Logic mastery strengthens comprehension and so improves home, work & study abilities .
Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits

Ages 14+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles
Forewords + leading chapters give original reasons, still valid, for site content & growth.

• Parents &  student advisors/teachers: the folder  Helping Your Child or Teen Learn  covers  Speaking Skills, Reading & Writing,  Preparing for Science &  Having Patience.  It also includes advice, resources and  pathways for mathematics education -  Mathematics booklets for  pre-k and grades 1 to 3 & grades 4 to 8, plus grade 6 to 9+ skill development guides/standards for arithmetic, algebra and  geometry.  
• Students & Teachers: See the  [Mathematics and Logic Starter Lessons etc for Ages 14+]. Site reviews may  lead you to explore the advice, directions and lessons here for the mathematics education of yourself or others. If one site element is not to your liking, try another. Bon Appetite. 

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

Page Sections: [Page Top] [Mathematics Starter Lessons for Ages 14+] [Forewords to Site Books] [Two Paths for Learning and Teaching ][Words for Parents, Education and School Authorities, Worldwide]  [Why Bother - Context and Motivation for Mathematics Education] [Common Needs with or  versus Technical Needs] [Elements of New Mathematics Education Program ] [More on Learning by Rote or With Understanding] [Horrible (Pointless) Course Design/Instruction] [Unreliable Teachers Certification Practices] [Two Gaps] [End Notes]

Site Review, one of many

The NSDL Scout Report for Mathematics, Engineering, & Technology -- Volume 1, Number 8 (May 24, 2002) Site Description: Math resources for both students and teachers are given on this site, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos with clear descriptions of many important concepts provide a good foundation for high school and college level mathematics. There are sample problems that can help students prepare for exams, or teachers can make their own assignments based on the problems. Everything presented on the site is not only educational, but interesting as well. There is certainly plenty of material; however, it is somewhat poorly organized. This does not take away from the quality of the information, though.   

Note from the Author:  In this vast website, I have explored and expressed skill development pathways including the filling what I see as technical gaps in skill development pathways, and including the development of alternate pathways and sequences for  course design and delivery.  See for example the site lessons on complex numbers and the leading calculus chapters 1 to 6 and 14 of  Volume 3.  The parent section of this site presents the latest effort to set forth a program for K0-12 skill development for mathematics, logic and quantitative skill development.  In order to set forth a program for secondary school mathematics from arithmetic to calculus, I had to understand what was or might be done in primary school mathematics and how. For that, I was to able to identify 18 mathematics booklets for ages 3+ to 14, that is, from preschool to grade 7 or 8 level, which provide a step by step development of skill and concepts - sufficient to serve as a base for secondary mathematics skill development from arithmetic to calculus. The secondary, observable skill development program I have given or outlined  includes smaller step and alternative routes that may be immediately useful in present day college and secondary mathematics programs from arithmetic to calculus.  The program represent a modern mathematics education with a twist - the axioms for real numbers (and complex number too) are not stated but implied from practices implicit, very much present and needed in earlier primary and secondary programs.   While the probability theory and functions skills and concepts are best expressed in terms of sets and operations on sets,  in accordance with the thought that notation and concepts should only be introduced when they aid and do not distract from skill development,   I still trying to identify the most advantageous moment in the program for the introduction of  sets and operation on them. Primary and secondary mathematics course design is normally inductive, with later skills and concepts depending on earlier ones, but not vice versa.  What I have done is to provide more, smaller and alternative  steps to make learning and teaching simpler and richer.  That addresses most technical difficulties in the exposition.  But there is still a question of ends and values for instruction.  The following question and its discussion identifies and recommends tangible, concrete, material ends, values and methods for instruction, If further  implies a refinement of programs here and elsewhere for primary and secondary mathematics, logic and quantitative skill development.  There-in lies a to-do for myself or others.  Writing has been an iterative affair in which possible elements or pieces of a mathematics education program have been explored and expressed offline since 1990 and online since 1995. The next iteration, logistics aside, will or should be a refinement implied by the following question.  Q. E. D.

What observable skills do you  want to see in mathematics, logic and quantitative skill education of yourself or others?   The question and the objectives it implies for learning and teaching are easy to understand. 

