Abstract
Many different methods of multidigit computation have been used historically and are now used around the world, but the context in which multidigit computation now occurs has changed. The worldwide availability of electronic calculators has decreased the need for complex computations. The emphasis now can be on understanding methods as well as performing them. This paper outlines a research program conducted over thirty years to find and test multidigit computation methods that are mathematically desirable and that many kinds of students and teachers can understand and explain. A nurturing Math Talk classroom environment, in which students made and explained math drawings supported sense-making by students and teachers. Powerful and simple math drawings were also developed and assessed. The methods and math drawings identified by this research for multidigit adding, subtracting, multiplying, and dividing are described. Examples are given of student explanations with the drawings. The criteria for deciding which methods are mathematically desirable are given, and the methods are judged by these criteria. Some methods that are common in various countries but that are difficult and may stimulate errors are described so that they might be replaced by the best methods identified by this research. How these methods fit the math standards of two different countries, the United States and China, is described. Sense-making about and using the identified best methods can reduce errors and engender understanding.
Highlights
Many different methods of multidigit computation have been used historically and are used around the world [1,2,3,4,5,6,7]
There was an increasing emphasis around the world on students and teachers understanding mathematics and on students inventing and sharing their methods. These changing contexts for computation led me to these research questions: 1. Are there multidigit computation methods that are mathematically desirable and accessible to many kinds of students and teachers? 2
Step 1: Gather and analyze multidigit computation methods: Multidigit computation methods were identified in the following kinds of sources: a. methods used around the world historically [7] b. textbooks from African, Asian, European, and North and South American countries; c. research articles and summaries [1,2,3,4,5,6]; d. conversations with colleagues and students at national and international conferences
Summary
Many different methods of multidigit computation have been used historically and are used around the world [1,2,3,4,5,6,7]. There was an increasing emphasis around the world on students and teachers understanding mathematics and on students inventing and sharing their methods. These changing contexts for computation led me to these research questions: 1. Are there multidigit computation methods that are mathematically desirable and accessible to many kinds of students and teachers? The results focus on two mathematically desirable and accessible methods for each kind of multidigit computation (addition, subtraction, multiplication, division). These methods are contrasted with methods considered “the standard algorithm” and taught in most textbooks in the United States. A nurturing Math Talk classroom environment supported sense-making by students and teachers [9]
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