Mathematics education is in the midst of a change because of the rapid pace of technological developments and the political issues surrounding it. Technology is considered a potentially valuable tool in the teaching and learning of Mathematics because it has the ability to empower both students and teachers. In some cases the use of technology is encouraged by the faster and more efficient means that can help a teacher do some tasks. In others it is motivated by the idea to communicate instructions in a way that will help to accomplish the learning goals of students. The National Council of Teachers of Mathematics states:
"Technology is an essential tool for learning Mathematics in the 21st century, and all schools must ensure that all their students have access to technology. Effective teachers maximize the potential of technology to develop students' understanding, stimulate their interest, and increase their proficiency in Mathematics. When technology is used strategically, it can provide access to Mathematics for all students."
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As stated in the UNESCO Institute for Statistics, Technical Paper no.2 'Guide to Measuring Information and Communication Technologies in Education', many policy reports have argued that societies are changing from industrial societies to 'information societies', in which the creation and dissemination of knowledge is of paramount importance (ERT, 1997). Indeed, ICT can play a key role in reshaping education to respond to current information that society needs.
UNESCO's principles on ICT in education show that in order to meet the international education goals by 2015 will require huge investments in teacher training institutions (UNESCO-UIS, 2006b). According to experts, this is a major challenge that the traditional face-to-face instruction will not be able to address.
ICT, if used appropriately, can make the teaching of Mathematics more:
ICT has the benefit of increasing the motivation of students towards the subject as well as making it more enjoyable to teach. It therefore has the potential to increase the number of students wanting to take the subject seriously.
There is plenty of evidence that teachers can cover topics in less time and yet more effectively when ICT is used. Looking round other subjects' resources, Mathematics appears to have by far the widest range of digital tools, especially with the use of the internet. These resources and tools are becoming the standard tool-box for the modern Mathematics teacher.
Oldknow and Taylor (date?) identify at least three reasons for promoting the integration of ICT in Mathematics learning in schools:
In terms of students, the use of ICT might:
Engage their attention and motivate them
Stimulate their curiosity
Encourage them to develop their problem-solving strategies
Provide models and images which aid them in concept formation
In terms of teachers, the use of ICT might:
Improve their efficiency
Release more time to address students individually
Be a stimulus to re-thinking their approach to Mathematics teaching
In terms of schools, the use of ICT might:
Reduce truancy, social disruption, etc.
The year 2000 was UNESCO's international year of Mathematics and in the United Kingdom it was promoted by the Department for Education and Employment as Maths Year 2000. Since then there have been significant changes resulting in the current drive to integrate ICT in the teaching and learning of Mathematics. The French government encouraged the introduction of ICT tools in Mathematics as a way of reducing the time available for Mathematics teaching. This shows that whether something becomes educationally inevitable depends also upon the socio-economic evolution of the purposes of education.
There has been a remarkably consistent acceptance of the educational benefits of ICT over the past twenty years, irrespective of which political party has been in power. As a matter of public policy the government has set challenging targets for the training of all teachers in the use of ICT to support teaching and learning.
Kieran and Drijvers (2006) emphasised the need for a strong connection between digital tool and paper-and-pencil techniques. They use a Task-Technique-Theory triad to depict the relationship between the tool techniques and conceptual understanding. The similarity between the two is an important criterion for effective ICT application. Moreover, activities with technological tools should not address procedural skills in isolation, but should offer means to link procedural techniques and symbol sense. Activity structures that exploit these opportunities can affect students' epistemologies and knowledge acquisition in a positive manner.
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In 2005, the Institute for Prospective Technological Studies (IPES) launched a project that aimed to assess the developments in e-Goverment, e-Health and e-learning in the, then ten new member states at the national level, one of which was Malta.
In the report of each country, the educational system was described together with the role played by e-learning within both the formal education system and other aspects of learning. Each report analysed, the key achievements, shortcomings, drivers and barriers in the development of e-learning in the country.
In addition, another report has been developed to review policy challenges and options for the potential development of e-Services and the Information Society towards the goals of Lisbon and i2010  . It promoted the positive contribution that information and communication technologies (ICT) can make to the economy, society and personal quality of life.
