1. Mathematics
Education:
Pedagogy
for
Transfer
of
Learning
from
School
to
the
Work
Place
[EDUC5001M]
Fatou
Kinneh
Sey
(200220649)
MA
Mathematics
Education
University
of
Leeds
School
of
Education
September
2010

2. 1
Acknowledgements
My sincere appreciation to the Head of School of Education, Mr.
Tom Roper for providing very important references; explaining very
complex concepts and also creating a very conducive and relaxed
atmosphere that allowed me to sound my ideas. However, I must
acknowledge that any errors in this thesis are entirely my
responsibility and not in anyway connected to him.
I also wish to thank the Ministry of Education, the Gambia, for
giving me the opportunity to study at the University of Leeds; an
experience that has given me the invaluable opportunity to “learn
how to learn” and view every material I read critically and get a
deeper understanding.
And finally, many thanks to my family and friends for feigning
interest when I talked incessantly about my topic.

3. 2
Mathematics Education: Pedagogy for Transfer of
Learning from School to the Work Place
1. Introduction......................................................................................................................3
2. Concept
of
Transfer.....................................................................................................11
2.1. Historical
Analyses
of
Transfer .................................................................................... 13
2.1.1.The
Theory
of
“Identical
Elements”........................................................................................ 14
2.1.2.Theory
of
Generalisation............................................................................................................. 15
2.1.3.Cognitive
Psychology
and
Transfer ........................................................................................ 17
2.2. Measures
of
Transfer........................................................................................................ 22
3. Transfer,
Social
Constructivism
and
Situated
Cognition................................27
4. Transfer
from
School
to
the
Work
Place..............................................................38
4.1. Mathematics
in
the
Work
Place.................................................................................... 39
4.2. Teaching
for
transfer
–
Functional
Mathematics.................................................. 45
5. CHAT,
Transfer
and
Teaching
Functional
Skills................................................52
6. Reference........................................................................................................................60

4. 3
Mathematics Education: Pedagogy for Transfer of
Learning from School to the Work Place
1. Introduction
There are many forces driving curricular reforms. Some of these include
economic imperatives, new applications of mathematics, and the effects of
technology. Consideration of the needs of these forces in mathematics
education makes it necessary that the curriculum and the methods being
employed in teaching/learning be examined to evaluate how well they serve the
needs of students and other stakeholders (e.g. Schneider, 2001, Schoenfeld,
2001, NCTM, 2000, DfES, 2005). Meeting these needs somehow implies regular
reviewing and updating of the curriculum and the research literature to
determine on the one hand, how these affect the mathematics content and, on
the other hand, how societal needs, for e.g. mathematics in the workplace,
should be taken into account in curricular reforms.
One of the reasons offered in the literature as to the reason why school
and college leavers are not doing so well in situations outside of school that
requires the use of mathematics is lack of “numeracy” (Cockcroft, 1982, Steen
2001, DfES, 2005), a term often used interchangeably with “mathematical
literacy” and “quantitative literacy” in the literature. Treffers (1991), for
example, claimed that learners’ level of numeracy might not be the result of
content taught (or not taught) but a function of the structural design of
teaching practices. However, instructional design and practices are only part of
the problem. Several issues, such as: defining numeracy, relating it to the

5. 4
nature of mathematics (as perceived by learners and most importantly
teachers), and the kind of competencies to be taught in schools or colleges to
allow learners to adapt and apply the mathematics they learn in school in
several contexts must be addressed. This is consistent with most of the views
presented in recent literature. For example, I came across the following
definitions of numeracy, which supports this philosophy:
Cockcroft (1982), stated that being
“Numerate should imply the possession of two attributes. The first is an “at-
homeness” with all those facets of mathematics that enable a person to cope
with the practical demands of everyday life. The second is the ability to
understand information to understand information presented in mathematical
terms. Taken together, these attributes imply that a numerate person should
understand some of the ways mathematics can be used for communication”.
Gal (1995) states:
The term numeracy describes the aggregation of skills, knowledge, beliefs,
dispositions, and habits of mind as well as general communicative and problem-
solving skills that people need in order to effectively handle real-world situations
or interpretive tasks with embedded mathematical or quantifiable elements.
Organization for Economic Cooperation and Development (OECD) (2000)
suggested that:
Mathematical literacy is an individuals ability, in dealing with the world, to
identify, to understand, to engage in, and to make well founded judgments about
the role that mathematics plays, as needed for that individuals current and future
life as a constructive, concerned, and reflective citizen.

6. 5
The PISA study (2003) define the term “mathematical literacy” as:
“The capacity to identify, to understand, and to engage in mathematics and to
make well-founded judgments about the role that mathematics plays, as
needed for the individual’ s current and future private life, occupational life,
social life with peers and relatives, and life as a constructive, concerned, and
reflective citizen.”
A common theme in all these definitions is the view that numeracy is not
limited to acquiring mathematical knowledge and skills. Rather, numeracy is
partly sociological, in the sense that it is heavily linked with its purpose and
value to society, and partly epistemological, concerned with the nature of
mathematics, as it is presented in the curriculum and implemented by teachers
in schools. The sociological as well as the epistemological view of numeracy,
mathematical knowledge or mathematics is consistent with Niss’s (1994) 5
faceted perspective of the nature of mathematics as (1) a pure science used to
develop, describe, and understand objects, phenomena, relationships etc; (2)
an applied science, used to and develop understand knowledge in other
domains; (3) as a system of tools for societal and technological practice
(“cultural techniques”); (4) as an educational subject; and (5) as a field of
aesthetics. On another level, the analysis of the definitions of numeracy yields
three common themes:
This suggests that in order to meet this need, the nature of mathematics
in school curriculum has to shift from the instrumentalist views of mathematics
as an accumulation of facts, rules and skills to be used in the pursuance of
some external end; or the Platonist view which sees mathematics as static but
unified body of certain knowledge; to a kind of mathematics that arises as a

7. 6
result of human enquiry or problem solving. As Ernest (1988) put it, the
problem-solving view of mathematics, is seeing mathematics as a dynamic and
continually expanding field of human creation and invention - a cultural
product. Moreover, problem-solving views of mathematics promote conceptual
development; emphasise the possible application of mathematics in other
disciplines; and also promote attitudes in students that increase their
willingness to apply ideas learnt in school in different contexts (Ernest 1988,
Prawat 1989). These different views of the nature of mathematics, that
teachers in particular have, influence classroom practices (Cohen and Ball,
1990, Fang, 1996). As a result, developing ‘numerate’ learners as implied by
the above definitions require the curricular reforms with aims, goals, purposes,
and rationales closely associated with societal needs. Relating mathematics
education to society brings up the second common theme that all the
definitions touched on; functional mathematics.
Functional mathematics, the second common theme pervading the
definitions, provides a strong basis for students entering the work force either
from high school or college to experience mathematics as problem solving.
Ideas of functional mathematics (which, I will discuss later) arose out of the
recognition that current mathematics curricula do not adequately equip people
to use and apply mathematics effectively in different situations. Hence,
functionality, in terms of curricular development, is a means of bridging the gap
between school mathematics and out of school mathematics or identifying the
areas of mathematics education that are inherent in employment for example.
As Schoenfeld (2001) noted, school mathematics is presented as a collection of
discrete, unrelated information; however, in view of the new perception of

8. 7
mathematics, learners should be provided tasks with enough scope to allow
adaptation and application of mathematics in several contexts. These are
referred to as enriched tasks.
Summarily, theme one is concerned with the re-conceptualization of the
nature of mathematics so that learners and teachers’ perception of the nature
mathematics presents a more jointed as opposed to disjointed view of the
different content areas; or a shift in its presentation, from a series of
procedures to be learnt to one geared towards achieving a balance between
procedural and conceptual development. And theme two’s concern is fostering
curriculum design that integrates the kind of mathematics practices found in
the work place in a bid to bridge the perceived boundaries between school
mathematics and out of school mathematics; as well as emphasize the
competencies that school mathematics hopes to foster in order to allow
learners to apply mathematics concepts acquired in schools in different
situations. The interrelationship between these two themes brings to bear a
third theme; the learners ability to apply the knowledge learnt in schools to
solve problems arising in similar or different contexts, in other words, transfer
of learning.
The third theme, implicitly, emerging from the definitions of numeracy, is
the issue of transfer of learning from school to wider society. Unfortunately,
transfer is generally believed be problematic in the mathematic education
community (or education in general). The consensus is that learners, more
often than not, lack the ability to transfer knowledge they have learnt in school
in contexts outside of school (e.g. Detterman, 1993; Lave, 1988). Like some of
driving forces behind curricular reforms (e.g. commerce and industry), current

9. 8
researches (e.g. Hoyles et al. 2002, Kent et al, 2007) lament the apparent lack
of value of students’ mathematics qualifications. These developments have
brought back focus on the concept of ‘transfer’ (among other issues), which,
like many of the ideas of cognitive psychology, has been relegated to the back
seat, so to speak, of research in mathematics education. However given its
obvious importance of learning transfer, I wish examine how mathematics
education can be structured to foster and facilitate transfer of learning to the
work place. To work towards developing this understanding, I organized the
thesis into five chapters:
Chapter 1 is this introduction, which explains the background and
rational of the study, as well as the overall structure of the thesis treating the
issues related to the complex notion of transfer.
Chapter 2, which follows from this introduction, is concerned with
characterizing the concept of transfer by, first of all, providing the definition of
transfer of learning; types of transfer and their measures; and concluding the
historical development of the concept of transfer and highlighting its problems
and concerns.
Chapter 3 examines the positions of two popular epistemologies (social
constructivism and situated cognition) on transfer and highlight why their key
principles embody the concept of transfer.
Chapter 4 covers transfer of learning from school to the workplace. My
aim for this section is to address the following questions: How do we teach for
transfer? What skills and knowledge are transferable from school to the work
place? What instructional and learning strategies promote/hinder transfer.

