Math for smart teachers

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Introducing OrthoMathetics to the scientific community like a new branch of education. OrthoMathetics defined as the science of correct learning, whose primary purpose is to prevent academic fraud and dishonesty, preventing students from copying answers on science problems and exercises. Emerges as a response to academic dishonesty or academic misconduct in any type of cheating that occurs in the schools everywhere.

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  • Emerges as a response to

    academic dishonesty or

    academic misconduct in any type

    of cheating that occurs in the

    schools everywhere.

    BY

  • Neftal Antnez H. Ph. D. Civil Engineer with a Doctorate Degree in Education

    Full time Teacher in the Faculty of Engineering of the UAGro and in CBTis No. 134

    Chilpancingo, Gro., Mxico

    Introducing OrthoMathetics to the scientific community, a new branch of education whose name is derived from Ortho (from the Greek word

    meaning "straight" or "correct") and Mathetics means the science of learning.

    ISBN-13:978-1481859462

    ISBN-10:1481859463

    BY

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    OrthoMathetics Neftali Antunez H.

    The teachers to teach less and the learner to learn more Comenius

    ISBN-13:978-1481859462

    ISBN-10:1481859463

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    OrthoMathetics For Teachers

    Stopping the Cheating in Science!

    NEFTALI ANTUNEZ H.

    Civil Engineer with a Masters Degree in Education

    Full time Teacher in the Faculty of Engineering of the

    Guerrero State Autonomous University (UAG)

    Chilpancingo, Gro., Mxico

    Introducing OrthoMathetics to the scientific community like a new branch of education.

    OrthoMathetics defined as the science of correct learning, whose primary purpose is to prevent academic fraud and dishonesty, preventing students from copying answers

    on science problems and exercises.

    Emerges as a response to academic dishonesty or academic misconduct in any type of cheating that occurs in the schools everywhere.

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    This page is intentionally left blank

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    INDEX

    Chapter 1

    Definition

    1.1 Roots of OrthoMathetics 5 1.10 Characteristics of the constructivist student

    16 1.11 Constructivist evaluation 17 1.12 Difference between exercise and problem

    18 1.2 Mathetics in literature 5 1.3 John Amos Comenius 6 1.3.1 Educational influence 6 1.4 Purposes of OrthoMathetics 8 1.5 History of academic dishonesty 8 1.6 Academic dishonesty today 9 1.7 Cheating 10 1.7.1 The Issue 11 1.8 Solutions? 13 1.9 Characteristics of the constructivist teacher

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    Chapter 2 OrthoMathetics Fundamentals 2.1 How to do problems with features of

    OrthoMathetics 21

    Chapter 3

    Applications to Algebra

    3.1 Operations with polynomials 59 3.2 Systems of Equations Linear with Two

    and Three Variables 87 3.3 Matrices and determinants 108 3.4 Roots of Equations 128

    Chapter 4

    Applications to Geometry and Trigonometry

    4.1 Arc Length 137 4.2 Right-angled Triangles 139 4.3 Non Right-angled Triangles 150

    Chapter 5

    Applications to Analytic Geometry

    5.1 Circle 161 5.2 Parabola 162

    Chapter 6

    Applications to Calculus

    6.1 Derivatives 167 6.2 Maxima and minima 168 6.3 Definite Integrals 184

    Chapter 7

    Applications to Physics

    7.1 Vectors 201 7.2 Coulombs Law and Electric Field 209 7.3 Conservation of mechanical energy

    215 7.4 Kirchhoffs Rules 219

    CHAPTERS

    Chapter 1 Definition 5 Chapter 2 OrthoMathetics Fundamentals

    20 Chapter 3 Applications to Algebra 59 Chapter 4 Applications to Geometry and

    Trigonometry 137 Chapter 5 Applications to Analytic

    Geometry 161 Chapter 6 Applications to Calculus 167 Chapter 7 Applications to Physics 201

    PARTS

    Part I Introduction 4 Part II OrthoMathetics Applications 58

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    Part I

    Introduction

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    Chapter 1

    Definition

    1.1 Roots of OrthoMathetics

    Introducing OrthoMathetics to the scientific community like a new branch of education, whose name is Derived from Ortho (from the Greek word meaning "straight" or "correct") and Mathetics means the science of learning. The term Mathetics was coined by John Amos Comenius (15921670) in his work Spicilegium didacticum, published in 1680. He understood Mathetics as the opposite of Didactic, the science of teaching. Mathetics considers and uses findings of current interest from pedagogical psychology, neurophysiology and information technology.

