An Applied Mathematics Program for Quantitative Skill Development, 14-09-2009

Summary:  This  14-09-2009 plan for an Applied Mathematics Program for Quantitative Skill Development K1-12 reflects ends, values and methods for work and study that the common person in the street may appreciate. The plan offers an empirically rigorous path for instruction from the development in primary school of counting skills to the senior high school or college level calculus.  In that development, the plan offer a context and motivation for the for axiomatic based of the modern mathematics curriculum the applied mathematics program is intended to replace.  The path  develops skills and concepts in small steps, progressively. And in contrast to the more recent and dominant constructivist view of knowledge, if not  skills, which locates the latter in the mind, in an unobservable and unverifiable manner,  the quantitative skill and concept development program focuses on the mastery of observable and hence verifiable skills and concepts.  

Addendum: 9-01-2010:  In my opinion:   (1)  Education in an art or discipline has to be result in observable and verifiable skills to be credible. In particular,  concrete ends, values & methods for work & study  may support that.  (2)  With some local variation,  primary school  and the first years of secondary school should be re-oriented (the first level below)  to focus on giving students an observable and empirical mastery of the routine mathematics & quantitative skills  the common person in the street needs or may need in daily life, immediately and in the long-term.  In that development, a mechanical mastery of skills and concepts may be sufficient. In that development, full explanations of why methods work should be available as reference,  but given only where that aids  and does not overwhelm students comprehension.  Young students may strongly believe that mathematics instructors are hired by local authorities to give them correct methods, methods that do not need justification.  After that, secondary school instruction (the second level below)  may cover separately, in sequence or with some overlap,   the  logical and mathematical skills  needed (a)   in calculus-free  trades with depth of explanation dependent on the students or trade;  needed  (b)  in pre-calculus  mathematics science,  technology and business courses;  and (c) needed in calculus.  Items (a) and (b) point to the cross-curricula presence and service of mathematics. Students may be given preparation for calculus as the foremost reason and context for the  development of observable  skills in  logic,  in exact arithmetic with whole numbers and fractions, of algebra,  of deductive geometry without and with coordinates,  of trigonometry and even complex numbers.   Again, education in an art or discipline has to be result in observable and verifiable skills to be credible.  The extent of explanation in that may vary in accordance with the skills and preferences of students and teachers:    Problem solving and proof writing skills both involve the careful use of rules and patterns, one at a time and one after another to arrive at results or conclusions, intermediated to final,  all with  steps done and recorded in an observable manner, for the sake of immediate or later verifications or correction by the doer and by peers:  teachers, parents or fellow students etc. That overlap and commonality with between  solving problems and proving assertions implies an emphasis on one is preparation for the other.  So in the development of skills and concepts,  explanations of why methods work should be available as a reference if not given and not required in class. 

A two level applied mathematics program for quantitative skill development from  primary school to senior high school or junior college level  is essentially defined below.    See what works and what can be improved.

Reflections 1981-2007 on how to fill inductive (progressive skill development) gaps in the content of sputnik motivated modern mathematics curricula led  to this  program.  The program here values  practical knowledge -  student mastery of  observable skills and concepts in an art or discipline.    The subjective constructivist view that knowledge is a private affair, located in the mind in an unobservable manner  is a partial truth -  A material form of constructivism may be recognized in objective or striving for objectivity skill development in many arts and disciplines. The ability to tell, invent and follow stories is transformed in education and research into the ability  to tell, invent and follow explanations and instructions (consistency helps)  as a  means to enrich, advance and provide a context for observable skill development, and a means to share what is the mind.   Skill and concept mastery has to be seen, tested or shown to be believed.  

To engage and motivate and provide a context for learning,  this two level program includes these ends, values and methods for work and study; by the first level ending with a focus on five application areas; and by the identification of second level  ends (a) to (d).  A base for non-routine or open problem solving is provided by those ends, values and methods alongside application areas whose mastery involves the learning of solution methods for routine problems - if such methods are not taught, every problem becomes non-routine.  Finally, site development of algebraic skills and concepts is progressive from the direct use of formulas to the forwards and backwards use of the rules of algebra and calculus.  

