Is it a lecture or a dialogue? given by the teacher? constructed with the pupils? A summary of a situation or the introduction to a sequence? It's hard to get to grips with the different conceptions of the lesson that circulate within the community of maths teachers. In this post, I propose a few ideas to help you get a clearer picture.
1. What's the point of the lesson phase in maths lessons?
2. Should concepts be introduced before the lesson phase?
3. How do you prepare a lesson when you're just starting out?
4. How do you bring the lesson to life in class?
1. What is the point of the lesson phase in maths lessons?
"I'm told that we have to follow the explicit teaching method, but at the same time I'm told that we have to introduce each concept to give them meaning. Isn't that contradictory?
Yes, a little. We shall see that models for teaching new concepts are sometimes contradictory and can lack plasticity. To clarify things, it's a good idea to start by remembering the objectives of a lesson. The lesson presents the knowledge that pupils need to retain and the skills that they need to learn. This phase is an opportunity for the teacher to validate the knowledge and skills and give them a special status: they become essential references for the class in the process of building a subject (in its transposition to school). The lesson also indicates to the pupil the formulations and statuses of the objects as they are considered by the disciplinary (school) community. It is therefore an ideal time to encourage pupils to become part of a subject culture with its own ways of saying, doing and thinking.
‘It doesn't help me, do I introduce the concepts before the lesson or not?’
2. Should the concepts be introduced before the lesson phase?
Rather than making a blanket decision about whether or not to use introductory situations in the classroom, I suggest you ask another question which I think is more relevant. Once we've written the lesson, we need to ask ourselves the following question: ‘How can we make this lesson as intelligible as possible?
"OK, thanks for the generalities, but I've got lessons to prepare. So how do I make the lesson intelligible?"
Let's look at two skills taught in secondary school:
a. Recognise two alternate-internal angles;
b. Produce a literal expression.
a. Recognising alternate-internal angles
The lesson corresponding to the skill ‘Recognise alternate-internal angles’ usually includes a vocabulary point on alternate-internal angles and the configuration of the two lines and the secant, as well as examples and perhaps counter-examples.
There are introductory situations. You could, for example, imagine a situation in which several geometric configurations are presented, each with two alternating-internal angles or two angles that are not alternating-internal. Below each of the configurations, it is indicated whether the angles are alternating-intercontiguous or not. The students are then asked to reconstruct the definition of two alternating-internal angles. This work of formulation and reformulation is relevant in itself, for the language exchanges - and hopefully the language progress - that it generates.
I used this introductory situation for several years in a row in 5e, but then I changed my mind when I realised that, for this visual recognition work, a dialogue moment with back and forth between the definition and the figure in the lesson achieved - it seems to me - the same results and was much less time-consuming. What's more, the pupils don't generally have any difficulty with the principle of alternating-internal angles and their recognition in elementary configurations. The difficulty arises when they have to identify alternating angles in complex shapes. It is therefore the training phase (guided and autonomous practice in the terms of explicit teaching), which seems primordial in this case for the success of this task.
b. Producing a literal expression
The situation is quite different for the skill ‘Produce a literal expression’. In general, this is the first paragraph of the lesson in the sequence on algebra. The lesson often starts with a vocabulary point on the notion of literal expression and variable. It then continues with examples of expressions produced representing relationships between geometric quantities and/or expressions derived from generalisations of patterns. Starting with this part of the lesson is not conducive to understanding the concept of ‘literal expression’, and for some pupils it may even be a break with maths: ‘Why are there some? What's the point? However, situations that introduce the production of literal expressions help to avoid this kind of question, because the correctly animated introductory situation has made the introduction of a letter and the expression intelligible, even natural. I suggest two at www.mathscours.com. The great classic in 5e: the bordered square; and another of the same kind in 4e: the banquet(https://www.mathscours.com/4chap7-calcul-litteral)
Behind the literal expression lie many new concepts for the pupils: the status of the letter, variable declaration, substitution of a variable by a number, expression structure, the role of the literal expression, equations of literal expressions, etc. The two situations mentioned above help the pupils to understand that the introduction of the letter and the expression makes it possible to move from the singular to the general, i.e. they can represent any number in a set. They can also see that two literal expressions can be different but equivalent, even if the pupils don't yet know how to transform one expression into another. In short, the introductory situation is essential if we want our pupils to be able to understand and therefore learn to manipulate and use literal expressions intelligently.
"But how do I know when I should propose an introductory situation to the pupils?
That's a matter for your own experience and teaching culture to guide you in your choices. In any case, whenever you write a lesson, I'd advise you to carefully determine the prerequisites and potential difficulties of the pupils, to carry out a didactic analysis in short. The accompanying documents for the subjects on the syllabus give recommendations that I think are relevant to these issues. In particular, they present introductory situations that are recognised by educational research as relevant (particularly the one on algebra).
‘Ok, ok, but how do you go about writing the lesson when you're just starting out?’
