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I am a math teacher from China, teaching a course in English.

Some students of mine are really good at finding answers for math problems designed in a quiz, however they are unable to write down full answers with details or explanations.

How do I decribe this situation in English? I am about to prepare an exam in which for those who do not give detailed answers would receive low scores. Is the following sentence ok: "Do not write only your answers (numbers), give me your detailed solutions."

quid
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Hoa
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    Show them the difference, via examples, in class. What matters is that they understand the level of justification that you want to see. The easiest way to communicate this is by showing them. – Dan Fox Mar 10 '19 at 08:39
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    Yeah, That is what I often do in class. However, I would like to know the English vocabularies to differentiate between "a number" and "a detaild explanation". Are they "an answer" vs. "a solution"? – Hoa Mar 10 '19 at 08:49
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    Are your students native English speakers? –  Mar 10 '19 at 16:07
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    Just here for a minute (been very busy the past week, more than expected), so I'll just comment, but something that worked for me is to explain that you can't give partial credit when little or no work is shown. And then make sure you give partial credit, and make sure that you give a clear explanation of what it's for when grading tests. Often times partial credit is simply grading according to a consistent rubric. I often used to hand out my rubric with briefly worked-out solutions when I returned graded tests to my students. – Dave L Renfro Mar 10 '19 at 19:50
  • Teach them GFSA method first: https://matheducators.stackexchange.com/a/14100/7930 – Rusty Core Mar 11 '19 at 01:15
  • @DaveLRenfro I really like the idea of handing out a rubric with worked-out solutions, as it helps student understand expectations. Out of curiosity, did you ever try to show your rubric to the students before the exam? If not, any reason why? (I'm not 100% certain what is meant by rubric, I'm assuming that the criteria are general enough that they do not reveal the test's contents, so disregard this comment if that wasn't the case). – orion2112 Mar 12 '19 at 07:18
  • @orion2112: See my answer to How can teachers reduce student suspicions in discrepancies and irregularities of grading?, especially the second paragraph. Regarding showing a rebric to students before the test, other than a rough idea sometimes of what to give credit for, I found it much simpler and less subject to later revision to develop the rubric details when I actually encountered the various issues during grading, that is, at the time when decisions on credit actually needed to be made. – Dave L Renfro Mar 12 '19 at 08:12

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The problem with the word "solution" is that it could mean the final answer or it could mean how the final answer was obtained, so I suggest that you don't use it.

How about the following?

Clearly show how you got your final answer if you want to get full credit for it.

If the phrase "full credit" is not familiar to them, you may use "all the points" or "the complete score."

I also suggest having the student make explicit what the final answer is.

Put your final answer in a box.

JRN
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    +1 Good suggestions. However, my editing instincts suggest maybe make "Clearly show how you got your final answer if you want to get full credit for it" a little less personal, perhaps something like "Full credit may not be awarded for answers that are not fully justified". The reason for "may not" is so that you're not always forced into following this, and to lessen having to justify to Student A why Student B (a friend of Student A, and they looked over both their graded tests together) got full credit when Student B omitted something you said in class needed to be done, or stuff like this. – Dave L Renfro Mar 10 '19 at 10:32
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    @DaveLRenfro, I agree that "Full credit may not be awarded for answers that are not fully justified" is less personal and that is how I usually write. (I think it is a good suggestion, and I encourage you to submit it as an answer.) However, my concern is that the students (who I assume do not have English as their first language) may find the active voice more understandable than the passive voice. – JRN Mar 10 '19 at 13:21
  • @DaveLRenfro, I also agree with your point on the use of "may." I believe that my suggested sentence ("Clearly show how you got your final answer if you want to get full credit for it.") does not explicitly state that full credit will not be given to those who do not show their working. – JRN Mar 10 '19 at 13:24
  • @DaveLRenfro, also, perhaps "may be not awarded" is closer to the intended meaning than "may not be awarded." – JRN Mar 10 '19 at 13:27
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(From a comment)

On American math exams, I see the phrase "justify your answers".

Gerald Edgar
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  • This is the best and most precise answer. The language on my exams is, "Write your answer in the space provided, and justify it with well-written math." – Daniel R. Collins Mar 14 '19 at 00:49
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There's no phrase that's going to reliably convey what you want - if it's not what they've been taught to do, they're not going to know what you want. Whatever phrase you use, you're going to have to back it up with a discussion about what it means and what the expectations are.

Henry Towsner
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The typical British English phrase would be "show your working", but of course if the students are not familiar with the idea, you need to explain what that means in practice.

