# Hassan’s internal number line

Hassan is a wonderful boy. He’s polite and has a great group of friends. But Hassan started Year 6 working significantly below his peers. His school history is of sustained underachievement with very little progress. He did not have an internalised number line with which to think about numbers, to the point where he could not reliably say which number out of two was biggest.

This post is an account of an intervention carried out by Amy Coyne. It is one of the most successful interventions I have seen and has resulted in vast improvements in Hassan’s ability to think about numbers. Here’s what happened:

These number cards were prepared: 53, 67, 54, 35, 76, 45

Two of the cards were presented to Hassan and, with the use of Numicon or dienes blocks or arrow cards, Amy modelled explaining which was the bigger number. Hassan picked this up fairly quickly, but to help him to retain this procedural knowledge, it was repeated little and often over the course of a few days.

A third card was added and Amy again modelled, using appropriate concrete equipment, how to order them. When he could consistently order three numbers, a further card was added until he could deal with ordering six cards. Using those six cards, Amy made seven different sequences:

53, 67, 54, 35, 76, 45

67, 35, 76, 54, 53, 45

76, 35, 67, 53, 45, 54

45, 53, 35, 76, 67, 54

54, 45, 76, 53, 35, 67

53, 54, 35, 76, 45, 67

45, 76, 54, 67, 53, 35

The cards were presented to Hassan in these orders, one set at a time, and Hassan was asked to order them. At first, with this slight change in task, he would place the numbers in fairly random order for each sequence. After completing each sequence, Amy ordered them with him, using concrete models when necessary. When Hassan was asked to read out the order, if he was incorrect, he often didn’t notice. However, when the sequence was read aloud to him, he could hear the error and would correct it.

This was repeated over several days for short periods of time. Sometimes this was in maths lessons and at other times it was not. The seven different sequences would be laid out in a straight line and he would pull the cards out and order them. As the days progressed, he could very quickly pick out card 35 and put it furthest left and also card 76 and put that furthest right – the smallest and greatest numbers. However the other cards in between were never placed consistently in order.

After a week of doing these sequences once or twice a day, he could order every sequence in the correct order. A new set of numbers was introduced: 12, 27, 45, 54, 59, 72

Hassan was very good at picking out the biggest and smallest numbers. The numbers in between were still more difficult for him. Amy modelled looking at the tens and units columns and this prompted him to order them correctly.

The cards were then mixed up, with more numbers being added one at a time to see if he could order them again. His confidence was growing and once he was happy with the order he had put them in, he was asked to read the numbers out to see if he could spot any errors himself. He often did and corrected them without Amy needing to intervene.

After a few days starting with six or more cards, he could reliably order them correctly every time. Next, some three digit numbers were added to make twelve cards overall. He was quickly able to deal with this progression. He was then given cards with multiples of ten to see if he could slot them into the correct places. He struggled a little with cards 10 and 20 but he placed multiples of ten more than twenty in the right places every time. If he needed to, he referred to a tape measure to check.

Once he was confident in ordering these numbers and could do it correctly every time, two numbers from the sequence were chosen. He was asked: ‘What are the smallest and biggest numbers that could go in the gap?’ This proved to be quite tricky for him and he would often say the number before the smallest card. This took him around a minute to process each time, and many answers were guesses. Amy modelled looking at a tape measure to find the two numbers (53 & 59). He then could see, using the tape, which numbers would come after 53 and before 59, and therefore the biggest and smallest that could go in the gap.

From here, Hassan is now working on adding and subtracting one digit numbers and multiples of ten from numbers in his card sets, with increasing success. Soon, the goal is that he can add and subtract any two digit number from any other.

Why this intervention worked, when other have failed

Spacing and interleaving

Regular short sessions, interspersed with other topics in maths lessons, with varied lengths of time in between those sessions has given Hassan time to internalise patterns of numbers and procedural knowledge for dealing with them.

Building knowledge of the number system

The more he practised recalling facts about numbers and procedures for how to think about them, the more successful he became. Each nugget of internalised knowledge enabled further memory development until he had internalised the basic number system.

Deliberate practice to mastery

The moment that Hassan understood and was successful did not signal the end of the intervention. It will continue until he never makes a mistake, even when tasks are altered.

Making links between ideas

Any new concepts were introduced alongside concepts that Hassan was familiar with.

Detailed dialogue between teacher and teaching assistant

Using video and observing ‘live’, the Amy and I talked about the nuances of the decisions that Hassan made to tweak tasks and feedback. This precise tailoring resulted in explanations, tasks and feedback which were accurately matched to Hassan’s needs.

This intervention was put in place because Hassan was working significantly below his peers at number. It was clear that he had not internalised a number line at the beginning of the year, but this shows that he now has. He will need more practice to cement his understanding but the progress that he has made has been good. We did not work on this with him for a week before the Christmas holiday. He had just over two weeks off school over Christmas and when he returned to school after the holiday, he could still deal with the number tasks accurately. Next, we are looking to see if the results are replicable with other children.

Details about the child have been changed to preserve anonymity.