In the Description part of this task, 13 students had used a logical connective before a Premise, but only 'and then' was used by more than one student (it was used twice). The situation is very different in this part of the task. In the Explanation, Table 4.37 shows that 'if' was used by over a quarter of the students to join a Premise to a Consequence. However, there do not appear to be any major differences between the groups of students using logical connectives before Premises. It seems that for this part of the Weigh Up task, that causality is shown in the logical connectives used before Premises rather than between Premises and Consequences.

In the Plan responses, many students used a Premise - Consequence - Elaborator, often using the phrase 'see if' as the beginning of the Consequence - Elaborator. In the Description responses, where students performed actions rather than describing what they might do, this phrase was not common. However, it reappeared in responses to the last part of the Weigh Up task, as in the example from a Year 8 girl from a low decile school, 'put another one in and see if it's um heavier or not'. Table 4.32 showed the distribution of the 38 students who used the Premise - Consequence 69 - Elaborator combination. The following table provides information on the distribution of students using logical connectives in front of elaborators.

The number of students using logical connectives before Elaborators is similar to those in the Plan responses. However, in this part of the task, there were many more students using a Consequence - Elaborator combination, so that, proportionally, Consequence - if - Elaborator was not used as often in this part of the task. If students did not use a logical connective before an Elaborator, the Elaborator would often be a relative clause such as 'which one's lightest' as in the following example from a Year 4 girl from a high decile school: 'and see which one's lightest'.

The other place where logical connectives were commonly used in other tasks was between a Consequence and a Conclusion (implicit or explicit). In this part of the task, there were 26 students who used a Conclusion. This is double the number of students who used this element in the Plan part of the task and significantly more than the three students who used it in the Description part of the task.

A total of 23 students used logical connectives between a Consequence and a Conclusion. As most students only used one Conclusion in their responses, it seems that Consequences are almost always connected to Conclusions with a logical connective. As girls and Year 8 students used Conclusions more often, it is not surprising to find that they also used more logical connectives in the Consequence - Conclusion combination.

In this part of the Weigh Up task, girls and Year 8 students were more likely to use logical connectives than any other groups. Students from high decile schools were slightly more likely to use logical connectives than students attending other decile level schools. There appears to be no difference in the use of logical connectives which were causal rather than narrative.

Table 4.40 shows the proportion of students in each group who used particular text elements or combinations. If students were accurate, regardless of their language fluency, they were more likely to use a Premise - Consequence combination than if they were only approaching accuracy. However, use of a Premise - Consequence - Elaborator or a Premise - Elaborator - Consequence - Elaborator combination does not seem to be related to the accuracy or the clarity of the response.

Conclusions were most likely to be used when students were approaching accuracy in their responses. This is not surprising, as students who did not provide complete details of each step would often finish their explanations with a broad statement such as 'so on like that' as in '… if B's lightest then it will be the second to lightest, and so on like that, until you think you know which one's lightest and heaviest'. Given the shared experience from doing the earlier parts of this task, this is an entirely sensible approach by students in responding to this part of the task.

As there was a limited number of Physical Consequences used in this part of the task, they do not appear to have any influence on accuracy or clarity of responses. On the other hand, it does appear that students who gave accurate answers were more likely to use Suppositions than those who were only approaching accuracy. This is interesting and suggests that students who were able to put forward propositions were more likely to be able to cope with hypothetically determining the order of the boxes. As was the case with the Description responses, it would seem that the combination of Premise - Consequence (with, to a lesser extent, an Elaborator) was more likely to be used by students providing an accurate response. It would also appear that Premise - Elaborator - Consequence combination was used by students who were deemed as giving moderately clear responses, regardless of accuracy. However, Physical Consequence - Conclusion combinations which had been commonly used by students providing accurate answer in the responses to the Description part of the task did not have the same significance in the Explanation section. This is not surprising given the decreased use of Physical Consequences in responses to this part of the task.

The Weigh Up task required students to give three different, mathematical explanations. Although there were some overlaps in how students structured their responses, there were also some differences. This section describes the differences and similarities between these explanations. Some of the trends evident across all parts of the task included the position of Elaborators in the text structures and who used them most frequently, as well as the use of Physical Consequences, which were text elements not found in the responses to other mathematics tasks. This section also looks at the most frequent combinations of text elements, such as Premise - Consequence - Elaborator and the use and frequency of different logical connectives. It finishes by commenting on the ways that students dealt with possible actions or events that arose in the Plan and the Explanation parts of this task. Being able to deal with possibilities is important, as it relates to the early stages of providing generalisations in mathematical explanations.

