Features of constructivism

Another view of learning is from a cognitive perspective, i.e., constructivism. An individual’s reasoning and cognitive growth is emphasized from perspectives of cognitive psychology (Fang & Chung, 2005; Voigt, 1994). Here, learning is interpreted as a growth in the internal cognitive areas (Wenger, 1998). Learning is typically described inside the mind of the individual from acquiring knowledge (Ford & Forman, 2006; Greeno, 2003; Peressini et al., 2004), or growth in conceptual understanding (Ford & Forman, 2006; Peressini et al., 2004). It is understood that knowledge is thought to be able to be transposed/generalised to other situations (Peressini et al., 2004) but the characteristics of tasks and contexts might affect the transformation of knowledge in other situations (Peressini et al., 2004).

Cognitive theorists argue that what is learned can also be independent of the context, even while learning takes place in a social context (Peressini et al., 2004). In contrast, some regard cognition as situated in the context, as a process of conceptual construction from reasoning information (Wenger, 1998). The focus here is on the “processing and transmission of information through communication, explanation, recombination, contrast, inference, and problem solving” (Wenger, 1998, p. 279). Prior experiences are significant and benefit students when making sense of new information (Wenger, 1998).

Cognitive psychology, on the other hand, concerns how children connect mathematics with their world in order to make sense out of both. It assumes that children bring knowledge and experiences to the classroom and when presented with a problem, through grappling with it and finally realizing that there are many possible paths that can be taken to arrive at a “satisfactory” solution, they develop their understanding.

The term constructivism has been interpreted from pedagogical, psychological, philosophical (Bettencourt, 1993) or sociological tendencies (Wood, Cobb & Yackel, 1991). For example, some scholars considered that constructivism is a theory of knowing (von Glasersfeld, 1993), a theory of knowledge (Bettencourt, 1993) related with personal construction (Wood et al., 1991), an epistemological theory, a theory about learning, teaching and administration of education (Matthews, 2000), and a theory of cognitive development (Confrey & Kazak, 2006; Greenes, 1995; Noddings, 1990). Therefore, cognitive enhancement is central for constructivist teaching (Kickbusch, 1996). An examination of the theory of constructivism reveals that learning is actively constructed by students (Cobb, 2007; Lesh, Doerr, Carmona & Hjalmarson, 2003) rather than passively received by teachers' transference (von Glasersfeld, 2005). So, the ownership of learning belongs to the learners and not to the teachers (Hong, Li & Lin, 2005; von Glasersfeld, 1993).

Students need to make sense of different ideas and activities and organize them into their own cognitive schemas, selecting and adapting (Boaler, 2002a; Confrey & Kazak, 2006) and reorganizing knowledge as part of their own constructions (Even & Tirosh, 2008). Their prior ideas affect the ways in which they make sense about new experiences (von Glasersfeld, 1995; Windschitl, 1999a) and these experiences are also influenced by the students’ social and cultural contexts (Windschitl, 1999b).

Constructivism is one possible way of thinking and knowing, and is a model that can never be claimed as “true” but more so a personal interpretation of reality (Confrey & Kazak, 2006; Hammersley, 2009; Lesh et al., 2003; Liu, 2004; Malara & Zan, 2008). Constructivism is one of the theories (for example, symbolic interactionism, the distributed view of intelligence) which emphasize student thinking development (Cobb, 2007). The recent development of constructivism was closely incorporated with a school of psychology and sought to explore the characteristics of learning (Lerman, 2001). Constructivism can serve to interpret the teaching or learning model and lead to the explanation of educational practices

such as individual development or analyses among groups within a specific unit (Confrey & Kazak, 2006); for example Gravemeijer’s work (1999).

According to Wenger (1998, p.279), “constructivist theories focus on the processes by which learners build their own mental structures when interacting with an environment”. Self-directed activities are favoured by teachers or researchers in classroom practices and lead to the development of students’ conceptual thinking abilities, especially in individual design and discovery (Papert, 1980; Wenger, 1998).

The different types of constructivism are individual/radical constructivism and social constructivism. The work of Piaget has great influences on constructivism and cognitive theorists (Confrey & Kazak, 2006); especially for individual constructivism (Scott, Cole & Engel, 1992; Smith, 1999). The followers of Piaget perceived constructivism as an individual learning independent from cultural and people influences (Scott et al., 1992; Smith, 1999). Although it is impossible to understand inside of a person’s mind, individual constructivists claimed that the creating models or metaphors of an individual’s thoughts enhance the ways to interpret learning (Smith, 1999). The majority of constructivists can be termed as close to individual and radical constructivism (von Glasersfeld, 1995; Smith, 1999).

