So What?

Towers, J., Martin, L. C., & Heater, B. (2013). Teaching and learning mathematics in the

collective. The Journal of Mathematical Behavior, 32(3), 424-433.

Purpose of article (What was studied/discussed): 
"In this paper we analyze and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented.” The article explores teaching and learning in highly interactive classrooms, which is something not all teachers and students are used to, especially in mathematical classrooms. The article also focuses on the role of the teacher in inquiry-oriented classrooms, which again dramatically differs from the original role of the teacher because they now act as a facilitator, rather than a lecturer. Finally, the article discusses the idea of positioning the work as a way of being with mathematics, in hopes that math will actually mean something to the students rather than just something else they have to learn. Overall, the authors claimed, “We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective.”

Important terms: Enactivism; Group learning; Group cognition

Results:
The authors focused on the idea of enactivism because it is in the interaction between the learner and environment that learning happens, not because of the learning environment (including the teacher) or the learning him/herself. The study then was based on the nature of collective mathematical understanding, which evolves from interaction and woven ideas. Data was collected in two high school classrooms (taught by the same teacher) in a single high school in a Canadian city.  "Our analysis reveals significant insights into the ways in which a collective might be orchestrated in the high-school setting."  The authors then clustered their findings into two groups: "(1) those that address the (teaching) structures necessary for initiating and sustaining such a collective, and (2) those that reveal the kind of relationship with mathematics that is fostered in this environment."

So What?

Should students primarily be in or out of their seats while learning?
The authors primarily focus on the set up of the classroom and were analyzed throughout the study. "Most days, the students are offered a problem and encouraged to get out of their desks to work on the many whiteboards.  Considerable numbers of whiteboard pens are made available to them and they jostle for position, some students writing on the board, others offering suggestions about what to write or draw.  Students often add to one another's drawings, or erase all or parts of a drawing someone else has created."  Personally, I love the idea of having students out of their seats and presenting their work to the rest of the class because this fosters higher-order thinking, questioning, and good discussion.  Do you think this classroom environment is effective or would it simply cause chaos and disturbance for other students?

Is the planning worth the product?

As a new teacher, I am certainly aware of how demanding this job is because we have to constantly be prepared, in ways that are primarily beneficial for our students. Do we have time to really design a plan of which fosters group work and collaboration or should we just stick to the basics to save time? “Planning for teaching and learning in this kind of space requires a preparation that goes well beyond the textbook and teachers’ guide. Sharon plans for teaching in a flexible manner, starting with the curriculum guide to ‘locate’ herself for the unit, drawing on a range of resources or creating her own problems and activities, and always reflecting on the previous lesson to determine where to begin the next.” I know that this type of classroom is more meaningful and engaging, but is all the planning worth the final outcomes?

Do you agree with the idea that “what is not prohibited is permitted”?
What can we interpret about a way of being with mathematics in a classroom like the one the authors described? “Students move freely around the room once the task is initiated.  There are natural limits but there is no densely woven, blocking, and stifling system of rules.” The authors stress on the idea that mathematical problem solving should not be something students need to struggle with on their own, but rather make it a public process and discuss with others. This type of interaction certainly calls for such freedom of movement around the classroom. In this type of classroom students are encouraged to look at other classmates’ work, by analyzing, questioning and challenging other students’ solutions. Do you agree with this idea of learning or do you consider it cheating?