In the Spring of last year, I wrote a note about what I envisioned the next stage of GeT: A Pencil could be. At that time, I noted that our first funding cycle was coming to an end at the end of August, and I indicated that we would write a grant asking for financial resources to support the development of a repository of primary and secondary resources for instructors to tinker with as they decide what to use in their GeT classes. We did write the grant but, unfortunately, it was turned down. We were advised to revise it and we did so this past June and July. A decision on it will likely come toward the end of 2024 or the beginning of 2025.
This state of affairs brings us to consider sustainability for a year between funding cycles. GeT: A Pencil has fulfilled an important role for GeT instructors. We are very active: producing an edited book, meeting in three working groups, offering biweekly seminars, and, in spite of the delay with this issue, publishing a newsletter. We also benefit from the participation of some high school geometry teachers, a feature that could grow as we move ahead toward the creation of resources. We want to maintain that positive momentum, but without funding, the work will need to continue being done voluntarily. In late June, we secured volunteer commitments from individuals to help take care of the basic functions of the community: organizing the GeT Seminar (e.g., scheduling speakers, sending invitations) and stewarding GeT: The News! (e.g., recruiting authors for the various sections).
We have also discussed how the working groups could be reshuffled and organized for next year, especially given the possibility that our revised grant proposal might get funded. The structure and focus of the working groups could help us not only sustain momentum but also prepare our community for the new funding cycle. For example, having developed the SLOs, our community is ready to start generating materials to teach the SLOs. Indeed, some of that has begun with the ESLO group, as members spent the 2023-2024 year discussing different tasks that might be used to review important content from high school geometry, hence supporting SLO3.
I envision the work ahead will consider resource development across three different dimensions:
What do we mean by resources to teach an SLO? Anything beyond tasks to give students?
What is involved in the development of a resource? Anything beyond writing them and publishing them?
How are we going to cover all the SLOs?
In regard to the first question, this is something that the community will need to eventually settle on. However, to get us started, it is useful to consider a distinction between primary and secondary resources. Primary resources include tasks used with students as well as texts that we might want students to read in order to be able to do those tasks. Some tasks may be self-explanatory but others may require knowing definitions or taking some statements as known, and yet other tasks may be done for the sake of introducing important concepts that we might want to write in the way the students are supposed to remember them. Hence, primary resources are everything that the students would see. Secondary resources, meanwhile, are everything else that might support the use of the primary resources. The answer key to problems or a rubric to grade students’ work on problems are initial examples. Secondary resources may also include a description of errors students make when doing the problems, or a roadmap of possibilities that the instructor could use as they respond to how students work on a task. This consideration of primary and secondary resources suggests that we might need some precision in terminology. I am proposing to use “module” to refer to a set of primary and secondary resources that can be used to aim at at least one SLO. Each module encompasses both primary and secondary resources.
With this clarification, the second question above can be restated as “what could be involved in the development of a module?” And indeed, the development of a module seems to involve much more than writing a task. It surely starts from having an idea of the task or tasks around which the module would be developed, but it would also be important to write the texts that students would need to read before and after engaging with the task/s. In addition, the suggestion that secondary resources would make the modules richer implies that module development involves more than writing. It could also involve anticipating student work, trying the modules out with our classes, collecting real but anonymized student work to include as secondary resources, examining the student work to inventory the difficulties students had in the task, and possibly using that information to improve the writing of the tasks and texts. The development of a module would surely take some time and it could use having access to classes where the instructor is willing to try the tasks. Furthermore, having a group of members of GeT: A Pencil involved in developing each module would make the work not only more fun and more of a learning experience but it would also bring to the work the variety of talents present in our community. It would be reasonable for this development to take a full year of meetings for each module.
This brings me to the third question. Our working group structure would allow us to gather individuals interested in working on developing specific modules. In the past, we have had three concurrent working groups. Moving forward, three groups could continue to run, and the scope of work of the new grant proposal could provide a suggestion of what they could be doing. At a recent meeting to discuss the sustainability of the community, an idea was proposed of choosing three different days and times in the week and asking community members to sign up for the day and time they would be most likely to make.Then, at their first meeting, each group could decide on one SLO to focus on, taking care not to choose one that has already been covered. Each working group could use the year ahead to focus on developing a module for their chosen SLO, and we could use the newsletter, community meetings, and the seminar for people to share across groups. That idea seems worth a try. We’ll be sharing a “whenisgood” calendar to identify three meeting times and will then ask you to sign up for the one that is best for your schedule.
To conclude, in the coming year we will not have any financial resources to offer conference support, incentives, or much staff support. Despite this, we are hopeful that some functions can be maintained at a smaller scale and on a voluntary basis. In particular, I am optimistic that the continuation of the seminars, the newsletter, and the working groups will support the continuity of our community while paving the way for new developments in future funding cycles. We hope you are able to join us for this next stage of GeT: A Pencil!
The emergent consensus our community has reached on the student learning outcomes for geometry courses for teachers allows us to come back to one of the aspirations of the initial working groups in GeT: A Pencil–the creation of a task repository. Indeed, the collaborative curation of resources for teaching GeT courses could be a next step for our community to cooperate toward the goal of increasing capacity for high school geometry instruction. Along those lines, we hope to submit a grant proposal to support this work; this essay describes some initial thoughts on how that work could be framed.
The notion of resources is key in this proposal. Trouche et al. (2020) elaborate what is meant by resources in the field of mathematics education thus:
a resource [is] anything likely to ‘re-source’ the teacher’s practice. It can be a textbook or a website …. Teachers search for resources, select resources and modify them; they use them in class, and this can lead to further modifications. … [T]his work is called teachers’ documentation work. (p. 1244)
Resources are tied to technologies inasmuch as they may be produced and modified with technologies. They are also tied to curriculum, inasmuch as curricula are canonical examples of resources, but the concept of resources in the context of instructional practice is more general. Furthermore, the resources approach spearheaded by Luc Trouche and his colleagues differs from approaches to curriculum use that emphasize central planning by authors and fidelity of implementation by teachers. It points to the dynamic interactions between teachers and resources⸺the latter informing the former, and the former modifying the latter, in ways similar to how Remi Llard (2005) had described some teachers’ interaction with curriculum materials as “participating with” curriculum materials (and contrasting with following, subverting, drawing on, or interpreting curriculum materials). Trouche et al. (2020) refer to the work that instructors do with resources as documentation, noting that the interaction with resources is projected in documents that, for example, attest to changes they made to resources or add observations made in the context of use that might inform future uses.
With the assistance of the concepts of resources and documentation, we can describe more concretely what we could be producing in the next few years. First, rather than producing a textbook, I argue that we should shape a system that can support the documentation of resources for GeT instruction. Further, we should begin to make that system work by putting out resources that might enable individual instructors to pick and choose what would be serviceable for the course they want to teach and support the work of documenting instructors’ use. Second, to support the discretionary use of resources and their documentation, I argue that the final destination for the publication of those resources needs to be a digital platform that permits both instructors tinkering with, editing, and augmenting resources, the use of those resources with their students, and the observation of patterns of instructors’ and students’ use by the community, particularly by those of us interested in understanding how instructors shape resources. Third, the digital publication of these resources also permits us to conceive of the resources that get put out as supporting various kinds of documentation. The customization of a set of curricular resources to compose a course of studies is one canonical documentation, but other traversals of the set of resources are possible to envision as we realize that the documentation process may generate other resources, derivative of the curricular resources.
To be clearer, if we think of a problem as a case of a resource that an instructor could choose to include in their course, other resources that support the use of that problem in class might be developed, linked, and consulted. For example, a set of tags that connect the problem to different SLOs can be one such secondary resource; hints for students that the instructor might provide access to are another example of a secondary resource. Information from research on students’ cognition that appraise the instructor of errors to be prepared to see in the context of GeT students’ work on the problem can be a third kind of secondary resource. Furthermore, materials derived from early uses of the resource (e.g., students’ work) can also be among these secondary resources, supporting other instructors’ anticipation of what students could produce.
This brings us back to the design of a system that can support this documentation work: The work the community may get to do includes specifying the workflows and task descriptions needed for the community to produce all of these resources. This may include many of the activities often thought of as part of curriculum development process (e.g., drafting, disseminating, piloting, assessing outcomes, revising, etc.) but with a focus more on community and systems development through the production of resources than on the production of a final set of resources. A community like ours can organize itself for the work of continuously improving its instructional resources; this can be our next step in the trajectory toward improving capacity for high school geometry instruction.
References
Remi Llard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211-246.
Trouche, L., Rocha, K., Gueudet, G., & Pepin, B. (2020). Transition to digital resources as a critical process in teachers’ trajectories: the case of Anna’s documentation work. ZDM–Mathematics Education, 52,1243–1257. https://doi.org/10.1007/s11858-020-01164-8
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.
