YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
|      | |     |
   \             /   
\    =   /

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Advice and Suggestions for Course Design and Delivery.

What to tell Students

 1.  Aim for Care and Precision

in reading, writing, reason and arithmetic.

Understanding that carefulness and precision are needed through will ease or avoid confusion and difficulty in work and studies.

Online logic chapters 2 to 5 from site book Three Skills for Algebra will test or develop care and precision in reading and writing.

Testing or developing precision reading and writing provides a first step in building skills and confidence for work and studies in many arts and disciplines.  It should also the first step for the greatest benefit from site appetizers and lessons.

While logic chapter 2 to 5 in the online book Three Skills for Algebra may lead to precision in reading, writing, reasoning, these online arithmetic review exercises could lead to precision in arithmetic. Many students are not aware of the need for results in arithmetic to repeatable and reproducible, and hence verifiable - that, is they can be checked.

The  arithmetic review exercises need not be done now. When  you try them, exercises involving operations or calculator buttons you have yet to meet can be skipped until later. 

The ability to figure well, without errors, that is in a repeatable, reproducible and hence verifiable manner, was once considered to be a sign of intelligence, a sign of skill and competence.  It is because a person who does arithmetic well has learnt to follow multi-step methods with decimals, fractions or both, and has learnt that lack of care in one step of a method usually leads to wrong or inaccurate results. So person is aware of the need for care and precision while following instructions or writing and giving them.  Whence mastery of logic and learning to figure well all imply or demonstrate greater care and precision for work and studies, and so point to fewer difficulties for both work and studies.  Take note. 

Tell Students: When you were young, to read, write and spell, you learnt all the alphabet not just some.  With regrets, algebra and higher mathematics demands mastery of times tables, arithmetic operations with decimals (addition, subtraction, multiplication, long division) and with fractions. During arithmetic mastery, you learnt or will learn that an error in one step of a multi-step calculation usually leads to wrong results. So each step, the work,  has to be done with care and precision. The same or similar care is needed in algebra, logic and beyond.

2.  Aim for Fraction and Algebra Skills and Sense

The careful and precise use of whole numbers and fractions in solving linear equations will provide a firmer base for further mathematics instruction.  Some steps can be followed or designed to avoid integers and signed numbers.  Emphasizing that solutions of linear equations can be checked before being submitted for correction or marking will eventually lead to greater student independence.  When a check fails, the mistake or mistakes fall between the start of the solution and the end of the check.

For all high school and older students, the unique site area solving linear equations introduces and illustrate fractional ops on line segments (sticks) to geometrically develop algebra and fraction sense and skills. That with solving triangular and essentially-one-unknown simultaneous equations begins a new, yet tried and  tested path to make solving word problems, substitution, the distributive law,  and solving simultaneous equations much easier learn and teach. The geometric approach is intended to promote the algebra approach, and it usually does.  Where it does not, students will need further help. Fractional operations on stick diagrams to build algebra and fraction skills and  sense is a site invention - a co-invention with an offline author.

While some students will want to leap into the algebraic way of solving linear equations, staying a little longer with or leaving and returning to the stick diagram approach will reinforce fractions skills and sense.  The fraction parts of the site area  Fractions,  Ratios, Rates,  Proportions & Units  offers fraction skills by rote, automation is important, and also fraction skill development with more operations on line segments to provide and perfect a thought-based mastery.  The latter is recommended only for (i) students who want to learn, and enrich or perfect their skills and knowledge, and (ii)  students who dislike or want to go beyond rote learning of methods.  That being said, drill, practice and correction with fractions and solving linear equations may build skills and confidence through the mastery of methods with repeatable, reproducible and hence verifiable results.

3.  Aim for a Greater Use of Words in Mathematics

Many people have difficulty with algebra. Even gifted or advanced students will have gaps in their understanding of the shorthand role of letters and symbols in mathematics. After all, arithmetic and algebraic expressions, the longer ones, are difficult or impossible to read aloud, symbol by symbol, bracket by bracket, and so on. And when expressions are read aloud, there is the frustrating possibility of confusion in the order of operations.  The reader may mean one thing, and the hearer may write something else. That is not good for precision in reading and writing in arithmetic and algebra. This difficulty in reading arithmetic and algebraic expressions aloud, including those appear in the algebraic described properties of arithmetic with numbers, leads to silence, a dearth of words,  in skill development and communication. We look at and digest mathematical expression in a glance instead of reading them aloud. Mathematics in and beyond arithmetic and algebra has been a silent discipline where written letters and symbols, together or separately, are used for communication. That makes learning and teaching harder. 

