Over at the First Excited State, the quasi-anonymous proprietor laments the tendency of basketball replays to focus on the shot rather than the play that set up the shot, and compares this to a maddening student habit:
Students in introductory physics classes inevitably place too much focus on the final numerical answer of the problem, which in reality is the least important part. I graded a quiz last week where I spent way too much time trying to decipher the numbers the students wrote down, because they placed the numbers in their equations rather than writing them clearly with the symbols representing the quantities in question. Grading problems written in this way is like trying to analyze a basketball play from the replay that only shows the shot. It can be done, but it requires more effort on the part of the grader. It's also a bigger risk for the student, because if they don't get everything right, it's harder for the grader to assign partial credit when there's no symbols to show exactly what they're doing.
This is something that I think most physics teachers try to drive home: the setup and the problem-solving process is more important than the final answer. Basketball coaches drive home the idea that focus on execution of the play is paramount and that the scoring will take care of itself. But the fans always focus on the shot, and the students always want you to tell them if they got the right answer.
I'm once again teaching introductory mechanics (out of a new textbook, though, which is both good and bad), and I'm once again going to be forced to grapple with this problem. Anybody who teaches a class in the physical sciences will-- students have an amazing insistence on plugging numbers in as soon as possible, and then manipulating six-digit decimal numbers for the rest of the problem, which makes it all but impossible to tell when they've transposed two digits in the second line of a fourteen-line problem, leading to an error that's off by 15% for no obvious reason.
As it happens, I do know a cure for this problem. It may be worse than the disease, though.
The only way I know of to get students not to plug numbers in from the very first step is to give them problems that don't have numbers in them. Rather than a 1000 kg car moving at 21.5 m/s, it's a car of mass M, moving at a speed v, and the end result is some algebraic expression.
This is the only method I know of for forcing students to do algebra. Even that isn't foolproof, as I've had students make up completely arbitrary numbers to plug in just so they could manipulate numbers through the whole problem. The ingeniousness of students looking for ways to get problems wrong knows no bounds.
The problem with this method, of course, is that students hate it with the burning passion of a million white-hot suns. If you think they get unhappy when they don't have the exact numerical answers to work toward, just wait until you see their reaction to no numbers at all.
My compromise is to work in one or two questions without numbers every test, and to always work example problems and questions asked in class algebraically. I also try to say at every opportunity that the exact numbers aren't important until the very end, but that just gets lost.
An interesting alternative approach, and one that I've tried with mixed success as a way of making lectures more participatory, is to assign problems where the students have to supply the necessary numbers themselves. A lot of newer textbooks include questions that ask students to estimate values for something based on their own experience. That way, there really isn't a numerical answer that they can check in the back of the book, but there's still the option of working with numbers all the way through, for the deeply symbol-phobic.
This is really only a problem for the intro classes, though. As soon as we get past the courses required by the engineering majors (say, the sophomore modern physics class I taught last term), I flip the balance in the other direction-- only one or two questions per test have numerical values all the way through, while the bulk of the credit is for questions with algebraic solutions. I figure, if the engineering students want to continue punching 47 things into their calculators all the time, that's somebody else's problem. Anybody who's going to major in physics, though, will need to know how to work in purely symbolic terms, and the sooner they get used to that (or change majors) the better.
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I've taken and TA'd for classes before in which the grading rule was that the correct numeric answer counts for one point, and the proper working of the algebraic expressions counted for the remainder. So, a twenty-point question in which a student starts by writing out the basic expressions with numbers already substituted and arrives at the answer gets one point, while a student who correctly manipulates the algebra but mishandles their calculator gets 19. I would say that the effectiveness of this system was mixed, and that as a student, I despised it and resisted it.
As a practicing scientist, though, I always work through the algebra first so that I can retrace my steps, calculate intermediate values if need be, and reuse the same work over and over again. If only students could be real scientists for a few years before studying science, they'd appreciate how they're taught!
What about making each question a two-part problem?
Part A (80% of the points): Derive an expression algebraically
Part B (20% of the points): Plug in these numbers into your expression above.
This wouldn't quite solve the problem, but it would give the number-crazy students something to look forward to at the end, and reinforce that the calculation is way more important than the numbers.
Hmmm, I was always the other way, (and far enough that I _also_ annoyed graders): I ran the whole way through with symbols, generally with subscripts, (there are three components, but concentration is always C...), then dumped a massive line calculation on them at the end.
