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whyslopes.com For Montreal Students: Head Start Math Tutoring is available from the site author. YOU are better than YOU think. Show yourself how:
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-/[]\- Logic
Mastery Do not leave here without it - 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 linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- 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. George Orwell: Is it nonsense for arts and disciplines based on and respected for carefully mastery of rules and methods, alone and combined, to face education reforms based on the supposition that mastery of rules and methods is not a sign of intelligence. Would you like to rewrite 1984 to include that angle? 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 twiddla.com to set up whiteboards to work with the webpage of your choice. Precalculus sites mathsisfun & purplemath are visually more appealling than this one. Do not go. 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.
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For calculus & pre-calculus: Visit the site calculus introduction, visit three annotated guides to calculus, or continue reading. Site previews of calculus will ease or avoid difficulties. Also see complex numbers - this link in particular give a unique veiwpoint, an alternate starting point for understanding and explaining where is the square root of negative one with the aid of polar and rectangular coordinates. Welcome to a big website. On the left are site lesson plans and on the right are site areas including four volumes 1A, 1B, 2 and 3 online in full with postscripts.
Quotes from Site Reviews
Volume 1, Elements of Reason, introduces site books and site objectives. The initial site was to provide appetizers or model lessons to ease or avoid difficulties due shortcomings in the development of algebra, logic and calculus. .
The current site aim in mathematics education is to define a clear and lean path instruction and self-instruction from elementary counting to advanced calculus. Visitors novice to expert with see different ways to understand and explain.
Math-free logic chapters in Volume 2 briefly, and the first part 1A of Volume 1A, Elements of Reason, more expansively, put aside mathematics to test and develop skills and confidence, the reading and writing with precision, all needed for studies and work in general and also for mathematics mastery.
Volume 2, Three Skills for Algebra and an essay, what is a variable, together remedy a silent and foremost shortcoming in mathematics from first use of algebraic formulas to calculus. Many or most arithmetic and algebraic expressions are better seen and read in a glance, and not aloud. This visual and silent nature of mathematics has led gifted student to master mathematics in nonverbal manner and confusion or lack of clarity in the verbal or spoken part of mathematics. A remedy begins in chapters 8 to 17 and in a postscript on what is a variable. See how describing or talking about numbers and quantities can becomee part of the common knowledge of mathematic before and then beside formal ideas in mathematics.
Volume 3, Why Slopes and More Math, in its first chapters, connect the study of slopes and factored polynomials and rational functions in senior high school courses to sign, zero and monoticity analysis of functions or ski hills y = f(x) while providing a more accessible route for introducing calculus and the full strength algebraic ways of writing and reasoning in it. Demanding the latter suddenly, a full strength, instead of being developed gradually identifies more than one shortcomings of calculus course design and delivery. Remedies appear in chapters 2 to 7, and chapter 14..Start with Volume 3 if you can, or visit the larger and wider Calculus Guide/Intro at this site. The intro includes more skills and concepts to go before, besides and after chapters 2 to 7 and 14.
When any shortcoming is recognized, when methods to ease or avoid difficulties are not rejected before being heard, new routes for learning and teaching can be explored and presented. Education and course desing committees, end your bureaucratic slumbers, open your eyes. Lesson Plans - Arithmetic to Calculus
DifficultiesDifficulties in learning and teaching mathematics and logic stem from
Logic mastery is a must for easing or avoiding difficulties at school and work due to imprecise reading and writing skills. Advice and directions below, and site lesson plans offer goals and objectives to adopt and exceed. Clear goals and objectives and lean, complete and full developments of skills and concepts are needed for mathematics and logic education and instruction.
Clear Goals and ObjectivesStudies and instruction with clear goals and objectives, and methods for reaching them, go further. Site lesson plans and site sections provide paths for this. Yet there is also the open question of motivation.
