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Discovering Trigonometric Identities in Geometric Representations

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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Mathematical Thinking and Learning Through Robotics Play for Children With Critical Illness: The Case of Amelia

Megan Nickels and Craig J. Cullen

A 14-year-old child with Acute Lymphoblastic Leukemia participated in 52 weeks of robotics task-based interviews. We present 3 of her tasks from Weeks 1, 20, and 46 along with an overview of the complete 52 weeks. We compare the data from the tasks to Brousseau's (1997) Theory of Didactical Situations of Mathematics to answer our research questions: Can robotics play support the devolution of a fundamental situation to an adidactic situation of mathematics for children who are critically ill? When children with critical illness engage in robotics play, what are the key features of the robotics phenomenon that support devolution to an adidactic situation? We found evidence of the robotics supporting the devolution of a fundamental situation to an adidactic situation of mathematics in each robotics task and evidence of 4 key features (thick authenticity, feedback enabling autonomy, connectivity, and competence) of robotics play that support this devolution.

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Technology: Reflecting and Moving Forward

Joshua T. Hertel and Craig J. Cullen

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Improving Our Criticism of Technology in Mathematics Education

Craig J. Cullen and Joshua T. Hertel

Rather than centering technology, we need to view tools as raw materials that students can use strategically to build mathematical knowledge.

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Construct It! Constructing Models of Relationships

Craig J. Cullen, Lawrence Ssebaggala, and Amanda L. Cullen

This activity focuses on reasoning about the relationship between two quantities and reflecting on how this is related to rate of change and functions.

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Technology Tips: Investigating Extrema with GeoGebra

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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All the Way Around a Circle: An Angle Lesson

Amanda L. Cullen, Carrie A. Lawton, Crystal S. Patterson, and Craig J. Cullen

In this lesson, third graders were asked how many degrees is a full rotation around a circle. After we gave students time and space to disagree, to make and test conjectures, and to explore, they reasoned about angle as turn and determined a full rotation is 360 degrees.

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Effects of Three Interventions on Children's Spatial Structuring and Coordination of Area Units

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.