[Re]imagining Student Mathematical Engagement through Problem Posing Pedagogy

Dr. Priyanka Agarwal’s dissertation for her Ph.D. in mathematics education at the University of California Irvine was funded by a CPM Dissertation Fellowship over 2018-2019 academic year. She is currently an Assistant Professor of Mathematics Education at the University of Wisconsin, Madison. Her research is focused on designing and examining inclusive mathematics learning environments through design collaborations with teachers. By combining micro-ethnographic, qualitative, and design-based research methods, she examines ways in which students make sense of mathematical concepts while embracing their personal and cultural ways of knowing and being.

The Study

Purpose and Context

Problem-posing practices are considered important for nurturing students’ inquiry, learning, and agency in mathematics. Although mathematical problem solving and rigor of mathematical tasks have received much attention in research, the role of problem-posing is still underrated in the work of math learners in schools. In this dissertation study, I partnered with a mathematics teacher to co-design and implement instructional lessons centered on mathematical problem-posing (as against just problem-solving), in low-tracked eighth-grade classes in a predominantly working class Latinx neighborhood school. Drawing on socially situated frameworks of learning, the study aimed to (1) understand how students gain entry to the practice of problem-posing, (2) investigate how learning processes unfold over time in interaction with peers, materials, and the teacher, and (3) shed light on the ways in which students negotiate their agency and social risks of posing a math problem. Research data included video-based observations, student written-work, reflections, and classroom artifacts collected in two different settings of task-based paired interviews and a classroom-based teaching experiment. The lessons were designed to align with and supplement the CPM curriculum the teacher was using in his classes.

Findings

The study findings reach beyond the past focus on the cognitive aspects of problem-posing and its linkages with individual creativity and ability. Instead, they offer dimensions of mathematical doubts as a novel characterization to support expansive and agentic forms of student problem-posing (Paper 1), an understanding of the sociocultural processes of productive posing (Paper 2), and conceptualization of collective agency and risk-taking in problem-posing (Paper 3). Together, the study findings elaborate on our understanding of how students become engaged problem-posers and the central role that student doubts and collective action play in supporting productive forms of posing. I also discuss how problem-posing practice is uniquely positioned to amplify capabilities, identities, and epistemic agency of students of color who get disproportionately sorted in remedial courses. The findings provide preliminary ideas of problem-posing pedagogy’s potential to challenge the deeply-rooted deficit discourse of race, poverty, and ability in education. The findings have implications for the design and analysis of inquiry-oriented learning environments in both formal and informal settings.

A Glimpse into Students’ Mathematical Problem Posing

Annotated Bibliography

Agarwal, P. (2019). Student participation and agency in mathematical problem posing. UC Irvine. ProQuest ID: Agarwal_uci_0030D_15948. Merritt ID: ark:/13030/m5x4042p. Retrieved from https://escholarship.org/uc/item/1s30n858

The dissertation constituted three papers, described below.

Paper 1 “The Dimensions of Mathematical Doubts and its Relation and Affordances for Problem-Posing”

In this paper, I conceptualize problem-posing using the notion of mathematical doubts and ask: What math doubts emerge when students explore open unstructured artifacts in relation to problem-posing? How does a pedagogical context afford or shape the surfacing of doubts? These questions reflect an effort to understand what I refer to as students’ epistemic needs about what students want to know and do, and their developing understanding of what can be known and done, and how to know it in the context of mathematics. Data from two different settings—task-based interviews (n=64) and classroom-based teaching experiment (n=57)—was used. I use student written work and video-recorded small-group student interactions to identify critical moments when a mathematical doubt emerged, what led to it, and the student talk and activity that preceded and followed the emergence of the doubt. After analyzing for possible patterns, I find three dimensions of students’ mathematical doubts that led to the student-posed problems—Pragmatic (finding purpose: What is it for?), Analytic (sensemaking: What is it?), and Transformative (questioning the established facts and reaching for new possibilities: Why is it the way it is? How can it be changed?).

A nuanced clarity of student mathematical doubts in the current study offers a window into the varied ways in which students interpret, reimagine, enact, and re-create the mathematical world, when given agency to do so. This view can be used to assert the value in designing learning environments and curricula that focus on cultivating students’ sense of how knowledge gets generated and who gets to shape it. The findings emphasize the constant dialogue with artifacts and peers as creating context for doubts to be surfaced, shared, and taken up. This context then becomes a space of shared ideas and mutual support allowing students a pathway to problem-posing practices. Since social spaces are not power-neutral, the doubts, however, may become visible and gain traction only if learning conditions allow students to share them with others and if the doubts that are shared are taken up for further exploration by teachers and peers. Thus, creating conditions that would allow students to pursue their doubts is also an act of nurturing students’ epistemic agency that are often denied to low-tracked students and the students of color as the literature on tracking and race suggests. By recognizing the ethics of considering doubts as something that not only belongs, but provides a foundation in a mathematics class, this study opens up a space for critical dialogue about what a mathematics problem is, who can be considered a problem-poser, and why we need problem-posing as a practice for school children. These theoretical considerations justify a stronger foundation for researchers and educators from which to mobilize the various dimensions of student doubts for the study of mathematical problem-posing and problem-solving.

