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Learners’ misconceptions in deductive geometry proofs and remedial strategies Benard Chigonga Department of Mathematics, Science and Technology Education, University of Limpopo, South Africa ABSTRACT The study sought to establish and describe the exiting Grade 11 mathematics learners’ knowledge of and misconceptions in geometry proof and proffer remedial strategies to turn misconceptions into teaching opportunities. A content analysis of learners’ responses to a circle geometry item in a National Senior Certificate Grade 11 mathematics paper 2 of November 2013 was conducted. The achievement test was taken by a cohort of 175 learners enrolled at a public secondary school in the Vhembe District of South Africa. Of these, 97 (55.4%) were girls and 78 (44.6%) were boys. Descriptive statistics was carried out on the quantitative data from the scoring rubric using SPSS version 22. Percentages, means, standard deviations, minimum and maximum of scores were used to indicate overall learners’ performance and the disaggregation of scores by gender and degree of success in solving the proof problem. Collection of incorrect or partially correct proofs were analysed qualitatively to identify patterns of learners’ misconceptions in the deductive geometry proof item. Only 1.7% of the learners performed well in the deductive proof item, while 98.3% evidenced misunderstandings or misconceptions which varied in complexity, suggesting proof development continues to be a problematic area for learners. Keywords: misconceptions, deductive reasoning, inductive reasoning, proof, deductive proof. Definition of keywords 1. Misconceptions are reasoning difficulties emanating from previous inadequate teaching, informal thinking, or poor remembrance. 2. Deductive reasoning is drawing conclusions from a logical chain of reasoning in which each step follows necessarily from the previous one (Simon, 1996). 3. Inductive reasoning is the process of arriving at a conjecture based on a set of observations. 4. Proof is a detailed explanation of how a mathematical statement follows logically from other statements already accepted as true. 5. Deductive proof is a step-by-step process of drawing conclusions based on previously known facts.
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1. INTRODUCTION AND BACKGROUND The role geometry plays in real life makes it a core component of mathematics that learners must understand and master (Luneta, 2015). For example, geometrical concepts such as triangles, circles, rectangles, lines, squares, areas and perimeters are used as traffic road signs meant to give control, information and danger warning signs (Siyepu & Mtonjeni, 2014). In South Africa geometry forms an important component of school mathematics curricula and its importance has been emphasised in the Curriculum and Assessment Policy Statement (CAPS). Also, from 2008 to 2013 geometry has been excluded from the core mathematics syllabus for Grade 12 and has been an optional topic tested in an optional paper (Mathematics Paper 3). Jansen and Dardagan (2014) concede that the issue of optionalising geometry was a way of marginalising South African learners from the development of advanced understanding of mathematics. Prior to the reintroduction of Geometry into the grade 11 core mathematics syllabus in 2013, it was a problem area to teach and learn in South Africa (Siyepu, 2012). Between 2009 and 2013, the majority of the students that opted for Mathematics Paper 3 did not do well in the examination (Department of Education, 2013). This suggest that the exclusion of geometry from the core mathematics syllabus for Grade 12 implied that most teachers were not qualified to teach Geometry. This is corroborated by Van Niekerk (1997: 112) who attributes the failure to teach geometry in South African schools to the fact that “... the majority of mathematics teachers are poorly trained”. The 2006 Trends in Mathematics and Science Study (TIMSS) report notes that mathematics teachers in South Africa were among the most frequently in-serviced (Reddy, 2006). This could be an indication of how low subject matter knowledge is among teachers of mathematics, which can have deleterious consequences on teachers’ pedagogical content knowledge. Hence, teachers’ knowledge of learners’ misconceptions in deductive proof should go a long way in equipping them for effective delivery and guidance of proof activities in their classroom. Van der Sandt (2007:2) concedes that in South Africa geometry is regarded as a ‘problematic topic’ at secondary school level. This implies that geometry education requires urgent attention now that it has been reintroduced into the core mathematics syllabus. Therefore, investigating the proportion of learners failing to solve a geometric proof problem and the misconceptions learners have in geometry proof item, a contribution is made towards the teaching and learning of Geometry in secondary schools. The learning of proof has been a major goal of mathematics curricula in many countries and for many generations (Otton, 2007). The two reasons for teaching proofs are to teach deductive reasoning as part of human culture, and to serve as a vehicle for verifying and showing the universality of mathematical statements (Gfeller, 2010). As such, “... proofs help learners to develop logical or critical thinking skills that are useful beyond the mathematics classroom” (Dickerson & Doerr, 2008: 408). The geometrical proofs learners learn in high school can prepare them for higher education studies in the Science, Technology, Engineering, and Mathematics (STEM) careers. The
