About LAMP - It Objectives
LAMP is an acronym for Logic and Applied Mathematics
Proposal. It is a proposal for adult and adolescent instruction that
covers mathematics from arithmetic to calculus.
LAMP divides skill and concept development between three phases:
- (Phase I) Mathematics for TCPITS
- (Phase II) Preparation for Calculus,
- (Phase III) Calculus
TCPITS is an acronym for the phrase: the
common person in the street
Learning and teaching in each LAMP will vary between
inclusive and comprehensive forms
- I-LAMP, the inclusive form, aims for an operational command of
skills and concepts with a thought-based development only when needed.
- C-LAMP, the most comprehensive form, aims for an operational
command of skills and concepts with a logically organized thought-based
development whenever possible for the sake of completeness, and with
references to compensate when not.
The challenge for mathematics education is to develop skills and concepts
in a way that students planning to end their studies gain the skills,
confidence and satisfaction sufficient to change their plans.
LAMP is based on inductive principles for instruction.
Those principles require large steps be clearly and explicitly
decomposed into smaller steps for skill and concept development, so
that student who cannot take larger steps may be given a simpler path
to follow when needed. Each student has a different background
and skill level on entering or continuing LAMP. In LAMP
includes directions of the form if the student has not master a skill
or concept then try the following steps. Where difficulties are common
or not, those inductive principles require LAMP to provide for smaller
steps to make the larger steps accessible and feasible.
Time will tell what is possible and what level of intelligence is
required to follow the most inclusive and accessible form of LAMP that
presently exists. Trial and error may tell suggest how to rearrange the
iterative coverage of skills and concepts to make LAMP more accessible.
There-in lies or shines the empirical approach to the further
implementation of LAMP.
The challenge for course design and the composition of course materials
is to make all as inclusive as possible so that students and teachers not
yet comfortable in mathematics have clear and readable explanations and
directions to follow.
Confidence in LAMP depends on whether or not it
delivers operational command of mathematics, the how and/or why,
to students, parents and teachers through steps, well-documented and
easily understood and followed, in instruction and
self-instruction.
LAMP includes several innovations to make key skills and concepts
easier. Where some students will be empirically satisfied that
methods give repeatable, reproducible and verifiable or refutable
(correctable) results, others will demand theoretical satisfaction - an
explanation of the why. LAMP aims to provide a maximal operational and
thought-based command of skills and concepts for minimal effort.
That includes removing artificial challenges - skills and concept not
necessary for the immediate and long-term objective of providing an
operational command of the how with a minimal, maximal or somewhere
in-between thought-based command of why. But minimal effort does not mean
no effort. Work is required. Students have to sit down and study. LAMP
has to provide material for students, parents and teachers - not
yet comfortable with its skills and concepts.
LAMP is empirical. LAMP in all its forms starting from column
methods for decimal arithmetic emphasizes clear and legible formats for
the performance oriented, very much observable and correctable expression
and development of ideas and results on paper, step by
step. By drawing and writing diagrams, words and
arithmetic or algebraic calculations on paper, one small step at a
time, students, their fellow and their teachers may rely on their
eyes to see what has been done, and to decide what comes next. If
mathematics be a mental exercise, it is one in which diagrams, words and
expressions drawn or written on paper record and express ideas as part of
dynamic, observable and readable record. Just as a carpenter works with
wood to build concrete objects, the mathematics students and
mathematician draws and writes on paper to provide a record of ideas and
thoughts in a concrete manner that can be immediate review by the
composer and the composer peers for correctness and completeness, or
compliance with prior rules and patterns. With that record, the composer
and others may see and continue prior chains of reason - the selection
and use of rules and patterns one at a time and one after another, alone
or in combination, with care or respect for limitations. With clear rules
and patterns to follow in the expression and application of skills and
concepts, students abilities in mathematics are judged by what they draw
and write. Performance or an operational command of mathematics needs to
be based on drill and practice sufficient to automate drawing and
figuring methods in an observable, legible, repeatable and reproducible
manner for the sake of immediate or later review by the author,
peers and teachers. Reaching that destination, one method at a
time, will involve work or drudgery, but it will also build the
skills and confidence of students, and so engage them as they learn how
to do mathematics in an observable way that can be reviewed and approved,
or corrected. There-in lies a means, a method and an end for mathematics
and logic instruction and self-instruction.
