Why Should Students Show Their Work?
From middle school through high school I had a lot of trouble with the classes that should have been my favorites, sometimes barely passing. My troubles could all be traced to two simple problems:
- I’d never do homework
- I’d never show my work
I’m pretty sure most of my teachers thought I was lazy and intransigent. At least that’s what my Physics teacher told me. But they were wrong. I was not lazy — I was always reading ahead in the books, I was doing problems that were not assigned because I thought they were fun. I did not do my homework because it was boring, and the thought never seriously crossed my mind to spend my time bored.
And it wasn’t intransigence that stopped me from showing my work on exams, it was more of a misunderstanding. Now I see that there are a few good reasons that teachers might ask students to show their work. Here are three of them:
- Method is more important than results
- Communication is more important than results
- Teachers are busy, and you’ll do better if your paper is easy to grade
I said that these were good reasons, but I didn’t make them sound very good. That’s because they sound pretty lame when presented to a “problem student”.
And it is pretty lame if “Method is more important than results” throughout your whole class. In life it is results that are important, and most methods I’ve learned fare poorly when faced with the complexities of the real world.
It is pretty lame if “Communication is more important than results” all of the time — you’ll just be training a bunch of useless BS-ers.
And it is too bad that teachers just don’t have the time to think hard about every response they grade, but how hard can you really think about 1500 Calculus problems a week? This is one problem that homeschoolers can sidestep. Well, maybe not all of us…
I’ll go into each of these reasons for showing work in a bit more detail, but first I want to dispense with a bad reason to ask kids to show their work that I hear all of the time. It goes something like this: “You need to show your work so the teacher knows you understand the material.”
Rubbish.
If a student can answer every question on an exam without showing a line of work, then the student understands the material — probably better than someone who needs to show a lot of work to get the same answers. Could there be holes in their understanding that weren’t picked up? Of course, but who’s fault is that? The student or the test writer?
Enough of that, let’s get on to the good stuff.
Method is more important than results
Sometimes in a course we learn new methods to solve old problems. Sometimes they seem better, sometimes just different. Sometimes the new method is just an automated version of the old one, the way the quadratic formula automates completing the square to solve quadratic equations.
Now imagine a student that has memorized the quadratic formula, but doesn’t know how to complete squares. He will ace any test asking him to solve quadratic equations, but he will be missing out on the real meat of the subject. How do you make sure that something like this doesn’t happen?
It is very tempting to say “Solve these problems by completing the square. Show your work!” on the top of the test, and I think that would be justified. But there may be some other ways around the problem
- Assuming they don’t know calculus, ask the students to find the minimum (or maximum) of a quadratic polynomial. Completing the square makes it easy.
- If they are are sharp and like to show off, have them plot circles, hyperbolas, and more from equations like
.
- Throw a quick example of something cool on the board that may seem way beyond the scope of the class. For completing the square I like examples from quantum mechanics because they sound really impressive, are amenable to cartoon presentations, and this simple trick from Algebra ends up being the crux of a number of problems.
These aren’t perfect of course, and different strategies work for different students. If a kid knows the quadratic formula, they might know the formula for the minimum of a quadratic polynomial, making approach (1) fail. Approach (2) will probably intimidate most Algebra I students. Approach (3) will only impress the geeks out there (who are the most likely to be having trouble), and it takes some care to present it so they get the point without being bewildered.
The point is, when you are teaching a method that really is important you need to find problems and examples that highlight the importance. If the method does nothing that the student can’t already do, I don’t think it’s really worth fussing over.
You might also try to make it explicit that you are studying methods, not problems. Ask a student to find different ways to solve a problem and describe their strengths and weaknesses. If a student really hates a method you are teaching, this will give them a chance to think hard about it and convince you what a waste of time it is. That wouldn’t be a bad thing.
Communication is more important than results
Communication is a crucial skill in life. Often it isn’t enough to know how to solve a problem, you need to convince others of your solution so they can help you implement it. Or if you’re an academic, all you are trying to do is share your ideas with the rest of the world. It is appropriate to teach technical communication in Math and Science courses, and when that is the goal “showing your work” is the whole point.
Learning to communicate technical information is difficult. If you are trying to teach your students to communicate, you should make this an important part of your course. When you give exams, you should make it clear when you are looking for an exposition and when you are looking for an answer. When you are looking for an exposition, you should demand more than the usual cryptic formated lines that usually count as “showing work” in Math classes. If you simply give the same old tests and think you’ll teach communication by docking points from kids who didn’t show their work, you are wrong.
So teach them to write and talk about Math and Science. Evaluate their communication skills. Just be upfront about what you are doing, and remember you are trying to teach them something much harder than simple understanding.
Teachers don’t have time to think about every answer
I never realized this until I started grading Calculus papers in grad school. I think I had about 50 students in two sections, and in a typical week they’d do 30 problems. That makes 1500 problems a week. If I spent one minute on each problem, that would be 25 hours just grading homework. And this was a relatively small amount of homework and a small number of students.
