GeT: A Pencil

Author: Amanda Brown

  • The value of contributing to the development of an edited volume: How I decided my participation matters

    Over the past few months, many of you have contributed chapters to an edited volume that the GeT:A Pencil Community will soon be submitting to the MAA. Some of you are still in the process of wrapping up a chapter. In the next month or so, the co-editorial team will be asking many of you, whether you contributed a chapter or not, to pitch in and provide reviews of those chapters. We certainly know how to keep you busy! When it comes to my own motivation for working on this project, including helping to write a chapter and, undoubtedly, reviewing chapters, it helps me to recall the reasons why I took on this project in the first place.

    In this newsletter article, I will share some of the reasons why I initially decided to participate in the development of an edited volume and why I continue to believe that the work we are doing can have a lasting impact on both individual authors and the collective field focused on understanding and improving the undergraduate geometry course for teachers.My hope is that if you are lacking motivation to finish up that chapter (we’ve all been there) or engage with the upcoming review process (which is not uncommon either), the reasons I outline here will help you reflect on your own reasons for contributing to the development of this volume.

    Oh, and for those of you who are just now hearing about this project, the edited volume will focus on providing readers with resources such as commentary, instructional materials, and ideas for improving the undergraduate geometry course for teachers.

    One reason I decided to get involved with the development of this edited volume is because of the opportunity for the GeT: a Pencil community to engage in synergistic collaboration with others in the field. By joining forces with a diverse group of contributors, each of us will have the opportunity for collaboration with other researchers and writers. By working together with a diverse group of individuals, we can tap into a collective pool of knowledge, expertise, and perspectives. This collaboration fosters a spirit of synergy, allowing us to build upon one another’s ideas, challenge assumptions, and explore new dimensions of scholarship related to the undergraduate geometry course for teachers. The result is a richer, more comprehensive body of work that surpasses the limitations of individual contributions.

    Another reason I decided to get involved with the development of this edited volume is that I knew these kinds of opportunities often allow contributors to amplify the impact of their own scholarship and perspectives. By being part of a collective work, each of us can gain visibility and recognition for our work within a larger context. As editors, our hope is that through careful curation and organization of chapters, we will ensure that each contribution complements and enhances the others, creating a cohesive and powerful whole that can help ensure greater reach for our individual work. This amplification of individual contributions elevates the reach and influence of authors, enabling their work to resonate with a broader audience and make a more significant impact.

    Additionally, I decided to get involved because work on edited collections often creates valuable opportunities for networking and professional growth. Collaborating with fellow authors and engaging with the editors has a way of fostering connections and relationships, something the GeT: a Pencil community has valued from the beginning. When considering whether I should get involved in this edited volume, I knew the opportunity to deepen and expand these connections both within the community and the field at large was too good to pass up. Even now, I am seeing ways the upcoming interactions around the chapters in this volume can lead to new collaborations, mentorship possibilities, and exposure to different areas of scholarship and perspectives. Furthermore, being part of an edited collection enhances an author’s professional portfolio, showcasing their ability to contribute to collective endeavors and demonstrating their commitment to advancing scholarship in our individual fields.

    A fourth reason why I decided to support the development of this edited volume was my awareness that an edited collection offers contributors a platform for intellectual engagement and inspiration. Interacting with diverse perspectives, exploring new ideas, and immersing oneself in the collective knowledge of the contributors can spark creativity and open new avenues of thought. The process of refining and adapting individual work to fit within the larger framework of the collection challenges authors to think critically and expand their intellectual horizons. The act of reading and writing educative reviews, as well as responding to reviewer comments (especially those with different perspectives), has been an incredible source of intellectual stimulation in my career. It has fostered personal growth, fueled my passion for scholarship, and inspired me to push the boundaries of my own knowledge and expertise. My hope is that this outcome will be the same for you as you set yourself to the tasks ahead related to writing and reading reviews of the chapters submitted to this compilation.

    One of the most rewarding aspects of contributing to the development of an edited collection is the opportunity to be part of a greater cause. In fact, when deciding whether or not my participation in this particular edited volume was worth the time investment, this greater cause to which we endeavor solidified my decision. From the beginning of the project, Pat and I have advocated for one particular cause: improving the capacity for teaching high school geometry. This edited volume focuses on a subject that, in my opinion, has a good chance of making progress on that cause by providing resources, addressing challenges, and advancing scholarship related to leveraging the recently proposed Student Learning Objectives (SLOs) for the undergraduate geometry course for teachers. By participating in the development of this volume, whether as an author or reviewer, I firmly believe individuals will have the chance to actively contribute to that greater cause.

