LAMP Lean Applied Mathematics Program
Skills Chapter 0 Introduction
Skills Chapter 1 Arithmetic
Skills Chapter 2 Geometry
Skills Chapter 3 Algebra
Ramblings - Introduction Algebra Essay
Ramblings - Extrinsic numbers theory
Skills Chapter 4 Logic
Skills Chapter 5 Calculus
A Introduction Objectives
B LAMP Introduction Curriculum Development Standards
C LAMP Introduction Culture in Mathematics Education
E LAMP Introduction Modern Mathematics
F LAMP Introduction Prerequisites
G LAMP Introduction Problem Solving Skills
H LAMP Introduction Instructional Concepts
I LAMP Introduction Study Habits
J LAMP Introduction Extrinsic Origins
K LAMP Musings Science Education
Appendix 1 primary and preschool mathematic
Appendix 2 primary school Arithmetic 01
A draft is a draft is a draft. With the
passage of time, diction, phrasing and musings should improve. The
tutor-teacher How-TOs above were posted online in the last week of
August, 2008. This site area appeared online on June 22, 2008. I will
making some webvideos in support of the How-TOs before reconsidering
this site area.
Postscript: I stopped work on LAMP
to reflect further ends, values and methods of mathematics education
from first steps in counting to calculus. The Sept 2009 Applied Maths Program for
quantitative skills development is the result, more complete, but with
some open questions.
LAMP is an acronym for Logic & Applied Mathematics Program. It
may shift course design and delivery in adult, secondary and college
education. Preparation for LAMP will provide a target and standard for
primary school instruction.
LAMP is an educational framework for instruction
and self-instruction of adults and teenagers in college, adult
education and secondary schools. Preparation for LAMP would occur
in primary school. LAMP aims for an operational command of
mathematics and logic. LAMP drill and practice demand that each
student work on paper, so that all steps are recorded and developed in
an observable manner for review and, if need-be correction or
refinement, by the student, fellow students, tutors, by
teachers and by parents.
Completeness: With a few exceptions, for each skill and
concept specified in LAMP, a development path is available with
the specification or in a whyslopes.com site folder.
Postscript Dec 12, 2008: Training versus
Education: In retrospect, LAMP is a program for
training students to master the tools and with them, the
observable practices of mathematics. The word training is
compatible with direct instruction. Here the instructor (trainer)
is expected to develop student skill with tools and practices, and then
to observe and correct student mastery of the practice.
Instruction may try to take large steps, but instruction or course
design should provide smaller and more steps for skill mastery if and
when student cannot follow or take the larger steps. So there may
primary and backup secondary methods for training - depending on the
needs, abilities and prior preparation of the student. The word
education in site is identified with training or direct instruction. An
art is discipline is viewed as a collection of tools and practices.
Those tools and practices may shift over time and vary
geographical. But there should be some continuity and
commonality, a common subset which identifies the
discipline. The objective of training in the art and disciplined
based is to provide students with a growing mastery of the tools and
practices in an observable, careful and confidence building manner. In
this training, intelligence is defined and recognized the careful and
then more and more skilful use of the rules and conventions to arrive
at results in an observable and verifiable or correctable manner. But
the constructivist mind-centered of learning and knowledge
presently dominates schools and faculties of education in many
countries (USA, the UK and Canada included) and calls for instructors
to provide students with food for thought in art and discipline,
so that student may construct and correct their own knowledge, without
the authoritative guidance or interference of instructors, all without
any possibility of instructors reliable observing or measuring students
learning and knowledge. That represent a shift in educational theory
from a focus on developing and checking observable skills and practices
to a focus on education as personal affair which occurs in the mind
student in an unobservable and unverifiable manner. That
represents a post-modern definition of education. It is subjective.
Most likely it is does not fit the training view of education that
might appear in mathematics, science, law, engineering,
construction, business and health practices and professions. LAMP is
contribution to the student training and direct instruction in
mathematics.
LAMP Ingredients
Six chapters identify LAMP components.
