Albert Einstein, a Famously Mediocre Student, Once

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 from year to year, as it provides for the generation of a list of written assignments that only need replication, so that time can be spared that would otherwise have been spent on modifying these assignments. 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 included in a student's portfolio, but it is a start. You need to consider the concept and build components into it that are appropriate for you and your students. (Ashe, 11)

Although a certain degree of leeway is always afforded teachers that wish to innovate, it often is difficult for these teachers to adopt sweeping changes. In certain respects, this is preferable: the interaction between parents associations in the community and school administrators has traditionally kept teachers in check as this insures that the ultimate authority over the welfare of students lies in the hands of their parents. Although this process is at odds with European systems that favor the national co-ordination of education plans, it allows for a great degree of dynamism in encouraging excellent education and community participation. We only see a breakdown in this process in troubled or low income areas, where the traditional social patterns have broken down and parents cannot be expected to maintain an active interest in their children's education or provide them with a supportive home environment.

The math curriculum at many schools has been hurt in recent years by the dearth of mathematics teachers available. This is due to many factors, although one can hypothesize that the skills that would have allowed one to become an excellent albeit modestly paid math teacher 30 years ago can now allow one to enter more well paying number intensive fields of employment, such as computer programming, statistical forecasting, derivatives modeling, and commodities trading. Many of the innovations that allow for such new fields as derivatives modeling, in which an investment analyst may create instruments that allow investors to predicate returns on an infinite array of risk structures, are new to the last 20 years. Whereas this has provided incentive for students to excel at math, it has robbed qualified people of their incentive to enter this field.

By far, the greatest problem concerning secondary education has been the collapse of any previous consensus as to its central purpose. This confusion has lead to the disintigraton of academic standards in the admission, curriculum, evaluation, and graduation of students.

This presents itself as a substantial problem because the quality of education has a direct bearing on the way in which society operates and performs. The modern secondary school, introduced in France in the early 1800's, was a vast institutional improvement in that it compelled the entire citizenry to reflect the civic values of the state. In a republican system, the mechanism which allows the nation to govern itself is predicated on the ability of its citizens to vote rationally. To do otherwise would be to encourage tyranny.

To admit that there is something wrong with the system of mathematics instruction is to turn on the basis for that instruction that seems normal to us because it is what we experienced. Math instruction represents the value that society places on such actions as borrowing and consumption. Proper understanding of such concepts allows them not only to consume goods but also to function in society: we must educate students in math so that they can manage a highly complex, technological, and interdependent world.

Math instruction represents the value that society places on such actions as borrowing and consumption. Proper understanding of such concepts allows them not only to consume goods but also to create valuable in a sense that is appealing to us. David Gardner, President of the University of California, and his colleagues on the National Commission on Excellence in Education, issued a speech, warning that:

Our Nation is at risk... [because] the educational foundations of our society are presently being eroded by a rising tide of mediocrity that threatens our very future as a Nation and a people....

If an unfriendly foreign power had attempted to impose on America the medicore educational performance that exists today, we might well have viewed it as an act of war....

Our society and its educational institutions seem to have lost sight of the basic purposes of schooling, and of the high expectations and disciplined effort needed to achieve them. (Gardner, 2002, pg. 2)

In order to identify some of the principal conditions that allowed the current situation mathematically oriented, it is first necessary to look at the contemporary 1st grade class. Teachers unions currently control the pay scale at the state level and have hastened a decline in the number of qualified math teachers. These unions advocate a more equal pay scale that does not favor one field of study over another in salary. To some extent, individual school districts have sought to mitigate this problem by offering 'fast-track' certification programs to new teachers. These certification programs represent a barrier to entry to those wishing to enter the teaching profession, which has grown since the beginning of the recession.

The quality of math teachers differs with different factors. These include the quality of the community, the pay that's offered, the general educational proficiency of people living in the region, and the age of the faculty. The combination of these factors makes for a diverse array of math professionals. Similarly, the demands placed on math faculties differ. Schools range from large, impersonal generalist institutions to small, alternative high schools and magnet schools with a stated disciplinary focus. Generally, one finds exceptional math professionals gravitating to small, selective institutions that offer a great deal of leeway. The benefits of these institutions, until recently only found in the most exclusive private schools, have been recognized and adopted by charter schools. While their methodologies cannot be considered universally applicable, certain practices adopted by such professionals are of interest to educators that work with gifted and above-average students. In this report, I will explore several of these methodologies, with an emphasis of those that have been implemented at the grade school level.

