Problem-Posing: Genuine Inquiry in a Primary Classroom     

Introduction and Rationale

Problem-posing is an integral part of problem-solving and some researchers would argue, it is actually more true to the work that mathematicians engage in (authors cited in English, 1998). English, Fox and Watters suggest that students need to develop both problem-solving and problem-posing abilities in order to Òfunction effectively in a world that is demanding more flexible, creative, and future-oriented mathematical thinkers and problem-solvers,Ó (2005, p. 156).

This paper shares findings from a collaborative inquiry with a class of grades 2 and 3 students, their teacher and a researcher. The purpose of the study was to examine what types of problems students pose, what inspires the problem-posing and how a collaborative, caring classroom community contributes to the problem-posing experiences. Considering an ethic of care (Noddings, 1984, 2005; Hackenberg, 2005; Long, 2006), studentsÕ interests and what they care about/for are placed at the forefront in this study. As adults both cared for the students and the mathematical problems they posed, a caring mathematical community developed in which students felt able to task risks, ask questions and pose problems.

 

Methodology

Over three months, the researcher participated in daily classroom activities, co-planned and taught with the classroom teacher and facilitated data collection and analysis. As co-researchers in this collaborative inquiry, the researcher, teacher and 22 grades 2 and 3 students shared in decision-making, data collection and analysis as much as possible.

Data collection involved a variety of qualitative research techniques in order to capture the multi-dimensions of the childrenÕs lived experiences. Observations of classroom experiences were recorded, self-reports were elicited from the children and their spontaneous narratives were captured on video and audio recordings.

Data analysis involved an continuous cycle of viewing and listening to the data, initially looking for global themes and patterns and then at a more detailed level looking for specific evidence to elucidate the themes and patterns that had emerged.

 

Sources of Data

In order to be aware of the background experiences of the participants, initial surveys were given to the teacher and students. At the beginning of the study, all participants (students, teacher/s and researcher) began entries in their own research journals. The journals were used to record the studentsÕ math ÒworkÓ, reflections, questions and wonderings. The first recording devices introduced into the classroom were digital cameras. There was a gradual introduction to the video and audio recording devices in the classroom as the students became more comfortable with their thoughts and actions being captured and recorded. The still images and some video recordings were used during the study for visual conversation starters and support, to help students ÒrecallÓ and describe events in their classroom. A final survey and some individual follow-up interviews were conducted at the end of the study.

 

Results and Findings

            After analysis of the multiple data sources, several findings emerged relating to what inspired young students to pose problems as well as findings in the area of the role of the classroom environment, including the role of the teacher, in creating the atmosphere and open-ness for opportunities for problem-posing.  This paper focuses on another finding important to this study, the types of problems the students posed, both in terms of mathematical content and structure.         

The problems students posed were first categorized into open and closed type problems. The closed problems in general were routine, word problems and involved number computations. The open problems were ÒwonderÓ type problems and followed some of the problem structures that had been modeled for the students. Examples of this structures include: ÒHow could youÉÓ ÒHow many different waysÉ?Ó and ÒI wonder ifÉ?Ó  Some of these open problems posed by the students during the study include: How do you measure the perimeter of a desk? How many different ways can you measure your hand? and What shapes can we see outside? Most of the closed problems involved number operations while the open problems focused on broader mathematical concepts and topics such as geometry and measurement.

In post-study analysis, it has become clear that it was essential to both provide students with modeling of a spirit of genuine inquiry that involved problem-posing and wondering but to also create opportunities or starting points for students to pose their own problems.

 

Conclusion and Significance of this Study          

            The mathematics education literature at the elementary level in the area of problem-posing, primarily focuses on the problems that teachers pose to their students. Studies involving studentsÕ posing their own problems involve contexts with significant constraints such as the structure of the problem and the content to be used, usually operation work with numbers (English, 1998). This study contributes a unique look at a primary classroom community and what mathematical engagement can look and feel like when opportunities for mathematical problem-posing are opened up. A classroom community that values its membersÕ thoughts and questions contributes to an environment that is conducive to genuine inquiry. Classroom teachers can learn to provide openings or starting points for mathematical inquiry and problem-posing that values studentsÕ interests and thus, seems to increase their level of engagement with the mathematics involved.

 

Priority Area: Professional Learning

            This session addresses professional learning as it suggests instructional practices and learning opportunities for students that focus on teachers considering mathematics through inquiry. The teacher plays an integral role in an inquiry-based classroom and a particular knowledge of mathematics and responsiveness to students is required in order to create a collaborative math community in the classroom.

 

Session Timeline

            In this session, a brief introduction to problem-posing and a description of the research study will be shared. The majority of allotted time will be used to share findings from the study. Using photographs and video clips to highlight the problem-posing experiences of the students in the classroom, attendees will be asked to make connections to classroom contexts and problem-posing opportunities they have experienced.

 

 

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