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Dragan Trninic

Swiss Federal Institute of Technology (ETH), Learning Sciences, Alumnus

University of California, Berkeley, SESAME, Graduate Student

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I study how people learn, focusing on the role of repetitive practice as an explorative activity. From this perspective, learning emerges through practice, often unexpectedly, as practitioners recognize anew something in their actions.

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Daniel D. Hutto

University of Wollongong

University of California, Santa Barbara

Ataturk University

University of British Columbia

University of California, Merced

David Pierre Leibovitz

Carleton University

Kansas State University

Shaun Gallagher

University of Memphis

Armando Marques-Guedes

UNL - New University of Lisbon

University of British Columbia

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Papers by Dragan Trninic

Bridging Cognitive Neuroscience and Education: Insights from EEG Recording During Mathematical Proof Evaluation

Trends in neuroscience and education, May 1, 2024

Math on cortex—enhanced delta phase synchrony in math experts during long and complex math demonstrations

Cerebral cortex, Jan 31, 2024

Rethinking failure in mathematics education: A historical appeal

Thinking Skills and Creativity, 2018

Although failure can be seen as unproductive or even counterproductive in mathematics classrooms,... more Although failure can be seen as unproductive or even counterproductive in mathematics classrooms, historical evidence suggests that negotiations around failure can act as a fountainhead of creative insight. To make this claim, we focus in particular on the historical development of mathematical formalisms. We survey recent scholarship supporting the notion that "failure" may be used productively in the classroom, and offer tentative suggestions as to how elements of "negotiable failure" may be implemented as an extension and elaboration of existing work on productive failure.

Paper presented at the annual meeting of the Jean Piaget Society, Jun 1, 2011

Summary: diSessa has recently called for “dialectical” approaches to the study of cognition and i... more Summary: diSessa has recently called for “dialectical” approaches to the study of cognition and instruction in mathematics and science, the principle being that neither cognitivist nor sociocultural approaches per seappear to have presented satisfactory narratives of learning processes in the disciplines, so that the field should develop integrated theoretical models that leverage and interweave the unique contributions of these historically parallel approaches, thus offering more complete, more nuanced descriptions (diSessa, 2008). ...

Comparing the effectiveness of preparatory activities that help undergraduate students learn from instruction

Learning and Instruction

This page was generated automatically upon download from the ETH Zurich Research Collection. For ... more

The Neural and Physiological Mechanisms of Learning through Productive Failure

Stepping out of rhythm: an embodied artifact for noticing rate of change

Educational Technology Research and Development, 2021

Making sense of data is fundamental to science. Yet, the form in which data are represented can m... more Making sense of data is fundamental to science. Yet, the form in which data are represented can make their interpretation challenging for students. We introduce an activity that transformed a data table into an embodied artifact. The activity helped 8th grade students interpret the data to find that cold water warmed “fast-then-slow,” in other words, that the rate of warming decreased over time. For the activity, students were invited to “act out” the temperature change by walking along a giant horizontal thermometer while their classmates clapped out the passage of time. To arrive at the correct temperature at the right time, students had to continually slow their pace. We offer a microgenetic analysis of the activity to illuminate how it helped participating students notice that the cold water’s temperature warmed fast-then-slow, developing an account of the students’ passage from action to understanding. We argue that embodied representations can be effectively coupled with tools...

Differential benefits of explicit failure-driven and success-driven scaffolding in problem-solving prior to instruction

Journal of Educational Psychology, 2020

Unscaffolded problem-solving before receiving instruction can give students opportunities to ente... more Unscaffolded problem-solving before receiving instruction can give students opportunities to entertain their exploratory hypotheses at the expense of experiencing initial failures. Prior literature has argued for the efficacy of such preparatory activities in preparing students to learn from instruction. Despite growing understanding of the underlying cognitive mechanisms, the pedagogical value of success or failure in initial problem-solving attempts is still unclear. We do not know yet whether some ways of succeeding or failing are more efficacious than others. We report empirical evidence from a classroom intervention (N = 221), where we designed scaffolds to explicitly push student problem-solving toward success via structuring, but also toward failure via problematizing. Our rationale for explicit failure scaffolding was rooted in facilitating problem-space exploration. We subsequently compared the differential preparatory effects of success-driven and failure-driven problem-solving on learning from follow-up instruction. Results suggested that failure-driven scaffolding (nudging students to generate suboptimal solutions) and success-driven scaffolding (nudging students to generate optimal solutions by giving them heuristics with low specificity) had similar outcomes on posttest assessments of conceptual understanding. Students exposed to failure-driven scaffolding, however, demonstrated higher quality of constructive reasoning. These trends were more salient for the learning concept with greater difficulty

