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The role geometry plays in real life makes it a core component of mathematics that students must understand and master. Conceptual knowledge of geometric concepts goes beyond the development of skills required to manipulate geometric shapes. This study is focused on errors students made when solving coordinate geometry problems in the final Grade 12 examination in South Africa. An analysis of 1000 scripts from the 2008 Mathematics examination was conducted. This entailed a detailed analysis of one Grade 12 geometry examination question. Van Hiele levels of geometrical thought were used as a lens to understand students’ knowledge of geometry. Studies show that Van Hiele levels are a good descriptor of current and future performance in geometry. This study revealed that whilst students in Grade 12 are expected to operate at level 3 and level 4, the majority were operating at level 2 of Van Hiele's hierarchy. The majority of students did not understand most of the basic concepts in Euclidian transformation. Most of the errors were conceptual and suggested that students did not understand the questions and did not know what to do as a result. It is also noted that when students lack conceptual knowledge the consequences are so severe that they hardly respond to the questions in the examination.

Geometry is the ‘study of shapes, their relationships, and their properties’ (Bassarear,

Research has also noted that geometry is difficult to teach as well as to learn. Coordinate or analytical geometry, for instance, requires not only geometrical knowledge, but also a vast amount of knowledge in working with coordinates on a 2D (two-dimensional) or 3D (three-dimensional) set of axes. These additional concepts make geometry more complex and require an intricate manner of thinking. Van der Sandt (

Piaget (

Van Hiele (

Understanding these levels enables teachers to identify the general directions of students’ learning and the level at which they are operating (Lim,

Van Hiele's levels provide teachers with a framework within which to conduct geometric activities by designing them with the assumptions of a particular level in mind and they are able to ask questions that are below or above a particular level (Lim,

The research question is: What were the most common error that students in Grade 12 displayed in the examination scripts on the geometry question?

School curricula worldwide cover four main learning outcomes in Geometry (Bahr et al.,

Analyse the characteristics, properties and relationships of two-dimensional and three-dimensional geometrical shapes (Euclidean Geometry).

Specify locations and describe spatial relationships using coordinate geometry and other representation systems (Coordinate Geometry).

Apply transformation and use symmetry to analyse mathematical situations (Transformation Geometry).

Use visualisation, spatial reasoning and geometric modelling to solve problems.

Research has delineated that errors occur mainly because students have difficulties in understanding the instructional strategies adopted by the teacher (Confrey,

According to Swan (

The most common errors in transformation geometry are the result of students operating at levels that are different to their teachers’. It is evident that people reasoning at different levels may not understand each other and this is true for teachers and students. A student reasoning at level

This study conducted an analysis of 1000 Grade 12 mathematics scripts. These were obtained from the Department of Education with this purpose in mind. The scripts were randomly selected from the entire 2008 batch of 108 000 scripts. The selection was not based on schools, but was merely an assortment of scripts. After being sampled the 1000 scripts were stratified into three groups according to student ability. Group 3 was made up of students who attained between 0% and 32% (

Students’ attempts to answer Question 3.

Question 3: Transformation geometry | Cognitive demand level (K/RP/CP/PS) | Van Hiele levels | Correctly answered | Partially correct | Incorrect | Not answered |
---|---|---|---|---|---|---|

3.1.1 |
K | 1 | 85,0% Group 1 |
0,0% Group 1 |
15,0% Group 1 |
0,0% Group 1 |

3.1.2 |
K | 2 | 85,8% Group 1 |
0,0% Group 1 |
14,2% Group 1 |
0,0% Group 1 |

3.2.1 |
K | 2 | 89,0% Group 1 |
0,0% Group 1 |
11,0% Group 1 |
0,0% Group 1 |

3.2.2 |
RP | 2 | 85,0% Group 1 |
0,0% Group 1 |
15,0% Group 1 |
0,0% Group 1 |

3.2.3 |
K | 2 | 93,0% Group 1 |
0,0% Group 1 |
7,0% Group 1 |
0,0% Group 1 |

3.2.4 |
CP | 3 | 84,0% Group 1 |
15,5% Group 1 |
0,0% Group 1 |
< 1,0% Group 1 |

3.2.5 |
RP | 4 | 44,0% Group 1 |
4,0% Group 1 |
47,0% Group 1 |
5,0% Group 1 |

CP, complex problems; K, knowledge; PS, problem solving; RP, routine procedures.

