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LA UREN B. RESNICK 7 quantitative skills needed to apply various tools of production and management, the ability to read complex material, and the ability to build and evaluate arguments. These abilities go well beyond the routinized skills of the old mass curriculum. In fact, they are much like the abilities demanded for college-bound students in the College Board's book, Academic Preparation for College (College Entrance Examination Board, 1983~. Yet teaching such competencies to the mass of students remains a considerable challenge. This, then, is part of what is new about the current drive for teaching higher order skills. The goals of increasing thinking and reasoning ability are old ones for educators. Such abilities have been the goals of some schools at least since the time of Plato. But these goals were part of the high literacy tradition; they did not, by and large, apply to the more recent schools for the masses. Although it is not new to include thinking, problem solving, and reasoning in Bomeone's school curriculum, it is new to include it in everyone's curriculum. It is new to take seriously the aspiration of making thinking and problem solving a regular part of a school program for all of the population, even minorities, even non-English speakers, even the poor. It is a new challenge to develop educational programs that assume that all individuals, not just an elite, can become competent thinkers. THE NATURE OF THINKING AND LEARNING: GOING BEYOND THE ROUTINE This challenge comes at a time when we also have new knowledge about the nature of thinking and strong hints about how thinking abilities are learned. In the last decade or two, cognitive science research has allowed us to look into the thinking mind, figuratively at least, and to specify more precisely the reasoning processes of both successful and less successful thinkers (Newell and Estes, 1983~. More recently, researchers have begun to investigate how the ability and the propensity to think well are acquired and maintainecl. These two bodies of research the nature of human thinking and on the acquisition of thinking and learning skills are beginning to make explicit what we mean by higher order skills and what means of cultivating such skills are most likely to be successful. This process of making explicit the abilities formerly left to the intuitions of gifted learners and teachers is precisely what we need to establish a scientific

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8 ED UCATION AND LEARNING TO THINK foundation for the new agenda of extending thinking and reasoning abilities to all segments of the population. The most important single message of modern research on the nature of thinking is that the kinds of activities traditionally associ- ated with thinking are not limited to advanced levels of development. Instead, these activities are an intimate part of even elementary lev- els of reading, mathematics, and other branches of learning when learning is proceeding well. In fact, the term Higher order" skills is probably itself fundamentally misleading, for it suggests that another set of skills, presumably called "lower order," needs to come first. This assumption that there is a sequence from Tower level activities that do not require much independent thinking or judgment to higher level ones that do-colors much educational theory and practice. Im- plicitly at least, it justifies long years of drill on the "basics" before thinking and problem solving are demanded. Cognitive research on the nature of basic skills such as reading and mathematics provides a fundamental challenge to this assumption. Indeed, research suggests that failure to cultivate aspects of thinking such as those listed in our working definition of higher order skills may be the source of major learning difficulties even in elementary school. Reading as a Higher Order Skill The process of understanding a written text, as it emerges in current psychological and artificial intelligence accounts, is one in which a reader uses a combination of what is written, what he or she already knows, axed various general processes (e.g., making inferences, noting connections, checking and organizing) to construct a plausible representation of what the author presumably had in mind (e.g., Just and Carpenter, 1980; Perfetti, 1985; vanDijk and Kintsch, 1983~. The mental representation constructed by the reader does not match the text itself, nor does the reader even try to match it, except under special circumstances. Instead, the reader tries to represent the situation the author had in mind or the argument the author hoped to build. The reader's representation omits details that do not seem central to the message. It also adds information needed to make the message coherent and sensible. The written text, then, is a vehicle that permits a partially common representation of some situation or argument to be constructed by two separate minds-the writer's and the reader's.