In order to understand the impact of a teaching intervention (e.g., a course or a professional development program) on what students gained from that experience, students are often administered a test before and after that intervention, and researchers study the gains computed by subtracting the pre-test scores from the post-test. Similarly, researchers use the difference in groups’ test scores to compare performance between different groups. This way of using a difference in test scores, however, requires an assumption that test items measure the same construct (e.g, knowledge) in the same way for different points in time (e.g., pre- and post-test) or different groups of participants (e.g., pre-service and practicing teachers).
To ensure this assumption of measurement invariance (i.e., the same test items measure the same construct in the same way) is important when comparing amounts of, or gains on, a construct, because the observed difference in the scores could be due to different types of constructs being measured by the same items rather than the difference in the same target construct. For example, different groups of participants could interpret the same wording differently due to their demographics or educational background. Similarly, participants could react differently to the same content of an item depending on when the item is provided to the participants. As such, we cannot take for granted that the use of the same assessment items guarantees that a set of assessment items is measuring the same thing across different groups of participants or over time. To validly compare a measured construct across groups or time points, it is recommended that a test of measurement invariance be performed. In other words, it is important to demonstrate that the way in which items are related to a target construct (e.g., MKT-G) is equivalent across the compared populations and over time. The statistical technique used to test this invariance in our study is called multi-group confirmatory factor analysis (Brown, 2006). In this note, we present how we used the measurement invariance tests to estimate the gain of GeT students’ mathematical knowledge for teaching geometry (MKT-G) before and after taking GeT courses and how their post-test score is different from practicing teachers’ MKT-G.
By using the 17 MKT-G items developed by Herbst’s research group, we examined the participating GeT students’ MKT-G growth over the duration of the course. Also, by scaling the growth using a distribution of practicing teachers’ MKT-G scores, we approximated GeT students’ growth in terms of in-service teachers’ years of experience. An assumption in estimating a construct (here, MKT-G) by using a set of responses is that the common variance among a set of responses to items is accounted for by the construct, and the relationship between the scale of an item score and the latent construct is a linear function. The slope of the linear function, where its horizontal axis represents the level of the latent construct and the vertical axis represents item score, is the item factor loading representing the magnitude of the relationship between the item and MKT-G. The intercept of the linear function is a predicted value of the item score when the level of MKT-G is zero. Thus, the equivalence in the way in which items are related to a targeted construct between the groups can be examined by testing the equality in the structure of the construct (configural invariance), factor loadings (metric invariance), and the item intercepts (scalar invariance). We tested the equivalence of item parameters simultaneously, not only between GeT students and practicing teachers but also between GeT students’ pre-test and their post-test.
The results derived from subsequent invariance tests suggested that the relationship of the items to the measured knowledge was at least partially equivalent between GeT students’ pre- and post-test, as well as between GeT students’ and practicing teachers. Here, partial equivalence means that we were able to establish the equivalence between the groups and time points after allowing unequal item parameters (item factor loadings or item intercepts) for 9 among 17 items. As we were able to establish comparable scales, we proceeded to calculate the GeT students’ MKT-G growth and compare the growth to the practicing teachers’ MKT-G.
The comparison of the scores suggested that, on average, GeT students scored about 0.25 SD units higher on the MKT-G test after completing the GeT course, but it was still 1.04 SD units below practicing teachers’ MKT-G who took the same test. This result implies the positive association between the college geometry courses designed for future teachers and mathematical knowledge for teaching geometry in terms of the growth in the knowledge of the students who took the courses. Additionally, examining the association contributes to research methodology by showing how to establish comparable scales of knowledge gains between two different teacher populations (e.g., pre-service teachers and in-service teachers).
Reference
Brown, T. A. (2006). Confirmatory factor analysis for applied research. Guilford.
I want to use this occasion to reflect a bit on what the development of a consensual list of Student Learning Outcomes (SLOs) represents vis-a-vis the field of mathematics teacher education and, in particular, its curricular history. I want to suggest that GeT: A Pencil has helped the larger mathematics education community make progress in identifying how it is that knowledge from the field of mathematics education (particularly the empirical notion of mathematical knowledge for teaching) can be reconciled with the geometric content that has traditionally been curated by mathematicians for its use in designing the curriculum of geometry courses for teachers.
For a long time, the question of what content should be covered in college geometry courses was one that involved the curricular organization of the mathematical domain of geometry. In this work of curriculum design, the considerations made concerned the history and scope of the subject and the different sequences in which the subject could be presented. In that sense, it was no different than how the content for other mathematics courses (e.g., calculus) might be experimented with. Facing the question of what geometry should be taught to teachers, this approach suggested the need to inquire within the mathematical domain of geometry and find topics and organizations of those topics that arguably would serve to educate future teachers. A premium was put on logical organization, aesthetic or historical value, and ease of understanding—not necessarily on readiness for use in teaching.
As research in mathematics education started to inquire about mathematics teachers’ knowledge, one important insight that emerged was that when teachers engage in the work of teaching, they also do mathematics. Tasks of teaching, such as creating or modifying problems for students or determining the mathematical qualities of work that students do (e.g., consistency, correctness, generality) involve the teacher in doing mathematics while teaching. Appropriately, scholars have called these mathematical tasks of teaching (e.g., Ball et al., 2008). While some of this mathematical work could clearly be filed under topics within the curricular organization of a mathematical domain (e.g., making up problems about isosceles triangles surely draws from knowledge of isosceles triangles that could be covered in geometry courses), other mathematical work teachers do could be described as so specialized to the work of teaching that it might not be common among people who otherwise knew a mathematical domain. For example, understanding why point P in Figure 2, below, cannot possibly be the center of a circle tangent to line a at A is a problem a teacher would need to solve when confronting this proposed solution to a construction problem by a student; yet wrong solutions to problems are rarely part of what one learns in mathematics courses.
This distinct mathematical knowledge for teaching is by no means as explicit as the knowledge documented in textbooks. One sees it manifested in actions (e.g., in teachers’ noticings or decisions), recognized by mathematically educated observers, and organized using resources from empirical research (e.g., typologies). One way in which researchers in mathematics education organized this knowledge–the domains of MKT offered by Ball et al. (2008)–was serviceable in demonstrating the diverse epistemological sources of the mathematical knowledge teachers use. The distinction between specialized content knowledge (SCK) and knowledge of content and students (KCS), for example, highlighted that while some knowledge special to teachers is purely mathematical (i.e., the truth or falsity of this knowledge could be established using mathematics alone; e.g., one can prove that point P can never be the center of a circle tangent to a at A), other knowledge also special to teachers might depend on blending mathematical and empirical knowledge (i.e., the incidence of particular error in a student population; e.g., how often does this idea come up when students are asked to construct a circle tangent to two intersecting lines? How hard is it for students to understand why the “solution” proposed in Figure 2 is incorrect?). This epistemological distinction among domains of mathematical knowledge for teaching has been useful for creating assessment instruments; it has helped create blueprints for the different types of items that need to be developed to tap the construct mathematical knowledge for teaching in all its aspects.
When we started the GeT Support project in 2017, the GRIP team had already invested some effort in the development of an assessment instrument whose items were useful to measure the amount of MKT-G (mathematical knowledge for teaching geometry) a teacher has (Herbst & Kosko, 2014). I note that the instrument measured the amount of MKT-G, taking this as a single construct; the instrument does not verify that the individual knows any concept in particular. This instrument had been used to assess the amount of MKT-G in a national sample of high school mathematics teachers, and part of the idea of the GeT Support project was that the same instrument might be of use to inform instructors and the public about the contribution GeT courses could make to increasing capacity for geometry instruction. In the context of the GeT Support project we have indeed been able to gather data that shows that the same instrument can detect changes in MKT-G students’ experience during the time they take a GeT course (see Ion, 2020 ; Ko, Ion, & Herbst, this issue). The items in that instrument illustrate the construct (MKT-G) in general; each item is situated in the context of a task of teaching geometry and presents a problem that a teacher may need to solve in that context. The items are related to the general theory of MKT (Ball et al., 2008) in that each of them avowedly assesses knowledge of one of four domains of MKT (CCK, SCK, KCS, KCT), and the items sample content from the high school geometry curriculum.
We showed examples of those items at our 2018 conference in Ann Arbor. Yet, not every GeT instructor recognized those items as examples of the knowledge that they taught their students in GeT courses. More importantly, it was clear to all, including ourselves, that the items themselves were not, by themselves, useful to think about the content for GeT courses. A reason for it is that the items are just very small bites of knowledge; they might suggest problems to solve, but they do not clearly point to the larger units in which the knowledge covered in a course gets structured. In that sense, they are hard to relate to the sort of curricular work described in the first paragraph.
As a mathematics teacher myself, one way I think of this different granularity of knowledge is with the knowledge exchange diagram shown in Figure 1. On the one hand, concepts and theorems are relatively large objects of knowledge or mathematical ideas that might be at stake in a course. These objects of knowledge are instantiated in many smaller problems or in particular actions in solving such problems. The work on problems and tasks, on the other hand, makes room over time for a large number of small things–noticings, intentions, ways of seeing, tricks, etc. (e.g., representing b as (ab)/a) that are part of the ways of doing mathematics, part of the mathematical sensibility. Some of the mathematics done in the context of problems and tasks may never receive a name or be taught by itself.
