Chapter 5. Processes of Instruction

Both the content and organization of lessons are generally planned in advance; they represent conscious decisions on the teacher's part. But not all that happens in classrooms is planned. Some processes only become evident as instruction unfolds and sometimes only through detailed analyses. In this chapter we present some additional analyses that describe, more fully, the nature of instruction in these three countries. Again, we remind the reader that these analyses are preliminary; much remains to be done.

DEVELOPING CONCEPTS AND METHODS

Earlier we distinguished two ways of including concepts in a lesson: A concept might simply be stated by the teacher or students, but not explained or derived; or, it might be developed (i.e., derived and/or explained) by the teacher or the teacher and students collaboratively in order to increase students' understanding of the concept. Our analyses indicated that development of concepts occurs more often in Germany and Japan than in the United States. Further analysis reveals, however, that there are some significant differences in the way in which concepts are developed in Germany and Japan. Development happens primarily during classwork in Germany, with most of the work being done by the teacher. In Japan, on the other hand, seatwork segments play a more critical role in the development of mathematical concepts, consistent with a strategy of giving students themselves more responsibility for the process.

Evidence for this conclusion is presented in figure 51. Recall that concepts were coded as "developed" if they were derived or explained by the teacher and/or the students in order to increase students' understanding. For each topic within each lesson, we first determined whether or not development of concepts was included. If it was, we next coded whether or not there were any seatwork segments within the topic/lesson. In panel (a) we show the average percentage of these topics within each lesson that included any seatwork at all. By this loose definition, the Japanese development segments included significantly more seatwork than did the German segments. Of course, the seatwork may not have been the part of the segment in which the development actually occurred. If we tighten the definition, as we have done in panel (b), we get a similar result. Here, we show only those topics for which development actually occurred during seatwork. The percentage in Japan was significantly higher than in either of the other two countries.

Figure 51

Average percentage of topics including development that (a) include at least some seatwork and (b) include actual development of concepts during a seatwork segment

(a)	(b)

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

THE USE OF INSTRUCTIONAL MATERIALS

We observed a wide variety of tools and materials being used in our samples of eighth-grade mathematics classrooms. The most commonly used instructional tools were the chalkboard and the overhead projector. Indeed, four lessons in Germany, none in Japan, and three in the United States used neither of these tools.¹

The percentage of lessons in each country in which the chalkboard and overhead projector were used is displayed in figure 52. (In some lessons, teachers used both.) The German and Japanese teachers used the chalkboard significantly more often than teachers in the United States. In contrast, U.S. teachers used the overhead projector more often than teachers in Japan or Germany, and German teachers used the overhead more than teachers in Japan.

Figure 52

Percentage of lessons in which chalkboard and overhead projector are used

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The percentage of lessons in which other kinds of materials were used is shown in figure 53. Most of the categories used are self-explanatory. The term "Manipulatives" refers to any concrete materials used to represent quantitative situations, such as paper circles, plastic triangles, unit blocks, or geoboards. Posters, used mostly in Japan, refer to prepared paper materials that are brought out and attached to the board during the lesson. Mathematics tools include objects specifically designed for use in solving mathematical problems. Examples of this category include rulers and graph paper.

Figure 53

Percentage of lessons in which various instructional materials were used

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

A number of cross-national differences emerged. Worksheets were significantly more common in Japan and the United States than in Germany. Textbooks, on the other hand, were seldom used in Japan but were rather common in Germany and the United States. Calculators were used primarily in the United States and rarely or never in the other two countries. Japanese teachers used significantly more mathematical tools and posters than did German and U.S. teachers.

A summary analysis in which we simply added up the total number of different materials categories represented in each lesson revealed that Japanese lessons had the most (average of 3.7) followed by the United States (3.1) and then Germany (2.6).²

Use of the Chalkboard

We discovered some differences in the way that chalkboards and overhead projectors are used in the three countries. One of these concerned the frequency with which students, as opposed to the teacher only, come to the front and use the chalkboard or overhead projector. In figure 54 we show (a) of the lessons in which the chalkboard is used at all, the percentage in which students actually use the chalkboard; and (b) of the lessons in which the overhead projector is used at all, the percentage in which students use it. The cross-country differences were not significant in the case of use of the chalkboard by students. Japanese students, on the other hand, did use the overhead projector significantly more often than did German students by this measure, and German students did so significantly more often than U.S. students.

