a database of acceptable answers for every question is a valid way to approach this, though it will require a fairly long "wear in" time and some fairly time-intensive human effort to set up. The point of this question is to decrease the amount of human interaction with the grading process while preserving a free-response format and the flexibility to change questions each year. Presently, I use a system like this and only take time on problems that aren't marked correct by the system. It is only slightly time-profitable. $\endgroup$ – Zediiiii Sep 30 '18 at 0:30

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