Because different answer formats (assigning to categories and choosing a number on a scale from 1-10) were used, it was necessary to find a method by which answers could be combined in an effort to derive composite scores for the main constructs. The following
solution was adopted: answers from both types of answer formats (interval vs. categorical data) were converted to a common scale ranging from 0-2. To illustrate how this was
achieved, it is useful to consider two examples of questions that were designed to tap parents’ perception of children’s understanding o f the cardinality principle with different
answer formats:
Categorical answer format:
15. Can s/he count backwards?
Yes with objects -
Score of 1
Yes without objects -
Score of 2
No -
Score of 0
If parents answer ‘Don’t know’ for this question, the cell is left empty, as this kind of
answer does not provide any information about the child’s level of performance or
knowledge. Answering “No” scores of 0, “Yes with objects” scores 1 and “Yes without objects” leads to a score of 2.
21. How confident are you that s/he knows that numbers refer to specific amounts?
Ordinal scales in question, such as 21 above, were divided into three sections. If parents’ answers were between 0-3, a score of 0 was assigned: answers between 4-5 received a
score o f 1, and answers between 7-10 on the scale corresponded to a score of 2.
It is clear that this conversion to a common scale is somewhat problematic since it is achieved post-hoc. Furthermore, the differences between 1 and 2 as derived from the
categorical answer formats is not strictly the same as the difference between values 1-2 converted from the interval scale answer formats. While it is desirable to have composite
scores, which are derived from questions with the same answer formats, in the present
study it was felt that this was the best compromise available in order to avoid having to
conduct the entire analysis on a question-by-question basis.
Composite scores (in the way described above) were derived for 1. “Counting”; 2. “Cardinality”; 3; “Addition and Subtraction”.
• The sum of answers to questions 1, 14 and 15 represent the “Counting” composite
score.
• The sum of answers to questions 17, 20, 21, 22 and 31 represent the “Cardinality”
composite score.
• The sum o f answers to questions 35,36,37,38,39 and 40 represent the “Addition
and Subtraction” composite score.
In addition to these composite scores, a ‘total composite’ score was computed by deriving
the sum of all the composite scores.
No composite score was computed for the fourth construct: “Everyday Number”. This
decision was taken in view o f several arguments: first, the questions under the heading “Everyday Number” tapped some possibly very different aspects of numerical
computation in everyday life, with the majority of questions being related to children’s
ability to deal with pocket money, estimate prices, etc. In addition to these questions, parents were asked whether their child could tell the time, demonstrate an awareness of
age differences, and whether or not s/he had a good understanding o f distances between places. Against this heterogeneity o f questions and the loose operational term “Everyday number”, it was felt that little coherent information could be derived from a composite
score that averages the answers provided to these various questions. In addition,
inspection o f the data quickly suggested that very few children in both groups actually received pocket money. Therefore the number of parents who answered questions
relating to pocket money was rather small. Finally, it was felt that some o f the questions in the “Everyday number” section of the questionnaire were ambiguous. For example.
Question 25 asks parents: “Have you experienced that s/he cannot deal with pocket
money very well”. This question is ambiguous because here a high score, in contrast with
Question 24, is a negative indication of the child’s ability. This sudden change o f the
semantics of the scale from 1-10 might have confused parents, leading them to think that
this question asked them to indicate whether their child can rather than cannot deal with pocket money. Indeed, inspection o f the scores suggests that parents who had indicated a
positive score on Question 24 also did so on Question 25, suggesting that the way the
question was phrased and the reversal of the scale was problematic. Therefore Question
25 was excluded from the analysis.
Taken together, the above considerations suggest that a composite score for “Everyday number” would not give a meaningful average of children’s ability and therefore a
number o f the questions under this heading will be analyzed separately in the Results
section below.
3.4.3 Other questions
In addition to the questions that fall under the four main constructs, parents were asked to
answer questions about other aspects o f their child’s number development, such as whether their child was better at reading or number or whether s/he performed equally
well at both. In addition parents were asked to indicate how much they themselves knew about their child’s number development at school, mathematical tuition at school, etc.
The analysis o f some, but not all, of these questions will be presented at the end of the results section below. Therefore the emphasis is on the four constructs and on other
questions that are directly related to the Questions and Hypothesis stated in 3.3 above.
3.4.4 Participants
In total 110 parents responded to the questionnaire. O f these, 43 were parents o f typically developing children and 67 were parents o f children with WS. For the sake of subsequent
analysis, both groups were further divided into subgroups according to their age. This
was achieved by inspecting the age distributions in both the typically developing and the
WS group and establishing age groups with roughly the same number of children in each
group. The details of these age groups o f both typically developing and WS children can be found in Table 3.1 below.
Table 3.1 Children’s background data
Typically Developing Children Williams syndrome Children 3.0-4.5 years 4.5-5.5 years 5.5-6.5 years 4.5-6.S years 6.5-8.5 years 8.5-10.5 years 10.5-15 years Number o f Participants 11 15 17 17 18 13 19 Gender (M=males F= female) 5 M , 6 F 7 M, 8 F 8 M , 9 F 7 M, l OF 1 2 M , 6 F 8 M , 5 F I I M, 8F Mean A ge (years; months) 3; 8 4;11 5;10 5;6 7;5 9;6 13;4 A ge Range 2 ; 9 - 4 ; 5 4;6-5;5 5;6-6;5 4;6-6;5 6;6-8;5 8;7-10;4 10;8-15;9
3.5 Results
3.5.1 Analysis o f "Counting"Data
At what age did children start counting?Before analysing the counting composite score data, the analysis o f an item on the
questionnaire that was not included in the composite score will be briefly described. Question 10 asked parents to indicate the age at which their child started to count. All 43
parents of typically developing children answered this question, while 50/67 parents of
children with WS provided an answer. While parents of typically developing children
indicated that, on average, their child started counting at 2 V2 years o f age, the average
age at which parents of children with WS thought their child started counting was 4 V2
years. An independent-samples t-test confirmed that, according to parental report, the
group of TD children started counting significantly earlier than the group o f children with
Counting Composite Scores
Initially the counting composite scores for each group were explored by plotting the
means for each group. The data can be found in Figure 3.1 below. These data were plotted as a proportion of the maximum score.
Figure 3.1 Mean Counting Com posite Score by Group