Student groups’ average Van Hiele levels of geometric thought on the questions.

It is worth noting that the sixth column (incorrect) of

Students were required to give the coordinates of the image of a point after reflection in the line

A formula not related to the question that was asked (E3.1.1A).

In this question, students were asked to give the coordinates of a point after a 90° rotation in a clockwise direction. This question also required students to operate at Van Hiele level 2. Nine percent of the sample group rotated D in an anti-clockwise direction. Twenty percent of the group reflected D about the

Typical student conceptual error on 3.2.2 (a) and the memorandum (b).

Below are three more examples of the various errors that students made regarding this question; they all show lack of knowledge of transformation geometry. The student's work on the immediate left is a reflection on the

Students lack of knowledge of rigid transformation (E3.2.2D).

This question required students to enlarge a polygon by a given factor and find the image of enlargement. More than half of the students answered the question correctly (method marks were often awarded following on from their answers to question 3.2.1). Students were asked to give the coordinates of D″ after enlarging the polygon by a factor of 3 through the origin. It was difficult to analyse the incorrect answers, as often the given values were not related to the coordinates of D or D′. This implies that students were not conceptually grounded on transformation that involved enlargement and whilst

Explanations of students’ average errors on each of the seven parts of Question 3 and the resultant Van Hiele classification.

Identified learners’ Van Hiele levels | Error analysis key on Question 3 | ||||
---|---|---|---|---|---|

1 | E3.1.1A | E3.1.1B | E3.1.1C | E3.1.1D | |

Student appears to have rotated P 180°: |
Coordinates have been swapped, but error made with signs, e.g. |
Student appears to have reflected P about the y-axis i.e. (√2; √3). | Inappropriate method used or coordinates unrelated to those of P. | ||

1 | E3.1.2A | E3.1.2B | E3.1.2C | E3.1.2D | |

Student appears to have reflected P about y = x, i.e. (√3; -√2). | Coordinates of P given, i.e. (-√2; √3). | Student appears to have reflected |
Unusual method or unrelated coordinates. | ||

1 | E3.2.1A | E3.2.1B | E3.1.2C | E3.1.2D | |

D′(3; -2) | D′(2; -3) | D′(-3; 2) | Unusual method or unrelated coordinates. | ||

2 | E3.2.2A | E3.2.2B | E3.2.2C | E3.2.2D | |

Polygon has not been rotated (it has been reflected on the x-axis). | Polygon has been rotated 90° anticlockwise. | The polygon has not been rotated (it has been reflected on y-axis). | Inappropriate transformation (e.g. rotation of 180°, shape not preserved, translation, rotation about point on the polygon or strange method). | ||

1 | E3.2.3A | E3.2.3B | |||

Coordinates D″ not related to D′. | Unusual method or unrelated coordinates. | ||||

2 | E3.2.4A | E3.2.4B | E3.2.4C | E3.2.4D | |

Student understands the concept of enlargement, i.e. |
Student understands the concept of rotation, i.e. |
Transformation rule not given (student has applied transformation to actual coordinates). | Inappropriate transformation. | ||

1 | E3.2.5A | E3.2.5B | E3.2.5C | E3.2.5D | E 3.2.5E |

Ratio of 1:6 | Ratio of 1:3 | Other ratios (1:4; 1:8; 1:16; 1:2). | Student understands the concept of 9× as being magnification but ratio expressed backwards, i.e. 9:1, or not given as a ratio. | Inappropriate method used. Level 1 |

For this question, students had to create a transformation rule for the combination of the two transformations in 3.2.1 and 3.2.3.

