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Craig J. Cullen and Tami S. Martin

Proving trigonometric identities are some students' least-favorite lessons. For us, those proofs are enjoyable puzzles for which the right algebraic manipulation leads to the desired outcome, but our students did not always find the same satisfaction in untangling those algebraic knots.

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Craig J. Cullen, Joshua T. Hertel and Sheryl John

Technology can be used to manipulate mathematical objects dynamically while also facilitating and testing mathematical conjectures. We view these types of authentic mathematical explorations as closely aligned to the work of mathematicians and a valuable component of our students' educational experience. This viewpoint is supported by NCTM and the Common Core State Standards for Mathematics (CCSSM).

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Amanda L. Cullen, Cheryl L. Eames, Craig J. Cullen, Jeffrey E. Barrett, Julie Sarama, Douglas H. Clements and Douglas W. Van Dine

We examine the effects of 3 interventions designed to support Grades 2–5 children's growth in measuring rectangular regions in different ways. We employed the microgenetic method to observe and describe conceptual transitions and investigate how they may have been prompted by the interventions. We compared the interventions with respect to children's learning and then examined patterns in observable behaviors before and after transitions to more sophisticated levels of thinking according to a learning trajectory for area measurement. Our findings indicate that creating a complete record of the structure of the 2-dimensional array—by drawing organized rows and columns of equal-sized unit squares—best supported children in conceptualizing how units were built, organized, and coordinated, leading to improved performance.