Copyright © University of Cambridge. All rights reserved.

'What's to Notice?' printed from

Show menu

The theme for this month is about 'actions on objects'. All of the activities we have devised involve some kind of noticing, and several of them involve new interactivities which we think promote lots of opportunities to notice.  John Mason, our guest editor this month, has written a lot about the 'discipline of noticing' and how important it is to the understanding of the deeper structures of mathematics. So what does this mean and how can we help our students get better at it - how can we 'educate awareness'?

Rather than think abstractly about this, let's consider what it looks like in some of this month's offerings.

The Add and Take-away Path  can be construed as an exercise in adding up and taking away.  One might also think of it as a very early introduction to vectors. But the task has been devised so that the problem solver is almost forced to notice something. Perhaps two totals being the same might be a coincidence, but if the children are encouraged to try some others they notice that so long as they fix the start and end of a path, the total will always be the same. To adults that may be obvious but to small children it is far from being so and they can be quite surprised.  The teacher's role here is in offering a structured task which has lots of flexibility so that the children can choose any routes they like. And then the big question - what do they notice and why do they think this happened? 
I really like Area and Perimeter. Children so often get these muddled up, usually because they meet them at different times in their school career,  so setting a task where they intentionally have to attend to both simultaneously is a good way of confronting the confusion. The usual questions students are asked are about knowing a formula and calculating the value of either area or perimeter but here the questions are of a different nature. They force the child to look at the relationship between the shape, its area and its perimeter. Long thin shapes may have a small area and a large perimeter for example; shapes with bits cut out may have a very small area and a very big perimeter -  so the structure and property of the shape itself become the important ideas. 
Missing Multipliers and the variants can be used as consolidation games.  In each case the students have to notice the commonalities, framed in the multiplication operation, so that they can deduce what whole number factors go where. Similar noticings have to take place in Finding Factors. Because many of the questions differ only slightly, asking which piece of the answer gives the most information demands noticing, which leads to a clearer understanding of the structure of the question. 

Whilst I was focused on this, two activities from previous months came to mind. 

Tilted squares is a task we published last month. The team have been using it a lot with students recently (you can see videos of this here), and have refined their thinking about how it might be used, and the sorts of questions to ask. The students are invited to perform a certain action (tilting a square) and asked what they notice. The teacher's role is in helping them to attend to similarities and differences - and in this case the way that these are recorded by the teacher on the board helps the students to notice in a way that they may not have done if they had recorded in other ways.
One of my favourite tools on the NRICH site is Shuffles. You'll need to read the instructions first, but as an interactivity to model actions, and successive actions, on objects, it's an engaging option. Lots of deciding what to attend to and how to record it. 
In all of these tasks  - and the others this month - 'actions on objects' change something. Learners get caught up in the detective work of trying to fathom what it is that makes a difference, and sometimes whether the order of actions matters. If you're short of something to do and want to refine your own noticing skill, try 18-hole Light Golf .  It could become addictive.