Chapter 2: Literature Review
2.7 Constructivism and Mathematical Problem Solving
use their prior experience to gain new information in the process of problem-solving, the teacher's role is to facilitate this collaboration.
the theoretical framework for this study is based on Constructivist approaches build upon Piaget and Vygotsky’s ideas of using higher-order thinking and questioning techniques. Thus, what follows is a description of cognitive constructivism and social constructivism that comprise the basis of the emergent constructivism that is underpinning the theoretical framework of this study.
2.7.1 Cognitive Constructivism
The aim of cognitive constructivism as a teaching method is to support students in assimilating new knowledge to the existing knowledge. Constructivists claim that knowledge is not absorbed by students from extrinsic resources, rather they create their own meaning in their minds based on their prior knowledge and experience (Ertmer &
Newby, 2013). Learning is seen as a process of active discovery as the knowledge is actively constructed. The teacher's role is to facilitate this discovery by guiding the students in their attempt to assimilate new knowledge to the old and to make modifications to the old to accommodate the new (GSI Teaching and Resource Center, 2016). According to Piaget, learning is an internal process that requires cognitive conflict to occur in the mind of the individual. The most important principle of Piaget’s theory is the principle of equilibration that takes place in the process of adaptation (Powell & Kalina, 2009). Cognitive teaching methods aim to help students assimilate new information to existing knowledge and enable them to make appropriate adjustments to their existing intellectual framework to absorb that information (GSI Teaching and Resource Center, 2016).
Teachers and curriculum developers need to understand the critical transition in the student’s cognitive development, how the students think and learn. Cognitive development is an active process when the students assimilate the new knowledge to
the old, they will be able to make suitable adjustments to their existing intellectual framework to accommodate the new knowledge. Consequently, learning is relative to the students’ stage of cognitive development (GSI Teaching and Resource Center, 2016).
According to Piaget, there are four main stages in the cognitive development of the child (Powell & Kalina, 2009). The “sensorimotor” stage during the first two years. The second stage is the “preoperational” and lasts until around the age of seven years. Then the next stage of “concrete operational” from seven to eleven years where the child begins to develop logic but only on concrete objects. The fourth and final stage is the formal operational stage, which is lasting from eleven years to the rest of the child's life. The child at this stage can approach mathematical problems intellectually in an organized way (Piaget, 1957).
Students at the age of 15 are generally in the fourth and final stage of Piaget’s four stages of cognitive development that is called the formal operational stage. PISA is devoted to 15 years, old students. Thus, the released items of PISA are suitable for the students’ stage of cognitive development. At this period, students are supposed to show improvement in their ability to think abstractly, use advanced reasoning skills, make hypotheses and inferences, and draw logical conclusions. Students at this stage of their life should be provided with new opportunities to adopt good thinking habits and mathematical practices. This stage is the focus of this research as it is important to understand what the abilities of the 15 years old students are. In this stage, the children could perform abstract intellectual operations, learn to formulate, and make abstract hypotheses. The children learn to appreciate others' opinion as well as their own.
In light of Piaget’s theory, Gredler (2001) stressed the importance of the teacher role in choosing and providing the students with problems that allow the students to provide different ways to their solution, and all the students should be engaged in a problem-solving process that has an impact in encouraging the students and forcing them to think.
2.7.2 Social Constructivism
Hyslop-Margison and Strobel (2008) stated that Social Constructivism is a learning theory with roots in cognitive constructivism (Piaget, 1957) and sociocultural theory (Vygotsky, 1987). Even that Vygotsky agrees with Piaget’s claim that learners respond to their interpretation of stimuli, not to external stimuli, however, he claimed that Piaget had ignored the social component of learning. Social Constructivism emphasizes that learning, like all cognitive functions, is dependent on interaction with others such as teachers, students, and parents (Vygotsky, 1999).
The process of learning mathematics requires the students to actively construct the meaning of concepts through individual re-construction of knowledge and social interaction with other students, and teachers (Belbase, 2014). Learning is seen as a contextual process that depends critically on its collaborative nature as the students learn from each other or with their teacher guidance (Schunk, 2012). Based on social constructivism, learning never comes from scratch, the successful learner embeds new learning within old to expand understanding to incorporate the new experience.