LAMP is not a cure-all. LAMP implementation requires that students
have the ability and inclination to sit down to meet and practice skills
and concepts, and to entertain, if not always allow, motivations offered
in LAMP or coming from a parent or instructor for the necessary practice
and drudgery. LAMP aims to provide an operational command of methods,
mathematical or logical, first for performing routine calculations, that
is, for solving common or routine problems, and second to develop general
problem solving and defensive (be prepared) critical thinking abilities
for matters met at home, at work and in society. The identification
of common or routine problems depends on society. Native,
aboriginal, first nation societies and other societies with brief contact
with the quantitative reasoning skills of larger, modern,
pollution-age, agricultural, manufacturing and city life may find larger
society schooling systems in need of adaptation to their needs.
Whence mathematics is not a universal language and rightly or wrongly,
the motivations for mathematics will have a large society flavour.
Part I - Critical Paths For Skill and Concept Development
Most elements of C-LAMPS are present online in this website www.whyslopes.com. To determine critical
paths for LAMP in its C-LAMP and I-LAMP forms, in parallel streams we
will identify and indicate a thought-based development all arithmetic,
algebra, geometry, logic and calculus skills and concepts which could be
part of C-LAMP, the comprehensive form of LAMP, along with their
thought-based development. For a critical path for C-LAMP, we will
identify the dependence, if any, of each skill and concept on elements of
the other streams. The one further stream not mention
consists of applications - mathematical methods for solving routine
problems in human affaires that depend on skills and concepts in
arithmetic, algebra, geometry, logic and calculus. The dependence
will determine how soon the applications and motivation for studying
mathematics and logic can be placed - the earlier the better.
The indication of thought-based development of skills and topics in
this site area will inversely proportional to the coverage in other
areas and if not in other site areas, to the difficulties that might
occur in that development. In other words, the indication
is greater and in more details for those skills and concepts not
treated in detail elsewhere or whose treatment is not obvious.
Six chapters 1 to 6 describe in detail and with some overlap
arithmetic, geometry, algebra, logic, calculus and application
streams of LAMP.
These streams may themselves be composed of parallel substreams. The
ordering of parallel streams and substreams there-in depends on which
logical dependencies are respected The logical dependencies will greater
in the more comprehensive forms and less in the more inclusive
forms. Ordering may also depend on pedagogical estimates of which
streams and substream blocks are easier to master.
Part II - Possibilities for I-LAMP and C-LAMP
Part I described how to provide a thought-based
development of skills and concepts from arithmetic to calculus in a
very detailed or full manner. The aim of part II is to assembly those
pieces into an inclusive and an comprehensive designs for LAMP based
instruction and self-instruction.
Chapters 1 to 6 in their definition of LAMP streams or substreams
set the stage for developing the most inclusive and the most comprhensive
forms of LAMP. As said above,
-
I-LAMP, the most inclusive and flexible form, and
C-LAMP, the most comprehensive, demanding and constrained form.
The inclusive form aims for an operational command of skills and
concepts with a thought-based development only when needed.
Where skills and concepts are described instead of derived, there
more be flexibility in sequencing than permitted in a more sequence
thought-based development. In other words, a critical path
diagram for non- C, LAMPs learning paths are always less
restrictive than C-LAMP.
-
C-LAMP, the most comprehensive form aims for an
operational command of skills and concepts with a logically organized
thought-based development whenever possible, and with references to
compensate when not. Chapters 1 to 6 describe and specify in writing,
the critical path diagram for C-LAMP.
Individual students, teachers and school may learn and teach LAMP between
these extremes. There-in lies a lower bound for instruction and, if
course materials are sufficiently clear, self-instruction too. In
all cases, critical path analysis of the dependencies indicated in
chapters 1 to 6 will possible routes for instruction
Part III: Reflections, Conclusions and Musings
Chapter 8 introduces further LAMP objectives, LAMP musings, and LAMP
possibilities including criteria for a mathematics curriculum or course
design to enjoy its seal of approval. Essentially, skill and
concept development must be self-contained so that that teachers
and students.
In retrospect, the end, values and methods of mathematics education are
culturally biased towards societies which have employed
quantitative skill and concepts from ancient to present times in
their agricultural, trading, manufacturing and political or
religious activities. Familiarity with those activities provides
a context and motivation for primary school mathematics and LAMP phases
in all or part.
Mathematics is not a universal language. Smaller societies or
groups that are surrounded by larger societies are caught in awkward
position where the educational systems of a surrounding society may be
needed for economic survival. There in lies a catch 22 for a surrounded
society. In following the education of the larger society, if the
latter is well-done, and not bureaucratically amiss, many of its
members may run the risk of assimilation, so that some will
returning to partially preserve the surrounded and threaten culture
through an adaptation of the larger societies education system.
