Learning Mathematics From Classroom Instruction: On Relating Lessons to Pupils' Interpretations

Learning Mathematics from Classroom Instruction: On Relating Lessons to Pupils'Interpretations

Clea Fernandez; Makoto Yoshida; James W. Stigler

The Journal of the Learning Sciences, Vol. 2, No. 4. (1992), pp. 333-365.

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THE JOURNAL OF THE LEARNING SCIENCES, 2(4), 333-365 Copyright o 1992, Lawrence Erlbaum Associates, Inc.

Learning Mathematics From Classroom Instruction: On Relating Lessons to

Pupils' Interpretations

Clea Fernandez and Makoto Yoshida University of Chicago

James W . Stigler University of California, Los Angeles

How do students learn mathematics from classroom instruction? We propose a framework in which we assume that a student must form a coherent mental representation of the events that take place in a lesson and then use this representation to construct new knowledge. The process of representing the events of a lesson as a coherent whole is assumed to be affected by characteristics of the lesson (e.g., the clarity with which goals are expressed), as well as by characteristics of the student trying to learn from the lesson (e.g., background knowledge, lesson schemas). This framework is applied (a) by assessing both the nature of the mental representations students form of lessons and what they have learned from the instruction and relating the two, (b) by manipulating the way a lesson is taught and seeing how this affects how it is represented, and (c) by seeing how students who differ in various ways represent the same lesson. A description of four empirical studies is supplied by way of illustration.

Whether or not one thinks that mathematics is best learned in traditional classrooms, with a single teacher guiding the activities of a whole class of students, one thing is certain: Most students throughout the world are taught mathematics in such contexts. Given this fact, it is surprising how little we know about the actual processes by which students construct meaning from their classroom experiences. There has been a great deal of research in the past 25 years that describes characteristics of classroom

Requests for reprints should be sent to James W. Stigler, Department of Psychology, University of California, Los Angeles, 405 Hildegard, Los Angeles, CA 90024-1563.

334 FERNANDEZ, YOSHIDA, STIGLER

lessons, and some of this research seeks to relate these characteristics to students' gains in achievement test scores (Brophy & Good, 1986; Dunkin & Biddle, 1974; Good, Grouws, & Ebmeier, 1983). There also is a large and fast-growing literature in which tools of cognitive science have been applied to the study of the nature and development of individuals' mathematical knowledge, without particular reference to instruction (Carpenter, Moser, & Romberg, 1982; Ginsburg, 1983). Yet largely missing have been studies that directly relate teaching and learning, seeking to explain how students represent classroom events and then use their representation of these events to construct mathematical knowledge.

There are, of course, exceptions to this generalization. There have been recent calls to unite the "two sciences" of teaching and learning (Romberg & Carpenter, 1986), and some researchers are heeding these calls. In one approach, investigators carefully describe students' preexisting knowledge, and then document changes in knowledge that occur as learners interact with classroom instruction. This approach is exemplified by the articles in a volume edited by Carpenter and Peterson (1986), in particular the article by Leinhardt (1986). Another approach is evident in studies that examine the unintended meanings students mistakenly extract from classroom instruction (e.g., Gelman, in press; Lave, Smith, & Butler, 1988; Schoenfeld, 1988). These studies offer compelling evidence of what many have feared for a long time: Students construe the goals and purposes of school mathematics lessons quite differently than teachers intend.

Despite these promising attempts to study learning from classroom instruction, progress has been slow. In our view there have been two major obstacles. First, we lack theoretical tools for describing lessons so that they relate to students' comprehension processes. Second, we lack techniques for monitoring students' interpretations of specific lesson content as the lesson unfolds. Most studies thus far have been based primarily on in-depth case observations of a small number of students and teachers, and none has offered a technique that could be used to extend the research to large samples, or to experimental as well as correlational research designs.

This article represents the beginning of our efforts to address these problems. We are not arguing in favor of a particular kind of instruction; rather, we are trying to study what we see as the most common kind of instruction: A single teacher leading a room full of students in whole-class instruction. Briefly, we assume that in order for students to construct knowledge from classroom instruction they must first make sense, or construct a coherent mental representation, of the events of the lesson. Thus, those characteristics of a lesson that make it easier to represent coherently will be those that ultimately facilitate content learning. In the first part of this article, we develop our ideas about what these characteristics are. We also present some evidence from analogous work conducted

LEARNING MATHEMATICS 335

on the psychology of text comprehension that suggests that these characteristics can indeed affect the coherence of the representations formed. We also present some preliminary evidence to suggest that when lessons are represented more coherently, learning is facilitated.

We then briefly describe a framework for thinking about how students learn from classroom instruction, and we suggest methods for both describing lessons and monitoring students' interpretations of lessons. Finally, we discuss four empirical studies that illustrate our approach.

LESSONS THAT MAKE SENSE TO STUDENTS

How does the way a lesson is taught affect students' ability to form a coherent representation of it? It is useful to begin with the analogy of a well-told story. A good story is more than just a sequence of events (Stein & Glenn, 1982). The events must be organized and interconnected such that each is logically connected to the events that precede and follow it. A good story has a beginning, middle, and end, and a consistent theme or themes that run throughout. A good story is hierarchical, with plots and subplots that make sense in relation to each other. A good story engages the reader by (a) narrating the challenges that face a protagonist, (b) the actions taken by the protagonist to resolve the challenges, and (c) the new challenges that arise as a result of these actions. A good story is easy to understand and remember because each event has meaning in relation to other events and to the protagonists' goals. In other words, a good story is told in such a way that it can easily be represented coherently by the comprehender.

We propose that lessons, like stories, will be more easily comprehended if they have certain characteristics. Just as the protagonist's goals serve to tie together the events of a story, so can the teacher's goals be used to integrate a sequence of instructional activities into a coherent whole. A lesson with multiple goals will be seen as coherent to the extent that the goals can be interrelated. Furthermore the activities and actions planned by the teacher will make sense to the extent that they are related to the goals of the lesson. A lesson with clear goals that motivate and interrelate the events of the lesson is a well-structured lesson that is likely to be represented coherently.

However, being well-structured, does not insure that students will infer relationships among events. Sometimes a teacher must explicitly point out the relationship between two activities or events. And there are often threats to students' successfully forming a coherent representation of the events taking place in a lesson. In the following sections, we discuss some of the strategies that teachers use, which affect the degree to which their lessons afford coherent representations by learners, as well as some of the threats to this process that are encountered in the classroom.


Effects of Lesson Structure

In order for a lesson to be readily perceived as coherent it must be structured to afford such a perception. But many teachers do not share our assumption that lessons that afford a coherent representation are more effective than those that do not. Stigler and Perry (1988) described a first-grade lesson in Chicago that consisted of three distinct segments: (a) 10 min spent on measurement, (b) 10 min spent on single-digit addition, and (c) 10 min spent on telling time. We often see such lessons, especially in the younger grades. The assumption appears to be that variety is what holds students' attention. Variety is important, but we wonder how students would interpret such a lesson. Would they be confused by the juxtaposition of topics? Would they come to expect that there should be no relationship between the different parts of a lesson? The challenge for teachers is to find techniques that can be used to organize a variety of experiences so that they can be perceived as forming a coherent whole.