The question further implies ends, value and standards for  skill development - tangible, concrete,  material and modern.  Those ends, values and standards may be employed to defend course design and delivery, and teaching training too, from the consequences of theories of "true" learning which say that knowledge is a private affair, located in the mind, apart from observable and mechanical skill mastery, as if the latter was a substandard end for schooling. In European, USA, and Canadian-style education, the dominance of the latter theory of  true learning explains why many teenagers lack observable skills in arithmetic, reading, writing and reason of the mechanical kind on arrival in college. 

Besides development of careful reading, writing, listening and talking skills,  education may explicitly emphasis skills serving common or likely needs first, skills with clear take home value as much as possible, before skills  serving more technical ends or needs. Primary school education in particular should strongly serve the 3R of Reading, wRiting and aRithmetic with the fourth R in Reason left for secondary school development in further reading, writing and mathematics courses. Observable skills with take home value sooner or later for  present day rural and urban life provides a practical goal to put first the mathematics education of students 3+ to 15 years of age. Support for that practical goal may overlap the  second or further goal of preparing students for the slightly more to much greater mathematical needs of  trades or college studies. All the foregoing provide a context for course design and delivery in which skill and concept includes smaller and clearer steps for skill development along side food for thought and the message that rule and pattern based processes have benefits and limits that need to be appreciated. Emphasizing observable and thus verifiable skill development provide tangible, concrete material goals for instruction, student-centered in a material manner if skills with the clearest or likeliest take eventual take-home value are put first, so that student obtain some material value from education.  

 Those who say true knowledge is a private affair, located and built  in the mind in an unobservable and unverifiable manner  may take their place in the immaterial and intangible spiritual education of students but they have should no  place in the design and delivery of courses and standards for skill development in subjects of a mechanical kind which employ rules and patterns in deed and thought on paper and beyond to arrive at observable and verifiable results for better and for worse.  Think about of pollution and processes that are not repeatable and reproducible in the long term -  processes in the which the activities benefit a few while actual or likely adverse affects are paid for unknowingly by third parties. 

Which way to go:    

In mathematic and logic,  site pages give a very direct,  detailed, answer  the question which way to go. 

• For you inner self, studies and instruction may provide food for thought, so that you may draw your own conclusions and construct your own views and opinions - no peer review necessary.
• For your public self, studies and instructions should provide observable skills, well-practiced, in accordance with ends, values and methods for work and study  

The latter values the skilful  mastery of rules, patterns & practices those whose results or steps can be seen and corrected one a time and one after another in a clear, material, observable and hence verifiable or correctable manner.  That is a must for training in arts and disciplines from cooking and carpentry to the empirical practices of science and technology (not all bad nor good).  More on Which Way to Go

Teachers & Course Designers: The key question  "what observable skills do you want" represents a viewpoint that instruction to have content and purpose out to aim for verifiable and skilful  mastery of rules, patterns and practices in arts and disciplines. Try to be skill development engineers.  Starting with reading, writing and arithmetic, practical and material consequences of instruction can been seen and measured. Instruction and course design may and should provide food for thought and reflection, but in material and observable arts and disciplines, instruction to be credible ought to develop observable and verifiable abilities.  In general, instruction and course design ought to serve the likely needs of life in the local community before  providing skills and food for thought which serves the technical needs of trades, professions and advanced studies.  In particular  greater motivation and context for mathematics as part of logic and  quantitative skill development follows from  serving t common or likely  needs   served first while technical needs (explicitly identified as such) should be be put second.  That would be student centered, skill development.  Site ideas for quantitative skill development are distributed over this page;  three site sections:  Help your Child/Teen Learn;  Mathematics Education essays, and the older  LAMP  program; and over site HOW-TOs (top level pages). The top level also  includes how-TOs for several skill development areas in mathematics.   Writing has been an iterative, zig-zagging,  process in which pages have been written in a blog-like manner to explore and express ideas while seeking a conclusion.  The conclusion essentially complete appears the parent section  Help your Child/Teen Learn.   Site steps and motivations for skill development give a lower bounds for observable skill development.   Further lower bounds is provided by this content oriented  New Zealand curriculum page, and  by these 1940-1960s works.  What further lower bounds are available for observable skill development, and how can they be raised in a practical and inclusive manner? The question poses some Pareto optimality issues - oops!

