ࡱ> 5@ bjbj22 gXX x x x $ ...P4/\/ TJ8800"0004:Y4m4 IIIIIII$KRM^Ix A3"4AAI00!JCCCA:"0x 0ICAIC CD:Fo rx 'G0,0 .—.MA&G GL$J0TJG General introduction Since a learners trajectory through the differentiated tracks that constitute the school is determined partially by his or her assessed performance at key branching points, any sociology of learning, broadly conceived, will need to address assessment. Clearly, to be equitable, modes of assessment should not cause children from some socio-cultural backgrounds to display their knowledge and understanding more or less readily than those from other backgrounds. Educators should want to avoid a situation, for example, where a child speaking language A, with good knowledge of some discipline, may be found to perform poorly in a test of this subject written in language B. Unfortunately, not all examples of bias in assessment are as clear cut as this. In this chapter I discuss a subtler example of potential bias arising in the use of realistic problems in tests. Furthermore, my examples will be drawn from a disciplinary context frequently portrayed as socially neutral, that of mathematics. I begin with a brief summary of Bernsteins concepts of classification and framing, and recognition and realization rules. This framework is then used to discuss some peculiar features of mathematics assessment items from the perspectives of both the test designer and the teste, and then some empirical findings from studies with which I have been involved in recent years. Bernstein In his original paper on the classification and framing of educational knowledge, Bernstein began by arguing that formal educational knowledge a major regulator of the structure of experience is realized through the three message systems: curriculum, pedagogy and evaluation. The first defines what counts as valid knowledge, the second what counts as valid transmission of knowledge, and the third what counts as a valid realization of this knowledge on the part of the taught (Bernstein 1973: 363) The concept of classification was introduced to describe the degree of permeability between the different disciplinary contexts making up a schools formal curriculum. In one type of educational knowledge code, the collection code, contents such as mathematics, science and history are kept apart, while in an integrated code the various contents are subordinate to some idea which reduces their isolation from each other (Bernstein 1973: 378). The concept of framing described aspects of pedagogy. At this time, Bernstein used frame to refer to the strength of the boundary between what may be transmitted and what may not be transmitted, in the pedagogical relationship (Bernstein 1973: 366), or, more specifically, to the degree of control teacher and pupil possess over the selection, organization, and pacing of the knowledge transmitted and received in the pedagogical relationship (Bernstein 1973: 366). His paper then uses these concepts to analyze a range of curriculum types, producing an insightful account of the possible implications of each type for the development of learners identities. It also includes some speculative discussion about the relation between educational and everyday knowledge under the collection code regime, and it is this discussion that is of most relevance to my topic here. In the context of an educational knowledge code where the values of classification and framing are both high, Bernstein argues that pupils must learn to work within a received frame. They must learn what questions can be put at any particular time since, because of the hierarchical ordering of the knowledge in time, certain questions raised may not enter into a particular frame (Bernstein 1973: 375). This is something that is soon learnt by both teachers and pupils (my italics). He goes on to describe educational knowledge as uncommonsense knowledge, freed from the particular and the local. He notes that this raises the question of the relation between the uncommonsense knowledge of the school and the commonsense knowledge, everyday community knowledge, of the pupil, his family and peer group (Bernstein 1973: 376). He suggests that the frames of the collection code, very early in the childs life socialize him into knowledge frames which discourage connections with everyday realities, or that there is a highly selective screening of the connection (my italics). Through such socialization, the pupil soon learns what of the outside may be brought into the pedagogical frame (my italics). He notes that this frame is sometimes weakened to allow everyday realities into the classroom, but usually for the less able (Bernstein 1973: 370/376). Arguably, there is some ambiguity in Bernsteins use here of the concepts of classification and frame. The discussion of the boundary between educational knowledge and commonsense knowledge makes use of frame even though it appears to be a matter of classification, as previously defined (something also noted by Atkinson 1985: 136). There seems no good reason for not describing the boundary between educational and everyday knowledge in terms of classification. In later work Bernstein adds to the discussion of classification and framing the concepts of recognition and realization rules. The need for these arises when he begins to offer an account of how the learner, or acquirer in his terms, is able to distinguish between the categories marked out by the boundaries in a highly classified set of discourses. If a learner is to be able to respond in appropriate ways in a particular disciplinary context, then he or she must be able to first recognize the context as an example of, say, school science, and then must have in his or her repertoire the capacity to respond in a legitimate way, as well as the required scientific knowledge and understanding. Bernstein models this in terms of the acquisition of rules. Recognition rules, which the learner infers from his or her previous experience of the nature of the specialized categories and the ways in which they are demarcated from one another, regulate what meanings may legitimately be put together (Bernstein 1990: 29). These rules concern relevance, and create the means of distinguishing between and so recognizing the speciality that constitutes a context (Bernstein 1990: 15). The realization rule, on the other hand, regulates how we put meanings together and how we make them public (Bernstein 1996: 32). In this chapter I shall be concerned with recognition rules. For a discussion of realization rules in relation to mathematics assessment, see Cooper and Dunne (2000: chapter 8). As we have seen, in the classification and framing paper, Bernstein suggested that pupils learn what he later called recognition rules early in their school careers. In 1996 he was making a comparable claim, but with the qualification implied by the use of many in the following quote: We may have the recognition rule