Threshold Concepts with Space and

Geometry (Stages 1-3)

Mark Grady August 2011

So what lies ahead?

• 2 minute recap on Threshold Concepts

• What are the Threshold Concepts related to Space and Geometry? Does NAPLAN inform us?

• Ideas to support the teaching of aspects of Space and Geometry

A 2 minute reminder of Threshold Concepts

• Threshold concepts have been described as a portal or gateway

• Transformative - once acquired it shifts perception of the subject

• Irreversible - once learners have come to see the world in terms of the threshold concept they cannot return to their former, more primitive, view

• Integrative - acquisition of the threshold concept illuminates the underlying inter-relatedness of aspects of the subject

• Troublesome - a threshold concept may be counterintuitive and initially very difficult for learners to accept. In grasping a threshold concept the learner moves to a new perception of the world that may be in conflict with previously held perceptions.

Threshold Concepts within Space and Geometry

• What do you consider to be Threshold Concepts within Space and Geometry?

• Discuss in pairs or small groups at your site (3 minutes)

What does NAPLAN tell us?

• Let’s investigate a few questions from 2008, 2009 and 2010.

• These are Yr 7 NAPLAN, but Stage 3 outcome questions

2010

2009

My view on Threshold Concepts and Space / Geometry

• There is no formal research on this, so it is my view (though discussed with a number of colleagues) Given the characteristics of Threshold Concepts :

• Once acquired it shifts perception of the subject • Once learners have come to see the world in terms of

the threshold concept they cannot return to their former, more primitive, view

• Acquisition of the threshold concept illuminates the underlying inter-relatedness of aspects of the subject

• Troublesome

• Not necessarily containing Threshold Concepts, but certainly full of troublesome concepts– Position when given an angle not measured from

North. See NAPLAN 2011 for a great example of this (smallest angle turn)

– Converting text into mathematics – metalanguage e.g. Position question above

– Recognising shapes in different orientations or views

– Which is a right angled triangle?

• Despite, in my view, no real Threshold Concepts in Stages 1 - 3 S and G, we have to carefully consider how best to program activities and experiences to best meet our students needs.

• That is a real strength of this course – a chance to discuss, reflect and adapt

Teaching Ideas• Euler’s formula: V + F – E = 2

– Although not formally in the syllabus to Stage 4, the discovery element of counting corners, faces and edges appears in earlier Stages. Polyhedra with straight edges

Diagram Name Corners Faces Edges

• Smart software – vortex• Hands on 3D shapes• Allow students to experience position, angle

questions as both practical and theoretical activities. Orienteering

• Sketching and describing from different views – give them 3D shapes to hold etc, then sketch. A bit of work with a hammer, a few nails (or glue) can make some great shapes.

• Spinning and stacking (Stage 2)• Viewfinder (Stage 2,3,4)• Nature by Numbers has some relevant parts – to

stimulate interest and discovery (SGES1.2)