The Disappearing “Advantage of Abstract Examples in Learning Math”

Cognitive Science, 2020

When teaching a novel mathematical concept, should we present learners with abstract or concrete ... more When teaching a novel mathematical concept, should we present learners with abstract or concrete examples? In this experiment, we conduct a critical replication and extension of a well-known study that argued for the general advantage of abstract examples (Kaminski, Sloutsky, & Heckler, 2008a). We demonstrate that theoretically motivated yet minor modifications of the learning design put this argument in question. A key finding from this study is that participants who trained with improved concrete examples performed as well as, or better than, participants who trained with abstract examples. We argue that the previously reported "advantage of abstract examples" manifested not because abstract examples are advantageous in general, but because the concrete condition employed suboptimal examples.

Productive dissent in learning communities

Instructional Science, 2018

Community" has become a commonplace term in the learning sciences. Alongside this popularization ... more

Instruction, repetition, discovery: restoring the historical educational role of practice

Instructional Science, 2017

This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion tha... more This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that "direct instruction" and "discovery" are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated through an empirical case of martial arts instruction. That repetitive practice can nonetheless be a fountainhead of discovery is explained using Bernstein's notion of repetition-withoutrepetition. Finally, we attend to parallels in canonical mathematics practice. Implications are discussed, with a focus on reconceptualizing direct instruction, repetition, and discovery as complementary and synergistic.

Toward an embodied-interaction design framework for mathematical concepts

Proceedings of the 10th International Conference on Interaction Design and Children, 2011

Hooks and Shifts: A Dialectical Study of Mediated Discovery

Technology, Knowledge and Learning, 2011

Bringing forth mathematical concepts: signifying sensorimotor enactment in fields of promoted action

Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition... more Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthro-pology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an ‘‘instru-mented field of promoted action’’. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disci-plinary semiotic systems. We exemplify our thesis with two selected excerpts from our videogr...

ICLS2012 Volume 1: Full Papers Embodied Artifacts and Conceptual Performances

Abstract: Learning scientists are only beginning to appreciate the potential of synergy between t... more Abstract: Learning scientists are only beginning to appreciate the potential of synergy between two concurrent developments—theory of embodied cognition and technology of embodied interaction. We characterize and evaluate this prospective synergy from a sociocultural perspective. First we analyze learning in explicitly embodied cultural practices (e.g., surfing), then analogize to the implicitly embodied practice of mathematics. We next contextualize this analogy via interpreting data collected in a design-based research study, in which twenty-two 9-to-11-year-olds developed notions of proportionality through participating in guided problem-solving activities in an embodied-interaction space. In both surfing and mathematics, we argue, learners develop “embodied artifacts, ” i.e. body-based and modular rehearsed actions. Embodied artifacts lend individuals entry into disciplinary competence via participation in action, refinement of operations, and integration into activity structure...

Pedagogical Tradeoffs Along Familiar and Generic Dimensions

The design and selection of pedagogically effective problem situations is a critical yet underthe... more The design and selection of pedagogically effective problem situations is a critical yet undertheorized facet of reform-oriented mathematics education. Drawing on instructional-design frameworks and cognitive-science theory, we propose a situation taxonomy centered on the dimensions “familiar ” and “generic ” and hypothesize the contrasting learning affordances of exponents thereof. 51 undergraduate students solved either a familiar or a generic version of a compound-probability problem, and subgroups thereof then participated in semi-structured clinical interviews. Familiar problems evoked common sense, yet their treatment was liable to be mathematically non-normative; generic problems triggered mathematically suitable treatments, yet these treatments were liable to remain opaque. We discuss implications of these tradeoffs with an eye to fostering mathematical-probabilistic literacy that is both powerful and grounded. BACKGROUND AND OBJECTIVES The study reported in this paper was m...

Adding up fine motor skills: developmental relations between manual dexterity and numerical abilities

We explore the relationship between mathematical skills and motor skills across three age groups ... more We explore the relationship between mathematical skills and motor skills across three age groups of normally developing children. The existence of such a relationship is postulated in classical accounts of human development. In contemporary research, the existence of a relationship between motor development and the development of abstract concepts may form a crucial piece of evidence for theories of embodied cognition. Existing studies suggest a link between fine motor skills and various numerical and mathematical tasks in young children; however, there are few attempts to measure the strength of this relationship across different ages. We use a cross-sectional design to investigate the link between fine motor and mathematical skills in children in Kindergarten, 2nd grade, and 4th grade. The results show that correlational patterns vary in the three ages; while in Kindergarten manual dexterity of the dominant hand is related to math skills, in 2nd grade the manual dexterity of the n...

