# Geometry

## Contrasting Cases in Geometry: Think Alouds with Students about Transformations

There is strong empirical evidence in support of learning from comparisons in mathematics education research (Rittle-Johnson & Star, 2007; Star, Pollack, et al., 2015; Star et al., 2016). Comparisons have produced gains in students’ procedural knowledge, flexibility, and conceptual knowledge of algebra (Lynch & Star, 2014; Star, Newton, et al., 2015; Star, Pollack, et al., 2015). The Animated Contrasting Cases in Geometry project seeks to extend this research and transform the learning of geometry for middle school students by designing a supplementary digital animated curriculum.

Author/Presenter

Erin E. Krupa

Brianna Bentley

Joshua P. Mannix

Year
2021
Short Description

There is strong empirical evidence in support of learning from comparisons in mathematics education research. The Animated Contrasting Cases in Geometry project seeks to extend this research and transform the learning of geometry for middle school students by designing a supplementary digital animated curriculum. This paper focuses on the Transformations unit, which is one of four units.

## Animated Contrasting Cases in Geometry

In this collection of materials, four geometric topics are covered in animated, digital materials.

Author/Presenter

Erin Krupa

Jon Star

Brianna Bentley

Josh Mannix

Year
2019
Short Description

In this collection of materials, four geometric topics are covered in animated, digital materials. There are also paper-based materials for the four geometric topics: Angles, Transformations, Pythagorean Theorem, and Volume. These topics are discussed in scenarios of contrasting cases, where two fictional students each present a unique method or solution strategy to the same problem. The goal is then to analyze both methods and discuss similarities and differences, strengths and weaknesses of each.

## Competencies and Behaviors Observed When Students Solve Geometry Proof Problems: An Interview Study with Smartpen Technology

This peer-reviewed research journal publication addresses one of the grant goals with respect to how students performed on a set of proof tasks. Student work was documented through the use of smartpen technology which allowed the researchers to "track" students' written work on the proof tasks as well as hear the students' explanations of their thinking about the tasks. Although the two tasks highlighted in this paper were relatively routine triangle congruent proofs, only 7 out of 23 of the sampled students were successful on both proofs.

Author/Presenter

Michelle Cirillo

Jenifer Hummer

Year
2021
Short Description

This peer-reviewed research journal publication addresses one of the grant goals with respect to how students performed on a set of proof tasks.

Resource(s)

## Engaging Students with Non-routine Geometry Proof Tasks

Students who earned high marks during the proof semester of a geometry course were interviewed to understand what high-achieving students actually took away from the treatment of proof in geometry. The findings suggest that students had turned proving into a rote task, whereby they expected to mark a diagram and prove two triangles congruent.

Author/Presenter

Michelle Cirillo

Year
2018
Short Description

Students who earned high marks during the proof semester of a geometry course were interviewed to understand what high-achieving students actually took away from the treatment of proof in geometry. The findings suggest that students had turned proving into a rote task, whereby they expected to mark a diagram and prove two triangles congruent.

## Building Mathematical Knowledge for Teaching Proof in Geometry

Presentation slides and handout from the 2019 NCTM Regional Conference in Nashville, TN.

Author/Presenter

Michelle Cirillo

Year
2019
Short Description

Presentation slides and handout from the 2019 NCTM Regional Conference in Nashville, TN.

Resource(s)

## Linear Algebra and Geometry

Linear Algebra and Geometry is organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field.

Author/Presenter

Al Cuoco

Kevin Waterman

Bowen Kerins

Elena Kaczorowski

Michelle Manes

Year
2019
Short Description

Linear Algebra and Geometry is aimed at preservice and practicing high school mathematics teachers and advanced high school students looking for an addition to or replacement for calculus. The materials are organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field.

## Addressing Misconceptions in Secondary Geometry Proof

Cirillo, M. & Hummer, J. (2019). Addressing misconceptions in secondary geometry proof. Mathematics Teacher, 112(6).

Author/Presenter

Michelle Cirillo

Jenifer Hummer

Year
2019
Short Description

Use these ideas to diagnose and address common conceptual obstacles that inhibit students’ success.

Resource(s)

## Thinking Scientifically in a Changing World

Shifting people’s judgments toward the scientific involves teaching them to purposefully evaluate connections between evidence and alternative explanations.

Lombardi, D. (2019). Thinking scientifically in a changing world. Science Brief: Psychological Science Agenda, 33(1). Retrieved from https://www.apa.org/science/about/psa/2019/01/changing-world.aspx

Author/Presenter

Doug Lombardi

Year
2019
Short Description

Shifting people’s judgments toward the scientific involves teaching them to purposefully evaluate connections between evidence and alternative explanations.

## Scaffolding Scientific Thinking: Students’ Evaluations and Judgments During Earth Science Knowledge Construction

Critical evaluation underpins the practices of science. In a three-year classroom-based research project, we developed and tested instructional scaffolds for Earth science content in which students evaluate lines of evidence with respect to alternative explanations of scientific phenomena (climate change, fracking and earthquakes, wetlands and land use, and formation of Earth’s Moon).

Author/Presenter

Doug Lombardi

Janelle M. Bailey

Elliot S. Bickel

Shondricka Burrell