Diagnostic Aids

A diagnostic aid for a problem is information that helps locate and identify errors. The most primitive diagnostic aid is the answer to the problem, and this may be sufficient for short problems. For longer or multi-step problems the result of an intermediate step often makes a good diagnostic.

A diagnostic can be used either by the student himself, or by a teacher or other helper.

Example

The example is taken from Products of Polynomials, Full Products problem 1.

Problem
Write (3 z^2 +5a)(z^{4}+2z^2-a) as a polynomial in z.
Answer
{ 3 z^6 + (6 +5 a)z^4 + 7 a z^2 - 5 a^2 }
Diagnostic
The raw output of the multiplying-out process is

z^6((3)(1)) +z^4((3)(2)+(5a)(1))+z^2((3)(-a)+(5a)(2))+((5a)(-a))
see Products of Sums for further detail on this particular example.

General ideas

  • Good diagnostics are nearly as important as good problems because the most targeted learning occurs when students understand and correct their own specific errors.
  • Diagnostics should not be complete solutions because this amounts to discarding all the student's work, right or not, and starting over.
  • Diagnostics should not presume the student will make a particular error because students are much more inventive than problem designers.
  • Diagnostics will often appear cryptic to someone who has not tried to work through the problem.
  • Diagnostics should encourage, by example, good practices.

Detail

In the example the diagnostic gives an intermediate step: the terms assembled to give the answer but before any arithmetic is done. If the student has a paper trail of his work he can localize where the error occurred. Specifically:

  • If the student's intermediate step agrees with the diagnostic then the error is in the arithmetic leading to the final answer.
  • If the student's intermediate step disagrees in only one place then the procedure is almost certainly correct and the problem is either a copy error or an error in combining exponents.
  • A student may use the correct procedure but not have an intermediate step that looks exactly like the diagnostic, for instance if he does the multiplications on the fly and writes down outcomes rather than unprocessed products. In this case a little processing of the diagnostic will yield something he can compare to his work. Combining steps often increases the error rate. In this case the format of the diagnostic provides a hint (separate different tasks) on how to organize the work to reduce errors. Note that only students who made an error get such a hint.
  • Finally a student may have an intermediate step nothing like the diagnostic. In this case the student can compare the form of the diagnostic with the original problem, with hints from reference material as necessary, and infer the method used. Many students learn well this way and effective diagnostics support this.
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