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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 1B Mathematics Curriculum Notes >> Foreword Next: [Chapter 1 Introduction.]   [1][2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]


Volume 1B, Mathematics Curriculum Notes

Four principles offer an inductive philosophy for the explanation and comprehension of math and reasoning skills. Three of the principles were met in a course on how to teach Nordic, that is cross-country skiing. The course was taught one weekend early in 1981, by an instructor-trainer from CANSKI, the CANadian association for Nordic SKIing in Flin Flon, Manitoba. Nordic ski instruction may begin with a lesson on how to put on the boots and attach them to the ski and also how to hold the ski poles – to be precise one holds not the poles, but their straps in way that will guide the poles.


understanding and explaining
reason and math
Volume 1

Alan M. Selby
Ph. D.

Printed in Canada
ISBN 0-9697564-6-1

There is a technique here, one that is not obvious. The course gave minute attention to the details which novice and even experienced skiers might not know. In this course on ski instruction, the more complicated movements or skills were deliberately preceded by simpler motions. Each of which was easy to describe, master and/or review separately. This course turned Nordic ski instruction into an art. The four principles follow.

  1. Each discipline needs to be 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.

  2. 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 again possibly why. Of course, differences of opinion exist on which disciplines should be taught or what pathways in them should be followed.

  3. 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, more precisely 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, a knowledge of or opinion on how the topics in question can be organized and taught. Here again opinions may differ.

  4. Each collection of mental and physical skills should 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. [1] To climb a ladder, the initial steps must be reachable, and each further step must be reachable from the one or ones before it, else failure occurs. Explanations should follow chains of reasons or persuasion which begin at the level of the student.

In mathematics education there are two barriers to comprehension to be lowered or removed. First, the algebraic or symbolic way of writing and thinking is better seen and read silently than read aloud or spoken. This has been an obstacle to the comprehension and communication of mathematical thought. Second, the deductive nature of formal mathematics exposition with its long chains of reason and preparation implies that concepts appearing at the end of a course are not comprehensible to students in the middle of the course nor at its beginning. Mathematics beyond the last concept mastered may seem impenetrable and mysterious.

To lower both barriers, students may be given lessons, easily described and repeated, which require a minimal formal comprehension of mathematics and logic while presenting ideas essential to deductive and to algebraic or symbolic thought. Recognizing, collecting and offering first such lessons may extend the common knowledge of mathematics beyond the mastery of arithmetic, counting and simple formulas that should be obtained in elementary school. This work identifies such lessons and indicates ideas for math and logic instruction from primary school to the start of college. Some of the ideas may be worth reading, repeating or refining, the three Rs that this author hopes for.

Alan Selby
Montreal 1996

Selby A, Volume 1B, Mathematics Curriculum Notes, 1996.

Postscript - February 2011

In two years of UK grammar school 1965-7, mathematics lessons consisted of given rules and patterns in algebra, trig and geometry, all given in what I presume was a mathematically correct manner. I was too young to know otherwise. In then next three years of English Quebec secondary schooling, rules and patterns were also given but in axiomatic structure. That is, rule and pattern mastery started from axioms - assumed patterns algebraically or geometrically put. The fine print in my Quebec high school textbooks emphasized or valued starting from a minimal set of axioms. The development was essentially logical. But there were four flaws in the Quebec portion of my secondary school and junior college education - nuances small and large.

  1. The arithmetic mastery of decimals and fractions met in my UK primary and secondary school days was required, but not explicitly sanctioned. That departed from the ideal in textbook fine print of building mathematical knowledge on explicitly given axioms

  2. The algebraic way of writing and reason was required to understand the shorthand role of letters and symbols in the axioms and in the further development of algebra, geometry and science in my UK and Quebec studies. But no course and the fine print in all of my textbooks did not provide any sanction for this shorthand role. While I found my own rationalization, self-constructed, I saw the instruction of myself and fellow students slowed by the silent assumption use of algebraic skill. Course design and delivery assumed it without clearly or explicitly discussing it. Talking about three skills for algebra, a lesson given in fall 1983, represented my second effort to address this flaw. The first was in a 1975 handout at a McGill University open house.

  3. The use of order pairs as coordinates in the plane ias in the analytic approach to geometry was mixed with synthetic Euclidean Geometry, the line, circle and triangle drawing approach. Having two approaches unreconciled departed from the fine-print promise in algebra of a minimal set of axioms.

