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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.

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.
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- 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

www.whyslopes.com >> Algebra Starter Lessons >> B Real Numbers Extrinsic Development >> 25 Mid-way 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 View

Ends and Values - a matter of choice

The 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 thought-based 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 know-how in mathematics may be met and mastered with comprehension or thought-based development, but for students or the common person in the streets, the take-home 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.

The site approach and Choice

Logic chapters 1 to 5 in Three Skills for Algebra end with a discussion of Islands and Divisions of Knowledge. The latter provides a metaphor for the organization of mathematics and the possibility of having different starting points for its development.

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 thought-based 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 take-home 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 take-home 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 take-home value, by rote if need-be. The second level, what is left, is dedicated to a thought-based development that does not begin with the modern mathematics mid-way axioms for secondary mathematics, but implies them. See site slow paths, computational and geometric, for the thought-based 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 take-home 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 pattern-based development of skills and concepts as possible preparation for college studies in mathematical fields.

The thought-based 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 diagram-free 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 two-level manner, with one level focusing on empirical rigour in skill mastery and the second level offer a thought-based development consistent first the need to sanction and extend common skills and know-how, with numbers, maps and plans.

Modern Mathematics Curricula,

In the modern mathematics curriculum, circa 1955-1990, 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 diagram-free, 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 take-home 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 take-home value. The modern mathematics curricula was designed to prepare students for college programs that required calculus or beyond, with context-free 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 Mid-way 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 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.
... 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.