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.
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.
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.
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.
This presntation addreses 4 research cquestions
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.
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.
Young Mathematicians (YM) is a design and development project that aims to broaden participation by addressing the need to provide young children with early mathematics experiences. In the coming year, we will test an intervention, developed in collaboration with teachers and families, that provides learning experiences and materials for teachers and families to support adult-child interaction and engagement in mathematics, promote school-home connections in mathematics, and address adult attitudes toward mathematics, while promoting childrens mathematical knowledge.
In prior work, BSCS studied STeLLA, a video-based analysis-of-practice professional learning (PL) model and found that it enhanced elementary science teacher and student outcomes. But the face-to-face model is difficult to scale. We present the results of a two-year design-based research study to translate the face-to-face PL into a facilitated online experience. The purpose is to create an effective, flexible, and cost-efficient PL model that will reach a broader audience of teachers.
Co-PI(s): Gillian Roehrig, University of Minnesota
The SDLC project has developed and studied curriculum modules for non-AP high school statistics to promote interest and skills in statistical thinking and data analysis among diverse high school populations. Modules engage students with social-justice-themed data investigations using large-scale socioeconomic data from the U.S. Census Bureau and student-friendly online data visualization tools. Current study findings show growth in student interest and skills in statistical thinking and data analysis following module use.
Science Coordinators Advancing a Framework For Outstanding Leadership Development (SCAFFOLD) develops and studies a PD program for District Science Coordinators (DSCs) in one Southeastern state. DSCs can have partial or full responsibility for supporting science teachers in their districts, but little is known about their training and impact on teachers. The goal is to determine the impact of DSCs on teachers and if they are in need of PD to enhance their work with teachers.
Co-PI(s): Brooke A. Whitworth, Clemson University