Here is a summary of some of the results of the NumberTalk project.

## Mastery of written and spoken number systems

This section refers to tasks examining the number system at a fairly basic level. We are discussing three tasks here: count aloud, transcoding and magnitude comparison.

### Count aloud

1. Count from 1 to 41

2. Count from 25 to 32

3. Count from 194 to 210

4. Count from 995 to 1010

5. Count backwards from 25

We expected that some children, in particular the language controls, may not have been able to count the larger numbers and therefore we did have a discontinue procedure in place. If the child clearly could not manage counting to 41, we only asked them to do the backwards counting. However, some children made errors on the first part of the counting task but wanted to continue and were allowed to do so.

The following graph illustrates the results of each group in terms of their performance over all five tasks. Count aloud

The boxes represent the middle 50% of all the values, from the 25th to the 75th percentile. The thick horizontal line represents the median; the midpoint in the range of performance. The vertical lines outside of the box connect the smallest and largest values which are not outliers, hence they give a representation of the variation of the values.

It can be seen that the age controls performed at a considerably higher level than the other two groups. The median performance for the LC and SLI groups is the same. We can see that the SLI group show the greatest variation in their ability to perform this task, with a range of between 0 and 100% correct.

It would appear that the ability to count fluently from 1 to 41 is very important. Only one child in the AC group made an error here, yet 21 LC's and 28 SLI's were unsuccessful. It is interesting to look at the types of errors made by the two groups.

LC 1

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, pause, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 40, 41

LC 2

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, thirforty, 41, 42, 43

Here two of the LC's are demonstrating a difficulty in counting over the decade boundary between 29 and 30.

SLI 1

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, missed, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 50, 51, 52

SLI 2

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 40, 50, 60, 70, 80, 90, 20, missed lots, 31, 32, 33, 34, 35, 36, 37, 38, 36, thirtysix-nine,30, 31, 32, 33, 34, 35, 36, 37, 38, 39 pause, 30, 32

With the examples from the SLI group, there are numbers missed and the problem of the decade boundary. It can also be seen that there can be confusion between the teen numbers and the ty numbers (e.g. 30 for 13). Listening to the recordings of the counting, it is clear that the speech output problems of some children in the SLI group interfere significantly with their ability to count aloud.

### Transcoding

Transcoding is the term used for translating from numerals to number words and vice versa. This task had three different forms: spoken to written ,written to spoken and spoken to print, and provides information about number system knowledge. A transcoding task allows us to examine how number production and number comprehension processes work separately.

The child was asked to write the numbers 1 to 10 on a sheet of paper. This ensured we could make allowances for errors in individual digit writing, provided that these were usual for that child. For example, young children might write a 3 backwards.

Spoken to written - the tester read out a series of numbers and the child was asked to write them down. The numbers used were 30, 500, 15, 308, 25050, 4500, 7200 and 6042.

The age controls performed at a higher level than the other two groups. However, this group also show the greatest variation in their performance. The least variation is found in the LC group, this may be accounted for since some of the numbers used were higher than it would be expected for the language controls to be familiar with. Although the SLI group show quite a lot of variation, in general their performance was more in line with their language controls.

Written to spoken - the tester showed a series of cards with numbers printed on them and asked the child to read out the number. The numbers used were 17, 305, 80, 400, 50042, 3051, 60000 and 4800.

The patterns of performance on the written to spoken transcoding were similar to that found for spoken to written. It was noted that the LC and SLI groups performed at a slightly higher level on this task. This is especially true for the LC group.

Spoken to print - The child was shown cards with four numbers printed on them, they were asked to point to a target number which was spoken. The four choices were devised so that they included phonologically similar numbers, reversed numbers and a distractor. E.g. when shown the following card, the target number would be 16.

Again, a similar pattern of performance on this transcoding task when compared with the other two. The performance was more varied in the SLI group, with some children performing at a high level, although generally performance in this group was more in line with their language controls.