To respond to this question students needed to operate at level 4 of the Van Hiele hierarchy according to

Students’ errors that were not related to the question (E3.2.5E).

_{1} = m(x – x_{1}

According to the curricula, transformation geometry (translation, reflection and rotation) is introduced and taught at primary school (Bassarear,

For instance, a student fixated on the natural shape of a trapezoid will fail to notice that all three figures in

Different orientations of trapezoids.

Teaching geometry to learners in a standardised way leaves them incapacitated when a change in the natural orientation of a figure is affected. For instance, in

Different orientations of a square and a pentagon.

The most common errors were procedural. Students were not able to engage with simple geometric relationships, reflections about a particular line, rotation of shapes in standard angles of 90°, 180° and 360° about a specific point. Every question answered had a higher occurrence of procedural errors than common mistakes or conceptual errors. Whilst there is no evidence from this study to back up this point, other research (Hansen et al.,

The study concurs with research (Centre for Development in Education,

The angle subtended by an arc at the centre is twice the angle subtended on the circle.

Lim (

This study confirmed, on the one hand, previous findings of the literature (a confirmatory study); on the other hand, it explored thinking in geometrical patterns and revealed a number of errors that Grade 12 South African students made in their final examinations. The analysis hinged on students’ answers to Question 3 and this question was mainly on coordinate geometry (specify locations and describe spatial relationships using coordinate geometry and other representational systems) and transformation geometry (apply transformation and use symmetry to analyse mathematical situations). These sections of geometry require learners to mainly operate on levels 1 to 4 of the Van Hiele hierarchy. However, the results revealed that the majority of students in Grade 12 operate at level 2 of Van Hiele's levels of geometrical thought. Whilst the research did not establish the main reason behind this, literature validates that most mathematics teachers are not grounded in instructional strategies that enable students to learn mathematics effectively. Hansen et al. (

Examining the literature and the results derived from the study the first question that this study helped to answer was: How to help students understand high school geometry. The study of geometry, like the other sections of mathematics, starts from early childhood. The first geometrical concepts form the basis for the rest of geometry in school curricula. Thus the best approach involves changing how mathematics and especially geometry is taught before high school. Some points to consider are:

Improve geometry teaching in the foundation and intermediate phases so that students’ Van Hiele levels of geometrical thought are brought up to at least to the level of abstract or relational.

Include more justifications, informal proofs and ‘why’ questions in geometry teaching during Foundation Phase and Intermediate Phase.

The Van Hiele levels explain the understanding of spatial ideas and how one thinks about them. The thinking process that one goes through when exposed to geometric contexts defines the levels of operation and they are not dependent on age (Battista,

While the levels are not age-dependent in the sense of the developmental stages of Piaget and a third grader or a high school student could be at level 0 […] age is certainly related to the amount and types of geometric experience that we have. Therefore, it is reasonable for all children in K-2 range to be at level 0. (p. 347)

Hence, one can expect children in the first grade to be in the first level of Van Hiele's hierarchy – the

I would like to acknowledge the Gauteng Department of Education who funded the main study of which this article is a part.

I declare that I have no financial or personal relationships that may have inappropriately influenced me in writing this article.

Given that P (-√2; √3) is on a Cartesian plane. Determine the coordinates of the image of P if:

P is reflected in the line

P is rotated about the origin through 180°.

Polygon ABCDE on a grid has coordinates A (1; 1), B (1; 2), C (2; 3), D (3; 2) and D (2; 2). Each of the points ABCDE on the grid is rotated 90° about the origin in a clockwise direction.

Write down the coordinates of D′, the image of D.

Sketch and label the vertices A′B′C′D′E′ on the image of ABCDE.

The polygon A′B′C′D′E′ is then enlarged through the origin by a factor 3 in order to give the polygon A″B″C″D″E″. Write down the coordinates of D″, the image of D′.

Write down the general transformation of a point (

Calculate the ratio of area ABCDE:area A″B″C″D″E″.