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  

   

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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.

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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  

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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

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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.

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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

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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

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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

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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.

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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

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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.

                                                                 

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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

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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

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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.

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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

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“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

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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.

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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

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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

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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

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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 &

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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

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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

  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

  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

  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

  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.

  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

  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

  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

  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.

  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).

  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:

  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

  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

  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

  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

  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.

  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

  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

  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

  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

  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

  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-

  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

  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

  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,

  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

  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

  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

  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

  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.

  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

  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

  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.

  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

  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

  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

  58  

activity systems (see Figure 3: NB. that real life cases can involved more two

networked systems).

The model of networked activity systems has important ramification for

studies on transfer. For instance drawing from the various researches discussed

in this thesis aimed at establishing effective mechanism for transfer, their main

challenge is characterizing transfer and developing pedagogic approach for

fostering. This stems very much from inability of these researches to integrate

theory and practice. From the earliest beliefs that knowledge is transferred

from task to task in the school context to more recent notions that what is

transferred are habits of mind to the situated cognition view that processes of

participation and apprenticeship in communities of practice is what is

transferred, there still remains a conclusive solution to the issue of transfer. In

a further attempt to explain why, I believe activity theory explicitly outlines a

practical solution to the dilemma of teaching for transfer, I will briefly look at

the notions of boundary objects, boundary crossing and translation,

presented in the literature and argue that their application in education offer

practical guidelines for teaching for transfer of learning from school to the

workplace.

Star et al. (1989) defined boundary objects, as concrete or abstract

objects, “ which are both plastic enough to adapt to local needs and the

constraints of the several parties employing them, yet robust enough to

maintain a common identity across sites (p. 393)”.   That is the concept of

boundary objects is used to explain the tools needed and the artifacts acquired

through participation in one activity system, which one then brings to bear in a

new activity system. These tools may consist of know-how, metacognitive

  59  

strategies or habits of thinking learners accumulated through their participation

in culturally mediated activity. Furthermore, the concept of boundary objects

helps in the analysis of what transpires when subjects from different social

worlds try to create new scientific knowledge. They establish a mutual modus

operandi or forms of practice, which involves or depends on communication as

well as the actor’s ability to reconcile special motives, goals, and actions (all of

which may not be overtly present) at the interface of interacting activity

systems. These special motives, goals, and actions, also referred to as shared

objects, may be methods or substantive factors, common to the interacting

activity systems. Subjects operating within these interacting worlds may need

to continuously redefine these shared objects to form new objects because

they mean different things to different worlds. Star et al. describe the process

of such reconciliation, which involves the culture of negotiation, debates,

triangulation and simplification of ideas in order for cooperation to take place,

as translation. The act of transporting ideas, concepts and instruments to and

from these interacting social worlds, is referred to as boundary crossing and

the subjects who make new connections and reconcile meanings across

communities of practice, are referred to as boundary crossers.

Even though Henderson’s (1991) and Star’s et al. (1989) studies focused

on workplace learning, their findings in relation to Engestrom expansive

learning offers insight into how functional mathematics curriculum could be

implemented in order to operationalise teaching for transfer. The basic idea, in

terms of transfer from school to the work place is that the role of boundary

objects is to facilitate boundary crossing between various communities,

communicating across different perspectives, and facilitating shared decision-

  60  

making. Consequently, the key idea that could be deduced from the work done

by these researchers is that teaching should closely mirror the kinds of activity

learners engage in their wider lives or in the workplace. This means that

instructional practices should begin with the identification of shared objects

within the interface of interacting systems and organizing activities in schools

aimed facilitating acquisition of such objects by learners. Drawing from this and

the discussion in earlier parts of this thesis, I can surmise that interaction

between complex systems such as school learning and workplace processes

cannot be understood from a single viewpoint, rather it requires an ecological

analyses as presented by CHAT; hence the rationale for establishing it as a

pedagogic framework for teaching for transfer or teaching in general.

However, further research based on CHAT paradigm needs to be

developed in order to study the concept of transfer in classroom settings or

promote transfer from school to the workplace. Such research may offer some

insight, more holistic than what is presented, into mechanism for transfer as

well as the role of teachers in cultivating habits of mind that encourage transfer

of learning among their students. The development of such a research program

is necessary if the issue of transfer is to be addressed. Furthermore since the

main issue of transfer is a lack of consensus in determining what elements are

transferred, I would suggest that research, based on CHAT paradigm, aimed at

illuminating the epistemological role of boundary objects from school to the

workplace is needed. This research will help clarify (a) the circumstances

artifacts/tools actually become boundary objects, and (b) how they may

facilitate effective communication between and within an activity system or

between activity systems.

  61  

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