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 Noun 1. reintroduction  an act of renewed introduction intro, introduction, presentation  formally making a person known to another or to the public , 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 OBSOLETE. This term is applied to those laws which have lost their efficacy, without being repealed, 2. A positive statute, unrepealed, can never be repealed by nonuser alone. 4 Yeates, Rep. 181; Id. 215; 1 Browne's Rep. Appx. 28; 13 Serg. & Rawle, 447. ? For those of us who have forgotten, what is a geoboard? Invented by English mathematician and pedagogist, Caleb Gattegno Caleb Gattegno (19111988) is best known for his innovative proposals for teaching and learning mathematics, foreign languages The Silent Way and reading Words in Color. Caleb Gattegno’s pedagogical approach (19111988), the geoboard was designed as a manipulative ma·nip·u·la·tive adj. Serving, tending, or having the power to manipulate. n. Any of various objects designed to be moved or arranged by hand as a means of developing motor skills or understanding abstractions, especially in tool for teaching primary geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts. in schools (Williams, 1999). Traditionally made out of plywood plywood, manufactured board composed of an odd number of thin sheets of wood glued together under pressure with grains of the successive layers at right angles. Laminated wood differs from plywood in that the grains of its sheets are parallel. 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 scaffold Temporary platform used to elevate and support workers and materials during work on a structure or machine. It consists of one or more wooden planks and is supported by either a timber or a tubular steel or aluminum frame; bamboo is used in parts of Asia. 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 problem solving Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. 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 isometric /iso·met·ric/ (met´rik) maintaining, or pertaining to, the same measure of length; of equal dimensions. i·so·met·ric adj. 1. 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 * coordination * 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 K6 mathematics syllabus A headnote; a short note preceding the text of a reported case that briefly summarizes the rulings of the court on the points decided in the case. The syllabus appears before the text of the opinion. document (Board of Studies New South Wales New South Wales, state (1991 pop. 5,164,549), 309,443 sq mi (801,457 sq km), SE Australia. It is bounded on the E by the Pacific Ocean. Sydney is the capital. The other principal urban centers are Newcastle, Wagga Wagga, Lismore, Wollongong, and Broken Hill. , 2002) classifies space and geometry as the study of spatial forms and is organised into three substrands: threedimensional space Threedimensional space is the physical universe we live in. The three dimensions are commonly called length, width, and breadth, although any three mutually perpendicular directions can serve as the three dimensions. Pictures are commonly two dimensional, they lack depth. , twodimensional space and position. It considers recognising, visualising and drawing shapes, and describing the features and properties of three and twodimensional 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 The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. (NCTM NCTM National Council of Teachers of Mathematics NCTM Nationally Certified Teacher of Music NCTM North Carolina Transportation Museum NCTM National Capital Trolley Museum NCTM Nationally Certified in Therapeutic Massage ) in their Principles and Standards for School Mathematics Principles and Standards for School Mathematics was a document produced by the National Council of Teachers of Mathematics [1] in 2000 to set forth a national vision for precollege mathematics education in the US and Canada. (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 mul·li·gan n. A golf shot not tallied against the score, granted in informal play after a poor shot especially from the tee. [Probably from the name Mulligan.] Noun 1. 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 Noun 1. spatial arrangement  the property possessed by an array of things that have space between them spacing placement, arrangement  the spatial property of the way in which something is placed; "the arrangement of the furniture"; "the placement of the 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 handson 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 handson 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 K6 Syllabus. Sydney: Author. Bobis, J., Mulligan, J. & Lowrie, T. (2004). Challenging Children to Think Mathematically. Sydney: Pearson Education Pearson Education is an international publisher of textbooks and other educational material, such as multimedia learning tools. Pearson Education is part of Pearson PLC. It is headquartered in Upper Saddle River, New Jersey. Australia. Carroll, J. (1992). Using the geoboard for teaching primary mathematics. In M. Horne & M. Supple supple Physical exam adjective Referring to free movement of a body part (Eds), Mathematics: Meeting the Challenge (pp. 283288). Brunswick: The Mathematical Association The Mathematical Association is a professional society concerned with mathematics education in the UK. It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1897. of Victoria. Clements, D. & Battista, M. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 420464). New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : 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),1517. Williams, S. (1999). Mathematicians Mathematicians by letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also
Hilary Scandrett University of Sydney The University of Sydney, established in Sydney in 1850, is the oldest university in Australia. It is a member of Australia's "Group of Eight" Australian universities that are highly ranked in terms of their research performance. (student) <hsca8836@mail.usyd.edu.au> 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 

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