Ends, Values and Methods for Mathematics Education K1-12

"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."
-- Chapter 6, Alice's Adventures in Wonderland,
Lewis Carroll 1832 - 1898.

For over three decades, I have been thinking about how to simplify mathematics education and address both its content and motivation difficulties. This presentation begins with non-technical ideas and finishes with technical ideas and references. The practical ends, values and methods outlined can be adapted to fit the mathematics and logic-language skill needs in multiple circumstances. The challenge questions are answered below. The scope of this big idea and proposal is not just for elementary school, that label on this submissiion is an accident. It is for elementary to senior high school or college level. This proposal aims to transform skill development methods and materials from counting to the start of calculus. The eventual impact is likely to be world-wide in developed and developing communities.

I have recursively explored, composed and collected ideas and methods for K1-12 skill development. With that research being done, the implementation phase is ready to begin. Corporate, academic and governmental partners and sponsors are now wanted to review, refine and generally help implement remedies and extra steps I have found to simplify and motivate learning and teaching. I would like to part of or lead a team of people, young and old, with technical know-how as well first hand experience of education and its difficulties. While I am able to continue the work alone, the implementation would be greater and quicker with more heads and hands. In that we may show how to weave the ends, values and methods into existing instructional materials and beyond that, write and support additional materials, all for more effective paths for building skills and confidence.

Which Way to Go - Setting the Stage

Education has room for both reflection and skill mastery. Theories of learning which say true knowledge is a private matter, personally constructed through reflection and located in the mind and beyond the observation of others and performance testing, offer a direction for education which is immaterial and apart from practical skill and know-how. The development of reading, writing, arithmetic and reasoning skills (4Rs) was once the foremost, practical and material objective of elementary and subsequent schooling. These skills allow to live, work and contribute to their communities, Those skill areas allow people to read and write at home, at work, in religion and in school in acquiring, using and recording know-how and knowledge. Counting, measuring and figuring aspects of arithmetic was and remains useful in was useful in cooking, agriculture, time-keeping, traveling, and buying and selling goods and services in general.

Practical know-how or abilities may be guided by rules and patterns, approximate to exact. Learning through practice or experience which is which provides an end for skill development in school or on the job. Over time, how some skills, rules and methods depend on earlier ones may be seen or emphasized. Thus with time, know-how may become structured. However, mastery of common skills and practices may be begin by following demonstrated or given rules and patterns by rote. For example, we rise, eat and sleep automatically by rote without fully understanding how and why our bodies function. Lights may likewise be turned on and off before mastery of electrical concepts in all or part. Similarly, students may see how to do arithmetic operations with only partial explanation of the why. Skill development does not require explanation in all or part of why. Learning physical and on-paper skills by rote may be a necessary and normally part of growing up. With regrets, to try to understand or explain everything would be too much for children, teenagers, teachers and their parents.

Practical know-how is often based on skills and practices. Some may be shown or given through demonstation of what to do. Others may be implied by rules and patterns. In physical and mental arts and disciplines in which activity or work consists of steps that can be seen, instruction may build skills and abilities by showing how to do the steps directly, or by introducing substeps to make the larger steps easier to learn, teach and do. The preparation of students for practical matters may show them how to communicate, reason and solve routine problems with data, steps or results recorded for the underlying method and skill to be seen and checked, as done or later, by the doer or others. While student thoughts are unobservable, their actions in deed and on paper can be seen and judged.

Education may provide students opportunities to reflect deliberately or accidentally. The methods and theories of mathematics, science and reason show how to use rules and patterns, one at a time and one after another, to arrive at results or conclusions. In this, some methods, theories and hence results and conclusion are more reliable than others. Some methods and theories may apply to different or overlapping domains, and in the process suggest different results. Practice and testing may materially decide which methods and results are the most accurate or reliable. The development of practices and know-how is not an internal matter, located in the mind. It is a matter than depends on material interaction with the environment and others.

Skills and know-how of the observable and material kind may not have the same standing in spiritual or constructivist eyes as true knowledge, but their explanation and development has value for life in the street and life in specialized occupations. The lack of certainty in some methods or approximations leaves room for thought and action.

The material aim of skill and know-how development and training is student performance in a reliable, repeatable and reproducible manner. Constructivist and cognitive theories of education which say "true knowledge" is a private matter, located in the mind, apart from observable testing and hence performance, are too immaterial, too impractical, for use in arts and disciplines which value and aim for observable and verifiable skill mastery. To be a cook, tailor, carpenter, driver, electrician, plumber, writer, statistician, mathematician, engineer, programmer or technician, observable and verifiable skills are required. Skill mastery needs to be seen to be believed.

The National Council of Teachers of Mathematics, 1989 Principles and Standards for School Mathematics, in advocating constructivism implies a retreat from the material to the immaterial and a promotion of student reflection, teacher encouraged, as a source of personal knowledge, which teachers should not judge. Constructivism with its aversion to testing, with its labelling of the latter as not being representative of student true knowledge - whatever is in their minds - is inconsistent with the want and tendency of educational authorities to measure and call for student performance. A remedy for the 1989 or present-day stance of the National Council of Teachers of Mathematics lie in the skill oriented mathematics education promoted in its earlier publication.

This 1953 publication says the following:

• The lack of correct concepts in arithmetic may be one of the great reasons for the difficulty algebra presents to so many of our students. Source: page 348.

• .. the position taken by this book is: We learn that mathematics which we are taught. ... There are some persons who say one who knows cannot teach for he cannot fathom the difficulties of his students. These persons say that as a teacher work with his students through a problematic situation which is new to both teacher and student, real learning takes place and then only. We believe this assumption to be entirely erroneous and assert that a teacher is a teacher is a learning engineer, a builder of minds that will solve problems. As such, the must first know the total mathematics he will teach, that is, the framework. Source: page 348.

• .. in a sense the teacher must be a master technician. He must know how to build any known kind of learning. .. must weigh, balance, and appraise the possible learning. ... know their relative worth both for the individual and for society. Source: page 349.

The terms learning engineer and master technician are noteworthy. They imply a practical skill-oriented approach to instruction from basic skills to the preparation of students for calculus-based college programs. Given that many primary and secondary school teachers engage in mathematics skill development are not fully trained in mathematics, or more have had an alienating experience in their school and college days, we need course materials to be clearly written by learning engineers in steps that most teachers, parents and students not too young can easily follow. The challenge today is not find those steps, they exist, but to employ them in course design, materials and delivery.

