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##  Teaching Mathematics to Gifted Students in a Mixed-Ability                      Classroom. ERIC Digest E594.                                                 

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 below. 
How Gifted Learners                  Differ from Classmates: 
1. Pace at which they learn 
2. Depth of             their understanding 
3. Interests that they hold (Maker,                       1982) 
Relationship to Mathematics Learning 
1. The sequential                nature of math content makes pacing an issue. 
2. Deeper levels of             understanding and abstraction are possible for most mathematical                topics, so differentiation becomes important. 
3. If the interest              is snuffed out early, the talent may not be                                     developed. 
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. 
* Give pre-assessments so that students who                already know the material do not have to repeat it but may be                   provided with instruction and activities that are meaningful. In the            elementary grades, gifted learners still need to know their basic               facts. If they do not, don't hold them back from other more complex             tasks, but continue to work concurrently on the basics. 
* Create              assessments that allow for differences in understanding, creativity,            and accomplishment; give students a chance to show what they have               learned. Ask students to explain their reasoning both orally and in             writing. 
* Choose textbooks that provide more enriched                        opportunities. Unfortunately, curriculum in this country is mainly              driven by textbooks, which are used about 80% of the time (Lockwood,            1992). Math textbooks often repeat topics from year to year in the              grades prior to algebra. Since most textbooks are written for the               general population, they are not always appropriate for the gifted.             Several series that hold promise for gifted learners have been                  developed recently under grants from the National Science                       Foundation; they emphasize constructivist learning and include                  concepts beyond the basics. 
* Use multiple resources. No single               text will adequately meet the needs of these learners. 
* Be                   flexible in your expectations about pacing for different students.              While some may be mastering basic skills, others may work on more               advanced problems. 
* Use inquiry-based, discovery learning                    approaches that emphasize open-ended problems with multiple                     solutions or multiple paths to solutions. Allow students to design              their own ways to find the answers to complex questions. Gifted                 students may discover more than you thought was possible. 
* Use               lots of higher-level questions in justification and discussion of               problems. Ask "why" and "what if" questions. 
* Provide units,                 activities, or problems that extend beyond the normal curriculum.               Offer challenging mathematical recreations such as puzzles and                  games. 
* Provide AP level courses in calculus, statistics, and                computer science or encourage prepared students to take classes at              local colleges if the supply of courses at the high school has been             exhausted. 
* Differentiate assignments. It is not appropriate to              give more problems of the same type to gifted students. You might               give students a choice of a regular assignment; a different, more               challenging one; or a task that is tailored to interests. 
* Expect            high level products (e.g., writing, proofs, projects, solutions to              challenging problems). 
* Provide opportunities to participate in              contests such as Mathematical Olympiads for the Elementary                      School (grades 4-6), Math Counts (grades 7-8), and the American                 Junior High School Mathematics Exam (grades 7-8) or the American                High School Mathematics Exam (grades 9-12). Give feedback to                    students on their solutions. After the contests, use some of the                problems as the basis for classroom discussions. 
* Provide access             to male and female mentors who represent diverse linguistic and                 cultural groups. They may be within the school system, volunteers               from the community, or experts who agree to respond to questions by             e-mail. Bring speakers into the classroom to explain how math has               opened doors in their professions and careers. 
* Provide some                 activities that can be done independently or in groups based on                 student choice. Be aware that if gifted students always work                    independently, they are gaining no more than they could do at home.             They also need appropriate instruction, interaction with other                  gifted students, and regular feedback from the teacher. 
* Provide             useful concrete experiences. Even though gifted learners may be                 capable of abstraction and may move from concrete to abstract more              rapidly, they still benefit from the use of manipulatives                       and "hands-on" activities. 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. 
*                  Teachers need training and support in recognizing and addressing the            needs of the mathematically gifted. 
* Teachers who teach                      mathematics to gifted learners need a strong background in                      mathematics content. If the school has only a few students with                 special needs and does not have such a teacher, a mentor from                   outside the school should be located to work with individuals. 
* A            coordinated curriculum plan needs to be in place so that the                    mathematical experiences for students are not duplicated or                     interrupted from one year to the next. 
* The school should have an            organized support system that includes resource books, technology,              and human resources. 
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. REFERENCESArchambault, 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., Berriozabal, 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. 
ERIC Digests are in the public domain and may be freely              reproduced and disseminated, but please acknowledge your source.                This digest was prepared with funding from the Office of Educational            Research and Improvement (OERI), U.S. Department of Education (ED)              under Contract No. ED-99-CO- 0026. The opinions expressed in this               publication do not necessarily reflect the positions or policies of             OSEP or the Department of Education.                                         **Title:** Teaching Mathematics to Gifted Students in a Mixed-Ability                      Classroom. ERIC Digest E594.                                                 **Document Type:** Information Analyses---ERIC Information Analysis Products (IAPs) (071); Information Analyses---ERIC Digests (Selected) in Full Text (073); **Available From:** ERIC Clearinghouse on Disabilities and Gifted Education, Council for            Exceptional Children, 1920 Association Dr., Reston, VA 20191-1589.              Tel: 800-328-0272 (Toll Free); e-mail: [email protected]; Web                 site: http://ericec.org.                                                     **Descriptors:**  Academically Gifted,  Discovery Learning, Elementary Secondary                 Education,  Gifted, Heterogeneous Grouping, Inclusive Schools,                   Individualized Instruction,  Mathematics Instruction, Problem                  Solving,  Teaching Methods                                                   **Identifiers:**  Differentiated Curriculum (Gifted), ERIC Digests                            
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