Our winter newsletter comes late this winter; it is officially spring! We wanted to wait until after our GeT: Together at RUME¹ pre-conference and workshop to publish this issue. Indeed, I wanted to wait until after the conference to write this note, in part because it is somewhat of a spring time for GeT: A Pencil also. Valuable ideas are springing up, and we are looking for a way to support them so they can bear fruit.

Several of us were at our pre-conference and workshop on February 22 and 23, and though we missed some of you, the work of each of the three working groups was well represented. Moreover, we made much progress talking across the different teams writing chapters for the book on The GeT Course. After the conference, we have continued to orient author teams to each other, especially when we see chapter proposals possibly covering comparable ground. This move pursues two purposes that can be expressed in James Carse’s (1987) distinction of finite and infinite games. Finite games are like matches of a sport; they are played to the end and have a well defined result. Infinite games are like the sport itself; they are played in order to perpetuate the game. As games in a sport attest to, playing a game often involves playing both a finite and an infinite game. Such is the game of democracy, each election is a finite game, but democracy itself gets perpetuated each election. 

Such is the case with our forthcoming book and the strategy to orient authors to each other. First, there is the finite game of writing and publishing the book, and to that end, we want to create mechanisms for each chapter to be unique. Along those lines, developing awareness of what others are writing is good for authors so that they can identify their chapter’s unique strengths and build on those. Second, and perhaps more importantly, there is the infinite (or, at least, indefinitely long) game of building and maintaining community around the GeT course to improve the teaching of geometry to future teachers. Writing and publishing the book and its chapters are a strategy to do that. The book is giving our community a chance to grow in numbers and to develop connections among members. In this sense, orienting authors to each other pursues the goal of building community and bringing more people in. I believe the pre-conference and workshop contributed to the playing of this infinite game as well. I want to revisit three reflections from these events that support the notion that beyond writing a book together, what we were doing was developing a stronger community.

First reflection: Developing narrative for GeT: A Pencil

In her reflections on the conference, Carolyn noted that one salient aspect for her was our collective realization of the need to develop and share a narrative of our community. It seems essential that, as we welcome new members, we can say more about where we are coming from and what we have been doing. Our concern with the geometry course for teachers is not the only defining aspect of our community; the way in which we are concerned with GeT courses also matters. Along those lines, the image of a pencil of lines² might assist; the notion that we come to be involved with GeT courses from different directions (as lines that converge to the same point though they have different directions) is a key definitional idea. It helps, for example, to assert that the community is pluralistic, and not because we have not yet found “the truth” about how to improve geometry courses for teachers but because we do not surmise that there is such a thing to be found. Instead, we take the differences we bring as riches that can be combined into compromises and consensus positions. And we take the building of a community that accepts those differences as riches and commits to working with them as more important than hitting a single best idea. 

In saying that, I am particularly aware of where I am coming from. As someone who does research in mathematics education, I have a particular way of looking at instruction that could be seen as the source of ideas on what “should” be done, but I do my best at second-guessing that instinct. I am willing to bring in information (e.g., what we know about the MKT-G test), ways of being in the world (e.g., creating, administering, and analyzing surveys is one way I know how to find things out), and some personal values (e.g., the aspiration that mathematics courses for teachers could improve prospective teachers’ knowledge of mathematics for teaching is something I believe is a valid concern) to our community. However, I am not willing to avail myself of the same prescriptive attitude many in the field of mathematics education take toward instructional practice and try to tell people what they should do. One thing we learn from research is that all the positive knowledge we may glean from research practice is eventually knowledge about models that reduce the complexity of real practice. There is quite a bit that researchers do not know, and the relationship between what is true and what should be done is always mediated by moral reasoning, on which researchers do not have a monopoly. Thus, I come to the GeT course also looking forward to learning from perspectives different from mine and not expecting that eventually people should think like I do. 

