Developing Talented ChildrenÕs Mathematical Ability through Visual and Spatial Learning Tasks Thomas Lowrie Introduction This paper looks at ways of extending mathematically talented children within the regular classroom environment. Many activities designed to stimulate children who display potential in mathematics are often repetitive, with analytical content that only challenges children for a limited time. Such students require more long term goals, with open-ended problems that encourage work on tasks that are separate from the main stream curricula they have already successfully interpreted and understood. A childÕs spatial and visual strategies are often overlooked when educators try to extend talented beginning mathematicians. This paper argues that a more balanced curriculum, that places increasing importance on spatial and visual development, will allow talented children to remain motivated and stimulated within the usual classroom setting. Extending Talented Mathematicians Recently there has been renewed interest in the provision of educational programs for gifted and talented children, particularly through ÒeliteÓ or ÒspecialÓ programs. Davis and Rimm (1989) when defining gifted and talented state that Òsome educators use the word gifted to describe the highly intelligent, Ôintellectually giftedÕ person and the word talented to refer to a person with specific skills and abilities - ÔtalentsÕ in just one of a few areasÓ (p.10). Such an open-ended definition is characteristic of many labelling techniques used in current programs. While it is not educationally sound to categorise children with only quantitative scores or measures , teachers are left wondering if particular children are in fact gifted, talented, or alternatively academically Òabove average.Ó Reinforced throughout this paper is the belief that it is advantageous to extend the gifted and talented within the usual classroom setting with open ended problems that stimulate learning. Particular attention is given to the development of a childÕs spatial ability, an area often overlooked when extending the gifted and talented. Extending competent children has always been a problem in the mathematics classroom. Not only should the work be productive for the children involved, but it must be educationally lasting. The children need to be motivated by the task, and encouraged to go beyond the confines of the classroom setting in pursuit of challenging solutions. They need to feel that they are learning, and that the extra work is both worthwhile and advantageous to their particular needs and interests. This, however, doesnÕt always eventuate. Children who finish their work quickly are often set brainteasers or extra numeration tasks to keep them occupied while the remainder of the class complete their work. Talented mathematicians often find tricks or relationship clues very quickly with these type of problems. Unfortunately this extra work becomes repetitious and time consuming. As Friedman (1984, p.12) explains, Òenrichment then becomes synonymous with ÔmoreÕ- more homework, more projects, more book reports, 30 practice examples instead of ten, and so forth.Ó Many talented mathematicians learn at an early age to take their time and so complete work assignments with the majority of the class. The workload and expectations already placed on teachers make it difficult for even the most willing professional to develop separate work programs for the gifted and talented. The classroom teacher, however, is ultimately responsible for the extension of the gifted and talented. While documents and policies are being developed to cater for individual needs (see the N.S.W Government, 1991 Government Strategy for the Education of Gifted and Talented Students ), aims and objectives of such programs seem ambitious in regard to time constraints, available resources and financial commitments. Extension tasks need to become open-ended, with problem solving activities that have no defined parameters, allowing for creativity and content choice. Work becomes more individualised. This type of open ended environment is not a new idea in education, for in fact competent teachers create such environments every day, across all other curriculum areas. Why not in mathematics? For too long mathematics has been a black and white subject, where being right or wrong has outweighed the problem-solving process. There is a need to create a learning environment that is more flexible, that allows teachers to be more creative in extending talented mathematicians, in more diverse and open-ended ways. A better balanced curriculum would enable teachers to create learning environments where process was significant, and right and wrong not always essential. Children with specialised needs, who require extension, can be easily overlooked if too much emphasis is placed on analysing and not enough on building a synthesis of ideas. As Dixon (1983) states, Òthis is perhaps more the case in spatial ability than any other areaÓ (p.x). What then is spatial ability, and how can spatial activities extend mathematically talented children? Spatial Ability and Mathematics Educators and psychologists have been trying to define spatial ability for the past three decades. Piaget and Inhelder (1967) proposed that a full spatial understanding of the child evolves around three different groups of interrelated skills- associated with the understanding of topological relationships, projective relationships, and Euclidean relationships. Piagetian research contributed to our understanding of spatial ability by focusing on the basic skills involved, and describing how children develop these skills. Factor analysts including McGee (1979), used the terminology pattern arrangement and transformation of orientation. Linn & Peterson (1985) maintained that three categories should be used when analysing spatially related studies including spatial perception, mental rotation, and spatial understanding. Visualisation and orientation are important to spatial thinking, because they involve the