Fractions, a pinch of arithmetic, two tears of literal arithmetic, a slice of functions, a spoonful of proportionality; cut all this up into curricula, and arrange it in a spiral; sprinkle in space and online arithmetic throughout the year... And you get a well-cooked progression. Oh, I skipped geometry, it's disgusting...
"I'm a teacher and I've known my class levels since yesterday and I start lessons tomorrow. I've got to work out the annual progressions for the year as well as preparing my first sequence... What the f**!
At the start of your career, you generally only find out which classes you'll be assigned at the last minute, a few days - a few hours? - before the start of the new school year. However, a year of teaching is organised, and it's essential to establish before the start of the year the order in which you're going to teach the concepts on the syllabus. In the jargon of the profession, the chronological list of sequences to be taught for a given class level is called a ‘progression’.
I'd like to suggest two points for discussion to follow or construct mathematics annual progressions:
1. What progression should you follow when you're starting out?
2. How do you work out a progression when you know your classes and colleagues in advance?
1. What progression should you follow when you're just starting out?
It's no mean feat to prepare a progression when you're just starting out, but very often it's already in place. In schools, middle and high school maths teams generally follow common progressions that they have drawn up themselves. A word of advice: in 99% of cases, it's better to follow your team's progress than your own personal progress. Even if the spiral nature of your colleagues' progression makes your head spin, or the long tunnels promised by a progression comprising just 9 sequences triggers your claustrophobia, it's still vital to share the same progression. Otherwise, the following year, you and your colleagues will be stuck with classes that don't have the same knowledge in mind. And that's a real pain...
‘Yes, but it creates diversity, and... ’
No, it's just hard to manage. I recommend that you always ask yourself if the bright idea that has you following a different progression from your team is worth it.
‘OK, OK, calm down, I get it, colleagues will get upset if I do that.’
For example, by the end of the first term, the maths team should have taught a certain list of sequences, whatever order they choose. So you get the idea.
"OK, but my colleagues tell me that they don't follow a common progression.
That's more of a problem. It'll be up to you to get the ball rolling at the end of the year, but for the moment, if you're just starting out, you're running out of time. I'd advise you to use a progression from one of your colleagues, and if that's not so easy, use one from a textbook or the internet (being careful about the quality of what you find on the internet, of course). You can find the progressions that my colleagues and I use by following the following link:
Don‘t hesitate to copy them.
‘Otherwise, I make my own progressions...’
Well, if it's your first or second year of teaching, it's a big risk. You don't have a precise knowledge of the content to be taught and you'll fall into a lot of pitfalls relating to the order of the concepts to be covered. I strongly recommend that you choose one developed by experienced teachers for your early career. You'll gradually get the hang of it.
2. What are the steps involved in building a progression (when you know your classes and colleagues in advance!)?
Successful progression involves structuring the content of the curriculum in a logical, balanced and rhythmic way over the school year, while taking into account the actual knowledge that the pupils have already acquired. Here are the 4 steps to follow to be - in my opinion - effective:
- Step 1: Build the progression as a team
- Step 2: List the learning objectives for each chapter
- Step 3: Draw up a list of sequences and decide on the learning objectives for the class level in question
- Step 4: Assign a duration to the sequences
Step 1: Build the progression as a team
A progression that achieves its objectives is the result of collective work by the maths team. In general, the teams meet in June, at the teaching council, and review their progress for the year.
‘I don't see the point, you can just take a progression from a textbook, they're well done...’.
Developing progressions is certainly based on the syllabus, but also on the pupils you've had in front of you for a year. By modifying the progressions from one year to the next, the maths team are able to rebalance the broad areas based on the difficulties experienced by the pupils they have diagnosed. For example, if we feel that the 4th grade chapter on spatial geometry is difficult to teach, it may be because it requires the 4th grade colleagues to review a lot of previous concepts with their students. In that case, it might be appropriate to adapt the sequence for 5th and perhaps 6th grade. Why not put this sequence at the very beginning of the year rather than at the end? Why not add one week of the year to spatial geometry and divide this chapter into two sequences, teaching them several months apart to encourage memorisation? What's more, pupils in 3e will probably not be up to speed with spatial geometry, so the progression in 3e will also have to be adapted.
‘But if you add a week to one chapter, you take it away from another...’.
Yes, that's why developing progression is a balancing act. And stable teams will tell you: if a progression is modified for the pupils of year n+1, the pupils of year n+1 will not have the same knowledge as the pupils of year n. That's the aim, but it's not enough. That's the aim, but it's also a risk: if you reinforce spatial geometry, you may be taking away some literal arithmetic. That's why it's essential to work out progressions collectively; these are not decisions to be taken lightly.
"OK, we'll play it collectively, but is there a method for working out progressions?
This brings us to the second step.
Step 2: List the learning objectives by chapter
By learning objective, I mean ‘knowing how to find the side length of a right-angled triangle using the Pythagorean theorem’ or ‘knowing how to add and subtract two fractions where the denominator of one is not a multiple of the denominator of the other’... And by chapter, I mean ‘major theme’. For example: trigonometry, functions, literal arithmetic, plane geometry, space geometry, etc. A chapter can form a teaching sequence (the reference unit for progression) or be divided into several sequences. Finally, don't get hung up on the distinction I'm proposing between chapter and sequence, it's probably very personal. And I'm not sure that the distinction has stabilised.
