Pedagogy

Introduction
The expectations of the global work place on Mathematics are increasing. As technological elements advance, living processes are done in a faster manner thereby dimensions in calculations geometrically progress and develop. A big chunk of these expectations rest on schools. The transfer of learning is from school to workplace is pivotal in making an individual productive to his community. This study will focus on discussing that a mix of learning pedagogies will help the transfer of learning

The first teachers
The millennium child is born with a computer, internet, an email address, an iPod, cellphones, a flickr photo album and electronic playstations. His learning with Barney the Purple Cartoon becomes his foundation for his stock knowledge.

The television is part of a childs everyday life. Studies show that children aged two and below watch television. In fact, twenty-six percent of US children younger than age two have TV in their bedrooms - often watched from the crib, and 36 percent of families leave the TV on almost all of the time, even when no one is watching, according to a 2000 Kaiser Family Foundation study. (Lotus, 2010) This means that one of four children, still inside their cribs access the television and learn from it.

Decades ago, it was Sesame Streets Count Dracula who taught the kids how to count amidst lightning and thunders. But now, there are a hundred kids channels to choose from. From these first teachers, the child becomes a student and later on a worker in a fast paced global workplace where the abacus is totally out of the picture.

The work place as an extension of the classroom
Human resource agents prefer workers who have high learning curves. Not all things are learned inside the classroom. Once out of the universities, bosses, supervisors, mentors become the new professors.

Continuous training is given from management down to employees. The employees are scored, assessed and evaluated. These trainings are included when deliberating about who keeps their jobs, and who has to go, who gets a raise or who stays in place. Therefore, an employee who has the ability to learn is an ideal employee when it comes to the training component. Teaching learning how to learn is the root mission of universities. In terms of transfer of learning, this is one of the crucial skills that a student has to master if they plan to contribute value to the bigger economic, social and political community.

Universities are crucial in preparing students to meet social expectations. Transfer of learning from the university to the workplace must meet community expectations. What is learning for if students cannot contribute value to the workplace (Haskell, 2000) Different factors affect transfer of learning. Transfer of learning is a challenge to universities, students and faculty. Transfer of learning is also a concern of the community. It takes a community to mold a child. Especially for mathematics, when everyone starting from the parents gives the child a scare about the subject, the community, the challenge of Math is more crucial because math is the language of inquiry. Math is a secondary concept and therefore is 95 intelligent learning. Transfer of math concepts from the schools to the workplace is a challenge because Math is a challenge by itself.

The workplace, no matter how small, is full of math concepts, calculations, miscalculations, budgeting, money exchange, abstractions, and other mathematical paradigms. The flattened world never sleeps in terms of breaking speed, optimizing resources, minimizing waste, calculating body mass index and stretching the twenty-four hour clocks. Math is a language of science and if science spells life, Math is life. Such is the expectations on a person when he computes his taxes, when he budgets his limited resource and when he computes his dreams and accomplishments. How well his school equips him with the faculties of Math can make or break his ambitions, being the person he can really be.

A research on the need for young people to engage into mathematical literacy skills has been done by Hoyles in 2003. In agreement with other recent studies, the research concludes that the country needs to rethink and look to upgrade mathematics provision for young people and to ensure that people have access to additional provision over their lifetimes. The research findings identify the central importance of what is termed mathematical literacy and it is argued that mathematical literacy represents a major skills gap in the present workforce. (Hoyles, 2003)

Objectives of the study
The paper is a discussion on the different position of theories on transfer of learning. Transfer of learning is the ability of student to use his school learning in the workplace. Transfer of learning is the process by which knowledge, skills and attitude help the student reach his desired career and quality of life after studying college.

Teaching theories affect the students confidence and competence in solving problems related to their work. Different teaching pedagogies approach the topic of transfer of learning in different ways. As society defines and redefines its learning styles, teachers gather strategies from these different teaching theories. Different learners adapt to these theories adjusting their learning styles to accommodate the learning goals.

The goal of all these pedagogies is for the most efficient and effective style of teaching to be partnered with the learning style of the student in order for the target learning to be taught. Retention is important in learning also. The higher the retention of the learning, the more chances that the transfer of learning will become effective.

The paper will also discuss issues related to transfer of learning its characteristics, instructional and learning strategies that promote or hinder it, role of technology on its development and its implications on current educational practices. These issues arise due to increasing realization that graduates are not entirely ready to take on the tasks that await them in the work place. They may have mastered concepts. They may have answered assessments in school with perfect marks but put into a different situation outside the classrooms, coping up mechanisms fail. Awareness of these issues will help the teacher and the learner prepare their curriculums, trainings and learning modules that will benefit not only the student but also the larger community.

Theories on Transfer of Learning.
Pedagogical theories have been evolving since the 1600 when thinkers started to inquire on how people learn. From these theory evolutions, communities have implemented their daily roles according to available theory design. Teaching mathematics has its own history of successes and failures depending on the kind of learning pedagogy are made available to professors and educators. Successful learning pedagogy requires teachers to understand how students learn and must have the capacity and autonomy to design, implement and assess educational activities that meet the needs of individual and all students. (Teo 2006) Learning pedagogies made available to teachers as theory and skill is crucial in the transfer of learning capability that the teacher eventually created in the lesson plans. The chapter describes these available pedagogical theories as they relate to the transfer of learning mathematics.

The Activity Theory
The workplace is full of action. From the moment a worker puts in his time card on the bundyclock at the start of work until he gets off work passing through the bundyclock again, the worker engages into the materials and processes of work. The activity theory assumes these materials and processes that the worker would encounter and takes on the curriculum from there. Activity theory presents a collection of basic ideas for conceptualizing both individual and collective practices as developmental processes of the context in which human activities normally takes place (Engestrm, 1987, Leontev, 1978). Designing curriculums using Activity Theory differs from the traditional approach

1.  The contents and outcomes of learning are not merely knowledge in texts and the heads of students but new forms of practical activity and artifacts constructed by students and teachers in the process of tackling real-life projects or problems - it is learning what is not yet known. (Mwanza 2010)

Mathematics needs primary concepts so that the student can visualize the secondary concept. The student cannot point to the concept of one, but could easily point to one apple. The artifacts used in Activity Theory are therefore crucial if the transfer of learning from classroom to workplace. Objects that engage the senses offer primary concepts and these concepts can bring about secondary concepts such as mathematical activities summation, multiplication, division or statistics analysis.

For example, speed can be taught by either writing the formula on the board or writing the formula at the same time illustrating the formula by the use of toy cars, a  made up runway using cartolina, having students measure the distance traveled and letting them jot down the time traveled using a stop watch. The activity surely be engaging. This simple experiment on speed can be transferred by the learner to his adult life when he engages the highway driving his own car.

2. Learning is driven by genuine developmental needs in human practices and institutions, manifested in disturbances, breakdowns, problems, and episodes of questioning the existing practice. (2010)
Activity Theory engages the learner into actual instances found in the workplace that puts an imbalance into the flow of processes. By making the student experience the activity, conflicts are highlighted so that problem solving can commence. This is crucial in learning because in the workplace, it is during the moments of disturbance when the faculties of problem solving are high. There are also instances when new processes need to be learned to solve emergent problems.

Math is essentially problem solving. Activity Theory helps in practicing the learner of the processes in problem solving. Transfer of learning increases as the process of problem solving is honed. The student becomes less fearful of the problem and at best is joyfully challenged to solve the problems encountered at the workplace. In accounting for example, as soon as there is a discrepancy in balances, it signals that a problem is at hand. The immediate thing that takes place is a redoing of the balancing process but for the second time, in a slower and more focused manner. The mathematical process of accounting is the same whether one accounts for boxes of printers delivered or carbon emissions by China.

3. Learning proceeds through complex cycles of learning actions in which new objects and motives are created and implemented, opening up wider possibilities for participants involved in that activity. This perspective on teaching and learning highlights the potential impact of new tools as vehicles for transforming activity procedures. (2010)

Activity Theory uses mastery of procedures to increase confidence of the student thereby enhancing the transfer of learning. It is impossible to memorize all possible procedures for a given field. But procedures involve steps that can be found in all types and kinds of activity. Participant involvement is crucial because the participants learning is the target of the activity.  Transfer of learning is strengthened because Activity Theory based instructions give participants varied chances to approach the problem either by the use of old tools or new techniques or better yet, being able to device new tools to solve problems.

