6.4 Emergence of Pertinent Issues
6.4.5 Recommendations of the study
The analysis of the data and the findings of this study lead to the following general recommendations to try and address some of the issues raised, and to help deal with misconceptions and errors. Addressing these issues can improve the teaching and learning of
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geometry and help improve performance in mathematics. The recommendations are presented in the discussions below.
Educators should use relevant vocabulary to try and address the issue of the language barrier.
Learners showed a lack of understanding for geometric terms like diagonals, right angles, co-interior angles, corresponding angles, etc. An understanding of geometric terms will help learners to understand the requirements of the questions which will make it easier for learners to cope with the demands of the question. Teachers should not only identify assumptions, hypothesis and concepts for geometric statements, but should also explain and show the role of definition and the meaning of the terms used. Teachers should equip learners with the relevant vocabulary in accordance with the van Hiele teaching model.
Learners should be exposed to different shapes, especially those with common properties and address issues on class inclusion.
This is recommended because learners showed a lack of knowledge on class inclusion, when they failed to relate squares to rectangles and to parallelograms. Exposure to many different shapes will help learners to compare their properties and come up with their own conclusions for example, a square is a rectangle because it has all the properties of a rectangle, therefore all squares are rectangles but not all rectangles are squares. Squares lie within the class of rectangles and rectangles lies within the class of parallelograms.
Learners must make conclusions that squares and rectangles are parallelograms but there are other parallelograms which are not squares and which are not rectangles.
Educators must give opportunities for learners to work with a diversity of registers of semiotic representation in geometry in order to enhance conceptual understanding of geometric terms.
Educators rarely take diversity of representations into account when teaching geometry, with most educators normally showing an over-reliance on geometric figures in explaining geometric concepts to learners. This makes it difficult for learners to apply geometric concepts when they are given geometric tasks in other forms of representations, like the natural language (definition), graphic representations or algebraic representations and this can lead to the formation of misconceptions. Educators need to expose their learners to a variety of representations so that they can learn to express themselves and to switch from one representation to another. This is supported by Ozerem (2012), who stated that for learners to understand geometric concepts and facts which will eventually improve their skills, educators need to use lots of images, words, explanations, principles and anecdotes which will make it easier to understand than memorizing geometric concepts.
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Educators should use or cite real life examples when teaching geometry like using road signs.
This recommendation was made because learners indicated that they find it hard to understand geometric concepts because geometry seems to be isolated from the real world and they don’t see how it relates to their everyday lives. Citing real life examples in the teaching and learning of geometry will help in making it more relevant and understandable. Freudenthal (1988) explains the realistic mathematics education approach as a theory which emphasises the teaching and learning of mathematics connected to reality as well as made up of human activity like road signs in the teaching and learning of geometry. Siyepu & Mtonjeni (2013) carried out a study on the use of road signs in teaching geometry and found that geometric concepts such as triangles, circles, rectangles, lines, squares, etc, can be taught using South African road signs. Their study also found that using real life examples enhance learners’ conceptual and critical thinking skills.
Educators should put effort into improving learners’ conceptual understanding.
Educators should use teaching methods which promote conceptual understanding rather than rote learning. The results of the study showed that some learners had misconceptions involving geometric concepts like that of angles on a straight line. Some applied geometric properties in incorrect situations and some even failed to solve items because of a lack of conceptual understanding. Geometry allows learners to analyse and interpret the world around them as well as equip them with tools they can apply to other areas of mathematics. Teaching for conceptual understanding means that the teachers will encourage learners to use concepts as much as possible and associate the new terms with diagrams, representations and symbols so that they can connect them easily with the newly learnt geometric terms. The teaching process must play a pivotal role in connecting newly learnt concepts with what is already known.
Educators must use manipulatives in teaching geometry to reduce the prevalence of misconceptions and errors and thus enhance conceptual understanding. The van Hiele theory strongly emphasises the use of manipulatives in geometry to facilitate the transition from one level to the next (van Hiele, 1999). Manipulatives are physical or concrete objects that are used as teaching tools to engage students in the hands-on learning of mathematics. Touching and manipulating concrete objects improves proficiency in knowing positions or locations in space and the structure and its properties.
Educators should design learning programmes and provide learning experiences which help learners improve their proof skills.
This recommendation is made because learners showed little or no knowledge of solving proof-type questions. Geometry teaching naturally aims to improve learners’ deductive thinking and reasoning through proofs. According to Moore (1994), there are seven major sources of learners’ difficulties in doing proofs and these are: no knowledge of definitions; having little intuitive understanding of concepts; inadequate concept images
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for doing proofs; being unable or unwilling to generate and use their own example; not knowing how to use the definitions to obtain the overall structure of proofs; being unable to understand and use mathematical language and notation, and not knowing how to begin proofs. Educators should discourage learners from memorizing proofs without understanding the inter-connectedness between one statement and the next. Educators are supposed to expose learners to many proof-type questions. Exposing learners to experiences involving proof should help the learners to perfect the art of responding to proof-type questions.