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Teaching Mathematics to Gifted Students in a Mixed-Ability Classroom
Author: Dana T. Johnson
Mathematically gifted students have needs that differ in nature from those of other students. They require some differentiated instruction, defined by Tomlinson (1995) as "consistently using a variety of instructional approaches to modify content, process, and/or products in response to learning readiness and interest of academically diverse students." Yet recent studies have found few instructional or curricular modifications in regular elementary classrooms (Archambault et al., 1993; Westberg, Archambault, Dobyns & Salvin, 1993). In grades 9-12, students may be able to select honors, advanced, and AP courses; however, even in these more homogeneously grouped classes there is a range of differences that need to be acknowledged.
Why should we do anything different for mathematically gifted students?
Gifted students differ from their classmates in three key areas that are especially important in mathematics. These are summarized in the table below.
Mathematically gifted students differ from the general group of students studying math in the following abilities: spontaneous formation of problems, flexibility in handling data, mental agility of fluency of ideas, data organization ability, originality of interpretation, ability to transfer ideas, and ability to generalize (Greenes, 1981). No list of characteristics of the mathematically gifted includes "computational proficiency," and yet at levels prior to Algebra I, this is commonly used as the criterion that determines who gets to move on to more interesting material. Furthermore, there is a myth that gifted students don't need special attention since it is easy for them to learn what they need to know. On the contrary, their needs dictate curriculum that is deeper, broader, and faster than what is delivered to other students.
Mathematics can be the gatekeeper for many areas of advanced study. In particular, few gifted girls recognize that most college majors leading to high level careers and professions require four years of high school math and science (Kerr, 1997). Students may drop out of math courses or turn toward other fields of interest if they experience too much repetition, not enough depth, or boredom due to slow pacing.
An Agenda for Action: Recommendations for School Mathematics of the 1980s (NCTM, 1989, p. 18) says, "the student most neglected, in terms of realizing full potential, is the gifted student of mathematics. Outstanding mathematical ability is a precious societal resource, sorely needed to maintain leadership in a technological world." By 1995, when the NCTM created a Task Force on the Mathematically Promising, not much had changed (Sheffield et al., 1995).
What do the Curriculum Standards of the National Council of Teachers of Mathematics (NCTM) say we should do about mathematically gifted students?
The NCTM Standards do not mention gifted students explicitly but recognize that students are not all the same. For all students, the Standards place a greater emphasis on areas that traditionally have been emphasized for the gifted. All students are now expected to complete a core curriculum that has shifted its emphasis away from computation and routine problem practice toward reasoning, real-world problem solving, communication, and connections. "The Standards propose that all students be guaranteed equal access to the same curricular topics; it does not suggest that all students should explore the content to the same depth or at the same level of formalism" (NCTM, 1989, p. 131). At the high school level, additional topics are suggested for "college-intending" students. The Report of the Task Force on the Mathematically Promising recognizes that there are special issues relating to the education of the mathematically promising student (Sheffield et al., 1995) and has made recommendations that include the development of new curricular standards, programs, and materials that encourage and challenge the mathematically promising.
What should be done to differentiate curriculum, instruction and assessment for the mathematically gifted in the regular classroom?
Historically there has been debate about the role of acceleration versus enrichment as the differentiation mode for mathematics. Most experts recommend a combination. The following are suggestions for differentiating for the mathematically gifted by using (1) assessment, (2) curriculum materials, (2) instructional techniques, and (4) grouping models. These opportunities should be made broadly available to any student with interest in taking advantage of them.
How can technology support the needs of the gifted?
Technology can provide a tool, an inspiration, or an independent learning environment for any student, but for the gifted it is often a means to reach the appropriate depth and breadth of curriculum and advanced product opportunities. Calculators can be used as an exploration tool to solve complex and interesting problems.
Computer programming is a higher level skill that enhances problem solving abilities and promotes careful reasoning and creativity. The use of a database, spreadsheet, graphic calculator, or scientific calculator can facilitate powerful data analysis. The World Wide Web is a vast and exciting source of problems, contests, enrichment, teacher resources, and information about mathematical ideas that are not addressed in textbooks. Technology is an area in which disadvantaged gifted students may be left out because of lack of access or confidence. It is essential that students who do not have access at home get the exposure at school so that they will not fall behind the experiences of other students.
What is the responsibility of schools and teachers in developing giftedness in mathematics?
Classroom teachers and school districts share the responsibility of addressing the needs of gifted students.
Regular mathematics classrooms that offer sufficiently challenging and broad experiences for gifted students have the potential to enrich the learning community as a whole since other students will be interested in attempting, perhaps with help, some of the more challenging tasks. If math classes offer diversity in assignments, products, and pacing and monitor student needs, all students will be able to work at their own challenge level.
Archambault, F. X., Westberg, K. L., Brown, S. W., Hallmark, B. W., Zhang, W., & Emmons, C. L. (1993). Classroom practices used with gifted third and fourth grade students. Journal for the Education of the Gifted, 16, 103-119.
Greenes, C. (1981). Identifying the gifted student in mathematics. Arithmetic Teacher, 28, 14-18.
Lockwood, A. T. (1992). The de facto curriculum. Focus in Change, 6. Maker, J. (1982). Curriculum development for the gifted. Rockville, MD: Aspen Systems Corporation.
Kerr, B. A. (1997). Developing talents in girls and young women. In N. Colangelo & G. A. Davis (Eds.), Handbook of gifted education (2nd ed., pp. 483-497). Boston: Allyn & Bacon.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Sheffield, L. J., Bennett, J., Berriozábal, M, DeArmond, M., Wertheimer, R. (1995) Report of the task force on the mathematically promising. Reston, VA: National Council of Teachers of Mathematics.
Tomlinson, C. A. (1995). Deciding to Differentiate Instruction in Middle School: One school's journey. Gifted Child Quarterly, 39, 77-87.
Westberg, K. L., Archambault, F. X., Dobyns, S. M. & Salvin, T. J. (1993) The classroom practices observation study. Journal for the Education of the Gifted, 16, 120-146.
Dana Johnson is a mathematics instructor at the College of William and Mary and also teaches enrichment classes through the Center for Gifted Education at the College.
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