Introduction to the study Geometry in South African high schools
about the construct that the instrument is supposed to measure matches their use in, for example, statistical analysis to determine if factor structure or scales relate to theory, correlations, and so on (American Educational Research Association/American Psychological Association/National Council on Measurement in Education [AERA/APA/NCME], 2014; Messick, 1980). As Thorndike (2005) points out, this definition shifts the traditional focus on the three-fold types of validity, namely, construct, criterion-referenced, and content validity, to the “evidence” and “use”
of the instrument.
1.2.3 Significance in mathematics education monitoring
The Department of Basic Education established the Dinaledi School Project in 2001 for the purpose of raising previously disadvantaged high school learners’ participation and performance in mathematics and science (Department of Basic Education [DBE], 2009). Part of the budget in the department provides these schools with resources (for example, textbooks and laboratories).
The ultimate intention is to improve mathematics and science results and thus increase the availability of key skills required in the economy (Department of Basic Education [DBE], 2009).
In monitoring the performance of these schools, the education officials take note research studies that focus on these schools (Department of Basic Education [DBE], 2009). The fact that this study made findings relating to SA#3 (as mentioned in the next section) in the CAPS document should draw the officials’ attention as to whether the stipulations of this aim were achieved. It is reasonable to believe that these officials will have access to this finding given that one of the conditions of approval of this study is that upon its completion, a brief summary of the findings, recommendations, or this thesis in its entirety must be submitted their research office.
Introduction to the study Geometry in South African high schools
reintroduction of proof into the CAPS mathematics curriculum reflected the notion that there is an appreciation of proof as the basis of mathematical knowledge. This notion finds support in Hersh’s (1997) claim that proof is an essential tool for promoting mathematical understanding. However, for many learners, proof is just a ritual without meaning (Ball, Hoyles, Jahnke, & Movshovitz- Hadar, 2002). This perspective is reinforced when learners are required to write proofs according to a certain scheme or solely with symbols.
In South Africa, as in most countries, the geometry curriculum includes Euclidean proof and analytical geometry. Whereas Euclidean geometry focuses on space and shape using a system of logical deductions, analytical geometry focuses on space and shape using algebra and a Cartesian coordinate system (Department of Basic Education [DBE], 2011; Uploaders, 2013). In this study geometry has been taken to be the mathematics of shape and space, which traditionally incorporates but is not limited to Euclidean geometry. This study focused exclusively on Euclidean geometry on the basis that learner performance in this area has been consistently poor compared to the other geometries just mentioned.
In the South African high school education system, Euclidean geometry is the place where learners should engage in formal deductive reasoning as they do proofs. As previously mentioned, functional understanding of proof, one of the Specific Aims advocated in CAPS for mathematics, is based on van Hiele’s (1986) broad theory of geometric thinking. Specifically, Euclidean proof (formal deduction) starts in Grade 10. In this grade, learners are expected to investigate, make conjectures, and prove the properties of the sides, angles, diagonals and areas of quadrilaterals;
namely, kite, parallelogram, rectangle, rhombus, square, and trapezium (Department of Basic Education [DBE], 2011). In addition, they are required not only to know that a single counterexample can disprove a conjecture, but also that numerous specific examples supporting a conjecture do not constitute a general proof. Accordingly, very few will contest the notion that Grade 10 instruction is assumed to have had an impact on learners’ functional understanding of proof in mathematics. Hence, this study investigated this understanding in Grade 11 learners.
Introduction to the study Geometry in South African high schools
However, the weakness in CAPS is that there appears to be a lack of explicit content on the functions of proof as well as the historical aspects of proof. As I argued earlier, it is precisely this absence of instruction on functional understanding of proof that seem to inhibit learners’
ability to construct proofs. By making the functions explicit, the intended curriculum can be realised. Support for this insistence arose out of Idris’ (2006) assertion that since functional understanding of proof is a largely conventional concept, its learning cannot take place without explicit instruction. Needless to say, this is not a suggestion that ability to prove is secondary but an attempt to underscore functional understanding as a prerequisite aspect of constructing Euclidean proof.
In primary schools, informal deductive elements are underscored while the formal deductive aspect is delayed until the FET phase. However, the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) not only underscores inductive proof, it also emphasises the didactic value of deductive proof by noting that all learners must be provided with the opportunity to ‘recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof’ (p. 56).
