Boundary Transitions Within, Across, and Beyond a Set of Digital Resources: Brokering in College Algebra

We address a problem of promoting instructional transformation in early undergraduate mathematics courses, via an intervention incorporating novel digital resources (“techtivities”), in conjunction with a faculty learning community (FLC). The techtivities can serve as boundary objects, in order to bridge different communities to which instructors belong. Appealing to Etienne Wenger’s Communities of Practice theory, we theorise a role of the instructor as a broker, facilitating “boundary transitions” within, across, and beyond a set of digital resources. By “boundary transition”, we mean a transition that is also a brokering move; instructors connect different communities as they draw links between items in their instruction. To ground our argument, we provide empirical evidence from an instructor, Rachel, whose boundary transitions served three functions: (1) to position the techtivities as something that count in the classroom community and connect to topics valued by the broader mathematics community; (2) to communicate to students that their reasoning matters more than whether they provide a correct answer, a practice promoted in the FLC; (3) to connect students’ responses to mathematical ideas discussed in the FLC, in which graphs represent a relationship between variables. Instructors’ boundary transitions can serve to legitimise novel digital resources within an existing course and thereby challenge the status quo in courses where skills and procedures may take precedence over reasoning and sense-making.

have textbooks with examples that privilege answer-finding rather than justification and reasoning (Mesa et al., 2012). Researchers have called for changes to this course, to push back against a status quo that focuses on the performance of skills and procedures, as well as the recall of prerequisite knowledge, which can disenfranchise students (Bhattacharya et al., 2020;Tunstall, 2018). For digital resources to promote instructional transformation (i.e. disrupt the status quo), it is important for them to become part of the fabric of "what counts" in college algebra (Olson & Johnson, 2022). Hence, instructional moves (e.g. Harris et al., 2012) that instructors make around digital resources can serve to legitimise (or marginalise) their status within a course. Our focus is on a particular type of instructional move: instructors' transitions, or shifts, related to digital resources.
By a transition, we mean an instructional move in which an instructor draws connections from one item to another. The items can take many forms, such as instructional tasks, digital resources, or student comments. We view instructors' transitions to actualise potential connectivities (Gueudet et al., 2018) offered by digital resources, which may extend beyond, across, or within a resource. Working within, an instructor may connect one or more parts of a digital activity. Extending across, an instructor may connect different digital activities based on distinguishing features. Reaching beyond, an instructor may connect digital activities to course objectives, content, or structures. Appealing to Communities of Practice theory (Wenger, 1998), we theorise a role for an instructor in facilitating transitions related to digital resources.
Two key elements of Communities of Practice theory are boundary objects and brokering. Boundary objects (Star & Griesemer, 1989) are those items, tangible or intangible, that can be part of and, hence, foster interconnections between different communities. A digital resource can serve as a boundary object when it is shared by different groups to which an instructor belongs, such as an FLC and their classroom community (e.g. Sinclair et al., 2020). Brokering is an act in which a person navigates membership across multiple communities. When an instructor's transition involves a boundary object, that transition is also a brokering move (Rasmussen et al., 2009;Zandieh et al., 2017). We call such a move a "boundary transition", because its function is to connect boundary objects to constructs and practices valued by different communities with whom the object is shared. Although a transition can involve brokering, it does not necessitate it, because an instructor may enact a transition without navigating different communities.
To organise this article, we begin by drawing on Etienne Wenger's Communities of Practice theory to explain how we position digital resources as boundary objects and to situate instructors' transitions related to those boundary objects as a type of brokering move. Then, we provide background on the techtivities and the intervention in which they are a part. To ground our argument, we report empirical data from an instructor, Rachel, who facilitated boundary transitions within, across, and beyond a set of digital resources (the techtivities) during four lessons in the same semester. We conclude with implications for theory and practice.

Boundary Objects, Brokering Moves, and Boundary Transitions
We draw on Communities of Practice theory to explain how instructors may transform instruction in conjunction with their participation in an FLC. According to Etienne Wenger (1998), practice is an on-going, social process, not something that one can hand down from one group to another. An FLC is a Community of Practice within a university setting, and, as with any community, an FLC does not exist in isolation. Faculty participate alongside other communities to which they belong, such as their local mathematics department and the broader mathematics community. There are boundaries between different communities, and faculty cross these boundaries as they navigate their membership in each (Akkerman & Bakker, 2011). Boundary objects, which span multiple communities, can serve as catalysts for instructors to cross boundaries between communities (Hanke et al., 2021). Instructors then may broker their membership in multiple communities as they facilitate interactions around boundary objects. Bakker and Akkerman (2014) put forward transformation as one potential outcome of such boundary crossing. Transformation can result in changes to the status quo, which may happen via emergent practices in a community.

