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Nine Steps or Milestones, A Base or
Directions for High School & College Level Mathematics (March 1, 2001)
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Professor Whyslopes:
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Site value lies in the difference
between its ideas and yours.
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If one site explanation is not to
your liking, try another. Each one is different.
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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.
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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.
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Some steps may seem too simple for you or too hard. The
simpler one's point to difficulties which others have had. The harder ones point
to material you should master by yourself or with help. Some steps or
sub-steps have to be mastered in sequence, others may be mastered in parallel.
See for yourself. Some steps describe material not yet written - plans
for tomorrow's student.
Teachers: the steps here point to a
redesign of high school or college mathematics which will allow more to go
further in an easier fashion.
- Step 1: Do all four sets of Arithmetic
Review Problems before or at the start of courses in
algebra, trigonometry and calculus. These review or reinforce skills for
doing arithmetic by hand or with calculators. They may hint at or
point to arithmetic or algebraic patterns in computations - situations where
two different calculations may give the same result.
Doing arithmetic by hand is important at least for calculations in
the absence of an electronic or mechanical calculator. Mastery of arithmetic
by hand requires decimal methods to be followed, one step at a time and one
step after another, to arrive at results. Here an error in one step makes
all the rest suspect or wrong. Knowing that is the first sign of
intelligence in following step-by-step methods in any domain, from cooking
to science. And except for errors, exact arithmetic with whole numbers and
fractions leads to repeatable and reproducible results, independent of the
human or mechanical computer.
Marking answers by hand shows the teacher or tutor errors in your
notation. Errors in notation indicate miscomprehension. Here the saying, a
stitch in time save nines applies. Early correction is best. In
general, you should ask someone not to do your homework, but to provide
feedback on your errors before you submit your work for correction or
marking. Doing this as a matter of habit would lessen the number of errors
that you make and lead to a better performance in class. Learn from
your errors before you are penalized for them.
Mastery of arithmetic with its methods that lead to repeatable and
reproducible is the first source of skill and confidence in
mathematics. Awareness that an error in one step makes all the rest suspect
is the first sign of careful thought.
- Step 2: Read the mathematics-free logic chapters 4,
6, 7, 8 and 12 in Volume 1A. These chapters introduce
the Euclidean way of reason with simple words and examples outside of
mathematics. Equivalently see chapters 1 to 6 in Volume
2. The Volume 2 version of chapter 4 in 1A is slightly shorter.
The (deductive) Euclidean way of writing and reason employs implication
rules, one at a time or one after another, to arrive at conclusions,
one at a time or one after another. In these chains of reason, as in
arithmetic, an error in one step makes all the rest suspect. And in
circumstances, where the implication rules in question apply, the
conclusions are independent of the thinker. The chapter on implication rules
(4 in volume 1A or 2 in volume 2)
is harder than the others. But it explains the difference between one
and two-way implication. Not seeing the difference is a source of error in
reading or writing or following step by step methods for arriving at
conclusions or results. Chapter 6 in 2 (12 in 1A) describes the division of
rule-based thought into islands and bodies of "knowledge" in each
of which, different starting points may make island points more accessible
or not.
The Euclidean way of reason has been previously been introduced within
mathematics courses with mathematical examples of chains of reason, examples
that may depend on the algebraic shorthand way of writing and reasoning, a
further skill that may be hard for many. The treatment here separates the
learning and teaching of Euclidean way of writing and writing
reasoning from the learning and teaching of the algebraic way of
writing and reasoning. The treatment in 4,
6, 7, 8 and 12 in Volume 1A gives an alternative which separates
the learning and teaching of Euclidean way of writing and reasoning from the
algebraic way of writing and reasoning. So you may learn each way of reason
separately before employing both together - divide and conquer.
Mastery of deductive reason with its methods that lead to repeatable
and reproducible conclusions could be the second source of skill
and confidence in or out of mathematics.
- Step 3: Read the essay What is a
Variable? and if possible, chapters 7 to 15, in Volume 2,
Three Skills for Algebra. The essay and chapters introduce the
algebraic way of writing and reasoning with words that have been missing in
course design and teacher training programs. The essay and three
skills given in the foreword clarify the notion of what is a variable before
and besides the uses of notation. The statement that a
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letter used in mathematics is a variable,
and vice versa, |
is half-right and is too misleading, but it is in common use.
There are two notions of What is a
Variable, one which can mastered before the use of letters and
symbols to represent them..
Algebraic and arithmetic expressions are often better read and seen
silently than spoken aloud. Words literally have missing for understanding
and explaining the shorthand role of notation, symbols and letters, in
arithmetic and beyond. But there is a need to talk about numbers, amount and
quantities, and about the two shorthand roles of notation. The first
role is in describing calculations that might be done. The second role is
saying or describing when two different calculations will give the
same result. Rules for the latter may be applied one at a time or one
after another to solve equations.
