Projects

"Kinemathics": Kinetically Induced Mathematical Learning

UCB Committee on Research: Faculty Research Grant, 2008-9 [$5k]

Working paper

Current instructional practices in mathematics education can by-and-large be said to focus on students’ development of competencies involving the production of textual, symbolical, and diagrammatic semiotic artifacts through activities such as computation, problem-solving, and logical argument. Research into the embodied nature of mathematical cognition, however, points to the central role of body-based experience in grounding mathematical concepts and enabling problem-solving. We are interested in how students recruit their body-based pre-articulated experiences as a basis for the reflective abstraction of embodied schemata that come to underlie their production of normative mathematical semiotic artifacts. A pilot study conducted by Abrahamson and Fuson (2005) demonstrated that students' cognitive difficulties in understanding proportional progression (e.g., the sequence of equivalent proportions 2:3, 4:6, 6:9, etc.) coincided with their physical difficulty of acting out such progressions with their hands (e.g., one hand grows by 2 units while the other hand simultaneously grows by 3). Our bold conjecture is that a lack of physical coordination might delimit the mental simulation of a concept, and therefore impede its mental construction. If we physically support the students in enacting the ambidexterity of proportional progression, they may be able to draw on their physical experience in developing a dynamic image of proportion, a potential prerequisite for conceptual understanding. Building on this conjecture, we plan to examine whether kinesthetically-induced experiences of proportional progression help students learn the mathematical concept.

Paradigmatic Didactical Mathematical Problematic Situations

In collaboration with Betina Zolkower, Ph.D., Brooklyn College, CUNY.

In the PDMPS project, we are implementing and researching an experimental design for pre-service teachers and future education researchers enrolled in graduate-level college courses on mathematics cognition, learning, and instruction. Central to this design are selected 'paradigmatic didactical-mathematical problematic situations,' i.e., unique activities evoked as contexts for collaborative inquiry into the epistemology, pedagogy, and practice of mathematics as well as into subject matter content. Our data include rich documentation from both the college classroom and the placement classrooms, where the student teachers are trying out the same problems. Building on functional-grammar analysis techniques, we are evaluating the conjecture that the curriculum’s value lies in the authenticity of the multi-disciplinary pragmatic approach it fosters in future teachers. We are also interested in potential tradeoffs inherent to a problem-focused curriculum.

Seeing Chance: Fostering Student Implicit Knowledge Towards Fluency in the Domain of Probability and Statistics

Seed funding from National Academy of Education/Spencer Foundation postdoctoral fellowship to Abrahamson, 2005-6 [$65k.]

Conducting and analyzing a set of one-on-one interviews with Grades 4 - 6 students engaged in probability-related activities. The broader project seeks to develop an empirically based integrated theoretical model of grounded mathematical learning that draws on literature of cognitive sciences, sociocultural theory, and cultural semiotics and applies these sources in the analysis of a corpus of data that includes video footage and artifacts from over 75 clinical interviews with elementary-school, middle-school, and college students who engaged individually in guided problem solving of a concrete situation pertaining to the study of probability (see The Real World as a Trick Question, below). A thematic question guiding the research is the prospect of creating instructional curricula -- including mixed-media materials, activities, and facilitation guidelines -- that would enable all learners to build upon their intuitive inferences a mathematically sound and sophisticated fluency with a range of semiotic tools. The research team approaches the ambitious objective of generating both theoretical models and instructional materials through the design-based research method, a conjecture-driven mode of inquiry that explores phenomena relating to the teaching and learning of mathematics by creating and facilitating innovative experimental units. Through iterated cycles of design, implementation, and mixed-methods analysis, the research group strives to articulate both what works and why it works, so as to accomplish a legacy of actual curricular units as well as theoretical models of learning and principled frameworks for instructional design (see previous and current design).

The Real World as a Trick Question: Mathematical Modeling, Knowledge, and Assessment

UCB Committee on Research: Junior Faculty Research Grant, 2006-7 [$6k]

Conducting and analyzing probability-related clinical interviews with college students majoring in statistics to explore issues of intuitive reasoning.

24 undergraduate/graduate students enrolled in mathematical programs participated in one-to-one interviews as part of a design-based research study of the cognition of probability. The students were asked to estimate outcome distributions of a very simple randomness generator consisting of an exposed bin full of marbles, half green and half blue, and a scooper—a 2-by-2 array of concavities—for drawing out exactly four marbles from the mix. This array formation (4-block) featured also in combinatorial-analysis materials and computer-based simulations of the probability experiment. Central to the design is the combinations tower, an assembly of the 16 unique outcomes in the form of a 1:4:6:4:1 "picto-barchart," i.e., with the outcomes themselves, not just stark columns as in regular histograms. All students said that the relatively most common experimental outcome should have 2 green and 2 blue marbles, but only 10 students initiated combinatorial analysis as a means of warranting their guess, of whom only 4 conducted it successfully. For all students, the combinations tower constituted a context for coordinating between the sample space of the stochastic device and distributions of actual outcomes in experiments with this device. I argue for the utility of guided, situated problem solving for the learning and consolidation of probability concepts.(see full AERA 2007 paper).

Handing Down Mathematics: The Roles of Gesture in the Design, Teaching, and Learning of Ratio and Proportion

Analyzing videotaped classroom interactions to understand the roles of gesture in the design, teaching, and learning of mathematics.

Distributed Learning in Practice and Theory

Tackling distributed-learning theoretical models from a complexity-studies perspective to frame the design and implementation of agent-based models and their extensions that support participatory simulations in mathematics classrooms; Using agent-based modeling to study and develop theoretical models of group learning (read sample paper).

The Three M's: Imagination, Embodiment, and Mathematics

Research into the mechanisms and potential agency of imagination in mathematical reasoning (read abstract).

Fractal Village

Design-based research utilizing a critical and constructionist pedagogical philosophy in an alternative high school setting to study mathematical agency, computational literacy, and identity.