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Welcome to the mathematics education revisited (essays, rants and
proposals) site section. In the past essays and rants used to be spread
though out site sections. Now they are all located in this site
area. Section material represent reflections and reactions to what has
been done in the past. When I first heard of constructivism in the
1990s, I read its then and revised principles and standards (NCTM
emitted) to see what proposed in the hope of supporting local trends with
a few methods (lessons that had worked) because I did not have an
academic position, and prospect for one were dim. Instead I found
myself in a counter-trend position with ends and values for mathematics
education strikingly different from dominant position and more closely
aligned what is called the mathematically correct. At the
same time, constructivist calls to engage students, to provide rich
learning problems and environments and to empower students through such
environments had a strong appeal. They represented an ideal to
strive.
My exploration and expression of ideas and methods for
instruction stemmed from an observation of steps too large or gaps and
eventually inconsistencies in skill and concept course design that the
constructivist movement inherited from earlier efforts. I had some
remedies which I felt compelled to explore and report. While the
latter was motivated by the difficulties of my students 1983-89 while I
taught in college posts, the remedies posted online 199502010 are too
late for past students, but may be timely for their grand-children. That
is another case of better late than never.
That being said, the constructivist principles or
standard that true knowledge is a private affair, located in the mind,
apart from reliable observation and testing, and not to be challenged
because everyone's ideas and reflection should be respected, appears to
be in contradiction with the old-fashion ends and values that called
for observable skill development in a tangible or observable
manner. Each of us in a daily lives from cooking to driving has
and follow skills or routines to obtain repeatable and reproducible
results. In that, there is an element of rote learning. In rules and
practices are given and followed carefully (we hope) to avoid mistakes.
Performance, know-how, observable skill development, does not require
comprehension. They may be learnt by rote. For material ends and
subjects in education, some practices may be learnt and combined by
rote. But there may be intelligence or observable skill in learning to
how apply implications, rules and practices carefully (avoid mistakes)
and in mastering the further super-practice of how to combine
implications, rules and practices to obtain more, all in an observable
and recorded manner for peer- or self-correction.
The site two level framework POMME for Progressive (meaning
step by step) or Practical Observable Motivated
Mathematics Education is based on teachers and students being given or
offered clear ends, values and methods for skill development via
small and alternative step likely to work and likely to ease or avoid
common fears and difficulties. The focus of instruction is on
credible and reliable skill development. True skill development has
to been seen and reliable to be credible. And this student
confidence and self-esteem comes learning to do. The rich learning
environment and problems currently sought to support constructivist may
also lead to projects and paths that enrich and motivate observable and
verifiable know-how development. POMME tangible ends and values
provide a context and motivation for its methods, while it methods
support the ends and values. In that, awareness of the domino
effect of approximations and errors in figuring and reasoning may be both
a method and also an end and value for instruction. So ends, values and
methods for instruction may overlap.
Before the advent of POMME, many ideas were explored in formative
essays. The essay still include reflections yet to be included in the
implementation of POMME. Just for the record, this site has existed
for a three to four years or not more.
Mathematics Education Rants and Essays
Section Content.
About this Site
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About POMME: In mathematics education, the
question of which way to go is answered here by a statement and
balance of ends, values and methods in a two level framework POMME
for Progressive (meaning step by step) or
Practical Observable Motivated
Mathematics Education. The first level for the
education serves common needs of life in the streets, or at home or
at work, in many societies - developing to de-industrializing. The
first level focuses on student centered skill
development. The core of the second level represents
preparation for college programs in scientific
fields.
The second level core is subject centered in that
students learn skills and concepts that may have intellectual
value, but more importantly to society, prepares some not all for
work in mathematical arts and disciplines - engineering,
technology, business and money matters, and/or mathematics
education itself.
The second level core is not inclusive where
observable student performance in mathematics and other disciplines
is employed for student selection by college programs. That
selection may be described positively via talk of competitive
spirit and giving the most able a chance to continue in the few
space available in higher education. It may be described negatively
as giving preference to some, not all, in terms of
opportunity.A similar question of competition versus
inclusion is met in the question of whether or not athletic
activities are for competition or general health.
The first level aim to develop know-how and
work habits with take-home value as long as that development
remains simple. We may may also say the role of the second level
is provide the less simple know-how with take-home value, and do
that as early as possible for the sake of student centered
instruction. That may offset content concerns, that is the
mathematical demands of further instruction and/or competition
for places there-in. While I prefer inclusion (helping as
many as possible) to competition (selecting the few), I dislike
more the existing state of secondary mathematics in which topics
and skill development are not explicitly connected to common
needs (even if they be years or decades later) nor explicitly to
the needs of college programs in scientific and other fields. The
ends and values of the two level framework may not be optimal,
but if that is all we have to offer, they should be stated
clearly and strongly. That being said, students gain
self-esteem or confidence by being shown what to do in an
observable ways, ways chosen to aid skill development and not to
distract from it. The confidence and self-esteem that may
follow from learning to do (see site methods to make learning to
do simpler and richer) may compensate where the ends and values
of instruction are not perfect.
Besides ends and values, POMME or site material
includes content innovations and different starting points for
skill and know-how development. Those starting or entry points are
chosen to facilitate, optimize and not impede the development of
skills and practices, those appearing in daily life and those
appearing in scientific fields. Different starting points and
starter lessons, all motivated by explanations of common content
difficulties apart from context, make quantitative skills and
know-how development richer and easier for learning and teaching.