### Magnitude comparison

This task addresses the understanding of place value knowledge: i.e. how the position of a number determines its magnitude.

This task was presented using a laptop computer. The child was asked to decide which of two presented numbers was the larger, and to indicate their choice by pressing one of two buttons, the left hand button if the number on the left hand side was their choice. The buttons were positioned beneath the presented numbers and were made clearly visible with 'smiley sun' stickers.

The transparent trials were formed such that a decision could be made correctly based on the digit occupying the same place in each pair. For example with 30 vs. 60 only the decades are relevant, with 24 vs. 21, only the units are different. The misleads comprise of pairs in which the non-target item contains a unit value greater than the value of the units in the target item, e.g. 77 vs.69, thus both digits needs to be evaluated. The reversible comparisons involved simply reversing the digits, e.g. 72 vs. 27.

There were 48 trials in total, 6 blocks each comprising 8 trials. The trial types were as follows:

• Block 1. - two digit transparent
• Block 2. - two digit mislead or reversible
• Block 3. - three digit transparent
• Block 4. - three digit mislead or reversible
• Block 5. - four or five digit transparent
• Block 6. - four or five digit mislead or reversible

The yellow line across the graph indicates the 50% chance level of performance for this task. Looking at performance, the age control group are making fewer errors and show less variation in accuracy than either of the other two groups. The SLI group are performing at almost age appropriate level, and at a higher level than their performance on the counting and transcoding tasks.

Looking at response time, that is the time it took from onset of the stimulus pair to a decision being made, there is a great deal more variation in the language control group. The AC and SLI pattern of response time closely resemble each other.

## Arithmetic skills and knowledge

This section refers to tasks assessing knowledge of number facts, skill in arithmetic and grasp of arithmetic principles, such as commutativity.

### Number facts

The children were told that we wanted to know what 'sums' they could retrieve the answer for without having to work it out. These sums were presented on flash cards. There were two sections: doubles, for example, 2 + 2, and single digit additions, for example, 4 + 3.

Essentially all groups showed a high level of knowledge of doubles. Knowledge of single digit addition number facts was extremely varied and particularly low for the LC and SLI groups.

### Sums

The children were given a series of simple arithmetic sums. They were asked to solve these using any strategy they wished, and counters were provided if they wanted to use them.

All groups showed a slight advantage for addition over subtraction. Whereas the AC group were typically all correct, the other two groups' performance was much more variable.

### Commutativity

This task tested whether the children could demonstrate a working understanding of the principle of commutativity; that if a + b = c, then b + a must also = c. It has been argued that the understanding of such conceptual principles guides children's problem solving.

This is the task in which the SLI group is performing much more in line with their age controls. All three groups show a wide variability in performance.

## Understanding of mathematical language

### Test of Relational Comprehension

This test specifically addresses the issue of mathematical language, testing understanding of terms such as 'more', 'least, etc. Without a complete understanding of these terms it is possible that a lack of performance on a mathematical task may only be a reflection of poor language comprehension, and not be informative at all regarding the mathematical ability, or understanding, of the child performing the task.

The Test of Relational Comprehension (TRC) presented the child with a series of pictures, from which they were required to point to the picture most appropriate for the question asked. The test had two forms, easy and hard. The words used for the easy section included: longest, smallest, most, least, and fewest. The words used for the hard section included: Longer, smaller, more, less, and fewer. The following are examples from the test:

Here the child was asked to point to the bowl with the most ice-cream.

Here the child was asked to point to the picture in which the pink bowl has more ice-cream than the blue bowl.

There is a very clear similarity between the SLI group and the LC group; with both groups showing considerable variability.

### Seriation - verbal and non-verbal

This task required the children to place five, differently sized, members of a squirrel family in size order. The children were free to move the different cards around to assist them in completing this task. If a child was not correct on the first attempt, a second chance was given. If still unsuccessful, the tester put the squirrels in the correct order.

