- Paradigmatic Didactical Mathematical Problematic Situations
- The Real World as a Trick Question: Mathematical Modeling, Knowledge, and Assessment
- Handing Down Mathematics: The Roles of Gesture in the Design, Teaching, and Learning of Ratio and Proportion
- Distributed Learning in Practice and Theory
- The Three M's: Imagination, Embodiment, and Mathematics
- Fractal Village

**Paradigmatic Didactical Mathematical Problematic Situations**

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

In the PDMPS project, we designed, implemented, and evaluated an experimental curriculum 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 tried out the same problems. We investigated the conjecture that the curriculum's value lies in the authenticity of the multi-disciplinary pragmatic approach it fosters in future teachers. We were also interested in potential tradeoffs inherent to a problem-focused curriculum.

*Graduate students and pre-service teachers working on the Pyramid Problem: learning how to "midwife the birth of algebra"*

Zolkower, B., & Abrahamson, D. (2009). *Studying paradigmatic didactical-mathematical situations: Design and implementation of an experimental graduate level course for pre-service mathematics teachers and doctoral students*. Paper presented at the annual meeting of the American Educational Research Association, San Diego, April 13 – 17.

ABSTRACT: We discuss results from the implementation of 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 tried out the same problems. Building on functional-grammar analysis techniques, we argue that the curriculum’s value lies in the authenticity of the multi-disciplinary pragmatic approach it fosters in future teachers. We discuss potential tradeoffs inherent to a problem-focused curriculum.

Brar, R., Galpern, A. J., & Abrahamson, D. (2006). Lost in translation: The 'bean snare’ as a case of the situated-symbolic divide. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), *The Twenty-Eighth Conference of the North-American Chapter of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 390-391). Mérida, Yucatán, México: Universidad Pedagógica Nacional.

EXCERPT: ...The third author designed the Bean Snare to spark discussion of the complexity of constructivist design, teaching, and learning, i.e., subtle interactions of content and context as well as multi-media, multi-modal, and multi-representational aspects of collaborative reasoning about a situated mathematical problem. Note how the presentation surreptitiously leads us down the garden path to a mathematically incorrect statement....

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

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

This research direction, part of the Seeing Chance project, consisted of conducting and analyzing probability-related clinical interviews with college students majoring in statistics to explore issues of intuitive reasoning.

Abrahamson, D. (2007, April). *The real world as a trick question: Undergraduate statistics majors’ construction-based modeling of probability.* Paper presented at the annual meeting of the American Education Research Association, Chicago, IL.

ABSTRACT: 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.

**Handing Down Mathematics: The Roles of Gesture in Design, Teaching, and Learning**

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

**Publications**

Abrahamson, D. (2007). Handling problems: Embodied reasoning in situated mathematics. In T. Lamberg & L. Wiest (Eds.), *Proceedings of the Twenty Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 219 – 226). Stateline (Lake Tahoe), NV: University of Nevada, Reno.

ABSTRACT: Fifty 4th -17th -grade students participated in individual interviews oriented toward probabilistic intuition. Participants were given a boxful of equal numbers of green and blue marbles, mixed, and a device for scooping 4 ordered marbles and asked to predict the most common sample. Students replied that the outcome with the highest relative frequency would have 2 green and 2 blue marbles. Their verbal reasoning was accompanied by a deictic–metaphoric gesture to the left then right, as if they were separating the colors in the box. Gesture, I argue, bridges direct intuitive grasps of situations to conscious reflection, thus concretizing the prereflective, possibly grounding it in material form such that it emerges as conducive to further elaboration in mimetic symbolic form. Situated mathematical reasoning transpires largely as embodied negotiation among kinesthetic image schemas afforded by available material resources and epistemic forms.

15 | 18 |

25 | 30 |

Fuson, K. C., & Abrahamson, D. (2005). Understanding ratio and proportion as an example of the Apprehending Zone and Conceptual-Phase problem-solving models. In J. I. D. Campbell (Ed.), *Handbook of mathematical cognition* (pp. 213-234). New York: Psychology Press.

