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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
June 3, 2006.
Constructivism Flaws
Postscript (July 4, 2006): Constructivism puts forth a smokescreen. It is against rote learning, it calls for critical thinking, and its calls for authentic, genuine and realistic problems and situations for students to discover and form their own knowledge. Yet (first but), the calls and conscience given by constructivism for educators to follow are not yet supported by documented methods (recipes) for implementations - those are to follow by trial and error in classrooms at student expense. Moreover (the second but) constructivism is inconsistent with mathematics, science and legal principles with (i) its support of subjectivity, the understanding constructed by a student, wishful or not, should not be corrected by teachers, so testing is wrong. (ii) its view that critical thinking is required but reliance on deductive reason, rule and pattern based, is not important; (iii) its view that mastery of rule and pattern based thought in mathematics, science and technology, and law too, is not a sign of intelligence; and consistent with item (iii) is (iv) its characterization of direct instruction (most higher education included) as a form of rote learning. A post-Luddite form of constructivism is needed.
Earlier Postscript: Constructivism in calling for students to be engaged and develop critical thinking skills with the aid of authentic, realistic or genuine situations echo calls of earlier reform movement in education - no objections there. Yet in mathematics, the figuring skills and the Euclidean Model for reason, the clear and direct, logical and therefore verifiable or correctable development of ideas or results (statistics aside). Here the constructive cognitive views of education appears to be conflict with the theoretical and empirical development of mathematics and science where people propose and nature corrects. Students of nature follow a behaviorist path in which explanations are constructed and empirically corrected in a repeatable and reproducible fashion not in the mind, but on paper with the aid of observations. While the prior knowledge or awareness of students can give a context for a lesson in mathematics or science or technology, teachers act in place of nature in correcting students construction, those provided on paper and not in the mind, in the hope that the same errors will not be repeated. The constructivism objection to testing (teachers can not read the minds of students, and success on tests does not guarantee further tests) casts education as act of fantasy rather than an empirical art. The constructivism objection to correction of student errors because we should respect the individual formation of knowledge, no matter how absurd, also casts the same shadow. While constructivism may develop methods and calls worth supporting, key of elements of constructivism are empirically unsound. Many schools of education in emphasizing a constructivist form for mathematics instruction do so at the cost of neglecting or rejecting the content, the rule- and pattern-based skill and knowledge essential for calculus and for learning in general in well practiced or empirically established arts and disciplines.
While people should construct their own knowledge as much as possible, in empirical and/or mathematical arts and disciplines, students need to meet rules and patterns previously found, whose discovery was not obvious, and whose verification in the classroom may be partial. whose application needs to be practiced, so that skill and confidence follows in a repeatable and reproducible manner. Empirical arts and disciplines rely on nature to correct errors and identify the limits of current theories or explanations. In engineering and science, students are in the business of meeting methods and theories that can be used for design and prediction, or creating such theories. Every such method or theory is a gamble as failed predictions point to the need for an adaptation or rejection of the method or theory, while verified predictions may confirm (make more likely) but not prove the validity of an empirical theory. Mathematics knowledge appears to be empirical (a function of our senses) and appears to be logical (a function of ideas recorded and developed on paper). In mathematics, there two standards for correctness. First, in arithmetic, results should be repeatable and reproducible, and therefore verifiable. Second in the theoretical development and justification of methods and ideas, the latter need to be develop by direct and indirect chains of implication rules, staring from given axioms (assume patterns, gambles) in a repeatable, repeatable and therefore verifiable manner.
The constructivist educational reform movement in calling for students to be engaged by authentic, genuine, authentic problems and situations echoes previous calls for reforms. Constructivist methods for engaging students via exploring and developing their prior knowledge of a subject are worth following. Yet constructivist rejection of tests or measurement of student abilities as being unreliable and being judgmental is empirically unsound. While students minds cannot be read, while measurement or observation of a skill or talent today is not guarantee that a student will maintain the same level of mastery tomorrow, and while tests may sample the student skills (a form of statistical quality control) instead of being comprehensive, education is an empirical art. The verification of student skills is a statistical affair. Reliability and feedback (correction) rises with the number of well-put observations. Training or skill development in empirical and logic based arts and disciplines aims for students mastery or display of skills and concepts in a repeatable and reproducible and hence verifiable fashion. Formation and evaluation in empirical arts and mathematics is or should be based on observation of student skills and practice. Performance standards need to be respected. The constructivist viewpoint that education should proceed by discovery, that knowledge is subjective, and hence not for correction, is inconsistent with the rule and pattern organization and development of mathematics and empirical arts and disciplines. The implication that personal knowledge and deeds are not for correction inconsistent with the judgmental, Euclidean rule and pattern based codification of geometry, modern mathematics and law. While constructivist methods are worth noting, the constructivist movement needs to be reconstructed in a rational fashion. Irrational parts, parts inconsistent with the hard sciences, need to be excised.
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.
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