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During Inquiry Maths workshops, teachers often tell me that the most challenging moment of a lesson comes after the students have posed questions about the prompt. In their attempt to build a community of inquiry, the teachers aim to respond in a way that acknowledges and takes account of students' contributions.
However, teachers often face the pressure to remain within the limits of a formal curriculum and the related time pressure to 'cover the content' - pressures that can compete with the wish to follow new lines of inquiry.
According to Inoue and Buczynski's research paper about inquiry in mathematics classrooms, the point of the lesson at which the teacher has to respond to a student's input can give rise to stumbling blocks.
The response acts to obstruct inquiry when the teacher fails to engage with ideas or to channel them in a meaningful direction.
This is a danger not only for pre-service teachers (such as those who feature in the research paper), but also for experienced practitioners under curriculum and time pressures.
Stumbling blocks in the classroom
Stumbling blocks in the classroom
The paper reminded me of Ania Maxwell's description of her first inquiry. After attending an Inquiry Maths workshop, Ania took the decision to launch into inquiry at the first opportunity. She relates what happened below, describing the stumbling blocks that arose in her classroom.
The questions and observations of Ania's year 8 students to the prompt.
Ania, who was 2i/c in a secondary school mathematics department, used the area prompt with her year 8 class. The students had middle to low prior attainment.
Ania reports that the prompt "brought out so many misconceptions" that she would not have known about if she "just launched into instruction." However, the emergence of the misconceptions left Ania in a quandary, "How can I expect the students to understand more complex ideas without dispelling their misconceptions?"
Ania had prepared to use the resource on the website, but "discovered how unconfident my students were even with that task." On reflection, she feels she was not prepared enough and needed more resources to address the different directions the inquiry took.
Ania was very glad when a student asked how to work out the area of a circle using a formula so that she could direct the class to follow that line of inquiry and "resort to [her] usual style of teaching."
Stumbling blocks
Stumbling blocks
In their paper, Inoue and Buczynski define stumbling blocks as "moments of difficulty [when] the teacher responded to an instructional situation in such a way that derailed the inquiry-based goals of the lesson and created moments that significantly undermined the quality of the inquiry lesson" (p. 14). In the lessons they analysed, the researchers identified 13 types of stumbling block (see list below).
Interestingly, the stumbling blocks were 'nested'. The appearance of one contributed to the emergence of another. An example from the paper occurred when the teacher was faced with no student response to a question she posed. Not knowing how to respond (the first stumbling block), the teacher asked leading questions without giving the students an opportunity to make sense of the mathematical concept (the second stumbling block).
The researchers comment that, "It was often the case that [the teacher] had not considered possible learner responses, and caught off guard, did not know how to respond.... The children who contributed these thought-provoking answers did not receive any meaningful response or validation of their ideas from the teacher or their peers" (p. 20).
The description in the paper resonates with Ania's account. She felt under-prepared, finding it difficult in the moment to respond in a way that acknowledged the students' contributions and upheld her intention to promote inquiry. Even though she was relieved when a student asked a question (about a formula for the area of a circle) that was easier to deal with, Ania ends by saying that from that point in the lesson, the inquiry "felt like mayhem."
Stumbling blocks to inquiry identified by Inoue and Buczynski in their research on mathematics classrooms.
Avoiding stumbling blocks
Avoiding stumbling blocks
Inoue and Buczynski's advice to avoid stumbling blocks, derived from research with trainee teachers, is relevant to all. Teachers need to be able to validate students' responses, even if they are unexpected, and use them to navigate towards curriculum aims through incorporating learners' prior knowledge into the lesson. Achieving that relies on, firstly, trying to anticipate possibilities in students' diverse responses and, secondly, giving pedagogically meaningful explanations that bridge mathematical content to students' thinking. The key, claim the researchers, "seems to be a deep understanding of how children think and might react to concepts" (p. 20).
Returning to Ania's lesson, the students' responses to the prompt are wide-ranging (see picture above). Knowledge about the meaning of 'area' and how to calculate the area of a rectangle already exists in the class. The teacher might draw on that prior learning to discuss the concepts of size, which appears twice in the students' questions, and dimensions and also the idea of measuring areas to verify that they are equal.
The initial line of inquiry could then be restricted to finding a rectangle and triangle with equal areas, giving all students the opportunity to re-connect with those concepts. Only after that preparatory stage would the inquiry consider the questions about working out the area of a circle and the relationship between perimeter and area.
Of course, the students' responses are easier to sequence in hindsight. In the Inquiry Maths model the teacher uses the regulatory cards to avoid the stumbling blocks of unanticipated and diverse student responses. The cards act to slow down this part of the lesson and give all participants time to reflect on the course and aims of the inquiry.
Andrew Blair, June 2021
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