ࡱ> %` \bjbjٕ . %) hhh|$O$O$O8\OdO|(PP"PPPQQQ$h4hsQQss PPs .PhPshCP|P O$O%z+J0(u+~ChCdQ U]8e!lQQQ%jQQQ(ssss|||9`L|||`L|||   Mathematical tasks in textbooks: Developing an analytical tool based on connectivity Presenter: Dr Birgit Pepin, University of Manchester, UK Address for correspondence: Dr Birgit Pepin, The University of Manchester, School of Education, Manchester, M13 9PL, UK. Telephone: +44(0)161 275 7128 Fax: +44(0)161 275 3484 Email:  HYPERLINK "mailto:birgit.pepin@manchester.ac.uk" birgit.pepin@manchester.ac.uk Key words: connectivity and understanding; textbook analysis; textbook tasks; understanding in mathematics Abstract Textbooks and other materials are used extensively in mathematics lessons and they influence to a large extent how students think about mathematics and come to understand its meaning. It is therefore important to analyse such materials and their use in order to ensure that what is offered to students is a rich, coherent and connected learning experience. In this paper, we use the concept of connectivity as an analytical tool to develop a framework for analyzing textbook tasks. In order to do this, we present a review of the research available relating to tasks, the analysis of those tasks, and the connections, connectedness and mathematical knowledge offered through tasks and worked examples. From this, we identify the need to make connections explicit; to support the making of connections through multiple representations, and the need for tasks in textbooks which satisfy particular features. The paper ends with the identification, based on the literature we examine, of a set of criteria to apply to tasks in the analysis of textbooks. Introduction Students spend much of their time in classrooms exposed to and working with prepared materials, such as textbooks, worksheets and, more recently, information and communications technology (ICT) materials. Teachers often rely heavily on textbooks in their day-to-day teaching, and they decide what to teach, how to teach it, and the kinds of tasks and exercises to assign to their students. It is reasonable to argue, therefore, that such materials are an important part of the context in which pupils and teachers work. In recognition of the central importance of such documents, the framework for the Third International Mathematics and Science Study (TIMSS) included large-scale cross-national analyses of mathematics curricula and textbooks as part of its examination of mathematics education and attainment in almost 50 nations. It is not surprising, then, that considerable attention has focused on textbooks, including the economic and political circumstances of their production (Apple, 1986, 1992), their linguistic features (Castell et al, 1989) and their sociological features (Dowling, 1996). Concerns have been expressed about the quality of textbooks and about their persuasive influence, and it appears that the textbook content, and how it is used, is a significant influence on students opportunity to learn and their subsequent achievement (Robitaille and Travers, 1992). It is also commonly assumed that textbooks are one of the main sources for the content covered and the pedagogical styles used in classrooms (Valverde et al, 2002). During the past decades there has been much concern and discussion about student conceptual understanding and their expertise in terms of mathematical thinking, reasoning and problem-solving (for example National Research Council, 1989; Hiebert and Carpenter, 1992). The underlying goals for addressing these concerns have been to enhance student understanding of mathematics and to help them to develop their capacities to move beyond procedural knowledge to think mathematically. Hiebert et al (1997) argue that classrooms that facilitate mathematical understanding share some core common features. Their framework consists of five dimensions that are said to work together to shape classrooms into particular kinds of learning environments: (a) the nature of the learning tasks; (b) the role of the teacher; (c) the social culture of the classroom; (d) the kind of mathematical tools that are available; and (e) the accessibility of mathematics for every student (p.2). For the purpose of this paper, the interesting feature is the nature of the learning tasks (Hiebert et al, 1997). Furthermore, it is important that students have frequent opportunities to engage in dynamic mathematical activity that is grounded in rich, worthwhile mathematical tasks (Henningsen & Stein, 1997, p.525), and it is argued that this is an essential component for understanding in order for connections to be made by the learner (Hiebert et al, 1997). Thus, of central importance in this article is the concept of academic or mathematical task (Doyle, 1986, 1988), a concept which is used here as an analytical tool for examining subject matter as a classroom process rather than simply as a context variable in the study of learning. The purpose of this paper is to develop an analytical tool for analysing tasks in mathematics textbooks. This is done by giving a general overview of research in the areas of textbooks internationally in relation to mathematical tasks, and subsequently to develop from that literature a framework for the analysis of textbook tasks. International focus on textbook research In order to explore possible influencing factors in cross-national differences in student mathematical achievement, it has been suggested that the curriculum is one of the key factors (McKnight et al, 1987, Schmidt, et al, 1997), and amongst those curricular factors the textbook has been identified as having potentially a large effect (Schmidt et al, 1997, Mayer, Sims and Tajika, 