«SE 062 649 ED 431 621 Stigler, James W.; Gonzales, Patrick; Kwanaka, Takako; AUTHOR Knoll, Steffen; Serrano, Ana The TIMSS Videotape Classroom Study: ...»
For purposes of analysis we defined two groups of U.S. teachers: One group (N=22) responded "a lot" when asked the degree to which the videotaped lesson was in accord with current ideas (i.e., those responding "a fair amount" were excluded from this analysis); the other group (N=19) answered either "a little" or "not at all." We can call the first group Reformers, the second, Non-Reformers. We compared the classrooms of these two groups of teachers on all of the variables discussed earlier in this report.
Overall, the analyses revealed very few significant differences between the two groups of teachers.
Although this lack of differences may be due, in part, to the lack of statistical power given the small size of our sample, we do not believe this is the primary reason. In our own viewing of the tapes, we did not see a strong distinction between these groups.
Statistically significant differences did emerge, however, in organization of the lesson and materials.
Organization of the Lesson As a group, Reformers spent a higher percentage of their time in seatwork than did Non-Reformers.
Reformers spent 43.1 percent of lesson time in seatwork, compared with 28.6 percent for Non-Reformers.
(The average for all U.S. teachers was 37.3 percent.)` ' Standard errors for Reformers and Non-Reformers were 3.99 and 5.26, respectively.
The two groups also differed in the kind of seatwork in the lesson: individual, group, or a combination of the two. In figure 84 we show the percentage of lessons in the two groups that included individual seatwork only, group seatwork only, both kinds of seatwork, or no seatwork. We can see that the Reformers were far more likely to include both individual and group seatwork in their lessons than were the Non-Reformers, while they were less likely to use individual seatwork only.
Instructional Materials Finally, we found one significant difference in the tendency of U.S. Reformers and Non-Reformers to use certain types of materials in the classroom. Specifically, Reformers were less likely than Non-Reformers to use textbooks in the lesson: 18 percent of lessons for Reformers, and 63 percent for Non-Reformers.'
REFORM IN THE CLASSROOM: QUALITATIVE ANALYSESOther than these areas of differences, we have little quantitative evidence that reform teachers in the United States differ much from those who claim not to be reformers. Most of the comparisons were not significant. But it is useful to look more qualitatively at the lessons taught by Reformers and NonReformers. Our conclusion is interesting: It is true that teachers who cite features of instruction, such as the use of real-world problems or cooperative learning, do implement such features in the lessons we videotaped. However, these features alone do not necessarily indicate the quality of instruction as intended by the NCTM standards, and in fact may only bear a superficial relationship to the quality of instrucStandard errors for Reformers and Non-Reformers were 8.56 and 8.44, respectively.
tion. Similarly, a high quality lesson could be constructed that did not contain these features. Quality of mathematical activity depends on how features are implemented. Let us explore a few examples.
Example 1: Airplane on a String (US-060) US-060 was taught by a teacher who judged it to be a good example of current ideas about the teaching and learning of mathematics. The lesson included a real-world problem situation and a period of cooperative group work. It also included a writing assignment in which students were asked to reflect on what they learned in the lesson. We agreed with the teacher's assessment of the lesson, and we will try to explain why.
The lesson started with the teacher asking for a volunteer to come to the front of the room and swing a model airplane on the end of a string around her head in a circular motion (figure 85). Everyone appeared attentive, and enthusiasm was high as students wondered what the airplane on the string would have to do with mathematics.
SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.
The teacher then posed a question: How fast is the plane going? Many students gave answers, which the teacher wrote on the overhead projector: "About thirty miles per hour?' "Fifteen miles per hour?' "Two-hundred thirty?' And so on. Why are the answers in miles per hour, the teacher wondered aloud.
What would be a rate we could measure? Revolutions per minute? Per second? If we measured such things, could we convert them to miles per hour? Discussion of these questions took up the first 10 minutes of the lesson.
After the teacher could see that most students in the class understood the problem and had thought through some of the issues involved in solving the problem, she had them work in groups for the next 30 minutes. Each group had its own airplane and string, and each worked to measure the speed of the plane. The general strategy adopted by all groups was the same: Measure the radius of the circle from the plane to the point around which the plane was swung; then, calculate the circumference in inches, count the number of times the plane traversed the circumference in a set amount of time, and get an answer in inches per second or per minute; finally, convert the answer into miles per hour. Students appeared to be very involved in the activity.
As students worked, the teacher circulated and posed a new question to each group: If a bird were sitting on the string, halfway between the plane and the center of the circle, would it be traveling the same speed as the plane, faster, or slower? Groups actively began to discuss this possibility, with differing opinions offered and justified. One student noticed that her hand was not completely still as she twirled the plane, and that this must have affected the radius of the circle. The teacher asked the group to consider how they might need to adjust their method to take this problem into account.
Thirty-nine minutes into the lesson students convened again as a class. Over the next 12 minutes all groups presented their answers, then two groups presented their solution methods to the class. Someone brought up the bird: how fast would the bird be going? The teacher told them that they would discuss this tomorrow, then handed out a writing assignment for homework. The writing assignment asked students to describe the problem they had worked on, then summarize the approach their group took to solving the problem. It also asked them to write about the role they played, specifically, in the group's work.
