M. Katherine Gavin
The University of Connecticut
In 1989 the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics with the hope of revolutionizing mathematics curriculum for K-12 students. Inherent in the Standards is the consensus that students need to learn more mathematics, learn new kinds of mathematics, and learn mathematics in a different way. The major thrust of the Standards is problem solving. “Problem solving (which includes the ways in which problems are represented, the meanings of the language of mathematics, and the ways in which one conjectures and reasons) must be central to schooling so that students can explore, create, accommodate to changed conditions, and actively create new knowledge over the course of their lives” (NCTM, 1989, p. 4). If one picks up a curriculum guide for a gifted/talented program in mathematics, one is apt to find a similar goal. In fact, it is true that the call for revision in the Standards is a call for the use of many skills we formerly considered the domain of gifted programs: problem solving, reasoning, communicating mathematically, creative thinking, and making connections between math and the real world. Topics formerly seen as enrichment for the gifted, such as probability and statistics, are now included as essential to the basic curriculum.
The Standards emphasize that the content outlined is for all students. “Our expectation is that all students must have an opportunity to encounter typical problem situations related to important mathematical topics” (NCTM, 1989, p. 9). Why then do we need to provide programs for high ability students? In defense of the Standards, it is important to note that they clearly state that all students are not alike. “We recognize that students exhibit different talents, abilities, achievements, needs, and interests in relationship to mathematics” (NCTM, 1989, p. 9). In the updated draft of their position paper on the Provisions for Mathematically Talented and Gifted Students, NCTM recommends that all mathematically talented students “have access to appropriate curricula and instruction that contributes to developing positive attitudes, furthering their mathematical interests, and encouraging their continuing participation in the study of mathematics” (NCTM, in preparation). In light of these recommendations, let us examine why and how a program for mathematically talented students should be developed.
First, it is important to look at the characteristics of highly able math students to recognize the types of mathematical experiences they will need. This is not a cut-and-dried procedure, because different students make use of different talents at different times. Keeping this in mind, characteristics these students might exhibit would include fast-learning pace, keen observation skills, curiosity and understanding about quantitative information, analytical reasoning skills, flexibility and reversibility of mental processes, energy and persistence in solving problems, ability to transfer learning to novel situations, ability to visualize patterns and spatial relationships, and a mathematical perception of the world (House, 1987). These students certainly need to explore math as problem solving, reasoning, communication, and connections (the hallmarks of the Standards), but they need much more. They need to be on the cutting edge of mathematical and technological discoveries. We have a responsibility to prepare them to become our future mathematicians and leaders in business and science.
Now, the question is how do we address the needs of these students. There are many ways to escalate the level of advancement in each standard. Depending on the talents of the students, the curriculum can be upgraded in terms of pace, depth, breadth, areas of interest, or level of intellectual dialogue. For the precocious student, acceleration through summer programs, course skipping, early college entrance, and curriculum compacting is appropriate. Julian Stanley has been instrumental in developing the Talent Search as a means of identifying students, ages twelve or older, to participate in such programs and has conducted extensive research which verifies the success of these programs (Stanley, 1991). Recently this identification procedure has been extended to students in grades three to five (Lupkowski-Shoplik & Assouline, 1993).
It is important to realize that acceleration of gifted students into a program that does not provide the challenges they need is not the answer! Enrichment is a necessary partner to ensure a stimulating math program. In fact, sometimes enrichment alone may be appropriate to develop the particular talents of the student. This does not mean giving students mind benders or logic puzzles after they have finished their math work. It must be much more-focusing on a well-planned, yet flexible and integrated program of instruction. The depth of the subject matter must be extended with interesting research questions, independent study projects, and simulation activities which include the use of technology. This will encourage students to apply their knowledge to other subject areas and life situations. Mentors and internship programs can further extend this application. The breadth of the curriculum needs to be expanded with introduction to exciting fields such as chaos theory and fractals. Students’ interest levels need to be tapped as they become mathematicians, discovering theorems and creating theory. We must dispel the notion that mathematics begins and ends with the Greeks, when, in fact, most of the mathematics known in the world today has been discovered in the last 50 years! (Sheffield, in preparation). Students need to go beyond problem solving to problem posing and finally to creating problems. It is only at this highest level of creation that students will begin to realize their true potential and experience the excitement of mathematical discovery and research. Throughout this process, we need to encourage intellectual dialogue among students of high ability, and be willing, as teachers, to become co-investigators in explorations stimulated by these discussions. Research has shown that this type of interaction invigorates these students and provides the necessary groundwork for mathematical inspiration (Sowell, 1993).
During the middle of a semester, I was asked to team teach a precalculus course already in progress. The teacher’s comments to me, when discussing the class, included “using” the boy in the back row as a resource for difficult questions or problems. What an injustice to this young man! In America 2000: An Education Strategy (1991), we are given the following imperative: “By the year 2000, U.S. students will be first in the world in science and mathematics achievement.” If we are to live up to this commitment, we must continually challenge our mathematically talented students, for it is these students who have the awesome responsibility of shaping the mathematics and science of the future.