The key principles of constructivism proposes that learners build personal interpretation of the world based on experiences and interactions with knowledge that is embeded in the learning context in which it is used. Learning which is viewed from social constructivism or social learning theories of situated cognition focuses on learners' prior knowledge and how they construct their understanding based on their contexts or learning culture (Vygotsky, 1978). Social learning theories advocate that students master new learning approaches through interacting with others (Doise, 1990) as knowledge and understanding develop in relationship with the social context (Fickel, 2002). The theories support learning as a social and cultural activity mediated by the social and environmental factors around the learners that stimulate their learning so that growth occurs in the cognitive, psychomotor and affective domains.
The instructional or learning strategies that are proposed under constructivism include constructivist teaching, collaborative learning and problem-based learning. In research reported by Helgeson (1994), most but not all, cases using inquiry-oriented curricula resulted in significant gains in problem-solving skills, and gains in achievement or attitudes towards science.
Constructivism informs educators that "learning is constructed, not only in an individual's head, but in the interactions among individuals or between individuals and materials as these occur over time" (Marshall, 1994).
Nariansamy in Ijeh (2003): For many years mathematics was taught in what is referred to as the traditional way with the teacher transmitting all the knowledge and the child passively accepting it without question. In the traditional mathematics classroom, where the teacher only shows how and what is to be done, there is little discussion; pupils are seldom given chance to ask questions if they do not understand something. Often children, who already built up a fear of mathematics, feel afraid of the teacher and the reaction of peers if they do not understand. On the other hand, a mathematics classroom where meaningful teaching and learning takes place provides a powerful means of communication between the teacher and the student of among the students themselves. In contrast, the traditional mathematics classroom is ironically a place where the children's opinions are never heard.
Since 1980, however, the theory of constructivism has been advocated as an effective way of learning and teaching mathematics. According to this theory, learners actively construct their knowledge with the focus on a problem-centered approach based on constructivist perspectives. Constructivists believe that learning is the discovery and transformation of complex information and that traditional teacher-centered instruction of predetermined plans, skills and content is inappropriate (Nicaise & Barnes, 1996). Furthermore, they suggest that situations and social activities shape understanding. They are critical of traditional teachers when they do not provide students with essential contextual features of learning, thus forcing students to rely on superficial, surface-level features of problems without the abilities to apply or use knowledge. Nicaise & Barnes, (1996) suggest that learning occurs within the world students experience and that when they deal with problems and situations simulating and representing authenticity, they learn more. The following section discusses constructivist learning theories and problem-solving and problem-centered approaches to teaching and learning mathematics.
1. Constructivist Perspective on Teaching and Learning
Constructivism is an epistemology that views knowledge as being constructed by learners from their prior experience. The learner interacts with his/her environment and thus gains an understanding of its features and characteristics. The learner constructs his/her own conceptualizations and finds his/her own solutions to problems, mastering autonomy and independence. According to constructivism, learning is the result of individual mental construction, whereby the learner learns by dint of matching new against given information and establishing meaningful connections, rather than by internalizing mere factoids to be regurgitated later on. Thanasoulas (2002) notes that in constructivist thinking, learning is inescapably affected by the context and the beliefs and attitudes of the learner. Here, learners are given more latitude in becoming effective problem solvers, identifying and evaluating problems, as well as deciphering ways in which to transfer their learning to these problems.
Constructivist learning is based on students' active participation in problem-solving and critical thinking regarding a learning activity that they find relevant and engaging. They are "constructing" their own knowledge by testing ideas and approaches based on their prior knowledge and experience, applying these to a new situation and integrating the new knowledge gained with pre-existing intellectual constructs. In this view, knowledge is gained by an active process of construction rather than by passive assimilation of information or rote memorization. This view of learning sharply contrasts with one in which learning is the passive transmission of information from one individual (teacher) to another (student), a view in which reception, not construction, is the key.
