What you will study
The module comprises eight core units and a final optional unit. Each unit has mathematical content and pedagogical content.
Unit 1: Organising and classifying
Mathematical content: triangles and quadrilaterals; shape properties (perpendicular sides, parallel sides, equal sides and angles).
Pedagogical content: properties – organising and classifying (shape definition; discrete and inclusive classifications); Van Hiele levels of geometric reasoning; emphasising and ignoring; figural concept.
Unit 2: Conjecturing and convincing
Mathematical content: triangles and quadrilaterals; congruence and similarity: symmetry; proving.
Pedagogical content: conjecturing and convincing (examples and non-examples; emphasising and ignoring; conventions in geometric notation).
Unit 3: Static and dynamic representations of geometric figures
Mathematical content: drawing and constructing geometric figures using squared paper, ruler and compasses and paper folding; constructing geometric figures using Dynamic Geometry Software; using measures of sides and angles to justify shape properties (and understand this is different from proof).
Pedagogical content: static and dynamic representations; soft and robust constructions.
Unit 4: Invariance and change
Mathematical content: lengths, angles, areas, volumes; Pythagoras theorem.
Pedagogical content: Invariance and change (conventions; another and another (examples))
Unit 5: Representing abstract concepts in geometry
Mathematical content: concrete manipulatives, diagrams, co-ordinates, Dynamic Geometry Software (DGS), mental imagery, verbal; constructions of figures; plans and elevations; coordinates; properties (and representations) of 3D shapes; reflecting on what geometric thinking is being worked on and how we recognise it?
Pedagogical content: representing abstract concepts (organising and classifying); learner constructed examples; conjecturing and convincing; generalising; doing and undoing; invariance and change; figural concept (Fischbein); concept image (Tall and Vinner).
Unit 6: Transforming shapes in two and three dimensions
Mathematical content: reflections, rotations, translations, enlargements; tiling patterns (infinity).
Pedagogical content: transformations (doing and undoing plus previous pedagogic ideas).
Unit 7: Circles, reasoning and proving
Mathematical content: use of diagrams both static and dynamic; angles subtended on a chord; cyclic quadrilaterals.
Pedagogical content: circles and circle theorems (say what you see; DGS: invariance and change; convince: use of diagrams and isosceles triangles).
Unit 8: Trigonometry
Mathematical content: ratio; similarity; graphing trig functions; unit circle to generate trig values.
Pedagogical content: trigonometry (representations; solving physical problems; context).
Unit 9: Geometry and algebra – making the connection
This unit provides optional study focusing on the links between algebra and geometry.
Mathematical content: links to algebra (algebraic equations; trig functions and identities; Pythagorean triples).
Pedagogical content: work linking the geometry and algebra modules.
You will learn
- Become familiar with the field of geometry and the use of analytic frameworks for understanding geometric thinking and learning.
- Apply a range of approaches to geometric problems in your own mathematics and in interpreting learners’ geometrical activity.
- Formulate approaches to teaching and critically evaluate evidence from observations.
- Communicate geometric thinking, including adapting problems to suit different learners and purposes.
- Develop a personal perspective on issues covered in the module and reflect on developments in your thinking.
- Communicate and write accurately and clearly, using the conventions of academic writing.
- Use dynamic geometry software to support the learning of geometry.