Appetizers and Lessons for Mathematics and Reason 
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22. Why Math
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Odds & Ends

Group I

1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24.  The  20 Times Table
25. Algebra Formulas
26. On Learning Maths
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More

Group II 

Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths

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Would you like to show yourself or others how to be  algebra power users

 

Why study or master SOME mathematics

Would you like to become an algebra power user? Professor WhySlopes shouts his methods for algebra skill development are second to none. Students and teachers should try them. They are different. 100% satisfaction not certain.

The aim here is to say why and how to study mathematics without giving too little nor too much motivation. Education, yours or that of others, is not yet a tidy affair. The advice below may appear with some repetition at this website.

Most people learn mathematics until circumstances force to them to stop, or until the subject becomes too hard or until they lose interest. Failure or near failure is one way to halt learning in a subject, and leave a last impression not worth repeating.  Mathematics courses, being compulsory, are not designed to leave a good last impression. Mathematics courses, being compulsory, are designed to cover topics. One by one, the topics need not be important or of immediate use, but altogether or cumulatively, the topics provide or point to a skill, a mastery of mathematics.

Despite these adverse circumstances, reasons for studying mathematics and making it compulsory exist. 

  Two Site Reviews  
  • Magellan, the McKinley Internet Directory, 1996: 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.   (Magellan is no longer online)
  • The World-Wide Web Virtual Library Education by Country - Canada 1, 2005. Why Slopes: Appetizers and Lessons for Math and Reason. This online classroom offers appetizers and lessons for math from arithmetic to calculus or why slopes; for deductive reason (logic) and critical thinking; and for learning in general. Included here are opinions on the communication of skills and mathematics instruction. The logic appetizers are math free. Each appetizer is different. If one is not to your liking try another. Most are from three books on understanding and explaining math and reason.

may encourage a visit to site entrance www.whyslopes.com.  

The ability to read, write and figure well is a required  for many disciplines including mathematics. Imprecision in reading leads you to not understanding and not recognizing errors in the writings of others and yourself.  Students who read, write and figure well are easier to teach and will do better than others in all subjects. To show that you are teachable is one reason to learn mathematics.  But there are others reason.

  • Logic and mathematics starting with arithmetic onwards may show you how to follow steps, one at a time and one after another, for arriving at results or conclusions, one at a time and after another. Learning that an error in one step make all the following steps and results or conclusions wrong or a least suspect (errors could cancel if you are lucky) is a step towards cautious wisdom or intelligence. This wisdom or intelligence applies to all subjects.

  • Mathematics or any other rule and pattern based discipline may show through experience and trial or error, how to solve problems first  by following given methods and later, if needed,  by combining and exploring different methods, by trial and error, opportunistically or with some advance knowledge of what may work.  I call this the jigsaw puzzle approach.

  •  Mastering it in one subject, say mathematics, gives you a wisdom, applicable to all. Understanding or least using patterns one at a time and one after another, while figuring, while sewing, while cooking, while building a model car or airplane, or while rebuilding a mechanical instrument such a bike or car, all points to a  jigsaw puzzle approach to problem solving, applicable in all subjects, circumstance permitting.  Tackling easy to challenging cross-word puzzles also demand a trial and error approach to problem solving.  Here the early clues and entries in the crossword puzzles may help in later ones. Playing games of chance, checkers or chess may also lead to an interest and practice in problem solving.

More Reasons For Mastering SOME Mathematics

  • Mastering arithmetic by hand or with a calculator is needed in the calculating weights, measures and amounts (money included) that appear in daily life.  If you can do arithmetic and estimate the results of calculations in your head, then you can catch or double check the figuring of others or your electronic calculator.  Incorrect numbers that appear in one step of a calculation make all the rest wrong. Tax forms give step by step instructions for calculating your taxes with arithmetic and a minimal use of formulas because government assume no competence in algebra. Arithmetic and not algebra is required for computing your taxes.  That is good to know :) And it is possible to have a thought-based comprehension of why methods for arithmetic work - a comprehension I would like to see offered or given in school.

  • Algebra may begin with formulas or methods for calculating areas, perimeters and volumes of common geometric objects in the line, plane or space. But there is more to algebra than following steps in a calculation, evaluating a formula or programming a calculator to evaluate a formula for you.  Rules (assumptions) in algebra say when different calculations will give the same result. Applying these rules one at a time and one after another allows you to solve problems algebraically and to algebraically obtain formulas for calculating numbers and quantities.  There is more to algebra than just doing arithmetic or being given a formula and numbers to use in it.  Algebra at full strength involves the thought-based derivation of formulas, that is, of explanations why they work.

  • For  life, now or in the future, you should meet and master formulas or methods for calculating areas, perimeters and volumes, and you should meet and understand formulas or computation methods needed for loans, pensions and investments,  for shop keeping or buying and selling with markups or markdowns. This understanding should go beyond using the formulas. You should understand how the formulas may be obtained or justified. With regrets, you may take several high school and college mathematics courses without covering the simple formulas and methods for money computations. That leaves you unable to compare precisely different options for earning, investing or borrowing money. Slight differences between different options may cost you years of work. Caveat Emptor.  Understand the origins of formulas in money computations and beyond, helps avoid costly errors in their use. Again, algebra at full strength involves the thought-based derivation of formulas, that is, of explanations why they work.

  • Beyond money computations and simple formulas for areas, volumes, weights and measures, algebraic calculations are not needed or not commonly used. Most calculations can be done without comprehension of why they work.   But the further study and use of accounting (money matters), carpentry, engineering, science and computers  involves formulas or calculating methods based on and described with algebra, geometry, trigonometry and calculus. Here the why is important to understand the computational theories given and why they work or don't. 