Mathematics is composed of basic tenets that can be used to solve the same problem. A child can arrive at the answer 9 either by adding four and five, subtracting on one from ten, multiplying three by three or getting the square root of 81. These are different activities and procedures but all these arrive to the same solution number 9. Transfer learning via the Activity Theory therefore is ensured because the confidence of the worker increases while more tools are accessible to him to solve the problem in the work area.

Shared objects used in Activity Theory are key in the smooth flow of the transfer of learning. Meaningful transfer of learning takes place through interaction between activity systems. The school and workplace engage in collaborative interaction in which both activity systems learn something from each other. What is transferred is not packages of intact knowledge and skills instead developmental transfer involves an active reconstruction of the skills and knowledge to be transferred. Teachers and students are used as change agents in the various transformations and redesign projects at work organizations. Thus, students and their teachers act as mediators and boundary crossers between the school and workplaces. The basic theoretical idea of developmental transfer can be described in Figure 1.

Figure 1. Developmental Learning using Activity Theory Conceptual Framework.
Source httpwww.edu.helsinki.fiactivitypagesresearchtransfer
Developmental transfer and expanse learning occurs during transference. The student gathers possible shared objects found between Activity System A and Activity System B. If there are more objects shared, learning and transfer is successful. If there are less objects shared, the student will lean on the Activity System he is more familiar with and with intelligence be able to cross the boundaries. The objective of the student is to have wider boundary zones that will enable him to work in more kinds of Activity Systems.

Basic mathematical operations are general boundary zones. As the student moves into more specific domains, the student will have common boundary zones within his chosen domain. Seeking out common experience is a good way to explore boundary zones. When the Coppenhagen Summit brought in news about carbon emissions, urgently being reduced to arrest global warming, calculating carbon emissions became an issue among countries. The student then goes back to his mathematics and chemistry domains to gather concepts so that he cross the boundary zones where shared knowledge about the issue is discussed. Therefore, if the student has these cognitive maps of mathematics and chemistry, then he can relate to the issues of carbon emission and would be able to converse with peers.

Constructivism
The best way to recall what constructivism means is by getting the meaning from the root word, construct. Like building a skyscraper, the constructivist concept believes that one cannot get to the fourth floor if there is no first floor yet. Its the same in explorations that a human mind is excitedly adapted to. A student will increase his chances of success if he has more faculties to help him explore. And as he is able to explore and acquire new finds, these new finds will automatically be included in his list of tools for further explorations. The process of acquiring sets of tools for transfer of learning is called schema building.

In the process of schema building, we all have to be explorers, since the constructivist principle, embodied in the present model, tells us that the conceptual knowledge cannot be communicated directly. It has to be constructed anew by every learner in his own mind. (Skemp pp. 203)

Constructivism is a philosophy of learning founded on the premise that, by reflecting on our experiences, we construct our own understanding of the world we live in. Each of us generates our own rules and mental models, which we use to make sense of our experiences. Learning, therefore, is simply the process of adjusting our mental models to accommodate new experiences. (IDRC 2010) Constructivism is guided by these principles

Learning is a search for meaning. Therefore, learning must start with the issues around which students are actively trying to construct meaning. (2010)

Most math learners cannot understand the topics of math because it is full of abstractions and therefore is not meaningful to the learner. Memorizing the area of the circle in geometry will have less learning retention compared to actually measuring the area of a plate and figuring how much donuts can cover the plates entire surface. Once the lesson approach mathematics with meaning, the transfer of learning will be easier because a person only acquires something when it is relevant to her.

A simple example would be measuring the perimeter of shapes like a square. The learner at school computes for this to pass his subject. When he grows up, he can use the concept and transfer this schema to building his porch. He can use this tool as he buys curtains for his office. He can compute how far he has to travel when weeding the edges of his farm. From these activities, the transfer of learning is enacted.

Meaning requires understanding wholes as well as parts. And parts must be understood in the context of wholes. Therefore, the learning process focuses on primary concepts, not isolated facts. (2010)

Constructivism advocates contextual learning. Due to the role of meaning, lessons using the constructivist theories elaborate on parts and wholes. Concepts do not sit in a vacuum and therefore the learner is taught to explore the whole in relation to the parts and vice versa. The simple ratio and proportion concepts of math exemplify this constructivist approach to learning. Concepts of ratios become more meaningful when taken in the context of sharing and appropriations. Since everything is connected in this world, a teacher can figure how a subject no matter how big be subsumed by a bigger idea. The same idea is true for small concepts that they are not so minute that they cannot be broken down into relevant parts.

The opinion of constructivism on transfer of learning uses the context of learning. The exercises taught about money, from the value of money to how it is broken down into pennies and coins to how money exchange happens is transferred to the workplace when the adult recalls these concepts while budgeting his salary, traveling to a different country and he has to visit the foreign exchange shop. What can be called as the part is the skill of money exchange. The context can be whole which in our example is the environment where money is used. It can be the workplace on salary day, the trip to Europe, or the shopping center. These environments add meaning to the math skill.

3. In order to teach well, we must understand the mental models that students use to perceive the world and the assumptions they make to support those models. (2010)

Constructivism builds on the mental models stored by the learners. The elements accessible to the students learning are used to deliver concepts and act on math problems in the classroom to the workplace. Some examples of transferrable mental models used to help the learner construct are the process of problem solving, the process of collecting data and the process of building shapes.

In Mathematics, the mere use of examples that are accessible to the imagination of the students is used to illustrate concepts. Problem solving for the speed and momentum of a Ferrari is assuming that the students know what a Ferrari is. Not all balls bounce and so it can be weak example in explaining gravity.

Transfer of learning using the constructivist approach will be helpful because the students are taught how to learn. They are guided by the constructivist approach on how to build from existing concepts so that no matter how vague the objects of the existing problem to be dealt with, the learner knows that it is part of a whole and therefore exploring how people perceive the object can jump start analysis of the problem which hopefully will lead to a solution.

The purpose of learning is for an individual to construct his or her own meaning, not just memorize the right answers and regurgitate someone elses meaning. Since education is inherently interdisciplinary, the only valuable way to measure learning is to make the assessment part of the learning process, ensuring it provides students with information on the quality of their learning. (2010)

Most of the time, assessment signifies the end of learning. In the constructivist approach, the assessment is but part of the learning process because in simple terms, the student learns from the mistakes articulated in the assessments. Since Math is very exact, assessment would seem rigid, with no room for further learning. This notion is negated by the fact that a student eventually needs to find the correct answer and so has to go back to the drawing board or ask the teacher how to come about the correct answer. This going back is the future learning that comes about from realizing mistakes. The teacher and learner must approach mistakes in a more positive manner, that of a chance to learn from the mistake. In a constructivist classroom, the right answer alone is not the point. The point of learning is the right process.

There are constraints to this approach. Bettencourt (1993) gives four constraints on knowledge that can be constructed ones previous constructions, interactions with others, ones experience, fit with the rest of ones knowledge. But given these limitations, the student learning with the constructivist approach is more likely to be more confident in using his learning in the workplace. Being aware that one has building blocks, a set of tools and the right exploratory open-minded attitude, engaging problems in that workplace will be meaningful encounter, even if the situation is totally new to the employee.

Socio-culturalism
The global learning place is filled with ethnic, cultural and ethical dilemmas. This is where socio-cultural strategies come in strong in making the transfer of learning a success. Being reared in a mutli-racial campus can be an advantage because big multinational work settings interact with numerous cultural groups. The internet itself is a place where on could practice communicating and interacting with different cultures. Socio-cultural approach in the pedagogy helps the student realize that learning is not confined in a vacuum. Cultural differences bring about varied dimensions in interaction and relations.

Socio cultural theories are rooted in constructivism but they focus on the role of community and environment in the creation of knowledge as opposed to the constructivist focus on internal negotiation of meaning. They acquiesce that meaning can vary but contend that it is defined by the community of practitioners, which uses it. Thus, knowledge resides in communities. Meaning-making is the result of active participation in socially, culturally, historically, and politically situated contexts. (Pantel 1997) Strictly, the socio-cultural approach prioritizes the learning of a collective community rather than the individuals own knowledge inside the community he is related to.

A feature in changing trends in research in mathematics education during recent years has been the growing interest in and focus on the social and cultural context of the mathematics classroom. (Bishop 1988)

Mathematics in the applied fields relates to the community. Math is used to develop a communitys welfare. As the individual learns the bigger contexts of his environment, math becomes more meaningful and relevant. Curriculums that use this pedagogy increase the efficiency of the transfer of learning because lessons try to approximate the community wherein the students will find himself after his studies.

