Math to the Third Power (Making Math Meaningful!)
By: Julia Watson, Ph.D., Exemplars Consultant and Gifted and Talented Specialist
Have a student identified as gifted in mathematics? Hmmm, what to do … ? Give more problems? Jump to the next book?
Because each learner comes to his/her school experience at a differing level of strength and need, a Response to Intervention (RtI) approach is suggested. Using data to identify levels of service, a continuum of programming options must be matched to student needs in order to most appropriately reach mathematically promising students.
For some students, in-class differentiation can meet student needs. In this case, the suggested modifications included with Exemplars tasks make it easy for teachers to provide mathematical challenges for a wide range of learning needs. In situations where a student is one or more grade levels above his/her assigned math class, more intensive programming options must be considered.
Dr. L.J. Sheffield, author of Extending the Challenge in Mathematics (Corwin Press, 2003), describes a continuum of mathematical thinking along which one can place students in a relationship to their level of mathematicalthinking; the continuum ranges from non-mathematical thinkers to the most sophisticated mathematical thinkers:
innnumeraters > doers > computers > consumers > problem solvers > problem posers > creators
In order to raise mathematical thinking to the highest level, Dr. Sheffield has identified five components necessary for profound mathematical problem solving; students must be able to relate, investigate, create, evaluate and communicate mathematical ideas.
Programming for mathematically promising students should include a combination of enrichment and acceleration, as these students are capable of processing math concepts with less practice and at a faster rate than other students. Instruction should be at an accelerated pace and the curriculum should have increased challenge and rigor focusing on the depth and complexity within mathematics. The tasks within Exemplars can be used in place of the regular curriculum (more depth and complexity) or as an extension of the current curriculum as the tasks have been aligned to NCTM and Common Core standards.
The concept of cluster grouping should be applied to meet the needs of gifted students. The ongoing support of “mental mates/peers” can help students as they face and work through challenges. Together, students can wrestle with and discuss complex problems to develop a greater understanding of mathematical concepts. Cluster grouping can support adjustment to the increased demands of complex problems and students can be encouraged to share their results and thinking with students who “speak the same language.”
Student enrichment can be provided through the addition of higher-level problem-solving activities and application of mathematics to real-life problem-solving situations. Look for problem-based learning opportunities in local news media or in everyday life and surroundings. For example, in problems concerning:
- water resources
- wildlife population management
- transportation routes and traffic pattern
- design of buildings
- parks or playgrounds
There is an opportunity to develop public opinion polls and surveys. These can be used for modeling and statistical analysis as well as to develop the concepts of geometry. Math activities should apply information from other content areas to increase student interest and involvement.
Teachers of mathematically promising students need to know the content of mathematics and understand how their students develop mathematical concepts over time. Teachers should serve as facilitators to help students develop the necessary skills to discuss their thoughts about the concepts they used to solve the problem and how these concepts relate to other curricular areas. Strong teachers of mathematics provide students with activities that allow students to show what they really know about mathematics. Problems that have multiple entry points, multiple pathways to solutions and/or multiple solutions should be commonplace.
In mathematics classrooms, teachers must create a “culture of opportunity” where students have the time for and feel free to explore the wonder and beauty of mathematics. In this environment, students expect to be challenged and are encouraged to develop their own problem-solving strategies. Additionally, students need opportunities to learn how to evaluate and critique the solutions of others in a collaborative way. Such classrooms require discussions among students regarding their solutions to problems; it takes a savvy teacher to know when to step in to clarify or to ask students to share solutions that enhance and clarify understanding for others.
Rubrics and anchor papers from Exemplars can assist students in more accurately rating and improving performance in order to increase achievement and success.
Making Math Meaningful through high-end problems and challenges for gifted learners using Exemplars is an exciting opportunity for students as well as teachers of mathematics.