Part 2. Alternative Strategies for Teaching Math Employed at the Secondary School Level.

Many new methodologies of instruction have been developed at the elementary school level and successfully implemented. These include generalist innovations that can be applied to any field of study, math-specific innovations, and inter-disciplinary innovations that seek to incorporate math and other subjects into a centrally themed and mutually complementary method of instruction. Despite these differences, many of the innovations suggested share characteristics, many of which are predicated on handing more autonomy to the student to explore mathematical or generalist questions on his or her own. Others include initiatives designed to bring students together to problem solve, role play, or complete a project. These not only allow for different methods by which students are able to comprehend information, but prepare students for the office environment by allowing them to replicate methods used in the professional world to complete projects.

In Interaction and Human Development, Jerome Brunner and other authors contend that a key aspect of learning often missed by conventional theorists is the process by which a child teaches himself, and that this process is most readily apparent in children who taught themselves how to speak English. He warns us to avoid the temptation to fall prey to what Piaget termed "magical thinking," whereby we attribute to the effects of instruction what is properly an achievement of the child. (Brunner, 1989) Brunner sees interactions between a developing child and an instructor in terms of several dimensions or aspects of intellectual and social development that utilize both formal and informal instruction. Brunner contends that interactions between a developing child and those who possess more knowledge lead to the creation of "sociosensory-motor structures" (Brunner, 1989) and to the formation of new means-ends procedures. These interactions also lead to the development of the child's ability to self-regulate and cause the child to self-identify as a learner.

Brunner begins by acknowledging the bi-directional nature of child development, and says that this is not at odds with the idea that children are constructive architects of their own understanding. Brunner feels that the aspects of self-correction and self-organization in linguistic development provide a clear example of self-perfection and self-organization that can be methodologically transposed to other fields of study and areas of cognitive development. Not only does Brunner characterize learning as bi-directional, but pays careful attention to the ability of a wide array of diverse students to help each other learn by sharing their respective competencies. In the traditional environment, children are often taught to work alone rather than in groups, especially with respect to tests where critical evaluations remain at the individual level. By sharing learning projects, not only does Brunner feel that children gain the ability to communicate effectively with others, but they also share the ability to learn and different aspects of the learning faculties that they employ to gain and retain new information.

The expression "zone of proximal development" is used to describe the gap that exists between a mere core competency that one can develop on one's own and one that required the assistance or structure of a network. Observational, experimental and interventional studies have all revealed examples of self-correction. Art education is one of the few areas of child development where it can be said that a child has an almost fully independent to develop his understanding of a new set of concepts. With art or music composition, a skill must be mastered in order to create effectively, but this represents only a very small percentage of the successful completion of the task at hand. It can be said that a child's artistic ability is a self-perfecting process in that an intuitive sense develops as to whether or not art pieces generated by the student measure up to certain unspecified or loosely specified criteria. In the end, the child gains a general sense of external satisfaction, dissatisfaction or indifference similar to that, which is experienced when a child that is undergoing the process of learning language feels when applying his new knowledge base.

Bruner feels that the feedback that a child receives is far from unstructured, and gives a child a sense of awareness that leads him to better understand the nature of his surroundings. "There are crucial systematic features in the relation of a language-learning child and the adults with whom the child interacts. Indeed, there may be aspects of prelinguistic interaction that are not so much precursors of language as they are factors that actually predispose the child to language use." (Bruner, 1989)

In an interview conducted last year, Gardner stressed the importance of interdisciplinary education, calling for education in the 21st century to reflect the necessity of engaging in activity that lets you "bring together different sources of knowledge, to be able to synthesize them, judge them, and so on." He explains this need by saying,

So much work now at the forefront of society is problem based. As we have increasing expertise in different realms, at the very least, one has to learn to work with people who have different expertise, and maybe, optimally, acquire more than one for m of expertise yourself. I wrote a book called The Disciplined Mind, and I argued there about the difference between multi-disciplinary work, where a number of disciplines discuss the same topic, and interdisciplinary work, where individuals from different disciplines actively work together to solve a problem that no discipline can attack on its own. (Gardner, 2002, pg. 3)

