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

**An example of a topic coded as CONCEPT can be seen in GR-074. The lesson deals with calculations involving polygons. CONCEPT was coded because solution methods were stated abstractly, as formulas, rather than using the specific numbers of a particular problem. Here is an excerpt from the lesson transcript:**

Another example of a topic coded as CONCEPT is found in JP-040. The topic of this lesson is similarity. The teacher first presents the definition of similar figures and, together with the students, generates examples. The teacher then demonstrates that operations of multiplication and division can be used but not addition and subtraction. Finally, the students derive a triangle's similarity conditions from its congruence conditions. Students are asked to memorize the conditions for similarity.

APPLICATIONS are coded when there is no explicit mention of mathematical concepts. In GR-096, for example, the topic is representation of data. The teacher reviews different types of charts by using illustrations as examples (figure 14).

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

Teacher and students talk briefly about the names of different kinds of charts, as well as what different values in the charts represent. The application begins when the teacher hands out data in tabular form and asks students, working in groups, to create their own charts from the data.

As shown in figure 15, U.S. lessons had significantly lower percentages of topics that consisted of concepts only than did either German or Japanese lessons. And, U.S. and German lessons had a higher percentage of topics that consisted of applications only than did Japanese lessons.

## WERE CONCEPTS STATED OR DEVELOPED?

When concepts were included in the lesson, they could be stated or they could be developed. A concept was coded as STATED if it was simply provided by the teacher or students but not explained or derived. For example, the teacher, in the course of solving a problem at the board, might simply remind the students of the Pythagorean theorem (e.g., "The formula for finding the length of the hypotenuse of a right triangle is a2 +1)2 = c2") in order to guide the solution of the problem. The focus here is on the mathematical information itself rather than on the process of deriving it. A concept was coded as DEVELOPED when it was derived and/or explained by the teacher or the teacher and students collaboratively in order to increase students' understanding of the concept. The form of the derivation could be through proof, experimentation, or both.US-068 provides an example of a concept being developed through experimentation. The topic of the lesson is the relationship between circumference and Pi. The teacher begins the lesson by defining terms such as circumference and diameter. Students then break into groups to work with circular objects. They measure the objects' circumferences (C) and diameters (D) with a measuring tape (figure 16). They then divide C by D and examine their answers. In a subsequent class discussion, the teacher uses the commonality across answers as a basis for defining Pi.

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

US-061 also engages students in a concrete example. This time, however, the concept is coded as The topic is data representation. The teacher asks all students to write their shoe size on a STATED.

piece of paper and asks them to stand in order according to shoe size from smallest to largest (figure 17). The teacher then defines several statistical terms (e.g., range and mean) and asks the students to calculate the statistics using their shoe sizes. This is not coded as development because the concepts are not derived, only stated and applied in the example.

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

The results of coding concepts as STATED versus DEVELOPED are shown in figure 18. Concepts were significantly more likely to be STATED in the U.S. lessons than in both Germany and Japan. Conversely, concepts were more likely to be DEVELOPED in German and Japanese lessons than in the U.S. lessons.

Did Applications Increase in Complexity?

Finally, in topics that contained applications, we were interested in the relationship among the application problems. Specifically, when a topic contained more than one problem, were the problems just multiple examples of the same level of complexity, or did they increase in complexity over the course of the lesson?

Change in complexitycategorized as either INCREASE or SAME/DECREASEwas coded only if there was more than one application problem within the same topic in a lesson. Increase in complexity was coded when either a procedural or conceptual difficulty was added from one application problem to another. Increasing procedural difficulties generally consisted of additional operations (i.e., calculations previously executed separately were now combined in one application). Increasing conceptual difficulties generally consisted of added mathematical information (i.e., previously learned concepts now had to be modified for a new application using more mathematical information).

US-018 provides a good example of increasing complexity across two application problems. The lesson deals with area and circumference of a circle. In the first problem students are asked to find the area of the shaded region between the circle and the square (figure 19). In order to solve this problem the students had to find the area of the circle and then subtract that from the area of the polygon.

Students next were asked to find the area of a semicircle in which a triangle was inscribed (figure

20) and then the area of the region not covered by the triangle. This problem, although it employed some of the same concepts of the previous one, is clearly more complex. To solve it, students had to find the hypotenuse of the triangle (which is also the diameter of the circle), calculate the area of the semicircle, calculate the area of the triangle, and then subtract the area of the triangle from the area of the semicircle.

The results of coding this distinction are shown in figure 21. The Japanese applications were significantly more likely to increase in complexity than those in Germany. There was no difference in this regard between the United States and the other two nations.