Many students and many parents do not see the need for education. Societies where education is compulsory need to go beyond the chant or mantra that education is good for students, and provide proof or evidence that parents and students will appreciate. Say why. Without it, education becomes a bureaucratic exercise, a demoralized one, in which teenagers are herded together to spend time in school, but not to learn. Societies and school boards which participate in compulsory instruction, attendance required, have an obligation to reach out to parents to explain why. Otherwise, education beyond primary school in industrialized and rich or relatively rich de-industrializing countries becomes questionable. Why bother? Quality and not quantity is needed in course design and delivery. Online Books And More Site Areas[Online Books and More Site Areas] [Study Tip] [Directions for High School Mathematics - Calculus Preparation] [Curriculum Shifts - Shorter, Better, Stronger]
Site Books Again:Volume 1, Elements of Reason, introduces site books, volumes online in full with postscripts, and points to a context and objectives for the books and further site material.
Volume 2, Three Skills for Algebra and an essay, what is a variable, in retrospect fill a shortcoming in how mathematics has been understood and explained from first use of algebraic formulas to calculus. The ability to describe numbers and quantities with words apart from and along side symbols has been side-tracked by the prevalence of arithmetic and algebraic expressions better seen, read, evaluated and read in silence in mathematics from formulas to calculus. Talking about numbers and quantities, that is, the first skill for algebra in Volume 2, is a pre- and co-algebraic talent whose emphasis aids learning and teaching from algebra to calculus.
Volume 3, Why Slopes and More Maths begins with a ski slope viewpoint of curves y = f(x) in the plane to informally introduce derivative-based analysis of where functions increase and decrease. This introduction, pre-calculus material, gradually, rather than suddenly, develops algebraic reasoning skills demanded in calculus. The same volume introduces a decimal viewpoint of error control as a context for the decimal or decimal-free definition of limits, convergence and continuity. These two innovations, fresh or recycled, ease or avoid algebra shocks in calculus instruction and studies, and in real analysis as well, and so remedy a shortcomings in the exposition.
Tips for Parents and Teachers
A general plan or redesign for secondary mathematics follow.
See too the advice and directions below. In this plan or redesign for secondary mathematics and calculus, years I to VI, the third year III is the year of motivation, a year which provides a context and reward for the study of mathematics. During years IV and V they may be more links to applications in consumer mathematics along side the use of algebra and/or trig in physics or chemistry. The year of calculus sets the stage for further application of mathematics in accounting and business, and in physics (engineering included) for efficiently describing and modeling investment and physical situations. Meeting the latter without calculus, a habit in university courses for students who have not mastered calculus, is laborious and most inefficient. The route to calculus is long but is it a door opener. Year III in this route, the year of applications of coordinates, algebra and geometry represents rest and recreation and motivation, to reward students for their effort so far, and to set the stage for further work.
Theories (skills and concepts) seen without examples give a vacuous knowledge. Mathematics mastery in particular further requires numerical and geometry drawing experience from examples and practice to put theory in context. Plans for reform given without examples to show how are vacuous in part and may be hazardous to education - a current complaint.
If education reform was a drug, it would be tried and tested, and clearly documented, before general distribution and general prescription. Reform in haste, repent at leisure. The end of streaming and the retention of enriched topics (apart from Euclidean geometry) leads to a fat curriculum where too much is suppose to be covered while teachers and students are overwhelmed by delivery style changes. Experts in pedagogical principles and generalities , apart from the elimination of Euclidean geometry, inherit, inflate and thus compromise or sabotage course objectives. Demanding too much leads to confusion instead of clarity, and to ineffectiveness. Course design and themes should be simplified. Site lessons and lesson plans point to cut and more effective paths for instruction. Sit Down and Study
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Thinking Part of Mathematics and Logic: 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 appears after inductive mastery of logic, that is mastery of implication rules If A then B and their use. The second kind follows the use of implication rules and definitions and assumptions, one at a time and one after another, to arrive at logical conclusion in a repeatable, reproducible and therefore verifiable manner. |
Volume 1, Elements of Reason, introduces all site volumes.
[Online Books and More Site Areas] [Study Tip] [Directions for High School Mathematics - Calculus Preparation] [Curriculum Shifts - Shorter, Better, Stronger] [References]
Preparation for calculus provides the motivation for many skills and topics in high school mathematics courses. Preparation for calculus is good preparation for most, if not all, arts and subjects at work and school that require some mathematics and logic.
