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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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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.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

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?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + 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 human-made 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

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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 Ideas

This 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 1950-89, but not those of the same organization 1990-2010. 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 Assumption

No 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 well-understood.


Differences 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 Instruction

For learning and teaching in all disciplines, the following overlapping principles appear self-evident, at least once stated.

  • Each discipline needs to be taught or presented, so that students understand what they are learning and why. Without a knowledge or an opinion of why, students may lose interest and not go further. The why could be approximate a little uncertainty leaves room for thought.

  • Pathways through easily described and repeated ideas may extend knowledge of any discipline, area of thought or belief. One or more paths through easily described and easily repeated topics may allow those who travel further to tell others willing to listen, what to expect and possibly why. Of course, differences of opinion exist on which disciplines should be taught or what pathways in them should be followed.

  • Awkwardness with an idea or skill often signals difficulty with previous ones. It may indicate at least one earlier skill has been missed or forgotten. When an awkwardness is felt or seen, learners should go or be taken back to practice the missing skills, and possibly the ones just before them. This retreat aims to restore confidence and build skills, so that the learner can go further. This requires a diagnostic skill and a knowledge of or opinion on how the topics in question can be organized and taught. Here again, opinions may differ.

  • Each collection of mental and physical skills could be organized into a ladder-like sequences of steps with the basic ones first and the more advanced ones second. Learning in any subject stumbles when a first or succeeding step is not easily reachable from those before them. To climb a ladder, the initial steps must be reachable, and each further step must be reachable from the one or ones before it.

    Explanations should follow chains of reason or persuasion which begin at the level of the student before advancing further.

Remark 1. An alternative to ladder-like structure is a tree-like 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 tree-like 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 cross-country ski instructor. The course was taught one weekend early in 1981, by an instructor-trainer from CANSKI, the CANadian association for Nordic, that is cross-country, 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 cross-country ski instruction into an art. Arts of this kind are required in other areas of instruction.

Plots and Subplots

A 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 Classroom

A 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]

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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 head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based 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 story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - 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. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

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.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.

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