aTask and acceptable answers

This task came from the 1997 administration and asked students to indicate which of two boxes of Pebbles was better value for money. Results were selected from 6 boys and 6 girls of high, middle and low decile schools. There was no special Pasifika group in 1997.

Both Year 4 and Year 8 students were asked the first three questions. Year 8 students were asked the additional questions numbered 4 - 6.

We chose to compare responses on the first three questions that were given to both age groups. Logically, this common task requires a comparison of two dimensions, weight or mass and price, and three mathematical calculations: 50g+50g=100g, 60c+60c=$1.20, and $1.20 < $1.30. This required the students to combine two values, price and weight, and select the one with the better ratio. Asking for better value implies selecting less cost for the same amount, or same cost for a greater amount.

The major difference between the subgroups that were asked this question was that only 7 of 36 Year 4 students were accurate in their reasoning for why one box was better value for money, and 27 of 36 of Year 8 students could give appropriate answers. More than half of the Year 4 students attended to only one dimension, usually price, and several of those who noted both dimensions either did not compare them or were inaccurate in the necessary calculation. Some indicated that they did not believe that value for money depended on a comparison, one saying 'because they 21 probably wouldn't cheat on you if they were good' (Year 4 girl from a middle decile school).

Table 3.1 below shows the focus of the answers given by Year 4 students. In addition to the 21 (48%) who attended to only one element in the problem, there were 9 students who attended to both but did not have a suitable way of making a direct comparison, such as doubling the price of the 50 gram box and comparing that price to the price of the larger box which held twice the weight.

An answer that incorporated both weight and price but did not integrate them adequately was:

Because it's less in money and um, like if you get them for kids to eat this is much small, this is less (Year 4 girl from a low decile school)

The concept of value for money as the result of a calculation appears to be difficult for this age group and particularly for middle and low decile students.

Both successful and unsuccessful students came from schools in all decile ranges. However, of those who were successful a higher proportion of students came from high decile schools than middle or low decile schools. The concept was considerably more familiar to students in Year 8, as shown in Table 3.2

In this age range, 75% of the students gave an appropriate answer. The next highest percentage of students was those who identified both cost and size as important but did not explain clearly what gave value for money. Although correct students came from schools in all deciles, all students who attended to only one element were from 22 low decile schools and most of students who mentioned both elements but did not clearly express value for money were from middle or low decile schools. It appeared that both cognitive and linguistic issues contributed to this.

A good example of an answer that did give value for money was:

Fifty grams… um this one [Q] Because if you times that by two it’s a hundred grams and, um, sixty plus sixty is a hundred and twenty, so that’s a dollar twenty and selling that for a dollar thirty. (Year 8 girl from a middle decile school)

The largest differences in use of linguistic structures were between Year 4 and Year 8 students, although there were also marked differences by gender.

Because of the size of the groups analysed (only 6 students per cell), apparently large differences between the small subgroups could be caused by one or two students who had different response patterns. This was the case for the ratio of middle decile Year 4 students using a mathematical object in their clauses, in comparison to low and high decile groups.

Year 4 students used a mathematical object as an agent nine times as often as they used a person as an agent. This use was usually a reference to 'this box', 'that', or 'it'. When only two boxes were being compared this was all the reference that was required for clarity. There were few references to 'the bigger box' while most were to 'that one' or 'it'. Year 8 students also used a high proportion of references to the boxes as an agent using these terms, but were more likely to use a personal pronoun as the agent of their clauses.

The ratio of mathematical agent to personal pronoun as agent shows an irregular pattern across deciles, with middle decile students using a lower proportion of personal pronouns than the low and high decile students. This may have been because the middle decile students said less in response to this question than did the low and high decile students.

This one because if you've got, you can get two of these for a lower price of that and you get the same amount, you can buy two of these you get a hundred grams and it's only a dollar twenty. Q Because that you just, because if you bought two you'd times that by two, sixty cents, it's a dollar twenty and net fifty net grams by two you get a hundred gram net (Year 8 boy attending a high decile school)

The older students who were more likely to use mathematical calculations were much more likely than the younger ones to use a number or a mathematical operation as the agent of a clause than were the younger students.