which enables us to distinguish the speciality of the context but we may still be unable to produce legitimate communication. Many children of the marginal classes may indeed have a recognition rule, that is, they can recognize the power relations in which they are involved, and their position in them, but they may not possess the realization rule. If they do not possess the realization rule, they cannot then speak the expected legitimate text (Bernstein 1996: 32). In fact, several studies carried out by Bernstein and co-workers had shown that many children from working class backgrounds tend to employ different recognition and realization rules than children from higher social origins. An example will make this clearer for the reader unfamiliar with Bernsteins work. Holland (1981), working with Bernstein, presented primary age children from various socio-economic backgrounds with a set of 24 colored photographs of food items. These could be organized through context-independent principles, as shown below: animal: roast beef, pork chop, sausages, hamburgers, sardines, fish fingers vegetable: lettuce, green beans, peas, boiled potatoes, chips, baked beans animal products: milk, butter, cheese, iced cream, boiled egg, fried egg cereal: bread, cakes, biscuits, rice, rice crispies, spaghetti rings As part of the work, children were asked to group the items and explain their grouping. The instruction was: Here are some pictures of food. What we would like you to do with them is put the ones together you think go together. You can use all of them or you can use only some of them (Bernstein 1996: 33). Social class differences were found in childrens initial responses. Middle class children were more likely to use general principles of classification, e.g. The same kind of thing, both made from milk or Those two you get from the sea. Working class children were more likely to refer to their everyday life, e.g. Thats what we have for Sunday dinner. Given a second chance to classify the pictures, middle class children often switched to a classification derived from everyday life, while working class children remained in the everyday mode of response. It should be stressed that the differences between the social class groups in Hollands work were not differences of kind, but of tendency. Some working class children did use non-everyday classifications. Bernsteins account of these findings in terms of classification and framing appears in his The Structuring of Pedagogic Discourse (Bernstein 1990). He argues that, although the spoken instructions are characterized by low values on both classification and framing (-Cl-Fr), the middle class children initially ignore this, and transpose the -Cl-Fr to +Cl+Fr. The working class children, on the other hand, read the instruction at its face value. He argues that this observed difference is not one of cognitive facility but of a difference in the recognition and realization rules the children used to read the context and create their text (Bernstein 1990: 55-56). Those familiar with Bourdieus work will be able to see that these differences might also and without reference to rules - be described in terms of his concepts of habitus and feel for the game (Bourdieu 1990; see Cooper 1998b for an application of these concepts in the context of mathematics assessment). Having introduced the concepts of Bernsteins that I wish to use, I now turn to the discussion of mathematics assessment and, in particular, to the ways in which realistic references in mathematical test items might be expected to raise difficulties of recognition for testes. I then present some findings from previous research which suggest that the class differences Bernstein has discussed have an effect in the context of mathematics problem solving. Overview of realistic mathematics problems There is a very long history of testing mathematical understanding and competence through problems which embed a mathematical operation in some textually represented everyday context. Two important factors vary. One is the extent and/or depth of the contextual reference. The other is whether the question setter actually wants factors implicit in the context to be attended to or ignored. In the past, and frequently today, the contextual reference is of a token nature and is not intended to be taken seriously. This was certainly the case for the items below, which I have taken from an Advanced level text I studied in the late 1960s. They appeared in Porters Further Elementary Analysis, in a chapter introducing the topic of permutations and combinations. Examples 1c begin: In how many ways can: 6 prefects be chosen from 9 boys? 4 clerks be chosen from 15 for promotion? A committee of 3 be chosen from 10 aldermen? 3 books be chosen from 7 different books? 11 players be chosen from 15 boys? . It is unlikely that any of the sixth form pupils (aged 16 to 19 years) tackling these items will have paid any attention to the people or objects appearing in them. In each case what is intended is likely to have been clear to them. The people and objects are understood to serve as mere tokens. At this time, the small minority of pupils who entered the English sixth form will have understood this implicitly. These pupils, in Bernsteins terms, will by this stage have learned what questions can and cant or, perhaps, should and shouldnt - be asked within this particular frame of pure mathematics. Certainly, drawing on my own experience as a pupil studying this text, I can recall no point ever being made by any pupil or teacher about the people or events referred to in these items. The way the problems are presented provides no cues to do other than treat the people and events referred to as an irrelevance. Unlike more recent problems (see below) there is no explicit descriptive reference to a surrounding context in which these choices are taking place. If we look at three more recent examples, we will see that, especially for younger pupils not yet so well inducted into the frames of reference of school mathematics, the recognition of what is an appropriate response may be likely to prove more difficult. Since the 1970s, when the abstract approach that flowered briefly within school mathematics in the 1960s began to be rejected (Cooper 1985), and especially since the publication of the Cockcroft Report of 1982, the fashion in English school mathematics apparently has shifted towards favoring the use of broader and deeper contexts in the teaching and assessing of the subject. The apparently here is important, as we shall see, in considering the first two of the examples to be discussed below. Three Examples Three examples have been chosen to show different and conflicting - ways in which a requirement to think realistically might be intended to be read. They can serve to illustrate the problems of recognition that may arise for pupils from the current lack of clarity over realism