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Introduction and Acknowledgement: This book is about a research conducted on student inventions, for teachers and math educators. It reports on how students involved themselves in calculating their work mentally, with little or no use of paper and pencil. Mental computation, in this research, was used to encourage students think more deeply about numbers and number relationship. Sharing mental computation brings about healthy mathematical classroom discourse which refers to ways in which students and teachers think, develop knowledge and express themselves in the classroom. The findings were organised around descriptive moments. This book describes how students, sharing mental computation were absorbed in mathematical conversation when they were to think mathematically and engaged to use their own strategies. The author drew on the epistemology of constructivism in problem solving, to understand how student develop conceptual understanding of mathematics when given the opportunity to explain their own strategies. To Teachers: Mental work looks like just recall of knowledge to many teachers thus avoid its implementation. This study enlightens teachers about how to implement it.