    Throughout my participation in this book project, a second worthwhile cause has become increasingly apparent: advancing the means for undergraduate mathematics instructors to see and intervene in their own instructional practices in order to collectively improve the capacity for teaching undergraduate mathematics courses for teachers. When I mention “advancing the means,” I refer to the belief that the development of this volume plays a crucial role in what I perceive to be an novel and emergent way of working together to support instructors of undergraduate geometry courses in taking a central role in improving their own practice.

    For me, the development and proposal of the SLOs for guiding GeT courses represent the first step in that process. Why? Because that development happened through an instructor-led effort where the community of instructors was open to contributions from any instructor interested in participating over the course of 2+ years in the writing of the first version of the SLOs. 

    However, that alone was not enough. The willingness of the authors of those SLOs to open their work for critique through the proposal of this book was an important second step not to be missed. It is so common to want to protect what we endeavor in. In my opinion, the group’s willingness to pick a different approach——to maintain an open stance toward the perspectives of newcomers—is one of the most beautiful, brave, and democratic things I have witnessed in my professional life.

    Finally, a third crucial step in this new way of working is the willingness of the broader community to respond to the call for proposals for this volume centered on those SLOs. The number of submissions we received for this book took us by surprise. When we started, we hoped we would have enough contributions to make a book, but we were not sure because academics often find it difficult to slow down long enough to read one another’s work carefully enough to build off of it. Now we have so many chapter contributions that we feel a little strange calling this thing a book anymore; it feels more like a volume!

    In this way, we are beginning to wonder if this volume is more than a volume, or if it is not playing its part in revealing a novel way forward for supporting meaningful collaboration among university instructors of mathematics. To that end, I now see how the time I have contributed towards the development of this book is part of something much bigger than I initially anticipated. Specifically, I see the potential for the development of this volume to serve as part of a collective effort to generate scholarship, inspire change, and contribute to a process of improvement that can serve as a roadmap for future efforts to enhance undergraduate mathematics.

    The satisfaction and fulfillment that have come from this last possibility—knowing that I am contributing to a meaningful endeavor that may help us rethink the way we work together—along with the other benefits I outlined herein, provide me with a sense of purpose that motivates me to deepen and further my own professional journey. It also motivates me to complete my chapter (yes, I am among those who needed an extension). My hope is that something I have shared connects with your “why” and helps to motivate you to power through the reviews I will soon be sending to your mailbox! Wishing you a productive and relaxing summer.

  • Embracing a Developmental Review Process: Fostering community and supporting contributors to the GeT Course Book

    I am excited to be writing to you with an update about the upcoming book entitled The GeT Course: Resources and Objectives for the Geometry Courses for Teachers. Since the RUME conference, the co-editor team has been busy finalizing the review criteria for the submissions that are due in May. At the conference, some of you shared concerns you had about the review process, including worries about what kind of review criteria might be used for handling such a diverse group of authors, as well as the need for rigorous criteria to ensure quality contributions. We want to assure you that we have taken these concerns into account and have developed a set of developmental review criteria that will encourage reviewers to provide feedback that not only assesses the quality of the contribution but also aims to support contributors’ professional growth.

    We recognize that the authors of this book come from diverse backgrounds and may have different standards for peer review. Writing for such a diverse audience may make the review process feel intimidating for some. We do not want it to feel that way. Our hope is that the review criteria supports the kinds of interactions that give all our contributors a learning opportunity to improve their writing for a more diverse audience and creates a book that embraces the diversity of our community. To construct the review criteria, the editorial team has drawn inspiration from several sources, including the review criteria from popular journals such as PRIMUS and Mathematics Teacher Educator which aim to support individuals engaged in teaching teachers in undergraduate mathematics and teacher education courses. We have also paid particular attention to developing criteria that encourage reviewers to produce what Sandra Crespo, a former editor of Mathematics Teacher Educator, has called an educative rather than evaluative review. With that, we thought we could use this opportunity to share a bit more about what we mean by terms like a developmental or educative review. 

