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  Home Page Welcome. The leading elements of Online books Three Skills For Algebra, and Why Slopes and More Math. stem from early college lessons that amused and informed recent high school students, science, engineering and education students, and adults in remedial to advanced mathematics lessons. Three Skills For Algebra begins with logic to test and sharpen reading and writing, continues with a high school level arithmetic review exercises to find gaps in solution writing skills, and continues again with algebra review chapters to strengthen algebraic abilities. The chapters and exercises were written to fill gaps in their senior high schoool command of mathematics The leading elements of Why Slopes and More Math gives a light technical context for the senior high school study of slopes, of factored polynomials and of endpoint and interior maxima and minima of functions through sign analysis of formulas for slopes and derivatives of functions, formulas given in factored form. Thus leading elements of online book Why Slopes and More Math begins with the algebraically easy part of a first course in calculus, the part that comes after algebraically harder decussion of limits, continuity and convergence. The leading elements of both books together show how different starting points make calculus and senior highs chool mathematics easier. Top Level LinksLessons and Lesson Ideas. Site Introduction For Teachers and Tutors. Site Introduction For Teachers and Tutors. Site Reviews. The Math Forum and Site Content. Road Safety Questions. SecondaryMathematicsPrerequisites. § Mathematics Skill Development Framework: § Secondary Mathematics  A Practical Approach: § Quebec Material: § Work and Study Tips: § Volume 1 Elements of Reason: §  Volume 1A Pattern Based Reason: §  Volume 1B Mathematics Curriculum Notes: § Volume 2 Three Skills For Algebra: § Volume 3 Why Slopes  A Calculus Intro Etc: § Advanced Calculus  Volume 3 Appendices: § 70 Calculus Starter Lessons: § Skills with take home value: § Arithmetic and Number Theory Skills: § Algebra Starter Lessons: § Geometry  maps plans trigonometry vectors: § More Algebra: § Mathematics Skills Year by Year: § Parent Center: § Direct Current Electric Circuits: § Francais: § Archives: Folder Content: 7 pages and 21 subfolders: . Page Contents: Oral Development  words have been missing  Visual Introductions of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act  A Stronger Base for Modern Mathematics  Which Way to Go, Why and How
"Would you tell me, please, which way I ought to go from
here?"
Site lesson and lesson ideas in implying step by step a stronger
and more effective path for direct instruction may mute or lessen the
claims in education physchology or constructivism 1990 onward that direct
instruction does not provide an effective way to teach mathematics  the
fears and troubles of too many or most being evidence of that.
Ends and Values for LearningThe question why learn or study a subject or a topic appears is a sign of intelligence. Some students have parents who say mathematics mastery is important. But many have parents who in recalling their experience express a dislike for mathematics after primary school. But if we combine ends and values from earlier times, we may arrive at overlapping sets of ends and values for learning and teaching primary and high school mathematics. The first ends reflect the actual or potential needs of adult or daily life, and in trades and activities that do not require common studies. The third end reflect the needs of calculusbased college programs and of advanced, senior high school science courses. The first two ends are more immediate than the third end. For the first two ends, if not the third, overpreparation is better than underpreparation to students and their families earn their livelihoods and to rationally defend their interests in a world where daily behaviour, and contracts involving money matters or income have huge consequences for individuals and their families. For mathematics and logic instruction, preparing children and teenagers to earn income as adults may meet employers needs but more importantly, it may meet the needs of students and their families for income from employment or selfemployment, and defending their own interests while changing jobs or being fired. High school, trade school, undergraduate university programs and graduate university programs may open doors for worthwhile employment, but depending on economic times, education too long may also distract from gainful or worthwhile work. There are no simple answers. Where does education for the sake of students begin and where does it end? It may begin by showing students early how to handle money matters in daily and adult life from not going into debt while buying or selling to evaluating the immediate or longterm value of a mortgage, a pension plan or the income stream and benefits of a job with or without benefits may help them face or avoid common situations and difficulties.
