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Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices. Site's Best LessonsFor Logic
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. For Arithmetic
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style. For Algebra
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct? |
www.whyslopes.com >> Algebra Starter Lessons >> B Real Numbers Extrinsic Development >> 25 Mid-way Convergence to Axiomatic Approach Next: [26 More Less Greater Than Comparison.] Previous: [24 Signed Numbers - Arithmmetic Properties.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25][26] [27] [28] [29] 25. Convergence to Axiomatic ViewEnds and Values - a matter of choiceThe ends and values of mathematics education, or logical and quantitative skill development may vary between students. Teachers who have been suddenly assigned a mathematics courses, despite a lack of background in mathematics or a quantitive discipline, may not value the logical development of mathematics. For many students and teachers, mathematics appears to be collection of facts and methods to learn and teach without any attempt to obtain or provide a thought-based development. One grade 8 student on observing my attempts to explain and justify mathematical methods instead of teaching them as facts told me that mathematics teachers were hired only to present mathematical correct methods, and thus I was doing my job properly, the methods I was covering would not need explanation nor justification. For many students, the notion that mathematics has a logical structure, one in which ideas are developed and derived in place of being given, is odd and not necessary. Indeed, they may be partially correct. The common know-how in mathematics may be met and mastered with comprehension or thought-based development, but for students or the common person in the streets, the take-home value of an operational command of counting, figuring and measuring skills with take home value in a repeatable and reproducible manner is more important than full or partial comprehension of why methods work in the first years of mathematics education and for students who may or may not go further in mathematics.
Site pages show with two or more paths how the existence of real numbers and their arithmetic properties can be derived from common practices assumptions about numbers and geometry with maps and plans. The demonstrations appear to be empirically and pedagogical sound, given the need to introduce skills, patterns and even axioms in an inductive manner. Site pages also provide a systematics introduction to algebra, or the shorthand role of letters and symbols. Nostalgia or attraction to the rigour of modern mathematics means the demonstration were written in a thought-based manner with as much rigour as possible. But there is a difficulty. Too much explanation may overwhelm skill mastery. Moreover, mastery of skills with care to avoid the domino effect of errors has great take-home value in which full or partial comprehension of why is optional. The foregoing suggests ends and values for instruction that support a rigourous development of skills, step by step, because of the take-home value with explanations why being available and present where they do not overwhelm. Site pages are part of a two level approach POMME. The first level and part of the second are dedicated to providing skills and concepts with take-home value, by rote if need-be. The second level, what is left, is dedicated to a thought-based development that does not begin with the modern mathematics mid-way axioms for secondary mathematics, but implies them. See site slow paths, computational and geometric, for the thought-based development of numbers and their properties from counting to the properties of real and complex numbers. The paths may not be given in classes where students have mixed ends and values - some wanting mathematics with take-home value only - some wanting to continue onto college programs in disciplines requiring or best taught with a command of calculus. The second level in full, as presented here, values thought- or pattern-based development of skills and concepts as possible preparation for college studies in mathematical fields. The thought-based development of numbers and their properties from counting to the properties of real and complex number, with the subsequent assumption of those properties as axioms for the further logical development of mathematics implies a partial convergence of the site two level approach POMME for quantitative and logical skill development with modern mathematics curricula. The modern mathematics curricula I saw began well at the start of senior high school mathematics, but soon departed from pure mathematics with the employment of a diagrams in the introduction of trigonometry, analytic geometry, and calculus to develop methods and prove theorems. The diagram-free development, a possibility in university mathematics, would be too difficult and have no context in senior high school mathematics. Whence some departure is needed - in for a penny, in for pound. The site development provides a departure in a two-level manner, with one level focusing on empirical rigour in skill mastery and the second level offer a thought-based development consistent first the need to sanction and extend common skills and know-how, with numbers, maps and plans. Modern Mathematics Curricula,In the modern mathematics curriculum, circa 1955-1990, the existence of the real numbers and the satisfaction of above properties were given as assumptions or axioms. That provides a simple starting point for a logical development of secondary and college mathematics. A justification of the axioms might then be seen by students who enter mathematics studies in university. In particular, assumptions for set existence and "safe" set construction provide an axiomatic codification, Euclidean style, for pure mathematics. For rigour, the approach sould be context- and diagram-free, a rigour not possible before university level studies in pure mathematics. As said, the modern mathematics curricula depart with the employment of diagrams in the introduction of trigonometry, analytic geometry, and calculus to develop methods and prove theorems. For all students, and many teachers, axioms for real numbers and within them, rational numbers, integers, natural numbers and whole numbers, the axioms will appear and will have to be accepted without explanation. But the axioms were not chosen to continue and sanction common knowledge and practices with decimals and diagrams which would have had take-home value. The axioms for real numbers provided a view of numbers that did not explicitly sanction and support common skills in counting, figuring and measuring with maps, plans and decimals. The modern mathematics curricula was not designed to meet the needs of students who would have benefited from mathematics with take-home value. The modern mathematics curricula was designed to prepare students for college programs that required calculus or beyond, with context-free development being an objective. Axioms and further development of mathematics did not sanction earlier number skills and sense with fractions and decimals. For many students and many of their teachers, the modern mathematics curricula was further flawed in that the secondary level axioms were described algebraically with out a systematics introduction of the shorthand role of letters and symbols. Whence the deductive axiomatic development of mathematics was beyond the reach of students and teachers for whom the algebraic way of reasoning with letters and symbols on paper was not a natural talent. The slower and more detailed systematic development of algebraic reasoning in site pages points to a remedy, one that requires less natural talent. www.whyslopes.com >> Algebra Starter Lessons >> B Real Numbers Extrinsic Development >> 25 Mid-way Convergence to Axiomatic Approach Next: [26 More Less Greater Than Comparison.] Previous: [24 Signed Numbers - Arithmmetic Properties.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25][26] [27] [28] [29] |
Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic? Site Reviews1996 - Magellan, the McKinley Internet Directory: Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it. 2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
For Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale. For Calculus
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. |