Besides scaffolding, fading and cognitive apprenticeship, socio-culturalism uses collaborative learning. By sharing ideas, reflection and interacting with classmates during problem solving, the community comes alive. The discussion that leads to the solution is given due credit because in the real world, problems are not stand alone in ones office but is related to the work of another colleague in the field. Purchasing departments in a factory affect inventory, affect sales and marketing and affect the retail prices of products and services. Teams in the workplace continue to learn how companies can evolve through changing times therefore the learner who can communicate with the community is equipped with evolutionary skills at the workplace.

Wenger furthers these meaning learning with the help of the community as communities of practice. Over time, this collective learning results in practices that reflect both the pursuit of our enterprises and the attendant social relations. These practices are thus the property of a kind of community created over time by the sustained pursuit of a shared enterprise. It makes sense, therefore to call these kinds of communities communities of practice. (Wenger 1998) With practice, the learner increases his chances of adapting his learning to the his place of work because at school, using the socio-cultural approach, much practice with these communities of practice is done.

The community of Math uses the artistic language of numbers. Math is a language art. It has its own universal symbols, more universal than English, French or Spanish.  All knowledge is, we believe, like language. Its constituent parts index the world and so are inextricably a product of the activity and situations in which they are produced. A concept, for example, will continually evolve with each new occasion of use, because new situations, negotiations, and activities inevitably recast it in a new, more densely textured form. So a concept, like the meaning of a word, is always under construction. (Brown 1989)

While students learn mathematics, they acquire this language of the sciences. This language has meaning, context and use. And like any kind of language, the more it is used, the more mastery is acquired. Therefore, when mathematics is needed in the workplace, the mind will turn on the language and soon the worker will be able to converse like how he did in the classrooms. Essentially, learning the language that is mathematics enables a student to be part of the mathematics community wherein he can speak the language, be understood and be able to express and share his own ideas.

Situated Cognition
Apprenticeship, thesis work, practicum are some of the strategies that situated cognition employ to deliver competencies. In these curriculums, the student is brought to actual situations where the student needs to physically do the tasks involved. Agriculture students are required to do their thesis that may involve planting to actual harvesting of produce. When these students are deployed to the farms, they would recognize basic things like soil, planting, harvesting, irrigation and pests. This is also true with doctors who need to spend hours in the hospital to fulfill duties. But the time they become practicing doctors, they would easily be familiar with the environment of the hospital including patients, disease and hospital administrators.

Research on situated cognition highlight the concept of communities of practice, where practice is one of the crucial elements in the transfer of knowledge. Situated learning (Greeno, 1989 Brown, Collins,  Duguid, 1989) is a stance holding that inquiries into learning and cognition must take serious account of social interaction and physical activity. A unifying concept emerging from situated learning research is communities of practice--the idea that learning is constituted through the sharing of purposeful, patterned activity (Lave  Wenger, 1989). This idea stresses practice and community equally. Knowledge is seen as practical capability for doing and making. Meaning is seen as a construction of a social unit that shares a stake in a common situation. As a consequence, learning is seen as a capability for increased participation in communally experienced situations--a dual affair of constructing identity and constructing understanding (Wenger, 1990).

Universities are aware that much practice of knowledge must happen in the academe. Computing as a skill has to be practiced. But practice with meaning is the better approach rather than making arithmetic into a mundane habit. This is where the community becomes relevant because community offers the context that is not in the mathematical formulas. Situated cognition works well when the context offers more objects for the learner to hold on to during transference.

Thus, learning organizations are born. Members in an organization are brought together to solve problems together. Together they are organized to attend to a particular situation. A learning organization needs to do so to understand the challenges it faces, recognize opportunities, and maintain a competitive edge. (Parcon 2009)

There are five critical elements to the definition given by Parcon
A learning organization learns consciously it introduces a necessary level of intent and commitment to the process of learning.

A learning organization learns continually, not just consciously.

A learning organization highlights experience as a source of learning it emphasizes the means and ability to exploit its track record, using field operations as a primary source of learning, while drawing from elsewhere.
A learning organization improves practice the litmus test fro whether learning has, in fact, occurred lies in the extent to which its practice has actually improved.

A learning organizations is built around people -- their know-what, know-how, and know-why are central to the undertaking.

A student upon leaving the campus enters into learning organizations when they find their careers. It is into these learning organizations that students are required to transfer their learning in the objective of continuing the learning process with a learning organization.

Researches on mathematics acquisition through situated learning were made by Lave. Lave (1988) used the results of her research about the grocery shopping activity, to criticize the traditional belief that mathematics is an abstract and powerful tool, whose knowledge is easily transferred from one situation to another. From her perspective, mathematics teaching in school is conceived as the acquisition of abilities that subsequently can be transferred to other practices. The abstract concept of division for example is very powerful that it can used in each moment of ones life. The day is divided into two time chunks called day and night. The meals are divided into breakfast, lunch and supper. The house is divided into the number of rooms. So the concept of division, though abstract is concretized into many instances.

Behaviourism
Behaviourism seems like an old concept but much of the pedagogy studies started with psychological experiments of this approach. Behaviorism, as a learning theory, can be traced back to Aristotle, whose essay Memory focused on associations being made between events such as lightning and thunder. Other philosophers that followed Aristotles thoughts are Hobbs (1650), Hume (1740), Brown (1820), Bain (1855) and Ebbinghause (1885) (Black, 1995). (Mergel 1998)

The stimulus response  (S-R) approach assumes that given the correct motivation, required behavior will follow. There are positive reinforcement and negative reinforcements to be used to control behaviour. Skinners experiment on animal behaviour enabled the scientists to realize that giving food that becomes the reward can be used to teach lessons. Behaviorism teaches habit learning. Once the process is learnt, it then becomes a habit and therefore is wholly acquired by the learner. A behaviorist teaching style in mathematics education tends to stress practices that emphasize rote learning and memorization of formulas, single solutions, and adherence to procedures and drill. Teaching is seen as a matter of enunciating objectives, providing the means to reach those objectives and using constant repetition in class for skill acquisition (Leder, 1994). Wood, Cobb, and Yackel (1991) argued that such approaches lead to passive modes of learning. (Handal, 2003)

Unfortunately for the subject of mathematics, habit learning or rote learning teaches less adaptability.  Once the stimulus is gone, or the environment changed, the learner will not be able to get his goal. Transfer of learning is low with the behaviourist approach.

At best, theory development is active. The figure below shows the evolution of these learning theories as one is a reaction to the other. From the 1800 to the present, psychologists are actively in search of the mind and how the mind learns so that teaching can be effective. From Behaviorism to Constructivism, scientists are able to understand how the mind works. The theory development helps professionals and practitioners to adjust their tools and strategies to attain educational goals.

Issues on Transfer of Learning

What characterizes transfer of learning
Transfer of Learning is dependent on the theory used to deliver learning targets
Transfer of learning is affected by the schools of thought the teacher employs. Behaviorists focus on stimulus and response that brings about habit learning. With this mode of teaching, transfer of learning is limited to the stimulus and responses that the student was subjected to. The habits that he acquired must be encompassing so that he can handle more types of problems when he gets to the real world.

Habit learning is used when a child needs to memorize the number system or the multiplication table. Habit learning occurs when information is stored unconsciously, through repetition and trial-and-error learning. These memories are believed to be retained in a different region of the brain, called the basal ganglia. In monkeys with lesions in the hippocampus, it had been shown that in contrast to humans with similar hippocampal lesions due to injury or disease who have difficulty learning certain tasks over a certain time period, the monkeys can learn the tasks at a normal rate, apparently as habits. (Bayley 2005) But take out the stimulus, the response cannot be achieved. Vary the stimulus to something that the monkey has not seen, heard or felt, there will be difficulty to perform the response.

Statistics formula for example is memorized and by the force of habit can be learned by the students enough so they can bring it with them and use these formulas come taxation time. But there are more theories to work with to achieve learning targets.

Students from the constructivist classrooms have more chances of survival in the workplace. The lessons learned would have more meaning outside the classroom walls because even without stimulus, the need to explore, build and create is inherent in the student. The environment will approximate familiarity giving him confidence to step out of his box and relate with new environments with new challenges.