Some things I would try/think about (I'm sure you've thought of these too)
1. Hand out a written out example of how you want them to solve problems, and say "make it look like that". I've seen several examples of things that "they must have seen this before" and kids just haven't (or didn't recognize it).
2. Structure the grading so they do the right thing.
eg (5pt problem)
1pt write down correct equation
1pt identify variables and unknown
1pt solve eqn algebraicly for unk
1pt (not sure...maybe work or add it to another area, or a graph/diagram)
1pt answer
I love algebraic manipulation way way more than computation. Some people are just too set in their old, high school ways...
As a student, (a senior math major, so I'm not in physics) I wholeheartedly prefer when given a computational problem, being allowed to do whatever method gets me the correct answer so long as this method is shown in full, which would include writing down what all my equations, definitions of curves, etc. are before doing an integral on the complex plane, for instance, and justifying anything not-trivial with the right theorems. (The only computational course I have right now ,is Complex Variables.)
The one exception I see to this is when the entire point of a specific problem is to teach a particular method. However I don't think in general it's a good idea to force massive amounts of algebra on everyone, including the engineers for whom I have a little sympathy. I do think requiring rigour that forces justification of each step, such that it's not just plugging 47 things into a TI-89, is awesome. (For the record, I've done basically everything in college with an 83.)
Anyway, that's my take. Don't penalize students for method as long as they're rigorous, or they ignore the specific point of the problem.
I like the other Brian's suggestion, and might give it a slight twist. Tell students they will get the numbers to plug in only after they've submitted the correct algebra. That way the carrot is still out there.
The other thing that I remember from freshman physics is units. If I did not meticulously carry the units through all equations, I often made errors. Not sure if that makes the numbers more or less important or has no effect, but I found it very helpful in both getting the right answer and in convincing the teacher/grader I knew what I was about.
OTOH, I had an advanced matrix compsci instructor that, on the first day of class, filled 6 of 14 blackboards (age alert) with the greek alphabet, upper and lower case. He then announced we would need to have this memorized by next class. Drop city.
Teaching science, including university Astronomy and Physics, I was willing to give at least half credit for back-of-envelope calculations. But if the answer was off by an order of magnitude, at most 1/4 credit. The right algebraic work, even without numbers plugged in, half credit. Even right drawing, or purely verbal narrative, half credit.
I agree that it's labor intensive frustration to reconstruct a student's thought processes from illegible prematurely numeric computations. I would sometimes do so, show where I thought the student went off the tracks, and ask them not to do it that way again.
We MUST know what's going on in their heads. What unspoken assumptions? What weird axioms? What incorrect algorithm? What metaphors? What learning style.
But we try to make it easier. Good homework, quizzes, exams, and Socratic dialogue help. But it is HARD. And a few good teachers such as Chad is not a solution that scales up to solving our national crisis.
My last observation, put first for emphasis. Don't assume that their engineering professors will expect anything different than you expect in an upper division class. Talk to them! You might learn that they want an exam solution that neatly lists each variable's value, then works out a final expression from first principles, evaluating that expression at the end.
As for your solution, are you sure it is only a million white-hot suns? ;-)
I like hybrid problems, where some things are given numerically and others are given symbolically so the answer is numbers and symbols, but there is another solution: Give a significant part of the points for the setup of the problem, starting from the appropriate first-principles equation, and punish unmercifully any numerical error (including rounding and omitting units) or algebra error in the remaining part of the calculation. This gets at annoying tendency number 2 (more serious, IMHO), which is a total focus on the final equation in your derivation rather than the first one. [* footnote] They don't like this either, but not as much as they dislike doing actual algebra.
I also punish things like 3X = 6 = x = 2. Unmercifully. If they complain, I tell them to complain to their math teacher that let them get away with nonsense pretending to be algebra, not me.
BTW, I give points back if they tell me they know the answer is wrong (and why) but can't find the mistake under exam time pressure.
[*] I think this focus comes from the twisted notion that all of us know the 1000 equations needed to solve the 1000 problems that come up in intro physics. Poof! Magic Equation! Answer! I lay this on the existence of the "Teachers Edition" in K-8 (maybe even K-12) where there really is one equation for every problem, and that is all the teacher knows about math. They really don't think that *this* (what we do in class, drawing pictures, etc) is how we solve every problem. Ditto (squared) in calculus class.
