### by Livia Mitson (April 2020)

### “Build a better mousetrap, and the world will beat a path to your door”

The problem with being a teacher and being familiar with the material that you teach, is that you can’t see how it looks to the learner. To you it is easy and obvious. To the learners, it is not, and they cannot understand how you do the maths so easily (they think you must be an incredible mathematician). To you, it is simple, and you cannot understand why they don’t get it.

This issue was thrown into sharp relief for me when our year 7s began to learn algebra for the first time, and did so online (during the coronavirus school closures). Everyone, teachers and pupils, were trying to get used to a new platform. Many people were struggling with massive changes to their lives.

We teach our year 7s in mixed ability groups. This has both good sides and bad sides. One of the great things is that we are able to create a classroom culture where students are able to show enthusiasm and creativity in maths. We often focus on showing many different ways of representing a problem, and spend time discussing which method is best, and looking for possible problems with methods.

We also try to make links between areas of maths. The students often find this relatively easy when these are familiar concepts that we are extending – fractions, decimals and percentages say – but when we come to topics where we are expanding the range of concepts that they are familiar with, they find it harder to make connections.

This is understandable. Here, we are working on the edges of their knowledge, and the connections and links are much more sparse compared to the knowledge that is better embedded, where they have had the time and the teaching to specifically make those links in many, many directions.

This is really highlighted by having to work online. On many online platforms, the jump between assignments/homeworks or even just between questions is such that if students are working at the edge of their knowledge, they quickly are unable to follow what is going on.

They need smaller steps.

In the physical classroom, these smaller steps can be provided by the children themselves having discussions about particular prompts or starting points; they can be provided by the teacher noticing that several children need additional explanation and gathering them together as a group. They can be provided by the teacher giving several methods (including some wrong ones) and asking the class to discuss as a group which methods give a correct answer, and to compare the methods.

In the online classroom, this is harder. If you are teaching live lessons using bespoke software, you can still have class discussions, and paired discussions. So you can put in smaller steps. But it still doesn’t have the excitement and energy that you can get in a classroom when the whole class are on a maths hunt to solve a puzzle.

If you are, as we currently are, setting tasks on an online platform, then this becomes much more of a problem. The basic concept for the lesson shown in the picture is that a’s, b’s and c’s are different and you can only add the same letters. This raises the obvious question of why are they different? What is it about algebra that means I can’t add different letters?

Ideally I’d like to teach this in a way that unfolds very clearly from maths the students already know to being able to simplify 18a + 12b – 6a – 4b + 5c – 2c. In the physical classroom I have several number investigations that I use and then towards the end of the investigation we generalise with letters… but that takes a good few lessons and a lot of repetition in lots of different contexts.

If anybody has a better mousetrap, suggestions to livia@cambridgemathshub.org (or as comments below this blog).

The world will beat a path to your door!

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