NEEDED RESEARCH: Mathematics for the TalentedAuthor(s): Alan OsborneSource: The Arithmetic Teacher, Vol. 28, No. 6 (February 1981), pp. 24-25Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191801 .

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NEEDED RESEARCH Mathematics for 1he Talented

by Alan Osborne

It is appropriate that interest in the development and nurture of creative mathematical talents be renewed after several years of limited attention to and concern for the special needs of the gifted. Do we have sufficient re- search to guide decisions about what should be taught to talented children, when it should be taught, or in what setting? Do we know enough to make prudent commitments of resources? If not, what type of research is needed to improve the state of our decision mak- ing?

Several interesting and useful com- pilations of research about the talented (Passow 1979; Keating 1976; Getzels and Dillon 1973) describe alternatives in designing programs and specify some characteristics of talented young- sters. However, most fail to account for the unique characteristics of mathe- matics and mathematical thinking. Mathematical thinking is sufficiently different from other types of in- tellectual activities to require consid- eration of special types of research questions. The distinctive character- istics of doing mathematics might ap- propriately be taken into account when designing programs and activities for talented learners if better information were available. A sample of research problems that relate specifically to the characteristics of mathematical think- ing and the talented is discussed in the paragraphs that follow.

Selection Identifying children who should partic- ipate in programs and activities for the talented is a difficult responsibility. Currently efforts at identification are based primarily on measures of achievement, general mental ability, and, to some extent, personality char-

ol lan Osborne, Ohio State University in Colum- bus.

acteristics. Achievement and general mental ability are good evidence to consider since bright children learn more faster and children who exhibit ability in one domain usually have comparable abilities in other domains. Most achievement and general ability tests, however, do not measure the fol- lowing categories of behavior, which are indicative of mathematical and problem-solving power.

(1) An appreciation of uniqueness, ex- istence, and universality. Teaching at the junior and senior high school levels convinced me that talented students have an almost innate sense of the power that comes from being able to base problem-solving strategies and ar- guments on the recognition that a set of objects all share a characteristic or that none of a collection have a charac- teristic. Recognition of existence and uniqueness evolves into the selection of patterns of reasoning that use indirect arguments and give particular signifi- cance to the roles of examples and counter examples in thinking.

(2) An inclination to see mathematics in the ordinary and commonplace. Kru- tetskii (1976) identifies, as a character- istic of mathematically talented chil- dren in Russia, the inclination and desire to "see" mathematics in every- day life situations and settings. For the young child, this is often exhibited in the propensity to ask questions about the numbers that can be associated with a setting; for example, wondering whether there are more nails than screws in a house. When the child ma- tures, the behaviors encompass trying to impose a pattern or a structure on events, facts, or situations.

(3) Flexibility in considering alterna- tives in problem solving and a willing- ness to curtail behaviors and processes (alternatives) that do not appear produc-

tive. For many children, all mathemati- cal ideas and processes appear equally important. The talented child not only has many ideas, but also exhibits the ability to sort out the more powerful and productive ideas and processes.

None of these three types of behav- iors is readily determined by an exami- nation of the achievement of a child. They are indicative of growing mathe- matical power and, indeed, are evi- dence of mathematical sophistication and maturity after they are refined. They can best be discovered in a set- ting that allows direct observation of the child doing mathematics. We have little evidence concerning how these behaviors are first exhibited by the child or how they develop. Research that would expand the understanding of these uniquely mathematical behav- iors would enhance the capability for accurately identifying and selecting talented children in mathematics.

A final note on selection: To what ex- tent are these mathematical behaviors learned? If they are learned, then the child who has not encountered the be- haviors in a mathematical setting where they can be used to good effect is likely to be overlooked in the selec- tion process. Arnold Ross, who has conducted programs for the mathemat- ically talented for more than two dec- ades, first at Notre Dame and more re- cently at Ohio State University, believes that these types of mathemati- cal behaviors are learned. If Ross is right, using the behaviors for selection before the child has the opportunity to grow mathematically may ignore a considerable pool of talented young- sters. The behaviors may be suffi- ciently subtle in the early stages of de- velopment that it would take a specialist to identify them. Do achieve- ment tests, general ability tests, and teacher reports have sufficient power to

24 Arithmetic Teacher

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allow the use of such mathematical be- haviors as a selection criterion?

Program Design During the past we may well have been asking the wrong questions about the design of programs for gifted children. The traditionally important questions are more concerned with the type and extent of resources, staff, and children than with what happens to the child's thinking and performance. Alterna- tively, we should be looking at more fundamental questions concerning pro- gram designs.