The attempt to place the main ICT tools in an educational context will mainly depend on the extent to which a school already has ICT resources and what it sees as the fundamental purpose.
A major government-funded project in the United Kingdom called 'MathsAlive', was carried out in 2000/1. In this project a high level of ICT was used in the Year 7 (ages 11 - 12) Mathematics curriculum in twenty pilot classes. The Interactive Whiteboard was used in many lessons including activities involving the whole class interacting with it.
The evaluation of the project indicated that students were aware of the ways in which the resources were helping them. They also reported that students had more motivation in Mathematics. Furthermore, teachers felt all their planning was worthwhile because the lessons were rewarding. The project succeeded in its aim of using ICT to enhance the teaching and learning of Mathematics.
The document 'ICT and Pupil Motivation', (December 2002, Becta Evidence and Research Team) presents a selection of research with regards to students' motivation and ICT. Evidence suggests that technology in conjunction with effective teaching can increase motivation and make learning more interactive. This eventually leads to more positive students' attitudes and interest in learning.
A study (Cox, 1997) of primary and secondary pupils' use of ICT and their attitudes towards it and subjects which make use ICT shows that the regular use of ICT across different curriculum subjects can have a beneficial motivational influence on student learning. Responses include an increased commitment to the learning task, their enhanced enjoyment, interest and sense of achievement in learning when using ICT as well as their self-esteem. More than 75% of secondary school students agreed or strongly agreed that using computers made their subject more interesting and more than 50% agreed that using ICT helped them understand their subjects better.
Three predominantly important sets of questions for Mathematics teachers that any ICT tool addresses are:
It can be used to help teach context, to develop concepts, to increase knowledge, to improve understanding, to practice and reinforce skills, etc.
It can be used to compute results, to produce tables, to draw graphs, to solve problems, to manipulate expressions, to compute statistics, etc.
It can help to produce materials more efficiently to keep records, to manage time, to communicate with others, to find resources, etc.
There exist also programs aimed at specific curriculum content. Perhaps the most frequently used examples are in the form of games or challenges where the interaction by an individual, or a group of pupils, with the computer, involves them in practising and applying some particular mathematical skills and/or knowledge.
A study by the Technological Pedagogical Content Knowledge (TPCK) framework of Mishra and Koehler (2006) found that the primary advantages of technology are related to the potential to engage students with varied visual representations and virtual manipulations which can help conceptual understanding. They emphasise that teachers need to be able to integrate technology with specific content in meaningful ways in order to enhance student learning.
Mishra and Koehler's (2006) framework provides a simple, diagrammatic representation of the complex interaction between a teachers' content knowledge, pedagogical models and technological expertise. The ways in which the three components blend are potentially unlimited depending upon the teachers' competence, confidence and beliefs with regards to the subject they are teaching.
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The ability to increase the availability of visual images in mathematics is important, since 'visualisation', as a key component of mathematical understanding, is increasingly being recognised (Arcavi, 2003; Sedig & Liang, 2006; Sedig & Sumner, 2006).
Particularly for Mathematics topics that are traditionally presented more abstractedly, namely Algebra, the use of such technology might give way to a pedagogical shift toward teaching styles that more readily acknowledge and use multiple representations. It is generally acknowledged that superior conceptual knowledge is characterised by students that have more than one way of understanding a particular concept (Brenner et al., 1997).
2.1.1 ICT in Algebra
The availability of ICT tools can evidently impact both on the way Algebra is taught and on the way it is applied.
The United Kingdom's Royal Society asked the Joint Mathematical Council of the United Kingdom to carry out a research on the teaching of Algebra. The report of a working group chaired by Prof. Rosamund Sutherland, 'Teaching and Learning Algebra pre-19', was published by the Royal Society in 1995. Section 1.5 contains the following:
"The growth of IT has made it possible for students to manipulate many different types of external representations on the screen, involving symbolic, graphical and tabular forms. It is now possible to manipulate graphical representations in ways which were not possible on paper. Harnessing this new power within Mathematics is the challenge for the 21st century."
Algebra learning is one the current subjects of worldwide discussions on education. Different opinions on goals, approaches, and achievements are at the heart of 'math war' debates (Klein 2007; Schoenfeld 2004). The 'Digital Tools for Algebra Education: Criteria and Evaluation' paper reports on a study in the Netherlands, set up to investigate in what way the use of ICT in secondary education might enhance the learning of Algebra.
