Triangulación empiria - teoría Categoría

LeFevre and Dixon (1986) found that students learning a procedural task prefer to use examples as a source of information and that written instructions tend to be ignored.

VanLehn (1986, 1990) has built a theory of children’s errors on the evidence he has gleaned that people prefer to use examples rather than written explanations. VanLehn (1986) has estimated that some 85% of children’s systematic errors are due to misunderstanding textbook explanations of problems.

Pirolli (1991; Pirolli & Anderson, 1985) found that novice programmers relied heavily on examples rather than instructions to help solve LISP recursion problems.

Carroll, Smith-Kerker, Ford, and Mazur-Rimetz (1987–1988) redesigned computer training manuals partly to take account of the fact that learners are put off by the “verbiage” in traditional training manuals.

ACTIVITY 7.2

Car A leaves a certain place at 10 a.m. travelling at 40 mph, and car B leaves at 11.30 a.m. travelling at 55 mph. How long does it take car B to overtake car A?

The equation to use is:

RateCarA×TimeCarA=RateCarB×TimeCarB

What figures would you use to replace the variables in the equation?

Third, novices may not understand the concepts embodied in the rule or principle, or may have misinterpreted them. For example, the equation in Activity 7.2 is based on the more general equation

Distance=Rate×Time. As both cars travel the same distance then DistanceCarA=DistanceCarB; and as the

distances are equal the Rate×Time for both cars must be equal too: hence the form of the equation. Now if you know something about algebra or mathematics in general, then that explanation might make sense and you can understand where the equation comes from. If you have little knowledge of mathematics, then the origin of the equation may be rather obscure. That is, you may not understand the concepts involved in the equation.

Fourth, trying to solve a current problem may force novices to extract more information from an earlier problem than they did at the time. If you saw how to solve the Fortress problem based on the Radiation problem, you may have been able to abstract out information from the Radiation problem that was more relevant to “divide and converge” problems.

Reimann and Schult (1996) point to three problems that the use of examples helps to overcome. These are the “interpretation problem”, the “control problem”, and the “generalisation problem”.

The interpretation problem. Examples show how theoretical principles can be interpreted and instantiated

in a problem (the second of Ross’s four roles). They show the relationship between a problem description and the concepts or principles they embody.

The control problem. At any one time in the middle of an algebra problem there may be a number of possible

operators that you can apply (related to the first of Ross’s roles). Examples show what specific operators apply and therefore demonstrate the specific solution procedure.

The generalisation problem. It is always difficult for novices to know what the salient aspects of a

problem (or a textbook for that matter) are. That is, they are poor at distinguishing between those surface features that are related to the structural ones and those that are irrelevant (see the discussion of Gentner’s structure-mapping theory in Chapter 5). Only the superficial features can be generalised over.

THE PROCESSES INVOLVED IN TEXTBOOK PROBLEM SOLVING

Various representations can be derived from a textual presentation of a problem. The first thing a student finds when confronted with a word problem to solve in a textbook is a piece of text. The first thing the student has to do, therefore, is to make sense of the text itself. This in turn requires several layers of representation.

First of all there are the individual words that compose the text. Understanding these comes through our semantic knowledge of the items in our mental lexicon—our mental dictionary. From the individual words and the context of the sentence, our overall understanding of the text of a problem is constructed, and so on. Kintsch (e.g., 1986, 1998; Nathan et al., 1992; Van Dijk & Kintsch, 1983) has argued that word problems require the solver to generate a number of different representations. The initial representation of the text is a propositional representation called the textbase. However, knowing what the text of a question means does not therefore entail an understanding of the problem. Kintsch (1986, p. 89) gives the example of trying to understand a computer manual:

all too often we seem to “understand” the manual all right but remain at a loss about what to do; more attention to the text as such would be of little help. The problem is not with the words and phrases, nor even with the overall structure of the text; indeed, we could memorize the text and still not know which button to press. The problem is with understanding the situation described by the text. Clearly understanding the text as such is not a sufficient condition for understanding what to do.