Positioning Digital Resources as Boundary Objects
By interacting with boundary objects, people can bridge different communities to which they belong (Star & Griesemer, 1989). These communities may intersect with others for whom that same item is not a boundary object. For example, an instructor may implement a novel task as a result of their participation in an FLC. The task may be a boundary object for the instructor, but not their students, who may not be brokering membership in different communities as they work on the task. While one community may intend for an item to serve as a boundary object, this alone is insufficient. Items become boundary objects when they are taken up by multiple communities (Star, 2010). The design of those items can impact their potential to become boundary objects -in particular, bringing together different perspectives that benefit from connection across constituent groups (Wenger, 1998).
Mathematics education researchers have pointed to a range of forms of boundary objects for practicing and prospective teachers. These include products of students' work (Bakker & Akkerman, 2014;Hanke et al., 2021), co-developed courses that integrate mathematics content and pedagogy (Goos & Bennison, 2018), digital touchscreen activities (Sinclair et al., 2020), interactive software tools , and paper-and-pencil tasks incorporating multiple representations (Robutti et al., 2021), as well as broader notions, such as number sense (Baccaglini-Frank & Maracci, 2015). When boundary objects are broader notions, they can provide grounding for novel tasks that may have less-clear connections to existing curricula.
Via working together with tangible boundary objects, researchers and teachers can make connections across different communities. These can include the classroom and the workplace (Bakker & Akkerman, 2014;Bakker et al., 2011), mathematics courses and mathematics education courses (Goos & Bennison, 2018;Hanke et al., 2021), and the classroom and an FLC (Robutti et al., 2021;Sinclair et al., 2020). It is valuable for teachers to engage with tangible boundary objects across intersecting communities (e.g. interact with a digital resource in an FLC, then implement in classroom instruction). Notably, such opportunities can afford teachers with space to question conventions presented in their curricular materials (Sinclair et al., 2020).

Instructors' Brokering Moves
Brokering is a form of connection, "provided by people who can introduce elements of one practice into another" (Wenger, 1998, p. 75). Hence, instructors who engage in brokering link aspects of different communities, because they hold membership in each. This brokering may happen in different settings, such as during classroom instruction (e.g. Zandieh et al., 2017) or in a professional meeting (e.g. Sinclair et al., 2020). In classroom instruction, when an instructor makes an instructional move that spans communities, either directly or indirectly, it is called a "brokering move" (Rasmussen et al., 2009;Zandieh et al., 2017). Being a member of multiple communities does not necessitate brokering (Wenger, 1998); it is an intentional act in which a person works to cross boundaries (Suchman, 1994) between different groups. While brokering can occur in a variety of settings, our focus is on the brokering that occurs during classroom instruction.
Analysing an instructor's practice in a collegiate mathematics classroom, Rasmussen et al. (2009) identified three categories of brokering moves, namely "creating a boundary encounter, bringing participants to the periphery, and interpreting between communities" (p. 211). A boundary encounter involves bringing different communities together via interaction with some boundary object, which could include indirect encounters between communities. From this point of view, students in a local classroom community could have indirect encounters with the broader mathematics community, via engaging in practices that are part of the work of mathematicians, such as modelling.
To bring participants to a periphery involved helping participants from one community to move closer to another, as if along a continuum. For example, an instructor could work to reconcile notions of function that students interpreted from their textbook (e.g. graphs of functions pass the "vertical line test"), with more nuanced notions addressing specialised relationships between variables. For a person to interpret between communities, they need to hold membership in different communities and to allow for one community to make meaning of ideas that emerge from another. For example, an instructor might explain how relationships between variables represented in a digital activity are a specialised type of function, which the broader mathematics community calls a "one-to-one function". Each of these examples illustrated how instructors' brokering moves could facilitate students' indirect encounters with other communities, such as an FLC or the broader mathematics community.

Boundary Transitions: a New Type of Brokering Move
Broadly, the function of a transition is to make links, which can occur within, across, or beyond classroom tasks or routines. When a task or routine serves as a boundary object, an instructor's transition can span communities and, hence, be a brokering move. We use the term "boundary transitions" to mean those transitions that also are brokering moves. With a boundary transition, instructors connect different communities as they draw links between items in their instruction.
We highlight three key aspects of boundary transitions. First, boundary transitions involve boundary objects, which may be tangible (e.g. digital resources) or intangible (e.g. the notion of function). Second, boundary transitions can include direct and/or indirect encounters between communities (Rasmussen et al., 2009). While it may not be possible for students in a classroom community to have direct encounters with an instructor's FLC or the broader mathematics community, they still can have indirect encounters across communities via their work in the classroom. Third, boundary transitions can take different forms, depending on the nature of the links between items. They can link different instructional items (transitions beyond), different forms of the same instructional item (transitions between), or parts of a single instructional item (transitions within). Through these aspects, we intend to illuminate the dual nature of a boundary transition, as both a transition and a brokering move.
For our purposes, we focus on instructors' boundary transitions related to novel digital resources, techtivities, introduced into an introductory undergraduate mathematics course, college algebra. While there are multiple communities possible for an instructor to span, we consider three: a classroom community, an FLC, and the broader mathematics community. These communities represent spaces in which instructors implement digital activities (classroom community), interact with instructors who also are implementing the activities (FLC), and connect beyond their local circles (broader mathematics community).