Similar Directions: The earlier site preparation for calculus page (written earlier) offers similar directions in three different ways - lean, wordy and very wordy. The words comment on the development of ideas in the classroom or historically.
Computer Games: If you play 3D computer games and want to write your own, you will need a good command of logic, fractions, algebra and geometry. The same advice applies if you want to enter a business, trade or science.
Follow the steps below alone or with help. The review of fractions etc in step 4 should come after steps 2 or 3. Other than that, which step to put first appears to be a matter of taste. Site areas which do not appear in these steps contain further material - optional reading. On first reading, focus on learning how, and leave explanations why for later.
Put logic First (if possible). Read the first logic
chapters in Volume 2. Logic mastery will, we hope, ease fears
and difficulties, or if you have none, enrich skills and
knowledge. Volume 1, Elements
of Reason, introduces all site volumes.
Master logic carefully to
develop precision reading and writings. Skills and knowledge are
easier to obtain when you are able to read precisely what is written, and do
not assume too much. Marks in all subjects are base on your written
work. Precision reading will help you recognize errors in your written work
- does it say precisely what you meant.
Secondary I
and II Material
Meet the
role of fractions in algebra: Explore the site area Solving
Linear Equation with stick diagrams to further develop your
algebra skills - those needed for solving problems in one or essentially one
unknown, and see how fractions of line segments, the sticks, are combined
(added, subtracted, multiplied and divided) exactly in the solution of
linear equations.
The site area [Solving Linear Equations with fractional operations on Stick Diagrams] develops algebra and fraction skills and sense together in way that can read before or besides the algebra chapters 8 to 14 in Three Skills for Algebra . Teachers & tutors should look at these Effective Algebra Lesson Plans for more material & suggestions for consolidating algebra and fraction skills & sense - a geometric view of the distributive law.
Test your algebra skills
and linear
equation problem solving skills.
Remark: Steps 1 to 4 may be covered in junior or
senior high school, the sooner the better. The following steps are for
senior high school students and older students in college or adult
education.
Review or Develop Algebra and Fraction Sense and Skills.
Read (i) the algebra
chapters 8 to 14 Volume 2, Three Skills for Algebra. Volume
1, Elements of Reason,
introduces all site volumes.
The shorthand role of letters and symbols is meaningless for
many people in school and out. But the shorthand role is easier
to grasp when we first learn
to talk about numbers and quantities, and how
they may vary, before the use of letters and symbols. Doing that
would make algebraic ways of writing and reasoning clearer in calculus and
all of high school mathematics.
Chapter 14, Compound Interest,
in Three Skills for Algebra, develops algebraic skills with the aid of a
calculator. Calculators are useful but success and precision in mathematics
requires efficiency with fractions without one.
Alternate Between Steps 3 and 4 if you wish. Each one has a
different taste. The addition of animated graphic make Solving
Linear Equation with stick diagrams easier than before.
If you spend grades 1 to 11 or 12 in
mathematics classes without mastering fractions sense and skills properly
and efficiently, you have been cheated - several hundred or thousand hours
of your time has been wasted.
Optional but Recommended: (i) Visit the fraction
pages in the site area, Fractions,
Ratios, Rates, Proportions & Units, to check your fraction sense
(step 4 could have helped in here) and to see the justification of methods
for adding, subtracting, dividing, multiplying and comparing fractions. (ii)
Develop an algebraic view of problem solving with units and with rates and
proportions, binary or multiple, direct, joint or inverse. (iii) learn how
to carry units through solutions in a way that relies more on mechanical
skill in algebra than on thought. Here is an algebraic perspective and
clarification of skills and concepts in junior high school mathematics,
which may be read after steps 1 to 4 above.
The site area Fractions,
Ratios, Rates, Proportions & Units view of junior high school
concepts may help teachers & tutors develop skills and concepts. Senior
high school students may explore this area to review and reform their
understanding. Area material needs to be rewritten to make it readable for
junior high school students. Writing is an iterative process in which the
first draft is not always best.