 First level  instruction may develop counting, measuring, arithmetic and geometric skills alone and through  five overlapping application areas: (i) time and date matters, (ii) money matters, (iii) measurement matters,  (iv) map and plan drawing and use, and (v)  basic chance and probability matters.   Preparation for and then a focus on the five application areas and the underlying arithmetic and geometric skills provides  a context and motivation (ends and values)  for studies, appreciated by parents, students (ages 5- 13) and teachers.

Formula evaluation and their empirical confirmation may appear as part of arithmetic and as a first & only hints of algebra  

Skill, confidence and context (motivation!) follows here by showing students how to use numerical and geometric methods to solve  routine or nearly routine problems, all  in ways that lead to students doing and showing work in an observable and  correctable form.  

Before second level instruction begins in earnest, first level instruction may continue  to  give students the greatest possible practical command of the application areas they are likely to meet in life, day after day, or year after year.  Lessons may cover and reveal application area skills needed in the kitchen, in travel, in scheduling, in division;  in working for a living; in visiting or  running a restaurant, a grocery store,  a clothing store, a zoo, a museum,  a cinema; in playing games and avoiding bad risks not only in games but also in running a small business; and so on. 

Numbers and geometry methods appear everywhere.  Mathematics lessons and course materials may mention and employ such appearances in the development and motivation of skills and concepts, all in an observable, repeatable, reproducible and verifiable fashion.  

Why methods work may be explained where that helps performance - for example in decimal methods for addition, if not subtraction.  But  methods may also be learnt by rote and confirmed empirically - see how they give repeatable and reproducible results. For example decimal methods for multiplication and long division should be learnt and taught by rote as the explanation would most likely overwhelm learners and many teachers.  Full explanations of why methods work are not needed for performance and confidence building.  The mastery of methods that yield useful results in application areas, those above or more, is sufficient in the first instance.  Problem solving where students combine patterns to arrive at result or further patterns sets the stage for understanding why methods work.  The ability to combing methods to arrive at results or conclusions may be learnt in problem solving before or besides any emphasize on  mastering explanations where combining patterns appears to suggest or imply further ones. 

Map and Plan Usage: A practical if not explicit mastery of similarity will be implicit in map and plan usage.  Drawing to scale permits length and area estimation as the number of map unit lengths in a length and the number of map unit squares in an area respectively equal the number real-life unit distances and real-life unit squares in the depicted or planned length or area. 

 In second level instruction, the foregoing dimensionless equality of the number of unit distances and squares in corresponding  lengths and areas  implies K and K² proportionality relations where K  = a map or plan scale factor. 

Logic Note: The study of logic (math-free) first could be part of first or second level skill development.  See the leading chapters of Volume 2.  The pre-algebraic description of sets, their unions, intersections and complements with rosters, Venn Diagrams and words might be an aid in level 1 or 2  to the development of logic and counting skills, and is required  in level 2 with the mastery of elementary probability and the graph view of functions.  Critical path analysis is needed to determine  what skills and concepts, the ends, values and methods of instruction require and when.  That may require skills and concepts with marginal real life application to be covered - preferable at the second level and not the first. 

An explicit focus on  five application areas above may provide students, parents and instructors an engaging context.  Within that framework, ends, values and methods for work and study may be offered.  Progressive skill development in these application areas is almost evident to instructors and parents. 

The five application areas are concrete. In developing them, course designers may talk among themselves about themes emphasized in NCTM and the UK English mathematics curricula, but those themes do not need to mentioned to teachers. It suffices instead to focus on the optimal or maximal development of this application areas to make skill development easier and simpler for instructors and parents to understand and support., and students too.  

