About LAMP - It Objectives
LAMP is an acronym for Logic and Applied Mathematics Proposal.
It is a proposal for adult and adolescent instruction that covers mathematics
from arithmetic to calculus.
LAMP divides skill and concept development between three
phases:
- (Phase I) Mathematics for TCPITS
- (Phase II) Preparation for Calculus,
- (Phase III)
Calculus
TCPITS is an acronym for the phrase: the common person
in the street
Learning and teaching in each LAMP will vary between inclusive
and comprehensive forms
- I-LAMP, the
inclusive form, aims for an operational command of skills and concepts with a
thought-based development only when needed.
- C-LAMP, the most comprehensive form, aims for an operational command of skills and concepts with a logically
organized thought-based development whenever possible for the sake of
completeness, and with references to
compensate when not.
The challenge for
mathematics education is to develop skills and concepts in a way that students
planning to end their studies gain the skills, confidence and satisfaction
sufficient to change their plans.
LAMP is based on inductive principles for instruction. Those
principles require large steps be clearly and explicitly decomposed into
smaller steps for skill and concept development, so that student who cannot
take larger steps may be given a simpler path to follow when needed.
Each student has a different background and skill level on entering or
continuing LAMP. In LAMP includes directions of the form if the
student has not master a skill or concept then try the following steps. Where difficulties
are common or not, those inductive principles require LAMP to provide for smaller steps to make
the larger steps accessible and feasible. Time will
tell what is possible and what level of intelligence is required to follow the
most inclusive and accessible form of LAMP that presently exists. Trial and
error may tell suggest how to rearrange the iterative coverage of skills and
concepts to make LAMP more accessible. There-in lies or shines the empirical
approach to the further implementation of LAMP.
The challenge for course design and the composition of course materials is
to make all as inclusive as possible so that students and teachers not yet
comfortable in mathematics have clear and readable explanations and directions
to follow.
Confidence in LAMP depends on
whether or not it delivers operational command of mathematics, the how
and/or why, to students, parents and teachers through steps, well-documented
and easily understood and followed, in instruction and
self-instruction.
LAMP includes several innovations to make key skills and concepts
easier. Where some students will be empirically satisfied that methods
give repeatable, reproducible and verifiable or refutable (correctable) results,
others will demand theoretical satisfaction - an explanation of the why. LAMP
aims to provide a maximal operational and thought-based command of skills and
concepts for minimal effort. That includes removing artificial challenges
- skills and concept not necessary for the immediate and long-term objective of
providing an operational command of the how with a minimal, maximal or somewhere
in-between thought-based command of why. But minimal effort does not mean no
effort. Work is required. Students have to sit down and study. LAMP has to
provide material for students, parents and teachers - not yet comfortable
with its skills and concepts.
LAMP is empirical. LAMP in all its forms starting from column methods
for decimal arithmetic emphasizes clear and legible formats for the performance
oriented, very much observable and correctable expression and development of
ideas and results on paper, step by step. By drawing and
writing diagrams, words and arithmetic or algebraic calculations on paper, one
small step at a time, students, their fellow and their teachers may rely
on their eyes to see what has been done, and to decide what comes next. If
mathematics be a mental exercise, it is one in which diagrams, words and
expressions drawn or written on paper record and express ideas as part of
dynamic, observable and readable record. Just as a carpenter works with wood to
build concrete objects, the mathematics students and mathematician draws and
writes on paper to provide a record of ideas and thoughts in a concrete manner
that can be immediate review by the composer and the composer peers for
correctness and completeness, or compliance with prior rules and patterns. With
that record, the composer and others may see and continue prior chains of reason
- the selection and use of rules and patterns one at a time and one after
another, alone or in combination, with care or respect for limitations. With
clear rules and patterns to follow in the expression and application of skills
and concepts, students abilities in mathematics are judged by what they draw and
write. Performance or an operational command of mathematics needs to be based on
drill and practice sufficient to automate drawing and figuring methods in an
observable, legible, repeatable and reproducible manner for the sake of
immediate or later review by the author, peers and teachers.
Reaching that destination, one method at a time, will involve work or drudgery,
but it will also build the skills and confidence of students, and so
engage them as they learn how to do mathematics in an observable way that can be
reviewed and approved, or corrected. There-in lies a means, a method and an end
for mathematics and logic instruction and self-instruction.