Whats Skills and Why

The questions of what skills to offer at home, in school and in college, all depends on the present and future state of society and within that, of the students and their families. The principle that all students should have equal chance to succeed in school is guided by and dependent on the views of their families and the state of the classrooms or schools they attend. Some parents may favour schooling to the fullest extent possible for a better future for their offspring. Some parents, those that did poorly themselves, may be skeptical. Some parents may be opposed to high school and even elementary education for all because of family needs or views on gender. Schooling in all or part may serve the hopes and values of parents for their children. But schooling may also serve and be imposed to serve the general ends and values of society. We may distinguish between student-centered and society-oriented skill development. The two developments overlap.

Basic mathematics and logic-language skills have take-home which may be appreciated by students, teachers and parents as serving everyday practical ends, actual or potential. Basic skill development may also be part of preparation for pre-college trades and professions. Good work habits may start with awareness and avoidance of the domino effect of errors included. Society-oriented skill development includes preparation for say calculus- or statistic-based college programs, programs in which entrance or success or not guarantee, programs whose skills are appreciated if student succeed. There the practical value is contingent on success with any intellectual value being a personal matter, valid for some but not all. The latter in retrospect may feel misled. In general, we may observe a progression for student- to society-centered skill development in schools. To serve the needs of all, not just the minority entering college studies, we pose the dynamic programming problem (technical term in optimization theory) of how to give each year of instruction as if it was the last chance to provide students know-how with take-home value. This dynamic optimization problem sets priorities. This problem imples ends, values and paths for instruction.

Student-Centered Instruction

In modern times, most parents see value in showing their children how to read, write, count, measure, figure and reason. These skills have immediate to potential take-home value for life at home, at work and on the street in telling and tracking time, in saving and spending money while buying and selling goods and services; in reading or using maps and plans; in direct and indirect counting and measuring of common objects and quantities; in following rules and patterns in making things and decisions; and in seeing risk in decision making when not all is certain in daily life and in games. Schooling may provide skills and practices in the foregoing areas in a minimal or rich manner, depending on student talents, teaching methods and depth or length of studies - the number of years available. Where parents are skeptical or time is short, a leaner development of the foregoing may be better.

Counting and measuring directly or through calculation would require mastery of integers, fractions, decimals and even signs in exact and approximate ways. Occupations which do not require college studies may enrich or refine the foregoing in accordance with their specific needs. For example, learning how to use maps, plans and diagrams drawn to scale is enough for everyday use and most trades in cloth making and construction. However, work in surveying, navigation and electricity would benefit from mastery of trigonometry. The latter subjects would also set the stage for calculus-based college programs. For students aiming for calculus-based programs, inclusion of skills and concepts of service to pre-college trades or directions might strengthen that preparation and help provide an alternative just in case their college plans fail in all or part. The main end for student-centered instruction is not how to prepare students for college, but how to help them as much as possible with their hopes and plans, successful or not. That objective or end would serve to motivate and engage students, their parents and their teachers. Parents too have to see value secondary education.

In general mastery and perfection of counting, measuring and figuring abilities has take-home value. That and how should be emphasized in the the mastery of arithmetic, of formula evaluation, and in the mastery of geometry in the form of maps, plans and diagrams drawn to scale. Primary and junior high school mathematics and perhaps logic-language skill teaching would follow the forementionned "dynamic" optimization principle of treating each year of instruction as if it was the last chance to provide learners skills with a mastery of methods and practices with take home value to themselves or theirr families. In poor communities, students may have to stop because of family circumstances. In richer communities, students may choose to stop because of alienation. The foregoing optimization principle coupled with critical path analysis of skill building steps would provide motivation to students and their families for continued studies while serving those who must or choose to halt for any reason.

Arithmetic skill is a sign of practical intelligence in society. It implies awareness of the domino effect of mistakes in figuring. Awareness of this domino effect and its avoidance gives an end, a value and stool for developing abilities and talents at home, at school and at work. instruction should emphasize the domino effect of care and mistakes, especially the latter as they occur. Further, student-centered instruction should also emphasize the ability to read and write with precision as part of its awareness of the domino effect of mistakes, campaign. For that, or in logic-language and mathematics development for the not too young, but as early as possible, instructors may talk about the difference between one- and two-way implications or conditions.

Seeing the difference and being aware of its impact on chains of reason or instruction represent asset for developing talents and avoiding mistakes at home, school and work. The difference may be mentioned repeatedly in late primary and further schooling until the difference is mastered. But where student failure to see the difference is likely to cause discomfort, instruction should leave its explanation for another day.

Society-Oriented Instruction

Skill and know-how development may help primary and high school graduates and early leavers work and get work. In learning how things work, and in employing their reading, writing and reasoning skills to learn more about life may be intellectual motivation for some. Beyond that, many societies say they want or need skilled and educated people for enterprise or work in commerce, science, technology, engineering and some mathematical disciplines. Just as public and private sport systems may search for and then groom individuals for competitive sports - some not well-paying, secondary school and college may be part of a slow selection process for individuals who have the time, patience and ability to enter college programs in art, engineering and science for their own sake, or to serve general and particular needs of society in careers and professions.

Where skills and concepts mastery serves concrete ends and values likely to be of service to students or their families, instruction is student centered. The dynamic optimization or programming principle may still be applied to treat each year of instruction as if it was the last chance to provide skills, concepts and practices with take-home value. As said above, that may set priorities, and even delay or offset the inclination of high school mathematics course to prepare students for destinations they will never reach. Putting practical skills and abilities first by covering every day mathematics including money matters as fully as possible would serve all - high school graduates and leavers of any age. The risks and benefits of money handling options in society from buying and selling services and goods to insurance and investment could be covered in ernest, preferably in a neutral manner where third party financial products are not described nor sold to students, so that academic intregity is thus defined and maintained.

After World War II, mathematicians due to the cold war politics and fears of the 1950s were concerned about the preparation of students for calculus-based college programs in science, engineering, engineering and mathematical disciplines - operations research included. See the references. In their writing, some acknowledged they were not considering the needs of terminal students, that is, those not heading say for college programs in science and engineering. Emphasizing every days mathematics including money matters as indicated above would serve the needs of both terminal and non-terminal students. It would provide common skills and know-how. In that the more advanced students would benefit from mastery of money related formulas for compound growth and geometric sums - forwards and backwards, algebraically, to the same level as those heading for calculus-based college studies.

