4.2 Data presentation and discussion
4.2.1 Effective teaching of mathematics
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and understand topics and programmes as stipulated in the curriculum and they should also understand the variety of instructional materials available for the programme.
By confusing the major content areas of mathematics with mathematical concepts and being uncertain when referring to the major content areas of mathematics showed that the respondents were not familiar with the Grade R curriculum which they were supposed to use every day when planning their teaching of mathematics. Teacher Moliehi, who failed to mention major content areas of mathematics but mentioned only mathematical concepts like matching and sorting, demonstrated that some teachers might be ignorant of the requirements of the curriculum; as a result they lack the knowledge and valuable information required for teaching mathematics. However, teacher Tselane was able to mention three major content areas of mathematics and provided relevant examples underneath each major content area of mathematics.
Major content areas of maths that I know are shapes, measurement, and number.
I think number has counting, addition; shapes has topics like square, rectangle and circle, and measurement has height, length (Teacher Tselane).
Her response indicated that she was partly familiar with the Grade R curriculum. As stated in Chapter Three, a requirement for sound teaching is that the Grade R curriculum and the Course Outline for Mathematics were analysed to determine if they contributed to teachers’
understanding of the teaching of mathematics in Grade R. Teacher Tselane’s response revealed that the two documents contributed partly to her understanding of the contents of the curriculum because she mentioned only three major content areas of mathematics which appear in both the Grade R curriculum and in the Course Outline of Mathematics document that is used for the mathematics/numeracy course. Her knowledge was not comprehensive as the literature clearly states that there are five major content areas of mathematics, namely number and operations, measurement, geometry, algebra and data analysis (NAEYC &
NCTM, 2002). I analysed the LCE Course Outline for Mathematics/numeracy course and found that the two major content areas are omitted. When analysing the documents, the findings showed that the Course Outline for Mathematics/Numeracy document as well as the curriculum document for Grade R used by in-service teachers at LCE offer training in only three major content areas of mathematics, namely number and operations, measurement and shapes, as mentioned by teacher Tselane. The other two major content areas algebra and data analysis are thus neglected in the training of teachers. This lack of training in essential subject content will affect their teaching of mathematics and it will deprive learners of an opportunity
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to be exposed to explicit mathematics that will lay a solid foundation for the learning of the subject in the latter grades (NAEYC & NCTM, 2002).
It was apparent that all the respondents understood that the teaching of mathematics in Grade R revolved around the teaching of the contents of the curriculum only, whereas the literature revealed that to effectively teach mathematics, teachers’ knowledge of the subject taught requires knowledge of the core components of competence and of effective teaching of that content (Mewborn, 2003; Gross man, 1990). In addition, Shulman (1987) suggests that the categories of teacher knowledge needed for effective teaching are subject matter knowledge, pedagogical content knowledge, and general pedagogical knowledge.
Because effective teaching requires teachers to have subject matter knowledge, I asked them to share what they understood about the differences between a triangle, a square and a rectangle. Teachers Itumeleng, Manyai and Tselane agreed that those shapes were different, even though their explanations of the differences were somewhat dissimilar. For instance,
Eee...! a square has four equal sides, rectangle has four sides as well and its two opposite sides are equal.(pause) a triangle has only three sides
(teacher Itumeleng).
A square has equal sides altogether. A rectangle has four sides and opposite sides are equal. Eee...! a triangle is different because it has only three sides.
(teacher Manyai).
These teachers explained the differences of the shapes in terms of their sides only. Teacher Tselane explained the differences of the shapes in more complex detail:
Two triangles with two equal sides, when merged together, form a square, and two triangles which its sides are not equal, when merged together they make a rectangle.
Her more elaborative explanation illuminated how the shapes related because it highlighted what happens when two triangles are merged.
Teacher Ntsoaki did not notice the differences between these shapes. Her response was I don’t think there is much difference because they are all shapes.
I was amazed to hear such and answer and, probing deeper, I asked, “Do you know these shapes that we are talking about?” Teacher Ntsoaki, somewhat uncertainly, responded;
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Yes, I know them. Eeerrr…! (Frowning) Triangle has three sides. Yaa! I think that’s how they differ.
Teacher Moliehi was also very uncertain as to how the shapes differ. She said:
They are not different. Yes, they may look different, but they are all shapes, maybe they may differ in terms of their sizes and colour.
The responses from all the teachers did not provide any satisfactory answer to the question, which revealed that the respondents had limited knowledge of the subject matter. Both teachers Ntsoaki and Moliehi lacked knowledge of the subject matter probably because of their lack of qualifications or from ignorance because the differences between those shapes are actually common content knowledge (CCK) which Ball et al. (2008, p.399) define as “the mathematical knowledge and skill used in settings other than teaching”. This implies that people may know the differences between those shapes even if they are not teachers. I expected the teachers to describe the differences in terms of their sides, corners and angles; as explained in the Curriculum for CECE programme (2007) for instance, a square differs from these other shapes because it has four straight sides with all sides equal. It also has four corners which are all right angles. A rectangle has four straight sides with opposite sides having the same length and corners, all right angles. A triangle has three straight sides which may be of equal or unequal lengths and its angles will be right, acute or obtuse angles (Curriculum for CECE programme, 2007).
The results based on the above question regarding the differences between the triangle, square and rectangle, clearly revealed that the majority of teachers lacked basic knowledge of the subject matter they were teaching. This implies that teachers are likely to experience challenges in terms of effective teaching of mathematics in Grade R, and that these challenges are consequently transferred to the learners. Shulman (1987,1986) insists that teachers should be competent in subject matter knowledge. They are required to go beyond the knowledge of the facts or concepts of a main content area and also understand the structures of the subject matter.