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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
For Instructor, Parents & Educational Authorities
Teachers: Site material stems from question met in 1965-9 of how to introduce algebra directly and clearly, a question answered in part by the two logic puzzles, three skills for algebra and why slope lessons in fall 1983 eurekas, a question answered fully and completed by the site preparation for calculus with the a little more material found in these algebra lesson plans, all based on inductive principles for instruction or the communication of skills. Mathematical induction and inductive principles for instruction both fail in similar ways. Such failures need to be avoided if an instruction is to be full and complete.
Site pages address the question of how to make skills and concepts, clearer and accessible, easier or less hard, in accordance with inductive principles for instruction met in 1981.While writing may continue, these Algebra Lesson Plans, October 2005, essentially complete the site mathematics program for skills and concepts from algebra and fractions to calculus.
Instruction in full accordance with inductive principles require for each skill and concept to be clearly and fully explained - get the point, quickly. Seeking, inventing and putting first tried and tested lessons, easily understood and repeated in the classroom, should be first priority in course design along side some clarity in the reason and motivation for course content. . The principles are very similar to those of mathematical induction. Knowledge of how induction may fail due to gaps is guide for lesson development.
The 1981 meeting was predated by 1965-9 question of how to make algebra accessible to my teachers and fellow students - General lack of skill in algebra on the part of fellow students and some of my teachers slowed learning in my classes, but while I was comfortable with the use of letters and symbols in reasoning algebraically, I lack the words to transfer my understanding to others, a sense of incompleteness followed, and I did not see in courses taken thereafter, a clear and reproducible introduction to algebra. But as student 198-1983, improving mathematics education, making it more accessible, was the responsibility of my betters, albeit I tried to address the algebra & logic problem in 1975 in voluntary contribution to a MgGill University open house, and in fall 1983, I invented three lessons, namely two logic puzzles, three skills for algebra and why slopes (a calculus preview) which effective for the most part then and thereafter. The last lesson is seen first in the name of Volume 3, Why Slopes and More Math, and later in the site domain name. A lot of work, mostly unpaid and unrecognized, has gone into thinking about mathematics education before and after site construction.
Barriers to Improving Instruction.
Ideas effective in the classroom fall 1983-89 could not be shared by this author as a college instructor. The same ideas are still not being used in classrooms due to a Chinese wall between lessons that work and education reform. The notion of seeking or inventing lessons easily understood and repeated to put first in mathematics course design is too vulgar, too much like common sense, for consideration in mathematics education reform 1970s onward in English speaking lands. The reason for writing December 1990 with an amateur status in education and mathematics was to report and explore ideas that worked. This website in its present state (October 30, 2005) not only report ideas, it defines a coherent program for the inductive to deductive development of skills and concepts from algebra and fractions to calculus in accordance with inductive principles for instruction.
Walk first, Run Later:
Site material supports direct instruction. Advocates of indirect instruction should begin with a a knowledge of how to develop skills and concepts directly and clearly, along side a clear definition of purpose to say what is essential and what is not in a discipline. Where a subject has not had a direct and clear explanation, indirect instruction faces a daunting challenge. Site material will ease that burden.
Education authorities facing a shortage of mathematics instructors should shrink the curriculum and number of hours spent on mathematics to focus on the needs of calculus, for the sake of effective instruction. Teaching less and doing that well may leave a thirst for knowledge and avoid alienating students. Hours taken from mathematics could be spent on physical exercise.
Student centered education is best served by learning how to explain material directly and leanly before trying to do so indirectly Instruction has to learn to walk before it runs in the constructivist style. Calls for engagement, authenticity, realistic, might be met in direct instruction by a focus on practical problems. Learning by discovery with an emphasis on problem solving is fine for teachers expert in that style. Such expertise should be documented -described and/or filmed - so that other teachers may follow.
While there is emphasize on indirect instruction in education for the sake of student motivation and engagement, the practitioner of indirect instruction needs to how to develop & define skills and concept directly and clearly. If the latter is not known or is not feasible, I fail to see how indirect instruction as in constructivism can succeed.