Chapter 4 describes more logic topics than needed. That poses the
question of what should be specified.
Chapter 5 does not specify a full course in calculus. Instead it offer
ideas to make learning and teaching calculus and beyond less
difficult. See Volumes 2 and 3.
Chapter 6 on applications will provide a description or list of
precalculus and then post-calculus skills and concepts to motivate and
reinforce quantitative skills and concepts. Before
mathematics education focuses on the needs of calculus, preparation for
it, can we provide mathematics lessons, easily understood and repeated,
with a context or motivation that will encourage skills and concept
development and perfection, satisfactory in itself for students who do
not have the time to continue in their mathematics education, and
satisfactory in a way that it will leave students with the urge to
continue or with respect for mathematics education that they will pass
on to their offspring? Today, in countries where school
attendance and in that mathematics education is compelled, many
students leave school with an aversion to mathematics that will be
passed on to their children. Thus bad or incomplete mathematics
education affects today's and tomorrows students -Oops!
Chapter 6, Applications: Quantitative Skills in the life of TCPITs
In instruction for a second language, the student may
be exposed to scenes from daily life, for example a train trip, a
restaurant visit, a day in the park. In the coverage of
those scenes, the books may provide a vocabulary that
applies. Likewise, in instruction for mathematics,
students may be exposed to common scenes and activities and after an
initial inquiry into their knowledge of the relate quantitative
skills and concept, instruction may continue to consolidate and/or
extend their knowledge of where is the mathematics in each scene or
activity. The aims of such scene coverage is to inform students
of what mathematics appears and to give them an operational command
of the mathematics in question in all or part.
Chapters 1 to 5 of LAMP will have
implications for primary school instruction of children and
pre-teens. The notion of studying scenes and
the mathematics there-in may provide motivation and a partial context
for learning and teaching from primary school to college
level. Chapter 6 will explore where is the mathematics,
the quantitative skills and even concepts, in daily starting from
preschool and primary school level. Phase 1 of LAMP,
mathematics for TCPITS before preparation for calculus begins,
depends on the width and breath of chapter 6. Chapter 6
may point to operational common of common place and commonly required
methods, arithmetic, geometric and then algebraic,
with explanations where required or in full in accordance with the
inclinations and abilities of students and teachers. Before worrying
about complicated problems - where the mathematics is not clear -
students need a practical, applied mathematics, oriented, of
algorithms for solving routine problem - benefits, origins and
limitations, included.
The above chapters point to a full framework and skeleton for a step by
step development of skill and concepts. LAMP construction may be
transformed into a wiki, so that readers may share their ideas.
In this first draft, explanations of how to develop a
step are more detailed in this draft when the explanation how is
missing in the rest of this site. The description of LAMP may go
through a few to several passes so that the development of all skills
and concepts is documented in a clear self-explanatory manner to
facilitate instruction and self-instruction. Once the technical
plans are complete, the expositional challenges, two of them,
then will be optimize material and its description to enable
instruction and/or self-instruction.
The LAMP Vision
LAMP reflects inductive methods for education in which larger steps
are decomposed into smaller steps for the sake of skill and concept
development. But the smaller steps are needed for skills and concept
perfection or for helping students for whom taking larger steps is
awkward or impossible. LAMP material when it fully developed should
be self-explanatory as much as possible, so that people required to teach
or learn mathematics have a reference for instruction and
self-instruction that is complete and accessible, as much as possible.
LAMP material at all levels may become easier to understand and follow
over time as different authors give more and more attention is given to
the development of skills and comprehension with the aid of words,
pictures and multimedia in its presentation, exercises and tests
included. LAMP material should be sufficient for an instructor with
good reading skills, not yet comfortable in mathematics, to cover the
most inclusive form of LAMP in class. LAMP material should be also
be clear and sufficient for tutors and parents to follow and understand
in the aid of their charges. LAMP material should be sufficient for
the self-instruction of teenagers with the will and ability to read
carefully to follow. That ability may be a function of age or maturity.
