There is an unspoken rule in American math education: all students should progress at the same rate. For example, gifted students who work hard should study calculus at about the same time as less talented students who put it little effort. Thus, almost all students study algebra in 8th or 9th grade, calculus in 11th or 12th grade, etc. To make this possible, most “honors” math courses are made artificially difficult with the addition of unnecessary or irrelevant problem types. Instead of allowing advanced and motivated students to progress more quickly and efficiently, the honors courses actually slow advanced students down. The results of this mindset are absurd. In many schools, the hardest algebra class is harder than the easiest calculus class (and yet, many teachers and administrators still maintain that 9th graders cannot study calculus). While a few honors algebra courses are made challenging in ways that effectively develop students’ cognitive abilities, many are just made difficult in ways that provides little benefit to the students. AVE’s programs are based on a drastically difficult mindset. Yes, our programs for advanced students are challenging. For example, AVE’s math programs are significantly more intensive than honors math programs found in schools. However, the challenges are designed to create the kind of cognitive development that allows our students to thoroughly surpass their peers. For example, at AVE, 8th and 9th graders have the option of studying calculus (AP BC Syllabus, the harder AP syllabus). What allows for these rapid gains in AVE’s cognitive approach to education. Through progressive levels of challenge, AVE helps students build the required cognitive skills that allow them to approach more difficult materials. For example, these methods help younger students develop the required spatial reasoning abilities for integral calculus. Under no circumstance is the subject “dumbed down” for younger students. Instead, the material is presented even more rigorously than it would be in an honors or AP class. To make this possible, the approach systematically develops the students’ cognitive abilities, making them able to master material considered too advanced for their years. With standard honors programs, you will move at about the same rate as others your age. With AVE’s accelerated math programs, you will leave them in the dust.
## 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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