# «SE 062 649 ED 431 621 Stigler, James W.; Gonzales, Patrick; Kwanaka, Takako; AUTHOR Knoll, Steffen; Serrano, Ana The TIMSS Videotape Classroom Study: ...»

At the other end of the scale, 11.7 percent Japanese teachers and 2.3 percent of German teachers reported feeling that the videotaped lesson was better than usual.

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

Four other questions probed teacher's judgments of how typical the videotaped lesson was of lessons they normally teach. On these questions, teachers rated the typicality of the teaching methods, the behavior of the students, the tools and materials, and the lesson as a whole. Teachers used a four-point scale, where 1 means very typical/similar to what usually happens, and 4 means completely atypical/very different from what usually happens. Responses to the four questions are summarized in figure 8. Again, the Japanese teachers, on each of the four questions, rated their lessons as less typical than did teachers in the other two countries.

NOTE: 1=very typical, 4=completely atypical.

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

Although Japanese teachers rated the lessons as significantly less typical than did German or U.S.

teachers, the overall ratings were not particularly troubling to us. In fact, when asked to rate the typicality of the lesson as a whole, most teachers in all cultures chose either "very typical" or "mostly typical." 95.6 percent of German teachers, 85.1 percent of Japanese teachers, and 97.4 percent of U.S. teachers responded in this way. In conclusion, the videotape study probably captured a fairly representative picture of what typically happens in eighth-grade mathematics classrooms in these three countries.

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53p Chapter 3. Mathematical Content of Lessons

## CONTENT: A PLACE TO BEGIN

Any observer of classroom instruction is struck by its complexity. Instruction is multidimensional.Instruction also unfolds quicldy in real time, comprised as it is of an unceasing flow of events. To focus on one dimension is to lose sight of the others, which is why video is so useful in the study of classroom instruction. With video, we can make multiple passes through the lesson, focusing in each pass on a particular dimension or layer of description.

For purposes of analysis, we can describe instruction in layers. Although these layers may differ in

**character across cultures, the layers themselves as analytical points of reference have validity across cultures. These layers include the following:**

Setting: What is the environment, both physical and social, in which the lesson takes place?

Content: What is the curricular content of the lesson? What does "mathematics" look like?

Participants: We can analyze the role of the teacher and the role of the student as separable layers of classroom instruction.

Organization: What is the social organization of the lesson (e.g., whole class, small groups, individuals) and how does the form of organization change over the course of the lesson? What is the functional organization of the lesson in terms of activities?

Scripts and Goals: What are the cultural scripts and goals that tie the parts of the lesson together? In other words, how are all of these layers put together to make a lesson?

In practice, these different layers are woven together by the teacher who, in all three cultures, takes primary responsibility for the lesson, both in planning and execution. In planning the lesson teachers rely on physical, intellectual, and cultural resources. Physical resources include the setting and the materials available. Intellectual resources include the teacher's own mathematical knowledge and other mathematical knowledge available to the teacher. Cultural resources include the shared cultural understandings of the students in terms of goals, assumptions, routines, and roles, and a set of participant structures that teachers and students together know how to construct in the classroom. These structures include such things as classwork and seatwork.

While recognizing the multidimensionality of classroom instruction, we have chosen to begin with content. It is difficult to draw the line between what is taught and how it is taught. Still, it is useful to examine content apart from the lesson in which it is embedded. The rationale for this is simple: No matter how good the teaching is, if a lesson does not include rich mathematical content, it is unlikely that many students will construct a deep understanding of mathematics from the lesson. In this section of the report we describe the mathematical content of lessons in each country.

## GENERAL DESCRIPTIONS OF CONTENT

Our first step in describing the mathematical content of each lesson was to apply the TIMSS content coding system. The complete system, which included 44 categories, is available in Robitaille, McKnight, Schmidt, Britton, Raizen, and Nicol (1993). All coding was done from the lesson tables. For each segment in the table, the coder wrote down the TIMSS code that best described the mathematical content.Each lesson was thus described using one or more content codes.

TIMSS content coding was checked by independent coders at Michigan State University (MSU)the same coders who had done the textbook coding for the TIMSS curriculum analysis. They, like the UCLA coders, based their analysis on the video lesson tables. There was perfect agreement between coders at UCLA and at MSU.

**In general, more topics were represented in the sample of U.S. videotapes (24 topics) than in the samples of German (18 topics) or Japanese (13 topics) tapes. The 44 TIMSS content coding system categories were further grouped into 10 major categories:**

Numbersincluding whole numbers, fractions and decimals; integers, rational, and real numbers; number theory; estimation and number sense.

1.2 Measurementincluding units, perimeter, area, and volume.

1.3 Geometry: Position, Visualization, and Shapeincluding two dimensional and three dimensional geometry.

1.4 Geometry: Symmetry, Congruence, and Similarityincluding transformations;

congruence and similarity; and constructions using straight-edge and compass.

Proportionalityincluding proportionality concepts and problems; slope and 1.5 trigonometry; and linear interpolation and extrapolation.

1.6 Functions, Relations, and Equationsincluding patterns, relations, and functions; and equations and formulas.

1.7 Data Representation, Probability, and Statisticsincluding data representation and analysis; uncertainty and probability.