Transfer of learning from situational approaches will be effective but limited to the situations introduced to the students. The transfer of learning may encounter difficulty once the situation changes and the changes are not included in the domain focused. However, when reflection is incorporated in every situational undertaking, there is a better chance that the student will be able to rise up to the problem solving challenge. This is so true in mathematics when the given becomes the solution and the solution becomes the given. Its almost the same situation but perspectives are different. Transference will be effective if critical thinking is also developed in each situation.

Transfer of learning under the socio-cultural approach will bring in a wider realm of considerations. Trained under this approach the student will look for learning collaborators, thinking communities to work with and cultural aspects to appreciate when dealing with problem solving. Since mathematics is more or less a universal language, transfer of learning using this approach would be more creative. Mathematics will be the bridge between differing cultures. And that by itself is an advantage of this approach.
Transfer of learning is enhanced by the process of the delivering learning targets

Transfer of learning is characterized by the process of delivering learning targets. Positive reinforcement versus negative reinforcement will affect retention. Some communities in the olden times inflict hardships to their students believing that the more pain that a student suffers, the more he will claim his successes. There are still schools that employ this type of belief. However, studies show that positive reinforcement has more sustainable results in learning retention. Like positive reinforcement, positive transference is the behavior of the student wherein he voluntarily uses what he learned in school in the workplace. If you punish a child for not doing his multiplication assignment, what he remembers first is the punishment.

Universities and colleges essentially have adults in their classrooms that are an essential factor in choosing teaching strategies. Adult learning must be considered when choosing the process of delivering learning targets. Positive reinforcement works for adults because the responsibility of learning rests on the students themselves. Though varying, their individual motivations are enough to get them to step one. As do all learners, adults need to be shown respect. Instructors must acknowledge the wealth of experiences that adult participants bring to the classroom. These adults should be treated as equals in experience and knowledge and allowed to voice their opinions freely in class. (Lieb, 1991)

Transfer of learning employs targets which involve expectations in the work place
Transfer of learning is characterized by the knowledge, skills and attitudes being transferred. An educator is aware of the subject matter to be taught. Based on the subject matter, the educator prepares his tools, process and motivational and engage activities. There are limitations to a teachers bag of tricks. When the subject matter is just too extensive to be taught in a limited given time, the educator goes back to lecturing, giving homework and asking the students to look up the subject themselves. Transfer of learning is thereby put to a hopeful thought.

In mathematics, there is solace for an educator who handles a homogenous group. Nurses studying mathematics will have a different curriculum compared to engineerings studying mathematics. Approximation of the math involved in the workplace helps the teacher have more focus on the tools he would use to deliver the subject matter. Physicist by far would have access to high mathematical computations far from a janitors imagination and visa versa. But there are realms that connect these roles. These functions are called quantitative literacy.

Arnold Packers essay What Mathematics Should Everyone Know and Be Able to Do, discussed important subjects adhering to an adults quantitative literacy. These are topics that could help the individual contribute to the community in as much as the list are the same topics that the educators need to focus on so that transfer of learning is efficient.

Packer notes that, the structure builds on the SCANS taxonomy, a set of competencies developed in 1991 by the Secretarys Commission on Achieving Necessary Skills (SCANS). These competenciesa better term is problem domainsare quite broad and were intended to accommodate a full range of situations from entry-level to CEO. The five SCANS domains and the subdomains that require quantitative literacy are
Planning problems. Allocating money (budgeting), time (scheduling), space, and staff
Systems and processes problems. Understanding, monitoring, and designing social, physical, or business systems.

Interpersonal problems. Working in teams, negotiating, teaching, and learning.

Information problems. Gathering and organizing data, evaluating data, and communicating, both in written and oral form.

Technology problems. Using, choosing, and maintaining equipment of any type. (Packer 2010)
Packer in his studies has defined these targets according to the roles one has to take after school. Competency exams in highschool are used to assess the skills of a student. The exams match these skills with immediate future undertakings of the student. These exams will help the teacher guide the student whether or not he will go to college. If he goes to college, the course that best suits him can be narrowed down. Then after college, the assessment of his skills can also guide him on his future undertaking whether he will pursue business or become an employee in a firm.

Figure 3 shows the different roles an adult takes in after school life. As a worker, consumer, citizen and persona role, different mathematical concepts are laid out to aid the adult in planning, systems and process analysis and design, interpersonal relations, information and technology competency. Though the skills discussed here are compartmentalized, the student in the field is expected to integrate these tools to deliver role expectations. Transference therefore can be measured according to this table. It can also be used to assess a students capacity to transfer his learning from one role to another.

Generally, industries look for key competencies in the workers of tomorrow. Specialized knowledge, which is taught very well in traditional higher education, and acquaintance with the enterprise in which one works are not sufficient, and more skills are necessary. Students, who are tomorrows employees, also need the abilty to work in teams communication and creative abilities the ability to recognize and understand problems from different viewpoints. These so-called key competencies are important and necessary for any staff member. (Peschges 1998)

Transfer of learning looks into the learning style of students
Learning styles enlighten teachers to lessen their frustrations when dealing with learners. As learning theories evolve, research on learning styles have evolved from the simple fast and slow learner to Kolbs learning theories, Gardners Multiple Intelligences. All these types of learning characterize the transfer of learning.
Carl Jung (1923) identified four basic human functions (1) the thinking function of organizing and analyzing in a logical fashion (2) the feeling function of personal and emotional reactions to experience (3) the sensation function of perceiving and reacting to immediate sensory information and (4) the intuition function of imagination and abstract thought. Subsequent learning style models have focused on perception and communication as key indicators of style.

Each student has their own learning style. It can be a mix of two or three of these learning styles. Learning styles affect transference because the way that the student process the transfer of learnt subjects is also the way they learn. A teacher teaches according to how he learns and therefore if the teacher is limited to one or two learning styles, chances are, he cannot transfer his knowledge and skills as effectively to the students who have different learning style compared to him.

If the lesson plan that delivers statistical computation subjects is delivered according to the learning style of the student, there are more chances that the transfer of learning will be successful. Mathematics is believed to be a thinkers haven. Thinkers enjoy abstractions, conceptualization and problem solving. In Kolbs Learning Theory, thinkers are opposite the feelers that rely on concrete experience. They need to be able to feel the problem and feel the solutions. Experiencing mathematics is crucial to their kind of transference. The watchers meanwhile enjoy learning by listening and watching presentations, lectures and movies. For them, to see is to believe and therefore field trips and mentorship stimulate their learning faculties. On the other hand, doers need hands on learning. Activity based lessons motivate their capacities to the fullest and transference is optimum.

Figure 4. Kolbs Learning Styles
What instructional strategies promote or hinder transfer of learning
Project-oriented learning strategies promote transfer of learning.
Educators can more or less approximate the problems that the student will encounter in the field. Given this assumption, the lessons can be prepared in such a way that problem sets are contextualized according to the environment of the future workplace. Workplace can be a business venture where entrepreneurial skills can be developed. It could also be a multi-national health firm providing wellness services and products. Given these varying fields of practice, the educator prepares project-oriented modules that increase the chances of transference.

Projects are complex tasks, based on designproblem-solving decision makinginvestigative activities that give students the opportunity to work relatively autonomously over extended periods of time and culminate in realistic products or presentations. (Thomas 2000) When students embark on a project, it usually consists of a holistic attack on the problem. The student starts with the given, ventures into the required and deals with possible solutions to arrive at an answer. Mathematics always begins with a given. Some trigonometry problems seems hard at first because theres only one given data to work on until the student realize the many concepts pre-taught to him such as basic conversions, the laws of gravity, the Pi 3.1416, formula of shapes, factors of speed, and other stock knowledge that can be used in completing the project.

The mere process of completion and project engagement is the key to transference. At the workplace project management invokes all the knowledge and skills that the student has learned. Using these skills, the worker is able to embark on small projects at first then on to bigger projects. What is beautiful in mathematics, whether the project is big or small, formula remains the same. A circle under the microscope or seen from the space stations will have the same formula for measuring its area. Therefore, transfer of learning is achieved. Learning more from existing or past projects will help the learner embark on bigger projects at bigger scales.

Project-based learning promotes transfer of learning because of its adaptability. When the project flowcycle is learned by the student he can always use this to other upcoming projects. Mastery of project management in the universities increase the students confidence in taking on projects especially when the projects in the university approximates the projects in the field.

The learning-by-doing approach to training is central to project oriented strategies. Learning-by-doing has high experiential capabilities. The target competency is delivered in hands on or apprenticeship mode. Given the target and training workshops, transfer of learning is ensured. After the transfer comes the transformation. The learning-by-doing concept collaborated with the 4Ts of learning can be very successful in instilling core values to the students given limited resources.