One of the primary techniques teachers can use to structure a mathematics lesson so that it affords a coherent representation is to pose a problem at the beginning of the lesson and then relate each subsequent event of the lesson to the initial problem. For example, a teacher planning a lesson on rounding numbers to the nearest 10 might start by posing a problem:

Suppose you go shopping. You only have 5 dollars and you don't have paper and pencil or a calculator, so you need t o keep track in your head of how much you are spending so that you will have enough when you get to the cash register. What do you do? (from Baranes, 1990)

Trying to solve this problem functions as a higherorder goal that can be used by the teacher to structure the whole lesson. Students might participate in an activity designed to demonstrate that rounded numbers (multiples of 10) are easier to use in mental calculation than are nonrounded numbers. They might practice counting by 10s, and then learn specific rules for rounding to the nearest 10. Each activity can be related, ultimately, to the solution of the problem posed at the beginning of the class. If the teacher highlights this relationship, the lesson will be perceived as highly coherent.

Starting with a problem is not the only way to structure a lesson so that it can be easily perceived as coherent, but it does provide a clear way for the teacher to connect the events that compose a lesson. Continuity of topic, in itself, is not enough. For example, in one third-grade geometry lesson we observed, the teacher began by stating that she wanted to pursue three objectives: (a) to review and describe some geometric shapes, (b) to categorize shapes, and (c) to pick out shapes from complex patterns. These


objectives are clearly relatable, but it is up to the teacher to show their relation as the lesson unfolds. In this case the teacher did not relate them, and so the lesson appeared as three unrelated activities, one after the other. Cues that might have helped students to infer continuities across seg-ments-utilizing the same materials or using the products of one activity as a basis for the next one-were not used in this case, and the result was a lesson that did not appear coherent.

The Role of Language

This last example leads us to another of the tools a teacher can use to facilitate students' ability to perceive a lesson as coherent, and that is language. The degree to which even a well-structured lesson is perceived as coherent can be enhanced if the teacher explicitly talks about the goals of the lesson and the lesson activities' relations to the goals.

One way teachers can use language to facilitate students' perception of instruction as coherent is to explicitly state the objectives at the beginning of a lesson. When teachers begin a lesson by directly performing actions intended to lead to the fulfillment of an objective without first explaining their objective, students' ability to perceive the lesson as coherent may be reduced. If children do not know why certain tasks are being carried out, the intended links between them may not be easily inferred. And because objectives often provide the logical link from one activity to the next, not stating objectives may prevent children from seeing that what they did at one point in the lesson is important for understanding what they are doing at a later point during the lesson.

Stigler and Perry (1988) described an American first-grade lesson on measurement. The lesson began with the teacher having children compare the lengths of different objects around them, stating which was longer and which shorter. Here is the transcript of what followed:

O.K., open your workbook to page 12. I want you to measure your desk in pencils, find out how many pencils it takes to go across your desk, and write the answer on the line in your workbooks. [Children carry out instructions.] O.K., the next line says to use green crayons, but we don't have green crayons so we are going to use blue crayons. Raise your hand if you don't have a blue crayon. [Teacher takes approximately 10 minutes to pass out blue crayons to students who raise their hands . . . I . Now write the number of blue crayons next to the line that says green crayon . . . O.K., now take out your centimeter ruler and measure the number of centimeters across your desk and write the number on the line on your workbooks. (p. 50)

This lesson is highly structured. However, it is doubtful that students perceived the structure because the teacher failed to discuss the connections

between the three activities that constituted the lesson. Even an advanced first-grade child would have difficulty, on his or her own, understanding the logic behind the sequence of events.

The structure of this lesson is interesting because it recapitulates, in a short space, a version of the historical events that led to the development of measurement techniques. First, objects are compared directly to determine which is longer. Then, the idea of a unit of measurement is introduced to solve a problem encountered when direct comparison is not possible. Finally, the idea of standard units is introduced as a way to overcome inconsistencies that arise when there is a need to compare objects separated by larger distances in time and space. Each new technology solves a problem generated by the prior technology.

It would have been possible for the teacher of this lesson to motivate each new activity by relating it to the previous activity. For example, after comparing the lengths of different objects in the classroom the teacher could have posed the following problem: "What if you wanted to compare the lengths of two objects that are not in the same place, say your desk here in school and your desk at home?" This problem would have been a natural way of introducing the need for a unit of measure, thus giving meaning to the activity of measuring multiple objects with the same pencil or crayon. Next, the disadvantage of nonstandard units of measures could have been illustrated by introducing the goal of wanting pairs of children to compare the lengths they found for their desks when using different crayons. This would have been a natural way of introducing the need for a standard unit of measure, thus giving meaning to the activity of measuring with rulers. Instead, there was no pause between activities, no attempt to explain the connections. From the students' point of view it may as well have been measurement followed by addition and telling time: Three separate activities not intended to make sense in relation to each other.

Teachers add meaning to a lesson when they skillfully link events that students cannot link on their own. Although a teacher might have explicitly stated an objective for an activity, it is also important to make sure that students understand what they have actually learned from the events taking place and how this relates to the objective they set out to achieve. We are reminded, again, of the lesson on shapes where children were told that they were going to categorize shapes in order to learn what characteristics different shapes have in common. Children took turns going to the board to place cut-out shapes in the categories parallelogram, polygon, and quadrilateral. Yet they never stopped to discuss what this activity was teaching them about similarities and differences among shapes. The extra step of reflecting on a lesson's activities and discussing their meaning is too often omitted in classroom instruction (see Lampert, 1988, for a discussion of the importance of this kind of talk in classrooms).


An advantage of using language to facilitate the coherence of students' representations is its flexibility. The structure of a lesson may be planned ahead of time, but students' responses (e.g., the spontaneous insights they offer in class discussion) and the direction the lesson actually takes when it is being implemented may deviate considerably from what was planned. In such cases, the skillful teacher can use language to draw together the unexpected events that arise so that they may be related, if possible, to the goals and flow of the lesson.

When unexpected events are not integrated into the flow of the lesson, the result is a lesson which is harder to interpret coherently, and perhaps a missed opportunity to exploit a student's novel insight. We observed one second-grade lesson in which students were being taught subtraction with renaming. Several times the teacher gave students examples which required borrowing. Each example followed the same script. The teacher would ask a child, for example, in the problem 23 - 17, "Can we subtract 7 units from 3 units?" The child would promptly respond "No." The teacher would then ask the whole class what should be done, to which the children, in unison, would say "Go borrow from our neighbor," referring to the 10s column. In one of these interchanges the teacher posed the problem 34 - 19, and asked a student if 9 units could be subtracted from 4 units. This time the script was violated: The student answered "Yes, it is minus 5." The teacher was clearly startled by this response. After an uncomfortably long silence, the teacher asked her question again to another child, and this time got the answer she was looking for. Because the teacher ignored the interesting response supplied by the first student, there was brief confusion and the flow of the lesson was interrupted. A more skillful teacher-and one with some flexibility in her own mathematical knowledge-might have used language to integrate this student's response into the lesson.

Threats to Coherent Representations

Structuring lessons and using language to make the structure explicit are two tools teachers can use so that their lessons easily afford a coherent representation. We have also observed lesson characteristics that, we assume, would impede students' ability to construct a coherent representation of the lesson. Chief among these characteristics are interruptions, which occur surprisingly frequently in American classrooms. Interruptions breach the flow of the lesson and challenge the teacher to maintain the unity of the lesson by clearly linking what occurred before and after the interruption so as to minimize its disruptive effect.

In one third-grade classroom we observed, a woman from the cafeteria entered each day during the math lesson to collect money for hot lunches. The lesson was interrupted for several minutes while children lined up to

give her their money. When she left, the teacher typically continued the lesson as if nothing had happened, but it was clear that the flow of the lesson had been disrupted and that some of the students had lost the thread of the lesson during this unnatural break. It might have facilitated students' ability to form a coherent representation of the lesson if the teacher had reminded the children of the objective they were pursuing before the interruption, how far they had gotten, and what they had left to do, rather than simply picking up where she had left off.