Critical Path Analysis

A. Respect Skill Dependencies

Which way to go in mathematics skill development depends on ends and values, and critical paths being known and respected. Course materials need to emphasize skill development ends, values, methods and dependencies. Ignorance is not bliss. The education and course outlines and materials given to primary and secondary instructor needs to outline and emphasize skill dependencies.

Times are changing. The ability to tie shoelaces and to tell time with an analogue clock was once a sign of readiness for pre-school or kindergarden. But in some places, shoelaces have been replaced by velcro. The associated dexterity obtained in learning to tie shoe laces has been lost. Likewise, the advent of digital clocks has lessened children exposure to a half and quarters after or before as fractions of an hour. Thus technology has the unforeseen consequence of weakening skills. The presence of calculators remove the need to use tables or slide rules for square roots and logarithm evaluation, forwards and backwards. The presence of calculators further removes or lessen the need at home and work to add, compare, subtract, multiply and divide fractions and decimals by hand, and in that doing and recording work in steps, one at a time, one after another. Thus the common need to do arithmetic with decimals and fractions is reduced. That reduces the initial role of arithmetic in demonstrating the domino effects of lack of care or errors in arithmetic, and more generally, in multi-step processes. All the foregoing activities may not have a great value in themselves, but in doing them like doing wieght lifting, trains the mind - makes it more agile and disciplined, a plus for further work and studies.

When exact and efficient arithmetic with limited calculator use is not emphasized, the subsequent use of calculators leads to approximations in figuring in place of exact arithmetic, and reduces the apparent need for students to master addition and times tables, forwards and backwards. There are consequences, especially if primary school teachers are unaware of how the first years of high school need to develop and perfect arithmetic skills and practices with decimals, fractions and signs. The path in skill development from learning to count, measure and figure through mastery of arithmetic, algebra, geometry and then calculus is then undermined. Parent and teachers at the primary and secondary school level need to be aware of how algebra, trigonometry and calculus mastery requires an efficient and exact mastery of arithmetic with decimals, fractions and signs. This requirement needs to be mentioned in course redesign and delivery to provide context and motivation for it - a hook to mention future needs or dependencies. Not doing so weakens or undermines cumulative skill development, where later skills and practices depend on earlier ones.

In course design and delivery, calculator-free arithmetic skills with addition and times tables, with decimals and fractions in general should be emphasized and retained as while their no other program that meets the requirements of calculus-based college programs in these skill areas, and also further introduces avoidance of errors in multistep method as an end, value and tool for building skills and confidence. Skipping skills has a down-stream effect. As long as skill development is valued for life in the streets or academic destinations, which way to needs to go depend on critical path analysis of skill development steps. Ideally the step are complete, none are missing, and the natural talented needed to follow them as a teacher or learner is minimized.

Steps too Large or Missing

In my high school days four decades ago, I shyly noticed steps and in retrospect substeps and nuances missing in algebra skill development. There was no talk and no rationalization of the shorthand roles of letters and symbols. They were simple use in the statement of formulas, the presentation and solution of equations, numerically or literally; and in the presentation of axioms for algebra, also known as properties of real numbers. Since entering graduate school in 1975, and even before, I have been thinking not only about mathematics skill development difficulties, but motivational ones too. Motivational difficulties as indicated above may be addressed by the dynamic optimization principles of emphasizing and putting first skills and practices with take-home value. The further mathematics and logic required for calculus- or statistic-based college programs may be emphasized second with little or no mention of that end for instruction before.

In 1981, after meeting met step-by-step skill development in a week-end courses on how to be a Nordic ski instructor, I was struck by the observation that my education in secondary school and university had emphasized and employed algebraic and deductive ways of writing and reasoning, but with a gap. The shorthand ways of writing and reasoning with letters and symbols was taken for granted with no clear no explicit explanation nor introduction. Mastery would follow from exposure, quickly or slowly, nor never. Instruction by exposure in place of directly can be seen today in mathematics from solving equations and using formulas backwards to the full-strength use and development of algebraic reasoning in calculus and its epsilonics. The underlying difficulty is due to steps, concepts and nuances not fully clarifed in course designs and material, olde to present day. Yet clarification is possible via the inclusion of extra and more gradual steps or concepts in course design and delivery.

Since my fall 1983 presentation of three lessons on three skills for algebra, on why study slopes, and the difference betweent one- and two-way implication rules I have been slowly seeking, collecting and composing more and more methods to fill and remedy technical gaps in the exposition of mathematics and logic. By fall 2007, most of my remedies for easing and avoiding algebra difficulties and algebra-related difficulties in calculus had been found and posted online. By then, I had seen that arithmetic skills need by algebra were not being not learnt. I had a general sense of what needed to be cover and how in arithmetic, algebra, logic and geometry to prepare students for calculus. The sense implied a technically path for skill and concepts, a path more gradual accessible, a path that would serve preparation for calculus well. But many of its steps were dry and technical. The question of how to fit the path or its steps into program with context and motivation remained. In secondary schools today, most students and teachers have no idea why skills and topics need to be mastered other than the bureaucratic reason: preparation for the next test or final examination. The lack of context in that is alienating not only for students who do poorly, but also for student who do well.

Context and Motivation

Since fall 2007, my thinking has evolved. Today, I suggest a deliberate focus on skills with actual or potential take-home value that parents and teachers, if not students, in late primary and early secondary school. That would be accompanied by a development of exact and efficient counting, measuring and figuring skills for their own sake; for sake of introducing the domino effects of diligence and errors; and for sake of exact and efficient arithmetic practices needed in algebra, geometry and calculus. Course material and outlines should mention all three reason for exact and efficient arithmetic skills.

In some countries where the school leaving age is four or five years later, there is a danger the take-home value is premature. But offering that context in the second or third year of secondary school, even if it be early, is better than not because the not has not take-home value, and there is no guarantee that students in question will stay in school for a later courses which covers the same material. In essence by age 14 or 15, before the mention of trigonometry and further subjects required by calculus, mathematics and logic-language instruction should try to provide and make common skills and practices with actual or potential value to students and their families in the short- and long-term. That should leave them with a good impression for them to take-home for themselves, and for their present or future families. That is to avoid or stop the vicious cycle where the bad impression of parents of mathematics in secondary school is transmitted to their teenage offspring.

