Grading student work in Math 251:05-10, spring 2006

Students should know how their work is graded. Then they can better understand the relative importance of various requests in the assignment. They also can more knowledgeably review the correctness of the grading.

The first Maple assignment
Here are the instructions the graders were given:
There will be a total of 10 points.

2 points for including all relevant Maple instructions
3 points: 1 point EACH for identifying each of the three vectors (pq and pr and v).
5 points for the labeled picture distributed as follows:
- 2 points for the picture. Only 1 point if the picture does not display the perpendicularity of v to the triangle T.
- 1 point for labeling p and q and r.
- 1 point for labeling T.
- 1 one for labeling v. The vector v should have its initial point at p otherwise this point is not earned and 1 point should be deducted from the possible initial 1 or 2 points for the picture.

The first workshop problems
Here is the body of a message I sent to the recitation instructors, who read and graded two-thirds of the workshop problems. I graded one section for each lecture. The contents of the message should provide some background on the grading.

I just finished grading the workshop problems for section ¶ taught by Mr. ¶¶. Here are some observations about the student work and my grading which almost surely will be true for all of the sections.

I gave each student a pair of numbers, (i,j), where i and j are integers in the range [0,5]. The first integer reflected the mathematical content, and the second integer reflected the exposition. I tried to be sure not to deduct points unless I could write specific comments about the work, so that someone earning, say, (3,3) would at least have some reasons (I hope valid!) for the deductions.
I read these papers very critically. I really concentrated. If I read something I could not understand, I did NOT attempt to guess at what the student meant. Instead I wrote a comment, and then all the comments were summarized in the grade I gave. I tried to be consistent in what I looked for in the answers.
I wanted people to explain the computations. The grade of (5,5) was not impossible to obtain: in this section, 4 students out of 20 got that grade. But I also gave grades of (1,0) and (2,2) and (1,3). I do not think that poor work should be given high grades. I gave one grade of (4,0): I told students that the mathematical work should be explained in complete English sentences, and I will grade what's handed in according to that standard!

Certainly we can accept more shoddy work on exams. Exams have severe restrictions on time and, consequently, on the kind of reflective responses we would like from students. There is no reason to accept low quality work on workshop writeups where students can take appropriate amounts of time. I want students to learn mathematics and communicating math, and part of the latter is especially exemplified in the workshop problems and even in the Maple labs.

The first exam

Problem	#1	#2	#3	#4	#5	#6	#7	#8	#9	Total
Max grade	8	12	10	12	12	12	12	12	12	96
Min grade	0	0	0	0	0	0	0	0	0	8
Mean grade	5.69	5.31	6.03	6.49	7.43	7.44	2.16	6.25	7.54	54.35
Median grade	6	2	6	6	8	9	0	6	8	54.5

140 students took this exam. Numerical grades will be retained for use in computing the final letter grade in the course. The different versions did not seem to have very different statistical results. Here are approximate letter grade assignments for this exam:

Letter equivalent	A	B+	B	C+	C	D	F
Range	[80,100]	[75,79]	[65,74]	[60,64]	[50,59]	[45,49]	[0,44]

Discussion of the grading

Some general background
Quoting from the formula sheet (even relevant formulas!) will not earn credit. Arithmetic errors and "small" errors (such as minor errors in differentiation) will be penalized minimally. The resulting work can be considered for full credit in the remainder of the problem unless the error makes the problem much easier.
Most of the problems on this exam were taken directly from the text, and were on the list of suggested textbook homework problems.

Problem 1 (8 points)
a) (5 points) 1 point each for finding two vectors (say pq and pr, for example). 3 points for finding the cross product.
b) (3 points) Students may use their answer (even if incorrect) to a) in this part if the answer is not trivial (such as the 0 vector or just <1,0,0>). -1 point for not knowing the the triangle's area should be half of the magnitude of the cross product.

Problem 2 (12 points)
2 points for computing the dot product of V and W correctly. 2 points for computing the magnitude of W correctly.
A correct value of V₁ gets 4 points, and a correct value of V₂ gets 4 more points.
Students certainly may use any of the formulas on the formula sheet, provided that the answers are correct. If a student makes a mistake early in the problem but does not trivialize the problem (for example, makes an error in computing the dot product) then the student will be penalized for that error but can earn all the points in the balance of the problem, working as if the earlier answer were correct.