It is important to point out conditions for the success of problem-solving transfer as experienced in project based situations. At the least, specific skills inherent in the domain must be at hand, accessible to the students cognitive awareness. The second condition is that the student has at the least experienced the problem in his immediate past. Third, the student must have acquired critical thinking skills so that he can summon his intelligence to engage on the problem to be solved.

Participatory learning environments promote transfer of learning.

The workplace is not a vacuum. There is hierarchy, structure, roles that connect to other roles, and role systems to follow. Transfer of learning benefits from participatory learning environments because it approximates most of the target workplace.

Participatory learning is part of the paradigm shift that took place in society as reaction to top down cultural movements. People centered principles have influenced the course of western culture over the last thirty years, often changing the bearings of education, business, public policy and international relief and development programs. These principles, larger humanist movements in the natural and social sciences and the emergence of post-modernism and chaos theory required organizations who were serious about adopting a people first orientation to change more than their tactics. It necessitated a paradigm shift. (Jennings, 2000) Realizations of the participatory approach trickled down to most of the societys undertakings including education.

Students who are adept at participatory processes acquire skills and talents that are useful to the participatory workplace. Essentially these environments promote teamwork, camaraderie, accountability, and learning organizations. Egger and Majeres (1998) enumerate the key principles of effective participation as Inclusion, Equal Partnership, Transparency, Sharing Power, Sharing Responsibility, Empowerment and Cooperation. These principles for effective participation can be applied to all aspects of the development process or project.

Though mathematics is not seen in these participatory principles directly, the student who learns these participatory principles in the classroom while solving a math problem will be able to transfer his learning in the workplace. For sure, he will be able to further articulate these participatory principles as he continues to practice the concepts outside the four corners of the classroom.

Reflective strategies enhance transfer of learning.

Reflection is the ability to take stock of the learning and put it into deep learning analysis putting ones own personal attitudes and beliefs and relating it to the learned subject. Reflective strategies enhance the transfer of learning because the process ensures that the learning is taken into the deep recesses of ones mind and heart. Lesson learned from experience goes to waste when the person misses out on reflecting on the topic.
The figure below illustrates the action  reflection model wherein the student after experimentation is taken to experience something new. From this experience, observations abound using sensory tools. Given these observations, reflection occurs that help the student come up with new concepts and realizations that can be experimented on again.

Figure 5. The Action-Reflection Model.
Transfer of learning is safe with reflective strategies. A student learns from his own reflections, whether he did well or did a mistake, the whole experience becomes a learning experience. Therefore, no lesson is wasted. It is more difficult to reflect on lecture modules than experiential modules though. Mathematics that is memorized does not hold as much retention as mathematics that is experienced. Even with mistakes, the actual experience of the mistake is remembered by the student, for life. From these mistakes, learning takes place.

Passive instructional strategies hinder transfer of learning.

A passive workplace is due to close down. In this rapid changing needs of global markets, teams that can evolve themselves to varied fluctuations in economic, socio political situations has more chances of survival. Passive instructional strategies in the classroom dont help prepare students to take on the global challenges inherent in the workplace. Figure 6 illustrates Edgar Dales Cone of Learning wherein the active and passive types of involvement are measured against the how much retention on the subject matter is achieved.

Figure 6. Edgar Dales Cone of Learning
As shown in the diagram, passive learning can only promise 50 retention brought about by lessons that bring students to watch a demonstration, view an exhibit, watch a movie, read, listen to the radio or attend an exhibit. These are types of passive forms of learning that can bring about learning but retention diminishes because in the first place, during these passive activities, the students reaction is not required. The student is merely an observer and therefore his engagement is cerebral, superficial.

But with active learning, retention can increase to 90 according to Dales studies. Doing the real thing obtains the highest retention level that a learner can achieve and this affects transference. Some students know that the smartest and most intelligent mathematician can be the worse teacher to teach math. Teachers who have not experienced what their subjects are will find it difficult to impart insights on the topic. These teachers will be relying on their tools as they start to lecture in class. A teacher who has experienced trigonometry will come to class ready to shoot the topic using different angles, approach, methodology, tools and discussion points.

A passive teacher will definitely have passive students and passive students will definitely end up with passive routine jobs that are less worthy to develop.

What learning strategies promotehinder transfer of learning
Experiential learning strategies promote transfer of learning.
Learning experiences abound in the classroom. Although its the same place, with the same amount of class hours spent and the same set of classmates and teachers, the subject matter changes. The experiential learning strategy promotes transfer of learning because the outside of the classroom, experience is all there is.
Bringing about the experience of mathematics in the classroom is a challenge but educators can bank on adults need to solve problems. Mathematics is after all is useful and all students know the wisdom and advantage of knowing Math concepts. The challenge lies in the teacher and how he delivers the strategy.
Developing a positive classroom can be done by (1) setting a positive climate for learning, (2) clarifying the purposes of the learner(s), (3) organizing and making available learning resources, (4) balancing intellectual and emotional components of learning, and (5) sharing feelings and thoughts with learners but not dominating. According to Rogers, learning is facilitated when (1) the student participates completely in the learning process and has control over its nature and direction, (2) it is primarily based upon direct confrontation with practical, social, personal or research problems, and (3) self-evaluation is the principal method of assessing progress or success. Rogers also emphasizes the importance of learning to learn and an openness to change. (Rogers, 1969)

Some examples of mathematical experience in the classroom are the lessons on measurements and how they are computed for conversion. Transforming the classroom into a market place of sorts where students can play different roles such as buyer, seller, grocery administrator, tax police, or produce vendor. Just like in theater where the stage can be set, the teacher can instill a lot of concepts given one imagined venue, with the students imagining their roles. The use of math is engaging so much so that it can bring about creativity in a student, as he is able to find solutions to the problems posted. If learning is fun, the student will look forward to learning in the workplace as well.

Some faculties would say that this is possible only with simple basic pre college math concepts. It becomes different with higher math altogether. On the contrary, the experience of mathematics in the higher grade levels and even up to the post college courses is more exciting because much experience can be put into the lesson. Adults have more experience compared to children and the teacher can take advantage of this wealth of experience when teaching higher math.

For example, computations on engineering can be cumbersome, long and frustrating however, engaging a group to finish engineering projects adds up to the challenge of collaboration. With hard work, engineering students will be happy to have arrived at solutions that can be an automobile design, an irrigation system or a plan for a house. Having attained the final solutions and final output of a collaborative project is in itself a reward for students of math.

Intelligent learning strategies promote transfer of learning
Math is an intelligent faculty. That is a given. Math expects a student to look at his existing cognitive map, use tools stored in that map to find solutions to the problems at hand. Intelligence is described as a cluster of related mental abilities, which together are very useful. Among these is the ability to learn in a way which is qualitatively different from habit learning. Intelligent learning consists, not in the memorizing of a collection of rules, but in the building up of knowledge structures from which a great variety of plans of actions can be derived as and when required. (Skemp, 1989 47)

Intelligent learning teaches the student ways where he can adapt his skills and knowledge with the given context. Adaptability is important in transfer of learning as discussed earlier. Intelligence-friendly classrooms nurture the adaptability talent of a student. In brief, intelligence-friendly classrooms are classrooms that celebrate the joy of the learners emotional and intellectual word, not through rhetoric and repetition, but through richness and relationships. (Fogarty, 1998)

Table 1. Theoretical Underpinnings of the intelligence pedagogy. (Fogarty 1998)
Traditional theory of general intelligence. Intelligence is inherited and unchanging.Piagets theory of developmental psychology. Intelligence is developmentally constructed in the mind by the learner and moves from concrete to abstract stages of understanding.Vygotskys theory of social mediation. Intelligence is a function of activity mediated through material tools, psychological tools, and other human beings.Feuersteins theory of structural cognitive modifiabilityIntelligence is a function of experience and can be changed through guided mediation.Gardners theory of multiple intelligences. Intelligence is made up of eight realms of knowing (verbal, visual, mathematical, musical, bodily, interpersonal, intrapersonal, naturalistic) for solving problems and creating products valued in a culture.Sternbergs successful intelligence. Intelligence is triarchic, with analytic, creative, and practical components that need to be balanced.Perkins theory of learnable intelligence.Intelligence is made up of neural, experiential, and reflective components that help us know our way around the good use of our minds.Costas theory of intelligence behaviors. Intelligence is composed of acquired habits or states of mind that are evident in such behaviors as persistence, flexibility, decreased impulsiveness, enjoyment of thinking, and reflectiveness.Golemans theory of emotional intelligence. Intelligence is both cognitive and emotional, with the emotional (self-awareness, self-regulation, motivation, empathy, and social skill) ruling over the cognitive.Coles theory of moral intelligence. Intelligence is composed of cognitive, psychological or emotional, and moral realms.