Students heading for calculusbased, college programs in business, if they avoid demanding high school science courses, will not see senior high school mathematics used before arriving in college. To compensate, longterm value needs to be emphasized. Emphasize how the calculations and logic of college level programs requiring calculus will be more difficult to follow or use and bend to our future requirements with a weak mastery of mathematics. Site volumes 2 and 3 in forming and reforming the views of students and teachers in senior high school mathematics as indicated above may inform and amuse, and in the process provide some context and motivation for the study of slopes, factored polynomials, function maxima and minima, and calculus too. Oral Development  Words have been missingPage Contents: Site Welcome  Links to Top Level Content  Which Way to Go, How and Why  Practical Ends and Values  the why  Oral Development  words have been missing  Visual Introductions of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act  A Stronger Base for Modern Mathematics  All the ideas described briefly below are explained in more detail in site algebra starter lessons and in site Volume 2, Three Skills and Algebra. The arithmetic related ideas could have been placed with site arithmetic lessons instead. Arithmetic and algebraic expressions are often to complicated to read aloud, term term by term. Diagrams too are better seen than "read aloud". Outside of mathematics, a picture is worth a thousand words. In mathematics, a symbol, an expression or a diagram better seen and grasped in silence may also be worth a hundred to a thousand words. There has been a great silence in arithmetic, algebra, geometry and calculus because mathematical ideas and methods are often better written and drawn in silence instead of being expressed and explained aloud. Yet we may deliberately use more words to introduce skills and concepts clearer, to talking unifying themes, and to improve communication in circumstances where writing or drawing is not possible. While demonstration how appears in site material, we will identify where the greater use of words is possible. There is more to mathematics than be given a formula, and numbers to use in it. But remember, pictures and diagrams too can be employed alone and besides words to make skills and concepts easier to learn and teach. Before and besides the role of letters and symbols in algebra, we may use words and numerical examples to talk about about and show how to calculate totals and products by adding and multipling subtotals and subproducts. We may also talk informally but precisely about counts and measures as being known or not, constant or not, forgotten or not, and variable or not. Many technical terms may be introduced and understood before and besides the letters and symbols. Moreover, to gossip or talk about people, places and activities, we need names, labels and phrases to identify them. In mathematics, names and descriptive phrases such as the compound growth formula, the rectangular area calculation, the distributive law and the Chinese Square Proof of the Pythagorean Theorem allow us to gossip and talk about calculations and further ideas in situations where symbols and diagrams cannot be formed nor read. Most formulas, methods and practices in mathematics and logic are named. For people wanting and able to talk about what they learning with others, learning the names becomes an asset and not a burden. In describing how to calculate averages and how to compute the perimeter of a polygon, word descriptions of how may be simpler or not to understand and explain than formulas. As a first example, the average of a set or sequence of numbers is given by their total divided divided by the number (count) of set or sequence elements. As a second example, the perimeter of a polygon is given by the sum of the lengths of it sides, or more briefly by the instruction: add the sides. As a third example, the total area of a region consisting of nonoverlapping subregions is given by a sum of subareas. In early mathematics instruction, how to compute this or that may be easier to understand and explain with words with the use of letters or symbols being more complicated. But for the compound interest or growth formula, for the quadratic formulas and later for the chain rule  do not worry what computations these phrases name or identify, the the letters and symbols in them are worth a thousand words. The greater use of words advocated for earlier instruction here is not possible in later instruction. So the silence will return. Using rules and formulas forwards and backward, and talking about it may end a further silence. Talking and writing about the forward and backward use of rules and formulas provides a unifying verbal theme for the study of logic, mathematics and science in school and college studies. Most if not all rules and formulas are not only used directly in a forward sense but also indirectly or backwards. Determing the constant in a proportionality relation uses the relationship, an equation, backwards. Once it is found, the proportionality relations may then be used or rewritten forwards and backwards to compute or express the value of one number or quantity in terms of others. The example