 Site pages in-between the thought-based development of skills and concepts may be read for  a do-this, do that approach to skill development. The latter by itself may build abilities and confidence, and set the stage for the former. . Besides ideas for easing and avoiding algebra difficulties from the first use of formulas to the multiple full strength use of algebra calculus,  site pages include methods fresh or recycled for redesigning secondary school mathematics from a thought based development of operations with decimals, fractions and signed numbers to an option, recommended, for  introducing complex numbers "rigourously"  before the study of periodic trigonometric functions.  The latter path has some easy consequences.  Site content stems from a sense that the algebraic way of writing and reasoning, that is the shorthand role of letters and symbols, was not clearly introduced. So algebra mastery has been harder than need-be or impossible for many.    The sense that the exposition or introduction of algebra was incomplete provided me the first example in mathematics education of how inductive principles for skill development, well-known before I met them, were followed incompletely in past course designs. Site material offers remedies.  

Then, there is a question of context and motivation. Simple put, most primary and early secondary mathematics quantitative skill development may serve common  needs in skill and concept development.  The common person in the street needs to know about arithmetic with decimals, fractions and percentages; about time and date matters, about buying and selling good and services - part of money matters; about units for counting and measuring, and their appearance in arithmetic,  about the use of maps and plans (or diagrams drawn to scale) for location, route planning and measurement. While advance mathematics employs deductive logic, the person in the street need not know about, but may need to know for decision making how to recognize the difference between a one-way and two-way implications.  The technical service of instruction to pre-college trades and college programs in science, commerce, mathematics, engineering and technology (multiple ends with multiple but different skill development needs), may begin in earnest after skill development for common needs has almost finished. With that, mathematics instruction unavoidably becomes more distant from the service of every day needs, exception to be emphasized.  For the example the study of quadratics is distant from everyday for all students not in sufficient advanced section of secondary physics. More generally, the study of polynomials does not have immediate applications even if they be artificial counterexamples. What is needed in course design and delivery is clear identification of the skills which serve common or likely needs,  and which skills is serve technical needs - and which technical needs.  Such clarity will offer a context for learning and teaching, even if that context honestly given is not motivation for all.  Current inquiries mathematics education seeking to provide rich and engaging problems for students may help with motivation. 

Page Sections: [Page Top] [Mathematics Starter Lessons for Ages 14+] [Forewords to Site Books] [Two Paths for Learning and Teaching ] [Why Bother - Context and Motivation for Mathematics Education] [Common Needs with or  versus Technical Needs] [Elements of New Mathematics Education Program ] [More on Learning by Rote or With Understanding] [Horrible (Pointless) Course Design/Instruction] [Unreliable Teachers Certification Practices] [Two Gaps] [End Notes]

Mathematics and Logic Starter Lessons etc Step by Step from Arithmetic to Calculus for Ages 14 to adult