Figure 1. The knowledge exchange
The exchange diagram opens up the possibility to realize that the same object of knowledge can have many different meanings depending on the various problems in which it is operational. For example, the theorem that says that a tangent is perpendicular to the radius of a circle at the point of tangency can be at stake in a number of tasks. It can be useful to come up with a method to construct a circle tangent to a given line; it can also be useful to argue why point P in the diagram below cannot possibly be the center of a circle tangent to line a at point A. In the first case, one uses the theorem to identify the locus of the center of the circle: It must be on the perpendicular to a at A. In the second case one uses it to feed a proof by contradiction: If PA was the radius of the circle tangent to a, then the angle PAO is right and the triangle OPA would be isosceles and with two right angles. The example illustrates how a given object of knowledge has various meanings that emerge in the context of different problems. This is not only the case for the knowledge we teach in mathematics courses, but I surmise it would be reasonably the case for mathematical knowledge for teaching.
Figure 2. P cannot be the center of the circle tangent to line a at A
The exchange diagram can be useful to explain how the notion of mathematical knowledge for teaching could inform curricular work in GeT courses. Problems like the ones in MKT items are candidate examples for the mathematical work in the red square on the left. Consider, for example, problems like those shown in Figures 3 and 4.
Mrs. Miyakawa wants to assign a proof problem using the diagram below. If she asks students to assume (i.e., take as given) that ABCD is a rectangle, E is the midpoint of DC , and AE ⏊BE , what could she ask them to prove?
In Mr. Desimone’s geometry class, kites were defined to be quadrilaterals with two distinct pairs of congruent adjacent sides. He then asked his students to draw a kite that has congruent diagonals and two pairs of congruent opposite angles. Andrea came to him in distress after a few minutes saying that she’d tried all sorts of angle measures and diagonal lengths and all she could come up with were squares. Mr. Desimone told his student teacher that Andrea did not understand what the definition of kite means. What do you think Mr. Desimone means by that?
Figure 3. Problem 15102
Figure 4. Problem 15503
The two problems in Figures 3 and 4 describe situations in which a geometry teacher might need to do some mathematics while teaching. Using the exchange diagram, they both belong in the red square. But what are the green circles that those problems exchange for? In other words, what is the knowledge at stake in those problems? The mathematical domain of geometry has had expository treatises like Euclid’s Elements, Hilbert’s Grundlagen, or Moise’s Elementary geometry from an advanced standpoint within which one can locate the knowledge at stake in problems, but MKT-G problems have not had similar resources. The difficulty is not that problems like those in Figures 3 and 4 cannot be classified into topics; they could be classified in, perhaps, even more than one topic. But where would they make a difference? Because they can be classified in multiple ways, it is unclear how one would build critical masses of them to serve the creation of problem sets and units of study. Indeed, many such items could be created. How would we know what to emphasize?
With the introduction of the SLO, the picture gets a little clearer. The SLOs have two important virtues that recommend them as candidates to be the green circles in the exchange diagram. First, they have come from a thoughtful process of negotiation and argument among a group of instructors that includes mathematicians and mathematics educators. Second, in arguing for them, both considerations of their stature in the domain of geometry and in the teaching of high school geometry have been made. As a result, if one looked at them only with the eyes of a mathematician or only with the eyes of a teacher educator, these SLOs might appear heterogeneous. I tend to think that is a good thing.
Having arrived at the SLOs and now having elaborations of the SLOs in various issues of this newsletter, our community has an important scaffold for the question of what content could or should be taught in GeT courses. The SLOs can, at a minimum, be a checklist; one could look through problem sets, notes, and syllabi and see where there are topics that match the SLOs. However, the SLOs can also be used generatively and along with other aspects of mathematical knowledge for teaching.
For example, one could see that problem 15503 (Figure 4) can be associated with the role of definitions in mathematics (SLO 5). One could also see that it deals with a mathematical task of teaching, interpreting what students do in response to problems, and that it involves specialized content knowledge about quadrilaterals. In fact, the topic of quadrilaterals might be a good host area to anchor the teaching of SLO 5; Usiskin (2007) wrote a monograph on definition using the classification of quadrilaterals that might support firming up these connections. An interesting issue, perhaps an idiosyncrasy, with quadrilaterals in high school geometry that reveals the value of this topic for the study of the role of definitions is that the HSG curriculum tends not to be consistent in how definitions of special quadrilaterals are read. While rectangles and rhombi are understood as defined inclusively (e.g., squares are rectangles), trapezoids are understood as defined exclusively (e.g., parallelograms are not trapezoids). One possibility afforded by this consideration is that instructors might use a unit on quadrilaterals to aim at the achievement of SLO 5 and, in that context, have their students work on a number of problems in which both quadrilaterals and definitions are addressed in the context of tasks of teaching.
Overall, I believe that the SLOs provide us with a way into creating a curriculum for GeT courses that can bridge the mathematical topics that have been used to organize geometry instruction in the past and the particular occasions in which a teacher might have to do mathematics while teaching geometry. Without downplaying the value of collaboration and consensus in forming a community of colleagues, an important achievement of the SLOs is to have helped us figure out how the idea of mathematical knowledge for teaching can become part of curriculum making.
References
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389-407.
Ion, M. (2020). Reporting on the MKT-G results from GeT Students. GeT: The News!, 1(3)
Ko, I., Ion, M., & Herbst, P. (this issue). Does our MKT-G instrument measure the same knowledge in the same way for GeT students and for practicing teachers? GeT: The News! 3(3)
This new edition of GeT: The News! arrives on the brink of the second anniversary of the COVID 19 pandemic–not that anybody wants to celebrate that birthday. However, I do celebrate that, in the midst of this enduring disruption, our community has continued to be involved in several joint projects. As we took stock of our 2021 activities when we met earlier in January, I realized how much the work of this community means to me.
The completion of the writing of the chapter titled “A cross-institutional faculty online learning community: Community-guided faculty development in teaching college geometry for teachers” was a particular highlight for me. It brought attention to something I have always believed in–that professionals can learn from each other as they collaborate on tasks that pertain to their usual work. In one way or another, this idea has been behind most of the GRIP’s efforts to create opportunities for professional learning. Rather than assuming that we know what instruction should look like and that we should tell that to instructors, we prefer to create work environments in which instructors can collaborate solving problems of practice and envisioning improvements. The text of the chapter, as well as the work that went into getting the chapter written, suggest that, at least with the GeT: A Pencil community, our approach to professional learning can work. I think of it as soft professional development; we may not know ahead of time what we are going to learn, but the actual learning that eventually takes place feels more authentic than what one usually gets in professional development workshops that follow a set agenda. The group-generated projects, namely the identification and elaboration of student learning objectives and the development of activities to teach geometry via transformations, have that kind of authenticity to them; yet the most important learning that has been taking place is of a social nature. We have learned to work together and complement each other, accepting the talents and the quirks that everybody brings.
When we started the GeT: A Pencil community three years ago, we brought together a group of instructors of geometry courses for teachers under the premise that an inter-institutional community might fulfill a need; because within each mathematics department there are usually few instructors who teach geometry, it is unlikely that learning communities specific to the subject matter can be organized within institutions. At the 2018 conference, we noticed how much of a divide there seemed to be between those trained in mathematics and those trained in mathematics education and suspected that collaboration might take effort. While our initiative to bring mathematicians and mathematics educators together is only one among many, it is good to see that after three years of working together, the community has not only endured but also grown in numbers, as well as in collective accomplishments.
At our meeting in early January one comment suggested that GeT: A Pencil has functioned as kind of a virtual mathematics department. I loved the metaphor. In fact, I wondered if it might suggest a path forward in the development of expertise for the mathematical preparation of teachers. Might similar communities be developed for instructors who teach mathematics classes for elementary teachers, for example?
The pandemic has touched our lives in many ways, but at least in regard to the work of GeT: A Pencil, it has not prevented us from continuing to learn from each other. Instead, as much of our regular work has also been happening through video conferencing, it seems that the difference between inter-institutional and intra-institutional collaboration has become less salient.
Over the past year, the Teaching GeT working group proposed that one way to contribute to reducing the variability in outcomes in the preparation of secondary geometry teachers would be to formulate and steward a set of ten student learning objectives (SLOs) that could be utilized by instructors of GeT courses. We recognize that the SLOs themselves are a work in progress and that at any one time we are dealing with a version of them. Precisely because of the open-text nature of the SLOs, it is important to identify the many sources of warrants that we could rely on in order to use the SLOs to build more specific curriculum or instruction, as well as improve the SLOs themselves. Important sources for the development of the SLOs have included: the mathematical domain of geometry and its history, instructors’ experiences teaching geometry courses and what they have seen their students do in those courses, policy documents for the teaching of geometry in K-12 and college, mathematics education scholarship, and instructors’ knowledge of research and practice in the teaching and learning of geometry at the secondary level. Those sources have supported lively discussions about what to include and how to prioritize possible inclusions. We at the GRIP thought that gathering students’ work on items that elicited knowledge of the SLOs could provide another kind of warrant to support discussions about the SLOs.