Figure 54

Percentage of lessons including (a) chalkboard or (b) overhead projector in which students come to the front and use it

(a)	(b)

NOTE: The overhead projector was used in only three Japanese lessons.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The second discovery was in the way the chalkboard is used. In Japan, the chalkboard is used in a highly structured way: Teachers appear to begin the lesson with a plan for what the chalkboard will look like at the end of the lesson, and by the end of the lesson we see a structured record or residue of the mathematics covered during the lesson (see figure 55). In the United States, in contrast, the use of the chalkboard appears more haphazard. Teachers write wherever there is free space and erase frequently to make room for what they want to put up next.

Figure 55

Example of chalkboard use from a Japanese lesson

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Objective support for this impression comes from an analysis of erasures during the lesson. We performed this analysis on the subset of 90 lessons used by the Math Content Group. We counted all of the tasks and situations that were represented on the chalkboard during the lesson, then looked, at the end of the lesson, to see what percentage remained. The results are shown in figure 56.

Figure 56

Percentage of tasks, situations, and PPDs (principles/properties/definitions) written on the chalkboard that were erased or remained on the chalkboard at the end of the lesson

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Analyses revealed that Japanese teachers left more information remaining on the board at the end of the lesson than did either German or U.S. teachers. It is interesting to consider the potential effect this practice might have on student comprehension of the lesson. If information is erased, it is no longer available to the student who may need more time to process it. Having the information available throughout the lesson, in an organized fashion, may provide a crucial resource to the student. Alternatively, students may absorb material on a chalkboard more completely if there is less information on it at a given time.

Use of Manipulatives

Although the Japanese teachers in our sample used manipulatives more frequently than teachers in the other countries, teachers in all countries did use them to some degree. But they were not always used in the same way across countries. One aspect in which we were interested was who used the manipulatives. In figure 57 we show the percentage of lessons in which manipulatives were used by the teacher only, the students only, or both teachers and students. The only significant difference was in the percentage of manipulatives used by the students only: Japanese lessons were significantly less likely to include manipulatives used by students only than were the U.S. lessons.

Figure 57

Average percentage of lessons where manipulatives were used in which the manipulatives were used by teacher, students, or both

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

PROCESSES DURING SEATWORK

It is difficult to infer what students are doing during classwork. They may appear to be listening to the teacher, but beyond that we have little information about the kind of thinking in which they are engaged. Seatwork is somewhat different in this respect: Students are generally given an explicit task to work on, and the task usually leads to some visible product. We wanted to describe the number of tasks and situations, as well as the kinds of tasks, students were assigned to work on during seatwork.

Tasks and Situations During Seatwork

It was possible to reliably identify tasks and situations engaged in seatwork. By examining the tasks and situations students were assigned to work on during each seatwork segment, we identified four distinct patterns:

• ONE TASK/ONE SITUATION--this would typically occur when the teacher had students do one example, then come back as a class to discuss it;
• MULTIPLE TASKS/ONE SITUATION--this would typically happen when students were given a single mathematical situation and asked to explore it from a variety of perspectives, performing multiple tasks;
• ONE TASK/MULTIPLE SITUATIONS--this typically happens when students, having just been taught how to perform a specific task, are asked to practice it in a number of exercises; and
• MULTIPLE TASKS/MULTIPLE SITUATIONS--this is a typical worksheet or problem set from the textbook in which students are asked to do a variety of exercises.

An example of ONE TASK/ONE SITUATION comes from lesson JP-007, which dealt with angles between parallel lines (figure 58). Presented with the diagram on the chalkboard, students were asked to find the angle (X) in the bend using any of the three methods that they had learned previously. After completing this task they reconvened to discuss their answers.

Figure 58

Excerpt from chalkboard of JP-007

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

An example of MULTIPLE TASKS/ONE SITUATION can be seen in US-012. Students are presented with a single situation, the equation x² + 14x ­ 43 = 0. They are told to solve the equation twice, the first time by completing the square, the second, by using the quadratic equation. In this example, each solution would be coded as a separate task.

ONE TASK/MULTIPLE SITUATIONS is exemplified in GR-103. Students are told to turn to page 95 of their textbooks and do exercises 12a, 12b, and 12c (figure 59). In each case the task is the same, namely, to solve the systems of linear equations.