1995). Textbooks are commonly charged precisely with the role of translating policy into pedagogy. They represent an interpretation of policy in terms of concrete actions of teaching and learning. Textbooks are the print resources most consistently used by teachers and their students in the course of their joint work. (Valverde et al, 2002, p.viii) Valverde et al (2002), in their research on textbooks and related to TIMSS (Third International Mathematics and Science Study), investigated textbooks and specifically how they go about the business of translating policy into pedagogy (p. viii). They further sought to explain how cross-national attainments in student achievement are related to features of educational policy and its implementation (p. viii). Clearly, one issue of pervading importance to the nations that participated in TIMSS was the quality of educational opportunities afforded to students to learn mathematics and science - and the instruments that optimise such quality. (Valverde et al, 2002, p.viii) Textbooks are a major source of provision of these educational opportunities. Romberg and Carpenter (1986), for example, noted that the textbook was consistently seen (in the US) as the authority on knowledge and the guide to learning [and that] many teachers see their job as converting the text. (p.25). Mayer et al (1995) suggest that US textbooks constitute a sort of de facto national curriculum . They, in turn, quote Armbruster and Ostertag (1993) in terms of the powerful role of textbooks in the American curriculum (p.69). Garner (1992), also mentioned, noted that textbooks serve as critical vehicles for knowledge acquisition in school and can replace teacher talk as the primary source of information (p.53). Glyn et al (1986) are quoted to have said that across many disciplines students experience a heavy reliance on textual materials for a great deal of their knowledge(p.245). In the European context the authors investigated mathematics textbooks (at lower secondary level) in England, France and Germany, and their use by teachers in the classroom (Pepin and Haggarty, 2001; Haggarty and Pepin, 2002). In a separate study they also explored mathematics teachers principles and pedagogic practices in England, France and Germany (Pepin, 1997, 1999, 2002), which helped to understand issues arising from the textbook study. The findings here not only pointed to a similar heavy reliance by teachers on the text, but also the ways in which many teachers further limited the learning opportunities offered in those texts. In other words it seemed, albeit from their limited research, that textbooks might offer better educational opportunities to pupils than those mediated by the teacher and those textbooks might simultaneously represent an upper bound on learning opportunity. Schoenfeld (1988) similarly argued that although good teaching could compensate for the inadequacies of texts, there is evidence to suggestthat it does not (p163). From extensive studies in many countries and from international comparisons, it is clear that textbooks are important artefacts in the classroom and that they strongly influence what happens in the classroom. Indeed, it can further be claimed that they are the mediators between the intent of curricular policy and the instruction that occurs in the classroom (Valverde et al, 2002, p.2). We now turn to the content of those textbooks. Textbooks and the importance of mathematical tasks It appears that textbooks, for better or worse, define and represent the subject for very many students, and they influence how those students experience mathematics. Textbooks provide children with opportunities to learn, and learn those things which are regarded as important by their government. Teachers mediate textbooks by choosing and affecting tasks, and in that sense student learning, by devising and structuring student work from textbooks. It has been shown that teachers use textbooks heavily for their selection of tasks (Kuhs and Freeman, 1979; Luke et al, 1989; Pepin and Haggarty, 2001). Furthermore, Doyle (1988) argues that the tasks teachers assign to students influence to a large extent how students come to understand the curriculum domain. Moreover, in his opinion, they serve as a context for student thinking not only during, but also after instruction. This premises that tasks, most likely chosen from textbooks, influence to a large extent how students think about mathematics and come to understand its meaning. Indeed, Henningsen and Stein (1997) argue that the tasks in which students engage provide the contexts in which they learn to think about subject matter, and different tasks may place different cognitive demands on students . Thus, the nature of tasks can potentially influence and structure the way students think and can serve to limit or to broaden their views of their subject matter with which they are engaged. Students develop their sense of what it means to do mathematics from their actual experiences with mathematics, and their primary opportunities to experience mathematics as a discipline are seated in the classroom activities in which they engage (p.525) Hiebert et al (1997) similarly argue that students also form their perceptions of what a subject is all about from the kinds of tasks they do. Students perceptions of the subject are built from the kind of work they do, not from the exhortations of the teacher. The tasks are critical. (p.17/18). In order to study tasks, it is necessary to examine the concept of task and subsequently select what is relevant for the mathematics classroom and in terms of textbooks. Doyle (1988, also 1983, 1986) defines academic tasks in terms of four components. He calls attention