Based on the SLOs v.0 produced by the Teaching GeT group, members of the GRIP Lab at the University of Michigan developed a set of open-ended assessment items that tap into GeT students’ attainment of the SLOs. The intention was to have each item elicit the knowledge named in one of the SLOs, though it was apparent that item responses might also provide evidence of knowledge of other SLOs. Following the genre of other MKT assessments (e.g., Ball et al., 2008; Herbst & Kosko, 2014; Hill et al., 2004), each item describes an event happening in a high school geometry classroom —in which the teacher needed to make a decision that required the knowledge named in that SLO. For example, the following item, designed to measure SLO 1 (Proofs), asked the participant to consider the following:
Unlike in the usual MKT-G items, the respondents did not receive a set of alternatives to choose from but were asked to compose an open-ended response and enter it in a text field.
The process through which the current set of items were created was loosely based on a set of recommended guidelines in developing measurement scales specified by DeVellis (2014, p. 105-152). In particular, as the constructs (SLOs) were already defined, the majority of the work involved scoping several items for each SLO, then choosing which of those scopes to turn into actual items, write those items, and put them through rounds of revision. The vetting of initial drafts of the items included considerations of whether the teaching scenario described in a given item (the student work, the decision the teacher had to make, etc.) seemed realistic and whether the item seemed likely to elicit a response that would be mainly driven by the participant’s knowledge named in a given SLO. In the end, two items for each SLO were chosen to be administered.
These items are a first, rapid prototype of what a summative assessment might look like, created to gather data to support our collective work on the SLOs. That is, we do not yet know enough about the items to use them for consequential tasks such as appraising an individual’s attainment of a specific SLO, an individual’s attainment of the SLOs in their totality, or a class’s average attainment of the SLOs as a proxy for the quality of the attained curriculum. The items target geometry knowledge by posing problems contextualized in tasks of teaching and make minimal assumptions about respondents’ knowledge of mathematics schooling, however, they are not intended to assess knowledge of pedagogy.
While not ready to be used in any formal assessment of students or evaluation of courses, the items support the process of stewarding the SLOs by prototyping what kind of items might be needed for our whole community to document our progress in student SLO attainment. So far, we have collected student responses from seven GeT courses from the Winter 2021 term. The responses we have collected can provide an empirical basis for our community to discuss and improve the SLOs; for example, the contents students might bring up in the item responses can resonate or not with the expectations we may have had about what it would mean to attain an SLO.
In order to engage the community in that conversation, we proposed a workshop where current and prospective members of GeT: A Pencil could come and review items and students’ responses to those items. Rather than work intensively over a few days like at a traditional conference workshop, and to make the workshop easier to attend, participants were asked to commit a couple of hours per week, every second week, over the summer and early fall term. For each item, they would discuss what the item seemed to assess in light of the responses and the SLOs. Participants were given access to more responses in a Canvas forum in which they continued to discuss the items. Finally, during the week of October 4th, participants had the opportunity to discuss the assessment more holistically.
In this volume and future iterations of GeT: The News!, we will provide articles that take a deeper dive into the items themselves. In these articles, we will provide an item and its intended SLO, our analysis a priori of the item, and what we heard from the instructors regarding the items, as well as how the students responded to the items in a categorized form. As we have learned from these workshops, there is much to be gained not only from the correct responses but from the incorrect or partially correct ones as well—which we will show through these writings.
References
Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. Teaching as the Learning Profession: Handbook of Policy and Practice, 1, 3–22.
DeVellis, R. (2014). Scale development: Theory and applications. Sage Publications. Thousand Oaks, CA.
Herbst, P., & Kosko, K. (2014). Mathematical Knowledge for Teaching and its Specificity to High School Geometry Instruction. In J.-J. Lo, K. R. Leatham, & L. R. Van Zoest (Eds.), Research Trends in Mathematics Teacher Education (pp. 23–45). Springer International Publishing.
Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. Elem. Sch. J., 105(1), 11-30.
The assessment items we developed last spring can be a resource in our collective work stewarding the Student Learning Objectives (SLOs). One way to operationalize that possibility could be to revise the items with the goal of incorporating them into a test, like the MKT-G. In such a scenario, we could envision that at some point the items could serve to provide instructors with information about how well their students attained each of the SLOs or all of them as a set. However, there is another possible use, which seems to us more compelling for the time being, and it hinges on a different interpretation of assessment: assessments that have formative, rather than summative, purposes.
Assessments like the MKT-G are often described as summative assessments; the information they provide feeds a single, synthetic score per student, that can be interpreted in terms of how much knowledge they gained over the course of the semester. As Schoenfeld (2015) says,
Summative assessments are examinations or performance opportunities the primary purpose of which is to assign students a score on the basis of their knowledge (p. 184).
In that sense, gains in the MKT-G test provide information comparable to course grades. However, educators also use assessment items for formative purposes. Schoenfeld (2015) writes that,
Formative assessments are examinations or performance opportunities the primary purpose of which is to provide student and teachers feedback about the student’s current state, while there are still opportunities for student improvement (p. 184).
We want to examine the possibility of using the SLO Assessment items for formative purposes, namely, to provide evidence to instructors and students about students’ understanding, with the goal of informing decisions the instructor might make during a class, such as what to do in the following class meeting.
Toward that end, the materials we have consulted and the discussions we have had during the summer of 2021 in the context of our assessment workshop can be quite valuable. They can help us consider what resources instructors would need in order to implement these items for formative assessment purposes. We initiate one consideration taking as an example item 15903. This item, which we transcribe below, was originally written to give students an opportunity to show evidence of attainment of SLO 9.
The student learning objective (SLO 9) about other geometries states:
SLO 9 Compare Euclidean geometry to other geometries such as hyperbolic or spherical geometry.
We took the examples provided in SLO 9 as suggesting a distinction between Euclidean and non-Euclidean, possibly appealing to the historical efforts to prove the parallel postulate. We assumed that if a geometry course would aim for students’ attainment of SLO 9, the class would likely have a discussion of the parallel postulate, its negation, and, possibly, Euclidean models of the different geometries that would ensue.
With that in mind, item 15903 reads:
Consistent with the genre of MKT items, the SLO items were all phrased in terms of a high school teachers’ work, so as to evoke the notion that mathematical knowledge for teaching is the knowledge a teacher needs to do their job. In this case, the notion that students could ask a question that involved comparison between geometries and that the teacher would need to answer such a question was meant to simulate for the prospective teachers taking the test the work that they would need to do later on if and when they became teachers. The text of the item, particularly “some students started wondering whether everything that is true in Euclidean geometry will turn out to be false in non-Euclidean geometries,” suggests that some contrasting truths might have been presented to the class. For example, the notion that, in Euclidean geometry, there is one and only one line parallel to a given line through a point not on that line might have been contrasted with how, in spherical geometry, there might not be any parallel line, or in hyperbolic geometry, there might be more than one such parallel line. Furthermore, for students to even think about the possibility that the different geometries would contradict each other everywhere, we thought it was likely that Mr. Thompson had discussed some contradictory facts across geometries. One fact that is often presented in history of mathematics textbooks is what the different geometries state for the sum of the angles in a triangle. In Euclidean geometry the triangle sum is a constant (180 degrees); in hyperbolic geometry it is not a constant, but it is always less than 180 degrees; and in elliptic geometry it is not a constant either, but is always more than 180 degrees. The context of a high school class afforded bringing in the thinking of adolescents and their penchant for playing with logical inferences by taking things to extremes. In writing the item, we thought it quite possible that after seeing two statements that said contradictory things about the same objects across different geometries, Mr. Thompson’s students might consider it reasonable to pose the question they posed to him.
Analysis a priori of item 15903
One immediate thing to notice in item 15903 is that the question the students asked Mr. Thompson implies a false generalization: even though some things which are true in Euclidean geometry are not true in non-Euclidean geometry, it is not the case that everything which is true in Euclidean geometry is false in other geometries. In order to show that generalization as false, Mr. Thompson would need a counterexample; a counterexample would be a statement which is true in Euclidean and non-Euclidean geometries. We thought item 15903 would address SLO 9 because if students had exposure to the difference between Euclidean and non-Euclidean geometries, it would show them the role the parallel postulate played in the emergence of non-Euclidean geometries. Students who had the opportunity to learn about non-Euclidean geometries might get to think of properties that rely neither on the parallel postulate nor on its alternatives to warrant their truth. The SAS criterion for triangle congruence is true in absolute geometry (geometry that satisfies Hilbert postulates except for the parallel postulate). Also, propositions that assert properties of incidence, separation, and betweenness would work across geometries.
