www.whyslopes.com || Fit Browser Window

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.

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.
- 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

Return to Page Top


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript A Story-Telling Next: [Postscript B More-on-Story-Telling-and-Reason.] Previous: [Chapter 24 Direct-and-Indirect-Reason.]   [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]

Appendix A. Story Telling
Key Talent

The ability to tell, invent and follow stories is transformed in education and research into the ability to tell, invent and follow explanations and instructions as a means to enrich, advance and provide a context for observable skill development, and a means to share what is the mind. In essence telling stories, one step at a time, and one step after another (in order perhaps) is the key to knowledge of folklore, culture and technical knowledge. We may judge stories as fictional or not, to greater or lessor extent. We judge the consistency of stories in accordance with our knowledge or view of what is possible or allowable. So a good alibi allows a detective to ease or remove suspicions about who committed a action. In writing or inventing a story, an author will attempt to avoid immediate or implied inconsistencies. What Louis Carroll's Alice sees in Wonderland or Through the Looking Glass provides an story to follow, one that is not realistic. Shakespeare and Moliere through their writing create pieces of fiction for us to follow on paper or even to act out. We may judge the characters and plots in stories and plays by our own ideas on what is not possible. But in telling stories, the tellers provide us with a imaginary world to follow and judge. Likewise in telling theories or providing instructions for the operation of a machine or the creation of a product, researchers and teachers are telling us stories, ones that may correspond to and be useful in practice. The composers of a story may adhere to rules and criteria for consistency, rules and criteria that say what is or should be possible or not, in order to decide or conclude what may be in the story or not. Chains of reason may be employed to find what a story implies or requires. The composer may include or not, those implications or requirements in the telling and extension of a story or theory.

The developers of empirical theories in science and technology tell many small technical stories which individually work and are practice in special cases, but not necessarily in all cases. High school and college chemistry for example include alternative theories for the identification of acids, salts and bases. The oxymoron acid-salt reflects the idea that what is an salt in one theory is an acid according to another (a litmus paper test say).

reality, imagination and fiction - exploring ideas, continued

The author of a story in a book or a play creates an imaginary world for us to visit in our minds. The story may or not be consistent with our knowledge of real life. More and less can be suggested in a story than occurs in real-life. Stories can be fictional, half-fictional, approximately true to life or true.

Stories may explain or describe how things came to be. Stories may provide lessons a for reader directly or through the words and interpretations of another. Stories may give us ideas of what to do or not. Stories have plots and chains of events or reasons to follow, real or not. Stories presented on stage as plays may include not only words but also actors and props to make the plot or reenactment easier to follow. Actors have scripts to follow. Actors are defined by their names, costumes and actions.

Most of us, many of us, have the ability to follow a story, its sequence of scenes with words and events, and to recognize what is real or pretend. We can learn stories, invent them and tell them to others via spoken and written words. Stories can be told and retold in ways that are almost repeatable and reproducible. Our knowledge of a culture may come from its stories and myths.

In cooking and construction, plans and recipes give or suggest sequences of steps or actions to take to arrive at results. The steps and the results should be repeatable and reproducible. Technical know-how is based on rules and patterns to follow plus some judgment as to when they can be applied. Trying to apply rules and patterns when items they require are missing usually leads to bad results.

In mathematics, science and technology, as in daily life, there are stories to follow. These stories, normally called theories, describe a situation (say what is what is assumed) and describe as a well assume methods for arriving at results or conclusions in a step by step way. The authors of these stories or theories would like their consistency with reality. A theory is inconsistent with reality if it says two exclusive events occur at the same time or if predictions based on it fail. Unfortunately, the author of theory to say what should happen may capture a pattern in theory that works in some circumstances, but not all. So a theory may be applicable and sufficiently consistent with reality to be useful in some circumstances - those it reflects - while failing in others.

Knowledge in mathematics and science and technology is based on theory and practice. A method or procedure describe in a lab or controlled circumstances how following certain steps will give a result. Those steps and the results, done carefully enough, appear to give repeatable and reproducible independent of the doer. Methods that work in practice may be described and accumulated, and used one at a time and one after another to follow steps and arrive at results, one at a time and one after another. A skilled practitioner may recognize when one method can replace another because it gives the same result or a more convenient result.

Geometry was codified in the works of Euclid, about 300 B. C. The codification consisted of assumptions or definitions about points, straight lines, circles, triangles and the geometric figures composed from the latter. The resulting theory or theories was presented not on stage, but on paper (a prop) with the aid of rulers and compass (more props) to provide construction methods and to suggest and describe results and conclusions one at a time and one after another. Students and teachers and philosophers could follow explanations one at a time and one after another in way that follow some or all of the strands of thought in Euclid's work. The codification provides a mechanical knowledge of geometry because each of us in following the steps should verify that the steps are valid, that the implication rules used in each step are justly applied.

The foregoing gives rule- and pattern-based chain of reasons independent of the followers and authors. All that provides a model for making and arriving at conclusions with rules and patterns not only in geometry, but also in other disciplines where rules and patterns are valued as guides. But this model for reason has its limitations.

Rules and patterns describing what we have observed, drawn from experience, are not absolute. We do not know if they are fully reliable, or we may not precisely when they apply, if at all. When rules and patterns are not reliable, a risk appears. What they suggest, one at a time and one after another, may not be consistent with reality. None the less, recognizing rules and patterns in a subject provides a means for accumulating know-how for arriving at results, and a limited know-why. The latter is given by the chain of reason or suggestion with rules and patterns, reliable or not, that led to a result. (Implication rules and suggestions in a theory may themselves rely on the need for a theory to be consistent. See above).


Selby A, Volume 1A, Pattern Based Reason, 1996.


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript A Story-Telling Next: [Postscript B More-on-Story-Telling-and-Reason.] Previous: [Chapter 24 Direct-and-Indirect-Reason.]   [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]

Return to Page Top

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.


Return to Page Top