Parents: Site pages on  Helping Your Child or Teen Learn  cover  Speaking Skills, Reading & Writing,  Preparing for Science,  ends, values and methods for work and study and parent-friendly children-teen mathematics booklets.
Students: The right-border link to site topics. The bottom border provides an annotated guide to site content. For grades 5 to 12, see Site Lessons Most Likely to Help. This a large site with a 1000+ pages.   
Instructors: The two level program POMME below offers ends & values for skill development. This algebra & logic subprogram (well put) & these Arithmetic/Number Theory Practices   identify skills to develop in ways that may enrich know-how and make the hard, less so.   

Schools and Colleges:  I can quickly show teachers (novice to expert in mathematical subjects)  how to improve students skills. Results may be striking. Make first contact, to discuss what is possible and what is not for in-service teacher training and for talks public to specialized.

Welcome. Online books and further material  may help skill development in college, secondary and primary school level mathematics.   Each appetizer or starter lesson is different. If one is not to your liking try another. 

Please direct your instructors here. This site is different and deliberately so. As a student, teachers and then educational writer, I sensed and then clearly saw gaps in the explanation of and motivation for skills and concepts, and I observed others make the hard less so in mathematics and physics. The first aim was to find ways to strengthen skill & concept development. That being done, the question of why learn or teach remained. The new two level program below answers the question of how and why learn or teach mathematics and logic. Even if you are doing well in mathematics learning or teaching, the difficulties of others slows your instruction.  In any event, site remedies for learning and teaching difficulties may help or speed your instruction. Site Volumes 2 and 3 stem from lessons that often worked well in class. Volume 1B set forth the question of how to address content and motivation problems - background information for interested parties. 

Site material shows how to learn or teach key skills & concepts.  Site material focuses  on (i) skill development and on (ii) the thought-based development or derivation of mathematical methods. Concentrate on skill development first if you like. 

Glitch: Where fonts in Volume 3 do not appear properly, view with internet explorer. Site pages were developed on PCs where those font problems did not appear. 

Help Elsewhere: Three text-based sites  mathsisfun,  purple math and themathpage are well-done.   The BBC also provides help (examples) in: mathematics and many other subjects for students.   The  Khan Academy has over a 1000 UTube videos on  mathematics etc. The span of topics in mathematics is good but  but equal signs that I would say are necessary are missing in some videos.   The Bright storm Flash Video Site:  (it requires a membership) for secondary  mathematics US style and some calculus lessons with an emphasis on the mechanics (the how, not the why), Brightstorm flash videos are neat and usually well-done except for notational lapses - doing calculations in place instead of doing one step per line, one step after another.  New Links: instructables.com Math_Help  Little multi step lessons in K5-8 level mathematics. That site offers many more multi-step lessons outside of mathematics.  The SOSmath site offers online cyberboard  for the discussion of secondary and college level mathematics and  mathematical subjects.

Mathematics Education Revisited  

Most people study mathematics until it becomes too hard, until they lose interest or both.  In many high school mathematics courses, the question of why this or that is present has the answer:  preparation for final examinations.  Ours is but to learn or teach without understanding why.

"Would you tell me, please, which way I ought to go from here?"
"That depends a good deal on where you want to get to," said the Cat.
"I don’t much care where--" said Alice.
"Then it doesn’t matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an explanation.
"Oh, you’re sure to do that," said the Cat, "if you only walk long enough."
(Alice's Adventures in Wonderland, Chapter 6)

The two level mathematics program below suggests which way to go, how and why. This  subprogram further says how teachers & tutors (or gifted students) may develop algebra skills and concepts from solving linear equations geometrically and talking about numbers and variables  to rearranging calculus. 

Drafts of the two level program POMME  June 2010 onward end  four decades of concern and two decades of  writing. The two level program reflects and reacts to what has and has not been done in past and present course designs - those available online or discuss in works dating back to the 1940-60 period. 