When we shared the item with members of GeT: A Pencil, here is what we heard:
One participant noted: “In the prompt, the student is wondering if everything that is true in Euclidean geometry will turn out to be false in non-Euclidean geometry, but the question is asking for an example of a response the teacher could give that would offer a theorem that is true in Euclidean geometry and is also true in non-Euclidean geometry.” The participant went on to note whether this will have an effect on the student responses, as the student is being asked to falsify a claim by giving an example of something that is true.
A second participant noted that students will need to understand what makes Euclidean geometry Euclidean and be familiar with examples of non-Euclidean geometries.
Another participant noted that students would need to have a working knowledge of at least a few examples of non-Euclidean geometry, including examples of at least one theorem that is true in both Euclidean and non-Euclidean geometries.
An additional participant noted that students should know what is meant by a “theorem.” “Specifically, they would have to give some thought to the following question: “is an axiom of a theory also one of its theorems?” (From the perspective of formal logic, the answer to this question is ‘Yes’, but from the perspective of standard usage the answer is ‘No’.)”
During our in-person discussion, this conversation continued about whether in the GeT course there is a need to distinguish between axioms and theorems. For example, is an axiom a theorem? Some GeT instructors shared that they make this explicit for their students, while others noted that they do not. Some instructors are concerned that their students would feel like they would not be allowed to state an axiom.
Another point of concern was whether or not the item needed to ask for a ‘theorem’ that is true in Euclidean and non-Euclidean geometry. One participant suggested that instead of asking “What example could Mr. Thompson offer of a theorem that is true in Euclidean geometry and in a non-Euclidean geometry”we could instead ask, “What example could Mr. Thompson offer of something that is true in Euclidean geometry and in a non-Euclidean geometry?”. This participant claims that this would target the same knowledge of the SLO and whether non-Euclidean means the negation of Euclidean.
The first two bullets seem to suggest the item has face validity as part of an assessment of whether students have attained SLO 9. The third, fourth, and fifth bullets, however, suggest sources of noise in the item. The notion of a theorem, for example, brings some noise. On the one hand, the third bullet contrasts theorems with axioms, noting that logically they are similar—indeed, both axioms and theorems are declarative statements, but they are different in usage. For the question, it is important that students would think of propositions that are proved within a theory making use of axioms (i.e., theorems) and not propositions that are postulated as true to build the theory (i.e., axioms). Otherwise, not only would the question be too easy (e.g., students could bring up the axiom that two points determine a line as an example that some statements are true across Euclidean and non-Euclidean geometries), but it would also fail to tap into the interpretation of the sense of logical necessity flowing from axioms to theorems called up by the question (i.e., a set of axioms defines a geometry by necessitating the truth of a set of theorems, but this does not mean that all the axioms are needed to prove all the theorems, and so if two theories have an overlapping set of axioms, as Euclidean and non-Euclidean geometries do, it is quite possible that some theorems would be true in both). To address the issue raised in the penultimate bullet, the question meant to signal, indeed, that providing an axiom would not be an answer to the question–as what students had said to Mr. Johnson, that such things would “turn out to be false,” pointed to the truth value of the proposition at stake being something dependent on something else rather than postulated by choice. On the other hand, the meaning of the word theorem is not merely that of a declarative proposition that has been proved. Theorems are special propositions, deserving of recognition; for example,they conclude an investigation or present a frequently used result. Along those lines, there are status differences among declarative propositions that can be proved—theorems, lemmas, propositions, observations, and corollaries may be logically created equal but are not mathematically equal. Indeed, some of them are provable but not proved because their proof is trivial. Even Euclid distinguished the statements he proved among theorems and scholia. However, theorems do not only have special status; they also often have names or shorthands that refer to them. This is especially on point here because many theorems of Euclidean geometry do not have names or shorthands that will make them memorable; students recognize the Pythagorean theorem and maybe the base angles and exterior angle theorems, but many of the theorems that would answer the question may not have enough of those accoutrements to be remembered as theorems. Furthermore, the properties of incidence, collinearity, and separation which could answer the question were not historically theorems for Euclid but rather assumptions that later geometers made explicit. Pasch’s Theorem, for example, says that if three points are not on a line and a line passes through the segment determined by two of them, the line will also pass through one of the two other segments determined by the three points. Within Hilbert’s axioms for Euclidean geometry, Pasch’s Theorem is a theorem which is true across geometries; yet when Pasch proposed it, he did so as a way to show the gaps in Euclid’s axioms—as the Theorem cannot really be proven from Euclid’s original axioms. Could students have brought up Pasch’s theorem as an example that Mr. Thompson could use? Maybe, but unlikely. The example of Pasch’s Theorem suggests that beyond the status differential among declarative propositions, there are historical developments used in distinguishing Euclidean and non-Euclidean geometry that could get in the way of students identifying a theorem that would be true across Euclidean and non-Euclidean geometry. Indeed, the status and historical confounds of the word theorem complicated the question too much; students who knew different geometries but did not have an example handy might be confused as to what would count as an example. It would be likely for students to answer that question correctly in a test if an example had been covered in their class but less likely if they had to come up with an example on their own.
The last bullet is particularly interesting in regard to the kind of assessment one is doing. The word “something” could be problematic in the context of a test or a written, summative assessment: Students might not necessarily think of declarative statements as the “somethings” to look for. The word something would also sound so informal and opaque that students who provided non answers (e.g., “sphere in both”) might have a point to complain about in the summative assessment setting. As a result, evaluating responses might be tricky. However, in the context of a formative assessment, done in class, there might be an opportunity for the instructor to start by asking for “something” and cue students to think of declarative statements that can be proved across different geometries as the target.
Analysis of student responses to the items
When we collected responses during Spring 2021, we found that, of the 42 student responses, 31 responses were non-trivial—that is, responses that provide some evidence of effort or knowledge of how to solve the problem. The student responses show a variety of ways in which students might relate to the question and to the distinction between Euclidean geometry and other geometries. After looking through the responses, we classify them in the following way:
Category 1: little to no evidence that the student was exposed to the knowledge of SLO 9 and some evidence that the student was swayed by an interpretation of the word “theorem,” which was more specific than just a provable declarative statement. Five students responded to this question by naming the Pythagorean Theorem (a “common” theorem).
In these responses, most of the responses simply wrote “the pythagorean theorem” or something very similar. One response (A6) noted, “the Pythagorean Theorem and resultant distant [sic] formula hold in both Euclidean and non-Euclidean geometry.” We are unsure whether the students had any exposure to the knowledge called forth by SLO 9 from these responses or if they were just naming a theorem that they knew; however, they did read the question clearly and provided a theorem, something that was not necessarily the case in other responses.
Category 2: some evidence that the student was exposed to the knowledge called forth by SLO 9. Three students showed knowledge of triangle angle sum properties in different geometries which we take as providing some evidence that the student was exposed to the knowledge called forth by SLO 9.
In these responses, there were references to the angle sum properties of triangles. The properties of how triangle sums differ across various non-Euclidean geometries is one of the basic facts covered in learning about those geometries. However, the knowledge the students recalled was generally incorrect. For example, responses A7 and A11 said something similar to “triangles sum to 180,” which is not true in the historical examples of non-Euclidean geometries. However, students’ responses were not limited to those geometries. Statement A51 (“A triangle’s angles still add up to 180 degrees in taxicab geometry.”) provided facts about triangles in taxicab geometries; this student attempted to give an example of a fact that is true in both Euclidean and a non-Euclidean geometry, but this fact is not true in the formulation of taxicab geometry where the sum of the angles is 4t-radians.
Category 3: little to some evidence that the student was exposed to the knowledge of SLO 9. Students named objects or properties of mathematical objects without any mention of explicit geometries.
Twelve responses were classified in this category. In these responses, students either named a mathematical object or properties about mathematical object(s) without explicitly naming the non-Euclidean geometry. Some examples of responses that name mathematical objects are A12 (“parallel lines”), A16 (“a straight line”), A19 (“sphere in both”), A26 (“hyperbolic shapes”), and A36 (“Parallel lines exist?…”). Some theorems can be proven about these objects, so it is possible that the students were remembering isolated bits of the knowledge associated with SLO 9, but the student did not provide a theorem nor a non-Euclidean geometry. One response, A35, states, “the fifth postulate is still true,” which is incorrect, as the fifth postulate is only true in Euclidean geometry. The rest of the responses A17 (“Theorem 1.2 that states two lines have at most one point in common.”), A23 (“A straight line segment can be drawn joining any two points.”), A28 (“the area”), A29 (“The angle between perpendicular lines remains 90 degrees in non-euclidean geometry.”), A39 (“the definition of a circle”), and A55 (“Def of line”) deal with properties or definitions of mathematical objects, yet do not name a non-Euclidean geometry.
Category 4: ample evidence that the student was exposed to the knowledge associated with SLO 9. These were (mostly) correct responses.