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LA UREN B. RESNICK 9 By their nature, normal, well-written texts are incomplete ex- pressions of the author's mental representation. They leave out some things essential to the representation on the assumption that read- ers will fill them in. If this assumption is not met, comprehension fails even if every word and every sentence has been individually understood. Usually, this process of filling in is so automatic that skilled readers are quite unaware they are doing it. Only when the flow of comprehension breaks down do competent readers become aware of their inferential and interpretive processes. Yet our mod- els of skilled comprehension suggest that inferences are being drawn and interpretations are being made throughout. And studies of eye movements during silent reading, of pause patterns as texts are read aloud, and of disruptions in comprehension caused by minor modifi- cations at key points in the text provide convincing evidence of the reader's inferential work even for quite simple texts. Four kinds of knowledge are called upon as readers construct meanings for texts. The first is linguistic knowledge: knowledge about how sentences are formed, rules of forward and backward reference, and the like. This knowledge is often only implicit, but readers depend on it to find common referents, to link agent to action to object, and to otherwise construct a representation of a coherent set of events and relationships. The second kind of knowledge is topical knowledge, that is, knowledge about the text's subject mat- ter. Like linguistic knowledge, topical knowledge is often used so automatically that readers are unaware of its contribution. Third, readers invoke knowledge about rules of inference. This knowledge, too, is likely to be implicit for the skilled reader. Finally, knowledge of conventional rhetorical structures often aids the process of text interpretation. An example drawn from the work of Walter Kintsch (1979) demonstrates the role of the first three kinds of knowledge in reading comprehension and shows how interactive they are: The Swazi tribe was at war with a neighboring tribe because of a dispute over some cattle. Among the warriors were two unmarried men named Kakra and his younger brother Gum. Kakra was killed in battle. According to tribal custom, Kakra was married subsequently to the woman Ami. The first three sentences of this short passage are un(lerstood so effortlessly that the reader does not notice the special linguistic work required to build a coherent representation. Yet some inference is re- quired. Note that the term "warriors in the second sentence has not

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10 EDUCATION AND LEARNING TO THINK appeared before. However, the definite article ~the" that precedes the term implies that warriors have been referred to previously. The skilled individual knows this linguistic rule, even if only implicitly. What Is more, such a reader infers the required referent by using topical knowledge: namely, that a war (which is referred to in the preceding sentence) is likely to involve warriors. Greater difficulty is encountered when the fourth sentence is reached. The sentence is puzzling. It seems anomalous, and even contradictory, in the context of the preceding sentences. To know that the final sentence is anomalous, the reacler must bring topical knowledge and rules of inference to bear. The reader knows, for ex- ample, that someone killed in battle is no longer alive. In addition, he or she is likely to assume that marriage requires a living bridegroom. This leads to the inference that it is impossible for Kakra to be mar- ried after the battle. Topical knowledge and rules of inference thus lead to the sense that the passage is incomprehensible. Yet topical knowledge can also provide the basis for resolving the comprehension problem. The knowledge needed relates to ghost marriage, a tribal custorr in which, when the oldest son of a family dies without heirs, his spirit is nevertheless married as planned, and his younger brother takes his place in the marriage bed until an heir is produced. In longer texts, knowledge about rhetorical structures also inter- acts with linguistic, topical, and inference rule knowledge. Narrative stories, for example, frequently conform to a prototypical structure in which, after a setting is described, an initiating event sets up a situation in which a character responds by setting a goal. In succes- sive episodes the character attempts to attain the goal, each attempt producing an outcome and a response to the outcome. Extensive research on story ~grammars" (see Stein and basso, 1982) has shown that people depend on this prototypical structure to under- stand and interpret stories. Readers are sensitive to the order in which categories of information are presented. They have difficulty recalling stories when information is given in an order other than that specified in the idealized story schema, and most important as evidence that this story schema plays a key role in understanding- people tend to recall story information in the order predictecl by the schema even if the version of the story they read or heard uses a nonstandard order. Expository texts, too, follow certain stanciard rhetorical forms. Structures such as compare/contrast, cause/effect, or problem/solution provide frameworks that support and sustain communication between author and reacler. When an author uses