According to constructivist learning theory, mathematical knowledge can not be transferred ready-made from one person (teacher) to another (student). It ought to be constructed by every individual learner. This theory maintains that students are active meaning-makers who continually construct their own meanings of ideas communicated to them. This is done in terms of their own existing knowledge base. This suggests that a student finds a new mathematical idea meaningful to the extent that he/she is able to form a new concept (Bezuidenhout, 1998).
Kamii (1994) states that: "Children have to go through a constructive process similar to our ancestors', at least in part, if they are to understand today's mathematics". Kamii goes on to say that, today's mathematics are the results of centuries of construction by adults, we deprive children of opportunities to do their own thinking. Students today invent the same kinds of procedures our ancestors did and need to go through a similar process of construction to become able to understand adults' mathematics.
Students' first methods (algorithms) are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestor did. By trying to bypass the constructive process, we prevent them from making sense of mathematics.
Reys, Suydam, Lindquist & Smith (1998) mention three basic tenets on which constructivism rests. There are:
Knowledge is not passively received; rather, knowledge is actively created or invented (constructed) by students.
Students create (construct) new mathematical knowledge by reflecting on their physical and mental activities.
Learning reflects a social process in which children engage in dialogue and discussion with themselves as well as others (including teachers) as they develop intellectually.
There are three types of constructivism that are applicable to mathematics education. There are known as:
Radical constructivism: According to this theory, knowledge can not simply be transferred ready-made from parent to child or from teacher to student but has to be actively built by each learner in his/her own mind (Glasersfeld, 1992). This implies that students usually deal with meanings, and when instructional programs fail to develop appropriate meanings, students create their own meanings. Ernest (1991) observes this type of constructivism lacks a social dimension in which the students learn dependently. Cobb, Yackel & Wood (1992) also contend that "the suggestion that students can be left to their own devices to construct the mathematical ways of knowing compatible with those of wider society is a contradiction of terms".
Social-constructivism: Ernest (1991) comes up with a new type of constructivism that is called social-constructivism which views mathematics as a social construction which means that students can better construct their knowledge when it is embedded in a social process. Through the use of language and social interchange (i.e. negotiation between the teacher and the students and among the students), individual knowledge (understanding) can be expressed, developed and contested.
Socio-constructivism: This type of constructivism is developed only in mathematics education. According to this theory, m is a creative human activity and mathematical learning occurs as students develop effective ways to solve problems. In connection with this, Jones (1997) notes: "Knowledge is the dynamic product of work of individuals operating in the communities, not a solid body of immutable facts and procedures independent of mathematicians. In this view, learning is considered more as a matter of meaning-making and of constructing one's own knowledge than of memorizing mathematical results and absorbing facts from the teacher's mind or the textbook; teaching is the facilitation of knowledge construction and not delivery of information.
Supporters of socio-constructivism theory claim that when individuals (learners as well as teacher) interact with one another in the classroom, they share their views and experiences and along the way knowledge is constructed. Knowledge is acquired through the sharing of their experiences. Therefore, it is socially constructed (Ernest, 1991; Stein, Silver & Smith, 1998).
Vygotsky holds the anti-realist position that the process of knowing is rather a disjunctive one involving the agency of other people and mediated by community and culture. He sees collaborative action to be shaped in childhood when the convergence of speech and practical activity occurs and entails the instrumental use of social speech. Although in adulthood social speech is internalized (it becomes thought), Vygotsky contends, it still preserves its intrinsic collaborative character (Kanselaar, 2002).
Vygotsky (in Nicaise & Barnes, 1996) articulated the importance of social discourse when he suggested that cognitive development depends on the child's social interaction with others, where language plays a central role in cognition. Vygotsky believes that social interaction guides students thinking and concept formation (schema). Conceptual growth occurs when students and teachers share different viewpoints and experiences and understanding changes in response to new perspectives and experiences (Nicaise & Barnes, 1996). The characteristics of socio-constructivism are:
mathematics should be taught through problem-solving;
students should interact with teachers and other students as well; and
students are stimulated to solve problems based on their own strategies (Cobb et at., 1992).