Ten responses were classified in this category. They include students who correctly provided a theorem that holds in both Euclidean geometry and a non-Euclidean geometry. Additionally, these students were explicit about which non-Euclidean geometry the theorem holds in. These responses include references to the intersections of lines in Euclidean and hyperbolic geometries (A22, A25, A32, A38). There were two responses that looked at properties of parallel lines across Euclidean and an explicit non-Euclidean geometry (A24, A49). Statements A15 (“The SAS theorem is used in both types of geometry.”) and A20 (“Proving congruency can work in both.”) about triangle congruence are on the right track. However, A20 could have been more specific about a theorem and a particular non-Euclidean geometry, and A15 needed to be clear which non-Euclidean geometry SAS theorem holds in (e.g., SAS congruence is true in hyperbolic and spherical geometry but not in taxicab geometry). Statements A13 (“area of a rectangle in taxicab geometry”) and A57 (“taxicab equilateral triangles are not always equiangular.”) are examples of correct statements that need to be rewritten to answer the question correctly. Lastly, one response was almost correct (A56), in which a student noted, “I would explore any theorem in hyperbolic geometry that doesn’t require the parallel axiom,” but the student was not explicit about which theorem to choose.
Instructor interpretations of the student responses
When members of GeT: A Pencil saw the student responses, we heard the following reactions/interpretations in the forum.
Forum participants thought it was remarkable that a large number of items (11) named postulates or definitions.
A24 (“In Euclidean and Hyperbolic, two lines perpendicular to the same lines are parallel.”) and A51 (“A triangle’s angles still add up to 180 degrees in taxicab geometry.”) name common theorems in hyperbolic and taxicab geometries, respectively.
All participants agreed that A24 was the strongest response: “in Euclidean and hyperbolic, two lines perpendicular to the same lines are parallel.”
One participant noted that A56 (“I would explore any theorem in hyperbolic geometry that doesn’t require the parallel axiom.”) is correct, while not providing a concrete theorem.
One participant noted that A17 (“Theorem 1.2 that states two lines have at most one point in common.”), A22 (“An example Mr. Thompson can give is that in both Euclidean Geometry and hyperbolic Geometry the intersection of two lines is at most one point, A32 (“In both Euclidean and hyperbolic geometry there is at most one point at the intersection of two lines.”), and A38 (“There is at most one point for the intersection of two lines in Euclidean and hyperbolic geometry.”) make the same claim, which is true in neutral geometry and is definitely a theorem (in the sense of “not an axiom”). A25 (“Lines that intersect in Euclidean geometry intersect [at] exactly one point. This can also occur in hyperbolic geometry.”) makes the same claim but is marred by the phrase “can also occur;” it should be “is also true.”
Instructors noticed students naming the Pythagorean Theorem, but noted that this is not a theorem that is true in neutral geometry.
Way Forward
As we move forward with this work, we want to hear your feedback and thoughts on what we have written here. We have heard from the participants in the workshop that these items or modifications of these items could serve as formative assessment tasks in the GeT courses. The GRIP team can serve as support, providing resources for the teaching of lessons using these tasks. Additionally, GeT instructors could work collaboratively, providing their students with the same tasks, and then come together to reflect and learn from each other on how the tasks helped elicit knowledge of the SLOs from their students.
We think that the analysis a priori as well as the categories of student responses may be helpful for instructors to use these items with formative purposes. We would love to know what you think about that and whether we could provide other things to support instructors as they use the items with their students. Additionally, if instructors who use items like 15903 collected their students’ work, it could help our community continue learning about possible student responses through scans of de-identified student work uploaded to our Canvas site.
References
Schoenfeld, A. H. (2015). Summative and formative assessments in mathematics supporting the goals of the common core standards. Theory Into Practice, 54(3), 183-194.
This past March, I was invited to speak about high school geometry to a college geometry class, one that we might describe as a Geometry for Teachers (GeT) class insofar as future teachers were an important, even if not the only, constituency. I was asked to talk about why high school students need to study geometry—something that may be taken for granted in identifying geometry as a mathematics class that future teachers need to take but may make sense to ask from the perspectives of both college mathematics students and the mathematicians who teach them. In our proposal to NSF to fund the GeT Support project, we noted that contemporary mathematical research has little to do with the geometry content students learn in high school, and the same might be said about the mathematical experiences of undergraduates. Thus, if the geometry content does little to cement future learning or research, it is worth asking what role high school geometry plays in high school students’ mathematical development.
This question of the purpose of the study of geometry is not new. At the end of the 19th century, as the high school curriculum was being formed in the United States, the idea that different scholarly disciplines were useful to train different mental faculties was in vogue. Geometry was then justified on account that it was expected to train logical reasoning. González and Herbst (2006) describe this logical argument along with others which emerged later during the 20th century. An intuitive argument was offered that described geometry as providing a language that students could use to relate to their everyday experiences in the material world. A utilitarian argument was offered during the second world war period that described geometry as providing useful resources for the world of work. And during the time of the New Math (in the 1950s and 1960s) a mathematical argument emerged that described geometry as providing an opportunity for students to experience what mathematicians do: define, conjecture, prove, and so on (González & Herbst, 2006). To some extent those four arguments (logical, intuitive, utilitarian, and mathematical) are still being made in different quarters; furthermore, we may come by a better answer to the question by integrating some of those arguments. While the logical argument was questioned early in the 20th century, both the intuitive and the mathematical argument have always felt complementarily compelling to me.
A modeling approach
In my talk to that class in March, I proposed that the geometry course provides students with opportunities to engage in the mathematical modeling of their experiences with space and shape. Herbst, Fujita, Halverscheid, & Weiss (2017) used this idea as the centerpiece of their graduate textbook on the teaching and learning of geometry. They offered Figure 1 below as a way to represent what could happen in high school geometry. The box in the center represents an envisioned modeling approach to high school geometry. The approach is informed by two sets of sources. On the one hand there is a source that we could associate with the intuitive argument named above: Real world objects and activities are often represented (i.e., named, described, depicted) using tokens that might be described as geometric. So, words—like line, square, and turn—and shapes are often used in describing how people manage space and shape. This does not mean that by virtue of their use of those tokens those people are doing mathematics in a way commensurate with that of mathematicians, but an argument that builds on embodiment and materiality could be envisioned to suggest engagement in physical activities and their enactive representation has some mathematical qualities. The diagram in Figure 1 suggests that high school geometry could build environments on top of those existing relationships between the real world and our cultural representations of it. These environments are described as “geometric models of representations of real world objects” which means at least two things. On the one hand, these models are particular interpretations of the primitive objects, relationships, and postulates of formal geometries such as Euclidean geometry. On the other hand, these models are environments for mathematical practice—environments in which the activities (e.g., calculation, construction) and dispositions (e.g., pondering whether a solution is unique) of mathematical practice can be mobilized to produce information that can be interpreted in terms of real world objects and activities. The arrow from Geometries to these models and the adjective geometric attached to models point to the aspiration that work within those models be guided by the mathematical sensibility that reigns in mathematical practice. Teachers and curriculum developers who are likely to know geometry as a mathematical domain can organize these environments in which students’ experiences with real world objects and activities and their representations are involved in the activities of mathematical practice and scrutinized with the sensibilities of mathematical practice. A quick example of these relationships is the modeling role the use of a straightedge plays in helping think of a straight line when confronted with concrete objects that might be described as forming a straight line—questions of incidence, betweenness, separation, parallelism, and so forth can be brought from the geometric theory of lines and specific practices with the straightedge may help interpret those questions in terms of concrete objects. The famous quote by Poincaré, “geometry is the art of reasoning well from badly-drawn figures” (cited in Bartocci, 2013), comes to mind along with the common practice of drawing in geometric problem solving. A more detailed example may help make the point clearer.
Figure 1. A modeling approach for secondary geometry (adapted from Herbst et al., 2017)
The pool problem
The pool problem serves as a more extended example of the modeling approach. This problem could be used relatively early in the high school geometry course, after students have learned some basic constructions (e.g., how to construct a perpendicular bisector to a segment) and how to prove triangles congruent. The problem can be used to develop students’ knowledge of a theorem about right triangles: that, in any right triangle, the midpoint of the hypotenuse is equidistant from the vertices. More importantly, the problem can be used to inscribe in high school geometry a simple example of the mathematical difference between sense and reference (Frege, 1997/1892): The same mathematical object (the same point in this case) can be the referent of two different ways of speaking (two different meanings or senses). This notion is pervasive in mathematics as two different procedures can construct the same mathematical object. The pool problem can be used to instill the disposition to ask whether two different construction procedures define the same object.
The pool problem says
Three swimmers are going to jump into a rectangular pool and race toward a buoy. One of them will do so from a corner of the pool, a second from a side of the pool adjacent to that corner, and the third from the other adjacent side. Suppose the swimmers are in position and you have to position the buoy. Where should you position the buoy to make the race fair?
