Only awareness is educable. Caleb Gattengo.
Have you ever found an amazing task for your pupils and really looked forward to teaching it – they will love doing this, we tell ourselves. You teach the lesson and you were right, your charges loved it. The frisson of excitement in the air says job well done. Never mind, those nagging doubts, the image of Suzy looking bewildered, those groups who were going through the motions and fulfilling the task while discussing Fortnite, and that the next lesson felt like wading through treacle. So we push those doubts from our heads, dispel those concerns – it was Suzy after all, and I’m sure they learned something and they had fun – look my pupil voice monitoring came back saying they loved maths.
I look back on some of my whoopee-doo lessons now and reflect, well what exactly did they learn? I remember investigating the chessboard challenge with Year Four (how many squares are there on a chessboard), dreaming that they would learn about relationships between square numbers. Molly did and she worked out some rules and relationships and I taught her about the square root. The others did drawing on tracing paper. Not my finest hour (or two).
Of course, I’m still learning to craft lessons, both alongside my colleagues and teachers, writing materials and, sometimes, just sometimes I am lucky enough to teach a class myself. As I read more and learn more about pedagogy and theory, my tasks take on subtle shifts in emphasis. Crafting lessons gives me great joy especially when I can see the snowball effect a few days or weeks later from support. Success breeds success and enjoyment of mathematics for teachers and children.
I know I only have to think about maths and my teachers have other subject plates to spin but, the fact is, many of the teachers I work with would and do find planning and teaching much more rewarding when it is guided by their learners and with the confidence of stronger subject knowledge – and by that I mean pedagogical and conceptual subject knowledge as well as what we were taught at school.
When presented with planning, either the teacher’s own or pre-populated, I always ask – What is it that you want the children to learn? At first, the response is often to describe what children will do and produce. In most cases the object of learning is in the planning – but even so the teacher has not quite identified what the object of learning is. Furthermore, it is often conflated with the context of learning. Once this is established we can move onto the critical aspects of that learning i.e. what it is important that the pupils notice about the object of learning, what it is and what it isn’t, any difficult points (misconceptions or possible prior learning weaknesses) and how it fits with other areas taught or to be taught.
For example recently some of our very kind schools agreed to run pilot diagnostic assessments for us in Year One. These were children from great schools taught by great teachers. Here is one the questions pupils were asked.
58% of the children who took part answered C. 37% of pupils answered A. It’s likely that by the end of Year One, children will have been exposed to ‘square’ since before they were at school. I find it significant however that over a third of well taught Year One pupils circled A. Discounting those behaviours which some children may have approached the assessments (answering randomly perhaps) the question aims to test the difficult points about ‘square’.
I’d love to have been there while these children were reasoning (or possibly not) which shape matched the word square, unfortunately I couldn’t so what follows are conjectures based on my experiences as a teacher.
Let’s keep this precise. If the object of learning is square, what have the 58% of children who were successful noticing about shape C? Perhaps…
- 4 square corners
- Equal length sides
- 4 straight lines
- 2 sets of parallel lines
- Squares can be orientated in different ways
Other critical aspects I’d want to tackle would be squares can be different sizes, have different surface features and can be faces on 3d shapes.
Granted that they will not use some of these words to describe a square. They notice that B and D cannot be squares because they do not have 4 sides, they discount A because of the longer length of two sides than the others.
Those that answered A are likely to have done so due to the orientation of C. Some may believe that C is an entirely different shape – they may not have seen a square at all but ticked A because it was the closest to their experience of seeing a square orientated flat on a side. C may have been labelled as ‘diamond’ by children.
I remember a similar experience with Isabella, a Year Five child in my class, who on being presented with a triangle on its apex, couldn’t name it, in fact she became quite belligerent that it wasn’t a triangle. Obviously I couldn’t go on to teach her about scalene, isosceles etc. finding angles in a triangle or finding the area of a triangle until she learned what a triangle was. Again, it’s not that she had never been shown triangles in different orientations, she had just discounted them in her mind as triangles. This showed me that even when we think we have brought a critical aspect of learning to the fore, we can’t assume pupils have noticed them. As a teacher I needed to find a way of allowing Isabella to discern the aspects that made the shape a triangle regardless of orientation. So I asked Isabella to draw me a triangle and another one and another one. I could see that all of the other critical aspects were intact. I span her whiteboard 180° and asked here to tell me what she had drawn. Isabella remained uncertain, so I showed her some not triangles and some triangles orientated differently. She came up with the definition for a triangle in her words. I asked her to tell me about herself. She described herself as having brown curly hair, dressed in school uniform, liked a laugh. She was Isabella. So I asked her what would have changed in her list of Isabellaness if I picked her up and placed her upside down. Thankfully she laughed and said she’d still be Isabella and made the link to the re-orientated triangles. She noticed that the classifying features of a shape/object did not change when re-orientated. Upside down triangles were no longer a problem for Isabella.
What could we show our Year One children that would help them attend to the critical aspects of square? Pointing to squares, colouring them in and finding them in shape hunts have their place and may be fun but we can’t be sure that they have discerned fundamental features. Showing contrast i.e. what isn’t vs is a square is often a helpful first step.
If we show them lots of squares only and ask them what a square is, children have nothing to compare square with and may not fully discern the relevant features. An activity with contrast in my experience is more likely to allow children to begin generalise about square. We also need to show them non-standard examples otherwise we risk limiting children to a restricted view which may cause unintended over generalisations later – as with Isabella.
We can further refine this in a further task by asking children to construct a square with an elastic loop, lollysticks or geostrips, listening to how they reason about how a square is made and then asking them to transform the square into an oblong. What changed? This question allows children to focus on the change in the transformation made between square and oblong on the length of sides. What stayed the same? And this question helps children to focus on the features that are common to all rectangles.
Odd one out or always, sometimes, never questions, well beloved by many teachers can also be harnessed as a task to further deepen understanding of square.
What is it that children are attending to when they make their choice and justify it? We can focus in on a specific feature of the shape say…orientation by asking;
Why might we say the green shape is the odd one out?
How could we make it more like the other shapes?
Finally if we can find a task where children assimilate all this learning then this allows children to develop connections and visualise the object of learning amongst other conditions.
Perhaps we could provide children with a range of building bricks and ask them to sort them according to whether they can see squares on the bricks or not. This helps children to connect squares with 2d shapes as faces on 3d shapes.
None of this is rocket science but keeping task design firmly focused on what needs to be learned and thinking about ways we can draw children’s attention to these through seeing and doing has allowed the children I teach to progress further with a greater understanding. Earlier in my career I definitely assumed too much, as a teacher I am an expert learner about squares. Assuming that if a child can point to the square and say square they know square is not necessarily true. After all Isabella got to Year 5 not really understanding triangle. This lead to over ambitious expectations of my learners being able to ‘see’ relationships between square numbers when they didn’t truly understand square numbers and were novices to the learning. Chessboard challenge is a great investigation, I would do it again, but not until my children were reasonably expert and not in Year 4. Their inexperience of square numbers and spotting number sequences and relationships meant that they could not possibly deal with the distracting bells, whistles and fireworks that the task presented. Prior learning must be in place or else how can children discern what we intend.
As a frequent observer in the classroom I do not ask myself, are the children enjoying this task but how well does the task allow children to focus on critical aspects of what is to be learned and how well are children noticing those critical aspects? This allows a much fuller subsequent learning conversation which focusses on children’s needs. Obviously I want the children to enjoy maths but, I believe by being successful mathematicians they will.
Gattegno, C. (1988) The Science of Education. Part 2B: The awareness of mathematization. New York: Educational Solutions.