Beyond those five application areas, the context for second level instruction is more remote from daily life and needs. None the less, a context and motivation may be offered in a hype-free manner that will appeal to some not.  At the second level,  the deliberate and fuller (gap-free)  progressive inductive skill development will make learning and teaching easier and more effective. Less alienation and more context for instruction should follow.  

The first level coverage of  money matters, may be extended and embedded  in the second level instruction development of algebra skills. 

First level instruction with it clearly described   ends, values and methods provides a context and motivation easily understood and engaging for most parents, students and parents. Progressive skill development for it almost self-evident. 

The second level reflects and formalizes site How-TOs and the progressive skill development indicated there-in and in the rest of this page and website.  

Second level instruction  (See Site How-TOs)  may  introduce students to the specialized numerical and geometric skills  (a) needed for common trades and professions, (b) needed for calculations in the physical sciences (arithmetic with units included), (c) needed for a greater comprehension of first level application areas and (d) needed to prepare students for an operational, if not theoretical, command of calculus in the senior high school or college level mathematics.  Mentioning these ends  may provide a context and motivation for studies,  easily understood and appreciated in general, but not necessarily in detail by  parents, students and teachers.    The extent to which preparation for ends  (a) and  (d) remains to be determined.   But ends (b), (c) and (d) are served below. 

Calculus demands a full-strength operational mastery of arithmetic, algebra, geometry, logic, trig and functions.  Probability theory may be included for its own sake and for the re-enforcement of algebra and exact arithmetic skills with fractions needed in calculus or preparation for it.  

When and where students find second level instruction not to their liking, either to begin or to continue, skill and confidence in the ends, values and methods of first level mathematics or quantitative skill development should leave a positive impression of mathematics in general. That being said, we hope, the attention to detail promoted in first level instruction should provide students with the skill and momentum to cover the basic arithmetic, algebra, logic parts and even more of second level instruction. 

Many  fail senior high school  and junior college calculus. Honesty in course design and delivery  should inform students of that to imply calculus and preparation for it is or will be very demanding.  

 While constructivist and pre-constructivist calls to engage students and to provide them with a rich, authentic, reality based, experiences for reflection and skill development are noted, and influence the form of first level instruction above. But in second level instruction there are topics which are introduced for technical ends, apart from immediate real-life examples and no immediate context.  That is unavoidable. The appearance of quadratics in economic supply and demand curves appears to be artificial.    Yet there is a big BUT overhanging preparation for calculus. The study of quadratics may be motivated by the study of falling and projectile motion in physics.  The study of quadratics and conics may be engaging for students of physics, but not for others who do not meet that part of physics.  Beyond that,  the further study of  polynomials and their addition, subtraction, multiplication, division and factorization have no immediate, real-life applications (applications called for by constructivist theorists). Instead, the topic is present only for the sake of first calculus and  beyond in mathematics, pure or applied.  

Progressive Skill Development In Algebra: While preparation for calculus with a nominal  pure math orientation has been part of high school mathematics, that the development of algebra skills in it has not been progressive.  The algebraic way of writing and reasoning has simple appeared and been required without progressive skill development.  That has made calculus and preparation for it harder than need-be.  Site coverage of solving linear equations, of three skills for algebra, of the forward and backward use of formulas numerically and algebraically, of  the appearance and manipulation of units in proportionality constants, of  the carrying of units through calculations, and algebraically,  and algebraic description of operations on fractions,  and of what is a variable, and of calculus appetizers or starter lessons,  altogether give a more oral, more progressive or inductively complete development and comprehension of the algebraic shorthand role of letters and symbols.  

Raising The Oral Aspect : Site remarks on talking about numbers and quantities, on naming key equations and identifying the latter with descriptive phrase (for example the radius-dependent circle area calculation formula) all point to greater & clearer  role for words in understanding and communicating algebraic reasoning skills. While the power of algebra is great,  the perimeter calculation rule "add the length of the sides" for polygons is easier to grasp for students with pre-algebraic mastery of maths.  And in an applied math departure from the values of pure math, the oral, slogan-like, description of associative laws for multiplication and addition is easier than algebraic versions for students, alone and in the expression of numbers and also polynomials in terms of products and sums.  Prior to algebra, slogan like rules for common calculations in arithmetic and in the plane may make earlier and later instruction easier - more oral as well.  

