THE BIG IDEAS
The Van Hiele Theory is five sequential levels of learning specifically related to geometry. The first three of these relate to primary education:
- During the first stage (recognition) shapes are viewed holistically and language is informal. The focus is on recognising shapes rather than attributes and locating these shapes in the students environment.
- The second stage is called analysis. This is where attributes are labeled and descriptions for shapes are produced.
- Ordering is where relationships between attributes and shapes are formed. This comes directly from experiences with concrete materials and justification is required for any statements made by the students.
This is important to note as a teacher because each stage requires concrete materials and a logical progression through the mathematical language model. The theory also highlights the links to number which will be discussed further in the thinking strategies.
THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
Concept
The general concept for geometry is that it is the study of shape, space and measurement. One way to introduce this to students is looking at concrete shapes in their environment around them, such as walking around the school and seeing what shapes we can find (e.g. windows look like squares, benches look like rectangles, the handles on the monkey bars look like triangles and the drum in music corner looks like a circle). To start an activity like this an introduction video of finding shapes in our environment (such as the one below) may be helpful.
Video: Shapes all around us song
There are five skills involved with geometry:
Visualising
Communication
Drawing and modeling
Sketching, scale drawing and modelling 2D and 3D shapes.
Thinking and reasoning
Includes logical skills such as classifying, analysing and seeking similarities and differences, reasoning and synthesising.
Applying geometric concepts & knowledge
Students are able to apply geometric concepts and knowledge to other mathematical areas (such as measurement) and reality.
Visualising
Students must understand the shape. For example, what is a circle? What are its attributes? What makes a circle
different from a square? What are its critical attributes?
Students should be encouraged to
talk and write about shapes and patterns and to describe figures and
relationships as accurately as possible via games and problem solving
activities.
Drawing and modeling
Sketching, scale drawing and modelling 2D and 3D shapes.
Thinking and reasoning
Includes logical skills such as classifying, analysing and seeking similarities and differences, reasoning and synthesising.
Applying geometric concepts & knowledge
Students are able to apply geometric concepts and knowledge to other mathematical areas (such as measurement) and reality.
Strategies
Strategies involved with geometry involve re-purposing the concepts learnt during pre-number and early number. These included:
Strategies involved with geometry involve re-purposing the concepts learnt during pre-number and early number. These included:
- Determining attributes
- Matching by attributes
- Sorting by attributes
- Comparing attributes
- Ordering attributes
- Patterning
For more information on these concepts click here.
THE LANGUAGE MODEL
The language model is used to display the relationship between the visual, verbal and symbolic elements of mathematics and forms 'stages' of learning.
Student language - During student language, all language used should be familiar to the child and accompanied with familiar objects.
Materials language - At this stage, there is still no introduction of mathematical terms. The language is very similar to student language however the visuals used have become more abstract.
Mathematics language - Moving away from 'stories', this stage introduces mathematical terms.
Symbolic language - This stage introduces symbols.
THE LANGUAGE MODEL FOR PLACE VALUE
THE MISCONCEPTION
One misconception is that some objects that are particularly flat (such as an A4 sheet of paper) are referred to as a plane shape. This is incorrect because even though it has very little width, it is still there. A child with this misconception many need more work discussing what makes a 3D (or solid) shape. I would do this by comparing a sheet of paper with a rectangular prism, and a drawn rectangle. I would ask the student to point at a solid shape and then point at a plane shape. Then we would discuss the difference between them and then, again, ask which shapes are solid and which shapes are plane. I would repeat this process with a variety of shapes.
THE ACARA LINK
Geometry is first introduced in the foundation year.
Strand: Measurement and Geometry
Substrand: Shape
Content descriptions: Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment
Elaborations:
- sorting and describing squares, circles, triangles, rectangles, spheres and cubes
Scootle resource ideas:
- STOP! that's an octagon is a short video about finding octagons in the environment, when an octagon is found (in a stop sign) the attributes are discussed in order to justify the shape.
- Learn to draw shapes with ziggy is a rap about line shapes.
- Shape overlays - Find and cut is an online game that requires students to cut one shape in order to create another.
THE RESOURCES AND IDEAS
- Maths Learning centre released a Geoboard app that allows students to create shapes digitally - this removes the possibility of bands breaking or students flinging rubber bands across the room.
- Complete the shape is online game that allows students to place shapes together to create a new shape.
- Kids educational games has a game called Basic 2D and 3D Shapes. It is an interactive explanation of the most common 2D and 3D shapes.
THE TEXTBOOK SUMMARY
- Starts in the foundation year
- Geometry is a natural site for including other skills, such as following directing and reasoning about shapes and their properties
- Geometry, together with measurement is one of the three essential contents strands of the Australian Curriculum
- Understanding the properties or attributes of objects and the relationships among different geometric objects is an important part of primary mathematics
- It is important to start the exploration with familiar 3D shapes such as balls and blocks as this provides the students with familiarity while providing the students with differences (such as a ball can role but a block can't, a ball has a curved surface and a block has a flat surface)
- Children need to describe and sort 3D shapes by their properties, such as by corners, edges or can roll/can't roll
- One difficult students have with 3D shapes is visualising the solids, it is essential to have models of these solids
- Students first recognise shapes in a holistic manner. If the student is only shown right angle triangles, that will be the only shape they will recognise as a triangle, if they are only shown green triangles, when asked to draw a triangle it will be coloured green
- Another part of geometry deals with location, movement, maps and plans which describe direction, distance and position. This includes positional language (Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al., 2012).
THE REFERENCES
ACU,. (2016). Learning Environment Online. Leo.acu.edu.au. Retrieved 3 March 2016, from http://leo.acu.edu.au/course/view.php?id=18458
Australia, E. (2016). Home - Scootle. Scootle.edu.au. Retrieved 3 April 2016, from https://www.scootle.edu.au/ec/p/home
Australian government,. (2016). Home - The Australian Curriculum v8.1.Australiancurriculum.edu.au. Retrieved 3 April 2016, from http://www.australiancurriculum.edu.au/
Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al. (2012). Helping children learn mathematics. Milton, QLD: John Wiley & Sons.
YouTube. (2016). Youtube.com. Retrieved 3 April 2016, from https://www.youtube.com/
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