SCAFFOLDING- FOR MATHEMATICS LEARNING

Educators have been talking much about constructivism as the learning theory for mathematics education since the last quarter of the last century. Has this new philosophy improved students’ performance in

mathematics?

This paper proposes scaffolding as a teaching strategy to enhance mathematics learning in the classrooms. Scaffolding is formulated from Vygotsky’s concept of the zone of proximal development. It emphasizes active participation or a greater degree of control from students over their learning. For successful scaffolding, five key features need to be addressed. These are:

 1. Students explain and justify their solutions.

 2. Teachers continuously assess students’ understanding.

 3. Teachers take into consideration students’ perspectives.

 4. Scaffolding tailor to the needs of students.

 5. Students take up or use the scaffolding.

When the scaffolding tendered is tailored to the needs of a student tackling a meaningful and challenging task, the student will be able to accomplish the task, which is otherwise impossible. However, teachers need to change their role in the classroom from the sole source of mathematical knowledge to facilitators in the development of students’ mathematical constructions, while employing scaffolding.

The 1990s had been a period of great change for mathematics education. A theory of learning called constructivism emerged. New curriculum documents were shaped by educators, which place more emphasis on mathematical constructions, rather than on contents. However, these educators avoided making recommendations about teaching approaches or strategies, which could help realizing this emphasis. Scaffolding as a teaching strategy could help to materialize the dream of these educators. In mathematics lessons where scaffolding is employed as a teaching strategy, the conventional assumptions about what it means to know mathematics are challenged. It becomes clear to the teachers that teaching is not only about teaching what is  conventionally called content, but also facilitating students’ mathematical constructions. Thus it is necessary for both teachers and students assuming different kind of roles and responsibilities to do different sorts of activities together.

Radical constructivism is increasingly being criticized for its limitations as a learning theory. Those educators who adhere to the Vygotskian School of Learning (Confrey, 1990, Steffe & Kieren, 1994, Lerman, 1996) suggest the extension of radical constructivism to social constructivism by incorporating ‘intersubjectivity’, which views mathematics learning as both a collective human activity and an individual constructive activity, rather than just an individual element for radical constructivism. Confrey (1990) says that

‘... the constructive process is subject to social influences. We do not think in isolation; our choice of problem, the language in which we cast the problem, our method of examining a problem, our choice of resource to solve the problem, and our acceptance of a level of rigor for a solution are all both social and individual processes.’ (p. 110)

In other words, there are two faces of mathematics. These are mathematics in students’ heads and mathematics in the students’ environment. The main concern of social constructivists is how to account for mathematics learning in the students’ environment.

Vygotsky’s school of thought probably has the most profound influence on the formation of the concept of scaffolding in the cognitive development of a child (Greenfield, 1984, Rogoff & Gardner, 1984, Stone, 1993). Vygotsky conceptualizes the idea of the zone of proximal development. He says that children who by

themselves are able to perform a task at a particular cognitive level, in cooperation with others and with adults will be able to perform at a higher level, and this difference between the two levels is the child’s ‘Zone of Proximal Development’. Vygotsky claims that

‘Every function in the child’s cultural development appears twice, on two levels. First on the social, and later on the psychological level; first, between people as an interpsychological category and then inside the child as an intrapsychological category.’ (1978, p. 128)

The process by which inter becomes intra is called internalization and involves more than the endowment of the child and more than the child can accomplish on his or her own, but it occurs within the child’s zone of proximal development. Hence Vygotsky proposes that the cognitive development in a child is social, which involves another person and the society as a whole. In other words, social interaction taking the form of dialogue or cues or gestures, plays an important role in concept formation.

This also outline six key functions of scaffolding:

 1. Recruitment: engaging the student in a meaningful and interesting task;

 2. Reduction in the degree of freedom: breaking the task into manageable components;

 3. Direction maintenance: keeping the students on-task and on-track to a solution;

 4. Marking critical features: accentuating key parts of the task;

 5. Frustration control: decreasing the stress of the task but not so far as to create total dependency on the tutor; and

 6. Demonstration: The tutor imitates attempted solution by the tutee, hoping that it will be imitated back by the tutee in a more appropriate form.

Greenfield (1984) defines the scaffold for building construction as follow:

‘The scaffold, as it is known in building construction, has five characteristics: It provides a support; it functions as a tool; it extends the range of a worker; it allows the worker to accomplish the task not

otherwise possible; and it is used selectively to aid the worker where needed.’ (p. 118)

Based on this definition, she puts forward the following idea of the scaffolding process in a learning situation.

‘… the teacher’s selective intervention provides a supportive tool for the learner, which extends his or her skills, thereby allowing the learner successfully to accomplish a task not otherwise possible. Put another way, the teacher structures an interaction by building on what he or she knows the learner can do. Scaffolding thus closes the gap between task requirement and the skill level of the learner …’ (p. 118)

Cambourne (1988) highlights the common interactions in scaffolding as:

Focusing on a gap to bridge in child skills/knowledge to accomplish a task. Extending by raising the skill level: asking questions like ‘What else will you (would you, could you) do?’ when the teacher is satisfied with the performance of the child.

Refocusing by encouraging clarification and justification by asking questions like ‘Is this what you are trying to say (do, write) or is it something else?’ when the teacher is confused or unclear about what the child is doing or saying.

Redirecting by offering new resources if there is a mismatch between the child’s intent and the message or in the expectations which the teacher holds for the child.

No matter how one defines scaffolding, it is ‘a metaphor for the temporary framework experts help create for novices in their attempts to solve problems’ (Lehr, 1985). 

Scaffolding exhibits the following key features:

• Scaffolding has the capacity to enhance the potential of an individual within his zone of development;

• It requires a meaningful and challenging task;

• It emphasizes active participation of a learner in tackling a task; and

• Scaffolding is developmental.

At the beginning , students either did not response to the teachers’ scaffolding questions, or students’ responses were confined to brief phrases and in single disconnected sentences. The teachers then amplified what they supposed their students might have meant. As the teachers emphasized students explaining and justifying their solutions, the students improved drastically in doing so. As such, the teachers and their students often had a shared understanding on an issue, which ensured successful scaffolding.

Thus scaffolding is a teaching strategy that can enhance mathematics learning and help implementing constructivism in the classrooms. However, five critical features need to be addressed for successful scaffolding. These are: 

(1) students explain and justify their solutions,

 (2) teachers continuously assess students’ understanding,

 (3) teachers take into consideration students’ perspectives,

(4) scaffolding tailor to the needs of students and

 (5) students take up or use the scaffolding. 

Finally, teachers need to reconceptualize their role as facilitators in the development of the students’ mathematical constructions rather than the sole source of mathematical knowledge while employing scaffolding in the classrooms.