I am teaching probability and statistics, a course for first-year students, for the first time this spring. I have been struggling with how to grade the students.

This course is unusual in that there is only a little mathematics in the course (we throw in all of the probability that we can, but it is still mainly statistics). This requires that I think like a statistician, which is new and somewhat painful.

It also makes designing a course more difficult. I have the basics of what I want to do, but—as mentioned above—how to grade the students is the most difficult part. I want to stick with an SBG approach, but I was not sure how to set up the standards.

In calculus last semester (and other courses in previous semesters), I had a general format of: “if you can do the basic skills from the course, you will get a C. To get a grade higher than C, you must demonstrated some conceptual understanding.”

I realized as I was brushing my teeth last night that this is completely and utterly backwards.

I want my C students to understand the concepts of the course, but not necessarily be able to do the computations and symbolic manipulations. My B students should, in addition to understanding the concepts, be able to do many of the computations and symbolic manipulations. My A students should, in addition to understanding the concepts, be able to do all of the computations AND demonstrate that they can do some self-guided work.

Here is my rationale for requiring understanding the concepts to get a C:

1. I am convinced that the concepts are easier in most college-level mathematics courses—students are better at drawing tangent lines on graphs of functions than they are at finding the equations of tangent lines.
2. The students who need the calculation and symbolic manipulation skills are the ones who are going to continue taking more mathematics (and related) courses. I am guessing that C students are less likely to continue taking these courses.
3. Computers can now do much of the calculation and symbolic manipulation, although the user has to understand the concepts to correctly enter the information.
4. Most importantly, the concepts are the most important part of the course! I want to explicitly encourage students to focus on the concepts—I don’t want, say, a calculus student to be able to get an A in the course by only having good algebra skills (a colleague yesterday complained to me about such students; I view this as a flaw in the grading system).

I am not happy about having this completely backwards, and I feel bad for my previous students. But I am happy that I now understand what I want.

The tough part is designing assessments that isolate concepts. But that is part of my job, and I find it fun to come up with such questions.

(image is “LED Light Bulb” by flickr user Wade Brooks, Creative Commons License)

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### 18 Responses to “Grading Epiphany (or, Bret was completely wrong about everything)”

1. Meg Kosowski Says:

Bret – for what it’s worth I think this is brilliant. I posted to your FB wall about it.

2. Andy "SuperFly" Rundquist Says:

Hi Bret, this is very interesting. I’ve never really had a problem with how you’ve described it before, but maybe those were for classes where the mechanics and the subtleties of the concepts we’re in the right order of difficulty. If I want to teach about the Euler-Lagrange approach to mechanics, I find it easy to get students to the point of crafting kinetic energy minus potential energy and hard to prove to them why we’re doing it in the first place. However, I really liked your example of the tangent line. It would seem there are some (lots?) of concepts that are easy but that the students get bogged down in the mechanics of implementing them. Could this be a topic-by-topic approach?

On the other hand, maybe “hard vs easy” isn’t the right way to think about it. I guess I’m not sure.

• bretbenesh Says:

Hi Andy,

This post was a huge oversimplification, and I am certain that this really should be a topic-by-topic approach (for instance, high school algebra should probably not be done with this approach). I can readily believe that this is not generally true in physics; it might not be generally true in mathematics, either, but I think that it is good for the classes I am teaching (calculus and multivariable calculus last semester, prob and stats this semester).

I think that, for calculus anyway, most of the topics are relatively easy conceptually, but difficult algebraically. Students can readily draw and identify things that are concave down, and they can probably even tell you that, say, you might want to consider selling a stock if it is “increasing and concave down.” But taking two derivatives can trip students up, and things get worse if they have to factor to find the roots of a derivative (as in finding local extrema).

I suppose that “hard vs easy” is really only a secondary idea, too. The point is to learn the concepts (or at least, I think that it should be). So maybe I should not have mentioned it.

3. Darci Kracht Says:

I don’t want to be a wet blanket, but I do think you’re wrong about this. True conceptual understanding is much deeper and more sophisticated than symbolic manipulation. On the other hand, it’s probably good to start by introducing students to the main idea than to start with the symbolic manipulation, but I think really understanding the idea takes a lot of hard work and time.