As was discussed in Chapter 1, Bills and Grey (2001) showed that students who used 'I' and 'you' in their responses were more likely to give accurate answers especially if they were also categorised as 'generic' or 'general'. This was because they were not referring specifically to themselves or the teacher administrator but rather to the general actions that any person would do. Given that Year 8 students were more likely to provide an accurate response and use 'you' in their responses, it seemed that a similar relationship was evident in our data. Bills and Grey did suggest that 'you' was common classroom talk and so it was students who were comfortable with the mathematics who were more likely to use classroom ways of discussing mathematics. However, more investigation into this was needed. This is discussed further in the next section on Text Structures.

In the processes, the ratio was one between mathematical and non-mathematical verbs. Mathematical verbs showed a relationship between things or doing mathematical actions such as 'measuring', 'estimating' and quasi-mathematical actions such as 'figuring out'. Non-mathematical verbs described other types of actions such as 'putting on', 'taking off' or personal actions such as 'saying', 'thinking', 'having' or static verbs such as 'is' in 'this is a box'. These final verbs need to be distinguished from the relationship verbs such as 'this is heavier' where the 'is' is describing a relationship and is very typical of the mathematical register. In Better Buy, as in Weigh up, relationship was the focus of the question. Students responded by using verbs that emphasised this relationship, usually in the form of 'it costs more', 'it's heavier', 'it's lighter', or 'it's second lightest'.

For both age groups the second most common classification of verbs were static ones, such as 'is'. These were used in statements such as 'this is better'. Similarly, for both 24 age groups mathematical verbs were the third most common category. There was a higher proportion of mathematical verbs for Year 8 students than Year 4 students. For Better Buy, the mathematical terms included 'costs', plus', 'equals', and 'times'. In the early years of school, students are encouraged to use everyday language for mathematical terms, but by Year 4 this is expected to have been phased out with mathematical terms being used instead (Learning Media, 1992). However, these transcripts showed that students continued to use both everyday and mathematical language, and for this task, everyday language, including pronouns for agents and static verbs were more prevalent than mathematical ones.

It is clear that students who are most likely to give clear descriptions of their accurate reasoning were those students from high decile schools, although a number of students from middle decile schools were also able to do this. Students from low decile schools were less likely to have reasoned the answer correctly and to have clear 25 language. Slightly more boys gave an accurate response than girls, whilst in the lowest level of accuracy, slightly more girls did not compare size and price than boys.

Although Table 3.5 provides some information about the differences between groups in the language they used to explain their answers, it was not sufficient to understand what were the essential parts of a clear, accurate response. It was decided therefore to investigate the features of clear language so that the typical structure of children's explanations could be identified. Using the ideas of Hasan (Halliday and Hasan, 1985), we looked for text elements and how they were combined to form a justification of their calculation.

From interrogating the data, it was clear that every student's explanation contained one or more or the following three features. These were Premise, Consequence and Conclusion. Premises are statements of ideas upon which the student's reasoning is built. These are the linguistic equivalences of Krummheuer's (1995) grounds which were described in Chapter 1. The students in responding to this task used two types of Premises. One was the repetition of a fact that was given in the question, such as 'It's fifty grams and that's 100 grams'. These were labelled as Factual Premises and were also seen in Motorway task. The other Premise was when a hypothetical situation was mooted, such as 'If I buy two of them …'. Descriptions of the ideas built on these Premises are labelled as consequences. For example:

The final feature of these explanations was a Conclusion. This is where the student made a reference to better value. Only nine students used an explicit Conclusion in their response. However, 25 other students used words such as 'more', 'only', 'but' to cue the listener to the fact that a comparison had been made. These were labelled as implicit Conclusions. Given that these were oral explanations where the context was shared between the child and the teacher administrator, it is to be expected that the listener would have to supply some background information to what they were being told (Halliday, 1985). It is perhaps more surprising that some students chose to be so explicit in their reasoning. If the Conclusion came before the Premise (and the Consequence), then the student was most likely pre-empting the question asking for their reasoning when they responded to the question about which box was better value.

The students used one of ten different combinations in giving their reasoning. Table 3.6 provides examples of each of these combinations and the number of students who used the different types of Premises. In the examples, Q stands for a question or prompt from the teacher administrator.