Teaching geometry

Rationale for teaching geometry• geometry has been an integral part of mathematics and a common

vehicle for teaching the critical skill of deductive reasoning.• It offers a physical context in which students can develop and refine

intuition, leading to the formulation and testing of hypotheses and ultimately resulting in the justification of arguments, both formally and informally. Geometry also describes changes in objects under such transformations as translation, rotation, reflection, and dilation.

• It helps students understand the structure of space and the nature of spatial relations.

• The measurement aspect of geometry provides a basis by which we quantify the world.

• Solving practical problems relies to some extent on approximate physical measurements but also rests on geometric properties that are exact in nature.

• Grounded in such certainty, geometry provides an excellent medium for the development of students’ ability to reason and produce thoughtful, logical arguments

example

• -Concepts involved in FET Euclidean Geometry

• -Difficulties & limitations connected with teaching Euclidean Geometry

• -Specific strategies and procedures for teaching Euclidean Geometry)

• -barriers to understanding Euclidean Geometry

• -teaching FET Euclidean Geometry with technology (DGE)

• Several frameworks for geometrical reasoning were proposed byresearch studies in the 1990’s that aimed at understanding theprocesses of teaching and learning geometry.

1. Jones (1998) suggests the van Hiele’s (1986) model of thinking ingeometry,

2. Fischbein’s (1993) theory of figural concepts, and3. Duval (1995) cognitive apprehensions for geometrical reasoning.

The van Hiele (1986) model is prominent among studies on geometryknowledge in South Africa. For example van der Sandt (2007), van derSandt and Nieuwoudt (2005), Atebe (2008) and Luneta (2014) employedthe van Hiele model of geometry thinking to study geometry knowledgeat primary, secondary and tertiary education.

The Duval model is of particular interest as it is more concerned withunderstanding the development of cognitive processes as revealed whensolving geometry problems (Duval, 1998, 2007). Duval (1995) suggests ananalytic theory for analysing thinking processes involved in a geometricactivity.

Learning and Teaching of Geometry:three cognitive processes to identify reasons for why and how geometry should be taught in school (Duval, 1995)

Visualization process : refers to the use of representations (e.g figures, images, diagrams, symbols) for illustration, exploration or verification of different geometric situations;

Construction process: related to actions for constructing a configuration according to restricted tools and geometrical requirements;

Reasoning process: related to discursive processes for proof and explanation.

Connectedness of the cognitive processes

• Arrow shows how the processes support another

Example of visualization and reasoning

Mathematical objects and meaning/interpretation

Geometric registers: symbolic, figural⊥;∥ perpendicular; parallel

∞ infinity

8 number

∀ for all/for every

A point/letter

∃ there exists

E point/letter

Duval’s cognitive model of geometrical reasoning: the role of a figure

• a figure gives us a figural representation of a geometrical situation which is shorter and easier to be understood than a representation with linguist speech.

• cognitive apprehensions of figures:Seeing, constructing and describing a geometrical figure and its properties

1. Perceptual apprehension

2. Sequential apprehension

3. Discursive apprehension

4. Operative apprehension

• It is about physical recognition (shape,representation, size, brightness, etc.) of aperceived figure. We should also discriminateand recognize sub-figures within the perceivedfigures since a relevant discrimination orrecognition of these sub-figure units may helpand give cues for problem solving ingeometrical situations.

perceptual apprehension

sequential apprehension

• It is about construction of a figure ordescription of its construction. Suchconstruction depends on technical constraintsand also mathematical properties sinceconstruction of a figure may merge differentfigural units. It is believed that constructioncan help recognition of relationships betweenmathematical properties and technicalconstraints.

discursive apprehension:

• It is about (a) the ability to connectconfiguration(s) with geometric principles, (b)the ability to provide good description,explanation, argumentation, deduction, use ofsymbols, reasoning depending on statementsmade on perceptual apprehension, and (c) theability to describe figures through geometriclanguage/narrative texts

operative apprehension: • It is about making modification of a given figure

in various ways to investigate others configurations:

• the mereological way: dividing the whole givenfigure into parts of various shapes and combinethese parts in another figure or sub-figures;

• the optic way: varying the size of the figures; you can make a shape larger or narrower, or slant, the shapes can appear differently

• the place way: varying the position or itsorientation.

Task

Study your question and showcase your understanding of

• The critical components of the question

• The actions required to complete the question

basing on Duval’s cognitive model of geometrical reasoning