I was lucky enough to have the opportunity to give a talk at the Herts for Learning Curriculum symposium recently, about effective curriculum models in primary mathematics. One of the points I asked schools to consider was about likely prior experience of the mathematical idea being learned. It felt like I spoke at a rate of knots, such was my desire to share much more than this. But this was one of the points that seemed to resonate most with the leaders in the room.
Many of the big ideas in mathematics are relevant in children’s life outside school. For example, whilst taking a walk to feed the ducks, children may notice that the swans are larger than the ducks. When they are asked if there are more swans or more ducks, they may say more swans as they are larger and take up more of their ‘picture’. But this idea is perhaps challenged when children count the swans and ducks and discover there are the same number of each. They may also notice that the ducks swim around but, as long as no more ducks fly in and no more fly off, the number of ducks stays the same. In this somewhat contrived, but hopefully believable scenario, a child might be experiencing the ideas of;
- cardinality (the value of the set counted)
- decomposition and hierarchy of number (numbers live inside other numbers but not all numbers)
- conservation (the set can be rearranged but the quantity is not changed)
- size does not affect discrete quantity;
- comparison and equality
Just because this is not happening in a controlled classroom, with bits of plastic, shouldn’t devalue the experience. In the example above, a child already has a schema for numerosity, children as young as 2 can work confidently with a set of 3 objects. A school curriculum must work hard not to undo a child’s natural sense of number by over focussing on numerals. We found some evidence of this happening over KS1 during our KS1 fluency project a few years ago. Year 2 children by the end of the year had decreased in their ability to ‘see’ number when compared to children in Year 1 at the same time of year. When surveying teachers, it became clear that the expectations of the KS1 test and not being allowed to use resources when sitting it had caused teachers to think that they needed to prepare the children for the test by not giving them the resources in the first place or to take them away too soon. This led to an over reliance on 1:1 counting to calculate and a diminishing use of number facts. It’s quite hard but not impossible to fix too and diminishing number sense really tells in LKS2.
Some of what is happening may be challenging the child's current schema, for example, bigger items does not necessarily mean a greater quantity of items. The child then (or maybe after a few more examples of this) accommodates this by adjusting their schema. Another adjustment might be that 6 ducks can be arranged in different ways.
Similarly building and expressing numberless pattern is often seen as something quite low level mathematically. Children will have been exposed to pattern for their whole lives when we consider their early environments such as daily routines, wallpaper, rhymes and faces. As humans we are born pattern seekers. How quickly we responded as babies to the pattern of a mother’s face or the bedtime routine.
Making the link between multiplication that the children have seen as numberless pattern means that the Year 3 teacher teaching multiplication can repeat and layer the number on top. We can ask children to find; how many objects make up the pattern unit (3 - pebble, stick, pebble), how many times the pattern unit is repeated, how many objects there are in the pattern altogether, how many there will be if I add or take away a pattern unit. Children are then experiencing how the units coordinate in the pattern and the effect of one lot more and one lot less.
Similarly in Year 6 when teaching ratio it’s perhaps unlikely that teachers will return to pattern as a way of helping children express ratio, but it can help children to see all of the parts involved. For every stick there are two pebbles, what might my pattern look like. What about if my pattern had sticks and pebbles in a 2:3 ratio? What might it look like?
The example I showed delegates on the day was taken from our materials - translation, position and coordinates on a grid.
The programme of study for National curriculum ’14 shows this being introduced in Year 4 in the first quadrant. But have children met position on a grid somewhere before?
Well yes – if they have had carpet places on a mat that’s essentially a grid. Yes again if they have a tray in an array of trays, names on trays aside, they probably know the position of theirs in relation to their friends. In Year 2, children are likely to have programmed robots to move around a grid. They may have looked at a map, perhaps in school perhaps at home. Some may even have played battleships.
All of these experiences can be harnessed to show that sometimes co-ordinates express position in terms of the grid square it is in and sometimes the point at which two gridlines intersect.
Could acknowledging that they’ve probably experienced something about what they are learning this week before, help to level the playing field? I observed a cracking example of this recently. In a Year 2 lesson, the children were going to learn about balance, comparison and mass. The very first thing the teacher did was ask who had been on a see-saw. Through this she quickly established with the children that if you stopped bouncing it, and were lighter than the other person you would be higher on the see-saw and vice versa. This she related directly to the balance scales before the children had a go for themselves. What that teacher did was level the playing field and lay maths on top of a shared experience. Moreover, it was an experience that all of the children could relate to, no matter how they were currently attaining in school mathematics. She also addressed an over-generalisation that many children held. The larger the object the heavier it is. Misconceptions are only based on the schema the child currently holds - probably every time they were up in the air on the see-saw it was because a bigger person was on the other end. You can see why they might draw those conclusions. It needed challenging and children needed the experience to ‘feel it’ and believe it for themselves thus adjusting their schema to accommodate the fact this is not always true. Simple, authentic, inclusive and effective.
So what are you about to teach and where might prior experiences come from?
Do you have a sense of where in the curriculum this learning has come from?
What might it have looked like?
Can the learning begin through a shared experience that is inclusive but not tenuous?
What do we do for the children who didn’t go to the park and count the ducks?