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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, 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
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
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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.
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 www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
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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Mathematics in Context
Where is it going
Mathematics in context
The study of mathematics year after year may seem endless and pointless for
many students and teachers.
The key elements in high school mathematics prepare for calculus. Calculus is
the subject which is used to justify calculations and formulas in mathematics
and other subjects. It requires an efficient command of arithmetic without a
calculator. It also requires in full logic and the algebraic way of writing and
reasoning. Most elements of geometry with and without coordinates, and
trigonometry too with triangles and unit circles are required in calculus.
The connection of decimals, ratios, rates, percentages to each other and
fractions is part of arithmetic. Mastery of Statistics (means, averages, pie
charges) exercises skills in arithmetic and knowledge with fractions.
While calculators are useful, the ability to work efficiently with fractions
with no calculator present is required for algebra or the shorthand roles of
letters and symbols in describing calculations and showing how to obtain them.
probability provides an opportunity to review and reinforce fraction sense
and efficient operation with fractions - must for algebra.
Most other elements of high school can be presented in ways that support and
strengthen the key elements. Dilatations, if taught provides practice with
algebra and arithmetic operations, and may be used to imply similarity of
diagrams in radial expansions, contractions and inversions about a point or
origin. Spatial sense or three dimensional geometry in secondary can be
linked to crystal shapes, to technical drawing or drafting and to arithmetic and
algebra via volume and surface formulas, and via Euler relation between the
vertices, faces and edges of polygons in plane.
Statistics in the lower high school curriculum provides exercises with
arithmetic, the proper and improper gathering and interpretation of data and
associated graphs. Recognizing what is improper is a base for critical thinking
when numbers and graphs are presented. The prerequisite and context for
statistical critical thinking is a proper and efficient command of arithmetic
without a calculator and familiarity with computations with methods or formulas
lead to repeatable and reproducible results. The latter leads the notion that
numbers do not lie, statistics might. But there are some statistical
methods for which may be present only because they are required for course final
examination - not an inspiring reason for mastering them.
Mastery of high school mathematics with its emphasis on arithmetic without a
calculator (we hope) and the algebraic way of writing and reasoning provides a
key to studies in the physical sciences, physic and chemistry included.
The difference between the advance and ordinary student of the phyical science
comes in part from mathematical skills. The advance course requires a mastery of
fractions, what they are and operations with them, exact solution methods for
linear equations in one to two unknowns, and mastery of the quadratic formula.
Anything less implies difficulty for the student.
Engaging and Inspiring Teachers
The prerequisite to an engaged student is an engaged, inspired and informed
teacher.
The foregoing may provide a context for high school mathematics – lines of
reason that tie the various elements together. The fact that five sixths of high
school students in secondary IV do not have the arithmetic skills or fraction
sense named a prerequisite for algebra in objectives for the first year of
high school points to a difficulty. The above context engages me and therefore
provides a framework for engaging students in the curriculum, so that it becomes
a more meaningful sequence of skills and concepts.
Engaging Students
One way to engage student could be to point out the appearance of
fractions and calculations in their daily life to give them more participation
and ownership of the lesson.
More generally to improve study skills and work habits would be to start this
is a guided discussion with students of how one learns or masters a subject,
discipline or collection of skills and knowledge in general. The discussion may
span lessons and separate times. A consensus on what they expect from the
teacher and how to favour those expectations by cooperating in their own
education is a goal. For example, cooking, swimming, body-building (like
it or not), nordic skiing, bicycling, drawing, writing, figuring (do arithmetic
without a calculator) are examples which students could discuss to provide them
insights into the learning and teaching process. Here the question for students
to considers is how they would teach or communicate a skill or concept, an
isolated one, or a sequence. The question of how to teach another an
organized set of skills or an organized body of notion may lead to students to
the model of skill and idea verification and development, one skill or idea, one
skill after another. That expectation gives a base for learning mathematics and
support for the statement of goals and standards to guide and focus their
work.
The site before you was developed to support direct instruction, the case
where the teacher explicitly controls the direction of a class and what is in
it. That may lead to a teacher or subject centered classroom. The
student-centered classroom calls for more students participation, more realistic
and more authentic examples and projects to engage the student's interest and
cooperation in his or her learning. The instructor in the latter case has
to provide or limit the examples and projects to the broad or narrow subject or
discipline or them under construction. The notion that a student can
rediscover key skills and concepts is false. Those key skills and concepts are
the product of many investigations and of thought on how to arrange the findings
into coherent islands and bodies of knowledge. None the less, some combination
of teacher led and student led activities which allow the student to beyond the
passive reception of ideas and concepts is called for. How is another question
which ideologues for pedagogical methods expect the instructor to
construct. Direct instruction requires less preparation than instructional
methods which call for student participation in that direct instruction only
requires content mastery in a discipline and lines of reason and skills to
present and verify. I would recommend teacher training colleges in the
first instance focus on content mastery to set a base for the more complicated
or more involved student centered approach.
Carrot and Stick
The workplace usually has a firm organization based on the meeting the
requirements of the employer by coercion, do this or else be fired - lose your
income. the school classroom in many English speaking lands do not
have a firm organization. Nominally, instruction is for the sake of the student.
However, education is often justified by the need for for better educated
population for the sake of a competitive or suffering economy.
For students, compulsory education for 10 years or so may be followed by the
uncertainty of the workplace: will the student be employed or not? will
education help? Education would be more authentic or appealing if students saw
favorable consequences, rewards or reasons for studying. Education for its own
sake may not be appealing to a student and the student's family without explicit
authentic and realistic advantages Not enough hope dims enthusiasm for
education.
The young student goes to school with enthusiasm, a curiosity and a will to
grow-up (so I imagine or recall). But belief in education is somewhat like
belief in Santa-Claus. It may be transient. In the end some students or
most serve time in high school, suspecting it may help in a general way,
but not being exactly sure how. Authentic consequences to education would help
prolong student enthusiasm and patience with being educated.
More notes - Personal
As a mathematics instructor, I enter a classroom where most students who
would take another mathematics course but for mathematics being required for a
diploma or for future studies. I have to motivate them and provide a context for
the task ahead, namely skill and concept verification and development reaching
for the goals and standards. The first section above mathematics in context
gives a context for my instruction. The call for inclusion of authentic,
realistic, cross-discipline examples, activities and themes cannot be criticized
save for the demand that I the teacher to find and invent ways to comply. The
callers do not provide a manual. At the same time, mastery of my subject
calls for students to see definitions and explanations, and do exercises to
confirm and reinforce those definitions and explanations and to build general
thinking or general problem solving skills in lines of reasoning that define the
discipline. Just as the body builder does exercises with no immediate
consequences, so must the mind-builder. I am looking for ways to engage
students to give them the goals and standards, and hence will to follow a
sequence of exercises, listening to definitions and explanations included, whose
net result in a repeatable and reproducible fashion for the patient at least is
mastery of mathematics and its logic. The next time I teach I hope to
maintain a written record via checklist or rubrics of the abilities of each of
my students, so that in dealing with each, I know at glance where the weaknesses
and strengths lie. Individualized or differentiated instruction may follow, time
permitting, or as needed.
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www.whyslopes.com
Mathematics Education Essays
57 or so
Area Entrance & Hub
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links
Help Me Learn/Teach;
- Algebra
words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course
Design & Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construction, 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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