• the sign and length of a  sum of signed numbers is given by the common sign and the sum of the lengths of the signed numbers.   Here we take a signed number to be given by  signed prefixed to an unsigned number or length.  Level 1 instruction introduces counting and measurement with unsigned or sign-free numbers. Saying how to compute the sum, defines it. 
• the sign and length of a sum of signed numbers with unlike signs are given by the sign of the longest and difference of lengths, the longest minus the shortest.  Saying how to compute the sum, defines it. 
• to calculate a product of signed numbers, prefix the product of the signs to the product of their lengths. The foregoing assumes the law of signs as part of the definition of products - saying how to compute the product, defines it.
• the angle and length of the  product of two arrows or points  in their plane are respectively given by the sum of their angles and the product of their lengths. 
• to find the perimeter of a polygon, sum the lengths of its sides. 
• The order of addition of signed numbers is both associative and  commutative: sums are independent of the grouping and subgrouping of terms in the sum.   (This property applies to addition and not subtraction. To apply this property to expressions involving subtraction. express each subtraction as an addition - the addition of an additive inverse.)
• The order of product of signed numbers is both  associative: products are independent of the grouping and subgrouping of the factors in the sum.   (This property applies to multiplication and not division.  To apply this property to expressions involving division,. express each division as an multiplication - the multiplication of a reciprocal or multiplicative inverse.)

The last two slogans in verbal  are necessary to "justify" addition and multiplication operations with polynomials.  With numbers, the foregoing slogans may be verified numerically before general use.  The identification of formulas by name (for example quadratic formula, compound interest or growth formula) or by descriptive phrase (for example,  the old product of adjacent sides, rectangle area formula, or for example the radius based circle area formula  or for example the radius- and diameter-based circle perimeter formula) all contribute to the oral element. A descriptive phrase as is or shortened may become the name.  Names, descriptive phrases and slogan may also be the base for telephone conversations.  

The first geometric development of algebra occurs in the presentation of formulas for lengths, areas and volumes where letters instead of denoting pure numbers, denote measures of geometric quantities easier for students to accept and grasp than phrases like "Let a,b, c and x denote numbers" -  real or otherwise. 

A 2nd Geometric Development of Algebra:   For unsigned numbers, the distributive law may be introduced geometrically as the notion that the area of a rectangle partitioned into rows and columns of subrectangles may be calculated directly or a sum  of the areas of the subrectangles.  The commutative law for sums and products of  pairs of unsigned numbers may be introduced geometrically as well. The aim is to develop an operational command of these properties and their algebraic description geometrically before empirically or deductively generalizing them. Technical Aside:  The foregoing  area calculations approach geometrically  extends & camouflages for easier digestion, product counting methods for rectangular arrays of discrete objects  as is or partitioned into sub-arrays, rectangular too. )

 The second level develops algebraic & deductive reasoning with deductive reasoning where possible, where its presence will not overwhelm students with technicalities, material left for inclusion in the third level, material that may appear in references or enriched reading for the second.   

After an algorithmic introduction of  arithmetic with signed numbers, students may be shown how to add and multiply points in the plane in a manner that extends arithmetic with signed numbers and visually introduces the complex numbers and square root of negative numbers,  and allows the tangent (sine and cosines too) of all angles to be defined.  With that, the slope of non-vertical lines may be identified with the tangent of angle of intersection with the horizontal axis before any further study of trigonometry.   

