January 06, 2006

Ads Build Up Value

This blog's title is an anagram of David Paul Ausubel, the educational psychologist who made serious play of the fact that you can't teach someone effectively until you know what they already know (and, by implicaton, don't know). In Educational Psychology: A Cognitive View (1968) he wrote: "The most important single factor influencing learning is what the learner knows. Ascertain this and teach accordingly". His dictum is often quoted but rarely acted upon, and I see no harm in running another advertisement to build up its value.

When students submit to any kind of assessment, they reveal information about what they know and don't know. This is often valuable information and usually it goes to waste (think of all those exams where the only feedback is a mystery number between 1 and 100). Does computer-aided asssesment (CAA) offer any remedies? Can it find out "what the learner knows" and then act accordingly? I believe the answer is "yes" and will try to convince you of this with a couple of simple examples.

If a factual(ish) question (eg Which philosopher might have said "Blogo ergo sum"?) is wrongly answered ("Renée Zellweger" perhaps), the same question can be repeated in a later test, with a friendly admonition in the feedback for a second wrong answer. Persistent weaknesses over a series of formative tests can be routinely reported back to the student.

For conceptual questions, CAA can do even better. We will take a simple example to illustrate how computer assessment can identify a student's problem and try to deal with it. Consider the arithmetic question:

Add one third to one half and express your answer as a fraction (i.e. a number of the form m/n).

There are many reasons for getting the wrong answer: a simple error of calculation (unlikely here though), or a failure to understand

(i) the nature of fractions (both as numbers and processes — does the symbol 1/2 mean the number 'one half' or the process of dividing by 2?) and
(ii) the rules for calculating with them.

Let's suppose a student submits the following wrong answer:

 \frac{1}{2} + \frac{1}{3} = \frac{2}{5}

It's a fair guess that they have used the following wrong rule (mal-rule) of adding the numerators and the denominators:

 \frac{a}{x} + \frac{b}{y} = \frac{a+b}{x+y}

We could reinforce this guess with another example, say two-thirds plus a fifth. If they come up with three-eighths, we can be pretty certain we've sussed out the mal-rule they're using.

Next we start to generate doubt by getting them to apply their rule to one half plus one half (giving two quarters, which one hopes they will know equals one half). Something funny going on here! A fluke exception? How about this then:

1 + \frac{1}{3} =  \frac{1}{1} + \frac{1}{3} = \frac{1+1}{1+3} =  \frac{2}{4} =   \frac{1}{2}

Having thoroughly undermined their faith in the mal-rule, we eventually return to the drawing board, asking the student how many sixths make a half (three) and how many sixths makes a third (two). If the penny drops that three of a kind plus two of a kind is five of a kind, the following equations take on meaning

 \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} =  \frac{3 +2}{6}= \frac{5}{6}

More examples will be needed to elicit a thorough understanding of the general rule for adding fractions (including unlearning the mal-rule) and yet more practice for the student to feel comfortable applying the rule as a fast and accurate reflex.

What I have just described is only one possible CAA response to just one of many misconceptions or mal-rules that may be revealed in the simple exercise of adding a half to a third. A student may be guessing or may have mislearnt the rule at an earlier stage of education. An effective face-to-face (human) tutor will patiently probe a student's mistakes and identify the root causes their failures, then painstakingly rebuild their knowledge and understanding on a sound foundation. Intelligent computer assessment can be programmed to do the same and it is nothing if not patient and painstaking.

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