    The idea of an developmental or educative review is not a new one. Peter Elbow, an English professor who devoted his career to democratizing writing, argued for the need for a more balanced approach to the review of academic writing. Elbow (2000) noted that as part of our training as academics, we are trained to take a critical stance toward new ideas—being as analytical and skeptical as possible. He called this the doubting game. Without discounting the importance of this kind of training, he advocated for the importance of embracing a different kind of stance towards new ideas—one that he called the believing game. In contrast to the doubting game, Elbow described the believing game as the “disciplined practice of trying to be as welcoming or accepting as possible to every idea we encounter: Not just listening to views different from our own and holding back from arguing with them… but actually trying to believe them” (Elbow, 2008, p. 2). He argued this kind of disposition as useful because in order to validly assess an new idea, even if we ultimately reject it, we must first dwell in it, really endeavoring to understand it, believe it, and see it from the viewpoint of the one who is offering the idea. Furthermore, if we only play the doubting game with other’s ideas, we run the risk of missing “what’s good in someone else’s idea” (ibid). 

    For our own purposes, embracing the believing game has value beyond helping us, as individuals, better assess the merit of new ideas. The believing game is also highly consistent with the values that have been part of GeT: A Pencil from the beginning—values which have supported the gathering of a community of individuals drawn from a variety of backgrounds to support one another and help each other grow professionally while working together to improve the capacity for teaching high school geometry.

    With this focus on the believing game, we have specified criteria intended to encourage reviewers to provide specific, constructive, and actionable feedback, highlighting strengths and opportunities for improvement in a way that helps authors to move forward. This mirrors the kind of feedback we have seen so many of you provide as you have walked through the difficult conversations about GeT courses that have informed the first version of the GeT SLOs. 

    We look forward to sharing more specific details about the review process soon, and we invite all of you to join us in this exciting endeavor of engaging in a process of review that not only worries about improving the book’s contributions but also endeavors to continue expanding our community of support for individuals that have taken the risk to contribute a chapter for the book and count themselves as among the stewards for the undergraduate geometry course for teaching. Together, through our efforts in this review process, we can create a resource that supports the teaching and learning of geometry for teachers across a wide range of settings and experiences.

    References

    Crespo, S. (Ed.). (2016). EDITORIAL Is It Educative? The Importance of Reviewers’ Feedback. Mathematics Teacher Educator, 4(2), 122-125.

    Elbow, P. (2009). The believing game or methodological believing. Journal for the Assembly for Expanded Perspectives on Learning, 14(3), 1.

    Elbow, P. (2000). Everyone can write: Essays toward a hopeful theory of writing and teaching writing. Oxford University Press on Demand.

  • Towards the Development of a Prospectus for a GeT: A Pencil Book

    In this past year, the GeT: A Pencil community has been very productive in terms of the dissemination of scholarship about the collaborative work happening within the community. For example, at the time of this writing the GeT Transformations Working Group has shared their work developing and co-teaching a series of lessons focused on the geometric transformations embedded in the Adinkra Symbols in a few outlets—including the 2022 RUME conference (https://www.gripumich.org/2022/04/24/get-a-pencil-represented-at-rume-2022/), the AMTE 2022 conference, and the AMS blog (https://blogs.ams.org/matheducation/2021/05/06/best-laid-co-plans-for-a-lesson-on-creating-a-mathematical-definition/#more-3605). Similarly, the Teaching GeT Working Group has shared their work focused on developing a set of Student Learning Outcomes (GeT SLOs, hereafter) in a variety of contexts—including numerous articles contributed to GeT: The News! and presentations at the 2022 AMTE and 2020 and 2022 RUME conferences (https://www.gripumich.org/2022/04/24/get-a-pencil-represented-at-rume-2022/). Also, several instructors have collaborated with members of the GRIP lab in efforts to share about the formation and development of the GeT: A Pencil community in higher education outlets—such as a chapter for the upcoming Handbook of STEM Faculty Development and a roundtable discussion at AERA (https://www.gripumich.org/2022/04/27/grip-at-the-2022-aera-annual-meeting/). 

