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Critical Study2

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Critical Study2

  1. 1. Mathematics  Education:  Pedagogy  for   Transfer  of  Learning  from  School  to  the   Work  Place   [EDUC5001M]   Fatou  Kinneh    Sey    (200220649)                 MA  Mathematics  Education   University  of  Leeds   School  of  Education     September  2010      
  2. 2.   1   Acknowledgements My sincere appreciation to the Head of School of Education, Mr. Tom Roper for providing very important references; explaining very complex concepts and also creating a very conducive and relaxed atmosphere that allowed me to sound my ideas. However, I must acknowledge that any errors in this thesis are entirely my responsibility and not in anyway connected to him. I also wish to thank the Ministry of Education, the Gambia, for giving me the opportunity to study at the University of Leeds; an experience that has given me the invaluable opportunity to “learn how to learn” and view every material I read critically and get a deeper understanding. And finally, many thanks to my family and friends for feigning interest when I talked incessantly about my topic.
  3. 3.   2   Mathematics Education: Pedagogy for Transfer of Learning from School to the Work Place 1. Introduction......................................................................................................................3     2. Concept  of  Transfer.....................................................................................................11     2.1. Historical  Analyses  of  Transfer .................................................................................... 13     2.1.1.The  Theory  of  “Identical  Elements”........................................................................................ 14   2.1.2.Theory  of  Generalisation............................................................................................................. 15   2.1.3.Cognitive  Psychology  and  Transfer ........................................................................................ 17     2.2. Measures  of  Transfer........................................................................................................ 22     3. Transfer,  Social  Constructivism  and  Situated  Cognition................................27     4. Transfer  from  School  to  the  Work  Place..............................................................38     4.1. Mathematics  in  the  Work  Place.................................................................................... 39   4.2. Teaching  for  transfer  –  Functional  Mathematics.................................................. 45     5. CHAT,  Transfer  and  Teaching  Functional  Skills................................................52     6. Reference........................................................................................................................60  
  4. 4.   3   Mathematics Education: Pedagogy for Transfer of Learning from School to the Work Place 1. Introduction   There are many forces driving curricular reforms. Some of these include economic imperatives, new applications of mathematics, and the effects of technology. Consideration of the needs of these forces in mathematics education makes it necessary that the curriculum and the methods being employed in teaching/learning be examined to evaluate how well they serve the needs of students and other stakeholders (e.g. Schneider, 2001, Schoenfeld, 2001, NCTM, 2000, DfES, 2005). Meeting these needs somehow implies regular reviewing and updating of the curriculum and the research literature to determine on the one hand, how these affect the mathematics content and, on the other hand, how societal needs, for e.g. mathematics in the workplace, should be taken into account in curricular reforms. One of the reasons offered in the literature as to the reason why school and college leavers are not doing so well in situations outside of school that requires the use of mathematics is lack of “numeracy” (Cockcroft, 1982, Steen 2001, DfES, 2005), a term often used interchangeably with “mathematical literacy” and “quantitative literacy” in the literature. Treffers (1991), for example, claimed that learners’ level of numeracy might not be the result of content taught (or not taught) but a function of the structural design of teaching practices. However, instructional design and practices are only part of the problem. Several issues, such as: defining numeracy, relating it to the
  5. 5.   4   nature of mathematics (as perceived by learners and most importantly teachers), and the kind of competencies to be taught in schools or colleges to allow learners to adapt and apply the mathematics they learn in school in several contexts must be addressed. This is consistent with most of the views presented in recent literature. For example, I came across the following definitions of numeracy, which supports this philosophy: Cockcroft (1982), stated that being “Numerate should imply the possession of two attributes. The first is an “at- homeness” with all those facets of mathematics that enable a person to cope with the practical demands of everyday life. The second is the ability to understand information to understand information presented in mathematical terms. Taken together, these attributes imply that a numerate person should understand some of the ways mathematics can be used for communication”. Gal (1995) states: The term numeracy describes the aggregation of skills, knowledge, beliefs, dispositions, and habits of mind as well as general communicative and problem- solving skills that people need in order to effectively handle real-world situations or interpretive tasks with embedded mathematical or quantifiable elements. Organization for Economic Cooperation and Development (OECD) (2000) suggested that: Mathematical literacy is an individuals ability, in dealing with the world, to identify, to understand, to engage in, and to make well founded judgments about the role that mathematics plays, as needed for that individuals current and future life as a constructive, concerned, and reflective citizen.
  6. 6.   5   The PISA study (2003) define the term “mathematical literacy” as: “The capacity to identify, to understand, and to engage in mathematics and to make well-founded judgments about the role that mathematics plays, as needed for the individual’ s current and future private life, occupational life, social life with peers and relatives, and life as a constructive, concerned, and reflective citizen.” A common theme in all these definitions is the view that numeracy is not limited to acquiring mathematical knowledge and skills. Rather, numeracy is partly sociological, in the sense that it is heavily linked with its purpose and value to society, and partly epistemological, concerned with the nature of mathematics, as it is presented in the curriculum and implemented by teachers in schools. The sociological as well as the epistemological view of numeracy, mathematical knowledge or mathematics is consistent with Niss’s (1994) 5 faceted perspective of the nature of mathematics as (1) a pure science used to develop, describe, and understand objects, phenomena, relationships etc; (2) an applied science, used to and develop understand knowledge in other domains; (3) as a system of tools for societal and technological practice (“cultural techniques”); (4) as an educational subject; and (5) as a field of aesthetics. On another level, the analysis of the definitions of numeracy yields three common themes: This suggests that in order to meet this need, the nature of mathematics in school curriculum has to shift from the instrumentalist views of mathematics as an accumulation of facts, rules and skills to be used in the pursuance of some external end; or the Platonist view which sees mathematics as static but unified body of certain knowledge; to a kind of mathematics that arises as a
  7. 7.   6   result of human enquiry or problem solving. As Ernest (1988) put it, the problem-solving view of mathematics, is seeing mathematics as a dynamic and continually expanding field of human creation and invention - a cultural product. Moreover, problem-solving views of mathematics promote conceptual development; emphasise the possible application of mathematics in other disciplines; and also promote attitudes in students that increase their willingness to apply ideas learnt in school in different contexts (Ernest 1988, Prawat 1989). These different views of the nature of mathematics, that teachers in particular have, influence classroom practices (Cohen and Ball, 1990, Fang, 1996). As a result, developing ‘numerate’ learners as implied by the above definitions require the curricular reforms with aims, goals, purposes, and rationales closely associated with societal needs. Relating mathematics education to society brings up the second common theme that all the definitions touched on; functional mathematics. Functional mathematics, the second common theme pervading the definitions, provides a strong basis for students entering the work force either from high school or college to experience mathematics as problem solving. Ideas of functional mathematics (which, I will discuss later) arose out of the recognition that current mathematics curricula do not adequately equip people to use and apply mathematics effectively in different situations. Hence, functionality, in terms of curricular development, is a means of bridging the gap between school mathematics and out of school mathematics or identifying the areas of mathematics education that are inherent in employment for example. As Schoenfeld (2001) noted, school mathematics is presented as a collection of discrete, unrelated information; however, in view of the new perception of
  8. 8.   7   mathematics, learners should be provided tasks with enough scope to allow adaptation and application of mathematics in several contexts. These are referred to as enriched tasks. Summarily, theme one is concerned with the re-conceptualization of the nature of mathematics so that learners and teachers’ perception of the nature mathematics presents a more jointed as opposed to disjointed view of the different content areas; or a shift in its presentation, from a series of procedures to be learnt to one geared towards achieving a balance between procedural and conceptual development. And theme two’s concern is fostering curriculum design that integrates the kind of mathematics practices found in the work place in a bid to bridge the perceived boundaries between school mathematics and out of school mathematics; as well as emphasize the competencies that school mathematics hopes to foster in order to allow learners to apply mathematics concepts acquired in schools in different situations. The interrelationship between these two themes brings to bear a third theme; the learners ability to apply the knowledge learnt in schools to solve problems arising in similar or different contexts, in other words, transfer of learning. The third theme, implicitly, emerging from the definitions of numeracy, is the issue of transfer of learning from school to wider society. Unfortunately, transfer is generally believed be problematic in the mathematic education community (or education in general). The consensus is that learners, more often than not, lack the ability to transfer knowledge they have learnt in school in contexts outside of school (e.g. Detterman, 1993; Lave, 1988). Like some of driving forces behind curricular reforms (e.g. commerce and industry), current
  9. 9.   8   researches (e.g. Hoyles et al. 2002, Kent et al, 2007) lament the apparent lack of value of students’ mathematics qualifications. These developments have brought back focus on the concept of ‘transfer’ (among other issues), which, like many of the ideas of cognitive psychology, has been relegated to the back seat, so to speak, of research in mathematics education. However given its obvious importance of learning transfer, I wish examine how mathematics education can be structured to foster and facilitate transfer of learning to the work place. To work towards developing this understanding, I organized the thesis into five chapters: Chapter 1 is this introduction, which explains the background and rational of the study, as well as the overall structure of the thesis treating the issues related to the complex notion of transfer. Chapter 2, which follows from this introduction, is concerned with characterizing the concept of transfer by, first of all, providing the definition of transfer of learning; types of transfer and their measures; and concluding the historical development of the concept of transfer and highlighting its problems and concerns. Chapter 3 examines the positions of two popular epistemologies (social constructivism and situated cognition) on transfer and highlight why their key principles embody the concept of transfer. Chapter 4 covers transfer of learning from school to the workplace. My aim for this section is to address the following questions: How do we teach for transfer? What skills and knowledge are transferable from school to the work place? What instructional and learning strategies promote/hinder transfer.
