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Welcome. The ends, means and values of mathematics education represent a
jigsaw puzzle with some pieces missing, or as yet undefined. This this site area
is a depository for some of the pieces, and ideas on how to find the missing
ones.
The June 2010 composition of POMME implies half of the essay below are
moot. . POMME, its outline and its ends and values, says how to put
the jigsaw pieces together, those previously online and those yet to be
written. Site growth and pruning will be begin with the composition of
lesson and the description of lesson plans or options for the mid-level
applications of POMME. Then the rest of site material, including some ideas
below, will be recombined or rethreaded to provide paths and in them a lower
bound for upper level secondary or early university instruction.
Most paths will develop key material (starter and further lessons) but some
paths given to reinforce and refine skills will be marked as optional, as
possibilities to consider or avoid. The question of which way to go is answer
here by a statement and balance of ends, values and methods. Here we avoid
ends and values for methods for supporting them do not exist.
About this Site
Paths for Mathematics & Science Education, an
2008-11-16 essay
Via the school of hard knocks, I have
seen that student may prefer rote learning and dislike explanations which
start with the supposition that explanations why are key to mathematics
mastery. Despite my aversion to rote learning, I see aspect of rote
learning in (a) Calculus instruction where students are told to learn to
do first and to worry about the theory second, and where technical proofs
in are inaccessible, or largely slowly, where the logical structure is
based on theorems stated with out proof, axioms or assumed patterns given
without motivations; and (b) before calculus in the mastery of
arithmetic, methods for addition and subtraction may be explained in
all or part while methods for multiplication and long division are
given without, and so learnt by rote.
When the mathematics teachers offers proofs in a
classroom or motivations or chains of reasons to support the use of method
or to reach the statement of a theorem, students may object on the grounds
that school authorities would not ask instructors to present false
formulas or false statements in class. Hence proofs and explanation why
are not needed. There-in lies a student argument for rote learning and
against the appearance of proofs in mathematics.
Mathematics education in general may provide
methods and statements to met and master via (I) rote -that is by
presentation of methods and patterns to use; OR via (II) inductive
learning based examples that draw on and provide experience,
hands-on or on paper, to provide comprehension in all or part (hand waving);
OR via (III) deductive learning that draw methods and conclusions from
chains of reason which formally or informally depend on earlier knowledge,
that earlier practices or axioms (explicitly assumed patterns).
While we may consider deductive education to be the highest form of
learning, that education is based on axioms or assumed patterns,
usually algebraically described, and which require mastery of the
algebraic shorthand way of writing and reasoning alongside the ability and
patience to follow chains of reason or implications. But this edifice
stands on axioms which are given, or drawn from experience - that of the
students or that of course designers and their ancestors in course design
and delivery. In essence there is a bootstrap operation. Deductive
codification begins with stated axioms, typically but not
necessarily given for rote learning. That being said, site lessons
via examples provide or indicate an inductive development of high school
mathematics in which axioms (assumed patterns) - those needed for a more
secure deductive development of the discipline - are drawn or suggested by
example based experience.
Mathematic education besides off-paper measurements and
perceptions begins with on paper mastery of counting, arithmetic and
drawing methods, methods that lead to observable, repeatable, reproducible
and if need-be, correctable, results and thus feedback to students. The
ability to apply a method in a careful step-by-step manner to obtain
repeatable, reproducible results is the first source of skill and
confidence in mathematics. The combination of methods, implications
rules A IF B included, in sequence one at a time and one after
another, forms chains of reason or action that can be followed to
again provide repeatable, reproducible, observable and correctable results
on paper or off. Developing the ability to combine rules and patterns to
obtain further ones, theorems included, is part of mathematics and key to
its deductive codification and foundation, modulo the limits of axiomatic
systems.