Students' ability to see a lesson as coherent may also be impeded if a lesson proceeds too quickly. Inferring the connections among events in a lesson takes time, and for some students it may take more time than for others. Even material that is structured to afford a coherent representation, if presented too quickly, may be difficult for students to learn.

LESSON CHARACTERISTICS, STUDENTS' INTERPRETATIONS, AND LEARNING

We have discussed characteristics of lessons that we assume affect students' ability to make sense of lesson events. But what evidence do we have to suggest that lessons that have these characteristics are in fact more likely to be represented coherently? We have very little direct evidence, but we do have some indirect evidence from analogous work on students' comprehension of textual materials. In addition, what evidence is there that coherent representations of lessons lead to learning? We address these questions in this section.

Text Comprehension: An Analogy

We have already suggested that it might be useful to think of lessons as analogous to stories: Lessons constitute a kind of structured "talk" just as do stories. In fact, the usefulness of the analogy has also been argued for by other researchers (W. Kintsch & Bates, 1977). Although very little research has focused on how lessons are represented, a great deal of research has focused on how stories and other texts are processed. To the extent that lessons and texts are comprehended in similar ways, it is useful to review what we know about how characteristics of texts affect comprehenders' ability to make sense of them and relate this to our discussion of the process of learning from classroom instruction. Aside from providing indirect evidence for the claim that certain lesson characteristics might facilitate students' ability to make sense of instruction, these studies can also serve as a source of possible techniques for studying students' representations of lessons.

Gernsbacher (1990) proposed a general theory of comprehension in which coherence figures prominently. Gernsbacher proposed that comprehension is a process of "structure building" in which the comprehender strives to construct a coherent mental representation of a text by linking within a mental structure incoming information to information previously heard. When new information that does not cohere cannot be mapped onto the existing structure, a new mental structure must be formed into which this information can be incorporated.

Gernsbacher (1990) cited a number of studies that support her general model. For example, the speed with which information is processed during comprehension of stories, as measured by reading times, depends on how well the new information coheres with previously processed information. The fact that processing slows down when less coherent information is encountered is consistent with the idea that a new mental structure must be formed in order to take in the information. Gernsbacher (1990) also cited evidence that when the coherence of stories is rendered very low by jumbling up the order in which events are narrated, comprehenders lose access more quickly to recently comprehended information. If new information can be mapped into an existing structure, then old information remains activated as well as new information. But if a new structure must be formed, as would be the case in jumbled stories, previously comprehended information is displaced by the information in the new structure.

Numerous other studies relate characteristics of a text to its representation. Research has shown, for example, that stories that maintain a consistent referent across sentences are better recalled than those in which the referent shifts (W. Kintsch, Kozminsky, Streby, McKoon, & Keenan, 1975). Similarly, it has been documented that subjects' recall of stories increases when appropriate contextual information, which makes it easier to see the relationship between different story events, is provided at the beginning of a passage (Bransford & Johnson, 1972). There is also evidence that narratives with a consistent point of view are read faster, and remembered better, than narratives in which the point of view changes (Black, Turner, & Bower, 1979). In general, poor recall and slow reading times are presumed to result from a difficulty on the reader's part to form a coherent representation of the story events. Quality of recall has also been used to study the effects of story characteristics on comprehension. For example, some researchers have looked at how jumbling up a story affects the way it is recalled (Mandler & Deforest, 1979; Poulsen, E. Kintsch, W. Kintsch, & Premack, 1979). Ability to make inferences is another measure used to study how texts are represented. Research has shown that having stories that provide goal information increases subjects' ability to respond to inference-probing questions about the story (Omanson, Warren, & Trabasso, 1978). Trabasso (1986) and Trabasso and Sperry (1985) also used


subjects' importance ratings of story statements as a way to infer characteristics of the representations subjects formed of stories. This technique has allowed them to study how characteristics of a story's structure (e.g., sequential vs. hierarchical) affects how it is comprehended.

Representations of Lessons and Content Learning

We have claimed that coherent representations of lessons should lead to greater learning. But what evidence do we have for the validity of this claim? Again our evidence is indirect and comes from cross-cultural studies of mathematics teaching and learning. Many of our ideas about which lesson characteristics more easily afford a coherent representation by learners emerged out of comparisons between American mathematics lessons and lessons taught in Japan, Taiwan, and China (Stevenson & Stigler, 1992; Stigler, Lee, & Stevenson, 1987; Stigler & Perry, 1988). Japanese and Chinese lessons seemed easier for us to represent coherently than lessons we observed in the United States, and many of the lesson characteristics that we described earlier do, in fact, differentiate Asian from American lessons. At the same time, Japanese and Chinese elementary school students have scored far higher than their American counterparts on tests of mathematics achievement (Stevenson & Stigler, 1992). Although we cannot prove that the higher achievement of Asian children results from their lessons having certain characteristics, the correlation certainly merits further investigation.

A FRAMEWORK FOR THINKING ABOUT LEARNING FROM CLASSROOM INSTRUCTION

The previous discussion leads us to propose a framework for thinking about how students learn from classroom instruction (see Figure 1). We assume that when a student experiences a lesson, he or she will form a mental representation of the events that occur in it. Learning from instruction entails extracting content knowledge from this mental representation of lesson events and integrating this information into the student's existing knowledge base. Therefore, a student's ability to learn from a lesson should depend, in part, on the nature of the mental representation he or she is able to form of the lesson. The nature of the mental representation a student forms of a lesson should be affected by characteristics of the lesson as well as by characteristics of the student interpreting the lesson.

Key among these lesson characteristics will be those that affect the degree to which the lesson affords a coherent representation. As we have already argued, these characteristics include many factors related to the way the


STUDENT EXPECTATIONS

LESSON EVENT CONTENT INPUT REPRESENTATION LEARNING

STUDENT /CONTENT KNOWLEDGE

FIGURE 1 A framework for studying how students learn from classroom instruction.

lesson is structured and the types of explanations given or omitted, as well as other particularities of the instructional session, such as its pace. Also, students who show evidence of forming more coherent representations of a lesson may also show evidence of greater learning of the content being taught.

Although we have not mentioned the role of individual differences in our discussion of students' comprehension of lessons, this is clearly an issue that must be addressed in a model of lesson comprehension. We have not meant to imply that lessons that afford a coherent representation do so for all learners. As Gelman (in press) pointed out: "The mind constructs representations on the basis of what it brings to the learning setting as much as what it is offered." Neither have we meant to say that lessons that are difficult to represent coherently cannot be coherently represented by some learners. Such a lesson might be made sense of by a clever student.

One student characteristic that may have an effect on the coherence of the mental representation that may be formed of a lesson is the student's background knowledge about the topic being taught. It is unlikely that low-knowledge students will form coherent representations of lessons because they will not interrelate events as the teacher intended; rather, the lesson is likely to be represented as a series of disjointed events or even misconstrued. Furthermore, the difference between higher and lower knowledge students may be greater when they are faced with lessons that do not lend themselves easily to coherent representations. In this case it should be particularly difficult for low-knowledge students to form coherent representations because they are given very little support from the lessons themselves.


