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.

Problems With Place Value

Place value is so much more than Base 10 blocks.
This is a common theme has emerged in a number of recent Collaborative Inquiry sessions where we focused on what students know and don't know about place value. A lot of questions from textbooks and worksheets tend be of the type shown below:

The danger with these types of questions is that I have seen students get the right answers but have no firm understanding of quantity whatsoever. They might learn a strategy such as 'The first digit goes in the first space, the second digit goes in the second space and so on...' or 'The digit on the right tells you how many little blocks, the one next to it tells you how many rods, and the one next to that tells you how many flats and so on...' If base 10 blocks are the only representation used then there is a real danger that students will develop misconceptions such as on that was highlighted by Sue Willis in First Steps in Mathematics. Grade 4 students were correctly able to identify that 4 rods and 3 smalls were 43 (below).   

However, when asked how many there would be if they were cut into individual pieces, two-thirds of Grade 4 students said “I don’t know, I would have to count them.” 
Similarly, Grade 6-7 students described a large cube as ‘the thousands cube’ but thought that if it was cut up, there would be 600 small cubes (100 each side).  This sort of error explains why some students think that 24+25 is 13 as it is simply 2+4+2+5.

With this in mind, we visited a number of classes to try some different sort of place value and quantity questions. We set out to deliberately bring out any student misconceptions by asking them to order a set of numbers such as:

547, 600 - 3 twenties, 5 tens 7 ones 4 hundreds, 5 hundreds 23 tens, 4 fifties

As the students began to order these, it quickly became clear that many were making errors based on a simplified grasp of standard partitioning. As one teacher said "Boy, we've done way too much of that...".
So in addition to the bog-standard standard partitioning questions, we realised the importance of asking questions like:

• Write 57 in at least three different ways
• 1000 take away 47 tens is the same as how many tens?
• How many 20s in 100? 500? 1000?
• How many 25s in 500? 1000?

This will give students a deeper and more flexible understanding of quantity and place value which will be necessary if they are to have good number sense.