Pedagogy

Chapter 1Introduction
Mathematics, as a concept, has evolved from simple arithmetic functions, which lasted up to 500 B.C., into a field that encompassed multiple disciplines (Devlin 2004). Freudenthal (1991) argued that mathematics is a convergence of four arts or a quadrivium, which is defined as a composite of arithmetic, astronomy, music, and geometry at a level higher than trivium (rhetoric, dialectic, grammar). On the other hand, a philosophical perspective on mathematics defined the concept in the following manner

In general, mathematics can be described as the search for structures and patterns that bring order and simplicity to our universe. It may be said that the object or beginning point of a mathematical study is not as important as the patterns and coherence that emerge. And it is these patterns and coherence that give mathematics its power, because they often bring clarity to a completely different object or process to another branch of mathematics, another science, or to society at large. (Berlinghoff, Grant,  Skrien 2001 1-2).

From the definition of Berlinghoff, Grant, and Skrien (2001), mathematics is presented as a field of inquiry that is composed of four different disciplines. In fact, mathematics is considered as the point where the four different fields convene and is considered to be of a higher esteem than other fields of inquiry. As of the present, mathematics is used in different fields, including economics, psychology, and science to name a few. The field, with its variety of use, has proved to be of worth to different disciplines and is no longer isolated and confined within the realm of hard sciences, as the definition above suggests.

The task of defining mathematics has already spurred several inquiries into the limits of the field. In relation to this, there are also several issues that emerged in relation to mathematics education, which deals with teaching the concepts, theories, and knowledge regarding mathematics. The following section deals with a brief discussion on existing and emerging issues regarding mathematics education.

1.1 Issues in Mathematics Education
Mathematics education is beset with several issues, especially in relation to the instruction of mathematics in the classroom. Among the issues gathered involve the demathematisation process that is brought about by the introduction of technology in mathematical processes (Ernest, Geer,  Sriraman, 2009) the preoccupation of teachers with lectures that result to the neglect towards ideas and techniques learned without formal schooling and the strategies of transferring the same (Powell  Frankenstein, 1997) differential student learning, achievement, and persistence along ethnic and racial lines (Martin, 2000 5) and gender and cultural issues regarding mathematics education (Atweh, Forgasz,  Nebres, 2001 Costello, 1991). The issues enumerated above shows the intricate nature of mathematics and the challenges faced by the teachers and the learners.

First, the field of mathematics education is challenged by the results brought about by the process of demathematisation. As cited in Gellert and Jablonka (2009 20-1),

This term also refers to the trivialisation and devaluation which accompany the development of materialized mathematics mathematical skills and knowledge acquired in schools and which in former time served as a prerequisite of vocation and daily life lose their importance, and become superfluous as machines better execute most of these mathematical operations.

The characterization of demathematisation involves three elements. First, it shows the decreasing importance of mathematical skills and knowledge that are traditionally derived from schools. Likewise, the said skills and knowledge are, in the past, considered to be a part of the fundamentals that are of use to an individual in personal and professional activities. Consequently, there is a lower rate of significance placed over the acquisition of the skills because of the availability of alternatives. As a result, a population of students is produced with lesser abilities of applying and understanding the theorems and concepts behind the mathematical processes involved in arriving at a particular situation. Likewise, learners also experience a relative perception of difficulty when it comes to the idea of manual computations and explanation of mathematical steps involved. Second, a transfer of mathematical processes occurs from the human beings to the machines that are created by human beings to perform the needed tasks. The shift occurred because of the need for efficiency, in terms of effort and time, in dealing with the task. For example, a person would need several minutes to compute for the mean of over a hundred numbers. However, the time spent to complete the very same task can be reduced to a few seconds using a computer and a variety of other tools made available to the public. To make things more convenient, these tools are made accessible and user-friendly to increase its appeal to the public. In relation to this, there is also an accompanying shift of instruction from lectures that focus on the manual methods of computation to the lectures that focus on the use of the tools. Nonetheless, it is important to note that the shift in instruction and use of technology vary deeply depending on the rate and capacity of a particular country or educational institution to adapt to the said changes. Third, the perception that the use of machines is a better approach to solving mathematical problems is present. With the thought of machines being a better tool, individuals, at both the academic and professional level tend to discount the importance of manual tools of mathematics. Thus, it results to a reduction in the frequency of use and importance within the classroom setting. The third factor plays a significant role in the first two elements because it is the driving force behind the continuous process of demathematisation and the use of machines.

Second, mathematics education also realizes problems with regard to the preoccupation of teachers with lecture (Powell  Frankenstein, 1997). While lectures play an important role in the learning process, the strategies and techniques that are inherently important for the field of mathematics are equally of great use for the students. The strategies related to mathematics that are learned beyond the perimeters of the school are called ethnomathematics (Nunes, 1992). The techniques learned from outside the school is important because of the significant role it plays in the ability of the learner to apply theories and better understand the concepts through experiential learning (House of Commons, 2005). With this, the teachers are beset with the problem of using the techniques learned from ethnomathematics into the formal curriculum (Charlesworth, 2005). In addition, teachers also encounter problems when the strategies are not applicable, or are different from, the concepts included within the lecture that the teacher follows (Charlesworth 2005). Charlesworth (2005) offers a solution where teachers should conduct lectures that involve experiences that can be encountered by the learner in everyday activities. By offering problems encountered outside the classrooms, teachers are given the opportunity to observe the strategies used by the students in arriving at the solutions (Charlesworth 2005). The observations and the strategies offered in the lectures are introduced by the teacher as alternative solutions to the problem, which also shows the application of the techniques to everyday experiences outside the classroom. However, the solution provided is not the absolute answer to the issue being discussed. Still, there are factors that affect the ability of the teachers in integrating the techniques obtained outside the classroom, such as the diversity of the learners, the level of understanding, and the access of the learners to the experiences used to name a few.

Third, mathematics education is also challenged by the issues of differential student learning, culture, and achievement, which is observed by Martin (2000) for the case of African-American mathematics classes. The achievement, ability of students, and culture are three significant factors because it affects the learning relationship that occurs between the students and the teachers. Likewise, it adds intricacies to the approach that the teachers need to employ in order to transfer the information from the textbooks to the learners because of the added consideration for the different needs of the learners. In addition to this, Martin (2000) noted that the existing literature that illustrates the relationship between culture and learning shows that the former is deterministic in nature. By being deterministic, culture disregards human efforts and motivation that is present along with culture (Martin 2000). The African-American experience offers an important addition to the growing list of the issues in the field of mathematics education. The set of issues listed by Martin (2000) in the African-American classroom shows that there are both specific problems that communities around the world face. However, there are also general problems, as shown before, that is shared by members of the academic community regarding mathematics education. In relation to the list of issues in African-American classrooms, the teachers are given the additional task of being sensitive towards determining the level of achievement and learning needs of the individual students in order to align the lectures with the said two factors. Aside from this, the epistemic community should also be aware of the role that culture plays inside the classroom, especially in the increasing pace of globalization where diversity of culture has become an apparent characteristic of schools across the globe. The issue of culture and its impact in mathematics education is discussed further in the succeeding section.

Fourth, multi-culture and gender issues affect mathematics education in respective ways (Atweh, Forgasz,  Nebres, 2001 Costello, 1991). Teachers of mathematics deny the importance of culture in mathematics and further argue that, among all disciplines, it is the one that proves to be culture-free (Costello, 1991). As a result, teachers do not recognize the need for integrating the different cultural backgrounds of the learners during the learning process. Lectures and approaches are aligned towards the mainstream culture, which varies depending on the dominant culture of the classroom or the teacher. With this, the teacher fails to address the distinct features that can affect the learning abilities of children from different cultures, such as language, beliefs, and traditions. For example, students coming from a country that does not use English as a medium of instruction experience difficulties in terms of communicating their ideas and understanding the concepts taught by the teacher. Likewise, the teacher who disregards the presence of different cultures in the classroom fails to encourage interaction within the students, which is an essential factor in the learning of the students because interaction among the learners lead to additional avenues for gaining knowledge among the students. In addition to this, the students miss the opportunity of gaining cultural awareness in preparation for the culturally-diverse workplace, which the students encounter after their graduation.