axiomatic structure of Geometry requires teaching for understanding through the sequential process of exploration, inductive and deductive reasoning (Connolly, 2010). For example, in an exploratory lesson on the triangle sum property, learners would generate several different triangles, record each of the angle measurements and sum them in a table and notice a pattern. 2
This would lead to a conjecture that all the angles in a triangle sum to 180° (inductive reasoning). This inductive reasoning process does not represent a valid proof. However, it suggests a possible mathematical truth worthy of further investigation. De Villiers (1998) propose that deductive proof is valuable to explain why the observed property hold true for all cases. The deductive proof is a step-by-step process of drawing conclusions based on previously known facts. Therefore, to prove that the angles inside any triangle sum up to 180° depends on the following known facts: 1. If two parallel lines are cut by a transversal line, then alternate interior angles (alt. int.∠𝑠) are equal and corresponding angles (corr.∠𝑠) are equal. 2. Angles on a straight line add up to180° (∠𝑠 on a line). The deductive proof might go something like this:
Although deductive proof seems rather simple, it can go wrong in more than one way. The premises (known facts) used in deductive proof are the most important part of the entire process of deductive proof (Simon, 1996). If they are incorrect, the foundation of the whole line of reasoning is faulty, and nothing can be reliably concluded. Deductive proof is effective when all of the premises are true, and each step in the process follows logically from the previous step (Simon, 1996). Therefore, investigating the exiting Grade 11 mathematics learners’ knowledge of and misconceptions in geometry proof and proffer remedial strategies, a contribution can be made to the improving of the teaching and learning of geometry proofs in secondary schools. Significantly, such information can inform teachers as to the instructional support in deductive proof that learners need, as they engage with geometry proof in Grade 12. The findings are also crucial to the Grade 11 teachers as the study delineates common conceptual and procedural misconceptions in geometry proof that they look out for when teaching the topic. 2. PURPOSE OF THE STUDY The purpose of the study was to establish and describe the nature of Grade 11 mathematics learners’ misconceptions in deductive geometry proofs and proffer remedial strategies to turn the misconceptions into teaching and learning opportunities.
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3. RESEARCH QUESTIONS The study responded to the following research questions: a) What is the proportion of learners failing to solve a deductive geometric proof problem? b) What misconceptions do learners have in deductive geometry proofs? c) How can these misconceptions be turned into teaching and learning opportunities? 4. SIGNIFICANCE OF THE STUDY Investigating the nature of learners’ misconceptions in deductive geometry proofs, a contribution can be made to the improving of the teaching and learning of deductive geometry proofs in secondary schools. 5. THEORETICAL BACKGROUND Ding and Jones (2006) support Piaget (1971) that children’s geometrical understanding develops with age and that for children to create ideas about shapes they need physical interaction with objects. Van Hiele (1986, 1999) on the other hand tried to analyse the various aspects involved in the learning of geometry and space. Van Hiele’s (1986) theory of geometric thought describes five different levels of understanding through which learners progress when learning geometry. The basis of the theory is the idea that a learner’s growth in geometry takes place in terms of distinguishable levels of thinking. In the first level (visualisation and recognition), learners can identify a shape, but are not able to provide its properties. The shape is judged only by its appearance. The second level (analysis) is descriptive: learners are able to identify particular properties of shapes, but not in a logical order. The third level (abstraction and relationships) is informal and deductive: learners can combine shapes and their properties to provide a precise definition as well as relate the shape to other shapes. The fourth level is formally deductive: learners apply formal deductive arguments such as in proofs. The fifth level (rigour and axiomatics) is characterised by formal reasoning about mathematical systems by manipulating geometric statements such as axioms, definitions, and theorems. Van Hiele’s levels provide teachers with a framework within which to plan geometric activities (Lim, 2011). Besides, the levels are also a good predictor of learners’ current and future performance in Geometry (Lim, 2011). As such, Van Hiele’s levels of geometrical thought were the guiding principles for studying exiting Grade 11 learners’ knowledge of geometry proof and for determining the level at which the learners in the sample operated. 