1. Ends, values and methods for work and study. 
2. Decimal and Integer Arithmetic Web Video Lessons- with exercises thought Based Development of skills and concepts.
3. Fractions - Thought Based development of what is a fraction and five operations on them: addition, subtraction, comparison, multiplication and division.
4. Fractions with Units. Arithmetic skill with units are introduce in a do-this, do that manner that may serve calculations with rates and proportionality constants in mathematics and the physical sciences.
5. Basic Number Theory: Primes & Composites Primes Factorization Theorem GCMs and LCMs from Primes   Prime Factorization Aids Prime Factorization Examples Counting Whole No. Factors  N-th  Roots and Primes Fractions & Decimals Fractions as Decimals 1 = 0.999 Recurring
6. Solving Linear Equations - Then algebra skills are introduced and strongly linked with arithmetic skills with the aid of fractional operation on stick diagrams.  Solution of triangle systems and systems in essentially one unknown may make word problems and solution of systems  of linear equation in two or more unknown easier to master.
7. Formulas forwards & Backwards. The mastery formulas and rules in mathematics and logic usually begins with their direct  or forward use. But once that is done, most if not all rules and formulas are used indirectly or backwards.  Talking about that provides words to describe a key part or key theme in the mathematics education of students 13 or 14+.
8. Proportionality Relations, Back- & For-wards. The study of proportionality relations (direct, inverse, joint) involves the  backward and forward use of the those relations (see theme 5) and involves arithmetic with fractions with units.
9. Mathematics Free Logic Chapters (chapter 2 essential) may lead to greater precision in reading and writing,  two must for work and study at all levels, while setting the stage for proofs and the necessity to see the difference  between saying A if B and saying A if and only if B. See the difference may help students understand the small print in agreements or contracts they will meet sooner or later in life.
10. Euclidean-Geometry The lean intro employ implication rules if B then A directly, one at a time, one after to further indicate  the role of direct logic in mathematics. The discussion of similar or proportional triangles provides a coordinate-free viewpoint and development for the discussion of right triangle trigonometry.
11. The first site coverage of Slopes and Lines involve algebraic and geometric reasoning, and the forward and backward use of  equations for lines. The graph of one quantity y versus another quantity x is a non-vertical straight line when y = mx + b  for some constant b and some proportionality constant m = the slope. When the quantities y and x have different  units of measurement, the coefficient m is a fraction with units.
12. Maps, Plans, Similarity & Trig Practices with maps and plans drawn to scale without and with coordinates lead to a  coordinate viewpoint and development of geometry and trigonometry for acute angles and in general. Graphs of periodic  trig functions appear here. Along a second treatment of slopes and lines. 
13. The Pythagorean Theorem and its Chinese Square Dissection proof (algebraic form) is given in Chapter 20 of Three Skills for Algebra.
14. Radian measure is introduced in Chapter 20 of Three Skills for Algebra. Radian measure of angles is needed to simplify formulas for derivatives (slopes to) trig functions. 
15. Electric Circuits Etc  - Miscellaneous notes (off the cuff) notes.
16. Calculus Preview: Why Study Slopes - Advanced Motivation. Besides studying and graphing constant rates of changes, we can graph functions y = f(x).  This first calculus preview or starter lesson understandable with a knowledge of slopes and the notion of how to graph relations y =f(x) previews an  elementary part of calculus where slope or derivative sign analysis may be employed to locate interior and endpoints maxima and minima.  The algebraic way of writing and reasoning is employed at full strength in calculus, and this step helps develop the necessary ability.
17. Complex numbers - the geometric approach here employs Cartesian or rectangular coordinates to define addition and polar coordinates to define operation  on points or numbers in the plane. Geometric observation or geometric reasoning implies the midpoint calculation commutes with rotation leads to a  simple proof of the distributive law. 
18. For n-th Roots of Unity and Regular n-gons: See Maps, Plans, Similarity & Trig
19. Chapter 18 in Three Skills for Algebra in long manner attempts to explain, justify and rationalize the shorthand role of letters and symbols (notation) in the  description and application of arithmetic rules and patterns, algebraically described.
20. What is a Variable? Introduction = Variation between Examples = Variation of Letters A letter denotes a variable Cases of Double Variation Three Notions of a Variable Constants, Parameters & Variables =Talking about numbers - Dependent or Independent Variable, a Matter of Choice
21. Quadratics in 10 Steps: The algebraic way of writing and reasoning is employed at full strength in the derivation of the quadratic formula. The 10 steps offer a simpler treatment, one that hopefully makes the full treatment accessible. If you are planning to study calculus, seeing and understanding the derivation of the quadratic formula is good preparation for the full strength ultrastrong use of algebra in calculus.
22. Polynomials. site lessons point to a geometric approach for introducing multiplication, addition, subtraction and division of polynomials. 
    The approach is correct only in special cases, but the approach makes the operations themselves easier to learn. Thought-based justification of the operations can come later. .
23. Application of Factored Polynomials. Besides studying and graphing constant rates of changes, we can graph functions y = f(x). 
    This second calculus preview or starter lesson understandable with a knowledge of slopes and the notion of how to graph relations y =f(x)
     covers an elementary part of calculus where slope or derivative sign analysis for factored polynomials may be employed to locate interior
     and endpoints maxima and minima. The algebraic way of writing and reasoning is employed at full strength in calculus, and
     this step helps develop the necessary ability.
24. Exponents, Radicals & logs - several pages. Here is a one page summary - the old version.
25. More Number Theory: Remainder Arithmetic I & Remainder Arithmetic II
26. Functions - Forwards & Backwards - A very full and detailed development. For inverse trig functions, see the site  Maps, Plans, Similarity & Trig as well.
27. Calculus: More Advice, Directions, theory and an incomplete set of Examples:  Here is a growing site section to support calculus - Limits, derivatives and integrals.  Section growth has been and will be motivated by student request to explain concepts and methods. 
28. Real Analysis - A decimal viewpoint of real analysis with proofs of theorem normally stated without proof in first courses on calculus, plus some ideas for extras.