    OrthoMathetics defined as the science of correct learning, whose primary purpose is to prevent academic fraud and dishonesty, preventing students from copying answers on science problems and exercises. Promoting individual real learning, where each of the students have to appropriate the knowledge, because it can not be copied because each student has an unique answer to exercise or problem. The main interest is that this project will contribute to the improvement of scientific education as a new approach to the traditional teaching.

    1.2 Mathetics in literature1

    Seymour Papert, MIT mathematician, educator, and author, explains the rationale behind the term mathetics in Chapter 5 (A Word for Learning) of his book, The Childrens Machine. The origin of the word, according to Papert, is not from "mathematics," but from the Greek, mathmatikos, which means "disposed to learn." He feels this word (or one like it) should become as much part of the vocabulary about education as is the word pedagogy or instructional design. In Chapter 6 of The Childrens Machine, Papert mentions six case studies, and all six have their own accompanying learning moral and they all continue his discussion of his views of mathetics. Case study 2 looks at people who use mathematics to change and alter their recipes while cooking. His emphasis here is the use of mathematical knowledge without formal instruction, which he considers to be the central mathetics moral of the study. Papert states "The central epistemological moral is that we all used concrete forms of reasoning. The central mathetics moral is that in doing this we demonstrated we had learned to do something mathematical without instruction and even despite having been taught to proceed differently" (p. 115).[1] Paperts 1980 book, Mindstorms: Children, Computers, and

    1 From Wikipedia, the free encyclopedia

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    Powerful Ideas, discusses the mathetics approach to learning. By using a mathetic approach, Papert feels that independent learning and creative thinking are being encouraged. The mathetic approach is a strong advocate of learning by doing. Many proponents of the mathetic approach feel that the best, and maybe the only, way to learn is by self discovery.

    1.3 John Amos Comenius (15921670)2

    John Amos Comenius (Czech: Jan Amos Komensk; Slovak: Jn Amos Komensk; German: Johann Amos Comenius; Polish: Jan Amos Komeski; Hungarian: Comenius mos Jnos; Latinized: Iohannes Amos Comenius) (28 March 1592 4 November 1670) was a Czech teacher, educator, and writer. He served as the last bishop of Unity of the Brethren, and became a religious refugee and one of the earliest champions of universal education, a concept eventually set forth in his book Didactica Magna. He is considered the father of modern education. He lived and worked in many different countries in Europe, including Sweden, the Polish-Lithuanian Commonwealth,Transylvania, the Holy Roman Empire, England, the Netherlands, and Royal Hungary.

    1.3.1 Educational influence

    The most permanent influence exerted by Comenius was in practical educational work. Few men since his day have had a greater influence, though for the greater part of the eighteenth century and the early part of the nineteenth there was little recognition of his relationship to the current advance in educational thought and practice. The practical educational influence of Comenius was threefold. He was first a teacher and an organizer of schools, not only among his own people, but later in Sweden, and to a slight extent in Holland. In his Didactica Magna (Great Didactic), he outlined a system of schools that is the exact counterpart of the existing American system of kindergarten, elementary school, secondary school, college, and university.

    2 From Wikipedia, the free encyclopedia

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    In the second place, the influence of Comenius was in formulating the general theory of education. In this respect he is the forerunner of Rousseau, Pestalozzi, Froebel, etc., and is the first to formulate that idea of education according to nature so influential during the latter part of the eighteenth and early part of the nineteenth century. The influence of Comenius on educational thought is comparable with that of his contemporaries, Bacon and Descartes, on science and philosophy. In fact, he was largely influenced by the thought

    of these two; and his importance is largely due to the fact that he first applied or attempted to apply in a systematic manner the principles of thought and of investigation, newly formulated by those philosophers, to the organization of education in all its aspects. The summary of this attempt is given in the Didactica Magna, completed about 1631, though not published until several years later.

    The third aspect of his educational influence was that on the subject matter and method of education, exerted through a series of textbooks of an entirely new nature. The first-published of these was the Janua Linguarum Reserata (The Gate of Languages Unlocked), issued in 1631. This was followed later by a more elementary text, the Vestibulum, and a more advanced one, the Atrium, and other texts. In 1657 was published the Orbis Sensualium Pictus probably the most renowned and most widely circulated of school textbooks. It was also the first successful application of illustrations to the work of teaching, though not, as often stated, the first illustrated book for children.

