Academic journal article Australian Mathematics Teacher

# Making the Most of Unanticipated Opportunities

Academic journal article Australian Mathematics Teacher

# Making the Most of Unanticipated Opportunities

## Article excerpt

In a grade 9 (ages 14-15) introductory algebra lesson, the class is exploring how the m and b in the equation y = mx + b are related to the graph of a linear function. During a class discussion about these relationships, a student asks if it is possible for the graph to have "two dots on the y-axis"--that is, whether is it possible for the linear function to have more than one y-intercept.

* How would you characterise this query? What is going on mathematically?

* What might be gained and what might be lost by pursuing this student idea at this moment?

How might you as the teacher act on the student's query?

Following calls for teaching to engage students in mathematics (e.g., NCTM, 2000), many authors have argued for the importance of focussing on student thinking and using it to build mathematical understanding (e.g., Breyfogle & Herbel-Eisenmann, 2004; Cavey & Mahavier, 2010; Foster, 2011; Stockero & Van Zoest, 2011). Smith, Hughes, Engle and Stein (2009) shared five practices for orchestrating classroom discourse around students' thinking:

1. Anticipating;

2. Monitoring;

3. Selecting;

4. Sequencing; and

5. Connecting.

Their work emphasises the importance of teachers actively planning to elicit and make use of student input. Despite the best planning, though, not all student input can be anticipated or will fall within the teacher's plan for the lesson. As in the vignette above, teachers are often faced with student input that interrupts the flow of a lesson and requires a decision about how to respond.

In the vignette, the student question introduces a topic--the definition of a function--that was not part of the plan for the lesson. The unanticipated question requires the teacher to decide whether the topic is worth departing from the plan to pursue at this time and, if so, how this might best be done. Within the complexity of classroom interactions, how does a teacher recognise student input that is worth pursuing? If a student's contribution seems worth pursuing, how can it best be capitalised on? To begin to address these questions, we share research results that characterise unanticipated student mathematical input worth pursuing and teacher decisions in response to it that use students' ideas to enhance their understanding of mathematics.

Framing the work

A goal of our ongoing research is to better understand how student mathematical thinking that becomes public in a classroom can be used to support the learning of mathematics content and practices (e.g., Stockero & Van Zoest, 2011). Although there are often instances of student thinking that the teacher has intentionally cultivated to emerge at a particular time through a given task or a posed question, we were interested in learning more about instances that were not planned. We defined pivotal teaching moments (PTMs) as instances in a classroom lesson in which an interruption in the flow of the lesson provides the teacher an opportunity to modify their teaching in order to extend or change the nature of students' mathematical understanding. Here, we draw on our study of PTMs (as reported in Stockero & Van Zoest, 2013) to consider the potential of unanticipated student ideas that emerge during class discussions.

To better understand unanticipated student input, we examined over 45 hours of video of mathematics teaching in six teachers' classrooms. At the time of the data collection, the participating teachers were teaching mathematics in grades 8-12 (ages 13-18) in a variety of US school settings, including a rural school with a large Hispanic population, an urban school where approximately half the students were African-American, a suburban school with a predominately white population, and an alternative school for students who had not been successful in their local school. …

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