  10. 10.   9   In chapter 5, I want to propose cultural historical activity theory (CHAT) as a possible research and pedagogic framework for studying and bringing together all the complex variables present in classroom settings in efforts to understand the best practices for teaching for transfer. Since traditional approaches have never take into account the influence of individual factors, the social context in which learning takes place or the environmental factors such as the impact of tools use, rules and regulations in a classroom community or learner characteristics such as motivation or interest all together, I will argue in this chapter that Engestrom’s (1999, 2005) CHAT, as a dynamic model, is particularly appropriate for studying/analyzing the concept of transfer. In conclusion, I highlighted, in this introduction, some of the factors (economic, it’s application and the impact of technology) driving curricular reforms in mathematics education, among which, the most important may be the need to prepare a generation of learners that can function effectively in a world that increasingly requires numerate individuals, in everyday situations, democracy and work place. Whilst these needs are not clearly defined by the mathematics community or society at large, some of the definitions of numeracy presented in the literature are indicative of precisely what those needs are. That is, in a broad sense, the development of mathematics curricula and teaching practices that enable learners to adapt and apply the mathematics they learn in school in different situations. According to Wake (2005), for learners to acquire the ability to apply, mostly basic mathematics to address complex problems of the work place, they will need to be provided with experiences to access richer and more diverse tasks than are currently provided by the mathematics curriculum. This
  11. 11.   10   development gave rise to, in UK for example, functional mathematics (which is discussed in chapter 4) aimed at bridging the gap between school mathematics and the mathematics. Functional mathematics, more than any teaching approach, makes teaching for transfer of learning an explicit educational goal. In the next chapter, I will consider what transfer is, its historical development and its relation to current influential theories of learning.                                                                  
  12. 12.   11   2. Concept  of  Transfer   In broad terms, transfer of learning can be defined as the learner’s ability to apply previous knowledge, or past experiences in similar or novel situations (Darling-Hammond and Bransford, 2005). However, there is an increasing inclination amongst the proponents of transfer, to emphasize thinking habits (which include motivation, collaborative teamwork, communication etc.). For example, according to Salomon & Perkins (1989), transfer is the ability to carry over knowledge, skills, understanding and habits of thinking from one learning situation to another. While recently, researchers focus on how general attitudes (affective factors and motivation) (Bransford et al., 2000), context (Greeno, 1997), social interactions or participation (Olivera and Straus, 2004) and other environmental factors affect the individual’s ability to transfer learning. Apart from the disparities in defining transfer of learning, there is also contention amongst researchers (even amongst proponents of transfer) as to its measures, means of achieving it, and theory of transfer. For example, some theorists concluded it rarely happens (e.g. Detterman, 1993); while others (e.g. Hammer et al, 2005; Lave, 1988; Lave and Wenger, 1991) do not believe it is possible; but some believe that transfer occurs all the time because the underlying principle of all learning is transfer (e.g. Dyson, 1999). Also, some current theories of learning posit that transfer is not possible (for example, social constructivism and situated cognition) while others, (behaviorism, cognitive psychology), despite supporting the concept of transfer, have very different ideas of it. Furthermore, despite the importance of transfer of learning, or the implicit assumptions in instructional settings that it will occur, the history of research on transfer paints a different picture and suggests that
  13. 13.   12   transfer rarely occurs. For example, Reed, Ernst and Banerji (1974, cited in McKeough et al., 1995), observed very little transfer from one river crossing problem to another and also Hayes and Simon (1977) reported similar failures of transfer in the study of problem isomorphs which are problems that are structurally similar (in approach or concept) but with very different surface features. They discovered that subjects more often than not do not recognize the connection between one isomorph and another and hence do not carry over strategies they have acquired while working from one to the other. However, they also highlighted the transient nature of failures of transfer by demonstrating that failures to transfer could be changed to successes simply by pointing out the relationship between the source and target tasks to the subjects. Amidst the failures, there were a few successes of transfer, among them was Judd‘s (1908) study which showed empirical evidence of children transferring learning of the principle of refraction; Katona’s (1940, cited in McKeough et al., 1995) success of demonstrating transfer of strategies for solving card tricks and match-sticks problems; and Palinscar and Brown’s (1984) report that when children are taught to self-monitor and self-direct themselves during reading in what they referred to as ‘reciprocal teaching’, they transfer comprehension strategies across a variety of settings. Campione et al. (1991), also using reciprocal teaching, showed that learners transfer these metacognitive strategies in other text-mediated areas of learning such as social studies and mathematics. Overall, the literature on transfer, including its theory, measures and mechanisms is very chaotic. Consequently, in an effort to further explain the
  14. 14.   13   concept of transfer, I will present a historical analysis of transfer of learning from the perspective of behaviorism and cognitive psychology as well current theories of learning that contradict it; in a bid to explain how and why transfer takes place (or not) from one situation to another. First, I start with earliest attempts to scientifically measure the phenomena of transfer to current research efforts aimed at advancing the concept of transfer and its application in mathematics education. 2.1. Historical  Analyses  of  Transfer   Transfer theory emerged from the empiricist perspective, which assumes that learner is a passive agent who transfers knowledge and facts learnt from a presumed “original learning situation” to new situations. The earliest research on assessing the quality of people’s learning experiences was carried out by Thorndike and Woodworth (1901, cited in Bransford et al., 2000). Their aim was to test the assumption of ‘formal discipline’ that learning Latin develops general skills of learning and awareness in other areas of studies. They generally failed to find positive impact of one sort of learning on another. In a subsequent study, Thorndike (1923, cited in Bransford et al., 2000) compared the performance in other academic subjects of students who had taken Latin with those who had not and found no advantage of Latin studies whatsoever. Thorndike and Woodworth (1901) concluded that transfer depended on the number of ‘identical elements’ between source and target tasks and situations.
  15. 15.   14   2.1.1. The  Theory  of  “Identical  Elements”   The hypothesis of “identical elements” proposed that transfer is dependent on the number of similarities, which exist, either in content, technique or context, between the old and novel situations and not due to the development of “general skill” or “mental muscle” as posited by theory of formal discipline. The theory of identical elements (elements were assumed to be specific facts and skills) identified positive and negative transfer, lateral and vertical transfer, and near and far transfer as the different types of transfers which I will come back to later in the discussion. These definitions of the types of transfer (for example near and far) suggest a “transfer distance”, which provide indication of how different tasks and their contexts are. That is how similar tasks and contexts are said to be near one another and dissimilar ones far from one another. However, without objective measures of the perceived distance or similarity between tasks, the theory of identical elements could not explain sufficiently the concept of transfer. Moreover, by acknowledging that “they spoke of elements ‘without any rigor’, but they meant mental and environmental objects and events "differing in any respect whatsoever" (Butterfield et al., 1989*1 , p. 7), they become even more unspecific about what was an element, as result, reducing the effectiveness of the original theory of identical elements. At least in the original version (though also not without problems) it can be easily deduced that to promote transfer, teachers need to make their classroom materials (i.e., stimuli) and the ways pupils are taught to use them (i.e., responses) as similar as possible to the stimuli and needed responses in their pupils' wider lives (Engelmann and Carnine, 1982). But in redefining the transfer ‘elements’ to be
  16. 16.   15   “mental and environmental objects and events”, which is basically anything, makes it virtually impossible to interpret Thorndike’s revised version of ‘identical elements’ in terms of teaching and learning. In this light, it is therefore not surprising that his contemporary, Judd, challenged his theory of identical elements and proposed the theory of generalisation. 2.1.2. Theory  of  Generalisation   Judd (1908, cited in Bransford et al., 2000) argued that transfer has less to do with the subject matter or the content and more to do with “general principles” of which one learns in a situation, and that the transfer issue must be considered in relation to learner’s characteristics such as prior understanding and strategies. As a result, his “theory of generalization” posits that transfer occurs when one is able to abstract invariances (in terms of surface features, concepts or strategies) across situations. In order to illustrate that the “theory of generalization” rather than the theory of “identical elements” is the means of achieving transfer, he carried out a research comparing the effects of “learning a procedure” with “learning with understanding”. He had two groups of children; one group received an explanation of refraction of light, which occurs when light is incident on an interface between two materials (in this case air and water), at an arbitrary angle, its direction is altered. As the speed of light is reduced in the denser medium, its wavelength is shortened proportionately and it bends. This bending of light causes objects to appear to be in different location than they actually are. The other group only practiced dart throwing, without any instruction on the principle of refraction. Initially, both groups did equally well on the practice task, which involved a target 12 inches under water
  17. 17.   16   but when the target was changed to 4 inches, the group that had been instructed about the abstract principle of refraction were able to transfer their learning when the conditions changed. Judd concluded they performed much better because they understood what they were doing, as a result they could adjust their behaviour to the new task. They were able to organise the information in the source and the target situations according to similarities in their basic components in other words, according to structural similarities – that is similarities in surface features, concept, or strategies. That is, by recognising that the phenomena of refraction occurs (a) when light travels between two media, and (b) the denser media shortens the length of the light, the participants who received instructions where able to recognise the similarities in structure, i.e., surface features and concepts between the principle of refraction and shooting targets submerged underwater. Using this knowledge, they were able to adjust their aims to hit the targets. This is one of the earliest successes of research demonstrating the feasibility of transfer. Moreover, the attention given on structural similarities is also recognized as a major development beyond stimulus-response psychology. Transfer of learning in the structural perspective takes place when learners already have an understanding of the procedures and concepts of the original problem, which they bring to bear in the target task. And, in emphasizing that transfer is facilitated by learner’s prior knowledge and the use of strategies (not the subject matter) such as directing learners’ focus on conceptual similarities across tasks, Judd’s model can be considered to be a precursor to the cognitive perspective on transfer (Haskell 2001). In the following subsection, I outline the cognitive psychology perspective of transfer.