6. METHODOLOGY a. Design The study sought to establish and describe the exiting Grade 11 mathematics learners’ knowledge of geometry proof, determine the nature of misconceptions in geometry proof and proffer remedial strategies to turn around the misconceptions into teaching opportunities. For that reason, an explanatory sequential mixed methods design was used in this study (Creswell, 2014). This design is a mixed methods strategy that involved collecting quantitative data, 4
analysing results, and then using the results to inform the sampling procedure for qualitative data collection (Creswell, 2014). Collection of quantitative data for investigating the degree of success of learners in solving the geometric proof problem and qualitative data for the analysis of learners’ misconceptions provided an understanding of the Grade 11 learners’ challenges in geometry proof. b. Participants Participants in the achievement test were 175 Grade 11 mathematics learners enrolled at a public secondary school in the Vhembe District of South Africa. Of these, 97 (55.4%) were girls and 78 (44.6%) were boys. Participants were learners between the ages of 15-22 years whose language of instruction was English as a second or foreign language. c. Instrument Table 2: Scoring rubric used for the analysis of learners’ responses. Mark Description of performance allocated 0 Lack of basic geometrical knowledge and vocabulary or lack of appropriate geometrical frame of reference. 1 Recognition of some helpful facts but inability to make a logical deduction. 2 Ability to notice helpful facts and make an inference, but inability to organise information in coherent chain of arguments from givens to conclusions. 3 Ability to notice helpful facts and make some inferences, but inability to be economic or precise, excess facts and/or imprecise labelling used leading to circuitous or clouded chains of argumentation. 4 Ability to notice helpful facts, make inferences and coherent chain of arguments from givens to conclusions efficiently. (Adapted from Ndlovu & Mji, 2012) d. Data collection Using the scoring rubric (Table 2), a quantitative content analysis of responses to the following circle geometry item (Figure 1) in a National Senior Certificate Grade 11 mathematics paper 2 of November 2013 was conducted. In Figure 1, 𝑆𝑊𝑇𝐷is a cyclic quadrilateral; 𝐾𝑆𝑇is a straight line and; 𝑇𝑆is a tangent to circle𝑅𝑆𝑃𝐷. Prove that𝑇𝑊 ∥ 𝑃𝑆. (In Figure 1, some angles are numbered to make alternative labelling possible, ^ 𝑆). e.g., 𝐷1 = 𝑅𝐷
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Also, a qualitative content analysis of 50 proofs that were incorrect or partially correct was done to identify a particular pattern of responses. Those proofs that did not fall in any pattern of responses were discarded. From that collection examples were selected for discussion. The test item was selected in order to theorise about the learner’s level of geometric thought development for purposes of suggesting remedial strategies. e. Data analysis Descriptive statistics was carried out on the quantitative data from the scoring rubric (Gall, Gall, & Borg, 2007). These data were analysed with the help of SPSS version 22. Percentages, means, standard deviations, minimum and maximum of scores were used to indicate overall learners’ performance and the disaggregation of scores by gender and degree of success in solving the proof problem. Data were illustrated using tables in order to show the key features of the data in a more interpretable manner (Johnson & Christensen, 2008). The collection of 50 incorrect or partially correct proofs was analysed qualitatively and categories of misconceptions were developed inductively out of the learners’ responses (Mayring, 2014). f. Ethical issues Permission to carry out the study at the selected site was not necessary since the author was the Grade 11 mathematics teacher at the time of data collection. During the process of data collection and processing anonymity and confidentiality were observed. Also insincerity and manipulation were guarded against. 7. RESULTS AND DISCUSSION a. Quantitative results There were more girl learners (55.4%) than boys. However, this difference was not statistically significant at 𝑝 < 0.05 because the Chi -square value was 1.077 for 1 degree of freedom. In addition, Table 3 shows that the mean score for girls was higher than that for boys. However, this difference was not statistically significant. The overall standard deviations indicate greater dispersion among scores for boys than those for girls.