1.8 Elementary Analysisincluding infinite processes and change.

1.9 Validation and Structureincluding validation and justification; structuring

## and abstracting.

1.10 Other Content In figure 9, we show the (unweighted) percentage of lessons in our sample that included content belonging to each of the 10 major categories. Our purpose in presenting these data is to better describe the mathematical content of our sample of videotapes; the resulting differences across the three samples should not necessarily be generalized to the populations of eighth-grade classrooms in the three countries. It is for this reason that no statistical test was done on this variable.

There are some clear differences in the percentage of lessons in our sample that were devoted to various topics across the three countries. The most frequent topic in the U.S. sample (about 40 percent of the lessons) was Numbers, which included such topics as whole number operations, fractions, and decimals. In the German sample, the two most common topics were Geometry (Position, Visualization, and Shape), and Functions, Relations, and Equations. In the Japanese sample, the most common was Geometry

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(Symmetry Congruence, and Similarity), followed by Validation and Structure. Recall, however, that the emphasis on geometry in Japan is partly a result of bias in our sampling procedure.1.6 1.8 1.9 1.4 1.10 1.7 1.5 1.3 1.2.1 NOTE: 1.1=Numbers; 1.2=Measurement; 1.3=Geometry (Position, Visualization, Shape); 1.4=Geometry (Symmetry, Congruence, Similarity); 1.5=Proportionality; 1.6=Functions, Relations, Equations; 1.7=Data Representation, Probability, Statistics; 1.8=Elementary Analysis; 1.9=Validation and Structure; 1.10=Other.

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

## HOW ADVANCED IS THE CONTENT BY INTERNATIONAL STANDARDS?

It is not possible, a priori, to say that one topic is more complex than another. However, it is possible to make an empirical judgment of how advanced a topic is based on its placement in mathematics curricula around the world. We were able to make use of the TIMSS curriculum analyses, conducted by William Schmidt and his colleagues at MSU, to make such a judgment. The TIMSS content codes for each lesson (Robitaille et al., 1993) were assigned a number indicating the modal grade level at which the majority of the 41 countries studied gave the most concentrated curricular attention to the topic.Average level for each lesson was obtained by averaging the MSU index for all topics coded for the lesson.

The average grade level of topics covered in the video sample, as indicated by the MSU index, is shown in figure 10. In terms of this index, the average grade level of topics covered in the U.S. sample was significantly different than in Germany and Japan. Based on the MSU index of international stanBEST COPY AVAILABLE dards, the mathematical content of the U.S. lessons in the videotape study was at a seventh-grade level, whereas the German and Japanese lessons fell at the high eighth- or even ninth-grade level.

## A CLOSER LOOK AT CONTENT

Teacher's Goal for the Lesson We begin a more detailed look at content by examining teachers' responses on the teacher questionnaire.We asked teachers whether the content of the lesson was all review, all new, or somewhere in between.

Responses to this question are shown in figure 11. Analyses indicated that the distribution of responses in Japan differed significantly from that in both Germany and the United States.

Mathematical Skillsresponses that emphasized the teaching of how to solve specific kinds of problems, use of standard formulas, etc.

Mathematical Thinkingresponses that emphasized students' exploration, development, and comprehension of mathematical concepts, or the discovery of multiple solutions to a problem.

Social/Motivationalresponses that emphasized non-mathematical goals, such as "listening to others," or the creation of interest in some aspect of mathematics.

Test Preparationresponses that focused on preparing for an upcoming test.

Indeterminableresponses that were not possible to categorize, usually because they were too vague or incomplete.

Number of Topics and Topic Segments per Lesson Having determined which topics are taught in each country, we proceeded to divide lessons into topic segments (i.e., the points in the lesson at which the topic shifts from one to another). By topic, we here refer to the TIMSS content topics, which are broadly defined. A shift in topic is a clear shift in the primary content of the lesson and is usually marked by an announcement by the teacher. For example, in lesson GR-080 (00:28:48) the teacher says: "Okay let's still start something new today. Your school exercise book please." This is a clear signal that the topic is about to change, thus marking the beginning of a new topic segment.

If a lesson has only one topic then it will, by definition, have only one topic segment. However, many lessons had more than one topic. In this case, the number of topic segments might equal the number of topics (i.e., there could be one segment for each topic) or it might exceed the number of topics if, for example, the lesson alternated from a segment of one topic, to a segment of another, to another segment of the original topic.

Students work on discussing the inequalities they had done for homework until, at 35:12, the teacher changes the focus back to proportionality, assigning exercises from the textbook until the end of class.

Thus, the first and third topic segments focused on proportionality, the second segment, on inequalities.

As shown in figure 13, U.S. lessons contained significantly more topics and topic segments than did Japanese lessons. Also, German lessons contained more topics and topic segments than did Japanese lessons.

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CONCEPT was coded when the only treatment of the topic in the lesson involved the presentation of information, through either a statement or a derivation of general mathematical principles, properties, or definitions (e.g., formulas and theorems), or statement or derivation of a method for solving a class of problems. A concept can be presented through a concrete example or abstractly.

It can be introduced for the first time or simply be restated.

APPLICATION was coded when the only treatment of the topic in the lesson was as an application to the solving of a specific mathematical problem.

Mathematical concepts were not explicitly stated or discussed. The emphasis was on developing skills for solving specific types of problems.

BOTH was coded when a topic included, somewhere in the lesson, both concepts and applications. (If a mathematical concept was stated that did not directly relate to the topic, it was not counted as a concept.)