Aside from culture, gender issues also abound the field of mathematics education, especially in the area of performance assessment among students. Atweh, Forgasz, and Nebres (2001) argued that assessments of performance in mathematics are usually based on male standards, which makes the performance of the female rather inferior compared to the male. For this particular situation, there is the tendency for female students to hold the perception that mathematics is a male-dominated area (Atweh, Forgasz,  Nebres, 2001). When this happens, female students exhibit lower levels of participation and performance and have a greater impact on their choice of career (Atweh, Forgasz,  Nebres, 2001). The result that affects the participation, performance, and career choices of female students adds greater weight to the issue because it creates a wider gap in the abilities of students to comprehend mathematics and immediately lessens the chance of females to succeed in professions related to mathematics. In addition to this, it might also decrease the willingness of the female students to engage in math activities both within and outside the classrooms and workplace. With regard to the use of technology, the use of computers in mathematics is also associated with the male population, which also affects the attitude of females towards computers as tools in mathematics (Atweh, Forgasz,  Nebres, 2001). In relation to this, female students hold a decreased level of interest and motivation in the use of technology associated with the males (Atweh, Forgasz,  Nebres, 2001).

1.2. Challenges and Demand of Technology Obtaining a Competent Workforce
A competent workforce is characterized as a group of professional employees that is responsive to mission requirements, with continued review and training to improve individual and system performance (United States General Accounting Office US GAO 1999 4). From the definition, a competent workforce is described as the group of employees who are able to respond to the demands of the organizations mission through the use of continuous training. The purpose of the training, from this particular statement, is to improve the performance of both the individual and the system. Indeed, the competency of the workforce is seen to rest on the ability of the employees to comply with the requirements through the help of training. Likewise, both the system and the individuals are simultaneously improved because of the important interaction that occurs between the two. In fact, the human beings serve as an important component in the operation of the systems, whether technology-based or not, within the organizations because of the human intervention often required to control the system.

More importantly, the importance of a competent workforce is critical especially in cases where the organization deals with technological changes and is bound by the need for introducing technology-driven operations. In fact, the workforce is affected in changes where there is a need for reassignment, retraining, or cross-training when it comes to technology-related changes (Kirmeyer 2000). Training, in essence, is one of the solutions that the organizations provide when faced with the challenges of a workforce that is adept with emerging technologies in their industries. As mentioned earlier, a competent workforce is defined by training that comes according to the need to improve both the processes and the individual. Thus, training to cope with technological changes remains as one of the responses that the workforce has in order to maintain competency within the field. However, there are still several problems encountered by the industries in recruiting a competent workforce that adequately responds to the demands and challenges of technology within the workplace. The significance of the problem is related to the growing importance of technology within the industries and in the workplace, more specifically in the processes involved in the production of goods and services (National Cooperative Highway Research Program 1999).

First, there is a tendency for employees to resist technology-driven change because of perceived negative implications on employment. Oftentimes, the introduction of change in the workplace is brought about by the need for efficient and effective processes. However, it also entails a reduction in the number of employees needed to complete a particular task. As a result, employees often respond to technology in the workplace with opposition and resistance, which makes it harder to solicit their participation for this task.

Second, the academic training of the employees does not include aspects of technology, which requires additional time for the employees to gain knowledge on the particular technology being used. The familiarity with technological changes usually involves training until the employee is able to navigate and operate it without encountering significant problems. The time and efforts spent for training usually entails additional costs for the company, which includes materials, manpower, and wastes due to errors, to name a few. The costs mentioned should have been avoided if the workforce is equipped with the necessary skills, knowledge, and abilities earned through proper training in the academe.

Third, there is a continuous change experienced in the field of technology, which requires the need for constant training and research on the possible technologies that emerge within the industry. The constant research and development in the realm of technology requires the workforce to be flexible with the changes entailed. Thus, the organization should spend the necessary time and effort in training and recruitment in order to ensure that the employed individuals are flexible and adept to the technological changes within the industry.

1.3. Importance of Preparing a Technology-Informed Generation
The significance of preparing a generation that is knowledgeable and prepared to address the demands of technology is highlighted by the increasing demands of technology in the workplace. Enabling the generation to become adept with technology within the workplace is important because it gives competitive advantage, reduces the costs incurred by the organizations, and increases the competitiveness of a particular country. First, a generation that is well-versed in technology-related sources in the workplace has a competitive edge over the others within the labour market because they are able to provide the demands of organizations. As a result, a higher degree of match between the jobs required and the skills available is attained, which means that a greater part of the labour supply is able to address the requirements of the demand for labour. Second, preparing the generation for the demands and challenges of technology in the workplace allows the organizations to save costs and time, which can be otherwise incurred from training and coaching. Likewise, organizations would have lesser difficulties in recruitment and selection of individuals who are deemed appropriate for the job. According to the principles of scientific management, it is necessary to find an employee based on hisher ability to carry out the responsibilities associated with a particular job (Morgan, 1998). Third, the nations also benefit from a technology-ready generation because it is an essential characteristic that defines a competent workforce, which redounds to the ability of the industries within a country to compete in the global marketplace. The reasons mentioned are general points that justify the need for a generation who are capable of understanding and working with technology.

1.4. Outline of the Paper
The paper is composed of five sections that deal with different relevant topics. The sections are a) Introduction, b) Section 1 Concept of Transfer, c) Section 2 Transfer of Learning from School to the Workplace, d) Section 3 Role of Technology in Fostering Transfer, and e) Section 4 Implication in Mathematics Education. The chapters are arranged according to the order previously stated.

The introduction provides pertinent information that serve as foundations for the focus of the paper. The discussions revolve mainly on the issues present in the field of mathematics education, the challenges and demands of technology in the workplace, the significance of preparing a generation of workforce that can address these challenges, and the outline of the entire paper. The next section, Section 1 Concept of Transfer, deals with defining and characterizing transfer in relation to learning. The relevant theories that deal with transfer are discussed to show the theoretical foundations of transfer. Likewise, strategies are shown to guide the transfer of learning. The third part of the paper, Section 2 Transfer of Learning from School to the Workplace, contains discussions related to teaching for transfer and the instruments used in the workplace that are related to mathematics and education. The fourth part of the paper, Section 3 Role of Technology in Fostering Transfer, presents information regarding the role of technology in transfer, especially within the workplace. The section deals with the aspects of transfer where technology is involved to show the relevant relationship between the two. The last chapter, Section 4 Implication in Mathematics Education, shows the relevant implications of the concepts discussed in the previous chapters to the field of mathematics education. Likewise, it provides relevant suggestions on future research endeavours.

Chapter 2 Concept of Transfer
Transfer learning plays an important role for an individual to succeed in the workplace. According to Wills (1998), teaching and training, regardless of how good, are recognized as a waste of time if individuals will not be able to apply the things that they have learn in the workplace. He further asserted that teaching and training fail to have a substantial impact to the society if individuals cannot put into practice their skills, especially in the workplace. In this sense, it is important that transfer learning is given due attention and importance, especially in relation to the way by which individuals apply specific skills that they have learn in their performance in the workplace. In relation to this, the main topic of this current study deals with pedagogy for transfer of learning from school to the work place. Being the case, in order to better understand the idea of transfer learning from the school to the workplace, it is necessary that the transfer learning should be properly defined in order to have a deeper understanding of its meaning and its corresponding implications. In relation to this, the theoretical foundations that support the concept of transfer learning will also be given due attention to be able to see how different theories of education support the idea of transfer learning. Furthermore, the characteristics of transfer learning will also be discussed in order to identify when this process actually takes place.      

