This
section sets out the results of the data analysis for the sample
of 40 children in each of the four tasks. All page references are
from Crooks and Flockton (2002).






Task
1 Addition Examples 
The
five addition tasks are examples of the following types of addition
found in school mathematics programmes and materials; (p.14) 
1. 
vertical addition, two addends of single digits (5 and 8) 
2. 
vertical addition, five addends of single digits (6, 3, 8, 7, and
4) 
3. 
vertical addition, two addends of double digits (42 and 35) without
renaming 
4. 
vertical addition, two addends of double digits (87 and 56) with renaming

5. 
vertical
addition, two addends of three digits (327 and 436) with renaming



The
focus of the task was “adding without a calculator” (p. 14) and the
instructions to the children were to write their answers in the designated
boxes and to use the shaded area for their working. The number of
correct, incorrect and no responses for each addition question is
set out in the table below. 



Question 
Correct 
% 
Incorrect 
% 
No
response 
% 
1

33

82.5

4

10.0

3

7.5 
2 
27 
67.5

8

20.0

5

12.5 
3

28

70.0

6

15.0

6

15.0 
4

22

55.0

12

30.0

6

15.0 
5

19

47.5

14

35.0

7

17.5



Table
1 Results of Addition Examples 


Within
the sample of 40, the percentages of correct answers were similar
to the NEMP 2001 data. Similarly, there was a fall off for correct
responses in questions 4 and 5 as noted in the commentary section
that children “struggled when required to rename” (2001, p. 14).
In our sample there was a relatively high percentage of no responses.
A further analysis
of the incorrect answers revealed the following errors in strategies.

(i)
Renaming: the errors associated with renaming were 
• 
the 'carried' number added to incorrect column/place, often the left
hand column regardless. 
• 
having carried to correct place but then calculated as a larger number,
eg renaming 1+2 as 12 rather than 3 (possibly using a subtraction
form of renaming). 
(ii)
Mixing addition and subtraction, particularly in questions 4 and 5. 
(iii)
Adjusting the answer to have the same number of digits as addends
(NB this is often the case with subtraction) 
(iv)
Other errors identified were 
• 
misread numbers in the addends 
• 
incorrect basic fact with addition in one of the columns 
• 
Possible missed addends with adding more than two numbers (eg task
2) 
The
errors identified were examples of errors in addition and in the use
of the addition algorithm. The addition of two digit or three digit
numbers appeared to compound errors in renaming. The presentation
of addends in a vertical form was also identified as problematic.
This vertical format may cue children to use other operations such
as subtraction including bugs in the renaming process. 


Task
2 Speedo 
New
to the NEMP study in 2001, this series of questions used a visual
representation of an odometer common to cars and referred to as
a speedometer. This task is an example of a traditional place value
task set in a context of a speedometer. The information provided
in the tasks was as follows:
A trip meter
on a speedo shows how many kilometres a car travels. (p. 17).
The first four
questions required adding respectively one, ten, one hundred and
one thousand more kms to a benchmark number, 1996. The addition
of ten and one hundred required renaming of one or more place. The
final four questions required subtracting respectively one, ten,
one hundred and one thousand kms from 3402. Only the subtraction
of 10 required renaming. The number of correct, incorrect and no
responses for each Speedo question is set out in the table on the
following page: 



Question 
Correct 
% 
Incorrect 
% 
No
response 
% 
1 
18 
45.0 
19 
47.5 
3 
7.5 
2 
4 
10.0 
31 
77.5 
5 
12.5 
3 
4 
10.0 
31 
77.5 
5 
15.0 
4 
12 
30.0 
21 
52.5 
7 
17.5 
5 
14 
35.0 
18 
45.0 
8 
20.0 
6 
3 
7.5 
30 
75.0 
7 
17.5 
7 
7 
17.5 
26 
65.0 
7 
17.5 
8 
9 
22.5 
23 
57.5 
8 
20.0 


Table
2 Results of Speedo 


Note
that the percentages of correct responses were very similar to the
results from NEMP 2001. There was less than 20% of 'no response'
for all questions and there were indications of uncertainty such
as evidence of some children rubbing out many of the answers or
writing the benchmark number 1996 as their answer for all 4 questions.
The three questions requiring renaming (questions 2, 3 and 6) had
a high percentage of incorrect responses. One common renaming error
was carrying the digits to the left hand 'place', correct for renaming
two digit numbers but not for larger numbers. Other strategies to
avoid renaming were to include an extra place, transposing some
of the places when adding to the 'nines' place, or to add on numbers
at both ends.
The nature of
this task may have been problematic for year 4 children in a different
ways. The context may have assumed some prior familiarity with a
trip meter that changes as kms are traveled. An odometer is a dynamic
system that shows the number changing, with two places changing
when a 9 changes to a 10. Yet the visual presentation was a static
image. The language load in this question may also have posed difficulties
as the terms 'more than' and 'before' are possibly confusing for
some Year 4 children. 