When I started lecturing, I felt strongly that spending more time preparing for my lectures was more important than spending that time grading. A bad lecture was a disaster for everyone, good lectures seemed to improve students performance. But thoughtful notes on homework problems took ages to write and had little noticeable effect. If I wanted to get a point across to a particular student, I needed to talk with them in my office hours.
So I would breeze through my grading quickly, scanning to make sure things “looked right”. I would usually give full credit for right answers, but partial credit went disproportionately to the students who would show their work the “right” way.
After that I realized that the secret to good grades is easy: think about what the grader expects to see on the paper and show them exactly that. Try to make your homework the answer key that the grader will use to measure all of the others, and you’ll ace your course. This is just a reality you have to deal with if you go to school and care about grades.
For teachers who want to avoid this problem, my only suggestion is to assign less homework and assign it very carefully. If you can’t eliminate the problem, at least you can mitigate it.
Working Unschooling into School
All of these suggestions I’m making really boil down to this: try to work a little more unschooling into Math and Science lessons. Let kids pursue their interests and don’t tie them down with busywork. If you want to know whether they understand something, just ask them discerning questions about the subject. If you want them to be interested in using a specific tool, show them how that tool is more useful than what they’ve already got. If you want them to learn how to communicate, then get them writing and talking and thinking about their feedback. More drills and stricter answer formatting rules will never do the job.
Oh, and cut back on the homework.
It’s been brought to my attention that I really seem to have something against homework, and that I don’t make much allowance for people who learn differently from me. Some kids need homework and structure, even if I didn’t.
I won’t dispute that one bit. I happen to think that in a free and supportive learning environment those kids who need structure will ask for it. They will make their own homework or ask you for suggestions. This is what I see in my kids anyway, and they are the ones I’m trying to educate for now.
I’m sure there are exceptions, and I really don’t know what I’d do if my kids didn’t seem interested in anything.
In any case, my intention here was to make you think a little bit about the dogma that kids should show their work, and that it is right to penalize kids for doing things the wrong way. Maybe that approach is good for some kids all of the time, or all kids some of the time, but it definitely doesn’t work for all kids all of the time.
Rolfe Schmidt
December 5, 2007 at 9:59 am
Hey Rolfe! I’m still here, somewhere under a mound of papers.
I agree with you about showing work. If it were up to me, they would do much more in their heads than they do on paper. However, the longer I teach, the more I ask them to show work. Here are some of the reasons.
1. Unfortunately, many students will try to get by without learning anything at all. If you assign 10 homework problems whose answers are single numbers (e.g. x = 32, x = 19.5, etc.) then it invites copying on a much larger scale. In my experience, asking to see the work actually cuts down on the copying problem.
2. I really like to give partial credit, but I can’t do it if I can’t see where the answer came from. So, I don’t usually require a specific method — as long as their method is correct, I will sort through it and give whatever credit I can. (It takes MUCH longer to grade like this, but I can’t seem to be happy any other way.) If no work is shown, then I don’t give partial credit, because I can’t see what caused the error.
3. Kids who are learning for the sake of learning will ask for structure when they need it, but kids who have no intention of learning anything if they can help it will not. Getting them to show their work is the only way I have of encouraging them to think.
4. Many of them will rely on calculators way too much if I don’t make them show me what they’re doing — they can download programs for their calculators that will do slope, simplify radicals, etc. Plus, the calculator will do permutations, combinations, and all sorts of things.
I’d rather be more easy-going with the whole “show your work” thing — but I keep feeling forced to make them show their work. With more willing students, it would be a whole different story.
But you made a lot of valid points — I just wish it could be more like that in public school.
Alane Tentoni
December 6, 2007 at 1:44 pm
Hi Alane, Good to hear from you! I hope you can make your way out from under that mound of papers…
Yes, I sidestepped the cheating issue. I was certain it was happening in the Calculus classes I graded, but I never found a satisfactory way to deal with it. In the end, we just weighted the exams pretty heavily so cheating on homework wouldn’t pay.
I also didn’t consider the situation where it is your job to teach kids something they don’t want to learn. That is a tough job that I haven’t really had to face yet, and I see your point. Making kids repeatedly show how to go through the steps of a method will get them to remember it whether they want to or not. I just wish it didn’t have to happen that way.
So we have a few more good reasons to ask kids to show their work.
I’m impressed that you take so much care grading your papers. I wish I’d had teachers like you, and I completely understand about wanting to give partial credit. It always pained me when I had to give low marks to a student who I knew had a good grasp of the material.
The important thing is to be mindful of your situation so you know when to relax and when to be demanding.