Why Bother - Context and Motivations for Mathematics Education  - which ones are  convincing?

Page Sections: [Page Top] [Mathematics Starter Lessons for Ages 14+] [Forewords to Site Books] [Two Paths for Learning and Teaching ] [Why Bother - Context and Motivation for Mathematics Education] [Common Needs with or  versus Technical Needs] [Elements of New Mathematics Education Program ] [More on Learning by Rote or With Understanding] [Horrible (Pointless) Course Design/Instruction] [Unreliable Teachers Certification Practices] [Two Gaps] [End Notes]

Key Questions: What  observable skills, if any, do you want to see in the mathematics education of yourself or others?  How should they learnt or taught, why and when? 

Site lessons and directions for learning and teaching  provides direct,  very detailed  answers to Alice's question at least in the subject of mathematics education. The motivation for that stems from observation of  common fears and difficulties,  and from course design and delivery in school systems which may give students a decade or more of mathematics lessons with no observable result except for an absence of skill and confidence. That is absurd.

Site books and topics span many topics seen in mathematics and logic.  The presentation has two motivations.  The first is make the hard easier to learn and teach.  As a student and teacher, I sensed or saw some gaps in skill and concept development. Here are my remedies.  The second motivation is to provide a coherent thought-based  or logical development of skills and topics. All is done at least in part via the presentation of starter and further lessons, fresh or recycled, in site sections.  The thought-based development of skills and concepts given or outlined in site pages may help those who want to understand as well as do.  

Posing the question of what observable skills should met and mastered posits a viewpoint of education in which the ability to obtain results and to express ideas in a visible form for peer or teacher review and interaction has great value - is an end for instruction.  While education may lead to private thoughts with great freedom,  the material world demands skilful mastery of rules and practices from application to, if possible, the ability to develop that observable mastery in others. 

Site pages not only provide goals for education, site pages also say how to meet the goals with the aid of appetizers and lessons, fresh or recycled.   Starter lessons and alternative routes may make skill and concept easier to learn and teach. The net result is an alternative curriculum for secondary mathematics education from arithmetic to calculus plus answers and questions about what be met and mastered in the service of common needs of daily life at home, at work and study, and after that in the service of technical needs of trades and professions, or mathematics itself. 

Current Context

• Primary School Level:  Practical and common needs are served by learning to count, do arithmetic carefully,   master time and date matters,  master money matters for buying and selling goods, work and calculate with measures, use maps and plans to find or estimate lengths, areas, angles and location.  Learning about the domino effect of errors in calculations leads to incorrect results should imply better work and study habits in situations where rules and patterns need to be applied carefully, one at a time, one after another, to arrive at good results.  The care  required to figure well is an observable sign of diligence or wits of the practical kind.  
• Current Secondary School level - Preparation for Final Examination:  After primary school, many students and teachers do not know why skills and topics are covered, except that their mastery is likely to be required by final examinations.  There is something rotten in that. (The site author as a teacher has had to teach course which contained material not of service to students, but still required for graduation. 
• University level Science and Engineering Instruction;  Courses are demanding. Students in being admitted are given  the chance to succeed and to prove that they are able, but success is not guaranteed and indeed about half the students in university level calculus will fail or drop-out. Science and Engineering programs  use mathematics course to select among the able,  those willing to sit down and study. 

Past Context

The initial educational aim of developing reading, writing and arithmetic skills prepares students for adult life. In well-off societies, five to 12 years of schooling is required by laws for the sake of child and their futures.  In poorer societies, going to school is a privilege and not a right nor an obligation. That is unfortunate. Reading, writing and arithmetic was the first aim of primary level schooling.  Further education of young teens,  or apprenticeships in the workplace,  has had the tasks of preparing students for trades or  for further studies or the task of polishing social skills manners or the task of keeping people in school instead of being idle on the streets.