The way a teacher understands what intelligence is, is crucial in his objective to teach math and make knowledge transferrable. If intelligence is experience, then the teacher would use experiential tools to teach. Baking bread is an experience where measurements are taught. If intelligence is composed of habits, the teacher has to subject students to continuous math drills. After 20 items of solving for x, the student eventually acquires the skill of looking for the unknown X that in the workplace setting can be the reason why sales are low, why inventory are disappearing or why the boss is angry.

Workers today are almost spoonfed due to technological gadgets that he has access to, compared to decades ago when there was no internet and mobile phones available. But these gadgets enable millennium workers to do more intelligent work rather than handling routine jobs that can be implemented by a machine. Thus efficiency processes were invented with intelligent learning. Examples of these processes are the Lean Process Solutions, Six-Sigma, the Kaizen type of resource management or the 5 S and 7 W types of work ethics. Work environments in this modern age are lead by intelligent solutions therefore the expectation on workers to have intelligent skills is urgent.

Memorization hinders transfer of learning.

Memorization hinders transfer of learning because
It is limiting. If the student does not have the ability to memorize large chunks of knowledge, he will be less adaptive and less useful in the workplace. The frustration of not being able to cope will feed his low self-esteem until he entirely slips out of the lesson until he will be trailing behind the class. As students age, their memory capacity declines therefore memorizing becomes dependent on the memory recall ability of a person. The older one gets, the less he would be able to remember details especially if they are relevant. Getting old is not only the issue with loss of memory. Medical procedures and medication can also affect this faculty in adults.

Memorization is not meaningful therefore acquisition of the things memorized will not be prioritized.
Remembering birthdays of all your friends is a challenge. But birthdays of parents, sibling, spouse and children are a must. This is another limitation of memorizing topics that do not have meaning to the student. They are forgotten as soon as they are memorized.

Only a handful of learners have the genius of having a photographic memory.
Though memorizing is a tool of learning and is crucial to habit learning, it is not adept in learning mathematics. A student can memorize all types of mathematical formulas but if the student does not know when to use these formulas, these memorizing skills will surely go to waste. Table 2 shows the difference between memorizing and understanding.
Table 2. Comparison between Memorizing vs. Understanding

MEMORIZINGUNDERSTANDINGTries to learn ideas and concepts word for word only.Converts ideas and concepts into own words.Difficult to explain ideas to someone else other than word for word.Able to use own words to explain something clearlyDifficult to see how ideas apply in real-life situations or case studies.Can apply ideas to real life situations or case studies.Relevance of ideas outside the classroom is difficult to see and are typically not sought.Seeks connections between knowledge from the classroom and the outside world.Does not see differences, similarities, and implications of ideas.Can identify differences, similarities between ideas and implications of these ideas.Interprets ideas literally.Realizes that there can be figurative as well as literal interpretations of ideas.Strives for rote learning and has trouble solving problems when numbers or components changed.Strives for understand and can solve problems even when numbers or components are changed.Believes there is 1 right answer for every question.

Accepts that there may be more than 1 right answer to a question depending on circumstances.Has trouble seeing beyond the basic concept or idea.

Can see meaning, effects, results, consequences beyond the basic idea or concept.Copyright  Dennis H. Congos, Certified Supplemental Instruction Trainer. University of Central Florida, Orlando, FL

Memorization is not all that negative. The student may start with memorization but with intelligent learning, the student will eventually understand the concepts. With understanding comes the ability of the student for transference. Figure 7 shows the inverted relationship of the need to memorize and understanding. A student needs low memory skills when their understanding becomes deep. When their concepts become good mental models, they can access these cognitive maps easily.

Figure 7. Relation of Understanding and the need to memorize.
Passive learning hinders transfer of learning.
Passive learning cannot support the transfer of knowledge because

It does not activate thinking skills that will lead to intelligent learning. With passive learning, the mind is taught to absorb what is offered. Passive learning does not require the student to react to the lesson. The student accepts the targets and need not question or delve into possibilities of the topic.

It encourages complacency because the student becomes satisfied when he attains required learning, not venturing into future learning that he can do given his newly acquired set of tools. Complacency leads to extinction. Many challenges that companies face are the ever-changing landscape of economics. Companies are always on their toes to be pro-active to the challenges of their environments.

It does not encourage collaboration that is very useful in the workplace
Role of technology on transfer of learning
Technology supports simulations.

Learners today are visual and tactile. Given the gadgets they are born with, students today do not have a problem exploring buttons on these technical gadgets. They are also adept at exploring software that are essentially simply configured according to the if and then pretexts set.

Take for example the play stations and computer games that children play. Adults play it as well. Computer games are essentially simulations where the gamer is given a task, a problem or a challenge and a set of tools to start with. From this, the gamer plays against the time if hes playing alone or with a gang, also online, or gamers across the globe. These simulations effectively engage the gamer motivating him to go for the goal.

Simulations help the learner in transferences while technology help the teacher bring about simulations. There are limitations though. Technologies do not guarantee effective learning, however. Inappropriate uses of technology can hinder learningfor example, if students spend most of their time picking fonts and colors for multimedia reports instead of planning, writing, and revising their ideas. And everyone knows how much time students can waste surfing the Internet. Yet many aspects of technology make it easier to create environments that fit the principles of learning discussed throughout this report. (CBASSE, 1999)
Computer Assisted Learning helps individual learning styles.

Computers have been used to teach language arts. To date, it has been used to teach mathematics also. Computer Assisted Learning makes use of the computer to help educators and learners introduce topics, facilitate practice, design individual simulations and make assessment faster. Since the computers can be programmed according to the needs of the learners, learning styles are can be considered when designing the computer software where instructions will be programmed. Additionally, a survey of adult learners in British Columbia who were exposed to several computer-assisted learning systems revealed that the most positive comments about learning gains were made with reference to mathematics (Thomas  Buck, 1994).

The role of computers is to take the pressure off the teachers. There are standards on the teacher student ratio that universities try to meet however due to economic reasons, a lecturer in the tertiary schools stands in front of an auditorium for one hour mindless of the rate of acquisition the students are experiencing or not. But with the help of computers as instructor assistant, the learner is again the center of the learning process.
Sloan states that Computer Assisted Learning (CAL) is being widely used because CAL can be adjusted to each learners style and help learners overcome their learning weaknesses. Sloan maintains that students learn in a variety of methods but that each student has a preferred learning style. And as such, good course design must be developed to be flexible enough to meet each students preferred learning style. (Henke, 2001)
Universities in the new millennium are equipped with computers and educational gadgetry that help students. The laptops of today are retrofitted with wifi, educational software and multi-media peripherals that help student learning. Some universities offer lay away plans so that more students can acquire the handy notebooks and subnotebooks. Apple for example, offer student discounts to most of their computer products. And acquisition of a laptop is a must in the schools of tomorrow due to the feature of computer-assisted learning systems as expounded by Yoder.

Another important feature of computer-assisted learning systems is that they can easily accommodate the major learning styles. Visual learners are provided with opportunities to read words and see graphics that are carefully presented on uncluttered pages with particular attention paid to color and number of words per screen. Students wear headphones and are stimulated auditorially as a clear voice articulates key points. Most importantly, tactile and kinesthetic learners are able to use a keyboard and mouse to manipulate data and produce answers. (Yoder, 2010)

The issues that challenge computer assisted learning strategies have always been about availability of resources. Comparing costs of computers decades ago, the machine nowadays is more affordable. There are other more specialized computers that require a big investment. Specialized software come with specialized computers. But there are a lot of ways how institutions are able to acquire these computers. Sometimes the target companies where graduates will hopefully go to donates to the colleges. A lot of discounts are given to educational institutions. At the onset of computer learning, parent and community associations agree to put a premium to education by pooling resources so that the school can offer computer courses and use computers and internet for learning.

