page_title: Singapore math
section_title: Three-step learning process
hierarchical_section_title: Singapore math / Features / Three-step learning process
caption_attribution_description: English: A part-whole model used to solve an addition problem. This pictorial approach is typically used as a problem-solving tool in Singapore Math.
context_page_description: Singapore math is a teaching method based on the national mathematics curriculum used for kindergarten through sixth grade in Singapore. The term was coined in the United States to describe an approach originally developed in Singapore to teach students to learn and master fewer mathematical concepts at greater detail as well as having them learn these concepts using a three-step learning process: concrete, pictorial, and abstract. In the concrete step, students engage in hands-on learning experiences using concrete objects such as chips, dice, or paper clips. This is followed by drawing pictorial representations of mathematical concepts. Students then solve mathematical problems in an abstract way by using numbers and symbols.
  The development of Singapore math began in the 1980s when the country's Ministry of Education developed its own mathematics textbooks that focused on problem solving and heuristic model drawing. Outside Singapore, these textbooks were adopted by several schools in the United States and in other countries such as Canada, Israel, the Philippines and the United Kingdom.
context_section_description: Singapore math teaches students mathematical concepts in a three-step learning process: concrete, pictorial, and abstract. This learning process was based on the work of an American psychologist, Jerome Bruner. In the 1960s, Bruner found that people learn in three stages by first handling real objects before transitioning to pictures and then to symbols. The Singapore government later adapted this approach to their math curriculum in the 1980s.
  The first of the three steps is concrete, wherein students learn while handling objects such as chips, dice, or paper clips. Students learn to count these objects (e.g., paper clips) by physically lining them up in a row. They then learn basic arithmetic operations such as addition or subtraction by physically adding or removing the objects from each row.
  Students then transition to the pictorial step by drawing diagrams called "bar-models" to represent specific quantities of an object. This involves drawing a rectangular bar to represent a specific quantity. For instance, if a short bar represents five paper clips, a bar that is twice as long would represent ten. By visualizing the difference between the two bars, students learn to solve problems of addition by adding one bar to the other, which will, in this instance, produce an answer of fifteen paper clips. They can use this method to solve other mathematical problems involving subtraction, multiplication, and division. Bar modeling is far more efficient than the "guess-and-check" approach, in which students simply guess combinations of numbers until they stumble onto the solution.
  Once students have learned to solve mathematical problems using bar modeling, they begin to solve mathematical problems with exclusively abstract tools: numbers and symbols.
caption: A bar model used to solve an addition problem. This pictorial approach is typically used as a problem-solving tool in Singapore math.
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