Teachers: The new mathematics education essay  which way to go, how and why   ends with  reflections on limits to the self-contained development of mathematics and science skills and concept in education - an issue that may be addressed later or not at all. )

POMME stands for progressive (that is step by step) observable, motivated, mathematics education. The word observable reflects one end and value. Skill mastery in many disciplines needs to be seen in order to be credible or to be confirmed or to be corrected. Reliable skill development by rote or with comprehension of why is the aim.  Students have to learn to avoid the domino effect of errors and approximations in figuring and reasoning, and in activities beyond mathematics.  POMME reflects and reacts to ideas written in English simply because that is my mother tongue.

In English language countries, legal systems now advocate plain language in contracts.  The first level, skill development ends, values and methods  for children and young teens appear in plain language intended to be clear for most adults. But for second development, the ends are described in plain language while comprehension of the technical ends or details require more and more knowledge of secondary mathematics and calculus. That being said, the presents of technical innovations (smaller steps, more steps and alternative steps) for skill development may make advance skills and concepts easier to understand and explain, with no loss of rigour - modulo that possible at the pre-university level. 

Skill development paths in level II are based on methods for direct instruction,  proven in class or similar to proven ones. Those methods employ smaller, clearer and sometime alternative steps, likely to work. The methods reflects and are strongly supported by end and values,  based on telling students about the domino effect of errors and approximation in multi-step methods, and take as end and value avoidance of that domino effect. Learning to do in a reliable manner may lead in self-filling manner to greater skills and confidence. That demand and expectation differentiates POMME from other which separate  self-esteem from learning to do in a reliable and repeatable manner. While many learning difficulties will no doubt defeat the program, its technical steps, not all new, will make skill development easier for teachers and learners. 

Young students who complain there are too many letters in the alphabet have to be told that all  letters have to met and mastered in order for them to read and write.  Skill development, whether it be in reading, writing and arithmetic, or beyond makes demands.  While some parts of education may provide food for thought and reflection with results that may be located in the mind in an unobservable manner, mathematics education in the form of  skill development and engineering  has more concrete aim of showing students how to follow paths and to solve problems in a manner that others have introduced. 

The selection of mathematics skill development  ends, values and methods  easily understood and repeated, is not just for students, but also for instructors. Many instructors without formally training in mathematics have to give mathematics lessons. Primary and junior high school course design based on the six  application areas and the work & logic values below may make earlier  skill development more effective -  easier to understand and deliver, with a context that adults and then student may appreciate.    In mathematics education reform, ease of exposition and skill development based on clearly described ends, values and methods may allow teachers to deliver observable and verifiable skill development - make the subject more teachable. In the program, the aim is to make observable and verifiable skill development easier for students and instructors.  

In practical subjects, confidence follows from learning to do with comprehension if possible, by rote if need-be, all with practice first, theory second.  That order will serve common needs. The foregoing is to give room for systematic skill and know-how development where students are given practices to learn and do without excluding space for reflection or situations that provide food for thought. 

Coherent ends, the values and  methods for skill and know-how development, technical innovation included, may improve current course delivery. Proof of that is given by this  algebra and logic subprogram.  But altogether, site innovations imply a new curriculum and a new lower bound for course design and delivery.  

Education authorities on five continents should quietly assign their best mind in mathematics to review and refine the two level program, and the support for it (not yet complete) in site material. Variants of the program may be generated to met local and changing conditions, until such time where the two level structure ceases to have value.  

Primary and Junior High School Mathematics - Level I (student centered skill development)

The first level for students 14 and under, 9 years long say,  represents student centered, skill and know-how development. It focuses on ends, values and application areas which  clearly serving the likely and then less likely common needs of students and their present or future families. That provides an initial context and motivation for skill and know-how development, one that may come with general advice on quantitative decision making and risk avoidance. Worldwide, English language course design in mathematics is based on university oriented themes and strands whose long-term value is too distant for adults - parents of children and young teens - to appreciate.  In contrast, primary school mathematics with lessons on time and date matters, geometry, money matters, and arithmetic, which all serve, even without saying so, common or likely needs easily understood and repeated.   

For the primary and junior secondary level mathematics education of children and young teens,  six application areas together with ends and values for work and study give a clearer way to go.  The program here serves common or likely needs of daily life in the streets of cities, towns and villages . Adults and older students will recognize ends and values that in retrospect will be self-evident - worthwhile supporting in their education of themselves and others. 

