Erin Pomponio

Montclair State University

pomponioe1@montclair.edu

Steven Greenstein, PhD

Montclair State University

greensteins@montclair.edu

Denish Akuom

Montclair State University

akuomd1@montclair.edu

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• Preservice elementary teachers have been characterized as coming to teacher preparation with limited conceptions of mathematics and a model of mathematics teaching that largely appeals to rules and procedures. This model is not consistent with a pedagogy that supports learning mathematics with understanding.

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• We present a pedagogically genuine Making (Halverson & Sheridan, 2014) experience within mathematics teacher preparation that we hypothesized would be formative for the development of a responsive, inquiry-oriented pedagogy.

Teacher Learning Through Making

The Hypothesis

The Issue

We set out to incorporate a Making experience into a specialized content course for prospective and practicing elementary mathematics teachers (PMTs).

Our Research

Constructivism (Piaget, 1970) proposes that knowledge is actively constructed by a learner, with Constructionism (Papert, 1980) adding the dimension that the knowledge be constructed through the process of making a shareable object.

Learning by Design (Koehler & Mishra, 2005) provides a venue for characterizing the interplay between a designer’s knowledge, experiences, intentions, and other resources as they are invoked during the iterative design of the shareable object.

A Constructionist, Learning-by-Design Task

The Task: “Digitally design, 3D print, and assess the efficacy of an original manipulative with a child to support their learning of mathematics.”

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teachermakers.com

Our Trajectory for Investigating Knowledge and Identity Development

The Question

How do learners make sense of and coordinate meanings across multiple representations of fraction division?

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“Dolly” and “Lyle”

Dolly is a participant in the larger study, and also a participant-researcher on this project. Lyle is her dad.

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Dolly conducted a problem-solving interview with Lyle focused on their meanings for fraction division using the “Fraction Orange” manipulative, which she designed.

Dolly and Lyle set off onto a problem solving journey together with a fraction orange Dolly designed and a flip-and-multiply algorithm. The 13 minutes of twists and turns in the productive struggle of their problem solving were intriguing.

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Proulx’s (2013) findings from an enactivist analysis of mathematics strategy development suggest that the nature of the processes at play are dynamic, emergent, and contingent on “an ongoing loop” (p. 319) of interactions between the problem and the solvers.

So we took an enactivist perspective and a revelatory case study approach (Yin, 2009) to understand how it happened...

An Enactivist Perspective

Enactivist theory of cognition (Maturana & Varela, 1987; Varela, Rosch, & Thompson, 1992)

Enactive theories endeavor to understand experience as it unfolds in an embodied subject situated in an environment.

Enactivist Principles*

1. Perception consists in perceptually guided action.
2. Cognitive structures emerge from the recurrent sensorimotor patterns that enable action to be perceptually guided.

Cognitivism: Cognitive structures are internal states that represent properties of the environment. Meaning derives in the service of adaptive behavior.

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Enactivism: As the organism interacts with – adapts its perceptual orientation to – its environment, it iteratively consolidates effective sensorimotor patterns into cognitive structures. These emergent cognitive structures are meaningful in relation to “adequate action” – the sense that they bear on the organism’s success or failure in its self-organizing.

Knowing is doing.

*The Embodied Mind (Varela, Rosch, & Thompson, 1992)

Enactivist Concepts

• Structural coupling: An organism and its environment co-adapt through recursive and repeated interactions.
• Structural determinism: Fit is dynamic and contingent upon unique histories of recurrent interactions and structural changes that are determined by an organism’s structure.

In addition: We use harmony and dissonance to emphasize that fit (or lack of fit) is an internally “felt dimension of experience” (Petitmengin, 2017, p. 144) that drives problem solving.

A Semiotic Perspective

Peirce’s contribution: The relationship between a sign and an object is NOT straightforward. The meaning of a sign is made through interpretations of it.

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“Signs… have the epistemological function to represent objects for an interpretant and to mediate between object and interpretant to make objects accessible to the mind” (Sáenz-Ludlow, 2006, p. 187).

Peirce’s (1998) triad of sign relations

Two thirds

2 shaded parts out of 3

Semiotic Interference

The “interpretant involves meaning making: it is the result of trying to make sense of the relationship… [between] the object and the representamen” (Presmeg, 2006, p. 170).

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Semiotic interference is used to analyze the process of meaning making across multiple artifacts whenever “the interpretant of a sign whose object belongs to the context of [one] artifact is translated by a student in a new sign whose object belongs to the context of another artifact” (p. 3:–58). (diagram next slide)

(Maffia & Maracci, 2019)

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Semiotic interference (Maffia & Maracci, 2019) provides a window into the chaining of signs (Presmeg, 2006; Bartolini Bussi & Mariotti, 2008) as learners negotiate their interpretations and translate meanings from personalized signs to generalized mathematical signs.

