One of our most popular programs, AVE’s Intensive Math Tutoring has turned D students into A students, and moved students from the very bottom to the very top of their class. It has become common to accept that students who really struggle with math will never be able to excel at math. After all, many students who really struggle with math have deep-seated mathematical cognitive deficiencies, right? Right. Absolutely, one hundred percent right. Major deficiencies in math are rarely caused entirely by a lack of knowledge, a tendency to forget specific formulas, anxiety, or carelessness. Instead, they are caused by deep seated deficiencies in cognitive ability that can be innate or caused by critical problems in early math education. The process has turned D students into A students, and moved students from the absolute bottom of their classes to the absolute top. While the approach is more intensive than standard approaches, it is far more rewarding. ## Deficiencies vs. DifferencesMost math courses in America deluge students with a mix of vital and unimportant information. The “best” math students are often the ones who are able to recognize what information is important and to deemphasize the rest. Thus, thoroughness and diligence on the part of a math student can actually backfire; many innately hardworking students attempt to learn all of the material thoroughly, instead of focusing exclusively on the key concepts. While this may lead to excellent quiz grades, it often causes horrendous performance on exams and standardized tests. These exams test a student’s ability to apply a small number of concepts effectively; students whose minds are full of dozens of specific formulas and problem types find themselves overwhelmed and confused. Although they may study several times as hard for exams as excellent math students, their exam grades are much lower. Many such students do not actually suffer from any cognitive deficiency; they simply think about math in a different way. In fact, these different approaches are often in significant ways superior to the cognitive methods favored in most classrooms. For example, a student who learns a topic thoroughly rather than in terms of a small number of concepts is in a position to develop an understanding that surpasses the understanding of the “stronger” math student. AVE’s approach treats deficiencies as deficiencies, and differences as differences. Individually tailored programs take full advantage of strengths inherent in different learning styles, while building underdeveloped cognitive abilities. As students learn to use their unique strengths while building their relevant cognitive skills, they are able to compete with and eventually surpass their peers.
## Frequently Asked QuestionsDoes the Curriculum affect all of AVE's math tutoring? If students move too quickly, is there a danger that they will not retain fundamental information? Is the curriculum appropriate for home-schooled students? Is there a danger that the student will learn the same thing in two different ways if he or she is taking a similar math class in school? Is the Curriculum the same for every student? ## Additional Information for Education ProfessionalsThe Curriculum reflects AVE's view that mathematics is most importantly a means to develop cognitive skills. This belief contrasts with the view of mathematics solely as an underpinning of the natural sciences. Thus, trigonometry is not introduced so that the student may solve problems involving the height of a mountain. Instead, a problem about the height of a mountain may be used to develop specific problem-solving and reasoning skills. The Curriculum is also developed around the cognitive processes of the student, rather than the structure of the discipline. Topics are presented in an order and manner that best facilitates long-term understanding. This is most significantly evident in two ways. First, the order of presentation of topics occasionally differs from those used by earlier methods. Secondly, fundamentals of several related topics are introduced before esoteric aspects of any topics are examined. For example, students learn the fundamentals of limits, derivatives, and integrals before they learn methods to find the derivative of an inverse function. This method creates a conceptual framework in which such abstruse topics can be understood, appreciated, and incorporated into a permanent understanding of the method. As part of its focus on developing reasoning and problem solving skills, the Curriculum develops and understanding of mathematics from the bottom up. This means that students learn core principles and methods, and learn to connect them to solve increasingly challenging problems. This contrasts with methods in which students learn a set of problem types and associated formulas, and must choose the right formula or sequence of steps for each problem. AVE feels that such methods encourage rote, temporary memorization, rather than a long-term permanent understanding. AVE also avoids the more extreme version of this method, in which the student learns to identify the problem type and execute a set of calculator steps. Despite its numerous innovations, the Curriculum is markedly traditional in many ways. Mathematical proof forms a significant part of the study of algebra, geometry, and calculus. Secondly, calculators are not used at all until precalculus, and even then are used extremely sparingly. This reflects the Curriculums goal of developing problem solving skills and a strong mathematical foundation. Third, the bottom up approach is drill intensive at times. The somewhat popular belief that a student of mathematics has only to learn the "big picture", and that facility with basic methods is not necessary, is not one shared by AVE. The relationships and patterns associated with more fundamental parts of math are similar in important ways to those associated with higher levels of math. Adding numerical fractions are similar in obvious ways to adding more complicated. Additionally, students insufficiently comfortable with basic fundamentals may not be able to thoroughly understand more advanced topics. Instead of developing a complete understanding of the topic, students may end up memorizing enough formulas and rules of thumb to get by. Finally, students who have not integrated basic principles and formulas into their understandings may be overwhelmed when they see problems that require the simultaneous use of several of these. A student who does not know basic area formulas may find a problem that requires the using interconnections between area formulas and the Pythagorean theorem to be completely overwhelming. Such a student would be forced to consciously hold all these formulas in his conscious mind while attempting to solve the problem. The difficulty he would face would be akin to the difficulty of memorizing three separate phone numbers while mentally subtracting two large numbers. One concern shared by many excellent educators is that if math is made too easy, talented students will not be challenged. Thus, their problem-solving and reasoning skills will not develop as much as they could. AVE shares this concern. Because the Curriculum introduces advanced topics much sooner than other curricula, students have to struggle. However, rather than struggling with topics made artificially difficult, they struggle with topics intrinsically difficult. A related concern is that a student's cognitive development may not be ready for advanced topics. This concern is also shared by AVE, and is addressed in two ways. First, methods are used to accelerate the development of cognitive skills necessary to study advanced topics. Secondly, the manner of presentation is at times adjusted to allow younger students to learn more advanced methods. The former method is generally preferred; when the second method is used, the topic is revisited when the students' cognitive skills are appropriately developed. |
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