LAMP origins
This proposal for LAMP is stems from inductive principles for
instruction met in 1981; from lessons three skills for
algebra, a geometric calculus preview and a logic puzzle on the
difference between one- and two- implications invented for my students
in fall 1983; on twenty minutes or so of a Richard Feynmann lecture,
one of three public presentations at McGill in fall 1989 lecture, in
which he describe his subject as the addition and multiplication of
arrows in the plane; from the example of guest speakers in mathematics
(analysis) at McGill University 1975-83 who show that the exposition of
mathematics could be advanced (the hard made clearer) from secondary to
research level subjects; from the exploration at this website www.whyslopes.com of ways to understand
and explain methods I learnt and met by rote; and from recognition of
inductive gaps or
shortcomings in the Modern Mathematics Curricula of the period 1955
to 1980 or so, a curricula I met as a secondary student in
1967.
From 1967 to 1983 as a student and then as a teacher, I saw gaps in the
exposition of my subject but my own studies distracted from the active
pursuit of remedies. From fall 1983 invention of lesson on three
skills for algebra, a geometric preview of calculus and the logic puzzle
to 1889, I held university professor and college instructor post of short
duration in which the expression of interest or ideas in and for
mathematics education was impolite. Writing began in the last few
days of 1990 with the aim of informally reporting multiple ideas to
educational authorities for review and refinement by my peers and betters
in a hit and run manner - the academic job market was difficult and I did
not expect a career in it. Yet I was driven to investigate and
consider mathematics education due to the difficulties students
first due to the challenges of calculus - my perception that its syllabus
could be re-arranged to make the hard easier, and later due to the
realization that the secondary school preparation of high school
mathematics might benefit from my fall 1983 lessons and their
expansion.
On meeting the 1989 and the later year 2000 standards and principles of
the US National Council of Teachers of Mathematics, I read them in
the hope of finding a reason to stop writing of seeing my exploration of
ideas and concepts was redundant. From say 1992 to the present I
have not read Mathematics Education Journals due to their focus on
delivery style matters or petite content issues, while finding
considerations of content issues appeared to be as difficult as looking
for a needle in haystack.
In retrospect the modern mathematics curricula of the period 1955 to
1989 which I once favored and tried to make more accessible or
inclusive in my writings did not address the algebra exposition gap
that existed in its time and before, and its introduced further gaps
and barriers in mathematics instruction as its nominally support of the
rigorous and context-free (intrinsic) development of Modern Pure
Mathematics was not rigorous, included extrinsic mathematics in its
development of Trigonometry, Calculus and before that Euclidean
Geometry, while inconsistently allowing the decimal representation of
numbers in arithmetic, but not sanctioning that representation nor
exploiting it in its development of mathematics from algebra to
calculus. In other words, the Modern Mathematics Curricula, the
implementation I saw in my education, took steps too large in the
development of algebraic skills and concepts alone and in further
subjects, required decimal arithmetic but avoided decimals in the
statement of its axioms for real numbers and in its formal discussion
of limits, continuity and convergence in calculus, a decision and
epsilon-delta abstraction that complicated comprehension; while the use
of diagrams in trigonometry to define sine, cosine and tangent
functions and the use of diagrams to imply that 1 was the limiting
value of (1/x) sin (x) as angle x measured in radians approaches
zero fell into a gray area - the extrinsic handwaving development
of mathematics which departed from the instrinsic (axiomatic)
development of Modern Mathematics.
LAMP Advantages
Algebra: The LAMP proposal provides a clearer exposition of
algebra with the aid of geometry and words before and besides
symbols. LAMP provides smaller and clearer steps from the start
of algebra to the multiple full-strength deployment of algebra in
advanced calculus. That by itself should justify the study of
LAMP ingredients in the rest of this website www.whyslopes.com, if not the adoption
of LAMP itself.
Mathematics Extrinsically
Modern mathematics curricula were not only
intrinsic, they were extrinsic as well in contradiction to a
nominal intrinsic nature, a nature with full compliant, a
pedagogical impossibility.
Before the advent of modern mathematics and its
intrinsic, context-free and logical development of skills and
concepts from arithmetic to calculus and beyond, many elements of
mathematics were well-known. They were implied by drawing and
figuring practices in say an empirical manner with some
uncertainty due to the possibility of drawing misleading diagrams
and more uncertainty due to the lack of clear or absolute logical
structure for figuring from arithmetic to calculus. LAMP
pushes for an operational command and an operational (extrinsic)
development of skills and concepts.