Semiotic interference in a semiotic chain (Maffia & Maracci, 2019, p. 3–59)

Data Analysis

• Line-by-line analysis of the video transcripts...
• The enactivist concepts of structural coupling and structural determinism were used to analyze the directed interactions where Dolly and Lyle aimed to coordinate meanings across two representations.
• Peirce’s (1998) triad of sign relations and Maffia and Maracci’s (2019) concept of semiotic interference were used to analyze their emergent processes of meaning making across those representations.

Findings: Three Excerpts

• A First Dissonance

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• The Messiness of Multiple Representations

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• A Crowning Achievement

The Drive for Harmony Among Multiple Representations of Fraction Division

The posed problem: ½ ÷ ¼

“First Dissonance”: In the drive for harmony, coupling is disrupted.

The Drive for Harmony Among Multiple Representations of Fraction Division

The posed problem: ½ ÷ ¼

The messiness of multiple representations

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In this next excerpt...

• Dolly enacts her knowing (interpretant) of a half (object) in the orange (representamen).
• She interprets the posed problem, ½ ÷ ¼, as “How many quarters go into a half?”
• Lyle references the orange, makes his own interpretations, and determines that two quarter pieces fit into a half.
• But then he shifts his attention to the algorithm on the page and changes his answer.
• Next, he aims to resolve the dissonance by finding an interpretation of the whole that would be harmonious.

A prolonged moment of semiotic interference

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• Dolly enacts her knowing (interpretant) of a half (object) in the orange (representamen).
• She interprets the posed problem, ½ ÷ ¼, as “How many quarters go into a half?”
• Lyle references the orange, makes his own interpretations, and determines that two quarter pieces fit into a half.
• But then he shifts his attention to the algorithm, privileges it, experiences dissonance, and changes his answer.
• He then aims to resolve the dissonance by finding an interpretation of the whole that would be harmonious.

The messiness of multiple representations

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With this question, Dolly expresses what is must feel like as she and Lyle navigate spirals of semiotic interference across:

• ... different representations: the orange and the algorithm
• … their wonderings about objects: What is a whole? What is division? What is 4⁄1?
• … their interpretations of relationships between artifacts and objects across multiple signs: What is the whole across these different representations? What does ½ ÷ ¼ mean and how does it relate to an enactment with the orange of “How many quarters go into a half?”

“Why is this so hard?”

A Final Harmony of Meaning Between Signs

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• the meaning of “4/2=2/1” in the division algorithm
• a determination of a unit whole
• interpretations of ½ and ¼ in relation to that whole
• a meaning made for division
• achieving their ultimate objective of making sense of ½ ÷ ¼ in coordination with these other sense-making achievements

Enchained Meanings in Dolly and Lyle's Problem Solving:

A Final Harmony of Meaning Between Signs

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• the meaning of “4/2=2/1” in the division algorithm
• a determination of a unit whole
• interpretations of ½ and ¼ in relation to that whole
• a meaning made for division
• achieving their ultimate objective of making sense of ½ ÷ ¼ in coordination with these other sense-making achievements

Enchained Meanings in Dolly and Lyle's Problem Solving:

A Final Harmony of Meaning Between Signs

​

• the meaning of “4/2=2/1” in the division algorithm
• a determination of a unit whole
• interpretations of ½ and ¼ in relation to that whole
• a meaning made for division
• achieving their ultimate objective of making sense of ½ ÷ ¼ in coordination with these other sense-making achievements

Enchained Meanings in Dolly and Lyle's Problem Solving:

What We Set Out to Do

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• Research Question: How do learners make sense of and coordinate meanings across multiple representations of fraction division?

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• Revelatory case study approach: We took enactivist and semiotic perspectives to understand the twists and turns in problem solving.

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What We Learned

• It’s prudent to appreciate that problem solving is a complex, dynamic, contingent, and emergent phenomenon.
• Multiple representations afford the opportunity for meaning making through embodied engagement with mathematical ideas.
• Inter-actions with multiple representations of challenging ideas may make ways for deep (and felt) ways of knowing and doing mathematics.
• Structural couplings with meaning-less mathematics may be disrupted in the presence of “sensible” material resources and an open-ended task, and situated in an exploratory and inquiry-driven environment.

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The process by which learners make sense of mathematics by connecting representations is emergent and actually quite messy.

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So there’s clearly a need to create pedagogical and material resources that honor and enable this kind of mathematical activity to occur in classrooms.

Why It Matters

Why it Might Matter

Whereas teachers’ professional noticing is about student thinking...

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In a noticing kinda way, imagine teachers being attuned to the felt harmony and dissonance of their students’ sense-making activity...

Come visit us.

Erin Pomponio

pomponioe1@montclair.edu

Steven Greenstein, PhD

greensteins@montclair.edu

Denish Akuom

akuomd1@montclair.edu