On Assessment, IV

Last week, I had ended with an interpretation for an A–D grading scale, shown below (here is a link to last week’s post for reference).

I remind you that this scale is not ideal; the purpose was to come up with some system of assigning grades which wasn’t punitive, but rather which motivated students to learn concepts rather than to avoid losing points on exams.

We continue with a discussion of how to use such a system in practice.

“Now let’s consider this in the context of an exam.  The first part of an exam is a skills portion, with, say, ten short problems of roughly equal length.  Expectations for this part of the exam [meaning a grade of B] are seven problems “essentially” correct, and four problems completely correct.  These expectations are written on the exam for students to see.

Are these expectations too low?  Perhaps.  But then an A [refer to the chart below] means eight problems “essentially” correct, with at least five completely correct. Of course we must ask what it means for a problem to be “essentially” correct — but when in doubt, err on behalf of the student.  (Students rarely suggest that their scores be lowered.)

Then grading is actually somewhat easier, and grades can be assigned as follows, with the abbreviations EC and CC meaning essentially correct and completely correct, respectively (for simplicity, a grade of C is assigned for all other cases not accounted for):

Now this eliminates the need for partial credit — but does require a judgment as to what “essentially correct” means.

This also makes grading much easier.  I would suggest that each problem be marked as “EC,” “CC,” or left blank.  Few comments, if any are necessary.  This is the approach I have taken in BC Fast-Track, and it encourages further learning as it leaves the student in the position of needing to work through their mistakes.

I would have students keep a section of their notebooks for exams and revisions, and there they can keep their reworked problems, should they choose to do so.  Then — as I did in BC Fast-Track — students could visit me periodically with their notebooks and I can take a look at their ongoing progress.  This “additional” work, if sufficiently well done, could boost their grade at the end of the semester.

I think this could have the same effect it did in BC Fast-Track — exams were easier and more enjoyable to grade.  But there were more discussions in my office about reworked exams and sources of error that were initiated by the students themselves, and these discussions were not about points, but about concepts.

Now what about the part of the exam which is intended to be more conceptual?  Let us suppose that there are three problems, roughly comparable in length, and of various difficulties.  Then grades might be assigned as follows:

More details about how this would fit in a classroom environment may be found in a later document [I cannot recall which document is being referred to here].  But this system allows for a more qualitative approach to grading.  Performance expectations are also clearer, but such expectations depend critically upon the nature of the problems given.  Moreover, grades are not assigned punitively, but the emphasis is on doing problems completely and correctly.

For an example, below could be a set of ten skills problems and three conceptual questions for a basic assessment on the rules of differentiation.  This would a 70-minute assessment.  Given expectations for completely correct problems, I think this is reasonable.

Skills questions:

1.  Evaluate

$\lim_{x\to\pi/4}\dfrac{\tan x-1}{x-\pi/4}.$

2.  If $f(x)=e^x\cos x,$ find $f^\prime(x).$

3.  Find

$\dfrac d{dx}\dfrac 7{\sqrt{x^3}}.$

4.  Using the quotient rule, find

$\dfrac d{dx}\dfrac{x^3}{\sin x},$

simplifying as much as possible.

5.  Find the derivative of $f(x)=\left(\sin\sqrt x\right)^{\!2}.$

6.  Find the equation of the tangent line to $h(x)=\sec(2x)$ at $x=\pi/6.$

7.  Using a definition of the derivative, find the derivative of $p(x)=x^2-x.$

8.  Assume that $f$ and $g$ are differentiable functions. Find

$\dfrac d{dx}f(g(x^2)).$

9.  Find the derivative of $q(x)=x^2e^x\cot x.$

10.  Let $f$ be the greatest integer function. Using the definition of the derivative, determine whether or not the derivative exists at $x=0.$

Conceptual questions:

1.  Using the product rule, find

$\dfrac d{dx}f(x)(g(x))^{-1}.$

2.  Suppose that the line $y=6x+a$ is tangent to both $f(x)=x^2+b$ and $g(x)=x^3+3x.$ Find $a$ and $b.$
3.  Suppose that $f$ is a differentiable function. Discuss the following limit:
$\lim_{h\to0}\dfrac{f(x+2h)-f(x-h)}{3h}.$