Algebra and Trigonometry at Watershed
Individual Pacing
We believe that much of the problem with math education stems from the "one size fits all" pacing that often occurs in schools. Students who take longer to grasp mathematical concepts often get lost when the pace is too fast for them to keep up. They are swept along before they have mastered the skills. What results is frustration and the "I hate math" mindset.
On the other end of the scale are the students who can progress rapidly and quickly get bored waiting tor the class to move on to the next set of skills. They are champing at the bit to get moving, but aren't allowed to move on.
At Watershed, we are using a primarily tutorial based approach for the teaching of Algebra and Trigonometry. Students can work at their own pace, while still having a mentor in the room to help them overcome obstacles and to provide coaching. The ALEKS system is a thoughtfully designed web-based program that requires students to work on problems with pencil and paper, and to answer their questions by entering the answer, not just choosing among multiple choice answers. They are provided with immediate feedback: if their answer is incorrect, they'll know immediately, and can try again.
Assessment and Credit
ALEKS has a built in assessment scheme that routinely tests students on their knowledge. After every 10% of new knowledge learned, students are tested on the entire body of knowledge since the beginning of the course. This means that students can't "cram and flush" for a test. Retention is an important goal.
We require that students attain mastery over 75% of the course material with an additional 15% exposure, for a total of 90% exposure in order to receive credit for the course of study. This may take two years for some students, but others will complete the course in one academic year or less. We are comfortable with students working at their own pace.
Occasionally, we come across students who don't do well with the ALEKS model. We strive to design individual programs for those students to facilitate their learning.
Geometry at Watershed
Geometry is taught in the sophomore year in a more traditional classroom setting. We've found that daily contact with the geometry teacher yields a much more rewarding and enriching experience for students
One of the big themes of Geometry is working back and forth between inductive and deductive reasoning. We derive geometric formulas from objects to build our confidence that the formulas are based in reality and that patterns were discovered by a person and can be rediscovered. It begins simply enough, using a piece of string and a ruler to measure the circumference and diameter of circular objects and discovering an approximate value for Pi. Creating a paper cone that fits perfectly into a tin can and filling it with water can prove that the volume of a cone is one third the volume of a cylinder with the same height and radius. We test the formulas for surface area and volume of a sphere by comparing actual measurements of an orange and its peel to the theoretical values obtained with formulas.
Students gradually add Geometric formulas and theorems to their repertoire. Algebra is used to derive new formulas from the ones students already know. The most involved of these is applying Cavalieri’s principle to derive the formulas for the volume and surface area of a sphere. They memorize 15 formulas for area, surface area, and volume, are able to demonstrate 7 constructions (including sliding a shape with a vector, rotating a shape, and constructing the center of a circle) and are able to apply 40 theorems to formal two column congruence proofs.
Other topics include creating tessellations, designing Celtic knots, and looking at some of the intriguing properties of Mobius strips, the Golden Rectangle, and fractals. The second term includes a project in architecture. Work with coordinate geometry and similar triangles leads to applications of basic trigonometry.
Geometry Assessment
Tests are given in class at the end of each topic ( usually every three weeks). This gives feedback on what was understood and where more clarification is needed. The process of effectively studying for and calmly taking tests in itself is a useful skill. If a student scores less than 80% on a topic test, they make it up. The mid year and final test are cumulative, some topics are required but in some sections students can focus on what they know best.