Site book Three Skills for Algebra  with its online postscripts point to a greater and clearer use of words in mathematics in two different, but mutually supporting ways,

1. Greater Use of Words in Describing Numbers, Amounts and Quantities:  Algebra chapters 8 and 9 identifies our ability to describe numbers, amounts and quantities with written or spoken words before or besides symbols.  The online postscripts use words to explain what is a variable, constant, or parameter, and do so without the use of symbols  in a manner that students can grasp years before functions, a senior high school topic, are met.  The online postscript brings mathematics in closer alignment with physics where numbers or quantities that may vary in one sense or another, are called variables.
   

2. Greater Use of Words in Common Operations on Equations and Formulas: Chapter 14 on  compound interest or growth formula introduces the direct and indirect use of formulas or equations, and shows the difference between algebraic and numerical solutions, and point to the ability of the algebraic solution to give the numerical solution in full, or just the end results.

   For all formulas in high school and college mathematics, we may now identify (A) direct and indirect use, and the (B) numerical and algebraic solutions that may be possible in the indirect use.  The repeated use of  two phrases in dealing with formulas, one at a time and one after another, gives voice to a previously unnamed and hence hidden operations and themes in mathematics learning and teaching. Remembering the phrases (A) and (B) while you study or  teach will make the methods of algebraic ways of reasoning clearer and provide a focus or two for their study.

Words and names are powerful. Once an mathematical object or operation is named, and clearly described with words,  we can use  names and words, again and again to point out recurring patterns. The foregoing lessen the silence that accompanies arithmetic and algebraic expressions, formulas included, because they are so awkward to read aloud term by term, parentheses by parentheses. The foregoing introduces a new avenue for mathematical learning and teaching.  Formula, operations and properties of real numbers, etc, known and  named can be mentioned and discussed without being present in written form.  The result is or will be more written or spoken communication in mathematics based on words to supplement or go beyond expressions better seen and read silently at glance, than read aloud in a way that communicates order of operations clearly and precisely.

4.  Develop More Algebraic Thinking Skills, Those Needed for Calculus

Preparation for the full strength use of algebraic ways of writing and thinking in Calculus. The algebraic way of writing and reasoning is required at full strength in calculus. This full strength requirement, and algebra shock, is too sudden and terminal for many students.  This aim (and the previous ones) point to a remedy, one that many, not all students have enjoyed.

The following methods or path for easing or avoiding algebra shock in calculus may be seen at the start of calculus and prior to that, in course in analytic geometry after or as part of the discussion of slopes to straight lines and the factorization of polynomials alone or in the numerators and denominators of quotients (rational functions).

Calculus gives the best framework for understanding calculations met in business, science and engineering. Describing the same calculation without calculus is long and shallower process. Shortcomings in the development o algebraic skills and concepts, those needed for calculus, led schools and course design to fill student with topics not needed for calculus and supposedly simpler. Good preparation for college mathematics (or calculus)  requires mastery of most, but not all the topics, you meet in high school mathematics: exact arithmetic with whole numbers and fractions, algebra, geometry without and with coordinates, and trig.

Calculus in the first instance is a subject of slope and rate related calculations, as is or reversed, with applications.

The online version of site Volume 3,  Why Slopes and More Mathematics, includes a geometric calculus preview before a more algebraic perspective in chapters 2 to 6 . The  geometric calculus preview explain how slope related calculations, forward, not reversed, appear in calculus.  That gives context or explains why slopes appear repeatedly  in earlier high school and college mathematics. In an courses  where slopes and then polynomials and rational functions are met, the geometric calculus preview and chapters 2 to 6 could be used to (i)  understand and explain  extreme points and identify where factored or easily factored polynomials and rational functions are increasing or decreasing; and (ii) to develop students algebraic reasoning concepts and skills.  The foregoing also provides a way to ease or avoid difficulties in the first and further weeks of calculus.

 Calculus is also the subject in which the many facets of what is a variable, constant and parameters appear as well. That being said, chapters 2 to 6 in Why Slopes and More Mathematics provide a slow and effective, induction, into the algebraic way of writing and reasoning required.

The chapters with the aid of slope interpretation identify interior and end-point extreme points (maximums and minimums). Polynomial and rational function formulas given for slopes (they are not computed in these chapters), and given in factor formed, are used in slope sign analysis.  The sign analysis of these factored polynomials and rational functions indicate the intervals where a function y = fix) is increasing or decreasing, and thus indicates the interior or end-point location of extreme points.  By skipping over lengthy discussion of limits and derivative calculations to the sign analysis of derivatives or slopes, these chapters provide a context for the skipped material while developing the algebraic maturirty needed to understand the skipped material.

