Constructivism is a cognitive learning theory with a distinct focus on the mental processes that construct meaning. According to Van de Walle (2007) and Olivier (1992) the general principles of constructivism are based largely on Piaget’s processes of assimilation and accommodation; where assimilation refers to the use of existing schemas to give meaning to experiences while accommodation is the process of altering existing ways of viewing things or ideas that contradict or do not fit into existing schemas.
Hanley (1994) describes a classroom based on the traditional model of teaching as a “one- person show with a captive but often comatose audience”. She explains that classes are usually driven by “teacher-talk” and that there is a fixed world of knowledge that the learner must come to know. Hanley further remarks that “teachers serve as pipelines and seek to transfer their thoughts and meanings to the passive student, leaving little room for student- initiated questions, independent thought or interaction between students”.
According to Orton (1994, 2004), the evolution of constructivism does not imply a rejection of earlier attempts to facilitate more effective learning within a cognitive learning environment.
He argues that “it is a misunderstanding of constructivism to suggest that there is little the
teacher can do to facilitate learning simply because the construction must be carried out by the learner”. Although literature contains many references to ‘inquiry learning’ as being best for the construction of understanding, the teacher still has to organize it. Orton points out that constructivism appears to suggest that the teacher needs to provide the “scaffolding” which allows the learner to progress, and it requires great skill to provide the best scaffolding for each learner. He remarks that “a consistent policy of complete non-intervention by the teacher is therefore certainly not likely to be the best way to promote the construction of knowledge.
However, a policy of non-intervention with a certain child at a particular point in time or with a particular group might be appropriate, especially when the responsibility of learning has been fully accepted by the child or the group”.
Echoing similar sentiments, Murphy (1997) observes that an important concept of constructivism is that of “scaffolding which is a process of guiding the learner from what is presently known to what is to be known”. She points out that scaffolding allows learners to perform tasks that would ordinarily be slightly beyond their ability without the assistance and guidance from the teacher.
Within constructivism, the teacher acts as a facilitator of knowledge and must ensure that the learner has the ability to construct knowledge in their own minds through the process of discovery and problem-solving. According to Mouton (1996), a mathematical modelling strategy is in line with current constructivist perspectives on learning and is seen as a vehicle to a better understanding of mathematics. This is one of the primary reasons that the researcher has positioned this research study within this particular theoretical framework.
According to Van de Walle (2007) a commonly accepted goal among mathematics educators is that learners should understand mathematics, and the most widely accepted theory, which is constructivism, suggests that children must be active participants in the development of their own understanding. Lerman (1989); Njisane (1992) and Clements and Battista (1990) concur that in a constructivist teaching and learning environment, knowledge is actively constructed by the learner, and not passively received from the environment. Furthermore Van de Walle (2007) points out that constructivism provide us with insights concerning how children learn mathematics and guides us to use instructional strategies that begin with children rather than with ourselves. He finally notes that “constructivism rejects the notion that children are blank slates…… they do not absorb ideas as teachers present them, rather, children are creators of their own knowledge”.
Within the constructivist paradigm, teachers should create opportunities for children to create new mathematical knowledge by reflecting on the things that they do (their physical actions) and the ways that they think (their mental actions). According to Doerr (2007), a modelling approach to teaching mathematics calls for a major reversal in the usual roles of teachers and students, whereby students need to do more evaluating of their own ideas and teachers need to create opportunities where this evaluation can productively occur.
Constructivism makes provision for the fact that each individual person interprets and makes sense of the world in his or her own way. Within a constructivist teaching and learning environment, the learner should be able to make sense of a real-world problem in his or her own way. The interpretations that learners make will depend on their experience and upon social interaction with other people.
Developing learner’s personal mathematical ideas is very important to the constructivist teacher when the teacher encourages learners to use various methods for solving problems.
Through interaction with mathematical tasks and with others, the learner’s own intuitive mathematical thinking gradually becomes more abstract and powerful. The researcher believes that mathematical modelling as a teaching strategy is the key to developing autonomous and self-motivated learners.
The following is a summary of some of the characteristics of a constructivist teacher as outlined by Brooks and Brooks (1993):
• Become one of many resources that the student may learn from, not the primary source of information.
• Engage students in experiences that challenge previous conceptions of their existing knowledge.
• Allow student responses to drive lessons and seek elaboration of students’ initial responses. Allow student some thinking time after posing questions.
• Encourage the spirit of questioning by asking thoughtful, open-ended questions.
Encourage thoughtful discussion among students.
• Use cognitive terminology such as “classify”, “analyse” and “create” when framing tasks.
• Encourage and accept student autonomy and initiative. Be willing to let go of classroom control.
• Use raw data and primary sources, along with manipulative, interactive physical materials.
• Don’t separate knowing from the process of finding out.
• Insist on clear expression from students. When students can communicate their understanding, then they have truly learned.
Jonassen (1991) suggests that educators create constructivist learning environments for their learners and has provided the following list of principles as a guideline:
• Create real-world environments that employ the context in which learning is relevant;
• Focus on realist approaches to solving real-world problems;
• The instructor is a coach and analyzer of the strategies used to solve these problems;
• Stress conceptual interrelatedness, providing multiple representations or perspectives on the content;
• Instructional goals and objectives should be negotiated and not imposed;
• Evaluation should serve as a self-analysis tool;
• Provide tools and environments that help learners interpret the multiple perspectives of the world;
• Learning should be internally controlled and mediated by the learner.
In providing support for Jonassen’s (1991) principles listed above, Wilson and Cole (1991) outline the following concepts as central to constructivist design.
• The educator should embed learning in a rich authentic problem-solving environment;
• He should provide for authentic versus academic contexts for learning;
• There should be provision for learner control; and
• The educator should use errors as a mechanism to provide feedback on learners’
understanding.