Function Notation: As part of the development of algebraic skills, an operational plug in command of function notation could be tied to the evaluation of formulas and polynomials. The values of functions may be a number or a point in the plane.  Function notation should also be use in the development of probability theory, with the arguments being sets or their elements. The chains of  substitutions, first introduced in solving linear equations and then perhaps in manipulating polynomials, may also be described in function notation to introduce and illustrate function composition/substitution.   Exercises showing that Associativity of chains of substitution (the independence of results on the order of substitutions)  will illustrate and imply by example the Associativity of  composition, that is, function substitution operations. Function and operations on them may also be presented and discussed in terms of programming a computer or calculator, structural or OOPs, etc.  

Five Operations on Polynomials: The site explanation of arithmetic with polynomials is deductively incomplete, but empirically sound. Geometric ideas (geometric versions of counting principles) are employed to introduce the multiplication and addition of polynomials p(x) in a manner justified only for non-negative coefficients and a non-negative argument (variable) x.  The resulting algorithms are then applied to arbitrary polynomials p(x) in a manner that leads to repeatable, reproducible and hence verifiable, confidence building results.  In the foregoing, very general laws for the associativity of multiplication and addition may be described orally, the algebraic versions being too complicated for students to follow before high level studies, if any, in pure maths.  

See the technical notes for second level instruction to learn more.

Preparation for calculus and calculus is may be served by some, not all of the material in  site volumes 2 and 3,  by site folders on 

3.- Fractions-Rates-Proportions-Units - 
5.  Solving Linear Equations  04-2005
6.-Euclidean-Geometry/Complex No.s 
7.  Analytic Geometry/Functions 2006
9.  Complex Numbers More 2001.   
10  Exponents, Radicals & logs. 2008
11. Calculus  2005.

For third level  college instruction beyond calculus if not in calculus,  the Real  Analysis 1995 appendices of Volume 3  provides a bridge between the common decimal view of arithmetic and the decimal-free view of arithmetic and continuity in modern mathematics.  

End Notes for Instructors

Chorus: The ends and values one chooses for work and education determines  preferences and choices in course design and delivery.

Parents are the first teachers of students.  When course design is incomprehensible to parents, parental support for education is diminished. Educational theorists who say it takes a village to educate a child should slow down and introduce methods for education easily understood and appreciated by parents. Anything different, anything incomprehensible to parents, may alienate students and their parents from instruction. 

• The mid-1950s onward, modern mathematics curricula in Europe and North America from first steps in counting to calculus were too narrow (or too pure)  to support and sanction the common use of decimals, drawings in society, in science and technology, and in secondary school introduction of trigonometry, functions & calculus.  The implementation did not fill and amplified the jump in the deep-end gap n the introduction of algebra. The modern maths curricula overwhelmed people with its technical viewpoint, and was too dry and technical.
  
• Constructivist educational theory 1990 onward  in holding knowledge and comprehension is a private  matter, located in the mind and not observable nor testable in any reliable, manner. That  shifts instruction away from the deliberate development of observable skills and towards an immaterial, nebulous, subjective view of knowledge inconsistent with the striving for objectivity present in modern day rules and regulations of society at large and inconsistent with the striving for objectivity present in science, engineering and mathematics.  Aside from this essential issue, the calls in constructivist for authentic, genuine problems and rich learning experiences are fine, but for the absence of examples or methods to support the calls. Constructivism may say what is right or optimal for education, but so far since 1990 in mathematics it has done so without giving pathways easily understood and followed by instructors. Ends and values for instruction without methods to support them may be vacuous and also overwhelming.  .

The exploration and expression of ideas for the site applied math program began in the last days of 1990 with the aim of supporting modern maths course design, and not replacing it.  Writing began without a knowledge of  constructivism.  While constructivist calls for engaging students and providing real learning experiences are fine,  the constructivist tenet that knowledge is located in the mind of students in an unobservable and unverifiable manner shifts the ends and values of education from the material to the immaterial. In contrast, the inductive principles for progressive skill development in Volume 1B, Mathematics Curriculum Notes, and the applied mathematics program sketched above represents a concrete, material view of skill development with the first level mathematics serving the needs of the common person in the streets of paved or unpaved  community, and with second level mathematics serving the needs or hopes of associated with secondary and further studies.  First level skill development may be locally adapted, continued and extended to include most if not of the immediate quantitative skill requirements in life, daily or over the years, in  communities, small to large.  

