The features and relationships of reasoning, proving and understanding proof in number patterns

Three issues about students' reasoning, proving and understanding proof in number patterns are investigated in this paper. The first is to elaborate the features of junior high students' reasoning on linear and quadratic number patterns. The second is to study the relationships between 9th graders' justification of mathematical statements about number patterns and their understanding of proof and disproof. The third is to evaluate how reasoning on number patterns is related to constructing proofs. Students in this study were nationally sampled by means of two stages. Some new findings which have not been discovered in some past researches are reported here. These findings include (1) checking geometric number patterns appears to have different positions between the tasks of the linear and the quadratic expressions; (2) proof with the algebraic mode is easy to know but hard to do; (3) disproof with only one counterexample is hard to know but easy to do; (4) arguments with empirical mode or specific symbols were hard for students to validate but very convincing for them; and (5) reasoning on number patterns is supportive for proving in number patterns, and reasoning on number patterns and proof in algebra should be designed as complementary activities for developing algebraic thinking.