Teacher Practice

Eliminating counterexamples: A case study intervention for improving adolescents’ ability to critique direct arguments

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples.

Author/Presenter

Carolyn Maher

Year
2020
Short Description

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’ conceptions about the validity of a direct argument after the students received instruction on the ECE framework. We describe student activities in response to the intervention, and we identify students’ conceptions that are inconsistent with canonical notions of mathematical proving and appear to be barriers to using the ECE framework.

Eliminating counterexamples: An intervention for improving adolescents’ contrapositive reasoning

Students’ difficulties with contrapositive reasoning are well documented. Lack of intuition about contrapositive reasoning and lack of a meta-argument for the logical equivalence between a conditional claim and its contrapositive may contribute to students’ struggles. This case study investigated the effectiveness of the eliminating counterexamples intervention in improving students’ ability to construct, critique, and validate contrapositive arguments in a U.S. eighth-grade mathematics classroom.

Author/Presenter

David Yopp

Lead Organization(s)
Year
2020
Short Description

Students’ difficulties with contrapositive reasoning are well documented. Lack of intuition about contrapositive reasoning and lack of a meta-argument for the logical equivalence between a conditional claim and its contrapositive may contribute to students’ struggles. This case study investigated the effectiveness of the eliminating counterexamples intervention in improving students’ ability to construct, critique, and validate contrapositive arguments in a U.S. eighth-grade mathematics classroom. The intervention involved constructing descriptions of all possible counterexamples to a conditional claim and its contrapositive, comparing the two descriptions, noting that the descriptions are the same barring the order of phrases, and finding a counterexample to show the claim is false or viably arguing that no counterexample exists.

Resource(s)

“Well That's How the Kids Feel!”—Epistemic Empathy as a Driver of Responsive Teaching

While research shows that responsive teaching fosters students' disciplinary learning and equitable opportunities for participation, there is yet much to know about how teachers come to be responsive to their students' experiences in the science classroom. In this work, we set out to examine whether and how engaging teachers as learners in doing science may support responsive instructional practices.

Author/Presenter

Lama Z. Jaber

Vesal Dini

David Hammer

Lead Organization(s)
Year
2021
Short Description

In this article, the authors present evidence from teachers' reflections that this stability was supported by the teachers' intellectual and emotional experiences as learners. Specifically, they argue that engaging in extended scientific inquiry provided a basis for the teachers having epistemic empathy for their students—their tuning into and appreciating their students' intellectual and emotional experiences in science, which in turn supported teachers' responsiveness in the classroom.

Talk is the Ticket to Teaching Math to English Learners

This article describes one mathematics professional development program designed to support all K-5 students' engagement in productive mathematical discussions, in particular emergent multilingual learners.

Malzahn, K., Sztajn, P., & Heck, D. (October, 2019). Talk is the ticket to teaching math to English learners. The Learning Professional, 40(5).

Author/Presenter

Kristen Malzahn

Paola Sztajn

Daniel Heck

Year
2019
Short Description

This article describes one mathematics professional development program designed to support all K-5 students' engagement in productive mathematical discussions, in particular emergent multilingual learners.

Decomposing Practice in Teacher Professional Development: Examining Sequences of Learning Activities

In this paper, we analyze a PD design, examining its activities and the sequencing of professional learning tasks. We use a theoretical framework typically used in pre-service teacher education to understand the design of one PD program. Our overarching goal is to theorize about how to design PD and sequence professional learning tasks for practicing teachers.

Author/Presenter

Paola Sztajn

Daniel J. Heck

Kristen A. Malzahn

Lara K. Dick

Year
2020
Short Description

In this paper, authors analyze a PD design, examining its activities and the sequencing of professional learning tasks.

Controlled Implementations: Teaching Practice to Practicing Mathematics Teachers

In this chapter, we use the Framework for Teaching Practice (Grossman, et al., 2009) as a conceptual tool for analzying the design of professional development. Although initially developed to examine the education of prospective teachers, we contend that this framework is appropriate for analyzing and supporting the design of professional development. The framework consists of three elements: decompositions, representations, and approximations of practice.

Author/Presenter

Paola Sztajn

Lara Dick

Reema Alnizami

Dan Heck

Kristen Malzahn

Year
2020
Short Description

In this chapter, authors use the Framework for Teaching Practice (Grossman, et al., 2009) as a conceptual tool for analzying the design of professional development.

In the Classrooms of Newly Hired Secondary Science Teachers: The Consequences of Teaching In-field or Out-of-field

Science teachers must sometimes teach outside of their expertise, and this type of teaching assignment is referred to as being out-of-field. Among newly hired teachers, this type of assignment may have a detrimental impact in the development of their instruction. This study explored the classroom instruction of 17 newly hired teachers who were teaching both in-field and out-of-field in the physical sciences during their first three years.

Author/Presenter

Jessica B. Napier

Julie A. Luft

Harleen Singh

Lead Organization(s)
Year
2020
Short Description

Science teachers must sometimes teach outside of their expertise, and this type of teaching assignment is referred to as being out-of-field. Among newly hired teachers, this type of assignment may have a detrimental impact in the development of their instruction. This study explored the classroom instruction of 17 newly hired teachers who were teaching both in-field and out-of-field in the physical sciences during their first three years.

Out-of-Field Teaching in Science

Special issue of the Journal of Science Teacher Education focused on out-of-field teaching in science.

Luft, J. A., Hobbs. L., & Hanuscin, D. (Eds.) (2020). Special issue: Out-of-field teaching in science. Journal of Science Teacher Education, 31(7), 719-820.

Author/Presenter

Julie A. Luft

Linda Hobbs

Deborah Hanuscin

Lead Organization(s)
Year
2020
Short Description

Special issue of the Journal of Science Teacher Education focused on out-of-field teaching in science.

Out-of-Field Teaching in Science

Special issue of the Journal of Science Teacher Education focused on out-of-field teaching in science.

Luft, J. A., Hobbs. L., & Hanuscin, D. (Eds.) (2020). Special issue: Out-of-field teaching in science. Journal of Science Teacher Education, 31(7), 719-820.

Author/Presenter

Julie A. Luft

Linda Hobbs

Deborah Hanuscin

Lead Organization(s)
Year
2020
Short Description

Special issue of the Journal of Science Teacher Education focused on out-of-field teaching in science.

Activating Math Talk: 11 Purposeful Techniques for Your Elementary Students

Many mathematics teachers agree that engaging students in high quality discourse is important for their conceptual learning, but successfully promoting such discourse in elementary classrooms—with attention to the needs of every learner—can be a challenge. Activating Math Talk tackles this challenge by bringing practical, math-specific, productive discourse techniques that are applicable to any lesson or curriculum.

Author/Presenter

Paola Sztajn

Daniel Heck

Kristen Malzahn

Year
2021
Short Description

Activating Math Talk is a resource for promoting high-quality math discourse in grades K-5. The content of the chapters and discussion questions within are meant to spark conversations among teachers, teacher leaders, administrators, and education faculty about how to get all students, in particular emergent multilingual learners, talking about math in productive ways. It presents both a theoretical and practical lens and offers useful frameworks, techniques, and other supports for mathematics instruction.