Teacher Practice

Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof

Lead Organization(s): 
Year: 
2020
Short Description: 

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes. In each section, organized by grade band, authors adopt particular conceptions of argumentation, justification, and proof, and they analyse one data set from each perspective. In addition, each section includes a synthesis chapter from an expert in the field to bring to the fore potential implications, as well as new questions, raised by the analyses. Finally, a culminating section considers the use of each conception across grade bands and data sets.

Domain appropriateness and skepticism in viable argumentation

Lead Organization(s): 
Year: 
2020
Short Description: 

Several recent studies have focused on helping students understand the limitations of empirical arguments (e.g., Stylianides, G. J. & Stylianides, A. J., 2009, Brown, 2014). One view is that students use empirical argumentation because they hold empirical proof schemes—they are convinced a general claim is true by checking a few cases (Harel & Sowder, 1998). Some researchers have sought to unseat students’ empirical proof schemes by developing students’ skepticism, their uncertainty about the truth of a general claim in the face of confirming (but not exhaustive) evidence (e.g., Brown, 2014; Stylianides, G. J. & Stylianides, A. J., 2009). With sufficient skepticism, students would seek more secure, non-empirical arguments to convince themselves that a general claim is true. We take a different perspective, seeking to develop students’ awareness of domain appropriateness (DA), whether the argument type is appropriate to the domain of the claim. In particular, DA entails understanding that an empirical check of a proper subset of cases in a claim’s domain does not (i) guarantee the claim is true and does not (ii) provide an argument that is acceptable in the mathematical or classroom community, although checking all cases does both (i) and (ii). DA is distinct from skepticism; it is not concerned with students’ confidence about the truth of a general claim. We studied how ten 8th graders developed DA through classroom experiences that were part of a broader project focused on developing viable argumentation. 

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

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

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.

“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.

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