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I am currently marking a set of math assignment at university, and I am wondering about the following general point: in a question such as "Find the solutions to the equation $x^2 - 3 = 0$", does the term "find" in and of itself imply that the reasoning that led to finding the solution must be laid out as part of a perfect answer? I.e. does the phrasing of the question imply that $x = \pm\sqrt{3}$ is insufficient to completely and appropriately solve the problem at hand?

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  • $\begingroup$ What does the grading scheme that the lecturer gave you say about that? $\endgroup$ – Polygnome May 6 at 9:03
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    $\begingroup$ I own this exact same shirt, and a year ago I wore it (for the first few minutes, over my other "dress shirt") at the beginning of one of several item writer workshop segments I gave for a company that I do math/quant contract work with, for a couple of graduate/professional high stakes tests. That particular segment was on the importance of being aware of possible misinterpretations. $\endgroup$ – Dave L Renfro May 6 at 11:33
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    $\begingroup$ What you're asking should be included in your grading instructions, which should include what work needs to be shown, what is partial credit to be based on, what to do with a correct final answer in which their prior work is incorrect in a major way but their work still results in the correct answer (i.e. they should be getting $4$ by adding $2$ and $2,$ but their work shows that they got $4$ by multiplying $2$ and $2),$ etc. $\endgroup$ – Dave L Renfro May 6 at 11:41
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    $\begingroup$ It's ambiguous and I think the instructor should be asked to clarify if work should be shown. $\endgroup$ – Amy B May 6 at 11:49
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    $\begingroup$ In and of itself, "find" does not imply that the reasoning needs to be shown. Of course, in the context of a class in which the instructor has announced general rules about such matters, the situation might be quite different, but that's not what you asked about. $\endgroup$ – Andreas Blass May 6 at 17:45
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Does the “Find” in questions such as “Find the solution” imply the process of finding must be laid out?

No. No single word or phrase is ever going to convey the idea that a student should solve the problem and explain the solution process in some particular level of detail. An entirely appropriate response to "Find all solutions to [equation]" is "Okay, I found all of the solutions. Now what?" Alternatively:

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You cannot, therefore, rely on a single word or phrase to convey your intentions. Rather, you must establish expectations, both early and often. Such expectations should become part of the culture of the class.

If you expect students to "show their work," tell them so. Start early in the term by giving examples of correctly written solutions, and non-examples of poor solutions. Mark formative assessments with a critical eye, and point out anywhere that steps you consider vital are skipped. When giving formative assessments, include language which clearly communicates to the students what the expectation is.

For example, in a precalculus course which I have taught multiple times, students are told over and over again that they must show how they arrived at a solution, but that they are not required to simplify. The syllabus for that course (which is very nearly a textbook) contains hundreds of problems, about a third of which have completely worked out solutions in the style expected on exams. Prior to each exam, the students are given a sample exam—significant numbers of office hours and a small amount of class time are devoted to working through those problems, including how they should be written up. Finally, every exam contains an instruction of the form

You must justify your answers—describe the process which leads to your solution. You will not receive credit for correct answers if insufficient work is shown. You will not lose points for correct statements (even if they are not relevant to the problem), so while keeping argument simple is a valuable skill, it is a good idea to err on the side of writing more, rather than less, on this exam. You do not need to simplify your results.

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does the term "find" in and of itself imply that the reasoning that led to finding the solution must be laid out as part of a perfect answer

No, definitely not.

The conventions for what's expected in an answer very wildly across individual schools, classes, and professors. No single word or phrase is going to convey what's expected of students dealing with a new assignment; if you want particular information included in the answers, students need to be told that explicitly.

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    $\begingroup$ Oi... I started writing an answer before going to teach, then came back to finish it. After posting, I see that we both used the same phrase "no single word or phrase..." Great minds? (+1) $\endgroup$ – Xander Henderson May 6 at 23:12
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I think instead of trying to get some legalistic interpretation of the words (that will apply to all classes/instructors), you have to look at the individual teacher/class expectations.

For instance, it is understood on AP tests, that the questions are "essay questions". So find the general solution for y'' + 2y' + y = 25x requires a solution that shows the method.

Another example is Navy Nuclear Power School, where the acronym "NWNC" stood for "no work, no credit". All it takes is a couple marking of that, to drive the requested behavior.

I think in general, most teachers and classes require that students show the process for getting the answer. (For the legalistic pedants...Yes, and more than guessing. Yes, more than plugging the answer back in and showing it works.)

But in any case, the way to resolve this is to ask the instructor what he expects. Then mark versus that expectation. There's no way we can anticipate every instructor's quirks. Nor any reason to think that every instructor wants cats skinned in the same fashion.

P.s. And yes, it is a judgment call what is a complete answer. And yes, it is more work to have to look at different methods (by the student) to answer a given question, So? This is how the normal, non-Euclidean world works. Not everything is an exact equation, when doing complex practical tasks, like grading.

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    $\begingroup$ Thank you for your comment -- although I'm taken aback by your somewhat passive aggressive post scriptum. I never insinuated that I wanted to make my grading work as simple as possible. $\endgroup$ – MacRance May 6 at 22:09

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