Figure 59

Excerpt from textbook page used in GR-103

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Finally, an example of MULTIPLE TASKS/MULTIPLE SITUATIONS can be seen in US-016. The teacher hands out a worksheet with problems and asks students to work on them during seatwork (figure 60):

Figure 60

Problems from worksheet used in US-016

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The average percentage of seatwork time spent working in these four patterns is shown in figure 61. U.S. students spent significantly more time working on MULTIPLE TASKS/MULTIPLE SITUATIONS than did Japanese students. Japanese students spent more time working on MULTIPLE TASKS/ONE SITUATION than did German or U.S. students. German students spent more time than Japanese students working on ONE TASK/MULTIPLE SITUATIONS.

Figure 61

Average percentage of time in seatwork/working on task/situation segments spent working on four different patterns of tasks and situations in each country

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

These patterns may affect students' experience during seatwork. For example, in Japan, where students generally work on only one situation during a seatwork segment (72 percent of the time), the students may experience the work as more unified or coherent than do U.S. students, who tend to work on multiple situations (64 percent of the time). Alternatively, students may develop a more coherent sense of a concept when presented with a variety of tasks and situations to approach.

Performance Expectations

What kinds of tasks were students working on during seatwork? We coded tasks into three mutually exclusive categories:

• PRACTICE ROUTINE PROCEDURES;
• INVENT NEW SOLUTIONS/THINK; and
• APPLY CONCEPTS IN NEW SITUATIONS.

PRACTICE ROUTINE PROCEDURES was coded to describe tasks in which students were asked to apply known solution methods or procedures to the solution of routine problems. Generally, the function of these seatwork segments was to practice previously learned information. For example, in GR-033 the class first goes over the solutions to two linear equations that had been assigned for homework. After sharing the solution to two equations that were homework, teacher and students together solve one more equation as a practice example : x + 2 (x - 3) = 5x - 4 (2x - 9). Then the teacher assigns two more equations for seatwork:

(1) 60 - 8 (6 - 2x) = 44

(2) 8x + 12 + (5x - 8 ) = 10x - (3 - 2x - 8)

These seatwork tasks were coded as Practice Routine Procedures.

INVENT NEW SOLUTIONS/THINK was coded to describe tasks in which students had to create or invent solution methods, proofs, or procedures on their own, or in which the main task was to think or reason. The expectation, in these cases, was that different students would come up with different solution methods.

An example of this category can be seen in JP-034 (figure 62). The topic of the lesson is similarity of two-dimensional figures. In a preliminary discussion, students are asked to think of objects that have the same shape but different sizes. After a number of objects are listed, the teacher presents quadrilateral ABCD on the board (on the left), together with a similar quadrilateral that is expanded twofold (on the right). She asks students, during seatwork, to think of as many methods as they can to expand the figure on the left into the one on the right.

Figure 62

Excerpt from chalkboard in JP-034

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Only one seatwork segment in the U.S. data was coded as INVENT NEW SOLUTIONS/THINK. The lesson, US-033, dealt with common fractions. The teacher gives definitions of equivalent and improper fractions, then illustrates each definition with an example. At this point the teacher tells students to come up with their own definition for "proper fraction":

It is important to note that performance expectations cannot be coded simply by analyzing the problem. It is also necessary to see what students do, both while solving the problem and afterwards. When seatwork is followed by students sharing alternative solution methods, this generally indicates that students were to invent their own solutions to the problem.

We have labeled the third category of performance expectations as APPLY CONCEPTS IN NEW SITUATIONS because most of the tasks coded into this category involved transferring a known concept or procedure into a new situation. In point of fact, however, we coded this category whenever the seatwork task did not fall into one of the other two categories. An example of this code can be seen in JP-012, which dealt with geometric transformations (figure 63). First, the teacher uses the display to review with students the fact that any two triangles between the same two parallel lines will have the same area.

Figure 63

Excerpt from computer monitor used in JP-012

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Then the teacher assigns students to work on the following problem: The border between Eda's land and Azusa's land is bent (figure 64). How can we straighten the border without changing the area of either person's land? We coded this as APPLY CONCEPTS IN NEW SITUATIONS because the teacher suggested which concept students should apply to the solution of the problem.

Figure 64

Excerpt from chalkboard in JP-012

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The average percentage of seatwork time spent in each of the three kinds of tasks is shown in figure 65.