  18. 18.   17   2.1.3. Cognitive  Psychology  and  Transfer   Unlike the behaviourist, the cognitivist acknowledged the importance of stimuli as represented by the learner (as opposed to an experimenter or a teacher) and on responses generated by their covert knowledge and strategies. Drawing from Inhelder and Piaget’s (1958) theorizing that knowledge is organised into mental schemata, cognitivists developed an information processing framework to explain learners’ ability to organize, retrieve and process relevant prior knowledge in order to solve new problems. In other words, they developed the idea of schemata to represent the elements in transfer theory. Cognitive theory of elements and mechanisms (or schema theory) proposed that three fundamental concerns must be addressed in order to teach for transfer. That is determining how students select particular knowledge and skills to use in new situations, from all that they have learnt, selecting the appropriate knowledge and skills to use in a particular context and choosing alternative strategies, knowledge or skills if previously selected ones are not useful. In a bid to address these concerns, cognitivists not only identified three elements that fit readily into transfer theory, but also built mental models to show how the elements worked together to produce behavior. The three elements are stimulus representations, strategies that operate on those representations, and knowledge of how to use the results of those strategic operations as integral to transfer. For example research on the formal reasoning problem of predicting the behavior of a balance scale provides an example of what cognitivists mean by representations, strategies, and knowledge. Weights on either side of a fulcrum and their distances from the fulcrum must be taken into account in order to
  19. 19.   18   obtain a solution of balance scale problems. Young children represent balance scales by considering only the weight, that is the side with more weight goes down, whereas older children, took advantage of their more advance knowledge and represent both weight and distance in their solutions. That is by taking into account that, the side with more weight goes down if its weight is not compensated by placing the objects at greater distance on the other side of the fulcrum (Siegler, 1976; Ferretti et al., 1985). The older children used their prior knowledge to devise more sophisticated strategies. These findings suggest that children of different ages have different mental models of balance scale problems, which is constrained by their knowledge, which in turn constrain the strategies they adapt in problem solving. This also suggests the possibility that learning and transfer may be promoted by teaching younger or less expert students, the mental models of older or more expert students. Butterfield and Ferretti (1984) showed that young children could be taught to use the representations, strategies, and knowledge applied by much older people to solve balance scale problems. Moreover, they were also able to transfer the strategies they have learnt to similar untaught tasks (Butterfield & Nelson, 1989*2 ). Similarly, McDermott et al. (1987) showed that when college students were shown strategies used by expert physicists, such as sketching graphs to generate mental models of position, velocity and acceleration of objects, their understanding of mechanics problems is increased. These examples also show that learners are able to appropriate mental models, strategies and thinking of more able students or experts and transfer them in other context when given the opportunity to learn. Equally, they also
  20. 20.   19   explain the relationship between stimuli representation, knowledge and strategies – that is more experienced learners have more sophisticated representation and strategies. However the means by which learners know which strategies or knowledge to draw on in problem situations is not evident in the examples. The cognitivist posits that choosing effective knowledge and strategies, requires metacognitive awareness (Brown, 1987; Butterfield & Nelson, 1989*2 ). An example of metacognitive awareness would be on encountering a balance scale problem in the previous example, older learners where able to tell in advance that changing the distance of the objects on either side of the fulcrum may allow them balance the scale. In this respect their choices were influenced by what they knew in advance (previous schema), and the knowledge that a particular strategy may solve a problem or knowledge of how much effort it will require as opposed to the trial and error methodology adopted by less experienced learners. In short, the metacognition model of transfer emphasises awareness of one’s thinking and actions. It also partly depends on learners’ willingness to seek out structural or conceptual similarities based on their past experiences. Despite some of the success reported by behaviourists and cognitive psychologists researchers on transfer, some researchers (for example, Ellis, 1965; Detterman, 1993) criticized those earlier successes arguing that apart from the lack of scientific rigor and the potential for bias (as the researchers conducted the test), those successes were based on analyses that did not reflect “natural circumstances” or real learning processes and situations. They suggested that any proposed theory of transfer should not be considered in
  21. 21.   20   isolation to a student’s purposes, existing knowledge, strategies, attitudes, motivation and classroom settings. Moreover, Thorndike’s “theory of identical elements”, for example, focuses only on developing procedural knowledge and not conceptual knowledge and even where it works, it only results in near transfer (Detterman, 1993), which is not the main or only focus of research on transfer. Far transfer, which none of these studies have been able to produce, allows knowledge to be applied across tasks or situations (e.g. school to workplace) and thus should be the main concern of education. Nevertheless the theory of identical elements is still a prevalent educational strategy. For example, the idea that “practice makes perfect” (in the tradition of Thorndike, Skinner and Watson) continues to be a fundamental aspect of transfer in mathematics education. Research such as Anderson and Singley’s (1985) and Silver’s (1981) reported findings supporting Thorndike’s claim. Also, Ericsson and Smith (1991) showed that expert performance in areas such as chess, physics, problem solving, and medical diagnosis, depended on a large knowledge base of specialized knowledge and skills. Equally, Judd’s work on abstraction and generalisation also has not lost influence amongst practitioners and researchers alike in mathematics education. Recent works on analogical problem solving in which the learner is first provided with conceptual representation of a problem or strategies to tackle the problem and later on introduced to a similar problem (surface features, principles/concepts or approach) is a derivation of Judd’s theory of generalisation. Learners have to recognise the similarities in structure in order to solve the problem. Examples of such experiments are The Tower of Hanoi (Kotovsky & Fallside, 1989), and Duncker’s radiation problem (see Gick &
  22. 22.   21   Holyoak, 1980, 1983). Also, Perry (1991) found that fourth and fifth grade students showed considerable amounts of transfer between mathematics problems when they received instructions on conceptual principles rather than procedures. So contrary to the claims of critiques of earlier studies on transfer, recent studies done in real classroom context are also substantiating the concept of transfer as proposed by Judd and Thorndike. Furthermore, instead of dismissing the concept of transfer, these studies confirm that transfer is not unachievable but rare and that different amounts or types of transfer occur depending on the amount of practice or the instruction participants received. Additionally, it strikingly odd for most learning theories (e.g. radical and social constructivism) to embrace the fact that learners bring their past experiences to bear in new learning (which is essentially transfer) and at the same time deny the concept of transfer. This paradox may be due to the association they make between transfer and the “conduit metaphor” presented by behaviourists. This metaphor views language as tool for transferring thoughts, ideas or skill from one person to another. That is, in writing or speaking people transform their thoughts or feeling in the words; these words (containing the thoughts or feelings) are then conveyed to others, who then through reading, writing or listening, extract the thoughts or ideas contained in the words. The conduit metaphor, like most socio-cultural models of learning, suggests two planes of interactions. However, there is no consensus regarding epistemology. But a closer look at some of the alternative concepts (e.g. ZPD, appropriation), which I will discuss later, reveals that their underlying principle is transfer. As I stated earlier on, measures of transfer is also another murky area amongst researchers. The typological or taxonomical approaches are most
