I am a mathematician. I am a college professor. I am a mother. From all three perspectives I have been following with interest the controversy over the current mathematics education reform. Last year I had an experience that finally brought clarity. My husband, who is also a mathematician, and I had a sabbatical at the University of Utrecht in the Netherlands. We enrolled our eight year-old son, Robert, in a local Dutch school. In doing so we were unconsciously starting a very interesting experiment. At home Robert had been experiencing a traditional mathematics curriculum where a great deal of time and effort is spent on learning the carrying and borrowing algorithms for addition and subtraction. The mathematics curriculum at his Dutch school was very different. The students were working on problems at the same level, but they were encouraged to develop their own techniques for doing the problems. They were not taught the carrying and borrowing algorithms. This approach has been used successfully in Holland for almost thirty years.At the same time Robert was adapting to a new curriculum, I was studying at the Freudenthal Institute at the University of Utrecht—a world-renowned center for research on mathematics education. I was learning that the curriculum he was experiencing is called Realistic Mathematics Education (RME). In RME, the mathematics is introduced in the context of a carefully chosen problem. 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. Students exchange strategies in the classroom and learn from and adopt each other’s methods. I also learned that much research has been done on this approach, that it is based on what we know about child development and the development of numeracy, and that it is this body of research that is driving the math education reform in our country.
When we first arrived in the Netherlands and I began to learn about RME, I spent a little time quizzing Robert on how he would solve a few addition and subtraction problems. I was shocked by the rigid attitude he had developed at his school in the U.S. When asked to do any addition problem with summands larger than 20 he would always invoke the addition algorithm. He would sometimes make mistakes and then report an answer that made no sense. He was putting all his confidence in the procedure and little in his own ability to reason about what might be a sensible answer. When I suggested there was a simpler way he could think about the problem he became upset and told me, “You can’t do that!”
After a few months in Holland, I began to see an amazing difference in Robert’s number sense. He was able to do the same problems more quickly, more accurately, and with much more confidence. For example, I asked him to solve 702 minus 635. He explained, “700 minus 600 is 100. The difference between 2 and 35 is 33, and 100 minus 33 is 67.” When he tried using the algorithm he made a borrowing error and became very frustrated. I asked him to compute 23 times 12. He explained, “23 times 10 is 230, 23 times 2 is 46, 230 plus 46 is 276.” This multiplication problem was much harder than anything in the curriculum at home. I was very impressed with the flexibility and range of methods he had developed in only a few months.
What happened to Robert in those few months has had a profound effect on my perception of learning and on Robert’s understanding of mathematics. My child learned to think. He learned he could think. He was encouraged to think. He learned to see mathematics as creative and pleasurable. This independent attitude towards mathematics will remain with him forever and serve him well. It is this fact that has convinced me of the value of de-emphasizing algorithms in the elementary years. Unfortunately, Robert is once again back in a school that focuses on the teaching of algorithms. The other day as we were driving to soccer, out of the blue Robert asked from the back seat, “Mommy, wouldn’t it be crazy to do 5000 minus 637 using borrowing?” I smiled proudly at him and said, “Yes, honey, it would.”