Mathematics and logic training or education in junior high school and before may begin with clear ends, values and methods, easily understood and repeated, and serving likely or common needs of daily life at home and in work. That can be in a way that  prepares for further studies. And that can be done to provide a favorable image of image of mathematics skill development to those who stop .Where those six application areas have take-home value,  we may say or hope instruction is student oriented in a material sense. Even before a child goes to school,  most of these application areas will appear in home life and so have some take-home value. Where possible skills and know-how with the greatest take-home value are put first,  but over time the remaining skills and know-how will have less and less immediate value and more and more long-term value or potential as part of general preparation for adult life and also for further instruction. 

Six  application areas with formats for numerical and geometric reason that make steps or results observable and hence correctable,  and explicit mention of the domino effects of mistakes and approximations in numerical and geometric methods provides a context, values and motivation easily understood and appreciated by people in communities worldwide.   That should  provides a strong skill and value base not for the daily use of quantitative skills, but also for the advanced level of secondary mathematics. The six application areas will be of greatness benefit to those students who in mastering skills and know-how, continually look for opportunities to apply them.  

Six Application Areas

In communities where the following application areas  arise,  the areas provide clear ends and context for instruction.  

1. Time and Date matters: Every one has a birthday.  Year round, daily life is governed by seasons and the day of the week, what time to rise or eat, what time to sleep, when to work and study and pray. School and work life are run or organized around time or schedules.  Events occur before, after or the same time.  Life in families and communities may run by clocks, time of day and calendars.
   
   Material to support the following areas are scheduled to appear online by September, 2010. 
    
2. Money Matters:   People buy and sell goods and services. People work for a living, There are budgets to balance in private life and in business. Money is counted, added, compared,  subtracted, multiplied and divided.  Life in families and communities often involves money matters. Talking about ends, values and methods for handling money at home, and in buying goods and services, or balancing a personal or business budget would provide a context and motivation for care and diligence in arithmetic with unsigned and even signed numbers.  The latter can be used to represent amounts due and amounts owed - assets and debts. Students may shown how calculate net worth by adding subtotals.  See corresponding remark below.   
    
3. Geometry Matters:    Maps and plans drawn to scale (or not) are everywhere.  In going to school and in traveling children may see maps and plans. Larger schools,  colleges and workplaces may  use building plans or maps to provide directions.  And in travel and in construction, we may use maps drawn to scale to estimate or calculate lengths and even areas, and thus solve problems without or before any knowledge of higher mathematics in the form of  trigonometry.  Solving problems with maps drawn to scale has take-home value, and may be emphasized and illustrated in geography and science lessons.  Reading maps and plans, understanding  contour levels, are all parts of quantitative skill development. The site area on maps, plans and geometry represents an senior high school mathematics version of the junior high school geometry matters to appear here.
   
   In the foregoing students may learn on paper to  evaluate a formula for an area or volume given the necessary lengths and measures. They should also be able determine those lengths or measures from hands-on experience with the actual figure or scale drawings of it. Instruction should point out applications robustly and fully. 
   
   Nuance:  Triangle construction algorithms ASA, SAS, SSS and even AA(A)  will be included among methods for drawing triangles, rectangles and the circles to scale, 100% included.  That makes explicit the common practice in which figures are drawn more or less to scale in the exposition of geometry.
     
4. More Counting & Measurement Matters:  Besides counts and measures of time, dollars and length,  master measures of area and volume, and of  mass and weight.  In buying and selling goods and services, and in making things - that includes cooking and construction - people use and combine measures alone and in proportions.  Examples include speed and the cost of a unit (a prequel to the discussion of per  unit rates) of a good or service.   
   
   Counting & Measuring Skills 
   

   1. Express Measure & Counts in terms of given and alternate units of each, respectively.  

   2. Add, Compare and Subtract Counts and Measures - when possible.

   3. Multiply: Form the product of both with a number.  

   4. Divide Measures & Counts by another measure or number.

   Exposure more measurement matters in the home and in shopping will depend on the family and community life of students.  Due to the variation,  skill development will have provide the missing experience and a context for it. 
   

