Albert Einstein, a famously mediocre student, once commented that "It is little short of a miracle that modern methods of instruction have not completely strangled the holy curiosity of inquiry." Many educational theorists and gifted teachers have taken this to heart, and endeavored to create learning environments that reflect innovations that are both intuitive and ingenious. Unfortunately, we often see these same innovations stifled at the High School level. Whereas the elementary grades have always seen as a test market for innovation and have lead to such new methodologies as multimedia education, the use of role play, and a team approach to the comprehension of new subject matter, the High School environment is seen as a sacred cow where entrenched methodologies are not to be tampered with. From a sociological perspective, High Schools serve to propel a community's best students to the University level where they will ultimately develop professional characteristics that will allow them to return value to their home town or city. From a pragmatic perspective, administers are quick to replicate methodologies that are proven to effect a positive result in measurable terms; in a good school these measures could consist of SAT scores and the caliber of university placement; in a struggling school, administrators must insure that students are able to pass basic state tests and that drop-out rates remain low. While these goals are admirable and in some respects immutable, they only serve to underscore the need for innovations designed to enhance the learning environment. Part 1. Current Teaching and Testing Methodologies in High School Mathematics Classes

As I have stated, the perceived general needs of the high school can be seen as duo-fold: to provide an education that encourages excellence to exceptional students, and to provide an education that encourages competency to average students. Based on the size, location and level of heterogeneity at any particular school, these needs attract varying degrees of attention. The former often receives the most interest from individual practitioners: the personality of one with exceptional mathematic capabilities will often resonate with that of the teacher.

Several organizations provide leadership to the mathematics community. These include the NCTM, Mathematics Association of America (MAA), American Mathematical Association of Two-Year Colleges (AMATYC), and MSEB. Together, these organizations provide a body of principles and standards adopted by most math teachers. Among these is a list of "Assessment Standards" that most mathematics teachers consider to be integral to their teaching methodologies, as they provide direction as to how to pursue a mathematic curriculum. Mathematics programs are assessed according to the success of the overall program, whether students are learning, how well the established mathematical goals are met, if students are capable of applying the mathematical knowledge in other areas of the curriculum and life, when students are enticed to study more mathematics, the worthiness and usefulness of the content, and if the program is teachable and learnable.

Teachers are encouraged to ensure that all students learn to enjoy mathematics. In order to assess achievement of students in a classroom, teachers: determine the progress of each student, ascertain the status of all of the students, and know the extent to which content and skills are mastered. Tests are used as diagnostic instruments. The number of questions that are asked is dictated by the ability of the body of questions to enable the teacher to accurately assess whether or not a student has grasped a concept. This process is described in "Teaching Secondary Mathematics" by Jerry Ashe:

If one question is asked, you have little certainty about whether or not a student has mastered the material. Asking two questions dealing with the concept is better, but how sure can you be? If a student gets one of the two right, what do you know? You could give another test, assess other work the student has done, or talk with the student about the issue, but those each take time. Multiply the time required by the number of times you possibly will need to do something like this times the number of students you will be dealing with and you begin to see some constraints. (Ashe, 61)

Ashe recommends that the teacher's testing methodology should reflect the most accurate effort of determining whether or not a student had mastered the material.

Ashe makes recommendations for making sure that the material is conveyed in a timely fashion so as to meet the goals dictated by the curriculum. He suggests the repetition of curricula...

He suggests that if a teacher is familiar with common errors that are made on his or her assignments and tests, that he may more adequately assess the means that a student used in order to complete the assignment, and where an error might have been made. This is why he suggests the use of a multiple choice format for the generation of diagnostics.
Ashe suggests that the teacher observes student behavior during class time. He recommends that the teacher assign individual or group work during class and that this work. In business culture, this process is known as "management by walking around," whereby a manager walks around and observes the work of his employees so as to assert a positive, passive role as both an observer and as a guide so that work is completed accurately and in a timely fashion. Ashe recommends that a teacher watch the facial expressions and body language of students. He recommends that the teacher act not only as an instructor but also as a careful auditor of motives:

When a particular student asks a question, is it sincere or an attempt to get some means of praise from you? Is the student a flexible thinker who is willing to try different approaches to the same question? Is the student asking merely as an attempt to lead you away from the objective at hand? (Ashe, 60)

Ashe suggests that a teacher conduct interviews with students so as to accurately determine whether or not a student has grasped assigned information. Here he breaks with his relative orthodoxy. He suggests that written tests might not necessarily paint an accurate assessment of a student's abilities. However, he mentions that this only be used with some students and be carefully considered.

One tool often employed by teachers in assessing student abilities is a checklist. This allows the teacher to determine whether or not a student needs a more thorough understanding of the material and places a student's learning capabilities and speed in the light of an assessment of other students. In addition to notes about a student's ability, a teacher may make notes about a student's preference with math and ability to work with others for the mutual completion of a task. Like tests, Ashe suggests that these checklists can be used over and over again, from year to year.

Common in high school math classes are the use of 'norm-referenced' tests and standardized tests. These tests are especially easy to use because they are usually supplied by the school system or the text book with answers. These tests are the most familiar to students. Despite these tests' ability to determine the level of knowledge that a student maintains, they aren't as accurate at determining the method that a student learns so that a teacher may cater to that method.

The concept of a student portfolio is another method commonly employed by instructors in order to teach math. This methodology was first developed for English Composition and art classes, but has made its way into other disciplines because of its versatility in allowing teachers to meet student needs. Because a student commonly has four mathematics teachers over the course of his or her high school career and troubled students are likely to have even more, the portfolio serves as a method by which teachers can gain a comprehensive understanding of a student's strengths and weaknesses.

A portfolio usually contains examples of the best works of a student that the student collects his or herself. The teacher provides the students with guidelines as to which completed assignments merit conclusion. Although examples of poorly completed work might be useful, it is considered more important that the student actively participate in the creation of this portfolio. The inclusion of poorly completed assignments would discourage this and might de-motivate the student. Some education experts advocate the regular completion of a journal. However, as useful as this is to educators who wish to track performance, it is hard to mandate this or to compel students to complete such a journal without including it in the grading process. In addition to topics covered in the coursework, other student work can be included. According to Ashe,

Demonstrations of the ability to use and interpret results generated through graphing calculators, spreadsheets, symbol-manipulating/function-plotting software, and dynamic geometry software would be appropriate. This is not an exhaustive list of elements that could be…