5. Make Limits, Convergence and Continuity Easier

Bring Back, O Bring Back, the Bonny Decimals.

Calculus and mathematics instruction became more complicated in the second half of the 20th Century due to the course design which employed decimals to represent numbers and to do calculations, but which also did not favor nor mention them in presenting theory.  In particular axioms for real numbers (algebraically described properties of arithmetic with real numbers represented by decimals)  do not mention decimals in high school mathematics despite coverage of scientific notation and decimal-baed accuracy or estimates in arithmetic. Then in calculus, hard to grasp, extremely algebraic, decimal free explanations of limits, continuity and convergence are inaccessible.  As a mathematics students, the site author Professor Whyslopes,  struggled repeatedly to understand the decimal approach in technical and intuitive manner.  For many students, that decimal free approach is a very large barrier to understanding. So today, calculus courses may teach students the properties of limits, continuity and convergence by example (rote learning) and not attempt to provide any theoretical framework.

The modern mathematics description of limits, continuity and convergence is often decimal free.  The modern mathematics curricula of the 1950's and 1960s, echoes of which still appear in some or all classrooms today, did not mention nor sanction the decimal representation of real numbers and so did not sanction methods of decimal arithmetic, while still employing in numerical results. Moreover the decimal perspective of limits, continuity and convergence was not mentioned or sanctioned as well.

Limits, Continuity and Convergence in Calculus.

A decimal viewpoint of limits, continuity, and convergence, and the associated question of limited or unlimited error control in function evaluation or computations,  is sufficient for most students and its provide an model which also makes the decimal -free viewpoints easier to understand and grasp - provides a context for the latter.  Therefore chapter 14 in Why Slopes and More Mathematics introduce the decimal viewpoint while the appendices to this volume push (or review) the decimal into advanced calculus or real analysis. That provides the proofs of theorems often given without in first and further courses in calculus. 

6. Aim for a Greater Use of Complex Numbers

This aim points to a change in course design and delivery at the secondary and tertiary (college level).  Senior high school and college students may use these underlying ideas in their self-instruction. Course design changes indicated here will most likely not occur in their school days.

A simple and clear way to understand and explain complex numbers (site starter lesson, pre-calculus level) is to introduce addition of points in the plane using rectangular coordinates; to introduce their multiplication via polar coordinates; and then to assume or geometrically imply the arithmetic properties of complex numbers. Implicit here is the assumption, that every point in the plane has both rectangular and polar coordinates.  From the numerical properties of complex numbers, algebraically described, we can obtain several easy consequences: a new proof or confirmation of the Pythagorean theorem, the properties of trigonometric functions; and a geometric, complex number development of trigonometry. Details are given in the site area on complex numbers. All the foregoing suggests simpler path for high school trigonometry and simpler, complex number developments of trig expressions for dot and cross-products of vectors in the coordinate plane.  University level schools of engineering and science will appreciate the shortcuts. They can be also be used in senior high school mathematics before calculus if time permits besides the other curriculum obligations.

Remark 1: Teachers will find several approaches to the derivation of the arithmetic field properties of complex numbers in site pages.  All assume points in the planes drawn on paper can be represented by ordered pairs of rectangular and polar coordinates.  All stem from different assumptions about Euclidean geometry, the use of coordinates and/or real numbers. The most extreme route is to (i) assume  decimals can be used as coordinates, (ii) assume the completeness of this representation by accepting infinite decimals expansions as coordinates, and (iii) imply or derive the properties of real numbers from assumptions about geometry and the use of unit lengths and directions to define coordinates systems.  The latter route employs geometry instead of set theory to represent and obtain the properties of real numbers.  The latter route met first could provide a context for modern mathematics even though modern is context-free in another sense.