10. 9
In chapter 5, I want to propose cultural historical activity theory (CHAT)
as a possible research and pedagogic framework for studying and bringing
together all the complex variables present in classroom settings in efforts to
understand the best practices for teaching for transfer. Since traditional
approaches have never take into account the influence of individual factors, the
social context in which learning takes place or the environmental factors such
as the impact of tools use, rules and regulations in a classroom community or
learner characteristics such as motivation or interest all together, I will argue in
this chapter that Engestrom’s (1999, 2005) CHAT, as a dynamic model, is
particularly appropriate for studying/analyzing the concept of transfer.
In conclusion, I highlighted, in this introduction, some of the factors
(economic, it’s application and the impact of technology) driving curricular
reforms in mathematics education, among which, the most important may be
the need to prepare a generation of learners that can function effectively in a
world that increasingly requires numerate individuals, in everyday situations,
democracy and work place. Whilst these needs are not clearly defined by the
mathematics community or society at large, some of the definitions of
numeracy presented in the literature are indicative of precisely what those
needs are. That is, in a broad sense, the development of mathematics curricula
and teaching practices that enable learners to adapt and apply the mathematics
they learn in school in different situations.
According to Wake (2005), for learners to acquire the ability to apply,
mostly basic mathematics to address complex problems of the work place, they
will need to be provided with experiences to access richer and more diverse
tasks than are currently provided by the mathematics curriculum. This

11. 10
development gave rise to, in UK for example, functional mathematics (which is
discussed in chapter 4) aimed at bridging the gap between school mathematics
and the mathematics. Functional mathematics, more than any teaching
approach, makes teaching for transfer of learning an explicit educational goal.
In the next chapter, I will consider what transfer is, its historical development
and its relation to current influential theories of learning.

12. 11
2. Concept
of
Transfer
In broad terms, transfer of learning can be defined as the learner’s ability to
apply previous knowledge, or past experiences in similar or novel situations
(Darling-Hammond and Bransford, 2005). However, there is an increasing
inclination amongst the proponents of transfer, to emphasize thinking habits
(which include motivation, collaborative teamwork, communication etc.). For
example, according to Salomon & Perkins (1989), transfer is the ability to carry
over knowledge, skills, understanding and habits of thinking from one learning
situation to another. While recently, researchers focus on how general attitudes
(affective factors and motivation) (Bransford et al., 2000), context (Greeno,
1997), social interactions or participation (Olivera and Straus, 2004) and other
environmental factors affect the individual’s ability to transfer learning.
Apart from the disparities in defining transfer of learning, there is also
contention amongst researchers (even amongst proponents of transfer) as to
its measures, means of achieving it, and theory of transfer. For example, some
theorists concluded it rarely happens (e.g. Detterman, 1993); while others (e.g.
Hammer et al, 2005; Lave, 1988; Lave and Wenger, 1991) do not believe it is
possible; but some believe that transfer occurs all the time because the
underlying principle of all learning is transfer (e.g. Dyson, 1999).
Also, some current theories of learning posit that transfer is not possible
(for example, social constructivism and situated cognition) while others,
(behaviorism, cognitive psychology), despite supporting the concept of transfer,
have very different ideas of it. Furthermore, despite the importance of transfer
of learning, or the implicit assumptions in instructional settings that it will occur,
the history of research on transfer paints a different picture and suggests that

13. 12
transfer rarely occurs. For example, Reed, Ernst and Banerji (1974, cited in
McKeough et al., 1995), observed very little transfer from one river crossing
problem to another and also Hayes and Simon (1977) reported similar failures
of transfer in the study of problem isomorphs which are problems that are
structurally similar (in approach or concept) but with very different surface
features. They discovered that subjects more often than not do not recognize
the connection between one isomorph and another and hence do not carry over
strategies they have acquired while working from one to the other. However,
they also highlighted the transient nature of failures of transfer by
demonstrating that failures to transfer could be changed to successes simply by
pointing out the relationship between the source and target tasks to the
subjects.
Amidst the failures, there were a few successes of transfer, among them
was Judd‘s (1908) study which showed empirical evidence of children
transferring learning of the principle of refraction; Katona’s (1940, cited in
McKeough et al., 1995) success of demonstrating transfer of strategies for
solving card tricks and match-sticks problems; and Palinscar and Brown’s
(1984) report that when children are taught to self-monitor and self-direct
themselves during reading in what they referred to as ‘reciprocal teaching’, they
transfer comprehension strategies across a variety of settings. Campione et al.
(1991), also using reciprocal teaching, showed that learners transfer these
metacognitive strategies in other text-mediated areas of learning such as social
studies and mathematics.
Overall, the literature on transfer, including its theory, measures and
mechanisms is very chaotic. Consequently, in an effort to further explain the

14. 13
concept of transfer, I will present a historical analysis of transfer of learning
from the perspective of behaviorism and cognitive psychology as well current
theories of learning that contradict it; in a bid to explain how and why transfer
takes place (or not) from one situation to another. First, I start with earliest
attempts to scientifically measure the phenomena of transfer to current
research efforts aimed at advancing the concept of transfer and its application
in mathematics education.
2.1. Historical
Analyses
of
Transfer
Transfer theory emerged from the empiricist perspective, which assumes that
learner is a passive agent who transfers knowledge and facts learnt from a
presumed “original learning situation” to new situations. The earliest research
on assessing the quality of people’s learning experiences was carried out by
Thorndike and Woodworth (1901, cited in Bransford et al., 2000). Their aim
was to test the assumption of ‘formal discipline’ that learning Latin develops
general skills of learning and awareness in other areas of studies.
They generally failed to find positive impact of one sort of learning on
another. In a subsequent study, Thorndike (1923, cited in Bransford et al.,
2000) compared the performance in other academic subjects of students who
had taken Latin with those who had not and found no advantage of Latin
studies whatsoever. Thorndike and Woodworth (1901) concluded that transfer
depended on the number of ‘identical elements’ between source and target
tasks and situations.

15. 14
2.1.1. The
Theory
of
“Identical
Elements”
The hypothesis of “identical elements” proposed that transfer is dependent on
the number of similarities, which exist, either in content, technique or context,
between the old and novel situations and not due to the development of
“general skill” or “mental muscle” as posited by theory of formal discipline. The
theory of identical elements (elements were assumed to be specific facts and
skills) identified positive and negative transfer, lateral and vertical transfer, and
near and far transfer as the different types of transfers which I will come back
to later in the discussion. These definitions of the types of transfer (for example
near and far) suggest a “transfer distance”, which provide indication of how
different tasks and their contexts are. That is how similar tasks and contexts
are said to be near one another and dissimilar ones far from one another.
However, without objective measures of the perceived distance or similarity
between tasks, the theory of identical elements could not explain sufficiently
the concept of transfer.
Moreover, by acknowledging that “they spoke of elements ‘without any
rigor’, but they meant mental and environmental objects and events "differing
in any respect whatsoever" (Butterfield et al., 1989*1
, p. 7), they become even
more unspecific about what was an element, as result, reducing the
effectiveness of the original theory of identical elements. At least in the original
version (though also not without problems) it can be easily deduced that to
promote transfer, teachers need to make their classroom materials (i.e.,
stimuli) and the ways pupils are taught to use them (i.e., responses) as similar
as possible to the stimuli and needed responses in their pupils' wider lives
(Engelmann and Carnine, 1982). But in redefining the transfer ‘elements’ to be

16. 15
“mental and environmental objects and events”, which is basically anything,
makes it virtually impossible to interpret Thorndike’s revised version of ‘identical
elements’ in terms of teaching and learning. In this light, it is therefore not
surprising that his contemporary, Judd, challenged his theory of identical
elements and proposed the theory of generalisation.
2.1.2. Theory
of
Generalisation
Judd (1908, cited in Bransford et al., 2000) argued that transfer has less to do
with the subject matter or the content and more to do with “general principles”
of which one learns in a situation, and that the transfer issue must be
considered in relation to learner’s characteristics such as prior understanding
and strategies. As a result, his “theory of generalization” posits that transfer
occurs when one is able to abstract invariances (in terms of surface features,
concepts or strategies) across situations. In order to illustrate that the “theory
of generalization” rather than the theory of “identical elements” is the means of
achieving transfer, he carried out a research comparing the effects of “learning
a procedure” with “learning with understanding”. He had two groups of
children; one group received an explanation of refraction of light, which occurs
when light is incident on an interface between two materials (in this case air
and water), at an arbitrary angle, its direction is altered. As the speed of light is
reduced in the denser medium, its wavelength is shortened proportionately and
it bends. This bending of light causes objects to appear to be in different
location than they actually are. The other group only practiced dart throwing,
without any instruction on the principle of refraction. Initially, both groups did
equally well on the practice task, which involved a target 12 inches under water

