The most common errors and possible misconceptions resulting to such in Space

With regard to Space and Shapes (Geometry) learners struggled with the following concepts: Angles, Recognising and naming 2-D Shapes, properties of shapes and attributes of 3-D shapes i.e. faces, vertices and edges.

At the pre-recognition level children perceive the shapes but they are unable to identify amongst and distinguish amongst them.

These levels can help us to understand what children understand about shapes The levels can guide teachers in providing appropriate learning opportunities for learners.

7.2.4.1 Concept of angles

Almost every subject of geometry requires a good knowledge of angle which is one of the basic concepts of geometry (Biber, Tuna & Korkmaz, 2013). Angles are classified in order of size as: acute angle, right angle, obtuse angle, straight angle, reflex angle and revolution angle. Literature points with regard to angles, learners develop misconceptions if they fail to understand angle as dynamic and they may be well unable to order angles of different sizes correctly because they are focusing on irrelevant pieces of information (Mooney et al. 2012). Most learners in this study seemed to have difficulties in identifying and naming angles formed when lines meet from the 2-D shape. From the analysis of learners’ work on space and shapes, this study found that 66% of the sample could not recognise and name obtuse and the 56% from the same sample could not recognise and name right angle from the 2-D shape. With regard to van Hiele levels of geometric thought, learners in this study were

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operating below the expected first three levels of geometric thoughts i.e. level 1 (visualisation) and level 2 (Descriptive/analytic). Instead, such of the learners appeared to be operating at level 0 (pre-recognition) whereby the focus was on some visual cues when they when naming the angles shown from the 2-D shape. The errors of naming angles using salient visual cues as in the following example of one the learners written work;

were found to be common amongst 44% of the sample.

Some learners (26% of the sample) generalised the name of the angles valid only to specific measure of a turn over different measure turn measures as in the following examples of some of the learners written work;

Research (Reys et al. 2012; Biber et al. 2013) attributes learners’ ability to name angles and shape with inability to recognise them to rote learning were by learners are taught geometric names without being given opportunities to explore the attribute. The findings in this study concur with those found in one of the study which sought to determine the learning levels, mistakes, and misconceptions of the grade 8 learners on the subject of angles in geometry (Biber et al. 2013). Haylock (2006) recommends teachers to give learners activities granting opportunities to explore various angles such as cutting out pictures from magazines, mark angles on them and display them in set as acute, right angle, obtuse, and so on.

7.2.4.2 Recognizing and naming the 2-D shapes

Learners are expected to be able to recognize and name the 2-D shape before they exit the intermediate phase. In their study of learners’ ability in understanding basic shapes, Researchers (Wu & Ma, 2006) found that most learners have problems in identifying the 2-shapes. This study also found that when given a composite diagram composed of the 2-D shapes, rectangle, trapezium, hexagon and octagon, some learners (22% of the sample) failed to identify any of the shape from the diagram. With regard to the van Hiele’s leve of geometric thought, such learners were operating at level 0 (prerecognition) seeming to lack necessary vocabulary for 2-D shapes and some resorted in to naming the shapes using the visual cues salient to them as in the

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following examples of some of the learners’ written work;

and .

According to literature, learners often have difficulty in seeing figures within a figure (Reys et al. 2012). One of the main reasons for such difficulty is presentation of the shapes as regular shapes and always sitting on one side or upright orientations. That according to Mooney et al. (2012) limits learners’ conceptual understanding of shapes Some of the learners (24% of the sample) identified only one of the 2-D shapes correctly and 28% of the learner sample named only two 2-D shape correctly. The common error observed this group of learners’ work was naming the shapes which were not shown on the composite diagrams as in the following examples of some of the learners’ written work;

, and

Such seem to have correct vocabulary but cannot visualise the shapes. Briggs (2013); Reys et al (2012) attribute these error types to teaching learners the geometric names without giving them opportunity to explore the attributes. Jorgensen & Dole (2011) recommends teachers to represent shapes in a range of representation.

7.2.4.3 Properties of shapes

The concept of properties of shapes was found to be one of the areas with more errors than other concepts assessed in Space and shapes. This is partly because understanding the properties of shape demands learners to operate at more advanced geometric thinking i.e. third level (Abstraction and relationship) whereby attention is more on the relationship between parts of the shapes to define attributes which in turn are interrelated to find set of attributes for specific class of shapes (Luneta, 2015; Reys et al, 2012). The findings in this study revealed that more learners were seemed to be operating way below such expected level (Abstract-relational) with most of them completing the properties two of the two shapes, rectangle and parallelogram, making

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the statements more irrelevant and even meaningless for the two shapes as in the following examples of some of the learners’ written work;

, .

