Using CAS for testing

The ETS recently posed a survey concerning the use of a symbolic manipulation calculator on their exams. Here are my thoughts on the survey.

 

First, the survey seemed to be aimed at determining whether a symbolic manipulation calculator (SMC) would give someone with a SMC an advantage on the standard questions that appear on the ETS exams (AP, SAT etc.). The questions did not seem at all the kind were designed for SMC use. Too bad.

 

The argument is always along the line of if using a graphing calculator (GC) or a SMC helps people learn more or better or deeper. That's different than what should be expected or required on an exam of one's own or of the ETS. So I'd like to discuss two questions to see the difference between good testing practice and using SMCs to do and learn math.

 

2003 AB 76 gave a velocity function v(t) = 3+ 4.1cos(0.9t) and asked for the acceleration at t = 4. The point of the questions was to see if the student know the relationship between velocity and acceleration. The problem could be done be hand; this requires the use of the derivative of a constant rule, derivative of a sum rule, the constant multiple rule, the derivative of the cosine and the chain rule. To do this by hand would necessitate the use of "easier" numbers and one of the special trig values. A student who makes a mistake on any of these gets the wrong answer and leaves the examiner unclear as to whether the student knows that the derivative of velocity is  acceleration. Therefore without the use of a GC the question is pointless.

 

But you say all those other things are important; a calculus student should know them. I agree -- but they were tested elsewhere on the same test. The velocity - acceleration thing was the point here.

 

2003 AB 87 gave f ' (x) = sqrt(x)/(1 + x + x^2) and asked for the x-coordinate of the point of inflection. The fastest and, I am sure, the intended method of solution is the use the GC to graph the given derivative and identify its extreme value which corresponds to a point of inflection. This concept is often tested and can be found on several free-response tests.    

 

The questions could be done by graphing f '(x) by hand, but even with a simpler function with easier arithmetic this is time consuming and not practical. You could also find the second derivative and find its zeros by hand. Again not really practical with the given function. It is correct approach but one that does not tell us that the student knows the graphical relationship between the graph of derivative and the POI.

 

Using a SMC you can more easily find the second derivative and its zeros. So you can get the answer, but even that takes more time and thought than the intended method. So here the SMC will work but is a disadvantage time wise. And, as before, does not tell the examiner if the student knows the relationship between the extreme values of the first derivative and the POI.

 

The point is that technology can make testing more focused on what you want to test. Using more complicated functions and less "easy" numbers is a way of pointing the student to his GC or SMC hoping he will show that he understands the concept being tested.

 

Whether you choose to let students use a GC or SMC on your tests is up to you; you can certainly test these and other concepts without them. Certainly you can write questions for which a SMC will not be of any help. Or you can use the SMC to focus on the concept you are interested in. I think that's the point in testing with a SMC.

 

But the other reason to use a SMC or other technology in your class and requiring your students to know how to use it is to help them learn math better. I could live with no technology on test, but not with no technology in class. The reason to require technology use on tests is to be sure it is used in class.