Mathematical skills of eighth graders from 20 countries were recently analyzed to provide insight into problems with educational practice. Kikumi K. Tatsuoka and James E. Corter, Columbia University, and Curtis Tatsuoka, George Washington University, used a new diagnostic- testing approach to compare mathematics achievement on the Third International Math and Science Study-Revised (TIMSS-R).

Twenty-three specific content, knowledge and processing sub-skills were measured using items on the TIMSS test. When average mastery levels for each sub-skill were compared for these 20 countries, clear differences among the countries were found in patterns of sub-skill achievement. U.S. students were strong in some content and quantitative reading skills, but weak in geometry. Unexpectedly, these researchers discovered that success in geometry was found to be highly associated with important mathematical thinking skills.

Tatsuoka et al. used a technology called the space-rule method to analyze patterns of scores on groups of individual items on the TIMSS test to measure knowledge and skills that are not directly observable. They compared performance differences in content, processes and skills.

### US students scored low in geometry

U.S. students did not perform well on geometry and higher-level mathematical thinking skills, although they had relatively high achievement in algebra, estimation, and quantitative and logical reading. It was unexpected that algebra did not correlate with important mathematical thinking skills. Instead, it appears that geometry might be a gateway skill to higher-order reasoning skills, just as algebra is considered a gateway skill in applied math and technical fields. The emphasis on teaching algebra to eighth graders in the United States may be misplaced.

Among the 20 countries studied, there were four distinct achievement clusters. The top group included Singapore, Korea and Hong Kong, followed by Japan, Belgium, the Czech Republic and the Netherlands. The second group included Italy, Jordan and Russia. The third group consisted of England and Finland with Australia, Canada and the United States clustered at the lower end. The lowest group included Turkey, Indonesia, Chile, the Philippines and finally Israel. Three out of the four groups share common cultural similarities or languages. These researchers question whether these similarities reflect common school and curriculum practices that influence students’ achievement.

Tatsuoka et al. believe that the investigation of cognitive processing, mathematical thinking skills and knowledge can lead to new insights that might improve educational practice and help diagnose students’ achievement.

These results show that the highest-achieving countries in the eighth-grade TIMSS-R math test achieved their high rankings in different ways. For example, Singapore students obtained the best performance mainly by showing excellence in reading and computational skills, while Japanese students achieved high scores by demonstrating excellent higher-order thinking skills. Belgian students were among the highest because of their strength in fractions and proportional reasoning skills. Korean and Hong Kong students showed relatively balanced knowledge and skills. Industrialized countries that were not high achievers, like Canada, England and the United States, tended to show weaker scores in higher-level mathematical thinking skills. In particular, the United States did not perform well in geometry, which was highly correlated with other important math skills such as proportional reasoning, judgment in the application of knowledge, concepts and properties, and managing data.

These findings suggest that the curriculum in the United States should put more emphasis on teaching geometry, and through that, to teach logical reasoning and higher-level judgmental skills. While algebra is related to good computational skills, it does not appear to help students acquire higher order thinking skills. These researchers suggest that the elementary algebra appropriate for students this age may not engage them in higher-order thinking.

In addition, aspects of the U.S. curriculum, textbooks and teaching practices may also be related to these weaknesses. There are significant differences in U.S. textbooks and those used by higher-scoring countries. U.S. texts are less focused, and cover far more topics than is typical internationally.

### Differences in educational practices

The TIMSS-R videos of mathematics classrooms also reveal significant differences in educational practice between countries. For example, teachers asked different kinds of questions. Japanese teachers emphasized divergent thinking in problem solving and the solution of open-ended problems. Students were asked to solve non-routine problems entirely on their own, using any method they chose. Japanese students were encouraged to think about how to solve the problem rather than actually solving it. Teachers let students go through the process of identifying a problem, investigating solution methods, sharing individual thoughts and collectively arriving at a conclusion. Math instruction in the United States emphasized mastery of principles and procedures and the production of correct answers, despite reform ideas that encourage teachers to shift their instruction toward non-routine problem solving.

*“Patterns of Diagnosed Mathematical Content and Process Skills in TIMSS-R Across a Sample of 20 Countries”, American Educational Research Journal, Volume 41, Number 4, Winter 2004, pp. 901-926.*

**Published in ERN April 2005 Volume 18 Number 4**