Elaborators are text elements which provide further information about the element in which it is embedded or which precedes it.Elaborators were mostly relative clauses which began with 'which' or 'that' or a subordinate clause, such as one which began with 'if'. In this task, Consequences are the text elements that are most likely to be followed by an Elaborator. Physical Consequences and Conclusions are the elements which are the least likely to be used with an Elaborator. This can be seen in Table 4.41.

In the responses to the Explanation section, which were the longest ones, more students used more elements followed by Elaborators that either of the other two responses. However, most of this increase was as a result of more students using Premises with Elaborators at the beginning of their responses.

In Tables 4.5, 4.18 and 4.31, there were no significant differences in the numbers of students who used Elaborators according to gender. However, as evident in Table 4.42, once the distribution of students who used Elaborators with different text elements was examined, some differences became apparent.

Table 4.42 shows that, on the whole, combinations with elaborators were more likely to be used by girls than by boys. This was particularly so in the combinations of an Elaborator with a Premise or a Physical Consequence. However, Elaborators with Conclusions were most likely to be used by boys, although the numbers of students who used this combination were quite small. The distribution of Elaborators used with Consequences and Suppositions was dependent on the section of the task. When boys did use proportionally more Elaborators with other elements than girls, the differences were not as great as when girls used more Elaborators than boys.

The use of Physical Consequences is, unsurprisingly, closely related to when students were actually performing actions. This shows up in the large numbers of students who used Physical Consequences in their response to the Plan and to the Description parts of this task. There is also a relationship between the combination of Physical Consequence and Conclusion and clarity of language in the responses to the Description. 94% of students who were deemed to have given an accurate and clear response to this task included this combination in their response.

As was discussed in Chapter 1, Bills (2002) suggested that particular words in students' descriptions of mental computations correlated with students providing accurate answers. In considering the results from the three parts of this task, it can be said that certain combinations of text elements are also related to students providing accurate responses. The Premise - Consequence - Elaborator was used by a significant number of students in responding to each part of the task. In the responses to the Plan, 40% of students used this combination. In the Description part of the task, 26% of students used it and in the Explanation part of the task 53% of students used it. Students who were considered to be accurate were more likely to use this combination than students who were only approaching accuracy. Although Tables 4.13, 4.26 and 4.40 suggest that Premise - Consequence combinations were important, they also show that it was more likely that these combinations were followed by an Elaborator than not.

As well as the Premise - Consequence - Elaborator combination being important, in the Description part of the task, students who gave accurate and clear responses almost all used a Physical Consequence - Consequence combination. In mathematical explanations, when physical actions are needed, it would benefit students if they learnt to use this combination both to support the clarity of their language but also to help their thinking. This is because they move from just describing the consequences of their actions to drawing a logical consequence from the results of that action.

There also seemed to be some distinct trends in the use of logical connectives across the different parts of this task, which are related to how different text combinations were used. Certainly when the Physical Consequence - Consequence combination was used in the Description responses, more often than not this was joined with 'so' suggesting a causal relationship between the ideas expressed in the Physical Consequence and the Consequence.

Causality is not shown through logical connectives joining Premises to Consequences as had been the case in some of the other tasks. Girls and Year 8 students were the most likely to use a logical connective in this position, with 'and' being the most common logical connective. Instead, causality was marked through the use of 'if' in front of a Premise or an Elaborator in the Premise - Consequence - Elaborator combination. If 'if' was not used, then the causality had to be implied from the relative clause which was the Elaborator. For example, in 'take two and measure them first, and see which one's the heaviest, and then put the heaviest one up on the heaviest end', 'which one's the heaviest' is the relative clause which was coded as an Elaborator. It presumes that the outcome of finding which box is the heavier is easily done and so it can be incorporated into the following suggestion which was 'and then put the heaviest one up on the heaviest end'. Thus the causality is between the proposed action and what follows. Contrast this with 'then weigh them all, all of them to see if that's the lightest, and then just do it to all of them'. In this case the 'if' at the beginning of the Elaborator marks this causal relationship in a more pronounced way. However, more has to be presumed from the original action for the final result to be valid. Both sorts of Elaborators allow students to deal with possibilities in a succinct way so that further actions can then be described. It would seem that students used the Premise - Consequence - Elaborator combination as one way of dealing with possibilities.

In order to discuss in a clear and concise manner the possible ordering of the four boxes, students needed ways of talking about possibilities. This has many links to the ability to use generalisations, which are important aspects of mathematical explanations and justifications. In order to interpret and use them correctly, students need to know the constraints on an equation being true.