The question of what to teach if the current year might the last year of instruction justifies the early inclusion of skills and practices with take-home value apparent to students, their teachers and adults in general. Following this, secondary school instruction may stop, save for courses that emphasize more and more basic skills and practices with take-home value to maintain and even refine such values. Those courses would be help students who might otherwise be alienated by preparation for calculus and the dry nature of preparation topics. For these students, mathematics education that serves skills and practices which has no immediate or long-term is best avoided. It may be better to leave a favourable impression and a sense that more could have been learnt that to impose instruction is skill and topics that have no intellectual nor practical value for the trade or academic destination of the students at hand. Leave them with a little thirst for learning more instead of overwhelming and alienating them. With that less may be best.

I would include as earlyt as possible for each student, a wordy and math-free development of logic to develop precision reading and writing, two must for work and study, and a precursor to the use of implication rules and chains of reason, forwards and backwards, in and outside of mathematics. Outside, knowing the difference between one and two-implications may help students avoid bad agreements or contracts. That may prevent them from being cheated. The word cheated sound strong, but it use in class would be motivation for logic mastery.

For students who are succeeding and enjoying mathematics because of that, and for students who need more skills for their wanted trade or academic destination, mathematics instruction may continue to cover more and more algebra, more geometry - trigonometry included and further logic as preparation for calculus- and even statistic-based college programs. I would restrict logic to the direct use of implications, use of the contrapositive and the difference between one and two-way implications in the mathematical preparation for calculus. My site development of Euclidean and Analytics geometry in its preparation of students for calculus is straight-forward because of the avoidance of proof by contradiction.

Finally in preparing students for students for calculus, we need to mention in course design and materials, that the forward and backward use of compound interest and geometric sums provides a hook or context for the further study of exponentials, logarithms and mathematical inductions. All students able to follow it or heading for calculus-based programs in accounting and commerce may appreciate the context. A further context for the study of exponentials, logarithms and probability is provided by compound growth in populations and the study of genetics. A further context for using linear functions and quadratics, alone or together is provided by projectile motion in physics. And students of chemistry will appreciate learning how to do how to calculate with pure and denominate numbers - that is numbers and units of measurement. But those students enrolled in preparation for calculus but not heading for science and engineering programs in college will not see any cross-curricular applications of most of the technical topics in that preparation. For these students, the only reason for skill and topic mastery will be preparation for final examinations, and if they are told, preparation for calculus. The why slopes and why factor polynomials calculus starter-preview lessons at my site will help provide a context and alleviate some student alienation. Course materials for students and teachers need to explicitly mention the cross-curricula application. Modern day calls for providing students with rich learning environments and non-routine problems in real life solve may succeed in the hands of well-trained instructors, but the same calls in the hands of course designers and textbook designer unfamilar with what calculus requires may distract from calculus preparation. Calls to provide richer and more motivating problems, authentic-geniune-realistic, need to take into account the critical path that K1-12 mathematics has to include.

Rote Learning and Two Kinds of Rigour: Empirical and Deductive

My formation as mathematician called and thirsted for deductive reasoning as much as possible in its development. However, skill development from late primary school to college level in arithmetic, algebra, logic and calculus may be defined and provided by showing students how to do and record work in steps that can be seen for confirmation or correction, as done or later. Such steps can be taught and learnt by rote in a repeatable, reproducible manner that can be seen and judged. Many skills in many disciplines can be learnt and taught in this manner. That provides one level of rigour, empirical, with or without full comprehension. Over time, seeing how some skills and concepts depend on others weaves a web or structure to be seen and even shared. The web or island of knowledge may have multiple entry points. Instructors may choose the easiest for their students. In the case of deductive rigour, work done and recorded in steps is accompanied by reasons or implication rules that justify each step. The reasons or implications that can be recorded with or along side each step as part of the work. With that skill and practice mastery moves from rote and empirical to deductive. In case of senior high school mathematics, if not calculus, we need only to provide students with the spirit of this deductive development. The details may be provided as reference or given in accompanying or later courses in pure mathematics.

Alternate Routes and Gradual Steps

In my graduate school days, I saw guest speakers in mathematics and physics make the hard easier in elementary to recent research level topics. But mathematicians in writing their dissertations and papers are expected to be original. When my investigation of methods to ease or avoid algebra difficulties began, I did not see in the recent mathematics education literature any recognition of the gap that my three skills for algebra lesson addressed. What I saw were papers advocating or comparing different ways to do arithmetic, ways whose only difference was cosmetic. What I saw were papers advocating constructivism instead of revisiting content matters. Since the National Council of Teachers of Mathematics publication of 1989 Principles and Standards for School Mathematics, I have not seen any substantive discussion of skill development, its methods and the selection and technical form of mathematics topics K1-12. The latter was implied by the consideration of mathematicians in the 1950-60s. Since then algebra difficulties have been taken for granted from the forward and backward use of formulas and equations to calculus and its epsilonics. I do not know of any other mathematician who has looked at the technical elements and choices skill development from counting pre-school to calculus in senior high school. So the ends, values and critical path I am in the process of developing and finishing is unique, and likely to simplify mathematics education - make it easier for students and teachers. The proof is in the details. At the secondary level, those details are mostlt online at my website www.whyslopes.com.

What I would like to see is a team of mathematicians to review and refine my work. My technical know-how was enough to provide the path. But in the presenting the path I have avoided making historical remarks or comments. The smaller and extra steps I have posted online offer a more gradual more accessible and more paths to a first course in calculus through the arithmetic, algebra and geometry. In the coverage of geometry, I unify the analytic and Euclidean perspectives of similarity of objects in the plane, and rejuvenate a well-known geometric approach to complex numbers by introducing a very simple proof of its distributive law. There-in lies an option to place the development of complex numbers before the study of unit circle periodic functions. In this, the equality of two different ways to multiply complex numbers, one using real and imaginary parts, the other using angles and moduli, makes the further development of trigonometry and the derivation of trigonometric formulas for dot- and cross-products of vectors in the plane.

