Should students get full credit for getting the correct answer (without work)?

Question

Pre-algebra

If the student is taking this branch of mathematics, they are expected to show their work because they're expected to solve specific problems in a certain way. Ex, when they're solving for a variable they're supposed to manually find the value of x by isolating it rather than entering the left and right hand of the equal sign in the calculator (in slope-intersect form) and finding the intersection point. If they only get the correct answer without showing work, we give them 2 points. 8 more points if work is shown.

Algebra

Students in algebra are expected to know how to get the correct answer. We don't care how they do it, as long as it's the correct answer. Full credit is given for the correct answer, basically.

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Are we being too "harsh" on our pre-algebra students? Should we let them get full credit for the correct answer or should we make sure they're ready for algebra with our current system?

Answer 1

Personally, I'd say "no."

The students who don't write out the steps in their algebra classes appear in calculus thinking that they should be able to write down all the answers without any intermediate steps. I even have some students in Calculus 2 who think that there is some kind of value in not writing down the steps. None of these students can complete any calculus problems without errors.

I'd prefer that in all math classes, student submitted solutions should be solutions: I should be able to read them and be convinced that the final answer is correct. "x = 7" out of nowhere isn't convincing anyone, and I don't see it as a useful submission.

In my opinion, the earlier this standard becomes expected, the better.

Answer 2

Short answer: No.

That said, I'm of the opinion that the common phrase "show your work" is really a misstatement of what we're aiming for. The point is neither to just evidence sweat, toil, and tears, nor purely as proof of non-cheating. Rather, the point of written mathematics is to prove, explain and convince other human beings of something. As Benjamin Pierce put it, a keystone idea that we should be making clear to our students:

Mathematics is the science that draws necessary conclusions.

So, we should not be developing students to be in the habit of pronouncing some answer by fiat, that might as well be delivered by ESP or divine providence, and expect that it will be accepted or used by colleagues or fellow citizens. (Nor expect that they can deal with skeptics by means of mere rhetoric, intimidation, or call to authority.) Rather, we should be sharing the scientific method with our students, that they can convince others around them by means of clear written evidence and logic.

To my mind, that is the real point of writing more than the final answer; the student should be writing a series of logical statements, in the grammar of mathematical statements, that will convince a reader that their answer must be correct. And if wrong: this process of writing allows the interlocutor an opportunity to participate, clarify, refine, and possibly improve our thinking. Many or most students fail to see that mathematical equations are actually assertive statements, directly equivalent to natural-language sentences (but more concise). And yet, to my mind, this is far more important than the answer to any particular question.

After lengthy consideration, the direction on all of my tests doesn't say "show your work", rather it says, "justify your work with well-written math", and of course I grade on that (uniformly regardless of level: from remedial arithmetic to college algebra and statistics).

Never mind pre-algebra, it's even more important to assess writing at the level of algebra and above. Frankly, I'm of the opinion that the only reason this is being eliminated from classrooms is some combination of the instructors being lazy, unaware, or the institutions engaging in fraud to "fake" students through math courses. The end result is a disastrous misunderstanding of math, and schools effectively giving up on students being able to even hope that they can master it.

Answer 3

My last undergraduate exam (many years ago...) was for the course "Microeconomic Theory II" - not a 4th-year course, but we had freedom to schedule. I was aiming to get an overall "Excellent" grade in my BA (= above 8.5/10 as grades are measured in my country). Moreover, this was my last exam, so I wanted it to be a triumph (vanity is never far away...). We knew that the exam would require drawing diagrams, but also performing basic algebraic calculations (finding the extremum of a function -cost minimization, utility maximization, etc). So I went in with pencil, ruler, eraser, sharpener, millimetre paper to draw the diagrams, and glue-tape to glue them on the exam papers. But no calculator - we were allowed calculators only in Statistics, to perform linear regression.

And indeed, there were the diagrams, and I was excited to draw them in such a high-quality (and flashy) manner, and of course, there was the "find the extremum" part. I did -and the end result was not a nice looking, round number. I got suspicious: we knew the professors were usually giving numerical exercises with nice round solutions - as a gift to the students, and perhaps a little easier (to the eye) to grade afterwards. So I re-checked the whole calculations twice (so in all, I did them three times). I could not find anything wrong, so I thought, "hey, this time, no nice and round solution". Apart from this little worry, I was pretty sure I had answered everything perfectly (i.e. completely and correctly).
After the exam ended I realized what I had done, three times in a row: I had "divided a multiplication": there was in front of my eyes something like "$3 \times 4$" ($=12$) and I have repeatedly calculated it as $3/4$ (=$0.75$). Down goes your triumphant final exam...