3. How do you prepare a lesson when you're just starting out?
When you prepared your sequence, you had to list the learning objectives: there are between 3 and 6 per sequence. You can find the ones that my colleagues and I listed for level 4, for example, by following this link: https: //www.mathscours.com/quatrieme.
A lesson in collège generally corresponds to a learning objective. It's customary for a lesson written during a session to be no longer than half a page in a large notebook at collège. The text of the lesson should be concise, but rigorously written in language appropriate to the level of the class. Pupils should be able to find their way around the lesson easily; they should be able to find the definition or property as well as the example or method they will use to solve the practice exercises.
It is sometimes advisable to build the lesson with the pupils, as this helps them to remember it. I don't know, but research hasn't proved it. What is certain is that saying and writing a lesson all by yourself on the blackboard in secondary school is not effective and does not encourage most pupils to take it on board. The lesson benefits from being informed by a dialogue between you and the class. The lesson prepared in advance by the teacher can be modified during the session according to the students' comments, but in my opinion it is essential that it is written down rigorously in advance, otherwise improvisation will inevitably produce errors, awkward wording or inconsistency in the articulation of concepts.
‘Yes, but it's difficult to get down to the level of the pupils, to write maths rigorously in a language suited to their class level...’.
Yes, but you have the pupils' textbooks. I recommend using two of them at the beginning, so that you can compare the wording, the examples chosen, etc. without getting lost. And then, check with the official syllabuses, the annual progression benchmarks and the end-of-cycle expectations that you're on the right track.
‘OK, but the pupils write slowly, so I prefer to give them a gap-fill text for the lesson, otherwise we'll spend the whole hour on it...’ Ideally, the pupils would benefit from having a gap-fill text.
Ideally, students would benefit from writing each assessment in full by hand. The researchers who look after our school have noted the inflation of handouts at school. From a very early age, pupils are given sticky notes and fill-in-the-blanks worksheets to fill in as lessons. The aim is to save time; it's true that getting a class to copy a text takes time. That said, the widespread use of this practice has probably made pupils slower to write than they used to be. It is therefore recommended that they write out the entire lesson.
‘It's still hard, we don't even have time to finish the syllabus...’.
Yes, it's true that the maths syllabus for cycle 4 and 2de is difficult (impossible?) to finish properly with 3h30 per week. It's probably less the case for the cycle 3 programme, where the passage of time is probably the main constraint for a collège maths teacher. It forces us to constantly arbitrate between different activities. That's why, even if we consider that taking the time to get students to write is essential, we often find ourselves handing out worksheets to be completed for fear that the students' mathematical activity time in class will be too brief. It's a question of finding the right balance.
Firstly, some of the lessons can be written out in full. For the others, as a compromise, you can set yourself a few limits that you don't want to cross. For example, it is preferable to leave it up to the students to :
write down all the definitions and properties (in full!). It's important that the pupils get used to their syntax, which is also part of the culture of the subject.
write down the formulae with their associated equality.
writing examples, particularly because of the modelling nature of their writing.
Furthermore, I'm not sure that there's any point in using cards with ‘little holes’, i.e. cards with a word to fill in here and there. The teacher's intention in drawing them up is generally to keep the pupils' attention while the lesson is being read and discussed, but is it effective? If we're in a hurry, it seems to me preferable to pre-write the headings, comments and instructions for the pupils and have them write the rest.
‘And in class: I write the lesson, I explain it, and then I do the practice and reinvestment exercises from the chapter?’
5. How can the lesson be put into practice in class?
Yes, but be careful: the lesson written during the session must correspond to the objective of the session. You don't write the whole lesson in one go. The lesson generally lasts no longer than 20-25 minutes in a session. It should be based on an introductory situation or an explanation, depending on the case. Particular attention should be paid to formulations and new words, with the class playing a game of formulation and reformulation. You don't introduce a new word without thinking about it. It's a good idea to work on its etymology, homonyms and synonyms and, above all, to get the pupils to use it and to give them plenty of time, in a dialogue lesson, to reformulate any clumsy statements they make.
Furthermore, in maths at collège, the lesson is often considered by weak pupils to be useless for revision. Assessments are made up of exercises, so for them, the important thing is to do the exercises again, or even to do the exercises corrected in class. The role of the lesson as a support for success in the exercises is important to explain. The lesson includes examples and methods. Here's a typical answer from a secondary school pupil who is weak in maths to the question ‘What's the point of the examples and methods written in the lesson? Nothing, it's already been corrected’. The key is to explain the central role of examples and methods. They are the ones that model the teacher's expectations for solving the exercises in the sequence. They should therefore be given priority when revising a lesson.
4. To sum up
- List the objectives of the lesson;
- Write a lesson paragraph corresponding to each objective, using two textbooks;
- This paragraph should be no longer than half a page in middle school and the lesson should not last more than 20-25 minutes in a session;
- Consider how to make this paragraph as intelligible as possible (introductory situation, explanation, etc.).
- Explain to the pupils the role of the lesson and its components (properties, examples and methods).
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