Anne Ray ‘Horizontal Enrichment: Avoiding the “Why Do I Have to do More Work?” Question’ International School of Islamabad Islamabad, Pakistan. [email protected] Advanced Math Project STEP ONE……… 1st, begin by deciding what kind of project you wish to do. Some possibilities include: PEOPLE of MATH Study a famous mathematician and strive to understand both what he or she discovered and why it is important. The math department has biographical “starter packets.” PLACES of MATH Investigating mathematics is done differently part in some other region of the world or how the focus of mathematics might be different from the focus of math with which you are familiar. The library has a few books on the math of other regions and there is a lot of information on the web. Also, there is a book titled Math Elsewhere that you might be able to find. IDEAS of Math Consider some major concept of math, such as: zero, or coordinate graphing, or the Golden Ratio or pi, or logarithms, or the use of unknowns; and try to follow the history of this concept back in time and understand how it evolved and why it became important to the future progress of math. This is a research project, that can be pursued both on the internet and in popular math literature. CONNECTIONS to Math Investigate how math relates to some other field of interest, such as art, music, or sports. Design a project that will help you better understand the connection between mathematic and some other interest of yours, and help you gain the skills necessary to become more proficient at both. For example, your might wish to better understand sports statistical analysis and how the “handicapping” works, or you might wish to learn how a computer program animates a figure and write such a program, or consider the question of how music might motivates mathematical reasoning… Step Two………… 2nd, Complete enough preliminary investigation to confirm that the topic is appropriate for your level of math understanding . Will you learn new concepts or at least new approaches to concepts that are important? If the topic is appropriate, begin developing a list of the questions you will be considering as you continue with the project. Some of these should be “essential questions.” You should write out these questions and prioritize them. Step Three……….Now, begin your research and/or investigation. Decide how you would like to demonstrate what you gain from this experience. Will you create a movie, a Power point, a Smart board interactive lesson, a physical model or something else. Write out a clear description of how you wish to share what you learn and by what date you will complete the project. Step Four……..Proceed with your research and/or investigations, modifying your questions and your goals as necessary. Be sure to discuss all modifications to your goals with your teacher in a timely manner. Do not wait until the due date. Step Five……..Complete you project with the goal of making the enthusiasm with which you pursued your questions, contagious. Teach what you have learned and/or discovered in an interesting and motivational way! After you’ve finished, congratulate yourself on your proven ability to work independently! Advanced Math ……..1 Proposal a Program for Advanced Middle Level Math Students at The International School of Islamabad Anne C Ray Advanced Math ……..2 Math education, especially prior to calculus, in accordance with the principles of NCTM, and my personal beliefs about math education, be taught in a constructivist way. Wherever it is meaningful and possible, the separate strands of math, as well as math and all other subjects should be integrated. Authentic learning and real problem solving encourages a greater depth of understanding than the mere manipulation of memorized algorithms. Identification of the Need for a Program Small schools often face the challenge of providing appropriate courses for all students, while seeking to integrate certain components of the curriculum, and maintaining minimum class sizes. This is a particularly challenging situation when certain students’ needs are well met in all but one subject area, but that subject area happens to be where their strongest abilities resides. In these cases, students, their parents and the school administration, have to determine placement in consideration of the scheduling limitations of the school, as distinct from the individual needs of particular students. It is beneficial, according to generally accepted educational theory, that schools such as The International School of Islamabad strive to help students synthesize their learning in separate subject areas, especially before the secondary level, by offering integrated classes, such as math and science, and that the school emphasizes constructivist modes of learning. Unfortunately, this also has the affect of weakening the appropriateness of what certain advanced math students are offered. Providing an individualized program in math can Advanced Math ……..3 ameliorate this problem. For this reason, I have designed an Advanced Math Standing Program for Middle Level students at The International School of Islamabad. Middle Level students at The International School of Islamabad are now placed in heterogeneous math classes in both sixth grade and seventh grade. In eighth grade, they are placed in General Math or Algebra I, but this greatly hinders the eighth grade teacher’s ability to integrate math and science. It should be noted that each year several transfer students have had Prealgebra in seventh grade, or in rare cases, in sixth grade, and these students are challenging to place appropriately. The curriculum of what is labeled “Prealgebra” varies greatly from one school to another and without appropriate testing it impossible to know if a student in adequately prepared to begin Algebra I. (As a placement test for algebra I, we presently use the Eighth Grade General Math Final Exam.) Added to this is the near impossibility of scheduling seventh and eighth grade math at the same time. Our present curriculum of math education has many positive benefits in terms of integration and constructivist exploration, but we are not at present providing adequate challenge to our most able math students, and middle level is a time in educational development when students need to be stretched and encouraged to begin teaching Advanced Math ……..4 themselves how to pursue answers to their own questions. Establishing a well structured, but ultimately self-directed, program of advanced math standing, with clear, but flexible requirements, could serve to provide some of this needed intellectual stimulation, while also serving as a system of recognition of talents. Proposed Advanced Math Standing Entrance Requirements The program I propose will be designed for those students who demonstrate all three components of the enrichment triad – above average abilities, task commitment and creativity, but admission into the program will be based on self-selection. Teachers can encourage certain students to apply and discourage others, but ultimately, it will be a student’s individual choice to seek this status or not. Enticements to apply will include: special recognition on the student’s transcript when and if all requirement are met, the privilege of taking pretests and potentially beginning excused from most of the regular assignments connected when students successfully “test out,” the freedom to work on project(s) of the student’s own choice, an optional semiweekly lunch session with other advanced math students to discuss their projects, and the honor of presenting the results of individual research projects at a special “Math Night” for parents and other community members. In my experience, students with the ability to work more independently and at an advanced level leap at opportunities for greater freedom in what they learn and how they learn it. If students hesitate to accept the additional challenges and responsibilities involved, their math teacher and/or advisor should approach them, and parents can also Advanced Math ……..5 be enlisted to provide additional encouragement. If a student still does not want to participate, despite having an exceptional mathematical ability, their choice will be honored. Likewise, students who seem unlikely to succeed at completing all the requirements, but wish to try for this honor, will be permitted to do so, without any fear of embarrassment if they select to leave the program before accomplishing all of the goals. Their attempts to do so will have no negative affect on their grade or class status, and all efforts they make while participating will be of obvious benefit to them. Designing Individualize Programs (IAP) Clarity and flexibility are the keys to this program. Students might excel in their geometric reasoning, while moving less confidently with symbolic algebraic reasoning, or visa versa. They may seek an extremely abstract reasoning project, such as striving to understand and expand on Buffon’s Needle (a theorem about Pi); or conversely, they might investigate the slope of the entrance stairways around the city by measuring the rise and run and then calculating the rate of change, in order to understand the importance of slope in architecture. Along with designing a project, they will also be required to maintain a high average, and use their talents to serve their community by tutoring others, creating learning tools or representing the school in math competitions. (See Appendix A – Requirements for Advanced Math Standing.) Students will create the rubrics for their special projects with guidance from their teachers. They will have to plan how to meet their specific goals. Prior to creating the rubric, students will need to do some preliminary researching and formulate questions Advanced Math ……..6 about the topic they are investigating. Students will be provided with pre–researching tools, but also be allowed to explore outside of the topics and books listed.