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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
Constructivism & Direct Instruction
Site material is written by a former college level instructor in the habit of trying to be the sage on stage (easier said than done in the past, but now more easily done) telling students what is expected by providing clear objectives, and then trying to develop those skills directly and clearly. Today, there is another fashion in education, especially at the primary and secondary level in which instructors are asked to find activities, situations and applications, which lead students to discover and become interested in mathematics, and so meet the objectives that used to be or could have stated and developed directly if not always clearly. I would like to see the new fashion combined with a clear end of course summary and consolidation of the objectives, so that student see how those objectives could have reach directly. Then students would enjoy the best of direct and indirect instruction. The prerequisite and safety-net for the new fashion is of course the ability to explain or develop all skills and concepts directly and clearly, with mastery of none left to chance.
A subject is not understood until students & teachers can be shown how to develop its skills and concepts inductively and deductively in others in a repeatable and reproducible, and therefore reproducible manner. The logical development or present and changing structure of modern scientific or technical discipline, mathematics included, resembles that of a sequence of constructions in which earlier construction serve as scaffolding for the current structure, scaffolding that may not remain as part of the current self-supporting edifice or construction, and so the history of the discipline is lost.
Direct instruction in the past has had it flaws. The observation or conviction that mathematics mastery is a natural talent points to an old incompleteness in the explanation or exposition of skills and concepts. Site coverage or introduction of logic, algebra and even calculus fills in a few gaps in the exposition to provide a firmer base for learning and teaching, the lack of which has slowed or harmed course design.
Most marks in mathematics come from written work. Errors in notation, logic and comprehension, incoherency and confusion, can be observed. Students need to learn the definitions and methods in mathematics so that they can combine the definitions and rules, one at a time and one after, carefully, precisely, logically and creatively to arrive at results that are repeatable and reproducible, and hence verifiable. Instructors have a duty to catch &gently correct errors in notation and comprehension indicated by written work, so student may learn from their mistakes, and learn how to recognize mistakes their reasoning and how to test their conclusions. While the mathematics teacher cannot read the mind of student to see inner workings, the student can demonstrate skills and knowledge through written work and becoming a tutor or teacher.
Behaviorist theories imply people learn from making and repeating mistakes to lessen the frequency of such mistakes. Behaviorist theory is consistent with many forms of direct instruction. This site advocates an inductive & empirical paths for direct instruction which are clearly-defined repeatable and reproducible, independent of the teacher and learner. This form is complete when it includes paths for developing and reinforcing skills and concepts with correction a necessary evil, gently applied for the sake of the student, so that mastery of no skill is left to chance in accordance with scheme for skill and knowledge correction and perfection.
In contrast, constructivist theories of education say students should build their own knowledge via teacher designed activities without being corrected as all knowledge is subjective, dependent on the learner, as correction may damage self-esteem, and correction may does not guarantee mistakes. Constructivist theories are thus in disagreement with the domains of knowledge that try to be objective, that is mathematics and science in particular. And in many schools, the interaction of teenagers leads to greater damage to self-esteem than the correction of errors in a students written work. Constructivism in its call for activities that develop and maintain interest and skills may serve mathematics education, but guidance in terms of statement of objectives at end of term at least is also need to obtain worthwhile, repeatable and reproducible results.
Cognitive or constructivist theories for education have many faces. The most appealing face calls for activities, the introduction of situations and applications, in which students see the need for skills and concepts, and want to learn them. That is an appealing alternative, if one can find it, to direct instruction. That positive aspect of constructivism could be combined with direct instruction. Activities involving situations or applications may engage students and give a context and driving force for meeting the objectives of direct instruction.
In education as an empirical art, constructivism may be admired for its calls to engage students and to artfully include students in the development and ownership or mastery of skills and concepts, but the face of constructivism which says all knowledge is subjective runs contrary to initial premise of public education in which authorities, parents included, point to and demand mastery of a common body of knowledge.
A second face, one I oppose, emphasizes and favors the subjective nature of knowledge, and say such knowledge is not for testing nor correction because testing is not reliable/. This face literally push aside or turns upside down the empirical method for discovery and verification of method-based knowledge in science, technology and business in which observable, repeatable and reproducible methods are sought. In education in my view is an empirical art. In it, students first learn via trial, error and reason to give teachers what the teacher or syllabus wants before or besides demonstrating their creative skills There is a common body of rules and patterns to be met and mastered, directly or not.
A third face of constructivism lies in its advocacy - the existence of textbooks and advocates without domain expertise who prescribe for all domains the constructivist approach for application before methods for the latter have tried and tested in any. This dogmatism in its application is not appealing and a recipe for disaster. The call for research-based methods in education that advocates the widespread use of peer-reviewed conjectures in the classroom without testing, or regardless of the results of testing, is impractical. Just as drug companies are asked to do field testing, so should pushers of educational reform.
Education based on constructivism research (or more generally, peer reviewed and approved conjectures) need to be tried and tested in schools before wide implementation. Calls for research-based practices in the classroom are hollow when the research consists of peer-reviewed conjectures. If an educational practice, the implementation of a principle, has not be described and tested in a repeated and reproducible manner, it should not be in the classroom.
A fourth face of constructivism within mathematics lies in its advocacy of or collusion with the use of calculators and technology. While technology can facilitate calculations and remove the drudgery of calculation with large quantities of data, the mastery of mathematics at the college level and the applications of mathematics in society at large still require an intellectual mastery of operations with whole numbers and fractions. Advance mathematics, say calculus and beyond, requires a mastery of functions, trig, algebra, geometry, logic and numbers in primary and secondary schools. Creeping reforms which push aside or de-emphasize the efficient mastery of fraction (fraction sense and operation) and associated number theory (primes, lcd, gcd) dilute the arithmetic base on which algebra, trig and calculus rely. The ignorant de-emphasize of arithmetic skill development has a downstream dulling effect on high school mathematics education. Subject experts have no say.
On the surface, constructivism includes many fine calls for action, but underneath, the constructivist viewpoint that knowledge is subjective and in the mind of student only, so that having students write tests is wrong from their viewpoint of respecting what each person thinks, and wrong from the viewpoint that testing does not guarantee understanding. So there is anti-empirical view of skill and pattern based knowledge at odds with the empirical nature of science and technology, and the right-wrong nature of some parts of mathematics. Regardless of whether or not constructivist viewpoints have been peer reviewed for the sake, dare we say, of objectivity in constructivism, Professors of Mathematics Education need a course in the empirical origins of mathematics, science and technology to appreciate the role of trial and error (and correction) in the development of rule and pattern based intelligence in many arts and disciplines. Mastery of rules and patterns is one of the aim of skill-based education in mathematics, logic and most further students. The mastery may be creative in a combinatorial sense or rule- and pattern-writing and -testing sense. The mastery may also lead to paths and results that are repeatable and reproducible, or appear to be. While not all is certain nor objective in the rule and pattern based deed and thought, the departure from objectivity advocated or implied collectively by Professors of Mathematics Education and Mathematics Education Societies with a constructivist bent knowingly or not may be in conflict with the views of mathematics professors - experts in the subject and adverse to the subjective nature of constructivism.
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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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
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