2.1 Definition of Transfer of Learning
Transfer as an issue deals with how knowledge acquired in one situation applies (or fails to apply) in other situations (Singley and Anderson 1989 1). Transfer is recognized as an important problem in the area of learning. A study about learning requires giving due considerations to the problems involved in transfer (Carraher and Schliemann 2009). Transfer of learning is defined as the ultimate aim of teaching (Cree and Macaulay 2000 1). Transfer of learning has been gaining popularity in the field of the academe, especially its relevance in a number of various fields such as psychology, philosophy, schooling, adult education, vocational, and professional education. In the United Kingdom, one of the main objectives of the professional qualification gives due emphasis on the concept of transfer learning among its graduates (Cree and Macaulay 2000). In line with this, the most common definition of the actual process of transfer learning means a prior learning affecting new learning or performance (Marini and Genereux 1995  2). Marini and Genereux (1995) further explains their definition of transfer learning in the context of the instructional setting, they emphasize that the concept of transfer includes three elements, learner, task, and context
The new learning or performance can differ from original learning in terms of the task involved (as when students apply what they have learned on practice problems to solving a new problem), and or the context involved (as when students apply their classroom learning to performing tasks at home or work). The basic elements involved in transfer are thus the learner, the instructional tasks (including learning materials and practice problems), the instructional context (the physical and social setting, including the instruction and support provided by the teacher, the behavior sic of other students, and the norms and expectations inherent in the setting, the transfer task, and the transfer context.

According to Cree and Macaulay (1998), the concept of transfer is not a new phenomenon, as the assumptions about the transfer of learning have long existed in the development of educational programs. One of the cases that is most relevant to transfer learning is recognized virtue of a classical education, which is grounded on the belief that learning the subjects of Latin or mathematics are requirements for the study of any other discipline because these aforementioned subjects train the mind of individuals and the developed skills are transferable (Thorndyke 1924).  

In the same manner, Aha, Molineaux, and Sukthankar (2009), supported the idea that transfer of learning entails applying previous knowledge and skills to solve new problems. They define transfer learning (TL) as the practice of recognizing and applying knowledge and skills learned from one or more previous tasks to more efficiently andor effectively learn to solve novel tasks (in new domains) (McGinty and Wilson 200930). The perspective of Aha et al. regarding transfer of learning emphasizes that this process is extremely useful not only to individuals but also to the organizations that they are working for. Transfer of learning can substantially decrease the number of trainingcases required to achieve a specified level of problem-solving competence in one domain by means of competently applying the knowledge gained from solving problems for various, but related, tasks (Aha et al 2009).

The definition of transfer learning by Aha et al. is also the same as the perspective of Cunningham, Dawes, and Bennett when it comes to this topic. Based on the work of Cunning et al. (2004), the simplest definition of transfer learning is the degree to which behavior will be repeated in a new situation (12). In this sense, transfer of learning is actually an important issue in the areas of education and training. Cunningham et al., argue that work-based learning often makes problem almost redundant. This kind of adage is observable in situation wherein an individual was able to learn how to solve a problem at work. However, this scenario does not involve transfer learning because he or he was able to learn precisely what he or she needs to know in order address that specific problem only. Knowledge learn in addressing problems in the office can be regarded as redundant because solving one problems at the workplace does not necessarily entail that it will be applicable in other problems in the workplace. Being the case, transfer of learning is most applicable in the field of education and training. In this scenario, individuals learn things on a course and apply the concept transfer of learning to real situations at the workplace (Cunningham et al. 2004).

In the studies of Detterman and Sternberg (1993), they discuss the issues of transfer in relation to the workplace. The research of Detterman and Sternberg suggests that in the United States of America, useful transfer of training to the workplace is as low as 10 percent. As a result, billions of dollars are wasted annually because of the model, which supports people to expose themselves to new information, ideas, and skills that are not use or applicable in the workplace. In addition, there are also instances wherein employees have to undergo further study and training because the knowledge and skills that they have are not enough to address the problems in the workplace. In the same manner, there are also some employees who actually lack the necessary capabilities, as they cannot properly transfer the knowledge and skills that they learn from school. The failure of properly implementing transfer of learning, especially in the academe, adversely affects individuals because they cannot satisfactorily perform in the workplace and organizations also lose resources from such incompetence (Cunningham et al. 2004).

Wills (1998) defines transfer of learning as the post-training application of the newly acquired knowledge and skills to improve the business (217). He emphasizes the importance of transfer learning in contributing to the success of business organizations. The aforementioned definition of Wills, does not really pertain to the acquisition of knowledge and skills only in the academe because he also accommodates the idea that individuals can also learn within the workplace through training and seminars. However, Wills main concern in the successful application of the acquired knowledge and skills of individuals towards the robustness of business processes and operations. In this sense, Wills is more focused on the outcome or benefit of transfer learning in the overall success of the business (Wills 1998).

The different definitions of transfer learning coming from various authors give emphasis on the different aspects of the topic. Transfer learning is defined as an issue, which deals with the relevance of the transfer of knowledge and skills from one situation to another. There are also definition of transfer learning that emphasizes this process in the context of education and training wherein it is deemed to be most important, as individuals should learn the necessary knowledge and skills to be used in the workplace during their time as students. Moreover, transfer learning is also perceived as a factor that should heighten the contribution of individuals to the workplace because the importance and success of transfer learning is only seen on the way by which it contributes to the robustness of the organization. The definitions of transfer learning, which are discussed, may give importance on different aspects of the topic but these definitions still have its commonality and that is highlighted in the idea that transfer learning should allow individuals to apply the knowledge and skills that they have acquired from the academe or in another source in the workplace. The real importance of transfer learning is most observable on the way by which individuals can be able to apply the ideas and competencies they learn into different situations.

2.2 Theoretical Foundations
Theories give the necessary foundation that guides the teaching and learning among individuals. Theories are regarded as a body of knowledge, which help in explaining and understanding different phenomena. In the case of transfer learning, it is necessary that theories related with this topic are applied in order to understand the way by which transfer of learning takes place within educational institutions and the workplace. In line with this, transfer theory will be discussed and other theoretical foundations that are relevant with the subject of transfer learning.

The Transfer theory emerged from the empiricist perspective, which recognized the learner as a passive agent whose learning is based on the similarity between a presumed original learning situation and the current situation. Similarly, the empiricist position is also illustrated in the hypothesis of identical elements of Thorndlike and Woodworth. Identical elements were regarded as objective, physical features that are constant to situations. In relation to this, Thorndlikes stance regarding the subject of mathematics education also coincides with his belief about identical elements. As a result, he suggests that students should be given mathematical problems that are similar with those they will possibly encounter in the workplace. The proposal of Thorndlike could be exemplified in the use of mathematical problems with Roman numerals wherein he argues that this kind of mathematical problem should not be given to students because it does not represent the problems that they need to solve when they are already in the workplace (Carraher and Scliemann 2009).

The transfer theory, especially in the perspective of Thorndike greatly embodies the behavioral learning theory because of the importance given to the concept of identical elements. Based on the perspective of behavioral theory, learning is identified as an observable change in the behavior of an individual. Behavioral theory is a very complex model of learning because it has many implications and relevance not only in education but also in other areas outside of the educational field (Redman 2007). One of the most important elements of behavioral theory is the transfer of learning. The term transfer learning refers to the application of something learned in some specific situation to a novel or new setting (Orlich et al. 200934). The perspective of behavioral theory when it comes to transfer learning is similar to the concept of identical elements because what is previously learned can actually be applied in other different situations. In relation to this, the behavioral theory gives a theoretical foundation for various approaches to teaching and curriculum construction. As such, the theory also has a huge influence in transfer learning, which can be exemplified by direct instruction. Studies have shown that direct instruction enable the transfer of skills across different learners and subject areas. The techniques used in direct instruction improve on-task learning time, thinking skills, problem solving, computer literacy, writing skills, and science learning (Orlich et al.  2009).      

On the other hand, other researchers do not give much emphasis on the idea of surface similarity but rather they focus on structural or conceptual similarities across different tasks. The major tenets of activity theory are used as a basis in order to show the emphasis given to structural or conceptual similarities. Activity theory is derived from Soviet psychology, which explains that mental processes and characteristics are conceived of as products of human activity with particular emphasis on the concept of age- and situation-specific dominant activity (Andrews, et al. 2001 457). Applying the activity theory entails that examiners should include activities which predominate within a specific age range and embodies everyday life situations in their activities. Being the case, activity theory emphasizes that transfer learning is possible and it can happen by focusing on learning as the dominant activity rather than merely giving importance to the results of learning (Andrews et al. 2001).    