Some see the core of Algebra education to be the development of strategic problem-solving and reasoning skills, symbol sense and flexibility, rather than procedural fluency (National Mathematics Advisory Panel 2007). This position is expressed in the Discussion Document of the 12th International Commission on Mathematical Instruction (ICMI) study on Algebra education:
"An algebra curriculum that serves its students well in the coming century may look very different from an ideal curriculum from some years ago. The increased availability of computers and calculators will change what mathematics is useful as well as changing how mathematics is done. At the same time as challenging the content of what is taught, the technological revolution is also providing rich prospects for teaching and is offering students new paths to understanding"
(Stacey and Chick 2000, p. 216)
The above quote raises many questions with regards the use of technology in algebra education. What types of skills are needed because of the availability of technological tools? If the teaching and learning of algebra might benefit from the integration of technology, the subsequent question must be what type of technology to use, and what criteria determine this choice. Many different types of software tools are available, each providing opportunities and constraints for different activity structures. It is not easy to foresee the effects and to decide effectively which tools to integrate in the learning process, depending on the goals of the learning process of Algebra.
2.2 Learning Theories for Learning Environments
Learning is defined as a process that brings together cognitive, emotional, and environmental influences and experiences for acquiring, enhancing, or making changes in one's knowledge, skills, values, and world views (Illeris, 2000; Ormorod, 1995).
There are three main frameworks of learning theories that attempt to describe how people learn: behaviourism, cognitivism, and constructivism. Behaviourism focuses on changes in behaviour of a person, shaped by the reinforcement of the environment. Cognitive theories look beyond behaviour and consider how human memory works to promote learning. Last but not least, constructivism views learning as a process in which the learner actively constructs or builds new ideas or concepts.
Constructivism is a theory that argues that humans create knowledge and meaning from an interaction between their experiences and their ideas. During infancy, it is an interaction between their experiences and their sensory-motor or behaviour-patterns. Jean Piaget called these systems of knowledge schemata. Piaget's theory of constructivist learning has had a high impact on learning theories and teaching methods in education and is a principal argument of many education reform movements.
Constructivist learning, therefore, is very personal, where learned concepts, rules, and general principles may, as a result, be applied in a practical real-world context. Constructivism therefore constitutes, of active learning, discovery learning, and knowledge building. Regardless of the variety, constructivism promotes a student's free exploration within a given structure. The teacher acts more as a facilitator who encourages the students to discover principles for themselves and to construct knowledge by working to solve realistic problems. Aspects of constructivism can be found in self-directed learning, transformational learning, experiential learning and reflective practice.
In past centuries, constructivist ideas were not widely valued due to the perception that children's play was seen as aimless and of little importance. Jean Piaget did not agree with these traditional views, however. He saw play as an important and necessary part of the student's cognitive development and provided scientific evidence for his views.
Jean Piaget suggested that through processes of accommodation and assimilation, individuals construct new knowledge from their experiences. When individuals assimilate, they embed the new experience into an already existing framework without changing that same framework. According to the theory, accommodation is the process of reframing one's mental representation of the external world to fit new experiences. Accommodation can be understood as the mechanism by which failure leads to learning: when we act on the expectation that the world operates in one way and it violates our expectations, we often fail, but by accommodating this new experience and reframing our model of the way the world works, we learn from the experience of failure, or others' failure.
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2.2.2 Social Constructivism
In recent decades, constructivist theorists have extended the traditional focus on individual learning to address collaborative and social dimensions of learning. Social constructivists hypothesize that knowledge is constructed when individuals engage socially in talk and activity about shared problems or tasks. Learning is seen as 'the process by which individuals are introduced to a culture by more skilled members' (Driver et al., 1994).
Social constructivism, strongly influenced by Vygotsky's (1978) work, suggests that knowledge is first constructed in a social context and is then appropriated by individuals (Bruning et al., 1999; M. Cole, 1991; Eggan & Kauchak, 2004).
Constructivist theorists emphasize that individuals make meanings through the interactions with each other and with the environment they live in. Knowledge is thus a product of humans and is socially and culturally constructed (Ernest 1991; Prawat and Floden 1994).
