The focus of this report is children's construction and analogical transfer of mathematical knowledge during novel problem solving, as reflected in their strategies for dealing with isomorphic combinatorial problems presented in "hands-on" and written form. Case studies of 9-year-olds, one low and one high achieving in school mathematics, serve to illustrate a general progression through three identified stages of strategy construction (non-planning stage, transitional stage, and odometer stage). The important role of domain-general strategies in this development is highlighted. It was found that achievement level in school mathematics does not predict children's attainment of the third stage, as evidenced by the low-achieving student's construction of sophisticated combinatorial knowledge and the high-achieving student's failure to do so. Children's ability to recognize structural correspondence between two isomorphic problem sets and the extent to which this facilitates problem solution are also reported. The study concludes that: (1) Children can construct important mathematical ideas through solving novel problems; (2) Level of achievement in school mathematics is not a reliable predictor of ability to solve novel problems; (3) Bright students' ability to generate ideas for themselves can be inhibited by formal mathematical rules; and (4) Assessment of students' mathematical competence must include a range of novel problems. (Contains 72 references.) (MDH)