In the 1980s, the island nation of Singapore decided to reform their math curriculum. The country's Ministry of Education brought together the best teachers to developed its own mathematics textbooks that focused on problem solving and heuristic (investigative) model drawing. I happened to be living in Singapore at the time, and watched with interest as they embarked on this new pedagogy. I say “pedagogy” because Singapore looked at how best to teach math, not necessarily focusing on the topics.
Many people argue that math is math, and to a certain degree they are correct. The principles and rules of math don´t change; however, the best way to teach these principles has been under scrutiny since the space race of the 1960s and the invention of “new math," which wasn't really new. Over the past few decades, Singapore has placed at the top of the heap every year on international tests such as Trends in International Mathematics and Science Study (TIMSS, grades 4 and 8).
According to the National Association of Mathematics Teachers, the best mathematics results come from classroom environments that do the following:
- View classrooms as mathematical communities, not collections of individuals
- Use logic and mathematical evidence to verify results rather than relying on the teacher as authority
- Emphasize mathematical reasoning rather than memorizing procedures
- Focus on conjecture, inventing, and problem solving instead of mechanical answer finding
- Make connections among the ideas and applications of mathematics, rather than seeing them in isolated concepts and procedures
The Singapore Math approach embraces these elements, building from the Concrete (hands-on) to the Pictorial (bar modeling) to the Abstract, or CPA. This highly effective approach to teaching, first developed by American psychologist Jerome Bruner, develops a deep and sustainable understanding of mathematics.
The CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects. In the CPA framework, every abstract concept is first introduced using interactive physical materials. Children often find math difficult because it is abstract, so the CPA approach helps children learn new ideas and build on their existing knowledge by introducing those abstract concepts in a more familiar and tangible way.
For example, if a problem involves adding pieces of fruit, children can first handle actual fruit (the concrete stage).
Pictorial is the “seeing” stage. In it, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures, diagrams, or models that represent the objects from the problem.
Abstract is the “symbolic” stage, where children use abstract symbols to model problems. Students will not progress to this stage until they have demonstrated that they have a solid understanding of the concrete and pictorial stages of the problem.
Of course, such an approach, like all effective teaching, depends on having highly trained teachers to guide young people. We are bringing in Singapore math specialists to work with our teachers in August and employing a full time Singapore Math coordinator (Zoe Zarrabi) to mentor teachers all year long. We want our students to enjoy math and become confident and skilled in mathematical problem solving.
To learn more, view this video created by our Singapore Math Coordinator, Zoe Zarrabi.