According to the 21st year book of the National Council of Teachers of Mathematics, 1953,  instructors and course designers should be learning engineers presumably for skill and concept mastery.  The viewpoint of education that says true knowledge is a private affair, located in the mind, apart from observable and verifiable skill development shifts education from the material to the immaterial and does not favour observable and verifiable skill mastery.  Oops. 

Aiming for observable and hence verifiable & correctable  mastery of skills and methods gives a tangible, material, concrete goals and pathways for instruction and self-instruction - lean, critical or just in time, as you like.  A do-this, do-that approach for instruction from elementary to advance levels with the focus on skill development, one small step at a time, one small step after another, could build confidence and give a viable, accessible, operational command of mathematics and logic at many levels. 

 Competence, communication, reason and problem solving can all be described concrete in terms of observable.  And in terms of skills, student centered education would mean providing skills and ends with take home value that serve common needs first, and in building abilities for work and study provide confidence and self-esteem.  Reality Check:  Economic conditions that provide employment at the end of a short or prolonged stay in secondary and college systems would help as well with self-esteem. It takes a village and a viable, sustainable, economic prospects to raise a child with confidence and hope.  

While some people complain that too people are not doing real work,  the use of machines and energy in farming, fishing, construction and mining implies fewer hands are needed to do perform physical labor needed to provide food, shelter clothes, medicine and construction. That leaves more time for idle time, office work and goods and services, optional or essential. Many strive to be part of the flow, control and design of goods and services, essential and optional as individuals or as employees or employers in private and public affaires. The main problem facing society is the provision or lubrication of goods and services, all in a way that will not lead to a population whose demands exceeds local or global resources, or to an impoverished population largely apart from the flow of goods and services.  Failure to plan is planning for failure.  While governments should not be in full charge of economics due to the nature of bureaucratic decision making, governments need to provide limits and to provide safety nets or emergency rules and plans to provide rations and basic necessities.  Idle hands are avoided via people working in service industries (education, health, government) in societies where mechanization and greater productivity implies more can be produced with fewer people gainfully employed.   

Common Versus or With Technical Needs

Mathematics and logic education may serve both common and technical needs.  Common needs are presently served by the development (we hope) in primary school mathematics. 

Primary & Junior High School Mathematics

As said above, practical and common needs are served by learning to count, do arithmetic carefully,   master time and date matters,  master money matters for buying and selling goods, work and calculate with measures, use maps and plans to find or estimate lengths, areas, angles and location.  Learning about the domino effect of errors in calculations leads to incorrect results should imply better work and study habits in situations where rules and patterns need to be applied carefully, one at a time, one after another, to arrive at good results.  The care  required to figure well is an observable sign of diligence or wits of the practical kind.  

A practical knowledge of geometry would entail students have experience with  measuring actual distances and angles in their environments and in construction projects, and also in doing geometric figuring, surveying and navigation with maps and plans using land areas or objects drawn to scale. That practical knowledge of drawing to scale (similarity applications) may be provided by any formal discussion of similarity and trigonometry.  

Games of chance and risk present in daily life provide a context for introducing probability theory.  Money matters may range from knowing the monetary value of paper and coins, their use in buying and selling goods and services with percent mark-ups,  discounts and commissions. Money matters may also include cautions regarding the balancing of personal, household and business income and costs.  

Money matters of the more technical kind would employ compound interest and geometric sum formulas forwards and backwards.  The full coverage of probability and money matters falls in the category of serving common needs that have technical prerequisites.   The latter should be presented in a lean or minimal manner, so that the technical support an a practical or operational command, and all further details are absent or delayed until later study.. For example, an geometric summation formula may be mastered  numerically with no general explanation of why the formula works.  Explanations may be left for later study, or provided as a reference where covering in class would be overwhelming for the students and/or instructors present.  Skills and concepts with the greatest immediate or likely take-home value or benefit should be put first and as early as possible too, given the risk that students will drop out. Do that may lessen the risk, or at least provide mathematics education with take-home value to those students. 

