Acquisition of Arithmetic (Learning and Mathematics)
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H. Ginsburg; Math Forum  
In a chapter entitled "Learning to Count. Computing with Written Numbers. Mistakes" in the 1977 book, Children's Arithmetic: How They Learn It and How You Teach It, Ginsburg draws heavily on the idea of assimilation  the incorporation of new ideas into an existing body of knowledge  to explain how children acquire or misacquire arithmetical skills and concepts. He looks at both the informal, concrete understanding of basic concepts that children acquire before entering school and the abstract, formal concepts and computations they are expected to learn in the classroom. The difference between such formal and informal knowledge often results in a gap between the ability to do paperandpencil calculations and intuitive understanding; sometimes students actually have strong informal abilities not indicated by their performance on school tasks, and sometimes they master formal algorithms without understanding the concepts behind them. Ginsburg focuses on students just entering school, but his ideas generalize to older students, for example calculus students who can take derivatives but can't explain the problems or their answers. A geometry.precollege newsgroup discussion.  


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