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Table 3: Overall learners’ performance Subject Girls Boys Overall
Mean score 1.577 1.551 1.564
Standard deviation 0.797 0.872 0.389
Minimum 0 0 0
Maximum 4 4 4
Total 97 78 175
% 55.4 44.6 100
Table 4 shows a further disaggregation of scores by gender and degree of success in solving the deductive geometric proof problem. Table 4: Performance analysis by score and gender Score 4 3 2 1 0 Total
Girls 1 4 59 19 14 97
% 1.0 4.1 60.8 19.6 14.5 100
Boys 2 3 42 20 11 78
% 2.6 3.8 53.8 25.6 14.2 100
Total no. of learners 3 7 101 39 25 175
% 1.7* 4 57.7 22.3 14.3 100
From Table 4, only 1.7% of learners evinced the ability to notice helpful facts, make inferences and coherent chain of arguments from givens to conclusions efficiently. These 3 learners (1.7%) can be classified as operating at Van Hiele’s level 4 of geometry thinking. 98.3% demonstrated misunderstandings or misconceptions which varied in complexity, indicating that the performance of learners in geometry proof item was low. This suggests that 172 learners are possibly operating at Van Hiele’s level 3 or less of geometry thinking. As a result, proof development continues to be a problematic area for learners as evidenced by this study. Suggestions are that: learners have to acquire a body of geometric content knowledge and; the activation and the utilization of this knowledge during the construction of proof need to be guided by general problem-solving and geometry reasoning skills (Chinnappan, Ekanayake & Brown, 2012). Therefore, geometry teaching needs to consider the interactive role of the three knowledge components (geometry content knowledge, general problem-solving skills and geometry reasoning skills). This will help learners develop higher levels of competency in the development of geometry proofs (Chinnappan et al., 2012). However, the design and incorporation of such knowledge components into a learning support environment are important issues for future research. b. Qualitative results Three categories of misconceptions were identified in the investigation. Two-column proofs, one giving the step-by-step of the proof and the other providing the reason, was evident in learners’ layout of the proofs as is eminent in the following examples.
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i. A misconception that ‘writing known concepts/theorems and properties of given figures equals a proof’ This misconception reveals lack of knowledge of what a proof of a mathematical statement entails. This leads to the inability to choose relevant concepts and properties that explain how that statement follows logically from them. Type I misconception: Learner 1 (Opa)’s response. Given: (1) 𝑆𝑊𝑇𝐷 is a cyclic quadrilateral (2) 𝑇𝑆 is tangent to circle 𝑅𝑆𝑃𝐷 RTP: 𝑇𝑊 ∥ 𝑃𝑆 Statement Reason Comment (True) 𝐿1 : 𝐷2 = 𝑇2 ∠s subtended by chord 𝑆𝑊 Given (Restating a given) 𝐿2 : 𝑇𝑆 is a tangent Tan-chord theorem (True) 𝐿3 : ∠𝑃𝑆𝑇 = 𝐷2 (True but irrelevant) 𝐿4 : ∠𝐷𝑆𝑃 = 𝐷1 Alternate ∠𝑠 (No reason given) 𝐿5 : ∴ 𝑇𝑊 ∥ 𝑃𝑆 (NB: 𝐿𝑛 refers to Line 𝑛 of the proof and pseudo names are used in this article).
