Using geoboards in primary mathematics: going ... going ... gone? Hilary Scandrett reminds us of geoboards and asks whether or not they still have a place in today's classrooms.

Until recently, I had forgotten what a geoboard was and how it was used. Upon reintroduction, fond memories of using it as an investigative tool in primary school, were triggered. It seems that geoboards have also been forgotten by mathematics educators and there is little reference to them in recent literature. This led me to question: Are geoboards a forgotten tool? Are they still relevant in today's classrooms? Has modern technology replaced geoboards and made them obsolete?

For those of us who have forgotten, what is a geoboard?

Invented by English mathematician and pedagogist, Caleb Gattegno (1911-1988), the geoboard was designed as a manipulative tool for teaching primary geometry in schools (Williams, 1999). Traditionally made out of plywood and nails, geoboards today are usually made out of plastic and come in a variety of different sizes and colours. Rubber bands are placed around the nails or pegs to form different shapes (see Figure 1). As a learning tool, it provides a means to act upon the world and can be used as a cognitive scaffold that facilitates the extension of knowledge (Salomon & Perkins, 1998, in McInerney & McInerney, 2002).

[FIGURE 1 OMITTED]

How can geoboards be used in teaching?

The geoboard is versatile and can be used at all levels for teaching and learning about different areas of mathematics. It has been found to be a particularly useful aid for investigational and problem solving approaches (Carroll, 1992). There is no set sequence to use with geoboards when using them to teach a mathematical concept and so, is an easy tool to incorporate into mathematic units and learning sequences. Like every tool, however, time needs to be allowed for free play, so that students have the opportunity to explore and experiment with new equipment. Another advantage of the geoboard is its design, as it allows for even young children, and those who may experience difficulty in drawing shapes, to construct and investigate the properties of plane shapes (Carroll, 1992).

Carroll (1992) suggests that geoboards can be used in different areas of mathematics. It is suggested that geoboards be used in conjunction with isometric dot paper, so that exploration can be furthered and work can be recorded easily. The areas of mathematics in which geoboards can be used in include:

* plane shapes
* translation
* rotation
* reflection
* similarity
* co-ordination
* counting
* right angles
* pattern
* classification
* scaling
* position
* congruence
* area
* perimeter.

From this it can be seen that geoboards, can particularly support learning in the measurement, space and geometry strands of the primary mathematics curriculum. The following example illustrates the versatility of geoboards and how they can be used to develop students' understanding in the strands of space and geometry.

The K-6 mathematics syllabus document (Board of Studies New South Wales, 2002) classifies space and geometry as the study of spatial forms and is organised into three substrands: three-dimensional space, two-dimensional space and position. It considers recognising, visualising and drawing shapes, and describing the features and properties of three and two-dimensional objects, as important and critical skills for students to acquire. The development of geometric understanding as set out by the syllabus document, incorporates the first three levels of van Hiele's theory (Clements & Battista, 1992). Table 1 describes these three levels and provides examples of activities which can be used to assist students' progress through the levels.

From Table 1, it can be seen that geoboards can be used to support all three levels of geometric thought and of course there are many other activities that could be done. Furthermore, through using geoboards, students can not only work towards space and geometry outcomes, but also be engaged in working mathematically (Board of Studies New South Wales, 2002).

Have computers made geoboards obsolete?

The National Council of Teachers of Mathematics (NCTM) in their Principles and Standards for School Mathematics (2000) provide insight into the ways that calculators and computers are reshaping the mathematical landscape. They suggest that the appropriate and responsible use of technology can enhance student learning. Bobis, Mulligan and Lowrie (2004), support this view and further suggest that, with appropriate software, the computer can become a very powerful tool that enables students to manipulate spatial arrangements and construct visual images that would be usually limited by their drawing capabilities. Many of the activities in Table 1, for example, could be undertaken using software or websites which feature geoboards as a virtual manipulative.

Technology, however, cannot replace the mathematics teacher, nor can it be used as the sole resource for developing basic understandings and intuitions. Instead the teacher must make prudent decisions about when and how to use technology and should ensure that the technology is enhancing students' mathematical thinking (NCTM, 2000). This view is endorsed by Way (2006) who recommends that hands-on activities should be used to help students form mental images before commencing abstract tasks on the computer.

Conclusions

Overall geoboards have the potential to develop students' understandings in the mathematical strands of measurement, space and geometry. This learning can be further enhanced when students, under the guidance of their teacher, have the opportunity to engage in the hands-on experience of using geoboards, followed up by the more abstract experiences accessible through technology. Geoboards should not be forgotten in the mathematics classroom, but like other tools, should be used to engage students and facilitate their learning.

References

Board of Studies New South Wales. (2002). Mathematics K-6 Syllabus. Sydney: Author.

Bobis, J., Mulligan, J. & Lowrie, T. (2004). Challenging Children to Think Mathematically. Sydney: Pearson Education Australia.

Carroll, J. (1992). Using the geoboard for teaching primary mathematics. In M. Horne & M. Supple (Eds), Mathematics: Meeting the Challenge (pp. 283-288). Brunswick: The Mathematical Association of Victoria.

Clements, D. & Battista, M. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 420-464). New York: Macmillan Publishing Company.

McInerney, D. & McInerney, V. (2002). Educational Psychology: Constructing Learning. Sydney: Pearson Education Australia.

National Council of Teacher of Mathematics. (2000). Principles and Standards of School Mathematics. Retrieved 25 August 2007 from http://standards.nctm.org

Way, J. (2006). Hot ideas: Polygons on the computer. Australian Primary Mathematics Classroom, 11(2),15-17.

Williams, S. (1999). Mathematicians of the African Diaspara. Retrieved 12 May 2008 from: www.math.buffalo.edu/mad/special/ Gattegno_Galeb1911-1988.htm/

Hilary Scandrett

University of Sydney (student)

<[e-mail]>

Table 1. Activities appropriate to van Hiele's levels of geometric
thought.

LEVEL           DESCRIPTION                  ACTIVITIES

One:            * Identifies and operates    * Making shapes on the
Visualisation     on shapes and other          geoboard, followed by
                  geometric configurations     discussion
                  according to their         * Make as many as you can
                  appearance                   of the same shape on
                * Reasoning is dominated       the geoboard, differing
                  by perception                in size and position
                * Objects are recognised     * Ask students to follow
                  visually "as the same        instructions and then
                  shape"                       ask them what shape
                                               they have made

Two:            * Recognises and             * Present differing
Descriptive/      characterises shapes by      shapes on the geoboard
Analytic          their properties             discuss the ways in
                * See figures as wholes,       which in which the
                  but now as collections       shapes are similar and
                  of properties rather         different
                  visual gestalts            * Ask students to make a
                * Properties are               shape with a certain
                  established                  property; e.g., four
                  experimentally by            sides and discuss the
                  observing, measuring,        similarities and
                  drawing and modelling        differences in the
                                               shapes made

Three:          * Forms abstract             * Ask students to sort
Abstract/         definitions                  shapes on the geoboard
Relational      * Distinguishes between        according to their
                  necessary and sufficient     properties (allow
                  sets of conditions for a     children to select the
                  concept                      criteria)
                * Reasons with the           * Reflecting and rotating
                  properties of classes of     shapes
                  figures                    * Investigating the
                * Reorganises ideas by         symmetry of shapes,
                  interrelating properties     using mirrors
                  of figures and classes
                  of figures