5. Matters of Chance.  In decision making, not all is certain.   Risks are present even for people who avoid games.  Learning about chance and probability may help avoid situations or decision where the risk are high, or help  in making decisions that lower risk and make the chances of success greater. Again,  due to the variation,  skill development will have provide the missing experience and a context for it.
   
   The Counting and Arithmetic Methods that the above application areas will have to be taught as well.  The application areas will require some  arithmetic, including decimals,  fractions and perhaps, for the description of rates, use of fractions with units.  Site to do: Suggest how and how much.   Primes and Prime Decomposition are needed for exact arithmetic in algebra, and they speed and improve fraction skills. But there is a question: Is their mastery a plus for Level I or should  this topic be delayed to Level II. 
   
6. Logic Mastery:  There is more to logic than the use of implication rules  IF A THEN B, forwards and backwards, alone or in sequence, to arrive at conclusions. Logic skill development begins with an awareness of the domino effect of mistakes in following steps or instructions, and care to avoid that effect.  Logic mastery may and should include doing and recording data (or inputs) and the steps of methods so what is done may be seen and confirmed or corrected. Logic skill development further includes approaching problems and puzzles with a systematic or deliberate exploration of what might fit or work, via trial and error - the solution of jigsaws with edges first provides examples. Logic is in part mechanical in that one tries to apply existing skills and know-how directly and carefully. But the habit of always looking for pieces that fit or methods that may work introduces creativity of a combinatorial and opportunistic kind. And logic mastery ends with an awareness of what is possible and what is not in the careful or diligence application of rules and patterns, and in the verification of recorded steps of oneself or others. .... 
   
   Precision in reading and writing may follow here from study of logic and from an awareness of the domino effect of errors. There is a conflict here between advocating logic skill and values because of  benefit for easing and avoiding learning difficulties, and the possibility that the some logic lessons will overwhelm the too young. 

Skills and concepts in the areas may be developed  to build  appreciation for and confidence in mathematical methods.  

Parents and Teachers: the folder  Helping Your Child or Teen Learn  includes this mathematics booklet sort list (with brief descriptions).  For parents, the booklets include exercises which your child or young teen may attempt with parental supervision. For teachers, the booklets indicate skill development paths that may serve as a basis for lesson planning and skill verification. Skill development with verification and correction as needed is the first aim of practical mathematics instruction.

Even before a child goes to school,  most of these application areas will appear in home life and so have some take-home value. Where possible skills and know-how with the greatest take-home value are put first,  but over time the remaining skills and know-how will have less and less immediate value and more and more long-term value or potential as part of general preparation for adult life and also for further instruction. The above application areas will be of greatness benefit to those students who in mastering skills and know-how, continually look for opportunities to apply them.  

The aim of instruction in the above areas and in the parts of arithmetic they require is to develop observable and verifiable skills and  know-how.  As part of that, we will show learners how to do and record numerical and geometric steps in learn but sufficient show work formats that allow steps and results, from start to end, to seen and confirmed or corrected.  Emphasizing that care and patience, and precision too, are needed to avoid the domino effect of errors and approximations in short and long chains of reason will be emphasized. The ability to figure well, and to read and write with precision, is an observable sign of intelligence of the practical kind for work and studies in general.  

Ends and Values

The organization of primary and junior high school math lesson around common needs presence in the applications areas  including the  ends and values for work in a logical manner provide a clearer paths for adults - teachers and parents - to follow for  observable and reliable (confirmable and correctable) skill and know-how development.  Quantitative skills, logic and chance may appear as part of a discussion of careful decision-making and risk/gullibility avoidance, a discussion that may have take-home value for students and their present or future families. The latter discussion may be followed by 14 year olds, even if further years in school may delay it utility.  To address that, we may say those further years are for ...., and those further years may include or end with a review of the discussion and the six application areas, so that the discussion and its requirements will not be forgotten. 

