Myths are stories we share over time and in community that serve to illuminate worldviews. Several myths exist about what it means to do and learn mathematics, many of which constrain possibilities for teaching and learning [1
]. This paper explores a myth that has dominated assumptions about learning mathematics with others; that off-task behavior in classrooms is detrimental to learning. As Wagner and Herbel-Eisenmann state,
“changing the stories (or myths) told about mathematics is necessary for changing the way mathematics is done and the way it is taught. We emphasize the need for change to combat the sense of repression often associated with mathematics.”
The vision of learning and doing mathematics with others includes notions of strict focus and attention to the task at hand. Part of what fuels this myth, as it plays out in mathematics in particular, is that learning and doing mathematics with others can evoke an Enlightenment-era vision of disembodied rational minds, engaged in the search for universal truth. This vision centers reason as the primary source of authority, where inductive and deductive logics enable new insights and give rise to a long-lasting and still held faith in the human capacity to attain objective knowledge that transcends even humanity. The idea that bringing in disruptions or distractions to the table could be productive simply does not ring true.
This paper explores the idea that off-task participation has an important role to play in collaborative mathematics classrooms. To do so, I first explore extant research in mathematics education, grounded in sociocultural/situative frameworks (i.e., [3
]) to argue that the emphasis on mathematics forms of talk has helped obscure inquiry into other forms of participation and their functions for productive collaborative mathematics activity. I then bring in a smaller set of new work that draws on positioning theory (and similar approaches) to expand our understanding of the relations between participation, identity, and learning posited by situative theories of learning. From a positioning perspective, described further below, interactions serve as identity resources that can be leveraged to navigate access to learning opportunities and other important group dynamics. This expanded view of learning and doing mathematics together is then applied to the case of collaborative problem solving, and, in particular, the kinds of on-task and off-task talk students engage in while doing mathematics together. I recast the oft-maligned role of off-task interactions as central to and supportive of learning mathematics together. I close with the implications of confronting this myth for new research.
3. Off-Task Interactions as Positional Resources in Collaborative Mathematics
While on-task participation draws primarily from storylines about schooling and school mathematics (e.g., what a “good student” looks and sounds like, and so on), off-task participation can draw on a broader range of storylines, including those of friendship and popular culture, potentially making a greater range of positional resources available to students to leverage as they interact with one another and the task [19
]. In this sense, off-task interactions are not simply the lack of directly engaging in the mathematical task, but rather the bringing together of multiple storylines, identities, and relations onto the collaborative activity and, as such, may be drawn into the shared regulation of activity.
There’s some precedent to consider this possibility. Sullivan and Wilson found that students used playful talk to negotiate status during collaboration [40
]. They draw on Vygotsky who “argued that play arises in young children as a response to desires that cannot be immediately gratified [41
]. To cope, children invent imaginary situations…in an effort to fulfill their unrealizable wishes. This behavior is what Vygotsky considered the foundation of play,” [40
] (pp. 7–8). In playful talk, students use imaginary situations unavailable in the official collaborative learning activity to position themselves or others as either more or less capable. For example, a student with low status engaged in playful talk to position herself with competence, while a high-status student engaged in playful talk to position others as less competent and, in doing so, maintained his status. These acts of positioning functioned to influence opportunities to learn within the group. They also found that playful talk served to establish and maintain group cohesion and thus achieved higher levels of coordination.
Esmonde and Langer-Osuna found that off-task interactions—which drew on discourses of romance and friendship, which were racialized and gendered—regulated group members’ participation in small group mathematics discussions [19
]. Students’ off-task conversations functioned to resist domination by a peer with high academic status, enabling shared intellectual contributions to the mathematics discussion. Off-task conversations supported shared participation in large part because two students who were relatively marginalized from engagement in the mathematics utilized off-task conversations to position themselves with social power, which functioned to resist the third student who had been dominating the mathematical work. That is, off-task interactions offered students the ability to position themselves with relatively greater power than was available through on-task interactions. While on-task interactions drew from school math discourses, off-task interactions primarily drew from discourses of youth popular culture, friendship, and romance. Students utilized these discourses as positional resources, enabling new pathways into the collaboration.
In the example provided above, students drew on storylines beyond the scope of mathematics to manage participation in ways that math talk alone could not do. This demonstrates that students may be able to find relatively more powerful positions, such as positions of competence through storylines not originating with participating in school tasks [36
]. Langer-Osuna followed a focal student, Terrance, over an academic year as he constructed positional identities of himself as a learner, starting with a resistant identity and shifting to a productive and engaged identity [26
]. To do so, Terrance drew on a variety of school-based and out-of-school-based storylines as he engaged with peers in collaborative learning activities. The discourses that organized much of the off-task interactions, such as youth popular culture or the armed forces, offered Terrance positional identities that ultimately supported engagement in mathematical work. Off-task interactions can thus enable new positions by drawing on alternate storylines.
