aIntroduction

Research on problems like this has drawn attention to the fact that students are usually asked to suspend their real world logic when attending to mathematics word problems. Nesher (1981, cited in Verchaffel et al.) calls word problems 'a very special type of text'. Verchaffel et al. refer to their 'stereotyped nature' (p. 144). Students learn to respond to them in a manner that looks for the numbers, performs some operation, and may ignore the sense of them. This corpus of literature usually refers to written or textbook word problems. For some of these problems, the student's task is to strip away the unnecessary story aspects, then operate with the numbers in the manner expected, and finally restate the answer in terms of the story context. Thus 'twenty-seven apples plus thirteen apples' is 27 + 13 = 40, although to give a complete answer students would need to say '40 apples'. For problems such as the bus problem in Figure 2, the task is to operate in a similar manner but the students must consider the context again when giving the answer in terms of buses: not '31 buses remainder 12', or '31.333 buses', but 32 buses.

Operating with mathematics problems in this manner is part of the didactic contract between teacher and student. Both parties have to acknowledge that the words in a mathematics problem are often irrelevant in some manner. Sierpinska defines this didactic contract as,

The rules and strategies of the game between teacher and the student-milieu, which are specific of the knowledge taught. Sierpinska (1999). This definition draws attention to the fact that the rules or didactic contract may be different for mathematics than for other subjects, such as creative writing.

Another use of word problems in mathematics is to require students to provide a context in order to make sure that they understand the mathematics given in a numerical problem. One example of this appears in the Chelsea test of numerical operations, in which students are asked to write word problems that fit numerical problems, such as 35 ÷ 5 (Brown, 1981). This is a less common experience in the didactic contract. For this type of problem a fanciful story would be thoroughly acceptable if it fit the mathematical conventions (e.g., if 35 clouds had to be divided among 5 angels, how many clouds would each angel have?).

Difficulties with stories in teaching mathematics may arise because students do not differentiate the expectations for stories in this subject from stories in other school work. In reading, writing, and drama, students are often asked to read, make up or portray a story for the sake of the story itself, not to coat a mathematics problem. They are read stories by their teachers which may be purely fanciful, and may have no underlying meaning other than to entertain. These stories have different conventions to those observed in mathematical problems. They may start with 'Once upon a time', an introduction that immediately cues readers or listeners into the knowledge that a fantasy story is to follow. Characters are to be defined and there is a plot in which these characters interact. A good story builds interest or suspense and has some sort of a climax that precedes the ending, possibly one in which characters 'live happily ever after'. Stories of this type take up more of a young child's day than do the stories invented to coat a mathematical problem. This may set up confusion for students who need to know what type of story is expected in which subject, because the didactic contracts are different.

An additional source of confusion in the didactic contract reflected here was that, although these were school-type problems, they were not presented in a classroom. They were given in an interview with an unfamiliar teacher administrator. Although the interviewer was expected to stick to a script, the nature of the interaction led both parties to interact following rules that were more closely allied to the rules of conversation than to the classroom contract. The conventions of conversation involve a different contract, in which certain types of statement elicit different responses. If the teacher-administrator used a personal pronoun like 'you', the student could legitimately think that she was supposed to respond with 'I'. (see, e.g., Sacks, 1992).

For example, when introducing one of the problems in this study, one about cars going down a motorway, the following conversation took place:

This conversation was appropriate to the adult-child interaction but unnecessary for the mathematics of the problem.

Most Year 8 students appeared to understand the didactic contract for both the Motorway and the Bank Account tasks. They appeared to understand that the didactic contract was different for different types of problems. For the first problem they were expected to adopt the procedure of ignoring the story and dealing with the numbers. For the second problem they were expected to make up a story to explain a graph. However, several Year 4 children displayed confusion about the rules governing the use of stories in these two mathematics problems.

In the first problem, Motorway, they were expected to peel away the story about cars going down a busy motorway in one and nine minutes and attend to the numbers mentally multiplying 9 times 98, and then possibly add the term 'cars' at the end of their answer, although there was no requirement for this. For the second item, Bank Account, they were expected to create a story to match a graph that described the mathematics represented, but despite the fact that the graph was probably fanciful for a student, the story that they made up should not be too fanciful.

One appropriate answer to the Motorway problem was:

888 [I: Explain to me how you got your answer] Well, I just went up to 98 plus 2 is 100 and 9 100s are 900 take away… 2 nines… (Low decile non-Pacific boy).

