The Functional Analysis of Informal Models

RETURN
edited 1/23/19

We carry around in our heads a variety of models which guide our practice. These models tend to be quite simplistic.

This set of introductory exercises will enable the user to specify such informal models in a more explicit functional relation. This may help us in determining the validity of such models or in making obvious their shortcomings.

The expression "A is a function of B and C" translates into informal English as "A depends uniquely on B and C." This does not permit more than one value of A to be determined by a unique pair of values from B and C, e.g. (b, c).*

So, if we believe that learning, (L), depends uniquely on a combination of ability, (Ab), and time spent studying, (T), we might venture the formalization. L= f(Ab, T).

Consider the chart below:

 STUDENT NEED AND TYPE N= f(Ab, Ach) Ability Achievement Need Type Group A 50 95 -45 Overachiever Group B 50 50 0 Normal Group C 50 15 35 Underachiever

The chart appears to present a functional relation between the output variable N, need, and the input variables, Ab, ability, and Ach, achievement: N = f(Ab, Ach). In fact, is specifies that relationship in the following way:

N = Ability-Achievement

This simple formula catches what many seem to be talking about when they discuss overachievers and underachievers. But is something important missing? Are we making important assumptions here that L, Ab and T are measurable variables and mutually independent? Having the functional relationship specified allows us to press the inquiry.

For example, consider John whose ability measure is 600 and whose achievement measure is 550. Compare him with Ed, whose ability measure is 75 and whose achievement is 25. By the formula, they are both underachievers with an equal need of 50. Does this ring true? Does this mathematical possibility square with our intuitions about human abilities and needs? Should we adjust the functional formula? (Is there an assumption that the relationships are linear? Or that the variables are not interactive?)

Here are some other suggested functions that relate common concerns in a specific manner. For each one consider the extent to it captures our common intuitions and supports our practice. Can you specify a formula that relates the variables?

1. Ability is a function of Potential and Learning, A= f(P, L).

What we call Potential, though seldom made clear, places an upper limit on Learning, therefore, also on Ability. Most people would find the claim, "John's abilities surpass his potential,"as confusing, if not incomprehensible.

2. Performance is a function of Ability and Motivation, P= f(A, M).

Clearly, someone may have an ability and not feel motivated to display it. Or conversely, one may have high motivation, yet have an underdeveloped ability. We would not expect a display of high quality performance.

3. Achievement is a function of of Performance. Ach= f(Per)

The less skilled the performance, the less likely it meets its goals.

4. Learning is a function of Study, Concentration and Ability. L= f(S, C, Ab)

We often explain low levels of learning but pointing out a low level in any one of the independent variables. Each one is commonly thought to be individually necessary and jointly sufficient to account for the quality of learning.

It might be important to consider under what restricted conditions such simple formulae might work. What background conditions might we assume constant which might otherwise be influential on the output variable?

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*This does not rule out that there may be alternative causal hypotheses, e.g. A may also be a function of B and D. It also does not rule out that other values of B and C, b' different from b, and c' different from c' might also determine the same value of a.