• Darci Kracht Says:

To clarify what I mean in the context of your example: I don’t think being able to draw a tangent line is really demonstrating understanding of it. The concept of the limiting process necessary to really define a tangent line is extremely subtle and takes a long time to understand. As part of the “calculus reform” movement in the early 90′s, I taught out of Ostebee and Zorn as well as the Harvard Consortium book, both of which took a concept-first approach. My experience with that is that the students still essentially memorized (or tried to memorize) step-by-step methods of solving problems/answering questions. However, maybe you will be more successful in implementing this approach than I was. (For one thing, you probably have a higher calibur of student than we do here at Kent State.) I do agree, also, that students get lost in the mechanics of finding limits and equations of tangent lines and lose sight of what they’re supposed to be doing.

• bretbenesh Says:

Be a wet blanket, Darci! I asked you to!

I agree completely that, for us, conceptual understanding is more difficult. However, I hypothesize that for many (most?) of my students, conceptual understanding is easier. This is because their algebra skills are not as good as I think they should be coming into the class; this creates a problem, because they need to learn algebra skills (not explicitly taught in my courses, for the most part) AND a minimal amount of conceptual material to do the computations. I think that the algebra skills is just a barrier for some students—there is just too much algebra for some of the students to learn in one semester (less than one semester, actually, since they still need to have enough time to apply the algebra to the course content).

I do agree with you that understanding takes a lot of hard work and time. This is why, for my students with poor algebra/computation skills, I would rather have them focus solely on the understanding. Having them focus on both algebra and concepts leads to little improvement in either area, I suspect.

Note that I am only talking about my C-ish students here; my students with a solid algebra background will still be working on both concepts and skills. The main difference for them is that they are going to be more explicitly told to focus on the concepts, which isn’t done seriously in most of the math classes I have observed (I see a lot of instructors verbally tell students that the concepts are important, but then grade on symbolic manipulation).

I expect a retort from you; thanks for challenging me!

4. Aaron Weinberg Says:

Can “skills” and “concepts” really be separated? One of my favorite quotes (from Brian Rotman):

Mathematical signs play a creative rather than merely descriptive function in mathematical practice. Those things which are ‘described’… and the means by which they are described… are mutually constitutive: each causes the presence of the other; so that mathematicians at the same time think their scribbles and scribble their thoughts.

• bretbenesh Says:

I largely agree with you, although I think that this is not so much the case in the lower-level mathematics course. I can easily imagine a student who can understand what a p-value means, and maybe can even tell you what test to use to get the p-value for a given experiment, but has great difficulty in creating p-values.

What do you think?

5. Chris Norby Says:

I think I agree. When doing basic skills quizzes, I found it easy to memorize the algorithm to solve a given type of problem and perform it without really knowing why. This was also supported by the fact that the category of problem was explicitly stated. Thus, to get a ‘C’, I could have gotten by without knowing the ‘deeper’ concepts and without the ability to determine, say, when to take a flux integral vs. a line integral.

If you can find a way of evaluating conceptual understanding in a way that does not depend on basic skills, that would be powerful. Unless a student is continuing education, she/he will benefit much more from the concept than the skill, especially (as you mentioned) with the increasing ability of software to do the skills part.

In class, you said that setting up the problem was the hardest (and most important) part, but then the quizzes were graded on its evaluation as well. I’d be interested to see if most of the wrong answers were because of setting up the problem incorrectly vs. making an algebra mistake. I’d penalize the former much more than the latter, since in the ‘real world’ we’d just let a computer take care of the evaluation for us.

• bretbenesh Says:

Yeah, I should have separated out the “setting up” from the “computation.” I think that I meant to do this, but lost track of it (note the two standards for “Parametrizing Surfaces” and “Computer Surface Integrals”).

So you were a different case, since you had the computational skills to deal with all of the problems. The reality is that many students had a tough time with skills quizzes, even though the quiz explicitly said what type of problem it was (this was much more true in Calc I than Calc III, but still somewhat true). So many students struggled with that, even though you didn’t. This fact helps me to justify making it a B or A level skill (in spite of the fact that it seems easier to people who are fluent with algebra and computation).

I think that I could have evaluated conceptual understanding without too much difficulty. Really, I could have just taken the clicker questions and put them on quizzes. So I could given a contour graph and asked students to draw in the gradient at a given point. Students would need to know that the gradient points in the direction of greatest increase, but would not need to know how to compute it.

Sorry about that! You were one of those who suffered because of my backwards policy.

• Chris Norby Says:

I wouldn’t say I ‘suffered’ at all, I just had to adjust (which I think is a good skill to be forced to learn). And I really like the idea of putting clicker-type questions on the quizzes, for what it’s worth. That would also increase continuity between the evaluation and instruction sections of the course.