Fractions are needed for algebra and beyond. In modern times, that is
today, we see and will see more and more cognitive experts and
curriculum advisors suggest the replacement of fractions and algebra
skills and sense development with calculator push-button
exercises in which the intellectual component of mathematics
instruction is eliminated to provide a child- and technology- centered
learning environment. Yet arithmetic mastery was and remains a sign of
intelligence in work and study.
Check & Consolidate your Arithmetic Skills. Do
asap, the first set of
arithmetic problems, chapter 7 of Volume 2, Three
Skills for Algebra, See too Simplification
of square roots. Logic
mastery asap is recommended for greatest benefit from site pages.
Aim for a logic-based mastery of mathematics after
arithmetic. That being said, arithmetic can be learnt by rote,
know-how without the know-why, provided you put aside your calculator and
learn the times and addition tables and learn to do arithmetic with
fractions and decmals (add, multiply, divide and subtract) in
an objective, efficient and automatic manner - arithmetic results
should be repeatable and reproducible, and you should know that an error
in one step makes all the rest wrong. Once you have a
logic-based mastery of mathematics after arithmetic, you can if you want
retreat to develop a deeper, logic-based understanding of
arithmetic, a retreat that could become easier, and a retreat that can be
woven in to the explanation of further mathematics for skill perfection
and enrichment.
Master Geometry without and with coordinates: Site
areas on Euclidean
Geometry and Analytic
Geometry offer senior high school students and teachers lean
logic-based development and connections of plane geometry, plane
trigonometry and functions of one variable. The site coverage of Analytic
Geometry does not include all that calculus requires, but is a start,
and the missing material can be found elsewhere.)
| Remark A: The treatment of Euclidean Geometry is not full, but it is enough to provide a logic-based consolidation of the skills and concepts seen in junior and high school mathematics, those needed to develop analytic geometry and calculus. The treatment of Analytic Geometry assumes results of the site treatment Euclidean Geometry with the assumption that real numbers alone or in ordered pairs may provide coordinates for lines and planes in space. The result is a logical, coordinate based, development of the key skills and concepts in analytic geometry, plane trigonometry and functions. The reliance seen here on geometric diagrams can be replaced and will be in studies of modern pure mathematics. Or, we could use the alternate route in Remark B. | Remark B: Step 6 follows the traditional path of defining trigonometric functions for acute angles with the aid of similarity postulates before defining them for all angles. This complex numbers introduction leads to trigonometry in general for all angles, with right-angle triangle, similarity based, trigonometry coming last. For the brave, that gives faster route for developing the senior high school mathematics which calculus and electrical studies requires. This route is leaner in that its reduces the need for Euclidean Geometry to a discussion of similarity principles. |
| Remark C: In the modern mathematics curricula of the late 1950s and 1960s, sputnik inspired, there is a fuller treatment of coordinate-free Euclidean geometry along side a general emphasis on logic. Geometric proofs were challenging - not student friendly. So Geometry was eliminated. But Euclidean Geometry was the traditional place for the emphasis of logic and Euclidean model for reason. Site logic and Pattern Based Reason chapters present the Euclidean model in a math-free way and do so to develop better study skills - or the precision reading and writing better work and study skills demand. | |
Test your arithmetic and Algebraic Skills: Try the remaining
problem sets in Chapter 7 of Volume 2. Get someone to identify all
errors in your answers in notation and comprehension, so you can learn from
your mistakes.
Optional: Explore the Number
Theory Site Area. Here is a mix of easy and challenging lessons, some in
sequence. If one lesson or sequence is not to your liking, try another.
Secondary VI
& VII Material
Meet or Revisit Calculus: Begins with the why slopes geometric
preview before the more algebraic
why slopes preview chapters in Volume 3. Then explore more of the site Calculus
Introduction. Volume 1, Elements
of Reason, introduces all site volumes.
