2.4 Some of the challenges in teaching and learning geometry at grade 12
2.5.3 Learning phases
The learning phases are phases that a student should go through in each level in order to move from one level to the next. Progress from one level to the next involves five phases (Mayberry, 1983; Hoffer, 1983; van Hiele, 1986). Each phase involves a higher level of thinking. The students’ progress from one level to the next is the result of purposeful
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instruction organised into five phases of “sequenced activities that emphasise exploration, discussion, and integration” (Teppo, 1991:212). Each instructional learning stage builds upon and adds to the thinking of the previous level (Genz, 2006). As a result, the instruction at each learning phase fully and clearly defines that which was implied at the previous phase. In short, the latter implies that the learning phases are useful in designing learning and instructional activities. What follows is a discussion about the phases within a level and the teacher’s role in providing instruction that enables this learning:
• Information/Inquiry
The students become acquainted with the context domain (Clements & Battista, 1992; van Hiele, 1986). The teacher sets an environment in which the conversation takes place between the teacher and the students about the topic to be studied.
Consequently, this process causes the student to discover a certain structure (Fuys et al., 1988; Presmeg, 1991). During this phase, questions are asked and observations are made by the teacher and students about the objects of the study. This helps the teacher to evaluate students’ responses and to determine students’ prior knowledge about the topic. Clements and Battista (1992) further explain that the teacher learns how students interpret the language and provides information to bring students to purposeful action and perception.
• Directed Orientation
In this phase, students become acquainted with the objects from which geometric ideas are abstracted. The students begin to realise what direction their learning is taking. This helps the students to become familiar with “the principal connection of the network of relations to be formed” (van Hiele, 1986:177). In short, this implies that the students are becoming familiar with the structures of the topic such as the figures, vocabulary, symbols, definitions, properties and relations. The teacher’s role is to direct students’ activity by guiding them in appropriate explorations (Clements & Battista, 1992). This
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activity helps the students to explore the field of investigation using the material, for example, by folding, measuring, and looking for symmetry (Mason, 1998). Therefore teachers should choose materials and tasks in which the targeted concepts and procedures are salient.
• Explicitation/Explanation
In this phase, the students have gained insights in working with the structures of the topic. Students become explicitly aware of their geometric conceptualisations, describe these conceptualisations in their own language and learn some of the traditional mathematical language for the subject matter (Clements & Battista, 1992; Mason, 1998).
According to van Hiele (1986), during this phase the students learn to speak the technical language. This means that the students are supposed to make their observations explicit and begin to use accurate and appropriate vocabulary with the help of the teacher (Fuys et al., 1988; Mason, 1998).
• Free Orientation
In this phase, students solve problems in which the solution requires the synthesis and utilisation of those concepts and relations previously elaborated (Clements & Battista, 1992). Therefore, students prepare themselves for multi-step tasks in addition to the one- step tasks they were familiar with. Van Hiele (1986) points out that it can be said that this is the further development of the second phase in which the student, for example, learns to find his or her way in a network of relations with the help of the connections he or she has at his or her disposal. Fuys et al. (1988) and Presmeg (1991) support the above statement by stating that the field of investigation or network of relations is still largely unknown at this stage, but the student is given more complex tasks to find his or her way round this field. A student might know about the properties for a new shape, for example, a kite. The teacher’s role is to select appropriate materials and geometric problems – with multiple solution paths, to give instructions to permit various performances and to
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encourage students to reflect and elaborate on these problems and their solutions, and to introduce terms, concepts and relevant problem-solving processes as needed.
• Integration
According to van Hiele (1986), the teaching process comes to an end with this final phase indicating that the students have reached a new level of thought, and have increased their thought level in the new subject matter. This means that a student summarises all that he or she has learnt about the subject, reflects on his or her actions and thus obtains an overview of the whole network or field that has been explored, for example, summarises and synthesises the properties of a figure (Fuys et al., 1988). In this phase, the language and conceptualisations of mathematics are used to describe the network (Clements & Battista, 1992). Hoffer (1983) elaborates that the teacher provides summaries of some of the main points of the subject that are already known by the students to help this process. In other words, this phase represents the stage where the teaching-learning process is evaluated.
The van Hiele levels of thinking with the help of the learning phases put an emphasis on conceptual and procedural knowledge. A brief discussion on the implications of the levels and learning phases on the instruction of geometry follows in the next section.