LAMP is not a cure-all. LAMP implementation requires that students have
the ability and inclination to sit down to meet and practice skills and
concepts, and to entertain, if not always allow, motivations offered in LAMP or
coming from a parent or instructor for the necessary practice and drudgery. LAMP
aims to provide an operational command of methods, mathematical or logical,
first for performing routine calculations, that is, for solving common or
routine problems, and second to develop general problem solving and defensive
(be prepared) critical thinking abilities for matters met at home, at work and
in society. The identification of common or routine problems depends on
society. Native, aboriginal, first nation societies and other societies
with brief contact with the quantitative reasoning skills of larger,
modern, pollution-age, agricultural, manufacturing and city life may find larger
society schooling systems in need of adaptation to their needs. Whence
mathematics is not a universal language and rightly or wrongly, the motivations
for mathematics will have a large society flavour.
Part I - Critical Paths For Skill and Concept Development
Most elements of C-LAMPS are present online in this website www.whyslopes.com.
To determine critical paths for LAMP in its C-LAMP and I-LAMP forms, in parallel
streams we will identify and indicate a thought-based development all
arithmetic, algebra, geometry, logic and calculus skills and concepts which
could be part of C-LAMP, the comprehensive form of LAMP, along with their
thought-based development. For a critical path for C-LAMP, we will
identify the dependence, if any, of each skill and concept on elements of the
other streams. The one further stream not mention consists of
applications - mathematical methods for solving routine problems in human
affaires that depend on skills and concepts in arithmetic, algebra, geometry,
logic and calculus. The dependence will determine how soon the
applications and motivation for studying mathematics and logic can be placed -
the earlier the better.
The indication of thought-based development of skills and
topics in this site area will inversely proportional to the coverage in other
areas and if not in other site areas, to the difficulties that might occur in
that development. In other words, the indication is greater and in
more details for those skills and concepts not treated in detail elsewhere or
whose treatment is not obvious.
Six chapters 1 to 6 describe in detail and with some
overlap arithmetic, geometry, algebra, logic, calculus and application
streams of LAMP.
These streams may themselves be composed of parallel substreams. The ordering
of parallel streams and substreams there-in depends on which logical dependencies are
respected The logical dependencies will greater in the more comprehensive forms
and less in the more inclusive forms. Ordering may also depend
on pedagogical estimates of which streams and substream blocks are easier to
master.
Part II - Possibilities for I-LAMP and
C-LAMP
Part I described how to provide a thought-based development of skills and
concepts from arithmetic to calculus in a very detailed or full manner. The aim
of part II is to assembly those pieces into an inclusive and an comprehensive
designs for LAMP based instruction and self-instruction.
Chapters 1 to 6 in their definition of LAMP streams or
substreams set the stage for developing the most inclusive and the most
comprhensive forms of LAMP. As
said above,
-
I-LAMP, the most
inclusive and flexible form, and C-LAMP, the most comprehensive, demanding and
constrained form. The
inclusive 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 sequence thought-based development. In other
words, a critical path diagram for non- C, LAMPs learning paths
are always less restrictive than C-LAMP.
-
C-LAMP, the most comprehensive 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 specify in writing, the
critical path diagram for C-LAMP.
Individual students, teachers and school may learn and teach
LAMP between these extremes. There-in lies a lower bound for instruction and, if
course materials are sufficiently clear, self-instruction too. In all
cases, critical path analysis of the dependencies indicated in chapters 1 to 6
will possible routes for instruction
Part III: Reflections, Conclusions and Musings
Chapter 8 introduces further LAMP objectives, LAMP musings,
and LAMP possibilities including
criteria for a mathematics curriculum or course design to enjoy its seal of
approval. Essentially, skill and concept development must be
self-contained so that that teachers and students.
In
retrospect, the end, values and methods of mathematics education are culturally
biased towards societies which have employed quantitative skill and
concepts from ancient to present times in their agricultural, trading, manufacturing and political or
religious activities. Familiarity with those
activities provides a context and motivation for primary school mathematics and
LAMP phases in all or part.
Mathematics is not a universal language. Smaller societies or groups
that are surrounded by larger societies are caught in awkward position where the
educational systems of a surrounding society may be needed for economic survival.
There in lies a catch 22 for a surrounded society. In following the education
of the larger society, if the latter is well-done, and not bureaucratically
amiss, many of its members may run the risk of assimilation, so that
some will returning to partially preserve the surrounded and threaten culture through an
adaptation of the larger societies education
system.