recognition of detailed information. Spatial ability, however, goes beyond this because it involves the analysis of structural relationships, so that operational thought can take place. Spatial ability is more than pictorial representations, it becomes abstracted, manipulative imagery. Dixon (1983) qualifies this by stating that Òspatial thinking occurs when visualisation and rational thought are applied togetherÓ (p.56). The difficulty in defining spatial abilities and visual imagery, particularly when relating findings to intelligence and education, are documented by Gardner (1983) who says: Children may know there way around many areas of the neighbourhood or town and, in fact, never fail to find what they are looking for. Yet they often will lack the capacity to provide a map, a sketch, or an overall verbal account of the relationship among several spots. Representing their piecemeal knowledge in another format or symbol system proves an elusive part of the spatial intelligence. Or perhaps one could say: while childrenÕs spatial understanding develops apace, the expression of this understanding via another intelligence or symbolic code remains difficult. (p.180) Clements (1983) claims that educators need to clarify their meaning of spatial and visual problems, and develop strategies that clearly promote spatial tasks. He recommends that Òmathematics education researchers should be more flexible in their use of the Òspatial abilityÓ variable; that, in fact, no advantage is to be gained from using so-called well- established tests of a variableÓ (Clements, 1983, p.8). While educators have not been able to agree on a common definition of spatial ability, most agree that such skills or abilities are important in the total education of the child. The authors of the NCTMÕs Curriculum and Evaluation Standards for School Mathematics (1989, p.4) suggest that Òspatial understandings are necessary for interpreting, understanding, and appreciating our inherently geometric world.Ó Spatial abilities are essential to many of the tasks a person performs from navigating an alternate route from work to home, to receiving a tennis serve, to adjusting furniture in your lounge room. Spatial abilities and perceptions are also important because of their relationship to most technical-scientific occupations and especially to the study of mathematics, science, art, and engineering. Wheatley (1977) believes that spatial abilities should be integrated into all curriculum areas. Del Grande (1990) states that spatial ability and geometry are dependent, and therefore improvement in one leads to improvement in the other. If educational curricula can promote and develop spatial abilities in not just geometry, but in all areas of the curriculum, not only will childrenÕs mathematical abilities improve but an ability to solve everyday problems using visual methods will develop. An increase in spatial awareness will promote greater learning flexibility. In an infant classroom children are usually provided with a range of verbal and visual stimuli. The older children get, the more the curriculum is weighted toward verbal, analytical activities, with visual perception and imagery decreasing, particularly mathematics. Dreyfus (1991) states that Òwe should aim for more balance: we should aim for integration of visual, verbal and algebraic thinking. Before one can aim for integration, however, one needs balance; and in order to achieve balance, visual reasoning needs to be given equal status and attention as algebraic reasonÓ (p.46). Tracey (1990) maintains however, that spatial skills may be developed through extra curricular activities involving sport, model construction, art and manipulation. Mitchell & Burton (1984) found that active play can also help spatial ability in a useful manner. Activities including music, physical development, art projects, and constructing mobiles can positively develop childrenÕs perception of space. Teachers then, should provide talented mathematicians with activities that can develop their mathematical abilities. Children can be motivated by open-ended activities, especially if they are specifically geared toward individual content preferences. Works would not be overly repetitious, and children should be encouraged to pursue interests after school. Extending Talented Children Spatially If they are to extend talented beginning mathematicians, teachers and educators should not be restricted to implementing programs based around standard mathematical concepts and content. Areas of geometry, space and problem solving should be developed from a variety of sources, using a more holistic approach to mathematics education. What new approaches should be used to encourage and extend talented mathematicians, without burdening teachers with new programs that require additional work for a minority of the class? The author feels that the children in both primary and secondary grades should be encourage to develop interests in art, photography, architecture, and music when they are obviously advanced in overall mathematical ability. Such work would not only help mathematical understanding in an area that is poorly treated in most mathematical curriculums (including visual and spatial sense), but motivate children to pursue interests in extracurricular activities. The line between Òschool workÓ and ÒlearningÓ would hopefully be less apparent. Examples of activities that would be used to develop spatial ability in the areas of art, photography, architecture and music are listed in Table 1. Table 1 Activities that promote and extend Visual-Spatial Abilities for Talented Mathematicians ArtArchitecture* study of cultural art eg. Aboriginal art * perspective in art * use of colour in various art periods * painting and drawing computer tools * tessellations in 2D and 3D space* scale drawing * designing 2D house plans * constructing and drawing 3D house plans * creating models of houses, parks * drawing 2D plans of fun parks, schools * CAD and other computer softwarePhotographyMusic* thematic representations eg. life * perspective work with B/W photos * studying techniques * close-up, background, wide angle images.