Figure 65

Average percentage of seatwork time spent in three kinds of tasks

 

NOTE: Percentages may not sum to 100.0 due to rounding.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Japan differed significantly from the other two countries, spending less time on practice of routine procedures during seatwork and more time inventing new solutions or thinking about mathematical problems.

CLASSROOM DISCOURSE

In this section we will present the coding categories and results from the discourse coding that has been completed thus far. We stress that these analyses of discourse are quite preliminary. However, they do provide a foundation on which to build subsequent analyses.

We report two sets of analyses. The first set (First-Pass Coding) utilized the full sample of lessons but sampled 30 utterances to represent each lesson. (See section "Coding of Discourse" for more details on how utterances were sampled.) The second set of analyses (Second-Pass Coding) utilized 30 lessons in each country (the same 90-lesson subsample used by the Math Content Group), but analyzed all of the utterances in these lessons.

First-Pass Coding: Categorizing Utterances

The unit of analysis for first-pass discourse coding was the utterance. An utterance was defined as a sentence or phrase that serves a single goal or function.

The first step was to categorize each utterance during public discourse into 1 of 12 mutually exclusive categories. Six of the categories were for teacher utterances, 5 for student utterances, and 1 (Other) for both teacher and student utterances. These categories are briefly described in figure 66.

Figure 66

Categories used for first-pass coding of utterances during public discourse

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Elicitations (E) were further subdivided into five mutually exclusive categories, presented in figure 67.

Figure 67

Subcategories of elicitations

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Finally, Content Elicitations (EC) were further subdivided into three mutually exclusive categories, as outlined in figure 68.

Figure 68

Subcategories of content elicitations

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

First-Pass Coding: Results of the Sampling Study

In the sampling study, recall, the total number of utterances coded was held constant at 30 per lesson. However, the number of these produced by teachers versus students could vary. In all three countries, teachers talked more than students, whether measured in terms of utterances or words. In figure 69, we show the average percentage of coded utterances made by the teacher and the average percentage of the total words spoken by the teacher in the 30-utterance corpus. When comparing utterances of teachers relative to those of students, German teachers talked less than U.S. teachers, who talked less than Japanese teachers. When we look at words, German teachers still talked significantly less than teachers in the other two countries, but U.S. and Japanese teachers are not distinguishable in this regard.

Figure 69

Average percentage of utterances and words spoken by teachers in each country

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

We look next at the kinds of utterances produced by teachers in the three countries. In figure 70 we show the average number of utterances (out of 30) made by teachers broken down by category. Japanese and U.S. teachers produced significantly more Information utterances than did German teachers. U.S. teachers produced significantly fewer Elicitations than German teachers, and German teachers produced significantly more Uptakes than both Japanese and U.S. teachers. German and U.S. teachers produced more Teacher Responses to student elicitations than did teachers in Japan.

Figure 70

Average number of utterances (out of 30 sampled per lesson) coded into each of six teacher utterance categories

 

NOTE: Numbers less than 0.05 are rounded to 0.0.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The distribution of student utterances is shown figure 71. The German lessons contained significantly more Student Responses than did lessons in the other two countries. German and U.S. lessons contained significantly more Student Elicitations and Student Information utterances than did Japanese lessons.

Figure 71

Average number of utterances (out of 30 sampled) coded into each of five student utterance categories

 

NOTE: All values less than 0.05 are rounded to 0.0.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Not only did the German students respond more frequently than Japanese and U.S. students but they also spoke at greater length during each response, as indexed by the number of words in the response. As depicted in figure 72, U.S. student responses are significantly shorter than responses produced by German students.

Figure 72

Average length of student responses as measured by number of words

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Let us next examine the types of Elicitations teachers produced in the three countries (figure 73). German teachers produced more Content Elicitations than did Japanese and U.S. teachers. Japanese teachers produced more Interactional Elicitations than did U.S. teachers. U.S. teachers produced more Metacognitive Elicitations than did the German teachers.

Figure 73

Average number of utterances (out of 30 sampled per lesson) coded into each of five categories of teacher elicitations

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

We were most interested in the Content Elicitations because these elicitations generate much of the mathematical content that is discussed in the lesson. In the next figure (figure 74) we show the average number of Content Elicitations (out of the 30 utterances sampled per lesson) that were coded as Name/State, Yes/No, and Describe/Explain. German teachers asked significantly more Name/State questions than did either Japanese or U.S. teachers; U.S. teachers asked significantly more Yes/No questions than did Japanese teachers; and, German and Japanese teachers asked significantly more Describe/Explain questions than did U.S. teachers.

