Teaching Procedures, page 3
Principles for Learning Procedural Skills
How is a procedural skill learned? Let's consider the procedure for adding fractions, since everyone is familiar with it. Its steps include:
Branching and Linear Procedures
Procedures can be branching or linear. This is a branching procedure, because different steps are used for different problems (cases). A branching procedure always has at least one decision step, and every decision step has at least two branches. Steps 1 and 5 are decision steps (shown in diamonds). There are two branches from step 1. One includes steps 2 and 3, whereas the other skips them. Every branching procedure has several paths to the end of the procedure. There are four paths through this procedure:
- Path 1: Steps 1 - 4 - 5
- Path 2: Steps 1 - 4 - 5 - 6
- Path 3: Steps 1 - 2 - 3 - 4 - 5
- Path 4: What is the fourth path? Steps ___________________________
A linear procedure has no decision steps and hence no branches. Therefore, it only has one path.
Variations in Size
Another important aspect of procedure learning is that procedures can vary tremendously in size. For example, we could identify the procedure for solving mathematical problems of any type. Such a procedural task would have many decision steps, which would direct the performer to different "branches" (different sets of steps) for each kind of problem: addition, multiplication, etc.; whole numbers, fractions, equations, etc.; and so forth. Obviously this will be a very big (long) procedure. On the opposite extreme, we could identify the procedure for subtracting single-digit numbers, which is basically just one step: recall the appropriate subtraction facts. It is interesting to note that this latter procedure would be a part of the former.
Procedures Within Procedures
In essence, each step is itself a procedure which needs to be learned on the skill-application level. Furthermore, as we saw earlier in this course, virtually any step can be broken down into substeps. Steps 2, 3, and 6 above would probably need to be further broken down into substeps to teach the procedure for adding fractions. The only way you can tell how far to break a procedure's steps down into simpler substeps (and substeps into even simpler subsubsteps, etc.) is to first determine what your learners already know. Then you keep breaking down steps and substeps until each is expressed at the learners' current level of knowledge.
Generalizing from One Procedure to Another
For the sake of simplicity, let's assume that the learners already know enough to be able to learn from the steps as stated above. Then how is the procedure learned? It is possible to memorize the statement of the procedure. It is also possible to memorize one particular performance of the procedure. But what we really want is for the students to be able to generalize to "previously unencountered" cases.
The need to generalize is based on yet another important aspect of procedures: the amount of variation from one performance (or instance) to another. Not all procedures have a lot of variation. Some procedures are always performed exactly the same way. Perhaps the name which comes closest to characterizing this type of procedure is recipe. "If you've seen one, you've seen 'em all." If you've learned one instance, you've learned them all.
Dimensions of Divergence
In reality, there is a continuum of procedures ranging from non-divergent, simple procedures such as recipes, to very highly divergent (or complex) procedures on the other extreme. This has important implications for your instruction, especially for divergence of examples and practice. Dimensions of divergence are the all the ways a procedure can vary.
Generalizing is required whenever a procedure has divergence or variation (is done differently) from one performance to another. The ways in which the performances vary are called the variable characteristics of the procedure. Variable characteristics for a procedure are the "inputs," or conditions under which you perform your actions.(1) For example, the fractions to be added are the inputs to our procedure, and they could have common denominators, or one's denominator could be a multiple of the other's, or the two denominators could have a common factor, or the denominators could be none of the above.
These types of problems are called equivalence classes, because all problems within an equivalence class are performed using the same path (steps) in the procedure Template:Reference needed. Each equivalence class will use a different set of steps than other equivalence classes. There may also be some variable characteristics of the inputs which don't require that the procedure be performed differently. For example, the procedure for adding 1/4 and 1/4 is the same as the procedure for adding 1/12 and 1/12; it doesn't matter that the denominator is a one-digit or two-digit number. One must learn to generalize across these differences, but it is usually not difficult to do so.
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- Procedure Using by Charles M. Reigeluth. Used by Permission.