here may not be familar to you if you have not seen them, but by talking about the forward and backward use of rules, formulas and proportionality relations, the backward use will be expected and not be another surprise for students weak and strong of mathematics, logic and science. This forwards and backwards use is common pattern previously met and mastered case by case in silence. Talking and writing about it introduces or extends the oral dimension of skill and concept development. Visual Development of Algebra and CalculusPage Contents: Site Welcome  Links to Top Level Content  Which Way to Go, How and Why  Practical Ends and Values  the why  Oral Development  words have been missing  Visual Development of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act  A Stronger Base for Modern Mathematics  Site algebra starter lessons and the online chapters of Volume 2, Three Skills for Algebra, material, show how to learn and teach skills and concepts with words, forwards and backwards. Algebra starter lessons include a geometric, stick diagram introduction for solving linear equations in a way that visually proves or improves fraction skills and sense. Here fractional operations on stick diagrams are suppose to make the algebraic solution of linear equations easier to grasp. However, in entertaining a group of students during a one hour, substitute teaching assignment, one keen student could not make the transition from solving with stick diagrams to solving algebraically. It was not my place to give him extra instruction. He may have been better served by more stick diagram examples, or by a leap to the algebraic method. I cannot say. Geometry too can help with the introduction of calculus and in providing motivation or context for the study of slopes (remember the domain name is whyslopes.com) and the study of factored polynomials alone and in ratios (rational expressions). See site Volume 3, Why Slopes and More Mathematics, online in full with a fall 1983 why slopes prequel. Volume 3 in a preview of calculus provide geometric motivation for the study of slopes and factored polynomial to the location of maxima and minima of functions. The site introduction of complex numbers is geometric instead of algebraic. It follows or reinvents a path in a 1951 book on Secondary Mathematics (possibilities) by Howard Fuhr, a mathematician who masqueraded as a mathematics education professor at Columbia University and who as part of the NCTM leadership in the 1960s help develop and implement the collegeoriented Modern Mathematics Programs for skill and concept development in primary and high school mathematics from counting to calculus. The level of rigour in this geometric introduction of complex numbers is not less than that in the geometric introduction of trigonometry using triangles and/or the unit circles drawn in a Cartesian plane. The big steps in modern mathematics programs were too hard for many to follow. Site material offer smaller steps to compensate. Before modern mathematics programs, instruction had a greater focus on skills and concepts with value for daily and adult life  work, mortgages and investments included. The discussion of ends and values above suggests preparation for daily and adult life as much as possible first and foremost, and on preparation for college second while emphasing anything in the latter that could have takehome value. More Concept and Skill Development NotesSite composition was driven by a search to remedy the skill and concept development difficulties stemming from steps too big and steps without clear value for students and their families in earlier programs in mathematics and logic education  programs which aimed for student mastery of selected skills and concepts. In consequence, site lessons and lesson ideas include many expositional innovations to aid skill and concept development. Most, if not all, are mathematically correct, with a few small departures from earlier views to make instruction simpler. In calculus and secondary mathematics, late primary mathematics too, there are many different starting points for instruction. For example, the site development of prime numbers begins with a definition that is not the most general but with a definition that is likely the easiest for students to understand and apply. For a second example, the site essay on what is a variable, by talking or writing about numbers and quantities varying in one sense or another, we provide a prequel to the later, more formal and more algebraically advanced view of what is a variable, a prequel that is easily understood because it is wordy and prealgebraic. For more examples, see the site geometric development of complex numbers before trig, and see light calculus preview in Volume 3, Why Slopes and More Math, and see, still in Volume 3, the decimal prequel to the epsilondelta view or definitions of limits and continuity. The choice of starting point need not reflect the conventions of higher mathematics, conventions that may be arbitrary despited being widely accepted. Instead, the choice of starting point may reflect the objective of making skill and concept development simpler for students and their teachers. The harder starting points may be left to advanced studies involving fewer students and teachers.