Within an axiomatic deductive framework, each of
it statements is subject to testing or the following question. Can we find
a direct or indirect chains of reason from the axioms - the explicit
assumptions of the framework or code - that imply the statement or
its negation. Those chains of reason provide a proof. In
English common law, the accused is assumed to be innocence and the
accusation or suspicion false until there is clear enough proof of guilt -
a convenience for the defendant or accused. In the French civil law, the
accusation or formally statement suspicion is assume to be guilty until
proof of innocence is available - a convenience for the prosecutor. In
contrast, in mathematics a conjecture or formally stated suspicion
is not presumed to be false or true.
In mathematics education, a rule or theorem that a
student is asked to meet and master is likely to be true, modulo the
axioms and practices of deductive logic in the discipline. The
student objection that following the chains of reason and verifying each
step there-in is not needed represents a plug-and-play approach to
mathematics and division of labour between the developers of mathematics
and those who might apply, including the students who object to meeting
and following proofs, or providing them. An operational and
empirical command of mathematics for the home and for the workplace
is possible in a repeatable, reproducible and observable manner by rote
learning, and without proofs. That operational and empirical command of
the necessary mathematical methods should be an aim of mathematics
education. In it, the key to deductive reason, the ability to
combine rules and patterns to imply further one might be present in that
operational command. Then a theoretical command of mathematics in
addition requires mastery of the proofs, or the ability to produce proofs
within a given set of axiomatic and logical practices. Full comprehension
of the axiomatic method may be based, as indicated above, on drawing the
axioms themselves from experience of students - that provided by a careful
selection and discussion of examples, along side a deliberate and earlier
development of algebraic, shorthand, ways of writing and reasoning.
Those could be prerequisites to the axiomatic development. For
classrooms full of students (a) wanting to learn how to to do and
nothing more, and (b) wanting to understand as well, course design and
materials may cater to (a) in class, and provide full or fuller support
for (b) in further material for reading or viewing. There-in lies the site
objective.
Mathematics is art or discipline whose methods do have
to be plug and play. Thought-based development is possible. In contrast,
the methods of science and technology can be described in the
classroom, and supported in part by basic instruments -
mechanical preferred - and simple experiment that do not involve
plug-and-play components. The question for science education is how to
minimize the role of plug-and-play components in the science lab to
maximize the hands-on experience and evidence for scientific process in
the high school and/or college lab. Unlike mathematics, where plug
and play can be avoided, science education and labs must rely on
plug and play elements - elements that are beyond the reach of the
classroom lab. That irks. |
- Site History and Content
- through site reviews 1995 onward.
- Site Eurekas - Site Highlights, an
old view
- How this Site Differs
- Reactions to Site
Material - comments & questions, good and bad.
- Site Origins
- About Site Lesson Plans
- Another tour of Site Content
Challenges for Education Reform
- Five Decades make a difference
- Managing Reform - Assigning
Responsibilities. (Should anyone be responsible? Should anyone be in
charge? Is reform headless?)
- Mathematics in Context -
What Context?
- What Should
be Learnt and When?
- Grouping
Students - Streaming?
- Learning Takes Time and Effort
- Making the Hard Easier but
Ignoring how and so missing the Point
- Hook, Line and
Sinker - Mathematics Education Inconsistencies - Reform in North America
- More on Mathematics Education:
Covers: For a leaner curriculum, Education an empirical art,
More on testing, Constructivism versus Empirical Methods.
- Four Skeptical
Essays on Constructivism Revisited - Incompleteness
- Euclidean Model for
Development. Damage Reversal
- Educational Follies -
Learning By Discovery incomplete, cannot work, compound difficulties.
- An Educational
Inconsistency.
- 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.
Ideas and Principles For Instruction and Educational Reform
- Inductive Principles For
Instruction - systematic skill and concept development.
- 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 artificially and artfully develop the
algebraic way of writing and reasoning, and thus make mathematics fairer.
- Apprenticeship
in art, trades and disciplines, a classical view.
- Education is an Empirical
Art
- Key Notes and Themes
- Three Remarks
- For a Leaner
Mathematics Curriculum
- Need for a
Mixed Mathematics Curricula
- Extent and Need
for Quantitative Skills depends on your society
- Ways to be a Better
Instructor - Ideas and Methods - try with caution
- Four Ways to
Improve Education Reform, and avoid disaster.