LAMP origins
This proposal for LAMP is stems from inductive
principles for instruction met in 1981; from lessons three
skills for algebra, a geometric calculus preview and a logic puzzle on the
difference between one- and two- implications invented for my students in fall
1983; on twenty minutes or so of a Richard Feynmann lecture, one of three
public presentations at McGill in fall 1989 lecture, in which he describe his
subject as the addition and multiplication of arrows in the plane; from the
example of guest speakers in mathematics (analysis) at McGill University
1975-83 who show that the exposition of mathematics could be advanced (the
hard made clearer) from secondary to research level subjects; from the
exploration at this website www.whyslopes.com
of ways to understand and explain methods I learnt and met by rote; and from
recognition of inductive
gaps or shortcomings in the Modern Mathematics Curricula of the period
1955 to 1980 or so, a curricula I met as a secondary student in
1967.
From 1967 to 1983 as a student and then as a teacher, I saw gaps in the
exposition of my subject but my own studies distracted from the active pursuit
of remedies. From fall 1983 invention of lesson on three skills for
algebra, a geometric preview of calculus and the logic puzzle to 1889, I held
university professor and college instructor post of short duration in which the
expression of interest or ideas in and for mathematics education was
impolite. Writing began in the last few days of 1990 with the aim of
informally reporting multiple ideas to educational authorities for review and
refinement by my peers and betters in a hit and run manner - the academic job
market was difficult and I did not expect a career in it. Yet I was driven
to investigate and consider mathematics education due to the difficulties
students first due to the challenges of calculus - my perception that its
syllabus could be re-arranged to make the hard easier, and later due to the
realization that the secondary school preparation of high school mathematics
might benefit from my fall 1983 lessons and their expansion.
On meeting the 1989 and the later year 2000 standards and principles of the
US National Council of Teachers of Mathematics, I read them in the hope of
finding a reason to stop writing of seeing my exploration of ideas and concepts
was redundant. From say 1992 to the present I have not read Mathematics
Education Journals due to their focus on delivery style matters or petite
content issues, while finding considerations of content issues appeared to
be as difficult as looking for a needle in haystack.
In retrospect the modern mathematics curricula of the period 1955 to 1989
which I once favored and tried to make more accessible or inclusive in my
writings did not address the algebra exposition gap that existed in its time
and before, and its introduced further gaps and barriers in mathematics
instruction as its nominally support of the rigorous and context-free (intrinsic)
development of Modern Pure Mathematics was not rigorous, included extrinsic
mathematics in its development of Trigonometry, Calculus and before that
Euclidean Geometry, while inconsistently allowing the decimal representation
of numbers in arithmetic, but not sanctioning that representation nor
exploiting it in its development of mathematics from algebra to
calculus. In other words, the Modern Mathematics Curricula, the
implementation I saw in my education, took steps too large in the development
of algebraic skills and concepts alone and in further subjects, required
decimal arithmetic but avoided decimals in the statement of its axioms for
real numbers and in its formal discussion of limits, continuity and
convergence in calculus, a decision and epsilon-delta abstraction that
complicated comprehension; while the use of diagrams in trigonometry to define
sine, cosine and tangent functions and the use of diagrams to imply that 1 was
the limiting value of (1/x) sin (x) as angle x measured in radians
approaches zero fell into a gray area - the extrinsic handwaving
development of mathematics which departed from the instrinsic (axiomatic)
development of Modern Mathematics.
LAMP Advantages
Algebra: The LAMP proposal provides a clearer exposition of algebra
with the aid of geometry and words before and besides symbols. LAMP
provides smaller and clearer steps from the start of algebra to the multiple
full-strength deployment of algebra in advanced calculus. That by itself
should justify the study of LAMP ingredients in the rest of this website www.whyslopes.com,
if not the adoption of LAMP itself.
Mathematics Extrinsically
Modern mathematics curricula were not only intrinsic,
they were extrinsic as well in contradiction to a nominal intrinsic
nature, a nature with full compliant, a pedagogical
impossibility.
Before the advent of modern mathematics and its intrinsic,
context-free and logical development of skills and concepts from
arithmetic to calculus and beyond, many elements of mathematics were
well-known. They were implied by drawing and figuring practices in say an
empirical manner with some uncertainty due to the possibility of
drawing misleading diagrams and more uncertainty due to the lack of clear
or absolute logical structure for figuring from arithmetic to
calculus. LAMP pushes for an operational command and an operational
(extrinsic) development of skills and concepts.