Design as an Object-to-think-with: Semiotic Potential Emerges Through Collaborative Reflective Conversation with Material

We chart a historical analysis of a collaborative design-based research project investigating the... more We chart a historical analysis of a collaborative design-based research project investigating the emergence of mathematical meaning from embodied interaction with a technological trackingsystem supporting the learning of proportionality. Recounting iterative cycles of a conceptually critical perceptual feedback element, we articulate three interconnected images of researchbased designers: (a) Janus the two-headed keeper of passageways who sees artifacts alternately as a student would or as an expert would; (b) an investigator searching to explicate design decisions coherently in light of learning-sciences theory; and (c) a reflective practitioner who embraces tradeoffs and is open to constructive criticism and to implementing radical changes to design and theory. Ultimately, we posit, we as researchers are continuously developing professional vision for our own design even as the design changes.

Evocation and Enactment: Understanding Trajectories of Conceptual Development in Artifact-Mediated Situations

When instructional designers develop content-targeted pedagogical situations, their practice can ... more When instructional designers develop content-targeted pedagogical situations, their practice can be theorized as engineering students' development of conceptual schemes. To account for the contributions of students' prior schemes and situated experiences towards the development of conceptual schemes, we suggest a distinction between the evocation of schemes and enactment of situations. The former suggests that the design of an instructional situation can activate students' prior schemes. The latter suggests that the structure of students' activity in an instructional situation determines the nature of newly constructed schemes. We contextualize our views using empirical data from case studies involving students' interactions with educational technology designed for embodying and grounding the concept of proportionality.

Exploring the efficacy of practice before instruction

Bridging Cognitive Neuroscience and Education: Insights from EEG Recording During Mathematical Proof Evaluation

Trends in neuroscience and education, May 1, 2024

Math on cortex—enhanced delta phase synchrony in math experts during long and complex math demonstrations

Cerebral cortex, Jan 31, 2024

Rethinking failure in mathematics education: A historical appeal

Thinking Skills and Creativity, 2018

Although failure can be seen as unproductive or even counterproductive in mathematics classrooms,... more Although failure can be seen as unproductive or even counterproductive in mathematics classrooms, historical evidence suggests that negotiations around failure can act as a fountainhead of creative insight. To make this claim, we focus in particular on the historical development of mathematical formalisms. We survey recent scholarship supporting the notion that "failure" may be used productively in the classroom, and offer tentative suggestions as to how elements of "negotiable failure" may be implemented as an extension and elaboration of existing work on productive failure.

Paper presented at the annual meeting of the Jean Piaget Society, Jun 1, 2011

Summary: diSessa has recently called for “dialectical” approaches to the study of cognition and i... more Summary: diSessa has recently called for “dialectical” approaches to the study of cognition and instruction in mathematics and science, the principle being that neither cognitivist nor sociocultural approaches per seappear to have presented satisfactory narratives of learning processes in the disciplines, so that the field should develop integrated theoretical models that leverage and interweave the unique contributions of these historically parallel approaches, thus offering more complete, more nuanced descriptions (diSessa, 2008). ...

Comparing the effectiveness of preparatory activities that help undergraduate students learn from instruction

Learning and Instruction

This page was generated automatically upon download from the ETH Zurich Research Collection. For ... more

The Neural and Physiological Mechanisms of Learning through Productive Failure

Stepping out of rhythm: an embodied artifact for noticing rate of change

Educational Technology Research and Development, 2021

Making sense of data is fundamental to science. Yet, the form in which data are represented can m... more Making sense of data is fundamental to science. Yet, the form in which data are represented can make their interpretation challenging for students. We introduce an activity that transformed a data table into an embodied artifact. The activity helped 8th grade students interpret the data to find that cold water warmed “fast-then-slow,” in other words, that the rate of warming decreased over time. For the activity, students were invited to “act out” the temperature change by walking along a giant horizontal thermometer while their classmates clapped out the passage of time. To arrive at the correct temperature at the right time, students had to continually slow their pace. We offer a microgenetic analysis of the activity to illuminate how it helped participating students notice that the cold water’s temperature warmed fast-then-slow, developing an account of the students’ passage from action to understanding. We argue that embodied representations can be effectively coupled with tools...

