Scaling Up

One of the biggest hurdles to mathematical understanding is moving out of additive thinking into multiplicative thinking. There are many reasons why students get stuck in an additive phase so what can we as teachers do to move them into a multiplicative phase?
I worked with a Grade 9 Applied teacher recently who noticed from her diagnostic tasks that many of her students could not think multiplicatively. As they were about to begin some work on ratios this was going to be a problem. We decided to adopt a concrete-diagrammatic-symbolic approach to move students on from additive thinking. 
We began with a simple problem:

The weights of two dogs as puppies and fully grown are shown:

Which dog grew more?

Without exception, the students said that they grew by the same amount (i.e. 6 kg). They were looking at how much weight had been ADDED.
So we then asked them, is there another way of thinking about this. After a bit, one student noticed that the first dog had DOUBLED in weight whilst the second dog had not increased by the same rate.

This was the platform we needed to build on.

I was clear with them: we need to learn how to compare things not just by addition but also by multiplication. I told the that we were going to do some activities that would help them how to see things in terms of multiplication and not just addition, and that this would make them better mathematicians. 
I also told them that we were going to do this in three steps: concretely, then diagrammatically, then symbolically. 

Each student was then given a set of cuisenaire rods.
I told them to find two orange ones and put them end-to-end. "If one of these is 10, how much will two be?" "20!" came the instant reply.
I then told them to put a yellow rod directly below the two orange ones (and showed this using the mathies.ca Relational Rods tool). I then asked them to estimate how many rods would be need to match the two orange rods. 

After they made some suggestions, I asked them to find out and then tell me how much a yellow rod was worth: they were able to tell me that it was 5.

I then asked them to write a number sentence for what they had just done. Over half wrote 5+5+5+5=20 so I then asked them to write a number sentence without using an addition sign. This nudged them toward multiplication and they wrote 5x4=20.

Again, I was clear with them: this is the goal of today's lesson...to think multiplicatively.
Next, I asked the students to do this again but this time with the purple rod. Seeing the students carefully lining up the rods to make sure they were equal to the two orange rods (and the four yellow rods) made me realise that maybe this is the experience that they had missed out on: the actual concrete act of creating equality using equal groups.

They wrote 4x5=20 without any prompting. One student then noticed something: "I can write it another way without using addition. If you split the rods up again you are dividing the 20 so you can write them using divisions!"

This led to related facts:

4×5=20

5×4=20

20÷5=4

20÷4=5

Next, I told them that as they were grasping this so well, it was time to scale up: now we need to use larger numbers and that these would be better modelled with diagrams. So I asked them to write a set of related facts for this diagram:

From this alone, they were able to write:

20×6=120

6×20=120

120÷6=20

120÷20=6

No-one wrote 20+100=120. We were seeing the students shift away from additive thinking.
Curious I wrote down my favourite math fact on the board:

37×3=111

and told them that we were about to scale up again. I asked them to complete the set of related facts which they were able to do even though they had not learned the 37-times table!

We then split them into visibly random groups and gave them a problem to try:

Two people do some decorating. Ann worked for 2 hours, Bill worked for one hour. Together they were paid $30. How much should each person get?

As the groups worked on this, it was clear that they realised that it would be unfair for the people to be paid the same amount. Most groups got the sense that Ann should get paid twice as much as Bill and used different ways to come up with an answer. We summarised their their thinking by using a bar model approach:

This allowed them to see the 'three-ness' of this problem and allowed them to see that each hour block is equivalent to $30÷3 or $10. We then challenged them with the following set of problems and encouraged them to use bar models to show their thinking.

It was pleasing to see many of them successfully use the bar models to solve the problems (though I wish I took more pictures of their work).

When I asked how they felt about this concrete-diagrammatic-symbolic approach at the end of the lesson, the students told me that it really helped them. 

Sometimes it takes just a well-timed nudge to move students on.