Note here that some geometric concepts are being used to describe the concrete situation: the pool is rectangular; the word adjacent is used to refer to the sides of the pool that make one of its right angles, etc. Finally, the notion that the race will be fair alludes to equidistance. That reading of the problem using geometry along with existing habits to study the geometry of the microspace (i.e., at a scale much smaller to that of the human body) might suggest an initial diagrammatic modeling to represent the problem: Students could draw a rectangle and plot three points for the swimmers. This initial modeling decision can be quite consequential in affording opportunities to project mathematical sensibilities onto the work; it can be advantageous, from a mathematical point of view, to have rectangles of different sizes and aspect ratios, rectangles drawn free hand, and rectangles constructed with straightedges. The expectation is that mathematical sensibilities and mathematical practice will help organize superficial variations among those representations. Poincaré’s proposition that “geometry is the art of reasoning well from badly-drawn figures” suggests that to reveal the geometry, it would be valuable if the diagram had some inaccuracies to begin with (such as the strokes representing the sides of the pool not being exactly straight or the angle representing the corner of the pool not being exactly 90 degrees; see figure 2).
Figure 2. An initial diagram of the pool problem
The drawing in Figure 2 represents the pool situation using geometry. Can students use it and what they know in order to make inferences about the situation? First of all, there are concrete answers that can be made available immediately: The distances between any point picked at will inside the rectangle can be measured with rulers, and the point may be moved so as to make the distances closer to each other. These answers can be revealed useless by virtue of the modeling choices: The distance that will be equal for one set of swimmer positions will not necessarily work for a different set of swimmer positions, and even if those positions were controlled, the particular measure of the distances among the points representing swimmer positions and prize location would unclearly prescribe how to locate the prize position in the actual pool. Thus, early modeling choices, using the tokens of intuitive geometry (vocabulary and imagery), might reveal the need for something else. In what follows I adopt the position of a teacher anticipating how one could think of the problem with a class: I consider my audience in terms of what they are expected to know (hence timing when and how I bring in what I know).
If we started from considering only two swimmers first, specifically the swimmer at the corner and a swimmer at one of the sides, it would be easy to see that the midpoint of the segment determined by the two swimmers is equidistant from them (and the pool context also suggests that would not be a reasonable place to locate the prize as such midpoint would not be in the pool!). But many other points in the pool would be equidistant from those endpoints. As students have learned to construct the perpendicular bisector of a segment, they might also know that this line is the locus of all points equidistant from the endpoints of the segment. And if they didn’t yet know that, this might be a good time for them to come to know it. It might help to ask questions like: Since the midpoint of the segment does not work, what would be a point inside the pool equidistant from the endpoints? Students might or might not bring the perpendicular bisector as a resource to think about other points. If they didn’t, one could ask how they would pick the point they are looking for, aiming to get explicit instructions. In our work with this problem in actual high school classrooms, we have seen students gravitate to the perpendicular bisector directly or to ideas germane to its construction (e.g., using the compass to make an arc from each of the endpoints and picking the point of intersection of the arcs). In case the students had thought of the perpendicular bisectors, the question, “how could we prove that a point chosen on the perpendicular bisector is equidistant from the endpoints?” might get to a partial conclusion of this simpler problem. In case the students had not thought of the perpendicular bisectors and instead constructed a single point, the question, “is there any other point that would work?” might move the discussion toward the end of characterizing all the points that would be reasonable locations for the prize.
Once that simpler problem has been solved, we can bring in the third swimmer. The ways of locating possible places for the prize in the simpler problem could help find possible locations for a prize that would be equidistant from the second and third swimmer. The question, “is there a spot that is equidistant from the first and the second swimmer and equidistant from the second and third swimmer?” could get students to think of the intersection between the two perpendicular bisectors. The question, “[how] do we know whether the first and the third swimmer are equidistant from the point of intersection of these two perpendicular bisectors?” is also relevant here and could bring awareness of the transitivity of congruence. In that event, we have found a solution to the problem, but we have not arrived yet at the statement of the theorem as no attention has been given to the midpoint of the hypotenuse; not even a right triangle is visible. Indeed, the midpoint of the hypotenuse and the intersection of the perpendicular bisectors of the legs of a right triangle are two different meanings (two different senses in Frege’s terminology) of the same referent. A virtue of this problem, and of this way of approaching the problem, is that it delays attention to the referent, making room, as we discuss below, for proof.
While one solution to the original problem has been proposed, it is mathematically sensible to ask whether that is the only one. Furthermore, the fact that we have found the one by doing some things with the diagram instead of others, it is also sensible from a diagramming action perspective to ask whether we would get a different point had we made other choices. In some cases, the question may yield simple, even trivial, answers, and in other cases, the question could lead to compelling stretches of mathematical practice. For example, if instead of considering first the two points on one side and second the two points on the other side, we altered that order, would the solution change? Students might be quick to note that the intersection of the same two perpendicular bisectors is a single point, no matter which perpendicular bisector is drawn first. The other possible choice is, or can be made to be, however, less immediate.
What if the first two points used for the simpler problem did not include a swimmer at the corner but were points on the sides of the pool? Because the segment determined by those two points has not been drawn yet, thinking about their perpendicular bisector might not be immediately obvious to students. The simpler problem warrants thinking about it, as the midpoint of that segment would be one reasonable place to position the prize if the swimmers jumped into the pool from the sides. One solution would be the midpoint of that segment. Yet, the real world situation also suggests some discomfort with it. In Figure 3, we can note that diving toward the midpoint of the segment between them would require swimmers to form an acute angle with the side of the pool and that it would feel more comfortable if that angle was larger. Thus, a practical reason might justify asking where, other than at the midpoint of the segment between the swimmers, are all the points equidistant to them. Based on what was considered in the first solution, students would likely gravitate toward thinking of the perpendicular bisector too, but this observation about the angle might importantly seed a preference toward a solution farther from the midpoint of the segment. Because we are not merely interested in solving the problem but in constructing the materials that matter in the theorem at stake (and more, as all of this matters in understanding why the circumcenter is unique), that preference is desirable; it helps create the conditions for students to understand that the referent of all these constructions is unique through rational means, by reasoning well about badly-drawn diagrams.
Figure 3. Two swimmers diving from the sides
Now it would be reasonable to bring in the third swimmer, diving from the corner. Again, one could consider two different perpendicular bisectors. Considering only one would provide a solution but leave open the question of whether choosing the other perpendicular bisector would provide a different solution. Figure 4 shows what this could look like in a diagram deliberately chosen to favor the posing of these questions. A deliberately chosen, badly drawn diagram would be one in which one could see the three different meanings of the location of the prize (three different construction procedures) as if they pointed to three different referents.
Figure 4a. The different possible solutions
Figure 4b. Shouldn’t X be on the bisector of AC?
A superficial appraisal of figure 4a would avail the question, “are we saying that there are three different points equidistant from the three swimmers?” As the last two solutions were found on the perpendicular bisector of AC, the first point found (X; see Figure 4a) might appear as the odd one out and allow the questions: “Is X really equidistant from A and C? What would need to be true about triangles AMX and CMX if AX was congruent to CX?” If available, the reciprocal property of the perpendicular bisector (if a point is equidistant from two points, it is on the perpendicular bisector of the segment formed by the points) could come in handy to infer that X should be on the perpendicular bisector of AC. If not, the conclusion could be reached by noting that triangles AMX and CMX should be congruent by side-side-side (see figure 4b).
If not only Z and Y but also X are determined to be on the perpendicular bisector of AC, the students could be invited to draw the figure again, paying attention to locating X, Y, and Z. This is another case in which sense and reference are different; while one could talk about those as potentially different points, there would be no way of drawing them as distinct points that lie on a single line, even in a badly-drawn diagram!
Figure 5.M and X must be the same!
However, one could still ask, “What about M?” If one now considers triangles, say BMR and AMR, they would need to be congruent by side-angle-side, making BM ≅ AM. Similarly, because BMS and CMS are congruent triangles, BM ≅ CM. This all would suggest M would also be in the intersection of all three lines and the only point that can be used to place the prize. X, Y, Z, and M would be one and the same point.
Back to the modeling approach
Badly drawn diagrams, such as those in Figures 4a, 4b, and 5, are doing things that formal, axiomatic geometry cannot do. By offering a diagrammatic interpretation of geometric concepts and relationships, they activate a source of intuitive feedback to the questions, assertions, and predictions that may come from the mathematical sensibility. This mathematical sensibility is one capable of endowing a badly drawn diagram with some properties and looking for it to produce other possible properties. This mathematical sensibility is one that will be ready to question the consistency of both classes of properties; and it is essential in order for reasoning with diagrams to model mathematically the relationships between real world objects and activities (including the activities of drawing, folding, and moving about in space) and the geometric representations of those objects and activities. As the narration above suggests, for the high school geometry class to have access to that mathematical sensibility not only well-chosen tasks are needed but also a teacher who is disposed to both ignore publicly what they know about geometry while embodying the dispositions to know that are part of the mathematical sensibility. In such a context, one could see the high school geometry class as a place where students are apprenticed into the practice of mathematicians by working with the intuitive ideas of geometry. High school geometry could be the place in which one learns to resolve the logical contradictions that arise from drawing bad diagrams and thinking with badly drawn diagrams. This modeling approach builds on earlier arguments for why students should study geometry. The importance of engaging in the practices of mathematicians is affirmed as a way to solve problems by making representations whose attributes are ascribed and verified rationally as opposed to empirically. Geometric content is valuable not because geometry is a preferred example of a mathematical system of postulates, theorems, and proofs, but because experiences managing space at different scales provide sources for representations (embodied, iconic, symbolic) that can help pose problems and provide feedback on thinking. Logical reasoning is called upon to support mathematical practice, as well as to confront the feedback from various representations. And the capacity to mathematically model experiences with shape and space, while contextualized in geometry, might support modeling in work contexts, where it can be useful to anticipate rationally the solutions to problems of managing space and shape.