From the textbase students have to develop a representation of the situation described in the text. This is a mental model which Van Dijk and Kinstch (1983) termed a situation model. For problem solving to be successful, the solver has to generate all the necessary inferences in order to build a representation of the problem that is useful enough to solve it. This, in turn, means that novices have to have enough domain- relevant knowledge to do so.

In a later formulation of the theory, Nathan, Kintsch, & Young (1992) divided the situation model into two. They explicitly distinguished between the situation model and the problem model. The situation model includes elaborated inferences generated from an understanding of the text. Such inferences might include the fact that if two cars leave from the same point at different times and the second car overtakes the first then both cars will have travelled the same distance at that point. The fact that both cars travelled the same distance may not be explicitly mentioned in the text. Nathan et al. (1992, p. 335) also add that:

because of the added demands of inference making, readers will make inferences only when they seem necessary. Poor problem solvers will tend to omit them from their representations, and so they will omit the associated equations (supporting relations) from their solutions to story problems. Problem solvers who reason situationally will tend to include these inference-based equations

The other representational form proposed by Nathan et al. is the problem model which includes formal knowledge about the arithmetic structure derived from the text, for example, or the operating procedure constructed from information in the text. The ability to make inferences from texts in order to derive a useful problem model depends on the relevant prior domain knowledge of the learner.

Kinstch (1986) argues that the text determines what situation model is constructed and how it is constructed. The situation model is important for learning and the textbase is important for remembering text (bear in mind that the situation model and problem model are conflated here). In a study of problem solving and retrieval of earlier problems, he found that recall of word problems that had already been solved was determined both by the properties of the textbase and the model constructed to solve a problem. It was the situation model that provided recall of earlier problems and not a reproduction of the textbase. Learning, according to Kintsch, depended on the problem model constructed from examples, and remembering depended on the coherence of the text. For example, common terms repeated in succeeding sentences lead to greater coherence and greater recall (Kintsch & Van Dijk, 1978). He argued that it was easier (at least for children) to form an appropriate situation model if there is a concrete, familiar structure. However, other studies (e.g., Chen & Daehler, 1989; Novick, 1990) have shown that this is not the whole story. Problem- solving transfer by adults and children from abstract representations can also take place (see Chapter 4).

The distinction between a propositional (textbase) representation of a text and the elaborated situation model was examined by Tardieu, Ehrlich, and Gyselinck (1992). They argued that novices and experts in a particular domain would not differ in the propositional representation they derived from a text, but that there would be differences between the two groups in the situation model (here again the situation model and the problem model are synonymous). Tardieu et al. found that there was no difference between experts and novices on their ability to paraphrase a text (i.e. they both generated much the same textbase) but experts performed better on inference questions than novices (they had derived different situation models from the textbase).

These hierarchical forms of representation have two implications for how novices understand textbook explanations and examples. First, as they are unfamiliar with the domain, they tend to have only a propositional representation of the surface features of the examples. Using examples to solve further problems means matching propositions and is unlikely to be guided by an understanding of the deeper

relational structure. Second, novices may not know enough to make necessary elaborative inferences to generate a complete situation or problem model.

The next section presents an example of a study where the students were unable to generate a complete situation or problem model.

LABORATORY STUDIES OF WITHIN-DOMAIN AND TEXTBOOK PROBLEM SOLVING

Figure 7.2 represents a hierarchy of “Rate” problems. The lower down the hierarcy, the more specific or concrete the problem becomes. The distance between any two nodes in the hierarchy represents a crude measure of the amount of transfer that would be involved between them. Generally speaking solving problems using examples in textbooks usually involves problems that would be adjacent in the hierarchy.

Reed, Dempster, and Ettinger (1985) describe four experiments in which one example problem and solution is presented and the student is thereafter expected to solve a transfer problem, or a problem whose solution procedure was unrelated to the practice problem. In the terminology of Reed et al., the transfer problems were called “equivalent” or “similar”. We will look at the experiments in general and at some of

the algebra word problems in particular with a view to discovering just what the solution explanations that were provided failed to explain.

STUDY BOX 7.2