The teacher's response to the students' responses to the prompt
Each pair of students has posed a question or shared something they have noticed or wondered about the prompt. The teacher has written the contributions on the board. Before deciding how to proceed with the inquiry (and, indeed, deciding how to decide), the teacher has the opportunity to respond.
In taking that opportunity, the teacher has three aims in mind:
firstly, to convince students that their ideas will be treated as a basis for inquiry;
secondly, to clear up points that will obstruct inquiry if they are not dealt with immediately; and
thirdly, to introduce (or remind the class of) a concept or procedure that has not yet arisen in the lesson, but is integral to the progress of the inquiry.
Two distinctions
When responding to students' questions and conjectures, I find two distinctions helpful.
1. Exploratory and summative speech
O'Connor identified two types of speech that teachers use in response to students' ideas. During exploratory speech the teacher might overlook imprecise speculations and pass over errors in calculations or meet them, not with direct corrections, but with counter-examples that provoke fresh thinking: "When we are in the heavy lifting and framing stages of developing new ideas, stopping to correct every flaw is disruptive to the real work" (p. 177).
However, when the focus of discussion is clearly defined and stable, the teacher "tightens the criterion levels for precision and correctness" (p. 178) by attending to students' mistakes and re-wording their ideas using formal mathematical language.
2. Arbitrary and necessary knowledge
Hewitt distinguishes between arbitrary and necessary knowledge. Students can only come to know arbitrary knowledge from an external source, which, in the classroom, is often the teacher. The conventions in the mathematics curriculum are based on choices made in the development of the discipline. To those learning about them, the commonly-accepted conventions may seen to be the result of arbitrary decisions.
However, it is possible for students to infer necessary knowledge from what they already know. That is not to say that all students will infer the knowledge, only that it is possible.
Case study
This month we have published Tariq Rasool's sectors of a circle inquiry.
The students in his year 8 mixed attainment class made these responses to the prompt (arranged in an increasing level of mathematical sophistication):
The teacher's response
Responses 1 and 3 refer to arbitrary knowledge, which the teacher should deal with immediately using summative speech:
"Good question! O stands for 'origin' (and marks the point of intersection of the two perpendicular axes on a coordinate plane)." The teacher might include the part in parentheses to link the prompt to prior learning.
"Yes, a quarter of a circle is called a quadrant (just as a quarter of the coordinate plane is called a quadrant)."
Response 2, in noticing a changeable property of the diagram, is important for generating new lines of inquiry. The teacher aims to plant an idea in students' minds with exploratory speech and a rhetorical question:
"That's right. I wonder what would happen if the angle was not a right angle. Would the areas still be equal?"
Response 5 refers to prior learning about the formula for the area of the circle. It is important to draw out the knowledge explicitly so all students have the opportunity to understand its relevance and incorporate it into the inquiry:
"Good question. Please remind us about the formula and tell us how you would use it."
Response 7, which gives a method for calculating the area of a quadrant, answers the question in response 4. Whether a method had arisen or not in the students' responses, the teacher would refrain from answering the question directly. That is because the answer is necessary knowledge, which students can derive from their existing knowledge.
Responses 4, 6, and 8 give rise to a main, and most probably, the first line of inquiry. By collecting the responses together and leaving them until the end, the teacher aims to give impetus to the idea of testing different cases.
The teacher would employ a combination of exploratory speech to draw out possible values of x that could be tested and summative speech to make sure that students' understanding of the variable is secure:
Exploratory
"So x could stand for any number. Can you give me some examples?"
"As x is the length of the radius, are there any values x cannot be in this context?"
Summative
"x > 0"
"x cannot take different values at the same time. So if x = 5, then the radius of the quadrant is 10."
To initiate a structured inquiry in which students practise finding the areas of a circle and a quadrant, the teacher might call on a student to model the calculations for a particular case.
In response 9 a student speculates about the truth of the statement in the prompt and even begins to use spatial reasoning to justify the claim. Such speculation must be met with exploratory speech to maintain students' curiosity about the prompt:
"That's an interesting idea! I hadn't thought about the region created by the overlap of the shape. Is the remaining space in the quadrant three times the area of the region? Let's see what we can come up with by using different values of x."
Andrew Blair, April 2024