  23. 23.   22   widely used in the literature. Below, I will discuss some of these classifications and their influence on teaching and learning. 2.2. Measures  of  Transfer   In the literature, transfer is classified according to either its types (see, e.g., Detterman, 1993; Salomon & Perkins, 1989; Singley & Anderson, 1989) or the perceived levels of transfer that have occurred (see, e.g., Haskell 2001). However, these classifications offer no explanation as to why the classifications are different, similar or what information is transferred. This makes it very difficult to appreciate the usefulness of the models to mathematics education or education in general. As noted earlier on, the theory of identical elements identified positive and negative transfer, lateral and vertical transfer, and near and far transfer as the different types of transfers. Below I make an attempt to describe, with help from examples, what they are, their differences, similarities and the kinds of information transferred, from one type of transfer to another as well as their implications in teaching or learning. Firstly, positive or negative transfer refers to when learning in one situation improves or hinders learning in another situation, respectively (Cormier and Hagman, 1987). For example when elementary students learn multiplication, they are often limited to examples involving positive whole numbers as a result, they develop the idea that multiplying numbers results in a bigger number. They bring this misconception to bear when learning to multiply fractions or decimals to the extent of misconstruing, for example when multiplying ½ x ½ = ¼, that the ½’s on left-hand side of the equation are less than the ¼ because one has to multiply them to get the ¼. In this way, a
  24. 24.   23   teacher’s failure to bring in counter examples when reviewing previously content resulted in negative transfer. Negative transfer usually occurs in early stages of learning and with experience, learners do correct it. Consequently, promoting positive transfer is the primary concern of education, as such, strategies aimed at achieving it, are the main focus of education. Lateral transfer also referred to as horizontal transfer, happens when the learning in one context, is applied at the same level in a new context. For example, if a learner knows that 7 x 3 yields 21 and use this knowledge to work out how much 3 bars of soaps, priced at £7 each cost. In this way, past learning is transferred an identical level either in approach or concept. However, vertical transfer occurs whenever learning necessitates prerequisite skills. For example, learners are usually introduced to calculating areas rectangles, squares, then circles and ellipses before that of composite figures. The hierarchical nature of school curriculum is derived from this conception of transfer. As a result, knowledge, either procedural or conceptual, is arranged from easier to difficult or according to complexity. Near transfer occurs when we transfer previous knowledge to new situations closely similar to, yet not identical to, initial situations (Perkins and Salomon, 1989). For instance if a student learns how to calculate the mean in mathematics and later on apply this knowledge in environmental studies to calculate the mean amount of rainfall in a particular period, that is regarded as near transfer. However, far transfer takes place, if ability to multiply in school allows a learner, doing the family’s monthly shopping at the local market to work out how many packets of soap he/she needs to buy to last a month, if his/her family exhaust 5 bars of soap a week; and the total cost of the
  25. 25.   24   purchase if each packet is priced at £7 each and contains 6 soaps. The key emphasis of far transfer is the extension of skill and knowledge between contexts that, on the surface, seem isolated from and unfamiliar to one another. One way of differentiating between near and far transfer is through comparing school-learned events and out-of-school events (Butterfield and Nelson 1989*2 ). School-learned events are recognized to be an example of near transfer (Mestre 2005) because the conditions involved for it are present and the same for both original and target tasks. But applying the knowledge and skills acquired in school-learned events to out-of-school problems exemplify far transfer because the context and the goals are different in the two situations (Haskell 2001). Far transfer, which is the most desirable construct, is the most elusive out of all the other types of transfer. So far research on transfer has only succeeded in demonstrating near transfer. The shift from behaviourist to cognitivist perspective of transfer emphasised transfer of concepts and complex skills as opposed to basic knowledge and basic skills. As a result, transfer is reconceptualised as a conscious process as opposed to reflexive application of routine skills. Perkins and Salomon (1989) coined the terms high road (or conscious application of strategies or principles) and low road transfer (automatic or unconscious application of strategies or principles), to account for the goals and motivational element of transfer. They defined high road transfer as the “explicit conscious formulation of abstraction in one situation that allows making a connection to another” (p. 118) and low road transfer as “the spontaneous, automatic transfer of highly practiced skills, with little need for reflective thinking” (p. 118). Similarly, Sternberg and Frensch (1993) argued that motivation, or intent
  26. 26.   25   to transfer, determines the extent of the learner's active involvement and attitude toward learning. Thus the identification and teaching of strategic knowledge (e.g., metacognitive and problem-solving skills) that are applicable to a broad range of tasks, is key to achieving transfer (Pressley et al., 1987). Moreover, motivation to transfer also referred to as ‘spirit of transfer’ (Haskell, 2001), involves possessing positive dispositions towards learning that allows students to process information in ways that facilitate transfer. These positive dispositions include traits, like high motivation, risk-taking attitudes, mindfulness or attentiveness, and a sense of responsibility for learning (Pea, 1988) Thus far, the behaviourists emphasise the importance of possessing a deep foundation of factual and procedural knowledge (Thorndike, 1901) and a need to design instruction to develop conceptual understanding (Judd, 1908) in facilitating transfer. And the cognitivists propose that in addition to having a deep foundation of factual knowledge and procedures, learners need to be taught strategies of organizing these resources in ways that facilitate their retrieval and application. These metacognitive strategies involve awareness of what they know or do not know, and knowledge of when and why to access certain information in problem situations. Similar to behaviourists, cognitivists also posit that learners’ previous knowledge and experiences are important factors. Most of these ideas crop up in almost all current theories of learning. However, the behaviourist legacy of transfer seems to hinder almost all possibilities of establishing transfer as a credible research construct. However, if we get past that and have a closer look at the conditions which these successes where reported or even, why the reported failures occurred, it becomes clear
  27. 27.   26   what conditions and mechanisms are necessary for transfer. May be then, the debate aimed at establishing whether transfer is feasible or not, or the issue of developing the appropriate metaphor, will make way for the development of instructional contexts and tasks to facilitate transfer. Nevertheless, the worst setback received by research work promoting the need to teach for transfer may be attributed to work done by proponents of social constructivism, and situated cognition. Social constructivists and early proponents of situated cognition (Lave, 1988, Lave and Wenger 1991) posit that knowledge does not transfer between tasks or situations. These two schools of thought mostly influence current researches in mathematics education. The interest in social constructivism represent a shift from a focus of intellectual processes within the individual, as in the classic research of Piaget, to a concern with social cognition very much influenced by increased interest in Vygotsky (Butterworth, 1992). In this respect, the emphasis of research is on the ways in which thought processes and cognitive processes are socially situated. Equally, socio-culturalist like Lave and Wenger stressed an inextricable link between contextual constraints and the acquisition of knowledge. This has resulted in a merger, if you like, between, context, social, and cognition and suggests that cognition is situated in a social and physical context and is rarely decontextualised. In the next section, I will discuss not only the similarities and differences between ideas of social constructivist and situated cognitivist but also highlight that their key ideas regarding learning intrinsically embody the notion of transfer.