• bretbenesh Says:

I think that you “suffered” in the sense that the course had some flaws. Of course, EVERY course has flaws, so that really isn’t surprising.

Also, you make a good point about aligning what we do in class with the quizzes. That just seems like a good thing to do, anyway. Bret

• Chris Norby Says:

I think you may have actually known this all along (perhaps it was more of a re-piphany)!

From one of your comments under the “Oral Exams Summary” post from 4/13/12:

“The biggest barrier to mathematical literacy in the U.S. [is that] we have favored mechanical computation over simple understanding for too long.”

• bretbenesh Says:

Hi Chris,

This is very kind of you, but I cannot let myself off of the hook that easily. In fact, this probably makes things worse.

There is very little original in saying that we overemphasize mechanical computation; a lot of mathematics professors will say that, I think. The epiphany was this: I can (and should) change my grading policies to reflect this!

So basically, I knew what I should be doing; I was just being too dumb to actually try doing it.

As I said: a lot of people would say that we favor mechanical computation too much. As far as I can tell, most of these people do not actually grade according what they say. I might find out this semester that there is a very good reason for that, forcing me to return to the standard way of grading until I think of another experiment to try.

Thanks for trying to make me feel better, though! Bret

6. Mark L. Lewis Says:

Some of this probably depends on the audience. If you are teaching calculus to engineers (or other people who are going to be applying it), it is more important that they are able to do the calculations, and less important that they understand the concepts (this is certainly the view of engineering faculty, I am not sure I agree). On the other hand, if the audience is mostly math majors, then certainly the concepts are far more important. Of course, the hope is that if they understand then concepts, then the computation should be easy. (I think this is also a faulty assumption.) In the real world, if we really want our students to understand math, then we need to emphasize both the concepts and the the computations.

• bretbenesh Says:

Hi Mark,

Thanks for the comment. I definitely agree that this could depend on the audience. I suppose I should have stated my assumptions: my calculus class is primarily for liberal arts students (but that includes math, physics, and computer science majors). Things get complicated if there are engineering students. I imagine there might be some other students who need computation more than concepts, although none come to mind right now.

“Of course, the hope is that if they understand then concepts, then the computation should be easy. (I think this is also a faulty assumption.)”

I agree that is faulty, and I will give you a faulty-er assumption that I had: I thought that the computations would easier regardless of whether they knew the concepts. This was only an implicit assumption, and I would have said it is wrong if anyone had asked me about which is easier. But that is the assumption that I had all throughout fall semester, leading to my previous grading scheme. I suppose this is the case of my judgment being clouded by the fact that I am a mathematics professor, and the computations are trivial for mathematics professors. I need to remember that most people do not end up being mathematics professors.

I agree that both the concepts and calculations need to be emphasized, and that there is a limit to one’s conceptual knowledge if one cannot do computations. But—for now—I think I am happy to give a C to any student who demonstrates a decent understanding of the concepts, even if they cannot do many of the computations.

Good day!
Bret

7. Joshua Bowman (@Thalesdisciple) Says:

Sorry to come late to this conversation. I’ve been thinking about this post since I read it last week, and I’ve gone back and forth on how much I agree with it. Mostly I agree, particularly as I reflect on the grades I’ve given in the past—they basically match the description you give for students that earn As, Bs, or Cs. On the other hand, I’ve just looked at the evaluations from last semester, and the most consistent criticism I received (presumably from the students who struggled most in the class) was that I spent too much time discussing the concepts and not enough time demonstrating how to work specific problems step-by-step. So it seems the students disagree that concepts are more basic than computations.

Perhaps a variant of this paradigm is in order. I was inspired by Paul Zorn’s talk at JMM on communicating mathematics, in which he referenced Terence Tao’s “stages of math learning”: pre-rigorous (in which the ideas are understood at a purely informal, intuitive level), rigorous (during which the ideas are placed on firm scientific ground), and post-rigorous (at which point the learner can reflect back, again through a prism of intuition, on the deep understanding she or he has gained). Tao attributes these, roughly, to time periods from early undergraduate to late graduate studies, but in our classes we may seem them in microcosm. (Tao’s post is here: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/) That is, conceptual understanding occurs both before AND after computational facility, and it may be useful to distinguish the different types of intuition.

• bretbenesh Says:

Hi Joshua,