    These texts were all based on the same fundamental ideas: (1) learning foreign languages through the vernacular; (2) obtaining ideas through objects rather than words; (3) starting with objects most familiar to the child to introduce him to both the new language and the more remote world of objects: (4) giving the child a comprehensive knowledge of his environment, physical and social, as well as instruction in religious, moral, and classical subjects; (5) making this acquisition of a compendium of knowledge a pleasure rather than a task; and (6) making instruction universal. While the formulation of many of these ideas is open to criticism from more recent points of view, and while the naturalistic conception of education is one based on crude analogies, the importance of the Comenian

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    influence in education has now been recognized for half a century. The educational writings of Comenius comprise more than forty titles. In 1892 the three-hundredth anniversary of Comenius was very generally celebrated by educators, and at that time the Comenian Society for the study and publication of his works was formed.

    1.4 Purposes of OrthoMathetics

    Emerges as a response to academic dishonesty or academic misconduct in any type of cheating, which occurs in the schools everywhere. Academic dishonesty has been documented in most every type of educational setting, from elementary school to graduate school, and has been met with varying degrees of approbation throughout history. Today, educated society tends to take a very negative view of academic dishonesty. This project is especially directed to the teachers of science in the world. For this reason, dont make an emphasis on theory and full development of the examples, is not because that is not its main objective. We only give examples so that the teacher can develop their own exercises from the examples presented here.

    This is a dynamic book, which will grow because will be adding more examples by the same author, or, by the contribution of thousands of teachers in the world. Can send their examples to my email: antunezsoftware@gmail.com, or better yet, write their own books according to the rules of OrthoMathetics. Some titles of books could be: OrthoMathetics for Algebra, OrthoMathetics for Calculus, OrthoMathetics for Physics, OrthoMathetics for Chemistry, OrthoMathetics for Numerical Methods, etc. All that is asked is that in the preface, prologue or introduction, you specify that OrthoMathetics was created by the author of this book.

    1.5 History of academic dishonesty3

    In antiquity, the notion of intellectual property did not exist. Ideas were the common property of the literate elite. Books were published by hand-copying them. Scholars freely made digests or commentaries on other works, which could contain as much or as little original material as the author desired. There was no standard system of citation, because printingand its resulting fixed paginationwas in the future. Scholars were an elite and a small group, who knew and generally trusted each other. This system continued through the European Middle Ages. Education was in Latin and occasionally Greek. Some scholars were monks, lived in monasteries, and spent much of their time copying manuscripts. Other scholars were in urban universities connected to the Roman Catholic Church.

    3 From Wikipedia, the free encyclopedia

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    Academic dishonesty dates back to the first tests. Scholars note that cheating was prevalent on the Chinese civil service exams thousands of years ago, even when cheating carried the penalty of death for both examinee and examiner. In the late 19th and early 20th centuries, cheating was widespread at college campuses in the United States, and was not considered dishonorable among students. It has been estimated that as many as two-thirds of students cheated at some point of their college careers at the turn of the 20th century. Fraternities often operated so-called essay mills, where term papers were kept on file and could be resubmitted over and over again by different students, often with the only change being the name on the paper.[citation needed] As higher education in the U.S. trended towards meritocracy, however, a greater emphasis was put on anti-cheating policies, and the newly diverse student bodies tended to arrive with a more negative view of academic dishonesty. Unluckily, in some areas academic dishonesty is widely spread and people who do not cheat represent a minority between the class.

    1.6 Academic dishonesty today4

    Academic dishonesty is endemic in all levels of education. In the United States, studies show that 20% of students started cheating in the first grade. Similarly, other studies reveal that currently in the U.S., 56% of middle school students and 70% of high school students have cheated.

    Students are not the only ones to cheat in an academic setting. A study among North Carolina school teachers found that some 35 percent of respondents said they had witnessed their colleagues cheating in one form or another. The rise of high-stakes testing and the consequences of the results on the teacher is cited as a reason why a teacher might want to inflate the results of their students.

    The first scholarly studies in the 1960s of academic dishonesty in higher education found that nationally in the U.S., somewhere between 50%-70% of college students had cheated at least once. While nationally, these rates of cheating in the U.S. remain stable today, there are large disparities between different schools, depending on the size, selectivity, and anti-cheating pol...