LAMP in the classroom should aim to make self-instruction an option but
not force it.
LAMP in many forms:
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I-LAMP, the most inclusive and flexible form,
aims for an operational command of skills and concepts with a
thought-based development only when needed. Where skills and
concepts are described instead of derived, there more be flexibility
in sequencing than permitted in a more sequenced thought-based
development.
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C-LAMP, the most comprehensive or complete form aims
for an operational command of skills and concepts with a logically
organized thought-based development whenever possible, and with
references to compensate when not. Chapters 1 to 6 describe and imply
the critical paths diagram for C-LAMP.
Individual students, teachers and school will cover LAMP phases between
these extremes. Or, school boards and course designer may prescribe
paths between these two extremes. In all cases, critical
path analysis of the dependencies indicated in chapters 1 to 6 will
possible routes for instruction. When time is nt critical, ease of
development or mastery may be a factor in sequencing skills and
concepts.
When students follow a path that is not C-LAMP, some may be yet be
prepared to digest the missing explanations for the sake of
completeness.
LAMP in 3 phases:
Each form or implementation of LAMP is expected to have three phases
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Arithmetic, Algebra and Logic Skill Development and Mathematics for
TCPITs: Besides preparation for Phase 2, Phase 1 will
focus on everyday mathematics for TCPITs. That is, Phase 1 will focus
on ideas and methods for solving or addressing routine problems
in every day life, for the development of good work habits, in order to
provide a context and motivation for the study of mathematics beyond
primary school. Before LAMP begins, Primary school instruction
should prepare for Phase 1. See Chapter 6 -
Applications, or Quantitative Skills and Concepts for TCPITs.
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Preparation for Calculus: Phase 2 of LAMP (preparation for
calculus) consist of all topics required by calculus.
Phase 2 by itself may be covered in college, in adult education and
in senior high school mathematics before calculus.
When skills and concepts that are only required
for calculus, is it proper to present them to students without saying
so? When skills and concepts that are only required for calculus,
is it proper to require their study by students whose futures will not
benefit from calculus or from the preparation for calculus. That
being said, covering skills and concepts in a fashion easily understood
and mastered by students, given their earlier operational command
of mathematics, may be a tool to retain and expand earlier skills and
concepts without being an imposition.
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Calculus Mastery: Phase 3 of LAMP focuses on calculus.
Again, the Phase 1 aim is to give TCPITs, the common person in the
street, a practical mastery and appreciation of mathematics. There
in lies a place for the description of easily understood applications,
routine and not, of mathematics, to build skills, confidence and
motivation.
The aim in phase 1 is to provide mathematics lessons, easily understood
and repeated, with a context or motivation that will encourage skills
and concept development and perfection, satisfactory in itself for
students who do not have the time to continue in their mathematics
education, and satisfactory in a way that it will leave students with
the urge to continue or with respect for mathematics education that
they will pass on to their offspring.
That being said, arithmetic, algebra and logic skill development is a
pre-requisite to the Phase 2 Preparation for Calculus with the
added aim of teaching students the importance of applying methods, step
by step, carefully, in order to obtain repeatable and reproducible
results for home, work and study.
For people who avoid phase 2, there will no mingling of phase 1 and
phase 2 material. That being, students planning to take phase 2 may see
phase 1 material on roots, logs and exponentials delayed until after
the phase 2 introduction of polynomials. Phase 2 material easier
than the advanced elements of phase 1 may be include in phase 1, time
permitting.
The Phase 2 aim is to cover topics in mathematics needed by
calculus. In this coverage, LAMP materials will very clear that the
main reason for a full and proper coverage of those topics is preparation
for calculus. Then the motivation for learning is clear. There-in
lies a remedy for students and teachers today meeting phase 2 topics
without knowing why.
Phase 3 aims is to rearrange differential and integral calculus to make
learning and teaching simpler and more effective. A good part of
that re-arrangement is implied by Volume 2 and 3, and in site area More
Calculus. Lipman Ber's Calculus book may provide a further context,
or at least background information. Phase 3 specifications,
when or if fully done in chapter 5, will clarify matter further.
