A lesson in mathematics lessons - update from Project IMPULS

Thanks to Belle Cottingham for this guest blog post on her recent trip to Japan. Belle is a participant of Project IMPULS 2017 and one of the authors of Rising Stars Mathematics.

I have just come back from Japan having been lucky enough to be selected to take part in Project IMPULS, a two week immersion programme designed to give an insight into the learning practices adopted in Japan and methods utilised to enhance learning experiences for all participants.

What is Project IMPULS?

Project IMPULS is an international project based in the Mathematics Education Department of Tokyo Gakugei University, Tokyo. Participants in the Project are chosen from a global pool of applicants, with a total of 30 attending in 2017 (of which 7 were from the UK).

The aim of the project is two-fold. Firstly, international visitors are given unique and unrivalled access to professors, teachers, students and schools, in order that they gain a full understanding of the Japanese curriculum, the problem solving approach to learning and of course lesson study. Secondly, to add value and assist members of the University who prepare significant and influential research examining the mechanisms involved in Japanese lesson study to maximize its impact on the schools in Japan and ensure that the process is constantly evolving and improving.

What is Lesson Study?

Whilst it is difficult to summarise a teaching practice that has evolved over 120 years, Lesson Study is commonplace across Japanese schools (so much so that Japanese educators often make reference to it being “like the air”; felt everywhere and influencing everything). Lesson Studies take place once a month for every class (in every school, the classes rotate so that each class and each teacher has a turn). 

The Lesson Study process includes:

  1. Goal Setting. Teachers and their colleagues develop long term goals for their students, drawing from historic plans and feedback and tailoring the same as appropriate for the class.

  2. Lesson Planning. This is very carefully detailed process that has been allowed to evolve over time, and identifies the key concepts in play and produces a broad script to allow lessons to maintain a structure.

  3. Research Lesson. This is ”structured problem solving” or teaching mathematics through Problem Solving. Other teachers, and very respected educators observe the session (in the school lesson study, there are around 20 – 30 observers, whereas in a district lesson study, there could be around 100 observers from the whole district).

  4. Post Lesson Discussion. The lesson is discussed in detail between teachers and observers and feedback is provided on ways to improve, enhance or approach the lessons in the future.The teacher, observers and students are all focused and driven towards continual improvement of themselves and the system and so actively appear to participate (and take on feedback) to the greater good.

Less is sometimes more

One thing that all the lessons I saw had in common was that they were built around one single task which was set in a context that the children were familiar with.

All the tasks were focused on developing the mathematical thinking in children rather than teaching them to calculate. I didn’t see any worksheets with lists of questions. The focus of the lesson was not about finding the answer, instead the process was centered on how to get to a position where you could answer the question, and where each student thoroughly understood the concepts involved.

I remember looking at the teaching plan on my first lesson and thinking ‘How is the teacher going to make this lesson last 45 minutes?’  Little did I know that 45 minutes was not going to be enough. Rather than covering a lot of surface, the tasks went deep into the mathematical thinking underlying the concept.

For example, I saw a Year 6 lesson which involved dividing fractions. The teacher merely explained what was to be achieved and then let the children work independently. The children that were unsure went to speak to the teacher quietly, who gently provided guidance and assistance (in the form of further questions to prompt or guide, but never giving an answer) to help structure thought processes and where necessary to prompt further dialogue. What amazed me was that children provided not one, but at least three different ways on how to divide fractions and attempted to generalise the rule of dividing fractions. All by themselves! All in 45 minutes!

After my first lesson, I realised that 45 minutes was definitely not enough time!! Usually there were two lessons allocated to each task, to allow for follow up and conclusion. In the first one a problem was set and children decided on the method they would use. On the second lesson, children explored the mathematics they would use. Each of the possible methods that children provided was in the teaching plan that the teacher had developed. How to deal with the unexcepted answers was what made each lesson different and in the post lesson discussion, ideas and suggestions were given on how to improve each lesson. There were lots of questions (“why”, “what”, “how”, “when”) asked from the teacher and the children before, during and after the task.

‘Why do children need to learn to divide fractions? What context should be used? How can the children relate to that?’

