Conference Program/Abstracts

Abstracts - Theoretical Reports

PAR for the Course: Developing Mathematical Authority

Daniel Reinholz

Abstract: Perceived mathematical authority plays an important role in how students engage in mathematical interactions, and ultimately how they learn mathematics. This paper elaborates the concept of mathematical authority (Engle, 2011) by introducing two concepts: scope and relationality. This elaborated view is applied to a number of peer-interactions in a specialized peer-assessment context. In this context, self-perceived authority influenced the way feedback was framed (as either questions or assertions).

Adapting Model Analysis for the Study of Proof Schemes

David Miller and Todd Cadwalladerolsker

Abstract: This theoretical paper describes model analysis and our adaptation of this method to the study of proof schemes in a transition to proof course. Model analysis accounts for the fact that students may hold more than one idea or conception at a time, and may use different ideas and concepts in response to different situations. Model analysis is uniquely suited to study students' proof schemes, as students often hold multiple, sometimes conflicting proof schemes, which they may use at different times. Model analysis treats each student’s complete set of responses as a data point, rather than treating each individual response as a separate data point. Thus, model analysis can capture information on the self-consistency of a student’s responses. Data was collected in a Transition to Proof course and analyzed using both traditional descriptive statistics and model analysis. We find that model analysis offers significant insights not offered by traditional analysis.

Implications of Realistic Mathematics Education for Analyzing Student Learning

Estrella Johnson

Abstract: The primary goal of this work is to articulate a theoretical foundation based on Realistic Mathematics Education (RME) that can support the analysis of student learning, both individual and collective, by documenting changes in local activity. To do so, I will build on previous work on the analytic implications of the Emergent Perspective, such as Rasmussen and Stephan’s (2008) analytic approach to documenting the establishment of classroom mathematical practices. The Emergent Perspective is broadly consistent with RME, but the existing analytic methods related to the Emergent Perspective fail to draw on the theoretical constructs provided by RME. For instance, current analytic methods fail to draw on the RME Emergent Models heuristic to inform the analysis of the development of mathematical practices related to models of/for student mathematical activity. Here I will be explicitly considering the roles that RME constructs could play in analytic processes consistent with the Emergent Perspective.

Developing Hypothetical Learning Trajectories for Teachers’ Developing Knowledge of the Test Statistic in Hypothesis Testing

Jason Mark Dolor

Abstract: In the past decade, educators and statisticians have made new suggestions for teaching undergraduate statistics, in light of these new recommendations it is important to (re)evaluate how individuals come to understand statistical concepts and how such research should impact curricular efforts. One concept that plays a major role in introductory statistics is hypothesis testing and the computation of the test statistic to draw conclusions in a hypothesis test. This proposal presents a theoretical approach through the development of a hypothetical learning trajectory of hypothesis testing by utilizing sampling distributions as the building block of coming to understand statistical inference. In addition, this proposal presents a way this hypothetical learning trajectory may support the development of research-based curricula that foster an understanding of the test statistic and its role in hypothesis testing.

Does/Should Theory Building Have a Place in the Mathematics Curriculum?

Hyman Bass

Abstract: Mathematicians distinguish two modes of their practice – problem solving and theory building. While problem solving has a robust presence in the mathematics curriculum, it is less clear whether theory building does, or should, have such a place. I will report on a curricular design to support a kind of simulation of mathematical theory building. It is based on the notion of a “common structure problem set” (CSPS). This is a small set of mathematical problems with a two-part assignment: I. Solve the problems; and II. Find and articulate a mathematical structure common to all of them. Some examples will be presented and analyzed. Relations of this construct to earlier ideas in the literature will be presented, in particular to the notion of isomorphic problems, and cognitive transfer. Designing effective instructional enactments of a CSPS is still very much in an experimental stage, and feedback about this would be welcome.

Developing an Explication Analytical Lens for Proof-oriented Mathematical Activity

Paul Christian Dawkins

Abstract: Sjogren (2010) suggested that formal proof could be understood as an explication (Carnap, 1950) of informal proof. Explication describes the supplanting of an intuitive or unscientific concept by a scientific or formal concept. I clarify and extend Sjogren’s claim by applying Carnap’s criteria for explication (similarity, exactness, and fruitfulness) to definitions, theorems, axioms, and proofs. I synthesize a range of proof-oriented research constructs into one overarching framework for representing and analyzing students’ proving activity. I also explain how the analytical framework is useful for understanding student difficulties by outlining some results from an undergraduate, neutral axiomatic geometry course. I argue that mathematical contexts like geometry in which students have strong spatial and experiential intuitions may require successful semantic style reasoning. This demands that students’ construct rich ties between different representation systems (verbal, symbolic, logical, imagistic) justifying explication as a reasonable analytical lens for this and similar proof-oriented courses.

The Action, Process, Object, and Schema Theory of Sampling

Neil Hatfield

Abstract: This paper puts forth a new theoretical perspective for students’ understanding of sampling. The Action, Process, Object, and Schema Theory for Sampling serves as a potential bridge between Saldanha’s and Thompson’s Multiplicative Conception of Sampling and APOS Theory. This theoretical perspective provides one potential way to describe the development of a student’s conception of sampling. Additionally this perspective differs from most other perspectives in that it does not focus on the sample size the student uses or the sampling method, but rather how the student understands sampling in terms of a sampling distribution.

Illustrating a Theory of Pedagogical Content Knowledge for Secondary and Post-Secondary Mathematics Instruction

Shandy Hauk, Allison Toney, Billy Jackson, Reshmi Nair, and Jengq-Jong Tsay

Abstract: The accepted framing of pedagogical content knowledge (PCK) as mathematical knowledge for teaching has centered on the question: What mathematical reasoning, insight, understanding, and skills are required for a person to teach mathematics? Many have worked to address this question, particularly among K-8 teachers. What about teachers with broader mathematics knowledge (e.g., from algebra to proof-based understandings of topics in advanced mathematics)? There is a need for examples and theory in the context of teachers with greater mathematical preparation and older students with varied and complex experiences in learning mathematics. This theory development piece offers background and examples for an extended theory of PCK as the interplay among conceptually-rich mathematical understandings, experience of teaching, and multiple culturally-mediated classroom interactions.