Remark: The introduction points to simpler ways to
cover the first steps in calculus. Those simpler ways are for all. The
algebraic way of writing and reasoning is usually required suddenly in
calculus. The previews here and the latter decimal view of limits,
continuity and convergence provides a more accessible and less algebraic
demanding or shocking approach to calculus.Then the introduction includes
enriched material - the proofs that are often omitted in first courses.
Innovations here make the proofs easier to understand, but not simple. The
enriched material is for people who do not like to accept mathematical
methods without proof. The site area Real-Analysis-Decimal-View
(advance calculus) and the calculus introduction at this site emphasize an
error-control decimal view of limits, continuity, convergence.
Remark The Modern Mathematics movement of the 1950s
and 60s made calculus algebraically hard or inaccessible need-be by
following a decimal-free view prevalent in pure mathematics. Here is a
correction sufficient for students outside of pure mathematics that may
provide a stepping stone and context for the decimal-free, epsilon-delta
view of pure mathematics.
Remark: Steps 5 onward can be followed or explored in any order you like.
Learners at all levels need someone to review their written work for mistakes in notation and comprehension in order to learn from their mistakes. Every time someone (on your side) identifies a mistake, say thank you because now you know not to make that mistake again. Do not worry, your helper will be employed in identifying further mistakes. It is a win-win situation.
Volume 1, Elements of Reason, introduces all site volumes.
[Online Books and More Site Areas] [Study Tip] [Directions for High School Mathematics - Calculus Preparation] [Curriculum Shifts - Shorter, Better, Stronger] [References]
Site innovations for mathematics and logic education were initially developed to fill skill and concept gaps and flaws sensed in the high school exposition of modern mathematics curricula prevalent from mid-1950s to the 1980s in schools and colleges. However, exploration and refinement of ideas for learning and teaching points to an alternative thought-based development of high school mathematics (algebra, geometry, trig and functions) needed for calculus. The net result may be fewer but more effectives hours in high school mathematics.
These curriculum shifts could be the basis for a leaner and more effective mathematics instruction.
Two Shifts - clearer and effective ways to
develop algebra and fraction skills and sense: The puzzle of how to
introduce the algebraic way of writing and reasoning clearly and directly
was first met by in high school days 1965-70. Difficulties of fellow
students and instructor in understanding and explaining algebra slowed
the site author's education. The first algebra
chapters in the 1995-6 Volume 2, Three
Skills for Algebra, point to a solution - a greater verbalization in
mathematics in which the overlooked ability of describing or talking about
numbers and quantities is recognized and emphasized. That is before and then
besides the introduction of letters and symbols in algebra as
placeholders for numbers and quantities in calculations or their
description. The spring 2005 site area Solving
Linear Equations with fractional operations on stick diagrams also
introduces algebra in a parallel approach to the foregoing, which comes
first is a matter of taste, while consolidating fraction sense and
skills. The two approaches together provide a solid base for algebra
for students starting their teenage years, or later remedial
instruction. Algebra
self-instruction alone or with help allows student to
benefit immediately. For self-instruction, the algebra
chapters in Volume 2 are recommended first. Volume 1, Elements
of Reason, introduces all site volumes.
Another Shift - Complex Numbers & Easy
Consequences: Vectors & coordinates, polar &
rectangular, are used in a very simple, logical development of complex
numbers., one that implies a quick, logic-based development of senior
high school mathematics (and the use of complex number methods with ei
in technical and engineering schools.)
Technical note: Assumption that the head to tail addition of
vector described displacements in the line or plane is independent of our
choice of rectangular coordinate systems implies the distributive law for
real and complex numbers. In other words the geometric assumption that
the coordinate description of sum of displacements gives a new logical
development of the properties of real and complex
numbers in ways that simplify and provide a base for analytic geometry
and trigonometry - that favored in university program without
explanation. This logical development based on geometry covariance, an
idea that appears in relativity, provides an axiomatic
shift for mathematics education with consequence for high school and
college studies. See the logic chapter Islands
and Divisions of Knowledge for thoughts on multiple starting or entry
points in the deductive arrangement of ideas. Self-instruction in complex
numbers alone or with help allows student to benefit
immediately At the college level in engineering and physics, the
properties of complex numbers and benefits for trig via the cis
function were often presented as efficient shortcuts without proof. Here is
a justification that may accelerate college and high school instruction.