Addressing Students and Family Alienation Page Contents: Site Welcome  Links to Top Level Content  Which Way to Go, How and Why  Practical Ends and Values  the why  Oral Development  words have been missing  Visual Introductions of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act  A Stronger Base for Modern Mathematics  More on Ends and ValuesMathematics after primary school has been difficult and without immediate value for one generation after another. While some students have parents who did well or who encourage skill and concept development, other students have parents who may say mathematics after arithmetic is a waste of time. High school and college students may attend courses because those courses are required. In high school and college, students who base their efforts only on whether or not their teachers are pleasing have a shallow context and motivation for learning. Students for whom doing well in tests and finals is the only motivation also have shallow reason for learning. Cultural values for learning may appeal to some. But practical ends and values may appeal to more. In primary school, students and their families may see the 4Rs (reading, writing, reasoning and arithmetic) as being useful in adult and daily life. Therein lies content and motivation. But at the junior and midhigh school level, some mathematics and logic lessons are of actual or potential service to daily and adult life for decisionmaking and moneymatters at home and the work place. Other lessons only have longterm value for college programs that some students may never enter or complete. Instruction may lean to the first kind of lessons initially to provide ends and values easily understood and appreciated by students and their families. Emphasizing the more useful methods and concepts first may help retain student motivation and also help those who have leave school early. But eventually, high school and college mathematics has less and less takehome value besides more and more value for future studies or courses that students may not see. Here again, instruction may focus on the takehome value, when present to provide motivation. At the precalculus level, instruction should focus on two kinds of skills and concepts, those that have actual or potential takehome value for daily & adult life, and for precollege trades and activities; and those that prepare students for a light and then deeper command of calculus. In the former, I would include a setbased development of probability theory. In both streams, I might include matrix operations but not linear programming. The latter can be left to college programs in commerce, science, engineering and technology. I would restrict high school mathematics to computations and proofs that are lead to repeatable and reproducible results, and to the computation of averages useful in small business for estimating demand for products and services being sold. Further elements of descriptive statistics, I would leave to college studies, or to high school courses on critical thinking. The recommended focus may mean fewer topics are taught. For students not heading for calculusbased studies, less with a focus of skills and concepts with takehome value may be best. In the preparation of students for calculus and senior high school mathematics, multiple topics with no shortterm value may be met. That shortterm value will vary between students. Students in courses required to prepare for calculus who do take mathematically demanding, senior high school courses will see more shortterm value. In general, calculus and preparation for calculus is a long demanding path which many find difficult or hard to complete. But, here is a plug, site Volumes 2 and 3, make the path easier and throught calculus preview make calculus and precalculus easier and more appealing. Course Design Balancing ActStill More on Ends and Values Page Contents: Site Welcome  Links to Top Level Content  Which Way to Go, How and Why  Practical Ends and Values  the why  Oral Development  words have been missing  Visual Introductions of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act  A Stronger Base for Modern Mathematics  To serve the skill and concept needs of the common person in the street, we need to put first those skills and concepts with actual or potential value for daily and adult life. Then students may attend school and go home with methods that help themselves or their families in money and other matters. Near the end of school coverage of arithmetic, geometric and logic (or reading and writing) skills and concepts with actual or potential service for daily and adult life, more algebra and higher level geometry skills may be introduced to revisit and reinforce the foregoing service while being of service to more trades and activities at the precalculus level, and also being of service or preparation for senior high school science courses and perhaps later studies in calculus. The multiple ends and values in the foregoing need to be balanced. The balance may depend on the local or immediate needs of students and their families, that is, how long students are likely to remain in school; on whether or not, they are likely to see all all ends and values served; and on whether or not, the students are quick or slow learners. The concept and skill development standards and principles for instruction in resultsoriented arts and