In summary, constructivism may be viewed as a way of thinking and knowing, where knowledge is a personal construction (Cobb, 2007), and interpretation of reality rather than an objective truth (Hammersley, 2009; Malara & Zan, 2008; von Glasersfeld, 1993). This theory places a focus on cognitive, epistemological and knowledge development (Matthews, 2000; von Glasersfeld, 1993). Further, constructivism as it applies to teaching and learning, has a student-oriented focus. The ownership of learning belongs to students rather than the teacher (Hong, Li & Lin, 2005; von Glasersfeld, 1993).

Social constructivism is highly influenced by Vygotskian’s and Bruner’s concepts (Hartas, 2010). Lerman (2001) comments that there are differences between these

two scholars. He stated that Vygotsky emphasizes sociocultural views of learning that generates a meaning closely associated with culture while Bruner highlights the importance of actions in learning and emphasizes the behaviour of exploring meanings in culture. Social constructivists apply cognitive perspectives to interpret individuals’ development in social interactions (Lesh & Doerr, 2003). Individuals, based on their experiences and previous knowledge, actively construct knowledge, especially concepts and hypotheses (Ernest, 1991), through interacting with people or cultural and social worlds (Hartas, 2010). Opportunities for learning occur during social interaction/dialogues such as teacher-student and student-student dialogues, students' explanations and justifications (Hong, Li & Lin, 2005; Ernest, 1991; Wood et al., 1991), argument, negotiation and mediation that will produce a consensus or a social form of knowledge (Confrey & Kazak, 2006; Jaworski, 1994).

It should be noted that, mathematical discussion has been emphasized in constructivist teaching (Richardson, 2003; Threlfall, 1996), a social perspective of learning (Peressini et al., 2004;Van der Lindendagger & Renshaw, 2004), and in recent education reforms such as in Taiwan or the USA (NCTM, 2000; Taiwan Ministry of Education, 2001). A social perspective on learning recognizes the importance of students presenting a collective form of knowledge through discourse in classrooms (Driver, Asoko, Leach, Mortimer & Scott, 1994; Wood, 1999). Discourse is not a tool to shape ideas into some ‘material’ actions (expected content knowledge), but rather a collective form of inference (Pontecorvo & Girardet, 1993, p. 366; O’Connor, 1998). Discourse also presents ways of thinking and serves as a social knowledge construction (McNair, 1998; Pontecorvo & Girardet, 1993; O’Connor, 1998); especially a synthesis on connecting core mathematical concepts (Romberg, Carpenter & Dremock, 2005). Mathematics learning has been considered as “a trajectory of participation in the practices of mathematical discourse and thinking” (Boaler & Greeno, 2000, p. 172). To some extent, each classroom is a unique social environment, and teachers use discourses to deliver their goals/lessons (O’Connor, 1998).

One remarkable character of constructivist teaching is that an individual or group generates “meaning–making” through the process of classroom conversations (Richardson, 2003, p. 1623). Classroom discussions have been recognized as important elements to improve students' mathematical conceptions (Wood, 1999) through spoken and written communication (Taiwan Ministry of Education, 2001; NCTM, 2000).

In conclusion, social constructivism places a focus on the fact that students learn via social interactions (Hartas, 2010; Hong, Li & Lin, 2005; Lesh & Doerr, 2003), through constructing their knowledge and interacting with social dialogues among students and the teacher (Hong, Li & Lin, 2005). Students are engaged in activities which allow them to select, adapt and make sense of ideas and activities into their own cognitive schemas (Boaler, 2002a; Confrey & Kazak, 2006). Thus, this environment provides the impetus for students to actively construct their own learning through social dialogues rather than passively receive teachers' transference (Cobb, 2007; von Glasersfeld, 2005). Their arguments and negotiations produce a consensus or a social form of knowledge (Confrey & Kazak, 2006). Students’ previous learning experiences and the influence of their social and cultural contexts also affect their learning (Windschitl, 1999a).

