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Appetizers and Lessons for Mathematics and Reason
by A. Selby, Ph. D.   Feedback & Questions

 20 pages in French: Algèbre  
 Définition d'une variable  
La raison basée sur les  règles et modelés

www.whyslopes.com > Volume 1B,  Mathematics Curriculum Notes,  1996  >   Foreword     [ Next ]

Every master of mathematical induction knows how induction may fail in and by analogy in education.

Links to Book Forewords:
1. Elements of Reason
1A. Pattern Based Reason
1B. Mathematics Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes and More Math

Foreword 

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. 

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

Links to Book Forewords:
1. Elements of Reason
1A. Pattern Based Reason
1B. Mathematics Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes and More Math

Note: The jumpmath program unconnected to whyslopes.com says the following: One feature distinguishes our workbooks from regular math textbooks, however: in the JUMP workbooks, teachers are consistently shown how to help students who are having trouble moving forward by breaking mathematical concepts and operations into the most basic elements of understanding and perception on page 2 of the Jump Teacher Manual - Fractions, one of three plus pdf files in the Jump publication page. The quote italized here complies with inductive principles given above.

Next: Three Remarks (Postscripts)


 

Mathematics
Curriculum
Notes
 Volume 1B
Printed in Canada
ISBN 0-9697564-6-1
Volume 1 = 1A+1B
bounded together

Foreword
1. Introduction [4]
2 For & Against Math
3 Algebra [3]
4 Why Slopes & Complex No. [2]
5 References - Past Efforts
6 Euclidean Logic
7 Geometry in 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition [3]
11 Primary School Math [13]
12 Four Phases

Book Entrance

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/
||
 -/[]\- 
||
   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   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
Tour their  forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing,  Preparing for Science,  ends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5 wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter    

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

 1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s
3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory, 
6.    Solving Linear Equations  
7  Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes. 
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19. Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21. Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
 Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
 Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
- Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
 These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.

 

www.whyslopes.com > Volume 1B,  Mathematics Curriculum Notes,  1996   >   Foreword     [ Next ]

Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep
A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition   (8 lessons)
Comparison to Subtraction  (9 lessons)
Multiplication ( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples. 
-Prime Factorization Aids  (Learn how to find factors quickly)
-Prime Factorization Examples 
-Counting & Generating. Factors. 
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
 Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences   
Units in calculations Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way  & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations. 
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for Algebra.  Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: -  Operations on Polynomials: Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.  		Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
 - Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check
-  Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in  Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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