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Characteristics of Highly Able Math Students
The MCPS Policy on Gifted and Talented Education states that, "In grades prekindergarten–8, accelerated and enriched curricula will be provided to all students who have the capability or motivation to accept the challenge of such a program. This curriculum will be rigorous and challenging and matched to the abilities, achievement levels, and interests of high ability students. There will be opportunities and expectations for students to learn at an accelerated pace, to learn in depth, and to learn integrated themes and connections between disciplines…"
Defining "Mathematical Promise"
In order to discuss ways to identify and program for gifted and promising mathematics students, mathematical promise must be defined. The National Council of Teachers of Mathematics (NCTM) Task Force on the Mathematically Promising defined mathematical promise as a function of--
- belief, and
- experience or opportunity.
These variables are not fixed and need to be developed so that success for these promising students can be maximized. This definition includes the students who have been traditionally identified as gifted, talented, precocious, and so on, and it adds students who have been traditionally excluded from previous definitions of gifted and talented and therefore excluded from rich mathematical opportunities. This definition acknowledges that students who are mathematically promising, have a large range of abilities and a continuum of needs that should be met." (Richard Wertheimer)
Characteristics to Look for When Identifying Mathematically Gifted Students
There are many characteristics to consider when identifying which students are mathematically gifted. The following descriptors of characteristics of highly able mathematics students should be viewed as examples of indicators of potential. Few students will exhibit all characteristics and these characteristics can emerge at different times as the child develops cognitively, socio-emotionally, and physically.
The highly able mathematics student should independently demonstrate the ability to:
- display mathematical thinking and have a keen awareness for quantitative information in the world around them.
- think logically and symbolically about quantitative, spatial, and abstract relationships.
- perceive, visualize, and generalize numeric and non-numeric patterns and relationships.
- reason analytically, deductively, and inductively.
- reverse reasoning processes and switch methods in a flexible yet systematic manner.
- work, communicate, and justify matheatical concepts in creative and intuitive ways, both verbally and in writing.
- transfer learning to novel situations.
- formulate probing mathematical questions that extend or apply concepts.
- persist in their search for solutions to complex, "messy," or "ill-defined" tasks.
- organize information and data in a variety of ways and to disregard irrelevant data.
- grasp mathematical concepts and strategies quickly, with good retention, and to relate mathematical concepts within and across content areas and real-life situations.
- solve problems with multiple and/or alternative solutions.
- use mathematics with self-assurance.
- take risks with mathematical concepts and strategies.
- apply a more extensive and in-depth knowledge of a variety of major mathematical topics.
- apply estimation and mental computation strategies.
It is important to realize that these variables are not fixed and need to be continually developed.
How to Identify Mathematically Gifted Students
Unfortunately, there is no single method for identifying gifted and talented students nor for assessing their performance. Ways of identifying mathematically promising students include:
Observation—while the students are working, particularly in problem solving situations of increasing difficulty or those designed to elicit the characteristics listed above.
Portfolios—students need access to exemplars from other students and the scoring rubric should include:
- patterns noted and generalized
- predictions made and verified
- interesting related problems posed and investigated
- measures of creativity—
- fluency (number of different solutions)
- flexibility (variety of solutions)
- originality (uniqueness of solutions)
- elegance (clarity of expression)
Questioning—individually, in small group, or whole class:
- Student interview
- Parent information
- Student interest/peer survey
- On-going assessment
- PADI diagnostic instruments, such as Rating Student Potential Teacher Checklist
- Diagnostic Thinking Tasks
- Math logs or journals
How to Teach Mathematically Gifted Students
When planning instruction for gifted and mathematically promising students, there are questions that need to be asked:
- Do the opportunities provide for the wide range of abilities, beliefs, motivation, and experiences of students who have mathematical promise regardless of their socioeconomic and ethnic backgrounds, and do the opportunities meet their continuum of needs?
- Are curriculum, instruction, and assessment qualitatively different and designed to meet the differing needs of promising students?
- Are there resources, projects, problems, and means of assessment that allow for differences in the level of depth of understanding and engagement?
- Are there appropriate oportunities in mathematics that have clearly defined, comprehensive, integrated goals—that are not simply isolated activities?
- Are the opportunities available to all interested students and in all schools?
As is stated in the Gifted and Talented Policy, "Children with special abilities and talents are part of the human mosaic in our schools and communities. They typically learn at a pace and depth that set them apart from the majority of their same-age peers. Because they have the potential to perform at high levels of accomplishment and have unique affective and learning style need when compared to others of their age, they require instructional and curricular adjustments that can create a better match between their identified needs and the educational services they typically receive."
Sections excerpted from Developing Mathematically Promising Students, edited by Linda Jensen Sheffield, National Council of Teachers of Mathematics, Reston, Virginia.