Technology enhances situational learning
Calculators get rid of the cumbersome of calculations making the learner have more focus on the situation.
Numerous studies show that calculators have improved performance of students in math. A study in Germany was conducted and the paper reports on the effects the use of a pocket calculator-based computer algebra system (CAS) has on the performance in mathematics of grade 11 students in Germany. A project started at 8 of about one hundred upper secondary schools in the federal state of Thuringia in 1999 3 years later the former restrictions on the use of technology in math education were lifted. In 2004, more than a quarter of all Thuringian upper secondary schools used CAS in math classes. Beginning in 2000, a test was carried out each year to compare the performance of CAS and non-CAS students (from different control schools). In 70 of the cases CAS students performed better than, and in the remaining 30 they performed as well as, non-CAS students. There is evidence that students in advanced courses benefit more from using CAS than students in basic courses. (Schmidt, 2010)

At first objections against the calculators were aplenty. Educators felt that loss in calculation mastery, multiplication memorization would dampen computing abilities of the student if they continually use calculators. Evidently, there are calculators that are programmable and could compute hard physics or statistics problems. It was in recent years that educators finally realize that calculators were able to save time from these cumbersome calculations. Skemp in 1989 explains that these calculators release us from the drudgery of acquiring speed and accuracy in doing complicated calculations. They do not release us from the task of knowing what are the appropriate calculations to do, or whether the answer makes sense. But they make more time available for learning with emphasis on understanding, and thereby help us to meet this obligation.

Multi-media visualize the situation.
Computers can create graphic elements that can help the students visualize the situation needing solutions. This is where multi-media tools come in handy. The role they play is to help the student get a tactile feel and help them imagine what forces are at play. Instead of consuming effort trying to imagine a new situation, graphic resolutions can aide the student in defining the situation. Research on the use of Geometers Sketchpad was done to how this particular technology can help mathematics students.

This article presents a case study in which researcher-practitioner collaboration took place to promote effective use of technology in an urban elementary school mathematics classroom. Data were primarily gathered through classroom observations and interviews. The aim of this study was twofold. First, to increase our understanding of the effects of teacher-researcher collaboration on the perspective of an experienced mathematics teacher who, for the first time, was teaching geometry with a dynamic computer software program, the Geometers Sketchpad. Second, to get the reactions of both the teacher and the students about whether or not Geometers Sketchpad had enhanced the teaching and learning of geometry. The data suggest that, although the teacher was worried at first that she might be replaced by the technology, she found that she is a vital part of ensuring that the technology is an effective tool for teaching and learning. Furthermore, the data also suggest that the students liked using technology and felt it enhanced their learning of the mathematics they were taught. (Yanik, 2010)

Virtual access enables the student to see in real time the actual work place that can be used as a venue for problem sets in mathematics.

The internet superhighway has yet to tapped. Numerous possibilities on the virtual superhighway is continuously being developed. Math tutorials can now be done via Skype and Wacom technologies. Math tutors can be sitting across the globe while discussing algebraic equations with their tutees. Cameras hooked up with the laptops can take real time videos where learning can take place. Sights and sounds are real time and therefore give an experience to students using the technology.

Virtual reality motivates young learners while for some teachers, venturing into this learning tool is yet, a learning by itself. When it comes to teachers learning and valuing the effective use of new technologies, some schools are discovering that the kinds of training programs offered in the past may not represent the most generative method of reaching a full range of teachers and their students. The key term is generative - meaning that behaviors and daily practice will be changed for the better as a consequence of the professional development experience. (McKenzie, 2001)

Technology enhances feedback and monitoring
Feedback and monitoring are essential in learning as well as in transfer learning. Feedback in mathematics is urgent. Like language, mistakes must be corrected during the time it was committed so that correction becomes meaningful and retention can follow. Table 3 shows how different approaches implement feedback. Given this listing, choosing appropriate technology to deliver feedback systems can be done in a more efficient manner.
Table 3. How approaches implement feedback mechanisms.
Behaviorism

Behaviorists would engineer feedback in the form of positive and negative reinforcers for learner behaviors, with the goal of encouraging desired behavior and discouraging undesired behavior. Software which punishes users for wrong answers and rewards for right answers is one example.Social  SituationalObserves consequences to models. Social learning feedback can take the form of learners having the opportunity to observe others (real or video or cartoon etc.) modeling behavior and experiencing consequences. This kind of feedback helps learners decide whether or not to engage in such behaviors themselves.ConstructivismChecks what knowledge was constructed. Constructivists want to understand what kind of knowledge constructions are happening within the learner, even though there is no emphasis on right or wrong.Collaborative LearningCompares notes with other learners. Collaborative learners float ideas to others and gauge their reactions, and listen to what others are thinking in order to compare it to their own ideas. Sometimes peer review can be set up to encourage further thinking.Source Michigan State University 2007

Decades ago the time delay of feedback from homeworks negates the chance of learning. The teacher spends time checking these assignments that by the time it returns to the student, a new topic occupies the mind. The mistakes of past exercises are not meaningful anymore. Teachers tend to adhere to easy to check assessments that have low communicability in terms of telling the teacher how far and deep students insights are on the lessons.

Now, however, electronic mail, computer conferencing, and the World Wide Web increase opportunities for students and faculty to converse and exchange work much more speedily than before, and more thoughtfully and safely than when confronting each other in a classroom or faculty office. Total communication increases and, for many students, the result seems more intimate, protected, and convenient than the more intimidating demands of face-to-face communication with faculty. (Chickering 1996)

Implication on current educational practices
Overly contextualized curriculum may strengthen specialization but threaten adaptability.
Educational practices are conscious of transfer of learning. Educators who are field practitioners know the rudiments of the workplace. Some universities have realized the wisdom of getting professors who are not only researchers in the fields but at the same time practice their expertise. This conscious effort may get these administrators over excited into putting the context of the workplace into the classroom. When the student cant adapt these concepts to other context, the danger of overly contextualization is evident.

Success in transfer depends on analyzing situations and determining which skills are relevant. Certainly an important dimension differentiating situations for an individual is the amount of domain-specific knowledge he or she has in each situation. Some skills or principles may be relevant in domains in which one has much previous knowledge others may be relevant for domains in which one has little prior knowledge. (McKeachie, 1987)

Educators therefore need to know when to contextualize and when to refer to basic knowledge and skills. To increase transference, the objects that are fed to the students cognitive map are specific but at the same time general. An active learner teacher will be able to lead his students towards this attitude of learning where the approach is simple yet challenging. Mathematics is a domain that cuts across general domains. Success in transferring mathematical knowledge relies on the student finding familiar principles contained in domains that he will encounter. It is not the context that makes it successful. The mind explores the meaning behind the context. If the student is able to find meaning in the context then transference is successful.

Over contextualizing is just object overloading. The same is true with memorizing when the student can take in a thousand objects but retention is just 100.

Development of new teaching tools must approximate real work place situations in a general perspective.
There are hundreds of teaching tools, exercises, work plans and problem sets that teachers refer to when making lesson plans. Selection is a responsibility. Teachers select teaching tools that are appropriate. There are limitations though to this selection. The tools may not be available. The tools may be available but confidence of the teacher in using it is low. Funds are low and so the teacher has to scrounge for visuals, manipulatives and reference books thus maximizing existing low resource to the detriment of the learner.

Another limitation is that the teacher is not abreast with the current situation of the workplace. There was a time when teachers were afraid of using computers while computers were already being used in the field of work. Reaction on the use of technology for lessons from teachers became an issue because it challenged their faculties to learn the technology faster than their students.

Umbach in 1997 observed that in only 20 years, microcomputers have gone from a hobbyists toy to an everyday tool in business, education, and even home life. The last few years have seen explosive growth in the Internet and in its role in communications, commerce, and education. The rapid growth and development of this technology has challenged millions of people in many occupations to master often difficult tools and concepts, and will continue to do so as prices drop, power increases, and applications proliferate.

Thus far, most educators who use technology to implement the alternative types of pedagogy and curriculum are pioneers people who see continuous change and growth as an integral part of their profession and who are willing to swim against the tide of conventional operating proceduresoften at considerable personal cost. However, to achieve large-scale shifts in standard educational practices, many more teachers must alter their pedagogical approaches and schools management, institutional structure, and relationship to the community must change in fundamental ways. (Dede 2010)

Professors must reflect the attitudes of a holistic approach to technology education to better understand the evolutionary context of the modern workplace. Characteristics of a holistic approach in technology education is summarized in Table 3.