Modern mathematics starting with the ZF axioms 1903-5
provide a more secure but in retrospect not an absolutely certain
framework for the theoretical or logical development of mathematics from
assumptions about sets to an intrinsic (axiomatic), context
free/independent codification of the properties of real and complex
numbers, and a development of calculus methods. Following that path
requires a strong algebraic-deductive maturity.
The Modern Mathematics curricula 1955 onward introduced
the second part of that path into secondary and primary mathematics with
axioms for real numbers as an intrinsic starting point. But the
exposition of geometry, analytic and Euclidean, and the development of
trig and calculus also involved the use of geometric reasoning or diagrams
in a manner that departed from the pure intrinsic development. That
reasoning and diagrams represent an extrinsic viewpoint of
mathematics. Further the decimal representation of real numbers was
required for arithmetic in algebra, trig and calculus but not mentioned
and hence not explicitly sanctioned in the modern mathematics
curricula assumptions about real numbers. LAMP is an
extrinsic successor with a more inclusive and consistent thought-based
development
LAMP extrapolates from counting, drawing and
figure practices, an extrinsic development of real and complex
numbers, of trigonometry, of calculus in manner equal in rigor to that
present in the extrinsic parts of the 1955 onward, modern mathematics
curricula. The development also includes an applied mathematics or
extrinsic view of the decimal representation of real numbers. The
discussion of limits, continuity and convergence in calculus begins with
that decimal representation and the question of error control in the
calculation of expressions or functions. LAMP includes set notation
and concepts where that aids comprehensions. See its development of
counting methods, probability theory and function concepts.
C-LAMP after extrinsically deriving properties of real
(and complex numbers) describes those properties as axioms in algebraic
manner using set notation and concepts. LAMP treatment of trigonometry
like the modern mathematics curricula is based on an extrinsic viewpoints.
But LAMP relies on its extrinsic development of two ways to multiply
complex numbers to speed the development of unit circle and right triangle
trigonometry, and in particular speed the derivation and verification of
trig identities, make them less of a challenge and more
accessible-inclusive. Easy consequences of the two ways includes another
proof of the Pythagorean theorem, and trigonometric formulas for dot and
cross-products of vectors in the plane when the latter defined using
rectangular coordinates. On a less rigorous note, LAMP exploits the
idea that partition of a rectangle into sub-rectangles gives two ways to
compute the original rectangles areas to geometric describe and generalize
distributive laws for multiplication of non-negative, and so suggest
column methods for the multiplication of polynomials and the
multiplication of whole numbers using decimal representations.
From arithmetic to calculus, C-LAMP offers an operational command of
skills and concepts, along with extrinsic thought-based development of
those skills and comprehension.
The thought-based development of LAMP is based on
suggesting and using rules and patterns, one at a time and one after
another, alone or in combination. LAMPS logic mains deals with the
direct use rules and patterns of the form B IF A (one way
implications) and of the form B IF and ONLY IF B (two-way
implications). The terms one-way and two-way stem from a 1989 lesson
in which a student name Flo made a comparison with one- and two-way
streets. I said YES and adopted her implied terminology for the rest
of that lesson. LAMP students will be encouraged to see the
difference between the two forms. Here I advocate using the form B
IF A in place of or besides the for equivalent form IF A THEN B to help
students see the difference between the one- and two-way implications.
Here I may use the word WHEN as an alternative to IF.
Capitalization is optional.
The indirect use of logic in LAMP is limited to the
contrapositive form (Not A) IF (Not B) for implication rules B
IF A in (a) the discussion of the zero product law for whole,
real and complex numbers in the precalculus element of LAMP and in (b) the
calculus divergence of a series test: If the n-term does not tend to zero
as n increases then the series diverges.
LAMP employs the decimal representation of nonzero whole numbers and the
idea that the area of a rectangular will not be zero to first suggest that
the product of nonzero numbers is nonzero, and hence to suggest from
the polar
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Communication, Reason
and Problem Solving
LAMP and constructivism differ.