17. 16
but when the target was changed to 4 inches, the group that had been
instructed about the abstract principle of refraction were able to transfer their
learning when the conditions changed. Judd concluded they performed much
better because they understood what they were doing, as a result they could
adjust their behaviour to the new task. They were able to organise the
information in the source and the target situations according to similarities in
their basic components in other words, according to structural similarities – that
is similarities in surface features, concept, or strategies. That is, by recognising
that the phenomena of refraction occurs (a) when light travels between two
media, and (b) the denser media shortens the length of the light, the
participants who received instructions where able to recognise the similarities in
structure, i.e., surface features and concepts between the principle of refraction
and shooting targets submerged underwater. Using this knowledge, they were
able to adjust their aims to hit the targets. This is one of the earliest successes
of research demonstrating the feasibility of transfer. Moreover, the attention
given on structural similarities is also recognized as a major development
beyond stimulus-response psychology.
Transfer of learning in the structural perspective takes place when
learners already have an understanding of the procedures and concepts of the
original problem, which they bring to bear in the target task. And, in
emphasizing that transfer is facilitated by learner’s prior knowledge and the use
of strategies (not the subject matter) such as directing learners’ focus on
conceptual similarities across tasks, Judd’s model can be considered to be a
precursor to the cognitive perspective on transfer (Haskell 2001). In the
following subsection, I outline the cognitive psychology perspective of transfer.

18. 17
2.1.3. Cognitive
Psychology
and
Transfer
Unlike the behaviourist, the cognitivist acknowledged the importance of stimuli
as represented by the learner (as opposed to an experimenter or a teacher)
and on responses generated by their covert knowledge and strategies. Drawing
from Inhelder and Piaget’s (1958) theorizing that knowledge is organised into
mental schemata, cognitivists developed an information processing framework
to explain learners’ ability to organize, retrieve and process relevant prior
knowledge in order to solve new problems. In other words, they developed the
idea of schemata to represent the elements in transfer theory. Cognitive theory
of elements and mechanisms (or schema theory) proposed that three
fundamental concerns must be addressed in order to teach for transfer. That is
determining how students select particular knowledge and skills to use in new
situations, from all that they have learnt, selecting the appropriate knowledge
and skills to use in a particular context and choosing alternative strategies,
knowledge or skills if previously selected ones are not useful. In a bid to
address these concerns, cognitivists not only identified three elements that fit
readily into transfer theory, but also built mental models to show how the
elements worked together to produce behavior. The three elements are
stimulus representations, strategies that operate on those representations, and
knowledge of how to use the results of those strategic operations as integral to
transfer.
For example research on the formal reasoning problem of predicting the
behavior of a balance scale provides an example of what cognitivists mean by
representations, strategies, and knowledge. Weights on either side of a fulcrum
and their distances from the fulcrum must be taken into account in order to

19. 18
obtain a solution of balance scale problems. Young children represent balance
scales by considering only the weight, that is the side with more weight goes
down, whereas older children, took advantage of their more advance
knowledge and represent both weight and distance in their solutions. That is by
taking into account that, the side with more weight goes down if its weight is
not compensated by placing the objects at greater distance on the other side of
the fulcrum (Siegler, 1976; Ferretti et al., 1985). The older children used their
prior knowledge to devise more sophisticated strategies.
These findings suggest that children of different ages have different
mental models of balance scale problems, which is constrained by their
knowledge, which in turn constrain the strategies they adapt in problem
solving. This also suggests the possibility that learning and transfer may be
promoted by teaching younger or less expert students, the mental models of
older or more expert students. Butterfield and Ferretti (1984) showed that
young children could be taught to use the representations, strategies, and
knowledge applied by much older people to solve balance scale problems.
Moreover, they were also able to transfer the strategies they have learnt to
similar untaught tasks (Butterfield & Nelson, 1989*2
). Similarly, McDermott et
al. (1987) showed that when college students were shown strategies used by
expert physicists, such as sketching graphs to generate mental models of
position, velocity and acceleration of objects, their understanding of mechanics
problems is increased.
These examples also show that learners are able to appropriate mental
models, strategies and thinking of more able students or experts and transfer
them in other context when given the opportunity to learn. Equally, they also

20. 19
explain the relationship between stimuli representation, knowledge and
strategies – that is more experienced learners have more sophisticated
representation and strategies. However the means by which learners know
which strategies or knowledge to draw on in problem situations is not evident in
the examples. The cognitivist posits that choosing effective knowledge and
strategies, requires metacognitive awareness (Brown, 1987; Butterfield &
Nelson, 1989*2
).
An example of metacognitive awareness would be on encountering a
balance scale problem in the previous example, older learners where able to tell
in advance that changing the distance of the objects on either side of the
fulcrum may allow them balance the scale. In this respect their choices were
influenced by what they knew in advance (previous schema), and the
knowledge that a particular strategy may solve a problem or knowledge of how
much effort it will require as opposed to the trial and error methodology
adopted by less experienced learners. In short, the metacognition model of
transfer emphasises awareness of one’s thinking and actions. It also partly
depends on learners’ willingness to seek out structural or conceptual similarities
based on their past experiences.
Despite some of the success reported by behaviourists and cognitive
psychologists researchers on transfer, some researchers (for example, Ellis,
1965; Detterman, 1993) criticized those earlier successes arguing that apart
from the lack of scientific rigor and the potential for bias (as the researchers
conducted the test), those successes were based on analyses that did not
reflect “natural circumstances” or real learning processes and situations. They
suggested that any proposed theory of transfer should not be considered in

21. 20
isolation to a student’s purposes, existing knowledge, strategies, attitudes,
motivation and classroom settings. Moreover, Thorndike’s “theory of identical
elements”, for example, focuses only on developing procedural knowledge and
not conceptual knowledge and even where it works, it only results in near
transfer (Detterman, 1993), which is not the main or only focus of research on
transfer. Far transfer, which none of these studies have been able to produce,
allows knowledge to be applied across tasks or situations (e.g. school to
workplace) and thus should be the main concern of education.
Nevertheless the theory of identical elements is still a prevalent
educational strategy. For example, the idea that “practice makes perfect” (in
the tradition of Thorndike, Skinner and Watson) continues to be a fundamental
aspect of transfer in mathematics education. Research such as Anderson and
Singley’s (1985) and Silver’s (1981) reported findings supporting Thorndike’s
claim. Also, Ericsson and Smith (1991) showed that expert performance in
areas such as chess, physics, problem solving, and medical diagnosis,
depended on a large knowledge base of specialized knowledge and skills.
Equally, Judd’s work on abstraction and generalisation also has not lost
influence amongst practitioners and researchers alike in mathematics
education. Recent works on analogical problem solving in which the learner is
first provided with conceptual representation of a problem or strategies to
tackle the problem and later on introduced to a similar problem (surface
features, principles/concepts or approach) is a derivation of Judd’s theory of
generalisation. Learners have to recognise the similarities in structure in order
to solve the problem. Examples of such experiments are The Tower of Hanoi
(Kotovsky & Fallside, 1989), and Duncker’s radiation problem (see Gick &

22. 21
Holyoak, 1980, 1983). Also, Perry (1991) found that fourth and fifth grade
students showed considerable amounts of transfer between mathematics
problems when they received instructions on conceptual principles rather than
procedures. So contrary to the claims of critiques of earlier studies on transfer,
recent studies done in real classroom context are also substantiating the
concept of transfer as proposed by Judd and Thorndike. Furthermore, instead
of dismissing the concept of transfer, these studies confirm that transfer is not
unachievable but rare and that different amounts or types of transfer occur
depending on the amount of practice or the instruction participants received.
Additionally, it strikingly odd for most learning theories (e.g. radical and social
constructivism) to embrace the fact that learners bring their past experiences to
bear in new learning (which is essentially transfer) and at the same time deny
the concept of transfer. This paradox may be due to the association they make
between transfer and the “conduit metaphor” presented by behaviourists. This
metaphor views language as tool for transferring thoughts, ideas or skill from
one person to another. That is, in writing or speaking people transform their
thoughts or feeling in the words; these words (containing the thoughts or
feelings) are then conveyed to others, who then through reading, writing or
listening, extract the thoughts or ideas contained in the words. The conduit
metaphor, like most socio-cultural models of learning, suggests two planes of
interactions. However, there is no consensus regarding epistemology. But a
closer look at some of the alternative concepts (e.g. ZPD, appropriation), which
I will discuss later, reveals that their underlying principle is transfer.
As I stated earlier on, measures of transfer is also another murky area
amongst researchers. The typological or taxonomical approaches are most