Gardner relates this process with that by which education is not only made to focus on the context of a global society (especially with respect to ecology and economics) but also is seen as a process which takes place over a lifetime rather than being limited to children and young adults. To this, Brunner would add that we must train children to educate themselves so that they may adapt to novel situations rather than looking to a specified authority within the context of a top-down hierarchy. Gardner cites the widespread adoption of inter-disciplinary methodologies at the adult educational level, equating those that specialize in just one discipline with "repairmen."

Gardner stresses that despite his universalist outlook, he is very much a proponent of individualization. He claims that the reason that he pushes for individualization is due to his conviction that people have different kinds of minds and have different strengths and epistemologies. Gardner believes that even where there is a singular curriculum, that it is pointless to teach everyone in the same way and assess everyone using the same set of criteria. Here Gardner's faith in technology and the social propensity for self-determination in the United States is apparent:

Computers will make it possible. Even if we wanted everyone to learn algebra, there is no reason why you would have to learn the same way as I do as long as we can both understand the ideas of algebra and use it effectively; so that is 'Individualism as means." In a few of my books I admit that if I was a czar, I would be happy to have a single curriculum...I believe that at least in a country that is as complicated as the United States, there are such deep differences among people in what they believe should be learned, what they think the curriculum should be that I don't think everyone would ever agree on a viable curriculum. (Gardner, 2002, pg. 2)

Many of the ideas that Gardner developed about the American-ness of his curriculum he developed during extensive travels in China, Western Europe, and the Soviet Union in the 1980's. He notes that many command economies mandate that students exhibit technical mastery at the expense of an intuitive gnosis of the subject matter. This propensity, which is at its most obvious in soviet literature classes where every analysis could be said to fall under the heading of "communist critical theory," is also apparent in subtler forms in art classes in China and most lyceum-style baccalaureate-oriented education in France and other European countries. In To Open Minds, where he gives the example of an art class where all students are required to paint a precisely identical goldfish based on the painting made by the teacher and the same goldfish as presented in a textbook. Of this, Gardner says:

By the end of the class, every student had produced at least one goldfish. The goldfish looked like the model; and, in that sense, the class was a singular success. But it epitomized the problems in Chinese arts education that have concerned so many observers: how does one go beyond slavish artwork where there is a prescribed procedure and a canonical "right" end product, to the point where one can try something new, modify schemes, combine them in novel and provocative ways? (Gardner, 2002, pg. 2)

Math curricula have often fallen into this 'goldfish' trap, although this is more common in other countries than it is in the United States. In the United States students are taught to master a rational approach to problem solving, something that is often ignored in Eastern cultures. Gardner shows some classical liberal traits in his excoriation of top-down approaches to education as being typical of totalitarian governments and command economies. Perhaps more importantly, however, he believes that education should exemplify Plato's criterion of making people want to do what they have to do. He places a lot of stress on the importance of not doing harm, what he notes that Isaiah Berlin referred to as "negative liberties;" the freedom to enjoy a certain liberty from something rather than the right to enjoy an entitlement.

Gardner explains the origins of his theory on multiple intelligences by saying:

Whenever I study something, I study..how it develops in children, how it breaks down, how it existed in different cultures. What led to my theory of multiple intelligences was bringing together these different perspectives. While they don't ensure accuracy, they sort of bounce against one another, and when they all seem to be saying the same thing, then to me that's a sign of validity. (Gardner, 2002, pg. 11)

Gardner's educational philosophy is ideally suited to small, intimate environments where teachers are allowed an opportunity to develop a close personal relationship with their students in order to properly cater to every child's individual needs. They are best suited to teachers who don't express a high degree of favoritism, as the individualized approach to teaching would make it more possible for teachers to overlook particular students if the teacher was given to whimsical favoritism. One could claim the individuation of individuation to be anti-egalitarian in that it abandons universal standards. However, it can also be said that traditional methodologies lack the ability to cater to the special needs of every student and rewards students that mesh well with the prescribed metrics rather than noting the given strengths of particular students. It is for these reasons that such methodologies are more effectively employed at the elementary school level rather than in middle or high school. Middle school and high school are more competitive, with students openly competing for a scarce number of positions in a select number of honors classes. Such students would be adverse to educational programs that would be characterized by its critics as prone to subjectivity. It is in the field of elementary education that multi-lateral education is poised to experience the most leeway and the least resistance.

