*In this monthly column, we speak with a notable member of the mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Dr. Alayne Armstrong.
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*Alayne Armstrong joined the Faculty of Education of the University of Regina in July 2016 as an Assistant Professor in Mathematics Education. She completed her PhD in 2013 in the Department of Curriculum and Pedagogy at the University of British Columbia (UBC), where she was a SSHRC Doctoral Scholar. Her Masters and Bachelor of Education degrees were also obtained from UBC, and she has additional degrees from the University of Manitoba and Queen’s University. Prior to joining the University of Regina, Alayne was a classroom teacher in the Coquitlam School District in the Lower Mainland of British Columbia, and she also taught undergraduate education courses in math methods and inquiry at UBC. She is currently getting to know Regina and enjoying the friendly people, the pelicans and muskrats, and the big blue sky.*

*First things first, thank you for taking the time for this conversation! *

*As you had spent 19 years teaching in the K-12 public school system prior to joining the faculty at the University of Regina in the summer of 2016, your research undoubtedly draws from a wealth of experience in the classroom. Whose work influenced you during your time as a teacher? Then, which gap in the research**, or which classroom experiences, urged you to transition into the domain of educational research?*

I taught a variety of subjects while I was in the school system, including drama, home economics, and art, so there were many different influences over the years. In terms of teaching mathematics, the work of John Van de Walle (e.g., Van De Walle, John A. & Folk, S. [2005/2007]) had a big impact on me in terms of showing that it was valuable to try a wide range of activities to help students grasp mathematical concepts, including ones that I’d normally use in other subject areas. Along that vein, the work of Gary Tsuruda (e.g., Tsuruda, 1994), an educator (now retired) and math consultant from California, helped me to see connections between students’ mathematical thinking and the act of writing in problem solving.

[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]I was intrigued by how non-linear my students’ learning was—they seemed to move forwards, backwards, forwards again, even sideways at times.[/perfectpullquote]I remember being intrigued by how non-linear my adolescent students’ learning was—that they seemed to move forwards, backwards, forwards again, even sideways at times – and I initially applied for graduate studies at UBC with the intent of researching ways teachers could most effectively support this kind of learning. Then I became interested in how this learning manifested itself in small group dynamics.

*Among the more general themes in your work, group work in the mathematics classroom and the notion of authorship have emerged as central (e.g., Armstrong, 2005, 2013, 2015). In particular, you call for a shift in authority in the mathematics classroom from external sources (e.g., a textbook) to students. As you write,*

A knowledge-making community counters the role of textbook as the authority in the classroom. Responsibility passes to the students to break away from playing the role of empty vessels waiting to be filled with facts and formulae, and instead to make meaning of mathematics for themselves. (Armstrong, 2015, p. 4)

*In this sense, does your work align with the theory of constructivism, or does the focus on groups of students, rather than individuals, necessitate a different lens? *

The quote you’ve selected does suggest some version of constructivism, although I’ve always found constructivism, in the classical sense, to be such a lonely concept – the learner seems so isolated from the world around her. My work in general would probably fit better under the umbrella of enactivism, because what learners do affects, and are affected by, the groups (or systems) in which they are embedded.

*In your work, you have referenced the distinction between the notions of cooperative, collaborative, and collective groups (Armstrong, 2015). How do you distinguish between these types of group action? Does the notion of authorship change depending on the particular character a group takes on at a particular point in time?*

Groups can be defined in different ways—for instance, by the type of task they’ve been assigned, or by the length of time members stay together as a group. I’ve been looking at groups in terms of how cohesive their behaviour is. The more the interests and actions of the members align, the more the groups can be considered as learning agents in their own right. “Collections” have the least social cohesion—for instance, a group of people who happen to be waiting for the same elevator—while “collectives,” like a jazz trio that is jamming together, have the most.

[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Teachers and students have the power not only to choose what texts to use as references, but also to consider themselves as authors in the mathematics they do.[/perfectpullquote]Who is deemed to have authorship can be a complicated thing. There is the notion of author/ity that Povey, Burton, Angier, and Boylan (1999) describe, where the presence of the “/” in the altered term “author/ity” points to the presence of a person, an author, who perhaps wields the power that the unaltered term “authority” conveys. Following this view, mathematics texts are recognized as having been authored by someone; they weren’t just handed down from the heavens somehow! Teachers and students have the power not only to choose what mathematics texts to use as references, but also to consider themselves as authors in the mathematics that they themselves do, just as they would in other subject areas (for instance, if they were writing up science reports or drafting stories).*How is the notion of collectivity related to that of flow (discussed in Armstrong, 2008), and is this a sustainable state for a group? What role might a teacher play in sustaining cohesive activity within a group of students working on a mathematical task?*

Drawing on R. Keith Sawyer’s work (e.g., Sawyer, 2003), group flow can be considered a level of peak performance for a collective, and it occurs when the collective’s behaviour is at its most synchronous and cohesive and is acting as if it were of “one mind.” I don’t believe it is sustainable for very long, just as a flow state for an individual person only lasts for a limited amount of time. And that’s probably a good thing—it takes a lot of energy and effort to retain that high level of focus. As well, one of the benefits of being in a group comes through the diversity of its members and the variety of ideas and actions that are available; in a flow state, convergence is key, so that kind of diversity isn’t in play in those moments.