Missing the Point
Mathematics education reform and teaching training has missed the mark in focusing on new pedagogical styles while old difficulties in explaining concepts directly and clearly remain, where inductive principles for instruction are not respected.
The pushing of technology - spreadsheets and calculator programs in high school mathematics has been a distraction from good preparation for calculus and a false cure or distraction from the discussion of mathematics education shortcomings. The student who cannot do mathematics without a calculator has been misled. His or time has been wasted.
Too Much Hope, not enough reason: Architects risk failure or costly overruns when they insist upon previous untried or unproven methods for building their projects. The constructivist approach to education has some great banners, great calls for action and thus a great design, but the enthusiasm for constructivist is not tempered by the caution. It is based on the assumption or faith that in implementation existing teachers & teachers in training will become constructivist adepts by edict.
Unethical: Hope or promise-based education reform today can be compared to the early days of drug testing in which early results held promise and the need justified speed while ethical procedures for testing were not established. The constructivists are pushing a solution that has never fully implemented nor tested in mathematics, apart from ethical consideration, the consideration of the risk of failure.
While the standards or objectives were being written and rewritten, and held out as a model before the how-to had been fully defined. The constructivists today has saying the results since 1990 do not truly represent their movement because of poor implementation or improperly formed instructors does not acknowledge the incomplete state of their program while demanding and insisting on further experimentation.
The constructivist themselves in emphasizing the subjective nature of knowledge, respect for individual conclusions or constructions, are pushing aside the logical structure of mathematics in which chains of reason in logic and computation should lead to objective result, right or wrong, independent of the individual. A group of anarchist, irrationality, have been in control of the mathematics education standards and despite their calls for authentic, realistic and problem solving skills and situations, have made the standards content-free.
More Missing the Point
The US National Council of Mathematics Teachers, an unfortunate influence in Canada too, was taken over by psychologists a few years before the publication of its 1989 Principles and Standards. The latter called for a dramatic change from direct instruction to indirect instruction in which student would be led to construct their own deductive (?) comprehension. The NCTM calls for more realistic, more engaging, more authentic, more deductive, more logical and more accessible student centred instruction sound very good. But the 1989 standards and the more recent year 2000 standards are long on talk of the Orwellian 1984 kind, and short on action. The standard mention mathematics and call for activities for students to develop their own comprehension. But where is the model? Where are activities documented? Has any been testing been done and documented, so that others may repeat the successes.
The standards have pushed aside the question of what should be taught and replace it by the question of how mathematics should be learnt, that is, the advocacy of indirect instruction in which teachers not only have to understand mathematics well, a problem for many in the first place, but also have to understand to it in a superior fashion in order to replace direct explanations by indirect activities that are suppose to lead students to an understanding which should be respected even if it is wrong by pre-1989 standards of mathematicians. That is folly.
Student centered education sounds great, but its introduction should not throw-out the education. Calls for more realistic, more engaging, more authentic, more deductive, more logical and more accessible, and thus student friendly or if not centered education can be applied to direct instruction.
By themselves, the standard are more complicated to read and follow for a mathematician or a person well-versed in the subject. than old-fashioned, textbooks which provide a full explanation of high school mathematics. Focusing on the pedagogy while not covering what is to be taught betrays the preparation of mathematics instruction. It is putting the horse before the cart. For clarity in defining mathematics standards, what should be taught apart from pedagogy, see the Mathematics portion of English National Curriculum,
Mathematics curriculums should identify what is be taught in one document, and explore or document methods how in another. But the definition of the curriculum is too complicated. There are too many cooks, too many steps, too much, Too many Too many artificial, committee and bureaucratically determined criteria, that individual judgment (a constructivist aim) is suspended or squeezed in development textbooks and manuals for instruction. Mathematics course design would be better off with the competition between textbooks written in accordance with the judgment of individual authors, experts in mathematics and pedagogy, a competition based on ease of use and readability.
If the constructivist movement called for indirect method of teaching to be invented before direct methods worked, would that make sense.
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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