POMME and site material stands on and reacts to earlier paths for
course design, on their strengths and weaknesses. Where the initial
aim in writing was to address a few initial concern or content
gaps, over time writing drifted to a more daring and bold endeavor
of providing not only more and more different pathways for skill
development, but also to provide context. Without that,
mathematics skill development year after year, no matter how
refined or polished, serves no ends and values.
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Site History and
Content - through site reviews 1995 onward.
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Site Eurekas - Site Highlights,
an old view
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Site Origins
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About Site Lesson
Plans - Another tour of Site Content
Lesson Plans, Aims and Goals (Ends, Values and Means?)
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Three Aims
for Students - Ends and Values
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Three Goals
for Mathematics Education, etc - Ends, Values, Unifying Themes
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Lessening or
Avoiding Algebra Difficulties
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Algebra Lesson
Plans
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Algebra,
Geometrically
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Mathematics
Curriculum Shifts
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Advice and Suggestions for
Course Design and Delivery
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Teaching Tips - from fractions
to Calculus
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Math Education Perils
(Arithmetic, Algebra, Calculus)
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Talk the algebra talk
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First Year High
School Math - Lesson Plans with Fraction Focus
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Second Year High
School Math - Lesson Plans with an algebra focus
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Third Year High
School Math - Lesson Plans with a Focus on Slopes
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How Letters Appear in
Mathematics
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Map, Plans and Drawings, a
multi-year project
Links
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Links -
Just a few.
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Activities to Engage
Students - links to explore
Ideas and Principles For Instruction and Educational Reform
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Inductive Principles For
Instruction - systematic skill and concept development.
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Fairness
in Education - requires systematic development of all skills and
concepts.
Can education be fair if students are tested on natural
talents instead of developed ones? Mastery of a skill, say
the algebraic way of writing and reasoning, is regarded as a natural
talent when and only when we do not know how to systematically
develop that skill or concept. Site material reduces the number of
natural talents required in the mastery of mathematics. Find
the four skills for algebra in chapters 8 to 14 of Volume 2,
Three Skills for
Algebra, to see how to develop the
algebraic way of writing and reasoning, and thus make mathematics
fairer.
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Apprenticeship in art, trades and disciplines, a classical view.
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Education is an
Empirical Art
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Key Notes and Themes
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For a Leaner
Mathematics Curriculum
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Need for
a Mixed Mathematics Curricula
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Extent and
Need for Quantitative Skills depends on your society
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Ways to be a Better
Instructor - Ideas and Methods - try with caution
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Four Ways
to Improve Education Reform, and avoid disaster.
Logic and Reason in Mathematics
Mixing Rote & Thought-Based
Development
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Cultivating
Intelligence - Why value careful mastery of rules and patterns,
steps and methods, practices, in a repeatable and reproducible manner.
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Multiply
Kinds of Reason in mathematics - Essay I
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Multiply
Kinds of Reason in Mathematic- Essay IIs - On the hierarchical
development of rules and patterns, steps and methods, and practices in
pure and applied mathematics (mixed mathematics). What is proof? What
options are there for a thought-based development and verification of
college and pre-college mathematics?
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Theory of Knowledge -
Stories, Longer and longer
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Formal or Informal
Peer Review
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Education
in Mathematics, Science and Technology - All based on empirical
verification and empirical skill development and verification. But in
mathematics we can offer a full thought-based development while in
science and technology, we can introduce the scientific method and
introduce lab equipment, but can only provide a full-thought based
development through visits to the lab and library. The lab alone is
insufficient.
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Maths
Instruction in General - Three Goals A B and C to Set for Student,
Supporting those goals and why rewrite the curriculum
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Operational
Viewpoint - Aim for an Operational Command of Mathematics First.-
For students with no immediate interest in the know-why, a focus on the
practice, an operational command of key skills and concepts may make
comprehension later of the know-why easier and more appealing. The
calculus teacher may says to students - learn to do now and to
understand later.
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How to Set
Standards for textbooks and course materials - Need for
Inspection by University Domain experts outside of Education Faculties
to ensure bureaucratic course design and textbook composition does not
lead to nonsense in mathematics education.
Teacher Training
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Teacher Certification
Issues and Cautions
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Math Ed.
Professors - Mathematics Background of, Trust but verify - see transcripts. Knowledge should be
above or beyond calculus. If that is not the case, explain why not.
Archives:
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About this site - old version
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Old Site Entrance -
2010-05-1
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Yet Another Old Site Entrance
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Maths Ed
Stopping Rule - 2010-04-25
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A New
Mathematics Curriculum - 2010-04-29
Challenges for Education Reform
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Five Decades make a
difference
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Managing Reform - Assigning
Responsibilities. (Should anyone be responsible? Should anyone be
in charge? Is reform headless?)
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Mathematics in Context
- What Context?
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What
Should be Learnt and When?
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Grouping
Students - Streaming?
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Learning Takes Time and
Effort
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Making the Hard Easier but
Ignoring how and so missing the Point
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Hook, Line and
Sinker - Mathematics Education Inconsistencies - Reform in
North America
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More on Mathematics
Education: Covers: For a leaner curriculum, Education an
empirical art, More on testing, Constructivism versus Empirical
Methods.
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Four Skeptical
Essays on Constructivism Revisited - Incompleteness
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Euclidean Model
for Development. Damage Reversal
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Educational Follies -
Learning By Discovery incomplete, cannot work, compound difficulties.
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An Educational
Inconsistency.
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Modern Education
But teaching by indirect instruction requires not only a knowledge of
what can be taught directly, but also a knowledge of how to explain all
elements indirectly. Anything less invites or compound difficulties.
Ouch.
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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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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