*5th-grade student discussing relation between mathematical operation (multiplication) and representation (multiplication table): "Every time you times 5 times one of the other number, it goes up five"*

Abrahamson, D. (2004). Embodied spatial articulation: A gesture perspective on student negotiation between kinesthetic schemas and epistemic forms in learning mathematics. In D. E. McDougall & J. A. Ross (Eds.), *Proceedings of the Twenty Sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 791 – 797). Toronto, Ontario: Preney.

ABSTRACT: Two parallel strands in mathematics-education research—one that delineates students’ embodied schemas supporting their mathematical cognition and the other that focuses on the mediation of cultural knowledge through mathematical tools—could converge through examining reciprocities between schemas and tools. Using a gesture-based methodology that attends to students’ hand movements as they communicate their understanding, data examples from design research in two domains illustrate students’ spontaneous spatial articulation of embodied cognition. Such embodied spatial articulation could be essential for deep understanding of content, because in performing these articulations, students may be negotiating between their dynamic image-based intuitive understanding of a concept and the static formal mathematical formats of representing the concept. Implications for mathematics education are drawn.

**Distributed Learning in Practice and Theory**

At the Center for Connected Learning and Computer-Based Modeling (Uri Wilensky, Director), we were: (a) building agent-based computer models as learning tools for students to simulate scientific, mathematical, and social phenomena; and (b) analyzing emprical data from our design experiments to develop theoretical models of cognition, teaching, and learning. These theoretical models were by and large expressed in diagrams and words. One day it occurred to us that we coudl combine these two activities -- building models, modeling behavior -- in a new way. And so we created agent-based virtual models to simulate our diagram-based theoretical models. At first, we built the "I'm Game!" model to compare predictions from Piagetian and Vygotskian models of learning. This led to a welcome pushback from sociocultural theorists Mike Cole and Jim Levin who raised the gauntlet and improved on our model, demonstrating how the zone of proximal development wins big time... But then, inspired by Paulo Blikstein's emerging ideas of bifocal modeling, we decided to use real and simulated data side by side. This led to a series of computational-modeling studies of Piagetian interviews as well as classroom interactions, where we ran actual and simulated data in parallel so as to tweak the theoretical assumptions underlying our simulations.

Thus at the CCL this direction of research tackled 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. At its broadest, this line of research uses agent-based modeling to study and develop theoretical models of individual group learning.

**Publications**

Blikstein, P., Wilensky, U., & Abrahamson, D. (2009). *Towards a framework for cognitive research using agent-based modeling and complexity sciences. In M. Jacobson (Symposium Chair), M. Kapur (Organizer), & N. Sibelli (Discussant), Complexity, learning, and research: Under the microscope, new kinds of microscopes, and seeing differently*. Paper presented at the annual meeting of the American Educational Research Association, San Diego, April 13-17.

OPENING PARAGRAPH: Complexity sciences and agent-based modeling has been increasingly used by scientists to study a wide range of phenomena such as the interactions of species in an ecosystem, the collisions of molecules in a chemical reaction, or the food-gathering behavior of insects (Bonabeau, 1999; Wilensky & Reisman, 2006). Such phenomena, in which the elements within the system (molecules, or ants) have multiple behaviors and a large number of interaction patterns, have been termed complex and are collectively studied in a relatively young interdisciplinary field called complex systems or complexity studies (Holland, 1995). Typical of complex phenomena is that the cumulative (‘aggregate’) patterns or behaviors at the macro level are not premeditated or directly actuated by any of the “lower-level” micro elements. For example, flocking birds do not intend to construct an arrow-shaped structure (Figure 1), or molecules in a gas are not aware of the Maxwell-Boltzmann distribution. Rather, each element (“agent”) follows its local rules, and the overall pattern arises as epiphenomenal to these multiple local behaviors—the overall pattern emerges. In the mid-nineties, researchers started to realize that agent-based modeling could have a significant impact in education (Resnick & Wilensky, 1993; Wilensky & Resnick, 1995). For instance, to study the behavior of a chemical reaction, the student would observe and articulate only at the behavior of individual molecules — the chemical reaction is construed as emerging from the myriad interactions of these molecular “agents.” Once the modeler assigns agents their local, “micro” rules, the model can be set into motion and the modeler can watch the overall patterns that emerge.