III. Learning Concepts and Mathematics Education

Conceptualizations of student learning reflect a broader trend in mathematics instruction: teaching toward understanding. This isn't a new concept: reform movements that valued understanding-based mathematics education were first introduced at the turn of the 20th century. In these early reform movements, mathematicians derived notions of understanding from the way in which they understood and taught mathematics. However, newer initiatives differ from these early models in that an emerging research base about the learning process can be used to determine what is meant by learning and what it means to teach for understanding. This research attempts to convey how students construct meaning for mathematical concepts or processes and how this form of self-education can be supported by the classroom.

Proponents of this system are quick to point out that it is generative and relies on the intuitive connections between thinking patterns rather than isolated skills. When students perceive topics as isolated skills, they fail to apply these skills to contexts other than that presented in the classroom environment. In effect, skills mastered in the classroom retain almost none of their value after a student has left the class; it could be said that the knowledge conveyed to students lacks liquidity. In that High School is the culmination of what might be called 'universal' education in that nearly everyone attends high school, it seems impractical or elitist that skills provided in this environment should only serve the purpose of preparing students for college interest exams. Promoting a more general understanding of mathematical concepts and problem solving allows students to develop their ability to interact as consumers, borrowers, problem solvers and general practitioners of mathematics.

Virtually all complex ideas can be understood in a number of different ways. It follows that it is most appropriate to think of understanding as a process that can be thought of as emerging or developing. This is to be contrasted with traditional views of understanding, which hold it to be an objective or set of objectives. Educational theorists have begun to categorize mental activity as it relates to understanding, instead of categorizing 'levels' of knowledge and placing students in an array of classes that assumes levels of aptitude.

In Fostering Cognitive Growth: a Perspective from Research on Mathematics Learning and Instruction, Erik de Corte maintains that mathematics teaching should be seen through the eyes of a professor who wishes to convey a mathematical form of reasoning to students. De Corte maintains that with a proper math education, one's ability to think can be improved. He notes that there is a broad consensus that the major categories of mathematical aptitude that underlie skilled problem solving are domain-specific knowledge, heuristic methods, metacognitive knowledge and skills, and affective components. Attention to these categories is most notable in elementary school curricula. Whereas the acquisition of procedural computational skills has allowed students to succeed in the past, educators are now emphasizing other means of learning and problem solving, which include domain-specific knowledge, heuristic methods, meta-cognitive knowledge and skills, and affective components such as beliefs and emotions.

Domain-specific knowledge is that which involves facts, symbols, conventions, definitions, formulas, algorithms, concepts, and rules. These constitute the content of a subject-matter field. It has been discovered that expert problem solvers master a large, well-organized, and easily accessible knowledge base. However, even the most careful attention to this type of knowledge will not acknowledge the importance of mastering an understanding of underlying concepts that facilitate problem solving. This methodology is more focused on the ability of students to memorize and often leads to the serialization of misconceptions through acceptance.

Heuristic methods are ones that emphasize systematic strategies for project analysis. Heuristic methods are centered around an approach to learning rather than the individual interests of young students. These methods advocate a systematic approach to the completion of objectives. Sometimes a heuristic method will start by breaking a problem down into known and unknown data and deriving an answer from that which is known to the student. Erik de Corte cites the following example:

store sells two kinds of fruit juice: Bottle A costs 16 Belgian francs for 20 centiliters, and Bottle B. is priced 19 francs for 25 centiliters. What is the best buy, assuming that both kinds of juice are of equal quality?" In solving this problem, a student might think of a related task solved before, such as comparing the price of potatoes in sacks of different weights. Through the analogy of figuring out the price per kilogram of each kind of potatoes, the student might decide also to decompose the present problem by calculating first the price per liter for each type of fruit juice, and then comparing both prices, which is of course a routine task.