I got a 10/10. I am not saying that this was a fair grade (because there were maybe other students that performed at the same level as me without making the silly mistake), but obviously the professor saw that it was a silly mistake, and decided to ignore it, (impressed, perhaps, by the unexpectedly executed diagrams). But, my point is, that the only reason he could see it as a silly mistake, was because I had written down all the steps leading to the solution.

So while the correctness of the end-result is important, it does not really convey anything about what the student knows, if it is presented alone (and leaving also aside issues of cheating, etc). It is the arrival method that gives the instructor something to evaluate (the journey and not the destination, as it is said in other contexts...). So I would agree that "answers out of nowhere" should get a "nowhere" grade.

Answer 4

Great Question.

In school, and in math especially the goal is to model systems. Wether the system be large, or complex. Super applicable or abstract. Our goal as educators is to teach students how to think deeply about the concepts, and to apply the skill sets.

For me, mandating showing work allows me a new dimension of teaching. It demonstrates an understanding of the conceptual, and of being able to follow a modeling process. For you as a teacher, it will allow you to differentiate your teaching to meet the needs of students who are lacking understanding or follow through in a particular area.

Showing work is equally helpful for the instructor and student. For the student, it forces them to compartmentalize their work in a framework that they can communicate to others. It also allows them to track mistakes, organize methods for problem solving, and build a framework that is applicable in all areas of life. For the teacher is allows you insight into the needs of the student, and a map for how your students are thinking differently.

And on the practical note, habit building is crucial. And in any field in industry these students will need to be able to document their work in a way that successfully communicates how they've solved the problem.

I'm an Undergraduate researcher, and this is something one of my favorite advisors told me, when I didn't think I needed to be thorough in the way I documented my findings. "If you can tell someone the answer, but you can't explain the process, or teach anyone to get there. More often than not your solution falls on deaf ears" Essentially. Communication is key.

The best Mathematician I've ever met, Dave Prince, always told me, "What works, is Work!". So if you don't show it then it doesn't work. We would give 20% for correct solution, and 80% for process.

Answer 5

I'll agree with what's been said, and say no.

Showing the work means showing that you've grasped the concept itself. Being able to produce the answer is more often than not the easy part. Often times, the answer is immediately apparent when reading the question, and one might be tempted to "show off" and simply state that I can tell you the answer without doing any work, and hope that the reader will be impressed by this.

It is impressive. It means that you've gained a very intuitive understanding of the concept. But with that, you should be able to tell the reader exactly why it is so intuitive and obvious. That's the really impressive part, and therefore should be the thing that awards full credit.

Answer 6

It depends on the mark allocation but in my opinion I would say no. I've seen before that a learner somehow arrives at the correct answer in homework but none of the work is correct. At the end of the day we are assessing a number of things, not just the answer. A learner must demonstrate the appropriate skills and knowledge when answering questions.

Answer 7

Professors worldwide should use the procedure as the basis of grading. Three reasons.

1. It allows the examiner to catch potential misunderstandings/mistakes, and it gives him/her the opportunity to signal them to the student.
2. It rewards students in a way such that the work is more valuable than simple fraud (it can be easy to copy a classmate's answer, but it degrades the whole methodology, turning it into something similar to a multiple selection exam).
3. As mathematics builds up, topics from first courses start getting trivial, and can be forgotten by students. Sometimes, a small error generated by this can ruin the answer, and deprive a student from a good grade, even if the actual course's theory is correct in the solution. Grading by procedure allows the student to receive a good (if not the whole) grade, because he has demonstrated that he/she understands and applies the actual course theory.

Answer 8

I say you are not being too harsh on the pre-algebra students because showing the work is the key to success. After all, for most students, algebra can be very confusing. Those who don't show their work and get the right answers should probably be in the Algebra class, a class more advanced where it appears they already understand basic manipulations.

Be careful on assuming that just because a student in Algebra gets the correct answer that they know how to get it.