Remark 2 : Pure mathematics from the algebraic statement of axioms (assumed patterns)  provides a thought-based development or codification of concepts and statement in mathematics with no logical dependence on suggestive drawings nor physical argument. So pure mathematics is said to be context-free , but there is a context for this context-free development.  In contrast to pure mathematics, applied or mixed mathematics employ suggestive drawings and physical arguments to introduce and employ coordinates or other device from pure mathematics to model objects in space and time.  That steps beyond pure mathematics, while assuming those calculations in the model which can be done in a mathematics or context free (formally or purely) way in order not to accidental introduce further physical assumption into the calculations or the results.   Finally, primary, secondary and tertiary mathematics from learning to count and recognizing geometric shapes to the definition of trig functions using right triangles and unit circles in a coordinate plane is not part of pure mathematics. Before pure mathematics can begin, mathematics education has to inductively (use of manipulatives and suggestive drawings appears here)  develop the necessary  numerical, algebraic and deductive skills and sense.  The set theory axioms of pure mathematics imply the axioms about real numbers which appeared in modern mathematics course design.  But there was a mistake. Modern mathematics course design did not mention decimals and the emphasis on context-free development was inconsistent with the high school and college development of Euclidean Geometry, Trig and Calculus with the aid of suggestive drawings. 

Modern mathematics course design inserted set language into course design - that description awkward in parts can be still be retained and refined as it provides a finer or more precise language for the development of some skills and concepts.  How will be explained

While the logical development of pure mathematics is and should be context-free to avoid dependence on suggestive drawings or physical assumptions, both of which can be useful but misleading, pure mathematics itself codifies the properties of real numbers used as a coordinates in mixed or applied mathematics.  Comprehension of how and why provides a context for the avoidance of suggestive drawings and physical arguments in pure mathematics.

7.  Understand Three Kinds of Reason in Mathematics

There are three kinds of rule-based intelligence in mathematics, logic and most pattern-based subjects.

• The first  kind met in primary school arithmetic consists of skills with repeatable, reproducible and therefore verifiable results - results that are then considered right or wrong.

• The second kind also met in primary school consists of pattern or rule recognition. The development or exploitation of the ability to recognize or suggest simply patterns in order to predict the next element in a sequence. If the prediction fails, another pattern is required. 

• The third kind, assumption-based, deductive reason, appears after inductive mastery of logic, that is mastery of implication rules If A then B and their use. The third kind follows the use of implication rules and definitions and assumptions, one at a time and one after another, to arrive at logical conclusions. Here chains of reason how to be posed in a readable,  repeatable, reproducible and therefore verifiable manner. 

For third kind of thinking in mathematics, there was a search for secure  assumptions, so that deductive reason could proceed in a consistent and reliable manner.   Unfortunately, uncertainty results in mathematical logic imply more can suggested than proven in mathematical theories which are not finite. So the assumptions made for the third kind of reason stem from experience or trial and error over time. That identifies modern pure mathematics as another empirical art.  But mathematics by providing a format for measurement and  calculations  remains the queen of science, a queen in the hierarchy of empirical arts.

Pre-coordinate  Euclidean geometry, the original model for pure reason in mathematics, with its assumptions and deductive chains of reason is still worth presenting in part if not in full in high school mathematics in a selective manner to build algebraic-deductive skills and geometric skills and sense. However, the empirical nature of pre-coordinate and hence coordinate-free Euclidean Geometry is implied by diagrams with subtle faults that imply incorrect conclusions - subtleties detected with the use of coordinates in advance mathematics courses. 

www.whyslopes.com
Mathematics Education Essays
57 or so 

Area Entrance & Hub
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.

Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I  - Fraction Focus
Sec II -  algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links

Help Me Learn/Teach;

1. Algebra
   words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)
2. Arithmetic
   - exercises
   - with fractions
   - videos on primes, lcm, gcm,lcd, square roots etc
3. Calculus - geometric preview, algebraic preview,
   3 study guides,
   much more
4. Complex numbers
   -starter lesson with java applet - easy consequences for trig & vectors in the plane
5. Education
   - Empirical Course Design & Delivery
6. Fractions
   - alone
   - by rote
   - with algebra
   - videos

7. Functions - introduction
   hindsight - composition aka
   substitution -
8. Geometry, Euclidean - Correspondence of triangles,  Triangle construction,  duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
   
9. Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
10. Logic
    - First Steps -
    Symbols in Logic -
     Occurrence & Truth Tables - Indirect Reason -Indirect Reason More
11. Proportionality
    - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions
12. Real Analysis
    - Decimal View of concepts and of proofs
13. Rules &Patterns in Science, Technology & Society - Pattern Based Reason
14. Mathematical Reasoning, empirical, inductive or deductive
15. Units
    - in rates & slopes & (?) derivatives
    - in ratios & proportions - slopes & rates included
16. Complex Numbers & Vectors & Trig
    -  trig expression for dot & cross - cosine law

[Top of this Page][[Site Exit] [ Back ] [ Area Entrance & Hub ] [ Next ]
[Comments, Reactions, Feedback]
www.whyslopes.com

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby, a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.