My description below of how the teacher ‘presents’ the mathematics to their students has been left somewhat vague, as a deep focus upon that is not my motivation in writing this piece; subsequent lesson accounts will more directly address the mathematics under consideration.
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After some administrative matters, the teacher describes the focus of today’s lesson and shares how they view it in relation to that which has come before and that which is still to come in their lessons. They share that the focus is on finding percentages of a given positive integer, without the use of a calculator, and the teacher shares with the class that they are going to build upon something most of them have just answered - finding 10% of 60.
Given that the teacher has observed that at least some students did not demonstrate that they were able to obtain the correct value to this question, what does the teacher then do with this information? How does it inform what they offer next, if it does? Is the identity of the students who did not display ‘6’ and/or the methods that the teacher may have been looking for retained for further action?
The teacher then demonstrates how to find 30% of 60, by writing a model solution on the board. They emphasise how they are laying out their working, that this is crucial and that they expect all of the students to produce similar when they attempt their own questions. This demonstration is completed without any interaction with the class, although the teacher is regularly turning from the board to make what appears to be eye contact with students and their eyes move in a way that suggests they are scanning over all of the students.
How has this demonstration been arrived upon - is it a method the teacher has arrived at themselves, is it a departmentally agreed upon strategy and wording that all teachers follow, is it following a ‘script’ from an external source that teachers are given to ‘deliver’? To what extent may the answer to this question have impacted upon the teacher’s thinking about the mathematics under consideration and their own teaching of it? Does this matter?
Whilst building upon how 10% of a positive integer is obtained, the mathematics here is presented as a set of rules to learn and follow. Is that true for how one obtains percentages of positive integers, or is this instead something students could have worked at potentially reasoning for themselves? Does the difference matter?
There seems to be an expectation here that students fully focus on what the teacher is saying, doing and writing - or that the teacher is convinced that this is happening - so that they follow in order to understand. Is this how understanding is derived - by following a set of instructions? Are questions or discussion thus a hindrance to be stopped rather than a potential aid to learning?
With this example left on the board, the teacher then writes down an additional question: what is 40% of 70? They state that they are going to repeat the process with similar working out, but this time with the class’ help. Several hands go up; the teacher asks for the students to put their hands down and states that they instead want everyone to think about what the steps are and that they will then ask specific students what to do. The teacher then proceeds to do this, with their inputs to the students taking the form of “what do we do next?”. There are no questions posed as to why particular steps are undertaken, nor questioning on why particular results are obtained. The teacher’s inputs are generally posed to the whole class, with a few seconds passing before a particular student is picked to answer. If the teacher accepts what is said as correct, another student is asked to repeat it and it is then written on the board. If a student says that they do not know, another student is asked instead, and it appears that if they offer what the teacher wants, the initial student is asked to repeat. If this second student also states they do not know, the teacher refers back to their first example and asks directed questions to the second student with the aim that they share similar for the question currently being undertaken. If a student offers an answer the teacher does not accept, the teacher tells them they are mistaken, and seeks out an answer they find acceptable from another student with the first then asked to repeat.
What are the advantages and disadvantages of not allowing any students to volunteer answers? What is gained and lost by a teacher entirely deciding who they are going to call upon to answer? How do teachers in this position make decisions on who they are going to ask, and how much variety is there in their choices from lesson to lesson?
How may students feel in response to the knowledge that they may be asked to answer questions at any point, no matter how ready they feel? Does this matter?
In the scenario described here, if a student says that which the teacher wants to hear, can the teacher reach a conclusion about the student’s understanding? If a student repeats back another student’s words, what does that mean about their understanding? Is this different from a student repeating back a teacher’s words?
If a teacher hears what they want to hear, does that mean learning is happening?
Is there an implicit direction in the background here that students in the class are discouraged from asking questions about what they are learning, and their understanding of it?
Next: moving away from whole class teaching
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