¶ … movie Stand and Deliver (Menendez & Musca, 1988), which is based on the true story of Jamie Escalante, an individual who overcame ethnic, cultural, and socioeconomic issues to become a highly successful mathematics teacher. Discuss the beliefs he held and the strategies he employed in his classroom that contributed to high achievement levels in his students.
The final report of the National Mathematics Advisory Panel (2008) presents a three-pronged argument for an effective math curricula: 1) It must foster the successful mathematical performance of students in algebra and beyond; 2) it must be taught by experienced teachers of mathematics who instructional strategies that are research-based; and, 3) the instruction of the math curriculum must accomplish the "mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts" (National Mathematics Advisory Panel, 2008, p. xiv). Jamie Escalante began teaching before this report was released, but he knew from experience -- and instinct -- that students who do not achieve mastery of foundational concepts of mathematics will face unforgiving -- perhaps harsh -- consequences in their lives (Won, 2010).
Frontrunner Excellence. A report published in 2006 by the National Council of Teachers of Mathematics made recommendations for math curriculum that are currently being implemented in a majority of U.S. states as the Common Core State Standards. One recommendation has been particularly influential and is pivotal to curriculum, instruction, and assessment aligned with the Common Core: Math curriculum and instruction should cover fewer topics at greater depth. Best practices in mathematics instruction have established the need to ensure students have sufficient time to learn concepts deeply so that they can build on the learning in subsequent grades and so that redundancy does not need to be built into the grade level curriculum and instruction. Rather, math curricula are designed to ensure continuity across the grade-level instruction, with each deep instruction provided at each grade. This successful approach has been adopted by a majority of nations where students are top-performers in mathematics.
An important outcome of these comprehensive reports on mathematics instruction in the U.S. is the recommendation that the issue of identifying which of two primary methods of math instruction is superior be laid to rest. For decades, there has been an ongoing debate about math instruction that is teacher-directed vs. student-centered. The fundamental difference between the two approaches is based on how much freedom students have to explore mathematical concepts (Barley, et al., 2002). In the traditional instructional approach to teaching mathematics, students are shown how to solve problems, complete multiple drill exercises, and take tests that don't vary from the formulaic approach to problem solving. The reports conclude that both approaches have merit, and that, according to the chairman of the panel, Dr. Larry Faulkner, "There is no basis in research for favoring teacher-based or student-centered instruction. People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other."
Escalante understood the relation of motivation to success -- for his students and for the teachers in the math department (Marzell, 2012). He was clearly an outlier with respect to his peers, and was prone to making statements at departmental meetings such as, "Students will rise to the level of expectations you are holding them." Comments like this did not endear Escalante to the other teachers in the school, nor did it help when he precisely cut to the core of the situation. When asked by the school principal what the school needs to be successful and accomplish their accreditation audit, Escalante said plainly that, "All we need is ganas" which is Spanish for desire and drive.
Research-based Instruction. Several meta-analyses of instructional strategies have led researchers to put forth four approaches that support solid performance in mathematics (Swanson, 2009; Wietzel, et al., 2003). The promising approaches include: 1) Systematic and explicit instruction; 2) Self-instruction; 3) Peer tutoring; and 4) Visual representation. Of these, Jaime Escalante appeared to favor and use all by the peer tutoring, which makes sense considering the lack of opportunity the students in his classes would have had to share in this manner. Escalante's students lived in barrios and studied in schools that created disadvantage rather than support.
Observing the mathematics instruction given by the character Jamie Escalante in the film Stand and Deliver, it is apparent that he employed a number of the strategies that have become best practice (Steedly, et al., 2012). Escalante provided systematic and explicit instruction through his step-by-step approach that took his students methodically through a specific instructional sequence. Escalante used rote repetition to help students achieve mastery of basic mathematical rules....
To wit: "A negative times a negative equals a positive." Escalante explicitly and systematically taught his students to apply certain strategies that would help them master advanced mathematical concepts.
Escalante's high standards and continual pressure on the students to perform at their highest levels provided a foundation from which the students managed their own learning when doing homework and studying during summer and after-school class sessions. The power of effective visual representation was not lost on Escalante, and he instinctively understood the value of using this strategy to teach mathematical concepts to his students (Berkas, et al., 2007). Escalante used manipulatives and visualization to help students grasp mathematical concepts. Recall Escalante's lesson the concept of zero in which he encouraged the students to picture what happens when they dig a hole in the sand and fill it up again.
Motivation and Method Acting. Escalante peppered his lessons with encouragement and confidence building "chants." Some of his favorite and frequently voiced words of encouragement were:
"You can do it." "You're the best. You guys are the best."
"It's going to be a piece of cake -- upside down. Step-by-step."
"Math is a great equalizer."
"You are the true dreamers, and dreams accomplish wonderful things."