23. 22
widely used in the literature. Below, I will discuss some of these classifications
and their influence on teaching and learning.
2.2. Measures
of
Transfer
In the literature, transfer is classified according to either its types (see, e.g.,
Detterman, 1993; Salomon & Perkins, 1989; Singley & Anderson, 1989) or the
perceived levels of transfer that have occurred (see, e.g., Haskell 2001).
However, these classifications offer no explanation as to why the classifications
are different, similar or what information is transferred. This makes it very
difficult to appreciate the usefulness of the models to mathematics education or
education in general. As noted earlier on, the theory of identical elements
identified positive and negative transfer, lateral and vertical transfer, and near
and far transfer as the different types of transfers. Below I make an attempt to
describe, with help from examples, what they are, their differences, similarities
and the kinds of information transferred, from one type of transfer to another
as well as their implications in teaching or learning.
Firstly, positive or negative transfer refers to when learning in one
situation improves or hinders learning in another situation, respectively
(Cormier and Hagman, 1987). For example when elementary students learn
multiplication, they are often limited to examples involving positive whole
numbers as a result, they develop the idea that multiplying numbers results in a
bigger number. They bring this misconception to bear when learning to multiply
fractions or decimals to the extent of misconstruing, for example when
multiplying ½ x ½ = ¼, that the ½’s on left-hand side of the equation are less
than the ¼ because one has to multiply them to get the ¼. In this way, a

24. 23
teacher’s failure to bring in counter examples when reviewing previously
content resulted in negative transfer. Negative transfer usually occurs in early
stages of learning and with experience, learners do correct it. Consequently,
promoting positive transfer is the primary concern of education, as such,
strategies aimed at achieving it, are the main focus of education.
Lateral transfer also referred to as horizontal transfer, happens when the
learning in one context, is applied at the same level in a new context. For
example, if a learner knows that 7 x 3 yields 21 and use this knowledge to work
out how much 3 bars of soaps, priced at £7 each cost. In this way, past
learning is transferred an identical level either in approach or concept.
However, vertical transfer occurs whenever learning necessitates prerequisite
skills. For example, learners are usually introduced to calculating areas
rectangles, squares, then circles and ellipses before that of composite figures.
The hierarchical nature of school curriculum is derived from this conception of
transfer. As a result, knowledge, either procedural or conceptual, is arranged
from easier to difficult or according to complexity.
Near transfer occurs when we transfer previous knowledge to new
situations closely similar to, yet not identical to, initial situations (Perkins and
Salomon, 1989). For instance if a student learns how to calculate the mean in
mathematics and later on apply this knowledge in environmental studies to
calculate the mean amount of rainfall in a particular period, that is regarded as
near transfer. However, far transfer takes place, if ability to multiply in school
allows a learner, doing the family’s monthly shopping at the local market to
work out how many packets of soap he/she needs to buy to last a month, if
his/her family exhaust 5 bars of soap a week; and the total cost of the

25. 24
purchase if each packet is priced at £7 each and contains 6 soaps. The key
emphasis of far transfer is the extension of skill and knowledge between
contexts that, on the surface, seem isolated from and unfamiliar to one
another. One way of differentiating between near and far transfer is through
comparing school-learned events and out-of-school events (Butterfield and
Nelson 1989*2
). School-learned events are recognized to be an example of near
transfer (Mestre 2005) because the conditions involved for it are present and
the same for both original and target tasks. But applying the knowledge and
skills acquired in school-learned events to out-of-school problems exemplify far
transfer because the context and the goals are different in the two situations
(Haskell 2001). Far transfer, which is the most desirable construct, is the most
elusive out of all the other types of transfer. So far research on transfer has
only succeeded in demonstrating near transfer.
The shift from behaviourist to cognitivist perspective of transfer
emphasised transfer of concepts and complex skills as opposed to basic
knowledge and basic skills. As a result, transfer is reconceptualised as a
conscious process as opposed to reflexive application of routine skills. Perkins
and Salomon (1989) coined the terms high road (or conscious application of
strategies or principles) and low road transfer (automatic or unconscious
application of strategies or principles), to account for the goals and motivational
element of transfer. They defined high road transfer as the “explicit conscious
formulation of abstraction in one situation that allows making a connection to
another” (p. 118) and low road transfer as “the spontaneous, automatic
transfer of highly practiced skills, with little need for reflective thinking” (p.
118). Similarly, Sternberg and Frensch (1993) argued that motivation, or intent

26. 25
to transfer, determines the extent of the learner's active involvement and
attitude toward learning. Thus the identification and teaching of strategic
knowledge (e.g., metacognitive and problem-solving skills) that are applicable
to a broad range of tasks, is key to achieving transfer (Pressley et al., 1987).
Moreover, motivation to transfer also referred to as ‘spirit of transfer’ (Haskell,
2001), involves possessing positive dispositions towards learning that allows
students to process information in ways that facilitate transfer. These positive
dispositions include traits, like high motivation, risk-taking attitudes,
mindfulness or attentiveness, and a sense of responsibility for learning (Pea,
1988)
Thus far, the behaviourists emphasise the importance of possessing a
deep foundation of factual and procedural knowledge (Thorndike, 1901) and a
need to design instruction to develop conceptual understanding (Judd, 1908) in
facilitating transfer. And the cognitivists propose that in addition to having a
deep foundation of factual knowledge and procedures, learners need to be
taught strategies of organizing these resources in ways that facilitate their
retrieval and application. These metacognitive strategies involve awareness of
what they know or do not know, and knowledge of when and why to access
certain information in problem situations. Similar to behaviourists, cognitivists
also posit that learners’ previous knowledge and experiences are important
factors. Most of these ideas crop up in almost all current theories of learning.
However, the behaviourist legacy of transfer seems to hinder almost all
possibilities of establishing transfer as a credible research construct. However, if
we get past that and have a closer look at the conditions which these successes
where reported or even, why the reported failures occurred, it becomes clear

27. 26
what conditions and mechanisms are necessary for transfer. May be then, the
debate aimed at establishing whether transfer is feasible or not, or the issue of
developing the appropriate metaphor, will make way for the development of
instructional contexts and tasks to facilitate transfer.
Nevertheless, the worst setback received by research work promoting
the need to teach for transfer may be attributed to work done by proponents of
social constructivism, and situated cognition. Social constructivists and early
proponents of situated cognition (Lave, 1988, Lave and Wenger 1991) posit
that knowledge does not transfer between tasks or situations. These two
schools of thought mostly influence current researches in mathematics
education. The interest in social constructivism represent a shift from a focus of
intellectual processes within the individual, as in the classic research of Piaget,
to a concern with social cognition very much influenced by increased interest in
Vygotsky (Butterworth, 1992). In this respect, the emphasis of research is on
the ways in which thought processes and cognitive processes are socially
situated. Equally, socio-culturalist like Lave and Wenger stressed an inextricable
link between contextual constraints and the acquisition of knowledge. This has
resulted in a merger, if you like, between, context, social, and cognition and
suggests that cognition is situated in a social and physical context and is rarely
decontextualised.
In the next section, I will discuss not only the similarities and differences
between ideas of social constructivist and situated cognitivist but also highlight
that their key ideas regarding learning intrinsically embody the notion of
transfer.

28. 27
3. Transfer,
Social
Constructivism
and
Situated
Cognition
Two schools of thought – social constructivism and situated cognition, mostly
influence current researches in mathematics education. In this section, I will
show why some of their key notions, such as learning through social interaction
or learning through participation in practices; zone of proximal development;
and the concept of appropriation may possibly be “old ideas in new clothing”
when compared to some of the ideas offered by advocates of transfer and
those expressed by Dewey (1925/1981), a renowned philosopher of education
whose ideas on learning are consistent with ideas of behaviorism and cognitive
psychology. In order to demonstrate these similarities, I will start with the
notion of social interactions in learning as presented by social constructivists
and situated cognitivists. Even though these two theories have similar ideas,
they have a slight difference in epistemology. For example, social constructivists
argue that learning should be:
"Viewed as an active, constructive process in which students attempt to resolve
problems that arise as they participate in the mathematical practices of the
classroom. Such a view emphasizes that the learning-teaching process is
interactive in nature and involves the implicit and explicit negotiation of
mathematical meanings. In the course of these negotiations, the teacher and
students elaborate the taken-as-shared mathematical reality that constitutes the
basis for their ongoing communication" (Cobb, Yackel, & Wood, 1992).
This suggests that meanings and knowledge are shaped and evolve through
negotiation within the communicating groups, involving for example, a teacher
and learners or learner and more knowledgeable peers. The nature of the

29. 28
learner's social interaction with knowledgeable members of the society is very
important in this paradigm. Without it, learners may not acquire social meaning
of important symbol systems and learn how to use them. From the premise that
people develop their thinking abilities by interacting with experts, I deduce that
any personal meanings arising from teaching and learning are shaped by and
conveyed (transferred) through these interpersonal interactions.
Similar to social constructivism, situated cognition also posits students
collaborate with one another and experts toward some shared understanding.
For instance, both theories maintain that learning should be situated in “zones
of proximal development” just beyond what a student can accomplish alone and
should employ peer or a teacher scaffolding to extend learners’ capabilities.
Equally, both theories maintain that learning may occur through intrapersonal
communication. That is, thought process or communication with one's self is
the means by which knowledge or skills are constructed. In this perspective,
the individual provides feedback to him or herself in an ongoing internal
process. This involves not only a willingness to be part of a community of
practice, but also involves a process of self-assessment and self-regulatory
activities similar to metacognitive strategies proposed by Judd and cognitive
psychologists as necessary for transfer. However, the difference with social
constructivism and situated cognition resides in their respective epistemology.
In contrast to the social constructivism, which begins with the acquisition of
theory and then work towards practice, situated cognition emphasises that
activity and enculturation is integral to learning. The development of knowledge
or skills arises through the persons situating or immersing themselves in
practice, at first by carrying out simple tasks or through “legitimate peripheral

30. 29
participation” (Lave and Wenger, 1991) and gradually, proceed to more
complex and demanding tasks as they establish there identities in the
community of practice. This is referred to as learning by apprenticeship. The
following quote from Brown et al (1989) sums up quite neatly the premise upon
which situated cognition theory is based:
“A theory of situated cognition suggests that activity and perception are
importantly and epistemologically prior - at a non-conceptual level - to
conceptualization and that it is on them that more attention needs to be
focused. An epistemology that begins with activity and perception, which are
first and foremost embedded in the world, may simply bypass the classical
problem of reference --- of mediating conceptual representations.” (p. 35)
In this regard, the critical distinction between social constructivism and situated
cognition is that people learn by doing. Initially, they may not have fully
grasped the concept involved, but through immersing themselves in the
practice, they will eventually abstract the conceptual representations embedded
within it. In other words, concepts are embedded in the world and people learn
through engaging in the activities alongside more able individuals, abstract
representations of real situations are developed and through them, come
conceptual understanding (Brown et al, 1989). Again, this is strikingly similar to
the both the theory of identical elements and theory of generalisation.
Another difference between social constructivism and situated cognition
is that in the apprenticeship model of the latter, no one specifically sets out to
instill knowledge and skills uniformly into a group of learners. The more
inclusive process of generating identities, or desire to become a master
practitioner, is both a result of and motivation for participation. In fact, unlike

31. 30
social constructivism, it is rarely the case that the individual apprentice has
someone to teach him or her in order to learn. Conversely, it is the ongoing
everyday activity that provides structuring resources for learning. Thus
gradually increasing participation provides the scaffolding. Again this is similar
to practicing procedures/skills and knowledge suggested by Thorndike.
Overall, the point being made is that despite the differences in
epistemology between social constructivism and situated cognition, their
treatment of the role of social interactions in mediating cognitive activities
suggest that knowledge is shaped by and conveyed through these interactions,
which is in a nutshell, transfer. Having considered the role of social interactions
from the social constructivist and situated cognitivist perspectives, I will now
relate Dewey’s (1925/1981) philosophy of education, which I refer to as social
behaviourism, to the social constructivism and situated cognition. But first, I
want to explain why I believe Dewey’s philosophy could be referred to as social
behaviourism and also strongly makes a case for transfer.
In his paper entitled “Deweyans Pragmatism and the Epistemology of
Contemporary Social”, Garrison (1995, pp. 718-720) declared that the field of
education should consider behaviorism as one way of understanding social
constructivism and situated cognition. As a result, drawing from what Richard
Rorty (1979) calls “epistemological behaviorism” or what W. V. O. Quine (1969)
referred to as "behavioral criteria for the truth" and the writing of Dewey
himself, Garrison (1995) argues that Dewey may be identified as a behaviorist
and coined the term “Dewey’s pragmatic social behaviorism” to describe his
philosophy. A fundamental principle of this brand of behaviourism is the
rejection of the notion of knowledge as merely conceptual representation.

32. 31
Rather, people acquire knowledge based on the evidence of other people's
overt behaviours or activities. As Dewey put it,
“meaning is not indeed a psychic existence; it is primarily a property of
behavior, and secondarily of objects. But the behavior of which it is a quality is
a distinctive behavior; cooperative in that response to another's act involves
contemporaneous response to a thing as entering into the other's behavior, and
this on both sides.” (Dewey, 1925/1981, p.141).
Dewey extends the behaviourist’s view on learning to include, social
interactions, cooperation and communication. In fact this quote, shows not only
the considerable influence of behaviours on cognitive activity but also suggests
that through his/her behaviour, the individual shapes and is shaped by the
world. Note that the dialectical nature of stimulus and response is akin to the
intersubjectivity involved in meaning making in other competing theories of
learning, but in this case the focus is on actions. Since I also strongly believe
that we most of the time infer what goes on in the individual’s mind through
their behaviour, I will henceforth refer to some of the ideas presented in
Dewey’s book “Experience and nature” as “Dewey’s Social Behaviourism”. That
notwithstanding, I will begin the comparison of Dewey’s social behaviorism,
social constructivism and situated cognition with the following quote from
Dewey (1925/1981):
"Through speech a person dramatically identifies himself with potential acts and
deeds; he plays many roles, not in successive stages of life but in a
contemporaneously enacted drama. Thus mind emerges" (p. 135).

33. 32
This shows that for Dewey, meaning is a social construction and is associated
with both activity and the gradual acquisition of the characteristics and norms
of a culture or group by an individual. Also, based on the above quote, note
that for Dewey, there is no need to determine the order in which activity or
concept formation engender meaning; because it all happens simultaneously.
In this regard, Dewey’s, social behaviourism could be said to unify the social
constructivist’s and situated cognition’s epistemologies.
Equally important to Dewey, was the idea that mind should be seen to
denote "the whole system of meanings as they are embodied in the workings of
organic life.... Mind is contextual…" (p. 230). This suggests there is no
discontinuity between the individual and the world and that the mind emerges
as the individual learns to participate in social activities involving labor, tools,
and language. Which is essentially both a situative and social constructivist
perspective, which places the locus of meaning making in social interaction or
in participation in practice. In a nutshell, Dewey’s social behaviorism has much
in common with both situated cognition and social constructivism. All these
theories posit that learning is mediated by social interactions and mediation
implies that some knowledge or skill is conveyed, and again this implies
transfer.
Other notions shared by situated cognition and social constructivism is
the zone of proximal development, a metaphorical space in which to study the
interaction between teacher and learners, and scaffolding which describes the
processes involved in allowing the child to advance through the ZPD (or
progress from what a he/she can do without help and what he/she can do with
help). Vygotsky defined the zone of proximal development as:

34. 33
“the distance between the actual developmental level as determined by
independent problem solving and the level of potential development as
determined through problem solving under adult guidance or in collaboration
with more capable peers”. (Vygotsky, 1978, p. 86)
This suggested that, according to Vygotsky, a child’s learning is advanced
through the provision of experiences, which are in his/her ZPD, and that in
interactive situations, children can participate in activities that are more
complex than those they can master on their own. Bruner (1985) introduced the
term ‘scaffolding’ in relation to ZPD to explain the process through which the
child's learning is guided through focused questions and positive
interactions in order to advance their knowledge from lower level of the ZPD
to a higher level. Scaffolding is achieved through semiotic mediation (Vygotsky,
1978, p. 40); that is when the direct impulse of the learner to react to a
stimulus is inhibited through the intentional introduction of a sign, by an
external material (e.g. a teacher, textbook or computer), thus controlling
their behavior from outside. The Socratic method/dialogue (see Renyi, 1967),
which encourages learners to reflect and think independently and critically,
embodies the concept of scaffolding. It is characterized by starting with the
concrete experience and through probing; the learner gains insights by
explicitly relating any statement made to his/her personal experience.
Some education experts criticized the notion of scaffolding for its
tendency to share the passive unidirectional transmission of information
characteristic of the conduit metaphor of communication. For example wertsch

35. 34
(1991) stated scaffolding invokes the conduit metaphor, which describes the
transference of ideas or schemata from one person to another by means of
human communication. He argued that the conduit metaphor fails to realize all
of the dialogical possibilities between teacher and student, which may involve
the tension caused by the learner’s need to engage in the existing practice and
their need establish their own identities. To incorporate this dialogical
relationship, Lave and Wenger (1991) reinterpreted the scaffolding from a
collectivist or societal perspective by putting "more emphasis on connecting
issues of socio-cultural transformation with the changing relations between new
comers and old-timers in the context of a changing shared practice" (p. 49). In
this regard, scaffolding refers to the process through which a person’s identity
is established or evolves as they participate in practice. Granted, scaffolding
may not be monological in this perspective, but transfer still underpins means
by which the change in relationships between new comers and old timers is
brought about. This is due to the fact that in any social group, knowledge or
standards are derived from established modes of communication or
interactions. As a result, it may be difficult to regard any meaning constructed
from such activities as objective. Rather, from social constructivists and
situative perspectives, individuals appropriate knowledge and skills due to their
involvement in social practices. Appropriation also invokes the concept of
transfer.
Newman et al. (1989) indicated that appropriation was proposed by
Leont’ev, a colleague of Vygotsky as an alternative to Piagetian concept of
“assimilation”; which describes the process through which new experiences are
modified to fit schemes previously developed. The concept of “appropriation” is