Blikstein, P., Abrahamson, D., & Wilensky, U. (2008). The classroom as a complex adaptive system: an agent-based framework to investigate students' emergent collective behaviors. In G. Kanselaar, J. v. Merriënboer, P. Kirschner, & T. d. Jong (Eds.), *Proceedings of the Eighth International Conference of the Learning Sciences (ICLS2008)* (Vol. 3, pp. 12-13). Utrecht, The Netherlands: ISLS.

ABSTRACT: This study applies agent-based modeling methodology to investigate individual and social factors underlying inequitable participation patterns observed in a real classroom in which an experimental collaborative activity was implemented. We created agent-based simulations of simplified collaborative activities and qualitatively compared results from running the model with the classroom data. We found that collaboration pedagogy emphasizing group performance may forsake individual learning, due to preference for shortterm group efficacy over individual long-term learning. The study may inform professional development and pedagogical policy.

Blikstein, P., Abrahamson, D., & Wilensky, U. (2008, March). *Groupwork as a complex adaptive system: A methodology to model, understand, and design classroom strategies for collaborative learning*. Paper presented at the annual conference of the American Education Research Association, New York, March 24-28.

OPENING PARAGRAPH: Agent-based modeling (ABM) has been increasingly used by scientists to study a wide range of phenomena such as species in an ecosystem or molecules in a chemical reaction (Bonabeau, 1999; Wilensky & Reisman, 2006). Such phenomena, in which the elements within the system have multiple behaviors and a large number of interaction patterns, have been termed complex and are studied in the field called complex systems (Holland, 1995). Typical of complex phenomena is that the cumulative (‘aggregate’) patterns at the macro level are not premeditated by the “lower-level” micro-elements. For example, flocking birds do not intend to construct an arrow-shaped structure (Figure 1), or molecules are not aware of the Maxwell-Boltzmann distribution. Rather, each element (“agent”) follows its local rules, and the overall pattern emerges as epiphenomenal to these multiple local behaviors. In the mid-nineties, researchers realized that ABM could have a significant impact in education ((Resnick & Wilensky, 1993; Wilensky & Resnick, 1995). To study the behavior of a chemical reaction, students would observe and articulate only at the behavior of individual molecules — the chemical reaction is construed as emerging from the myriad interactions of these molecular “agents.”

Abrahamson, D., Blikstein, P., & Wilensky, U. (2007). Classroom model, model classroom: Computer-supported methodology for investigating collaborative-learning pedagogy. In C. Chinn, G. Erkens, & S. Puntambekar (Eds.), *Proceedings of the Computer Supported Collaborative Learning Conference (CSCL)* (Vol. 8, Part 1, pp. 46-55). NJ: Rutgers University. CD ROM.

ABSTRACT: We have been exploring the potential of agent-based modeling methodology for social-science research and, specifically, for illuminating theoretical complementarities of cognitive and socio-constructivist conceptualizations of learning (e.g., Abrahamson & Wilensky, 2005a). The current study advances our research by applying our methodology to pedagogy research: we investigate individual and social factors underlying outcomes of implementing collaborative inquiry classroom practice. Using bifocal modeling (Blikstein & Wilensky, 2006a), we juxtapose agent-based simulations of collaborative problem solving with real-classroom data of students’ collaboration in a demographically diverse middle-school mathematics classroom (Abrahamson & Wilensky, 2005b). We validate the computer model by comparing outcomes from running the simulation with outcomes of the real intervention. Findings are that collaboration pedagogy emphasizing group performance may forsake individual learning, because stable division-of-labor patterns emerge due to utilitarian preference of short-term production over long-term learning (Axelrod, 1997). The study may inform professional development and pedagogical policy.

Abrahamson, D., Wilensky, U., & Levin, J. (2007, April). *Agent-based modeling as a bridge between cognitive and social perspectives on learning*. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL, Chicago, IL.

ABSTRACT (Symposium): The symposium presents exemplars of the potential power of complexity-studies methodology, embodied in agent-based modeling, for engaging in research on psychological phenomena involving individual learners in social contexts. Working in the NetLogo environment (Wilensky, 1999), we explore the debate between Piagetian and Vygotskiian accounts of learning, the development of student reasoning on Piagetian conservation tasks, and student’s implicit argumentation strategies in science inquiry. Also, we reflect on the artifacts of ABM as mediators of distributed research effort. ABM lenses could enable education researchers to explore, articulate, and share intuitions we have struggled to study and express -- that individual intelligent behavior emerges from multicomponential cognitive interactions and, one “level up,” that individuals and communities are interdependent through myriad dynamic reciprocities.