It has been said that a progression is a chronologically organised list of sequences. True, but the point is to go into a bit of detail, which will complicate the work. To be able to draw up a progression, you need to know the learning objectives behind the titles of the sequences. At collège, there are generally 4 or 5 per sequence. It doesn't really make sense to draw up a progression by throwing sequence names and a duration on a sheet of paper. In fact, teaching the factoring technique in the ‘Literal arithmetic’ sequence in Year 5 - which is not generally the case - makes the sequence much more cumbersome. In this case, it would be beneficial to split it in two. In short, it's essential to list the learning objectives before starting to discuss the progressions, otherwise there's a big risk that the development work will be superficial.
‘When it's your first or second year of teaching, detailing all the objectives for each sequence is a mammoth task, and I'm already drowning in it...’ In fact, you'll do it when you've finished your first year of teaching.
In fact, you'll do it as you go along for the first few years, sequence by sequence. But doing this work with your colleagues is much more fun and avoids a lot of blind spots. What's more, everyone will have learned something in the end ! It's often during these discussions of detail that we discover that a concept is not taught in the same way in the same school, which can have repercussions on pupils' learning. There are several techniques for simplifying fractions, for example. Do you know which one is taught by your colleagues depending on the class level? It's best to get your facts straight. But don't panic, teams don't start work from nowhere. Lists of learning objectives by sequence already exist in many textbooks, where the chapters are structured according to learning objectives, but also in many progressions such as these: https: //www.mathscours.com/espace-enseignants. There are also official documents, such as the annual progression benchmarks, which can help to break down progressions by cycle: https: //eduscol.education.fr/137/reperes-annuels-de-progression-et-attendus-de-fin-d-annee-du-cp-la-3e
Step 3: Draw up a list of sequences and decide on the learning objectives for the class level in question
Let's assume that you now have a list of your chapters and the associated learning objectives. Now, with the help of the maths syllabus and other accompanying documents, and also the collective experience of the maths team, you can cut up the sequences neatly and think about where you're going to place them in the year. In the jargon, we tend to talk about spiral progression or curricular progression, when the chapters are systematically divided into several sequences. There was a time, before 2008 I believe, when annual progressions only included around 9 chapters. Today, it's more like 15. Some colleagues even opt for an extreme division, to encourage long-term memorisation. I'd still advise you to start simply. If there are too many scissors cuts in your chapters, you risk losing the thread, not to mention your students.
You should also try to vary the pleasures. I'd advise you to avoid offering two sequences in the same subject area in a row. Broadly speaking, you should alternate between numerical and geometrical elements.
"I'm going to use one chapter as a common thread throughout the year. I'll sprinkle it into all the other sequences...’.
Students need a structured lesson to learn. It's not just a question of understanding when the teacher corrects the exercise, but of learning (soon a post on the distinction between understanding and learning). A sequence is made up of introductory situations, a lesson, practice exercises and reinvestment exercises. If the skills in a chapter are never institutionalised, it will be very difficult for part of the class. Of course, there can be exceptions, but these should not be abused;it is possible to deal with a concept throughout the year without a lesson on it, but not an entire chapter. For example, you might think that you're going to use quick activities throughout the year to focus on writing calculations online in 6th and 5th grade to prepare for literal arithmetic. But you need to ask yourself whether there is a teaching reason for this choice. If there isn't, it's probably just a way of speeding up and sweeping a concept under the carpet. It should therefore be avoided. This is often the case with spatial geometry, which is sprinkled throughout the exercises (‘Oh yes, in exercise 15 of chapter 3, there's a volume calculation!’), and I can confirm from experience that there won't be much left in the pupils' heads at the end of the year.
Step 4: Allocate a duration to the sequences
Taking into account the different teaching objectives is essential. I'd say that teaching subtraction of relative numbers in 5th grade takes 3 full sessions. But you can't devote that time to all the objectives, otherwise you'd need twice as many hours of maths - Yes, please, please... Experience will tell you how many hours to allocate to each objective. So, if you're just starting out, ask your colleagues. Don't forget, either, that in a sequence, there's the assessment and a whole host of other events: the 6e2 going to the cinema on Monday, for example... So you need to leave yourself a margin.
Deciding how long sequences should last is a delicate balancing act, because in cycle 4 or 2de, you're running out of time all year round.
"What if I'm behind schedule?
You can take out the ‘if’, you're always behind, but the delay has to be reasonable. Arriving at the 4th sequence of the year out of 15 at the end of the year is a problem that needs to be tackled head on. There will be trade-offs to be made. Four weeks in a sequence is the upper limit, it's already too much.
"But they have to understand...
At the beginning, and even afterwards, it can be difficult to move on to the next stage when part of the class fails to achieve a learning objective. The idea of persevering is a noble one, but in practice it's untenable. When the learning objectives are not met, you have to know when to let things rest and pick them up again later. It's best to avoid getting bogged down, which is demoralising for you and your students. If a concept is having trouble getting through, putting it back on the syllabus for an assessment 3 months later, for example, and taking just one session to practise it with the students, can have good results.
In any case, I wish you good progress!
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