Lesson Plans, Aims and Goals (Ends, Values and Means?)
- Three Aims for
Students - Ends and Values
- Three Goals
for Mathematics Education, etc - Ends, Values, Unifying Themes
- Lessening or
Avoiding Algebra Difficulties
- Algebra Lesson
Plans
- Algebra,
Geometrically
- Mathematics Curriculum
Shifts
- Advice and Suggestions for
Course Design and Delivery
- Teaching Tips - from fractions to
Calculus
- Math Education Perils
(Arithmetic, Algebra, Calculus)
- Talk the algebra talk
- First Year High School
Math - Lesson Plans with Fraction Focus
- Second Year High School
Math - Lesson Plans with an algebra focus
- Third Year High School
Math - Lesson Plans with a Focus on Slopes
- Math Wall Posters
- How Letters Appear in Mathematics
- Map, Plans and Drawings, a multi-year
project
Links
- Links - Just
a few.
- Activities to Engage
Students - links to explore
Logic and Reason in Mathematics
Mixing Rote & Thought-Based Development
- Cultivating Intelligence
- Why value careful mastery of rules and patterns, steps and methods,
practices, in a repeatable and reproducible manner.
- Multiply Kinds
of Reason in mathematics - Essay I
- 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?
- Theory of Knowledge -
Stories, Longer and longer
- Formal or Informal
Peer Review
- 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.
- Maths Instruction
in General - Three Goals A B and C to Set for Student, Supporting those
goals and why rewrite the curriculum
- 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.
- 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
- Teacher Certification Issues
and Cautions
- Math Ed. Professors
- Training, Education of
Archives:
- About this site - old version
- Old Site Entrance -
2010-05-1
- Yet Another Old Site Entrance
- Maths Ed Stopping Rule
- 2010-04-25
- A New Mathematics Curriculum
- 2010-04-29
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Mathematics
Education Essays
Group I
Site Reviews
Permissions
Applied Maths Program K1-12
Old Site Entrance 2010-05-13
Old Site Entrance
Words For Teachers
A New Maths Curriculum
Maths Ed Stopping Rule
Words For Teachers
Mathematics Ed. References
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.
Why Bother
Three Goals to Set for students
Implied
About
Lean Effective
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Group II
Which Way to Go
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
In for a Penny
Constructivism and Cognitive Theory
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
Education, Empirical Art
Five Decades make a difference
Damage Reversal
Math Wall Posters
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
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For
Senior
High School & Calculus Students
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Words to clearly
introduce algebra and variables
have been missing in course design. For people who cannot do
algebra,
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the missing words may
explain or ease their difficulties. Volume 2 ,Three
Skills for Algebra, in Chapters
8 to 14 & 18 etc, puts words before symbols to
providing the missing words in a way that enrich the
comprehension of all. Those words form the middle part of a algebra
(and logic) lessons aimed at helping or improving all
of high school mathematics and also calculus course
design & delivery.
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For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide. If your
calculus questions is not answered here, submit
it. Over time, that may complete the site development of
calculus.
For Parents: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly maths
skill development booklets for ages 4-14.
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Mostly
For High
School
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students. Question for Tutors: When do
you use and when you skip the stick diagram method
here?
Fraction
Skills, thought-based development, Ages 10 to 14 may need a
tutor. Students who have to understand in order
to do may like the development in all or part.
For Senior
High School Mathematics & Calculus
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
Tutors & Instructors:
These lessons introduce skills differently Would you
recommend them?
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More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
for- & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- Free
Advice and Directions for teaching primary & high school maths
will be given in online meeting place with voice &
whiteboard.
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students. Offering ideas to change
education makes this site different. Nothing
ventured, nothing gained. Site material is
mathematically correct, and where not, please report
errors. The two level program POMME in the site
entrance implies multiple paths for instruction. Supporting
those paths in turn implies a clear destination for
site development and perhaps a new name.
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