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Goals for Mathematics Education
The story of the race in which an overconfident hare is
beaten to the finish line by a slower but non-stop plodding tortoise
gives a lesson on determination for all students, slow or gifted.
See For Greater Clarity
in Mathematics Course Design after the three
goals.
1. From Logic to Precision Reading and Writing
For the first goal, and for the greatest help from site pages to
your work and studies, start with and see if you enjoy logic chapters 2 to 5 in the
online book Three Skills
for Algebra
Logic chapters
2 to 5 aim to ease or avoid difficulties in school and at
work by making you aware of the need for care and precision in
reading. Once you can read exactly or precisely, you will
have a better chance of seeing the mistakes in your writing - the
difference between what you meant to say and what you said. Precision
writing allows one to state the maximum possible without becoming
inexact or wrong. Parents: If your teen has difficulties
in learning, once a year, try to guide your teen, through these
logic chapters, as many
as possible, or ask your teen's school to test and check logic
skills.
For literate math-phobics, reading online logic chapters 2 to 5 may
postpone mathematics studies for a few hours while building skills
and confidence for work and studies in general and for mathematics
education too
Mathematics studies may be further postponed by
reading Pattern Based
Reason, an online book, the source of the logic chapters. The
question of what rules or patterns are reliable and which ones apply,
if any, leaves room for thought and worry.
2. From missing words or links, to unifying themes
For the second goal,
learning how to describe numbers and equations with words to make
algebra mastery stronger and easier to obtain
site lessons will change how mathematics is understood and
explained.
There has been a silence, a lack of words, in
mathematics because arithmetic and algebraic expressions are difficult
to read aloud in a way that listeners understand or follow the order of
operations. So expressions of many kinds are better read in
silence and understood nonverbally in a glance. There-in lies the
silence.
With the passage of time, the following program for reducing the
silence, or adding words to lessen its effect, may be improved. But
here is the first draft, likely to be immediately effective in college
and high school instruction.
Online algebra chapters
8 to 14 in site volume Three Skills for Algebra
show (i) how to describe numbers with words apart from and
besides symbols, that is the first skill; and show (ii) how to
use describe formula usage as direct or indirect, and describe
indirect use as a numerical or algebraic solution. Item (ii) in
retrospect points to a fourth skill for algebra in the volume.
The second skill is given by our ability to describe calculations
with words and formulas. The third skills is given by ability to change
the description of how numbers and quantities are computed via
replacement or substitution operations. Learning the four skills
or emphasizing as unifying themes in high school and college will
change mathematics instruction. The online essay What is a Variable
provides a word-based and word-only explanation, and so illustrates the
first skills for algebra, our ability to describe numbers, amounts and
quantities with words alone, before any formalism of modern
mathematics, indeed before the use of any symbol.
Recognizing our ability to describe numbers, amounts and quantities, and
to describe operations on formulas or equations provides themes to unify
mathematics education from algebra to calculus. I kid you not.
To lessen the silence, we use names, descriptive phrases and more words
in mathematics by naming equations and formulas (old hat), by learning
how to describe numbers, amounts and quantities before or besides the
use of letters to stand for them (a first site contribution to
mathematics education), and by learning short descriptive phrases for
common operations on equations and formulas - direct and indirect use,
numerical or algebraic solution. There-in lies another site
contribution to mathematics education.
3. From Geometric Quantities to Algebraic Sense
For the third goal, we observe people have greater comfort and less
panic in algebra when letters are used in geometric formulas to stand for
lengths, areas and volumes which can be drawn or sketched. Those
geometric formulas can be evaluated when the geometric quantities in them
are known, given or measured. But the formulas are understandable,
they provide arithmetic recipes to follow, even when the geometric
quantities or numbers appear in drawings, but with values not yet given
or measured. There in lies a start for algebra - several steps to
provide an geometric start for algebraic skills and concepts.
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The site three column method for solving linear equations with stick
diagrams to go from use of letters denoting a length in
formulas for area and perimeters to acceptance we hope of a
letter denoting a number in equation.
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Geometric
(area) views of the distributive law leads to methods for
multiplying and adding polynomials and also the decimals
representation of whole numbers.
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This complex number starter
lesson represents the last of several efforts in site pages, the
first appears in Volume 3, Why Slopes and More Math, to provide a
concrete coordinate-based development of complex numbers. Easy
consequence imply trig formulas for dot- and cross-product, and
algebraic proof of the cosine law.
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Geometric & algebraic
previews of calculus to explain why slopes & factored
polynomials appear, and doing so provide a gradual, rather than
sudden start to the full strength use of algebra in calculus.
Thinking Part of Mathematics
and Logic: There are three kinds of rule-based intelligence in
mathematics, logic and most pattern-based subjects. The first
kind met in primary school arithmetic consists of skills with
repeatable, reproducible and therefore verifiable results - results
that are then considered right or wrong. The second kind also met in
primary school consists of pattern or rule recognition. The development
or exploitation of the ability to recognize or suggest simply patterns
in order to predict the next element in a sequence. If the prediction
fails, another pattern is required. The third kind appears after
inductive mastery of logic, that is mastery of implication rules If A
then B and their use. The second kind follows the use of implication
rules and definitions and assumptions, one at a time and one after
another, to arrive at logical conclusion in a repeatable, reproducible
and therefore verifiable manner.
Pure modern mathematics depends on or assumes the ability of students to
think logically and algebraically. Pure modern mathematics does not
require geometric diagrams nor physical assumptions for its thought-based
development from axioms (assumed patterns, algebraically described) about
sets and numbers.
To develop student ability to think logically and algebraically, and to
introduce the geometric concepts in trig and calculus, concrete geometric
diagrams if not physical assumptions are required. And the modern
mathematics high school and college mathematics of the last half of the
20th Century while inspired by the axiomatic structure of pure
mathematics followed in practice, impure mathematics programs of study,
with some contortions or routes more complicated than need-be in course
design. The latter was due to the conflict, not then fully understood,
between the demands of pure mathematics and the demands of mixed
mathematics, say trigonometry, calculus and pre-coordinate synthetic,
Euclidean geometry. That was the situation outside of the United
Kingdom in other parts of Europe and also in Canada and the rest of
North America.
Greater clarity in course design from primary school to college calculus
level could follow by using manipulatives, physical objects and geometric
suggestions and assumptions to develop an accessible, mixed mathematics
curricula which leads to results in a repeatable, reproducible and
therefore verifiable manner, in an empirical, if not pure manner.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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