Teacher A introduced the lesson by telling the learners that they were going to learn about circle geometry. She then wrote the heading ‘Circle Geometry’ on the chalkboard and underlined it. Although Teacher A might have believed that at Grade 11, learners are expected to have an understanding of circles, she wanted to make sure that her assumptions were accurate. The teacher therefore conversed at length with the learners throughout the early introduction of the lesson. This is crucial to van Hiele phase one of instruction as it states that it is by communicating with the learners that we learn about their pre-knowledge of the subject matter. I observed how the teacher encouraged the learners when she asked them to label as many parts of the circle as they possibly could. In this way, the teacher learned exactly what the learners knew and did not know, and how she would go about teaching them. Teacher A asserted in the interview that “learners do not really know everything they claim to know so, as a teacher I must
make sure that I do not get a wrong impression of their existing knowledge.” In the van Hiele
phases of instruction, the information phase is necessary for the introduction part of the lesson. Therefore, it is important that the teacher establishes the learners’ pre-knowledge as accurately and as early as possible.
The teacher further asserted that she knew that the learners might label many parts of the circle to impress her or the researcher in their class without really thinking about their own learning. “Learning without thinking is not proper learning” exclaimed Teacher A. I can therefore conclude at this juncture that the events of the classroom observations during the information phase aligned well with what the teacher said in the interviews. The teacher openly practiced what she said she would. I observed her lesson closely and could see how she organized her instructions according to phase one’s features.
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4.3.2 Teacher B
Like Teacher A, Teacher B also introduced the lesson by letting the learners know that they were going to learn about circle geometry. The teacher informed the learners that they were expected to have in their possession a fully equipped mathematical set in order for them to cope during the lesson. The teacher further informed the learners that all drawings were to be done in pencil only and that they should sharpen their pencils to avoid drawings lines that were not clear and thus might be inaccurate.
The teacher assumed that informing the learners about the necessity of using a pencil in geometrical constructions is an important aspect of the van Hiele phases of instruction. “You told
me about the phases of instruction, so I think that the sooner the learners learn about the stationery required in circle geometry the better. In fact, this is the information phase and I was just trying to inform them.” I concur with Teacher B. The information phase is aimed exactly at
telling the learners what is expected of them. This includes the technicalities of geometric constructions.
I asked Teacher B to enlighten me more on his experiences with the information phase. Teacher B stated that it was I who introduced him to the van Hiele phases of instruction and that the first thing that a teacher should do is to inform the learners about the subject matter to be studied i.e. circle geometry. Teacher B explained that in addition, as required in the information phase, he needed to find out what the learners knew about circle geometry. “That is why I began the lesson
by asking the learners to draw and label all the circle geometry concepts that they could remember”.
Teacher B added that the information phase of the van Hiele phases of instruction is very crucial as it is during this phase that learners are made aware of what is that they had to learn. The teacher therefore seized the opportunity to discover the learners’ existing knowledge. “I know
that most of them could label parts of the circle accurately but they were unable to apply them at this stage. That was the reason why I only asked them to label and not say much about the concepts as I still wanted to find out what they knew about circle geometry.”
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4.3.3 Teacher C
Similarly, Teacher C also introduced her lesson by informing the learners about the content to be studied. She then asked a few questions to test the learners’ pre-knowledge of circles in general before leading them to basic competencies of the lesson. The teacher told the class that they were going to learn about circle geometry concepts and angle properties of circles. “Today we are
going to look at geometry. We are going to talk about angles in circles but before we proceed, we need to know what a circle actually is.”
The teacher asked the learners to define a circle. She listened to their different definitions and used these to summarize a definition of a circle that she then wrote on the chalkboard. The teacher also asked learners to name and define circle geometry concepts such as radius, diameter, circumference, chord, arc etc. before she drew and labeled a circle with these concepts. She encouraged the learners to participate actively in the classroom activities by giving each learner an equal chance to engage.
I asked the teacher to describe her actions during the information phase in relation to what happened in the classroom. The teacher alleged that she was aware that learners in Grade 11 would be familiar with circle geometry concepts and a few theorems which is why she started by asking them to name and define circle geometry concepts as a way of introducing the lesson. “I
knew that they learned geometry and circles in previous grades, I also know that they applied those concepts during problem-solving.” It therefore remained the teacher’s role during the
information phase to determine how much learners knew about circle geometry to enable her to locate an appropriate area of instruction for the remaining van Hiele phases of instruction. I noted that the information phase helped the teacher to introduce her lesson while at the same time assessing the learners’ pre-knowledge as well as gaining insight for the way forward.
4.3.4 Synthesis of Teacher A, B and C
It is evident that all three teachers were interested determining their learners’ pre-knowledge of circle geometry. All teachers started their lessons by asking the learners to describe what they know about circles and specific parts of the circle. Inquiring about the learners’ background knowledge of the subject matter is embedded within van Hiele’s phases of instruction framework. Therefore, this engagement from the teachers ascertaining the learners’ prior
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knowledge and informing the learners about the direction the lesson will take fits in exactly with the van Hiele requirements (Crowley 1987, p. 5).
Although all three teachers used the designed teaching programme to plan and teach their lessons, I noticed several differences among their practice. This is because the three teachers used different teaching methodologies to present their lessons. Teacher A asked the learners to draw a circle and they filled in circle geometry concepts together. Teacher B asked learners to draw a circle and fill in all the circle geometry concepts that they could remember. Learners did so without the teacher’s help. Teacher C on the other hand asked learners to talk about concepts first and then draw the circle later. Even though the three teachers’ approaches were different, they all reached a similar objective of obtaining information about the learners’ existing knowledge of circle geometry.
4.4 DIRECTED ORIENTATION PHASE