Models of effective mathematics instruction

iStock_000016075406XSmallThe 1995 TIMSS (Third International Mathematics and Science Study) Video Study of mathematics teaching led many educators to conclude that copying Japanese teaching methods would produce high achievement. A larger and more ambitious study of math and science teaching, conducted in 1999, studied teaching in seven high-achieving countries. This expanded study of mathematics teaching included Australia, the Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland and the United States. The analysis of this study was recently released by the research group conducting the study.

Like the 1995 study, the 1999 study videotaped randomly selected eighth-grade mathematics lessons. The 1999 study included 638 lessons collected from the seven countries. In addition, the 1995 lessons from Japan were reanalyzed for the current study. All videotapes were analyzed by an international team of researchers representing all the countries in the study. The challenge was to develop a reliable and consistent way of analyzing the lessons that would capture both the similarities and the differences that might influence students’ mathematics learning. Lesson features were coded in three categories: the structure and organization of lessons, the mathematical content, and the way in which lesson content affected how students worked.

Major findings

All the countries shared a number of teaching features. At least 80 percent of lesson time in every country was spent on problems. Most of the lessons devoted some time to whole-class discussion and some to individual student work. Teachers in all the countries did most of the talking, with a ratio of at least 8 teacher words for every student word.

This study reveals that there is more than one effective way to teach mathematics. In this more detailed and expanded analysis, Japan retains its distinctive style. In Japan, students spent three to four times as long working on one problem than students in the other countries. This allowed Japanese students to engage in different kinds of learning experiences such as proving or verifying mathematical statements. Japanese students spent a greater percentage of their individual work time doing things other than practicing procedures. They spent more time analyzing new problems and developing new solution methods. The pattern of teaching in Japan involved introducing a new problem for the day and asking students to work on it; discussing solutions; and then presenting one or two more problems. This pattern was not seen in any other high-achieving country.

Three types of statements of math problems were identified: those emphasizing procedures, those making connections to underlying concepts, and those asking students to define concepts. Countries differed significantly in the percentage of problems that focused on procedural aspects of mathematics. The Czech Republic, Hong Kong and the U.S. all used very high percentages of problems focusing on procedure.

How the U.S. teaching differs from the highest-achieving countries

A close-up analysis of the kind of mathematics presented and how it was used with students reveals a few features that distinguish the highest-achieving countries. Except for the Czech Republic and the U.S., all the higher-achieving countries spent more time working on new content than reviewing old. The U.S. spent about equal time on new work and reviewing previously learned work.

Closer analysis also revealed that although lessons are presented as a certain type of problem (procedural, conceptual or defining), problems can be transformed into something different as they are worked. For example, while 17 percent of problems in the U.S. are presented as conceptual (making connections between ideas, facts, and procedures), virtually none of these conceptual problems in the U.S. were discussed in a way that made mathematical concepts, connections or relationships visible to students. Teachers in the highest-achieving countries paid more attention to concept development than teachers in the U.S.

Further analysis was made by reviewing country-blind written records of a random sample of 20 lessons from each country. Researchers analyzed the degree to which math concepts or procedures were developed during the lesson. Development required that mathematical reasons be given for the results presented. Forty percent of U.S. lessons were completely undeveloped. Students either practiced procedures or gave answers with no apparent development in their understanding of the procedures or concepts involved. No other country had more than 15 percent of its lessons undeveloped in this way. This was a striking difference between U.S. math teaching and instruction in higher-achieving countries.

Variety of approaches yield good results

This in-depth study of math instruction demonstrates the complexity of teaching and learning. The Japanese method of instruction remains unique and highly successful, but analysis of other high-achieving countries reveals that other practices can also produce high achievement. These videos from other countries are a source of alternatives to study. These researchers conclude that high performance on international tests is not a sufficiently defined learning goal to determine the selection of teaching methods. Many different methods are associated with high achievement. More precisely defined goals are needed to analyze the potential benefits of different teaching strategies. This analysis makes clear that it is not just the presence or absence of certain features of teaching that defines classroom practice, but how those features are implemented.

The U.S. falls in the middle among higher-achieving countries in terms of the quality of the problems posed. However, when teachers work problems with their classes, the U.S. is at the bottom because little attention is paid to the conceptual underpinnings of the mathematics. Combined with the relatively light emphasis on new content, it becomes clear that eighthgrade mathematics teaching in the U.S. is more limited than in higher-achieving countries, with a smaller range of learning experiences provided to students.

It is not enough to say that our students should be presented with challenging and conceptually rich problems. How instruction is implemented is equally important. These researchers suggest that our biggest problem is the lack of on-going, system-wide professional development for teachers. There is growing evidence that continuous teacher learning is the key to improving practice.

Videotapes of four lessons from each country a re available on a CD-ROM that includes commentary by the teachers and other educators from each country. Worksheets, textbook pages, lesson plans and the text of the lesson in English and the native language are provided. For information on obtaining the videotapes and supplementary information, see:

“Understanding and Improving Mathematics Teaching: Highlights from the TIMSS 1999 Video Study”, Phi Delta Kappan, Volume 84, Number 10, June 2003, pp. 768-775.

Published in ERN September 2003 Volume 16 Number 6

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