Background: the Techtivities and the Intervention
"Techtivities" is a portmanteau that Gary Olson invented to blend the ideas of technology and activities. The techtivities are interactive digital activities designed in the freely available Desmos platform (Desmos, n.d.). Their design aligns with practices promoted in an FLC for college algebra instructors (e.g. make room for students' reasoning), as well as key course topics (e.g. functions) valued by the broader mathematics community. Hence, the techtivities have potential to serve as boundary objects.

The Techtivities: Five Design Principles
Too often, students in college algebra can experience mathematics as an exercise in finding correct answers and performing given procedures (Mesa et al., 2012), and instructional efforts to actualise reform remain limited (Tunstall, 2018). With the five design principles, we articulate how instructors' implementation of the techtivities has potential to push back against the status quo.
The first is to privilege students' thinking and reasoning, rather than answer finding. Through this principle, we intend to raise the status of reasoning, so that it is something more than "explain how you got your answer". The second is to provide students with feedback and a space to reflect. For instance, after inputting a response or sketching a graph, students can see others' responses or a computer graph sketch. The third is to foreground exploration and to background accuracy. This principle works along with the first two, for example, it is okay if students sketch a graph that looks different from the computer graph. The point is for students to explore and represent relationships between variables, as well as to question attributes they may notice in a computer graph (e.g. curvature, corners, cusps).
The fourth is to incorporate variation against a background of invariance (Kullberg et al., 2017;Marton, 2015), meaning that we intentionally vary some aspects (e.g. the attributes represented on graph axes), while keeping others invariant (e.g. the task situation). The fifth is to broaden and challenge what counts in mathematics: For instance, students might think that graphs only represent functions when they pass a vertical line test. Through the techtivities, we challenge this notion, by incorporating univariate functions such that a variable represented on the horizontal axis is a function of a variable represented on the vertical axis (see Figure 1).
Boundary objects represent a "nexus of perspectives", and it is in this meeting of perspectives where the objects can become meaningful for different communities (Wenger, 1998). As boundary objects, there are different kinds of communities that the techtivities can bring together. One is an FLC and a local classroom community. Another is different theoretical communities, because there are different theoretical perspectives informing the design of the techtivites (Johnson et al., 2020). Our use of variation theory (Kullberg et al., 2017;Marton, 2015) underlies our decision to incorporate variation against a background of invariance. Specifically, students are to sketch two different graphs to represent the same relationship between attributes (see also Moore et al., 2014). With this move, we aim to promote students' discernment (Marton, 2015), or separation, of aspects of the co-ordinate plane, such as graph axes.
In the theory of quantitative reasoning (Thompson, 2011;Thompson & Carlson, 2017), Patrick Thompson explains a form of reasoning -covariational reasoning -that we intend to promote via students' work on the techtivities. The term "quantity" refers to a person's conception of an attribute as something possible to measure (Thompson, 1994). When students engage in covariational reasoning, they can conceive of relationships between quantities. Together, these perspectives afford our design to promote students' conceptions alongside their work to sketch and interpret graphs in the Cartesian plane, a socially shared representation system.

Elements of a Techtivity
Each techtivity has four main elements (Johnson et al., 2020). First, students are to view a video animation and then consider attributes in the animation that may be possible to measure. Second, students are to represent change in individual attributes via sliders on axes of a Cartesian plane (Fig. 2). The sliders are inspired by Thompson's (2002) recommendation for students to use their fingers as tools to represent change in variables. Third, students are to sketch a graph to represent a relationship between variables. Fourth, students are to reflect upon their thinking related to the activity (Fig. 1). In each techtivity, the second through fourth parts repeat, with the variables represented on different axes in the Cartesian plane. Figure 3 shows both graphs for one of the Ferris wheel techtivities.
Each techtivity includes a set of interactive screens that students work through, which is typical for activities created in the Desmos platform. Across the screens, students can sketch graphs, manipulate points, and type responses. Figures 1 and 2 each show a single screen from a techtivity. Furthermore, there is a toolkit for instructors to facilitate students' work during a lesson. We highlight two features: pacing and anonymised. Instructors may pace students' work on an activity by limiting the number of screens which students can access. When students type responses or sketch graphs, instructors can anonymise students' names to encourage sharing of ideas. Notably, the anonymised names are those of famous mathematicians, both historical and contemporary, which can foster conversation.