within mathematics education. I have assumed that we can infer question designers intentions from associated marking schemes where these exist. The first example (intended for nine year-olds) is taken from the initial exemplar items supplied by Englands Qualification and Curriculum Authority (QCA) to illustrate the nature of the World Class Tests then under development in the UK. This item, which received national prominence in The Times Educational Supplement coverage of these tests (TES 17/11/00), began by stating that Tom compares the weight of apples, bananas and pears. He finds that seven pears weigh the same as four bananas and five bananas weigh the same as six apples. The question then asks first, which single piece of fruit weighs the most? and second, which single piece of fruit weighs the least? The World Class tests are aimed at the top ten percent of the ability range, and so we have an item here intended to be answered by the most able nine year-olds in English schools. The required answers are banana and pear. It is not difficult to find a way of generating these answers. Using an algebraic approach, we might proceed as follows. Solution 1 to the fruit item Let p be the weight of each pear, etc. Then we have 7p = 4b and 5b = 6a. From the first of these, we have b>p. From the second, we have b>a. Therefore b is greater than both a and p. Therefore the banana is the heaviest. We also have p =4/7 of b, and a = 5/6 of b. Therefore, since 4/7 < 5/6, we have p less than a. Therefore, since both are less than b, we have the pear as the least heavy. However, to proceed in his way we need to make the assumption that each pear equals each other pear in weight, etc. An alternative solution might proceed without making this assumption: Solution 2 to the fruit item This problem is insoluble, unless we assume that all of the bananas (and similarly all of the pears, and all of the apples) are identical to each other in weight. It must be a trick question, and, as one of those thought by my teachers to be amongst the top ten per cent in ability, I must be expected to spot this! Ill say theres no unique solution. We can see that the teste has a simple choice to make here - whether or not to employ everyday considerations in developing a solution. There is a serious issue here. The rhetoric surrounding these tests has suggested that the teste will be intelligent, flexible and creative, and that problems will often be set in everyday, if sometimes unfamiliar, contexts. One press release from the QCA announced of the mathematics tests: The maths tests aim to encourage deeper thinking about topics that children will have covered in normal lessons. The questions are set in everyday contexts that children will have experienced or thought about. The problem solving questions require children to apply their existing knowledge, skills and experience to unfamiliar situations. They are set in the context of science, design and technology and maths and children need some knowledge of these subjects to tackle the questions. (QCA 2001) The rhetoric e.g. deeper thinking, everyday contexts, etc. - suggests that a realistic response to problems might sometimes at least be appropriate. However, drawing on our knowledge of the required answers, and using Bernsteins descriptive terms, we can see that the item under discussion here notwithstanding the context of weighing fruit - is actually intended to be part of a strongly classified category of activity. This must be so, since otherwise it would seem entirely rational for an intelligent, creative teste, expecting everyday problems, to offer the second wrong solution. What is actually required here is a teste well socialized, at the age of nine, into the frame of reference of school mathematics and able, therefore, to recognize that knowledge drawn from everyday life about fruit must not be brought to bear. (I realize that in these days of supermarket-led homogeneity a particular childs experience of fruit may not contradict entirely the assumption of equality set out in the first solution, but this does not, I think, affect the illustrative point being made here. Indeed, that childrens experience of the relative weights of fruit will vary as a consequence of their particular contingent experience of shopping would seem a problem in its own right.) My second example is one I have discussed previously. An early national test item for 13-14 year-olds concerned a lift (SEAC 1992). A sign was reproduced from this lift, stating, This lift can carry up to 14 people. This was preceded by the sentence, This is the sign in a lift at an office block and followed by In the morning rush, 269 people want to go up in this lift. How many times must it go up. Here, in comparison with the Advanced level examples discussed above, or the fruit item, we have more detail. We are told the place and the time of day, allowing us to infer that the lift will be in some demand. A single answer is allowed in the official marking scheme. This is 20. The teste is supposed to divide 269 by 14, obtaining something like 19.21428, and then, having noted that this non-whole number of lift trips is not meaningful, is required to round this up to 20. As I have argued elsewhere (Cooper 1992), this requires the child to introduce a particular realistic consideration. However, any child introducing more general realistic considerations (and arguing, for example, that it is impossible to provide a precise answer, given the fact that the lift may be less or more than full on some trips) will fail to gain the mark on offer. Once again, we have an item which clearly forms part of a strongly classified activity, notwithstanding its contextual references outside of mathematics. In Bernsteins terms, a quite subtle act of recognition is required. The cues supplied by the context must not be taken as a sign that fully realistic reasoning about a relatively open system is required. For my third and last example I want a problem where the real is actually intended to be taken fully into account in the solution of the problem. Such a problem is likely to step outside the closed system of mathematics or, in Bernsteins terms, to transgress the boundary between mathematics and other disciplines or between mathematics and everyday knowledge (or, of course, to be exhaustive, to involve a major redefinition of what counts as mathematics). School problems which seem, at first glance, to fit this description often require the children to use their knowledge of arithmetic - and measurement in particular in respect of some materials or objects drawn from everyday life. Best buy problems, in which children are asked to discover through a series of calculations on the dimensions of, say, real cereal packets, that it is cheaper per unit of weight to buy larger packets, provide one example. This might seem to meet the requirement of bringing the real in, but it is still the case