Acknowledgements. Summary. Introduction. Mathematical Induction. Mathematical induction, logical deduction, and empirical induction. Poincare's analysis of mathematical induction. Frege's and Russell's analysis of mathematical induction. Critical review. Conclusion. Reasoning by Mathematical Induction: Piaget's Critique. Piaget on the analysis of Frege and Russell. Piaget on Poincare's analysis. Inhelder and Piaget's (1963) study. Conclusion. Research on the Development of Children's Reasoning. Reasoning. Deduction. Induction. Mathematical induction. Reasoning, Reasons and Responses. Critical method. Rationale for a critical method. Children's Reasoning by Mathematical Induction. Method. Results: statistical analysis of responses. Results: epistemological analysis of reasons for responses. Conservation task. Teams tasks. Recurrence task. Discussion. Hypotheses. Developmental mechanism. Educational Implications in a Constructivist Model of Education. Piaget's model of education. Educational implications in a constructivist pedagogy. Commentary by Damon Berridge. References. Boxes, figures, tables.