* mathematical relationships in music * dynamics with art analogies * linear musical presentations with computer graphics Using Architectural Techniques to Promote Visual-Spatial Abilities in a Year 6 Classroom From the proposed examples in Table 1, an open-ended project was designed to extend eight talented year 6 mathematicians. These children had displayed achievement levels Òwell aboveÓ average in most mathematical areas throughout the year, and had performed admirably in an earlier mathematics competition. The children were required to: 1. design a two-dimensional (2-D) house plan (to scale) displaying widows, doors, covered areas, roof line, and position on land. 2. draw three-dimensional (3-D) elevation views (eg front and side) of the house based on the two-dimensional (2-d) plan. This task had to correspond with the previous task. 3. construct a model of the house using appropriate materials. The children were asked to complete the project in five weeks. Work was productive, with motivation remaining high throughout the five week period. Children even went to the trouble of visiting display centres on weekends in the hope of gaining additional inspiration. Class time was allocated to the project, although work at home was also encouraged. While the children were obviously extended by the activity, additional evaluation of their work was needed to delineate the type of spatial and visual processes were required to complete such a project. At the completion of the task the children were asked about strategies, approaches and techniques displayed in the development of the three activities. The thought and detail that went into each task could be best illustrated in a response from Chad. I was doing rough copies of the house until they started getting better and when I got the right one I started drawing it on the drawing board. When I did all the angles dad took it to work and blew it up, I cut a piece of balsa wood the size of the paper, and I began to build the two the 2D representation. I then put support frames on my the roof like the ones getting built near my house. I then imagined the lines of the house became real walls, then built the house by estimating where the windows would be. Comments from both Janelle and Justin indicated that the children used visual and spatial skills to solve problems throughout the project. Janelle tried to create a mental image of a house she had only once visited to produce 2-D and 3-D drafts of here plan. While Justin was somewhat more analytical, he made the walls Òstand up in (his) mindÓ to create a visual account of what the 3-D design would look like. My aunties got a house in Perth. I drew one side of the house and I visualised that I was there. I then imagined myself in the backyard looking at the back of the house, and drew it from there. (Janelle, Year 6) I drew little bits of houses I liked, then stuck them together with sticky tape. I knew where the windows were so I just made them stand up in my mind. Instead of the walls being flat you could see them standing up. I had to draw the house in perspective, like an arrow shape, making the font bigger and when it got further back I did half the scale which made it smaller. (Justin, Year 6) Claire created spatial images that were both large and small scale, to transform the 2-D image into a 3-D model. The visual imagery she uses to solve problems with space indicate that such an activity has required Claire to use visual and spatial processes. I agree with Bishop (1989) in emphasising the fact that visual representation should become essential in all aspects of the mathematics classroom. Claire is a competent designer who obviously tries to call on such visual skills when performing these tasks. Such skills need to be encouraged more fully in all mathematic classrooms. I thought of a plan and imagined myself walking through the house. I imagined standing in the middle, turning around viewing each room looking at it. I then pretended to walk around the outside of the house, putting in widows and doors. I thought about how builders build a house and start from the ground up. (Claire, Year 6) Balance Within the Curriculum Educators appreciate that there is more to mathematics than its scientific, pure form. Only a very small percentage of school leavers choose a career in pure mathematics, but extension work given to even small children is usually analytically based. Teachers know that work with number, operations, equations and formulas are only part of the education process, but feel that enrichment must extend children in these areas. Recently a grade 3 teacher proudly informed me that a child in his class was actually solving year 6 operations, and wanted access to high school material. Before long the child would also be able to master analytical problems from the grade 7 curriculum. Could the child, however, accomplish geometric or spatial tasks from a much lower level? This is where problems arise. Because most analytical, verbal exercises are easy to produce, and even easier to evaluate and correct, the burden on teachers to extend children is not great. The black and white syndrome raises its ugly head. A more balanced, holistic view of mathematics education, where talented pupils are encouraged to pursue interests, both within and outside school time, should be a prime educational objective. Mathematical concepts and understandings can be improved and developed without teaching ÔmathematicsÕ. Visual and Spatial reasoning must be given more ÒvalueÓ in mathematics education. References Bishop, A. J. (1989). Review of research on visualization in mathematics education. Focus on Learning Problems in Mathematics, 11(1), 7-16. Caplan,P. J., MacPherson, G.M., & Tobin,P. (1985). Do sex-related differences in spatial abilities exist? A multilevel critique of new data. American Psychologist, 40, 786-799. Clements, M. A. (1983). 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