An appropriate answer to the Bank Account problem was one that gave the daily balance and explained deposits and withdrawals.

One appropriate answer to this question was:

He's ah just got his bank account and he got ten dollars with it, his um mum put in another ten dollars and he's got twenty dollars, but he ah, didn't do all the jobs in his house so um, and he wanted to buy something, he didn't get his pocket money for not doing the jobs in the house so he took some out of his money, and ah he left the his bank account alone for another day, and then he decided he'd that he um, wanted something else which was better than the last one and so he got out some more money. (Year 8 non-Pasifika boy from a low decile school)

Another appropriate answer from a student who was less fluent was:

And there was ten dollars, then he decided to go back on Tuesday, then he had twenty dollars, then he went back, then he went on Wednesday and he still had twenty dollars...on Thursday, at his work, it went down a little, cos he did a little...what should I say (whispers to himself)..mistake, and Friday he had a day off cos he was sick, and then on Saturday he had ten dollars. (Year 4 Pacific boy from a low decile school)

The conversational conventions of both of these questions require the student to, 'explain' or 'tell me'.

Our interest in relation to the first problem is in students who held to the conventions of story telling rather than stripping these away and responding to the mathematics in these stories. Our interest in the second problem was different, in that students were given a mathematical representation and asked to make up a story about it. It provides another attempt to make mathematics relevant to real life without it actually being relevant, because it is hard to imagine a student with a bank balance that changes daily in this manner. The conventions of making up a story for this situation would be that the story has to relate closely to the graph, explaining each change, and that it should relate events that could make the bar on the graph higher or lower on different days, but, also, that it does not need to contain extraneous information and does not need to start with 'Once upon a time…'

Thus 33 of 36 Year 8 students followed the didactic contract, either dealing in mathematics or saying that they did not know. The three who did not give mathematical excuses used proto-quantitative terms, like 'lots of cars' and 'too many cars'. The latter response appeared to come from a student whose first language was not English. Two of these students had an answer that was close to the correct one (810, 950) and the other student was well out (100).

Among the Year 4 students most of the stories were told by low decile students, and more often by girls (6) than by boys (2). All of the students who told stories had incorrect answers. A question arises here: why did some students who were uncertain said that they did not know (10 students) and while some made up stories (8 students)? The students who told stories may have been responding to the conversational conventions covered by Sacks (1992) and trying to please the teacherinterviewer.

Below are the answers that were judged to be stories that did not reflect the mathematical content. They were more elaborate than the protoquantitative explanations given by Year 8 students.

Although four of these answers are given by students of Pacific descent, only (g) gives a story that has evidence of English being an additional language once allowances have been made for the disjointed nature of speech in conversation. This is evident in the lack of a tense marker in 'they busy' and 'they waiting'. It is possible that this boy meant traffic lights when he said 'waiting at the traffic'.

The fact that all of these stories followed either answers or statements indicating that the student didn't know suggests that they did understand the part of the didactic contract that indicated that they were to give a numerical answer. These answers also follow the conversational convention that 'How many…' was to be answered by a number. As they did not know the correct answer or were unable to describe how they achieved it their stories appeared to be an attempt to follow the conventions of conversation. An adult asks you a question after telling a story about cars on a motorway. Therefore it is reasonable to attend the story aspect, especially if you do not understand the mathematical aspect. In a landmark book, John Holt (1964) outlines the strategies that students use when they are not able to complete a problem. When students are aware that they cannot do a problem, they use a variety of strategies to save face. In a direct conversation, this means answering some aspect of the interviewer's question. This can be done by talking about cars, motorways, and people being busy and there being lots of traffic, as is normal in conversation about traffic. The response of 'don't know' may come from either a confident student who knows the limits of his or her knowledge, or from a student with little confidence. Facial expression and the slope of their shoulders usually distinguish which of these this is. Students' main goal is to get away from the unpleasant situation of being in the spotlight. They use what he calls a 'safe policy' (Holt, 1964, p.5).

Several answers include words that are proto-quantitative, that is, that refer to quantities in an inexact manner. These include: 'lots of'; 'half' (used in an inexact manner to mean some); 'heaps of'; and 'a lot of'. There are several elements in the stories that elaborate on the imaginative context. These children add bridges, visitors, people getting angry and not being able to wait, getting to work by 4 o'clock, traffic lights, and lack of a speed limit. All of these are elements of a story that would do well in creative writing, but are not part of the expectations for a mathematics story.