The attention given on structural or concept similarities is also recognized as a major development beyond stimulus response psychology. Transfer learning in the structural perspective takes place when learners already have an understanding of the original problem. The structuralist approach was further developed in the 1940s, when Gestalt theorists further expanded the focus on understanding. In doing so, Gestalt theorists made a distinction between blind transfer and transfer, which is depended upon the structural characteristics of a specified task. The related experiments regarding the tenets of the Gestalt theory shows that original and target tasks still have its similarities. Usually, the new task only has few differences with the original task (Andrews at al. 2001).

Transfer learning further evolved during the 70s and 80s, when researchers gives emphasis on the rules involve in solving problems that are similar to puzzles. The experiments done in this area shows that the participants cannot solve problems that are totally dissimilar from the original problem that they have addressed despite the fact that the same logical structure should be used in the new problem. The examiners explain that the participants fail to solve the new task because they are not instructed to refer to the learned procedures that they have applied in the previous problem. In this sense, it is necessary that the logical structure of one problem is properly reinforced upon the participants in order for them to solve new tasks requires the same logical structure (Carraher and Schliemann 2009).

Aside from the activity theory and the behavioral theory, there are also other bodies of knowledge that do not directly support transfer learning but some of the major principles of these theories are actually related and can accommodate the concept of transfer learning. The constructivist, socio-culturalist, and situated cognition approaches have their respective tenets which can be related in order to further explore and support the transfer of learning.

Based on the theory of constructivism, individuals create a unique mental image by means of combining pre-existing information with the information that comes from his or her sense organs. The constructivist approach asserts that learning is an outcome of the process by which the learner matches new information against pre-existing information and integrating these together in order to produce meaningful connections. In the constructivist thinking, learners are provided with more latitude in order for the learner to become more efficient in problem solving, identifying and evaluating problems, and deciphering ways for them to transfer their previous learning to be able to address current problems (Federal Aviation Administration 2009).
In relation to this, the research study in the book of Reigeluth (1999), discusses the retention and transfer tests for a lesson on the formation of lightning.  The result of the study shows that retention and transfer learning is important in the success of the students who did well in the tests, which is also recognized as a constructivist learning outcome. The constructivist learning emphasizes that learners attempt to make sense of the presented information by means of also giving due importance to the previous learning that an individual has, which can be related to the present information that is given (Reigeluth 1999). Due to this, it is quite observable that the constructivist theory also supports and uses knowledge and skills which are learn before in order to make sense of the new information that is provided. In line with this, the learner also has to apply previous learning to be able to address a current situation or problem. Despite the fact that the constructivist approach believes that a learner construct the way by which he or she understand and addresses new learning and problem, the knowledge and skills that are learn before are also essential in this process, which makes transfer of learning pivotal in the constructivist theory.

The socio-cultural theory is initially recognized as part of the other end of the spectrum in terms of agreeing with the concept of transfer learning. Socio-cultural researchers argue that knowledge and skills cannot transfer because they are so strongly embedded in and tied to the context in which they are acquired (Murphy and McCormick 2008). However, the contemporary perspective about transfer of learning has made it possible for the socio-cultural theory to become more accommodating on the idea of transfer learning. Current socio-cultural researchers have created various models that integrate socio-cultural issues with learning transfer. The socio-technical learning transfer model by Analoui highlights the importance of technical and social competence skills, which are necessary to the efficiency of transfer (Koster 2007). In the same manner, Lim and Wentlings international model of training is regarded as a unique model because it is one of the first approaches that gives due consideration to cultural factors when it comes to transfer learning, especially in its impact in the workplace. Due to the aforementioned models, socio-cultural contextual approach to transfer process was developed (Leberman et al. 2006).  

The situated cognition theory is also similar with the socio-cultural theory in a sense that the first proponents of this theory also do not support the concept of transfer learning. However, developments in the situated cognition approach allow other contemporary researchers to see the relevance of transfer learning with the situated cognition theory. One of the most significant contributors of contemporary ideas in the situated cognition theory is Geoffrey Saxe. Saxe shows exemplary optimism to transfer education as compared with the other theorists before him. Saxe used the candy sellers in Brazil in order to show the relevance of transfer education to the theory he is studying (Haskell 2001). The candy sellers use school algorithms within their calculations when they are interacting with their customers. In addition, he also points out that the sellers are also using the same knowledge and skills in selling candy in an attempt to solve school problems, which is the reason that they perform better in school as compared with non-sellers. The work of Saxe suggests that efficient and successful transfer is not likely to take place but nonetheless, it is possible to happen through a long process of continuous attempts to construction. Moreover, he also highlights that it is important for individuals to have a depth of understanding of their previous knowledge (Orthon 2004).

Transfer of education is not merely a process or operation in the field of education that tends to exists without any basis because the concept itself is grounded on different theories. The activity and behavioral theories clearly identifies the importance of transfer learning as a phenomenon that actually exists among individuals. Despite the fact that there are theories that initially contradict or do not directly support the idea of transfer learning, the development among these different approaches has also started to accommodate that transfer education is indeed possible. The theories of constructivism, socio-culturalism, and situated cognition also have its respective principles that supports transfer education, which further heightens the importance of transfer learning in the society.

2.3 Characteristics of Transfer
The concept of transfer of learning is difficult to understand without actually having the proper knowledge in order to identify whether transfer is actually taking place or not. The characteristics that describe transfer of education should be given due attention and importance because this will help in identifying the way by which transfer of learning actually happen in real life situations. In line with this, the characteristics of transfer will be properly discussed by taking into account the different types of transfer and its respective descriptions. Moreover, understanding the characteristics of transfer will aid in objectively assessing whether transfer should be taught or not.

Describing the different types of transfer substantially helps in illustrating the historical development of transfer and also the breadth and scope of the transfer issue. In relation to this, the types of transfer also describe the characteristics of transfer. The different types of transfer are lateral and vertical transfer, specific and nonspecific transfer, near and far transfer, and literal and figural transfer.

The differences between lateral and vertical transfer were identified by Gagne. Vertical transfer takes place when a skill or knowledge unit learned in one situation has a huge influence in acquiring more complex skill or knowledge unit in another situation at a later time. Gagne believes that simpler skills are a prerequisite of learning more complex skill. As such, instruction or teaching should be designed in such a way that it will take advantage of the hierarchical structure of information (Mestre 2005). On the other hand, lateral transfer was less described by Gagne. Lateral transfer is defined as a kind of generalization that spreads over a broad set of situations at roughly the same level of complexity (Mestre 2005 ix). The characteristics of lateral transfer is observable in situations wherein a child would learn the similarity between fractions and decimals, which is also the same with letter wherein it is still the same even though its appearance might change through a change of font (Mestre 2005).

The specific transfer describes situations wherein there is a well-defined similarity between the stimulus complex which is dealt with in one situation and the stimulus complex that is encountered in another situation. The idea of specific transfer is exemplified in the Hoffdings representation, which illustrates the concept of transfer. On the contrary, nonspecific transfer pertains to situations wherein there is no clear or obvious relationship between the stimulus properties of two learning events but acquiring one inevitably influences the acquisition of the other (Butterfield and Nelson 1991). The most common example of the nonspecific transfer is found in the behavioral learning era, which embodies the principle of learning to learn, which illustrates that one set of learning events could be shown in order to help in facilitating another set of learning events. Even though there are no obvious stimulus similarities between the two events, one event still substantially affect the other (Haskell 2001).

In the discussion of the specific and nonspecific transfer, the similarities and differences of stimulus properties are given due importance. In order to properly explain the distinction of the stimulus properties, some researchers decided to pertain to transfer events as either near of far transfer. Near transfer entails situations that involve a great deal of similarity between the conditions of original learning and transfer learning. The difference of far transfer is that there is only a limited similarity between two events (Butterfield and Nelson 1991). The usual way in order to point out the difference of near and far transfer is through the discussion of school-learned events and out-of-school events. School-learned events are recognized to be an example of near transfer because the conditions involved are present and the same for both events. However, applying the knowledge and skills acquired in school-learned events to out-of-school problems exemplify far transfer because the stimulus complexes are different in the two situations (Haskell 2001).