Yet another shift - calculus
re-arranged.: Calculus demands full mastery of logic, fraction
skills and sense, algebra, analytic geometry, trig and functions. That
demand provide a standard and goal for high school mathematics instruction
which needs to be emphasized as the coverage of more and more topics in high
school may distracts learning and teaching from the full mastery..
Even with that full mastery, calculus employs the algebraic way of writing
and reasoning at full strength. The site calculus
introduction employs geometric and algebraic previews, and decimal view
of error control in computations, to develop the multiple full
strength uses of the algebraic way of writing and reasoning
gradually and systematically in ways that should eliminate or avoid some
calculus perils, and allow more to go further. Calculus
self-instruction alone or with help allows student to benefit
immediately. Note in a recently seen discussion of the modern
mathematics curricula of the 1960's, there is mention of a slope-oriented
analysis which site geometric and algebraic previews may duplicate. If that
is the case, site previews are re-inventions and not new.
Numbers, Geometrically Induced.
Modern Mathematics Curricula in the mid-1950s and earlier 1960s gave an
axiomatic view of algebra and geometry. Artificial and authoritative
chains of reasons began with the assumption of algebraic and geometric
patterns and inconsistently followed the axiomatic development of pure
mathematics.- diagrams were used to explore geometry and trigonometry
while the decimal representation of real numbers, continuity and
convergence was banished. Further more, the algebraically shorthand
way of writing and reasoning was employed and taught by example rather
than discussed or introduced. The ability to talk about and describe
numbers and quantities was not recognized. The axioms were not for
questioning by students and teachers. Now leap forward three decades
to the constructivist approach in which students and teachers are left to
explore the artificial structure of modern mathematics without
clear, definitive guides since the latter would allow for authentic
learning - the subjective construction of comprehension, and without
testing because (i) subjective learning (individual comprehension) should
not be criticized and because (ii) success on a test one day does not
guarantee success on another test. And at the same time, constructivism at
the secondary school level builds on and vaguely and without
conviction follows the patterns and flaws present in the modern
mathematics curricula of three decades before. That being said, at the
primary school level, constructivism calls for inductive paths for
learning in which hands on experience suggests the properties of whole
numbers and fractions. The latter by a stroke of luck coincides and
even motivates the following curriculum shift.
Use geometric experience and assumptions in the small to suggest by
interpolation and extrapolation the existence of real and complex
numbers, their field properties, and the applied mathematics use of
coordinates to model and represent 1, 2 and 3D locations and objects.
Here the invariance or relativistic assumption that the choice of unit
length in measurement and in the coordinate representation of vectors should
not affect the result of vector addition implies the distribution of
multiplication over addition. Here the field properties of real
and complex numbers can be deduced or induced from geometric considerations
in a mixed mathematics manner that gives a practical alternative to
axiom-based development and codification of pure mathematics while also
providing a mixed mathematics context for the latter, and while mimicking or
rewriting the path to understanding and explaining real and complex numbers
before the latter. Formal set-language description of the field properties,
their assumption, could begin after their geometric induction or derivation.
Expert Instruction (Mastery Learning): In classes, grades of 50%, 65% or 80% in a sequence of assignments and tests say how well you are doing, but do not say what you have missed. If the teacher or marker identifies and correct all mistakes in your answers, you can learn from your mistakes, and you know what you missed. In my classes, I intend to make a checklist of skills and topics, so that I can record which ones have been mastered to report to student a grade - the percentage of skills and topics which appear to be mastered, and to track and report what remains to be reviewed by the student or re-taught. Efficient learning (more gain for less effort) might follow. But I am advocating here what I have yet to do in class, an expert approach to learning and teaching. Tutors too can be hired to follow this approach instead of being hired to improve marks.