In this task, both the Plan and the Explanation sections required students to describe general ways of ordering the boxes. Anthony and Walshaw (2002) suggested that there was a need 'to discern generality in students' informal utterances' and to understand '[t]he interplay between generalisation and justification' (p. 52). By analysing their responses, it is possible to understand how typical primary students limited their generalisations so that they remained true. The next few paragraphs describe first the Premise - Consequence - Elaborator combination and how the logical connectives and other aspects of this combination limit when the possibilities were true. In many cases, this combination of text elements could be considered as examples of Bills and Grey's (2001) 'general' responses, as they provided a rule which would cover most instances. Then Suppositions, especially students' use of 'say' could be related to Bills and Grey's 'generic' responses, as they were used to provide examples of how these rules operated. Students who were only able to talk about the specific boxes in front of them would be considered as giving 'particular' responses. From Tables 4.13, 4.26, 4.40, there would seem to be a relationship between the use of Premise - Consequence - Elaborator combinations and/or the use of Suppositions with the responses which were deemed accurate and clear.

Tables 4.13, 4.26 and 4.40 show that more students used the Premise - Consequence - Elaborator combination when giving their Explanations, with the least number of students doing so when giving their responses to the Description part of the task. This supports the belief that students used this combination so that they could talk about the possibilities in ordering the boxes. The Premise - Consequence - Elaborator allows an action to be proposed and the consequent result of that action to be carried forward into the step, even though the result may not be definite. As was described earlier, sometimes a logical connective such as 'if' is used to mark the conditions under which the following action would be true. However, a relative clause, such as 'which is heavier', was also used where the conditions for the statement to be considered true are within the object itself.

Students also used Suppositions, such as 'say', to mark a possibility. Although the number of students using Suppositions was similar, the expressions used did differ across each section of Weigh Up. In responses to the Plan part, four students used 'say'. In their responses to the Description part only one student used 'say', whilst in their Explanation responses, five students used 'say'. By using 'say', students marked the fact that it was a possibility rather than a fact, as in the following: 'so say it was D, you put it in front of it'. 'Suppose' and 'imagine' were more Standard English words which also indicate that what was following was a proposition rather than a fact. However, only two students throughout the task used these expressions. It is interesting to note that these expressions which mark possibilities are verbs.

As well as verbs, adverbs such as 'perhaps' and 'probably' were also used occasionally to indicate a possibility. Although four students used these expressions in their responses to the Plan and another two students used them in their responses to the Description, no student used them in their responses to the Explanation. This would suggest that these adverbs have only a limited role in describing possibilities. However, when the actual examples such as the one from a Year 8 girl from a high decile school are examined this is not the case. The example was 'say B's the heaviest so A's lighter, I'd do that later, I'd perhaps put D, perhaps B's still heavier, so throw out D, then C, perhaps C's heavier'. Here 'perhaps' could be replaced by 'say' and the meaning would remain the same.

The large numbers of students who used 'you think' in the responses to the Explanation seemed to have used them like 'say'. This can be seen in the following example from a Year 8 boy from a high decile school: 'See how heavy it is in your hands first and then you put the two that you think are the lightest in'. Once again an action is suggested and the consequent result, although not a certainty, is carried forward into the following action. Although 'think' as a verb is also used to mark a possibility rather than an actual fact, the inclusion of the pronoun suggests that a person, 'you' could actually change the certainty of the actual result. 'Say' puts forward a possibility so that a proposed action can be described which could be used with any set of four boxes. 'You think' on the other hand is used to be more definite about those particular boxes.

However, the use of 'I think' in the responses although also coded as Suppositions does not appear to mark possibilities. Instead, it is used to highlight uncertainty about actual results of actions such as in the following from a Year 8 boy attending a high decile school: 'I think that that one is heavier than that one'.

Being able to describe possibilities is one way of being able to define the conditions under which certain statements are true. This is an important component of being able to describe mathematical generalisations. Although Esty (1992) suggested that this is mainly done through the use of logical connectives, this would seem to be only one way that primary school children do this. Although they can provide information on the conditions under which a statement is true through the use of 'if' at the beginning of an Elaborator in a Premise - Consequence - Elaborator combination, possibilities can also be described through the use of relative clauses as Elaborators in these combinations or by the use of 'say', 'perhaps' or 'probably' as markers of possibilities.

The Weigh Up task was a long, extended task with three related parts. Students' responses to these different parts showed that the structure of mathematical explanations do differ as the task requirements differ. Although the Premise was a text element found in all other tasks, some students, in responding to the Description, did not include a Premise. Instead a new element, Physical Consequence, was more commonly used by students in responding to this part of the task. It would also seem that students who were considered to give accurate and clear responses to this part of the task were most likely to combine Physical Consequences with Consequences. In the other two sections of the task, a Premise - Consequence - Elaborator combination was used by many students who gave clear, accurate responses. This has implications for the types of structures that may support students' being able to provide clear accurate responses, but there is also a need to make students aware of the type of tasks which are most likely to require different text combinations.