Lessons from lessons

01 "Do you plan ahead?"

Our CLR Visit to Japan to further explore mathematics education started at a school with which we have worked over the last five years or so: the Tsukuba University attached Elementary School. This weekend 14th/15th February they are hosting a "national lesson study event". Given the expertise in teaching mathematics at the school this has attracted some three and a half thousand teachers. Over the two days of the weekend these teachers will be able to carefully consider their teaching in considerable detail. Their deliberations will be informed by a form of lesson study that we could never aim to replicate in the U.K. or any of the European countries represented by the participants in our group: England, Scotland, Northern Ireland, Germany, Italy, Spain and Switzerland (an international school).

At the end of the first lesson in which a class of first grade pupils was considering ideas of subtraction, one of the visiting teachers from Japan asked the teacher who had just taught the lesson, "Do you plan ahead so that things conclude in the sixth grade?"

The simple answer is of course "yes" - but possibly more correctly the answer is, "no: we plan ahead so that we can build on everything we do in mathematics so that things conclude at the end of the school curriculum". Having been privileged enough to have seen many maths lessons in Japan, and "studied" their text books, in today's single lesson I could see many didactical moves that I have seen time and again in many lessons and which I see built on throughout the grades.

To provide an example. Today's Task:

"On Valentine's Day (today) I received five chocolates and 3 cookies. What is the difference?"

This is a "Google Translate" version of the task which was written in Japanese and which became formulated as 5 -3 as far as the mathematical discussion developed.

I believe that in this particular case, the context may have been more effective if the task had been formulated somewhat differently to this, but my purpose of using this example is to focus on how the teacher re- presented this on the board. See how the five chocolates are presented as five yellow blocks and the three cookies as three white blocks. They are aligned horizontally with three yellow blocks divetly above the three white blocks as the pupils putt it "holding hands". The "difference" between the chocolates and cookies as represented by the different coloured blocks turns out to be the two yellow blocks at the left-hand side of the illustration.

What struck me here was how this re-presentation is consistent with how the pupils will work mathematically throughout their mathematics lessons. In the very first stages of counting objects they will re-present them as blocks with one-to-one correspondence with the original objects whether they be ducks, flowers, books, or..... The blocks will be aligned horizontally as here directly (above the images of the objects being counted). The horizontal alignment is important/essential as the formality of the mathematics is developed towards 'working with number lines. Discrete objects that are countable are eventually abstracted more fully to give continuous number lines that are often used to provide positions and lengths associated with measurable quantities such as length and time.

Eventually double number lines will emerge and be used as pupils will explore relationships between quantities. Ideas of difference will build further to consider directed number , again using direction along the number line to provide insight into positive and negative numbers. Much later these two- dimensional 'vectors 'will be extended into three-dimensional space. I hope that this is giving some insight into how there is a connectedness and coherence across the curriculum to how our Japanese colleagues work with their pupils to explore mathematical ideas.

The re-presentation of a meaningful problem contained a whole hidden depth of thinking that I see throughout lessons in Japan. Pupils gain insight into mathematics in ways that are carefully thought through.

As teachers we have much to learn from expertise such as this. There is much in these lessons that requires careful interpretation to our untrained eyes.

02 Framing a task: focussing thinking

What follows is from just a small part of a lesson observed in Japan during an open-house lesson study event. This was with a grade 5 class in an elementary school - equivalent to Year 6 in the U. K.

The task: The scenario was about thinking about the discount of a product on a shopping trip. Two different discount coupons can be used. Coupon A provides a discount of 20%; coupon B provides a fixed reduction of 60 Yen. The teacher asked which gives better value of a product costing 300 Yen.

The double number line below soon emerged for coupon A, and was drawn on the board by one of the class.

That the two coupons led to the same final price was soon determined by the pupils. But the double number line became an important focus of the lesson. The teacher asked the question "which part of the number line is the discount?"

This in the post-lesson discussion he argued was very much significant. He wished to focus thinking not on the solution, the better value, but on the fact that with coupon A the discount varies in relation to the cost of the product. This understanding, the teacher argued,is an important life-skill for pupils, even at this early age, to understand.

It may be surprising for U.K. readers to notice that the pupils readily worked with understanding that a reduction of 20 % meant that is equivalent to a reduction of 0.2 and that the price is therefore 0-8 of the original price. These values can be seen in the structure of the double number line diagram and pupils' attention was drawn by the teacher to the representation of the discount.

The teacher raised the question of whether or not the discount is always the same when you use coupon A and coupon B.

After a while the teacher drew on the work of a girl who had worked through the problem with the cost of the product being 400 yen, or a cost of 200 yen. That is with a cost of more or less than 300 yen,

As another pupil pointed out "it [the discount] is better if you buy more expensive stuff." Another pupil suggested working with a product costy 100 million yen. I (I like this idea of testing out your thinking with extreme Values).

As the teacher pointed out in the post-lesson discussion he was keen, when talking about the situation, to ensure that at the forefront of pupils' thinking was the relationship between the numbers involved and what they mean in the reality. When writing down 60, for example, he wrote this as 60 yen discount, and made sure that pupils were aware of which part of the double number line diagram represented this discount. He was keen to use the double number line in ways that allowed him to understand pupils' thinking,

In the post-lesson discussion some questions were raised about how the situation might be explored further. For example, if you could use both coupons, should you apply coupon A before B or vice versa? What could you buy with zero cost?

Finally, the teacher considered, how he had framed the task at the start of the lesson. By focussing on the discount rather than the price, and which coupon gave the cheaper result, he directed pupils to think more carefully on how the structure of the mathematical model (as represented by the double number line) and thereality are interrelated. This contrasts with many situations in lessons I observe that all too readily focus on "answer getting" rather than understanding mathematical structure.

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