Accordingly, Magajna (2011) asserts that the two systems of reasoning in (school) geometry – one based on empirical observation (informal proof) and the other based on deduction (formal proof) – are essential and mutually support each other. In addition, empirical evidence merely gives a sense that something ought to be true (Sundström, 2003). Empirical evidence refers to the testing of a conjecture using numbers after gaining conviction and confidence about the truth of the conjecture (Hanna, 1995). Reference to truth in this project implies contingent truth rather than absolute or infallible truth given that proving is a human activity and humans are prone to making mistakes despite the best efforts to avoid them. However, the problem with viewing proof as a means to make a convincing argument is essentially a return to its everyday usage which engenders semantic contamination (Reid & Knipping, 2010).
Introduction to the study Geometry in South African high schools
Herbst (2002) argues that proof is valuable in mathematics education not only as an opportunity for learners to engage in a process of mathematical reasoning, but more importantly, as a necessary aspect of knowledge construction. This new curriculum at FET phase advocated for teaching that involves not only the “how” of mathematics, but also the “why.” For learners, CAPS discouraged the learning of procedures and proofs without a good understanding of why they were important as lack of understanding left them ill-equipped to use their knowledge in later life (Department of Basic Education [DBE], 2011). Reintroducing Euclidean geometry as a compulsory component of mathematics in 2011 seemed to suggest that curriculum designers acknowledged that Euclidean deserves a place in the high school curriculum. According to Adler (2010), this reintroduction was a response to an outcry at universities about the widening gap between school mathematics and tertiary education with a mathematical content. In my view, this development is clear departure from previous perspectives which can reasonably be attributed in large part, to the realisation that:
An informed view of the role of proof in mathematics leads one to the conclusion that proof should be part of any mathematics curriculum that attempts to reflect mathematics itself. (Hanna, 1995, p.
42)
As Jahnke (2010) points out, the importance attached to proof in the curriculum arose from the perspective that its functions provide a more comprehensive image of the nature of mathematics.
Hence, I contend that the pressure on schools to improve pass rates in mathematics examinations encourages the pursuit of rote acquisition of mathematical knowledge thus distorting the nature of mathematics and also undermining some of the Specific Aims in the CAPS document. The Department of Basic Education (DBE) identified eight Specific Aims:
SA♯1: To develop fluency in computation skills without relying on the usage of calculators.
SA♯2: Mathematical modeling is an important focal point of the curriculum. Real life problems should be incorporated into all sections whenever appropriate. Examples used should be realistic and not contrived. Contextual problems should include issues relating to health, social, economic, cultural, scientific, political and environmental issues whenever possible.
Introduction to the study Geometry in South African high schools
SA♯3: To provide the opportunity to develop in learners the ability to be methodical, to generalize, make conjectures and try to justify or prove them.
SA♯4: To be able to understand and work with number system.
SA♯5: To show Mathematics as a human creation by including the history of Mathematics.
SA♯6: To promote accessibility of Mathematical content to all learners. It could be achieved by catering for learners with different needs.
SA♯7: To develop problem-solving and cognitive skills. Teaching should not be limited to “how” but should rather feature the “when” and “why” of problem types. Learning procedures and proofs without a good understanding of why they are important will leave learners ill-equipped to use their knowledge in later life.
SA♯8: To prepare the learners for further education and training as well as the world of work.
While these are all important aims, only (italised) three of them were relevant for this study; SA♯3, SA#5, and SA♯7. These three aims seem to reflect an internal view of mathematics which emphasises that the processes of mathematics are fallible. In SA♯3, conjecturing and generalising are stressed before engagement in formal proof. In SA#5, the internal view of mathematics is underscored. Functional understanding is the focus of SA♯7, which also stresses the explanatory function of proof in mathematics. Taking SA#7 into account and the fact that a third of the Grade 12 (sometimes loosely known as “matric”) second paper examination consisted of Euclidean geometry, making it the component with the highest weighting in the overall assessment of this paper, I think it is reasonable to conclude that the curriculum planners placed value on holding informed functional understanding of proof in mathematics.
Although others may disagree, my opinion is that the South African curriculum assumes that by placing emphasis on making learners understand why a mathematical proposition is true or by merely doing proofs learners will come to understand the functions of proof in mathematics.
However, learners’ performance in proof is not only evidence that this assumption is unsubstantiated but also a reflection of defective Euclidean geometry instruction. Hence, it would
Introduction to the study An overview of the theories in this study
be sensible for future research to examine both preservice and practicing mathematics teachers’
perspectives of the functions of proof in mathematics.
While I believe that making Euclidean proof compulsory again is indicative of a willingness by curriculum planners to embrace the functions that proof performs in CAPS mathematics, of concern is the absence of an explicit mentioning of the functional dimensions of proof. This absence can also be detected in the school textbooks that the education authorities recommended for enacting the mathematics curriculum.