References
Bartocci, C. (2013). “Reasoning well from badly drawn figures”: the birth of algebraic topology. Lettera Matematica, 1(1-2), 13-22.
Frege, G. (1997). On Sinn and Bedeutung In M. Beaney (Ed.), The Frege reader (M. Black, trans., pp. 151-171). Blackwell. (Original work published 1892)
González, G. and Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students supposed to study geometry in the twentieth century? International Journal for the History of Mathematics Education, 1(1), 7-33.
Herbst, P., Fujita, T., Halverscheid, S., and Weiss, M. (2017). The learning and teaching of secondary school geometry: A modeling perspective. New York: Routledge.
One of the goals of the GeT: A Pencil community is to improve the instructional capacity for high school geometry. Geometry courses for teachers can be instrumental in preparing teachers with that instructional capacity. It is pertinent to ask what evidence we can use to steer that mission. Unlike in the K-12 environment where curriculum development and implementation are used to ensure that instruction meets standards, the culture of college instruction is one founded on academic freedom; and instructors often take pride in the development of their own course materials. In this context where instructors may be doing different things, it is important to understand what could be meant by improvement and how we could know that we are improving instructional capacity.
The approach to improvement science espoused by Bryk et al. (2015) counters the usual paradigm of evaluation research, often focused on establishing the main effects of interventions, controlling for implementation fidelity. Bryk et al. (2015) consider it sensible that interventions will vary across sites as they attend to characteristics of their context. This aligns well with the situation in which each instructor of Geometry for Teachers designs and implements their course: Instructors know the students they usually have, their mathematical backgrounds, and other elements of their professional preparation. It would not be sensible to try to make all Geometry for Teachers courses alike.
However, based on analyses of healthcare operations, Bryk et al. (2015) propose that an alternative guide for improvement would try to reduce the variability in the outcomes of education interventions. In healthcare, outcomes might include various measures of patients’ health, such as the time-to-hospital-discharge by condition treated or the number of changes in treatment needed to achieve recovery. In terms of time to discharge, for example, we know that recovery depends not only on the efficiency of medical teams, but also on the condition and the patient’s comorbidities. Thus, to say that a healthcare provider is doing good quality work, it would not be sensible to expect the time to discharge to reduce to 0 or for all patients to take the same amount of time to recover. Yet, if predictions of the time to discharge gave a very large time interval, this large variability might suggest a possible focus for improvement. Instructional capacity for geometry teaching may also be seen as amenable for this kind of improvement. We know that there will always be things a teacher needs to know that they did not learn in our courses; we also know that there will always be things they had the opportunity to learn in our courses and yet they didn’t. How can we think about reducing the variability of outcomes of geometry courses for teachers in ways that allow us to gauge the improvement of our collective efforts to increase instructional capacity? What are some options for outcome variables whose variability we could aim to reduce?
One way of assessing the variability of outcomes could be to count credits among graduates, such as how many geometry courses a teacher had in their preparation. In the past, the number of mathematics courses a teacher had taken was believed to have an influence on teacher performance, but research has not been conclusive (Begle, 1979; Monk, 1994). In our experience surveying practicing high school geometry teachers, however, this number has shown very little variability to begin with, to the point that it does not even make sense to ask what its effects are on other teacher variables (e.g., the amount of mathematical knowledge for teaching).
A second way of assessing the reduction in variability of outcomes comes from the availability of scores in our MKT-G test (Herbst & Kosko, 2014; Ko & Herbst, 2020). The GRIP lab has surveyed a nationally distributed sample of practicing high school teachers using this instrument. We have also been using the MKT-G test at the beginning and at the end of the GeT course for students of instructors of the GeT: A Pencil community. As a result of implementing the test for several semesters and across several courses, we have gotten a sense of how much mathematical knowledge for teaching geometry students have when they start the course and how much they have when they end. On average they start at a score -1.10 and they end at -0.92 both below the standardized mean of practicing geometry teachers. Our current data suggests that the average students’ increase in MKT-G is about 0.18 standard deviations. As experience teaching geometry correlates with MKT-G scores, that increase of 0.18 SDs is equivalent to the growth that a teacher would have in 2.5 years of experience. While one way to think about improvement might urge us to try and increase that difference beyond 0.18 SD, we also know that such an increase is likely to be bounded as there is only so much learning that can happen in a semester.
The improvement approach would suggest that we look instead at whether the variability in outcomes reduces over time. For example, when students arrive at our GeT courses, what they know of geometry for teaching may vary widely depending on their prior experiences. At the end of the course, we would expect everybody to know more. But rather than only looking at this average growth, we could also look at the variability of individual growth. What is the variation in growth among the students in a class over the semesters? What is the variation in growth among the students of instructors in our community over the semesters? While increasing the amount of MKT-G score improvement is desirable, reducing the variability among those increases could help us argue that geometry for teachers courses are associated with predictable gains among individual prospective teachers.
Yet the MKT-G test is built on a conception of instructional capacity to do tasks of teaching high school geometry, in the context of instructional situations from the high school geometry course. As not all GeT courses are focused on that material, the MKT-G is not necessarily aligned with a shared conception of desirable outcomes by instructors of the course. It is reasonable to look for reductions in the variability in the growth of MKT-G scores but that by itself might not have the key to how to increase instructional capacity.
The recent effort by the Teaching GeT working group to develop a shared list of student learning outcomes (SLOs) helps move toward understanding what variability in outcomes we might want to aim to reduce. In this issue of the newsletter, Nat Miller introduces the effort and lists the student learning outcomes the Teaching GeT group developed over the 2019-2020 academic year. Another note, by Sharon Vestal, provides a commentary on the first of these SLOs. We are eager to publish commentaries on all of these SLOs, as well as written responses to published elaborations, or possibly complementary elaborations. The effort to identify SLOs rests on the notion that while we might be foolish to expect all GeT courses to be the same, mathematics departments in high schools, parents, and high school students are entitled to expect their geometry teachers to have some competencies. GeT instructors could choose many materials and pedagogical approaches to achieve those outcomes and as long as those outcomes are achieved, our community could stand behind any one of our graduates. But in order to be able to assess our improvement using these SLOs, it is really important to develop consensus on these SLOs. We hope the notes included in this newsletter and the following ones will help us move toward such consensus. It would then be sensible to survey students in regard to whether they have had, in their GeT courses, opportunities to learn aligned with each SLO. The notion of improvement in the sense of reduction in the variability of outcomes could be understood as having more and more students indicate that they have had opportunities to learn the same SLOs even if the ways in which those opportunities were provided varied.
As we develop consensus, it also seems important for us to try to integrate the developing SLOs into the framework of the MKT-G test, as we can use successive administrations of the test to calibrate and phase in new items that, over time, might also inform the assessment of growth among GeT students. MKT-G items usually pose a mathematics problem in the context of a task that a high school geometry teacher may need to do. Creating problems for their students, preparing materials for lessons, understanding what students do in response to problems, crafting explanations for key ideas, providing definitions, etc. are tasks a teacher has to do routinely. We hope that as the SLOs develop we might also hear suggestions as to how to create assessment items that might tap into the knowledge named in the SLOs. We anticipate that those items might help bring closer together the objectives GeT instructors have for their courses and the knowledge needed for teaching high school geometry. This effort may therefore help us track more accurately how we are improving instructional capacity for high school geometry.
References
Begle, E.G. (1979). Critical variables in mathematics education: Findings from a survey of the empirical literature. Washington, DC: Mathematical Association of America and National Council of Teachers of Mathematics.
Bryk, A. S., Gomez, L. M., Grunow, A., & LeMahieu, P. G. (2015). Learning to improve: How America’s schools can get better at getting better. Harvard Education Press.
Herbst, P., & Kosko, K. (2014). Mathematical knowledge for teaching and its specificity to high school geometry instruction. In J. Lo, K. R. Leatham, & L. R. Van Zoest (Eds.), Research Trends in Mathematics Teacher Education (pp. 23-45). New York, NY: Springer.
Ko, I., & Herbst, P. (2020). Subject matter knowledge of geometry needed in tasks of teaching: Relationship to prior geometry teaching experience. Forthcoming in Journal for Research in Mathematics Education, 51(5)
Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125-145.