‘Why does the first fraction stay the same but the second one changes to upside down? Why do you divide the first fraction by the second one? Can you do it in a different way? Why?’ Slowly but surely, with each question, came a greater understanding, and the knowledge gained seemed that much greater because the children took ownership of it, because the ideas were theirs!!!  

Since coming back to UK, I was curious to compare and contrast experiences and so ran a similar task with a top set of Year 10’s GCSE students, and also a group of Year 12 students doing A level maths. I knew that they knew how to divide fractions, but did they know why it happened that way?

‘Why when dividing a fraction, do you multiply the first fraction with the reciprocal of the second one?” I asked.
The answers I received were: ‘Because my teacher said so……. That’s the rule……. That’s what we learned….’

But does it matter that our top students don’t know why they are doing calculations, or what these calculations actually mean in practice?  Is it important to have a deep and strong foundation of the mathematical concepts rather than memorised steps? Does our curriculum and teaching encourage our children to think independently, to gain a greater understand and depth to our knowledge? Does our curriculum provide time for multiple answers to each question? How are we supporting our teachers in using tasks that develop mathematical thinking? How important is the foundation we give our children in Primary, in their later learning development?

In a world where calculators and technology can do pretty much anything, how important it is to give our children, the future of our society, the power to think, rationalize, problem solve? 

This is all great but does it actually work?

Frankly, Japanese results speak for themselves. They are consistently ranked in the top 5 countries according to TIMSS results. But what is more impressive than the actual results, is that when you analyse the TIMMS results, it shows that Japanese students have an incredible aptitude for solving questions they have never seen before. As can be seen from the illustration below, though only 54% of the material from the test had actually been taught in schools (different countries have different curriculums), children achieved an average of 69%, so basically they could solve questions they hadn’t even been taught because of their mathematical thinking/ reasoning. Is this an important skill in life? Well, a problem wouldn’t be a problem if we knew the answer beforehand!


The Japan trip has made me reflect on my teaching, writing and all the choices I make in my work. I know that change takes time and it is not easy to implement. However, I feel that if we all reflect on why we set the tasks we set and work together to improve, we can make that change happen. Learning should not be a tick box exercise, and discussions around learning should not obsess over self-interest or ego or be a blaming game. It should be a place from where improvement can happen.

My biggest fear having been lucky to get this extraordinary experience is that I come back and nothing changes. That I am unable to effect subtle perception shifts, and engage people and help them see what I see. I am not for a second suggesting that the UK should morph into Japan (after all there is a lot to admire about the UK, even if recently this does not always feel so), or that I can flick a switch and people will suddenly become more respectful, attentive and polite, but this does not mean we shouldn’t strive to evolve and improve, not just as individuals but as a whole. For me this constant movement forwards, this relentless need to improve (even though you could argue that it already works in Japan), so evident in Japan, is what we must reintroduce to the UK and be proud of. Why should striving for excellence by considered wrong?

As teachers, authors and educators we are uniquely placed to effect change in the world. Not by some insidious nature, not because of political motivations or self-interest, but because by helping learners help themselves, empowering each and even one of them and letting them believe that however insignificant their input seems, when added up collectively it becomes a powerful tool.
I am lucky to work in the profession I do, and even if I can only help shape one mind, and help that mind to open up the endless opportunities inside themselves, then surely this alone is worth its weight in gold. I will of course aim a little higher than just one!!!

A Brief Note

I wouldn’t feel comfortable writing about Japan without mentioning the people involved, those I encountered in Tokyo and of course the students. Quite simply, in all my travels, I have never been made to feel more welcome. It is certainly unusual to see politeness and hospitality on the level demonstrated throughout my stay.

Japan is an incredibly beautiful and mystical place, and the humbleness and respect that seems to coarse through everything and everyone is incredibly inspiring. The patient approach to life, deep sense of self and the collective trust in each other was noticeable everywhere, and incredibly endearing.

To find more about my journey in Japan visit my blog

To participate in project Impuls visit http://lessonstudyuk.wikispaces.com/

Belle is one of the authors of Rising Stars Mathematics. Download your free sample unit to try out today!


mathematics, maths

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