2. Problem-solving and Problem-centered Approaches to teaching and learning mathematics
a) Problem-solving approach
A problem-solving approach is an approach to teaching mathematics. With this approach the focus is on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterized by the teacher helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing and verifying (Lester, Masingila, Mau, Lambdin, dos Santon & Raymond in Taplin, 2007). According to Taplin's (2007) review of research reports, specific characteristics of a problem-solving approach include:
- interactions between students mutually as well as teachers and students;
- mathematical dialogue and consensus between students;
- teachers providing just enough information to establish background/intent of the problem and students clarifying, interpreting and attempting to construct one or more solution processes;
- teachers accepting right/wrong answers in a non-evaluative way;
- teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems;
- teachers knowing when it is appropriate to intervene and when to step back and let the pupils make their own way; and
- the possibility of using such an approach to encourage students to make generalizations about rules and concepts, a process which is central to mathematics.
b) Problem-centered approach
A problem-centered approach is also an approach to mathematics education that is based on problem-solving. We could just as easily have called this a learner-centered approach or, to use the more formal term, constructivist; it follows the theory that learning occurs when students construct their own knowledge. In problem-centered mathematics instruction, students construct their own understanding of mathematics through solving reality-based problems, presenting their solutions and learning from one another's methods. The learner interprets the problem conditions in the light of his/her repertoire of experiences (knowledge and strategies previously assimilated). The teacher provides the necessary scaffolding during this process.
Problem-centered approach theory opposes the view that mathematics is a ready-made system of rules and procedures to be learned; a static body of knowledge. According to this theory, mathematics is a human activity and students must engage in a way similar to the genetic development of the object. Supporters of this theory hold that students should not be considered as passive recipients of ready-made mathematics, but rather that education should guide the students towards using opportunities to invent (re-invent) mathematics by doing it themselves (Ndlovu, 2004). Students should be given the opportunity to experience their mathematical knowledge as the product of their own mathematical activity.
In a problem-centered approach, instruction begins with reality-based problems, dilemmas and open-ended questions. The learners acquire knowledge from the solution of problems. They engage in a variety of problem situations and along the process learn mathematical content (Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier & Wearne, 1996). They also use mathematical knowledge to solve real life problems.
c) The role of social interaction
The problem-centered classroom is a place where problem posing and problem-solving takes place. These processes are characterized by invention, explanation, negotiation, sharing and evaluation (Nakin, 2003). As Murray, Olivier & Human (1993) point out in this regard, social interaction creates the opportunity for children to talk about thinking and encourages reflection; students learn not only from their own constructions but also from one another and through interaction with the teacher. The opportunity to exchange, discuss and evaluate one's own ideas and the ideas of others encourages decentration (the diminution of egocentricity), thereby leading to a more critical and realistic view of the self and others (Piaget in Post, 1980).
d) The role of the teacher
In a problem-centered classroom, the role of the teacher is no longer that of transmitter of knowledge to students, but rather a facililator of their learning. He/she has "the role of selecting and posing appropriate sequences of problems as opportunities for learning, of sharing information when it is necessary for tackling problems, and of facilitating the establishment of a classroom culture in which pupils work on novel problems individually and interactively, and discuss and reflect in their own answers and methods" (Hiebert, Carpenter, Fennema, Fusson, Human, Murray, Olivier & Wearne, 1997). Casey (1997) compares traditional views with current views on the roles of teachers and learners in learning as follows: "The old teaching paradigm implies that learning only happens when the teacher puts information into children's heads. The new paradigm does not imply that the teacher is unnecessary... a knowledgeable teacher who acts as a guide, facilitator, or fellow learner is essential".
