Compression of objects through categorization.

Van Hiele (1959) formulated a theory which consists of five levels to describe the learning of geometry. The first level is known as visualization which indicates that a child is able to classify geometry shapes based on holistic appearance. At this level, visual prototypes are developed in children minds. The second level is known as analysis and the objects of thought are classes of shapes therefore a child at this level will be able to analyze the properties of a shape. For instance, the child might say, “A triangle is a figure with three sides and three angles.” The third level is known as abstraction which indicates the objects of thought are geometric properties. At this level, a child can relate different set of properties and see the implications of one set of properties to another. For instance, a child

19 might say, “All squares are rectangles but not all rectangles are squares.”, this is because the child understand the properties of square and rectangle and sees a square is just a specific example of rectangle.

Level four is known as deduction where the object of thought is deductive reasoning (Euclidean proof). At this level, Euclidean geometry is used to deduce other properties based on a figure with certain property and the understanding of geometric ideas is limited to the objects in Euclidean plane. For instance, the idea of congruent triangle arises through the laying one triangle over the other triangle. The final level which is level 5 is known as rigor. At this level, students understand that definitions are arbitrary and need not to refer to any real world concrete object so that the students can study non‐Euclidean geometries.

Over the years, different researchers have renamed these levels for their research contexts. Hoffer (1981) renamed these 5 levels as recognition, analysis, ordering, deduction and rigor. Meanwhile Clements & Battista (1992) renamed these as recognition, descriptive/analytic, abstract/relational, Euclidean deduction and rigor. Tall (2013) renamed these as recognition, description, definition, Euclidean proof and rigor. 2.2.3Distillationofthetheories.

Section 2.2 describes general theories of mathematical thinking covering all the domains of mathematics. However, they do not put much emphasis in explaining the different stages of compression in mathematics. Meanwhile the theories in section 2.2.1 focus on the explanation of how the

20 mathematical concepts are compressed from one level to another level, in particular the compression of operation to concept. On the other hand, the theories in section 2.2.2 focus on the explanation of different stages of compression through sensory input.

There is no perfect theory in this world. Different theories have different emphases to serve different purposes. For instance, Jean Piaget’s stage theory focuses on the readiness of humans for learning based on their biological development from sensori‐motor beginnings through concrete operational and formal operational stages of development. Meanwhile, Van Hiele (1959) proposed a theory to explain the learning of geometry. Bruner (1966) focuses on different kinds of representation of information from enactive through iconic and symbolic. Skemp (1979) focuses on the humans’ innate ability to make sense of mathematics through perception, action and reflection and whether the student’s understanding is instrumental (in terms of rote learning) or relational (Skemp, 1976). Efraim Fischbein (1987) focuses on three approaches to mathematics: intuitive, algorithmic and formal. Tall (2004) focuses on three distinct developments of mathematical thinking by proposing three worlds of mathematics, one focusing on the perception, recognition and construction of objects and their properties, one focusing on operations that are compressed into manipulable symbols, and the increasing sophistication through reasoning that leads to the highest level of axiomatic formal mathematics in university and in mathematical research.

21 Part of the theoretical framework in this study was formulated by blending these theories together in a coherent way in order to explain how humans make sense of mathematics. This gave rise to the three modes of making sense through perception,operationand reason, where operations are seen generally as actions that are performed with a specific purpose e.g. for construction in geometry or through symbolic operations in arithmetic and algebra. The aspects of operations in embodiment and symbolism are blended together in the study of trigonometry. In the following table, we consider the roles of perception, action (which becomes more sophisticated as operation) and reason.

Author Proposed framework Emphasis

Bruner (1966) Iconic (Perception) Enactive (Action) Symbolic (Reason) Representation of information Skemp (1979) Perception (Perception) Action (Action) Reflection (Reason) Humans’ ability to make sense Liebeck (1984) Experience (Perception/ Action) Language (Reason) Picture (Perception) Symbol (Operation) Sequence of abstraction Fischbein (1987) Intuitive (Perception) Algorithmic (Operation) Formal (Reason) Approaches to mathematics Tall (2004) Embodiment (Perception) Symbolism (Operation) Formalism (Reason) Three distinct types of thinking in maths Cottrill et al (1996) APOS

(ActionProcessObjectSchema)

Sequence of compression of operation into a concept Sfard (1991) Operational (Operation) Structural (Perception) Two different types of mathematics Van Hiele (1959)

Successive levels in geometry

involving perceptions, geometric operations and reason

Explain the hierarchy of

learning geometry Gray & Tall

(1994)

Procept

(operationsbecomementalobjectsrepresentedassymbols withpropertiesthataresubjecttoreason)

Symbol as process and

concept Table 2.1 Links between the proposed theoretical framework to other theories.

22 Based on Table 2.1, it can be noticed that the proposed theoretical framework in this study corresponds to other theories up to certain extent. The words in italics indicate the proposed mode which corresponds to different parts of other theories. However, while these modes may focus on on perception,operationor reason,usually they involve a blend of all three. For instance, the three worlds of Tall (2004) all involve perception, operation and reason while the main emphasis in conceptual embodiment is on perception of objects and their resulting properties found by operation and reason, operational symbolism is based on operations and their properties represented symbolically and axiomatic formal theory is based on verbal definition and reason, often suggested by perceptions and operations.