The question was on the screen:

One year 6 child said: ‘The empty box is in the middle so you do the inverse. You have to add the numbers together’.

This got me thinking about how children build on their early concepts of number to be able deal with problems like this, which I’ll call ‘empty box problems’.

The underlying pattern of additive reasoning is the relationships between the parts and the whole. Getting children to think and talk about the whole and parts using concrete manipulatives early on should lay the foundations for them to internalise this underlying pattern. Every time children think and talk about number bonds, they can be practising identifying the whole, breaking it into parts and then recombining to make the whole once more.

Alongside talking about the whole and parts, children should begin to generate worded statements whilst manipulating cubes or Numicon, for example. At this point it is important to experiment with rearranging the words in the statement. They should get to know that ‘four add two is equal to six’ and ‘six is equal to four add two’ are statements that are saying the same thing. Some discussion around what is the same and what is different about these two statements would be worthwhile.

When children are then shown how this looks abstractly with numerals and the equals sign, this would hopefully go some way towards avoiding the misconception that the equals sign means that ‘the answer is next’.

In the examples used so far, the whole and each of the parts have been ‘known’. Using the same manipulatives and language patterns, children can be introduced to unknowns. It seems sensible to begin with giving children the parts and using the word ‘something’ to show that the whole is unknown, i.e., four add two is equal to something. Some modelling alongside a clear explanation followed by plenty of practice should see children get used to the language patterns needed to think about the concept with clarity. The next step is to show children the whole and one of the parts, using the word ‘something’ to replace the unknown part. All of this talk and manipulation of objects is intended to support children to develop a concept of additive reasoning where they do not have the misconception that ‘inverse’ means ‘do the opposite’.

More sophisticated additive reasoning is the understanding of the inverse relationship between addition and subtraction. Children need to fully understand that two or more parts can be equal to the whole. From this, they need to internalise the underlying patterns: that Part + Part = Whole and that Whole – Part = Part. From this, they should be able to work out the full range of calculations that represent one bar model. Again, it is important to vary the placement of the = sign.

One more way to get children to think about the whole and the parts is to use bar models for calculation practice rather than simply writing a calculation for children to work out. When done like this, children have to decide what calculation to do to work out the unknown. Children often exhibit misconceptions such as ‘when you subtract, the biggest number goes first’. These can be addressed using the underlying patterns; adding parts together makes the whole and, when you subtract, you always subtract from the whole. When unknowns are introduced, they can be substituted into these basic patterns:

Part + Something = Whole Part + □ = Whole 35 + □ = 72

Something + Part = Whole □ + Part = Whole □ + 35 = 72

Whole – Something = Part Whole – □ = Part 72 – □ = 35

Something – Part = Part □ – Part = Part □ – 35 = 37

Knowing these patterns will help children to able to analyse problem types in order to decide on the calculation needed. An additive reasoning bar model with one unknown generates both an addition statement and a subtraction statement. Showing children empty box problems pictorially, they can talk through the calculations that can be read from the bar model, using the word ‘something’ to represent the unknown. The next step is to show children abstract empty box problems and get them to map it onto a blank bar model. They should be drawing on their knowledge that the whole is equal to the sum of the parts and that when you subtract, you always start with the whole. Eventually, the hope is that the language alone should suffice to work out how to solve empty box problems, with children no longer needing the bars.

Which brings us back to that year 6 child. Of course, children will develop misconceptions as they make sense of what is shown and explained to them. By expecting them to think and talk about additive reasoning in the ways described above, it should go some way to building sound conceptual understanding.

I teach my son using Zig Engelman’s DI approach. He chooses to make the same point with ‘number families’ with two ‘small numbers’ and one ‘big number’. The child will then manipulate number families in one way or another every lesson, literally for years! My son is in yr3 and with two years of number families under his belt he now gives word problems the merest glance before going ‘oh good, this one is an add’ or whatever. Despite this confidence he is still asked to construct number families for newer word problems and calculations are often still presented as number families. I think what is distinctive isn’t so much the use of a family rather than a bar chart but the careful sequencing and the way some of these calculations must be done every single day. The way practice of old material is part of each lesson in a carefully planned manner means that even though it took a while, gaining real understanding is almost unavoidable due to overlearning.

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