LAMP development recognizes that students and instructors may have
private thoughts and that direct mind reading is not possible. But
people can draw and write their ideas on paper and so provide an
observable, repeatable and reproducible record of their thoughts for peer
review, correction and refinement. While we may speak, hear, read
and write words in an essential one dimensional, discrete, sequential
manner, the use of diagrams and column methods in mathematics, and
the reading and writing of arithmetic and algebraic expressions with
fractions, superscripts and subscripts, record and communicates
ideas in a two-dimensional manner, a manner awkward to read aloud in a
sequentially, yet a manner that can be seen, if not understood, in a
glance. The drawing of diagrams and the drawing or writing of
expressions with horizontal and vertical elements provides a visual record
of ideas and 2D visual extension of our memories. Whence steps too
complicated for an individual to do mentally may be done and recorded on
paper for all, including the doer, to observe and check. While
memory and mental agility is a plus in any subject, mathematics is based
on the communication of ideas on paper using words, diagrams and linear or
two-dimensional, arithmetic and algebraic expressions, ideas that can then
be followed and checked by the doer and others for the sake of immediate
or future self- or peer-review and, if need-be,
correction.
Mathematics is a discipline or art form based on an observable forms of
communication of ideas and reason. Instruction can suggest or impose
standards to aid and speed communication and reasoning. And that
good notation and clever or standard presentation may aid the on-paper
record and mastery of mathematics. The constructivist suggestion
that student thoughts are for reading ignores the deliberate and
systematic development and verification of concrete, observable, on-paper
communication and reasoning. In this, drill and practice may be
needed, will be needed, for students to see the need to follow steps in a
method for solving a routine problem in a way that can be reviewed later
or immediately by the doer and by peers in the form of fellow students,
tutors, parents and teachers. Meeting methods or strategies for
solving problems of a routine nature is one motivation for mathematics and
logic education. Precision in reading and writing is further
necessary to understand, select and apply a strategy in logical manner.
Meeting methods or strategies for solving many problems of a routine
nature in repeatable, reproducible and defendable manner (see
communication0 provides a base for routine and non-routine problem
solving in an observable manner that the doer and peers may review to
approve or correct. The application and development of
patterns, rules and laws in society is based on the latter. Familiarity
with earlier methods, their origins, benefits and limitations, provides a
better starting point than ignorance for problem solving.
Non-routine methods for solving problems should be explored only when
routine methods are not known or are not satisfactory. That being
said, the problem solving abilities of students can be tested and expanded
by giving routine problems just beyond the reach of their present
knowledge for them to investigate. Finally, open problems in society
may still abound and may be given to develop the critical thinking and to
develop an appreciation of the limitations of existing lines of thought,
but there is a still a need to provide drill and practice with routine
problem solving methods, so student can tackle common or routine problems
in an automatic manner, without going into research mode.
Constructivism with its assertions that student
comprehension is not observable; that testing is not reliable - students
may forget; that following rules and patterns to arrive at conclusions is
not a genuine nor reliable form of reason all point to a pre- or
post-empirical view of knowledge and the formal or informal peer review
process present in disciplines striving for objectivity: law, science,
mathematics, technology. Allowing constructivism with those elements
dominant in charge of education in the latter disciplines is like putting
the fox in charge of the hen house, and telling it to guard the
chickens. There-in lies a contradiction. On the other hand, constructivist
ideas for engaging students may be worth exploring and testing in an
empirical fashion.
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coordinate introduction of
products, that the product of nonzero complex numbers is also nonzero. C-LAMP's
lean treatment of Euclidean geometry, assumptions and theorem employ one
and two-way implications. That being said, mastering the difference
between one and two-way implications may lead students to precision or
greater precision in reading and writing in their studies and in their
present or future workplaces.
The math-free discussion of logic in LAMP, an optional part,
identifies proof by absurdity with an elimination of possibilities due
to their immediate or implied inconsistency with prior knowledge or
axioms. The proof by inconsistency argument appear to be of value in (a)
criminal detective work - an alibi eliminates a suspect from further
suspicion; scientific detective work where a hypothesis or its
consequence are inconsistent with prior knowledge or expectations; and
(c) in mathematics where tentative assumptions or their consequences are
inconsistent with prior assumptions. In reading a few detective stories
in the hope of finding literary examples of proof by absurdity,
contradiction or inconsistency, I saw endings based on revelations
instead logic, endings which the chief inquirer would reveal facts or
observation not previously found in the text to imply conclusions to
arrive at conclusions, all in a I told you so manner. The
exposition of logic in LAMP may lead students to appreciate the
benefits, origins and LIMITATIONS of rules and patterns.
Finally, LAMP may develop the algebraic-logical maturity necessary for
the calculus development and beyond that, for an optional study of modern
mathematics by university students.
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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 Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
[ Area Intro ] [ Next ]
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