According to social constructivism, learning is neither connected to the external world nor the learner’s mind, but it exists as the outcome of mental contradictions that result from group interactions with the environment (Schunk, 2012).
Moreover, learning as seen in social constructivism is based on real-life adaptive problem solving which results from social interaction through shared experience and discussion with others where the learner adapts rules to make sense of the world based on matching the new ideas against existing knowledge. Kukla (2000) stated that societies together create the properties of the world as the reality-based on social constructivism is seen as constructed not discovered through human activity.
Vygotsky stressed the social learning leads to cognitive development. An important concept that is essential to this socially mediated cognitive development is the “Zone of Proximal Development” (ZPD) that has been introduced as "the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem-solving under adult guidance, or in collaboration with more capable peers" (Vygotsky, 1978, p. 86).
In the ZPD there are two levels: the actual development level that indicates where the student has already reached, and to what level is the student capable of solving problems independently. The level of potential development is the ZPD level where the students can reach under the guidance of the teacher or by collaboration with other students. In the ZPD, the students will be able to solve problems and understand topics that they are not able to solve or understand at their level of actual development. This means that the level of potential development is the level at which learning happens.
The teachers should be aware that on one hand unless students are challenged and active in their learning, they will lose attention, while on the other hand, if the students face a great challenge, they might just give up. Vygotsky (1978) explained that “learning which is oriented toward developmental levels that have already been reached is ineffective from the viewpoint of the child’s overall development” (p. 89).
Therefore, according to Vygotsky, the intellectual development theory implies that teachers should organize learning to be just above the level of the actual developmental level of the individual student through the interaction of peer tutoring, collaboration, and small groups.
Vygotsky believes that the education role is to provide the students with suitable exercises within their ZPD to encourage and improve their learning. The teacher’s role is facilitating the students learning as they share knowledge through social interaction that is considered an effective way of developing skills (Dixon- Dixon-Krauss, 1996). Vygotsky's theories imply the importance of collaborative learning as the group members have different levels of ability where the more advanced peers can help less advanced members function within their zone of proximal development (McLeod, 2019).
The ZPD and the term “scaffolding” are mentioned together like two faces for the same coin in literature as ZPD is defined as the area where a child can solve a problem with the help (scaffolding) of an adult or more competent peer (McLeod, 2019), while Wood, Bruner, and Ross (1976) defined scaffolding as the process to help and lead a child or novice through the ZPD to solve a task or achieve a goal that is not reachable without others' help. In the classroom, scaffolding strategies could be modeling a skill, providing hints or cues, and adapting material or activity (Copple &
Bredekamp, 2009).
2.7.3 Emergent Constructivism
In response to educational reforms, the constructivist approach to teaching and learning is supported by many math educators (Draper, 2002). This research is framed
by the emergent perspective of constructivism that draws on both Piaget and Vygotsky.
For examining the mathematical growth of students, the emergent perspective is suitable "as it occurs in the social context of the classroom" (Cobb & Yackel, 1996, p.
176). The knowledge, in the emergent perspective, is personally constructed in the medium of social interaction and that individual and social dimensions of learning to complement each other (Tobin & Tippins, 1993; Murray, 1992; Cobb & Yackel, 1996). Students have an opportunity for mathematical learning when they try to understand other people's interpretations and when they try to compare their own solutions with those of others (Cobb & Yackel, 1996). From an emerging perspective, mathematics is a social activity as well as an individual activity. When negotiating criteria in the classroom, the teacher plays a pivotal role in initiating and guiding these standards, but the individual student has an active role in this formation as well (Cobb
& Yackel, 1996).
Constructivists depend on teaching practices that are rich in conversation and constructivists understand that experience, environment play important roles in learning where learners create their own knowledge based on interaction with other people (Draper, 2002). Classroom discourse is one of the seven NCTM Standards for teaching and learning mathematics (NCTM, 2007). The NCTM states, "In practice, students' actual opportunities for learning depend, to a considerable degree, on the kind of discourse that the teacher orchestrates" (p. 32). Findell (1996) stated that for student- centered math classrooms, the teacher should play the role of questioner and problem poser, so students make sense of their learning.