Thanks to Deputy Head Teacher, Nick Hart, for the latest maths blog post where he looks at the importance of getting the basics right in number bonds.
The inability to recall or apply number bonds continues to be one of the reasons why some children have not yet developed sound mathematical understanding of number and so, however much we think we’ve got it cracked, our teaching of number bonds needs to be continually evaluated and refined.
Many concrete representations in the early years are objects such as bears. These are wonderful for getting to grips with counting. When it’s time for children to start conceptualising how maths works, a better concrete representation is needed. Five frames do just the job. When children are learning pairs of parts that make a whole of 5, arranging five frames clearly represents one of the first mathematical patterns that children come across:
This pattern is easily replicable by children and it’s a natural instinct for them to arrange objects in this way. The important bit is what they hear teachers say and what they say themselves whilst arranging the objects. In the first instance, “Five and zero make five; four and one make five; three and two make five…” works just fine. However if we can encourage, at this stage, the development of the language of additive reasoning, then we could develop even more secure foundations for following concepts:
The whole is 5 and the parts are 3 and 2. Representing counters like this on a 5 frame and the beginnings of a bar model can help children to grasp the idea that a whole is made up of parts.
When five frames are filled this way with different coloured counters, they act as worked examples – complete mathematical problems. The next stage of development will be to make a part unknown. In the image below, the whole is 5 and one of the parts is 1. The other part is unknown in the sense that the counters haven’t been placed on the 5 frame yet:
Whereas before, teachers will have modeled saying both known parts and the whole (for example, three and two make five), now the unknown part should be made clear: “One and something make five.” Cue eager children filling the empty spaces with counters, counting them as they go and proudly saying that the unknown part is 4.
Next, a pictorial representation of the counters on a five frame can help to consolidate children’s understanding and get them to begin to work more efficiently. An image of the one counter on a five frame and a colouring pencil is all that’s needed. Again though, it’s the talk that’s important. Children will immediately want to draw colourful circles in the empty boxes, but that’s the task not the learning. When the teacher models how they want the child to think, they’ll say something like: “Look! A five frame. The whole is five and one part is one. One and something make five. Let’s count as we draw counters in the empty spaces. One, two, three, four. One and four make five.” If teachers think aloud like that, children will imitate.
Finally we’ll show children an even more efficient way of recording their mathematical thinking – writing digits instead of drawing counters. Once more, it matters what the teacher says whilst modeling: “We always write part + part = whole. The first part was one *writes a one*; add *writes the addition symbol*; the second part was 4 *writes a four*; is equal to five *writes a 5*.”
The same concrete to pictorial to abstract representations can be used to teach number bonds beyond five, this time using two five frames. We simply make both numbers in an addition calculation then ‘make 5’ by filling up one of the five frames:
This is a vital concept to embed early on which underpins a lot of later calculation – the concept of bridging through a landmark number when adding. This is also the time to work on subitising. Once the child has filled a five frame by moving counters, we do not want them to count all the counters from the beginning. By now, we want them to know that a full five frame is five, and count the remaining counters from there. Eventually, they’ll subitise the whole and not need to count at all.
This is a great opportunity to work on another key concept – the meaning of the equals sign:
Hold on a minute! An addition symbol either side of the equals sign? But equals means the answer is coming. This misconception is so common and can be addressed from the minute children are writing mathematical statements.
What works for the five frame works for the ten frame. Learning number bonds to 10:
Learning number bonds within 20:
Multiple ten frames can become a little unwieldy but luckily, diennes do exactly the same job for learning number bonds to 100. The whole is 100 and the parts can be represented with 10s and 1s:
The whole is 100, one part is 24 and the other part is as yet unknown. 24 add something is 100. By filling up the 100 square with ones then tens, children work on the strategy of counting up and over time internalise number bonds to 100. This is easily progressed to pictorial representation in just the same way as with five and ten frames – provide a hundred square and some colouring pencils.
The key to all this is the focus on conceptual understanding through multiple representations and mathematically sound teacher talk. Without those, we have at best children who learn number bonds by rote and cannot apply that knowledge to calculating or problem solving and at worst children of 11 years old that do not know how to appropriate number bonds at all. With sound conceptual representations and great teacher talk, all children will develop tools for thinking and have many opportunities to internalise facts in a meaningful way.