Let’s consider this problem in a standard introductory course where “solving for the variable” is first introduced.
$$\mbox{Show all your work and solve for x: }x + 3 = 7$$

And a student writes nothing more than $$x = 4$$

Now, the solution is correct. But the student did not “show any work”. So, what to do from an assessment standpoint? There are a few schools of thought on this:

1. The solution is marked as completely wrong since no work was shown.
2. The solution is marked as partially wrong since even though the solution is correct, the process by which it was found was not written.
3. The solution is marked as completely correct since the solution is, in fact, correct.

So, is there a universally correct way to handle this? I can see reasonable arguments for all three approaches. Let’s a take a look at typical (anecdotal) pros and cons for each approach.

Approach 1

To borrow from the wanderer’s philosophy: “The journey is more important than the destination.” In other words, it’s irrelevant that the solution is correct, what matters more is that the process by which the solution was found is sound. Some will go a bit further to say, “sound and efficient”. Some still further to say, “sound and optimal”. Others would say “sound and conventional”.

Of the approaches most inline with the wanderer is a simple requirement that the method be demonstrated and sound. The efficiency of the method is irrelevant and what the teacher / assessor (sometimes the same person) wants to know is the story of the wandering. Thus, from the standpoint of teaching, providing feedback, and assessment, it becomes null and void since the process is never shown. And since the method by which the solution is left a secret, there can be no credit awarded. If anything, this approach should be inline with all those who decry “answer-getting” in mathematics — just show the work! Mathematics isn’t about just churning through algorithms and processes. It’s a thinker’s subject and that’s what is to be measured. Hence, no work, no credit.

Approach 2

Awarding partial credit for a correct answer with no work shown is a grading approach that most instructors would likely regard as a fair middle ground. Students also tend to not bicker too much about this, provided that the partial credit awarded is fair (often considered to be 50%).

Approach 3

Some view this approach in grading as promoting guessing, while those in favor tend to see a correct solution as a correct solution, regardless of methodology. Students like this approach as it is more forgiving than the other two grading approaches and as it tends to be in better alignment with students’ typical understanding of the point of the problem — to get the correct solution.

Advanced students, or at least those who are capable of “seeing the solution”, also like this approach to grading since the part of showing the work is simply tedium.

The tricky part with this approach is in making sure that the problems posed don’t actually lend themselves to pure guesswork. A common design flaw in elementary math courses is the prevalence of problems with (small) integer only coefficients and solutions. Adopting a “if the answer is correct then full marks are awarded” policy can actually be a poor assessment philosophy if the problems posed are susceptible to pure guess work. For a larger discussion on this, see Fractions Are Part Of The Whole Algebra Thing.

General Comments

The bitter pill for students to swallow with Approach 1 and Approach 2 is that students often view the zero points (or partial points) awarded as purely a result of bureaucratic failure, not one of mathematical failure. After all, the answer was correct. And this is often a point of division between a student’s objective and an instructor’s.

Where things begin to get dicey is when we toss in requirements that the work shown be “efficient”, “optimal”, “conventional”, etc.; then we’re no longer wandering, but instead we are marching. And here is often where the “show all your work” policy tends to go awry. For the case that no work whatsoever is shown, perhaps the instructor can argue that the exercise wasn’t about the final result, but more about the steps taken to arrive at the final result. However, it becomes a bit murky when the requirement is to show a specific procedure.

Though not the original scenario given above, a characteristic of the pro “show your work” crowd is to err on the side of militancy when work is shown. For example, the preferred method is to subtract three from both sides (and to say it as such) and then to show that \(x = 4\). But what if the student subtracted 7 from both sides and then subtracted \(x\) from both sides to obtain \(-4 = -x\) which then would yield \(x = 4\)? Some instructors still ding students since this isn’t an optimal procedure. This is over the top and is a generally harmful way to assess. If the student takes the long way around the barn, then that can warrant a further conversation and demonstration on an efficient approach, but doesn’t warrant a scoring penalty. Here is an example of such extremal grading behavior.

Instructors should generally be interested in what a student’s thought process is more so than whether or not they obtained the correct answer. And for this reason, I think the burden is actually on the instructor to construct problems that are resistant to guessing. In the event that the student does actually guess the correct the solution, the instructor ought view that as a design flaw in the problem rather than an execution flaw on the student’s part. In other words, it’s “ask better questions rather than demanding procedural compliance.”