Furthermore, mathematical classroom discussions afford opportunities to students to present their mathematical ideas through expressions, agreements, and disagreements (Peressini et al., 2004), while engaging in “sense-making” and problem solving practices (Boaler & Greeno, 2000, p.172). Class discussion is a continuous negotiation between members (Pontecorvo & Girardet, 1993; O’Connor, 1998). Students can practice evaluating their own work and that of others to make sense or arguments during class discourse (in small group time or in whole-class discussions) (Lamberg, 2013; Lampert, 2001). The conceptual structure of subjective mathematical knowledge is achieved through the functions of language (Ernest, 1991). Through this process, students are likely to identify conflict and restructure their own thinking (Hiebert & Wearne, 1993). As students understand and learn about the discourse, they will improve their own mathematical dialogue (Rittenhouse, 1998). Moreover, students’ higher-order

thinking skills, including skills of discovering, reasoning, organizing and arguing (Torff, 2003), can be achieved in mathematical class discussions.

Opportunities in classroom discourses offered chances for students to assess their understanding in solving problems (Webb, 1991), and chances for receiving support from others (a teacher/students) for misunderstood or incomplete answers (O’Connor, 1998; Webb, 1991). Students have opportunities to control the pace and content of the teaching activities (Webb, 1991).

Thus, opportunities for class discussions are offered to allow students to contribute to “the judgement of validity, and to generate questions and ideas” (Boaler & Greeno, 2000, p. 189). As Resnick (1988) described it, whole class discussion is likely to employ a large group as a medium to empower individual students to formulate their ideas for conflict and development of ideas. The strategy of students sharing or explaining provide opportunities for other to get further clarifications and understanding (Franke et al., 2007). This strategy will therefore bridge the growth of “connected knowing” among individuals (Boaler & Greeno, 2000). The classroom base knowledge will be enriched (Brown & Campione, 1994) and will lead to the development of collective public knowledge (Ball & Bass, 2000b; Franke et al., 2007; Kazemi & Stipek, 2001). Student discourses also can be regarded as verbal forms of thought about relations of mathematical ideas, reasoning, asking questions, making of plans (Franke, et al., 2007) and correlated with students’ ability to use conceptual knowledge while explaining a phenomenon (Van Boxtel et al., 1997).

Classroom discourse is therefore regarded as the key principle for the educational design and instructional tools (Cazden, 2001). Researchers believe that “Students in these learning communities are capable of deep, sustained, complex thinking, both in whole-class discussions and in their small groups” (Brown & Campione, 1994, p. 261). Lively open-class discussions represent normal class patterns (Pirie, 1988) that benefit the development of a student’s mathematical understanding (Boaler & Greeno, 2000; Cazden, 2001; Franke et al., 2007).

mathematical reasoning (Hunter, 2008). Further, teachers can know students’ thinking from class conversations and this is essential for teaching for understanding (Franke et al., 2007). Thus, class discourse can also be regarded as an important part of ongoing classroom evaluations (Kahan, Cooper & Bethea, 2003). In addition, the teacher can generally teach students not only mathematics but also how to study mathematics, by asking students to reason, to explain, to interpret the assumptions of their peers, and to explore mathematics together (Lampert, 2001). Another benefit of exploratory talk is that class discussions are also able to foster students’ participation in thinking (reasoning) in the whole class discussion (Nathan & Kim, 2009), such as shown in Nathan & Kim’s work (2009), and Hunter’s (2008) work.

The characteristics of instruction that promote classroom discourse are not well documented in the literature (Franke et al., 2007). However, some key elements that foster class discourse have been pointed out by several scholars. Generally, teachers are mindful to allow conversations to serve as a source of students’ ideas (Walther, 1982; Lampert, 1990a). To discuss this in detail, in order to guide class discourse, a teacher needs to (1) select and offer discussion questions, (2) coach, explain, respond and challenge students’ conversations, (3) address mathematical meaning or norms in time, and (4) maintain the engagement of all students (Franke et al., 2007; Peressini et al., 2004). Another detail that could be added to support class discourse is that of problem posing by teachers to provide a range of answers; not just right or wrong (Franke et al., 2007; Lamberg, 2013; Lampert, 2000). In addition the teacher can allow some time for students to explore their own ideas as well as those of others (Hunter, 2005; Nathan & Knuth, 2003), question students’ thinking (Ford & Forman, 2006; Lampert, 2001), explore students’ mistakes to offer chances for them to reflect on their learning by explaining and challenging their own arguments (Ford & Forman, 2006), and managing the coverage of the content (Lampert, 2001) Students gaining ownership of their learning will better manage the coverage of the content to be learnt (Lampert, 2001). As a result, through discourse (class discussion), a teacher can grasp the mathematical needs of the class and understand students’ mathematical thought. Specifically, they can find out what students know, their misconceptions, and how these misconceptions might have developed (Franke et