  28. 28.   27   3. Transfer,  Social  Constructivism  and  Situated  Cognition   Two schools of thought – social constructivism and situated cognition, mostly influence current researches in mathematics education. In this section, I will show why some of their key notions, such as learning through social interaction or learning through participation in practices; zone of proximal development; and the concept of appropriation may possibly be “old ideas in new clothing” when compared to some of the ideas offered by advocates of transfer and those expressed by Dewey (1925/1981), a renowned philosopher of education whose ideas on learning are consistent with ideas of behaviorism and cognitive psychology. In order to demonstrate these similarities, I will start with the notion of social interactions in learning as presented by social constructivists and situated cognitivists. Even though these two theories have similar ideas, they have a slight difference in epistemology. For example, social constructivists argue that learning should be: "Viewed as an active, constructive process in which students attempt to resolve problems that arise as they participate in the mathematical practices of the classroom. Such a view emphasizes that the learning-teaching process is interactive in nature and involves the implicit and explicit negotiation of mathematical meanings. In the course of these negotiations, the teacher and students elaborate the taken-as-shared mathematical reality that constitutes the basis for their ongoing communication" (Cobb, Yackel, & Wood, 1992). This suggests that meanings and knowledge are shaped and evolve through negotiation within the communicating groups, involving for example, a teacher and learners or learner and more knowledgeable peers. The nature of the
  29. 29.   28   learner's social interaction with knowledgeable members of the society is very important in this paradigm. Without it, learners may not acquire social meaning of important symbol systems and learn how to use them. From the premise that people develop their thinking abilities by interacting with experts, I deduce that any personal meanings arising from teaching and learning are shaped by and conveyed (transferred) through these interpersonal interactions. Similar to social constructivism, situated cognition also posits students collaborate with one another and experts toward some shared understanding. For instance, both theories maintain that learning should be situated in “zones of proximal development” just beyond what a student can accomplish alone and should employ peer or a teacher scaffolding to extend learners’ capabilities. Equally, both theories maintain that learning may occur through intrapersonal communication. That is, thought process or communication with one's self is the means by which knowledge or skills are constructed. In this perspective, the individual provides feedback to him or herself in an ongoing internal process. This involves not only a willingness to be part of a community of practice, but also involves a process of self-assessment and self-regulatory activities similar to metacognitive strategies proposed by Judd and cognitive psychologists as necessary for transfer. However, the difference with social constructivism and situated cognition resides in their respective epistemology. In contrast to the social constructivism, which begins with the acquisition of theory and then work towards practice, situated cognition emphasises that activity and enculturation is integral to learning. The development of knowledge or skills arises through the persons situating or immersing themselves in practice, at first by carrying out simple tasks or through “legitimate peripheral
  30. 30.   29   participation” (Lave and Wenger, 1991) and gradually, proceed to more complex and demanding tasks as they establish there identities in the community of practice. This is referred to as learning by apprenticeship. The following quote from Brown et al (1989) sums up quite neatly the premise upon which situated cognition theory is based: “A theory of situated cognition suggests that activity and perception are importantly and epistemologically prior - at a non-conceptual level - to conceptualization and that it is on them that more attention needs to be focused. An epistemology that begins with activity and perception, which are first and foremost embedded in the world, may simply bypass the classical problem of reference --- of mediating conceptual representations.” (p. 35) In this regard, the critical distinction between social constructivism and situated cognition is that people learn by doing. Initially, they may not have fully grasped the concept involved, but through immersing themselves in the practice, they will eventually abstract the conceptual representations embedded within it. In other words, concepts are embedded in the world and people learn through engaging in the activities alongside more able individuals, abstract representations of real situations are developed and through them, come conceptual understanding (Brown et al, 1989). Again, this is strikingly similar to the both the theory of identical elements and theory of generalisation. Another difference between social constructivism and situated cognition is that in the apprenticeship model of the latter, no one specifically sets out to instill knowledge and skills uniformly into a group of learners. The more inclusive process of generating identities, or desire to become a master practitioner, is both a result of and motivation for participation. In fact, unlike
  31. 31.   30   social constructivism, it is rarely the case that the individual apprentice has someone to teach him or her in order to learn. Conversely, it is the ongoing everyday activity that provides structuring resources for learning. Thus gradually increasing participation provides the scaffolding. Again this is similar to practicing procedures/skills and knowledge suggested by Thorndike. Overall, the point being made is that despite the differences in epistemology between social constructivism and situated cognition, their treatment of the role of social interactions in mediating cognitive activities suggest that knowledge is shaped by and conveyed through these interactions, which is in a nutshell, transfer. Having considered the role of social interactions from the social constructivist and situated cognitivist perspectives, I will now relate Dewey’s (1925/1981) philosophy of education, which I refer to as social behaviourism, to the social constructivism and situated cognition. But first, I want to explain why I believe Dewey’s philosophy could be referred to as social behaviourism and also strongly makes a case for transfer. In his paper entitled “Deweyans Pragmatism and the Epistemology of Contemporary Social”, Garrison (1995, pp. 718-720) declared that the field of education should consider behaviorism as one way of understanding social constructivism and situated cognition. As a result, drawing from what Richard Rorty (1979) calls “epistemological behaviorism” or what W. V. O. Quine (1969) referred to as "behavioral criteria for the truth" and the writing of Dewey himself, Garrison (1995) argues that Dewey may be identified as a behaviorist and coined the term “Dewey’s pragmatic social behaviorism” to describe his philosophy. A fundamental principle of this brand of behaviourism is the rejection of the notion of knowledge as merely conceptual representation.
  32. 32.   31   Rather, people acquire knowledge based on the evidence of other people's overt behaviours or activities. As Dewey put it, “meaning is not indeed a psychic existence; it is primarily a property of behavior, and secondarily of objects. But the behavior of which it is a quality is a distinctive behavior; cooperative in that response to another's act involves contemporaneous response to a thing as entering into the other's behavior, and this on both sides.” (Dewey, 1925/1981, p.141). Dewey extends the behaviourist’s view on learning to include, social interactions, cooperation and communication. In fact this quote, shows not only the considerable influence of behaviours on cognitive activity but also suggests that through his/her behaviour, the individual shapes and is shaped by the world. Note that the dialectical nature of stimulus and response is akin to the intersubjectivity involved in meaning making in other competing theories of learning, but in this case the focus is on actions. Since I also strongly believe that we most of the time infer what goes on in the individual’s mind through their behaviour, I will henceforth refer to some of the ideas presented in Dewey’s book “Experience and nature” as “Dewey’s Social Behaviourism”. That notwithstanding, I will begin the comparison of Dewey’s social behaviorism, social constructivism and situated cognition with the following quote from Dewey (1925/1981): "Through speech a person dramatically identifies himself with potential acts and deeds; he plays many roles, not in successive stages of life but in a contemporaneously enacted drama. Thus mind emerges" (p. 135).
  33. 33.   32   This shows that for Dewey, meaning is a social construction and is associated with both activity and the gradual acquisition of the characteristics and norms of a culture or group by an individual. Also, based on the above quote, note that for Dewey, there is no need to determine the order in which activity or concept formation engender meaning; because it all happens simultaneously. In this regard, Dewey’s, social behaviourism could be said to unify the social constructivist’s and situated cognition’s epistemologies. Equally important to Dewey, was the idea that mind should be seen to denote "the whole system of meanings as they are embodied in the workings of organic life.... Mind is contextual…" (p. 230). This suggests there is no discontinuity between the individual and the world and that the mind emerges as the individual learns to participate in social activities involving labor, tools, and language. Which is essentially both a situative and social constructivist perspective, which places the locus of meaning making in social interaction or in participation in practice. In a nutshell, Dewey’s social behaviorism has much in common with both situated cognition and social constructivism. All these theories posit that learning is mediated by social interactions and mediation implies that some knowledge or skill is conveyed, and again this implies transfer. Other notions shared by situated cognition and social constructivism is the zone of proximal development, a metaphorical space in which to study the interaction between teacher and learners, and scaffolding which describes the processes involved in allowing the child to advance through the ZPD (or progress from what a he/she can do without help and what he/she can do with help). Vygotsky defined the zone of proximal development as:
  34. 34.   33   “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers”. (Vygotsky, 1978, p. 86) This suggested that, according to Vygotsky, a child’s learning is advanced through the provision of experiences, which are in his/her ZPD, and that in interactive situations, children can participate in activities that are more complex than those they can master on their own. Bruner (1985) introduced the term ‘scaffolding’ in relation to ZPD to explain the process through which the child's learning is guided through focused questions and positive interactions in order to advance their knowledge from lower level of the ZPD to a higher level. Scaffolding is achieved through semiotic mediation (Vygotsky, 1978, p. 40); that is when the direct impulse of the learner to react to a stimulus is inhibited through the intentional introduction of a sign, by an external material (e.g. a teacher, textbook or computer), thus controlling their behavior from outside. The Socratic method/dialogue (see Renyi, 1967), which encourages learners to reflect and think independently and critically, embodies the concept of scaffolding. It is characterized by starting with the concrete experience and through probing; the learner gains insights by explicitly relating any statement made to his/her personal experience. Some education experts criticized the notion of scaffolding for its tendency to share the passive unidirectional transmission of information characteristic of the conduit metaphor of communication. For example wertsch