Yep... prepare a quick Powerpoint slide that reveals all of the specific numbers to be plugged in for calculation after the student has done the algebra that solves the problem in general. Now, don't show that slide to the class until only ten minutes remain in the test period.
Actually, you wouldn't have to do this with every problem on the test. Just pick a couple of average-difficulty problems for this treatment. Force them to do it the right way on a couple of examples, then drop a hint that they should do them all this way.
Or, tell everybody that if they touch their calculators before the final ten minutes of the test, you'll break their fingers. Maybe not on every test, but how about on, say, just the second test of the semester, try banning calculators until forty minutes have elapsed so that the students are forced to spend almost the entire exam time getting all their algebraic ducks in a row for a brief spasm of key-poking at the end.
Ban calculators outright. Then tell 'em that they can leave the final formula with the correct values inserted, but un-plugged-and-chugged.
Who cares if they can work a calculator?
As an upperclassman physics major, I far prefer it when no numbers are given. If you plug in specific numbers into the equations, not only have you done extra work, you've just narrowed the applicability of your solution to one specific case. It seems so pointless.
I had a prof -- dim memory suggests a geology prof -- who gave 8 point exam questions when calculation was required. Correct symbolic solution was 8 points, with part marks based on where and how badly it went wrong.
A correct numeric solution was 10/8, and an incorrect numeric solution was 0/8. Attempting a numeric solution meant no part marks; the symbolic answer and the numeric answer both had to be correct. (There was persnickety insistence on significant figures, too.)
The burning hatred of a million suns part was about right, too, but it did make you decide if you trusted your symbolic answer.
One advantage of having realistic numbers to plug in (at least when the problem is describing something familiar), is that students have the chance to use their intuition about the appropriate range or scale of the answer to catch errors. For example, I tutored a calculus student recently who was working on a problem involving finding the area inside a flower-shape described using polar coordinates. Since she had graphed the function, I asked her to draw a circle (of radius 1) around it and estimate how much of the circle the shape took up. Then when she solved the problem and found that her solution was off by an order of magnitude, she was able to see right away that she had made a mistake.
Unfortunately I have never had a student who actually did this kind of error-checking without my prompting.
I preach to my students about the utility of obtaining algebraic solutions first and entering numerical values last. BTW, I teach Mechanical Engineering, and the good engineers I know and work with also harangue our students to obtain symbolic solutions first.
After years of talking to the wind, I realized that I had to be very explicit in my expectations and to reinforce those expectations. Specifically, I started giving quizzes that only had symbolic solutions. Students were warned in advance and I even gave them a practice quiz. I used this approach on quizzes, and then used problems with numerical solution as well as symbolic solutions on the midterm and final exam. It didn't make them like the symbolic solutions more, but it took away any argument that they were surprised by an unfamiliar format.
But there is more than just requiring symbolic solutions. Way more.
Qualitative reasoning is a higher level thinking practiced by folks who have mastered a discipline. It involves manipulating equations and symbols to predict a trend without needing any numerical solutions. This is much harder than just obtaining a solution in symbolic format. For the typical physics or engineering exam or homework problem, a student can start by identifying a relevant formula and then solving for the one variable that he/she is not given in the problem statement.
I got this idea from an article by Lillian McDermot. In particular, see section V.A.1 on page 1130.
Some older examples from an undergraduate fluid mechanics class are Quiz 1, Quiz 2, and Quiz 3 on this page. N.B. I've been an administrator for a couple years and haven't taught that class since 2006. A more recent example is our blender experiment.
Sure, students will resist efforts to stretch their thinking. I believe one of the first steps to address the resistance is to make the effort to explicitly teach a different way to approaching problems. We need to be persistent. Some students will always resist, but others will learn if we give them a consistent message and plenty of chances to practice the new skill.
My teachers and profs always required full working with symbols for full credit, and I LIKED it, dammit! Much easier than crunching numbers for 17 lines.
(A friend of mine used to answer math problems with just the [correct] answer. For the working, he would put "Steve Wonder Method, trust me sir." Amazingly, this worked for him, but probably only because he always did get it right in his head.)
While it is obviously good practice to first derive an expression and then find the solution, I am reminded of someone (I can't remember who) telling me that the numerical answer is vital - in a real life problem if you get this is wrong then you are in trouble, even if you understand the physics.
Of course, you are more likely to find the correct answer if you do the manipulations before inserting numerical values.