36. 35
concerned with what individuals may take from interactions with other
individuals in a cultural context. This suggests that in the educational context
for example, the social relation between teacher and learner is an integral
aspect of meaning construction. In this respect, the process of teaching and
learning may even be viewed as not a relation between individuals and
knowledge, rather as individual's introduction into an existing culture. For
example, in collective activities, like exchanges at Oksapmin trade stores,
described by Saxe (1999); individuals are engaged in constructing
communications about quantity and in accomplishing quantity related problems.
Saxe wrote
“such occasions provide opportunities for reciprocally appropriating at least
superficial features of one anothers’ constructions. In such appropriations, new
forms are born as particular representations become valued and
institutionalized as regularized ways of representing and accomplishing
problems linked to collective practices” (unnumbered).
Furthermore, by acknowledging that individuals appropriate understanding
through cultural contact, social constructivist and situated cognitivists are in
essence validating the concept of transfer. For example, from an educational
perspective, the term appropriation suggests that learners are cultural beings
(who relate to the ideas, customs, and social behavior of a society) and the
interactions between themselves or between them and teachers are cultural
experiences. As a result, all understanding derived from such encounters are
essentially culturally defined. As such, it may difficult for any socio-culturalist to
deny that somehow, cultural experiences - for example, learning in the social

37. 36
context - fundamentally imply transfer. Similarly, in an attempt to explain the
meaning and the origins of the term appropriation (Bartolini Bussi, 1994,)
declared that:
“For Vygotsky, the process of learning is not separated from the process of
teaching: the Russian word obuchenie, which is used throughout Vygotsky's
work, means literally the process of transmission and appropriation of
knowledge, capacities, abilities, and methods of humanity's knowing activity; it
is a bilateral process, that is realized by both the teacher and the learner”(p.
125)
Drawing from the arguments in this section, I can comfortably conclude that
even within the theories that supposedly deny validity of transfer as an
educational construct, they describe processes that essentially embody its
concept. As a result, there should not be any reason for mathematics education
not go ahead and develop a research framework for teaching or learning
transfer. As it is, research on transfer is fragmented to say the least and
considerable efforts need to be put in reviewing the empirical evidences
produced by various theories on transfer in a bid to create a unified framework
for teaching for transfer. Equally, more researchers (e.g., Lester, 2005; Cobb,
2007) are calling for the adoption of multiple perspectives in matters of
teaching and learning in other to account for the multifaceted nature of
classroom settings. As such, analyses of theories or research findings should
reflect the interplay between psychological, sociological and contextual aspects
of learning. I think this view can be applied to research on transfer in general.
Consequently, adopting a conceptual framework such as the cultural historical
activity theory (CHAT) may be a more effective framework for examining

38. 37
educational practices. It is a broad approach that can be used to develop
conceptual tools for analysing many of the theoretical and methodological
issues on transfer from school to the work place. In my final chapter, I will
highlight the potential of CHAT as a pedagogic framework for teaching for
transfer.
In chapter 2, I presented, the concept of transfer by highlighting some
of the practical and theoretical issues associated with it and mapping out its
historical development. In so doing, I have not only shown the importance of
making teaching for transfer an explicit educational goal, but also how research
trends on transfer has gradually evolved to accommodate three fundamental
elements. These elements are the (a) tasks (procedural vs. conceptual
approach); (b) learner characteristics such as dispositions, teaching of
metacognitive strategies, and promoting their willingness to transfer learning;
and (c) considering the transfer context which includes impact of the social
participation or instructional settings, and the quality of the instruction or
teacher intervention. In a bid to argue for teaching for transfer, I have shown
(in this chapter – chapt. 3) that even with learning theories that supposedly
doubt the feasibility of transfer; it underpins the alternative notions they
presented regarding how we come to know. In the next chapter, I will consider
transfer of knowledge from school to the workplace, in terms of the learner,
tasks and context.

39. 38
4. Transfer
from
School
to
the
Work
Place
In Chapter 1, I highlighted the development of functional mathematics in
United Kingdom as a means bridging the gap between school mathematics and
out of school mathematics and equipping learners with the desirable
mathematical competencies in their wider lives. However this begs the
questions as to what level of mathematics to teach learners. For example, in
terms of pedagogy, how do we accommodate learners, who may want to
pursue mathematics studies in higher education and those whose mathematics
use may very much be subsumed by technology use in the workplace? Does
technology use interfere with mathematics learning? Should teaching continue
to be led by traditional, routine or expository approaches or should problem
solving and more investigatory approaches to learning be adopted? Should
traditional mathematics with its formal tasks and problems be the basis of the
curriculum, or should it be presented in realistic, authentic, or ethno-
mathematical contexts? I will attempt to answer these questions in relations to
the transfer of learning in this chapter.
Through out this essay, I maintained that the ultimate goal of education
is to help students transfer what they have learned in school to everyday
settings and workplace. And from the historical development of transfer
outlined in chapter two, it is clear that transfer between situations is a function
of the common elements or similarities in strategies, principles, surface features
of tasks, context etc. between original and target situations. These common
elements may also involve learners’ disposition such as their willingness to
transfer. These factors identified for transfer are in their own right very
complex. Consequently, it becomes evident that if educators aim to teach for

40. 39
transfer of learning from school to other settings, it is important not only to
study the elements identified by transfer theories, but also understand the non-
school environments in which students are required to function. Since the
focus of this study is transfer of learning from school to the work place, I will
start by briefly outlining the role of mathematics in the work place and the key
differences between school learning and work place learning identified by
research.
4.1. Mathematics
in
the
Work
Place
Mathematics plays a significant role in the workplace and continues to be a
factor of increasing importance in the workplace (Hoyles et al., 2002; Kent et
al., 2007). However, mathematics competencies required in the workplace are
better defined as numeracy (Steen, 2001). This is because unlike school
mathematics, numeracy, is so heavily embedded in practice that it may be
difficult for the casual observer to discern the underlying mathematics. Equally,
numeracy, which more closely defines the mathematics used in the work place,
is anchored in real data, offers solutions to problems about real situations. In
other words, mathematics in the work place is framed by the work situation and
practice, and, in many cases, by the use of IT tools. The purpose of
mathematics in industries also varies because of the diversity of processes
involved. Hoyles et al. (2002) identified the following mathematical skills and
competencies recurring in the sectors (i.e. electronic engineering and
optoelectronics; financial services (Retail); food processing; health care,
packaging; pharmaceuticals; and tourism) they studied. In their report entitled

41. 40
mathematics skill in the workplace, Hoyles et al. (p.11), summarised numeracy
as involving:
a) Analytical, flexible, fast and often multi-step calculation and estimation in the
context of the work (with and without the use of IT tools)
b) Complex modeling (of variables, relationships, thresholds and constraints),
c) Interpretation of, and transformations between, different representations of
numerical data (graphical and symbolic)
d) Systematic and precise data-entry techniques and monitoring
e) Extrapolating trends and monitoring models across different types of work
f) Concise clear communication of judgments
g) Recognising anomalous effects and erroneous answers.
From this we can deduce that the role of mathematics lies mainly on the
decision-making process and interpretation of results. It also involves the
practical application of rational numbers and the metric measurement system
with contextualised approximations and estimations in critical calculations, often
with other workers. And more often than not, numeracy goes hand in hand
with non-mathematical skills such as for example, planning, organising,
cooperating, communicating effectively and teamwork. This is in marked
contrast with school learning, which is often an abstract, rule-bound, individual
activity, with one correct answer (usually a number, an algebraic expression, or
a standard graph), and where mistakes are without penalties.
Several other studies also reported striking differences between school
and out of school methods. For instance, abstract reasoning is often
emphasized in school, whereas contextualized reasoning is often used in
everyday settings (Resnick, 1987). This may explain why, unlike tasks requiring

42. 41
only the reproduction of skills or facts, problems concerning application tasks,
problem solving and scientific argumentation in mathematics education poses
considerable difficulties for most students (e.g. TIMSS/PISA Studies). These
studies reported that reasoning could be improved when abstract logical
arguments are embodied in concrete contexts. In their well-known study of
people in a weight watchers program, Lave et al (1984) showed how thinking,
is not only situated in the physical, social and cultural context, but also derives
its form from these variables. One example is of a man who needed three-
fourths of two-thirds of a cup of cottage cheese to create a dish he was
cooking. Instead of multiplying the fractions, as any student would do in a
school context, he measured two-thirds of a cup of cottage cheese, removed
that amount from the measuring cup and then patted the cheese into a round
shape. He then divided patted cheese into quarters, and used three of the
quarters. Abstractions never featured in his method. Likewise, grocery store
shoppers use non-school mathematics under standard supermarket and
simulated conditions (Lave, 1988). Similar examples of contextualized reasoning
have also been found in educational tasks. For example Brazilian market sellers
make arithmetical calculations more effectively at their market stalls than in the
classrooms (Carraher, Carraher and Schliemann, 1985).
These findings suggest that the individual’s interpretation of tasks is
partly determined by the context and the goal of the activity. That is since
different institutions have their own body of cultural knowledge, goals,
conventions or rules and ways of communicating, learning must be
contextualised to reflect these variables. On the other hand, these examples
also suggest potential problems with contextualised reasoning and that transfer