A. Diligence, Being Careful 

People who figure well do so to avoid the domino effect of mistakes, where an error in one step makes all that follow wrong or suspect.  Experience in the above areas and arithmetic there-in  exposes students to the domino effect of errors and approximations in calculations and reasoning.

The value of avoiding the domino effect of errors and approximations may be introduced in arithmetic, but beyond that can be emphasized as value common to most rule and pattern based arts and disciplines. 

B. More Diligence - the duty

Once a skill is practiced to the point of mastery, students will be expected to maintain that mastery. While knowing the alphabet or the difference between left and right may be difficult for some, we still expect it.  Arithmetic like the alphabet and spelling taught and learnt in a half-hearted manner is insufficient.  Clear ends and values need to be set.  Awareness of the domino effect and its avoidance demands student remember what they have seen - a personal duty and responsibility. If we do not ask for it nor expect it, some or all students will suffer.

C. Still More Diligence -  tell a story

Skill mastery to be credible has to be observable. Notation or formats that permit steps or results to be done and recorded in an observable manner record and aid step.  Just as lean column  or place value methods for arithmetic made the latter accessible, notation in general needs to be chosen leanly to aid performance and in particular make it observable, in a step by step manner. Skills and know-how needed to be seen to be verified or corrected.  The work done needs to tell a story or leave a trail for the doer and others to follow and check.

Level I, End Notes

Rote Learning Permitted
in paths to comprehension

Site areas explore the thought-based development of skills and concepts give an alternative to rote learning, but the development of practical skills and know-how, the alternative should be available but not imposed to the extent that explanations overwhelm. 

At one extreme, students will insist that instructor are hired to give methods that work, so explanations of why are not needed. At the other extreme, my case,  students will refuse or having difficulty in accepting and using methods that is not explained. In general, explanations are or should be included where they help skill and concept mastery, but where  they may or begin to overwhelm learning and teaching, explanations should not be given. When to give and when to limit explanation may depend on students.  An operational development of skills and concepts is sufficient if explanations are available in reference material for keen students, or in class for the subgroup of students who insist on explanations why method work as a condition of acceptance. 

Comprehension skills may mature. While  students learn to carefully apply the rules and methods alone and in combination,  one at a time, one after another,  the combination of rules and methods in sequence generates and so explains further ones. That together with  show-work formats for doing and recording the steps or intermediate results in geometric and numerical figuring for the sake of confirmation or correction sets the stage for proof of correctness, and  a fuller, if not full, thought-based development in senior high school mathematics.   

While an operational command of mathematics may be had without explanations,  learning by rote may be lessened by watching for and collecting thought-based developments of skills and concept, developments which are easily understood and not overwhelming for most.  Deposits for that appear in site pages.

Just in Time Notation & Concepts

Notation and concepts like explanations should be selected to aid the operational command. In level I, the formalism and concepts of modern mathematics at the senior high school level and beyond should be introduced only when it aids skill development, or does not distract from it.    In level I, focus is on the needs of the many and in that providing them or all with a know-how that is practical,  that first has take-home value and secondary is of service to the level II pre-university stream.   The objective is to leave students with a reliable operational command that they will value in their lives and perhaps in the education of their children before, if at all, studying more mathematics.  Level I should be the base of pyramid, a base that first serves the common needs of the many. The minority who go on to study advance mathematics etc will benefit from that as well. 

More End Notes for level I & II

1. In counting objects, the latter may be grouped arbitrarily into non-overlapping subsets, so that the total number is the sum of subcounts.  The foregoing requires each object to belong to one and only one group. This grouping is an iterative affair. Some subsets themselves may be divided or or partitioned into sub-subsets, in a way that their subcounts are sums of sub-subcounts. In practice, all divisions and subdivision should lead to the same total count.  If not, census taking would be an impossibility.  In the case of accounting,  assets and debts represented by positive and negative monetary values may likewise be grouped and subgrouped for the calculation of subtotals and totals.  The foregoing common practices are present in the application areas in counting and in the addition of unsigned and signed numbers. And on a technical note, these common practices are precursors too and generalizations of the commutative and associative laws, and even distributive laws, met and formalized in senior high school mathematics.  But in the service of common or likely needs, we will teach those practices, and save the statement and comprehension of the laws for later instruction. Here we treat mathematics as a service subject, one whose first duty is to build and strengthen the common knowledge. 
   

