Response: On Claims That Answer the Wrong Questions

representations in isolation from their meanings, the outcome can be that students learn a set of mechanical rules that can support their successful performance on tests requiring only manipulation of the notations, not meaningful use of the representations. On the other hand, it is perfectly consistent with the situative perspective that abstract representations can facilitate learning when students share the interpretive conventions that are intended in their use. Examples (cited by Anderson et al.) are the studies of hitting targets under water by Scholckow and Judd (Judd, 1908) and by Hendrickson and Schroeder (1941) which Greeno et al. (1993) discussed. As I mentioned previously, Greeno et al. provided a situative analysis based on the hypothesis that the meaningful abstract representation of refraction oriented students to properties of the situation in a way that resulted in their learning a more generalizable way of interacting with the material systems in the situation. Implications for Theory There is much that we do not understand about the ways that abstract representations function in activity. Part of the power of abstraction is in people's ability to use representations to orient their attention to properties and relations in situations that are important in activity, as in the refraction examples. Another crucial property of abstract representations is that the representational materials themselves-the symbolic or iconic expressions that are written or drawn-can be manipulated as objects, supporting explorations of possibilities and evaluations of relations such as implication or consistency between different statements. There has been a considerable amount of research into the processes of reasoning in some of the formal systems of logic and mathematics, but the question of how these systems can support learning in conceptual domains and reasoning in domains of application has been addressed much less. However, some promising beginnings have been achieved (e.g., diSessa, Hammer, & Sherin, 1991; Godfrey & O'Connor, 1995; Hall & Rubin, in press; Lampert, 1990a; Ochs et al., 1994). Implications for Discussions of Educational Practice Anderson et al. remarked that "Abstract instruction can be ineffective if what is taught in the classroom is not what is required on the job" (p. 8). That is a useful observation, but the issue goes deeper than job training. Abstract instruction can also be ineffective regarding some important purposes if what is taught in the classroom does not communicate important meanings and significance of symbolic expressions and procedures. An example in mathematics education is the learning that often occurs when students are taught to manipulate the notations of algebra without connecting them to their conceptual meanings. Algebraic formulas are intended to serve as representations of functions-that is, the symbols of formulas refer to numbers, variables, and arithmetic operations, and equaJANUARY/FEBRUARY 1997 13 This content downloaded from 207.46. 13.162 on Thu, 30 Jun 2016 05:36:53 UTC All use subject to http://about.jstor. org/terms tions express relations of equality or equivalence between functions (e.g., "3x 7 = x + 1" is a relation that holds if x is 4, and "2(3x 7) = 6x 14" is a relation that holds whatever x is). Valid algebraic operations can be stated and learned as a set of rules for transforming symbolic expressions, and those rules can be applied without understanding of their meanings. Many students learn rules of algebra in this way; evidence includes the frequent occurrence of systematic "mal-rules" in students' performance (e.g., Anderson, 1989; Lewis, 1981; Sleeman, Kelly, Martinak, Ward, & Moore, 1989; R. H. Wenger, 1987). An example is the transformation of an expression such as "2(3x 7)" to "6x 7," which uses a close procedural variant of the correct rule, but produces an expression with a quite different meaning. I have written this before (Greeno, 1989), but it bears repeating. Some teaching of mathematics is a good realization of Searle's (1980) parable of the Chinese room. In that parable, a person lives inside a room that has baskets of tokens of Chinese characters. The person does not know Chinese. However, the person does have a book of rules for transforming strings of Chinese characters into other strings of Chinese characters. People on the outside write sentences in Chinese on paper and pass them into the room. The person inside the room consults the book of rules and sends back strings of characters that are different from the ones that were passed in. The people on the outside know Chinese. When they write a string to pass into the room, they understand it as a question. When the person inside sends back another string, the people on the outside understand it as an answer, and because the rules are cleverly written, the answers are usually correct. By following the rules, the person in the room produces expressions that other people can interpret as the answers to questions that they wrote and passed into the room. But the person in the room does not understand the meanings of either the questions or the answers. Searle's parable was intended as a critique of the view in artificial intelligence that if a program can produce character strings that can be interpreted as answers to questions by people who understand the language, we should say that the program understands the language. Searle's conclusion, with which I agree, is that the kind of artificial intelligence that performs formal transformations on character strings to produce what human users of a language can recognize as answers to questions should not be said to understand the language. When we teach mathematics as a set of rules that operate on symbolic expressions without teaching the meanings of the expressions or the rules, we succeed in creating that kind of artificial intelligence in the minds of real students. The example of mathematics learning shows why we need to distinguish generality from abstraction. The problem with teaching mathematics as a set of rules is not just that some students learn to do it incorrectly. The deeper problem is that students who learn the correct rules may not learn to use mathematical expressions to represent mathematical concepts and relations between quantities. They have learned a set of abstractions, but what they have learned is not general. When the practices of schooling are focused on students' learning to perform correct transformations on symbolic expressions, the result can be to make that knowledge very specific, in the sense that it satisfies requirements of school activity without strengthening students' general reasoning and understanding. There is much that we need to understand about the ways that abstract representations can facilitate general learning. Anderson et al. mentioned several findings in which some generalization has been shown by using a combination of abstract representations and more concrete examples. Those examples are encouraging, although many psychological studies apply a standard of generality that is quite modest. Learning that enlarges the class of word problems that students can solve on tests is a valuable improvement in that it will help some students perform more successfully in their school activities. However, such tests are still located in school practice and do not necessarily tap students' understanding or use of general mathematical concepts. The important question is how to make school learning more beneficial beyond the classroom, providing students with general resources for reasoning both in and with the concepts of subject-matter domains. Anderson et al. recognized that there is a problem when they said that "relatively little time is spent relating algebraic expressions to the real-world situations they denote" (p. 9). I would add that much instruction could also be strengthened by relating expressions to the general concepts and principles that they denote. This is the problem that motivates much of the effort in mathematics education reform that is focused on "authentic" activities in school learning. As we proceed, I believe we will increase our theoretical understanding of what all this means, as well as our capabilities of arranging more productive learning activities for students.