The expectations students bring to bear on lessons can also have an effect on the coherence of the mental representations they form of lessons. There is quite a bit of evidence to suggest that children form expectations very quickly about everyday events in which they participate (Nelson, 1986), and events that make up the school day have been no exception (Stodolsky & Salk, 1991). It is quite possible that Japanese students expect lessons to afford coherent representations and therefore look for coherence in instruction. It is also possible that American students have come to expect lessons that do not have much internal logic and therefore are not as strongly driven by an attempt to understand lessons as series of related events. According to the framework being proposed here, such different expectations should lead to different representations of the same lesson, and should ultimately lead to differences in what is learned from the lesson.

PRELIMINARY EMPIRICAL FINDINGS

As we pointed out previously, two major obstacles to the study of learning from instruction have been the lack of theoretical tools for describing lessons and the lack of techniques for monitoring students' interpretations of lesson content. The framework just described points to ways in which both of these obstacles can be overcome. The theoretical underpinnings of this framework suggest that it may be fruitful to describe lessons in terms of the logical relationships that can be inferred between different lesson events and the goals of the lesson. The framework also suggests that the mental representations students form of lessons could be assessed for their coherence.

In our lab we have been experimenting with several possible research designs for applying our framework to the study of students' learning from instruction. We also have been experimenting with various techniques for assessing students' mental representations of lessons. So far we have carried out two types of studies. In one design, we videotape a mathematics lesson and show it to children who differ either in their expectations about lessons in general (e.g., Japanese vs. American children) or in their mathematical knowledge. By analyzing different students' responses to the same lesson, we can study the role that knowledge and lesson schemas play in forming mental representations of the lesson. In a second design, we have experimentally manipulated certain lesson characteristics, such as the explicit statement of goals, and studied the effects of the manipulations on lesson representation. To implement these studies, we edit a videotape of a lesson to create versions that vary in terms of their coherence. We then show these tapes to subjects, and measure both their representation of the lesson and what they learn.


Thus far we have tried several techniques for measuring the mental representations students form of lessons: recall of the events that comprise a lesson, ratings of how important different events are for understanding a lesson, and recognition memory for statements that either were or were not made in the course of a videotaped lesson.

In this section, we describe our first efforts to carry out research based on our framework. It is important to emphasize that the purpose of describing these studies is to illustrate our approach, not to present conclusive findings. Although we number the studies from one to four, they do not form a sequence or add up to a complete story; in fact they were conducted simultaneously. However, there are points of contact between the four studies, and where appropriate we discuss how one study addresses weaknesses of, or builds on questions raised in another study. Our main goal is to report our preliminary efforts to apply our theoretical framework in empirical research.

I. Students' Recall of a Math Lesson

In this study we compared the mental representations that high- and low-knowledge students formed of a lesson. We asked whether high- knowledge students formed a more coherent representation than their lower knowledge counterparts. In order to do this, we compared high- and low-knowledge students' recall of a videotaped lesson to see the degree to which their patterns of recall reflected awareness of the content relationships that the teacher was trying to convey.

We videotaped a 45-min third-grade lesson on geometric shapes. In this lesson the teacher and students worked on three different activities. First, they reviewed the names and descriptions of 12 basic shapes (square, rectangle, triangle, etc.). Next, they categorized shapes in order to see what characteristics they have in common. Finally, they looked for shapes embedded within intricate patterns.

We edited the tape of the lesson to create a 15-min stimulus lesson. This shortened version consisted of fewer examples than the original lesson, but the overall structure of the lesson remained intact. Care was taken not to remove any instructions supplied by the teacher because these were considered important for understanding the lesson.

Analyzing the Lesson

In order to assess the coherence of the mental representations students formed of the lesson, we had to first establish what a coherent representation of this lesson would look like. A lesson is represented coherently to the extent that the learner is able to relate the events of the lesson to each other


and to the goals of the lesson. Thus, to know whether someone's representation of a particular lesson is coherent, we need to know the possible relationships that can be inferred among its events. Our first step, therefore, was to devise a system for coding the relatability of the events in the lesson. For each event we were able to code how many other events it was relatable to and then separate the events into three categories: those that should be perceived as poorly embedded within the lesson (one to two links with other events), intermediately embedded (three to six links), or well-embedded (seven or more links).

Before describing how we did this, we want to briefly mention three general points about our analysis. First, we coded events as related if they were relatable, regardless of whether or not they were explicitly related in the lesson. Sometimes the relations were highly salient and would probably be seen by most learners; other times the relations were quite difficult to infer. By coding as related all events that could be related (even if only by the most sophisticated learners), we in effect constructed the most coherent representation that could possibly be formed of the lesson.

Second, whether or not two events are relatable often depends on the teacher's goal. For example, if the teacher has a discussion on how to find the perimeter of a rectangle and then says "Now let's find the area," the events to follow are not relatable to the events that just took place if the teacher's goal is simply to teach the formulas for finding perimeter and area of a rectangle. On the other hand, if the teacher's goal is to let children discover the relationship between the area and the perimeter of a rectangle, then the two parts of the lesson are relatable and make sense in relation to each other. Given the importance of goals for relating events, whenever the teacher did not explicitly state a goal, we had to infer one.

Third, not all relations between events must be represented for learning to occur. For example, some events are not relevant to the content of the lesson. Events such as a student asking to go the bathroom, or a teacher telling a student to go to the office because the student's dad has brought his or her lunch, are examples of these types of events. We assume that relationships between events that are not relevant to the content being taught are of no use to the student trying to learn mathematics from the lesson. Therefore, we decided not to include these types of events and relationships in our analysis. Similarly, not all the relationships that exist between relevant events seem important for learning. For example, two events, A and B, that occur one after the other in a lesson are related by virtue of their temporal contiguity. However, if temporal contiguity is their only relationship, representing the link should not be necessary for learning. So, for instance, suppose the teacher said "Tell me the perimeter of this rectangle," pointing to a rectangle on the board (Event A), and then, after a student answered (Event B), the teacher said, "Now let's review the


homework on fractions you did for today" (Event C). One would not expect a student who is trying to make sense of this lesson to relate events B and C, because the only relationship between these two events is that they are sequential.

To code relations, we first watched the lesson and divided it into events. For example, each time the teacher asked a question, made a point, posed a problem, instructed the students to do something, or stated a goal, we considered this an event. Similarly, every time a student responded to a question or carried out instructions, we also coded as an event. Many of the events that occurred in the lesson were verbal, but some were physical. An example of a physical event is a teacher pointing to a student as a way of asking for the student's response or a student going to the board to draw a shape. Events not relevant to the mathematical content were excluded from further analyses.

Next, we divided the relevant lesson events into categories. Four basic types of events were discernable in this lesson:

1. Goals-Events in which the teacher communicates to the students the goals for parts of the lesson, or the lesson as a whole. The teacher saying "Today we are going to describe shapes," is an example of a goal.

2. Settings-Events in which the teacher prepares what is needed to pursue the goals. For example, projecting a shape on a screen or passing out a sheet with problems are examples of setting events.

3 . Actions-Events in which the teacher elicits responses or reactions from the students. A teacher asking for the name of a shape or telling a student to come up to the board to place a shape in the parallelogram category are examples of actions.

4. Results-The outcomes of the elicitations that occur during actions. A teacher saying the name of the shape because no one answered, or a student putting a square in the parallelogram category are examples of results.

Next, we determined how the events in the lesson were related to each other. The kinds of relations we coded were determined by their perceived usefulness to a hypothetical learner. These are the eight types of relations we coded in this lesson:

1. Relations among goals. Two goals are relatable when one can be seen as a subgoal of the other. An example of two relatable goals is the teacher telling the students that she wants to compare two shapes (goal) and then telling them that she wants to first describe each of the two shapes (subgoal).