Table 3. Characteristics of the holistic approach in technology education
Teacher Learner Curriculum Pedagogy sees self as a process guide
sees self as an architect of learning experiences during which students seek
knowledge and expertise from a range of sources
identifies broadly with design and technology
is interested in contemporary issues of design and technology
values innovation and creativity
embraces change and anticipates a professional life of learning.
enjoys creative activities and learning about different forms of technology
evaluates and critiques own work and the work of others
sees technical skills as a means of realising designs and plans
is curious about technology and emerging developments in technology
relates technology to human contexts.
defines processes and transferable ideas
is expressed in the language ideas and concepts
contains a futures focus
accommodates flexible approaches to programming
models cognitive as well as physical processes
integrates theory and practice
recognises a progression of learning
uses negotiated tasks and activities
places importance on social context
uses both individual and group work .

Using these characteristics as guides, the teacher can be more confident in using technology for teaching and learning. Teachers need to realize that technology in the classroom is merely a tool. Technology motivates the learner because of its three dimensionalities and gadgetry tricks but the teacher has to make sure employing the right technology is crucial to retention. Razzle dazzle power point presentations can get a whoa from the classroom but over the years of using the power point programs, razzle dazzle merely gets the attention of the class for the first few seconds. The real benefit of these programs is the ease of presenting the modules. It can be prepared before the class, be used for the next classes and be updated in minutes. Decades ago, the teacher had to write the whole lesson on the blackboards over and over again.

And this is also true in work places because presentations to top managers are done using power point presentations. This is just one example of how transference happens using similar objects found in the classroom that is also used in the workplace. Why teach Pagemaker when the workplace is using InDesign Why survey land with crude compasses when a GPS is accessible. If the teacher uses the crude compass in surveying, chances are the student will have to familiarize himself with the GPS when he is hired by a big surveying firm.

Teachers not only teach the subjects but guide students into learning how to learn.

Human beings move in three-dimensional fields. They grow in maturity. They grow through time. They grow through geographic placements. These dimensions offer human beings the chance to achieve their maximum potentials as human beings. The need to learn is innate. The discovery of how to make fire from friction was possible because the human mind is capable of learning. Human needs motivate exploration, deduction, integration and experimentation. Discovery is both a motivation and a reward.

In teaching math with due effort on making sure that transference is high, teachers help the students in learning how to learn. The student is inside the academe for only a limited number of years. Some just get limited certificates for a few weeks or months. Curriculums have to ensure that students finish school with enough tools so that transfer learning can take place.

Mathematics is a living subject which seeks to understand patterns that permeate both the world around us and the mind within us. Although the language of mathematics is based on rules that must be learned, it is important for motivation that students move beyond rules to be able to express things in the language of mathematics. This transformation suggests changes both in curricular content and instructional style. It involves renewed effort to focus on
Seeking solutions, not just memorizing procedures
Exploring patterns, not just memorizing formulas
Formulating conjectures, not just doing exercises.

As teaching begins to reflect these emphases, students will have opportunities to study mathematics as an exploratory, dynamic, evolving discipline rather than as a rigid, absolute, closed body of laws to be memorized. They will be encouraged to see mathematics as a science, not as a canon, and to recognize that mathematics is really about patterns and not merely about numbers. (National Research Council, 1989, p. 84)

Learning how to learn is crucial to an individual. The professor or mentor is not going to be there forever. Sooner than later, the student eventually becomes the mentor.  Learning how to learn is important because this process will become the perpetual professor beside the perpetual student who ventures into learning experiences that makes him become an integrated person. This is the highest goal of Western cultures. An integrated person is someone whose goals, values, thoughts and actions are in harmony someone who belongs to a network of relationships someone who accepts a place within a system of mutual responsibilities and shared meanings. (Gardner, 2001)

Student assessment strategies must promote adaptability.
Assessments are crucial for the student, teacher, administrator and community. A new graduate brings her credentials when looking for a job. His past thesis or recommendations from professors is based on how he fared in his studies. This assessment bridges the university and the workplace. It also bridges the student and the new worker. Assessment therefore is a crucial element that describes transference.

Assessment involves the who, what, where, when, and how. Its been said that in life, timing is everything. As in life, assessments performed at crucial times in the learning process can spell the difference between gathering data to evaluate students and using assessments to enhance learning. Based on timing and purpose, four functions of assessment data are
Formative assessment provides diagnostic feedback to students and instructors at short-term intervals (e.g., during a class or on a weekly basis)
Summative assessment provides a description of students level of attainment upon completion of an activity, module, or course
Evaluative assessment provides instructors with curricular feedback (e.g., the value of a field trip or oral presentation technique)

Educative assessment integrated within learning activities themselves, educative assessment builds student (and faculty) insight and understandings about their own learning and teaching. (SP 2010)

Assessments measure retention of learning. In turn, retention is key to adaptability. Some assessment exams bring the learner to different situations. And ask the learner to apply concepts learned to the situation. Multiplication is evident in other tertiary concepts such as summation, ratio and proportion, conversion and even statistics and accounting. High rates of assessment of the multiplication concept relates to the students success in finding himself in the fields of engineering, business, physics or economics. Assessment is a pathway guiding the learner to where he can successfully transfer his learned skills.

Assessments measure learning capacities. This is evident in creative assessments. A mere true and false type of quiz will compute a students exact knowledge of the subject matter. Attaching the question why to the items on the quiz enables the teacher to know how deep the students insights are. Creative assessments can look into the how the students think and learn rather than just stockpile contents of their cognitive maps.

Educators need to regularly assess their assessment styles in order for assessments to be effective as venues where learning how to learn. A list of questions that can guide the assessment style are
Who conducts the assessment and why
What are the objectives of the assessment
Is the assessment creative If not, how can it be creative
Where will the assessment be conducted Why
How will the assessment be conducted
The implications on educational practices are summarized by Martin in 2003. Educators and educational institutions must
Teach students how to create and apply technology to serve their own purposes and to serve other people. This requires that they develop skills and capacities to design.
Teach students to consider the appropriateness of the technologies they use and how their use of technology may impact on other people and the environment.
Provide learning experiences in which students consider futures and the types of actions we must take today to produce futures that we may desire.
Provide a broad range of technological experiences and frequent opportunities for students to transfer their learning to new contexts.
Encourage students to think broadly about technology and to appreciate its role and impacts in the wider community.

Provide students with experiences that develop confidence and enthusiasm for learning about technology. (Learning to learn technology is a critical skill for all young people. It would seem that those who continue to learn about technology will be the ones who shape technological change into the future.)

Teach students to value the human ingenuity and creativity that has produced the products, systems, services or environments that are part of all cultures.

Summary
Transfer of learning is about the process of adapting knowledge, skills and attitudes learned in the classroom to the work place. Each student embarks on education to equip him with skills so that his life can become meaningful to himself, his family, his work, his community and to society in general. Mathematics is one of the essential subjects taught in school to equip the student with basic, functional, quantitative and numeric literacy. Unfortunately, teachers have been perennially challenged in the math transference.

Different learning pedagogies have different approaches in delivering mathematics skills. Transfer of learning is dependent firstly on existing concepts that the student has. From behaviorist to constructivist paradigms, teachers and students continually relate with the subject matters like mathematics to increase stock knowledge, skills and attitudes of the learner. Upon graduation, teachers and students sit and hope that they can hopefully transfer the learning to the workplace. A mix of these pedagogies with constructivist theory leading the way and an in-depth analysis of the students learning style, the teacher is able to construct the most effective lesson that promotes transfer of learning.

To complicate matters with regard to transfer of learning are urgent issues that needs to be addressed. The evolving communities cannot wait. The vacuum for good workers cannot get larger because then there will be an imbalance in the domains. Companies can only increase their training budgets so much. As it is, training is a given in the corporate structure and budgets. Universities must do their share in training and transferring.
There are instructional strategies that promote transfer of learning. There are strategies that hinder it. Active models reinforce adaptability while memorization doesnt help especially in the fields of mathematics. Technology helps transference because these are objects part of the realms of both academe and workplace. Technology helps deliver target learning in an efficient and effective way.

In conclusion, the objective of transference or the transfer of learning is the objective of all universities, to be able to mold character and competence into an individual so that he or she may be able to differentiate himself from his peers and integrate into his immediate community and become a self-actualized person. Constructivism increases the likelihood of transfer of learning because it equips the individual with readily accessible tools that can use to build more learning.

The most important part of Mathematics is not the calculations on the calculators nor the computations of the engineers designs. Mathematics teaches discipline, fortitude, and perseverance to know the truth. If individuals are able to do good work in their particular roles, then for sure, the transfer of learning has been successful.