Some learners resorted into completing the properties by merely copying the names of the shapes as in the following example of one of the learner’s written work;

By recognizing that learners are not seeing the interconnections between properties of shapes Dole & Jorgensen (2011) recommends teachers to organize learning experiences that have goal of enabling learners to see connection between the properties. One way of doing this according to Briggs (2013) organizing play environments with models for children to explore shapes at early childhood and that should be continued throughout primary school. In addition, Luneta (2015) recommends both teachers at primary and secondary to be grounded in in the content of geometry to be able to teach in ways that equip learners with conceptual knowledge.

7.2.4.4 Concept of 3-D faces, vertices and edges

All the 3-D shapes have faces, edges and vertices. These three attributes are used for describing and sorting the 3-D shapes. This study found that some learners seemed to confuse the edges with the vertices as in the following example of one of the learner’s written work;

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and

Such errors were common amongst 16% of the sample.

Apart from the aforementioned errors, the study also found that some learners (28% of the sample) seemed to be acquainted to the three attributes but errors due miscounting resulted as in the following example;

and .

Such errors were considered carelessness errors in this study. Literature attributes such errors to failure to mark or use numbering when counting (Reys et al, 2012). There were also common errors of writing numbers with no discernible link to the three attributes. Learners with such errors seemed to have no meaning for the three attributes.

7.2.5 The most common errors and possible misconceptions resulting to such errors in Data Handling

Even though this study revealed a notable highest average percentage mark in Data Handling compared to other four content areas learners were assessed covered in the ANA paper, the analysis revealed that some of the learners have challenge in the following areas of Data Handling; Reading and interpreting data, and answering question on measure of central tendency i.e. Mode

7.2.5.1 Reading and interpreting data

Some learners in this study struggled to answer items/questions for which the answers were directly displayed on the pie chart. Such learners, for example, this learners’ work;

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, and

,

committed common errors of merely copying any information displayed on the pie chart. Learners’ levels of statistical thinking in this study were level 1 (Idiosyncratic) and level 2 (transitional). The items on Data Handling demanded level 3 (Quantitative) and level 4 (Analytical) thinking levels. Learners were also found to have a challenge in applying their fraction and percentage knowledge to interpret data from the pie chart. Even though 80% of the sample was able to write percentage of marbles Pete got from the pie chart, 20% of the sample seemed to have a challenge. The greatest challenge was on conversion of the number marbles Thandi got to fraction with most learners (56% of the sample), showing no success. Below are some of the examples of learners written responses;

and .

Literature attributes learners’ inability to read and interpret data to less of exposure to data interpretation at primary school owing to considerable time teachers spent in teaching mechanical skills of constructing the graphs and the charts (Jorgensen & Dole, 2011; Briggs, 2013). AS this study was only preoccupied with the analysis of learners’ written responses, further research involving interview to ascertain whether

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learners’ inability to interpret data is linked to reading and comprehending what each item is recommended.

7.3 CONCLUSIONS

The fundamental goal of this study was to investigate the most common errors and their possible underlying misconceptions contributing to low performance attained in the mathematics Grade 6 ANA. Investigation such as this one would help to get a better understanding of the difficulties learners are encountering in learning mathematics at primary school. My assumption in this study was that a better understanding of common errors and underlying misconceptions resulting into such errors in all the content areas covered in the 2014 ANA paper would help to shed a light on low performance in mathematics at selected primary schools in Johannesburg South District. This section summarises this study. It includes conclusions on common errors and their possible underlying misconceptions. The recommendations for future study stemming from this study are also included in this section.

The results of this study reveal that learners have committed numerous errors and some errors are common amongst most of the learners as they emanate from common misconceptions held by group of learners. The errors identified in this study across all the content areas covered in Mathematics 2014 ANA paper, reveal learners’ misunderstanding of various mathematical concepts.

 Learners in this study struggled with the concept of computation with whole numbers, concept place value and rounding off, addition and subtraction operation on fractions, comparing and ordering decimals, number theory, equivalence relation between fractions, decimals and percentage, and number line.

The concept of computation with whole numbers forms one of the fundamentals for mastering primary school mathematics and also determine the understanding for other concepts. The study found that more learners merely resorted into writing responses with no discernible link to the multiple digit numbers they were required to compute. Such errors are linked to learners’ lack basic number sense coupled with lack of basic addition, subtraction, multiplication and division facts.

 Learners struggled with the concept of equality, inverse relationship between operations, investigating and extending the geometric patterns, investigating patterns to determining the following three components i.e. input, the rule and the output.

 Learners struggle with time concept, reading scales, conversions from one unit to another, and comparison of lengths of different units.

 Learners struggled to identify angles, recognising and naming 2-D Shapes, properties of shapes and attributes of 3-D shapes i.e. faces, vertices and edges.