Chorus: The ends and values one chooses for work and education determines  preferences and choices in course design and delivery.

Caution: The exploration and expression of ideas in site pages began before ideas for the applied math program  solidified.    Presently,  the support in site pages for the above program is present in an incomplete and sub-optimal form, and a few digressions are present that may serve as food for thought without being part of the program.  Writing has been an iterative affair in which the exploration of different paths for instruction provided more than enough options for program definition, with some options, yet to be determined, to be cut.  So do not take everything literally.  The final cut is not done, and local variations in the implementation may be advisable.

The applied math program stands on the work of my instructors, live or in the textbooks they wrote. The applied math program weaves their work into and between site innovations.  If you are not sure of your teaching skills, one way to compensate is to look for and compose appetizers and lessons easily understood and repeated, likely to be effective in the classroom in easing fears and difficulties, and in enriching knowledge.  This site stems from the sense that inductive methods for progressive skill development were missing in mathematics education, particularly from the introduction of algebra and to its multiple full strength use in calculus. The site also stems from the example of guest speakers in maths and physics at McGill University, a university not all bad,  who showed how to make the hard easier or more accessible in high school, undergraduate and research level material. The aim in exploring and expressing different avenues or possibility for instruction offline and then 1991 to 2007 was to support and restore the modern maths curricula of the mid-1950s to the end of the 1980s to favour, by offering for it, progressive pathways for skill and concept development.  But ends and values for instruction were loosely present, but not clear.  Much of the modern maths high school programs represented preparation for calculus, technically overwhelming for students and teachers,. and not for all.  The contructivist movement with it fine calls for instruction, calls that imposed an overwhelming number of duties on instructors, did not pay attention to details, at least in the NCTM standards and principles. The latter appears to inherit course content without recognizing the calculus orientation of earlier course design, while putting aside the observable and verifiable skill development valued in earlier times.  

In fall 2007, I started to think about providing an alternative to the modern maths curricula.  See the site lamp folder. It offers a Lean Applied Math Program for secondary math instruction  based on progressive skill and concept development, but it does not offer ends and values needed to engage people in that skill development. The ends and values were not stated. Primary math instruction was not considered. So lamp was incomplete. While the constructivist view of knowledge of students as a private (subjective) affair  located in the mind  of students in a manner, beyond observation and beyond any form of objectivity, was in contradiction with the striving for objectivity view of practices and patterns in mathematics, science, technology, business and justice,  the constructivist calls for inclusive education and for engaging students remained appealing.  But a local school reform document call competence in communication, reason and problem solving in ways I could fathom materially, but which I could rewrite and express in concrete terms.  See the Ends, Values and Methods box above.  The constructivist call to engage students and build their skills and confidence is reflected in the identification of first level instruction, application areas. The importance of those areas is self-evident to most adults. And in second level instruction, the ends and values are long-term, not immediate, and some of the technical skills and topics which calculus requires are as yet difficult or impossible to connect with genuine, real-life, authentic problems or situations at all, or in a time-economic manner and  fashion accessible to students and teachers. Thus some parts of mathematics education will have only technical motivations. The summer of 2009 was spent thinking about how and what ends and values would be sufficient to motivate progressive, observable skill and concept development in an applied mathematics program.   The net result is the two level, applied mathematics program for quantitative skill development, outlined on this page. 

A Material form of  Constructivism: 

The ability to tell and follow stories may be transformed in education into the ability  to tell and follow explanations and instructions, a means to enrich, advance and provide a context for skill mastery, and a means to share what is the mind.  but not a viable tenet for empirical skill and knowledge mastery and sharing. 