The algebraic use of letters and symbols is often declared to be a
natural talent, too obvious to explain or impossible to explain directly. In
either case, the use is not taught or not understood. This traditional hole
in the description of mathematics makes learning and teaching
difficult or harder than need-be. Sorry.
Algebraic notation in the first instance is a
meta-language for describing arithmetic with expressions and formulas.
Algebraic notation in the second instances is a meta-meta-language for
saying when different expressions or formulas give the same result.
Mastery of three skills for algebra, two notions what is a variable, and
an awareness of the two roles of notation in algebra (describing
calculations and rules for saying when calculations are equal) could
be a third source of skill and confidence in mathematics. Learning and
teaching has been harder than need-be
- Step 4: Read (eventually) about [Functions
and Sets]
Set membership, union, intersection and complements form a language for a
precise description of mathematical ideas. Set theory is emphasized in
mathematics since the axiomatic method of analysis, more precisely the
arithmetic properties of integers, real numbers and functions can be
logically codified and described in it. Outside calculus, you may see the
role of set concepts in some presentations of analytic geometry. Lines,
planes, surfaces and sometimes solid objects are regarded as set of points.
Sets and set membership also have a role in combinatorics and counting, and
in the combinatorial based parts of probability. Combinatorics counts or is
concerned with the number of ways objects in sets can be grouped or placed
together.
The chapter [Functions and Sets] in
shows the equivalence of rule- and set-viewpoints of functions. That
is never done in Canadian and US course design which stems from the
set-theoretic codification or axiomization of mathematics Doing so
steps outside the codification.
- Step 5: Master the yet to be posted explanation or justification
of the decimal methods which you met and hopefully mastered before high
school.
The explanation and justification begins the observation the decimal
expansions of a whole number is equal to polynomials in powers of ten with
coefficients belonging to the set of digits 0 to 9. With simple
examples and later with mathematical induction, decimal methods for the
addition, subtraction, multiplication and division of whole numbers with
carries and borrows etc., may be justified. They may seen as examples
of calculations with polynomial with a special conversion or treatment of
coefficients when they not in the range 0 to 9.
This topic which links decimal arithmetic to the polynomials is not found in
modern course design. But one duty of mathematics education is to provide a
thought based understanding of its subject. This topic fills a common gap in
course design which leaves explained but not justified decimal methods for
arithmetic. Linking decimal arithmetic to the polynomials while
showing how the algebraic way of writing and reasoning can be used to
explain previously mastered skills, provide motivation and a further
context or reason for polynomial operations and the algebraic way of writing
and arriving at conclusion about calculations.
Understanding why decimal methods work could be a fourth source of skill
and confidence in mathematics. The demonstrations may illustrate
polynomial manipulations and beyond that, for enriched students, methods of
mathematical induction. (Leave the proofs of associative and
distributive/grouping properties needed in these demonstration to a later
course in mathematics, one given to students specializing in the
subject).
- Step 6: The Trigonometry & Complex Number
section of the subscriber area describes three ways to cover its
subjects. One way, the first, may
be enough for you.
The first way described here is for today's
students who have mastered trigonometry. It exploits knowledge of
trigonometric to arrive at a key property of complex numbers.
And trigonometry is not needed if you assume one key property of complex
numbers instead of deriving it.
The second and third ways in contrast or reverse the order in which complex
numbers and trigonometry. Both ways show how the addition and multiplication
of points in the place may geometry extend yours or another's knowledge of
arithmetic with unsigned numbers to a full command of real and complex
numbers. This command leads to a proof of the Pythagorean theorem and a
simpler treatment of trigonometry.. The third way goes beyond the second in
providing a knowledge of geometry axioms sufficient to justify the
assumption made in the second. The second way with its assumption of key
property (see the first way) could be for all students - students in
enriched instruction may see the third way
Following a mastery of complex numbers and trigonometry, the trigonometric
cosine and sine interpretation of dot- and cross-products follow,
and complex number based shortcuts for handling trigonometric
identities are easily justified. Engineering students may appreciate the
latter.
Putting mastery of complex numbers before trigonometry first provides a
quick or enriched way to understand and explain arithmetic with real and
complex numbers, and a simpler, quicker, starting part for trigonometry.
This could be a fifth source of skill and confidence in mathematics.
Learning and teaching has been harder than need-be.
Danger, Danger
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| The simpler, more effective, but non-traditional,
second and third ways to treat complex numbers and trigonometry
described may be employed for enriched studies and presented besides
traditional approaches. High school teachers in locations where there
are central examinations and a common approach to the teaching
high school mathematics will not able to present the second and third
way to ordinary students. The second and third will provide an easier
and better comprehension, but not prepare the student for examination,
unless it possible to present the second and third way quickly, and
then teach towards the examinations. That is a risk. |
- Step 7: See an Detailed Explanation of Error Control in Computations:
This step for tomorrow's high school student. The book Calculus by
Lipman Bers gives a presentation that can be adapted to
provide a discussion of significant digits and absolute error control in
arithmetic for the operations of addition, subtraction, multiplication and
division. This technical discussion of significant digits and absolute
error control in calculation is a concrete topic, one that refines or
justifies the non-technical discussion or presentation of rules for
estimating the number of significant digits, one that optionally can
be related to the polynomial perspective of decimal arithmetic in
which coefficients outside the range 0 to 9 are handled through carries or a
coefficient normalization process. This further discussion of significant
digits or accuracy in computations, a version yet to be posted online
here, could serve as preparation for the discussion of continuity,
limits and convergence in calculus.