I surmise that all our community members have a similar sense of what they know and can bring and also of what they do not know. Moreover, by coming to work together we are affirming some sense of interdependence, represented by the point to which all the lines converge; to make headway toward a common goal, the different lines need to find ways of getting closer to each other. Newcomers could be quite disruptive if they came across as someone who knows exactly what we need, and we might need tools to disabuse them of that thought without making them feel that we do not need them as participants or that what they have is not useful at all. Unfortunately, both the academy and the commercial world have conditioned us to expect that value is shown in the competition among products and that finding the best product is more important than how we find it. GeT: A Pencil has tried to do things differently, or maybe I should say, it has become a different kind of community, where the membership comes up with goals and means, and those are serviceable to maintaining and growing community where we look for complementarity among people and the riches they bring. I think we need good ways of impressing this aspect of our community onto others. Developing a narrative might help toward that.  

The need for a narrative was apparent in at least two ways. One was in Dorin’s proposal that the book provide an account of how it is that we landed on the need to develop some essential Student Learning Objectives (SLOs). Dorin articulated the conjecture that each of the SLOs has a trace in our earlier activities mapping the courses that each member of the community has been offering. This mapping exercise happened during our first year. We started with a map that I created where I tried to be inclusive of all sorts of things I had heard in our interviews, seen in the posters at the 2018 conference in Ann Arbor, or learned from research. I did that in a piece of software that allowed people to create their own mind map by editing the original mind map. It seems that the software was handy for people to subtract nodes and add new nodes as needed, and a nice set of very diverse maps was created. I was not at the discussions of those diverse maps, but what I heard was that out of that diversity came the impetus to develop the set of SLOs.

The second way in which the need for a narrative was apparent was in Erin’s advocacy that the complete set of elaborations on the SLOs should be included in the book. This seemed like a good idea to everyone I have talked to, especially because the book is being written in response to the current version of the SLOs, and so, having the current version in a place from where it can be cited seems really important. At the same time, part of the narrative about the SLOs is that they are a living document, and their existence on the website getapencil.org means to keep track of that living document over time (i.e., the site will archive old versions and post the newest version of the SLOs). It seems to me that the publication of the first version in the book is akin to the photo with which we often introduce a new family member to our acquaintances. As the child grows, new photos document what they look like, but old photos are still valuable to tell the life story. With this, I mean to say that for the SLOs to actually be a living document, we might need to embrace the idea that we will have periodic releases, comparable to an annual school portrait of a child. The first picture may be the publication in the book, and subsequent ones might be on the website or in other books that may emerge later on. They will all help us tell the story of how the SLOs exist as a living document. 

Second reflection: How does the living document continue?

A second reflection was about possible mechanisms to keep alive the discussion of SLOs as we enlarge the community. As I mentioned above, all three working groups were represented at the conference. One of the three groups, the ESLO group, represented by Michaela, Younggon, and Mara, has been meeting since October 2022 and includes both mathematics professors and secondary teachers. The acronym ESLO means Engaging with the Student Learning Objectives, and the group’s finite game is to provide a first set of commentaries on each of the SLOs. By the time of the conference, the group had managed to discuss only the first three SLOs and had not posted any commentaries on the website yet. However, it was already clear that they had things to say that could inspire changes in the SLOs. We heard, for example, that the current version of SLO 3, while oriented to meeting the content needs of high school geometry, had been written in such a way that it only covered process standards. The ESLO group was not satisfied with that.

The ESLO group had been having discussions of how particularly rich tasks could be used in GeT courses to bring up some of the content from high school geometry. I had been thinking about that too. In my own work teaching future secondary teachers, I have often used the angle bisectors of a quadrilateral task (to answer the question “what can be said about the angle bisectors of a quadrilateral?”) in teaching them instructional methods. This task has a lot of rich mathematical content that connects to several of the topics and processes of high school geometry. The question itself calls for reviewing what an angle bisector and a quadrilateral are. Considerations of what sorts of things could be said arise from tinkering with incidence questions (e.g., how many intersections can be created with 4 different lines?). An interesting contrast is often recalled with triangles where the angle bisectors always meet at a pointa; and one might then ask in what circumstances the angle bisectors of a quadrilateral do so. But one might also ask what figure do the intersection of the angle bisectors make when they don’t converge at a single point. All of that questioning makes good use of concepts of parallelism and congruence, the sum of the angles of a triangle and a quadrilateral, and properties of tangents to circles. The ESLO group discussed other tasks that might be used to review the high school geometry content.  