One of the instructions printed on a UK national exam paper in mathematics (at the level where a high enough grade in the exam would be a requirement for university admission) is

You should show sufficient working to make your methods clear. Answers without working may not gain full credit.

alephzero
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    In US English this would be "show your work". You should show sufficient work to make your methods clear. Answer without work may not gain full credit. – Amy B Mar 11 '19 at 07:22
  • If I saw show your work[ing], I would write out intermediate steps like a change of variables or simplying a polynomial, but not prose explanations. If you want the work justified (e.g., can use theorem X because condition Y is met), you need to explicitly ask for it. – Matt Krause Mar 11 '19 at 18:37
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    The current phrase used in British A-level papers (taken at 18) is "In this question you must show detailed reasoning", and is particularly common where a student could use a scientific calculator to jump straight to the answer. – dbmag9 Mar 12 '19 at 10:16
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Let me frame challenge the question.

Instead of focusing on the terminology how to "make the students provide details/explanations" you could also provide the numeric result up front and simply ask how they would solve this using technique "xyz". By providing the result up front it should be clear to anyone that just providing the solution will earn them no points.

If you would like to see specific intermediate steps you could also divide the problem into multiple parts (optionally provide the numeric answer for each part). Each part could also have a score attached to it so there's also no question about the grading. This has the benefit (although that may be subjective) that if a student gets stuck at the beginning they don't have to skip the complete question.

I realize that depending on the questions this might not be possible.

MSeifert
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  • +1 for you could also provide the numeric result up front and simply ask how they would solve this using technique "xyz" --- In many cases this might be all that is needed without giving the numerical/algebraic final result, but giving the final result is an interesting parallel to test questions in more advanced classes that ask the student to prove that such-and-such result is true. – Dave L Renfro Mar 11 '19 at 05:55
  • ...ask how they would solve... I did something like this recently on a quiz. However, instead of giving the answer, I forbade them to give a numerical answer. It was on the first page of the quiz where no calculator was allowed. For the directions, I wrote "For your answer to this problem, do not calculate a final answer (because this is the no-calculator part). Rather, you must write a single mathematical expression that represents the area, and include all units. I should be able to type your expression into my calculator, press "enter" exactly once, and get the correct area." – Nick C Mar 11 '19 at 06:22
  • @Nick C: I used to do this for volume of revolution problems on tests, and ask (as separate questions) evaluations of definite integrals elsewhere (generic type definite integrals, not anything resembling a volume of revolution integral to avoid cluing them what the set-ups would look like). Here are some examples I posted back in 23 February 2003 in sci.math. – Dave L Renfro Mar 11 '19 at 07:39
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    "If you would like to see specific intermediate steps you could also divide the problem into multiple parts (optionally provide the numeric answer for each part)." — most non-trivial problems can be solved using different approaches, so your steps may not align with the steps a student chooses. – Rusty Core Mar 11 '19 at 16:42
  • @RustyCore You're of course right. That's why I added the "if you would like to see". However more often than not questions are chosen so a particular (often newly learned) concept should be applied to solve them. In these cases having specific intermediate steps could "force" the student to actually apply that concept and/or provide a hint to the student how it can be solved. – MSeifert Mar 11 '19 at 16:49
  • I think asking them to re-trace your train of thought is akin putting their minds in shackles. IMO, providing their own steps is good enough, and if the method that you want them to use is more effective or generic, you can tackle this issue during review. – Rusty Core Mar 11 '19 at 16:54
  • @RustyCore Maybe we're thinking about different questions or you misunderstood what I intended to say. Most exam questions are aimed to find out if the students understand what they've been taught. In that sense my suggestion wasn't to "shakle their minds" but to clearly state what they're intended to do there. However I know that this shouldn't (and couldn't) be applied to all kinds of questions which is why the answer contains "If you would like to see" and "depending on the questions this might not be possible." – MSeifert Mar 11 '19 at 17:15
  • You suggested to offer answers for every part, and I am saying the parts may be different. Ok, a simple example: given two rectangles 3 by 2 and 4 by 2 cm find their total area. You can do it using the following steps: (1) Find area of the first rectangle: 32=6 (cm^2); (2) Find area of the second rectangle: 42 = 8 (cm^2); (3) Find total area: 6 + 8 = 14 (cm^2). Or you can use the following steps: (1) align the rectangles so they are touching with 2cm side. (2) Find the length of the new side: 3 + 4 = 7 (cm^2); (3) Find the area: 7 * 2 = 14 (cm^2) Which steps are you going to verify/enforce? – Rusty Core Mar 11 '19 at 17:25
  • @RustyCore In that specific case I wouldn't separate the question into multiple parts. I would maybe ask for the formula to compute the area of a rectangle with side-lengths of x and y and then ask to apply that formula to calculate the total area of both rectangles and provide the solution only for that part. – MSeifert Mar 11 '19 at 17:38
  • @RustyCore I think this discussion is probably not really constructive without specific examples. The question did not mention specific examples or topics so it's rather hard to provide tailored advise. It's okay if we disagree on that part of the answer, I didn't mean to recommend splitting all questions into multiple parts. But if you want to follow up on specific examples or ask when it might make sense to split a question into multiple parts (with solutions for each part) please ask a new question. – MSeifert Mar 11 '19 at 17:47
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When I see a student write down an answer with absolutely no work I will point out a problem to them. If I don’t know them personally, I do not know if they are a gifted student who is able to skip many or all steps, or if they simply sat near a smart student and copied just her circled answers.