43. 42
is also partly dependent on the degree to which learners can relate their
procedures to more general sets of solutions. For example, in the cottage
cheese example above, if the material was a liquid, will the man be able to
adapt his methods? It is not very clear from the research, but I would assume
the answer to that is no. For this reason teaching abstraction and generalisation
does have its merits. Consequently, context (from subject’s perspective) could
be reconceptualised to include whatever knowledge learners bring to bear from
past experiences, to make sense of novel situation.
Another contrast between school and work place environments is the
organization of work or division of labour. In school, assessment procedures put
much more emphasis on individual work, whereas in work environments,
teamwork or collaboration is preferred (Resnick, 1987). As a result,
opportunities should be created for students to experience working
collaboratively and share their ideas (Kent et al, 2007). Equally mathematics in
the work place is driven by the need for accuracy, especially in safety critical
systems (hospitals, nuclear plants etc.). Collaboration or teamwork is the
mechanism established to ensure accuracy. For example, calculations are
double-checked, and team and group work are fostered as part of workplace
practice.
Another difference between school mathematics and work place
mathematics is that a large part of the latter is observed to lie within
technology-related tools. Tools are used to solve routine problems in work
settings, compared with “mental work” in school settings (Resnick, 1987). Some
of the simple mathematical tools used in the workplace are the graphs, tables,
simulation, modeling, and calculators to name a few. Since most of these

44. 43
technologies are also present in some schools, mathematics curricula (for
example in UK and France) have been reformed in the hope of simulating
certain aspects of practice in everyday situations and the work place. This is an
important development because for me, it contradicts the beliefs of teachers
and some curriculum developers (in my country, the Gambia, for example) that
use of technology reduces learners’ knowledge of mathematics. Rather,
technology use in mathematics education transforms nature of the
mathematical skills or processes learnt in school to reflect mathematics use in
their wider lives or the work place.
For instance, in the workplace, workers need to able to transform the
data collected into a mathematical model (e.g. a graph, chart or tables) and
then interpret what this model means in terms of the work situations and then
report these findings to their supervisors, who then make the relevant decisions
out of the information arising from the analysed data. All these processes
require not only, computation skill, but also ability to analyse and interpret
information. Also certain work situations present a need to examine the effects
of changing variables in mathematical models for a large amount of data. For
example, in statistical models, multi-variables representing different thresholds
may be keyed in to examine their effect on the output. Computers and other
technologies can take this further through allowing faster computation, direct
manipulation of representations and visualisation of conceptual objects
(Jonassen, 2000). In some of these cases workers may not even be aware of
the mathematics involved in the work activity, all they need is the ability to
interpret what the figures on the monitor mean and make the corresponding
decision. In this regard, use of technology is instrumental and workers “black-

45. 44
boxed” from the meaning of the mathematical activity they are involved with
(Wake, 2005).
In education, the availability of these new technologies make it possible
for students in schools to use tools very much like those used by professionals
in workplaces. Also simulating a situation such as I just described may not
necessary require computers, which pose a problem for under-resourced
schools in my country. The emphasis here is on how the content is presented to
provide learners the opportunity to experience processes such as those found in
the workplace. For example, when investigating the relationship between
speed, distance and time, learners do not have to stop at learning to draw
distance-time graphs. The content could be extended to involve interpretation
of the information contained in graphs. Study of models, charts or graphical
representations allows learners to objectify their thoughts so that they can
reflect upon them. In this way, graphs, tables and symbolic notations in
mathematics can be described as cognitive tools that extend thinking. Through
analysing and interpreting models, learners in schools without ICT can
experience processes similar to those found in technologised workplace.
In summary, workplace mathematics (numeracy) suggest that the skills
required are basic and near lower high school level, but the fact that they are
applied in complex ways to ill-defined and continuously evolving problems
makes it difficult for school leavers. As a result, mathematics curricula should
incorporate the processes that learners need to learn in order to cross the
metaphorical boundary between school and the work place. As noted earlier,
some of these competencies are: ability to recognise situations in which
mathematics can be used; make sense of these situations; describe the

46. 45
situations using mathematics; analyse the mathematics, obtaining results and
solutions; interpret the mathematical outcomes in terms of the situation; and
communicate results and conclusions. In short workplace mathematics is based
on professional practice, which is basically functional knowledge. Functional
knowledge offers people the know-how to get things done.
The implication for teaching and learning is therefore, the creation of
opportunities for learners to experience sessions that have a significantly new
emphasis and focus on enriched tasks. That is tasks that simulate the scope,
complexities and contextual nature of workplace mathematics. In other words,
learners need to engage in activities that involve the application of
straightforward mathematical skills or concepts in complex contexts. This is
different from traditional mathematics teaching in which learners often do very
challenging mathematics in very simple contexts, or entirely out of context.
Conversely, this new approach is referred to as functional mathematics.
4.2. Teaching
for
transfer
–
Functional
Mathematics
In the introductory part of this thesis, I defined functionality, in terms of
curricular development and referred to it as a means of bridging the gap
between school mathematics and out-of-school mathematics. However, this
raises an important question as to the level of mathematics to teach students.
That is, since most researches reported that that the level of mathematics in
work context is often at lower high school level, should the curriculum be
reformed to teach only up to lower high school level? And if such an approach
is adopted, what about learners who may want to study mathematics in further
and higher education, or those who want to pursue careers such as

47. 46
engineering, which demands more advance mathematics. Or, how do we even
begin to determine the appropriate level of mathematics for individual
students?
Clearly no one, even teachers, can predict what mathematics their
learners will use as they move through their lives. As a result, a possible
solution will be to integrate functional mathematics into the curriculum; that is
(for lack of a better word) not to dumb-down the mathematics curriculum, and
at the same time, provide learners opportunities to experience use of
mathematics close to what occurs in the work place. In this respect, the
adoption of functional mathematics, an approach that has potential in
simulating the complexities inherent in work situations, will complement
mastery models prevalent in most education systems. In other words, the goal
of functional mathematics may be the creation of contexts in which theory and
application are intertwined to encourage meaningful, memorable and
internalisable learning. Traditional mathematics curricula, which often separate
pure mathematics from applied mathematics, are unlikely to achieve this goal.
Functional mathematics allows the learner to apply theoretical concepts.
According to Shulman (1997), establishing practice as a fundamental aspect of
school mathematics may overcome the deficiencies of theoretical learning.
These deficiencies include loss of learning or forgetting concepts learnt, illusion
of learning or thinking you have grasped the concepts and uselessness of
learning or the inability to apply what you have learnt. Research has shown
that most adults in professions that require the use of mathematics are faced
with such deficiencies. For example Hoyles et al. (2002) and Kent et al. (2007)
reported workers inability to apply the mathematics learnt in school. Equally,

48. 47
Cockcroft (1982) suggested that the traditional curriculum creates a severe
psychological impediment to the practice of mathematics in adult life.
Functional mathematics avoids many of these issues by focusing not only on
theory, but also on authentic mathematical practice. For example, Qualifications
and Curriculum Authority (2007) in UK proposed that functional mathematics
should be:
“considered in the broad sense of providing learners with the skills and abilities
they need to take an active and responsible role in their communities, everyday
life, the workplace and educational settings. Functional mathematics requires
learners to use mathematics in ways that make them effective and involved as
citizens, operate confidently and to convey their ideas and opinions clearly in a
wide range of contexts.”
That is, processes of functional skills are related to problem-solving and data
handling cycle, involving three important phases; representing, analysing and
knowledge of the appropriate mathematical procedures to use in a variety of
contexts. The proactive measures of the QCA to provide learners with functional
skills that transcends cultural boundaries (from school to everyday lives,
including work) provide schools a framework for organising tasks that mirror
the kinds of activities that take place in the workplace. It must be noted that
even when a classroom lesson is designed to simulate the workplace, it can
never completely capture the demands of actual practice. However, the current
indictment of traditional, routine or expository teaching in research examining
the transfer of knowledge from school to the workplace suggest that the
functional mathematics or the problem solving and more investigatory approach
to learning offers an effective solution. In Hoyles et al. (2002) report, it was

49. 48
emphasised that most employers’ lament the lack of initiative and problem-
solving skills of most graduate workers. For example in the packaging case
studies, a senior manager said the following about a work manager under his
supervision:
“…He could do the mechanics of it because he knew how to do that... but he
was not able to interpret it, to transform it into a trend or a problem solving
analysis. He wasn’t able to really use the information in a well-constructed
argument. He could not present an argument, which was fact-based, by, using
the numbers and the information that we do collect.” (p. 69)
This excerpt shows that even though, this employee knows the mathematics
involved, he was unable to perform when it is embedded in a real world
situation. He could not interpret what the results indicated or what decision to
take as a result of the analyses of information or data collected. More
importantly, it also shows the limitations of traditional mathematics curricula in
preparing learners for everyday situations or workplace. For example in my
country the Gambia, mathematics is rarely considered in relation to its
application outside of school. It is mostly presumed that once mathematics is
learnt, students automatically developed the ability to apply it to various
problems in their everyday lives or in their future roles in the workplace. This
limitation still persists in the newly developed Gambia national mathematics
curriculum, despite efforts to integrate applications and mathematical models.
This is mainly due to the fact that the emphasis of this new curriculum is on
application without much focus on the context in which such application arises.
That is rather than emphasizing the real life context in which such problems
may arise, they are often designed to teach particular skills or concepts without