Problem 3 (10 points)
a) (4 points) 2 points for recognizing the correct normal vector. 2 more points for writing the correct parametric equations.
b) (4 points) 2 points for recognizing the problem: that is, addressing the distance question correctly. 2 more points for getting a correct point (either correct point earns the points!).
c) (2 points) This is earned for writing the correct equation for the plane. 1 point off for either a wrong "point" on the plane or for a wrong normal vector. The student may use an incorrect point from b) and still earn 2 points here.

Problem 4 (12 points)
2 points for a correct r´(t).
2 point for computing (and recognizing the "need" for) |r´(t)|. Students do not need to "simplify" this expression, or any other in the problem.
2 points for a correct result for T(t) and then for T(1).
3 points for T´(t) (take off a point for each error but continue to read for other results!).
3 points for N(t) and then N(1): any "mess" (if correct!) will earn the points.
Quoting from the formula sheet (even relevant formulas!) will not earn credit.

Problem 5 (12 points)
4 points for the graph: the graph should be "unimodal": down then up, with good limits (2 points). And it should be 0 on an appropriate interval in between. (2 points). Since I'm convinced that the curve sketched is smooth 1 point will be deducted for graphs of curvature which seem offensively non-smooth to me.
4 points for the limits: the limit as s goes to + (or -) infinity are each worth 2 points.
4 points for the explanation. What's needed is an explanation, not a description in words of the graph of the limiting behavior. There must be some reasoning or explanation given. Ideally, the word circle should occur, and some relationship between the radius of a circle and the curvature of that circle. Then the relationship between circles A and B and the curve should be mentioned.
An acceptable explanation need not be long. Certainly some students received full credit with just one or two well-written sentences. When the word "it" was encountered, an effort was made to identify the referent (what the "it" means). If this identification was ambiguous or impossible, credit was reduced. I read what students wrote. Such phrases as "the curvature of a point" or "the definition of curvature is 1/R" are meaningless or incorrect, and don't earn credit.
Students whose only error is somehow systematically interchanging +infinity and -infinity will be penalized 2 points only.
Comment I meant to ask for an explanation of both the graph and the limits, but did not. Then I would have wanted a statement about why curvature is 0 for an interval around s=0 (the curve is a straight line segment there, and straight lines have curvature 0.) I didn't phrase the question as I should have, however!

Problem 6 (10 points)
1 point for computing f's value at (2,1).
2 points each (total: 4 points) for computing the first partial derivatives of f correctly.
2 points (1 each) for evaluating these first partial derivatives.
3 points for the linear approximation formula. It does not need to be explicitly stated. A numerical version is o.k. Students who insist upon doing numerical work incorrectly should lose a point. Realize that there is no need to do any numerical work in this (or any!) problem -- answers do not need to be simplified.

Problem 7 (12 points)
a) (6 points) 2 points for a correct z_x and 2 points for a correct z_y. 2 points for assembling these into a correct verification of the PDE. Working with a specific function f earns no points (!).
b) (6 points) 4 points for a correct use of both the product rule and the chain rule (-2 for one of them not correct). Then 1 point each for stating explicitly what the two functions (A and B) are (that is requested!).
Comment This problem had the worst results. Therefore either the chain rule wasn't taught well and/or it wasn't learned well. There will be a similar question on the next exam.

Problem 8 (12 points)
a) (6 points) 1 point for F(x,y,z). 1 point for the correct value of F at p. 2 points for the maximum directional derivative stated correctly and 2 points for the correct unit vector.
b) (6 points) C's value earns 1 point. Realizing that F at p is a normal vector earns 2 points (so incorrect data from a) can still earn these points). Writing the equation of the tangent plane earns 3 points. Students may use their result from a) if incorrect and non-trivial without penalty.

Problem 9 (12 points)
a) (4 points) 2 points for the correct computation of f (the correct computation of the partial derivatives alone earns 1 point -- assembling them into the gradient earns the other), and then 1 point for f(2,1,2) and 1 point for f(2,1,2). The gradient of this function should be a vector in R³, not in in R². A two-dimensional answer loses a point.
b) (4 points) 2 points for the correct computation of g (the correct computation of the partial derivatives alone earns 1 point -- assembling them into the gradient earns the other), and then 1 point for g(2,1,2) and 1 point for g(2,1,2).
c) (4 points) 2 points for realizing that the cross product (or something equivalent!) is needed were rewarded, and 2 points for the computation. Students could use their results from a) and b) even if incorrect without penalty provided that the resulting problem was non-trivial. A correct geometric observation such as "The tangent vector of the curve is perpendicular to both surface normals" would earn at least a point.

Maintained by greenfie@math.rutgers.edu and last modified 2/28/2006.