  35. 35.   34   (1991) stated scaffolding invokes the conduit metaphor, which describes the transference of ideas or schemata from one person to another by means of human communication. He argued that the conduit metaphor fails to realize all of the dialogical possibilities between teacher and student, which may involve the tension caused by the learner’s need to engage in the existing practice and their need establish their own identities. To incorporate this dialogical relationship, Lave and Wenger (1991) reinterpreted the scaffolding from a collectivist or societal perspective by putting "more emphasis on connecting issues of socio-cultural transformation with the changing relations between new comers and old-timers in the context of a changing shared practice" (p. 49). In this regard, scaffolding refers to the process through which a person’s identity is established or evolves as they participate in practice. Granted, scaffolding may not be monological in this perspective, but transfer still underpins means by which the change in relationships between new comers and old timers is brought about. This is due to the fact that in any social group, knowledge or standards are derived from established modes of communication or interactions. As a result, it may be difficult to regard any meaning constructed from such activities as objective. Rather, from social constructivists and situative perspectives, individuals appropriate knowledge and skills due to their involvement in social practices. Appropriation also invokes the concept of transfer. Newman et al. (1989) indicated that appropriation was proposed by Leont’ev, a colleague of Vygotsky as an alternative to Piagetian concept of “assimilation”; which describes the process through which new experiences are modified to fit schemes previously developed. The concept of “appropriation” is
  36. 36.   35   concerned with what individuals may take from interactions with other individuals in a cultural context. This suggests that in the educational context for example, the social relation between teacher and learner is an integral aspect of meaning construction. In this respect, the process of teaching and learning may even be viewed as not a relation between individuals and knowledge, rather as individual's introduction into an existing culture. For example, in collective activities, like exchanges at Oksapmin trade stores, described by Saxe (1999); individuals are engaged in constructing communications about quantity and in accomplishing quantity related problems. Saxe wrote “such occasions provide opportunities for reciprocally appropriating at least superficial features of one anothers’ constructions. In such appropriations, new forms are born as particular representations become valued and institutionalized as regularized ways of representing and accomplishing problems linked to collective practices” (unnumbered). Furthermore, by acknowledging that individuals appropriate understanding through cultural contact, social constructivist and situated cognitivists are in essence validating the concept of transfer. For example, from an educational perspective, the term appropriation suggests that learners are cultural beings (who relate to the ideas, customs, and social behavior of a society) and the interactions between themselves or between them and teachers are cultural experiences. As a result, all understanding derived from such encounters are essentially culturally defined. As such, it may difficult for any socio-culturalist to deny that somehow, cultural experiences - for example, learning in the social
  37. 37.   36   context - fundamentally imply transfer. Similarly, in an attempt to explain the meaning and the origins of the term appropriation (Bartolini Bussi, 1994,) declared that: “For Vygotsky, the process of learning is not separated from the process of teaching: the Russian word obuchenie, which is used throughout Vygotsky's work, means literally the process of transmission and appropriation of knowledge, capacities, abilities, and methods of humanity's knowing activity; it is a bilateral process, that is realized by both the teacher and the learner”(p. 125) Drawing from the arguments in this section, I can comfortably conclude that even within the theories that supposedly deny validity of transfer as an educational construct, they describe processes that essentially embody its concept. As a result, there should not be any reason for mathematics education not go ahead and develop a research framework for teaching or learning transfer. As it is, research on transfer is fragmented to say the least and considerable efforts need to be put in reviewing the empirical evidences produced by various theories on transfer in a bid to create a unified framework for teaching for transfer. Equally, more researchers (e.g., Lester, 2005; Cobb, 2007) are calling for the adoption of multiple perspectives in matters of teaching and learning in other to account for the multifaceted nature of classroom settings. As such, analyses of theories or research findings should reflect the interplay between psychological, sociological and contextual aspects of learning. I think this view can be applied to research on transfer in general. Consequently, adopting a conceptual framework such as the cultural historical activity theory (CHAT) may be a more effective framework for examining
  38. 38.   37   educational practices. It is a broad approach that can be used to develop conceptual tools for analysing many of the theoretical and methodological issues on transfer from school to the work place. In my final chapter, I will highlight the potential of CHAT as a pedagogic framework for teaching for transfer. In chapter 2, I presented, the concept of transfer by highlighting some of the practical and theoretical issues associated with it and mapping out its historical development. In so doing, I have not only shown the importance of making teaching for transfer an explicit educational goal, but also how research trends on transfer has gradually evolved to accommodate three fundamental elements. These elements are the (a) tasks (procedural vs. conceptual approach); (b) learner characteristics such as dispositions, teaching of metacognitive strategies, and promoting their willingness to transfer learning; and (c) considering the transfer context which includes impact of the social participation or instructional settings, and the quality of the instruction or teacher intervention. In a bid to argue for teaching for transfer, I have shown (in this chapter – chapt. 3) that even with learning theories that supposedly doubt the feasibility of transfer; it underpins the alternative notions they presented regarding how we come to know. In the next chapter, I will consider transfer of knowledge from school to the workplace, in terms of the learner, tasks and context.
  39. 39.   38   4. Transfer  from  School  to  the  Work  Place   In Chapter 1, I highlighted the development of functional mathematics in United Kingdom as a means bridging the gap between school mathematics and out of school mathematics and equipping learners with the desirable mathematical competencies in their wider lives. However this begs the questions as to what level of mathematics to teach learners. For example, in terms of pedagogy, how do we accommodate learners, who may want to pursue mathematics studies in higher education and those whose mathematics use may very much be subsumed by technology use in the workplace? Does technology use interfere with mathematics learning? Should teaching continue to be led by traditional, routine or expository approaches or should problem solving and more investigatory approaches to learning be adopted? Should traditional mathematics with its formal tasks and problems be the basis of the curriculum, or should it be presented in realistic, authentic, or ethno- mathematical contexts? I will attempt to answer these questions in relations to the transfer of learning in this chapter. Through out this essay, I maintained that the ultimate goal of education is to help students transfer what they have learned in school to everyday settings and workplace. And from the historical development of transfer outlined in chapter two, it is clear that transfer between situations is a function of the common elements or similarities in strategies, principles, surface features of tasks, context etc. between original and target situations. These common elements may also involve learners’ disposition such as their willingness to transfer. These factors identified for transfer are in their own right very complex. Consequently, it becomes evident that if educators aim to teach for
  40. 40.   39   transfer of learning from school to other settings, it is important not only to study the elements identified by transfer theories, but also understand the non- school environments in which students are required to function. Since the focus of this study is transfer of learning from school to the work place, I will start by briefly outlining the role of mathematics in the work place and the key differences between school learning and work place learning identified by research. 4.1. Mathematics  in  the  Work  Place   Mathematics plays a significant role in the workplace and continues to be a factor of increasing importance in the workplace (Hoyles et al., 2002; Kent et al., 2007). However, mathematics competencies required in the workplace are better defined as numeracy (Steen, 2001). This is because unlike school mathematics, numeracy, is so heavily embedded in practice that it may be difficult for the casual observer to discern the underlying mathematics. Equally, numeracy, which more closely defines the mathematics used in the work place, is anchored in real data, offers solutions to problems about real situations. In other words, mathematics in the work place is framed by the work situation and practice, and, in many cases, by the use of IT tools. The purpose of mathematics in industries also varies because of the diversity of processes involved. Hoyles et al. (2002) identified the following mathematical skills and competencies recurring in the sectors (i.e. electronic engineering and optoelectronics; financial services (Retail); food processing; health care, packaging; pharmaceuticals; and tourism) they studied. In their report entitled
  41. 41.   40   mathematics skill in the workplace, Hoyles et al. (p.11), summarised numeracy as involving: a) Analytical, flexible, fast and often multi-step calculation and estimation in the context of the work (with and without the use of IT tools) b) Complex modeling (of variables, relationships, thresholds and constraints), c) Interpretation of, and transformations between, different representations of numerical data (graphical and symbolic) d) Systematic and precise data-entry techniques and monitoring e) Extrapolating trends and monitoring models across different types of work f) Concise clear communication of judgments g) Recognising anomalous effects and erroneous answers. From this we can deduce that the role of mathematics lies mainly on the decision-making process and interpretation of results. It also involves the practical application of rational numbers and the metric measurement system with contextualised approximations and estimations in critical calculations, often with other workers. And more often than not, numeracy goes hand in hand with non-mathematical skills such as for example, planning, organising, cooperating, communicating effectively and teamwork. This is in marked contrast with school learning, which is often an abstract, rule-bound, individual activity, with one correct answer (usually a number, an algebraic expression, or a standard graph), and where mistakes are without penalties. Several other studies also reported striking differences between school and out of school methods. For instance, abstract reasoning is often emphasized in school, whereas contextualized reasoning is often used in everyday settings (Resnick, 1987). This may explain why, unlike tasks requiring
  42. 42.   41   only the reproduction of skills or facts, problems concerning application tasks, problem solving and scientific argumentation in mathematics education poses considerable difficulties for most students (e.g. TIMSS/PISA Studies). These studies reported that reasoning could be improved when abstract logical arguments are embodied in concrete contexts. In their well-known study of people in a weight watchers program, Lave et al (1984) showed how thinking, is not only situated in the physical, social and cultural context, but also derives its form from these variables. One example is of a man who needed three- fourths of two-thirds of a cup of cottage cheese to create a dish he was cooking. Instead of multiplying the fractions, as any student would do in a school context, he measured two-thirds of a cup of cottage cheese, removed that amount from the measuring cup and then patted the cheese into a round shape. He then divided patted cheese into quarters, and used three of the quarters. Abstractions never featured in his method. Likewise, grocery store shoppers use non-school mathematics under standard supermarket and simulated conditions (Lave, 1988). Similar examples of contextualized reasoning have also been found in educational tasks. For example Brazilian market sellers make arithmetical calculations more effectively at their market stalls than in the classrooms (Carraher, Carraher and Schliemann, 1985). These findings suggest that the individual’s interpretation of tasks is partly determined by the context and the goal of the activity. That is since different institutions have their own body of cultural knowledge, goals, conventions or rules and ways of communicating, learning must be contextualised to reflect these variables. On the other hand, these examples also suggest potential problems with contextualised reasoning and that transfer