Modern mathematics starting with the ZF axioms
1903-5 provide a more secure but in retrospect not an absolutely
certain framework for the theoretical or logical development of
mathematics from assumptions about sets to an intrinsic
(axiomatic), context free/independent codification of the
properties of real and complex numbers, and a development of
calculus methods. Following that path requires a strong
algebraic-deductive maturity.
The Modern Mathematics curricula 1955 onward
introduced the second part of that path into secondary and primary
mathematics with axioms for real numbers as an intrinsic starting
point. But the exposition of geometry, analytic and
Euclidean, and the development of trig and calculus also involved
the use of geometric reasoning or diagrams in a manner that
departed from the pure intrinsic development. That reasoning
and diagrams represent an extrinsic viewpoint of mathematics.
Further the decimal representation of real numbers was required for
arithmetic in algebra, trig and calculus but not mentioned and
hence not explicitly sanctioned in the modern mathematics
curricula assumptions about real numbers. LAMP is an extrinsic successor with a more inclusive and
consistent thought-based development
LAMP extrapolates from counting, drawing and
figure practices, an extrinsic development of real and
complex numbers, of trigonometry, of calculus in manner equal in
rigor to that present in the extrinsic parts of the 1955 onward,
modern mathematics curricula. The development also includes an
applied mathematics or extrinsic view of the decimal representation
of real numbers. The discussion of limits, continuity and
convergence in calculus begins with that decimal representation and
the question of error control in the calculation of expressions or
functions. LAMP includes set notation and concepts where that
aids comprehensions. See its development of counting methods,
probability theory and function concepts.
C-LAMP after extrinsically deriving properties of
real (and complex numbers) describes those properties as axioms in
algebraic manner using set notation and concepts. LAMP treatment of
trigonometry like the modern mathematics curricula is based on an
extrinsic viewpoints. But LAMP relies on its extrinsic development
of two ways to multiply complex numbers to speed the development of
unit circle and right triangle trigonometry, and in particular
speed the derivation and verification of trig identities, make them
less of a challenge and more accessible-inclusive. Easy
consequences of the two ways includes another proof of the
Pythagorean theorem, and trigonometric formulas for dot and
cross-products of vectors in the plane when the latter defined
using rectangular coordinates. On a less rigorous note, LAMP
exploits the idea that partition of a rectangle into sub-rectangles
gives two ways to compute the original rectangles areas to
geometric describe and generalize distributive laws for
multiplication of non-negative, and so suggest column methods for
the multiplication of polynomials and the multiplication of whole
numbers using decimal representations. From arithmetic
to calculus, C-LAMP offers an operational command of skills and
concepts, along with extrinsic thought-based development of those
skills and comprehension.
The thought-based development of LAMP is based on
suggesting and using rules and patterns, one at a time and one
after another, alone or in combination. LAMPS logic mains
deals with the direct use rules and patterns of the form B IF
A (one way implications) and of the form B IF and ONLY IF B
(two-way implications). The terms one-way and two-way stem
from a 1989 lesson in which a student name Flo made a comparison
with one- and two-way streets. I said YES and adopted her
implied terminology for the rest of that lesson. LAMP
students will be encouraged to see the difference between the two
forms. Here I advocate using the form B IF A in place of or
besides the for equivalent form IF A THEN B to help students see
the difference between the one- and two-way implications. Here I
may use the word WHEN as an alternative to IF.
Capitalization is optional.
The indirect use of logic in LAMP is limited to the
contrapositive form (Not A) IF (Not B) for implication
rules B IF A in (a) the discussion of the zero product
law for whole, real and complex numbers in the precalculus element
of LAMP and in (b) the calculus divergence of a series test: If the
n-term does not tend to zero as n increases then the series
diverges. LAMP employs the
decimal representation of nonzero whole numbers and the idea that
the area of a rectangular will not be zero to first suggest that
the product of nonzero numbers is nonzero, and hence to suggest
from the polar
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Communication, Reason
and Problem Solving
LAMP and constructivism differ.
LAMP development recognizes that students and instructors may have
private thoughts and that direct mind reading is not
possible. But people can draw and write their ideas on paper
and so provide an observable, repeatable and reproducible record of
their thoughts for peer review, correction and refinement.