disciplines, as espoused in site material, seek to provide students with an observable and verifiable knowhow of the ideas and methods currently forming and characterizing each art or discipline. The latter presents a moving targets as best practices in each may vary over time and place. But in a moving target, concept and skill mastery may be seen or empirically measured by student response to questions. In each such art and discipline, students are expected to retain knowhow and build on it in a progressive manner, with regression being a sign of weakness, or absence too long from practice in an art or discipline. Each art or discipline comes with different cultural and practical values, some more important than others in ways that may justify its instruction or not in each school or school system. Morover, course design and delivery needs to acknowledge that there are multiple intelligences in learning and teaching styles. A style that is suitable for instruction in the humanities where conclusions are highly subjective is not suitable for instruction in mathematics and science where the benefits, origins and limitations of ideas and methods should be shared. A Stronger Base for Modern MathematicsPage Contents: Site Welcome  Links to Top Level Content  Which Way to Go, How and Why  Practical Ends and Values  the why  Oral Development  words have been missing  Visual Introductions of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act A Stronger Base for Modern Mathematics  In modern mathematics programs for secondary mathematics education, direct instruction aimed at student mastery of given concepts and skills has been uncertain and unreliable due to steps too big or hard for most to follow, and due a collegeoriented choice of concepts and skills with value too longterm for students and their families. Those steps too big undermined course design and delivery. However, direct instruction can address its own problems by serving short and longterm ends and values in the selection and arrangement of course topics, and in offering smaller, more accessible and reliable steps for concept and skill mastery. The key question is whether or not remedies based on the smaller and alternative steps in site lessons and lesson ideas, alone or with the proposed ends and values above, will be effective.. Site lessons and lessons ideas from counting to calculus provide a foundation for college level studies of modern mathematics. Site lessons and lesson ideas offer student and their teachers a mastery of concepts and skills with comprehension, based on a redundant set of practices and axioms, whose redundancy can be explained and removed in college course in or leading to modern mathematics. The ends and value further offer reasons for mathematics and logic mastery that students and their families are more likely to appreciate before preparation for calculus becomes the main focus of instruction at the senior high school level. For calculus, Chapter 14 of site Volume 3, Why Slopes and More Mathematics, offers a decimal, error control development of limit and continuity concepts that may stand alone, or be used to make the epsilondelta development much easier to understand and explain. Site departures in early instruction from modern mathematics are intended to provide TCPITS an more accessible view, but they are also intended to develop the logical and algebraic maturity needed for college and senior high school students to study modern mathematics if they choose or where it appears in their programs of study. Indirect Instruction Benefits and LimitsIndirect instruction in mathematics has the advantage of enriching skills and concept mastery in classes where there is time for individual and group creativity, and where teachers not all trained in mathematics are shown how to provide problems and circumstances which studentsmay investigate to discover or build their own ccmprehension. But with or where teachers are not fully versed in mathematics or course content, it appears far simpler to provide instructors with lessons easily understood and repeated in class, which avoid the affects o steps too large, not for all, but for most to follow. For example, the verbal introduction of algebra in Volume 2, Three Skills For Algebra, and the first six or seven chapters of Volume 3, Why Slopes and More Mathematics, provide lessons and lesson idess, easily understood and repeated in class, and in the process may introduce learning difficulties of students with instructors at many levels. Here advocates of indirect instruction while declaring student mastery of given skills and concepts in direct instruction to be a substandard objective for mathematics education essentially kept the course design and content which direct instruction employed in its identification of skills and concepts for student mastery. Moreover, educational authorities at the precollege level retain final examinations for the yearly end of most high school mathematics courses. Final examination by their very nature test student mastery of chosen skill and concepts. Due to the continued presence of mathematics final examinations which tests student mastery of given skills and concept, fairness in student evaluation requires all the given skills and concepts be clearly explain, illustrated and checked first in class. So direct instruction is still required. For fairness, student mastery