that this is not the sole basis on which a real decision about purchasing might be made, since the purchaser would also have to consider other factors such as available storage space and the shelf life of the product (Lave 1988: 120). The real situation is actually less closed than it appears when set out in typical mathematics problems. Clues as to what an authentically open problem might look like can be found in Her Majestys Inspectors comments on mathematics, made soon after Cockcroft (1982) had reported: It is worth stressing to pupils that, in real life, mathematical solutions to problems have often to be judged by criteria of a non-mathematical nature, some of which may be political, moral or social. For example, the most direct route for a proposed new stretch of motorway might be unacceptable as it would cut across a heavily built-up area. (HMI 1985: 41) There is a boundary drawn here between mathematics and political, moral and social issues that most mathematicians would, I suspect, accept. There is also an implicit recognition that mathematical problems and real everyday problems are not coextensive, and that a wide range of non-mathematical factors will be relevant to everyday problems, even where mathematics has something to offer in their solution. In this perspective, mental operations that have formed a part of traditional school mathematics might be brought to bear as just one contributory element of a more integrated real problem solving process. More radically, at the level of school mathematics in particular, there have been various attempts made to redefine what is to count as mathematics, indeed to dissolve the boundary between mathematics and non-mathematics, usually under the slogan of critical mathematics. Gates (2002) includes several examples of critical problems. One concerns best buys and shows where a concern with authentically realistic problems might lead. Gates development of this genre provides my third exemplar item. The pupil is asked to: Consider how fair it is that larger sizes cost less (my italics). Gates also argues that, since traditional mathematics will unsurprisingly discover that food is cheaper if brought in larger sizes, it raises the question of why they make small sizes. The pupil is asked to explore a range of additional questions. Who actually buys the small sizes? What reason do people have for buying small packets rather than large ones? Is there any correlation between the shoppers income and the sizes of cereals purchased? How much does the packaging cost when you buy various foods? (Gates 2002: 224) Fairness of the sort implied in Gates item is a topic traditionally more likely to be treated in philosophy than mathematics. If this problem were to appear as part of a mathematics assessment then clearly, in Bernsteins terms, we would have seen a change in the degree of classification. The boundary between what are traditionally regarded as mathematical and non-mathematical issues would have become more permeable. Having now discussed a range of problems in order to illustrate the different ways that the boundary between traditional mathematical topics and the everyday world can be treated, I want to summarize the discussion especially concerning the problems such items cause for pupils with the help of the diagram in  REF _Ref13889938 \h  \* MERGEFORMAT Figure 9.1. This describes three levels of realistic consideration which the test designer intends to be brought to bear in the solution of problems presented in some apparently realistic context. Figure 9. SEQ Figure \* ARABIC 1: Three types of realistic problem with respect to the nature of any extra-mathematical thinking required of the teste (to gain full marks)  Box 1 is intended to contain those realistically contextualised mathematics problems which, in spite of the context, actually are intended by the test designer to require no extra-mathematical considerations to be brought to bear. As I have already argued, we can infer these intentions from marking schemes. Such problems would include the examples I quoted from the Advanced level text as well as the QCAs fruit item. As we move across the first boundary we pick up problems where one or more particular realistic considerations are seen as relevant by the test designer, but where the marking scheme makes it clear that introducing realistic considerations in general would not lead to an approved solution. The lift item is our current exemplar. As we cross the second boundary we move into the domain containing problems like that taken from Gates article. We might also find here some problems from the sciences which involve the use of a variety of mathematical techniques, including those associated with modeling. Of course, we can place problems in the appropriate box only if we can access the associated marking scheme. Furthermore it is clear that a change in the marking scheme can move a problem across a boundary. If, for example, the lift item were to allow a wide range of responses, with appropriate accompanying reasoning, as correct answers, then it would move from Box 2 to Box 3 in the diagram. The problem for the child is that he or she will not have access to the marking scheme for any item faced in an assessment context. Instead, tacitly or otherwise, testes have to draw on their previous experience of mathematics texts and tests to make a stab at recognizing what is intended in any particular case. While we can use marking schemes, retrospectively, to place items in one of the boxes with a fair degree of confidence, testes must use whatever cues are available in the item itself, in conjunction with previous experience of the peculiarities of the genre. We can note, at this point, that it is not clear that any simple, or single, recognition rule, to use Bernsteins terminology, will allow a child to infer correctly what elements of the real world should be introduced into his or her solution. Some Research Findings I now draw on previous research to confirm that children are likely to experience difficulties in reading test designers intentions and that the extent of these difficulties is likely, as we might expect from the results of such work as Hollands (1981) on the food classifying exercise, to vary with social class origin. I begin with an illustration of the ways in which children read problems in ways which are not those apparently intended by test designers. A problem in the English national tests for 10-11 year-olds (SEAC 1993) concerned a traffic survey. Children were told that the children in a school had conducted a traffic survey outside their school for one hour. This sentence was followed by a table of data reporting the number of cars, buses, lorries and vans that had passed in one hour (respectively 75, 8, 13, 26). Below this appeared: When waiting outside the school they try to decide on the likelihood that a lorry will go by in the next minute. And