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Using Connected Curriculum Project Modules - Checking answers

John Hannah

Getting students to check their answers is a feature of most CCP modules. This encourages students to engage in self-monitoring, to become more independent learners. It can also have an indirect benefit, encouraging deeper understanding or developing connections between ideas which might otherwise remain compartmentalised in the student's mind. For example, a solution to an equation can be checked by substituting it into the original equation (thus clarifying what it means to be a solution), or again, an integral can be checked by differentiating the answer to get the original integrand (thus illustrating the fundamental theorem of calculus).

How students reacted to this idea was best seen in the lab sessions themselves. Sometimes checking occurred without being asked for. For example, in Part 1 of the Spring Motion module, several students copied and pasted the data for the spring system incorrectly, missing the initial decimal point of the first reading. This made the first reading much larger than the others and the resulting plot of the data appeared to represent a constant zero function. A typical student query was "What's gone wrong here?" Here an expectation (presumably of oscillatory motion) has been thwarted, and the students have gone into checking mode, looking for the source of the problem.

A similar attitude was apparent in  Part 1 of the Second-Order Linear Homogeneous Differential Equations with Constant Coefficients. Step 3 of this Part looks at the differential equation y" + y = 0 with varying initial conditions:

Reset y(0) = 1, and change y'(0) first to 2, then to 3. How do the solutions change as y'(0) varies through positive values? Use the symbolic solution to explain what you see in the solution graphs.

Some students tried relatively large values for y'(0), and then the Maple command given in the accompanying worksheet:

DEplot(DE, y(t), t= 0..15,[[y(0)=y0,D(y)(0)=y1]], y=-4..4,  stepsize=0.1, linecolor=blue);

produces a plot where y appears to grow indefinitely. Again the self-monitoring attitude showed itself in the resulting query: What has gone wrong here? (Incidentally, modifying the y-range overcomes this problem.)

Sometimes, however, students were unsure about what checking actually meant. Part 11 of the Helper Application Tutorial module provides a case in point. The introduction explains that checking is part of the forthcoming task:

In this part we explore Maple's ability to solve the logistic equation

                               dy/dt = y (1 - y)

and to check the solution. Then we will adapt the solution procedure to an initial value problem with this same differential equation.

Step 1 uses Maple to find the general solution to this differential equation. In Step 2 the student is led through the calculations needed to check that the general solution does satisfy the original equation (although, perhaps significantly, the word check is not used):

Differentiate your solution expression with respect to t to get an explicit expression for dy/dt Then use your solution expression to find an explicit formula in t for y(1 - y). Is this formula the same as the one for dy/dt? You may want to simplify the output before you try to answer this.

In Step 3 an initial value is added to the problem, and Maple is again used to find the solution. Finally in Step 4 the student is asked:

What do you have to do to check the answer from the preceding step? Have you done it already? If not, can you get the checking technique from what you did in Step 2?

Although all students eventually got correct answers to Step 2 (with more or less help from the tutors), most of them checked the solution to the initial value problem in Step 4 by just substituting the proposed solution into the original differential equation, ignoring the initial value part of the problem.

Failing to check the initial condition here probably indicates some uncertainty about what checking actually means. On the other hand, substituting a particular solution into a differential equation after they have already checked that the general solution works may mean that they haven't understood what is meant by a general (or particular) solution. Another possibility is that Step 2 was not perceived as a check. Apart from the absence of the word check in the instructions for Step 2, the actual calculation of y(1 - y) was complicated by the fact that Maple did not store the solution in a variable called y. The purpose of Step 2 may have got lost in the intricacies of the calculations.

A similar uncertainty about checking surfaced in the Experiments With the Laplace Transform module. In Part 1 the student uses Maple to calculate the Laplace transforms of some standard functions, including exp(-at), cos(at), and sin(at). Then in Step 2 they are asked to verify that the Laplace transform of  df/dt  is  s F(s) - f(0)  for the functions exp(-at) and cos(at). Many students asked what they were supposed to do here, almost as if they did not know what was meant by the word verify.

As an aside here, it is interesting that most students ignored the accompanying advice:

If you use the results of Step 1, you should not need additional computer algebra calculations for this step.

and used Maple to do the calculations anyway. Bookman and Malone  noted student uncertainty about which tool to use for a particular calculation (see their Vignette 3). In this case it seemed that most students preferred to use Maple, although as a tutor I felt that they would learn more from a hand calculation -- and the advice quoted above suggests that the authors of the module felt this way too.

None of the students mentioned checking in the surveys, but I did get some positive feedback from individual students who had experienced the warm glow of knowing their answer was correct in the end-of-course exam.

John Hannah, "Using Connected Curriculum Project Modules - Checking answers," Convergence (December 2004)