 Individuals may follow whatever logic they please in forming and justifying their own ideas, tentatively or firmly. . But individuals through communication by example, with words and diagrams, etc, may share ideas or least point each other to ideas, and together agree about a logic and methods for arriving at conclusions together  - that represents the start of communal or social knowledge.  In that sense, empirical, inductive and deductive methods for striving for objectivity via an agree upon logic or methods may be viewed as part of social constructivism. The  agreed upon logic and methods for arriving at conclusions leads to common or overlapping construction of pattern based knowledge in intellectual and applied disciplines.  So constructivism or personal methods for arriving at ideas, tentative or firm, about the world around us may range from the very subjective in which everyone is the master of his or her own thoughts - to the striving for objectivity via the assumption and use of common methods for arriving at conclusions or concrete in a field of interest. Even direct instruction in providing students with rules and patterns to combine in an observable manner, with initial data and results, intermediate to last, shown in order to comply with a given or agreed upon logic may regarded as a directed form of constructivism, applied and empirical.  Compare and contrast that form of constructivism with the immaterial form which says students knowledge is and should be a private affair, unobservable and uncoupled with the  mastery and  testable development of observable skills and practices.  Constructivism is so broad and so subjective in it definition, that the applied mathematics program outlined on the right could be cast as material form of  constructivism.   See Volume 1A to learn more about this individually chosen, limited form of constructivism, its benefits and limitations.  

Remark: The casting of arts and disciplines which strive for objectivity in story or theory development  via agreed upon logics and methods that permit peer or instructor review,  as a form of material constructivism may not be the liking of constructivist who insist on that knowledge is a private matter, not observable and subjective.  But the choice to  build a common knowledge of practices and theory in a communicable, observable, repeatable and reproducible manner,  represent a subjective choice, one that can be shared and taught.   I propose that material form of constructivism  be recognized and formally adopted by the National Council of Teachers of Mathematics in the USA and by the framers of the English National Curriculum, the mathematics portion. I kid you not.

Second Level Instruction - Technical Notes

A. Pure and Applied Logic In level 2, the  use of logic in its deductive form of  is limited to the direct use of one-way and two way implication rules A IF B and  A IF & ONLY IF B, and the use of contrapositive form of the one way implications A IF B.  The elimination of imagined possibilities via proofs by contradiction (the appearance of chains of reason that lead to inconsistencies with previously accepted knowledge) is left to language skill development in parallel courses or to third level maths.  In pure maths, logic appears as long deductive chains of reason. But in both pure and empirical or applied maths, showing work and applying patterns one at a time, and possibly one after another, provides a form of rigour, a trail or proof  to compose or follow in the pure or impure justification and derivation of results, intermediate to final. The axiomatic development and codification of a discipline represents a purified form, and may try to build or secure a discipline in terms of its "most" reliable rules and patterns - a notion subject to refinement by others.

B.  More on Functions.  After an operational mastery of algebra and function notation, algebraic set or subset builder notation 
{ x \in B |  condition  p(x) = true}
 may appear.  The set-based representation of functions y = f(x) in terms of graphs in the plane and the use of sets  to define vertical or horizontal line rules for the calculation of functions forwards and backwards (direct and inverse) will appear as just another way of  thinking about functions, and in a departure from modern maths, not as a the starting point.  There-in lies a surprise for students in that the set view of function gives calculation rules apart from algebraic formulas.  Those graphs for which the vertical line rule fails may be classified as a relations or employed to define set- valued functions or worse, pre-set theoretic multi-valued functions. graphs for which the vertical line rule fails may come from solution sets

 S ={(x,y) \in IR² | P(x,y) = c }   

for  algebraic equations  P(x,y) = c. The  circle of radius 3 provides an example:

 S ={(x,y) \in IR² | x² + y² =3²}  

 Site folder  Analytic Geometry & Functions illustrates some  of the above ideas.

