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. Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. Early High School Algebra
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? Early High School GeometryMaps + Plans Use  Measurement use maps, plans and diagrams drawn to scale.  Coordinates  Use them not only for locating points but also for rotating and translating in the plane.  What is Similarity  another view of using maps, plans and diagrams drawn to scale in the plane and space. Many humanmade objects are similar by design.  7 Complex Numbers Appetizer. What is or where is the square root of 1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of 1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.  Geometric Notions with Ruler & Compass Constructions : 1 Initial Concepts & Terms 2 Angle, Vertex & Side Correspondence in Triangles 3 Triangle Isometry/Congruence 4 Side Side Side Method 5 Side Angle Side Method 6 Angle Bisection 7 Angle Side Angle Method 8 Isoceles Triangles 9 Line Segment Bisection 10 From point to line, Drop Perpendicular 11 How Side Side Side Fails 12 How Side Angle Side Fails 13 How Angle Side Angle Fails 
www.whyslopes.com >> Algebra Starter Lessons >> B Real Numbers Extrinsic Development >> 25 Midway 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 thoughtbased 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 knowhow in mathematics may be met and mastered with comprehension or thoughtbased development, but for students or the common person in the streets, the takehome 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 thoughtbased 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 takehome 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 takehome 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 takehome value, by rote if needbe. The second level, what is left, is dedicated to a thoughtbased development that does not begin with the modern mathematics midway axioms for secondary mathematics, but implies them. See site slow paths, computational and geometric, for the thoughtbased 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 takehome 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 patternbased development of skills and concepts as possible preparation for college studies in mathematical fields. The thoughtbased 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 diagramfree 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 twolevel manner, with one level focusing on empirical rigour in skill mastery and the second level offer a thoughtbased development consistent first the need to sanction and extend common skills and knowhow, with numbers, maps and plans. Modern Mathematics Curricula,In the modern mathematics curriculum, circa 19551990, 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 diagramfree, 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 takehome 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 takehome value. The modern mathematics curricula was designed to prepare students for college programs that required calculus or beyond, with contextfree 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 Midway 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? Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based ReasonOnline Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge. 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; patternbased 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
crossproducts, 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 howtos 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. ...
Senior High School Geometry

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions. Calculus Starter Lessons
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. 