82.5 67.9 54.3 45.7 32.1 17.5

Alternative Solution Methods Often the teacher's goal is to teach students how to solve a specific type of mathematics problem (e.g., an algebraic equation or a geometric construction). This can be accomplished by presenting one solution method and asking students to use it on similar problems or by encouraging the development of different methods and examining their relative advantages. We coded whether an alternative solution method was presented by the teacher, or by students, during the course of each lesson.

Panel (a) of figure 22 shows the percentage of lessons that included alternative solution methods of each type; Panel (b) shows the average number of alternative solution methods of each type presented in the lessons of the three countries. U.S. lessons included significantly more teacher-presented alternative solution methods than did Japan. Japanese lessons included significantly more student-presented alternative solution methods than did either German or U.S. lessons.

Principles, Properties, and Definitions As described earlier, when describing the content of each lesson we not only recorded tasks and situations but also the PRINCIPLES, PROPERTIES, and DEFINITIONS (PPD) that were stated in the lesson. Although these PPDs filled a relatively small percentage of lesson time, they nevertheless could be crucial for moving along the content of the lesson.

We were not able to reliably differentiate PRINCIPLES and PROPERTIES, but we were able to differentiate between these and DEFINITIONS. To be coded as a DEFINITION, a statement of mathematical information had to include a general statement (generic term) followed by the defining characteristics or properties. For example: A ray is a straight line (generic term) that has a beginning point but no end point (defining characteristics). A statement that was not a complete definition in this sense was coded as PRINCIPLE/PROPERTY. Any mathematical information stated in the lesson that was not coded as a DEFINITION was coded as PRINCIPLE/PROPERTY.

**An example of a DEFINITION is seen in the following excerpt from GR-022. The topic of this lesson is congruence of triangles. Here is a part of the lesson transcript:**

## BEST COPY AVAILABLE

**Another example is from the Japanese lesson JP-035, which deals with the topic of similarity of geometric figures. The teacher writes the word "similarity" on the board, then says:**

An example of PRINCIPLE/PROPERTY is from JP-039, which dealt with parallel lines and similarity. The teacher first reviewed four theorems on this topic that were covered in a previous lesson, then introduced the fifth theorem, namely, the midpoint connection theorem (figure 23). The theorem is clearly stated on the chalkboard. (Some of the writing on the chalkboard has been digitally enhanced to improve readability.)

NOTE: Some of the writing on the chalkboard has been digitally enhanced to improve readability.

**A few other examples of PRINCIPLES/PROPERTIES are found in the following teacher statements:**

Proofs Constructing proofs is an important mathematical activity because it provides a reasoned method of verification based on the accepted assumptions and observations of the discipline. Many reform documents, for example, those by the National Council of Teachers of Mathematics (1989, 1991), recommend that students should have increasing opportunities to examine and construct mathematical proofs. We coded a lesson as including a proof if an assumption was presented, a proof executed, and the assumption confirmed as correct. The proof could be presented by the teacher, by a student, or worked out collaboratively during classwork. As long as an assumption was presented and the strategy for proving it was discussed, it was still considered a proof, even if the proof was not executed. Likewise, if a proof was started but not completed due to running out of time in the lesson, we still coded the lesson as including a proof.

Analysis revealed that a greater percentage of the Japanese lessons included proofs than either the German or U.S. lessons. Indeed, 10 percent of German lessons included proofs while 53 percent of Japanese lessons included proofs. None of the U.S. lessons included proofs.' ' Standard errors for German, Japanese, and U.S. lessons are 4.32, 3.45, and 0, respectively.

## FINDINGS OF THE MATH CONTENT GROUP

We turn now to present the findings of the independent Math Content Group. Recall that this group analyzed the content of 30 lessons from each country, 15 algebra and 15 geometry. The Math Content Group based its analyses on the detailed descriptions of mathematical content contained in the lesson tables, as previously described. To reduce the likelihood of bias, tables were disguised (e.g., references to currency and other country-specific contents were altered) so that it was not possible to tell for certain which country the lessons came from. After analyses were complete, results were then tabulated by country Methods of Analysis Content descriptions in the lesson tables were subjected to a detailed series of analyses. The first step was to construct a directed graph representation of each lesson. The purpose of the directed graph was to show the content and flow of the lesson in shorthand notation so that patterns within and among lessons would become more apparent. In this graph, the content of each lesson was represented as a set of nodes (depicted by circles) and links among the nodes (depicted by arrows), depending on relationships existing among the nodes. Both nodes and links were then labeled according to the coding system developed by the group.Definitions of each code will be presented later. For now, however, it is useful to go through one lesson in detail, showing how the Math Content Group transformed the content description in the table into a directed graph. For convenience, we will use a Japanese lesson (JP-012) to illustrate this process.

The lesson table for JP-012 was presented earlier. The directed graph produced by the Math Content Group is shown in figure 25.

NOTE: PPD = Principle/Property/Definition; NR = Necessary Result; C+ = Increase in Complexity; HP = Helpful Process.