The distinction between literal and figural transfer is presented by Royer. Literal transfer includes the implementation of intact unit of knowledge in a new learning situation. An example of the literal transfer is the skill of calculating an area of the rectangle, which can also be applied in determining the size of a rug. On the other hand, there are also events that do not involve the transfer of intact skill because it represents the implementation of segmented world knowledge, as a means in order to think or learn about new problem or issue. According to Royer, the most obvious example figural transfer can be seen through figurative language such as metaphor and simile. In addition, activities that use analogy also characterize figural transfer because in order to obtain knowledge of one topic it is necessary to understand and learn about another topic (Mestre 2005).      

The different types of transfer show the various ways by which transfer of learning can actually occur in different situations. The description of each type of transfer constitute to the characteristics of transfer of learning. Knowing the respective characteristics of the types of transfer also gives the opportunity to properly identify the transfer of learning process and the various ways and situations by which knowledge and skills can be transferred to the workplace or in other field of context.

The previous discussions about transfer learning, especially its definition, theoretical foundation, and different types of transfer clearly show the importance of teaching transfer. Transfer learning should be given due attention and importance by educators and other stakeholders, especially in educational institutions. Most individuals who go to school have the primary motivation of acquiring a reputable and high-paying job once they graduated (Borck and Fawcett 1982). In order to assure students that they will indeed become productive citizens, who will be able to have good and successful careers after graduation, it is necessary to teach transfer learning. In addition, the knowledge and skills that are learned by individuals will only be really useful if they can apply it in different situations in order to address problems or learn other ideas and expertise. Moreover, as previously mentioned, the knowledge and skills learned by individuals will only be useful if it is applied and can actually contribute to the success of an organization and the learner as well (Wills 1998).

Chapter 3 Transfer of Learning from School to the Workplace
In Chapter 2 of the paper, the discussion revolves around the strategies used for teaching transfer, the type of skills and knowledge that are transferable from the school to the workplace, the instructional and learning strategies that promote or hinder transfer, and the possible application of mathematics in the workplace. The discussions are divided into the previously mentioned sections to facilitate a better understanding of the transfer of learning from the school to the workplace. In addition, the discussion explains the important theoretical foundations related to the transfer of learning between the institutions of school and workplace.

3.1 Teaching for Transfer
The term transfer is defined in the educational setting to be the process that occurs when an item learned in one context is applied and used in another context (Brown  Baird 1999 22). Teachers hold the notion that a particular area of knowledge is used in another but the process rarely happens and findings show that materials of instruction within the level of higher education is limited to certain institutions (Brown  Baird 1999). Moreover, the concept of transfer is usually brought about, to a greater degree, by the environment than by the conscious efforts of the learner (Sousa 2006). Within the educational setting, a greater part of the environment is attributed to the teachers, which makes this group the cause for transfer to occur (Sousa 2006). In other words, teachers are the agents of transfer within the classroom setting, which gives the responsibility of initiating strategies to enable the students to transfer the concepts learned from one setting to another.

Two types of transfer are enumerated in Brown and Baird (1999), which are near transfer and far transfer. The first of the two types occur when the context of use have similarities or are close in terms of characteristics, such as the use of ideas learned in two related subjects (Brown  Baird 1999). On the other hand, the second of the two types requires the ability to connect two types of subjects or contexts that are not related andor have fewer similarities with one another (Brown  Baird 1999). For example, the far transfer occurs when a learner is asked to apply a concept learned in mathematics to the concept of social work.

In order to enable the transfer of learning, Brown and Baird (1999) suggests that teachers should adopt certain steps that should be integrated into their teaching. The steps include a) building the learners self worth, b) creating motivation, c) using appropriate teaching methods, and d) giving the learner responsibility (Brown  Baird 1999 22). All of the steps involve inputs and efforts from the teachers. The first and second step requires the teachers to foster an environment that allows the learners to realize and improve the worth that they perceive of their selves together with the right amount of motivation. The third step involves the preparation and implementation of teaching strategies that are meant for transfer of learning. The strategies employed in the third step are comprised of the materials, approach, and other physical and non-physical resources used by the teachers to convey the knowledge, skills, and abilities that the students should have to enable them to transfer learning from one context to another. The last step concerns the responsibility that is passed from the teacher to the learner. In this step, the teacher can use various forms or techniques that teach the students the concept of responsibility and allows them to exercise such.

Another process is offered by Tileston and Darling (2009), which is a basic layout of the responsibilities of the teacher when it comes to teaching for transfer. In the process, teachers are offered three responsibilities, which should be accomplished together with the learner. First, the teacher should acquaint the learner with the opportunity for identifying past experiences related to learning that could aid in the present course of learning (Tileston  Darling 2009). Second, the teacher should facilitate the impact of the previous learning with the process involved in the current learning (Tileston  Darling 2009). Third, the teacher should direct the students towards enumerating the instances of future learning where the information gained in the current learning can be used (Tileston  Darling 2009). In all of the three steps listed, it is noticed that the teacher remains as a facilitator who ensures that the environment is enabling for the students to engage in the transfer of learning. Likewise, teachers are treated as mere agents for transfer to occur and leave most of the efforts to the students. By so doing, the students are able to carry out the activities and experience the transfer of learning according to personal experiences.

Aside from the processes, Cornford (2010) discussed two models of promoting transfer that guides the students and trainers in instructing learners about the practical applications of the theories learned inside the classroom. The two models that are compared and analyzed by Cornford (2010) are the Yelons MASS Model and the Cornfords Sequential Skill Practice Model. The Yelons MASS Model, with the MASS being an acronym, is an outcome of different theories combined together and is largely influenced by the field of cognitive psychology (Cornford 2010). The acronym MASS is known to refer to motivation, awareness, skill, and support (Cornford 2010). The four elements are considered to be the chief elements that guide the process of transfer that should bind instruction and training. Moreover, each of the four components has respective roles to play and possess certain characteristics that define their contribution on the transfer of learning. First, motivation should be present before, during, and after the instruction for the learners to have a better interest in the activities and topics discussed. The level of motivation present during the training is the responsibility of the trainer or the teacher. To facilitate in the identification of the learners level of motivation, Yelon offered a set of questions that the teacher or trainer should answer, together with the learner, prior to the activity. However, Yelon also recognizes the intrinsic forces within the organization that could affect the learners level of motivation. Second, awareness should also be present among the learners and this constitutes the level of understanding that the learners have regarding the areas where application of the gained knowledge is appropriate. Yelon offered two types of questions that should be asked to the learners and the teachers or trainer. Third, the element of skills focuses on the trainers and teachers and recognizes the need for cognitive skills that paves the way for effective learning. The teachers and trainers are borne with the responsibility to identify and use the necessary skills that aids in the learning processes, which also depends on the needs of the situation. Fourth, support should also be present within the work environment. Support is identified to be the

The second model in the article of Cornford (2010 93), Cornfords Sequential Skill Practice Model, is geared towards gaining the transfer of teaching skills from the micro-teaching setting to the natural classroom setting with the involvement of trainee, vocational education teachers. The model is specifically applicable for near transfer and is supported by empirical evidence from two different research (Cornford 2010). Moreover, the principles followed by the model are taken from the skill-learning and social-cognitive, which requires the acquisition of skills and principles and the presence of responsibility for identifying the appropriate instances where previous knowledge can be applied. The model rests on the assumption that the learners are active participants in the process and the teachers and trainers serve as facilitators and guide during the process (Cornford 2010).

A variety of models and processes are available for the teachers, trainers, and learners, as shown in the previous discussions. In line with the various suggestions on the transfer of teaching, McKeachie (1987) argued that the variety of models and processes suggested by different scholars are made available to the public but the implementation proves to be difficult. From the authors personal standpoint, McKeachie (1987) suggested that self-monitoring skills are essential factors that could have a possible contribute to the facilitation of transfer. Scholars, such as Pea and De Corte and Verschaffel, have shown that it is successfully possible to implement self-monitoring among learners and are important determinants for transfer (McKeachie 1987).

The significance of teaching for transfer is shown through its impact on student achievement. According to Darling (1999), student achievement is increased by 37 per cent when teaching for transfer is introduced. The increase in student achievement is attributed to the impact of the process to the students ability to expand knowledge and critical thinking, create new methods of thinking, and integrate learning as a personally relevant process (Tileston  Darlin 2009).