Teachers have to constantly assist and support individual learners to develop their cognition at their own level and pace. The teacher has to plan, set up, manage and evaluate the teaching and learning activities to benefit the total development of every individual in the classroom. He/she must be thoroughly organized in planning appropriate activities, providing opportunities and creating a classroom atmosphere with his/her learning objectives in mind. He/she should create learning environments containing multiple sources of information and multiple viewpoints where students think, explore and construct meaning (Nicaise & Barnes, 1996) and situations to develop creative thinking and develop a wide range of problem-centered activities and materials aid problem-solving development in learners. He/she should also encourage learners to think critically, adapt ideas that make sense to them, invent many different ways to solve problems and to expand and enhance the development of mathematical concepts through problem-solving activities.
Teachers guide learners to discover and develop mathematics skills, such as active inquiry and reflection, in order to analyze and synthesize information, solve problems and successfully construct new knowledge through creative participation and understanding. Progressive teachers facilitate learning by selecting and implementing suitable learning matter and by motivating learners to improve their personal skills and abilities through the use of different materials and tools, such as computers. Teachers observe and evaluate learners' progress and provide them with relevant feedback in this regard. They thus monitor and guide rather than dominate and direct learning activities (Bonk & King, 1998; Newby, Stepich & Russel, 2000).
d) The role of the learners
In the problem-centered approach, learners choose and share their methods (Hiebert et al., 1997). Learners should also be free to express themselves without fear of reprisal. Mistakes are often as constructive as the correct strategies in helping learners to understand the mathematics involved (Erickson, 1999; Hiebert et al., 1997). According to Hiebert et al. (1997), mistakes provide opportunities for examining errors in reasoning, and thereby raise learners' level of analysis. Learners should realize that learning means learning from others and must take advantage of others' ideas and results of their investigations.
Communication and Collaboration
Communication and collaboration are two of the essential processes in understanding and learning mathematics because they allow students to reflect, share and discuss their understandings of mathematical concepts and procedures (Ontario Ministry of Education, 2007). Gadanidis, Graham, McDougal and Roulet (2002) underline the importance of collaboration and communication by arguing "mathematical learning is a social activity that helps students learn from listening and sharing and also from watching the actions, movements, and manipulations of others". Paper-and-pencil method is the most preferred method for students to demonstrate their solution processes in face-to-face environments because students can easily share their solution procedures and describe what they did by drawing tables or sketching diagrams. As a consequence of this communication, they can improve their understandings and cognitive abilities. However, this approach is rather challenging in online environments, and the existence of "obstacles associated with this text based communication interfaces, where it is difficult to expressed ideas with mathematical language and graphical representation" may prevent effective collaboration (Gadanidis et at., 2002). This challenge will be the main focus of our study.
Collaborative learning has situated its roots in the theory of distributed cognition introduced by Lev Vygotsky since 1930s as meta-analyzed by Morgan, Brickell, and Harper (2008). Distributed cognition describes how people interact with their environment including each other in order to advance their cognitive abilities, but not capacities. Sweller (1999) explained the limits of cognitive capacity of individual by introducing cognitive load theory and possible solutions for the effective use of this limited capacity. Collaborative learning enhanced by peer interaction not only in face-to-face settings but also in online environments provided evidence for a two-way benefit for learners (Tseng and Tsai, 2007). They analyzed online peer assessment and the role of peer feedback and demonstrated that students learn better by getting feedback as well as by providing feedback.
This chapter describes the methodological rational for the study. There is a range of methodologies and methods available for researchers. In mathematics education researchers should make explicit the theories that influence their work since these theories influence both the ways in which they work in the classroom and the ways they analyze their data. The following research design is structured according to Crotty's (1998) suggested research processes.
Crotty (1998) argues that in developing a research design the researchers should answer two basic questions: firstly, what methodologies and methods will be employed in the research and secondly how this choice and use of methods and methodologies is testified. The second question deals not only with the purpose of the research but also with the researcher's understanding of reality (theoretical perspective) and about what human knowledge is and what it entails (epistemology) (Crotty, 1998). Thus the two initial questions have explained into four: what epistemology is embedded in the theoretical perspective, what theoretical perspective lies behind the methodology, what methodology controls our choice and use of methods and what methods are proposed to be used. These four elements are presented separately because each element is substantially different from the other (Crotty, 1998).