• Analysis of Opa’s proof Opa’s solution evidenced serious lack of knowledge of what a proof of a mathematical statement entails. She could accurately identify the equality of angles in the same segment (𝐿1 ) and the tan-chord theorem (𝐿3 ). This implies that she effectively put to use the given facts that 𝑇𝑆is a tangent to circle 𝑅𝑆𝑃𝐷 and that 𝑆𝑊𝑇𝐷 is a cyclic quadrilateral. Although the learner could not combine these facts towards the conclusion, there was evidence of knowledge of theory (properties, definitions, theorems). The ability to make conclusion was therefore evidently absent. This is so because she did not know that for her to conclude that 𝑇𝑊 ∥ 𝑃𝑆, she needed to either show that ∠𝑃𝑆𝑇 = 𝑇2 or∠𝑃𝑆𝑊 + (𝑊1 + 𝑊2 ) = 180° (𝑎𝑙𝑡 ∠𝑠 𝑒𝑞𝑢𝑎𝑙 or𝑐𝑜 − int ∠𝑠 𝑠𝑢𝑝𝑝, respectively). So combining 𝐿1 and𝐿3 , it can be seen that ∠𝑃𝑆𝑇 = 𝑇2 . Failure to know where to start and end with the proof can be attributed to the lack of teaching emphasis on the meaning of proof (Gagatsis & Demetriadou, 2001). Opa’s level of response was/could be classified as Van Hiele’s Level 2 (Van Hiele, 1986), signifying a set of relevant facts known by the individual. •Remedial strategy of Opa’s proof. The seemingly lack of knowledge of what a proof in the geometry of the circle, parallel lines or a combination of both entails calls attention to the need to emphasis on what to show when proving [for example] that: a quadrilateral is cyclic; a line is tangent to a circle; two lines are parallel; two line segments are equal in order to help learners like Opa. This initial step (what to show) is frequently overlooked as a starting point to build learners’ prior knowledge on how
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to do geometry proofs. The “what to show” when proving that two line segments are parallel is: ^1 = 180°,then 𝐼𝐽 ∥ 𝐾𝐿 (𝑐𝑜 − 𝑖𝑛𝑡 ∠𝑠 𝑠𝑢𝑝𝑝) [U pattern] 1. If 𝐼^1 + 𝐾
OR 2. If 𝐴^1 = 𝐹^1 , then 𝐴𝐵 ∥ 𝐹𝐺(𝑐𝑜𝑟𝑟∠𝑠 𝑒𝑞𝑢𝑎𝑙) [F pattern]
OR ^1, then 𝐴𝐶 ∥ 𝐷𝐹, (𝑎𝑙𝑡∠𝑠 𝑒𝑞𝑢𝑎𝑙) [Z pattern] 3. If 𝐶^1 = 𝐷
Any of these three conditions constitutes the “what to show” when proving that two line segments are parallel. This suggests that remedying similarly affected learners would mean familiarising them with the “what to show” before the “how to show” and the “conclusion”. ii. A misconception that ‘a set of properties or relationships observed true in the figure represents a proof’ This misconception indicates an obstacle to building up proofs and, consequently, to learning to prove. Type II misconception: Learner 2 (Sofia)’s response. Given: (1) 𝑆𝑊𝑇𝐷 is a cyclic quadrilateral (2) 𝑇𝑆 is tangent to circle 𝑅𝑆𝑃𝐷 RTP: 𝑇𝑊 ∥ 𝑃𝑆 Statement 𝐿1 : 𝑆𝑊 = 𝑆𝑊
𝐿2 : 𝑊1 + 𝑊2 + 𝐷2 + 𝐷3 = 180° 𝐿3 : ∠𝐷𝑆𝑃 + ∠𝑃𝑆𝑇 + ∠𝑇𝑆𝑊 + 𝑇1 + 𝑇2 = 180° 𝐿4 : 𝐷3 = ∠𝑇𝑆𝑊 𝐿5 : 𝐷2 = 𝑇2 𝐿6 : 𝐷2 = ∠𝑃𝑆𝑇 𝐿7 : ∴ 𝑇𝑊 ∥ 𝑃𝑆 9
Reason Common side in ∆ 𝑃𝑆𝑊 & ∆ 𝑇𝑊𝑆 Opp ∠s of a cyclic Opp ∠s of a cyclic
Comment (True but unhelpful fact) (True) (True)
∠s subtended by chord 𝑇𝑊 ∠s subtended by chord 𝑆𝑊 (Tan-chord theorem)
(True) (True) (True) (No reason given)
• Analysis of Sofia’s proof Sofia correctly deduced that, if a side is common to two shapes, then it is the same length (𝐿1 ). This was true but unhelpful given the demands of the question. She was also correctly aware that: opposite angles of a cyclic quadrilateral are supplementary (𝐿2 𝑎𝑛𝑑 𝐿3 ); angles in the same segment (𝐿4 𝑎𝑛𝑑 𝐿5 ) and; tan-chord theorem, but never used these facts as possible points of departure to build up a proof. Being able to identify properties or relationships in the geometric problem without connecting them in deductive arguments does not equal to a proof. Kim & Hannafin (2010) point out that learner difficulties emanating from limited prior knowledge and experience can lead to cognitive overload. The learner is aware of the properties or relationships in the geometric problem but did not know what to do with the facts, which was an obstacle to building up a proof. However, I posit that the learner’s failure to connect them in deductive arguments to be a part of such prior knowledge and experience