al., 2007; Romberg et al., 2005) and apply students’ responses to instruction (Romberg et al., 2005). This is demonstrated in Lampert’s work (2001). This will also benefit teachers’ question asking, to connect to students’ ideas and extract multiple strategies to assist the development of students’ mathematical proficiency (Franke et al., 2007).

Teachers may be called upon to perform different roles such as facilitators (BRAP, 2003), where they are engaged in fostering students’ participation and mathematical discourse amongst each other. This helps students to develop their comprehension and it helps them to use the discourse to deepen their mathematical understanding (Franke et al., 2007; Rittenhouse, 1998). Teachers may also function as mediators to reconcile differences in students’ inner knowledge and understanding of mathematics (Walther, 1982; Lampert, 1990a). Teacher talk will support and develop students’ mathematical command as they move from legitimate peripheral participation of class discussions to enhance engagement (Lave & Wenger, 1991; Rittenhouse, 1998).

On the other hand, the process of discourse lays the foundation to transform the classroom practices into a supportive learning community (BRAP, 2003; Hartas, 2010), to establish a collective understanding through the class discourse and students’ justification (Hunter, 2008) from the multiple input from the teacher and students. Besides this, seating arrangement can help to balance supportive social interaction and support to clarify students’ spoken ideas (Lampert, 2001). However, some challenges can arise from class discussions. For example, new students often find it difficult to make sense of what is being said, even at a normal rate of speed for conversations (Rittenhouse, 1998). Many scholars have discussed the two core elements: justifications and arguments inside classroom discourse that lift up high level of mathematical thinking and understanding. The next sections will further explore these two factors.

(i)Justifications and Arguments

A constructivist approach to teaching offers teachers several opportunities for students to engage in activities that require them to justify and establish reasonable arguments. The rich information (justifications) is contained in class

discourse while developing and explaining ideas in classes about their problem solving strategies (Webb, 1991;Wood et al., 1991). Justification can be defined as the value of something to be true or certain (Ball, 2003).

Mathematical arguments offer individuals opportunities for reasoning (Wood, Williams & McNeal, 2006), to criticise and justify ideas from a collective point of view and to generate new perspectives (Hunter, 2006b; Rojas-Drummond & Zapata, 2004; Wood et al., 2006) and conceptual understanding (Wood, 1999). Moreover, students can create a public knowledge from different forms of mathematical explanations in the class discourses that are aligned with the content and students’ inspections/inquiries. This will also develop the mathematical identities of students (Franke et al., 2007). In addition, mathematical content discussions and debates can also lead to the development of student autonomy (Hunter, 2006b) and competence (Hunter, 2006b; Lambdin & Walcott, 2007).

In conclusion, class discussions can foster mathematical arguments that benefit students’ mathematical understanding (Ball & Bass, 2000b; Boaler & Greeno, 2000; Franke et al., 2007; Lampert, 2001), knowledge (Franke et al., 2007; Wood et al., 2006) or reasoning (Hunter, 2006b). Moreover, informal discourse can enhance a higher-level of thinking (Franke et al., 2007; Hunter, 2008; Nathan & Kim, 2009; Wood et al., 2006). For example, Hunter (2008) reported that four teachers challenged students through questioning, in-depth explanations, and justification. This form of discourse led to the development of collective reasoning and views. Other studies also have indicated the positive relationship between classroom discourse and students’ learning outcomes (Hiebert & Wearne, 1993; O’Connor, 1998; Webb, 1991). For example, high achievement correlates with the behaviour of giving explanations to classmates (Webb, 1991).

(ii)Two Patterns of Classroom Discourse

Classroom discourse has been classified according to two models (Cobb, Yackel & Wood, 1993; Peressini et al,2004). One type is that arriving at a solution is the driving force for class discussions; typically found in the traditional school mathematics classrooms (Peressini et al, 2004). Classroom interactions can be

illustrated as three steps: the teacher starts first to pose a known-information question (Cobb et al., 1993;Peressini et al., 2004), students respond, and then the teacher evaluates the feedback (Peressini et al., 2004). These steps match an “IRE (initiate–respond–evaluate)” pattern (Cross, 2009, p.340).