to four aspects of work in class: the end product to be achieved; a set of conditions and resources available to accomplish the task; the operations involved to reach the goal state; the importance of the task. (Doyle, 1988, p.169) He also points to the problem that a task exists at several different levels at once (as announced by the teacher, or as interpreted by the student, for example). This multiplicity of meanings, clearly likely to complicate task research, alerts us to the idea that curriculum content can be represented in a variety of fundamentally different ways in the classroom. For example, if one looks simply at the level of cognitive mathematical content demand in each tasks, one may fail to recognise that the task itself might require the student to calculate answers to a well-structured exercise at one extreme, whereas at the other extreme, require the application of conceptual understanding to a real-world problem (Doyle, 1988). Thus, the way in which the student is being asked to work with the content determines to some extent the cognitive demand of the task. In his research on mathematics instruction Schoenfeld (1988) argues that students, despite gaining proficiency at certain mathematical procedures through the completion of tasks, gained a fragmented sense of the subject matter and understood few connections that tie together the procedures they had studied. Indeed, he argues that most textbooks present problems that can be solved without thinking about the underlying mathematics but by blindly applying the procedures that have just been studied (p163). Moreover, he claims that the students developed perspectives regarding the nature of mathematics that were not only inaccurate, but were likely to impede their acquisition and use of mathematical knowledge (p.145). We explore the idea of connectedness in a later section. The issue that texts, and tasks in texts, are not mere carriers of content, but can signify processes, is supported by Christiansen et al (1986). They ask the question What is a text? And analyse the interaction between reader and text, especially texts in mathematical textbooks. They claim that the problem of the interaction between text as a subjective scheme and text as an objectively given structure of content or information is not obvious. Indeed they add that it is necessary: that the textbook is not conceived of as a written lesson protocol, and that the pupils handling of the text is not thought of as mere reaction to stimuli presented, but that the text is introduced in the classroom as an independent object of activity, and that handling texts is specially introduced and trained. Only then will texts become exploratory tools instead of remaining mere straitjackets. Texts must develop into real means of communication and cognition on the pupils side also. (p.175) Textbook task analysis has also been carried out by cross-national comparativists (Stigler et al, 1986). These studies on mathematical problems have highlighted the importance of two dimensions for analysing mathematical problems: mathematical and contextual features. A third dimension that is likely to influence students understanding and perception of mathematics is the differences embedded in a problems performance requirements, such as the type of responses elicited (for example explanation) and cognitive aspects (for example procedural practice) (Li, 2000). The analysis of tasks In this section we will review the literature in terms of the different ways in which tasks could be analysed. These include general features of mathematical tasks to enhance learning; the notions of appropriateness of tasks and respectful tasks; cognitive demand in mathematical tasks; the contextual features and purposes of tasks; and, finally, connectedness and mathematical knowledge. (1) General features of mathematical tasks to enhance learning Kilpatrick et al (2001) give a comprehensive view of what they regard as successful mathematics learning. They coin the term mathematical proficiency to capture what they think it means for anyone to learn mathematics successfully. In their view, mathematical proficiency has five interwoven and interdependent strands. conceptual understanding - comprehension of mathematical concepts, operations, and relations procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately strategic competence - ability to formulate, represent, and solve mathematical problems adaptive reasoning - capacity for logical thought, reflection, explanation, and justification productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and ones own efficacy. (Kilpatrick et al, 2001, p.5) These are remarkably similar to the specific objectives identified by HMI (DES, 1985) categorised as the learning of: facts; skills; conceptual structures; general strategies; and personal qualities (p.7). In terms of instruction, Kilpatrick et al (2001) argue that the quality of instruction depends, for example, on the tasks selected for instruction and their cognitive demand. They further claim that the teachers expectations about the mathematics particular students are able to learn can powerfully influence the tasks the teacher poses for the students, the questions they are asked, in other words, their opportunities and motivation for learning. (Kilpatrick et al, 2001, p.9). This supports findings of Haggarty and Pepin (2002) in English, French and German classrooms, where teacher expectation influenced their choice and presentation of tasks to students. Hiebert et al (1997) reflect explicitly on the nature of tasks that may build mathematical understanding. They believe that students build mathematical understanding by reflecting and