of what entails a proof. In this instance, she did not know that for her to conclude that 𝑇𝑊 ∥ 𝑃𝑆, she needed to show that (𝑊1 + 𝑊2 ) + ∠𝑃𝑆𝑊 = 180° ( 𝑐𝑜 − int ∠𝑠 𝑠𝑢𝑝𝑝 ). From 𝑊1 + 𝑊2 + 𝐷2 + 𝐷3 = 180°(𝐿2 ) , substitute 𝐷2 with ∠𝑃𝑆𝑇 ( 𝐿6 ) and 𝐷3 with ∠𝑇𝑆𝑊 ( 𝐿4 ), to give 𝑊1 + 𝑊2 + ∠𝑃𝑆𝑇 + ∠𝑇𝑆𝑊 = 180°, i.e., 𝑊1 + 𝑊2 + ∠𝑃𝑆𝑊 = 180° (since ∠𝑃𝑆𝑊 = ∠𝑃𝑆𝑇 + ∠𝑇𝑆𝑊). Sofia’s level of response was also classified as Van Hiele’s Level 2 (Van Hiele, 1986), signifying the products of thought are relationships among properties of geometric objects. • Remedial strategy of Sofia’ proof. Sofia appeared to be a more redeemable case than Opa in that some of her statements were true. A remedial programme that starts with exploring what can be deduced from the givens to identify a collection of choices (frame of reference) could be appropriate in supporting such learners. For example, assuming that TW//PS, questions to generate a collection of choices could be: Which straight lines (transversals) intersect/meet with both parallel lines? What facts do we know or can we deduce about angles formed at the intersection of the parallel lines and the transversal(s)? This would entail refreshing the learners’ knowledge bank of properties and relationships between objects/properties (Van Hiele’s level 2). Scaffolding existing bank of geometrical knowledge of learners should not be done by conveying a ready-made deductive proof. Instead, effort should be made to provoke sense-making through questioning what can be deduced from the givens in order to identify frame of reference. iii. A misconception that ‘in a proof the word given justifies any statement derived from the figure (geometric problem)’ This misconception suggests a poor set of properties related to the geometric problem leading to an inability to make a logical deduction. This lack of awareness prevents the learner from connecting facts in the process of building up proofs. Type III Misconception: Learner 3 (John)’s response.
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John presented the following as his proof: Given: (1) 𝑆𝑊𝑇𝐷 is a cyclic quadrilateral (2) 𝑇𝑆 is tangent to circle 𝑅𝑆𝑃𝐷 RTP: 𝑇𝑊 ∥ 𝑃𝑆 Statement 𝐿1 : ∴ 𝑇𝑆is a tangent
Reason Given
𝐿2 : 𝑆𝑊𝑇𝐷 is a cyclic quad. 𝐿3 :𝐷2 = 𝑇2 𝐿4 : ∴ 𝐷2 𝐷3
Given Given
𝐿5 : ∠𝑇𝐷𝑆 = ∠𝑇𝑊𝑆 𝐿6 : 𝑊1 + 𝑊2 + 𝐷2 + 𝐷3 = 180° 𝐿7 : ∴ 𝑇𝑊 ∥ 𝑃𝑆
Given Given
Comment (A given stated as a conclusion, odd to start with ‘therefore’) (Restating a given) (True but it is not a given) (True if 𝑆𝑊 = 𝑇𝑊 : equal chords subtend equal angles) (True if 𝑇𝑆 was given as a diameter) (True but it is not a given) (Nothing to show how this is arrived at)
• Analysis of John’s proof John apparently did not use any of the givens. He simply restated them as conclusions (𝐿1 and 𝐿2 ) and thus could not develop his proof in a meaningful way. For John, anything he perceives correct in the geometric problem is thus given information. No statement is true simply because it appears to be true from a figure. Developing his proof, John concluded that 𝐷2 was equal to 𝐷3 (𝐿4 which wrongly implied that 𝑆𝑊 = 𝑇𝑊 and claimed that ∠𝑇𝐷𝑆 was equal to ∠𝑇𝑊𝑆 (𝐿5 ). The deduction and premise were incorrect. Using these incorrect facts he inferred that line segments 𝑇𝑊 and 𝑃𝑆 were parallel (𝐿7 ) without showing background knowledge of [for instance] relationship between parallel lines, transversals, and alternate angles. Earlier studies of how learners prove have also stressed the importance of maintaining the connections between proving and knowing (Herbst, 2002a). With these shortcomings John was unable to score any marks. His response could be classified as Van Hiele’s Level 1. • Remedial strategy of John’s proof. John’s imprecise designation of angles implied a weakness in communication skills as an obstacle to effective handling of deductive geometry proof. Learners such as John who seem to have some correct ideas in some instances, but cannot express themselves accurately on paper need to be assisted to gain precision in their references to geometric objects. They need to be encouraged to reflect on the meanings of the symbols they use and to search for unintended interpretations that may arise from them. John’s failure to justify statements with correct reasons (for example, 𝐿5 ) together with the failure to link steps of a proof implied a lack of effective argumentation skills for proof execution. Learners encountering such difficulties need to be encouraged to read their sequence of statements again and again and to critically pay attention to the coherence in their argument – a critical metacognitive skill.