    In light of these efforts, the GRIP Lab recently (January 2022) proposed the idea of advancing the community’s dissemination efforts with the publication of an edited book. Our initial suggestion to the community has been met with enthusiasm, both during the meeting and in the intervening time since that initial discussion. Pat and I have invited Dr. Nathaniel Miller and Dr. Laura Pyzdrowski to join us for some initial discussions regarding how we can get going with the project. 

    I would like to take this opportunity to report about those conversations and, in doing so, share some of our initial ideas. So far, our meetings have focused on the following topics: (1) what the content of such a book might be, (2) some potential publishers we might approach with the idea, and (3) a rough timeline for ensuring the project is completed in a reasonable timeframe. In what follows, I share details about the first item—our initial ideas about the content of the book—in hopes that you, as part of the community of individuals that read GeT: The News!, will provide feedback we can use to further shape our efforts to envisage what such a book might look like. We conceive of the book being comprised of contributions that fit roughly within six topics that I describe briefly below.

    Topic 1: Background about formation of GeT: A Pencil 

    This topic emerged out of our collective sense that it might be important to provide the reader with some context about the formation of the community. Some of the contributions that might fit within this first topic include: the complexity of the system of improvement that surrounds GeT courses from the perspective of multiple stakeholders; things we have learned about the existing variations in GeT courses from various instruments such as the MKT-G, syllabi collection, instructional logs, and end of course surveys; the ways that an online inter-institutional community can serve as a kind of “virtual mathematics department” focused on development of a course and instruction within a given course; and a retrospective description of the development of GeT: A Pencil. These contributions may help identify the value of the projects the community has undertaken. 

    Topic 2: Background and elaborations of the SLOs

    Contributions to this topic would complement, without simply duplicating, the work that has been ongoing over the last two plus years to articulate the SLOs. Unlike the future SLO website which will contain the “official” elaborations produced by the collective teaching GeT group, we envision the inclusion of this topic as providing opportunities for smaller teams of authors to provide more personal accounts regarding what a particular SLO means to them. In seeking contributions to this topic, we plan to encourage contributors to work in smaller groups in order to produce chapters about some of the diverging perspectives that have emerged during the development of and/or conversations about the SLOs—with the real possibility that there will be more than one chapter per SLO. Our hope is this will result in a focus area that encourages readers to engage with the larger conversations underlying the development of the SLOs by putting these multiple perspectives about the SLOs in conversation with one another. Unlike the SLOs and their elaborations, which speak with a collective voice and articulate a compromise arrived at by a certain date, we surmise that a diverse set of perspectives on the SLOs will help keep alive the various strands of discussion toward the goal of the SLOs being a living document. 

    Topic 3: Supporting the SLOs in instruction

    We hope that the inclusion of this topic would create an outlet for the plethora of efforts within the GeT: A Pencil community and possibly elsewhere to share instructional activities. For example, the inclusion of this topic area could create opportunities for individual or teams of authors to further develop some of the past GeT: The News! articles that have focused on the sharing of instructional activities. Also, this could create an outlet for some of the work on the Adinkra Lesson that has been ongoing in the Transformations Working Group or for expansion on the work that started back at the beginning of our community’s existence in the GeT task repository working group. Crucially, chapters contributing to this topic area will provide the authors with the opportunity to go beyond simply describing an activity, providing space to allow them to account for the activity in terms of how it can help serve in the instructional support of the SLOs. We imagine that both members and nonmembers might want to participate in writing these illustrations and that the editing process could help connect the writing to the SLOs, so that even if someone does not quite know how the activity they do can be connected to any SLO, the printed result will make that clearer. We think this can be a strategy for disseminating the SLOs as well as inviting commitment to the SLOs by others (in this case, authors that publish their activities).

    Topic 4: Assessing the SLOs

    For this topic area, we seek to include chapters focused on the assessment of the SLOs. This could include contributions focused on the process of constructing items for assessing the SLOs, what we might learn about students or GeT courses from the administration of such items, the uses of such items for various purposes within GeT courses (diagnostic, formative, summative), the construction of different types of rubrics for grading or scores those items (holistic, analytic), and various perspectives on such items and their use with GeT students. We have seen some examples of what this can be like in notes written by Michael Weiss in this and the previous issue of GeT: The News! Again, both members and nonmembers could share assessment activities they use and comment on how these target the SLOs.