communicating, and tasks should allow and encourage these processes. This means that such tasks should have the following features: First, the tasks must allow the students to treat the situations as problematic, as something they need to think about rather than as a prescription they need to follow. Second, what is problematic about the task should be the mathematics rather than other aspects of the situation. Finally, in order for students to work seriously on the task, it must offer students the chance to use skills and knowledge they already possess. Tasks that fit these criteria are tasks that can leave behind something of mathematical value for students. (p.18) Interestingly, Hiebert et al (1997) use the term appropriate for tasks which have the above mentioned characteristics. In the next section we discuss appropriate tasks as seen by other members of the education community. (2) Cognitive demand Smith and Steins work (1998) is useful in terms of positioning the nature of tasks. They argue that the nature of the task determines what students learn. They also use four hierarchical categories of cognitive demand to assess tasks: Memorisation Procedures without connections to concept or meaning Procedures with connections to concept or meaning Doing mathematics In another study Stein et al (1996) query whether tasks that start out as cognitively demanding may decline into less demanding activities over the course of the lesson. High level tasks are often perceived by students as less structured, more complex, and longer than they are normally exposed to. According to research by Doyle and others, students often perceive such tasks as ambiguous: they do not know what to do and how. In turn they often urge teachers to explain such tasks, thereby potentially eliminating some of the complexity and perhaps even cognitive demand of the task. However, Smith and Stein (1998) point out that although the decision to use a cognitively demanding task does not necessarily lead to high-level engagement by students, Starting with a good task does, however, appear to be a necessary condition [for high-level engagement], since low-level tasks almost never result in high-level engagement (p344) Nevertheless, Nicely (1985) asserts that mathematics textbooks have a poor track record in terms of higher order thinking skills (p.26). He and his colleagues developed an analytical tool to evaluate textbooks. This consists of four categories which enable analyses and classification of printed materials according to Type of content; Level of cognitive activity; Stage of mastery; and Mode of response. In terms of the cognitive level of student tasks he identified 27 verbs grouped into nine categories and arranged in an ordinal scale. Their levels of cognitive demand (reflected in cognitive verbs and codes, according to which textbooks problems were analysed) ranged from lower levels (no task, observe, read; recall, recognise, repeat) to higher levels (prove, solve, test, design; evaluate). This reminds the reader of Blooms taxonomy (1956) as a framework for classifying statements of what we expect or intend students to learn as a result of instruction (Krathwohl, 2001). Without revisiting the whole theory, it is sufficient for the purpose of this review to say that Blooms cognitive domain is organised into six hierarchical categories: knowledge; comprehension; application; analysis; synthesis; and evaluation. These are generally regarded as levels of difficulty and reflected in student action- for the sake of textbooks in verbs that supposedly reflect student behaviour: Knowledge: write, list, name; Comprehension: describe, summarise; Application: use, solve, apply; Analysis: compare/contrast, analyse; Synthesis: design, invent, develop; Evaluation: critique, justify. Blooms taxonomy has received criticism (see Braendstroem, 2005), not least because it is relatively dated. However, we believe that it is relevant and to some degree leads us back to the first four of the five strands of Kilpatrick et als mathematical proficiency identified above. (3) Contextual features and purposes of tasks The research literature shows increasing evidence of how mathematics is used in everyday activities (e.g. Carraher et al, 1985, 1987; Lave 1988). Assuming the pervasiveness of mathematics embedded in everyday activities, and the motivation it stimulates to get something done, an important issue for textbook task analysis is to what extent, and in which ways, these real world experiences are incorporated. In terms of learning, the argument is likely to centre around the idea that mathematical knowledge and understanding would be enhanced and become more coherent for learners if they could establish connections between the networks of out-of-school experiences and those of in-school mathematics, and these should become integrated. Problem-solving in one setting should be informed by strategies learnt in other settings. Skovsmose (2002) talks about learning milieus and distinguishes between three different paradigms of exercises: those with references to pure mathematics (context unembedded); those with reference to a semi-reality (context embedded tasks, albeit contracted realities); and real-life references (Skovsmose, 2002, p.119). His example of semi-reality is taken from Dowling (1998): Shopkeeper A sells dates for 85p per kilogram. B sells them at 1.2kg for 1. (a) Which shop is cheaper? (b) What is the difference between the prices charged by the two shopkeepers for 15kg of dates? Whilst this can be seen to have features similar to very many textbook tasks, the point Skovsmose makes about the task is that such a situation is artificial and is unlikely to lead to students awareness