While we may speak, hear, read and write words in an essential one
dimensional, discrete, sequential manner, the use of diagrams
and column methods in mathematics, and the reading and writing of
arithmetic and algebraic expressions with fractions, superscripts
and subscripts, record and communicates ideas in a
two-dimensional manner, a manner awkward to read aloud in a
sequentially, yet a manner that can be seen, if not understood, in
a glance. The drawing of diagrams and the drawing or writing
of expressions with horizontal and vertical elements provides a
visual record of ideas and 2D visual extension of our
memories. Whence steps too complicated for an individual to
do mentally may be done and recorded on paper for all, including
the doer, to observe and check. While memory and mental
agility is a plus in any subject, mathematics is based on the
communication of ideas on paper using words, diagrams and linear or
two-dimensional, arithmetic and algebraic expressions, ideas that
can then be followed and checked by the doer and others for the
sake of immediate or future self- or peer-review and, if
need-be, correction.
Mathematics is a discipline or art form based on an observable
forms of communication of ideas and reason. Instruction can suggest
or impose standards to aid and speed communication and
reasoning. And that good notation and clever or standard
presentation may aid the on-paper record and mastery of
mathematics. The constructivist suggestion that student
thoughts are for reading ignores the deliberate and systematic
development and verification of concrete, observable, on-paper
communication and reasoning. In this, drill and practice may
be needed, will be needed, for students to see the need to follow
steps in a method for solving a routine problem in a way that can
be reviewed later or immediately by the doer and by peers in the
form of fellow students, tutors, parents and teachers.
Meeting methods or strategies for solving problems of a routine
nature is one motivation for mathematics and logic education.
Precision in reading and writing is further necessary to
understand, select and apply a strategy in logical manner. Meeting
methods or strategies for solving many problems of a routine nature
in repeatable, reproducible and defendable manner (see
communication0 provides a base for routine and non-routine
problem solving in an observable manner that the doer and peers may
review to approve or correct. The application and
development of patterns, rules and laws in society is based on the
latter. Familiarity with earlier methods, their origins,
benefits and limitations, provides a better starting point than
ignorance for problem solving. Non-routine methods for
solving problems should be explored only when routine methods are
not known or are not satisfactory. That being said, the
problem solving abilities of students can be tested and expanded by
giving routine problems just beyond the reach of their present
knowledge for them to investigate. Finally, open problems in
society may still abound and may be given to develop the critical
thinking and to develop an appreciation of the limitations of
existing lines of thought, but there is a still a need to provide
drill and practice with routine problem solving methods, so student
can tackle common or routine problems in an automatic manner,
without going into research mode.
Constructivism with its assertions that student
comprehension is not observable; that testing is not reliable -
students may forget; that following rules and patterns to arrive at
conclusions is not a genuine nor reliable form of reason all point
to a pre- or post-empirical view of knowledge and the formal or
informal peer review process present in disciplines striving for
objectivity: law, science, mathematics, technology. Allowing
constructivism with those elements dominant in charge of education
in the latter disciplines is like putting the fox in charge of the
hen house, and telling it to guard the chickens. There-in
lies a contradiction. On the other hand, constructivist ideas
for engaging students may be worth exploring and testing in an
empirical fashion.
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coordinate introduction of products, that the product of nonzero
complex numbers is also nonzero. C-LAMP's lean treatment of
Euclidean geometry, assumptions and theorem employ one and two-way
implications. That being said, mastering the difference
between one and two-way implications may lead students to precision
or greater precision in reading and writing in their studies and in
their present or future workplaces.
The math-free discussion of logic in LAMP, an optional part,
identifies proof by absurdity with an elimination of
possibilities due to their immediate or implied inconsistency
with prior knowledge or axioms. The proof by inconsistency
argument appear to be of value in (a) criminal detective work -
an alibi eliminates a suspect from further suspicion; scientific
detective work where a hypothesis or its consequence are
inconsistent with prior knowledge or expectations; and (c) in
mathematics where tentative assumptions or their consequences are
inconsistent with prior assumptions. In reading a few detective
stories in the hope of finding literary examples of proof by
absurdity, contradiction or inconsistency, I saw endings
based on revelations instead logic, endings which the chief
inquirer would reveal facts or observation not previously found
in the text to imply conclusions to arrive at conclusions, all in
a I told you so manner. The exposition of logic in LAMP may
lead students to appreciate the benefits, origins and LIMITATIONS
of rules and patterns.
Finally, LAMP may develop the algebraic-logical maturity necessary
for the calculus development and beyond that, for an optional study
of modern mathematics by university students.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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