A few educational paradoxes: Constructivism philosophy is correct in saying that learning is an individual affair - no one else can master a subject for a student. Students have to sit down a study and develop or construct their own comprehension or explanations. That being said, science and technology went beyond the personal explanation of how matters worked with the aid of feed back in which explanations or theories could refuted (shown to be false) or supported in a relative but not absolute manner. Constructivist writings which insist comprehension individually developed is not for correction, testing and refutation do so in opposition to the empirical and logical growth and correction of rule and pattern based methods or knowledge in business society, science and technology, and so introduce mysticism into education practice. Constructivist philosophers who insist that teachers should not be authoritative in the classroom by testing and correcting student skills and comprehension, should not be authoritative in their own manner by insisting on constructivist methods and should not be misleading in characterizing direct instruction as a form of instruction by rote, instruction that opposes critical thinking by students. Calls and methods for education or instruction to be authentic, engaging, genuine and relevant predate constructivism. Studies developing those methods are worthwhile, but the opposition of constructivism to clarity in education, its continual support cognitive dissonance lies in direct contradiction with efforts in many arts and discipline and in higher mathematics to elucidate matters via clearly stated definitions and clearly stated axioms (assumed patterns).
Thought Based Development, Partial or Full. Science and technology are too complicated today to be developed via critical thinking in the classroom, by hands-on student experiments and activities. Here the history of science, its peer review process and accounts of the scientific methods in research and development, the origins of ideas and skills, provide a substitute. Many elements of science and technology need to be used or accepted or tested in a plug and play manner. That being said, mathematical methods and patterns in contrast can be developed, implied and suggested by paper and pencil calculations and drawings. So students can be given a full informal if not formal development during high school, if not early college days, of arithmetic, algebra, trig and its geometrical foundations, and some decimal-based calculus. Sets can be introduced during this, where useful, to aid probability calculations and to aid the description of functions (computation rules). An introduction of sets where useful and not forced for high school mathematics will aid any later set-based exposition of formal mathematics.
Mathematics, science and technology can be taught in a plug and play manner. Give students rules and patterns to follow alone or in combination, and provide means for them confirm or verify results - to see how the rules and patterns give a useful or applicable body of knowledge. Here no philosophy is required. Yet the scientific methods for the development of knowledge or recipes can be described and illustrated though historical examples of what was tried and what worked or did not. That provides a context for further scientific critical thinking and development in engineering, sceince, and mathematics. But explanations in mathematics can be structured further. The initial elements of college and precollege mathematics from arithmetic to calculus can be suggested or implied wiht paper and pencil experimens and reflections using diagrams and calculations, general or particular, algebraic or numberical.. There-in lieas a self-contained account sufficient for the initial impure exposition of mathematics in schools and colleges. Mathematics,science and technology have their own form of impersonal or objective crticial thinking in which students according to abilities try to learn key skills and pattersn, so that the latter can be used precisely and carefully in routine and non-routine situations. In this philosophy of education, students - apprentices all - learn and build upon key elements of previous practices to minimize trial and error, and at the same time, they learn the art of using combining earlier skills and knowledge in a plug and play to obtain repeatable, reproducible and thus verifiable resul. Constructivistin putting individual comprehension on a pedestal appear to in opposition to empirical practces in science, mathematics and engineering - the latter includes many skills and trades - in which students have to see the empirical benefits and limitations of methods. That being said, axiomatic practice in management and politics appears to be too dogmatic - too presumptive. That calls for a different kind of critical thinking than appears in mathematics, science and engineering. The constructivist call for critical thinking should not be applied dogmatically to all disciplines in school and out.
Start site exploration with pattern based reason or logic
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- by rote
- with algebra
- videos- Functions - introduction
hindsight - composition aka
substitution -- Geometry, Euclidean - Correspondence of triangles, Triangle construciton, duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
- Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
- Logic
- First Steps -
Symbols in Logic -
Occurrence & Truth Tables - Indirect Reason -Indirect Reason More- Proportionality
- Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions- Real Analysis
- Decimal View of concepts and of proofs- Rules &Patterns in Science, Technology & Society - Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes & (?) derivatives
- in ratios & proportions - slopes & rates included- Complex Numbers & Vectors & Trig
- trig expression for dot & cross - cosine law
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