Rolfe Schmidt
December 6, 2007 at 4:03 pm
homework is a *very* touchy issue.
i quit doing any myself in about 8th grade,
whereupon i discovered that they’d never fail me
in a “required” course if i kept showing up
(and talking a lot; this may have been optional
but as for me … i couldn’t help myself).
when i got to uni, i started up again grudgingly.
it’s perfectly clear to me now that the only way
to learn any doggone math is to do a lot of problems
but i sure didn’t believe it then and didn’t learn much.
finished a math major with mostly b’s and got to grad school.
where i learned to do homework whether it was graded or not:
a flat-out survival tactic. okay. now i’m a pro.
when i briefly had classes with actual math majors
i assigned, and mostly got, what seemed like a reasonable amount;
an unusual case where playing by the rules actually seemed to work.
then i was sent down to the minors, where over the years
i more or less forgot how this was done. (two-year) diploma mills
like mine provide pretty powerful disincentives for trying
to motivate remedial students … our actual job being to
teach them to blame themselves for their lack of earning power.
when i finally got to work this quarter with some “better” students
(a calc class instead of the highschool stuff that pays the bills)
i soon discovered that i wasn’t assigning *nearly enough*
(or grading the stuff i *did* assign fast enough …
like alane, when i *do* mark problems, i’m compelled
to do it in pretty careful detail …). and i mean not enough
even for the material i actually ended up trying to present;
never mind the absurd amount of stuff i was officially *supposed*
to’ve “covered”. there just don’t seem to be any rules that’ll apply
to all the different levels of the game. “just a little more
than what the students are perfectly willing to do anyway” is probably
about as close as i can get as of right now …
vlorbik
December 7, 2007 at 8:55 am
OK, I’ll back off on the homework issue. There are clearly times when it is needed.
And I do think I could have learned much more in middle school and high school if I had been given the “right” homework. As it was I spent too much time reading and too little time doing, giving me some false confidence.
I didn’t mention that I actually did all of my homework once I got to my Math major courses, and I was a pretty good student. When I got to grad school I was making my own homework — it was the only way I’d pass general exams. So my story is not too different from Vlorbik’s, at least until I dropped out of school to get married and work 80 hour weeks at a doomed dot com.
When I did teach, I had reasonably sharp students who were interested in getting good grades honestly even if they weren’t all interested in the material. So I don’t have very broad experience here.
I think your rule of assigning “just a little more than what the students are perfectly willing to do anyway” sounds pretty close.
Maybe I should have titled this post “How to convince a student like me to show their work”, it would have been more to the point. I always hated it when I’d fail a test without missing a single question, and that always happened because I wouldn’t show my work. It left me feeling like I was in some battle with my teachers to get through school. That could have been different if I had just seen that there were good reasons to show your work. Had I thought that I was learning to communicate or that I was mastering techniques that were useful well beyond the simple problems we had in class, I would have gotten much more out of my homework.
After hearing from Alane and Vlorbik, I feel bad about not grading homeworks so carefully. I don’t think I caused any great injustice — the students who knew what they were doing got the best grades, almost everyone else got B’s thanks to our mandated curve. But I was paying more attention to my thesis. That didn’t really work out for me…
So Vlorbik, your actual job is to teach “them to blame themselves for their lack of earning power”? You get to instill the American Dream! Why, if anyone can live the good life and rise to the top in America, then there must be something wrong with you if you don’t. It’s pretty clever.
Rolfe Schmidt
December 7, 2007 at 10:51 am
Hi Rolfe,
I didn’t do homework in high school math. I just checked the highest numbered problem. If I could do it, sweet. No reason to work back to the easier stuff. And I could almost always do it.
It was a combination of stubbornness and laziness that had me not showing my work. I can hold lots of stuff in my head, enjoyed doing it, and only wrote down when there was too much to keep track of, and the answer.
You’re equation #2, I completed the squares in my head, but looks like there’s no real solutions (unless I goofed).
For homework, I’ve written a bunch. Essentially I tell kids that they are responsible for all of it, any of it can appear on a test, and then I give them ‘credit’ for putting up correct problems, putting up incorrect problems, putting up questions they couldn’t start. I only check assignments for the appearance of completeness – they have every opportunity to take correct answers from the back of the book and good work from me or their classmates.
jd2718
December 9, 2007 at 2:39 pm
You caught me on equation #2. I like questions without answers too…
I guess I was stubborn about not showing my work, but I didn’t feel defiant at all. I just didn’t see the point. I’m still pretty bad about refusing to do what I’m told if I don’t have a good reason. Now I see plenty of reasons to think and write about “easy” problems.
I like your approach to homework, and I like the way you use puzzles in your classes too. I’m guessing that not many kids are bored in your classes, even if they are ready to move on to the next level.
One of my favorite teachers in college gave us a take-home exam one semester where he wrote one differential equation on the board and told us to discuss it. We had one month and were free to work together and to use any references we could find. I never worked harder for a class than that one.
Rolfe Schmidt
December 10, 2007 at 5:50 am