of and willingness to draw on their own intellectual capabilities when making mathematical decisions and judgements (Cobb and Yackel, 1998, p170) His gives an example of a real-life task which draws on the context of unemployment: Real-life based exercises provide a learning milieu For instance, figures concerning unemployment can be presented as part of the exercise, and based on such figures questions can be asked about the decrease or the increase of employment, comparisons can be made between different periods of time, different countries, etc. All figures referred to are real-life figures, and this provides a different condition for communication between teacher and students, as it now makes sense to question and to supplement the information given by the exercise. (p.121) Cummins developed a simple two-dimensional model to analyse tasks, where one axis represents the context-embeddedness, and the other the degree of conceptual demand of the task (Cummins and Swain, 1986). Hall (1996) used the Cummins matrix to investigate differentiation for bilingual children - in the secondary English and Science curriculum. In fact s/he mapped some of the common classroom cognitive processes onto the Cummins matrix, and claimed that it helped teachers when reflecting on their teaching. It is beyond the scope of this paper to examine the complexity of pupil learning relating to working with tasks in context, but it is useful to comment nevertheless that a students mathematical procedure and performance can be largely determined by the particular context used in a task, so that students interact with the context of a task in many different and unexpected ways and this interaction is, by its nature, individual (Boaler, 1993). Thus, it would be unhelpful to suggest that the inclusion of tasks in context necessarily leads to understanding. Boaler argues, nevertheless, that Students are constructing their own meaning in different situations and it is inappropriate to assume a generality of familiarity or understanding in presenting students with a 'context'. This acknowledgment does not preclude the use of contexts but suggests that a consideration of the individual nature of student's learning should precede decisions about the nature and variety of contexts used as well as the direction and freedom of tasks in allowing students to bring their own 'context' to the task P15 (5) Connections, connectedness and mathematical knowledge Connectedness is not a new idea in the field of education, or indeed in mathematics education. For the sake of this paper, we distinguish in this section between the issues raised by (a) the research literature, and (b) by the national guidelines of individual countries. (a) Connections in the research literature Hiebert and Carpenter (1992) believe that it is essential to make connections in mathematics if one intends to develop mathematical understanding. They emphasise the importance of learning with and for understanding. According to them, understanding can be defined as: the way information is represented and structured. A mathematical idea or fact is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and strength of the connections. (p67). For them, what is essential for facilitating student understanding involves a number of principles, amongst them that understanding can be characterised by the kinds of relationships or connections that have been constructed between ideas, facts, procedures, for example. They describe understanding in terms of the way an individuals internal representations are structured and connected, and also how these internal representations are structured and connected to external representations. These representations would include spoken language, or written symbols, or analogies, to name but a few. Thus, a mathematical idea or procedure or fact is understood if it is linked to existing networks with strong and numerous connections. This means that understanding is not an all or nothing phenomenon (Hiebert and Carpenter, 1992) and internal networks can be thought of as dynamic. It also emphasises the importance of past experiences for interpreting and understanding new experiences. As Ma (1999) argues, When it is composed of well-developed, interconnected knowledge packages, mathematical knowledge forms a network solidly supported by the structure of the subject (p120) Hiebert and Carpenter (ibid) further point out that if mathematical tasks are overly restrictive, students internal representations are severely constrained, and the networks they build are bounded by these constraints (p76). Further, the likelihood of transfer across settings becomes even more problematic (p79). Indeed Hiebert (1984) commented that many students did not connect the mathematical skills they possessed with the symbols and rules taught in school: I shall argue that it is the absence of these connections that induces the shift from intuitive and meaningful problem-solving approaches to mechanical and meaningless ones (Hiebert, 1984, p498) This in turn reminds the reader of ideas relating to instrumental and relational understanding (Skemp, 1976), with instrumental understanding relating to rules without reasons, which we might understand here as rules without connections. Indeed, Cooper (1989) suggested that when students assumed instrumental beliefs about mathematics they failed to develop genuine understanding: the connections simply were not there to be made. Research conducted at Kings College in the United Kingdom (Askew et al, 1997) revealed that highly effective primary teachers of numeracy paid attention to Connections between different aspects of mathematics: for example, addition and subtraction or