Figure 74

Average number of utterances (out of 30 sampled) coded into each of three categories of content elicitations

 

NOTE: Values less than 0.05 have been rounded to 0.0.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Second-Pass Coding Categories

Whereas the first-pass coding of discourse was based on a sample of 30 utterances from each lesson in the full data set, second-pass coding was based on all the utterances in each lesson, but only for the subsample of 90 lessons used by the Math Content Group. (These 90 lessons also were coded in the first-pass coding.) Several new codes were added for second-pass coding. Content Elicitations, Information statements, and Directions were further subdivided. In addition, we started the process of grouping utterances into higher-level categories we call Elicitation-Response Sequences. We will briefly describe each of these new codes.

1. Content Elicitations. In our reading of transcripts we discovered that Content Elicitations, regardless of type, can be further divided into two categories according to their function:

Elicitation of Factual Information [Ef] is defined as any elicitation that requests a piece of mathematical information in pursuit of a correct answer. The purpose of the elicitation is for the teacher to assess whether the students know the answer or whether they are able to produce the answer. The teacher is not interested in finding out a particular student's thinking, and the response could be given by any student, even by the teacher.

Elicitation of Individual Ideas [Ei] is defined as any elicitation that requests a student to report on their individual opinions, ideas, or thinking processes. There may be a mathematically correct answer to the elicitation, but the purpose of the inquiry is for the teacher to find out what individual students have in mind. Control of the response rests more with the student than with the question itself. There is no specific response that the teacher is pursuing, and therefore it is less likely that the response is evaluated by the teacher as right or wrong.

Several examples from the data will help to illustrate this distinction. In the first two examples content elicitations were coded as Factual Information. (Letters in brackets indicate both first- and second-pass discourse codes.)

(Example 1)

(Example 2)

It is often necessary to see what kind of uptake follows the response in order to code Factual Information versus Individual Idea. When the teacher does not provide such feedback, a coder must determine whether the expected response is something that is an objective mathematical fact or something over which the respondent has ownership, so that there is no "correct" answer to the elicitation. The most common circumstances in which Individual Idea elicitations occur is when the teacher collects a variety of responses from different students without providing evaluative feedback. Below are two examples.

(Example 3)

(Example 4)

 

[E][EC][DE][Ei]

T

What do you think about when you look at bread.

 

2. Further categorization of Information and Direction utterances. Information and Direction utterances were each further categorized into one of four mutually exclusive categories: Content related, Managerial, Disciplinary, and Other. Brief definitions of the four categories for Information and Directions are presented in figure 75.

Figure 75

Four subcategories of information and direction utterances

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

3. Coding of Elicitation-Response Sequences. Our next step in discourse coding was to define higher-level units we called Elicitation-Response Sequences. Discourse is organized and cannot be understood simply by characterizing the utterances. Some utterances are more important than others. Defining ER sequences was our first step in coding the organization of discourse.

Elicitation-Response Sequence [ER] was defined as a sequence of turns exchanged between the teacher and student(s) that begins with an initial elicitation and usually ends with a final uptake. The ER sequence is a cohesive unit of conversational exchange. ER sequences may consist of a single elicitation, response, and uptake, or they may consist of several of these three utterance types. They may also consist of a single Elicitation without a Response or Uptake, or of single Elicitation and Response without an Uptake. A new ER sequence begins when there is a new Elicitation. A new Elicitation is one that requests new information. Repetitions, redirections of the initial elicitation to other students, or clarifications are not considered new elicitations. A diagram representation of the ER sequence is shown in figure 76.

Figure 76

The elicitation-response sequence

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Following are some examples.

[Example 1]

[Example 2]

Results of Second-Pass Coding

We have not done many analyses of the second-pass coding. However, we can present a few results at this time.

In the sampling study, because we looked at only 30 utterances in each lesson, we were unable to say anything about the rate of talk in the classroom. In second-pass coding, even though the number of lessons analyzed is smaller (30 in each country), all utterances in each lesson were included in the analysis, giving us a more detailed sense of how talk occurs as the lesson unfolds. In all, for the 90 lessons, we entered more than 42,000 discourse codes.

