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 1: Arithmetic  (specifications complete)           
• Chapter 2: Geometry   (specifications complete)  
          Postscript for later reading: Extrinsic Number Theory
• Chapter 3: Algebra (specifications complete)
          Postscript for Later reading: Algebra Essay
• Chapter 4: Logic  (ideas for but specifications incomplete)
• Chapter 5: Calculus  (ideas for introducing key concepts only).
• Chapter 6: Applications (incomplete)

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:

• 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.   
  

• 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. 

1. Before LAMP - Preparing for LAMP in primary school, a question to resolve.
   

2. 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. 
   

   ---

3. 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.  
   

4. Evaluation - Here are tandards for the Evaluation of LAMP instruction that stem from inductive criteria for course design and delivery.
   

5. 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. 
   

6. 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. 

   ---

7. 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.  
   

8. 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.
   

9. 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. ... 
   

10. 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?

    ---

11. 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.  

1. Zero Saga: http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM#rDecatAnaly  Zero in Four Dimensions: Cultural, Historical, Mathematical, and Psychological Perspectives

2.  

People familiar with the mathematics education literature may suggest references.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  All Rights Reserved.