The Intervention: Situating the Techtivities
We implemented the techtivities as part of two US National Science Foundation-funded research projects. The first took place with college algebra instructors at a single university, while the second extended the work of the first project across multiple institutions. Participating instructors implemented the techtivities with their students and took part in an FLC in which they could connect with other instructors and develop a sense of the place, purpose, and process for implementing the techtivities with college algebra students. By place, we meant how the techtivities fit within existing curricular topics and practices in college algebra, such as function and modelling. By purpose, we referred to the "why" behind the techtivities, relating to the design principles of privileging reasoning, foregrounding exploration, and broadening what (and who) counts in mathematics. By process, we highlighted instructional moves to support the implementation of the techtivities.
Along with the FLC, Gary Olson and colleagues developed facilitation guides for instructors (e.g. Olson et al., 2019), which specified instructional moves and aligned those moves with the place and purpose for the techtivities. These moves were compatible with emerging efforts to revamp early undergraduate mathematics courses to build on students' strengths, rather than remediate their deficits (e.g. Bhattacharya et al., 2020). For example, one move was to allow students to change their mind in response to a question and then to reflect on what convinced them to make a change. Often, recommendations in the facilitation guides related to instructional moves of transitions, which could be within, across, or beyond the techtivities. Within a techtivity, one recommended move was for instructors to pause the activity before students received feedback in the form of a computer graph sketch, then tell students not to change their answer. Across the techtivities, one recommended move was for instructors to ask students to reflect upon ways in which key mathematical ideas, such as function, were represented in the situations. Beyond the techtivities, one recommended move was for instructors to reframe their language around function. Rather than asking students if something "was" a function, instructors were encouraged to ask whether y was a function of x and to connect such questions to attributes in the techtivities, such as height and distance (see also Olson & Johnson, 2022).

Methods
To provide empirical grounding for our argument, we report on an instructor, Rachel, who enacted boundary transitions within, across, and beyond a set of digital activities. We organise this section into three parts. First, we share our researcher positionality. Second, we describe the setting and source of data, to situate our analysis. Third, we discuss our analysis methods.

Researcher Positionality
Our author team is composed of two men and two women. Three are White, one is Filipino, and an indigenous citizen of the Navajo Nation. We seek to examine how our identities impact the way we see and interpret the world and how the world reflects that back on us. We bring life experiences from different social classes and US geographic locations. Each of us has classroom teaching and research experience, spanning elementary students through undergraduates. We respect the complexity in the work of teaching and acknowledge that instructors can hold different perspectives from our own. Accordingly, we have asked Rachel to calibrate our interpretations with her own recollections of the experience.

Setting and Source of Data
Our setting is a college algebra course, offered through the mathematics department at a university whose physical location is in the metropolitan area of a large city in the US Rocky Mountain region. The university serves a diverse student population, with 49% of students being first generation to college and 42% of students identifying as Black, Indigenous, or a Person of Colour. At this university, each section of college algebra is divided into two parts, lecture and recitation. Because of this set up, there are two instructors for any given section, one for lecture and another for recitation. Typically, lecture instructors are full-time or part-time faculty members, while recitation instructors are graduate students. During lectures, new topics are introduced, and during recitation, there are activities to support students' learning.
In the 2018-2019 academic year, the project team convened an FLC, in which college algebra instructors could learn to incorporate the techtivities into their course. Four times throughout each semester, participating instructors met together with members of the research team, to discuss the place, purpose, and process for implementing the techtivities. Rachel was a graduate student in the department of mathematical and statistical sciences; she felt comfortable with digital technologies and was open to using them in her instruction. In 2018-2019, Rachel began serving as a recitation instructor, and she participated in the FLC that year. After Spring 2019, the FLC continued in an informal capacity, facilitated by the instructor who served as course co-ordinator for college algebra. In the academic year 2019-2020, Rachel began serving as a lecture instructor, continuing until Spring 2021, when she again served as a recitation instructor, this time in a dual role as a recitation instructor and a coach for other recitation instructors. Rachel continued with the FLC through Fall 2021, after which time she began teaching a different course for the department of mathematical and statistical sciences.
Our sources of data included four video recorded lessons, spread out over the course of the Spring 2021 semester (see Table 1). In that semester, the course was held via synchronous video conference, due to the global pandemic. In the first and third lessons, 1 3 students worked on one techtivity. In the second and fourth lessons, students worked on two techtivities. Lessons ranged in length from 45 min to 1 h. Rachel would begin with instructions for the whole group, then assign students to work in small groups on a portion of a techtivity, and then return to the whole group. In each lesson, students had at least two interactions in small groups, separated by whole group instruction.
We selected to record Rachel's lessons because of her experience implementing the techtivities, as well as for her to serve as a model for other instructors implementing a course in a remote format. After watching these lessons, we noticed that Rachel made purposeful statements to transition between the techtivities and other course activities and ideas. We felt that such statements were important for positioning the techtivities as something more than just a novel addition to the course, and we encouraged other instructors to make such moves in their implementation. After reading the call for this special issue, we theorised Rachel's transitions, drawing on empirical evidence from the lessons to ground our theorising.