We thought this was in line with NCTM standards for several reasons. Students were engaged in a rich mathematical problem that appeared to be perceived as a real problem by most of the students in the class. The problem was closely tied to mathematical conceptscircumference and radius of a circle, and rate. The task encouraged students to make connections among these concepts and between these concepts and a real-world domain. Students were encouraged to come up with their own ways of solving the problem, and much of the lesson focused on discussing the validity of the methods they devised.
Finally, students were left to reflect on their activity and to ponder a new dimension, the addition of the bird to the string. There was a clear sense of what the next step would be as the class pursued the topic further.
Example 2: The Game of Pig (US-071) US-071 was also judged by the teacher to exemplify current ideas about the teaching and learning of mathematics. On a superficial level, the lesson had much in common with the one we just described: It included a hands-on learning experience, working in groups, and a writing assignment. However, we judged this to be less in line with NCTM standards than the previous lesson. Let us try to explore the reason for our judgment.
The lesson started by asking students to recall the game of Pig, which they had played previously.
Take 5 minutes, the teacher told the students, and write down everything you remember about the rules of the game.
The game of Pig is played with dice. Students work in groups. One student rolls a pair of dice, and all students who are playing receive the number of points that is the product of the two numbers rolled.
The process is repeated, and the number of points on each turn is added to each player's total. However, if a one is rolled, players receive no points for that turn; and if two ones are rolled, players lose all of their points and have to start over. Players may elect to stop playing at any time, in which case they are left with whatever number of points they have accumulated up to that point.
After students completed the 5-minute writing assignment, the teacher went over the rules of the game. She then told the students that today they would play the game twice, first with 6-sided dice, then with 10-sided dice. "We will try to see if you can get a higher score playing with 10-sided dice than with 6," the teacher remarked. She told students to prepare a score sheet, then handed out the 6-sided dice to each group of students and the play began. Halfway through the period the teacher handed out
the 10-sided dice. At the end of the lesson, students were given a 10-minute writing assignment:3
This teacher reported, on her questionnaire, that this lesson dealt with probability and uncertainty.
When asked what she wanted students to learn from the lesson she wrote: "How does theoretical probability compare with actual experimental data?" And indeed, it would seem possible to develop a lesson on probability that involved comparing 6-sided with 10-sided dice. However, when we watch the lesson and read the transcript in detail, we find no evidence that this lesson involved probability. Virtually all of the mathematical talk that went on in the lesson concerned multiplication of single-digit numbers and addition (i.e., those operations required to keep score). In our judgment, that was the mathematics that students were getting out of this lesson.
Twice the teacher brought up probability for discussion, but in each case, she failed to pursue the discussion for more than a few seconds. The first such instance occurred near the beginning of the lesson
when the teacher was explaining the rules of the game. She was discussing what happens when doubleones are rolled on the dice:
A double slash mark (//) on two succeeding turns in a transcript indicates overlap in speech. Thus, the // in the beginning of the turn at 40:29 indicates that this utterance (which was inaudible, as indicated by the blank space between the parentheses) started right after the teacher said "Venn diagram" in the preceding turn. An explanation of this and other transcription conventions is included as appendix F.
There was no attempt on the teacher's part to explain how knowing the probability of a dice throw might influence the playing of the game. No additional problem was posed for which students could use probability, and there was no discussion of probability throughout the rest of the lesson.
The other point at which the teacher asked a question that could have led to a discussion of probability happened 34 minutes into the lesson. The teacher suggested that students who kept ending up with a score of zero might want to reconsider their strategy. However, the point was dropped after this isolated comment and did not lead into any discussion of probability, game strategy, or the relation between the two. It is interesting that the teacher raised the question of strategy. One could, from there, potentially get into a discussion of probability theory. However, there really is no clear strategy for this game, which makes it harder for the teacher to get a discussion going about strategy.
At the end of the questionnaire we asked the teacher which part of the lesson exemplifies current
ideas about the teaching and learning of mathematics and why. This teacher wrote:
Students were involved in a hands-on, interactive activity. They were allowed to come to their own conclusions about their experience. They were required to communicate their experience to others, both verbally and in writing.
It is clear to us that the features this teacher uses to define high quality instruction can occur in the absence of deep mathematical engagement on the part of the students.
Example 3: A Non-Reformer (US-062) Let us briefly present one more example, this from a teacher who, although she claimed to be "very aware" of current ideas about the teaching and learning of mathematics, judged her videotaped lesson to be "only a little" in accord with these current ideas. The lesson dealt with factoring of polynomials, and discussed factoring in the context of slope and the solving of simultaneous equations. This teacher stated the goal of the lesson in terms of student understanding: "I wanted the students to arrive at an understanding of what factoring is, and to be able to use the language?' In fact, this lesson showed a great emphasis on student thinking and understanding and appeared to us to be in line with NCTM standards. It is an interesting example to consider because the teacher did not see it as in accord with current ideas.
At the beginning of the lesson students were given back a test they had taken the previous day that dealt with factoring of polynomials. She asked students to go over the tests in their groups, for 5 minutes, discuss the questions they got wrong, and then decide on one item, presumably the most problematic one, to present to the class. As the students deliberated, the teacher went from group to group answering questions and facilitating discussion.