then: Put a ring round how likely it is that a lorry will go by in the next minute, followed by the choices: certain, very likely, likely, unlikely, impossible. The same question was then asked of a car, with the same five choices again available for selection as the answer. There is much that can be said critically about the mathematical features of this item (Cooper 1998a), but that is not my focus here. What became clear in some exploratory work I carried out with a small sample of 15 children was that children varied in a key aspect of the method they used to solve this problem. Some employed the data made available in the table while others drew instead on their own everyday experience of roads and traffic to make their two choices. Here are some illustrations (Cooper 1998a). Jane, who had chosen unlikely for lorry, seems to have referred to her real world experience: BC: Why did you put unlikely for the lorry? Jane: Um, lorries go very slow, and I dont think theyd come round the corner, um, in a minute. BC: You dont think theyd come round the corner in . Jane: Um, they might not, theyre not very fast, and, um, you dont very often see lorries but you see lots and lots of cars. Graham, who had chosen very likely for the car, argued from his everyday experience: BC: ... What about the car? Why have you put very likely? Graham: Cos there are loads of cars. BC: Because? Graham: There are loads of cars all over the place. BC: There are loads of cars all over the place. Right, this says a traffic survey outside the school. Are there loads of cars? Graham: There are some just down there. [He gestures outside of the window.] BC: There are some just down there. What this road, outside? Graham: No, just down on another road. BC: What, about how far away? Graham: Um, theres the school ground, theres the school field here, and then theres the block of flats, and then - that road. Diane chose unlikely for the lorry, but the nature of her reasoning was quite different from Janes or Grahams: BC: Right, why have you chosen those two? Diane: Well, because in an hour, if there were, if there were 60 minutes in an hour, then if there were only 13 lorries in one hour, its not very likely that, um, youre going to get a lorry just in one minute. And the cars - cos there were 75 cars, more than one a minute, so its more likely that youd [pauses, stops]. In later work, employing more focused questioning with a larger sample, the relation to the childs social class background of the different responses given when children were asked to explain their answers was explored (Cooper and Dunne 2000). The results are shown in  REF _Ref13971687 \h  \* MERGEFORMAT Table 9.1. Children of 10-11 years of age from working class backgrounds were considerably more likely to employ everyday knowledge than were children from service class origins (see Cooper and Dunne 2000: 17-18 for details of these class groupings). In the case of this particular item, because the distribution of vehicles in the given data was similar to what the children had experienced in their everyday urban lives, they were not disadvantaged by this tendency. Indeed the mean marks achieved (out of two available) for the three social classes were 1.08 for the service class and 1.00 for the two other classes in the group test situation. I doubt that the test designers intended children to gain marks technically false positives - even when they had ignored, as many clearly had, the given data from the survey. Table 9. SEQ Table \* ARABIC 1: Distribution of response strategies to the traffic item in interview by social class (Cooper and Dunne 2000: 103) Uses given data aloneUses everyday knowledge and given dataUses everyday knowledge aloneTotalsService class (salariat)38101159Percentage64.416.918.6Intermediate class1610430Percentage53.333.313.3Working class1661032Percentage50.018.831.3Totals702625121Percentage57.921.520.7 We have here evidence that some of these children, notwithstanding Bernsteins claims about the early learning by children of rules concerning what is or is not relevant in any disciplinary context, seem not to have yet learned that, in the case of mathematics problems set in school, it is usual that everyday knowledge and experience should not be brought to bear when data have been provided. In his terms, some children have not yet understood the recognition rule for identifying this context and its specialized requirements, and these children are more likely, it seems, to be from working class backgrounds. It is easy to see that had the given data been out of line with everyday experience, and had the children behaved in the same ways, then the children who chose to respond by drawing on their everyday experience would have been likely to gain no marks. Another item examined in the research did lead to childrens purely mathematical competence being underestimated. Under the heading organizing a competition, children were asked to undertake a combinatorial problem concerning tennis pairs (SEAC 1993). Two bags were shown, each containing the names of three children written on rectangular cards. In one bag the names David, Rashid and Rob were shown; in the other Ann, Katy and Gita. Above the bags was written: David and Gitas group organize a mixed doubles tennis competition. They need to pair a boy with a girl. They put the three boys names into one bag and all the three girls names into another bag. Below the picture of the bags was written: Find all the possible ways that boys and girls can be paired. Write the pairs below. One pair is already shown. The box in which the pairs were expected to be written included Rob and Katy, printed in bold type. Now, the mathematical question embedded in this problem can be expressed, in esoteric terms, as: find the Cartesian product of the sets { David, Rashid, Rob} and { Ann, Katy, Gita}. The answer would be {David & Ann, David & Katy, David & Gita, Rashid & Ann, Rashid & Katy, Rashid & Gita, Rob & Ann, Rob & Katy, Rob & Gita}, i.e. nine pairs. The marking scheme for the item sets out nine pairs in a similar way: Rob & Katy Rob & Ann Rob & Gita Rashid & Katy Rashid and Ann Rashid & Gita David & Katy David & Ann David & Gita We can infer from the setting out of the pairs in this model answer that the test designers were not much interested in tennis per se, but rather in the abstracted operation of combining the names in a purely mathematical way. On examination this item seems to have the potential to confuse children, especially those more likely to draw on everyday knowledge in responding to problems (Cooper 1994). A tennis competition can begin once some particular set of pairs, rather than all conceivable pairs, has been selected. Furthermore, children will have observed televised draws for competitions like the FA Cup (the English annual knockout soccer competition) in which, once a teams name has been taken from a bag, it is not returned (Cooper 1994). For these reasons, I hypothesized not only that some children would offer just three pairs but also that this response might not constitute adequate evidence that they were incapable of producing the nine pairs required to gain the mark. In addition, given the sociological evidence that recognition rules vary across social classes, I expected working class children to be more likely to fall into this trap, losing marks as a consequence. My exploratory work supported the first of these claims. There was some confusion present in the childrens minds about what was actually required of them, since find all the possible ways that boys and girls can be paired can be read as requesting methods which might be used rather than the pairs themselves (see Cooper 1998a). However, of 15 children aged 10-11 years, 7 produced three pairs using all six names - when interviewed individually while solving this question (Cooper 1998a). Of these 7, 4 produced the remaining six pairs when asked to reconsider whether they had indeed produced all the possible pairs. An additional case of a child who had written nine pairs initially, Lynda, was particularly interesting. She explained that she had nearly stopped after writing three pairs. In fact, she had paused after writing three, looked up at me as if seeking a cue, and then continued, producing nine pairs. It seemed as though she was aware that both three and nine pairs might be meaningful answers within some frame of reference. In the later large-scale research, these phenomena reoccurred. Here, in the context of a sample of 123 10-11 year-olds, it was possible to explore the relationship of response to social class. Here, in contradistinction to the traffic item, there were class differences in marks achieved in both a group testing and an interview context (Cooper and Dunne 2000: 108). In the group test context, for example, 50% of the sample as a whole gained the mark for this item, but the figures were 58%, 47% and 39% for the service, intermediate and working class children respectively. As for method, working class and intermediate class children were more likely, as in the case of the traffic item, to respond in a realistic fashion to this item, with realistic responses here including both the production of three pairs and the production of more than three pairs set out in a way that might suggest that tennis playing could actually proceed (Cooper and Dunne 2000: 109-110). The critical question that arises is again whether some of the children who initially produced three pairs had done so as a result of failing, in Bernsteins terms, to recognize the speciality of the context, rather than as a consequence of being unable to undertake the required combinatorial operation successfully. This was explored, as it had been in the exploratory research, by asking the child during an extended individual interview, once they had completed the item, whether they had indeed found all the possible pairs. Of the 123 children, there were twelve cases where, in the interview, the initial production of three pairs was followed by the subsequent construction of another six, giving nine in total. Some ten percent of the sample were therefore, on my reading of this, failing to demonstrate initially a mathematical or combinatorial - competence they possessed as a result of their failing to recognize the speciality of the context. Just 3 of the 60 service class children fell into this trap. Of the 30 intermediate class children, 5 did so, and of the 33 working class children, 4 did so (Cooper and Dunne 2000: 114). The approximate percentages for each class are therefore 5%, 17% and 12% respectively. We seem to have in the tennis item a type of problem which, if used in tests, will tend to underestimate the distribution of purely combinatorial competence, and which may well do so differentially by social class. The discussion so far has focused on problems which embed mathematical operations in some everyday context. Of course, not all problems in mathematics tests do this. Another category of item is purer in the sense that problems do not refer outside of the purely mathematical to the everyday world in any of the senses marked out in  REF _Ref13889938 \h  \* MERGEFORMAT Figure 9.1. For example, an algebra item which makes no everyday reference might be: if n+12=15, then what is the value of n+10? A data handling item might just ask for a calculation of the mean of some numbers, without making any reference outside of the set of numbers to any context of origin. For example: calculate the mean of 12, 34, 55, 13, and 90. We have already seen a Cartesian product version of the tennis item. This could be made purer by replacing the names by, say, the letters a, b, c in one set and the numbers 1, 2, 3 in the other to give: find the Cartesian product of the sets {a, b, c} and {1, 2, 3}. Now, if it is indeed the case, as the evidence concerning the traffic and tennis items suggests, that children from certain class backgrounds are less likely to read some realistically contextualised problems in the way intended by test designers, then we might expect that, over a large sample of both realistically contextualised and not contextualised items, we would be able to discern some class differences in relative performance on the two categories of item. In the research reported in Cooper and Dunne (2000) this has been explored by comparing the percentages of the total marks available for each category of item achieved by 10-11 year-olds from the three social classes when they took three separate test papers in a group situation. These papers comprised two papers taken as a required part of the annual national testing regime in England, and a third paper taken some months earlier which had been created from earlier national test papers so as to include a mix of realistic items (including the traffic and tennis items discussed here) and non-realistic items. The detailed discussion of the findings in Cooper and Dunne (2000) shows clearly that working class children do less well on realistic items in comparison with non-realistic items than their service class counterparts and, importantly, that this is not apparently due to differences in measured intelligence on the side of the children or to differences in the linguistic demands of the two categories of items on the side of the problems (Cooper and Dunne 2000: 90-94). One way of demonstrating the possible consequences of the differences found is via a simulation of a selection process occurring at age 11 and based on the realistic and non-realistic scores respectively. If approximately a quarter of these children were to be selected for, say, an elite secondary school on the basis of their achieved scores, then, were the actually achieved realistic scores to be employed, some 12% of the working class children would win a place, while if the actually achieved scores on the