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## Isolate, Then Integrate

by Nina Shteingold
January 2001

The traditional mathematics curriculum in Russian schools consists of two separate subjects that are studied simultaneously: (1) algebra and geometry in middle school (grades 7–9) and (2) introduction to analysis and stereometry, or three-dimensional geometry, in high school (grades 10–11). For the second half of their time in school, all students study two mathematics subjects in parallel. For example, the students of middle school age usually have three hours of algebra and two hours of geometry per week. Yet the two subjects remain distinct; their contents are not well connected.

A typical mathematics lesson is organized around gaining skills. The teacher introduces a concept, presents an example of a problem with a solution, and then gives students practice in solving problems of the same type. A successful mathematics student classifies a problem in the curriculum by the method used to solve it. A method always exists, has been previously defined, and is almost always the unique method learned for a given problem. The student then uses that method—or, rather, locates an example to follow—while solving that problem. Of course, I am not talking here about nontraditional teaching, in which teachers define their own goals and develop their own methods; I am referring only to teaching that is supported by textbooks and teacher materials.

Until just a few years ago, a single curriculum was used by the whole country. Therefore, teachers can usually assume that students know the previously studied material in other subjects. For example, algebraic manipulations are constantly used—and thereby reviewed—even in a geometry class, and definite integrals are used to find volumes of solids. As a result of the single curriculum, students' mathematical knowledge is usually solid.

However, mathematics is a single body with many internal connections and with general ideas that unite this body. That image of mathematics, I believe, is not the one that typical Russian students have. Instead, they see mathematics as many facts carefully organized by topics; each fact is based on previously studied facts.

That description illustrates my opinion that school mathematics in Russia is not integrated. Even with the recent introduction of calculators and with vector algebra widely used in three-dimensional geometry, school mathematics in Russia still consists of a variety of well-structured but isolated topics, with students taught a collection of mathematics facts and skills.

Despite this curricular organization, some Russian students develop a view of mathematics as a united body of knowledge through a variety of extracurricular mathematical events. Such events include "mathematics circles," after-school mathematics meetings, long-distance enrichment mathematics classes, and mathematics Olympiads for interested, not necessarily advanced, students in grades 5 to 10. These activities are offered by the school system, by universities, or by private volunteers, and they are diverse. In these activities, students learn problem-solving methods that can be used in more than one area of mathematics. Or they might solve mathematics problems that are not easily classified by topic and cannot be solved by familiar methods. This type of activity serves as an antidote to the isolation of mathematics topics in school. See Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, and Ilia Itenberg (Washington, D.C.: American Mathematical Society, 1996).

Students who participate in these mathematics activities have an opportunity to reconsider the isolated-topic image of mathematics and see it as a more complexly interrelated body, over which they gain power by being in charge of choosing—or even creating—a solution method or representation of a problem.

I believe that this modified view is possible only because students already have skills to apply and are confident in using these skills. The Russian curriculum's focus on a single topic in a course is responsible for giving students these skills.

 Nina Shteingold has worked in mathematics education in both Russia and the United States. She is currently involved in several curriculum development projects at the Education Development Center, in Newton, Massachusetts.

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