The training received by students regarding transfer proves to be important in both the academic and professional realms. As mentioned earlier, teaching for transfer allows the students to gain a considerable percentage of increase in terms of academic achievement. Within the professional setting, the same students are able to relate the past, present, and future learning in an easier manner compared to those who are not able to undergo instructions on transfer of knowledge. Aside from time-related transfer, which occurs from the past to the future, transfer among two different settings are also important for students where an ability to transmit knowledge and experiences gained within the school is easily applied to the workplace. When the successful transfer of knowledge is achieved from the school to the workplace, the learner applies the relevant training received from the school and translates it into actions. Likewise, the same can be done when the learner moves from one workplace to another or when heshe returns to the university to gain further knowledge on a particular field of study.

3.2. Mathematics in the Workplace
 Mathematics, from the perspectives of the students, resides within the four walls of the classroom and is often referred to as school mathematics. On the other hand, mathematics that is applied and experienced in the workplace is referred to as workplace mathematics. The former is considered to be the foundation of the individuals knowledge regarding mathematics and the latter serves as the one of the purposes for the need of such type of knowledge.

Mathematics plays a significant role in the workplace and continues to be a factor of increasing importance in the workplace (Hoyles, Wolf, Molyneux-Hodgson,  Kent 2002). The purpose of mathematics in the different industries also varies because of the diversity of processes involved. A general summary of the roles that mathematics takes in the practical setting is provided in the AAMT report, which is cited by Fitzsimons (2002) and includes a) Serves as an elucidatory factor for the outcomes of a particular activity or endeavour and provides a guide for the decisions needed to achieve the same b) Identifies the instances or places that can be seen through the lens of mathematics that is followed by the identification and selection of the concepts and strategies to be employed c) Applies the needed mathematical concepts and strategies with certain modifications in order to meet the demands of a particular instance d) Guides the decision-making process in terms of the level of accuracy called for in a particular situation e) Deciphers the results and assesses the methods employed. In the summary, the role of mathematics lies mainly on the decision-making process and interpretation of results. Mathematics aids in the understanding of certain concepts from the qualitative perspective and further supports the choices that can be made with regard to the mathematical concepts and strategies that can be utilized for the other steps involved. The quality of being general is apparent because the only purpose of the summary is to show the relative importance of mathematics and to prove that it is indeed present within organizations. Likewise, it is to be noted that not all of the characteristics mentioned above are simultaneously present in the organization.

Orton (2004) noted that the link between workplace mathematics and school mathematics is broken but still placed emphasis on the development of advanced mathematical ideas within the workplace. In fact, the advanced mathematical ideas are transformed into practice and serves as the foundation for knowledge used in the workplace (Orton 2004). Likewise, Kent, Noss, Guile, Hoyles, and Bakker (2010) noted that most of the processes involved in the workplace are found in school mathematics but confounding variables make it difficult for the employees to apply the theoretical concepts into the workplace. More often than not, the context of use makes it difficult for the employees to determine the appropriate mathematical concept or tool to be used, which emphasises the need for a model-based approach to mathematics (Kent et al., 2010).
A large part of mathematics is observed to lie within technology-related tools (Orton 2004). The field of mathematics education serve as a critical element for the formulation and improvement in technology (Clarke 2003). Some of the simple mathematical tools used in the workplace are the graphs, tables, simulation, modelling, and calculators to name a few. In line with this, the burden of providing training for individuals who can handle the mathematised and technologised workplaces still remains within training and school mathematics (Orton 2004).

3.2.1. Spreadsheets and Tables
In terms of characteristics, spreadsheets and tables are essentially the same with particular differences on the use and degree of complexity. According to Parsons and Oja (2010 130), a spreadsheet is a tool that uses rows and columns of numbers to create a model or representation of a real situation. Similarly, tables are defined in Microsoft Encarta (2009) as the assembly of data placed under columns or rows that serves as a summary or shorter version of a particular set of information. A variety of tools can be used to create tables and spreadsheets and the most common can be found under the Microsoft Office package.

The spreadsheets and tables, while it can be used to contain qualitative data, are used to create an organized system of quantitative information to aid in the decision-making processes within an organization. In the article of Williams and Wake (2006), the use of a spreadsheet is discussed as the tool is used in order to aid the managers in making relevant decisions and in forecasting the input requirements for a particular gas day. In the situation offered, the managers are not necessarily equipped with the needed mathematical skills to run the spreadsheet because another employee is hired to complete the needed computations (Williams and Wake 2006). However, in most of the organizations, the managers should be able to understand the implications of certain mathematical concepts in order to understand the information offered by other employees and arrive at the appropriate decisions. Other uses of the spreadsheet are also noted, such as the tasks related to the financial documents of the company. For example, it can aid in the computation of the assets, liabilities, and equities for the entire company especially when large amounts are involved. Likewise, the information contained in the financial spreadsheet can be used to make relevant decisions in the human resources department or in cost-cutting purposes. In the case of the spreadsheet, the complexity of the tool, as shown in the article, can be a determinant for the ability of employees to deal with work-related situations and the capability to make sound decisions. Basically, the operation and interpretation of the information from its raw form depends greatly on the mathematical capabilities of the employees who directly and indirectly work with the spreadsheet.

On the other hand, the tables are relatively simple and shorter tools because of the need to serve as effective summaries of a greater set of information. Tables are popular because of the ease in entering and interpreting the data. Oftentimes, the data placed in tables are written and prepared before it is placed in the rows and columns and it is also a common practice to create a framework of the information by placing it in a table form prior to analysis or report. The interpretation of data found in tables, especially for quantitative forms of data, require knowledge in mathematics to understand the meaning of the numbers. Without the needed skills and knowledge, the information found in the tables is nothing but mere numbers to an employee. For example, tables can be used to summarize the monthly turnover rate in a particular department versus others. The information can be displayed in terms of percentage or raw numerical form. An employee, when presented with such kind of information, can interpret it by looking at the percentage of turnover in the department being studied and compare it by listing the departments that have higher or lower turnover rates. The manner of interpreting the data varies greatly on the purpose of the decisions to be made, which should be known to the employee prior to the analysis of information. The information contained in the tables can also be presented in a graphical manner, such as graphs.

3.2.2. Graphs
Graphs can be found in varying forms, complexity, and purpose. The most simple is the form used in the schools where students are asked to show the relationships among two or more quantified variables by drawing lines or plots along two different axes, which form right angles and has four quadrants. Different functions within the workplace are associated with the use of graphs and vary greatly depending on the needs of the organization. Likewise, the use of the graph also differs based on the complexity of the data and the purpose. For example, graphs can be used to visually represent the increasing trends of electricity expenses in a particular company. On a more complex scale, graphs can be used to forecast the cost-benefit ratio of a particular project and aid in the decisions of the organization. The essential tools needed in the construction and interpretation of the graph is the understanding of the data found in the source and to construct it in a graph according to the purpose. A variety of software products are available in the market to aid in the creation and interpretation of graphs, which also contain several functions depending on the needs of the user. However, the employees should still have the needed mathematical KSAs in order to determine the appropriateness of the graphs and to interpret the results based on the needed information. The significance of math when dealing with graphs is highlighted by the fact that the process requires the understanding the relationship between variables that are often used in the X and Y axes. The information obtained from the relationships of variables shown in a graph can sometimes be aided with the use of simulation to make it interactive.

3.2.3. Simulation
Simulation, in general terms, is defined as the process of imitating a certain phenomenon (outward appearance or behaviour) by using another device (Ohnari 1998 1). In mathematical simulations, numbers are introduced and showed outcomes based on the input of the users. In most business, the use of mathematical simulations is significant in showing results where certain variables are manipulated or controlled. The manipulation and control of variables enables the users to have an experience of the outcomes that can result using certain strategies. To a certain extent, it saves the organizations from waste of resources and failures, especially for situations when great risk is involved. For example, the company wants to know the impact of increasing the benefits of employees based on the production levels on a monthly basis. The simulation of the increase requires the employees to change the values for every variable involved and record the findings. The results of the simulation from the example given can be used to make decisions related to the percentage of increase given to the employees depending on the profits expected by the organization. Other uses of simulation are observed from different organizations and it depends greatly on the needs and capabilities of the employees. Likewise, familiarity with mathematical concepts and strategies is necessary because simulations still require an understanding of certain functions involved in the process. Likewise, the input of data involves mathematical theories and principles that are used in the process of simulation.