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LA UREN B. RESNICK 11 a familiar text structure, it serves as a kind of scaffolding for the reader's interpretive work. For example, structural markers like "on the other hand" and ~furthermore" are used to signal rhetorical functions. This broad analysis of comprehension as a meaning-imposing process that depends on the reader's knowledge of text structure as well as linguistic, topical, and inferential knowledge is common to all current cognitive theories of reading. Furthermore, when studies compare successful and less successful readers, the former always turn out both to possess more of these kinds of knowledge and to be more likely to use that knowledge spontaneously. Although there are important differences among theories with respect to specific aspects of these processes their timing, the kinds of cues that set them in motion, the ways in which knowledge is organized there are no clis- agreements regarding the general characterization of comprehension. Research still does not provide a clear answer about the extent to which meaning imposition proceeds strategically, in a deliberate, self-conscious fashion rather than automatically and unconsciously. Much evidence suggests that, for a skilled reader not totally new to the text's topic, most of the work to build a text representation proceeds quite unconsciously through processes of automatic acti- vation. The process slows down, requires deliberate attention, and becomes accessible to conscious awareness under special conditions: when there is an anomaly in the text or some unusual linguistic con- struction; when the topical domain is so unfamiliar that the reader lacks necessary prior knowledge for interpretation; when a particu- larly complicated chain of reasoning is presented; or when the reacler wants to study and remember the text rather than just understand] it (see chapters in Mand! et al., 1984, for a discussion of many of these issues). Some psychologists (e.g., Collins and Smith, 1982) believe that the same processes of self-questioning, summarizing, and the like go on in highly skilled reading as in more self-conscious reacting, but at a much faster rate. Other research (e.g., Neves anti Ar~derson, 1981; Newell and Rosenbloom, 1981) suggests that as readers develop automatic skills the nature of the process actually changes and cer- ta~n steps drop out. In any case, it is evident that educators ought to aim to produce both kinds of reading comprehension abilities among students: the ability to understand written texts automatically and with little effort, and the capacity to apply deliberate strategies for interpreting and remembering when the need arises. It is striking that the processes identified In cognitive research on

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12 EDUCATION AND LEARNING TO THINK reading comprehension are related to the techniques of textual exege- sis and analysis cornrnonly taught in high-level courses in literature, philosophy, and other disciplines in which multiple interpretations of texts are discussed as part of instruction. Cognitive theory, in other words, suggests that processes traditionally reserved for advanced students that is, for a minority who have developed skill and taste for interpretive mental work-might be taught to all readers, includ- ing young children and, perhaps especially, those who learn with difficulty. Cognitive research suggests that these processes are what we mean by reading comprehension. Not to teach them is to ignore the most important aspects of reading. This convergence of cogni- tive research on reading with traditional high literacy concerns offers some promise that the goal of extending high literacy standards to the mass educational system can be achieved. Meaning Construction in Mathematics A higher order interpretation of the basic mathematics curricu- lum is less straightforward than we have been able to propose for reading. Nevertheless, a close consideration of recent research on mathematical cognition suggests that in mathematics, as in reading, successful learners understand the task to be one of constructing meaning, of doing interpretive work rather than routine manipu- lations. In mathematics, the problem of imposing meaning takes a special form: making sense of formal symbols and rules that are often taught as if they were arbitrary conventions rather than expressions of fundamental regularities and relationships among quantities and physical entities. Recent research on mathematics learning points to an apparent paradox. We have abundant evidence that young children-even before attending school develop rather robust, although simple, mathematical concepts and that they are able to apply these con- cepts in a variety of practical situations. Yet school mathematics is decidedly difficult to learn for many children. Children's first and best-developed mathematical competence is counting (German and Gallistel, 1978~. Several investigations have shown that young chil- dren are able to use counting to solve informally a wide variety of arithmetic problems, including problems that they have difficulty solving in school (Carraher et al., 1985; Ginsburg, 1977). F`urther- more, an examination of shortcut procedures invented by children