Examples of the squirrel family in correct size order:

It can be seen that both the SLI and LC groups find this task more difficult than the AC group.

After completion of the ordering of the squirrels, the tester then asked the children to point to various members of the family. For example, "Can you point to the smallest squirrel?" and "Can you point to the third largest squirrel?"

For the pointing to the first smallest or largest (the squirrels on the outside of the row) the age control and SLI groups both performed at the 100% level, making no errors. In the language control group, 55 of the 59 tested got both trials correct, the remaining 4 made an error on one of the two trials.

Performance when pointing to the squirrels on the inside of the row (second, third and fourth smallest or biggest) was more varied.

Although this is a verbal test, it shows a similar pattern of performance across the groups as the non-verbal seriation.

The final seriation task involved showing the child a picture of 5, differently sized, items of clothing. The child was then asked to say which of the items a particular squirrel would wear, the tester pointed to the appropriate squirrel whilst asking the question. Below is an example of the gloves:

For selection of the appropriately sized item of clothing, performance was very high for the outside items (Baby and Dad). In the age control group 55 of the 59 (97.5%) children got all six trials correct, 2 children made 2 errors and 2 made 3 errors. In the language control group, 51 of 59 (97.2%) got all six trials correct, 7 made just one error and one child made 2 errors. In the SLI group 57 of the 60 (98.3%) were correct on all six trials, the remaining 3 children made only one error each.

Performance when selecting the clothing for the squirrels on the inside of the row (Nan, Mum and Granddad) was more varied.

The general pattern is quite clear across all the seriation tasks, irrespective of the involvement of language; that is the SLI group closely resemble the linguistic control group.

## Overall conclusions

The NumberTalk project has explored number skills, knowledge and principles in children with SLI to find out whether they show characteristic patterns of strengths and weaknesses in contrast to other children their age and children with similar linguistic skills. What makes it special is the combination of a large sample of children (180) with a large number of tasks, several of which were specifically developed for the project. Our analyses involve both examination of group differences and differences between individuals. Some findings are broadly in line with predictions based on previous research. Counting and arithmetic by children with SLI closely resembles that of younger children with similar language levels and is substantially below that of other children their age. Proficient counting involves fluent retrieval of the count list and understanding of the numeration system: a fluent counter can count up or down from any number whether it is 9, 99, or 999. For all children, how proficient they are at counting is a good indicator of their ability to do sums, compare numbers and read and write numbers. Children with SLI struggled to count. In contrast to their linguistic peers, who were on average over 2 years younger, they were less fluent and less likely to self-correct. What can be done to help children become more proficient at counting and how this will affect their mathematical development are questions we hope to pursue. Also, as we expected, children with SLI overall show similar levels of understanding of mathematical language to their much younger language peers. This suggests that they are at a very considerable disadvantage in classrooms that do not recognize their limitations. However we also found very considerable differences between children with SLI suggesting that we should be cautious about drawing conclusions based on group differences. In contrast to these findings, our test of children's grasp of the principle of commutativity indicates children with SLI were much more like other children who are the same age. How the development of skills and principles influence each other is much debated by researchers. It seems that children with SLI can develop a grasp of principles without relying so much on the support of skills as other children. Understanding how they do this would make a significant theoretical and professional contribution. We examined children's understanding of ordering by size using tasks that varied in their verbal demands. We expected that children with SLI might resemble their linguistic matches on the more verbally demanding task and their chronological matches on the less verbal task. Unexpectedly, the level of verbal demand made little difference: children with SLI had difficulty with both. We hope that future research will provide an opportunity to test explanations of this. Finally, on every task we found marked differences between children with SLI . We are currently exploring our data to assess whether these differences are related to their linguistic skills, nonverbal reasoning ability, or their spatial and auditory memory characteristics. We also are examining the possibility that they form subgroups that may be clinically distinguishable.