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## Geometry and Geometric Measurement

Geometry is the study of objects, motions, and relationships in a spatial environment. We use it to examine containers, buildings, cars, and playgrounds—familiar things that students see, touch, or move. Because students are naturally interested in these things, geometry can be a highly motivating topic.

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

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.

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## Delving Deeper: Transforming Shapes Physically and Analytically

Constructing formulas “from scratch” for calculating geometric measurements of shapes—for example, the area of a triangle—involves reasoning deductively and drawing connections between different methods (Usnick, Lamphere, and Bright 1992). Visual and manipulative models also play a role in helping students understand the underlying mathematics implicit in measurement and make sense of the numbers and operations in formulas.

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## Enhancing Teachers' Formative Assessment Practices Through Learning Trajectory-Based Professional Development

This research study was designed to evaluate the extent to which professional development (PD) designed around a learning trajectory (LT) on geometric measurement of area was successful in helping teachers use the LT to conduct formative assessment. Six 3rd-grade teachers from the Midwest participated in 20 hours of PD centered on the LT. Data to evaluate the PD were obtained from a set of questionnaire prompts administered before and after teachers' participation in the PD. The results suggest that teachers increased their ability to elicit and interpret student thinking and use assessment results to make instructional decisions. We consider the design and evaluation of this PD to be valuable for future efforts aiming to use LTs to support teachers in their formative assessment practices.

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## Supporting Teacher Learning through “Sketching Standards”

Common Core State Standards in Mathematics (draft). K–5 Geometric Measurement. Tucson, AZ : Institute for Mathematics and Education, University of Arizona . Maruta , Aki , and Chana Stewart . 2017 . “ Facilitating Mathematical Practices through

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## All the Way Around a Circle: An Angle Lesson

Department at Illinois State University in Normal. They are interested in children's understanding of geometric measurement and the use of technology in teaching and learning mathematics. Carrie Lawton , clawton@ilstu.edu , is a faculty member and doctoral

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## Chapter 5: Developing Deep Mathematical Understanding

outcome. Students who have conceptual understanding can identify patterns in geometric measurement formulas, leading to conjectures and generalizations. A deep knowledge of the geometry and measurement domain includes seeing connections to other domains