Secondary School -Technical Needs

Students of art and design and computer games may learn geometric views and projections.   Surveyors and navigators may need a knowledge of trigonometry functions of acute and obtuse angles.   Students heading into construction and electrical fields should know about the use of maps and plans drawn to scale with solution of triangles with trigonometry being a plus.   Those heading into electrical trades should know about  phasors or complex numbers in the description of alternating current. 

Students heading for university studies in engineering, science, commerce and  itself need to master the mathematical subject of calculus, elementary to advance level, in m The latter provide a technical language for the expressions and development of computational skills and concepts. Since students with different academic destination may be taught in the same high school courses or program,  the technical preparation of all possible destinations needs to be covered by course design in mathematics and/or science.  And in that strength or weakness in the destination subject and in mathematics may decide the academic destination or how far each student continues in each.   

Preparation for an operational command of calculus requires a mastery of the following.

• Arithmetic - exactly and efficiently without and with calculators 
• Arithmetic - Properties of Operations on Numbers
• Algebra (solving linear equations,  forward and backward use of geometric formulas etc.)
• Analytic and Euclidean geometry
• Slopes,  Lines and Direct Proportionality Relations
• Complex Numbers
• Logic (Direct and Indirect Use of Implications, Difference between one and two way implications,  Recognizing when one hypotheses is inconsistent with another.)
• Right Triangle Trig Ratios for acute and even obtuse angles
• Periodic Trig Functions
• Quadratics in one and two variables
• Polynomials and their Ratios
• Natural logarithms and exponentials, Powers and Radicals
• Sets and Counting
• Probability
• Vectors
• Mathematical terminology and practices - Functions included

The topic emphasized in bold above serve common needs directly or indirectly. For example providing the algebraic background for forward and backward use of formulas in money calculations (loans, mortgages, return on investment) and in refining or supplanting map and plan use skills  will help students handle common or likely problems in the daily lives of themselves and immediate or future families. Thus skill and concept  with some take- home value is present.  

If students heading for university studies are in a competitive environment where continuation is based on academic performance due to personal reaction to failure or due to elimination of students based on their performance, we should mix and emphasize first those skills and concepts with greatest take home-value with those skills and concepts easiest for students to master.

Given that students may halt their studies or become discourage at any time, there is no harm in attempting to put first or as early possible those elements of the preparation for calculus providing skills with the greatest take home value, that of greatest service to likely or common needs. After that, students and teachers should be made aware that further elements in the preparation are present because of the needs of calculus or further studies. Those topics and how they are of service should be labeled as such. 

Students of probability may appreciate the role of sets (and even functions)  in formulating probability theories and in aiding the identification and counting of possibilities. Remember a knowledge of probability is needed for estimating odds or chances of success and failure, and in that estimating expected returns or losses in cases decisions are being made in the face of uncertainty.  That uncertainty or risk  may be environmental and unavoidable. Or the uncertainty and risk may be due to playing games of chance. The latter may range from throwing dice, drawing a card, betting or making an investment.  Thus a knowledge of probability has take-home value in helping students or their families avoid  or minimize risks.  That being said, expected value of choices or outcome may justify some risk.

Students in physics, chemistry and biology courses may see 

• the forward and backward use of linear equations, quadratics and vectors in precalculus description and analysis of motions and forces. 
• the description of gas law and gravitational laws as formulas and proportionality relations that may used forwards and backward, along with some probability and geometric formulas to imply or suggest or interpret the laws (physical properties).
• the forward and backward use of logarithms, exponentials, powers and radicals in the description of compound and continuous growth and decay in biology, physics and commerce - geometric sums may be employed in the description of  the periodic stocking of ponds with fish or material that may be diluted or flow out. 
• the role of probability and counting methods in genetics.

The foregoing provides a further context for topics met in the preparation for calculus and beyond. 

Site coverage of complex numbers show how the latter may be introduced with a high school level of rigor before the introduction of periodic functions. The development is very simple. The development is based on a technical item - a very simple geometric proof of the distributive law for multiplication over addition.  Before the coverage of complex numbers, the properties of real numbers, those given as axioms in modern mathematics curricula, can be derived or implied by mathematical practices - those met and essential in the earlier development of quantitative skills in the service of common needs.  The foregoing development implies the continuity between earlier and later instruction - and a convergence with the modern mathematics secondary curricula of the 1960s met in  some American,  some European and some Asian schools. 

