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OnlineVolumes
1, Elements of Reason.
-with foreword for all volumes
1A. Pattern Based Reason
- striving for objectivity, etc
1B. Math Curriculum
Notes
inductive principles etc
2. Three Skills for
Algebra
- unifying themes + study skills
3. Why
Slopes & More Math - previews & starter lessons for
elementary & advanced calculus.
See Volume 2 and 3 if you are preparing kids for calculus.
More Site Areas
1. Help Your Child or Teen
Learn.
2. Linear Equations
& Fraction Skills - Sec I to V level
3. Fractions Ratios
Rates Proportions Units - Sec I & II
4. Euclidean
Geometry - Sec IV
5. Analytic Geometry -Sec
IV & V
6. Number Theory. Sec V
&VI
7. More Calculus Sec V
& VI
8. Complex Numbers Sec
II to VI
9. Qc Maths Education
10. Secondary IV(?) math
11.Real
Analysis College level
12. LaTeX2HotEqn College level
13. Electric Circuits Etc
Sec IV+
14 Français - Sec III +
15. www.whyslopes.com Entrance
Level Pages:
This Calculus Preview and Chapters 2 to 6 in Volume 3 offer lessons to make the hard easier at the start of calculus, or to provide a context for the study of slopes and factored polynomials before calculus.
Your IP
Address &
how to use it
Three Links for Teachers:
(i) First
Year High School Math - Lesson Plans with Fraction Focus
(ii) Second
Year High School Math - Lesson Plans with an algebra focus
(iii) Algebra Lesson
Plans
Parents: Site Area Helping Your Child or Teen Learn covers 1. Speaking Skills, 2. Reading & Writing, 3. Preparing for Science, 4. Math Work Books, 5.Books for Parents, 6. Mathematics for ages 6 to 14, 7. Having Patience -you'll need it. Chaperone your sons and daughters through jumpMath workbooks for grades 3 to 8 along side site lessons for grades 7 to college and material elsewhere. Parents and teachers need to say no for small things of little consequence to build and maintain authority to say no for larger matters. Parental authority: use it or lose it, but do not abuse it.
Lesson Plans and lessons
Secondary I - fractions
& allied concepts (decimals, percentages)
Secondary II - Algebra (arithmetic versus algebraic methods, backward
use of formulas and proportionality equations)
Secondary III - to come(?)
Secondary IV - Functions to Trig & Statistics
Algebra Lesson Notes & Ideas for All levels
Still More on Mathematics Education, Etc.
For a Leaner Curriculum
Where mathematics education reform is too bureaucratic or too rigid to consider ideas that should count, more generations of students will suffer from gaps in course design at the secondary & college level.
Education reform has led to more and more topics being included in secondary school mathematics. while old shortcomings linger and new ones born. A lean mathematics curriculum would focus on fraction and algebra skills and sense, 2D geometry with and without coordinates, trigonometric, and logic, all as preparation for calculus. A lean mathematics curriculum might include some application to demonstrate the usefulness of fractions, algebra and coordinates, and so invite the further studies. Preparation for calculus is key to college or university the thought-based as distinct from rote, study and comprehension of accounting, science and mathematics. In secondary school mathematics, statistics, 3d geometry, nets for 3D polyhedra, and transformation geometry are digressions for learning outside of mathematics, in say courses on social science, art or technical drawing, if need-be. Including these digression in core mathematics programs dilutes the preparation for calculus and calculus-based studies in mathematics Inclusions leads to a loss of focus in skill and knowledge development. Lean mathematics instruction could and should focus on mastery of fractions, algebra, 2d geometry with and without coordinates, logic, and trig.
Cut, cut, cut. Do the minimum well. Then enrich once the minimum is well-taught. Further cuts or shortening are possible by dropping artifacts in course design and delivery, topics not required for further skill and concept development. That being said, teachers still have to cover topics demanded by local school authorities. Site remedies may be woven into lessons to support and enrich local curricula, lean or not.