Vygotsky (1978) also drew attention to the amalgamation of the social and practical elements in learning by stating that the most significant moment in the course of intellectual development occurs when speech and practical activity, two previously completely independent lines of development, meet.
A fundamental characteristic of the role of the facilitator from the social constructivist point of view is that the instructor and the learners are equally involved in learning from each other as well (Holt and Willard-Holt, 2000). This means that the learning experience requires the teacher's culture, values and background as an essential part of the interaction between learners and tasks in the shaping of meaning. This creates a dynamic interaction between task, teacher and learner. The social constructivist model thus emphasizes the importance of the relationship between the student and the instructor in the learning process.
Some learning approaches that could adopt this interactive learning include reciprocal teaching, peer collaboration, cognitive apprenticeship, problem-based instruction, web quests, anchored instruction and other approaches that involve learning with others.
Therefore the notion of constructivist learning leads a change in the use of ICT in the classroom: from using it as "instruction tools" to enhance conventional teaching, to using ICT as "cognitive tools" to promote meaningful student learning that is active, constructive, intentional, authentic and co-operative (Jonassen, Peck & Wilson, 1999; Reeves, 1998).
2.3 Interactions in ICT
Like the Swiss army knife, the internet is a multi-purpose tool with several different functions. Both teachers and students can contribute information and knowledge instead of only looking it up.
"The benefits of sharing should be to improve the range of resources available so as to allow teachers more time to interact with their students."
(Andy Lane, Open University, UK, 8 January 2008)
ICT also has the potential to offer valuable support to the teacher of secondary Mathematics by providing a flexible and time-saving resource that can be used in different ways and at different times without repetition of the teacher's input. Moreover, it provides a means by which subject and pedagogic knowledge can be improved and kept up to date, such as to exchange ideas with peers outside the school. The internet allows for communication in both real-time and elapsed-time modes through on-line discussions, e-mail and forum. Feedback is crucial if we want technology to act as a learning environment in which students can engage in a process of practice or meaning making. If feedback on learning activities is actually used to modify teaching to meet the learner's needs, one of the conditions for formative assessment is fulfilled (Black and William 1998).
In fact, there are two main types of virtual communication tools:
Asynchronous tools: enable people to communicate at a time that is adequate for them. Individuals can post a message which can later be read and answered when the recipient comes on-line. An example is a message board which provide the facility for discussion under various topics. This allows individuals to respond to topics, sometimes referred to as threads, or to begin a new topic by posting a comment as well as a question. These messages are visible to everyone who has access to the message board.
Synchronous tools: enable people to communicate when they log onto the same system at the same time.
The majority of the students learn most effectively when they are working in a supportive environment where tutors and learners are working together on collaborative activities. In fact some theoretical models of learning focus on the importance of social processes, and this is a fundamental element in e-learning. Thus, it is a challenge that most teachers face when it comes to provide a learning process that facilitates social processes while achieving the learning outcomes. While it may be difficult for students to walk out of a traditional learning situation, it is easy to switch off the computer or move on to other activities. Therefore it is essential that there is easy access to help.
In a traditional classroom the teacher is often working from the front and the physical layout of the room identifies the central position of the teacher. In an on-line environment, the teacher takes the same virtual space as the students. The only difference is that the teacher is the administrator, having the technical rights to set up and organise this space.
One important point to take note of when communicating on-line is Netiquette. This is a term used to describe good practice of on-line communication. The following are some guidelines:
Thank, acknowledge and support people freely.
Acknowledge before differing.
Speak from own perspective.
Avoid flaming spirals.
Use emoticons to represent feelings.
CAPITAL letters are equivalent to shouting - don't use them.
Be cautious when quoting other messages while replying to them - don't over-quote as this leads to very long messages.
Send messages to the appropriate forum and thread.
Keep messages short. Longer messages need to be used as an attachment so people can choose to read them at leisure.
Don't respond to a message if you are experiencing strong negative feelings. Give yourself time to cool down. Then respond.
In conclusion to this, it can be noted that e-learning involves a new approach to time management as the students take part in asynchronous discussions as well as face-to-face lessons at school, besides independent study.