[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Being in a state of flow is very enjoyable, so the more flow experiences a student can have during mathematical activities, the better![/perfectpullquote]Still, being in a state of flow is very enjoyable, so the more flow experiences a student can have during mathematical activities, the better! In terms of a teacher’s role in encouraging group flow, a lot of what is in the literature about effective group work, in terms of balancing structure and challenge, would be applicable: the establishment of effective group routines; flexible leadership roles within the group so that various students can move in and out of these roles, depending on the activities; a safe and accepting environment where all group members feel comfortable that they can both share their ideas and have them fairly considered, etc.*In your view, (how) can collaboration and collectivity in the mathematics classroom coexist with the culture of individual accountability in our current school system?*

The short answer might be that they can coexist in mathematics classes in the same way that they co-exist in classes in other subject areas. For some activities, collaboration is very beneficial—diversity of ideas, immediate feedback, explaining one’s thinking, and questioning others about their ideas; for other tasks, individual work may be most effective so a student can go deep within the self to build understanding.

The long answer might be that there are a number of pressures related to school mathematics, particularly at the secondary level, that make the idea of collaboration suspect:

- the pressure of limited time, the amount of curriculum that must be “covered,” and the perception that collaboration simply takes too long;
- the perception of mathematics being a black or white subject, and that sharing ideas and possible solutions paths is a waste of time because your final answer is either right or it’s wrong;
- the definition of expertise in mathematics, where student learning comes only from listening to the teacher/expert rather than working with peers to explore and wrestle with concepts;
- the difficulty of grading group performance. Who gets the marks? How does the teacher know which group members did the work and who actually understands?

About a month later, I was at my school during nutrition break, and a teaching colleague asked me a probability question that one of his students had asked him. To my surprise, the question was related to that probability problem from the previous month, and to my greater surprise I was able to answer it in a manner that was clear and coherent enough that he was able to go back to his student with a satisfactory explanation. So, when would it have been most appropriate to evaluate my level of understanding—while I was working with the forum group, or individually during my lunch break (so the issue of group versus individual)? As well, there are boundaries to assessment related to time: Should I have been evaluated the day of the group session, or later, after I had been away from the problem for a month? Should I only have had one shot at being evaluated on my understanding of a certain concept, or should I have had another chance to be evaluated once I’ve had more time to process it?

In terms of accountability, group or individual, the issues of assessment and evaluation are complex. It has been interesting to see how the changes in assessment and evaluation practices in more recent years (formative and summative assessment, standards-based grading, etc.) have been affecting teaching practices in mathematics classrooms.

*Thank you, Dr. Armstrong, for taking the time for this conversation. We look forward to your upcoming work and to continuing the discussion in the future!*

*Ilona Vashchyshyn*

**References**

Armstrong, A. (2005). *Group flow in small groups of middle school mathematics students* (Master’s thesis). University of British Columbia. Retrieved from https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0092452

Armstrong, A. (2008). The fragility of group flow: The experiences of two small groups in a middle school mathematics classroom. *Journal of Mathematical Behavior, 27*, 101-115.

Armstrong, A. (2013). *Problem posing as storyline: Collective authoring of mathematics by small groups of middle school students *(Doctoral dissertation). University of British Columbia. Retrieved from https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0073575

Armstrong, A. (2014). Collective problem posing as an emergent phenomenon in middle school mathematics group discourse. *Proceedings of PME 38 and PME-NA 36, 2, *57-64.

Armstrong, A. (2015). Textual construction of middle school math students as “thinkers.” *Language and Literacy, 17*(3).

Niessen, S. (2016, February 11). New appointment for mathematics education. *Education News. *Retrieved from http://www2.uregina.ca/education/news/new-appointment-for-mathematics-education/

Povey, H., Burton, L., Angier, C., & Boylan, M. (1999). Learners as authors in the mathematics classroom. In L. Burton (Ed.), *Learning mathematics: From hierarchies to networks* (pp. 232-245). London: Falmer Press.

Sawyer, R. K. (2003). *Group creativity. Music, theater, collaboration*. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Tsuruda, G. (1994). *Putting it together*. Portsmouth, NH: Heinemann.

Van De Walle, John A. & Folk, S. (2005/2007). *Elementary and middle school mathematics*. Toronto: Pearson Education Canada Inc. (Canadian Edition)