  • Probability and statistics are further topics in mathematics, in fashion at the moment.  The calculation of odds, chances and probability involves algebra and may involve a knowledge of sets and functions for modeling and calculation. Modeling and calculation starting from assumptions may be done precisely, but an error or doubt in the assumptions makes all the modeling or calculation suspect.  Not all is certain. The uncertainty may begin with the assumptions made to calculate probabilities.  Statistics is useful in the  measurement or estimation of numbers and the error or variation in the estimates. Estimates may be given by average. A small variation or none in the estimate is best. A small variation in estimate may allow you say a given number or quantity will be near a certain value. In social situations in contrast to physical situations,  statistics for income,  productivity, the price of a car or house, does not concentrate around a single value. Large variation in a number or quantity that is observed implies that the calculation of averages give little or no information.  And in sports, averages without mention of variations in performance may appear as a source of admiration for professional athletes and as possible source for computing the odds of a team or horse winning the next game or competition. The initial motivation for calculating probability came from gambling in games of chance. Calculating the odds of winning might be enough discourage you from purchasing lottery tickets but for the hope such purchases may provide. This site author does not purchase lottery tickets, except in social circumstances where the expectation of losing is offset by the knowledge that the purchase benefits a charity.

Each item and skill in the further study of mathematics may not be important or useful by itself. Yet the items and skills in mathematics altogether, cumulatively, have a greater and greater use in obtaining and describing calculations, and in describing the calculations and assumptions that appear in many disciplines. Mathematics courses are designed to problem solving skills, rote or opportunistic, and to provide a growing knowledge of ideas and skills that altogether, if not individually, may be useful in further study. If you follow how to obtain and justify formulas for calculations with money,  the mastery of further ideas in mathematics involves similar and further ideas.  Each method of algebraic reason can be recycled and eventually will be if you move from topic to topic.

Theories without examples are vacuous

 Your task is watch for examples and read them if need-be whenever a theory is presented.  Theories full of abstract or remote ideas without examples to illustrate or apply them provide a vacuous or empty knowledge.   

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Professor Whyslopes:

  • Site value lies in the difference between its ideas and yours.  

  • If one site explanation is not to your liking, try another. Each one is different.

Two gaps

  • The Old Algebra Gap:  Algebra  appears with too few words of explanation in high school and college mathematics.  Online Volumes 2 and 3 offer remedies.   Chapters 8 to 12 in Volume 2  put more words into the explanation and comprehension of algebra.  Chapter 14 in Volume 2 with its explicit discussion of the direct and indirect use a formulas identifies a unifying theme for mathematics and logic - all rules and patterns will be used forward and backwards. Chapters 2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very challenging use of algebra in the high level mathematics: calculus.    Calculus requires earlier high school mathematics at full strength: (i) This logically complete but long lesson on  complex numbers shows how to simplify the senior  high school exposition of circular trig functions upto to formulas in the plane  for vectors dot and cross-products.  The lesson provides the route that would have been taken in course design if the key element of the lesson, a December 2009 invention,  had been available in the 1950s.  For further algebra skill development. See the site coverage of fraction with units, proportionality,  ratios and rates, polynomials, quadratics functions  and straight line slopes and equations.
  • The Arithmetic Gap: An exact and efficient mastery of arithmetic with decimals and fractions is best (required)  for the high level  study of mathematics alone and in science, technology and business.   Pages here on arithmetic with decimals and integers,  on  fractions and solving linear equations with fractional operations on stick diagrams may help fill the gap.  That exact and efficient command should be obtained in the last years of primary school and the first years of secondary school.   

 Skill mastery in mathematics has to be seen to believed.  To that end,  learn or teach how-to write and draw the steps in mathematical figuring or  reasoning  clearly. Do not try to save space by doing a sequence of step in one place. Instead, do or record the steps in sequence on a separate lines to make each step obvious and verifiable.   

 

 

Schools/Colleges:  Hire the site author, as an online instructor, as technical support for teachers, or advisor for curriculum review.    Site Reviews may serve as references.  See how online whiteboards with  voice and real-time writing make online help possible with board content printable.  Text or written work scanned or saved to a  pdf file may be  uploaded  for discussion in the whiteboard.  

www.whyslopes.com

Parents: Help your Child/Teen Learn

Online Volumes
 
(orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3 .Why.Slopes.&
.More.Math.1995

Math How-TOs etc  2008
1. Arithmetic
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus

Site Description/Reviews  by 3rd parties

Site  Math Lessons
1. Arithmetic Flash Videos  11-2008
2.  Algebra Videos (to appear)
3. Fractions and More 
4.. 
Solving Linear Equations  04-2005
5. Euclidean-Geometry To Complex No.s 
6.  Analytic Geometry/Functions 2006
7.  Number Theory. 2006-7
8.
  Exponents, Radicals & logs. 2008
9 Calculus  2005
10..Real  Analysis 1995
11 Electric Circuits Etc  2007
12. .Algebra, Odds & Ends, HS level-2001
13.Maps, Plans,  Similarity &Trig, with
Complex   Numbers
, 12-2009. 

For Math Instructors/Tutors/
Curriculum Committees


1. K0-11Applied Math Program Outline  
2. Mathematics education  essays 
3. LAMP - an earlier applied math program.
4.
(150 pages)

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Would you like to show yourself or others how to be an  algebra power users?

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