Which Way to Go - My BIG IDEA

Clear technical advances methods for mathematics and logic-language skill development K1-12 will have an impact:

1. they are easily understood and deployed by most adults at the primary school and junior school level, and followed by teachers with more technical backgrounds at the senior high school and calculus.

2. if they explain and remove old difficulties in the introduction of algebra and its full-strength use in calculus - epsilonics included.

3. if they introduce shorter routes for the geometric development of trigonometry at the senior high school and college level.

4. if they make efficient and exact arithmetic without or with a limited use of calculator easier to learn and teach at the senior primary and junior high school level.

Most if not details these unique advances are online at my website.

Ends and values for skill development K1-12 will also have an impact if:

1. if they respect skill dependencies - the contraints of critical path analysis, and

2. if they clearly offer a context and motivation easily understood, if not always appreciated, by adults and students.

Ideas for this are being consolidated.

While I may contradict some choice and preferences made by course design committees, I would recommend a two-level approach to building skills, confidence and context. Primary and junior high school mathematics would provide the counting, measuring, figuring and geometry skills and practices need in common application areas: time-date-calendar matters; money matters, chance matters; use of maps, plans and diagrams drawn to scale; and routine problem solving skills with guides for non-routine problems. Junior or early senior high school mathematics or language courses should also include deduction and chains of reason with implications; the difference between one- and two-way implications, and the equivalence of an implication with its contrapostive. The foregoing and avoidance of the domino effect of mistakes has value for precision in reading and writing, and in following or giving instructions at home, work and school. . It has intellectual in that it sets the stage for comprehension of Euclidean style logic in and outside of mathematics.

In the case of communities which are new to modern times, with parents or families not to keen about mathematics and logic-language skill development, the foregoing applications may be covered leanly, hopefully in a manner rich enough to set the stage for calculus preparation. Richer developments may be left to keener students for the sake of their intellectual development, or for the sake of distracting them while other students are following leaner routes. That distraction may backfire - lead the more able students to ask for more help from an instructor. It may be better to train keen and gifted to help slower students with their skill development. Doing so would also enrich the skill development of the helpers. In the case of communities adverse to skill development, the main point of arithemtics skill development would be to provide an awareness of the oft-mentioned domino effect of errors in arithmetic. That awareness by itself has take-home value for work in many fields, and it provides an end, value and tool for further skill development, a tool that may be exploited or required one day. Despite the hope for impact, there will be learning difficulties and circumstances which counter the best laid plans. Adaptation will be needed. To reason with students or their families is to convince them of the need for an ideas or action,

Appendix A: Technical Transformations, Small and Large

Elementary Arithmetic

Counting and measuring may be done directly or via arithmetic. For most people, decimal place and decimal methods for arithmetic can be met and mastered step by step with partial justification. Detailed explanations are available. But they are likely to overwhelm students and their adult tutors or teachers. In decimal arithmetic, student ages 9 to 12 may be shown how to do and record work in steps that can be seen and hence confirmed or corrected. Practice in this is valuable because it will reveal the domino effects of care and mistakes figuring. Care to avoid mistakes in multistep methods in arithmetic is a sign of practical intelligence. People who figure well are likely to watch and avoid the domino effects of mistakes in further multistep methods met at home, school and work. Avoiding the domino effect of mistakes provides an end, a tool and value for skill mastery in general.

Skill and confidence in arithmetic may come from learning to do by rote or with some comprehension in a way that leads to repeatable and reproducible results. This approach would be simpler for children, their teachers and their parents. Here the take-home value of learning to do exceeds the value of comprehension in full or part. Comprehension of why methods work can be left for later or skipped completely. But explanations of why should always be available in references for those students uncomfortable without.

Long Division Revisited

• In the context 23 ÷ 4 = 5 R 3, the expression has 5 R 3 has one meaning - here, 5 times 4 is three less than 23.

• In the context 33 ÷ 6 = 5 R 3, the expression has 5 R 3 has another meaning - here, 5 times 6 is 3 more than 30.

At the primary school level, the two meanings are easily understood from the context. But in further mathematics, we avoid expressions with ambiguous meaning. To remove the ambiguity or dependence on context for expression like 5 R 3, where is a simple remedy: avoid the remainder notation, and use mixed numbers to describe the result exactly

As part of the development of fractions, students may learn that 3 = ¾ of 4 = ¾ × 4. To avoid and end the use of the mathematical ambiguous notation 5 R 3 in primary school mathematics, I would rewrite 23 = 5 × 4 + 3 as

23 = 5 × 4 + 3 = 5 × 4 + ¾ × 4 = 5¾ × 4

Fractions

Arithmetic with fractions can be learnt after or besides arithmetic with decimals. The latter is a pre-requisite. Here again methods can be learnt by rote. However, methods for adding, comparing and subtracting fractions can be introduced and justified through raising terms. That is standard. Some students and teachers may find the explanation comforting. Other will find the explanation a source of discomfort. There is no pleasing all. That being said, raising terms can also be used to develop and justify fraction multiplication and division methods. So mastery with comprehension becomes an easy or easier option. Details appear in the fraction section of my website: www.whyslopes.com

Algebra

In algebra, the general use of letters and symbols beyond formula evaluation is a great mystery for many, year after year. The phrase "let x be a number" has no meaning nor context for many. However, in talking about drawings of circles and rectangles, in the three phrases

• let r be the radius,

• let D be the diameter, and

• let a be the length of a side

the letters stand for geometric lengths which can be seen and grasped. The terms square and cube in algebra echo the geometric origins of algebraic notation and concepts. Introducing the algebraic shorthand roles of letters and symbols geometrically is one technique presented at my site for easing or avoiding common fears and difficulties, and making algebra easier to learn and teach.

Since my high school days, I have thought there were gaps in the exposition of algebra - steps and in retrospect concepts too large or not explained- that made learning and teaching harder than need-be. Since then I have been looking for remedies. Multiple ideas to make the multiple shorthand roles of letters and symbols from solving linears equation and the forward & backward use of formulas to calculus appear at my site. Website reviews have been kind. The main difficulty in their dissemination is skeptism. Since the 1950s, mathematicians with the technical background to address technical difficulties or change skill development routes have not been studying technical alternatives for mathematics education. First rate mathematicians who might be invited to review elementary material may not see due to their talent how gaps in the exposition make their subject hard for others. However, in my school days, textbooks and courses assumed the algebraic way of writing and reasoning, it appeared for use without providing the rationalization I needed. I was forced and able in my then inarticulate ways to provide my own rationalization. But for that, I might have stopped. Fifteen years after meeting algebra, I started talking about three skills for algebra, and illustrating them with examples to provide a remedy. The remedy has since expanded. See algebra and calculus steps and chapters at my site for more details.

