Learning to Think: An American Third Grader Discovers Mathematics in Holland

By Torrence, Eve | Teaching Children Mathematics, October 2003 | Go to article overview

Learning to Think: An American Third Grader Discovers Mathematics in Holland


Torrence, Eve, Teaching Children Mathematics


As is often the case with academics who are parents of young children, my sabbatical abroad was not only a wonderful learning experience for me but also an opportunity for my children to experience life in a different country. For my eight-year-old son, it was a chance to experience a different education system.

My husband and I are mathematicians and we spent the first six months of 2002 in the Netherlands visiting the University of Utrecht (UU). While I was studying the Dutch philosophy of Realistic Mathematics Education (RME) at the Freudenthal Institute at UU, my son, Robert, was experiencing RME as a student in a Dutch primary school.

Robert was enrolled in a local school in the village in which we lived. When we arrived in January he spoke very little Dutch, but we hoped that he would pick up the language through immersion. Some of the first words he learned were numbers and he quickly learned to count in Dutch to ten, then one hundred, then one thousand. Because we knew that his ability to participate in many academic subjects would be limited, we brought school books from home and he often worked independently in the classroom. From the beginning, however, he did the mathematics lessons with his Dutch classmates.

Meanwhile, I was learning about RME. Founded by Hans Freudenthal, this philosophy is widely respected in the Netherlands, and the ideas of RME are common in many of the curriculum materials that the schools use. RME is particularly strong in the primary schools, where students develop numeracy and strong mental arithmetic skills over a period of many years. Many aspects of RME are similar to the ideas in Principles and Standards for School Mathematics (NCTM 2000), such as emphasizing the development of computational fluency and flexibility and the sharing of solution strategies among students.

In RME, a teacher introduces the mathematics in the context of a carefully chosen problem that is realistic to a child; the child can completely understand the problem and is motivated to try to solve it. In the process of trying to solve the problem, the child develops mathematics.

The teacher uses the method of guided reinvention, by which students are encouraged to develop their own informal methods for doing mathematics. The belief that students should invent their own solution strategies is central to the philosophy of RME. Students exchange solution strategies in the classroom and learn from and adopt one another's methods. Over time, the students accumulate a collection of flexible problem-solving strategies that they have developed as a class. The influence of RME in Robert's third-grade classroom in the Netherlands is noticeable in the informal arithmetic strategies he developed during our stay in Utrecht.

Robert's mathematics instruction during the first half of third grade in the United States was very different. The curriculum included many interesting topics such as patterns, geometry, measurement, probability, and statistics, which the teacher introduced with interesting and creative classroom activities. A great deal of time and effort, however, was spent on learning the carrying and borrowing algorithms for addition and subtraction. Every day, Robert brought home worksheets on which he practiced these algorithms.

When we first arrived in the Netherlands and I began to learn about RME, I quizzed Robert on how he would solve a few problems. I was shocked by his rigid attitude. When I asked him to do an addition problem with addends greater than twenty, he always invoked the addition algorithm. He sometimes would make mistakes and then report an answer that made no sense. He was putting all his confidence in the algorithm and little in his own ability to reason about what might be a sensible answer. When I suggested that a simpler way existed to think about the problem without the algorithm, he became upset and told me, "You can't do that! …

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