### Banishing the curve

My students asked me today if I grade on a curve. That was easy: No.

One of my biggest pet peeves in mathematics education is the curve. I don't know if it exists outside of math classes, and I don't really care. It's one of the worst, most damaging ideas out there.

Why, you ask? the curve gives points to people who haven't earned it, either because the teacher doesn't care (or know how) to write a good test, or because the teacher isn't interested in the students' learning the material. The message of the curve is, "Since many of you didn't perform to expectations, the material must be unlearnable. Our solution is to give you points, so that the scores correspond to a preconceived statistical model."

In other words: what we want to be true is more important than what is true.

You can conclude from this that I try to write tests that the students can actually perform. This doesn't mean that I don't ask questions to probe the students' understanding; rather, it means that if students don't do well on a test, I conclude that either:

- I wrote a bad test; or,
- I graded the test badly; or,
- I taught the material badly; or
- the students didn't study properly.

Grading any test is difficult enough; writing a good test is actually quite hard. I'll admit that I don't think I write very good tests, and I hate grading. However, my tests are adequate enough that I can determine who understands the material, and who doesn't. Oddly enough, the students who understand the material fall very neatly into two groups that correspond to the traditional A-B range; the students who don't understand the material fall messily into two groups that correspond to the traditional D-F range; the students who understand some material, but not all of it, fall more or less neatly into a group or two that corresponds to the traditional C range.

There's some room for maneuvering, of course: sometimes, one group of C students includes scores of 68 or 69. In this case, the student's long-term performance resolves any questions one might have. (This is why I never, ever put letter grades on a student's test: any one test by itself is not an adequate measure, and students often move from one group to another; four tests are much better.)

However, it really is amazing how these groups always work out. This is why my final exam (always comprehensive) is the one test that I do curve. I can keep high standards throughout the course, and force the students to learn the material in spite of themselves.

So, what do I do when the test scores reflect that students haven't learned the material? After all, I'm certainly not the super-great teacher who magically teaches his students successfully.

In this case, I give the tests back to the students, and tell them to rework the problems, using their book and notes to learn what they failed to learn the first time. I give them half the points in return.

It's amazing how well that works: most students really like the opportunity to show that they can learn the material, that they can earn points instead of having them given to them. They don't like the extra work, of course, but it goes a ways towards mollifying their frustration, especially since sometimes their failure to learn the material is, in fact, my own fault.

I'd be interested in hearing anyone else's opinions. This strategy of re-testing is not my idea, of course; I learned it from someone else.

## 7 comments:

It's amazing how well that works: most students really like the opportunity to show that they can learn the material, that they can earn points instead of having them given to them. They don't like the extra work, of course, but it goes a ways towards mollifying their frustration, especially since sometimes their failure to learn the material is, in fact, my own fault.

============================

It sounds like a great idea to me. I've never been subjected to the curve grading, and probably would have the same criticisms you made. (haven't thought about the subjec actually - that's why I won't add any comments).

Re the above idea, it sounds great, because it starts to work with several of the emotional and motivational dynamics that can hinder the performances of some students.

When teaching, I've always taken great care with elaborating fair questions and tests. When I am a student, I really question any professor that does not play a fair game with the type or focus of each exam question.

When teaching, I've always taken great care with elaborating fair questions and tests.That's increasingly hard for me to do. I can of course ask students simply to regurgitate the methods shown in class, which shows that they're good machines, but not that they're good human beings. I always try to ask questions that make them think; the trouble is that students in a math class usually look on that as "not fair, because we've never seen it before."

I think I understand what you mean. Although I don't know exactly what problems one encounters when formulating math tests, some basics are the same as other subjects. Are you asking your students to solve a problem that they have the math means to solve? They have learned how to solve such a problem? Or they haven't been taught something and they are being asked about it?

I have never studied much math, but from the tests I had, the questions were always variations on exercises that we had done before and on material we had covered in class. I actually don't recall ever having a math test that was asking things that had not been taught. I don't know if in higher levels of math, that happens, but in school math, it doesn't. If you had understood the classes and had done the endless homework exercises, you always went well, because you understood what was being asked and you knew the way to solve it. Obviously, sometimes you took a wrong route in trying to solve a problem, or made some other mistake, but overall I never encountered much issues with hard sciences questions. It's always the social sciences that I think are much harder to make good questions (specially if they are multiple choice). Written essays and papers are, in many ways, a better approach to test social sciences learning (beyond the basics), but it puts a huge correcting burden on the teacher.