2. Relations between goals and settings. A goal and a setting are relatable when the setting is created so that the goal can be achieved. An example of

a relatable goal and setting is the teacher saying the students are going to describe shapes (goal) and then drawing several shapes on the board (setting).

3 . Relations between goals and actions. A goal and an action are relatable if the action is carried out as a step toward achieving the goal. A teacher whose goal is to compare shapes might ask what is similar between a square and a rectangle (action). In this case the action and goal are seen as related.

4. Relations between results and goals. A result and a goal are relatable if the result can be perceived as the motivation for the teacher trying to achieve a certain goal. For example, if a student answers a question incorrectly, the teacher may decide, because of this failed result, to pursue the goal of reviewing certain material. Similarly, if students come up with the formula for solving the area of a rectangle, the teacher might try, because of this successful result, to get them to come up with the formula for the area of a triangle.

5. Relations between actions and results. An action is relatable to a result if the action can be perceived as having motivated the result or vice versa. For example, if a teacher asks a question (action), the answer that follows is usually relatable to her question (result). Similarly, if a teacher tells a student to do something (action), usually what the student does is relatable to the teacher's request (result). If a student answers a question or writes a solution on the board (result), the teacher might then ask for a clarification or explanation of what the child did (action).

6 . Relations between actions and settings. An action and a setting are relatable if the setting is perceived as necessary for the action to take place. For example, if a teacher divides the board into areas labeled parallelogram, quadrilateral, and so forth, and then asks where the square should be placed on the board, the question makes sense. If the teacher had not divided the board into areas, the question could not be interpreted.

7. Relations between settings and results. Settings and results are relatable when, in order for a result to be made sense of, a setting must be referred to or used. In the previous example, in order to make sense of a student's answer to the question about where to place the shape on the board, one must take into account the way the board was divided.

8. Relations among results. Two results are relatable when one can be seen as incorporating the other. For example, if a teacher asked a student to name two shapes that have the same number of sides and the student named a square and a rectangle, this result is an integration of two previous results: one student saying that a square has four equal sides and another student saying that a rectangle has four equal sides.

Although it might seem that perceiving some of these relations would be trivially easy, in many instances it is not, which is why making sense of


lessons can be difficult. For example, take the task of determining which sets of actions and results are relatable. As we have already argued, an interruption between an action and its result makes it hard for students to connect the two. Similarly, some lesson events are difficult to connect without the necessary background knowledge. If a teacher posed the problem 35 - 19 (Event A) and then asked how much is 5 - 9 (Event B), the question posed in event B is related to event A. Yet, if you did not know that 5 - 9 is a reasonable step in the solution of the larger problem, the connection between Events A and B might be hard to make.

Method

The subjects were 16 third-grade students in the classroom where we videotaped the lesson. Based on the teacher's judgments of the children's mathematical competence, we divided the students into two groups, one low knowledge (n = 9) and one high knowledge (n = 7). The classroom was in a public school located in an integrated, middle-income neighborhood in Chicago. All subjects had been present the day the lesson was taped.

Two weeks after the lesson, the experimenter returned to the classroom with the stimulus tape. After explaining that the tape had been edited, the children were shown the tape. Next, each child was called to another room and instructed: "Tell me everything you remember that happened in the lesson you just saw." All recall data was collected on the same day.

Each child's recall was transcribed and parsed into individual lesson-event descriptions. We were interested both in how children described lesson events and in which lesson events they chose to describe.

When we looked at how children described events, we found that the length and detail of the event descriptions varied greatly. For example, one child would describe an event as "She asked us a question," another as "She asked us something about a shape that she projected," and yet another as "She projected a bright red shape and asked us how many equal sides it had." Therefore we first coded the children's recall in terms of the level of detail employed in their descriptions of events, separating children's descriptions into generic descriptions and lesson-specific descriptions. De- scriptions of events were coded as generic if they were so general that they could have applied to many lessons. An example of a description coded as generic would be recalling that the teacher asked a question, or that she had children go to the board to do work, without specifying what the question or the work done at the board was. Descriptions were coded as lesson- specific if they related to the content of this particular lesson.

Next, we divided lesson-specific descriptions into those that were underspecified and those that were fully specified. Descriptions were coded as underspecified if they omitted information that would be important for understanding the relationship between the event being described and

another lesson event. For example, in recalling the lesson, a student might say that the teacher said a shape had parallel sides but not say that the shape was a rhombus. Knowing that the shape in question was a rhombus was necessary to understand why, during another part of the lesson, a rhombus was identified as a parallelogram. Therefore, this student's description would be considered underspecified. Fully specified descriptions did not need to incorporate an explicit statement of how events were related (and almost never did), but they did need to incorporate enough information to make it possible to infer the relationship. We devised this coding because we thought that describing lessons in terms of fully specified descriptions would be indicative of a greater understanding of how lesson events were interrelated than would be supplying underspecified descriptions.

Finally, we coded the children's recall protocols to determine which specific lesson events were recalled. Only fully specified lesson descriptions were coded since these were the only ones that contained enough detail to be unambiguously matched with a specific lesson event. We devised this coding to see whether children would be better able to recall events that were well-embedded in a coherent representation of the lesson than those that were poorly embedded. Such a tendency would indicate having understood the relationships being conveyed in the lesson because these same relationships determined embeddedness.

Results

Both high- and low-knowledge students recalled an average of 10% of the events of the lesson. About one fourth of the descriptions were generic, and three fourths were lesson-specific; this breakdown did not vary by level of mathematical knowledge. A two-way repeated-measures analysis of vari- ance (ANOVA) with low versus high knowledge as a between-subjects variable and level of detail (generic vs. lesson-specific) as a within-subjects variable revealed no main effect of knowledge, and no significant interaction of knowledge and level of detail. There was a significant main effect of level of detail, F(l, 14) = 13.50,p < .005.

Next, we examined the division within lesson-specific descriptions between those that were fully specified and those that were underspecified, and we found an interesting difference between the high- and low-knowledge groups. As shown in Figure 2, the vast majority of high- knowledge students' descriptions were fully specified, whereas only about half of the low-knowledge students' descriptions were fully specified. A repeated-measures ANOVA revealed a significant interaction between knowledge and level of specification, F(l, 14) = 8.089, p < .05.

In our next analysis, we looked at which specific events were recalled by high- and low-knowledge students, particularly the frequency with which

High Achievers Low Achievers FIGURE 2 Percent of events recalled by high and low achievers that were under-specified and fully specified.

they recalled poorly, intermediately, or well-embedded lesson events. As shown in Figure 3, high-knowledge students' recall was more sensitive to embeddedness of events than was low-knowledge students' recall. A repeated-measures ANOVA with low versus high knowledge as a between-subjects variable and degree of embeddedness as a within-subjects variable

. -

60 -

u 50 - Low Achievers a,--

4 0 -a, [r

30 -2 0 -

10 -0 . I I I

Low Medium High FIGURE 3 Percentage of fully specified events recalled as a function of ern-beddedness for high and low achievers.

2

revealed a significant Knowledge Level x Degree of Embeddedness interaction, F(2, 28) = 4.774, p < .05. Simple effects tests revealed that level of embeddedness had a highly significant effect on high-knowledge students' recall, F(2, 28) = 19.724, p < .001, but only a borderline effect on low-knowledge students' recall, F(2, 28) = 3.179, p < .06.