fractions, decimals and percentages; Connections between different representations of mathematics: moving between symbols, words, diagrams and objects; Connections with childrens methods: valuing these and being interested in childrens thinking but also sharing their methods. (Askew, 2001, p.114) This orientation was called a connectionist orientation, in contrast to the discovery and transmission orientations towards teaching numeracy which were shown as less effective as teaching approaches. Whilst it could be argued that teachers might mediate texts in such a way that they support students in making connections, it is by no means clear that they would (for example Schoenfeld, 1988, Haggarty and Pepin, 2002) or, indeed, that they could. For example, in terms of international and comparative research, Ma (1999) compared Chinese and US elementary teachers mathematical knowledge. She found that Chinese elementary teachers perceived mathematical concepts as interconnected, which was in contrast to US colleagues who perceived these concepts as arbitrary collections of facts and rules. She developed a notion of profound understanding of fundamental mathematics (PUFM), an argument for structured, connected and coherent knowledge (Ball et al, 2001), which is deep, broad and thorough (Ma, 1999, p 120) and this was seen as one of the factors for student enhanced mathematical performance. Research on multiple representations and case-based reasoning has demonstrated the crucial role of worked examples and concrete analogies in supporting students with their problem-solving skills (Mayer, 1987). Worked examples help to model problem-solving processes, and concrete analogies provide a means for connecting procedures to familiar experience (Mayer, 1987, in Mayer et al, 1995). Mayer et al also cite Grouws (1992) and Hiebert (1986) when they further argue that it is important to help students build connections between multiple representations. Multiple representations of problem-solving procedures might usefully include all of symbolic, verbal and visual representations, with content embedded within familiar situations so that the symbolic, verbal and visual are interconnected. In their study of textbooks in Japan and the United States, they found that the Japanese books devoted 81% of their space to explaining the solution procedure for worked-out examples compared with 36% in US books, and concluded that the cognitive modelling of problem solving processes supported the view of learning as knowledge construction. Mayer et al suggest that textbooks might follow either a view of learning as knowledge acquisition, thereby emphasising the product of problem solving, or a view of learning as knowledge construction, thereby emphasising the process of problem solving. In the former case, the textbook would have page space devoted to unexplained exercises involving symbol manipulation; in the latter, page space devoted to presenting and connecting multiple representations of step-by-step problem-solving processes through worked examples. Carroll (1992) argued that worked examples should serve as an extension of the teacher, providing scaffolding during practice at home or in class although Renkl et al (2000) suggest that in order to support a transition from the studying of the worked examples to unaided problem solving, there needs to be a fading out of solution steps. Nevertheless, it seems that the idea of connections can be extended to worked examples as well as other elements of the text, and we see that across the ideas we have presented, there is a strong case for the importance of providing opportunities for students to build rich connections in order to understand mathematics. (b) Connections in the national guidelines It is interesting to note that many official and potentially influential teaching guidelines mention the notion of connectedness as a vehicle for better understanding. For example, the Initial Teacher Training National Curriculum (ITTNC) in England (TTA, 1999) said that newly qualified teachers should know how to use formative, diagnostic and summative methods of assessing pupils progress in mathematics, including (ii) undertaking day-to-day and more formal assessment activities so that specific assessment of mathematical understanding can be carried out (and) (iii) preparing oral and written questions and setting up activities and tests which check for misconceptions and errors in mathematical knowledge and understanding and understanding of mathematical ideas and the connections [our emphasis] between mathematical ideas. (TTA, 1999, p.14, 9aii and iii) Although the English National Curriculum (NC) (DfEE, 1999) does not itself make explicit connections between topics, it says that At Key Stage 3 and 4, teaching should ensure that appropriate connections are made between the sections on number and algebra; shape, space and measures; and handling data. (p.6) Thus, the NC does not actually say how the ideas are connected, but tells teachers to make connections. In England the DfEE (2001), in their Key Stage Three National Strategy, suggest that Mathematics is not a group of isolated topics or learning objectives but an interconnected web of ideas, and the connections may not be at all obvious to pupils. Providing different examples and activities and expecting pupils to make links is not enough; pupils need to be shown them and reminded about work in earlier lessons. (DfEE 2001, 1.46) They also offer advice to teachers on how these connections may be made. As far as possible, present each topic as a whole, rather than fragmented progression of small steps Bring together related ideas across strands Help pupils to appreciate that important