non-realistic items were to be used, some 24% would do so. These results suggest that, from an equity perspective, it matters considerably, at least at the age of 10-11, which type of item is used to determine childrens mathematical knowledge and understanding. The earlier discussion of the traffic and tennis item, considered against the background of the relevant sociological work, suggests that an important part of any explanation of the class differences found in overall performance on the two types of item might need to be couched in terms of childrens differentiated understanding and use of recognition rules as defined by Bernstein, i.e. in terms of cultural or subcultural differences between children. Conclusion I have tried to show that Bernsteins general framework is a useful one for analyzing the problems caused for children by a lack of consistency in the world of mathematics education concerning the ways in which realistic considerations should or should not be brought to bear in the solution of realistically contextualised mathematics problems. However, I have also drawn on some research evidence to suggest that Bernsteins more specific claim that recognition rules are learned very early in the childs career needs to be questioned. Working class children in particular, at the age of 10-11 years, seem to have difficulty in reading the intentions of test designers who set mathematical questions in realistic contexts. As we might expect from Bernsteins general account of social class and culture, children from different socio-cultural origins seem to have learned at different rates how to recognize legitimately the specificity of the context of realistic school mathematics problems. Some children, at this stage, have more of a feel for the game, to use Bourdieus (1990) phrase, than others. This raises a worrying issue. Many school systems provide differentiated curricula at the secondary stage, and childrens previous performance is often used to place children in an appropriate stream or track. If the forms of assessment used at this transitional point continue to include the sorts of potentially confusing realistic item I have discussed here, and if working class children as in the research sample I have discussed - do relatively less well than others on this particular type of item, then children may be placed in streams on the basis of an underestimate of their purely mathematical competence. There is obviously an ironic aspect to this conclusion, since the introduction of realistic problems is thought by many to be especially helpful to working class children. I am aware that the work I have discussed needs to be replicated (though see Theule-Lubienski 2000 for some recent supporting evidence from the USA). However, its findings suggest that an important dilemma exists for mathematics educators who concern themselves with equity issues, but who also wish to utilize realistic contexts, especially as part of assessment regimes. They may believe strongly that realistic contexts are of especial value in the mathematical education of working class children (more relevant, more motivating, etc.) but there may be a substantial price to pay for this belief at critical points of assessment. At the very least, more thought needs to be given to the advantages and disadvantages of using realistic items in high stakes assessment. It may be that a different type of realistic item, with a wider range of answers allowed, may offer us one way to avoid this dilemma (Cooper and Harries 2002 discusses some early evidence concerning a revised version of the lift item) but, at the very least, mathematics educators need to reach a clearer consensus on the purpose and nature of realistic problems as well as exactly when they are or are not appropriate for the testing of mathematical knowledge per se. Are realistic problems to remain, as they often are now, merely disguised pure problems, residing in Box 1 of  REF _Ref13889938 \h  \* MERGEFORMAT Figure 9.1, or are they to migrate to Box 3 to become more genuinely open, with childrens capacity to bring realistic considerations in general to bear actually becoming one focus of assessment? The latter way forward would almost certainly require, as Bernsteins work has always suggested, a fairly fundamental change in the dominant educational knowledge code. References Atkinson, P. (1985) Language, Structure and Reproduction. London: Methuen Bernstein, B. (1973) On the classification and framing of educational knowledge, in R. Brown (Ed) Knowledge, Education and Cultural Change. London: Tavistock. Bernstein, B. (1990) The Structuring of Pedagogic Discourse. London: Routledge Bernstein, B. (1996) Pedagogy, Symbolic Control and Identity: Theory, Research, Critique. London: Taylor & Francis. Bourdieu, P. (1990) From rules to strategies, in P. Bourdieu In Other Words. Cambridge: Polity. Cockcroft, W.H. (1982) Mathematics Counts. London: HMSO. Cooper, B. (1985) Renegotiating Secondary School Mathematics: a Study of Curriculum Change and Stability. Basingstoke: Falmer Press. Cooper, B. (1992) Testing National Curriculum Mathematics: Some critical comments on the treatment of real contexts for mathematics, The Curriculum Journal, 3(3): 231-243. Cooper, B. (1994) Authentic testing in mathematics? The boundary between everyday and mathematical knowledge in National Curriculum testing in English schools, Assessment in Education: Principles, Policy and Practice, 1 (2): 143-166. Cooper, B. (1998a) Assessing National Curriculum Mathematics in England: Exploring childrens interpretation of Key Stage 2 tests in clinical interviews, Educational Studies in Mathematics, 35 (1): 19-49. Cooper, B. (1998b) Using Bernstein and Bourdieu to understand childrens difficulties with realistic mathematics testing: an exploratory study, International Journal of Qualitative Studies in Education, 11 (4): 511-532. Cooper B. and Dunne, M. (2000) Assessing Childrens Mathematical Knowledge: Social Class, Sex and Problem-Solving. Buckingham: Open University Press Cooper, B. & Harries, A.V. (2002) Childrens Responses to Contrasting Realistic Mathematics Problems: Just how realistic are children ready to be? Educational Studies in Mathematics, 49(1):1-23 Gates, P. (2002) Issues of equity in mathematics education: defining the problem, seeking solutions, in L. Haggarty (Ed) Teaching Mathematics in Secondary Schools: A Reader, London: Routledge Falmer. HMI (1985) Mathematics From 5 to 16. London: HMSO Holland, J. (1981) Social class and changes in orientation to meaning, Sociology, 15 (1): 1-18. Lave, J. (1988) Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge: Cambridge University Press. Qualifications and Curriculum Authority (QCA) (2001) ( HYPERLINK "http://www.qca.org.uk/news/press/20001113.asp" http://www.qca.org.uk/news/press/20001113.asp, Last Updated: 25 May 2001 17:27:44, downloaded 13/08/02) Schools Examinations And Assessment Council (1992) Mathematics Tests 1992, Key Stage 3, London: SEAC/University of London. Schools Examinations And Assessment Council (1993) Pilot Standard Tests: Key Stage 2: Mathematics, SEAC/University of Leeds. The Times Educational Supplement (17/11/00) Theule-Lubienski, S. (2000) Problem-solving as a means towards mathematics for all: an exploratory look through a class lens, Journal for Research in Mathematics Education, 31 (4): 454-482. Appeared in: Olssen, Mark, Ed. (2004); Culture and Learning: Access and Opportunity in the Classroom, Information Age Publishing, pp. 183-202 (Chapter 9). PAGE  PAGE 3  I would like to thank Tony Harries and Stephanie Cant for their helpful comments on an earlier draft of this chapter.  