E-assessment: why, what for and how

George Fernandez & Gary Fitz-Gerald
Australia

3   A DETAILED EXAMPLE

Simple Differential Equations

This example illustrates the difference in how to use an e-assessment approach instead of a traditional approach. In Engineering and Science courses a student’s ability to correctly and accurately determine the solution of a homogeneous, second order differential equation with real constant coefficients is a fundamental requirement. Such a topic is usually introduced in a first year calculus course by seeking solutions arising in RLC circuit theory or damped oscillatory behaviour in simple mechanical contexts. To be able to correctly solve such equations students need to be able to:

• write down the correct auxiliary (or characteristic) equation
• solve the resulting quadratic equation
• identify the roots as distinct and real, real and repeated or complex conjugates
• construct the corresponding basis of the solution space
• deduce the final solution.

The second and third of these items are skills that students normally bring from a successful completion of a prerequisite course at secondary school level. The remaining items build on this knowledge base to achieve the required outcome.

Traditional methods of formative assessment built around print media would involve a specific limited set of examples with answers usually to be found in another location, such as at the back of a text book. In e-assessment—armed with a robust Computer Aided Assessment package such as WebLearn—an infinite variety of such examples can be generated dynamically as required, marked almost immediately and, where necessary, with feedback tailored to misunderstandings identified by virtue of the incorrect answer that has been supplied. The randomness built into such systems also ensures that any one question is very unlikely to be repeated for different students, or for the same student at different attempts. Consequently, there is no opportunity for students to simply rely on their memory in order to obtain correct answers. This encourages the aspects of deeper understanding that instruction is trying to achieve.

In traditional forms of teaching, the most common approach is to give students an equation to solve, and address their questions and difficulties as they arise. In student-centred online learning, however, the formative assessment tasks to support learning of this topic should first address each of the five steps outlined above. This may be done in a number of ways. Assuming that the first step has been set up as a learning objective, the simplest way to test it is by providing a randomly generated second order constant coefficient differential equation, and requesting the corresponding auxiliary equation as a response. If incorrect, suitable feedback would point the student to available online notes or similar examples and invite more attempts. In cases where their provided answer demonstrates known misunderstandings, specific feedback to that effect can be provided.

Assuming now that the second step has been made as a learning objective under specific investigation, the simplest way to test it is to provide a randomly generated quadratic equation (with coefficients lying in sets for which ‘reasonable’ solutions will exist) and providing appropriate feedback if the supplied answers are incorrect. If so, feedback similarly to the previous case may be provided.

Additionally, in most instances the answers supplied after an incorrect response could also be analysed with regard to their correctness for the given response, and suitable feedback could be provided. Correctness or otherwise of each subsequent step in the calculation could also be provided, despite the calculation at this stage being incorrect. Although in e-assessment this is almost a trivial task, in a traditional approach this has massive resource implications, especially for large enrolment classes.

Assuming that these two combined represent the learning objective under investigation, the first and second steps may then be combined and appropriate positive feedback provided if:

• the auxiliary equation is incorrect: provide suitable feedback along the lines described above.
  Additionally, the answers supplied to this incorrect auxiliary equation could also be analysed
  with regard to their correctness for this equation and suitable feedback could be provided.
• the auxiliary equation is correct but incorrect roots are provided: provide suitable feedback
  along the lines described above.

• the auxiliary equation is incorrect: provide suitable feedback along the lines described above.
• the auxiliary equation is correct but incorrect roots are provided: provide suitable feedback along the lines described above.
• the identification of the characteristics of the roots may or may not be explicitly designated as a learning objective. If it is, and the answer supplied is incorrect: provide suitable feedback along the lines described above.
• the basis of the solution space is incorrect: provide suitable feedback along the lines described above. An error at this stage could be due to several reasons ranging from a totally wrong form of the proposed answer to a solution space that is not of order two.
• the final solution is not a suitable linear combination of the functions in the solution space:
  provide suitable feedback along the lines described above

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