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Description: Child uses the distributive property and a halving strategy to multiply 6 times 12. Purpose: This video clip is used to demonstrate to preservice teachers the complexity of thought used by many elementary school children to solve problems.

Description: Child uses her basic knowledge of fractions to add 1/2 + 3/4, despite never having been taught an algorithm for adding fractions. Purpose: This video clip is used to demonstrate how a deep conceptual understanding is generative and enables children to extend and apply their thinking to new situations.

Description: Child uses his understanding of division to think through 1 ÷ 1/3. He applies this understanding to a similar problem, 1 1/2 ÷ 1/3, but with limited success. Purpose: This video clip is used to demonstrate how children can make sense of difficult concepts and how partial understandings can exist at the same time. The second problem also challenges the understanding of some preservice teachers.

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