Talking about three skills for algebra and a fourth skill, the forward and backward use of formulas, expands the role of words in the exposition of mathematics. Indeed, one can expand the role off words further before algebra in the description of calculations that may done, and in the identification of calculations that give the same results. That too has an impact. Reference: www.whyslopes.com

Similarity Revisited plus Geometry via Maps and Plans

In the high school dicussion of geometry, all squares and circles may be declared to be similar; rectangles with the same aspect ratio may be declared to be similar; and triangles with corresponding angles equal and corresponding sides proportional are declared to be similar. High school mathematics may also cover similar shapes in space. The foregoing criteria for similarity may be met and mastered separately. But there is a unifying perspective, one that video game makers may appreciate. Two different regions in the plane or in space may be declared to be similar if in two different rectangular coordinate systems, each can be represented by the same set of coordinates. That is equivalent to saying one object can be mapped onto the other via a combination or composition of translations, rotations, reflections and dilatations. This technical perspective can be introduced slowly in primary and secondary school geometry by studying the properties of maps, plans and diagrams drawn to scale. The latter study actually gives a base for analytic and Euclidean geometry. For students who do not need to study trigonometry, mastery of skills and practices with maps, plans and diagrams drawn to scale might have some actual or potential take home value in junior high school. It also gives a base for later skill and concept development in trigonometry.

Complex Numbers Before Trigonometry

This book is interesting for its exploration of possibilities, its rigour, and its frequent mention of physical applications. I wonder if modern calls for cross-curricula development of mathematics and other disciplines recognize as such the possible interplay between high school mathematics and physics and/or the mathematics of finance, growth and decay.

This work is written by a Columbia Teaching College, Professor of Mathematic Education with great expertise in mathematics. The book explores possible routes for the development of geometry, linear and quadratic functions; numbers, constant, variable, equation and graphs; elementary curve tracing; loci and conic sections etc. Where are Mathematics Education Professors with a similar level of mathematics know-how, today?

His complex numbers and trigonometry chapter, pages 254-296, gives as an exercise for readers the task of giving a geometric proof of the distributive law for complex numbers when the latter are identified with points in a plane with addition defined via rectangular coordinates - or, real and imaginary parts, and multiplication defined via "polar coordinates" by multiplying moduli and adding angles. This approach and assumption of properties of real numbers immediately implies all the properties of complex numbers, except for distributive law. My short proof of it is online. It may be placed before the development of trigonometry in high school or college mathematics. That may transformed one or both levels of instruction.

Properties of complex numbers and the equivalence of two different way to compute products - derived from the distributive law or assumed - has easy consequences for the coverage of trigonometry and the derivation of formulas for dot- and cross-products of vectors in the plane. These consequences could be part of the transformation.

With this development of complex numbers and trigonometry has the potential to simplify the electrician study of amplitudes and phasor, and to simplify the education of students in science and engineering.

The geometric treatment of complex numbers dates from the time of Wessel and Gauss in the 1940s. The simple distributive proof of the distributive is short and neat enough to make complex numbers accessible to senior and perhaps junior high school students in way that simplifies trigonometry, preparation for calculus and calculus itself. Due to other priorities, I have not yet determined if my proof is a re-invention or not.

More Transformations

My independent and private research online at www.whyslopes.com describes and implies may different technical changes for secondary school calculus and preparation for it. There is a need to provide smaller and extra steps, easily understood and repeated, so that teachers seconded from others subjects to give mathematics courses and keen students can instruct themselves on how to develop and master skills. My work is now switching from the question of what to teach and how to the implementation phase. Writing this essay is part of the latter, or the identification of ends and values to provide a context and motivation for any implementation.

My online work is not the best possible, but it is the best I can provide. This competition provides an informal avenue for its evaluatiaon and then refinement. A more accessible path for high school mathematics and logic-language skill development appears to be feasible. With the documentation of most how-TOs or steps essentially done, the next phase for my research into secondary mathematics curriculum is to finish a critical path and just-time analysis of the steps. That requires clear and explicit ends and values for each step, and more explicit documentation of how steps may depend on each other, and how they may scheduled in accordance with that dependence and how learning and teaching maturity is required.

References: Mathematics Education Readings and References

Most of the technical decisions in curriculum design 1995-2010 has been revolved around the preparation of students for calculus-based college programs, and more recently, statistic-based ones as well. The needs of terminal students - those not heading for calculus-based college programs is not considered - has not been a priority for senior high school mathematics course design. Development of elementary skills and practices with take home value, so that they become become should serve the needs of all.

My knowledge of mathematics skill development is technical. In pedagogy, I subscribe to the notions of cumulative skill development with minimal dependence on natural talent, some always being required, and to the notion that most if not each year of instruction should be given as if it was the last chance to help students. That sets some priorities for instruction.

Coffee Table Books

1. Mathematics From the Birth of Numbers, by Gullberg Norton Company, New York & London, 1997, ISBN 0-39304002-X, QA21.G78 1996, 1002 +xxiii pages, well-illustrated. Very readable by masters of differential and integral calculus. A copy of it should be in every school where calculus or preparation for calculus is taught. If not, strongly suggest that one should be ordered.

2. The VNR Concise Encyclopedia of Mathematics by W. Gellert, H. Kustner, M. Hellwich & H. Kästner, Van Nostrand Reinhold Company, 1975 (or 1977). 450 West 33rd Street, New York, N.Y. 10001 (circa 1977) 750+ pages. ISBN: 0-442-22646-2 (hard cover) and ISBN: 0-442-22647-0 (paperback). Applications of mathematics in money computations, geometry, navigation, surveying and so on, are found in this encyclopedia makes this work a reference for subjects for further inquiry. This is a beautiful work. It has many colored pages and many diagrams. This work gives a broad overview of mathematical ideas from advanced high school to specialized studies in college or university. It contains many worked examples. Every high school math and science teacher should own or have access to a copy of this encyclopedia. So should every gifted student taking mathematics at the high school level and above. A copy of it should be in every college and community library. If not, strongly suggest that one should be ordered. Notes: This work is out-of-print. Demand or orders for it, could change that.