Analysis
We watched all four lessons, transcribing the portions of those lessons in which Rachel made statements to the whole group. A "statement" referred to an uninterrupted speech turn by Rachel. Our analysis had three passes. In our first pass, we read each of Rachel's statements. We identified those statements that were transitions, meaning they connected one lesson item to another. We then coded those statements by the type of transition (within, across, or beyond), allowing for single statements to have more than one type of transition. A transition within represented a link between one or more parts of a single techtivity. A transition across represented a link between one or more digital activities. A transition beyond represented a link between the techtivities and broader course topics or classroom routines. We selected this analytic lens in response to the call for this special issue, because it explained salient aspects of Rachel's instruction.
In our second pass, we organised Rachel's statements into categories of instructional moves (e.g. Harris et al., 2012) in which transitions occurred. There were three such categories: introductory statements situating each lesson; statements communicating expectations for students' participation; statements describing the Apr 21 Dynamic Tent, Changing Kite One-to-one function Each input maps to a unique output selection and/or interpretation of student responses. These categories of moves were representative of Rachel's instruction across all four lessons. In our third pass, we interpreted the statements through the lens of Wenger's Communities of Practice theory, explaining why these statements represented brokering moves, which we called "boundary transitions". First, we appealed to three different communities that Rachel linked in her statements: her classroom community; the FLC; and the broader mathematics community. Second, we described whether the boundary transitions were within, across, and/or beyond the techtivities. Third, we explained our perspective of the function of each boundary transition within Rachel's instruction. Table 2 links our three passes of analysis. We include three categories of instructional moves -communities being connected via those moves, types of boundary transitions within each category of instructional move, and functions of the boundary transitions within each category of move.

Results
We organise the results into three sections, reflecting the three categories of instructional moves shown in Table 2. For each category of move, we include one of Rachel's statements from each of the four lessons (see Tables 3, 4, and 5). The statements that we have selected are representative of the broader set of Rachel's statements across the lessons. We present the categories of instructional moves in the chronological order that they appeared during instruction. In each lesson, Rachel begins with an introductory statement, then sends students to small-group breakout rooms to work on a portion of the techtivity. When students return from breakout rooms, Rachel communicates her expectations for their participation and/or selects and interprets student responses to reflection questions included in the Techtivity (e.g. Fig. 1).