- Step 8: Explore Why Slopes, A Calculus
Preview, and then read Volume 3, Why Slopes and More Math, offline ( online in the subscriber area) to understand further why slopes are
met in courses before calculus, or explore the area for a learn or review
key ideas in calculus and beyond. Calculus in the first instance is the
subject of slope related computations, their reversal and interpretation.
This area rearranges the order of topics in calculus to put the simpler
one first and so gradually introducing and reinforcing the skills need for
further study, while avoiding calculus or algebra shocks, and providing
simple examples to lessen or avoid them. First and further courses in
calculus and real analysis switch back and forth between demanding the
algebraic way of writing and reasoning. at full strength or not. Before you
meet the associated calculus shock, explore Why
Slopes, A Calculus Preview
- Step 9. Consolidate your knowledge.
Volume 1B, Mathematics
Curriculum Notes offers a context for mathematics education,
yours or that of others, from elementary school to college in twelve
chapters. See the calculus, complex number and algebra appetizers outside
the subscriber area of this site first and optionally, most chapters of
Volume 1A - some may be too hard.
Consolidation lessons yet to be written has the following tasks.
First describe the algebraic, set theoretic codification of
arithmetic in which the existence of sets of real and complex numbers are
assumed, along with methods for their arithmetic, and algebraically
state assumptions (axioms) which describe properties of arithmetic with real
and/or complex numbers. The latter properties should have been assumed or
concluded from the earlier geometric discussion of complex numbers and
trigonometry.
Second, assume or describe how real number have decimal
expansions and assumes or describes the convergence of infinite
decimal expansions. The assumptions here handle and sanction the common
assumptions about decimals and provides continuity of high mathematics with
elementary school mathematics.
Retaining the decimal-free view of the set-theoretic codification of
mathematics in courses for students not specializing in the subject
obstructs and does not help the common knowledge. Not talking about
and not explicitly sanctioning the decimal expansion of real numbers has
made learning and teaching harder than need-be, and left a gap in the
exposition, a discontinuity between elementary and high school mathematics
in North American course designs 1960 to the present.
Third, delicately describes the arithmetic-based codification of
geometry in which real numbers may be used as coordinates for a line, and
ordered pairs or triplets represents points in a plane or space. That
introduces analytic geometry, a way to do or represent geometry without
depending on the drawing of suggestive diagrams to arrive at conclusions.
This is avoidance of diagrams for proofs, use in illustrations still
allowed, provide a thought-based codification of ruler- and compass-based
geometry.
The earlier introduction of complex numbers and trigonometry employed
suggestive diagrams and assumptions about ruler- and compass-construction.
This makes learning and teaching possible. The advanced viewpoint is
not for beginners. Yet the advanced viewpoint disowns the earlier,
suggestive-drawing, diagram-based introduction to complex numbers,
trigonometry and associated parts of calculus, and replaces it with
diagram-free, algebraic-arithmetic considerations that most will never see
and most students would never be able to follow in the first instance. That
is to say that the introduction to complex numbers, trigonometry and
calculus requires supports which the advance exposition, a more rigorous
approach removes but those same supports also provide a context for the
advance exposition.
Fourth, delicately explain that in advance courses, analytic
geometry combined with a decimal or decimal-free viewpoint of real numbers
can be used to define (to say how to compute) trigonometric functions
without the use of diagrams and also how to introduce complex numbers
and their arithmetic properties without the use of diagrams. Details of the
latter is a story or chain of reason for students in enriched or advanced
studies in mathematics, those who take a more rigorous course in real and
complex analysis. But the diagram-based explanation is enough for everyone
else.
Remark: In geometry based on coordinates or on ruler and
compass constructions, the introduction of trigonometric functions using the
ratio of sides to triangles or coordinates of points on unit circles
represents a large or smaller step away from the development of
mathematics from axioms about arithmetic or sets of real numbers. This makes
the previous or current development of trigonometry and calculus which
use trigonometry, impure. The alternative recommended here gives first and
openly, a complete and fully impure geometric development of trigonometry
and the properties of real and complex numbers. A switch to the pure
axiomatic set-based description of real or complex numbers, or their
arithmetic properties comes later.
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Odds & Ends
Group I
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
8. Complex No. Applet
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
27. Biology - Growth & Decay
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
Two Gaps
[ Back ] [ Next ]
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For
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High School & Calculus Students
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Words to clearly
introduce algebra and variables
have been missing in course design. For people who cannot do
algebra,
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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.
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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.
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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?
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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.
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