One of my greatest pet peeves is any teachers that starts the test question as "tell me in your own words what author X said..." - which often means they want you to repeat EXACTLY what author X wrote (ideas/facts) but you have to change around words to not sound like them.

I have often wanted to quote the author from memory and then go up to the prof and say, but these words are part of my dictionary too...

I hate my students. (Not really, but I want to.) I'll write more about this tomorrow or over the weekend, but: it's pretty clear why they were asking for a curve. At least I'm done marking what's right and wrong; now I have to assign scores. Ughhhhhhhh...

Alessandra:Are you asking your students to solve a problem that they have the math means to solve? They have learned how to solve such a problem? Or they haven't been taught something and they are being asked about it?Yes; for the most part; and no. In some cases, I should be asking them questions that they haven't seen before, because they're supposed to be learning how to do problem-solving, not how to be computational machines. We have computers for the computation.

I actually don't recall ever having a math test that was asking things that had not been taught. I don't know if in higher levels of math, that happens, but in school math, it doesn't.In higher levels of math, it's expected that you will see things you haven't been taught. At my level, that's your job: research. At high-level undergraduate levels, it's a matter of testing how well you understand what you

weretaught. The class I'm teaching is for engineers and scientists however, so I'm not asking them to think much beyond the basics. However, Iamgoing to ask them to think: otherwise they should attend a vocational school, not a university.So, for example: instead of asking them to diagram the phase plane for a given differential equation, I say: "Given the differential equation y'=y^2-2y+s, what values of s will give two, one, or no equilibrium points?" Or I ask, "Why is property such-and-such true?"

Notice that these are questions that their calculators and computers can't solve; these are questions that HUMAN BEINGS can (and should) solve.

One of my greatest pet peeves is any teachers that starts the test question as "tell me in your own words what author X said..." - which often means they want you to repeat EXACTLY what author X wrote (ideas/facts) but you have to change around words to not sound like them.I understand what the professor is aiming for, but in math that backfires. :-) Once I asked students to define something, and they tried to define it all sorts of ways except the

actual definition. When I pointed this out, they protested, "But in our other classes, teachers tell us not to repeat what the book says."It's kind of hard to explain to students indoctrinated on expressing themselves in their own words why they should, in fact, repeat exactly what the books says on definitions. They don't quite understand it, and actually I think the social sciences are hurting us here. How on earth do they approach definitions in the social sciences? I must determine this.

One of my greatest pet peeves is any teachers that starts the test question as "tell me in your own words what author X said..." - which often means they want you to repeat EXACTLY what author X wrote (ideas/facts) but you have to change around words to not sound like them.

-----------

I understand what the professor is aiming for, but in math that backfires. :-)

------------

Actually, I don't think you understood what the problem is, since I wasn't very clear.

The reason why teachers require the "say with your own words" is to try to see if the student has understood something, and just doesn't recite a definition (a bunch of words), without understanding what they mean. There is no problem there.

However, it can become just a synonym exercise. Because what some teachers are really asking for is the exact definition (just as you want in math). However, the teachers prohibit you from using some of the best words (which are the original ones). So, you word it differently, because if you word it as the original, the teacher can't tell if you understood or not, and does not bother to check in any other way. And then if your mastery of the English language is not enormous, sometimes you can get a lesser grade because you used a synonym against your will, which was not the word you would have preferred to use to articulate exactly one thought, and it all becomes a word game.

That's why a paper where you have to work a theme and apply or discuss the theoretical learning is in many ways better to see if a student learned something, rather than playing English language word substitution games with reciting definitions/ideas from other people "in your own words."

I hope it's more clear now.

So, for example: instead of asking them to diagram the phase plane for a given differential equation, I say: "Given the differential equation y'=y^2-2y+s, what values of s will give two, one, or no equilibrium points?"

-------------------

Right.

lol...

I agree that it is more work, but the bottom line is that there should be mastery of the material to get an A. The curriculum can be (semiquantitatively) described and so can mastery. If less than the curved amount achieve mastery than less As should result. And the converse.

A great example is the AP exam. Some schools may have lots of 5s and some schools nothing above a 2. But it's all graded from afar with no curve.

In universities, they should do the same thing for all large classes (multisection affairs). Even for smaller classes, you can still make judgments on mastery based on previous years or try to use tests from national societies (e.g. APS).

Post a Comment