ABSTRACT (paper): This paper is a proof-of-existence empirical paper. That said, it is also a methodological paper. The methodology is agent-based modeling, a computer-supported mode of inquiry into complex phenomena, such as weather fronts, market fluctuations, or participation patterns in a middleschool mathematics lesson. We have previously shown that agent-based simulation can express theoretical models of learning (Abrahamson & Wilensky, 2005; see also Smith & Conrey, 2007). In that paper, we claimed that a promising attribute of simulation-based research into learning is that it fosters scholarly critique and engaging collaboration. It is that claim that is herein proven to stand. The proof lies in the collaborative criticism offered by the third author, Jim Levin, to the first two authors, Dor Abrahamson and Uri Wilensky, concerning their agent-based simulation of learning, which was first presented at the 2005 annual meeting of the Jean Piaget Society and was then made available online, along with its underlying computational procedures that were laid out for scrutiny. The 2005 paper explicitly invited fellow researchers to critique the simulation and possibly modify it so as to accommodate their own perspectives and possibly enable their own investigations. Indeed, the critique received from Levin took a unique form—he improved the computer-based model such that it better simulates the target constructs. Such coconstructive critique, we argue, is a hallmark of the promise of agent-based modeling. So we submit that this proof of existence, if anecdotal validation, may be a harbinger of a new mode of research in the learning sciences and beyond—a mode that builds on constructionism (Papert, 1991): constructionist collaboration.

Blikstein, P., Abrahamson, D., & Wilensky, U. (2006, June). *Minsky, mind, and models: Juxtaposing agent-based computer simulations and clinical-interview data as a methodology for investigating cognitive-developmental theory.* Paper presented at the annual meeting of the Jean Piaget Society, Baltimore, MD.

ABSTRACT: We discuss an innovative application of computer-based simulations in the study of cognitive development. Our work builds on previous seminal contributions to field, in which theoretical models of cognition were implemented in the form of computer programs in attempt to predict human reasoning (Newell & Simon, 1972; Rose & Fischer, 1999). Our computer model can both be a useful vehicle to illustrate the Piagetian theoretical model or to simulate it departing from clinical interview data. We focused in the Piagetian conservation experiment, and collected and analyzed data from actual (not simulated) interviews. The interviews were videotaped, transcribed, and coded in terms of parameters of the computer simulation. The simulation was then fed with these coded data. We were able to perform different kinds of experiments: Playback the interview and the computer model side-by-side, trying to identify behavior patterns; Model validation: investigate whether the child’s decision-making process can be predicted by the model. We conclude that agent-based simulation, activated alongside real data, offers powerful methods for exploring the emergence of self-organized hierarchical organization in human cognition. We are currently exploring the entire combinatorial space of all hypothetical children’s initial mental states and activating the simulation per each of these states. From that perspective, our data of real participants become cases out of the combinatorial space.

Abrahamson, D., & Wilensky, U. (2005, June). *Piaget? Vygotsky? I’m game!: Agent-based modeling for psychology research.* Paper presented at the annual meeting of the Jean Piaget Society.

ABSTRACT: We discuss agent-based models (ABM) as research tools for developmental and social psychology. “Agents” are computer-based entities, e.g., “people.” The modeler assigns the agents real-world roles and rules, conducts simulation experiments in which the agents follow their rules, and observes real-time data. Agent-based models have some properties that can be very useful to psychology. Agent-based models are more dynamic and more expressive as compared to diagrammatic models. Also, simulations afford immediate feedback on the validity of the models. Agent-based modeling is useful for understanding complex phenomena, e.g., the dynamics of multiple individual learners interacting with their peers and with artifacts in their environment and the emergent group patterns arising over time from these multiple interactions. We demonstrate an agent-based simulation that we designed as a “thought experiment” that can shed light on the ongoing debate between two theories of learning, constructivism and social constructivism. The process of building the simulation and embodying these learning theories in explicit rules is an example of how agent-based modeling may help researchers in honing their theories. When “running” the models, unexpected consequences arise, and this leads to successive refinement of theory.

