Geometry in Nature

Total Budget:                        $ 2469.95

Proposal Summary

The project grant and unit were put together by University of Texas degree holders Larkin Campbell (B.A. Mathematics & Government), Reem Kattura (B.S. Chemical Engineering), and Christopher Rodriguez (B.S. Mathematics).  The project is based on studying geometry through nature.  This will give students the opportunity to study this mathematical discipline in an inquiry-based setting in order to make connections with the real world.  According to a study presented in “Experiencing School Mathematics” by Jo Boaler [1], students perform better in inquiry-based environments as opposed to traditional classroom settings.  The project will take place over four weeks and will incorporate numerous hands-on activities, which explore tessellations, Fibonacci numbers, the Golden Ratio, symmetry, and fractals in the natural world.  The students will also work on and research a geometrical concept in nature and present it to the rest of their class mates at the end of the unit.  The project will provide the students with an inquiry-based learning experience which will allow them to view mathematics differently.  It will also expose students to the discipline of geometry in a novel way which offers a deeper understanding of the concepts.  In addition to the mathematical concepts that this project will cover, students will gain practice in using technology through geometrical software, cooperation through group work, and public speaking skills through presentations.

Description

Students today are having problems realizing the relationships between mathematics and the real world.  They feel that mathematics is too abstract to have any importance in their daily lives [2].  For this reason, our students are leaving mathematics programs and delving into other disciplines that appear more concrete and applicable [3].  Our project is designed to expose students to the often overlooked mathematical relationships that are found in our natural world.  We will begin by presenting a video that will introduce the students to the contents they will be exposed to during the unit.  Over the four weeks, the students will not be asked strictly to copy down notes presented to them on the chalkboard.  Instead, the students will have the opportunity to work with a series of hands-on activities that they normally would not think of as dealing with mathematics.  These activities will also serve to immerse the students into the world of mathematics that surrounds them.  By working with snake-skins and honeycombs, the students will be able to explore tessellations.  Nautilus shells, flowers, and rabbit reproduction will present the Fibonacci sequence.  Taking measurements of the human body will allow students to discover the Golden Ratio.  Finally, by taking pictures within nature using digital cameras, students will explore fractals and symmetry.

To implement these activities, we will require various things.  Many resources required are currently provided by the school district, such as access to computers with internet, calculators, and projectors.  However, we are lacking Geometer’s Sketchpad and digital cameras.  Since our unit is based primarily around Geometry, we will need software that can demonstrate geometrical concepts and also be used during student activities and presentations.  Geometer’s Sketchpad is the best resource for this purpose.  Digital cameras allow for us to bring nature into the classroom for deeper exploration.

As the designers of this project, we feel we are best qualified to present this unit because we each obtained an Undergraduate degree at the University of Texas at Austin, one in Chemical Engineering and two with Mathematics degrees.  As participants in the UTeach program, we were exposed to many teaching environments, from developing and teaching lesson plans at the Coasts of Texas to enriching students’ educational experience in the field of Astronomy building and using telescopes.  Additionally, our experiences with Geometer’s Sketchpad in our upper division Mathematics courses have provided us with the knowledge we need to use in the classroom and to help our students take advantage of its numerous applications.

In doing this program, we suspect that students will become confused or frustrated by the way we present the material.  Students are generally taught in a very traditional way, which has them writing down notes and working through numerous problems to memorize rules and procedures.  Now we are suddenly asking the students to become independent learners.  This sudden shift in learning styles can pose problems in the classroom, but we believe that once students participate in the different activities and realize that they can have fun and learn at the same time, the students will adapt to this type of learning and enjoy our unit.  We have designed our activities in such a way as to be both engaging and worthwhile.  We believe that the students will feel the same way.

Rationale

Traditional mathematical education focuses on dry and algorithmic methods for solving problems.  Few connections are drawn from class work to the natural world, despite the fact that much of mathematics has been developed to explain our natural world and quantify relationships therein.  Such connections seem more evident to students in other disciplines, even though mathematics is just as embedded in the real world.  As a result, many students are drawn into other subjects because they seem less abstract.  NCTM has called for a reform in Mathematics such that the students can view mathematics as relevant.

            To compete with other disciplines, mathematics needs to be just as relevant, creative, and engaging.  Patterns in nature provide an abundant source of mathematical relationships and applications that can be explored by students in the classroom.  This unit gives students ample opportunity to discover this type of mathematics, including but not limited to Fibonacci Numbers, the Golden Ratio, fractals, and symmetry.  A direct result of this is the chance for the students to collect data on topics that interest them.  This unit is intended to allow the participating students to use progressive and innovative approaches to mathematical content, which will encourage a positive attitude towards mathematics.

Research done by Dr. Boaler on traditional and project-based schools has shown that the latter perform better on exams.  Therefore, this unit should benefit our students with their schooling in the long run.  Additionally, boys in general tend to enjoy a competitive environment while girls like a contextual and conceptual environment.  Our unit is designed such that it takes into consideration both types of personalities, therefore creating a balance and helping our students.  

Potential Impact

The goals of this project are to integrate natural objects and phenomena into the mathematics classroom as well as stimulate the students’ minds to think about mathematics in a different way.  By the end of the unit, students will be able to observe mathematical properties and relationships in nature such as sequences, fractals, and symmetry, and will use explicit formulas, recursive formulas, ratios, and other tools to model these properties.

By connecting topics in mathematics to the natural world, students will have a deeper understanding of these principles.  In addition, by working with technology such as Geometer’s Sketchpad, they will be able to create models of what they are learning about.  Working in small groups on a project will foster cooperation between students, and presenting their project will give students valuable public-speaking experience.  The presentation is designed to be a student-led lesson and therefore some students may even recognize a desire to become educators.

Three teachers, with an estimated four classes of 25 students will teach this unit; therefore over 300 students will participate annually.  In addition, because of the relatively low cost of supplies and the fact that geometric and algebraic elements are both important parts of the unit, we will be able to expand to other classrooms later on.  The teachers plan to present the project at CAMT so that other teachers can replicate the project at their schools.  We will also make project materials available on-line for others to view and submit how-to articles to The Mathematics Teacher to increase project impact.

Evaluation Plan

            Students will be continuously evaluated through their class work, participation, and brief in-class assessments.  Furthermore, the students will be required to research an extension of one of the topics and present what they have learned to the class.  The students must work in groups of 2 to teach a 10-minute lesson on the topic of their choosing.  They will be evaluated both by the content, presentation style, and group participation. 

In addition, at the end of the unit, a cumulative exam will be given encompassing the material covered.  Scores on the Geometry End-of-Course Exam as well as the Texas Assessment of Knowledge and Skills can also measure progress.

Project Calendar

Budget

§       The Geometer's Sketchpad(r):

Dynamic Geometry(r) Software for Exploring Mathematics Unlimited – User School Site License/Network Package

Total = $1499.95 + $10 shipping = $1509.95

§       6 Digital Cameras, approximately $160 each

Total = $ 160 * 6 = $ 960

Grand Total: $ 2469.95