High-knowledge students, when compared to low-knowledge students, often included in their descriptions of lesson events the information needed to link the events to other events that occurred in the lesson. This, together with the fact that high-knowledge students were more likely to recall lesson events that were well-embedded in a coherent representation of the lesson than those that were poorly embedded, provides preliminary evidence that high-knowledge students form more coherent representations of lessons than do low-knowledge classmates.

II. Japanese and American Students' Ratings of Events in a Mathematics Lesson

In the first study, we encountered several problems. The first related to using recall as a dependent measure. Children's recall of the lesson was very limited, as evidenced by the fact that they only recalled an average of 6.75 out of 65 total events, and that about one fourth of these events were described in such general terms that they could have applied to most mathematics lessons. Second, we felt there were disadvantages to using an edited (shortened) version of the lesson, even though having students watch the entire lesson would have made the procedure very long. Editing the lesson involves making a number of decisions about what to include and what not to include, and we felt it would be better to ask teachers to make these decisions.

In this next study we avoided these limitations: Children's mental representations were assessed by having them rate the importance of different lesson events for understanding the lesson. And, to avoid editing, we asked a teacher to design a 15-min lesson and teach it to the students.

Unlike the first study in which we compared high- and low-knowledge students' mental representations of a lesson, in this study we compared the mental representations that American and Japanese students formed of a lesson. In particular we wanted to see whether Japanese students, as compared to their American counterparts, formed more coherent representations, and whether these representations led to greater learning as measured by scores on pre- and posttests.

Method

The 15-min lesson we used in this study was taught and videotaped at the beginning of the school year in a fifth-grade classroom in Japan. The lesson


was designed to teach children how to solve problems such as: "There are 38 children in a class. There are 6 more boys than girls. How many boys and how many girls are in the class?"

A videotape of the lesson was used as the stimulus tape for the Japanese group of subjects. In order to create the stimulus tape for the American subjects, the audio track was dubbed over in English, and the parts in the original tape where the teacher wrote on the blackboard in Japanese were replaced by reenactions in which an actor who resembled the original teacher wrote in English. None of the 44 children or the handful of adults who viewed the lesson reported noticing the presence of this surrogate teacher. In these reenactions, care was taken to match the actions and the pacing of the real teacher as closely as possible to make the two tapes comparable in every aspect, including length.

We acknowledge that there are certain differences in the two tapes, such as subtle differences in the teacher's tone of voice, yet we feel that the essence of the Japanese lesson was captured in the American version.

As in the first study, we broke the lesson into goals, settings, actions, and results and determined how they would be related in a coherent representation of the lesson. We used this analysis as the basis for dividing events into well-embedded and poorly embedded events.

Fourth-grade students from Japan and the United States participated in the study. The American subjects (N = 24) were from two classrooms at a private school in Chicago. The Japanese group (N = 17) were from two classrooms at a private Japanese school in the United states.' The latter institution is a full-time school that follows the curriculum and calendar set by the Ministry of Education of Japan for elementary schools. All classes are taught entirely in Japanese by teachers who are certified in Japan by the Ministry of Education.

Procedure. Because we wanted to study subjects who had not been previously taught to solve these types of problems, the experiment was conducted toward the end of the fourth-grade school year. (This topic is taught at the beginning of fifth grade in Japan and much later in the United States.) Children were tested in groups as part of their class. The testing session consisted of the following four activities:

1 . The children were asked to try to solve a problem similar to those taught in the lesson. This served as a pretest measure.

'All subjects who were at cei!ing on the pretest were eliminated from the sample and are not included in these numbers. This was necessary because we also wanted to assess how much students learned from the videotaped lesson. Only two American subjects were eliminated for this reason, but nearly half the Japanese subjects were so eliminated, even though they had not been taught previously to solve this type of problem.


2. They were told that they were going to watch a tape in which the teacher would instruct them on how to solve problems such as the one they had just been given. They were asked to pretend that they were actually participating in the lesson, and to do as the teacher in the tape instructed them. To help them get involved in the lesson, materials handed out in the tape were also handed out by the experimenter. Most children became rather involved in the lesson. Quite a few even raised their hands emphat- ically, asking the teacher on the TV monitor to call on them.

3. They were given two more problems to solve as a posttest measure. 4. Subjects were presented 12 videotaped events in the same order in

which they occurred in the lesson. Events were presented one at a time, and separated from each other by blank tape. After viewing each event, subjects were asked to rate, on a scale ranging from unimportant (1) to very important ( 3 , how important they believed the event was for "understanding what the teacher in the videotape was trying to teach you."

How target events were selected. The target events consisted of six results and six actions. Half of the results and half of the actions selected were well embedded in the lesson's web of content relationships and half were poorly embedded. Care was taken to select events that were spread out across the lesson so as to avoid confounding event type and position in the lesson. Similarly, events were selected to avoid confounding event type with event length. In addition, only events closely equivalent in length across the Japanese and English versions were selected. We focused on actions and results because we noticed, in our observations of lessons, that most of the content of the lesson was conveyed in the results rather than in the actions. Imagine, for example, that a student attended to a teacher's questions but not to the answers. The resulting representation would be disjointed, because it would be difficult to link the questions to each other without the content contained in the answers to them. A student who has formed a coherent representation of a lesson, therefore, should judge results to be more important than actions because results are more salient in their representation of the lesson.

Results

Japanese students rated results as more important than actions, but American students did not (see Figure 4). A repeated-measures ANOVA with group as a between-subjects variable and event type and linkage as within-subjects variables confirmed that there was indeed a significant Group x Type interaction, F( l , 39) = 5.966, p < .05. Although the overall mean of the American ratings (2.8) was lower than that of the Japanese ratings (3.1), there did not appear to be any important differences in the


American 4th

-. . I I

Actions Results FIGURE 4 Importance ratings of actions and results by Japanese and American fourth graders.

distributions of the ratings. Both American and Japanese students used the full 5-point scale, and the standard deviations of the ratings were similar across the two groups (American SD = 1.2; Japanese SD = 1.3). (As expected there was a main effect of linkage, but no interaction of linkage with type or group.) Simple effects tests showed that the difference between actions and results for the Japanese sample was highly significant, F(1, 39) = 11.87, p < .001, but the difference was not significant for the American sample, F( l , 39) = 1.99, p = .17.

Although there was not a significant difference between the two groups on the pretest (.I8 out of 7 for the Japanese subjects and 1.1 for the American subjects), the Japanese group had a higher mean on the posttest (5.06) than the American group (3.58). A two-way ANOVA with group as a between-subjects variable and pre- and posttest scores as a within-subjects variable showed a significant interaction between these two variables, F(1, 39) = 6.37, p < .05, indicating that the Japanese students learned significantly more from the lesson than did the American students.

Our first analysis revealed that the Japanese students rated results as more important than actions, indicating that the Japanese students constructed a more coherent representation of the lesson than the American students. Our second analysis revealed that they learned more from the lesson. But do they represent the lesson more coherently because of a difference in lesson schemas or because of differences in their background knowledge of mathematics? Do Japanese emphasize the importance of results because they think this category of event is likely to be more

important, or because they understand the results better than their less knowledgeable American counterparts?

To address this question, we decided to add a group of American sixth graders (n = 14) to our sample. American sixth graders should be much more knowledgeable about mathematics than American fourth graders, and should at least approach the level of knowledge typical of Japanese fourth graders (cf. Stigler, Lee, & Stevenson, 1990). We reasoned that if the differences in the ratings of results and actions by the American and Japanese fourth graders were related to differences in mathematical knowledge, then American sixth graders' ratings should be similar to Japanese fourth graders'.