Women do get a belated mention. Item 17 is: in how many ways can a committee of 5 men and 3 ladies be formed from 12 men and 8 ladies?  In principle, a child might, of course, offer both solutions, with some justifying account.  The basic design of the research involved collecting test data from children and then subsequently interviewing children individually while they worked through a selection of the items they had attempted in the group test situation. The design, samples, methods and forms of analysis are fully described in Cooper and Dunne (2000). The work was funded by the United Kingdoms Economic and Social Research Council. Increasing degree of intended realism in required answer 3: Problems intended to require general extra-mathematical considerations 2: Problems intended to require particular extra-mathematical considerations 1: Problems intended to require no extra-mathematical considerations tuaot{9CEMR\N\"& #Ƕǧsasasasasasasasasasasasasas"h`hQ6OJQJ]mH sH h`hQOJQJmH sH hhWhQ5OJQJmH sH hQCJOJQJmH sH  hIhQhIhIOJQJmH sH  hhW6CJOJQJ]mH sH  hQ6CJOJQJ]mH sH -jhQ0J6CJOJQJU]mH sH  hI6CJOJQJ]mH sH $v 9:D>?{"|"##%%''d 9r $a$gdI$a$#%''' ((X(h(((?)*~--5566 6B6::=$===>> >>><@b@@@@AGHtIph`hOJQJmH sH h`hQCJmH sH )jh`hQ0JOJQJUmH sH "h`hQ6OJQJ]mH sH #h`hQ0JH*OJQJmH sH h`hQ5OJQJmH sH $h`hQCJOJQJaJmH sH h`hQOJQJmH sH h`hQCJ(' (X((((!-"-00222 6 6 6B6l66666;;>> >yB dd7^7yBzBCCCEEAFBF_FGGsItIgKhK}P~PVVXX"\#\\\>^ 9r 7^77^77^7tI\KgKJTTTTTXX4ZFZ\>^V`s`bbcegg h hhh(h)hzhhhhhhͻͻͻͻͻͻͻͻͨ͒ͨͻxeR$h`h)iCJOJQJaJmH sH $h`hQCJOJQJaJmH sH h`hQmH sH h`h^sHOJQJmH sH +jh`hQOJQJUmH sH %jh`hQOJQJUmH sH "h`hQ6OJQJ]mH sH h`hQOJQJmH sH h`B*OJQJmH phsH %h`hQB*OJQJmH phsH  >^?^``cceeffhhii)o*o^r_rvrssawbwyyy^gde X^ 9r hhii i iiiij'jHjJjkk2l 0^`0ɾ˾ξٿ(k20FW1cAѻѻѻѻѨєєєєхuєєєє_ѻ*h`hQ5CJOJQJ\aJmH sH h`hQ6CJOJQJaJh`hQCJOJQJaJ'h`hQ6CJOJQJaJmH sH $h`h,>CJOJQJaJmH sH *h`hQ6CJOJQJ]aJmH sH $h`hQCJOJQJaJmH sH hQCJOJQJmH sH hhWCJOJQJmH sH %CNJ* .h]h&`#$$ 9r 0^`0a$gd`$a$gd` 9r 0^`0gd` 0^`0 9r 0^`0#;)*S`a =ɺsZD4h`CJOJQJaJmH sH +h`0JB*CJOJQJaJmH phsH 1h`h`0JB*CJOJQJaJmH phsH 3jFh`h`CJOJQJUaJmH sH h`h`OJQJjh`h`OJQJU$h`h`CJOJQJaJmH sH h`h`CJOJQJaJhQCJOJQJaJmH sH 'h`hQ6CJOJQJaJmH sH $h`hQCJOJQJaJmH sH =S"Vز{{dZTZTPZTZh!x, h!x,0Jjh!x,0JU-h`h`B*CJOJQJaJmH phsH h`56OJQJ]h`h`5OJQJh`h`56OJQJ]hQCJOJQJaJmH sH h`CJOJQJaJmH sH *h`hQ6CJOJQJ]aJmH sH $h`hQCJOJQJaJmH sH 'h`hQ6CJOJQJaJmH sH   -/;COY|}ܹ๬๬๬਑-h`h`B*CJOJQJaJmH phsH hQh!x,6CJOJQJ]h!x,CJOJQJ h!x,6]h!x,mH sH jh!x,0JUhIh!x, h!x,0Jjh!x,0JUhI0JmHnHu*./|}$a$h]h&`#$$ 9r 0^`0a$gd`01hBP. A!"#S$S% 3 01hBP. A!"#$% {DyK  _Ref13889938{DyK  _Ref13971687$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T555/  / 44 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T55544 k$$If!vh5C 5T555#vC #vT#v#v#v:V k t5C 5T555/  44 k{DyK  _Ref13889938{DyK  _Ref13889938-DyK .http://www.qca.org.uk/news/press/20001113.aspyK \http://www.qca.org.uk/news/press/20001113.asp@@@ NormalCJ_HaJmH sH tH Z@Z Heading 1$<@&5CJ KH OJQJ\^JaJ \@\ Heading 2$<@& 56CJOJQJ\]^JaJV@V Heading 3$<@&5CJOJQJ\^JaJJ@J Heading 4$<@&5CJ\aJN@N Heading 5 <@&56CJ\]aJH@H Heading 6 <@&5CJ\aJ8@8 Heading 7$@&>*:@: Heading 8$@&5\J @J Heading 9 $dh@&>*CJOJQJDA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k@(No List >@>  Footnote TextCJaJ@&@@ Footnote ReferenceH*4 @4 Footer  9r .)@!. Page Number4B@24 Body Text6]@"@@ Caption xx5CJ\aJDC@RD Body Text Indent ^V@bV Normal Indent5$7$8$9DH$^CJaJ6P@r6 Body Text 2CJ*O* page-info6U@6 Hyperlink >*B*phD+@D  Endnote TextCJOJQJaJ>*@> Endnote ReferenceH*LQ@L Body Text 3$a$56\]mH sH bR@b Body Text Indent 27d^7CJOJQJmH sH t.?@x_:;<=*('& :;<=" Ͷ2v9:D>? {| X !%"%((*** . . .B.l.....3366 6y:z:;;;==A>B>_>??sAtAgChC}H~HNNPP"T#TTT>V?VXX[[]]^^``aa)g*g^j_jvjkkaoboqqqVrrssfsgssss tttu@u_uuuUvVvvww8|9|||| }+}2}3}L}O}R}U}X}Y}d}i}n}s}t}u}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}~ ~~~~~~~~PQDEĆΆن*+,-“lmWXOP[ȶɶʶ˶Ͷζٶ#ķ"WCNJ* ../|}000000h00000h00000000000000000000000000h00000000000000 00000x00000x00000000000000000000000000000000000h0000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000000000000000000000000000000000000h000000000h00000000000000000000000000@0@0@0@0@0@00 @0@0@0@00 000000000000000vJ*O900 O900O900O900O900 O900O900O900O900 O900 #tIhyτɾ=hknprt~'yB>^yτXtɅDlCilmoqsuvwxyz{|}j_`(``a axxyA|W|Y|1Xc` t ttX !!l*+,b$>ixa ?@*B(   h  ( %# m~#" N &   $ N '    $ N (    &$ 4 ) ($T * # l$ B S  ?a% ! 4 _Ref13889938 _Ref13971687`9| aZ|=,Y|v-Yx.Yx/Yv0Ylx1Ylx2Yx3Yx4Yx5Y|v6Yv7Yv8Y<v9Y|v:Yv;YvYv?Y w@Y$ wAYd wBY wCY wDY$ wEYd wFY wGY wHY$ wIYvJYLhh((mmָָ==JJbb}}$8oo      !"#$%&'()*+,.-/0123456789:;<$$69aa ggggGUkk+-""ss߸߸CCQQ ""hh,C}}    !"#$%&'()*+,.-/0123456789:;< *1isiresearchsoft-com/cwywcitation:6*urn:schemas-microsoft-com:office:smarttagsStreet>*urn:schemas-microsoft-com:office:smarttags PersonName;7*urn:schemas-microsoft-com:office:smarttagsaddress88*urn:schemas-microsoft-com:office:smarttagsdateh<*urn:schemas-microsoft-com:office:smarttagsCity0http://www.5iamas-microsoft-com:office:smarttagsV=*urn:schemas-microsoft-com:office:smarttagsplacehttp://www.5iantlavalamp.com/_;*urn:schemas-microsoft-com:office:smarttagscountry-regionhttp://www.5iantlavalamp.com/=*urn:schemas-microsoft-com:office:smarttags PlaceName=*urn:schemas-microsoft-com:office:smarttags PlaceType  1113172000200120022558DayMonthYear=<;=<8  76767611=;=;7676=<=<=<=<=<=<=<==<=<=<=<=8   8   =<=<=8  ==;  X ^ ""@*V*+.-.K.M.}......X6c6<<hCDEEJJQQ``qmxmqrrrrrisosvvx x9|A|_dؑӲٲٶ#ķոd/3"pu3333333333333333333333333333333333333333333eddxy|| }~ن,˶ٶ Paul Ernest-8D*hh^h`o()^`o()88^8`o()^`o(()^`o(()pp^p`o(()03^0`o(.@ @ ^@ `o(.  ^ `o(.- @h h^h`OJQJo(\,>!x,fr1E^sHhWe X`u|QIt.)iE v66tAgC``8|9|||| }+}2}3}L}O}R}U}X}Y}d}i}n}s}t}u}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}~ ~~~~~֋˶ w-* @0i?@{@UnknownGz Times New Roman5Symbol3& z Arial7&  VerdanaI& ??Arial Unicode MS"qh*&zf9Rja,B GS24dKF 3qH(?  General introBC Paul Ernest  Oh+'0  8 D P \hpxGeneral introoeneBCeCeCe Normal.dotr Paul Ernest6ulMicrosoft Word 10.0@_@n2p@x[^I@va,՜.+,D՜.+,L hp  University of Durhamo BKA General intro Title 8@ _PID_HLINKSAQF.http://www.qca.org.uk/news/press/20001113.asp  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry F<ΗData 1TablePNWordDocumentgSummaryInformation(DocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q