Musings and Reflections
Some LAMP area pages are labeled as musings to indicate a continuing
process of reflection on what should be done and how to make mathematics
and logic education clearer and to provide reasons for it.
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Before LAMP -
Preparing for LAMP in primary school, a question to resolve.
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LAMP_Implementation -
Ends, Means and Values Besides mastery of mathematical
methods through practice, by rote if need-be in basic instruction,
and through the thought-based development of skills and concepts in
both basic and advanced instruction, the LAMP program aims for
operational command of skills and concepts in a practical,
observable, repeatable, reproducible and verifiable
manner.
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Mathematics
Cultural Origins. While modern mathematics aims
to be context free for the sake of rigor, reasons for
mathematics study and mastery have cultural roots, roots that may
differ between societies or be absent in some.
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Evaluation
- Here are tandards for the Evaluation of LAMP instruction that stem
from inductive criteria for course design and delivery.
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Student Cooperation
-- Student cooperation is needed. LAMP requires students to sit down
and pay attention to detail. Anything less - years of study without
requiring attention to detail - will waste the time and energy
invested by students, teachers and society in education.
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LAMP and
First Nation Education - Food for thought, if not action.
Mathematics has cultural roots, roots that differ between societies
and may be absent in some. The applications (chapter 6) which
may appeal to student and provide a context for mathematics and logic
education in one society may not provide a context in another
society. That raises a problem of context and motivation when
mathematics instructors and mathematics courses design from one
society appear in another. There-in lies a mess to
consider.
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Mathematics
Extrinsic Origins - More on the extrinsic cultural origins
- the prelude to axiomatic or intrinsic developments. While the
modern mathematics curricula were motivated by the intrinsic
(axiomatic) development of pure mathematics, the modern mathematics
curricula themselves also involved the impure extrinsic, geometric,
development of skills and concepts in geometry itself, in
trigonometry and calculus. In retrospect, the skill and concepts
codified and logically derived in pure mathematics mostly have an
extrinsic origin - they are extracted, abstracted or extrapolated
from experience that appears to be repeatable and
reproducible.
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LAMP and Modern
Mathematics LAMP points to a consistent extrinsic development of
mathematics from arithmetic to calculus as a prelude to the study of
the very algebraic, pure mathematics logical codification and
development of skills and concepts.
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Instructional
Concepts - LAMP like the modern mathematics curricula of the
1950s provides a very structured view of mathematics. In that
view, mathematics is an art or discipline in which the steps or
reasons for results or conclusions are recorded and developed on
paper in observable and verifiable manner using methods which have
invented and passed-on or inherited as is or in transformed form.
...
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Problem
Solving Skills Routine to Non - Open problems are fine, but
should not students be given tools and standards for routine problem
solving as well, if not before?
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Science Education -
LAMP provides for a pen and paper, thought based development of
skills and concepts in mathematics from arithmetic to calculus.
In contrast, Science education presents and illustrates
principles, and provides information but cannot provide a
self-contained thought-based development. The role of labs in
introducing hypothesis testing in science should not be a sham.
Science instruction appears to be a mixture of
description, mathematical calculations and incomplete lab work couple
with a philosophy for the empirical development and testing of rule
and pattern based methods. Some thought is required to the logical
development of biology, chemistry and physic courses for teens and
adults in view of the difficulty or impossibility of verifying
theories and concepts in school science labs.
External References (draft collection):
The development of a curriculum in a self-contained, self-explanatory
manner, obviates a need to know about antecedents. That being said,
LAMP and site contents in general are technical consequences of (i) my
education in mathematics and logics as provided by books and instruction
in elementary to advanced mathematics; and of (ii) teaching
experiences. The following references are and will be of a
technical nature.
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Zero Saga: http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM#rDecatAnaly
Zero in Four Dimensions: Cultural, Historical, Mathematical, and
Psychological Perspectives
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People familiar with the mathematics education literature may suggest references.
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