First Steps in Developing Number Sense

One of my clearest memories of doing maths when I was 6 or 7 was in Mrs. White's class at St. Thomas More R.C. School. We used Cuisenaire rods to develop our understanding of number bonds up to ten. I'm pretty convinced that this laid the foundations for my good number sense and it is an experience I love repeating with young students from Kindergarten, Grade 1 and Grade 2. I start by asking the students to make a 'Staircase' as shown below:

Now this can be quite challenging for some students, but after this has been completed, I then get students to see that each coloured rod represents a quantity between 1 and 10. So I'll ask questions such as "Blue is what number? 5 is what colour". When the students know these they are then ready for the next step: I lay down an orange 10 red and ask:

Put two rods together that match the 10 rod.

This is my 5-year old son doing it for the first time:



When I was in Mrs. White's class, I do remember that before long, I was thinking "8+2 is 10" as I put the brown with the red. And this is what I see kids do now I try it myself with them: they are learning the facts by acting out the operation with the rods. The aim is that after a while they will not need the rods, that they will just 'see' that 7+3 is 10.

So what future knowledge will this connect to?
Well, firstly, it isn't too long before kids will see that 7+3 is the same as 3+7. In other words they will 'discover' the commutative law. I won't call it this though; I'll call it 'Dan's Rule' or 'Samantha's Rule' after whomever first notices it.
The rods also lead nicely into the idea of bar models by which we can represent the concrete (the rods) with a diagram and these in turn lead us to the abstract notation of related facts.

In fact, this development (concrete, diagrammatic, symbolic) is one which we must keep in mind when we are getting our students to develop their number sense. It will give students the opportunity to generalise their number sense into algebraic sense. If a question reads:

In Grade 3 there are 47 girls and 35 boys. How many more girls are there?

Students can represent this with a bar model and then use this to think about how this can be represented with any one of four number sentences:

In this case, some students might choose to do 47-35 but others might think 'What do I have to add to 35 to get 47'. Either way, you get 12.
And all of this thinking can begin in kindergarten with Cuisenaire rods.