During the summer, Diane had felt a sense of
power when she began to understand why some
of the arithmetic rules worked. Wanting this ad
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vantage for her students, she listened carefully
in the grade-level meetings as her colleagues
discussed ways to represent a number such as
345 with concrete objects or diagrams that would
illustrate the relative amounts represented by the
digits 3, 4, 5. In this way, the teachers could bet
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ter examine the concepts of “carrying and bor
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rowing” in addition and subtraction with their
class. Gradually Diane, who was used to simply
drilling the children in the steps of the sub
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traction algorithm, began to experiment with
concrete objects and diagrams to discuss the rea
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sons behind these steps with her students. A few
weeks later when I visited Diane’s class, she and
her students explained their work to me. Con
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sidering the effect of this work, Diane was es
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pecially delighted to see how the students’
un
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derstanding
of borrowing improved their ability
to subtract. In the past when she moved from
problems such as (345 minus 157) to (305 minus
157), it had seemed like a whole different process
to her students. Now “moving across the zero
to borrow from the hundreds seemed like no big
deal to the students.