Epistemology refers to the theory of knowledge embedded in the theoretical perspective (Crotty, 1998). This study is based on a social constructivist view of learning: pupils learn mathematics through active construction of their own knowledge and this can be facilitated in a computer environment through the interactive process of conjecture, feedback, critical thinking, discovery and collaboration (Howard et al., 1990). I consider constructivism to be the socially collective generation and construction of meanings rather than a meaning-making activity of the individual mind as Crotty (1998) claims. I do not take constructivism to highlight the unique experience of an individual that tends to resist the critical spirit (Crotty, 1998); in contrast, I ground my research on a social constructivist nature of knowledge in which: The meaning are negotiated socially and historically. In other words they are not simply imprinted on individuals but are formed through interaction with others (hence social constructivism) and through historical and cultural norms that operate in individuals' lives (Creswell, 2003).
Social constructivism claims that rather than being transmitted, knowledge is created or constructed by each learner (Leidner and Jarvenpaa, 1995); there is no knowledge independent of the meaning attributed to experience constructed by the learner (Hein, 1991). According to certain cognitive theories learning does not involve a passive reception of information; instead, the learning process can be regarded as an active construction of knowledge in a learner-centered instruction (Kapa, 1999). Constructivism claims that students can not be given knowledge; students learn best when they discover things, build their own theories and try them out rather than when they are simply told or instructed. Vygotsky argues that: Direct teaching of concepts is impossible and fruitless. A teacher who tries to do this accomplishes nothing but empty verbalism, a parrot-like repetition of words by the child, simulating a knowledge of the corresponding concepts but actually covering up a vacuum (Vygotsky, 1962).
But participating in social constructivism activity students have the opportunity not only to learn mathematical skills and procedures, but also to explain and justify their own thinking and discuss their observations (Silver, 1996). From a social constructivist perspective ICT offers teachers a powerful pedagogical tool-kit (O'Neill, 1998). Hoyles (1991) argues that in mathematics lessons involving computers, learning is achieved through social interaction for three reasons: the social nature of mathematics; the collaboration that computer based activities invite; the basis for viewing the computers as one of the partners of the discourse.
The theoretical perspective refers the philosophical stance informing the methodology, providing a context for the process followed and justifying its logic (Crotty, 1998). Recognizing the fact that there are multiple socially constructed realities this study adopts the interpretive paradigm and more specifically symbolic interactionism as the primary theoretical perspective.
The interpretive paradigm emerged in the social sciences to break away from the constraints imposed by positivism (Boghossian, 2006). The main aspiration of the interpretive paradigm is to understand "the subjective world of human experience" (Cohen et al., 2007). In order to maintain the integrity of the investigated phenomena efforts are made by the researchers to enter into the culture and find out the insider's perspective. Interpretive researchers see reality as a social construct and try to understand individuals' interpretations of the world around them (Bassey, 1992). Researchers work directly with experience and understanding in order to see the observer's view point and thus build the theory (Cohen et al., 2007).
As seen above, Creswell (2007) argues that the meanings constructed are negotiated socially and historically. The interpretivist approach looks for 'culturally derived and historically situated interpretations of the social life' while symbolic interactionism 'explores the understanding and meanings in culture as the meaningful matrix that guides our lives' (Crotty, 1998).
This is directly linked to the purpose of the research: to get inside the classroom and see how the computer software could be used as an aid in pupils' understanding of mathematics. Only through significant symbol, for example language and other symbolic tools which humans within a culture share and use to communicate, researchers can become aware of the insiders' perceptions and attitudes and interpret their meanings and intentions; hence symbolic interactionism (Cohen et al., 2007; Crotty, 1998).