of given skills and concepts requires both teacher and student awareness of how later skills and concepts stand on earlier ones. Otherwise, students will be promoted with the necessary background to succeed. When and where direct instruction has clear steps or lessons to provide student mastery of important skills and concepts, teachers and course designers may provide circumstances and pose questions to indirectly lead student to formulate ideas and skills and gain the experience on which direct instruction may stand. But where direct instruction lacks those clear steps and lessons, it is doubtful that indirect instruction will provide a practical and clearer path to to student mastery of the given skills and concepts. The ability to explain matters directly is likely a prerequisites to the ability to provide skill and concept mastery indirectly. Each program of instruction aim at mastery of given ideas and methods has varying degrees of success and failure, and of motivation and alienation for students and their families. In the case of modern mathematics programs for secondary mathematics and calculus, the step by step development was clear to some and due to the presence of steps too big, not for but for some, confusing for others. Site material in providing smaller steps allows steps too big to be recognized and gives remedies  full or not  to be tried and tested. Smaller steps should allow more to go further. Page Contents: Which Way to Go, How and Why  Practical Ends and Values  the why  Oral Development  words have been missing  Visual Introductions of Algebra and Calculus  More Concept and Skill Development Notes  Addressing Students and Family Alienation  Course Design Balancing Act  A Stronger Base for Modern Mathematics  Appendix  Closing WordsStill More on Ends and ValuesIn Canada and the USA five decades ago, that is, in the 1960s, modern mathematics programs set forth the substance or content of high school mathematics and calculus in a way that served the technical needs of college bound students, those heading for university programs in commerce, science and technology. Mathematical discussion leading to those modern mathematics programs acknowledged the collegeorientation and mentioned that the needs of student not college bound were not being addressed. However, for the sake of inclusivity, modern mathematics programs were adopted for all instead of the just college bound. The tradition of mathematics course design incomprehensible to parents  different from what they had seen in school or college may have begun then. From my perspective as a student and then as an instructor, the modern and premodern mathematics course design in algebra at least included steps too large not for all, but for most to follow. In primary and secondary schools, the collegeoriented selection of content in modern mathematics programs helped move course design and delivery away from serving the actual or potential needs of daily and adult life, and in doing so offered less context and motivation for mastery of given ideas and methods.
For senior high school mathematics, preparation for college programs of the technical or mathematical kind provides one context and motivation for collegebound students. Senior high school mathematics is of clear service to students headed for science or engineering take senior high school courses in physics, chemistry and biology. But students headed for college commerce programs  those best studied with the help of calculus  may or should be encourage to take senior high school courses in science. Otherwise, physicsbsaed examples in senior high school mathematics will have less value for them. To provide context and motivation for more students and their families, primary school and the leading years of high school may deliberately develop mathematical and logical skills, concepts and work habits with clear or identifiable actual or potential benefit in daily or adult life. In particular, Primary school and the leading years of secondary school may focus on basic skills in counting, figuring and measuring of actual or potential service to daily and adult life. Students and their families may value this focus. The leading years of secondary school may further introduce and emphasize skill, concepts and work habits of service in senior high school and college mathematics. In this standards for mastery of given skills and concepts may be maintained and supported by lesson and lesson ideas deliberately chosen and continually reviewed to make the hard less so, without loss of substance; and guided by the setting of final examinations set not by amateurs but by subject experts. Here multilevel course design and delivery requires knowledge of and respect of how later ideas and methods depend on earlier ones. In this, while high school mathmatics as a whole may prepare student for calculusbased college studies, the inclusion of skills and concepts of service, actual or potential, to daily and adult life; money handling included, or to precollege trades and professions could provide and sweeten the context of secondary mathematics for all. And in the preparation for calculus, set formalism may appear to set the stage for the later study perhaps of pure mathematics in college, but in a minimal manner, one that does not overwhelm. www.whyslopes.com  Home Page 
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. 