Skemp's theory-based approach to problem solving has students learning one new concept at a time in the context of activities that are interesting and engaging but low in mathematically irrelevant material. This facilitates the process of abstraction as progress is made through the relevant concept maps and the child's knowledge structure is developed. The carefully sequenced activities lead to the development of appropriate mathematical models which can be used to generate solutions to problems, which in turn are tested in the original problem situation. This approach contrasts strikingly with approaches that begin with high-noise problem situations and lead to a disorderly development of concepts and processes. For a thorough discussion of the importance of well-structured early concept development to facilitate intelligent problem solving, see Richard Skemp's position paper, "Theoretical Foundations of Problem Solving."
In SAIL through Mathematics students learn to solve problems in an orderly manner, building on a solid foundation. They are deliberately led from verbal problems to physical representations of the objects, numbers, and actions described in the number story (modelling), and from the latter to the mathematical statement, not directly from words to mathematical symbols. They learn how to produce numerical models in physical materials corresponding to given number stories, to manipulate these appropriately, and to interpret the result in the context of the number story. Later, they do this by using written symbols only. (For an example, see Number stories: abstracting number sentences in SAIL 1, pp 254-256.) They are encouraged to draw diagrams to show what they have done, as in Different questions, same answer. Why? (SAIL 1, pp.270-272), to show the connection between grouping and sharing in division. In Treasure chest (SAIL 2, pp. 183-185) they apply problem solving skills as they predict the best strategies for choosing their treasure. In other activities they are helped to invent their own real-world problems, applying the concepts they have learned in practical situations, as in Front window, rear window - make your own (SAIL 2, p. 143).