2. Logic mastery in a mathematics-free manner is an optional part be part of instruction before senior high school mathematics, a part whose presence or mastery will be required in senior high school mathematics because of its take-home and long-term value.  Logic mastery in mathematics or apart - say in a reading and writing course - may lead to greater care or precision in reading and writing, and so avoid or lessen difficulties in studies and in work.    The leading chapters of Volume 2, Three Skills for Algebra  develop deductive reason (logic mastery) in a math-free way. Altogether, those chapters hint at the partial Euclidean organization and codification of rule and pattern based arts and disciplines. 
   
   In particular, awareness of  the difference between say A if B and saying A if and only if B (or equivalent expressions) will sharpen reading and writing. And seeing how to chain implication rules together will help reasoning in general.  These two elements of logic and further elements may be introduced when students are ready for them - the age level for that may depend on the student. The net result should fewer difficulties in work and study, and better results in general.
   

3. Algebra mastery in the aforementioned instruction may be limited to the evaluation of formulas for perimeters, areas and volumes; and also for distance, time and speed.  Other formulas may be present. In the evaluation, we will require  each step done and recorded in an observable and hence confirmable or correctable manner. The presence and a vertical alignment of equal signs is recommended in doing and recording the steps. A common format here will be a source of skill and comfort for learning and teaching. It provides a common destination for skill development.  That being said, the notational or  shorthand role of letters and symbols here should aid skill development, and not overwhelm it. For example, the perimeter of a polygon may be defined and understood by saying how to calculate it with the phrase: add the lengths of the sides. Once that is understood and illustrated, the algebraic description with letters and symbols, subscripts and dot-dot-dot summation notation,  is secondary. The ability to find  the perimeter of a polygon is more important.  That being said, sans the dot-dot-dot notation, there is no harm and even a benefit to introducing letters and symbols, sans or with subscripts etc, to indicate or denote a definite series of quantities or measures (areas, lengths, times) that will be summed. 
   

4. Quantitative skills will be observable if students are shown how to follow  arithmetic and geometric practices in a show work manner, one in which most steps or intermediate results are done and recorded fully or almost so, for the doer or someone else to see and confirm or correct.  Mastery of show work formats in arithmetic etc provides an observable standard for students, parents and teachers to see and encourage.  That being said, the format should be lean and chosen so that the writing, the drawing aids the reasoning.  Notation that does not help the current level of work should be avoided.   
   

5. Explanations or the thought-based development of skills and practices, that is showing how the combination of skills and practices, or there steps, one at a time, one after another, should aid mastery and not overwhelm it.  In the case of arithmetic methods for addition,  talking about place value and carries may aid the comprehension and mastery.  But in the case of long division, talking about why methods work may overwhelm students and distract from the mechanical mastery of the method.   Where explanations of why may overwhelm students in class, teachers or tutors may focus on the mechanics - ensure the latter are mastered in an observable and hence confirmable or correctable manner - while for the sake of completeness, explanations why should be available as a reference.  Site material may serve.  Children and young teenagers may be content to be given a method that works, and see explanation in all or part (I learn this from the school of hard knocks) as not needed. After all, teachers are employed to provide methods that work.  That being said, upper level mathematics instruction may have the task of changing the values of students, so that explanations are appreciated.  And for that, explanations should be coherent, gap-free and make the hard as easy as possible. See site material.
   

6. Explanations are present in the show work format given and require in primary and junior high school skill development.  The format gives steps to follow and check. Chains of reason are present here in the backward sense that the results of a step may be correct if earlier steps were done correctly.  To check or test a result, a teacher or peer may say to a student, show me your work.  The work explains how results were obtained in an observable, confirmable or correctable manner.  Over time, students may go from following skills and practices, one at a time, and one after another in a given manner, to combing skills and practices to obtain and so imply further ones. With that, later skills and practices may depend on earlier ones.  Seeing and recording how in a show work format provides an explanation of the later one. So know-how based on skill and practices develops a logical structure. And may teaching or showing how to combine rules and practices in an observable, verifiable or correctable manner, the instruction allows students to generate rules and practices beyond the seeds or initial ones given during instruction.  All the foregoing provides a prequel to the show-reason or work format of proofs in higher level mathematics.  
   