  43. 43.   42   is also partly dependent on the degree to which learners can relate their procedures to more general sets of solutions. For example, in the cottage cheese example above, if the material was a liquid, will the man be able to adapt his methods? It is not very clear from the research, but I would assume the answer to that is no. For this reason teaching abstraction and generalisation does have its merits. Consequently, context (from subject’s perspective) could be reconceptualised to include whatever knowledge learners bring to bear from past experiences, to make sense of novel situation. Another contrast between school and work place environments is the organization of work or division of labour. In school, assessment procedures put much more emphasis on individual work, whereas in work environments, teamwork or collaboration is preferred (Resnick, 1987). As a result, opportunities should be created for students to experience working collaboratively and share their ideas (Kent et al, 2007). Equally mathematics in the work place is driven by the need for accuracy, especially in safety critical systems (hospitals, nuclear plants etc.). Collaboration or teamwork is the mechanism established to ensure accuracy. For example, calculations are double-checked, and team and group work are fostered as part of workplace practice. Another difference between school mathematics and work place mathematics is that a large part of the latter is observed to lie within technology-related tools. Tools are used to solve routine problems in work settings, compared with “mental work” in school settings (Resnick, 1987). Some of the simple mathematical tools used in the workplace are the graphs, tables, simulation, modeling, and calculators to name a few. Since most of these
  44. 44.   43   technologies are also present in some schools, mathematics curricula (for example in UK and France) have been reformed in the hope of simulating certain aspects of practice in everyday situations and the work place. This is an important development because for me, it contradicts the beliefs of teachers and some curriculum developers (in my country, the Gambia, for example) that use of technology reduces learners’ knowledge of mathematics. Rather, technology use in mathematics education transforms nature of the mathematical skills or processes learnt in school to reflect mathematics use in their wider lives or the work place. For instance, in the workplace, workers need to able to transform the data collected into a mathematical model (e.g. a graph, chart or tables) and then interpret what this model means in terms of the work situations and then report these findings to their supervisors, who then make the relevant decisions out of the information arising from the analysed data. All these processes require not only, computation skill, but also ability to analyse and interpret information. Also certain work situations present a need to examine the effects of changing variables in mathematical models for a large amount of data. For example, in statistical models, multi-variables representing different thresholds may be keyed in to examine their effect on the output. Computers and other technologies can take this further through allowing faster computation, direct manipulation of representations and visualisation of conceptual objects (Jonassen, 2000). In some of these cases workers may not even be aware of the mathematics involved in the work activity, all they need is the ability to interpret what the figures on the monitor mean and make the corresponding decision. In this regard, use of technology is instrumental and workers “black-
  45. 45.   44   boxed” from the meaning of the mathematical activity they are involved with (Wake, 2005). In education, the availability of these new technologies make it possible for students in schools to use tools very much like those used by professionals in workplaces. Also simulating a situation such as I just described may not necessary require computers, which pose a problem for under-resourced schools in my country. The emphasis here is on how the content is presented to provide learners the opportunity to experience processes such as those found in the workplace. For example, when investigating the relationship between speed, distance and time, learners do not have to stop at learning to draw distance-time graphs. The content could be extended to involve interpretation of the information contained in graphs. Study of models, charts or graphical representations allows learners to objectify their thoughts so that they can reflect upon them. In this way, graphs, tables and symbolic notations in mathematics can be described as cognitive tools that extend thinking. Through analysing and interpreting models, learners in schools without ICT can experience processes similar to those found in technologised workplace. In summary, workplace mathematics (numeracy) suggest that the skills required are basic and near lower high school level, but the fact that they are applied in complex ways to ill-defined and continuously evolving problems makes it difficult for school leavers. As a result, mathematics curricula should incorporate the processes that learners need to learn in order to cross the metaphorical boundary between school and the work place. As noted earlier, some of these competencies are: ability to recognise situations in which mathematics can be used; make sense of these situations; describe the
  46. 46.   45   situations using mathematics; analyse the mathematics, obtaining results and solutions; interpret the mathematical outcomes in terms of the situation; and communicate results and conclusions. In short workplace mathematics is based on professional practice, which is basically functional knowledge. Functional knowledge offers people the know-how to get things done. The implication for teaching and learning is therefore, the creation of opportunities for learners to experience sessions that have a significantly new emphasis and focus on enriched tasks. That is tasks that simulate the scope, complexities and contextual nature of workplace mathematics. In other words, learners need to engage in activities that involve the application of straightforward mathematical skills or concepts in complex contexts. This is different from traditional mathematics teaching in which learners often do very challenging mathematics in very simple contexts, or entirely out of context. Conversely, this new approach is referred to as functional mathematics. 4.2. Teaching  for  transfer  –  Functional  Mathematics   In the introductory part of this thesis, I defined functionality, in terms of curricular development and referred to it as a means of bridging the gap between school mathematics and out-of-school mathematics. However, this raises an important question as to the level of mathematics to teach students. That is, since most researches reported that that the level of mathematics in work context is often at lower high school level, should the curriculum be reformed to teach only up to lower high school level? And if such an approach is adopted, what about learners who may want to study mathematics in further and higher education, or those who want to pursue careers such as
  47. 47.   46   engineering, which demands more advance mathematics. Or, how do we even begin to determine the appropriate level of mathematics for individual students? Clearly no one, even teachers, can predict what mathematics their learners will use as they move through their lives. As a result, a possible solution will be to integrate functional mathematics into the curriculum; that is (for lack of a better word) not to dumb-down the mathematics curriculum, and at the same time, provide learners opportunities to experience use of mathematics close to what occurs in the work place. In this respect, the adoption of functional mathematics, an approach that has potential in simulating the complexities inherent in work situations, will complement mastery models prevalent in most education systems. In other words, the goal of functional mathematics may be the creation of contexts in which theory and application are intertwined to encourage meaningful, memorable and internalisable learning. Traditional mathematics curricula, which often separate pure mathematics from applied mathematics, are unlikely to achieve this goal. Functional mathematics allows the learner to apply theoretical concepts. According to Shulman (1997), establishing practice as a fundamental aspect of school mathematics may overcome the deficiencies of theoretical learning. These deficiencies include loss of learning or forgetting concepts learnt, illusion of learning or thinking you have grasped the concepts and uselessness of learning or the inability to apply what you have learnt. Research has shown that most adults in professions that require the use of mathematics are faced with such deficiencies. For example Hoyles et al. (2002) and Kent et al. (2007) reported workers inability to apply the mathematics learnt in school. Equally,
  48. 48.   47   Cockcroft (1982) suggested that the traditional curriculum creates a severe psychological impediment to the practice of mathematics in adult life. Functional mathematics avoids many of these issues by focusing not only on theory, but also on authentic mathematical practice. For example, Qualifications and Curriculum Authority (2007) in UK proposed that functional mathematics should be: “considered in the broad sense of providing learners with the skills and abilities they need to take an active and responsible role in their communities, everyday life, the workplace and educational settings. Functional mathematics requires learners to use mathematics in ways that make them effective and involved as citizens, operate confidently and to convey their ideas and opinions clearly in a wide range of contexts.” That is, processes of functional skills are related to problem-solving and data handling cycle, involving three important phases; representing, analysing and knowledge of the appropriate mathematical procedures to use in a variety of contexts. The proactive measures of the QCA to provide learners with functional skills that transcends cultural boundaries (from school to everyday lives, including work) provide schools a framework for organising tasks that mirror the kinds of activities that take place in the workplace. It must be noted that even when a classroom lesson is designed to simulate the workplace, it can never completely capture the demands of actual practice. However, the current indictment of traditional, routine or expository teaching in research examining the transfer of knowledge from school to the workplace suggest that the functional mathematics or the problem solving and more investigatory approach to learning offers an effective solution. In Hoyles et al. (2002) report, it was
  49. 49.   48   emphasised that most employers’ lament the lack of initiative and problem- solving skills of most graduate workers. For example in the packaging case studies, a senior manager said the following about a work manager under his supervision: “…He could do the mechanics of it because he knew how to do that... but he was not able to interpret it, to transform it into a trend or a problem solving analysis. He wasn’t able to really use the information in a well-constructed argument. He could not present an argument, which was fact-based, by, using the numbers and the information that we do collect.” (p. 69) This excerpt shows that even though, this employee knows the mathematics involved, he was unable to perform when it is embedded in a real world situation. He could not interpret what the results indicated or what decision to take as a result of the analyses of information or data collected. More importantly, it also shows the limitations of traditional mathematics curricula in preparing learners for everyday situations or workplace. For example in my country the Gambia, mathematics is rarely considered in relation to its application outside of school. It is mostly presumed that once mathematics is learnt, students automatically developed the ability to apply it to various problems in their everyday lives or in their future roles in the workplace. This limitation still persists in the newly developed Gambia national mathematics curriculum, despite efforts to integrate applications and mathematical models. This is mainly due to the fact that the emphasis of this new curriculum is on application without much focus on the context in which such application arises. That is rather than emphasizing the real life context in which such problems may arise, they are often designed to teach particular skills or concepts without
