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LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education
Mathematics education standards implied by calculus should
be a factor, not the only one, yet not a forgotten nor hidden one in course design
Area Intro
Introduction
Arithmetic
Geometry
Algebra
Logic
Calculus
Musings - More Ideas
More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving - Routine & Not
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
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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.
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
- 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.
- 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.
- 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.
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.
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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