3.2.4. Summary
The tools discussed above are some of the commonly used instruments in the workplace. The primary function of these is to aid in the decision-making process and to enable the organizations to gain efficiency and effectiveness in doing certain tasks. A variety of software products are produced in order to aid in the production and interpretation of the instruments, which is a response to the increasing demand for technology within the workplace. It is also noticed that all of the tools above require the use of skills related to mathematics and technology in order to accomplish the required tasks. The presence of mathematics and technology in the use of the instruments requires the employees to gain additional training and skills on these fields in order to gain competitive advantage and capabilities to address the pressing demands of the organizations.

Chapter 4 Role of technology in fostering transfer
Currently, the world is linked to technology that every aspect of peoples lives requires technology. In the daily lives of individuals it is impossible that technology is not present. Different materials that people utilize require technology in order to create higher quality for various products available in the market. Through the use of technology, every action that needs a great amount of time is lessened and is immediately created. Moreover, productivity is the most important aspect in which technology upheld thus, making every action faster and easier for everyone.

4.1 Mathematics in the Work Place
In the workplace, different changes already present. Filing of different materials is done within data bases to attain faster search and organization. Moreover, different kinds of files could be easily checked if needed. Thus, making it faster for different organizations to create updates are well as changes within their systems. Therefore, technology is direly needed to provide the best services possible. As a quote stated Time is gold. No person could revert time and every minute counts in the productivity of the company.

In addition to this, the aspect of technology has greatly provided impact on different industries. Various types of activities are process through the use of data bases which allows processing and filing at the same time. Hence, the technology allows the concept of hitting two birds with one stone. Thus this process shall make it faster to look upon the errors which had been committed by the person in charge.  Further more, the impact of technology could not be simply changed. Technology is already a way of life of many people as well as the industries at hand.

4.2 Academic Curricula and Development in the WorkplaceIn the years which had passed, changes had occurred. Priorities had been changed due to the needs of the society. Hence, in order to comply with the changes in the society, it is very important that the academic community brings forth a change in the education of students. In order to compete with other students, it is essential that professors bring in a perspective of technological advancements in order for students to learn their subject as well as relate to the changes which are currently implemented in the real world.  Hence, it is important for professors together with the school to create a link with the school as well as the work place to provide a stage of preparation in the future.

Different schools promote different programs to furnish the capabilities of students in the workplace. There are various programs that promote the interests of students to understand the programs in the real world. In the grade school level, there are computer subjects that are necessary for students to take. Given that most students have personal computers in their homes, it is only logical that computer subjects provide a much technical aspect of computers. Moreover, the level of learning of a student is developing through time. The high school level is much advanced in a sense that the learning is focused on different kinds of software that would help students in their course as well as through their selected careers in the future. In the book written by Thomas Friedman (2004) The World is Flat, he highlighted the importance of creating an academic program which will provide the proper learning for different students. Due to the changes which had occurred and the fast paced development which is currently experienced it is a need for every student to be flexible.  Flexibility is attained through the ability of various students to have knowledge in different kinds of subjects. It is very evident that trends shift every now and then. Hence, the needs of the industry rise and fall depending on its needs. Therefore, students must attain different kinds of knowledge to be successful in their careers.

4.3 Role of Technology
Hence, the role of technology is important most especially in education. Provided that technological advancements are a necessity in the work place, it is very significant that students must have a training of what they will do in their future work. In the part of an institution, it is very significant that they obtain different kinds of individuals who will provide important skills to have a much productive team. Therefore, individuals that are willing to work in the company must have different kinds of attributes that will entail the demands of the company. Furthermore, an individual must have various kinds of abilities that would let him be competitive with other interested applicants.

4.4 The Relationship of the Work place and Outsiders
In relation to this, it is important for the academic society to manifest the needs of the workplace. It is important that the academics of the students shall provide training that mirrors the possible work for students. In the article written by Julian Williams and Geoff Wake (2007), the authors presented that there is a need for a communication between workers as well as the outsiders. The outsiders that are being pertained are teachers, students and researchers. In order to create a much responsive and perfect explanation of the situation, there are different models to which addresses the importance of work and pedagogical dialogues. Thus, the authors include the metaphors utilized by different authors in order for the manifestation to be present. Hence, on another note, it is important that the relationship of the academic standards as well as the work standards to assume a responsive type of relationship with one another. As the academic requirements are provided, it must response to the need of the work place. As mentioned by Voyager, the academic standards hone the person to be capable of the work in the real world. Furthermore, during the 1900s it was stated that, the secretary of labor and Secretarys Commission on Achieving Necessary Skills (SCANS) identified workplace learning and achievement competence in basic skills, thinking skills, personal qualities, and the ability to productively use resources, interpersonal skills, information, systems and technology (Roulis, 2004, p. 20). The provided attitudes and skills are the generic skills which should be attained in order to comply with the requirements of the workplace.  Although these are all basic needs of workers, it was highlighted that the importance of the basic skills are significant for most work places.

In the article written by Wake and Geoff, they have related the written of Ponzi in their study. Ponzi had stated that there are problematic aspect of the workplace and the outsiders in terms of understanding of mathematics. Again, the article presents the importance of the workplace and the academic curricula in the subject of Mathematics.

In the course of time, it is very significant that different actions of the academic society shall respond to the need of the people in the work place. Furthermore, the application of technology had been one of the most important aspects of mathematics. As presented by the authors, there are various examples where in situations are given. The authors gave the examples of careers that show the need of mathematical computations. In the case of the engineer Dan who was the main example of the authors, it is very important to pay attention with the fact that the authors stated that the college mathematics is very different fro the mathematics in the work place. Given that the work place and its mathematical computations are based on a real situation. In another view point, College mathematics is similar to generic situations wherein the situations are highly controlled and given. As the term used by the authors, the work place is similar to a black box. The boxes provide the best answer to the situation however, the content is very unpredictable. Thus, it is not certain whether the box contains positive or negative aspects of the situation.

In addition to this, the concerns regarding mathematical language is heavily dependent on the experience of the person. There are acknowledged forms of language in college and later one these terminologies are integrated to the type of work which is done by the individuals.  Moreover, there are different types of adaptations that are causes a differentiation of language. Some workplaces simply adapt their own terminologies in order to make a situation much personal rather than professional. There are various manifestations of these language is very evident through the aspect of culture and realization of the need to have a language that would define the actions done by a certain group. Hence, there is a concept of adaptation of the culture as well as the mixture of the experiences of different people within the whole group.
In addition to this, different cultural manifestations are often held responsible to the changes of language in the work place. Although the view of the workplace is seen and understood by the society at hand, the execution of an experienced employee and a student is highly different from one another.

4.5  Academics, Workplace and Technology
The technological transfer had been one of the challenges present in the academic society as well as the work place. In order for students to be much competitive, there are variety of software and mathematical computations that are needed to be utilized.  Moreover, it is very important to realize that the academics are the basis of the work place. Although there are instances to which the workplace is requiring a higher demand. it is the academic society which forms a significant action towards the situation in the work place. Hence, through technological advancements such as the creation of various software a college student could assume that he or she is able to compete with different professionals within the industry.

4.6 Summary
In the work place it is very significant that changes that are present in the society. Technology is a huge factor in the mathematical perspectives and is required to develop a much better perspective. Hence, this section shows that the importance of technology in the work place is very relevant to the development of different industries.

Chapter 5 Implications in Mathematics Education
The implications of transfer learning to mathematics education can only be properly identified by taking into consideration two important contexts namely, pedagogy and workplace. The effects of transfer learning should be assessed in the field of education, especially in the way by means teachers or educators will be able to apply the concept of transfer. In the same manner, the effects of transfer of learning in the workplace should also be given utmost attention, especially on the way individuals apply mathematical knowledge and skills that they have previously learned in their respective jobs. Furthermore, it is also necessary to identify the developments in transfer learning, specifically the contemporary methodologies and techniques that are being used in order to enhance transfer in mathematical education.

