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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
High School Mathematics in Context Where is it going
Mathematics in context The study of mathematics year after year may seem endless and pointless for many students and teachers. The key elements in high school mathematics prepare for calculus. Calculus is the subject which is used to justify calculations and formulas in mathematics and other subjects. It requires an efficient command of arithmetic without a calculator. It also requires in full logic and the algebraic way of writing and reasoning. Most elements of geometry with and without coordinates, and trigonometry too with triangles and unit circles are required in calculus. The connection of decimals, ratios, rates, percentages to each other and fractions is part of arithmetic. Mastery of Statistics (means, averages, pie charges) exercises skills in arithmetic and knowledge with fractions. While calculators are useful, the ability to work efficiently with fractions with no calculator present is required for algebra or the shorthand roles of letters and symbols in describing calculations and showing how to obtain them. probability provides an opportunity to review and reinforce fraction sense and efficient operation with fractions - must for algebra. Most other elements of high school can be presented in ways that support and strengthen the key elements. Dilatations, if taught provides practice with algebra and arithmetic operations, and may be used to imply similarity of diagrams in radial expansions, contractions and inversions about a point or origin. Spatial sense or three dimensional geometry in secondary can be linked to crystal shapes, to technical drawing or drafting and to arithmetic and algebra via volume and surface formulas, and via Euler relation between the vertices, faces and edges of polygons in plane. Statistics in the lower high school curriculum provides exercises with arithmetic, the proper and improper gathering and interpretation of data and associated graphs. Recognizing what is improper is a base for critical thinking when numbers and graphs are presented. The prerequisite and context for statistical critical thinking is a proper and efficient command of arithmetic without a calculator and familiarity with computations with methods or formulas lead to repeatable and reproducible results. The latter leads the notion that numbers do not lie, statistics might. But there are some statistical methods for which may be present only because they are required for course final examination - not an inspiring reason for mastering them. Mastery of high school mathematics with its emphasis on arithmetic without a calculator (we hope) and the algebraic way of writing and reasoning provides a key to studies in the physical sciences, physic and chemistry included. The difference between the advance and ordinary student of the physical science comes in part from mathematical skills. The advance course requires a mastery of fractions, what they are and operations with them, exact solution methods for linear equations in one to two unknowns, and mastery of the quadratic formula. Anything less implies difficulty for the student. Engaging and Inspiring Teachers The foregoing may provide a context for high school mathematics – lines of reason that tie the various elements together. The fact that five sixths of high school students in secondary IV do not have the arithmetic skills or fraction sense named a prerequisite for algebra in objectives for the first year of high school points to a difficulty. The above context engages me and therefore provides a framework for engaging students in the curriculum, so that it becomes a more meaningful sequence of skills and concepts.
Engaging Students One way to engage student could be to point out the appearance of fractions and calculations in their daily life to give them more participation and ownership of the lesson.More generally to improve study skills and work habits would be to start this is a guided discussion with students of how one learns or masters a subject, discipline or collection of skills and knowledge in general. The discussion may span lessons and separate times. A consensus on what they expect from the teacher and how to favour those expectations by cooperating in their own education is a goal. For example, cooking, swimming, body-building (like it or not), nordic skiing, bicycling, drawing, writing, figuring (do arithmetic without a calculator) are examples which students could discuss to provide them insights into the learning and teaching process. Here the question for students to considers is how they would teach or communicate a skill or concept, an isolated one, or a sequence. The question of how to teach another an organized set of skills or an organized body of notion may lead to students to the model of skill and idea verification and development, one skill or idea, one skill after another. That expectation gives a base for learning mathematics and support for the statement of goals and standards to guide and focus their work. The site before you was developed to support direct instruction, the case where the teacher explicitly controls the direction of a class and what is in it. That may lead to a teacher or subject centered classroom. The student-centered classroom calls for more students participation, more realistic and more authentic examples and projects to engage the student's interest and cooperation in his or her learning. The instructor in the latter case has to provide or limit the examples and projects to the broad or narrow subject or discipline or them under construction. The notion that a student can rediscover key skills and concepts is false. Those key skills and concepts are the product of many investigations and of thought on how to arrange the findings into coherent islands and bodies of knowledge. None the less, some combination of teacher led and student led activities which allow the student to beyond the passive reception of ideas and concepts is called for. How is another question which ideologues for pedagogical methods expect the instructor to construct. Direct instruction requires less preparation than instructional methods which call for student participation in that direct instruction only requires content mastery in a discipline and lines of reason and skills to present and verify. I would recommend teacher training colleges in the first instance focus on content mastery to set a base for the more complicated or more involved student centered approach. Carrot and Stick The workplace usually has a firm organization based on the meeting the requirements of the employer by coercion, do this or else be fired - lose your income. the school classroom in many English speaking lands do not have a firm organization. Nominally, instruction is for the sake of the student. However, education is often justified by the need for for better educated population for the sake of a competitive or suffering economy. For students, compulsory education for 10 years or so may be followed by the uncertainty of the workplace: will the student be employed or not? will education help? Education would be more authentic or appealing if students saw favorable consequences, rewards or reasons for studying. Education for its own sake may not be appealing to a student and the student's family without explicit authentic and realistic advantages Not enough hope dims enthusiasm for education. The young student goes to school with enthusiasm, a curiosity and a will to grow-up (so I imagine or recall). But belief in education is somewhat like belief in Santa-Claus. It may be transient. In the end some students or most serve time in high school, suspecting it may help in a general way, but not being exactly sure how. Authentic consequences to education would help prolong student enthusiasm and patience with being educated. More notes - Personal As a mathematics instructor, I enter a classroom where most students who would take another mathematics course but for mathematics being required for a diploma or for future studies. I have to motivate them and provide a context for the task ahead, namely skill and concept verification and development reaching for the goals and standards. The first section above mathematics in context gives a context for my instruction. The call for inclusion of authentic, realistic, cross-discipline examples, activities and themes cannot be criticized save for the demand that I the teacher to find and invent ways to comply. The callers do not provide a manual. At the same time, mastery of my subject calls for students to see definitions and explanations, and do exercises to confirm and reinforce those definitions and explanations and to build general thinking or general problem solving skills in lines of reasoning that define the discipline. Just as the body builder does exercises with no immediate consequences, so must the mind-builder. I am looking for ways to engage students to give them the goals and standards, and hence will to follow a sequence of exercises, listening to definitions and explanations included, whose net result in a repeatable and reproducible fashion for the patient at least is mastery of mathematics and its logic. The next time I teach I hope to maintain a written record via checklist or rubrics of the abilities of each of my students, so that in dealing with each, I know at glance where the weaknesses and strengths lie. Individualized or differentiated instruction may follow, time permitting, or as needed. |
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