Academic journal article Australian Primary Mathematics Classroom

What's a Real 2D Shape? Designing Appropriate Geometric Instruction

Academic journal article Australian Primary Mathematics Classroom

What's a Real 2D Shape? Designing Appropriate Geometric Instruction

Article excerpt

This paper describes the process of designing and teaching an instructional unit on two-dimensional shapes for 5th grade students in two primary schools in the same local school district. Conscious of Van Hiele's (1999, p. 310) promotion of Piaget's tenet that no instruction '... is better than giving it at the wrong time' the first lesson was used to uncover children's level of geometric understanding. This exploration ensured that our geometry instruction was needs-led (Fox, 2000) and built upon current understandings. Therefore activities in the first lesson were carefully selected to ascertain the level within Van Hiele's hierarchy of geometric thinking at which students were located. According to this theory, students functioning at level 0 (the visual level) classify shapes based solely on overall appearance. For example, when presented with a rectangle they may say 'it is a rectangle because it looks like a door'. Students at level 1 (the descriptive level) identify the properties of a and use these for the purpose of classification. For example, when presented with a rectangle, they may classify it as a rectangle based on the number of sides and corners. Students functioning at level 2 (the informal deduction level) can deduce one property from another and make informal arguments to justify their conjecture, for example students may understand that squares are special types of rectangles (Van Hiele, 1999; Robichaux and Rodrigue, 2010). Research has reported that students may 'straddle' levels and that their tendency to judge shapes based on appearance may persist beyond elementary school (Shifter, 1999).

The first lesson consisted of a sequence of open-ended activities. This paper focuses on the first two of these activities, describing the activity as well as the observations made of children's geometric reasoning. The findings with regard to children's knowledge were quite revealing.

Considerations underpinning our selection of 2D shapes

In an effort to determine students' level of geometric thinking, a sequence of activities was developed requiring students to work with polygons (Mack, 2007). As many puzzles, posters, picture books and even elementary level textbooks focus on prototypical shapes such as the regular (all sides and angles equal) triangle and pentagon , we were aware that students may have limited or no experience working with non-prototypical shapes. Research indicates that the overuse of prototypical shapes can lead to difficulties in recognising particular shapes as belonging to a specific category (Fox, 2000; Edwards and Harper, 2010). For example, the first triangle in figure 1, the equilateral triangle, is a common triangle prototype. However if children are exposed only to this prototype, they may not recognise that the other shapes in figure 1 are triangles. Similarly, the over representation of shapes on a horizontal base leads to similar difficulties. The repeated presentation of a square on its horizontal base (see first shape in figure 2) may result in many children not recognising the second shape as being a square.

Therefore we chose to include non-prototypical shapes in our activities. It was intended that the activities would assess and extend students' geometric understandings. An important aspect of these activities is that they encourage children to communicate their reasoning which, in turn, allows us to make more informed instructional decisions when planning subsequent geometry lessons.

Activity 1: Taking a virtual tour

To introduce the lesson, children were invited by the teacher to go on a 'virtual' tour of their school in the search for polygons:

   One of my favourite hobbies is photography.
   I love taking photos of just about anything. I
   was asked by the principal to take photos of
   different areas of the school as she is creating
   a new brochure for parents. So I went around
   the building--inside and outside--taking
   photographs of everything. … 
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