Rachel's Introductions: Boundary Transitions Within, Across, and Beyond the Techtivities
Rachel opened each lesson with an introductory statement to situate the lesson within the broader scope of instruction. In each introduction, Rachel made connections across and beyond the digital activities. In the introduction to lesson 3, Rachel also made a connection within the digital activity, addressing an aspect of the activity recommended during the FLC. In Table 3, we included Rachel's introductory statement for each lesson.
In lesson 1, Rachel first made a connection between the techtivity and other digital activities in the same software platform (Desmos). Then, she made connections beyond the techtivity, to two aspects of classroom instruction, students' accountability ("This techtivity is going to count as your submission today") and to course topics ("We're going to get into working with different graphs"). In lesson 2, Rachel's introduction followed a similar pattern, this time drawing connections across the set of techtivities ("If you remember when we did that Ferris wheel activity, we are going to do something very similar"). By lesson  When you log in, enter your name as your first name and last initial. This techtivity is going to count as your submission today. You won't have anything physical to turn in. I'll post the link in the chat here, too. The direct link. That way everybody can get in to it. What this techtivity is going to teach us, or let us play around with, is how points on graphs can represent relationships between different variables, because we're going to get in to working with different graphs, like how we talked about symmetry, and that kind of stuff on Thursday. We're getting more into graphs, and what this is going to let us do is play around with graphs a little bit more. We're going to go into breakout rooms, but everyone has to work on it individually, because it counts as your submission 2 For recitation today, if you remember when we did that Ferris wheel activity, we are going to do something very similar to that again today. Another one of those techtivities. We are actually going to do two of them. So, in class on Thursday, we talked about functions and what it means for y to be a function of x, or x to be a function of y, and what these activities are going to do is allow us to explore the definition of a function a little bit more or what it means for something to be a function. So, we are going to start with one that's called "Cannon Man". I am going to put the link in the chat here, but the code is also in the assignment page for the recitation activity for today in the module. It looks like a couple more people just joined, so I will paste it again so they can see it. You don't have anything physical to turn in today; the only thing that counts for your submission for recitation is the participation in these activities. So, we are going to do it kind of similar to how we did the last one. We'll go to break out rooms work on it a bit come back together talk about it, go back into breakout rooms and we will keep doing that 3 So, we've already talked about how we're going to do some more Desmos activities today. And these are really to help us with this modelling of real-life things. You know, we've done some modelling with Desmos where we've found these best fit lines. So, we're sort of going to do the same thing and we're going to go back to our Ferris wheel. So, let me grab the link here, and I will post it in the chat, and then I just want to explain a couple things about it before we go into breakout rooms. Let me share my screen. So, when you join Desmos and enter your first name and last initial. This will be your first screen [shows image of Ferris wheel from the Desmos activity]. And in the first one we're going to be talking about the relationship between the total distance that green cart has traveled and also its distance from that center line. When we're talking about width, we want to imagine this vertical line going down through the middle of the Ferris wheel and the width is that distance from the center line. So, it's not the distance from that center point; it's the distance from that center line. Any questions before we get started? When you're in your breakout rooms, have one person share your screen so that way we can talk about this in groups 4 Today we're going to do the last two of these Desmos techtivities that we've done throughout the semester, and so we're finishing them up today. It's stuff like the Ferris wheel, and the Cannon Man, and the Toy Car. We're doing more of those style activities today. And we're doing them today because they kind of connect to what we've been talking about with inverses and one-to-one functions. So let me grab the link. The links are also in Canvas [the on-line learning management system] too. We're going to start with the Dynamic Tent activity. I'm about to put the link in the chat. And when you sign in to the activity, just use your first name and last initial. While everyone is signing in to that, I will make some breakout rooms 1 3 3, Rachel abbreviated her connection across the techtivities, which made sense given that students had become more familiar with them. Interesting, in lesson 3, Rachel added a component to the introduction, in which she made a connection within the techtivity, to explain how to interpret a novel attribute ("width") in the  [She then selects another student response.] "Both show width and total distance in relation to one another." These are all really good observations. These two graphs show the same relationship between width and total distance traveled, because if you think about each spot here [moving cursor to a maximum point on left graph shown in Fig. 6], it would be the same. The same spot on the Ferris wheel has a point on each of these graphs. [She moves cursor back and forth to indicate corresponding points on both graphs, illustrated by dotted line shown in Fig. 6 situation. In lesson 4, Rachel's provided a brief connection across the techtivities, then a connection beyond, to course topics valued by the broader mathematical community (one-to-one and inverse functions). Rachel's introductory statements addressed place, purpose, and process for the techtivities, which were ideas promoted in the FLC. Place included both connections to course topics (e.g. one-to-one function) and also challenges to presentation of such topics in curricular materials. One challenge was related to the viability of the vertical line test, common in many US textbooks (see Olson & Johnson, 2022). A Cartesian graph would pass the vertical line test, and, hence, represent a function (y as a function of x), if a vertical line drawn anywhere in a co-ordinate plane intersected the graph at most once (assuming y and x are represented on vertical and horizontal axes, respectively). However, the vertical line test would not identify graphs that represent x as a function of y. Hence, the project team advised instructors to describe functions in terms of relationships between variables. In lesson 2, Rachel linked the techtivity to a lesson in which they discussed "what it means for y to be a function of x, or x to be a function of y", providing evidence that she had begun to incorporate these descriptions in her instruction. Through these statements, Rachel opened opportunities for her students to have indirect encounters with the broader mathematics community, because they could draw connections between ideas developed in the techtivities to broader course topics.
One purpose for the techtivities was to present a scenario in which students could conceive of graph attributes as being possible to measure (e.g. Thompson, 1994Thompson, , 2011. For example, one of the Ferris wheel techtivities involved a Ferris wheel cart's total distance travelled around one revolution and its "width from the centre". In the Ferris wheel facilitation guide (Olson et al., 2019), instructors were encouraged to help students to think about a Ferris wheel cart's "width from the centre" as its distance from a vertical line of symmetry passing through the centre of the wheel. When Rachel told students a productive way to "imagine" the width, it evidenced her attention to this recommendation. In each of her introductory statements, Rachel drew connections between students' work with the techtivities and topics valued by other communities to whom she belonged.
Even as the techtivities became more familiar to students, their status as a boundary object remained, as evidenced by Rachel's description in lesson 4 ("We're doing more of those style activities today."). We interpret Rachel's use of the phrase "those style" to set the techtivities apart from other digital activities in the course. Furthermore, we note how Rachel qualified a transition beyond the techtivities in lesson 4, by saying that they "kind of connect" (italics added), providing further evidence of the boundary status of the techtivities. In our view, Rachel's introductory statements set the stage for a boundary encounter (Rasmussen et al., 2009) with the techtivities. With these introductory statements, Rachel communicates that the techtivities have status. They count in the classroom community and connect to topics valued by the broader mathematics community.

