## RESEARCH FINDINGS

This work is addressing the following research questions. Brief responses are provided.

The Making experience has provided opportunities for PMTs to embed their developing knowledge into the design of their tools, influencing not only their experience as a Maker, but also the depth of their mathematical understanding. Moreover, it also elicited distinct facets of content and pedagogical knowledge as well as their integration during authentic teacher learning moments, which supported practices like reasoning mathematically or persisting through the often messy process of teaching and learning mathematics.

The curricular module serves as a transitioning vehicle for PMTs between their teacher education setting and actual classrooms, supporting a diverse and multifaceted array of possibilities for them to draw from in their future teaching. In addition, by enacting pedagogies of care in conjunction with mathematical Making experiences, we offered our PMTs a shift in mindset that celebrated and re-centered marginalized individuals within the mathematical learning and Making communities.

The Making experience has afforded PMTs the opportunity to engage the complexities of navigating mathematical problem-solving tasks through multiple representations of ideas, emerging with deeper understandings of those ideas and a realization of the power of exploratory, tool-mediated mathematical activity on teachers and students alike. Our analysis of the nature of our PMTs’ design activity has revealed the diversity of design decisions, rationales, and mediating resources such an experience entails, as well as the importance of viewing learning as a situated process of a teacher becoming.

We’ve been exploring how PMTs learn to teach mathematics through their engagement in a pedagogically informative Making experience, and the roles that their identities played in their design decisions, mathematical activity, and pedagogy. Using a commognitive perspective, we examined our PMTs’ learning by observing changes in their discourse surrounding mathematics, identity, pedagogy, and design. We found that the Making experience was an effective venue for provoking all four of these discourses and that these discourses are actually intertwined. This finding further illustrates that one’s identity is as central to learning to teach mathematics as is their learning of mathematics, pedagogy, and design. More broadly, through the reflective and participatory creative processes associated with the Maker experience and the inquiry-oriented learning environment the teacher educators cultivated in the course, we found that the class provided the PMTs with an opportunity to remake personal relationships with mathematics and mathematics teaching rooted in exploration and creativity, and imbued with dynamic and deep understandings of mathematical ideas.

## PRESENTATIONS

This work is being disseminated through the following presentations.

*In this paper, we present the design of a tool and task that enabled two students to simultaneously embody both the creation of a graph and the covariational actions that the graph represents. Through a process of collaborative instrumentation, the students made meaningful connections between their motions and the embodied traces they created as they reasoned about the covarying quantities of height and time.*[Presentation]

*We explored the problem solving of two learners as they aimed to make sense of fraction division by coordinating meanings across two artifacts, one being a physical manipulative and the other a written expression of the standard algorithm. In addressing the question, “How do learners make sense of and coordinate meanings across multiple representations of mathematical ideas?” we took an enactivist perspective and used tools of semiotics to analyze the ways they navigated the dissonance that arose as they sought to achieve harmony in meanings across multiple representations of ideas. [Presentation]*

*In this presentation, we share what we’ve learned about how the designing and testing of 3-D printed manipulatives by prospective elementary mathematics teachers (PMTs) acts as a creative process that reveals the potential benefits of mathematical Making in teacher learning. [Presentation]*

*In this presentation, we share what we’ve learned about how the designing and testing of 3-D printed manipulatives by prospective elementary mathematics teachers (PMTs) acts as a creative process that reveals the potential benefits of mathematical Making in teacher learning. [Presentation]*

*We demonstrate that the Making experience afforded opportunities for enacting a caring pedagogy that elicited PMTs’ funds-of-knowledge, and that this approach provided a transitioning vehicle between the teacher education setting and actual classrooms.**[Presentation]*

*We share our findings that Making is an effective venue for provoking mathematizing, identifying, pedagogy and designing discourses; revealing their intertwined nature; and illustrating that one’s identity is as central to learning to teach mathematics as is their learning of mathematics, pedagogy, and design. [Presentation]*

*This work explores the emergent nature of mathematical activity mediated by multiple artifacts. Findings reveal the complexity of mediated engagement and have implications for the practice of learning mathematics through multiple representations. [Presentation]*

*Guided by the question, “How does enacting a caring pedagogy during a Making-centered experience impact and broaden opportunities for meaningful mathematics learning?” we suggest a pedagogy of caring, sharing, and Making as one way to celebrate and re-center individuals who are often marginalized from meaningful learning in spaces for Making.**[Presentation]*

*This paper shares findings that convey the diversity of design decisions, rationales, and mediating resources that entailed PMTs’ mathematical Making. [Presentation]*

*We use project data and personal narratives to make a case for the transformative power that design and Making have within the space of mathematics learning and teaching. [Presentation]*

## PUBLICATIONS

The work is also being disseminated through these publications.

