30.9 Macrotheory of Instruction

How does the concept of mathemagenic activities fit into a broad theoretical framework that deals with the impact of instruction on a large aggregate of learners? Rothkopf (1981) has proposed such a theory, centered around the notion of an instructive event as the basic counting unit of instructive experience. An instructive event is an informative presentation, event, or happening that is sufficient to induce a targeted competence in at least some learners in at least some situations at least some of the time. Many possible experiences may serve as instructive events for a given targeted competence. The competence to erase an entire line with a simple maneuver in a word-processing program may be produced by a paragraph of text in a manual, a personal demonstration, a labeled diagram, or clips from videotapes of a working writer or secretary. All of these instructive events support competence. All or some of these may be part of a training program. As may be seen in Figure 30-2, the likelihood of encountering an instructive event is the first major factor of the macrotheory for forecasting the number of individuals who will have the competence to erase an entire line with a simple maneuver. The likelihood of encountering an instructive event is determined by two variables: (1) the number of relevant instructive events-their density or redundancy in the instructional stream, and (2) compliance with directions and assignments (e.g., do students read what they are supposed to read?). Compliance is clearly an important variable in determining the number of instructive events that will be encountered. Compliance represents a class of overt mathemagenic activities and is affected both by consequencing and by modeled behaviors and modeled values.

Encountering an instructive event is not sufficient for the acquisition of competence. For that, the instructive event must be successfully processed. Three variables determine the likelihood of successfully processing an instructive event once it has been encountered. As may be seen in Figure 30-2, the three variables are: (1) the disparity between the representation of the instructional information and some canonical representation of the target competence, (2) persistence and topography of elicited mathemagenic activity, and (3) the instruction-relevant experience and knowledge of the learner. Disparity can be thought of as the logical distance between the instructional representation of the required knowledge and a simple hypothetical mental representation. It is a conceptualization of the obstacles that the mathemagenic processes must overcome in order to produce the required knowledge. In text for example, the information about erasing a line may be simply and directly stated, or it may be said in an involved indirect manner that requires much analysis and inference. Mathemagenic activity of a given topography and vigor may be sufficient for simple instructive representation but may fail when encountering a complex instructive event. Hence, there is a trade-off between disparity and mathemagenic activities. The last variable, instruction-relevant experience, also trades off with disparity and mathemagenic activities. Instruction-relevant experience refers to familiarity with elements of the instructive representation such as word knowledge or particular pictorial techniques. The simple metaphor of a successful leap across a ditch describes the interaction of the three variables. Disparity represents the width of the ditch. The persistence of mathemagenic activities describes the energy of the leap. Mathemagenic topography describes its direction. Finally, instruction-relevant experience refers to how little weight you are carrying when the leap is made: the more experience, the less weight.

The remaining aspects of the model deal with successful retrieval, and we do not need to concern ourselves with them here.

To sum up, mathemagenic activities are involved in two aspects of acquisition within the macrotheoretic model. Overt compliance influences the likelihood of encountering an instructive event. Covert mathemagenic activities determine the successful processing of instructive events once they are encountered.

The intervention and control variables for acquisition in the macromodel can range widely in molarity. For a given system, indicators of instructional-event density may range from actual counts of instructive events judged relevant for a particular competence in a particular body of materials to computer-based statistical approximations obtained from key word counts, to the salaries paid to teachers in systems where teachers are the main instructive-event generators. Disparity indicators may include judged. disparity and quantitative measures of sentence length and complexity. Schools can be evaluated with respect to measures that are taken to foster compliance, although socioeconomic and cultural characteristics of the student population are bound to play an important role. Vigorous management of compliance enhances achievements. The control of covert mathemagenic activities requires micromanagement for which technological aids to teachers are very useful. A computer-aided homework help system via telephone is an example of such technological aid.