Rachel's Expectations for Student Participation: Boundary Transitions Within and Across Techtivities
Across all four lessons, Rachel communicated expectations for students' participation in their work on the techtivities. These statements occurred before sending students to small group breakout rooms (lesson 1) and after students' return to whole group (lessons 2-4). In Table 4, we included one of Rachel's expectation statements from each lesson. In lessons 1-3, Rachel made explicit her expectations. By lesson 4, she implied those expectations through her reference to the nature of the activities.
In lessons 1 and 3, Rachel told students that it did not matter whether their graph was right or wrong; it was their reasoning that mattered. This move transitioned to an upcoming screen in the activity, when they would see a computer-generated graph that they could compare with their own. In lesson 2, Rachel also emphasised the importance of reasoning, telling students that it was okay to change their mind and that she wanted them to explain what convinced them to make the change. This move connected the wholegroup instruction to students' work in small groups, when they typed in their responses to a reflection question. Similar to her introductory statements, by lesson 4, Rachel treated the techtivities as a form of digital activity with which students were familiar. Connecting across the set of the activities ("We've done enough of these, so you know what's coming with the second graph"), she communicated her perspective of commonalities.
Rachel's statements in lessons 1-3 dovetailed with ideas promoted in the FLC and the facilitation guides (e.g. Olson et al., 2019), namely that instructors should promote and make space for students' reasoning. One recommendation was for instructors to anticipate that students may want to change their original graph to make it look like the graph shown on the computer. To address this, instructors were to communicate with students that they valued their reasoning more than whether the graph was accurate. Two years prior, in the Spring 2019 FLC, Rachel shared how she encouraged students to explore without fear of being penalised for sketching an inaccurate graph (Johnson et al., 2021). In these lessons, her communicating expectation statements indicated that she continued this practice, making her stance explicit across the lessons. In our view, these communicating expectation statements helped to bring students to a periphery (Rasmussen et al., 2009), to incorporate into the classroom community a practice promoted in the FLC. Rachel communicated what she valued in student participation; their reasoning on the techtivities matters more than whether they provide a correct answer.

Rachel's Selection and Interpretation of Student Comments: Boundary Transitions Within the Techtivities
Across all four lessons, Rachel selected and interpreted students' comments, connecting their comments to other parts of the techtivity. In lesson 1, a student volunteered to share their response with the class and Rachel interpreted that response. In lessons 2-4, Rachel selected students' comments from their typed responses and then stated why she thought those comments were valuable. These statements occurred when students returned to whole group, after having interacted in small-group breakout rooms. In Table 5, we included one of Rachel's interpretation statements from each lesson. Across all lessons, Rachel incorporated the Desmos anonymise feature, so that she could share student responses without identifying the student. In her interpretation of each student's response, she drew connections within the techtivity, from a reflection question to a graph representing a relationship between variables.
In each of the lessons, Rachel drew on student responses to make connections within a techtivity, between a reflection question at the end of the activity and graphs that students encountered during the activity. Across her interpretations, she emphasised how a single point on a graph represented a relationship between variables, which was a broad learning goal for the techtivities (Desmos, n.d.). In lesson 1, students were shown a single point on a co-ordinate plane, then were asked whether they agreed or disagreed with a hypothetical student who claimed that the point could represent more than one location for a cart on the Ferris wheel. Just before Rachel's statement, a student said that they disagreed, because, "it never went around that point again". While talking, the student pointed with their index finger and then made a counterclockwise circular motion, appearing to draw a Ferris wheel. In response, Rachel went back to one of the interactive graphs, pausing the graph at the point shown in Fig. 4. Claiming that the graph passed through the point a single time, she asserted that the point could represent only one location on the wheel.
In lesson 2, Rachel provided further evidence of her attention to function in a way consistent with what was valued by the FLC. Students were given two graphs which they had encountered earlier in the toy car techtivity (Fig. 5) and then asked whether they agreed or disagreed with a hypothetical student (Val), who claimed that both graphs showed a toy car's distance from the shrub as a function of the car's total distance traveled. Students typed responses to this question, and Rachel anonymised those responses via a feature of the Desmos platform. She selected "Brahmagupta's" response, who described a function in terms of a relationship between input and output. Anticipating that students may assume the input to be the variable represented on the horizontal axis, she showed how both graphs in Fig. 5 mapped an input (total distance travelled) to a single output (distance from the shrub). To do so, she moved the cursor from the axis representing total distance travelled to a corresponding point on the graph. The dotted arrows in Fig. 5 show her movements. Notably, the majority of students agreed with Val's statement (see Fig. 5), despite the graph on the right failing the vertical line test. Hence, students seemed able to interpret whether Cartesian graphs represented univariate functions in ways beyond performing a routine test.
In lesson 3, Rachel continued to provide evidence that she intended for students to discern that different looking graphs could represent the same relationship between variables, a learning goal for the techtivities addressed in the FLC. Again, students were given two graphs which they had encountered earlier in the techtivity (see Fig. 6). This time the hypothetical student claimed that both graphs showed the same relationship between a Ferris wheel cart's width from the centre and total distance travelled. Highlighting three students' responses, Rachel demonstrated how a single location on the Ferris wheel was represented in both graphs. She placed the cursor at a maximum point on the left graph and then moved the cursor horizontally (as shown by the dotted line in Fig. 6) to a point representing that same location on the right graph. In this lesson, even a broader majority of students agreed that both graphs could represent the same relationship between variables, suggesting that students were interpreting graphs in ways promoted by the FLC.
In lesson 4, Rachel continued to emphasise graphs as representing relationships between variables. In this lesson, students worked on the Dynamic Tent techtivity (see Fig. 2), in which a tent grew and then shrunk. At first, the height increased while the base decreased, until it reached a maximum height, after which the height decreased while the base increased. Hence, the computer sketch traced each point twice, such that a single point represented two instances in the animation, in which the tent had the same height and base. With the reflection question in Fig. 7, students were to consider whether different base lengths could have the same height. Overwhelmingly, students concluded that each base length had a different height. Highlighting a student's response, Rachel dragged a cursor from the vertical axis to the graph, as shown in Fig. 7, to demonstrate how each height had a unique base. Interesting to us, in this move, Rachel also illustrated how the relationship between variables in the tent situation represented an inverse function, because either argument would suffice (i.e. if each base has a unique height, then each height has a unique base). In both cases, only one distance from the shrub corresponds to a total distance traveled Across the lessons, Rachel has continued to align with ideas promoted in the FLC, namely that it is important for instructors to help students learn that graphs represent relationships between variables. In our view, these student-focused statements help to interpret between communities (Rasmussen et al., 2009), connecting student responses in the classroom community to perspectives on functions and variables promoted in the FLC. With her interpretation statements, Rachel communicates how students' responses illustrate how a single graph, or two different-looking graphs, can represent a relationship between variables.