    Topic 5: Sustaining the work around the SLOs

    We thought it might be important to include a topic that looks to the horizons of this work–naming work that is ahead and fleshing out what that might look like. As of right now, this is the least developed topic area, but we are open to ideas. We have some ideas about general terrain that could be named. We also have a few ideas for chapters related to the longer run implications of the SLOs on things like the MKT-G instrument. To be clear, as a group, we are not yet sure if this is its own topic area or something that could be subsumed into the next topic area, as a kind of commentary about the SLOs. But we thought in this early stage, we would at least share the idea about the topic and invite you to respond with ideas you might have in mind. Since the beginning of the community, we have regularly heard from many of you about ideas you have for the ways this work might be expanded and sustained in the years to come. Perhaps this topic area could be a place to start filling out those visions into fuller proposals. Let us know what you think.

    Topic 6: Reactions from the Stakeholder communities

    For this topic, it might be valuable to have reactions about the SLO work (or perhaps particular chapters from the edited collection) from those that might well represent the stakeholders of the system that contains the problem of improving the capacity for teaching high school geometry. This system includes not only the GeT course but also the institutions that make demands on and provide resources for the course as well as those institutions which stand to benefit from and feed the GeT course. In this topic area, we envisioned inviting contributions the following types of individuals: mathematicians that influence or have, in the past, influenced the GeT course; mathematics educators that have had some investment in the teaching and learning of K-16 geometry; authors of frequently used GeT textbooks; mathematics education researchers who have done basic research on children’s thinking about geometry; mathematicians or mathematics educators with a more international focus on geometry; individuals that have served in administrative roles in university mathematics departments; individuals who have stewarded teacher education credentialing processes at the university; individuals who have played a role in projects that look at improvement in ways different from the strictly institutional perspective; practitioners who have experience working in and with high school mathematics departments to improve geometry instruction; educational researchers who have methodological expertises in the areas of survey and assessment design; and possibly also recent graduates from teacher preparation programs who may have the ambition to write and could bring a fresh perspective. 

    In sharing these roughly sketched abstracts for the topics to be included in the book, our hope is that in the next month or so you will find ways to reach out to one or all of the four of us and provide your thoughts. This could include suggesting additional topic areas and ways to expand or collapse these topics, along with any other thoughts you might have. So, please, don’t be a stranger. We really do endeavor for this to be a product that serves our collective aims to disseminate about the work we have all been engaged in these last few years together. You can reach us at the following addresses: Amanda Brown (am******@***ch.edu), Pat Herbst (pg******@***ch.edu), Nat Miller (na**************@**co.edu), and Laura Pzydrowski (la***@********ki.ws). We are also planning to have a discussion about this project during the GeT: A Pencil Community meeting on Friday, June 10, 2022 @ 2:00 to 3:30 pm. We are hoping by scheduling it for the same time slot as our seminar series, we will enable as many community members as possible to attend and share their thoughts. 

  • Developing and Stewarding the Get: A Pencil Community

    One of the most commonly discussed goals of the GeT Support Project is working collectively to improve the instructional capacity for high school geometry. It makes sense for this to be a point of common discussion for us, as it is the goal to which the Get: A Pencil community is collectively committed. There is, however, a second, equally important goal of the GeT Support Project: developing an inter-institutional network of support for instructors of the undergraduate GeT course. As conveners of the GeT: A Pencil community, the GRIP lab has been keenly aware of our responsibility to observe and support the community’s progress toward this second goal. After all, without the establishment and maintenance of the GeT: A Pencil community, it will be quite difficult to make progress toward the goal of improving the instructional capacity for high school geometry. Because of this, many of our internal reflections, conversations, and decisions at the GRIP lab have been animated by this second goal. 

    Internally, we have found the forming–storming–norming–performing model of group development (Tuckman, 1965) useful for trying to understand both what we are observing in the community, as well as what changes in the community we should anticipate to be on the horizon. When we first began together as a community in June of 2018, Tuckman’s forming stage described our community well—with individuals engaging in introductions with a certain amount of eagerness and excitement as well as anxiety. Similarly, our first year together was fairly well accounted for by Tuckman’s storming phase of development. In this phase, it is typical for a group to experience some amount of conflict, competition, and drop offs in participation. This is because a storming group has moved past simply exchanging pleasantries into the identification of real issues facing the group. While these issues have a way of surfacing conflict, they also have a way of getting a group ready for the norming phase—with individuals electing to work together, in spite of their differences, to collectively seek the resolution of the issues they have identified.