mathematical ideas permeate different aspects of the subject Use opportunities for generalisation, proof and problem solving to help pupils to appreciate mathematics as a unified subject (DfEE, 2001, 1.46) The DfES Standards Unit (Swann, 2005) go a step further and have produced linking activities that are particularly designed to draw out connections across mathematical topics (p10). The argument is given that this has been done because: A common complaint from teachers is that learners find it difficult to transfer what they learn to similar situations. Learning appears compartmentalised and closely related concepts and notationsremain unconnected in learners minds (Swann, 2005, p10) Looking beyond English guidelines, the American National Council of Teachers of Mathematics (NCTM) claims that a central theme of Principles and Standards for School Mathematics is connections. Students develop a much richer understanding of mathematics and its applications when they can view the same phenomena from multiple mathematical perspectives. One way to have students see mathematics in this way is to use instructional materials that are intentionally designed to weave together different content strands. Another means of achieving content integration is to make sure that courses oriented toward any particular content area (such as algebra or geometry) contain many integrative problems problems that draw on a variety of aspects of mathematics, that are solvable using a variety of methods, and that students can assess in different ways. (NCTM, 2000) This resonates with the research literature provided by Hiebert and Carpenter (1992) in their emphasis on the richness of connections as a basis for developing a deep understanding. Earlier documents published by the National Council of Teachers of Mathematics (1989, 1991) and the National Research Council (1989) all point to the importance of students developing interconnected understandings of mathematical concepts, procedures and principles, and not simply to memorise and apply procedures (Stein et al, 1996). According to the Professional Standards for the Teaching of Mathematics (NCTM, 1991) one finds consistent recommendations for the exposure of students to meaningful and worthwhile mathematical tasks, tasks that make students think rather than simply repeating and using an already demonstrated algorithm. The American literature is quite explicit about connectedness. In their chapter on Becoming a mathematics teacher Brown and Borko (1992) referred to the different documents produced by the National Council of Teachers of Mathematics (in particular NCMT, 1989) when saying According to these standards [NCMT, 1989a, Curriculum and Evaluation Standards for School Mathematics], computational algorithms, manipulation of symbols, and memorising of rules must no longer dominate school mathematics; rather, mathematical reasoning, problem solving, communication, and connections must be central. At the heart of the Curriculum and Evaluation Standards is the commitment to developing the mathematical literacy and power of all students. (p.210) They assert that this departs from the traditional practice of mathematics teaching in the US. Teachers are expected to provide time for students to explore mathematics and to stimulate students to make mathematical connections. Summary and conclusions We have argued in this paper that textbooks are widely used in many classrooms, and students perceptions and understanding of mathematics are heavily influenced by the kinds of tasks offered in textbooks and the connections made in those textbooks. We have further argued that worked examples can provide a powerful method for modelling solution steps as well as providing opportunities for multiple representations of ideas through the inclusion of symbolic, verbal and pictorial representations. Although it would be tempting to argue that teachers mediate textbooks to ensure that tasks are appropriate and challenging, to ensure that worked examples do indeed model solution steps to multi-step examples, and to help students make connections between ideas, it is not clear from the evidence available that this is happening. We therefore have to ensure that textbooks attend to those features to ensure that students are offered appropriate kinds of learning opportunities. Indeed, if we allow textbooks to remain restrictive in what they offer, students internal representations are in danger of remaining limited and their understanding remains deficient. Our analysis of the literature has alerted us, therefore, to the need for textbooks to make connections explicit and to support the making of connection through multiple representations. Further, we have argued that as part of an analysis of textbooks, tasks in those textbooks should be scrutinised according to the extent to which they: emphasise relational rather than procedural or instrumental understanding make connections with what students already know make connections with the underlying concepts being learnt make connections within mathematics and across other subjects are embedded in contexts which help to make connections with real life make high cognitive demand on pupils connect different representations (analogies, worked examples) Further, more research needs to be carried out aimed at understanding how, by whom and why textbooks are used as they are; how, with what purpose and with what effect students have access to textbooks; and how teachers mediate textbooks in relation to the connections they emphasise, the cognitive demand of tasks resulting in their mediation, and the features of the textbook they draw on. 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