The mean ratings produced by the American sixth graders, compared to those produced by the Japanese fourth graders, are presented in Figure 5. Interestingly, the sixth-grade American students, like the fourth-grade Japanese, rate results more important than actions. A repeated-measures ANOVA found a significant main effect of results versus actions, F( l , 29) = 26.22, p < .0001, but no interaction between group and event type. (There also was a main effect for group, F(1, 29) = 10.41, p < .005, for which we have no immediate interpretation.)

We also found that the American sixth graders learned as much from the lesson as had the Japanese fourth graders, with a mean pretest score of 1.5 and a mean posttest score of 5.8. An ANOVA with group (Japanese fourth graders vs. American sixth graders) as a between-subject variable and pre-

American 6th

Actions Results FIGURE 5 Importance ratings of actions and results by Japanese fourth graders and American sixth graders.


and posttest scores as a within-subjects variable confirmed there was no significant interaction between group and gain in test scores.

Although in this study we set out to document cultural differences, we found that regardless of cultural background, higher knowledge subjects judged results as more important than actions, a pattern indicative of having formed a coherent mental representation of the lesson. We also found a relationship between forming such a representation and learning more from the lesson, as evidenced by higher pre- to posttest gain scores.

Ill. Experimentally Manipulating the Coherence of a Lesson: A Study by Baranes (1990)

One weakness of the second study was that mental representations were assessed with a post facto importance rating task. It is possible that high-knowledge students' ratings reflected their ability to manipulate representations after learning had occurred instead of the actual representations they were forming as they viewed the lesson. In other words, coherent representation of the lesson-as indicated by a student rating results as more important than actions-may not have caused learning, but instead may have resulted from the greater learning of higher knowledge students.

This shortcoming was addressed in the doctoral dissertation of Ruth Baranes (1990), in which recall was elicited on-line at various points in the lesson. Probing students' memories during their viewing of the lesson allowed a more direct assessment of the mental representations that high- and low-knowledge students formed as they processed the lesson. Baranes also added a new twist. Using video editing, she was able to manipulate the lesson systematically and thus manipulate the ease with which a coherent representation could be formed of the lesson. With the addition of pre- and posttests she was able to study (a) the effects different lesson inputs had on students' interpretations, and (b) the effects that students' interpretations had on their subsequent learning of the content of the lesson.

Method

Baranes, in collaboration with a teacher, designed and videotaped a second-grade lesson that dealt with rounding numbers to the nearest 10. The lesson was designed to be highly coherent: It consisted of eight segments, each of which was motivated by a clearly stated goal and followed the preceding segment in a logical manner. She created a second, only slightly less coherent, version of the lesson by editing out events that made the interrelationships among lesson events more salient. Specifically, some goals and results were removed, as well as some of the explanations that


made the relationships between different parts of the lesson explicit. The editing was so subtle that naive adults watching the two versions of the tape generally did not notice the differences.

Baranes showed her tapes to 42 second-grade students, half of whom were classified as high in mathematical knowledge and half as low based on teachers' ratings. An equal number of high- and low-knowledge students were randomly assigned to view either the more coherent or the less coherent version of the l e s s ~ n . ~ Subjects were instructed to watch the tape carefully because the experimenter was going to stop the tape at certain points to ask them questions. At the end of each eight lesson segment the tape was stopped and the students were asked to recall everything they could that happened in the segment they had just viewed and to explain what the teacher was trying to help them understand.

Results

Total amount recalled did not differ between high- and low-knowledge students or between students viewing the more and less coherent lessons. What differed across groups was the quality of recall, as evidenced by the proportion of actions and results recalled. The more knowledgeable children recalled a lower proportion of actions and a higher proportion of results than the less knowledgeable children. Similarly, children who viewed the more coherent lesson recalled a lower proportion of actions and a higher proportion of results than children who viewed the less coherent lesson. All of these differences were statistically significant.

Baranes also conducted a path analysis to explore the relationships among lesson input, prior knowledge, and the coherence of the mental representation formed of the lesson (as measured by the ratio of results to actions recalled); and the relationships of these three variables to learning (as measured by gain scores from pre- to posttest). The hypothesized causal model that guided her analysis is depicted in Figure 6 (adapted from Baranes, 1990). First, as expected given the results previously discussed, both lesson input and knowledge were found to significantly affect the ratio of results to actions recalled. Second, a more coherent representation of the lesson (as indicated by the results-to-action ratio) was found to have a significant effect on learning, even when controlling achievement level and coherence of input. Finally, achievement was found to influence learning directly as well as indirectly (through mental representation of the lesson), but coherence affected learning only indirectly, mediated by the mental representation of the lesson.

'The full design of Baranes's study was more complex than we describe. The interested reader is referred to Baranes (1990).


COHERENCE OF MENTAL LEARNINGREPRESENTATION - (TESTSCORE)(RESULTSIACTIONS)

ACHIEVEMENT LEVEL

FIGURE 6 Causal model of relationships studied by Baranes (1990).

The dissertation by Baranes (1990) revealed that although there were no differences in the number of lesson events recalled by subjects who viewed either version of the lesson, there were qualitative differences in their recall. Children who viewed the more coherent lesson focused more on the results of actions than on the actions themselves, a pattern indicating the formation of a more coherent representation of the lesson. Also, regardless of the version they viewed, high-knowledge students tended to encode more lesson results than did low-knowledge students, again pointing to the formation of a more coherent representation of the lesson by high-knowledge students. The path analysis by Baranes provided direct support for our framework: Both student and lesson characteristics affect how a lesson is represented, and this in turn has an effect on learning.

IV. Students' Recognition of Relevant and Irrelevant Teacher statements3

In the studies described thus far, we examined the effects of content knowledge and lesson coherence on the representations students form of lessons. In this study we explore the effect of a third variable we have hypothesized to affect students' mental representations of lessons: lesson schemas. We explore whether Japanese and American students bring different expectations to bear on their understanding of a lesson and whether this has an effect on their ability to learn what is being taught in a lesson. We also experiment in this study with yet another way of assessing

3This study is reported in more detail elsewhere; see Yoshida, Fernandez, and Stigler (1992).


the mental representations students form of lessons. We administer a recognition memory task to determine which aspects of the lesson students attend to; patterns of attention are then related to learning as measured by pre- and posttest scores. We reason that students trying to form a coherent representation of a lesson should attend to certain aspects of the lesson more than others.

Method

In this study we used the same videotaped lesson-both the English and Japanese versions -as in the second study.

All statements made by the teacher were categorized as either relevant or irrelevant for learning the content being taught. An example of a relevant statement would be, "To find the number of boys you subtract 6 from the number of girls." An example of an irrelevant statement would be, "Miriam, you are always answering." We reasoned that a student who understood the web of relationships that the teacher was trying to convey in the lesson would attend more to the relevant than to the irrelevant teacher statements, and thus find the relevant statements easier to recognize.

The American and Japanese subjects were from the same schools used in the second study. As in the second study, we started with a group of American fourth graders and a group of Japanese fourth graders, and then added a group of American sixth graders.

Procedure. The procedure followed in this study was identical to the one followed in the second study in every respect except that instead of asking students to judge the importance of events, we asked them to judge whether or not statements were actually made by the teacher. Students were presented, one by one, with a list of 24 statements. Half of the statements were actually made by the teacher, and half were not. Also, half of the teacher statements and half of the non-teacher statements were relevant for learning the mathematical content of the lesson and half were irrelevant. The statements were presented on an audiotape with a 4-sec interval between statements during which time students had to make their judgments. The statements were randomly ordered. They were read by a voice different from the voice of the teacher in the videotape. Subjects were told to make their decisions on the basis of content and not on the basis of tone of voice or any other surface cues.

