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Lesson Structure and Generalisation in a Shanghai Maths Classroom

 

 

A blog written by Mastery Specialist – Emma Brooksbank

 

 

Throughout our visit we have observed many lessons in both primary and secondary, and despite the topics being very different, the structure of the lessons has been the same. I am going to focus mainly on the secondary lessons here, but this same structure is applied in secondary school.

Review of Previous Learning

The lessons begin with a review of previous learning. This is sometimes in the form of skill questions, they sometimes recite previous knowledge or answer problems that require previous knowledge. They then introduce the new knowledge through a problem that needs solving. This can be a real life context, or a more complex problem. This is then used to explore the concept, sometimes moving away from that particular problem to clearly explain a concept. After this problem is solved they generalise what they have learned. This generalisation leads into the variation.

For example in our second school we watched a lesson on percentages. They had previously learned about how to calculate the percentage of an amount and they completed some problems using this knowledge. They then generalised this method and this was written on the board. The problem for the next concept posed involved finding one value as a percentage of another. I feel that in my lessons, I previously would have taught different aspects of percentages as separate methods and in different lessons. In this lesson they used the knowledge that they already had, referring back to the generalisation from last lesson;

Original amount x % = % of the original

Students then had the fluency to use this generalisation to figure out how to use it to find what percentage one amount was of another.

% = % of the original ÷ Original amount

This ability is only basic number fluency; the relationship between division and multiplication. Using this generalisation they could quickly work out how to solve other problems. As a result there is less content to actually learn, just the practice and exposure of using these generalisations in many different ways to solve problems.

The generalisation stays on the board for the rest of the lesson and is consistently referred to. At this point it is really clear that the variations staff have picked are really thought about. These variations are not always a change of number, or a slight change, they are a variation on the concept; how else can this generalisation be used. In this lesson the following are the questions used;

Problem 2.

A business in the double 11 promotion, the original price of another laptop 9000 yuan, now 7200 yuan, what’s the discount?

(This was done altogether, the teacher then gave them time to answer 3 variation questions themselves before discussion)

Exercise 1.

At the beginning of the new semester, the bookstore will give the students a discount. With the student card, you can enjoy 20% discount. Xiao Ming bought a set of books for 48 yuan after the discount and asked for the original price.

Exercise 2.

The original price of a set of sportswear is 380 yuan. If the price is reduced by 152 yuan, what percentage was the discount? If the price is reduced to 152 yuan, what percentage was the discount?

A lot of discussion then followed these examples, but the students were always coming back to;

Exercise 3.

If the price of a product is 3400 yuan after it is 15% off the price, how much cheaper is the price than the price?

A lot of discussion then followed these examples, but the students were always directed back to the generalisation to help them with the problem. For example in the second part of exercise 2, they don’t know what the reduced price was to input straight into the generalisation, so their attention was drawn to how they would find it out. Also there wasn’t a calculator in sight, and this wasn’t because they are human calculators, it was because the teachers had picked numbers such that the numeracy wasn’t the focus of the lesson and wouldn’t distract from the understanding of the key concept.

 Variation Doesn’t Slow the Lesson Down!

Another really key thing to note is the variation doesn’t slow the lesson down. The variation of the problem is used, one that can still be solved using the generalisation, rather than the variation of the numbers. Because of this, there is a real pace to the lessons, and it moves away from looking at a particular method for a particular type of question and allows students to build those connections that might not have been made had each type of question been taught separately. Students see a concept forwards, backwards, and with extra bits added into the mix.

The use of generalisation is also used in primary. They see the idea that a + b = b + a. Even though there is no expectation that they can manipulate algebraic expressions, they are still expected to understand from the generalisation the commutative relationship. This then means when they get to secondary school, they are used to seeing generalisation in this form.

The structure in every lesson is clear and clearly based on the use of generalisation;

– Review of previous knowledge

– Problem to help discover, explore and understand new knowledge

– Generalisation of new knowledge

– Variation of new knowledge

Round Up

Obviously this structure would need to be carefully applied and introduced slowly with our students back home, to help guide their thinking. However, I will definitely be using the generalisations, whether that be algebraically or using words, and getting students to manipulate them in my lessons, with a view to moving towards this lesson structure.