    In the norming phase of development, a group learns to resolve the kinds of personality conflicts that characterize the storming phase by learning to accept one another as they are in order to work collectively towards a common goal. Until recently, I felt pretty confident that the GeT: A Pencil community had been mostly operating within this phase. But in some of our recent reflections about the work people are doing within the GeT: A Pencil community, I felt it quite likely that we have turned the corner in some crucial ways—with the group engaging in activities more akin with what Tuckman described as the performing stage of a group. 

    In the performing stage, a group benefits from previously established norms and is able to achieve high levels of success towards meeting the goals they set out for themselves. Such a group is now able to operate in ways that are fairly autonomous, needing less of the kinds of supervision or organization they originally could not muster on their own—with the organization hierarchies that may have been crucial in the early stages being much less visible and prominent. I’d like to take this opportunity to illustrate my point by taking a brief tour of some of the work that is happening in the community.

    In the last year, the Teaching GeT Working Group has elected to set aside differences in their individual courses to work together towards a common set of student learning objectives (SLOs) for the undergraduate GeT course. Although the processes the group has used have been marked by consensus-seeking from the beginning, the group’s interactions make it clear that dissent is not only an allowable part of the process but a necessary and crucial aspect of their work for the continued refinement of the SLOs for broader dissemination and consumption. Some of the artifacts this group has produced can be seen in this issue’s articles entitled GeT Course Student Learning Outcome #8 and GeT Course Student Learning Outcome #10, as well as the Working Group Update provided by the group’s facilitator, Dr. Nat Miller. This group also recently gathered to find avenues to disseminate their work—managing to submit two conference proposals and one handbook chapter proposal in the last six months.

    Similarly, the Transformations Working Group has brought together individuals to think together what it might mean to teach geometry using a transformational approach. In this group we have seen the members grow in greater intimacy as they have voluntarily elected to open up their classrooms to one another through a series of virtual classroom observations. More than that, the group elected to do this using a process called lesson study in which the group collaboratively developed, implemented, observed, reflected on, and refined a set of lessons for teaching transformational ideas using the Adinkra patterns (see Babbitt et al., 2015). Remarkably, the choice to center a set of lessons on the Adinkra patterns emerged from an idea in the group, rather than from an area of expertise that any of the faculty members had previously developed. We see this as an important indicator of the group’s willingness to be vulnerable with one another and take risks—moving past the more typical norms of competing and showcasing that can sometimes go along with lesson study in the U.S. (Rappleye & Komatsu, 2017). Like the other working group, this group has begun to disseminate about their work—producing two conference proposals and posting a note on one of the AMS Blogs (Boyce et al, May of 2021). Dr. Julia St. Goar provides an update about the current happenings of the group in this issue’s Working Group Update.

    We also note that for the first time since the GeT: A Pencil community formed, we, as members of the GRIP lab, have been able to make time to facilitate a series of workshops within the community. Up until now, we have had our hands full handling the administration and organization of the community—expending most of our effort making sure that things are running smoothly. But this past year, at the request of the community, we engaged in the development of items for assessing the SLOs—piloting those items with a set of GeT students this spring. This summer, we felt the other two working groups were operating autonomously enough for us to have the needed bandwidth for our team to run a series of workshops focused on engaging instructors with those items as well as the responses we gathered to them this spring. More information about those items and what we are learning through those workshops can be found in this issue’s articles entitled “From Theory to Practice: Development of the SLO Items”(Ion & Herbst, 2021) and “A Deeper Dive into an SLO Item: Examining Students’ Ways of Reasoning about Relationships between Euclidean and Non-Euclidean Geometries ” (Herbst & Ion, 2021).

    Finally, alongside the evidence we see in these bodies of work, the interactions between instructors also provide evidence that we have, indeed, turned a corner as a community. In this issue, Dr. Carolyn Hetrick, a recent graduate whose research focuses on community organization and a postdoctoral scholar at the GRIP lab, shares about some of the changes she has observed in her re-entry into the community after being away for several years. 