3. Evolution of Mathematical Concepts, An Elementary Study, R L. Wilder, John Wiley & Sons 1968. Wilder is a past President of the American Mathematics Society. From the Jacket: This book attempts to explain how mathematics came into being from the types of numerals found in primitive cultures, and to determine the cultural forces that have governed its development.The realization that mathematical content evolves implies mathematics education content may evolve. The latter quote is liberating.

Secondary Mathematic Education - technical base etc.

There is a difference between discussion of delivery style and content matters. Delivery styles come and go quickly. Content matters change, but does so more slowly. The 1960s, 1950s and even the 1940s set the stage for the technical discussion and design of course content. That content lingers on today in pre-university calculus oriented courses. In some of the texts below we see discussion of the topics prior to the settling of conventions regarding the extent, if any, of their inclusion in the curriculum.

1. What is Mathematics, R. Courant & H. Robbins, Oxford University Press, Fourth Edition.Classic Work. This may be taken a prequel to the discussion in the 1950s of what should be taught in pre-university mathematics. Very readable for undergraduate students in mathematics.The geometric interpretation (or representation) of complex numbers assumes the addition theorems (angle sum formulas) for sine and cosines in order to show how to multiply complex numbers using moduli and angles. Compare and contrast that with the site development of complex numbers.

2. Secondary School Mathematics, J. J. Kinsella, published by The Center for Applied Research in Education, Inc., New York, 1965. It describes mathematics instruction from the early 1900s to the 1960s in North America. Many of its comments are still valid.

3. Program for College Preparatory Mathematics, Report of the Commission on Mathematics, College Entrance Examination Board, New York 1959, 63 + x pages. This basic outline of mathematics in grades 9 to 12 still echoes in US and Canadian courses. This booklet focuses on education for "university-capable" students with a few remarks on mathematics in the general education for those (most) not going. Appendices provide more details. The Preparation for college here means preparation for calculus and analytic geometry. There is a strong, college preparation orientation not just for engineering and the physical sciences, but also for mathematics itself. In that, the mathematical orientation (the striving for a logical rigour) may be too much. The rigour present in the diagram-free, algebraic-deductive axiomization of modern mathematics is lost in the classroom with the use of diagrams in the development and application of trigonometry and beyond calculus for the exposition of ideas. In order to avoid details that are too technical (overwhelming) for students and teachers, the classroom approach to mathematics has to introduce mathematical practices and tools likely to be of service in other disciplines in a manner that prepares for but does not provide the rigour of advanced studies. I object to the criticism of earlier curricula on the basis of standards for rigour that in retrospect, the new curricula in formation cannot meet. That is the pot calling the kettle black.

4. Program for College Preparatory Mathematics, Report of the Commission on Mathematics, APPENDICES, College Entrance Examination Board, New York 1959, 223 pages.In these appendices, there is a strong, college preparation orientation not just for engineering and the physical sciences, but also for mathematics itself.

5. New Thinking in School Mathematics, Organization for European Economic Cooperation, Office for Scientific and Technical Personnel, May 1961. The text discusses what should be in or out in mathematics skill development. The selection of topics appears to be college oriented. That being said, given the experience of the last five decades, I suggest common or likely needs of student in daily life, immediate or long-term, should be the first focus of quantitative skill development in say K-8 or 9, so mathematics instruction is concrete for teachers, parents and these students. That being said, we should weave advance level ideas into this early instruction only where that inclusion makes skill and concept development clearer, since the inclusion may be seen as needless overhead by teachers - those not familiar with the long-term value of that inclusion. Given the choice between two routes for skill development, both being of equal service for common or likely needs, the route which serves advanced mathematics most should be chosen.

6. Synopses for Modern Secondary School Mathematics, Organization for European Economic Cooperation, Office for Scientific and Technical Personnel, 1961. This cover secondary school education, 1961, European style for cycle I - ages 11 to 15, and cycle II, ages 15 to 18. Arithmetic, algebra, geometry, analysis are all covered from an advanced level, with preparation for university studies very much in evidence.

7. L' Enseignement des mathematiques: J. Paiget, Beth, J. Dieudonne, A. Lichnerowicz, G. Choquet, C. Gattegno, published by Delachaux & Niestle, Nechatel (Switzerland). Of interest here is the fact that this is a joint work of the pyschologist Paiget and first rate mathematicians with positions in France and the USA. This work connects Paiget with the very abstract Bourbaki school of mathematics in a way that implies an alliance, not opposition. That should be food for thought for present day interpreters of Paiget's work, constructivists included.

8. 1985 Curriculum Guidelines, Mathematics Intermediate and Senior Divisions, Grades 7 & 8, Grades 9 & 10 Advanced Level, Grades 11 & 12 Advanced Level, Ontario Academic Courses, Minister of Education, Ontario. The description in detail of skills and concepts is worth noting. It indicates a progression. This curriculum guides names or describes in detail skill and concepts, but does not specify the teaching technique for each. The curriculum clearly represent preparation for university or college level studies in mathematics science or business. That being said, I think students in grade 7 and 8 would benefit from a focus on the quantitative skills and concepts likely to be needed in daily life, sooner or later. That focus most likely occurs outside the advance level versions of grades 9 to 12. But the underlying subject matter would be of great benefit to pre-university students, and would give common ground between them and others not heading for university, including students who do leave high school before graduation.