  50. 50.   49   emphasizing authentic application from everyday life and work. Similar problems that prevail in the Gambia are also present in the west. Generally, the mathematics curriculum can be categorized as traditional, which is centered on algebra, functions and Euclidean geometry or as reformed which focuses on a more constructivist view that mathematical meanings arises from active engagement with contextualized problems. What both types of curricula (reformed and traditional) have in common is that topics are designed to move the learner along the path of arithmetic to calculus. Functional mathematics curriculum also follows the same path but also places more emphasis on realistic mathematic education (Treffers, 1993) – that is authentic problems centered on real life situations. The idea is that problems arising form authentic context expand learners’ understanding and allow them to see the interrelationship between mathematics concepts, skills and procedures and their application across different context. The fundamental principle is that mathematics learnt in school, no matter how specific, should have the potential to enhance mastery in other areas. Moreover, by embedding mathematics in practice, functional mathematics offers students both theory and know-how. Furthermore by advancing authentic contextualized content, engaging tasks, and active instruction functional mathematics can motivate students to link meaning with mathematics. Functional mathematics may also provide students the opportunity to recognize the diverse uses of mathematics in various walks of life as well as develop their knowledge from elementary to advance concepts which not only help them if they choose to study mathematics in higher education, but also prove useful in creating experiences that closely reflect the
  51. 51.   50   mathematical processes or activities that take in the learners’ wider lives and the work place (Forman and Steen, 1995). Similarly, De Corte (1999) suggest that curricula focused on functional mathematics design tasks that resemble those found in everyday life and work and encourage students to develop “systems thinking”. The term “systems thinking” is inspired by systems such as those found in commerce and industry, science, technology and society. In such complex systems, being functional in mathematics is a fundamental factor that influences performance. Systems thinking require learners to develop habits of mind that recognizes complexities hidden in situations (which may or may not involve technology use) subject to multiple inputs and diverse constraints (Kent et al, 2007). The underlying idea for developing “systems thinking” is to encourage students to learn to apply logic and careful reasoning in many situations. This concept of “systems thinking” suggest that schools and colleges are not the sole domains of learning, but the home, workplace and communities are also areas that learning takes place. For this reason, the significances of functional mathematics in integrating social practices such as those existing in students’ wider lives cannot be overemphasized. In this respect, functional mathematics offers great promise in extending the learning and the learner beyond the school and thus highlights the significance of teaching for transfer or the emphasis of prior experiential learning in students’ future endeavors. However pedagogic approaches for teaching functional skills may risk being ineffective if not based on an appropriate theoretical framework. For example the theories of learning discussed in this thesis have varied epistemologies, which have their strengths and weaknesses. However in
  52. 52.   51   considering what these learning theories have to say about how we come to know and relating these ideas to pedagogical research and practice in the complex classroom context, it becomes necessary to adopt a conceptual framework such as cultural historical activity theory (CHAT), which effectively frame most of the ideas from different theories of learning under one umbrella. For instance CHAT seamlessly brings together key elements of learning such the individual cognitive development, the situational and cultural factors. Additionally, notions of communities of practice, networks activity systems and boundary objects (which I will discuss later) in CHAT will help frame researchers or practitioners understanding of pedagogy in extended and complex contexts of learning/teaching. As a result in the next section, I will explore the potential of CHAT as a pedagogic framework for imparting functional skills and thus transfer.
  53. 53.   52   5. CHAT,  Transfer  and  Teaching  Functional  Skills   The main focus of this study is to explicate the key pedagogic issues that should be addressed in order to foster transfer of learning from school to the work. In my opinion, transfer of learning is means of encouraging lifelong learning in students or the development of learning that has the capability of meeting their present and future needs irrespective of whatever situation they find themselves. This means that a key focus of instruction should consist of the creation of conditions for the emergence and development of conceptual understanding and activities associated with mathematics as a domain and its function in students’ wider lives; thus the development of functional mathematics curricula. However, a key issues emerging from research work aimed at establishing the most effective strategies for facilitating transfer of learning are teaching facts and skills vs. developing conceptual understanding; contextualisation vs. decontextualisation of learning; individualized learning vs. collaborative learning, and active vs. passive learning. In reading for this thesis, I have concluded that the viewpoints of various theories of learning on the concept of transfer and their corresponding strategies for facilitating it are basically “systems of thought”. They all propose ideas on achieving transfer that may always be true within the theoretical framework they are considered; nevertheless, their key tenets only embody partial truths. For this reason, I maintain CHAT theoretical framework offers a sound framework for incorporating all these ideas as well as transcending the traditional approach to teaching for transfer, which either concentrate on individual learning in the school context or situated learning at the workplaces and never link the two or
  54. 54.   53   account for ecological factors. In addition, by modeling how transfer may take place through interaction between activity systems, CHAT holds great promise in research to understand transfer of learning from school to the workplace. In this chapter, I will explain CHAT and argue that it has great potential for analyzing as well as enhancing pedagogic practices aimed at fostering transfer. Activity theory was originally elaborated in the framework of cultural historical theory in 1978 and was applied to learning activity by Davydov (1988, 1996) and Engestrom (2005) among others. It is a conceptual framework consisting of a collection of basic ideas for conceptualizing both individual and collective practices in developmental processes. The first generation of activity theory inspired by Vygotsky and modeled on a triad consisting of subjects (an individual or group) who work towards an objective (object) in order to achieve an outcome. For example figure 1 shows a possible pedagogic research framework for teaching for transfer. In this scenario, the means by which the subjects (teachers) achieve objects (e.g. teaching for transfer) and outcomes (transfer of learning) is through meditational means (tool). These tools may be external (such as teaching and learning materials) or internal (strategies, plans etc.). This first generation model is especially powerful for anyone wishing to study what takes place in a person’s mind as they interact with their environment. Vygotsky managed to move away from the behaviorist’s tendency to study behaviors in action and extend the unit of analysis to take into account the thinking behind actions. As such, conceptual tools learners bring to bear when they carry out tasks and the way they use them becomes an integral aspect of educational research and learning. However, the limitation of the first
  55. 55.   54   generation (see figure 1) CHAT was that the unit of analysis remained individually focused. Engestrom further expanded the triad above to enable examination of an activity system not only at the level of the individual operating with tools but also at the level of the community. This development is referred to as the second-generation activity theory (see Figure 2). The unit of analysis changed from an object-oriented action mediated by cultural tools to take into account mediation by other people and social relations. In this way, the focus then shifted towards the study of the complex interrelations between the individual subject and his/her community; who are involved in a collective work activity, and whose main focus is to find a solution of a problem (object), which is mediated by tools and/or signs used in order to achieve the desired goal (outcome). The activity is constrained by cultural factors including conventions (rules and beliefs) and social organization (division of labor) within the immediate context.
  56. 56.   55   In contrast to traditional pedagogic framework, CHAT not only accounts for contextual and environmental factors within and learning systems but also identifies three mediating relationships for analyzing such settings. These mediating relationships in relation to pedagogy are: 1. Tool mediates between subject and object: these tools may be explicit tools such as language, knowledge and the physical artifacts needed by students to engage with tasks or tacit tools concerned with how do students manage emerging knowledge. 2. Rules mediate between community and subject: In terms of pedagogy, this involves two dimensions for creating supporting structures for effective learning. This involves encouraging students and teachers to interact in ways that allow them to build or appropriate knowledge through tasks, and from community and social relationships. 3. Division of labor mediates between community and object: This is concerned with who does what and may require the redefinition of the
  57. 57.   56   teacher’s role (from an instructor to mediator) or their beliefs about what constitute good teaching. On the learners’ part, active participation, questioning as well as autonomous learning strategies is important. Engestrom’s (1987) theory of expansive learning give details of how the second generation of activity theory brings together the necessary variables when developing a learning system. In this perspective, development is interpreted on the basis of expansive circles, which consist of processes of internalization and externalization as proposed by Vygotsky (1986). These two terms are instrumental in understanding how subjects are socialised by participation in the world (externalisation) and in the process internalise the same world by abstract symbol and inner speech or thought. In a later publication, Engestrom (2005) explains that expansive learning arises as a result of the interrelationship between internalisation and externalisation. He suggested that expansion starts by internalisation to allow the novice to pass through processes of socialisation. As they later become more competent members in the activity, the process of externalisation becomes gradually more dominant as members, driven by the need to resolve tensions and contradictions within the activity system, carry out processes of innovation. In this way, the internal tensions and contradictions of an activity system, much like Piaget’s concept of continuous cycle of disturbances, accommodation and assimilation, serve as the motive for change and development. CHAT as a pedagogic framework is especially powerful because instead of reliance on a particular theory of learning, it integrates pedagogical concepts from a various theoretical frameworks and thereby offers a much more holistic
  58. 58.   57   account or analyses of learning systems than any one theory of learning can ever achieve. From the pedagogical standpoint of CHAT, expansive learning makes completes sense to me, especially in terms of implementing a functional mathematics curriculum. For example, a key principle in Engestrom’s (1987) expansive learning is that students construct new forms of practical activity and/or artifacts in the process of tackling real-life projects or problems. This is exactly the pedagogic approach that functional mathematics curriculum also seeks to promote. Another focus of functional mathematics is the idea that learners should be able to apply knowledge learned in wider society. As I will explain below, activity theory may again prove useful in facilitating the realisation of this goal. The second generation was further still developed to establish the third generation activity theory by researchers who were interested in studying the interactions between different social worlds (for e.g. Star and Griesemer, 1989; Henderson 1991). The third generation activity theory represents networked

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