    Our community’s shift into the performing stage of development has some important implications for our future work—including the creation of some exciting opportunities we see emerging on the horizon. One of the opportunities that we envision is a conference with a different kind of participation from GeT instructors than we envisioned for the June 2018 conference. For example, we can imagine a future conference that is organized by a committee made up of members from the GeT: A Pencil community, rather than planned solely by the GRIP. If you are interested in potentially playing a role to help shape a future conference, let us know. 

    References

    Babbitt, W., Lachney, M., Bulley, E., & Eglash, R. (2015). Adinkra mathematics: A study of ethnocomputing in Ghana. Multidisciplinary Journal of Educational Research, 5(2), 110-135.

    Boyce, S., Ion, M., Lai, Y., McLeod, K., Pyzdrowski, L., Sears, R., & St. Goar, J. (2021, May 6). Best-Laid Co-Plans for a Lesson on Creating a Mathematical Definition. AMS Blogs: On Teaching and Learning Mathematics.

    Herbst P. & Ion M. (2021, November). A Deeper Dive into an SLO Item: Examining Students’ Ways of Reasoning about Relationships between Euclidean and Non-Euclidean Geometries. GeT: The News!, (3)1.

    Ion M. & Herbt P. (2021, November). From Theory to Practice: Development of the SLO Items. GeT: The News!, (3)1.

    Tuckman, B. (1965). Developmental sequence in small groups. Psychological Bulletin. 6(3), 384-399.

  • 2021: Looking Back, Looking Forward

    With 2020 in the rearview mirror, we take a moment to pause and reflect on some of what’s happened since the last newsletter. The rising incidence of COVID-19 infections brought with it a massive upheaval for American higher education institutions—forcing students, in many cases, to vacate dorms, and instructors to rapidly transition from hybrid to exclusively online forms of instruction. While this upending of “business as usual” has certainly come with its share of unique and difficult challenges, we have heard from you that it has also provided some surprising and uncommon opportunities. This fall, the ubiquitous use of online and hybrid instruction provided us the possibility and privilege of peering into some of your classrooms. We’d like to share some of what we learned.

    We would be remiss if we didn’t acknowledge the challenges that this time period has created for undergraduate students. For one, the disruptions to more routine university life has brought unique logistical challenges for undergraduates. In the moments between problems in one small group, we overheard one student sharing about the struggles she was having with the university’s decision to leave some courses in-person and others remote—putting her in a situation in which her parents insisted she save money and move home only to face several two-hour commutes over snowy roads for her one in-person class. We also observed ways that the sweeping changes to university life has left students feeling uncertain about their futures. In another small group exchange, two students discussed their fears regarding their ability to make progress in their teacher preparation programs with local districts and mentor teachers feeling too overwhelmed to make room for university students’ field placements. Also, we observed ways that these shifts have forced many students to blend their various roles (i.e., in addition to being students, they have familial obligations to their siblings, parents, and sometimes spouses or children) making it more challenging, at times, to focus on their role as students. For example, in the midst of a virtual whole class debate about whether or not Euclid’s third axiom holds in hyperbolic space, we observed one student splitting her attention between the small wiggly child in her lap and taking her turn in the debate—sometimes struggling (understandably) to keep her train of thought.

    Yet, in the face of these challenges, we were consistently impressed with the innovative ways that both instructors and students worked collectively to maintain some semblance of normalcy in instruction. We observed instructors and students making use of physical manipulatives to maintain students’ engagement with instructional activities. One instructor told us how, in order to avoid the risk of spreading the virus, she organized and distributed physical manipulatives at the start of the semester to enable students to gain familiarity with the kinds of resources they will eventually need to use in their work as high school teachers. Another instructor supported students over video conferencing to construct their own 450-degree cone out of construction paper and then used these cones in a series of investigations exploring the viability of Euclid’s postulates on the 450-degree cone. 

    Beyond the use of physical manipulatives, we also saw familiar technologies being leveraged in creative ways to support teaching and learning in the GeT courses we observed. Technologies as simple as document cameras, video cameras, and even social media were leveraged by instructors and students alike to not only cope with but also thrive in the current situation. On numerous occasions, we even watched as students helped one another and the instructor out by using social media platforms, such as Snapchat and Instagram, to share course files with other students. In some cases, instructors have shared how they have identified new practices during this time that they intend to keep—even after the pandemic is behind us. For example, one instructor shared her plans to continue taking advantage of newly-installed classroom cameras to record students’ mathematical presentations, enabling students to reflect on and improve their communication skills. Another instructor shared plans to maintain virtual office hours for students—noticing a substantial increase in students’ participation.