In contract to the Motorway problem where students needed to strip away the story elements and deal with the numbers, in the Bank Account item they were expected to take an unrealistic mathematical representation and create a story around it. Although this type of request of students is increasingly popular, especially for interpreting graphs, it is still relatively unusual to children who have not learned the conventions of stripping away story elements.

The graph is improbable in children's lives, as they are most unlikely to add small amounts to their bank account or deduct small amounts from it on a daily basis. They are also most unlikely to see a graph of a bank account. The stories of some students indicated that they had seen graphs of the amount of money raised, but that these graphs did not include deductions.

An adequate answer would be:

Okay, he starts off with ten dollars puts ten dollars in the bank and then he does some work and gets another ten dollars it goes up to twenty, on Wednesday he didn't do anything with it, on Thursday he needs to buy, lunch, or something and it cost him five dollars, and it goes down to .. oh, fifteen, and then Friday nothing, and Saturday he needed to buy something else that cost five dollars and it went down to ten. (Year 8 boy from high decile school)

This would be satisfactory because it mentioned all of the appropriate aspects of the graph, but did not include too much extraneous material.

Table 7.3 gives the types of stories told in response to this question. More categories are used to describe these stories because they differed in more ways. They ranged from stories that only indicated that the height of the bar was different on different days, through to ones that had no more than the necessary story elements, through to ones that still referred to the graph but had many elements. In assigning the stories to types, no attention was paid to accuracy.

With this problem it was also the case that the Year 8 students understood the didactic contract better than did the Year 4 students. None of the Year 8 students told stories with superfluous elements or ones that did not refer to the graph. Where they erred, it was on the side of brevity, merely describing the bars of the graph being higher or lower rather than indicating that money had been added or taken out. The brevity of their answers was clear can be seen when they said, for example,

Well, it's getting um lower um, Monday it's getting higher on Tuesday and Wednesday, lower on Thursday and Friday and Saturday it's lower. (Yr 8 boy from high decile school)

These students reduced their answer to attend only to the essential aspects of the graph, just as students in the Motorway problem had reduced the problem to the essential elements of '98 times 9'.

Among the Year 4 students, six described the changing length of the bars and 19 students told minimal stories. The 19 students who told minimal stories adhered to the didactic contract for this type of problem. Many of these stories indicated that they did not understand that it was the balance in the account that was changing, but that is 124 not the focus of this chapter. Of interest here are the 10 students who told stories that went well beyond the demands of the question and sometimes disregarded the graph altogether. These students all came from Year 4 groups except the boys from high decile schools, whose answers were more like those of Year 8 students. These six boys understood that the question required describing the mathematical representation only. The other groups of Year 4 students were less certain about these rules. Twelve Year 4 students, spread across all groups except high decile boys, relied more on story-telling conventions, as would be expected in school in their writing and reading, and possibly out-of-school settings.

The stories with no reference to the graph were:

All of these stories indicate that the students knew that banks were related to money and that money goes into banks and is taken out of them, usually by other people. The numbers of questions asked by the teacher administrators indicate the difficulty that they had in getting students to tell these stories. However, they do not indicate an understanding of a graph or even that it is the graph rather than the bank that is expected to be the subject of the story. Several of the answers do have the elements that suggest personal involvement in the story that they make up.

Three of the stories that included unnecessary elements were those below. For the sake of brevity, only the extra elements are included.

Um .. the robber is .. tooken .. fifteen dollars.[Q] They took twenty five dollars.[Q] .The same, on Tuesday.[Q] They took .. twenty dollars. …. (Yr 4 Pacific boy from low decile school)

These stories, which all referred to the graph, had some of the same story elements of the stories above that did not refer to the graph at all. They included robbers and a man who was not paid because he did not work.

They differ from two other such stories told by upper decile girls that are given below. These two stories indicate that these girls were well versed in the rules of story telling. These rules or conventions appeared to overrule the need to describe what had happened as displayed in a graph. The videotape of one of these transcripts below shows the girl looking into space or at the interviewer as she tells her story, rather than referring to the graph. She does start by referring to the graph, but after this initial reference to money being put in a bank, she is not concerned with the details of the graph. The teacher-administrator goes beyond the intended script in attempting to draw her attention back to the graph. The comments of both the girl and the teacher-interviewer are, therefore, included.

