Task Separation

Experience suggests that qualitatively different tasks interfere when they are combined, resulting in higher error rate. Students should be encouraged explicitly and by example to separate such tasks.

Example

Add (i.e. write in standard polynomial form) the polynomials in :

(1)

There are two tasks: collecting terms with the same exponents, and adding the coefficients. Collecting without adding gives:

(2)

Adding then gives

(3)

The collecting step avoids arithmetic thinking, even to the extent of using parentheses freely to avoid deciding whether they are really needed.

There is an almost irresistible temptation to combine these steps and do the (very simple) additions on the fly. Experienced teachers can handle this example but students trying to emulate them will often be frustrated by trivial arithmetic errors. Everyone has a threshold of complexity where this becomes a problem: see Products of sums for an example where even experts would be well-advised to avoid on-the-fly arithmetic.

Separating tasks should therefore be considered good practice. It should be relaxed only in simple instances when students demonstrate they can do it successfully.

Tasks to separate

• Translating word problems to symbolic form;
• Organizational tasks such as collecting terms or expanding a product of polynomials;
• Arithmetic and algebra.

These are general examples; teachers alert to this difficulty will find other dissonant tasks depending on topic and level.

Possible reasons

Task interference is an empirical observation resulting from an effort to explain frequency of trivial errors in some problems. We offer a “cognitive science” explanation, but the fact there is a problem does not depend on this being the correct or primary explanation.

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One of the earliest findings in functional brain scanning is that activity shifts dramatically when the subject is given an arithmetic problem. It seems — roughly speaking — that people develop “hard-wired” modules dedicated to arithmetic. Other tasks also use different parts of the brain. Algebra and translating word problems, for instance, probably involve different sides in the “right-brain/left-brain” dichotomy.

Combining tasks therefore requires switching activity between different brain areas. On-the-fly arithmetic for example requires repeatedly swapping the arithmetic module in and out. This swapping interrupts activity in both areas and often degrades performance.

Facility with different cognitive functions is highly variable and individual. Many students have trouble with translation of word problems. For them it is particularly important that focus be maintained on this task and not be interrupted by arithmetic. Similarly students whose arithmetic modules are not fully developed (not “automatic”) should avoid interrupting arithmetic tasks. However this is just a matter of threshold: as task complexity increases eventually everyone has trouble.