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8. CONCLUSION The article focused on reporting the degree of success of Grade 11 learners in performing geometry proof and their subsequent misconceptions. The findings of this study revealed that proof development continues to be a problematic area for learners. Suggestions are that teachers’ knowledge of learners’ misconceptions should go a long way in equipping them for effective delivery and guidance of proof activities in their classroom. As such, the phases of Van Hiele’s levels of geometric thinking have to be considered when designing instructional activities. These phases, as adapted from Fuys, Geddes, Lovett and Tischler (1988) and Presmeg (1991) are useful in designing activities in the following manner:
Information: The learner gets acquainted with the working domain/ field of exploration by using the material presented to him/her, for example, examines examples and nonexamples. This process causes him/her to ‘discover’ a certain structure. Guided/Directed Orientation: the learner explores the field of investigation using the material, for example, by folding, measuring, and looking for symmetry. Explicitation/Explanation: A learner becomes conscious of the network of relations, tries to express them in words and learns the required technical language for the subject matter, for example, expresses ideas about the properties of figures. Free Orientation: The field of investigation/network of relations is still largely unknown at this stage, but the learner is given more complex tasks to find his/her way round this field, for example, a learner might know about the properties of one kind of shape but is required to investigate the properties for a new shape, for example, a kite. The tasks should be designed so that they can be carried out in different ways. Integration: A learner summarises all that s/he has learned about the subject, reflects on his/her actions and thus obtains an overview of the whole network/field that has been explored, for example, summarises properties of a figure.
I can conclude that formative assessment, like homework, can be used to locate mistakes, to figure out why they were made and how to provide support to learners by way of explanation and tutoring (Fang, 2010). This approach can help teachers learn some pedagogical lessons from exploring the content of learners’ procedural knowledge and understanding (Pedrosa De Jesus, Neri De Souza, & Watts, 2005). In this case it is the procedural knowledge and understanding of deductive geometry proofs vis-à-vis a knowledge base of theory and procedural etiquette in proving. Since proof development continues to be a problematic area for learners as evidenced by this study (98.3% operating at Van Hiele’s level 3 or less), the investigation of knowledge components that learners bring to understanding and constructing geometry proofs could provide important insights into the above problematic area (Battista, 2007). Therefore, research has to identify the processes and associated domain knowledge that learners activate and bring to the solution context (Chinnappan et al., 2012). This article has limitation. As evidence of trustworthiness, this type of study relies on credibility. The findings from this approach are limited by their inattention to the broader meanings present in the data. This is so because learner debriefing was not done to shed more 12
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