Langer-Osuna, Gargroetzi, Chavez and Munson recently examined the functions of off-task interactions during collaborative mathematics problem solving [42
]. The research team analyzed 13 videos of 20 to 30 min collaborative mathematics problem-solving sessions among fourth graders during a classroom unit on place value [43
]. Fifty-six instances of off-task participation, such as conversations about games or movies, making play swords out of connecting cubes, or singing a popular song together, were identified and coded for their functions on the collaborative dynamics. The functions of off-task interactions were determined by the shift between the nature of the interactions prior to and subsequent to off-task interactions. For example, at a table of four students working in pairs, a focal student made several bids to engage the second partnership to work as a group of four. His bids were ignored and so he began telling a story about riding his horse in the online game Minecraft and burning down a village. As he did so, the gaze of other students at the table shifted up and toward the focal student, giving him their attention. Upon gaining their attention, he shifted back to the collaborative work, suggesting that they, “put all your tens in here!” His peers took up his suggestion and began to work together, discussing their contributions to the task thus far. This instance of off-task interaction was coded as functioning to recruit others into the collaboration. After several unsuccessful bids to grow a collaboration that would include both his immediate partner and another table mate, the student’s Minecraft story functioned to gain his peers’ attention, in particular their gaze. Once he had their gaze, he again made a bid for collaboration, suggesting they organize their 10 sticks in a basket that he held up. His peers took up his suggestion, thereby enabling him to recruit both girls into a collaboration with him and with each other.
Results of this study found that the majority of off-task instances—59%—function to support the collaborative process [42
]. The functions identified that served to grow or sustain the collaboration were: warming up to collaboration, gaining attention, recruiting others into the collaboration, gaining access to the collaboration for self, resisting concentrated authority in the group, and extending the task. Importantly, a fifth of instances served to bring more peers into the collaboration, whether recruiting others or gaining access for one self. These instances typically began after unsuccessful bids to work with others through the task itself. The off-task interactions created new opportunities for the students to engage with one another by disrupting the on-task dynamics and enabling students to either gain traction into and join existing collaborations or recruit peers into interaction through off-task activity, which subsequently shifted back on task, while maintaining the larger collaborative group. These instances typically occurred after a series of successful bids to either join or initiate collaborative work, suggesting that interactional pathways into the collaboration through on-task activity were restricted or more cumbersome. Students often utilized these discourses as positional resources, enabling relatively powerful positions that were otherwise unavailable through on-task interactions.
Why did off-task interactions succeed where on-task bids failed? To examine this question, Langer-Osuna, et al. additionally examined the positional identities made available through the off-task interactions [42
]. Specifically, they analyzed a subset of the instances that served to shift the group dynamics by growing or destabilizing the collaboration, as well as the “flops” that bid for shifts but were unsuccessful. These instances were further coded for the storylines invoked, the positional identity made available, and whether the bid was successful for shifting the dynamics. Results showed that when students drew on positional identities that did not afford them relatively greater power, their bids were as likely to be successful as unsuccessful. These positional identities included characters such as “victim” in the storyline of theft and “noodle server” in the storyline of patrons at a restaurant. However, when students positioned themselves powerfully within the storylines-imagined social hierarchies, their bids were nearly always successful. These positional identities included “warrior” in the storyline of conquering villages, and “maker” in the storyline of genius inventing. These findings suggest the possibility that off-task interactions functioned to support collaborative learning because students were better able to negotiate positions of power that drew from storylines beyond school mathematics and enabled access to their peers, resources, and the collaboration itself.
Herbel-Eisenmann, et al. call on mathematics education researchers to “intervene to shift these storylines and positionings and to have greater impact on policy, practice, and public perception in the future” [1
], (p. 103). In particular, they argue that researchers “need to identify and better understand historical, current, and pervasive storylines about mathematics education research (p. 110).” To this end, this paper offer a counter-story that can be theoretically and empirically fruitful for mathematics educators, that is, that identity is inherent to learning and doing mathematics together in classrooms and, as such, positional resources, including those made available during off-task interactions, merit serious attention by the field.
Though mathematics education research literature has historically and primarily focused on classroom talk that is explicitly mathematical in form, new work points to mathematics problem solving as social negotiations, including positional negotiations that serve to make claims about individuals in relation to mathematics activity. These negotiations partly determine who participates and in what ways, as well as what mathematical ideas are taken as true by the group. As such, interactions during collaborative mathematics problem solving serve not only as cognitive resources for shared thinking, but also as identity resources that can be leveraged to navigate access to learning opportunities and other important group dynamics. These functions of talk can draw on discourses beyond mathematics, per se. Indeed, off-task interactions can have the potential to offer a broader range of identity resources than made available strictly through on-task interactions.
With these ideas in mind, several new questions arise. What kinds of storylines are potentially productive, and how might teachers prime or otherwise make available such storylines for students? Rather than whether off-task participation is good or bad for learning, we might ask, when and in what ways is off-task participation beneficial or detrimental to learning and doing mathematics together? Further, how might classrooms be designed in ways that resist problematic and powerful storylines, such as racialized and gendered storylines about mathematics ability or enacting authority? Just as research in mathematics education has become increasingly interested in and concerned about the role of identity in mathematics teaching and learning, so, too, must teachers and teacher educators. How might we support teachers in noticing and effectively responding to student collaborative dynamics? When should teachers intervene and when should they step back and allow off-task talk to co-occur with on-task activity?