9. Secondary Mathematics, A Functional Approach for Teachers, H. F. Fehr, D. C Heath and Company Boston 1951.The book is interesting for its exploration of possibilities, it rigour, and it frequent mention of physical applications. I wonder if modern calls for cross-curricula development of mathematics and other disciplines recognize as such the possible interplay between high school mathematics and physics and/or the mathematics of finance, growth and decay. This work is written by a Professor of Mathematic Education who has great expertise in mathematics. The book explores possible routes for for the development of geometry, linear and quadratic functions; numbers, constant, variable, function, equation and graphs; elementary curve tracing; loci and the conic sections (a must read for me), etc. etc. The chapter (pages 254-296) on complex number systems and trigonometry gives as a exercise for students (!) the task of giving a geometric proof of the distributive law for complex numbers when multiplication is defined by multiplying moduli and adding angles. The site development of complex numbers gives a proof. It gives the most recent and simplest site proof of the distributive law, the simple proof I have looking for since seeing Feynman in 1976 describe physics in terms of adding and multiplying vectors in the plane. This simple proof of the distributive law for complex numbers, independent of trigonometry. Whence complex number methods may be employed to develop circular, periodic function, trigonometry. That implies a simplification of the high school development of trigonometry which I have seen in high schools and colleges since the mid-1960s. Thus the development gives methods, fresh and re-invented, the exact division is not clear to me, for making complex numbers, trig and vectors easier to learn and teach.

College Level Mathematics Education

1. Calculus, Lipman Bers, Holt, Rinehart and Winston 1969, SBN 03-065240-5A. Here the author, a leading mathematician of his time, favored the decimal viewpoint of real numbers, at least for students not in mathematics. To myself, the use of decimals in practice without mention nor sanction of them since the modern mathematics curricula in North America since the mid-1960s represents an inconsistency in course design and delivery. The remedy for that is to emphasize the decimal representation of numbers - integral, rational and irrational upto and including calculus and any epsilonics taught to advanced students there-in, with the decimal -free perspective briefly mentioned, if at all. Post-calculus courses in pure mathematics or its history may develop and stand-on the decimal-free view. See the Appendices and Chapter 14 of site Volume 3, Why Slopes and More Math.

2. How to Teach Mathematics, second edition, S. Kranz, American Mathematics Society, 1991. ISBN 0-8218-138-6 Here are recommendations for college level instruction. In an end of year replacement high school post, I tried to follow the recommendation of announcing one's marking scheme only to find at that the school required a new one, made-up at the last minute, by a school committee.

3. Committee on the Undergraduate Program in Mathematics: A Compendium of CUPM Recommendations, Volume I, Mathematical Association of America, circa 1972. This volume offers recommendations for Training of Teachers, Two Year Colleges and Basic Mathematics, Pre-Graduate Training. Here I suggest College and university level mathematics teacher programs meet or exceed the recommended training for the training of secondary mathematics instructors is met or exceeded in their hiring and promotion of professors of mathematics education for the formation of primary teachers and secondary level mathematics teachers. There-in lies a standard which providers of public funds to teacher training and certification programs may check and enforce. The design and delivery of mathematics teacher training and certification programs lacks academic integrity where mathematics education professors do have a command of senior high school mathematics and calculus. Without that standard or know-how checked and enforced, teaching training and teacher evaluation in the area of mathematics becomes a lottery with uncertain and unreliable results.

4. Committee on the Undergraduate Program in Mathematics: A Compendium of CUPM Recommendations, Volume II, Mathematical Association of America, circa 1972. This volume offers recommendations for college level programs in statistics, computing and applied mathematics.

5. Mathematics as a Service Subject, ICMI Study Series, Udine 1987, Cambridge University Press 1988, ISBN 0-521-35395-5 (Hardcover) and -9 Paperback. The title of the conference is what catches the eye. The authors were talking about college level instruction. However, the title represents a possible theme for earlier instruction as well.

Mathematics - Foundations, History, Logic, Philosophy

1. Mathematical Thought from Ancient to Modern Times, by Morris Kline, as three volumes (1990, published by Oxford University Press). It was first published as one book in 1972 by the same press. This work gives an overview of the discipline, the strands of reason and geometric thought that entered into it in rigorous and not so rigorous fashion. This work describes the changing nature of mathematics. Mathematics apart from geometry was not a deductive exercise. In particular, the symbolic reasoning of algebra, also called analysis from 1700 to 1900 was a tool with useful results: faith in it would follow usage. There was no rigorous and no precise thought-based foundation. The material underlying algebraic or symbolic analysis treatment of calculation, that is the concept of number (whole, fractional, negative, imaginary, complex) was only clarified gradually. This work describes mathematical knowledge before its deductive codification, that is, its derivation in an axiomatic framework for sets and arithmetic. This reference is more technical than the rest, and may need to be sampled rather than read from end to end in the first instance. Its eventual comprehension could be the target of a college student specializing in mathematics.

2. History and Philosophy of Modern Mathematics, Editors W. Aspray & P. Kitcher, Minnesota Studies in the Philosophy of Science, Volume XI, University of Minnesota Press, Minneapolis USA, paperback 1998, ISBN 0-8166-1567-5.

3. A Short Account of the History of Mathematics, W. W. Rouse Ball, 4th edition 1908, Dover Publication Inc, paperback 1960. ISBN 0-486-20630-0

4. A History of Mathematics, 1968 C. B. Boyer, Princeton Paperbacks, Princeton University Press 1985, ISBN 0-691-02391-3

5. Makers of Mathematics, S. Hollingdale, 1989 & 1991, Penguin Books ISBN 0-14-01922-8

6. The Nature and Growth of Modern Mathematics, 1970 E. E. Kramer, Princeton Paperbacks, Princeton University Press 1982. ISBN 0-691-02372-7

7. Number Theory and Its History, Oystein Ore 1948, Dover Publications 1988, ISBN 0486-65620-9

8. A Source Book in Mathematics, D. E. Smith, 1929, Dover Publications 1959. IBSN 0-486-64690-4

9. A History of Algebra from al-Khwarizmi to Emmy Noether, B. L. van der Waerden. Springer Verlag, ISBN 3-540-13610-X, 260+ pages. Page 178 says the following regarding complex numbers: Euler ... did not give a satisfactory definition. Clear, geometrical definitions ... were given by Caspar Wessel in 1997, by Jean Robert Argand in 1806, by John Warren in 1828, and by Carl Fredrick Gauss in 1831. ...William Rowen Hamilton defined (1843) the complex numbers as pairs of real numbers subject to ... rules of addition and multiplication. Augustin Cauchy interpreted (1847) the complex numbers as residue classes of polynomials,..., modolo x² +1

10. Foundations and Fundamental Concepts of Mathematics 1958, H. Eves, Dover Publications 1997, ISBN -0486-69609-X

11. Logic for Mathematicians, A. G. Hamilton, Cambridge University Press 1978, Revised edition 1988, ISBN 0-521-36865

- THE END -

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science