    We also saw instructors and students making use of math-specific technologies to overcome their diverse geographic locations. For example, in a hybrid GeT course, students were managing (in spite of their geographic distance) to collectively communicate and reason about proofs using dynamic geometry software. Crucially, students had learned to harness the functionalities of Geometer’s Sketchpad to remotely walk through the steps of their proofs—simultaneously highlighting statements in their proof and corresponding elements of the given diagram (see Figure 1). 

    Figure 1. GeT student remotely presents their proof using Geometer’s Sketchpad.  As the student gestures to a statement in the proof, the associated angle is highlighted in the diagram.

    The students’ used this functionality to participate in a core mathematical practice of listening and critiquing one another’s proofs. Also, this use of Sketchpad enabled a kind of dynamic form of mathematical communication in which a single gesture towards a proof statement is intrinsically linked with the corresponding aspect of the geometric diagram that we think has potential beyond this current season of disruption to normal instruction. In all cases, we saw instructors and students finding ways to use the tools they had available to communicate about new and powerful ideas  for students’ mathematical development.

    Overall, what we saw while “sitting in the back of your classrooms” left us curious about the future. We found ourselves wondering what kinds of pedagogical practices students in these courses were having opportunities to “apprentice” into.  We think it’s possible that this cohort of students is more ready than any previous cohort for engaging in online teaching, which may take the field a long way towards the goal of increasing the capacity for teaching high school geometry. We also wondered whether pre-service teachers would translate the kinds of practices they are learning, such as the use of Sketchpad to communicate dynamically about their proofs, into their future professional work. And while we, like you, long for things to return to some semblance of normal, we also found ourselves hopeful regarding the kind of resolve and commitment we saw in both GeT students and instructors. We think it’s possible that undergraduate mathematics instructors and their students will come out of this difficult season stronger and more ready to engage in mathematics education for the 21st century.

  • π Day

    When I first began my career as a high school geometry teacher (in 2000), neither I nor my colleagues had ever heard of “π Day”. In perusing one of the many practitioner journals, I learned about other secondary mathematics teachers celebrating March 14th with their students in a variety of ways, including pie-eating and baking contests as well as school-wide recitations of the digits of π. As a new teacher, I decided my students and I would join in the celebration. In that first year, only a handful of students seemed excited to participate, but by the time the second year rolled around, many more of my students had heard about the event and were on board. This enthusiasm for the celebration had its way of catching the attention of my peers, and by the time I reached my third year of teaching, I had two additional colleagues join in on the fun with their students. 

    It has been over a decade since I have worked as a high school teacher and I wonder sometimes what “π Day” looks like now, after it has had some time to gain prominence. It is certain that the public’s awareness of π Day has grown. For example, for some time now, my friends and family members (otherwise unaffiliated with mathematics) have taken it upon themselves to wish me a “Happy π Day”. On more than one occasion, one of these well-wishers has sent a pie to my home (which quickly disappeared as soon as my boys doubled down on the celebration). A few years ago, I and others at the GRIP lab participated in a π Day 5K (or 3.1 mile) run, though I opted out of the pie-eating contest that strangely came before the run.

    Beyond the general public’s awareness, I have also noticed local and national pie and pizza companies cashing in with their homages to π. With so much hype about the food-related activities of π Day, I have found myself wondering a bit about whether mathematics has taken a back seat to. . . well, eating pie!? While I am not opposed to adding another food holiday to the calendar, I do find myself hoping that the food celebration does not ultimately distract from the opportunity for the general public to grow its appreciation of mathematics.

    Perhaps I am not the only one with this concern. This year, the United Nations Educational, Scientific and Cultural Organization (UNESCO) has declared March 14 as International Day of Mathematics. Notably, π is missing from this title. With this declaration, I wonder whether UNESCO is asking the pie to take a back seat to the mathematics? And if so, I am all for it. But does this mean I won’t be getting pie sent to me this year? On second thought, maybe we should keep it π day after all!

    Do you celebrate π Day with your students? If so, please send us a note and/or picture that we can share with the community in the next newsletter!

    Members of the GRIP Lab complete a π Day 5k