*Operating from an enactivist theory of cognition, this work addresses the question, “What role might multiple artifacts play in learners’ emergent sense making as they act within an ecology of mathematical problem solving?” We do so by analyzing the ways in which two learners navigated the dissonance that arose as they simultaneously sought to create harmony in meanings between the flip-and-multiply algorithm for fraction division and a manipulative that one of them designed for engagement with fraction concepts. Our findings illuminate the role that multiple mathematical artifacts can play in establishing a notion of sense making in mathematics education as fundamentally grounded in embodied understandings.*

*In addressing the question, “How do learners make sense of and coordinate meanings across multiple representations of mathematical ideas?” we took an enactivist perspective and used tools of semiotics to analyze the ways they navigated the dissonance that arose as they sought to achieve harmony in meanings across multiple representations of ideas.*

*We share that the Making experience elicited distinct facets of content and pedagogical knowledge and their integration during authentic teacher learning moments, and that these supported reform-minded teaching practices in mathematics education.*

*Covariational reasoning and the creation and interpretation of graphs of covariational situations are important skills in math and science. Unfortunately, research shows that students often struggle to make meaningful connections between graphs and the covariational situations they represent. Educational activities designed to help students overcome this struggle tend to use either student-generated or automatically-generated graphs, and have students either act out covariational situations or more passively observe them. In this paper, we present the design of a tool and task that enabled two students to*simultaneously*embody both the creation of a graph and the covariational actions that the graph represents. Through a process of*collaborative instrumentation*, the students made meaningful connections between their motions and the embodied traces they created as they reasoned about the covarying quantities of height and time.*

*In this paper, we share research that explores the potential benefits of a novel Making experience within mathematics teacher preparation that we hypothesized would inform the pedagogical and curricular thinking of prospective teachers of elementary mathematics (PMTs). That experience had PMTs exploring at the intersection of content, pedagogy, and design to digitally design, 3D print, and share an original manipulative with a child to promote their mathematical thinking. We share several vignettes of our research that aim to discern some of the potential benefits the experience might offer PMTs. These take a variety of theoretical and methodological approaches at the intersections of teacher knowledge, identity, pedagogy, and design. Implications of our findings for teacher preparation and professional learning are provided throughout the paper and in its conclusion.*

- This paper reports on research that examines connections between the pedagogical/conceptual knowledge that prospective teachers embed in the designs of original manipulatives and how those designs mediate the pedagogical moves they make in teaching situations. We share findings from the analysis of prospective teachers’ design activity that conveys (1) the diversity of design decisions, rationales, and mediating resources that it entailed, and (2) how the designed manipulative act as anchors for their conceptual/pedagogical moves. The implications of these findings for teacher preparation and professional learning are considered. LINK

*This paper reports on research that examines connections between the pedagogical/conceptual knowledge that prospective teachers embed in the designs of original manipulatives and how those designs mediate the pedagogical moves they make in teaching situations.*[PDF]

*We explored the problem solving of two learners as they aimed to make sense of fraction division by coordinating meanings across two artifacts, one being a physical manipulative and the other a written expression of the standard algorithm. We took an enactivist perspective and used tools of semiotics to analyze the ways they navigated the dissonance that arose as they sought to achieve harmony in meanings across multiple r**epresentations of ideas.*[PDF]

*We demonstrate that the Making experience afforded opportunities for enacting a caring pedagogy that elicited PMTs’ funds-of-knowledge, and that this approach provided a transitioning vehicle between the teacher education setting and actual classrooms.**[PDF]*

- In this chapter, we share vignettes of many of our research projects that address our larger project’s broader research question:
*What are the potential benefits of a Making experience within mathematics teacher preparation?*These vignettes take a variety of theoretical and methodological approaches and address research questions at the intersections of teacher identity, teacher knowledge, pedagogy, and design. We conclude with a range of implications for both teacher preparation and professional learning.

*We share our findings that Making is an effective venue for provoking mathematizing, identifying, pedagogy and designing discourses; revealing their intertwined nature; and illustrating that one’s identity is as central to learning to teach mathematics as is their learning of mathematics, pedagogy, and design. [PDF]*

*This work explores the emergent nature of mathematical activity mediated by multiple artifacts. Findings reveal the complexity of mediated engagement and have implications for the practice of learning mathematics through multiple representations. [PDF]*

*Guided by the question, “How does enacting a caring pedagogy during a Making-centered experience impact and broaden opportunities for meaningful mathematics learning?” we suggest a pedagogy of caring, sharing, and Making as one way to celebrate and re-center individuals who are often marginalized from meaningful learning in spaces for Making.**[PDF]*

*This paper shares findings that convey the diversity of design decisions, rationales, and mediating resources that entailed PMTs’ mathematical Making. [PDF]*

*We use project data and personal narratives to make a case for the transformative power that design and Making have within the space of mathematics learning and teaching.*[PDF]

## Book Chapters

We also disseminated this work through these book chapters.

*We illuminate the processes of a teacher becoming as it is mediated by a variety of social and conceptual resources within teacher preparation. We do so through a revelatory case study of a prospective elementary teacher named “Moira” as she participates in a Making experience within a specialized mathematics content course for future elementary teachers.*

- In this chapter, we share research that explores the potential benefits of a novel Making experience within mathematics teacher preparation that we hypothesized would inform the pedagogical and curricular thinking of prospective teachers of elementary mathematics (PMTs). That experience tasks PMTs with digitally designing, 3D printing, and sharing an original manipulative with a child to support and promote their mathematical reasoning and understanding. With a focus on the design of new tools to support teaching and learning through the use of learner-centered design practices and digital fabrication technologies, this experience has prospective teachers exploring at the intersection of content, pedagogy, and design. We begin by sharing the findings of a pilot study that revealed a surprising breadth of teacher knowledge leveraged by PMTs through their Making activity. Those findings convinced us of the promise of a Making experience within mathematics teacher preparation. They also convinced us to pursue a research trajectory aimed at discerning what other benefits the experience might offer. We share several vignettes of research on that trajectory that take a variety of theoretical and methodological approaches to address research questions at the intersections of teacher identity, teacher knowledge, pedagogy, and design. We provide implications of our findings for teacher preparation and professional learning throughout the chapter and its conclusion.