50. 49
emphasizing authentic application from everyday life and work. Similar
problems that prevail in the Gambia are also present in the west. Generally, the
mathematics curriculum can be categorized as traditional, which is centered on
algebra, functions and Euclidean geometry or as reformed which focuses on a
more constructivist view that mathematical meanings arises from active
engagement with contextualized problems. What both types of curricula
(reformed and traditional) have in common is that topics are designed to move
the learner along the path of arithmetic to calculus. Functional mathematics
curriculum also follows the same path but also places more emphasis on
realistic mathematic education (Treffers, 1993) – that is authentic problems
centered on real life situations. The idea is that problems arising form authentic
context expand learners’ understanding and allow them to see the
interrelationship between mathematics concepts, skills and procedures and their
application across different context. The fundamental principle is that
mathematics learnt in school, no matter how specific, should have the potential
to enhance mastery in other areas. Moreover, by embedding mathematics in
practice, functional mathematics offers students both theory and know-how.
Furthermore by advancing authentic contextualized content, engaging tasks,
and active instruction functional mathematics can motivate students to link
meaning with mathematics. Functional mathematics may also provide students
the opportunity to recognize the diverse uses of mathematics in various walks
of life as well as develop their knowledge from elementary to advance concepts
which not only help them if they choose to study mathematics in higher
education, but also prove useful in creating experiences that closely reflect the

51. 50
mathematical processes or activities that take in the learners’ wider lives and
the work place (Forman and Steen, 1995).
Similarly, De Corte (1999) suggest that curricula focused on functional
mathematics design tasks that resemble those found in everyday life and work
and encourage students to develop “systems thinking”. The term “systems
thinking” is inspired by systems such as those found in commerce and industry,
science, technology and society. In such complex systems, being functional in
mathematics is a fundamental factor that influences performance. Systems
thinking require learners to develop habits of mind that recognizes complexities
hidden in situations (which may or may not involve technology use) subject to
multiple inputs and diverse constraints (Kent et al, 2007). The underlying idea
for developing “systems thinking” is to encourage students to learn to apply
logic and careful reasoning in many situations. This concept of “systems
thinking” suggest that schools and colleges are not the sole domains of
learning, but the home, workplace and communities are also areas that learning
takes place. For this reason, the significances of functional mathematics in
integrating social practices such as those existing in students’ wider lives cannot
be overemphasized. In this respect, functional mathematics offers great
promise in extending the learning and the learner beyond the school and thus
highlights the significance of teaching for transfer or the emphasis of prior
experiential learning in students’ future endeavors.
However pedagogic approaches for teaching functional skills may risk
being ineffective if not based on an appropriate theoretical framework. For
example the theories of learning discussed in this thesis have varied
epistemologies, which have their strengths and weaknesses. However in

52. 51
considering what these learning theories have to say about how we come to
know and relating these ideas to pedagogical research and practice in the
complex classroom context, it becomes necessary to adopt a conceptual
framework such as cultural historical activity theory (CHAT), which effectively
frame most of the ideas from different theories of learning under one umbrella.
For instance CHAT seamlessly brings together key elements of learning such
the individual cognitive development, the situational and cultural factors.
Additionally, notions of communities of practice, networks activity systems and
boundary objects (which I will discuss later) in CHAT will help frame
researchers or practitioners understanding of pedagogy in extended and
complex contexts of learning/teaching. As a result in the next section, I will
explore the potential of CHAT as a pedagogic framework for imparting
functional skills and thus transfer.

53. 52
5. CHAT,
Transfer
and
Teaching
Functional
Skills
The main focus of this study is to explicate the key pedagogic issues that
should be addressed in order to foster transfer of learning from school to the
work. In my opinion, transfer of learning is means of encouraging lifelong
learning in students or the development of learning that has the capability of
meeting their present and future needs irrespective of whatever situation they
find themselves. This means that a key focus of instruction should consist of
the creation of conditions for the emergence and development of conceptual
understanding and activities associated with mathematics as a domain and its
function in students’ wider lives; thus the development of functional
mathematics curricula.
However, a key issues emerging from research work aimed at
establishing the most effective strategies for facilitating transfer of learning are
teaching facts and skills vs. developing conceptual understanding;
contextualisation vs. decontextualisation of learning; individualized learning vs.
collaborative learning, and active vs. passive learning. In reading for this
thesis, I have concluded that the viewpoints of various theories of learning on
the concept of transfer and their corresponding strategies for facilitating it are
basically “systems of thought”. They all propose ideas on achieving transfer
that may always be true within the theoretical framework they are considered;
nevertheless, their key tenets only embody partial truths. For this reason, I
maintain CHAT theoretical framework offers a sound framework for
incorporating all these ideas as well as transcending the traditional approach to
teaching for transfer, which either concentrate on individual learning in the
school context or situated learning at the workplaces and never link the two or

54. 53
account for ecological factors. In addition, by modeling how transfer may take
place through interaction between activity systems, CHAT holds great promise
in research to understand transfer of learning from school to the workplace. In
this chapter, I will explain CHAT and argue that it has great potential for
analyzing as well as enhancing pedagogic practices aimed at fostering transfer.
Activity theory was originally elaborated in the framework of cultural
historical theory in 1978 and was applied to learning activity by Davydov (1988,
1996) and Engestrom (2005) among others. It is a conceptual framework
consisting of a collection of basic ideas for conceptualizing both individual and
collective practices in developmental processes. The first generation of activity
theory inspired by Vygotsky and modeled on a triad consisting of subjects (an
individual or group) who work towards an objective (object) in order to achieve
an outcome. For example figure 1 shows a possible pedagogic research
framework for teaching for transfer. In this scenario, the means by which the
subjects (teachers) achieve objects (e.g. teaching for transfer) and outcomes
(transfer of learning) is through meditational means (tool). These tools may be
external (such as teaching and learning materials) or internal (strategies, plans
etc.). This first generation model is especially powerful for anyone wishing to
study what takes place in a person’s mind as they interact with their
environment. Vygotsky managed to move away from the behaviorist’s tendency
to study behaviors in action and extend the unit of analysis to take into account
the thinking behind actions. As such, conceptual tools learners bring to bear
when they carry out tasks and the way they use them becomes an integral
aspect of educational research and learning. However, the limitation of the first

55. 54
generation (see figure 1) CHAT was that the unit of analysis remained
individually focused.
Engestrom further expanded the triad above to enable examination of an
activity system not only at the level of the individual operating with tools but
also at the level of the community. This development is referred to as the
second-generation activity theory (see Figure 2). The unit of analysis changed
from an object-oriented action mediated by cultural tools to take into account
mediation by other people and social relations. In this way, the focus then
shifted towards the study of the complex interrelations between the individual
subject and his/her community; who are involved in a collective work activity,
and whose main focus is to find a solution of a problem (object), which is
mediated by tools and/or signs used in order to achieve the desired goal
(outcome). The activity is constrained by cultural factors including conventions
(rules and beliefs) and social organization (division of labor) within the
immediate context.

56. 55
In contrast to traditional pedagogic framework, CHAT not only accounts for
contextual and environmental factors within and learning systems but also
identifies three mediating relationships for analyzing such settings. These
mediating relationships in relation to pedagogy are:
1. Tool mediates between subject and object: these tools may be explicit tools
such as language, knowledge and the physical artifacts needed by students
to engage with tasks or tacit tools concerned with how do students manage
emerging knowledge.
2. Rules mediate between community and subject: In terms of pedagogy, this
involves two dimensions for creating supporting structures for effective
learning. This involves encouraging students and teachers to interact in
ways that allow them to build or appropriate knowledge through tasks, and
from community and social relationships.
3. Division of labor mediates between community and object: This is
concerned with who does what and may require the redefinition of the

57. 56
teacher’s role (from an instructor to mediator) or their beliefs about what
constitute good teaching. On the learners’ part, active participation,
questioning as well as autonomous learning strategies is important.
Engestrom’s (1987) theory of expansive learning give details of how the second
generation of activity theory brings together the necessary variables when
developing a learning system. In this perspective, development is interpreted
on the basis of expansive circles, which consist of processes of internalization
and externalization as proposed by Vygotsky (1986). These two terms are
instrumental in understanding how subjects are socialised by participation in
the world (externalisation) and in the process internalise the same world by
abstract symbol and inner speech or thought. In a later publication, Engestrom
(2005) explains that expansive learning arises as a result of the
interrelationship between internalisation and externalisation. He suggested that
expansion starts by internalisation to allow the novice to pass through
processes of socialisation. As they later become more competent members in
the activity, the process of externalisation becomes gradually more dominant as
members, driven by the need to resolve tensions and contradictions within the
activity system, carry out processes of innovation. In this way, the internal
tensions and contradictions of an activity system, much like Piaget’s concept of
continuous cycle of disturbances, accommodation and assimilation, serve as the
motive for change and development.
CHAT as a pedagogic framework is especially powerful because instead
of reliance on a particular theory of learning, it integrates pedagogical concepts
from a various theoretical frameworks and thereby offers a much more holistic

58. 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