Education, An Empirical Art
In empirical arts, practices with repeatable and reproducible results come first, tested via trial and error, while theories and principles come later to summarize, to codify, to refine and even enlighten the practices. While practices or sequences of them in some empirical or hands-on arts in science, technology and business, assembly lines included, may comply with principles and standards, even be connected and organized and designed around said principles and standards, the forerunner to such organization consists of experience where principles and standards in formation and adaptation met reality - success and failure included.
Education is an empirical art. We may not read a student's mind, how a student thinks or links together skills and patterns, yet we can observe and test student performance, skill by skill, concept by concept, and encourage, but not guarantee, mastery of standard calculations and standard arguments or chains of reason in algebra, geometry and beyond. In some disciplines, not all, there are right and wrong answers due to methods that lead to repeatable and reproducible, and thus verifiable results independent of whom-ever applies the method. Learning how to apply and combine methods carefully to obtain reproducible and thus verifiable results is an old sign of intelligence in many old arts and disciplines in business, trades, science, engineering, technology and bureaucracy. The latter is subject to the limitations of rule and pattern based thought and practices, and the critical knowledge that not all is certain in empirical based thought and practice.
Critical thinking in science and technology begins with an awareness that what we hope for, dream of or construct in our minds remains speculation or faith IF or WHILE it or its consequence cannot be observe or tested directly, and thus corroborated if not confirmed. The foregoing is a rebuttal to the constructivist theory of learning, the part which opposes testing, the existence of questions with right or wrong answers, and which says student knowledge, if individually constructed, should not be contradicted. Empirically sound education must oppose wishful thinking. That being said, constructivist methods for engaging, authentic, genuine material and the development of critical thinking could be incorporated into education as an empirical art.
More on Testing. Knowledge empirically found or tested is relative and not absolute. Instruction which relies on testing skills and concepts can only identify errors in the mastery of the latter while correct responses only confirm, but do not guarantee mastery. But the level of student competence in a discipline defined by skills and concept mastery can be estimated from the degree of difficulty, the unlikelihood of correct responses if skills and concepts have not been mastered, and comprehensive of a test or series of test. Here individualized testing may be informative that mass testing. Empirical soundness of instruction and testing, the issue of lessons and associated tests with repeatable and reproducible results locally and beyond, should not be scrutinized in an absolute manner. Cognitive theory should look at education as an empirical art.
Constructivism versus Empirical Methods
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After all is said, I found myself advocating an empirical approach to course design and delivery, an approach which may be combined with constructivist educational methods, those that work regardless of flaws in empirically unsound constructivist principles or theories, - principles and theories which imply subjectivity in mathematics and science, and beyond, which emphasize the empirical weakness of testing in education, if not in general, in place of the empirical merits. Constructivism with its advocacy of critical thinking in criticizing testing is contradicting the empirical basis of science and technology, the readiness to test in order to eliminate errors and so favor some success.
While we cannot read a student mind to see what has been constructer or understood or not, or how, we can in good empirical form observe, correct and mark what is written or produced by students. Continuous testing, probing and observation of student performance is part of a continuous educational process. Through test feedback and/or direct explanations, students learn to avoid or discount wishful suppositions or constructs in contradiction with their environment in and out of school. Thus schooling can shape students minds rigidly. Or, schools can present rules and patterns of various arts and disciplines, and indicate the origins, benefits and limitations of rule and pattern based knowledge, the presence of uncertainty, the open ended nature of many situations or problems, a necessary disappointment for those of us nostalgic for certainty.
Putting constructivism subjective views of knowledge in charge of mathematics and science education is akin to rejecting the form of critical thinking in mathematics and science developed since the 14th century A.D. The placement invites cognitive dissonance (confusion) for all involved - students, teachers and parents. Bon Appetite. |
1B, Mathematics Curriculum Notes, Chapters 1 to 12
Chapter 11: Primary School Mathematics
-Inductive principles for course design & delivery require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for what was missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.
Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.
| for online sessions by chance when I am
online or appointment when I am off. The first session (saying hello) is
free. While talking online, we may scribble on Yahoo, MSN,
Skype or
whiteboards. The twiddla whiteboards has a built-in browser for students,
teachers and tutors in general to import webpages and explore/scribble on
them together. It also has audio in theory. [Session
length depends on supply and demand. Call during off-peak periods
for better service. ] |
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