# Zones of Teaching and Learning

### A summary of Norton & Anderson, 2008

## Constructivism

Constructivism is a theory of knowledge that looks at how humans generate knowledge and meaning from their relationships, interactions, experiences and ideas. It has been used as a rationale for using cooperative groups, technology, whole-class discussions, and various other teaching and learning techniques (Norton & D’Ambrosio, 2008). In all of their actions and experiences, students are constructing, even when they are taking notes or reiterating a teacher’s remarks (Noddings, 1990). Jean Piaget is considered to be the father of this epistemological view.

## ZPD: Zone of Proximal Development

The underlying assumption of the ZPD is that students learn through assistance, resulting in subsequent development that will enable them to independently interpret and solve such problems later (Vygotsky, 1986, p. 188; Norton & d'Ambrosio, 2008, p. 221). This means that teachers facilitate the learning process by guiding and scaffolding students through the steps required to solve problems, either independently or in small groups. Bruner (1985) explains that the teacher "serves the learner as a vicarious form of consciousness until he is able to master his own action through his own consciousness and control" (p. 24).

## ZPC: Zone of Potential Construction

In the ZPC, schemes describe students' cognitive structures that develop through the abstraction of actions and operations; they are teacher constructs that provide "a way to discuss the development of stable and predictable courses of action" (Norton & d'Ambrosio, 2008, p. 222; Confrey, 1994, p. 4). For ZPC, the teacher decides, based on inferences of the students, how to guide and order the educational tasks.

## Lev Vygotsky - ZPD Social constructivism - learning and development are collaborative activities. | ## Ernst von Glasersfeld Radical constructivism - the process of constructing knowledge is dependent on the individual's subjective interpretation of their experiences and ideas. | ## Leslie Steffe - ZPC ZPC is "determined by the modifications of a concept a student might make in, or as a result of, interactive communication in a mathematical environment" (Steffe, 1991, p. 193) |

## Ernst von Glasersfeld

## The Study

The pair of students being studied were given time to play with the software, thus allowing them to exercise their ways of operating. Then the students had goal-directed activities posed by either themselves or by the teacher regarding fractions and their equivalents. The first student, Hillary, already had a scheme for solving equivalent fractions but she was able to build off her knowledge to progressively solve more challenging problems. The second student, Will, was able to learn from Hillary (through assimilation) and how she approached questions. Although his schemes were not as strong as Hillary's, he was successful in creating procedural schemes for producing equivalent fractions.

Students work within their ZPD when teachers are actively questioning students to explain how they were able to demonstrate their understanding. Students work within their ZPC when they can use their constructed schemes and apply them to new situations, generally with the guidance of teachers. Assistance can take many forms from either a teacher or a capable (and collaborative) peer, including modeling, questioning, and praising (Norton & d'Ambrosio, 2008, p. 243).

Therefore, mathematics teachers should work within their students' ZPDs to understand when and how to provide assistance. Scaffolding might interfere with problem solving but assistance is necessary for learning and development. "Teachers need to build models of students' operations in order to decide *how* to assist (Norton & d'Ambrosio, 2008, p. 244)".

## References

*Culture, communication, and cognition: Vygotskian perspectives*(pp. 21-34). Cambridge, United Kingdom: Cambridge University Press.

Confrey, J. (1994). A theory of intellectual development, part I. *For the Learning of Mathematics, 15*(3), 2-8.

Noddings, N. (1990). Constructivism in mathematics education. In R. B. David, C. A. Maher, & N. Noddings (Eds.), *Constructivist views on the teaching and learning of mathematics (JRME monograph 4*) (pp. 7-18). Reston, VA: National Council of Teachers of mathematics.

Norton, A., & d'Ambrosio, B. S. (2008). ZPC and ZPD: Zones of teaching and learning. *Journal of research in mathematics education, *220 - 246).

Steffe, L. P. (1991). The constructivist teaching experiment: Illustrations and implications. In E. von Glasersfeld (Ed.), *Radical constructivism in mathematics education* (pp. 177-194). New York: Kluwer Academic Publishers.

Vygotsky, L. (1986) *Thought and language* (A. Kozulin, Ed.). Cambridge: Massachusetts Institute of Technology.