In my experience as a student, I knew many students actually cheated their way through much of Algebra. If asked how they got it, they would most likely be unable to answer.

In fact, Algebra is where cheating in math usually starts. Those who don't cheat in Algebra will either struggle, succeed, or lie somewhere in the middle. Those students, however, probably won't cheat in mathematics in the future.

Just some food for thought.

Answer 9

I think it's up to the teacher to decide which students are actually capable of doing the math they're doing by their own work, if they're leaving out steps. I say this because when I was in high school I hated the idea of "showing all my work" for tedious steps that I could do in my head. Most of my teachers understood early on that I was capable of actually doing this, and not just copying my answers.

I really appreciated that but looking back I can see why some teachers want every student to show work. They just want to be sure students aren't cheating, and that their students are learning to go step by step in their thought processes. But it's part of the teacher's job to get to know their students, and figure out who's actually capable of what, because sometimes the extra writing just feels like a waste to a student who's not challenged.

Answer 10

I agree with the consensus that the general answer is an emphatic No. As has been pointed, mathematics is fundamentally a study of valid means of reasoning, so the method is much more important than the answer itself. It helps to make this explicit to students. And in order to do that, it is necessary that the instructor think carefully about why this is the case. As the other answers explain this point very well, I won't elaborate further.

However, context does matter. When the steps are simple and the method is well-known from the past, it would be absurd to require a student to show all the trivial intermediate steps. For example, in a middle school algebra class, I'd expect a student solving $3x + 7 = 19$ to methodically subtract $7$ from each side, then divide each side by $3$ and only then to arrive at the answer $x = 4$. On the other hand, if a student in calculus course encountered this equation as part of a more complicated problem, I'd have no problem with them immediately deducing $x = 4$ from the equation. That is the sense in which I mean that context matters.

Answer 11

It depends on the goals you have for the assessment for which you are giving credit.

For some assignments, getting the right answer, or a certain amount of right answers, might be the only real goal you set out for students. For example, if you're giving an assignment to assess a kid's skills with doing multiplication of two-digit integers, you could give the kid 20 problems to do and 20 minutes in which to do them, then grade each one on whether it's right or wrong and don't think about the work. You could do something similar with any rote mechanical skill, e.g. Finding very simple derivatives. In this case, the goal of the assignment is to demonstrate evidence that the kid is sufficiently fluent in the mechanical processes that they can get a reasonable number of answers in a certain period of time. In that case, the work being done is not really all that necessary.

In other cases, the goal of the assignment has little to do with the answer and everything to do with the process and the explanations. I suspect this is where a lot of our assessments fall -- homework, projects, complex problems on exams, etc. In that case the goal of the assessment is not to get a right answer per se but to get an answer and then explain why it's right, and you're assessing the explanation and answer.

Both approaches to assessment could reasonably have a place in a math class, and one's not better than the other. They just measure different things. The important thing is to know what your objectives are, communicate these clearly to the students in advance, and then give credit in a way that's consistent with your goals.

Answer 12

A student should follow the directions given. On a standardized test, there are multiple choices (usually) and a student has the ability to beat the test while not necessarily knowing how to solve the problem the "correct" way. If the instructions for the exam or homework say "show all work," the unspoken implication is that it's one step shy of a two column proof. That may seem harsh from the student's perspective, but with the advent of the TI84 type of calculators, there's a lot that be plugged in, and the answer appears. Solutions to a quadratic equation? The calculator will offer the roots. Should the teacher simply accept "$x=4, x=8$" with no work? During this part of the algebra curriculum, for example, there are multiple ways to solve, and teachers can expect students to learn each way and solve by the method requested. Spell out the quadratic equation, work it out, and give the solutions, but if the test section said "solve by factoring," any or all points are lost.

Yesterday, I got back an exam I proctored, and one question was $\log_3 5 = x$. The answer was there, $1.46$, to $2$ digits as requested, but no work. I asked "did you do this on scrap paper?" hoping to just staple it to the test. He said he guessed and checked. Knowing that $\sqrt{3} \approx 1.73$, and so $3^{1.5} \approx 5.20$, he just hacked down to $1.46$. Clever, but he forgot how to manipulate logs. If I were to grade this, I'd have to ask how he showed any knowledge of logs, even though his gut was correct for exponents.