Many programs have used number theory as a vehicle for simulating mathematical behavior because signifi- cant mathematics is immediately below the surface of children's prior arith- metic experience and uses already de- veloped intuitions about number. Many areas of mathematics require considerable experience before signifi- cant problems can be meaningful or before supportive intuitions can be brought into play by the child. Beneath the surface is a research question of greater import: Have we assessed accu- rately the role of memory, intuition, and experience in the design and selec- tion of activities to nurture mathemati- cal talent?

Memory, a basic factor in learning, has generally been ignored by re- searchers in mathematics education. Recent studies in how experts and nov- ices differ in problem solving (Larkin, et al. 1980) suggest that memory fac- tors affect the quality and quantity of thinking. Thus, selecting content for which intuitions are lacking may be equivalent to imposing boundary con- ditions on what is appropriate content and processes for programmatic and instructional designs for the talented. That is, if the goals encompass proc- esses of thinking and problem solving, then children need the "stuff' to ma- nipulate in cognitive processing. Mem- ory and natural access to that memory are important if children are to use their cognitive structures efficiently in developing mathematical power.

Bruno Bettelheim, in an editorial in the old Saturday Evening Post during the immediate, post-Sputnik period of

curriculum reform and concern with the talented, noted that the child who is in a class with less able peers can do considerable daydreaming. He asked whether we know the effects on crea- tivity, of being able to let the imagina- tion roam in the unfettered manner as- sociated with daydreaming. We assume, with little evidence, that it is best to keep bright children busy with constant activities or acceleration. Do we know the effects of constant, struc- tured activities on the development of mathematical talent? What is the role of daydreaming in the development of mathematical creativity?

Piagetian psychology suggests that children in the early elementary school typically do not exhibit the character- istics associated with mature thought processes in mathematics or science. Yet, working with talented children of this age reveals some startling capabili- ties for doing mathematics. If talented children do not have certain character- istic mathematical behaviors consoli- dated, to what extent can we be justi- fied in providing the environment of a special program in mathematics? Is it a more effective use of limited resources to delay special attention to mathematics until children exhibit some of the think- ing and learning processes that would al- low their taking better advantage of the richer environment?

The NCTM's Agenda for Action rec- ommends that problem solving be the primary thrust of curricular experi- ences in mathematics throughout the school years. This is an appropriate orientation for mathematically talented students also. However, the role of problem solving in programs for the talented is not completely clear. We do not know the relative contribution of process learning, such as problem solv- ing, and the acquisition of knowledge, such as facts and structure, to the de- velopment of mathematical talent. My personal view is that problem-solving experiences are very important in the development of talent in children. However, I am struck by the number of times I observe that the knowledge of a stray bit of information stimulates the selection of the successful problem- solving strategy.

Many programs for the talented bring children together for group at-

tention to mathematics. The inter- action of the group provides an impor- tant factor of stimulation for many children. Anecdotal evidence of special programs for the talented, such as the Governor's Honors Program in Geor- gia, indicates that many talented chil- dren, particularly those from rural and impoverished urban areas, are appar- ently lonely and may have a poor self- concept because they are different. Do we have adequate information con- cerning the effects of talented chil- dren's interaction with each other? Many adult mathematicians, who are powerful problem solvers, in their childhood or youth had extended peri- ods of intensely personal, private en- counters with mathematics. What is an appropriate balance between the per- sonal encounter and the group attack on mathematics in programs for the tal- ented?

Research concerning the talented provides a special challenge for mathe- matics educators. I have tried to in- dicate some categories of problems that are both interesting and potentially useful. Most require expensive re- search techniques - case studies, exten- sive observation, and longitudinal analyses. Part of the problem in doing research in the area has been the fact that the funding of research and pro- grams for the talented has been so lim- ited that little research has been at- tempted. Consequently, practitioners have had to rely primarily on personal, professional judgment in designing programs and activities for the mathe- matically talented.

References

Getzels, J. W., and J. T. Dillon. "The Nature of Giftedness and the Education of the Gifted.'* R. M. W. Travers ed. Second Handbook of Re- search in Teaching. Chicago: Rand McNally and Company, 1973.

Keating, D. P. Intellectual Talent: Research and Development. Baltimore: Johns Hopkins Uni- versity. Press, 1976.

Krutetskn, V. A. The Psychology oj Mathemati- cal Abilities in Schoolchildren. Chicago: The University of Chicago Press, 1976.

Larkin, J., J. McDermott, D. P. Simon, and H. A. Simon. "Expert and Novice Perform- ance in Solving Physics Problems." Science 208 (June 1980): 1335-42.

Passow, A. H. (ed.). The Gifted and Talented: Their Education and Development. The 78th Yearbook of the NSSE. Chicago: National So- ciety for the Study of Education, 1979. •

February 1981 25

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