Differential benefits of explicit failure-driven and success-driven scaffolding in problem-solving prior to instruction

Journal of Educational Psychology, 2020

Unscaffolded problem-solving before receiving instruction can give students opportunities to ente... more Unscaffolded problem-solving before receiving instruction can give students opportunities to entertain their exploratory hypotheses at the expense of experiencing initial failures. Prior literature has argued for the efficacy of such preparatory activities in preparing students to learn from instruction. Despite growing understanding of the underlying cognitive mechanisms, the pedagogical value of success or failure in initial problem-solving attempts is still unclear. We do not know yet whether some ways of succeeding or failing are more efficacious than others. We report empirical evidence from a classroom intervention (N = 221), where we designed scaffolds to explicitly push student problem-solving toward success via structuring, but also toward failure via problematizing. Our rationale for explicit failure scaffolding was rooted in facilitating problem-space exploration. We subsequently compared the differential preparatory effects of success-driven and failure-driven problem-solving on learning from follow-up instruction. Results suggested that failure-driven scaffolding (nudging students to generate suboptimal solutions) and success-driven scaffolding (nudging students to generate optimal solutions by giving them heuristics with low specificity) had similar outcomes on posttest assessments of conceptual understanding. Students exposed to failure-driven scaffolding, however, demonstrated higher quality of constructive reasoning. These trends were more salient for the learning concept with greater difficulty

The Disappearing “Advantage of Abstract Examples in Learning Math”

Cognitive Science, 2020

When teaching a novel mathematical concept, should we present learners with abstract or concrete ... more When teaching a novel mathematical concept, should we present learners with abstract or concrete examples? In this experiment, we conduct a critical replication and extension of a well-known study that argued for the general advantage of abstract examples (Kaminski, Sloutsky, & Heckler, 2008a). We demonstrate that theoretically motivated yet minor modifications of the learning design put this argument in question. A key finding from this study is that participants who trained with improved concrete examples performed as well as, or better than, participants who trained with abstract examples. We argue that the previously reported "advantage of abstract examples" manifested not because abstract examples are advantageous in general, but because the concrete condition employed suboptimal examples.

Productive dissent in learning communities

Instructional Science, 2018

Community" has become a commonplace term in the learning sciences. Alongside this popularization ... more

Instruction, repetition, discovery: restoring the historical educational role of practice

Instructional Science, 2017

This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion tha... more This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that "direct instruction" and "discovery" are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated through an empirical case of martial arts instruction. That repetitive practice can nonetheless be a fountainhead of discovery is explained using Bernstein's notion of repetition-withoutrepetition. Finally, we attend to parallels in canonical mathematics practice. Implications are discussed, with a focus on reconceptualizing direct instruction, repetition, and discovery as complementary and synergistic.

Toward an embodied-interaction design framework for mathematical concepts

Proceedings of the 10th International Conference on Interaction Design and Children, 2011

Hooks and Shifts: A Dialectical Study of Mediated Discovery

Technology, Knowledge and Learning, 2011

Bringing forth mathematical concepts: signifying sensorimotor enactment in fields of promoted action

Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition... more Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthro-pology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an ‘‘instru-mented field of promoted action’’. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disci-plinary semiotic systems. We exemplify our thesis with two selected excerpts from our videogr...

ICLS2012 Volume 1: Full Papers Embodied Artifacts and Conceptual Performances

Abstract: Learning scientists are only beginning to appreciate the potential of synergy between t... more Abstract: Learning scientists are only beginning to appreciate the potential of synergy between two concurrent developments—theory of embodied cognition and technology of embodied interaction. We characterize and evaluate this prospective synergy from a sociocultural perspective. First we analyze learning in explicitly embodied cultural practices (e.g., surfing), then analogize to the implicitly embodied practice of mathematics. We next contextualize this analogy via interpreting data collected in a design-based research study, in which twenty-two 9-to-11-year-olds developed notions of proportionality through participating in guided problem-solving activities in an embodied-interaction space. In both surfing and mathematics, we argue, learners develop “embodied artifacts, ” i.e. body-based and modular rehearsed actions. Embodied artifacts lend individuals entry into disciplinary competence via participation in action, refinement of operations, and integration into activity structure...

Pedagogical Tradeoffs Along Familiar and Generic Dimensions

The design and selection of pedagogically effective problem situations is a critical yet underthe... more The design and selection of pedagogically effective problem situations is a critical yet undertheorized facet of reform-oriented mathematics education. Drawing on instructional-design frameworks and cognitive-science theory, we propose a situation taxonomy centered on the dimensions “familiar ” and “generic ” and hypothesize the contrasting learning affordances of exponents thereof. 51 undergraduate students solved either a familiar or a generic version of a compound-probability problem, and subgroups thereof then participated in semi-structured clinical interviews. Familiar problems evoked common sense, yet their treatment was liable to be mathematically non-normative; generic problems triggered mathematically suitable treatments, yet these treatments were liable to remain opaque. We discuss implications of these tradeoffs with an eye to fostering mathematical-probabilistic literacy that is both powerful and grounded. BACKGROUND AND OBJECTIVES The study reported in this paper was m...