  4. The use of drawings and diagrams, disowned in the algebraic course view of mathematics, was present in both high school level trigonometry and in later college level calculus. And geometric drawings were employed along side mathematically and algebraically deep utilization of epsilons and delta views of continuity and convergence, with the underlying theory avoiding decimals, while examples and illustrations employed or required decimals.

My strength and weakness as a student and a human being was and may still be a reflex to take everything literally. So I was disappointed with my high school and college education because the algebraic way of writing and reasoning was not introduced in a clear step by step manner. In all the courses I took and in all the textbooks I saw, this shorthand way of writing and reasoning was required while the effort to explain it was absent or, when present, insufficient. I was also disappointed by the espousal of an ideal, the consistent and full logical development of mathematics from axioms - assumed patterns. Course design and delivery failed to deliver. In retrospect, following graduate studies and doctoral degree in mathematics, and further thought, the ideals espoused represented the hopes and motivation of modern mathematics. But mathematicians if not mathematics educators since Godel in the 1930s were aware that the hopes were not feasible. None the less, the modern mathematics curricula echoed those hopes and emphasized a rigour in ways that many tried to take literally.

In retrospect, mathematics is an empirical subject. Its development is a mix of practice and theory. Given that, I first recommend K3-9 observable skills and practices with take home values be learnt and taught in manner that emphasizes their value, with explanations to aid mastery without overwhelming it, with the end of making students aware of the domino effect of errors in calculations and reasoning, and with the end of showing students how to do and record steps in manner they and others can do or check. With skills that have take home value, students may expect instructors to teach correct methods - methods that can be learnt by rote if the student wishes, methods for which explanations why are available for reading by students when or if they want.

For skills that have high take home value for life in the street or work, rigourous mastery is more important than comprehension. But in the preparation of students for college programs, those that may employ one variable calculus, or more. skill development needs to show students how to use and combine rules and patterns in applications, and in the development of further rules and patterns. The ability to apply and the ability to reproduce in all or part what has been shown is in itself an observable skill. Skills that can be seen can be described, confirmed or corrected. But the thought-based development of skills and concepts does not require a minimal set of axioms. Minimality here represent a value of higher mathematics education. The development requires a convenient and consistent set. This set may provides the opportunity for students to see how rules and patterns may be applied to obtain results and combined to obtain further rules and patterns. The set develops and sanctions decimal, algebraic, geometric and logic skills and practices, so that the flaws indicated above are avoided while the stage is set for further studies in college programs. That represents the current or last objective of site material.

The initial objective when writing began was not so large. Writing began with the inductive criteria for course design and delivery, with the question of how to motivate skill development, and with the hope of making the modern mathematics curricula of the years 1965-90 more accessible - less challenging to learn and teach. The initial aim was to report the inductive principles and a few appetizers and starter lessons to educational authorities, and then leave further work in this matter to others. However, I was not formally qualified to present ideas to educational authorities, or academic committees, more ideas followed, More over, in 1989, before I started writing, education had shifted from valuing skill development in reading, writing and arithmetic, mathematics included, to saying true knowledge is a personal affair, located in the mind, apart from observation and correction of teachers, and not associated with perfomance, that is, observable rule and pattern mastery. That subjective movement in education, led in US and Canadian mathematic education by the NCTM rtoday is the reverse of that espoused by the NCTM in the 1950s. Site material provides a rational alternative.

A K1-9 emphasize of skills and practices with current or potential take home value for work or life in the street represent student-centered skill development. The K7-12 parallel or subsequent emphasis on skills for one-variable calculus and college programs in technical fields represents college oriented skill development for careers - not guaranteed - that may benefit the student or society. Such instruction has intellectual and/or take-home value for some, not all. That is not ideal. But recognizing this situation represent a step forward from the situation in which secondary mathematics instruction is clouded in mystery, with the question why learn or teach this has the bureaucratic answer: preparation for final examinations. Moreover, this college oriented instruction can be offered, does not have to be taken nor required, once most skills with take-home value have been covered. The latter needs to be done first. It will be useful to all, including those students aiming for college studies who might other wise miss it. As a high school teacher, I once had to give a mathematics course required for graduation to a group of students who have benefited from a review and consolidation of skills with take-home value. Instead, their time was wasted because of government standards for education that forced the learning and teaching of topics with no academic nor take home value for the students in question. That situation needs to be addressed.


www.whyslopes.com >> - Volume 1B Mathematics Curriculum Notes >> Foreword Next: [Chapter 1 Introduction.]   [1][2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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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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