CHILDREN'S STRATEGIES IN NUMERACY ACTIVITY CHAPTER 4 : Results 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).

The five addition tasks are examples of the following types of addition found in school mathematics programmes and materials; (p.14)

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
incorrect basic fact with addition in one of the columns
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.

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.

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 100-98)
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).

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 one-to-one 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.

(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 number-related 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.  For further information and contact details for the Author    |    Contact USEE