C. Roots and Powers.  The site folder Exponents, Radicals & logs indicates how roots and powers may be expressed and calculated  with natural logarithmic and exponential functions. The algebraic description and assumption of properties of the exponential and logarithmic functions allows for a precise, precalculus, level treatment with calculators employed for approximate but not exact calculations.   Courses on calculus may then complete the story by showing how to define log and exponential functions via area under curves and inverse functions. 

D. Trig with Right Angle Triangles & with unit circles.  In first level maths, maps and plans drawn carefully to scale, the same in all directions,  may be employed to estimate lengths and areas for the following reason.  In each map and drawing, the number of map unit lengths and map unit areas is the same as that of the corresponding real-life or planned object.  Whence drawn and real-life measures are proportional.  In second level maths, with the advent of trig, many of the same estimation or calculations may be done with the aid of sketches and table- or calculator-given values of trig functions.  But the further development of trig on the unit circle for arbitrary angles and the measure of angles in terms of radians is yet another part of maths required by calculus, a part best done with diagrams in an applied math rather than a diagram-free, pure math manner. The  very simple and early introduction of complex numbers and the algebraic description of their arithmetic properties simplifies the development of trig identities in alone and in conjunction with trig formulas for dot and cross-products of points or vectors in the plane. 

E. Axioms for Real (& Complex) Numbers.  Instead of using axioms for real numbers as a starting point for second level  mathematics instruction, these axioms become a destination and optional starting point for the third level.  Here  the counting, measurement and geometric practices or assumptions tacit or explicit in the development of first and second level maths  explicitly  imply the algebraic field properties of real and complex numbers in an applied rather than pure math style.  Details of how appear for the most part in site folders on  Euclidean Geometry  & Complex No.s and in the site folder on  Number Theory,. A more explicit route left to a site update.   While the earlier innovations for skill and concept development may be immediately useful and acceptable, this last innovation  E,  call it optional part, reverses the logical development present in the modern maths curricula for secondary maths. In E,  the axioms for real numbers and for complex numbers too are implied and extended with explicit  and very practical assumptions about the decimal representation of real numbers.  The net result is an applied view of maths sufficient for most students, and of service in providing a prequel or context for third level studies in pure mathematics. 

Note:   From an empirical or applied math perspective, this appearance of the axioms for real and complex numbers as a destination and not as a starting point for level 2 instruction is both mathematically and pedagogically sound.  The axioms to be understood have to come after the development of student deductive and algebraic abilities. The derivation of the axioms from counting, measurement and geometric practices or patterns instead of their assumption makes their appearance less arbitrary. The appearance as a destination in the applied math program, an aside perhaps to the preparation for calculus there-in,   may  reflects or parallel the historical development where an empirical if not deductive knowledge of real and complex numbers came before the appearance of the ZF set theory axiomization and algebraic-deductive codification of modern mathematics.   I say the latter with some fear, given my dismal or incomplete knowledge of the history and the people in my discipline. With the completion of this applied program, I may find the time and interest to review the history.  Or, I may leave maths and mathematics education to others. 

F. Conics - critical path analysis.  The special study of conics (of service in theoretical physics, mechanics and multivariable calculus)  could be postponed and given between the exposition of differential calculus and the application of integral calculus to the calculation of volumes of revolution.  In that just in time  relocation, the algebraic skill level required and implied by success in differential calculus would make completing the square transformation in the special study of conics and their equations far easier for students. That relocation would make the derivation of equations for ellipses, parabolas and hyperbolas from definition in terms of locii easier as well.   While the latter are described as the intersection of a plane with a conic, I have yet to see in any pre-linear algebra math course, a simple geometric proof that the intersections have the special form attributed to them in pre-linear algebra maths courses. 

09-01-2010:  The coverage of Conics and Quadric Surfaces, pp 198-201, in the 1941 book What is Mathematics by Courant and Robbins, Oxford University Press, 14th printing 1969,  indicates how to derive the quadratic equations for ellipses and hyperbolic conic sections from geometric arguments involving the intersection of cones and planes. 

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