The other girl from a decile high school attended to the graph much of the time but also looks into space and smiles while adding details like buying sugar and cherries. She smiles at the teacher-administrator when adding some details, as though she is pleased with her creativity.

Oh, like do you...like do you, um, make a story up? [Q] ....But how do you know if it's a man or a girl? [Q] On day on Monday the man, the man went to his bank account and, he had...ten dollars in his bank account, so he, he, he left it to the next day to get some money out, and on Tuesday he had twenty dollars, so he um, twenty, thirty dollars altogether, so he, he got his ten dollars out and he went to the store and bought some sugar and some cherries, and then he, on Wednesday had twenty dollars as well, um…um...so he had forty dollars left...um....he left it in his bank account to save up, so he, he …said 'oh I'll just look through all my money, so he's and see how much I've got, so on Thursday, um, Thursday, um, he looked how much money he had, he had um...fifteen dollars, um, and that mean, meant he had...fifty-five dollars, and then he, he went to the store, the bank account the next day, and, and then he looked, um, how much money he had, and he had fifteen dollars again, and so he had seventy dollars, and then on Saturday he went there and he saw, he saw that the bank got robbed and his money wasn't there. (Year 4 girl from a high decile school)

These last two stories include several story-telling conventions that are not usually included in mathematics stories. The first student shows that she knows how to start a story, with 'Once there was', an equivalent of 'once upon a time'. She establishes her character and begins the plot with some background. If she had not been interrupted by the interviewer, it is possible that she would have told a complete story without reference to the specific nature of the graph. These extra questions break the train of story telling, but do not result in an accurate story for the graph.

The second girl needs first to establish who the characters in her story are. Having satisfied herself on this, she then starts her story. Her story is basically one long sequence with separate episodes joined with 'and' or 'so', but these lead to a surprise climax on Saturday when she discovered that the bank had been robbed. It even includes direct speech by its character. It includes mathematical misconceptions, such as an apparent belief that the money on each day is not the total but the amount that mysteriously arrives in the account. The story would be likely to earn praise and receive full marks in creative writing or story telling, but fails as a mathematical description.

Stories of the students from lower deciles included suggestions of personal poverty such as 'My mum hates to go to the bank, to get some power'. Others reflected communal ownership of money in money owned by 'the people' (P044-C2, P023- C3). This could reflect church or other money raising activities in which communities worked together. Several students associated the graph with only deposits or only withdrawals. In one case, this was a story of money being raised, a situation that they would have been more familiar with than with a graph of deposits and withdrawals.

These students responded primarily to the aspect of the question that required them to tell a story for the graph. Few appeared to have experience with a bank balance that changed daily. The agents in their stories were other people - the man, the people. One student asked, 'how do I know if it is a man or a girl?' Only one student used 'she' as person responsible for the changes.

Why did the students quoted above tell stories that were not closely related to the mathematical issue to hand? For both questions, the younger students told these stories while few of the older students did. Thus it is possible that they did not know the mathematics required and possible that they did not understand what was required of them under the didactic contract. In the absence of this knowledge, they responded to the rules of conversation and answered the interviewer literally, giving a story that would be appropriate in conversation.

In the case of the Motorway question, all students who told stories had incorrect answers. This was also true for the Bank Account answers that did not refer to the graph and for some of the stories with extraneous material in them, but this was not always the case.

Seen through the lens of the didactic contract, it appears that older students clearly understand the contract for both types of mathematical stories. They were aware that the stories are unnecessary shells that can be discarded in the case of the Motorway problem and kept to a bare minimum in the case of the Bank Account story. Among 127 younger, Year 4 children, the high decile boys were most aware of the requirements of the didactic contract. They may have had an advantage in that they were less interested in verbal fluency which could have interfered with succinctness. In both the Motorway problem and the Bank Account problem, the groups that told the most irrelevant stories were low decile, Pacific students, followed closely by non-Pacific low decile students. For both problems, they told stories in cases for which they did not know the answer but did know the conventions of story telling or at least had some personal experience to share. This sharing often came after repeated prompts or probes. Stories with extra elements, like robbers or what the money was needed for, occurred in answers that were correct, although these extra elements were not needed for the didactic contract in a mathematical context. Their creativity was a pleasure to see. It is rather a pity that, from the evidence of Year 8 students, this creative flair is likely to disappear in a few years time.