Escalante was not all sweetness and light; he was an unofficial proponent of tough love (Jesness, 2002). He didn't accept excuses from his students, using a sort of reverse psychology that seemed to cause the students to want to attend his class in response to Escalante's tenor of exclusivity (Jesness, 2002). Escalante held a key to the students' future success, and the students knew it. To the student in his first calculus class who believed he would never understand the math, Escalante provided the real-life demonstration the student had been clamoring for: Driving the student in the student's own car, Escalante made a bad show of driving when following the student's directions. Escalante summed the experience up for the student: "All you see is the turn. You don't see the road ahead."
When introducing the concept of zero to the students, he informed them, "Neither the Greeks nor the Romans were capable of using the concept of zero. It was your ancestors, the Mayan, who first contemplated the zero -- the absence of value. True story. You burros have math in your blood." Escalante's parents were both teachers and descendants of the Aymara in Bolivia. Escalante would occasionally boast that, "The Aymara knew math before the Greeks and Egyptians" (Schraff, 2008). He wanted his Hispanic students to be proud of their mathematical heritage, and understandably thought that this legacy would motivate the students to put more effort toward learning advanced math ("Hero," 2004).
Before he immigrated to the U.S., Escalante had been a professor of physics and mathematics for over a decade in Bolivia. As is typical with many immigrants, it was not possible for Escalante to step directly into his teaching role when he arrived in the U.S., so he basically had to jump through a seemingly endless number of hoops before he could once again use his remarkable talent as a teacher. He worked odd jobs -- including washing dishes in a restaurant -- earned a degree at an American college, and taught himself English. So Escalante knew well the value of an education and he had an affinity with the young students from East Los Angeles barrios. He pushed them nearly as hard as he pushed himself because he could well see the difference it would make in their lives. And Escalante "walked the talk" as few other teachers have; because of this, he experienced the rate busting within a peer group that often accompanies superior accomplishments. Escalante was said to have frequently asserted that both students and teachers had to work hard to reach high standards. Unfortunately, other educators did not welcome the legacy that Escalante created, choosing instead to focus on being adequate -- in effect, sentencing their students to a lifetime of sub-ordinary opportunity.
____. (2004, April 13). "Hero'" Teacher Escalante Addresses Students At Wittenberg Commencement May 9. Wittenberg University. Retrieved http://www4.wittenberg.edu/news/1998/commspeaker.shtml
____. (2008). National Mathematics Advisory Panel, Foundations for Success. The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education. Washington, D.C. Retrieved http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
Barley, Z., Lauer, P.A., Arens, S.A., Apthorp, H.S., Englert, K.S., Snow, D., & Akiba, M. (2002). Helping at-risk students…
Being able to "crunch the numbers" is an essential part of the manager's role. Too often managers feel uncomfortable working with numbers because of their limited mathematical background. This reduces their usefulness, however. Strong managers are not intimidated by the numbers, but rather view them as an essential component of the job. Therefore, part of the process of studying business management is to build the set of tools that
(Hilton, 26) in general, no mathematician would be willing to accept the solution to a problem without some sort of proof, and in the same way, no student of calculus would be ready to accept the resolution of a problem without the necessary proof. (Cadena; Travis; Norman, 77) It must be stated that Newton's mathematics that involved 'fluxions' was one of the first forms of the area defined as 'differential
Calculus and Definitions of Its Concepts Indefinite integration Indefinite integration is the act of reversing any process of differentiation. It is the process of obtaining a function from its derivative. It is also called anti-derivative of f. A function F. is an anti-derivative of f on an interval I, if F'(x) = f (x) for all x in I. A function of F (x) for which F'(x)=f (x), this means that for
Nevertheless, an individual may prefer to have this type of calculus removed for other reasons or otherwise as part of a long-term treatment regimen. For example, Bennett and Mccrochan note that, "When the American Dental Association later approved Warner-Lambert's mouthwash, Listerine, by stating that 'Listerine Antiseptic has been shown to help prevent and reduce supragingival plaque accumulation and gingivitis. . ., ' sales rose significantly" (1993:398). It remains unclear,
Mamikon even takes this simple observation about curves to establish a new relationship between the tractrix and exponential curves (Apostol & Mamikon 2002). Mamikon's visual understanding and explanation of calculus is not limited to two-diemnsional curves, nor does he concern himself only with new insights into mathematical relationships. In another paper, again published with Apostol, Mamikon established new proofs for Archimedes' discoveries concerning polyhedrons and their circumscribing prisms (Apostol &
The semi-minor and semi-major axis are easily determined, and can then be subbed into the standard equation for an ellipse. Taking the square root of y will result in a plus/minus, and discarding the minus erases the lower half of the ellipse. The long axis extends horizontally, and the short axis extends vertically. The x and y axis bisects the ellipse already, so both a and B. are available: