Lessons from lessons

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.