The first analysis of interest concerns the rate of talk in the classrooms of each country. In figure 77 we show (panel a) the average number of discourse codes in each lesson (excluding Elicitation-Response Sequences) divided by the number of minutes in Classwork. U.S. classrooms had a higher rate than Japanese classrooms. In panel (b) of the figure we show the average number of Elicitation-Response Sequences divided by the number of minutes of Classwork. The rate in the United States is highest, and in Japan, lowest, among the three countries.

Figure 77

(a) Average number of discourse codes per minute of classwork in the three countries; (b) average number of elicitation-response sequences per minute of classwork in the three countries

(a)	(b)

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Not only was the rate of talk slower in Japanese classrooms but also the length of ER sequences was greater than in the United States. The average number of discourse codes per ER sequence was 9.6 in Germany, 12.1 in Japan, and 7.2 in the United States. This means that Japanese teachers stuck with a question longer than did U.S. teachers before moving to the next question.³

Whereas in the sampling study we counted utterances regardless of their importance in the lesson discourse, in this analysis we were able to take into account the fact that not all utterances are equally important. Specifically, we assumed that the first elicitation in an Elicitation-Response Sequence will be more significant than the follow-up elicitations. The following two figures show the average percentage of initiating elicitations of ER sequences that were of various types. In figure 78 we look at First Elicitation/Content Elicitations that were judged to be eliciting a fact or correct answer. The incidence of Name/State elicitations did not differ across countries. Lessons in the United States were more likely to contain Yes/No elicitations than those in Germany; German and Japanese lessons were more likely to contain Describe/Explain elicitations than U.S. lessons.

Figure 78

Average percentage of initiating elicitations of elicitation-response sequences in each country: Content-related elicitations seeking facts

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

In figure 79 we show the same graph but for Content Elicitations judged to elicit individual student ideas. Analyses revealed no differences between the three countries.

Figure 79

Average percentage of initiating elicitations of elicitation-response sequences in each country: Content-related elicitations seeking individual ideas

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Explicit Linking and the Coherence of the Lesson

Language can serve many functions in a mathematics lesson. One of these is to explicitly link together ideas and experiences that the teacher wants students to understand in relation to each other. Using the subsample of 90 lessons coded by the Math Content Group, we coded two kinds of linking: Linking across lessons and linking within a single lesson. We defined linking as an explicit verbal reference by the teacher to ideas or events from another lesson or part of the lesson. The reference had to be concrete (i.e., referring to a particular time, not to some general idea). And, the reference had to be related to the current activity.

The results of this coding are displayed in figure 80. The highest incidence of both kinds of linking--across lessons and within lessons--was found in Japan. Indeed, teachers of Japanese lessons linked across lessons significantly more than did teachers of German lessons, and linked within lessons significantly more than teachers of both German and U.S. lessons.

Figure 80

Percentage of lessons that include explicit linking by the teacher (a) to ideas or events in a different lesson, and (b) to ideas or events in the current lesson

(a)	(b)

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Our observations of the videotapes reveal that the Japanese teachers appear to use linking in a systematic, almost routinized, way. They tend to start the lesson by recalling or reviewing what was done in the previous lesson. JP-036, for example, opened with the teacher saying, "Then, um yesterday the last part on ratio we did two characteristics, but characteristics that can be changed into a multiplied form, right? Please summarize what we practiced." The same lesson closed with the teacher saying, "Well we could not do concrete practice so I would like to do it using this in the next class period. I also want to review and go over number five in the next class period." Twenty-six of the 30 Japanese lessons included linking to a past lesson, and 19 included linking to a future lesson.

Twenty-nine of the 30 Japanese lessons included linking to different parts of the same lesson. In one example, the teacher referred back to a statement made by a student several minutes earlier: "Just now you've heard the opinion of your friend and um while using that as a reference you can continue on your computer or um you can continue on your computers by reading ahead yourselves or you can prove it while getting hints from the computers. Okay? Or you can prove it by um expressing your opinions um with your friends." These kinds of statements were far less common in German and U.S. lessons.

¹ These lessons were GR-16, GR-21, GR-34, GR-, US84-9, US-16, and US-42.

² Standard errors for Germany, Japan, and the United States were 0.10, 0.09, and 0.12, respectively.

³ Standard errors for Germany, Japan, and the United States were 1.25, 0.62, and 0.50, respectively.