Task
3 Money A 
This
was another contextual word problem and the context of money is often
assumed to be more familiar and therefore more accessible for children
of this age. The tasks were: (from p. 37) 
1. 
In a sale 1/4 is taken off the price of everything. How much will
you save on something which used to cost $2.00? 
2. 
$2.50 is divided equally between 2 children. How much will each child
get? 
3. 
Sonny bought 3 Play Station games at $98 each. How could he work out
how much he spent? 

A 
3 x $100, minus $2 
B 
3 x $100, minus $3 
C 
3 x $100, minus $6 
D 
3 x $100, minus $12 





The
number of correct, incorrect and no responses for each Money question
is set out in the table below. 






Question 
Correct 
% 
Incorrect 
% 
No
response 
% 
1

6

15.0

25

62.5

9

22.5 
2

20

50.0

14

35.0

6

15.0 
3

8

20.0

27

67.5

5

12.5 


Table
3 Results of Money 





Overall,
in our sample, 5 children (12.5%) did not respond to both questions
1 and 2, with 3 of these children not responding to all questions.
In the first question, 8 of the 25 incorrect answers were $1.50,
the discounted amount for the problem. This is similar to the finding
in the commentary that “many students in both years gave the discounted
price, not the amount of the discount.” (p. 37). Another common
incorrect response was $1.00 which halved the original dollar amount.
There were a few errors that seemed to be generated by doing something
with the numbers presented ie 1 and 4 from the fraction. For this
question, there were 9 errors that were classified as unknown.
For the second
question, the most common incorrect response was $1.50 (4 instances),
generated by dividing the dollar amount but leaving the cents intact.
Similarly, there were 2 instances of doubling the dollar amount
and leaving the cents intact. The remainder of incorrect responses
were classified as unknown although all except 2 were answers that
suggested the dollar amount was shared equally but the cents amount
had been divided in idiosyncratic ways.
The third task
was a multichoice question and the four options included the distributive
property for this calculation. The results were: (No responses were
5) 

A 
10 
25.0% 
(leaving the
2 dollars from 10098) 

B 
2 
5.0% 


C 
8 
20.0% 
(correct answer)
cf 21% from NEMP 

D 
15 
37.5% 
(less obvious
 using 96 instead of 98?) 

(no
response 12.5%) 





Word
problems set in a context can pose language demands of key signifiers
and terms. This was evident in these three questions; for example,
equally, divided, between (qtn 2), and minus (qtn 3). Year 4 children
may not have been familiar with multichoice tasks or with the process
for dividing quantities of less than one (ie the cents amount). 





Task
4 36 and 29 
The
task was presented verbally to each child as:
Here are two
numbers, 36 and 29. If you had to add the two numbers, and you
didn't have a calculator, how would you work it out? Try to think
of one way you could work it out, and explain it to me. Encourage
student to think of, and explain a way of working it out. They
are not asked to work out answers. If the student succeeds in
explaining one way ask: is there another way you could work out
36 plus 29? Explain to me how you would do it. (p. 20) (the students
are also shown a card with 36 and 29 written horizontally.)
Task Four is
an example of a decontextualised mental addition task where the
interviewer asks the child to explain how they worked it out. The
child 'thinks aloud' and their explanation is assumed to be the
calculation strategy. This particular example a more complex task
than 29 plus 36 because the larger number is the first addend. This
was a onetoone task and analysed from the transcript of videotaped
interview.
The researchers
used the NEMP marking schedule to categorise the first strategy
provided by each child. The most common strategy (35%) was the conventional
algorithm of adding the units first and carrying the ten. Two transcripts
are set out below to illustrate the use of this strategy.
If it's two
numbers, first I take, because tens and ones. In the one's column
there's 6, and in the other one, which is 29, in the one's column
which is 9 so I add 9 and 6 which is 15. Then I just have the
one, the ten in my mind and then it's 15, so I take the 10 off
15 and I just think 5. So in the one's column it's 5 when it's
added together. And 3 and 2 equals 5 plus one more ten is 60,
so it's 65.
Easy. Am
I adding it?
(Interviewer: Yes that's right.)
I'd go, write 29 up here (writes with finger on table), 36 down
there. Put the plus sign here and draw the line underneath. Then
I would go 9 plus 6 equals 15. I'd put the 5 down there and the
one up here. Then 2 plus 3 plus 1 equals 6 and it would equal
65 … I think.
The next most
common strategy (22.5%) was to add the tens first (30 plus 20) and
then to add the units, often referred to as 'front end' addition.
An example was:
Get two first
numbers, the tens.
(Interviewer: You mean the 3 and the 2?)
Yes put them together and that makes 50 then add the 9 and the
6 and that makes 65/
There were some
other numberrelated strategies such as the one below.
Well I would
take 1 off the 6 and put it on the 29 which makes it 30 and 35
and then I would add the 30 and 30 together which makes 60 and
seeing I've also got 5, it would be 65.
This strategy
is an example of recomposing 29+1 as 30 and followed by front end
addition (Sowder, 1988). 37.5% of responses were categorised as
other and included methods that were often descriptive (eg counting,
count on a ruler, use fingers, use equipment such as pencils, sticks,
coins, a calculator) rather than an explanation of a number related
strategy. A few children used other operations such as subtraction
and division. 