7. The use of maps and plans drawn to scale provides experience with similarity matters and practices, even before the concept of similarity is discussed.  Drawing to scale provides a practical method for solving missing lengths, missing angles and missing area problems in the plane.  Much can be done and explained, maximal so,  in the early study of mathematics with take home value before the say upper level and technical development of trigonometry for acute and/or further angles.
   

8. Numbers and Counts.  In the decimal counting system, single digits 0 to 9 represent a number or count of ones.   Multiple digit decimals like  
   
   432 = 4 hundreds & 3 tens & 2 ones. 
   
   represent a number in mixed units of counting. Abraham Lincoln's phrase  four score and ten also represent a number  4 twenties & one ten using mixed units of counting, one non-decimal. 
   
   In the decimal counting or number system, larger whole numbers are expressed in terms of mixed units of counting:  ones, tens, hundreds, thousands and so on.  Proper decimal fractions like
   
   0.563 = 5 tenths & 6 hundredths & 3 thousandths  or  563 thousandths
   
   are expressed as a sum of counts of tenths, hundredths and thousandths. Improper decimal fractions like
   
   3.45 = 3 ones & 4 tenths & 5 hundreths
   
   are expressed in terms of counts of powers of ten:
   
   ones, tens, hundreds, thousands etc 
   for the integral part, and
   
   tenths, hundredths, thousandths etc
   for the decimal fraction part. 
   
   In other words counts and mixed units of counting are part of our description of numbers, and when units of measures are present, also part of our description of measures. Counts too are present in the numerators of fractions.
   
   Invariance Principle for Common Needs: The method of counting & measuring and the choice of units for each count & each amounts does not affect their values.   
   
   This counting principle is present when children first learn to count and add.  Mathematical skill development could use it as is, or embedded it say set theory to derive the properties of unsigned numbers - integral & fractional - in senior high school and higher studies.  Doing so would provide continuity with earlier instruction. Skill developers or mathematician may debate whether or not this principle is extrinsic or not. 
   

9. Timing:  In some "rich" communities,  school systems keep students in school not for the sake of education, but for the sake of keeping teenagers away from employment or unemployment. Where students are keep in schools and promoted year after year in a bureaucratic manner but without observable and verified skill and know-how development that is a fraudulent form of education. The shock to self-esteem of not being promoted due to lack of skills needed for the next level is simply delayed and even compounded by promotion regardless of skill and talent. 
   

10. Education reform, year after year, decade after decade, will get somewhere, but that somewhere will not include observable and verifiable know-how as a source of skill and confidence while giving skills to learn (or rules and patterns to follow) is cast as a horrible form of rote learning.  

    Learning by imitating others, or following instructions step by step, is a necessary part of education, even if why the underlying skills or steps work is unknown. Pathways for mathematics and logic development may begin with skill mastery first and explanation second.  Rote learning is not all bad, albeit learning with explanations not too demanding is better. 

11. For Differentiated Instruction: In skill development, observable mistakes and difficulty should be seen as a learning experience or teachable moment.  Education to be child- or student centered, should formatively provide opportunities to make mistakes on the journey to skill development and perfection. Where education is compulsory, there is a question of how to make those formative mistakes, penalty-free, and so count against the learner at the end of a course.  There-in lies an argument for not counting course work in situations where students do better on final examinations. 
    
    The present model of recording marks for tests and assignments does not track what skills and concepts are missing or mastered. That impedes differentiated instruction. Here is a challenge for course delivery:  In giving a course, track the mistakes made by each student, and only assign and correct those questions corresponding to past mistakes and new material.  Just as medical charts follow a patience symptoms and reactions,  instructors too might chart for each student, the level of know-how, one observed skill after another.  

:

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.