Introduction to Geometry
Geometry is the study of shapes, tessellations, patterns and positions in both 2D and 3D concepts (Van de Walle, et al., 2014; Cornell University, n.d.). Furthermore geometry is a "network of concepts, ways of reasoning and representation systems" surrounding the study of shapes (Battista, 2007 in Van de Walle, 2014). This means that geometry is more than just 2D and 3D shapes, it is about how they are represented, positioned and patterned in real life contexts. In primary settings, students examine both 2D and 3D shapes through perceiving, describing, discussing and constructing these shapes (National Research Council Committee, 2009). Through this, the development of geometric thought takes place, particularly through the the Van Hiele theory of levels of geometric thought.
Geometry is the study of shapes, tessellations, patterns and positions in both 2D and 3D concepts (Van de Walle, et al., 2014; Cornell University, n.d.). Furthermore geometry is a "network of concepts, ways of reasoning and representation systems" surrounding the study of shapes (Battista, 2007 in Van de Walle, 2014). This means that geometry is more than just 2D and 3D shapes, it is about how they are represented, positioned and patterned in real life contexts. In primary settings, students examine both 2D and 3D shapes through perceiving, describing, discussing and constructing these shapes (National Research Council Committee, 2009). Through this, the development of geometric thought takes place, particularly through the the Van Hiele theory of levels of geometric thought.
(Van de Walle et al., 2014)
Level 0: Visualisation
Shapes are classified according to the visualisation that students see. Appearance is at the forefront of this level of thinking, with students placing higher focus on what the shapes look like, instead of the properties of the shapes. With this in mind, when students come across shapes that are positioned differently the student may not perceive that the original shape has just been altered (i.e. a square that has been rotated 45° can now appear to look like a diamond (Van de Walle, et al., 2014)).The classification of shapes at this level will be entirely dependent on the appearance and students can classify according to likeness and difference to other shapes (Van de Walle, et al., 2014). Level 1: Analysis At level 1, students are using their geometric thought to begin to identify basic properties of shapes as well as recognising that there are more than one shape in a class. Through this, students begin to understand that collections are made up of properties of shapes, and not just the irrelevant features (Van de Walle, et al., 2014). Yu, Barrett & Presmeg (2009) elaborate, stating that the identifying of shape properties is important for cognitive development. |
Level 2: Informal deduction At this stage of geometric thought, students begin to develop understanding of relationships between shapes and their properties. This means that students are able to work with shapes using very little information (Van de Walle, et al., 2014) and can appreciate that properties have a rule structure in place for formal logical reasoning. Questioning that encourages students to find out why shapes are made up as they are as pivotal to this level of geometric thought (Van de Walle, et al., 2014). |
Level 3: Deduction Geometric thought at this level extends on the basic reasoning strategies employed in level 2. According to Van de Walle et al. (2014), analysis of informal arguments takes place, where "axioms, definitions, theorems, corollaries and postulates" begin to develop. These enable the students to employ higher order thinking and develop conclusions surrounding shapes and geometric thought. This type of geometric deduction occurs in later high school years where students are beginning to branch into specialist mathematics subjects. |
Level 4: Rigor At the highest stage of the Van Hiele theory, students extend on the higher order thinking developed in high school mathematics. They begin to look at axiomatic systems themselves, rather than the deductions of geometry (Van de Walle, et al., 2014). This means that students begin to look at the basic premises of why certain things occur and how they occur like that. Additionally, there is further distinction between axiomatic systems and their relationships (Van de Walle, et al., 2014). It is assumed that this level of geometric though occurs in tertiary education and beyond. |
Transformations
Within the study of geometry, students learn how to manipulate shapes and points and lines within these shapes (Van de Walle, et al., 2014). Transformations also enable students to alter the position and size of shapes. By changing the shapes in any way enables students to work out if the transformations are congruent or similar.
Within the study of geometry, students learn how to manipulate shapes and points and lines within these shapes (Van de Walle, et al., 2014). Transformations also enable students to alter the position and size of shapes. By changing the shapes in any way enables students to work out if the transformations are congruent or similar.
Manipulation of shapes through transformation (MathsIsFun, n.d.)
Mapping and location
Mapping and location concepts within geometry elaborate on the positioning of shapes within a coordinate plane (Council of Chief State School Officers, 2000). They analyse the paths shapes take when transforming and students learn specialised reasoning regarding spatial understandings (Van de Walle, et al., 2014). As students engage with the concepts, they move onto more intricate processes using different axes within geometric reasoning (Haylock, 2013).
Mapping and location concepts within geometry elaborate on the positioning of shapes within a coordinate plane (Council of Chief State School Officers, 2000). They analyse the paths shapes take when transforming and students learn specialised reasoning regarding spatial understandings (Van de Walle, et al., 2014). As students engage with the concepts, they move onto more intricate processes using different axes within geometric reasoning (Haylock, 2013).
Different location and mapping styles within geometric processes (Van de Walle, et al., 2014; Coghlan, 2012)
Common student misconceptions
- Shape orientation may confuse students when beginning to learn geometric processes
- Students may believe that shapes can only be a certain type of shape and not fit into a class of shapes i.e. a square is a type of rectangle
- Students may not comprehend that shapes can be manipulated to create irregular shapes i.e. square to a parallelogram
- Student may not realise that shapes can transform in more than on way and can be mapped different on a grid
Mathematical language
Angles, shape, classify, compare, location, map, translation, transformation, rotate, straight, curved, round, long, short, congruent, similar, parallel, tessellations, axis, two-dimensional, three-dimensional, visualisation
(Adapted from Van de Walle, et al., 2014; Haylock, 2013)
Angles, shape, classify, compare, location, map, translation, transformation, rotate, straight, curved, round, long, short, congruent, similar, parallel, tessellations, axis, two-dimensional, three-dimensional, visualisation
(Adapted from Van de Walle, et al., 2014; Haylock, 2013)
Word problems
1. The radius of a circle is 5 cm. What is the circle's circumference?
2. A square has an area of 32 cm2 . What are the lengths of it's sides?
3. A cube has a surface area of 64 cm3. What is the volume of the cube?
1. The radius of a circle is 5 cm. What is the circle's circumference?
2. A square has an area of 32 cm2 . What are the lengths of it's sides?
3. A cube has a surface area of 64 cm3. What is the volume of the cube?
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