Research

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.

NCTM Presentation Line of "Good" Fit in Grade 8 Classrooms

Lead Organization(s): 
Year: 
2018
Short Description: 

This presntation addreses 4 research cquestions

 

What extant criteria do Grade 8 students use to choose the better line
of fit between two lines “fit” to a set of data, when both lines express
the trend of the data?
 
Is a residual criterion accessible and useful to Grade 8 students when
learning about line of fit?
 
How does introducing a residual criterion impact student
understanding of line of fit and their understanding mathematical
modeling process?
 
What stages of learning do students express as they engage in our
lesson?
Resource(s): 

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.

Teachers Collaborating in Communities of Mathematics Immersion

Beyond initial college preparation, secondary teachers in the United States have few professional opportunities to do and learn challenging mathematics, especially incollaboration with colleagues. The Mathematics Immersion for Secondary Teachers at Scale program engages sets of teachers in local school sites, connected synchronously and asynchronously to colleagues in other sites, in doing mathematics designed to promote experiences of mathematical immersion, community, and connection to the work of teaching.

Author/Presenter: 
Daniel J. Heck
Pippa Hoover
Evelyn M. Gordon
Matthew McLeod
Lead Organization(s): 
Year: 
2020
Short Description: 

The Mathematics Immersion for Secondary Teachers at Scale program engages sets of teachers in local school sites, connected synchronously and asynchronously to colleagues in other sites, in doing mathematics designed to promote experiences of mathematical immersion, community, and connection to the work of teaching. This study of two groups of sites over one year examines fidelity to the program as a model for systematically providing these opportunities, and the extent to which teacher participants experienced immersion, community, and connection in their collaborative work with the course facilitator and their local and distant colleagues.

Developing Transmedia Engineering Curricula Using Cognitive Tools to Impact Learning and the Development of STEM Identity

This paper examines the use of Imaginative Education (IE) to create an NGSS-aligned middle school engineering curriculum that supports transfer and the development of STEM identity. In IE, cognitive tools—such as developmentally appropriate narratives, mysteries and fantasies—are used to design learning environments that both engage learners and help them organize knowledge productively. We have combined IE with transmedia storytelling to develop two multi-week engineering units and six shorter engineering lessons.

Author/Presenter: 
Glenn W. Ellis
Jeremiah Pina
Rebecca Mazur
Al Rudnitsky
Beth McGinnis-Cavanaugh
Isabel Huff
Sonia Ellis
Crystal M. Ford
Kate Lytton
Kaia Claire Cormier
Year: 
2020
Short Description: 

This paper examines the use of Imaginative Education (IE) to create an NGSS-aligned middle school engineering curriculum that supports transfer and the development of STEM identity.

Resource(s): 

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