This ongoing series will offer descriptions of parts or entirety of lessons. These lessons may have been taught by myself, or be among those I have observed, or an amalgamation. The intention is not to pass judgment on that which is described, nor on myself as a practitioner or on others, but instead to question and learn.
I offer the following account of a lesson, which was structured according to a template; the lesson focus was upon ‘calculating percentages of amounts’. My questions regarding what was offered and observed are in italics.
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The students arrive at their mathematics lesson, where the teacher awaits to greet them at the entrance to the classroom. On the board are projected six short questions, which have been selected from the curriculum the students have, in theory, already encountered. It transpires that one question is related to the focus of today’s lesson, the others are not.
Is it important what students immediately encounter upon entry to a lesson? What is the purpose of an initial, opening task (using whatever naming language is applicable)?
With this kind of opening, does it matter which questions are chosen, and when? Are these a standard set of questions which all students are set, irrespective of their class or teacher, or are they chosen by the teacher for the class in front of them? Do the questions chosen link to one another from lesson-to-lesson, or do they appear to the students to be a random unrelated selection? Does this matter?
There is an expectation that the students sit in silence and answer these questions, on something which can be used to reveal their answers later, without assistance from the teacher or from other students. This is a standard opening routine to every lesson.
Are such routines important? Are they needed? If so, how and when - as similar as possible across the whole school, or across all lessons in a subject, or across all lessons by a specific teacher, or across all of the lessons by a specific teacher with a specific class? What if the teacher believes there is a need to deviate from this opening - are they automatically wrong?
What is the power of students working in silence in this situation? What is potentially lost by its insistence, and does this matter? What does a student do if they do not know how to approach answering a question, given they know no assistance is forthcoming?
After all students have arrived, the teacher takes the register and then circulates among the students. They have been praised for their efforts, or encouraged (it appears that the teacher attempts to do this discreetly) where required. The students are told how much time they have remaining to complete the task. Some students are not visibly writing, whilst others appear to be making themselves look busy.
Is it okay for students to ‘opt-out’ in this situation? Why might they be doing so?
When the stated time elapses, the teacher asks the students to reveal their answers as a class, one question at a time. As they proceed, the teacher offers brief feedback, in most cases either only providing an answer or a short instruction on how they obtained their answer. For two questions, after the teacher has seen the class' answers they comment that they will offer a fuller explanation after the answers to the rest of thel questions have been revealed. They do this, after apparently aiming to ensure all of the class are paying attention, spending a short period of time demonstrating how the answers are obtained. They do this without any interaction with any members of the class, with the exception of one student who after the demonstration puts their hand up and then shares that they still do not understand. The teacher explains this question does not relate to today’s lesson and that the student should not worry right now, but should aim to do some independent learning to ensure their understanding is improved.
What do the students gain and what do they lose from only knowing whether their answers are “correct” or not, along with at most only the briefest of explanation, and without any discussion with any other person about the mathematics? Is there an implicit belief on the teacher’s part that what is being offered should be nothing more than a reminder?
What might the teacher be looking for that allows them to reach a conclusion about how many and which questions deserve further consideration? Is this assessed during the revealing of answers or could it have been determined - or at least suspected - prior to the lesson?
What is the purpose of a teacher explaining without any interaction initiated by students, versus explanations where such interaction is allowed or encouraged?
When does the teacher consider it “okay” for at least some students to not understand something that is offered here? What is the potential difference in perception of this between the teacher and the student(s)? What may a student take away from a situation like this regarding themselves as learners, the subject as a whole, and their teacher? Does this matter?
What influences decisions about when to further explore mathematics that is ‘tangential’ to the planned focus of the lesson? If the teacher does not want such a ‘distraction’, should they have set questions that are away from the main scope of the lesson and thus carry a higher probability of the students being provided with such distraction?1
Is it possible that, no matter what the teacher says or does, the students are at this point mentally focused upon a different area of mathematics than that which the teacher wants?
If instead of mathematics being addressed in a lesson, students are encouraged (or in this case, left) to pursue explanations or understanding in their own time, how often is there evidence of this happening?
If a student has been unable to answer a majority of the questions set, what feelings could have been invoked about themselves as a mathematics student before they then encounter any new mathematics in the rest of the lesson? Does this matter?
What does this lesson opening implicitly tell students about how mathematics is learnt and its source? Does this matter?
Next: the main ‘focus’ of the lesson
1Mathematics has many beautiful aspects, one being that any prompt at all poses the opportunity for our minds to take us on journeys away from that which may be 'intended'!
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