Then I tell that student that a teacher is looking to see all the work, step-by-step, how they went from the question to the final answer. I remind them that if they have an incorrect answer, the teacher has no ability to offer partial credit, say three points out of the four points this question might be worth.

It might also help for you to provide two good examples of the process. Whatever level you are teaching, you should be able to find two different types of problems this would apply to and how a good answer is composed.

TLDR - please show all steps/work that shows how you calculated the answer.

JTP - Apologise to Monica
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I always state the number of points possible on exam problems. If you have this practice, then take it a step further. Suppose a 10-point problem says:

Find the volume of the rectangular prism shown below.

Ask yourself what you want to see a student do. In my class, I would want (based on class demonstrations, homework problems, etc.) them to write an expression that involves a basic formula (no numbers), followed by that expression where the particular lengths have been substituted into that formula (using the correct units), followed by the answer they obtain from their calculator, concluding with the correctly rounded answer, according to the significant digits given (again, with the correct units).

Instead of writing [10 points], I will often write something like [Correct numerical answer = 7 points, Correct units throughout = 1 point, Showed the formula = 1 point, Correctly rounded = 1 point]. This has the benefit of giving me a rubric to follow and publishing that to the student. If you're demonstrating all of these steps in class and awarding/withholding points for them consistently on homework/quizzes, then seeing this breakdown given on the exam should make sense to your class.

Nick C
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    Of course, this kind of leads a student by telling them exactly what they should already know to do. Maybe you could be prescriptive like this during the low-stakes quizzes and prepare them in advance that this is the expectation for the rest of the class, whether you write these steps or not. However, in my experience, writing it out like this makes it all the more obvious who can do the work and who can't. Even with these steps listed out, you'll still know if a student can't do them. – Nick C Mar 10 '19 at 22:41
  • Maybe you could be prescriptive like this during the low-stakes quizzes --- This seems to me an excellent suggestion for what is somewhat of an overkill approach, although my handing out rubrics when handing out graded tests is not all that much different. (Incidentally, my handing out of rubrics was mostly a way to minimize the many student questions about why 1 point was deducted here, or 2 points deducted there, despite the fact that I gave reasons.) Nonetheless, Nick C's method might be something to use in a really difficult situation when nothing else seems to work. – Dave L Renfro Mar 11 '19 at 05:51
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"Show your work" . . .

. . . is the phrase my instructors always used to declare that answers, absent the written steps involved in arriving at such an answer, are worth only partial credit. This was at the elementary, high-school and university levels.

For context, I attended school in the USA in the midwest in the 1970s through the 1980s.

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MrWonderful
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We use the word 'demonstrate' (or prove).

"Demonstrate/prove that the answer is correct". That way, the student must go through the deductive procedure when resolving math issues.

As a note, I only partially agree as a teacher to making everyone follow a manual-like demonstration. That is because sometimes students can find alternate unique undocumented ways to solve a math problem. So I split them into 3 categories: the ones finding a new way to solve the math problem (best of the students), the normal ones that reach the result by the manual and a special category that can directly put in the result without following an existing or new demonstrated procedure. This last category can be more complex to deal with - some can cheat and copy the result from someone else, in which case I do not take their answer into account but some are of very high skills and can put the correct result no matter what without copying it from someone. I found such skilled students that could state the correct result of any logarithmic and exponential-based problems. No reason not to grant them good scores.

Based on what I said above, I developed the scoring system.

Overmind
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"Your goal is to demonstrate that you understand this material."

Peter Saveliev
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    If I, as a student, were given this instruction, my response would be "Look, I got the right answer. How could I have managed that if I didn't understand the material?" I think that you need to be more explicit about your instructions. – Xander Henderson Aug 13 '19 at 18:16
  • @Xander Henderson It's a nice comeback, but you can find one easily for every suggestion given here. As for more detailed instructions, that's what the syllabus is for. – Peter Saveliev Aug 14 '19 at 00:28