Paper 2 “Characterizing Shifts in Student Participation in Mathematical Problem-Posing”

In paper 2, I characterize the processes that enable students to engage in increasingly more sophisticated processes of problem-posing, explicate the nature of these processes and conditions that afford or constrain them. In particular, I ask: How do students shift from the periphery of their doubts to engage more fully in posing mathematical problems? By coding, constantly comparing what emerged, revising the codes, and thematizing the various phases of student problem-posing in the two settings of task-based interviews (n=32 pairs) and classroom experiment (n=7 four-member groups), I found three processes through which students gradually increase their participation in the practice of problem-posing. I call these processes assembling, casting, and carving and shaping.

1. Assembling: Students assembling observations, doubts, and wonderings about the given task artifact to ask basic arithmetic or geometric problems that students could easily resolve

2. Casting: Students casting (posing) their mathematics problems using a pre-existing mold or example (i.e., posing a problem using an example of a problem they have seen before posed in the textbook or by the teacher). Not all students may do that but those who do it, do it prior to carving their own unique and original problems (as described in the next step).

3. Carving and Shaping: Students carve out promising ideas from multiple nascent doubts earlier explored and shape them into mathematical problems that are yet unresolved for students.

The three themes formed a constitutive set of practices of posing, that is, casting required students to also assemble their observations and doubts, and students who carved and shaped their doubts also initially created initial problems by assembling and casting.

Further, shifting towards more productive problem-posing processes was a result of how students negotiated meaning-making when they worked together with their peers and the design features of the learning environment. I highlight the role of artifacts (both pre-designed and student-constructed), fluid participation structure roles (e.g., team roles), collective argumentation, recursively revising the posed problems, and student-defined objectives for problem-posing as important for allowing students to make temporal shifts in their practices.

Paper 3 “Organizing for Collective Agency and Risk-taking in Mathematical Problem Posing”

In paper 3, I problematize the ease with which the literature talks about student agency in mathematical problem-posing and examine what is socially and disciplinarily at stake for students when doing this work. While problem posing practices may nurture student agency, it does not come without embracing social and disciplinary risk-taking by students. Problem posing often involve reframing the purpose of mathematical objects, questioning the validity of mathematical rules, and modifying the given assumptions to discover new patterns, all of which require intellectual courage. It may also require negotiating one’s social position and deeply-seated beliefs about who has authority to pose a problem, what is an acceptable math problem, and who is a good problem poser. I ask: How do groups of students exercise agency in the presence of social and disciplinary risks when working together to productively problem-pose? I find that when agency was investigated in relation to the social risks of posing, two sociocultural aspects of student activities emerged that allowed students to productively negotiate emerging risks: active noticing and foregoing control over one’s own ideas to pursue collective goals. Analyses of small-groups reveal that the social and disciplinary risks of problem-posing were always present, but their negotiations were most productive when socially-constructed relations through active listening and noticing to each other paved the way for individuals in the group to collectively shape each other’s nascent doubts and wonderings into meaningful problems.

Related Conference Proceeding:

Agarwal, P. (2020). Organizing for Collective Agency: Negotiating Social and Disciplinary Risks of Collaborative Learning. In Gresalfi, M. and Horn, I. S. (Eds.), The Interdisciplinarity of the Learning Sciences, 14th International Conference of the Learning Sciences (ICLS) 2020, Volume 1 (pp. 223-229). Nashville, Tennessee: International Society of the Learning Sciences. https://doi.dx.org/10.22318/icls2020.223

Presentations

Agarwal, P. (2020, April). Organizing for Collective Agency in Mathematical Problem-Posing. Accepted for presentation at the American Educational Research Association (AERA) Annual Meeting (canceled due to COVID-19), San Francisco, CA.

In this presentation, I outlined the findings about the processes through which students come to organize their individual efforts towards collective agency and its role for productive problem posing and solving.

Agarwal, P. (2018, November). Problem posing pedagogy: processes of students’ mathematical doubts and wonderings. Invited paper presented at the California Educational Research Association (CERA) annual conference, Anaheim, CA.

In this invited talk, I argued for a conceptualization of mathematical doubts as productive for student learning and discussed its relation and importance for students’ problem-posing.

Agarwal, P. (2019, November). Dimensions of student doubts and mathematical problem posing. Brief research report presented at the annual conference of Psychology of Mathematics Education-North American Chapter, St. Louis, MO.

In this presentation, I outlined the different dimensions of the mathematical doubts that emerged when students were asked to create their own problems.

Agarwal, P. (2018, October). Function, process, and development of students’ mathematical wondering and problem posing. Poster presented at Learning Sciences Graduate Students Conference, Nashville, TN.

In this poster presentation, I presented the findings about the collaborative learning processes that allow students to take up and productively transform their doubts and wonderings to mathematics problems for solving.