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 >>  Volume 1A Pattern Based Reason >> Chapter 2 Skill Development Next: [Chapter 3 What is in chapters 4 to 8.] Previous: [Chapter 1 Introduction.] [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] [30] Chapter 2. Communication of IdeasThis chapter describes the inductive educational principles or pedagogy which guided the organization and content of this book on reason and its mathematical companions. The principles are echoed in Volume 1B, Mathematics Curriculum Notes. They reflect the pedagogical principles of the NCTM in 195089, but not those of the same organization 19902010. Masters of mathematical induction in knowing how the latter may fail will easily recognize how a like failure may hit step by step skill development. Initial AssumptionNo area of knowledge is properly mastered until it can be readily explained to others. Each subject needs paths (or curricula) passing through easily described and easily repeated ideas and skills. Each such path permits those who have traveled along it to tell others what to expect and hopefully why. The existence of such paths may show that an area is wellunderstood. DifferencesDifferences in our alertness, how awake we are, imply that a lecture, a book, a picture, or a film is seen and understood differently by each of us, depending on the time of day, etc. All of us are witnesses. Some witnesses see more than others. In describing and explaining ideas to several people, we need to speak in a way that each listener (or witness) will understand as much as possible. When we speak to several people, our words will be understood by each differently. In communication, especially in teaching, these differences need to be considered. Principles for InstructionFor learning and teaching in all disciplines, the following overlapping principles appear selfevident, at least once stated.
Remark 1. An alternative to ladderlike structure is a treelike structure. Here skills and ideas are represented by the branches of a tree or bush. The tree can be climbed when those branches closest to the ground are accessible while the higher ones are accessible from branches below. For ease of exposition and comprehension, the organization of ideas into a flat treelike structures where each branch can be reached via a few lower branches or directly from the ground is to be preferred to the case in which most branches are high and can be reached only from the one just below it. This is to simply observe that short chains of reasoning are better for explanation and comprehension than longer ones. Remark 2. These words or thoughts on communication of skills echo a course on how to be a crosscountry ski instructor. The course was taught one weekend early in 1981, by an instructortrainer from CANSKI, the CANadian association for Nordic, that is crosscountry, SKIing. The course gave a piece by piece approach to instruction. The objective was to build both the confidence and ability of students. The course emphasized that difficulty with a skill signaled the need for a retreat to, or even before, previously mastered skills. The detailed structure provided turned crosscountry ski instruction into an art. Arts of this kind are required in other areas of instruction. Plots and SubplotsA writer normally offers a main plot along with a few or several subplots. The main plot and ideas should be obvious to everyone the first time. After the main plot is noticed, the subplots themselves and links between them may become apparent. On reading for the first time a book or an article, we grasp and master some of its ideas. The rest remains to be found. Others ideas and messages just pass us by. A second or third reading may help us see them. In the ClassroomA teacher has to explain ideas to students with different backgrounds and knowledge. One approach to this is to divide students into groups, and to speak to each group separately. A second approach is to speak to all in a way that speaks to each group at its own level without being too imprecise. A teacher may try to explain and broadcast ideas or knowledge at several levels at once. The intent here is to allow each listener to tune to the level most suited to him or her with reinforcing echoes from the other levels. Echoes can be provided by the repetition of words and phrases with similar, like or related meanings. They can be used one after another in a single sentence. Familiarity with one word or phrase leads to an understanding of the others. The latter in turn favors variety instead of monotony in speech. Multiple themes and multiple levels of meanings may challenge the listener or student and take some beyond what was expected of them. Some redundancy and repetition in communication is fine. Too much may be boring. www.whyslopes.com >>  Volume 1A Pattern Based Reason >> Chapter 2 Skill Development Next: [Chapter 3 What is in chapters 4 to 8.] Previous: [Chapter 1 Introduction.] [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] [30] 
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. 