In contrast, in the other type of classroom discourse, the students’ dialogue drives the mathematics teaching and learning flow in an inquiring classroom. Information-seeking questions are raised first from the teacher and it is expected that students give an explanation of their interpretation and problem solving (Peressini et al, 2004).

Other strategies when used appropriately are possible to increase the level of class discourse. This can be seen for example in cooperative groups or revoicing strategies for students’ mathematical conversations (involving explanation, rephrasing or reporting) (Franke et al., 2007). Teachers facilitate discourse around mathematical ideas through support and monitoring or extracting students’ ideas from discussions (Franke et al., 2007).

The above section has illustrated how different teaching styles lead to different class practices. The following section is going to introduce constructivist teaching, the role of a teacher and student in constructivism, advantages of constructivism, relevant studies long-term research and disadvantages of constructivism

(a) Constructivist Teaching

The constructivist learning approach, when applied to teaching, is aimed at producing life-long learners. It is intended to build up learners as skilled and thinking people (Hagg, 1991). However, constructivism as is applied to teaching, is relatively less developed than the views of constructivist learning (Prawat, 1992). This is also true for the factors that contribute to effective constructivist teaching which are still under investigation (Richardson, 2003).

Most research on developing constructivist pedagogy, concerns the relationship between teachers’ actions (including teachers’ beliefs, values, behaviour and activities) and students’ learning (Richardson, 2003). The other important area of

developing constructivist pedagogy is linked to theory building. Research experiences will release information of effective teaching practices/pedagogy to benefit teacher education and professional development (Richardson, 2003). Investigations of effective teaching practices might suggest to go back to the focus of classroom practices relating to teaching and learning (Boaler, 2002c). Researchers can start from a subject or a general level (Richardson, 2003). For example, some researchers have discussed the effective teaching practices with respect to students’ learning outcomes from the constructivist pedagogical perspectives, such as standardised tests (Boaler, 1997; Boaler & Staples, 2008; Hiebert & Wearne, 1993), students’ deep mathematics understanding (Ball & Bass, 2000b; Boaler, 1997; Boaler & Staples, 2008) and some disciplines of establishing constructivist classrooms (Boaler, 1997; Boaler & Staples, 2008; Malara & Zan, 2008). Moreover, teachers’ knowledge of the subject matter and their awareness of cultural issues are also addressed in the theory building of developing constructivist pedagogy (Richardson, 2003).

When constructivism is applied to teaching, it does not specify a particular model of instruction (Windschitl, 1999b). Constructivism states that students learned best through conducting their own approaches to problems in reaching mathematically competence (Lambdin & Walcott, 2007), and students will learn from different forms of instruction (Richardson, 2003). It is rather a set of beliefs, norms and practices that contribute to the culture in classrooms and in the school, but new relationships exist between teachers, students and mathematical ideas (Windschitl, 1999b). The constructivist view of learning and its application to teaching has the following characteristics:

 Teachers minimise their direct instruction or lecture mode (Simon & Schifter, 1991), and promote discussion and problem posing by students (Wheatley, 1991; Trotman, 1999).

 Teachers develop their own curricula according to their students' current conceptions or needs (Begg, 1996; Windschitl, 1999b). It is possible that curricula developed from theses are not driven by external curriculum such as school schemes or national syllabi (Steffe, 1990). Teachers need to be experienced in applying diverse strategies to help students’ understanding, such as explaining, demonstrating, and advising etc

(Windschitl, 1999b).

 Teachers encourage and facilitate discussion (Brooks & Martin, 1999; Trotman, 1999; Windschitl, 1999b) by creating a culture for inquiry (Windschitl, 1999b); guiding and framing an issue which is realistic and open-ended for students' discussion (Brooks & Martin, 1999; Threlfall, 1996; Windschitl, 1999b). Teachers select activities to facilitate discussions (Gravemeijer, 1994). Teachers allow a certain waiting time after giving questions (Brooks & Martin, 1999). It places an emphasis on students’ explaining their thoughts (Kazemi & Stipek, 2001). Some other reform studies also valued the waiting time, beside the advantages