Adding up fine motor skills: developmental relations between manual dexterity and numerical abilities

We explore the relationship between mathematical skills and motor skills across three age groups ... more We explore the relationship between mathematical skills and motor skills across three age groups of normally developing children. The existence of such a relationship is postulated in classical accounts of human development. In contemporary research, the existence of a relationship between motor development and the development of abstract concepts may form a crucial piece of evidence for theories of embodied cognition. Existing studies suggest a link between fine motor skills and various numerical and mathematical tasks in young children; however, there are few attempts to measure the strength of this relationship across different ages. We use a cross-sectional design to investigate the link between fine motor and mathematical skills in children in Kindergarten, 2nd grade, and 4th grade. The results show that correlational patterns vary in the three ages; while in Kindergarten manual dexterity of the dominant hand is related to math skills, in 2nd grade the manual dexterity of the n...

Design as an Object-to-think-with: Semiotic Potential Emerges Through Collaborative Reflective Conversation with Material

We chart a historical analysis of a collaborative design-based research project investigating the... more We chart a historical analysis of a collaborative design-based research project investigating the emergence of mathematical meaning from embodied interaction with a technological trackingsystem supporting the learning of proportionality. Recounting iterative cycles of a conceptually critical perceptual feedback element, we articulate three interconnected images of researchbased designers: (a) Janus the two-headed keeper of passageways who sees artifacts alternately as a student would or as an expert would; (b) an investigator searching to explicate design decisions coherently in light of learning-sciences theory; and (c) a reflective practitioner who embraces tradeoffs and is open to constructive criticism and to implementing radical changes to design and theory. Ultimately, we posit, we as researchers are continuously developing professional vision for our own design even as the design changes.

Evocation and Enactment: Understanding Trajectories of Conceptual Development in Artifact-Mediated Situations

When instructional designers develop content-targeted pedagogical situations, their practice can ... more When instructional designers develop content-targeted pedagogical situations, their practice can be theorized as engineering students' development of conceptual schemes. To account for the contributions of students' prior schemes and situated experiences towards the development of conceptual schemes, we suggest a distinction between the evocation of schemes and enactment of situations. The former suggests that the design of an instructional situation can activate students' prior schemes. The latter suggests that the structure of students' activity in an instructional situation determines the nature of newly constructed schemes. We contextualize our views using empirical data from case studies involving students' interactions with educational technology designed for embodying and grounding the concept of proportionality.

Exploring the efficacy of practice before instruction

IT'S NOT EASY BEING GREEN: EMBODIED ARTIFACTS AND THE GUIDED EMERGENCE OF MATHEMATICAL MEANING

by Daniel Reinholz, Dragan Trninic, and Dor Abrahamson

This on-going design-based research study focuses on Grade 4-6 students' guided task-based intera... more This on-going design-based research study focuses on Grade 4-6 students' guided task-based interaction with a novel computer-based hand-tracking system built to suggest the limitations of naïve additive schemes and create opportunities to develop core notions of proportionality as elaborations on these schemes, even before engaging numerical semiotic forms. Study participants struggled with canonical issues inherent to rational numbers. They formulated a string of insights leading up to a new type of equivalence class. Reported as a case study of Itamar, a 5 th -grade middle-achieving student, our analyses reveal emergence of conceptually critical mathematical meanings in an activity that initially bears little mathematical significance.

DESIGN AS AN OBJECT-TO-THINK-WITH: SEMIOTIC POTENTIAL EMERGES THROUGH COLLABORATIVE REFLECTIVE CONVERSATION WITH MATERIAL

by Daniel Reinholz and Dragan Trninic

We chart a historical analysis of a collaborative design-based research project investigating the... more We chart a historical analysis of a collaborative design-based research project investigating the emergence of mathematical meaning from embodied interaction with a technological trackingsystem supporting the learning of proportionality. Recounting iterative cycles of a conceptually critical perceptual feedback element, we articulate three interconnected images of researchbased designers: (a) Janus the two-headed keeper of passageways who sees artifacts alternately as a student would or as an expert would; (b) an investigator searching to explicate design decisions coherently in light of learning-sciences theory; and (c) a reflective practitioner who embraces tradeoffs and is open to constructive criticism and to implementing radical changes to design and theory. Ultimately, we posit, we as researchers are continuously developing professional vision for our own design even as the design changes.