Scoring. Subjects were given recognition scores for both relevant and irrelevant statements. Recognition was measured as hits minus false alarms to correct for response bias.

Results

The mean recognition scores for Japanese and American fourth graders for relevant and irrelevant teacher statements are plotted in the left-hand panel of Figure 7. A repeated-measures ANOVA confirmed what is apparent in the graph: There was a highly significant interaction between culture and relevant versus irrelevant, F( l , 42) = 13.717, p < .Owl. Although Japanese and American students did not differ in their ability to recognize lesson-relevant statements made by the teacher, a simple effects test revealed that American students were significantly better than Japanese students at recognizing the lesson-irrelevant statements, F( l , 81) = 44.240, p < .0001. The Japanese students' ability to recognize lesson-irrelevant statements did not differ significantly from chance, and they were significantly better at recognizing the relevant than the irrelevant statements, F( l , 42) = 23.883, p < .Owl.

As in the second study, we reasoned that these differences in patterns of recognition may be due either to differences in culture or to differences in knowledge between the two groups. As before, in order to have groups comparable in background knowledge, we added a group of American sixth graders (n = 16). When we compared American sixth graders and Japanese fourth graders, we discovered, as shown in the right-hand panel of Figure 7, that the American sixth graders were overall better at this recognition

Japanese American American 4th Graders 4th Graders 6th Graders

FIGURE 7 Japanese and American students' recognition of teacher's statements that are relevant or irrelevant to the content of the lesson.

task than either group of fourth graders but that their pattern of recognition was similar to that of the American fourth graders and significantly different from that of the Japanese students F(l , 32) = 15.328, p < .001. That is, the American sixth graders were equally good at recognizing relevant and irrelevant statements.

In order to determine if the Japanese pattern of differential recognition of relevant and irrelevant teacher statements was associated with greater learning, the American children were divided into those who were better at recognizing relevant statements (relevant - irrelevant > 0) and those who were better at recognizing irrelevant statements (relevant - irrelevant < 0). When we looked at these two groups' pre- and posttest scores, we found that the group that was better at recognizing relevant teacher statements had a mean gain score of 3.3, whereas the group that was better at recognizing irrelevant statements only had a mean gain of .87. When a two-way ANOVA with this pattern of recognition grouping and grade as between- subjects variables was carried out on the gain scores, a significant main effect for pattern of recognition grouping was found F(1, 38) = 6.377, p < .05. Furthermore, we found no significant interaction between grade and pattern of recognition grouping, meaning that higher gain scores were associated with better recognition of relevant statements for children in both grades.

Discussion

The differences in attentional patterns manifested by American and Japanese students in this study provide preliminary support for the claim that American and Japanese children have different lesson schemas. Perhaps Japanese students, by being continually exposed to lessons in which it is clear what is and is not relevant, quickly learn to ignore irrelevant information. American students, on the other hand, by being exposed to lessons in which it is not clear what is or is not irrelevant might develop the strategy of attending to all aspects of the activity taking place in the hope that eventually it will become clear what is and is not crucial. Also, because a lot of the irrelevant statements used in this study contained social information, it is possible that American children have come to think that one role of classroom interactions is to teach social information and that this role is as important as that of teaching the mathematics. Including social information probably leads American students to try to form much more complex representations of the lesson than those formed by the Japanese students, who focus on the mathematical content being taught and ignore social information. It is possible that trying to form a simpler representation that focuses just on content is a more manageable task that ultimately will lead to better learning of the mathematical content. These

results, of course, are tentative because only a Japanese lesson was employed; perhaps the atypicality of the Japanese lesson prompted Amer- ican students to manifest unusual patterns of attention.

CONCLUSION

In this article, we developed a framework that could guide future research on learning mathematics from classroom instruction. We began with the assumption that students use the mental representations they form of lessons as a basis for constructing new knowledge. Therefore, how lesson events are represented by students, particularly the degree to which those representations are coherent, should have a profound effect on students' ability to learn the content being taught. We also argued that lesson characteristics, student knowledge about the topic being taught, and student knowledge about how lessons unfold should all have an effect on a student's ability to form a coherent representation of a lesson.

Based on this framework, we suggested that a methodology for studying classroom learning of mathematics would be to assess the coherence of the mental representations students form of lessons and then relate this to the amount of learning manifested by these students, as well as to the coherence of the lesson input they were exposed to.

Our first efforts to work within the framework yielded preliminary, yet encouraging results. Although the dependent measures we used, the manner in which we analyzed lesson input, the lesson content presented, and the age groups tested all varied across studies, we found that our results tended to converge. In comparing high- and low-knowledge students, we found evidence that higher knowledge students form more coherent representations of lessons than do their lower knowledge counterparts. This finding is interesting in light of the fact that we found evidence in several of our studies of a relationship between the coherence of the representation formed of a lesson and the amount of learning manifested. In the third study, we manipulated lesson coherence and found support for our claim that coherent lessons lead to more coherent representations, which in turn lead to greater learning. Finally, when we compared students who should have very different lesson schemas (i.e., American vs. Japanese students), we found some evidence that the lesson schemas that students bring to bear on a lesson affect the representation they form of it.

In future studies, we would like to begin experimenting with on-line measures that allow us to assess, as a lesson unfolds, what inferences students are making, to what aspects of the lesson they are attending, and to what previously heard information they are linking incoming information.


ACKNOWLEDGMENTS

The research reported in this article is part of a collaborative research project with our Japanese colleagues Giyoo Hatano, Shizuko Amaiwa, and Hajime Yoshida. The project is funded, in part, by the Spencer Founda- tion.

We are grateful to Kevin Miller, Rochel Gelman, Keith Holyoak, Robert Seigler, Allan Collins, and James Greeno for reading earlier drafts of this article.

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Learning Mathematics from Classroom Instruction: On Relating Lessons to Pupils'InterpretationsClea Fernandez; Makoto Yoshida; James W. StiglerThe Journal of the Learning Sciences, Vol. 2, No. 4. (1992), pp. 333-365.Stable URL:http://links.jstor.org/sici?sici=1050-8406%281992%292%3A4%3C333%3ALMFCIO%3E2.0.CO%3B2-I

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Is There More Than One Way to Recall a Story?Jean M. Mandler; Marsha DeForestChild Development, Vol. 50, No. 3. (Sep., 1979), pp. 886-889.Stable URL:http://links.jstor.org/sici?sici=0009-3920%28197909%2950%3A3%3C886%3AITMTOW%3E2.0.CO%3B2-H

Mathematics Classrooms in Japan, Taiwan, and the United StatesJames W. Stigler; Shin-ying Lee; Harold W. StevensonChild Development, Vol. 58, No. 5, Special Issue on Schools and Development. (Oct., 1987), pp.1272-1285.Stable URL:http://links.jstor.org/sici?sici=0009-3920%28198710%2958%3A5%3C1272%3AMCIJTA%3E2.0.CO%3B2-9

Student Views about Learning Math and Social StudiesSusan S. Stodolsky; Scott Salk; Barbara GlaessnerAmerican Educational Research Journal, Vol. 28, No. 1. (Spring, 1991), pp. 89-116.Stable URL:http://links.jstor.org/sici?sici=0002-8312%28199121%2928%3A1%3C89%3ASVALMA%3E2.0.CO%3B2-G

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