Since its introduction in 2004, the Arvin Vohra Accelerated Math Curriculum has dramatically influenced AVE's mathematics teaching and tutoring strategy. The Curriculum increases the speed and rigor of mathematics education using a perspective that places the cognitive development of the student above all other concerns. It influences methods used to teach all levels of students, although it is used most often by highly talented and competitive students.
The Initial Development of the Curriculum
Although the Accelerated Math Curriculum is most often used by advanced students as a more efficient alternative to traditional honors math classes, its development began in much more desperate situations. The initial seed of the Curriculum came from working with several different students who had procrastinated excessively and had found themselves in somewhat desperate situations. At times a few days of work had been neglected before a major test. At other times weeks or months of work had been ignored. Arvin Vohra, the Curriculum's primary developer, repeatedly found himself working with students late at night, faced with the task of teaching several chapters worth of material in an hour or two. At other times, it wasn't the night before, but a class period before the test. Invariably, the student was not the type of math student who naturally understood the material quickly.
Fortunately, the right combination of creativity and desperation would generate some way to prepare the student for the exam. Through repeated trials of this kind, these methods became more numerous and refined. In fact, these methods were largely responsible for the initial success of Arvin Vohra Education.
As this process continued, a question began to emerge. Methods to teach below average students weeks of material in a few hours now existed. What would happen if these methods were used to teach above average students?
The initial results were good, but not great. Above average students generally saw things in a different way, and preferred a deeper level of understanding. Some of the methods which relied on easy ways to learn certain formulas just didn't appeal to the advanced students as well. Other methods, however, had exactly the desired results. Above average students learned the material twice as fast. They could absorb a week's worth of material from an Honors Math Class in an hour. However, there were some problems even here. Some of the rapidly learned material was also rapidly forgotten.
The methods entered a whole new round of modification and refinement. The ordering of the material changed, as did the manner of presentation. Rather than covering each topic in depth before moving to the next, foundations of several related topics were covered before any advanced details were added. For example, students learned basic factoring and its application to graphic, solving equations, and manipulating rational algebraic expressions were covered before advanced factoring methods were discussed. Students learned the fundamentals of limits, derivatives, and integrals before learning how to find the derivative of an inverse function. This gave students a conceptual framework that gave students perspectives with which such abstruse material could be understood and integrated.
After each progressive stage of development, the methods became more effective. The problems became fewer; the methods used to fix these problem areas became some of the Curriculum's greatest strengths.
Soon the methods became organized into a coherent whole. After a few structural adjustments the Arvin Vohra Accelerated Math Curriculum was ready.
Honors Math Classes vs. The Arvin Vohra Accelerated Math Curriculum
The Honors Math classes in place at most schools reflect a deep-seated belief that all students should move at approximately the same rate. Thus, more talented students may cover algebra in greater depth, but they will cover similar topics in a similar time frame.
This belief has lead to several anomalies in many schools. It is not uncommon for a school to have a calculus class that is easier than its Honors Algebra I class. Nor is it uncommon for honors math students to be deluged with information of passing relevance. Much of this information is ignored or forgotten, since strong math students generally learn to extract key principles from their classes. Students in honors math classes generally become well acquainted with the principle of diminishing returns; they often put in ten times the work of students in regular classes for a one percent gain in understanding.
AVE does not hold the belief that all students must progress at the same rate. Rather, AVE believes that a student's education should be as efficient and beneficial as possible.
This mindset guides most of the methods used in the Curriculum. The approaches used to form the foundations of understanding are extremely streamlined. With the Curriculum, students who do ten times as much work gain ten times as much understanding.
Current Methods and Approaches of the Accelerated Math Curriculum
In its current form, the curriculum covers Algebra I through AP Calculus BC. A heavy emphasis is placed on developing cognitive skills and understanding of fundamental principles. Drills build familiarity with fundamentals, while challenging problems refine this understanding and develop cognitive abilities.
Review of fundamentals continues throughout the process. Students are drilled on these fundamentals directly, and are required to use them to solve more complicated problems.
Most importantly, the curriculum first develops an understanding of the entire subject matter based on a few key concepts, and then adds details. This contrasts with traditional methods, which introduce fundamentals and details of each topic before moving on to the next. For example, the Curriculum introduces basic factoring, its applications graphing, and its application to manipulation of rational expression before introducing such esoteric factoring patterns as those of sums of cubes. This method gives students a foundation to build on, and a context in which to appreciate the details.
Additionally, the order of presentation more closely approximates innate learning processes than does a more traditional order. For example, derivatives are introduced before formal proofs of limits. Because students understand the applications and significance of limits before studying formal proofs, they are better able to remember and integrate knowledge of the proofs.
Frequently Asked Questions
Does the Curriculum affect all of AVE's math tutoring?
To some extent, the Curriculum affects all of AVE's math tutoring. However, when tutoring is used to help a student with a specific school class, the effect is less pronounced than when math is studied independently of a school class.
If students move too quickly, is there a danger that they will not retain fundamental information?
This was an issue during the earliest stages of development of the Curriculum. Later versions took this into account, and incorporated continual review of fundamentals. The current version places a heavy stress on fundamentals and reviews them repeatedly, ensuring that the student will retain and integrate them appropriately.
Is the curriculum appropriate for home-schooled students?
Although it is often combined with school education, the Curriculum is designed to stand on its own. Thus, it is 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?
The Curriculum may approach a topic differently than the student's school. If the school's method is effective, the Curriculum may be adjusted to incorporate the school's method. However, if the Curriculum's method is significantly more beneficial to the student, the student will probably learn the same material from more than one perspective. In most cases, an additional perspective is beneficial when approaching a math problem.
Is the Curriculum the same for every student?
Methods of presentation are generally tailored to the student, and order of presentation may be varied. Content varies less, though it may vary in unusual situations.
Additional Information for Education Professionals
The 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.