5.1 Pedagogy
Pedagogy or the study of becoming a teacher is one of the most important context that transfer of learning must be related to in order to clearly see its implications in mathematic education. The role of educators in teaching learners of the necessary math skills are greatly affected by also applying the transfer of education. In this sense, the different types of transfer are being used by educators in order to apply the concept of transfer in teaching mathematical educations.

Educators who apply transfer to mathematical education greatly follow the various educational theories that advocates transfer learning such as activity theory and behavioral theory. In relation to this, the concepts of Putnam substantially influence the way mathematics is being taught together with transfer learning. According to Putnam, it is necessary to use various mental images or representations of concepts in teaching mathematics. He suggests that children should experience different mathematical concepts that are properly represented in order for them to have a strong foundation of the mathematical skills that they have learned. In addition to this, Putnam also emphasizes that students should practice doing mathematics and talk about mathematical procedures and concepts (McKeachie 1987 709). Educators should realize that the different lessons about math that they are teaching their students whether it is about common elements, verbalization of meaningful understanding, analogical reasoning or other approaches, teachers still have to focus on the original learning of the student in order to ensure that the students understand the current or later lessons (McKeachie 1987).

Mathematics is often regarded as a subject that requires a strong foundation because students have to properly understand the basic concepts and skills in math in order to deal with the other more advance level of mathematical education. A student needs to master her addition and subtraction before she can perform multiplication and division because the principles of the former are necessary in understanding the latter. Being the case, educators have to make sure that there is indeed a transfer of learning that exists in teaching mathematics education, as it would be difficult for the student to properly comprehend and apply mathematical knowledge and skills (Carraher and Schliemann 2009).

The importance of transfer in mathematical education is not only observed in the transition of students from one level of mathematical competence to the next but rather it also involves real-life application. Mathematics is one of the most practical subjects that are being taught in different educational institutions because it can be used in different day-to-day activities. Math is used when an individual buys something in the mall, especially when he or she has to make sure that the right change is given. Math is also applied in the construction of houses, buildings, and other infrastructures. The practical application of mathematical knowledge and skills are endless, which is why it is necessary that transfer learning is taught in relation to mathematical education (Rektorys 1994). In line with this, Putnam highlights that learning mathematics should entail transfer learning to more advanced mathematics and also to different practical problems in the everyday life of individuals (McKeachie 1987).

Nevertheless, the transfer of learning has its respective problems, especially when it comes to the failure of students to recognize the transferability of their learning in events that are completely different from the original one. In addition, there are also individuals who fail to transfer their prior learning to formal education. In order for people, especially students to recognize the knowledge and processes that are needed in a certain dynamic situation, the interaction of several variables are necessary (McKeachie 1987). One of the most important variables is the educators who are responsible in teaching the students the necessary mathematical knowledge and skills. However, teachers should also be mindful that transfer of learning also takes place because this is necessary in order to ensure the continuity of the learning of the students and also, the practical application of the things learned. In doing so, another important variable must also be given due emphasis and those are the techniques, methodologies, tools, which are necessary in order to aid teachers that transfer of learning indeed takes place (Coben et al. 2000).      

In this sense, it is clear that educators indeed have a pivotal role in the success and efficiency of transfer learning. Teachers have a huge influence in making sure that transfer of learning actually takes place as well as the corresponding outcomes related to the transfer of mathematical knowledge and skills of students. Being the case, the field of pedagogy should give more attention to the concept of transfer learning. Further studies should be made in order to improve the way transfer learning can be further integrated in educational institutions. Educators should also be given the necessary knowledge and skills in order to efficiently teach transfer learning among their students.

5.2 Workplace
One of the primary motivating factors of almost any individual who go to school is to be able to get a reputable and high-paying job once they have finished their studies. Students believe that successful careers entail going to school and having a diploma, which will increase their chances in finding their dream job (Combs and Canfield 2007). The aforementioned adage is indeed true but aside from the thought of merely graduating from an educational institution, the knowledge and skills that the students learn from school also play an important role in order for him or her to get a job and actually make a successful career. As previously mentioned, things that can be learn in mathematical education are actually very practical because there is a wide range of areas wherein mathematics is applied in everyday life and this includes the workplace. Almost all fields of profession use mathematical knowledge and skills one way or another, which make transfer learning in mathematical education even more important (Kent et al. 2007).

According to the report of Hoyles, Wolf, Molyneaux-Hodgson and Kent (2002), who are among the reputable and credible personalities from the University of London, a key finding in their research study shows that mathematical literacy has a huge influence in the competitiveness and success of business organizations. The findings of the study stated that mathematical literacy is interdependent to technological based world-wide economy that exists nowadays. Mathematical literacy is needed for employees who work in the Information Technology (IT) industry. Furthermore, the results of the study also have an implication in adaptation of a more systematic approach to business success, which requires properly appreciating and compensating the pivotal role played by employees who have mathematical knowledge and skills (Kent et al. 2007).

The importance of mathematical skills in the workplace is changing nowadays. There is an increasing number of people who are involved in jobs that are requires mathematical skills. The presence of industries that are involved in further technological advancement like information technology and the likes also requires employees who have increasingly sophisticated mathematical skills. Being the case, there is a need for people, especially government leaders to rethink the level of mathematical education that they have and also, to upgrade the mathematics provision for young people and to make sure that they have equal access to additional provisions over their lifetimes (Hoyles et al. 2002).

The increasing importance of sophisticated mathematical knowledge and skills only strengthens the need for a more efficient implementation of transfer learning. Government officials should make the necessary changes and introduce the necessary policies and programs in order to make people more competent and competitive when it comes to mathematical education. In line with this, business organizations should also do their part in order to further improve the mathematical knowledge and skills of their employees by means of conducting training programs. The competence of employees with their mathematical literacy would surely benefit the organization and the employees as well.

 There have been efforts to further improve mathematical literacy among employees, by means of conducting different research studies that would help in designing the appropriate teaching strategy or tool, which will help improve the mathematical knowledge and skills of employees. In the study conducted by Pozzi, Noss, and Hoyles (1998), they examine the structuring of resources and the influence that it has on shaping mathematical activity. The research was conducted in order to gain a better understanding of mathematical epistemologies in the workplace. The researchers analyze two nursing activities, specifically drug administration and fluid balance monitoring. In the first activity, the researchers observed that the nurses conceptualize ratio and proportion problems. In the second activity, the nurses discuss conflicting strategies when monitoring the fluid balance of a patient. The activities of the nurses help the researchers to conclude that available resources give mathematical and professional meanings, which are used by the nurses. In line with this, the researchers were able to develop a design of viable didactical strategies in vocational mathematics teaching (Pozzi et al. 1998).

5.3 Recommendations
Transfer learning has indeed come a long way from merely a topic of theoretical study to actually a recognized process, which is being implemented nowadays. However, transfer of learning can still be substantially improved, especially in relation to mathematical education by means of conducting further studies to improve transfer in educational institutions and in the workplace. The important areas that must be given due attention involves different aspects of transfer learning.  First, visibility and awareness of the importance of mathematical literacy in the workplace should be increased. People should be well-informed that mathematical literacy used in the workplace has its benefits and that mathematical skills are also needed in jobs that are not obviously mathematical. Second, models for training and professional development should be created in order to heighten mathematical literacy. Different studies should be made in order to explore the means by which employers can identify the exact role of mathematical literacy in their workplace and also to create ways in order to develop mathematical literacy in different sectors in the society. Third, core concepts should be properly defined and identified in order to have a strong basis for mathematical literacy. The importance of transfer learning is most observable in this recommendation because pre-employment mathematical learning experiences should be studied in order to know how to develop and enhance mathematical literacy in the workplace. Lastly, employers should properly understand mathematical literacy, especially informing them of what they can expect from national qualifications. In doing so, closer link among educators, employers, and employees should be established (Hoyles at al. 2002).

The effects of transfer learning to mathematical educations entail improving and strengthening the participations of educators and employers. Transfer learning is clearly important for an individual and even for the society as a whole, which is why educators need to improve the way they teach transfer learning together with mathematical education, especially since the present demand requires more sophisticated mathematical knowledge and skills. In relation to this, employers also have to do their part in enhancing the mathematical capabilities of employees. Educators and employers should work together in order to develop teaching strategies that will meet the demands for more sophisticated mathematical literacy through the use of transfer learning.