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LAUREN B. RESNICK 13 suggests an implicit understanding of several basic arithmetic prin- ciples. For example, the min procedure (first clocumented by Groen and Parkman, 1972) is an addition strategy that involves setting a mental ~counter" at the larger of the two addends, regardless of whether it is the first or second, and then incrementing by the smaller. The child's use of such a procedure requires acknowledg- ment, at least implicitly, of the commutativity principle of addition. Several studies (e.g., Svenson and Hedenborg, 1979; Woods et al., 1975) have shown that children, starting at about age seven, solve subtraction problems by either counting down from the larger num- ber or counting up from the smaller number, whichever will require the fewest counts. This procedure reveals implicit knowledge of the complementarily of addition and subtraction, which in turn depends on thinking of the minuend (top number) as a whole, with a de- composition into the subtrahend and the difference. These examples and many others suggest that an intuitive understanding of many basic mathematical principles develops early and finds expression in various kinds of practical problem-solving tasks. There is substantial evidence that children's difficulty in learning school mathematics derives in large part from their failure to recog- nize and apply the relations between formal rules taught in school and their own independently developed mathematical intuitions. Part of the evidence lies in close analysis of the kinds of errors that children typically make in the course of learning arithmetic and, eventually, algebra. To an important degree, calculation errors derive not from random or careless ~slips" but from systematically applying incorrect procedures. These incorrect rules, of course, are not taught. Chil- dren invent them, as they do the shortcut strategies. By analyzing their incorrect rules we can understand what children are and are not attending to as they learn arithmetic. The most carefully stud- ied domain of arithmetic errors is subtraction. The kinds of errors (called "bugs" from their similarity to minicomputer programs with bugs in them) that children make have been carefully documented; these bugs serve as the basis for an artificial intelligence program (Brown and Van Lehn, 1980) that invents the same subtraction bugs children invent but does not invent the many other logically possible bugs not observed in children. Because the program's performance largely matches children's performance, its processes and knowledge base provide a theory of what children probably know and do that leads them to buggy inventions. According to the Brown and Van Lehn theory, children invent

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14 EDUCATIONANDLEARNINGTO THINK buggy procedures when they encounter problems for which they have no complete algorithm available. This may occur because they have not yet been taught what to do in special cases (for example, how does one borrow from a zero?) or because they have forgotten certain steps in procedures already taught. To respond, children engage in a form of problem solving: generating possible actions and testing them against a list of constraints. Although this is an intelligent problem- solving process, it produces errors because certain key constraints are missing from the test list. The missing constraints have to do with the meaning of the symbols; constraints regarding how the symbols ought to look on the page (e.g., only one digit per column, borrow marks in appropriate places) are largely obeyed. What is more, the program has no representation at all of the quantities that are involved; it only has rules for manipulating symbols. This suggests that children, like the program, solve arithmetic problerr~s by manipulating symbols while ignoring their meaning (Resnick, 1987~. We can reach the same conclusion from an analysis of the charac- teristic errors made by students learning decimal fractions (Hiebert and Wearne, 1985) and algebra (Matz, 1982; Resnick et al., 1987; Sleeman, 1983~. Research on algebra learning shows that when think- ing about transformation rules, students rarely refer either to quanti- tative relationships or to problem situations that could give meaning to algebra expressions. Not surprisingly, students are not very skillful at the process of ~mathematizing," that is, at constructing links be- tween formal algebraic expressions and the actual situations to which they refer (e.g., Clement, 1982~. All of this points to a conclusion that current mathematics education does not adequately engage students' interpretive and meaning-construction capacities. This conclusion is supported by data from national assessments (e.g., National Assess- ment of Educational Progress, 1983) showing declines in students' mathematics problem-solving skills even as calculation abilities rise. In short, most students learn mathematics as a routine skill; they do not develop higher order capacities for organizing and interpreting information. It seems likely that a less routinized approach to mathematics could produce substantial improvements in learning. Although the evidence is limited, it suggests that successful math learners engage in more metacognitive behaviors (e.g., checking their own under- standing of procedures, monitoring for consistency, trying to relate new material to prior knowledge) during math learning; they are also less likely to practice symbol manipulation rules without reference