In response to the brief feedback on SLO 3 provided at the conference, we heard back from the Teaching GeT group that SLO 3 had a rocky history and that even the original writing team was not completely sold on its current state. Out of this exchange came the thought that two new chapters in the book could help represent how the revision process will come about. In one of these chapters, members of the ESLO working group will articulate their critiques of the current version of SLO 3, and in the other chapter, members of the Teaching GeT group will respond, possibly with a proposal for a new version of SLO 3. Including this exchange in the book might help show the community at large how the living document idea could be brought to fruition. It will not just be a matter of adding or subtracting from the original document; it will take engagement with the prior work and incremental improvement upon it. This process illustrates the infinite game of community building.

How the community continues to exist

Amanda and I sometimes describe the development work we do in our projects (including GeT Support and ThEMaT IV and V) as soft professional development. We see ourselves as creating facilities (environments, tools, events) that allow people to learn through doing work together. Often, over time, change happens organically, but we do not see ourselves as leading anybody to make any particular change; we treasure, instead, the opportunity to build communities from which people will draw the support to do what makes sense to them. Because of this, I have a love/hate relationship with the expression professional development. On one hand, the expression is handy to put our work in a box and in the company of other projects. On the other hand, that company sometimes generates expectations that are unlikely to be the ones I would like to accept responsibility for. The latter includes the “I am trying to make you change” sort of stance toward participants; I try not to come across that way. The work I do is very unlike professional development, in the sense that there are no specified changes I pursue for the community to enact, and I do not consider myself an expert on what the community is or should be doing. Along those lines, I am really happy about how much we have been able to do together cooperatively and collaboratively. 

The word cooperation is often used to describe people doing things in parallel while making use of common resources. An example of this is how the work of the working groups occurs in parallel while the GeT Support project provides resources such as recordings and publication on the website and the Newsletter. The GRIP Lab also benefits from those resources in that, as some of you may have seen at RUME, one thing we are doing in our analyses of the transcripts from working group meetings is to investigate how the consensus develops. As you may surmise, the arguments that happen around instructional decisions are different than mathematical arguments; they include mathematical considerations but also other considerations, and they draw from the variety of funds of knowledge that members bring with them (in all cases the working groups include mathematicians and mathematics educators; ESLO also includes teachers with experience teaching high school geometry). We also are able to further our work documenting the teaching of geometry for secondary teachers when community members distribute our surveys and MKT-G tests. It has been great to see that side projects such as the Adinkra lesson study group or the study of using the FullProof software have developed around subgroups from within the community. At the RUME conference we could also attend each other’s presentations and learn about each other’s work. 

The word collaboration is often used for a different kind of joint work–not the sharing of the same resources and working in parallel but rather working together toward a common goal. The various writing projects we have done together illustrate this aspect of the community’s work. Sometimes we have the chance to coalesce into a single task. The book chapters we wrote recently and the forthcoming book illustrate this collaboration. Along those lines, I was happy to hear the warm reception of my idea of dedicating part of the new grant proposal to the development of an “uber-book” that includes tasks, videos, dynamic geometry sketches, and more, using an electronic book platform that permits user research. I think if we do get funded it will give us a great opportunity to collaborate. As we move closer to writing a new grant to support the community for another four or five years, it will be great to hear what other projects members would like to engage in to also sustain cooperative work. 

Conclusion

We are by no means done yet, but I can say that the work of GeT Support, developing GeT: A Pencil as a community has been a highlight of my career. But it is only spring! We can look ahead to milestones in the production of the GeT Book as a finite game. Receiving the chapter submissions at the end of May will be like half-time break, and then, after feedback and editing, we can look forward to the end of that finite game. But the infinite game of growing the community will continue. Keeping our narrative in mind, whereby it is not just the goal of improving capacity for high school geometry instruction but the pluralistic way in which we go about it, will be important. I surely hope more people will join us in our cooperative and collaborative work.

References

Carse, J. (1987). Finite and infinite games. Ballantine.

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¹NB: RUME is the annual conference of the Special Interest Group on Research in Undergraduate Mathematics Education of the Mathematical Association of America. See http://sigmaa.maa.org/rume

²A pencil of lines is the set of all lines that shares a common point.