Discussion
We put forward a new type of brokering move (Rasmussen et al., 2009), which we call a "boundary transition", to characterise an instructor's moves around a set of novel digital activities (techtivities). The transitions could take one of three forms: within, across, or beyond the techtivities. Boundary transitions included three different forms of statements, namely ones situating each lesson, communicating expectations for students' participation, and describing the selection and/or interpretation of student responses. Introductory statements included all three forms of boundary transitions: expectation statements included two forms (within, across), while interpretation statements included one form (within). When Rachel enacted boundary transitions within a techtivity, she drew links between her classroom community and ideas valued by the FLC.
In her boundary transitions across and beyond the techtivities, she drew links between her classroom community, the FLC, and the broader mathematics community. The boundary transitions served three functions. With the introductory statements, Rachel's boundary transitions positioned the techtivities as something that counted in the classroom community and connected to topics valued by the broader mathematics community. With the expectation statements, Rachel's boundary transitions communicated to students that their reasoning mattered more than whether they provided a correct answer. With the interpretation statements, Rachel's boundary transitions connected students' responses to a mathematical idea promoted in the FLC, that a single graph, or even different looking graphs, represented a relationship between variables. Together, these boundary transitions served to legitimise the status of the techtivities within an existing course.
As suggested by our analysis, boundary transitions can connect to other types of brokering moves (Rasmussen et al., 2009). In her introductory statements, Rachel's boundary transitions set up students for a boundary encounter with the techtivities, which were something different from their prior experience in the course. In her expectation statements, Rachel's boundary transitions helped to bring students to the periphery of the FLC, as she promoted practices addressed in the FLC (e.g. make room for students' reasoning). In her interpretation statements, she helped to interpret between communities, by discussing student comments in light of mathematical ideas valued by the FLC. Adding boundary transitions to the landscape of brokering moves, we illuminate how instructors may navigate between different forms of such moves.
The last design principle for the techtivities is to broaden and challenge what counts in mathematics. When digital resources serve as boundary objects, they can create space for instructors to question conventions presented in their curricular materials (Sinclair et al., 2020). This is particularly salient when instructors' curricular materials focus heavily on skills and procedures, such as those for college algebra (Mesa et al., 2012). Through Rachel's boundary transitions, she worked to broaden students' notions about graphs, emphasising how a single point on a graph represented a relationship between variables, a key understanding articulated by Thompson in his theory of quantitative reasoning (Thompson, 2011;Thompson & Carlson, 2017). Furthermore, she challenged the notion that answer-finding is the ultimate goal in mathematics ("So what I'm looking for is that you explain your reasoning, not if your graph was right or wrong" [lesson 1; Table 4]). Notably, a boundary transition is one move through which instructors can work to transform instruction via challenging the status quo.
Instructional transformation can have different grain sizes. It may be that of an individual instructor, who changed their practice from one year to the next. It also may refer to transformation to the status quo in mathematics courses, such as college algebra, where instructors cover feel pressure to cover content and fill gaps in students' knowledge. When novel digital resources have status as boundary objects, there is potential for instructors to leverage them to promote ideas and practices valued by an FLC, which may yet to be part of their local classroom community or mathematics department. In future studies, researchers can examine how instructors' interactions with boundary objects create space to question tacit assumptions in curricular materials and to connect such questioning to changes in practice.

Conclusion
Broadly, we are interested in how instructors take up novel digital activities in courses where skills and speed too often take precedence over reasoning and exploration. We offer "boundary transitions" as a move through which instructors may connect different communities to legitimise the status of such activities. We acknowledge that to position novel digital activities as something more than an "add-on" requires change at a systemic level, as well as at the individual instructor level. As we theorise the role of an instructor in taking up the techtivities, we do so with consideration of systems in which the instruction is taking place.