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

Research into the mechanisms and potential role of imagination in mathematical reasoning.

Abrahamson, D. (2006, June). *The three M's: Imagination, embodiment, and mathematics.* Paper presented at the annual conference of the Jean Piaget Society, June 1-3, Baltimore, MD.

ABSTRACT: The objective of this paper is to call for research into the mechanisms and potential agency of imagination in mathematical reasoning and to propose an agenda for such research. Drawing on a broad spectrum of resources in philosophy, the cognitive sciences, embodiment theory, and mathematics-education research, I conjecture that by learning mathematics in environments that support engagement of imagination, students could tap this powerful cognitive tool to support the construction and effective application of concepts. The research objectives are to: (a) investigate the roles of imagination in mathematical creativity, learning, and problem solving, e.g. exploring whether mathematicians’ images are idiosyncratic, culturally mediated, or some combination thereof; (b) ground an understanding of the roles of imagination in current cognitivescience theories and pedagogical perspectives; (c) develop methodology for evaluating students’ access to imagination as a cognitive resource and their imaginative engagement in learning activities; (d) outline principles for the design of objects and activities that encourage students both to engage in imaginative mathematical reasoning and, inter alia, to embrace imagination as an accessible cognitive resource; (e) design and build mathematical objects and create activities that encourage and guide utilization of imagination; and (f) research students’ learning in these designed activities.

**Fractal Village**

Dissertation Project, Sneha Veeragoudar Harrell

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

**Publications**

Veeragoudar Harrell, S., & Abrahamson, D. (2010). Second Life Unplugged: A design for fostering at-risk students' STEM agency. In H. Gazit, D. L. Garcia, G. LeMasers, & L. Morgado (Eds.) The metaverse assembled [Special issue]. *Journal of Virtual Worlds Research*.

ABSTRACT: At an alternative high school serving predominantly at-risk underrepresented students evicted from mainstream education, we designed and implemented Fractal Village, a critical-constructionist computational and mathematical pedagogy learning environment. Fractal Village, instantiated in the virtual world “Second Life,” constituted an empirical environment to research our emergent model of mathematical/computational agency (m/c) as well as an intervention aiming to foster such agency. Key research objectives were to: (1) study relations amongst cognitive, affective, material, technological, and social factors that would contribute to individual development of m/c agency; and (2) delineate design principles for fostering m/c agency. The student cohort engaged collaboratively in virtual world imaginative construction activities each manifesting generative themes (Freire, 1968), to which the designers-as-teachers tailored mathematical and computer-science concepts, such that students appropriated the STEM content apropos of tackling their own emergent construction problems. We argue that to build agency, students must develop both skills and dispositions—a spiraling inter-constructive growth—and articulate a developing methodology for fostering agency development. We conclude that we can, and must, engage at-risk youth by helping them to build STEM-oriented identities, engaging their a priori m/c agency, and customizing skills and dispositions-related classroom discursive supports.

Veeragoudar Harrell, S., & Abrahamson, D. (2007). Computational literacy and mathematics learning in a virtual world: Identity, embodiment, and empowered media engagement. In C. Chinn, G. Erkens, & S. Puntambekar (Eds.), *Proceedings of the Computer Supported Collaborative Learning (CSCL) Conference* (Vol. 8, Part 1, pp. 264 - 265). New Brunswick: NJ: Rutgers University. CD-ROM.

ABSTRACT: We are engaged in the on-going development of a computer-supported collaborative learning environment within a virtual world and use it as a setting for studies exploring relationships between student mathematical cognition, computational literacy, and identity. Our design research is informed by the work of Gee (video games), diSessa (computational literacy), Cole (mediated collaboration), Abrahamson (embodied design for mathematics learning), and Lee (cultural modeling). Within the constructed virtual ecology, we are conducting an ethnographic study of a technologically enabled learning environment with real students bearing virtual identities. The participants are physically remote but embody characters with personae of their own making in playful activities that foster intrinsic motivation and bear mathematical and computational integrity that transcends the medium. Collecting both real and virtual data of a group of at-risk urban high-school students working in Teen Second Life, we examine for changes in participants’ cognitive–affective dispositions toward mathematical practice and identity.