The inclusion in the preparation for calculus of modern mathematical concepts and notation in the form of sets and in the set-based description and codification needs to be lightly done, and to be included in a just in time manner where it speed and aids skill and concept development. Mathematics for the person in the street and the for student outside of mathematics may a subject where notation is employed where is helps instruction and not overwhelm students and teachers.  The underlying question: which is to be more important, that is, the aim of making mathematics simpler and more accessible, and not overwhelming,  or the aim of introducing and emphasizing modern mathematics notation and terms. The skill development program in the parents area Help your Child/Teen Learn gives a compromise - it propose a notational and conceptual light introduction and development of calculus - an operational viewpoint -  be given first. 

Skill development in mathematics and further disciplines to be student centered needs to be put first those skills and concepts with greatest take home value. In particular, primary and secondary school courses should provide students with an operational command of skills likely to be needed sooner or later at home or in the typical workplace.  That latter does not involve employment in science or engineering.

End Notes

Page Sections: [Page Top] [Mathematics Starter Lessons for Ages 14+] [Forewords to Site Books] [Two Paths for Learning and Teaching ] [Why Bother - Context and Motivation for Mathematics Education] [Common Needs with or  versus Technical Needs] [Elements of New Mathematics Education Program ] [More on Learning by Rote or With Understanding] [Horrible (Pointless) Course Design/Instruction] [Unreliable Teachers Certification Practices] [Two Gaps] [End Notes]

1. Learning how to do  and apply arithmetic carefully and fully with decimals, fractions and even signs is needed is needed in daily life, so much so, that learning how by rote is justified where explanations overwhelm.  The same may be said of map and plan usage, money matters, time and date matters and measurement matters - those involving length, time, amount alone or in  the description of rates and proportions.  For the common knowledge, a consistent, practical plug and play mastery of skills and concepts, or rules, methods and patterns is necessary and sufficient.  The aforementioned  ends, values and methods for work and study  should  help studies and instruction. While learning to do is important detailed explanations of how and why should be available to serve as  a reference or to aid in the instruction of serious students - those who want to understand the why as well as the how.     To make skill development student centered emphasize skill and concept with take home value - quantitative skills and concepts likely to serve common ends and needs, sooner or later, before emphasizing the advanced mathematics of interest to the few.  And in the latter, emphasize service of common needs when and where possible.  Mathematics education will be more inclusive if skill and concept is inductive with all steps clearly defined, and none too large or absent.

2. International efforts for course design and delivery include statistics and data collection methods as part of mathematics education.  The mathematics program above does not.  That is a deliberate choice.  The inclusion of statistics and data collection methods in primary and secondary mathematics has no immediate take-home value, distracts from the development of core skills and concepts in arithmetic, algebra, geometry and logic etc needs no distraction, and includes material whose thought-based development is fuzzy or beyond the reach of students and teachers.
   

3. Mathematics education in general is inductive with later skills and concepts built on earlier ones. But there are gaps or holes in the introduction of algebra - the shorthand role of letters and symbols obvious to those gifted in mathematics is not full and clearly developed and rationalized in secondary mathematics from first use of formulas to calculus. Those holes or gaps first sense in my 1966-70 secondary school days were finally address in 1975 hand-outs I gave a McGill University mathematics department open house and further addressed in fall 1983 lessons on Three Skills for Algebra and why slopes, while I waiting for formal graduation. See the algebra guide above.  That being said, the drift away from arithmetic skill development in primary schools has led to an arithmetic gap - students not knowing their addition and times table, and beyond that method for exact and efficient arithmetic with decimals and fractions. That drift  undermines skill development in algebra itself and all further high school mathematics: trigonometry, analytic geometry, functions, calculus.  
   

4. The site program for skill development is a successor for an earlier attempt  LAMP (Lean Applied Mathematics Program - obsolete) and to site mathematics education essays.  Site growth - the exploring and expression of different ideas and options - set the stage for the lean core mathematics program or plan described above.  The latter in turn sets the stage for the growth and editing of site material to support and even refine the program.  Writing is an iterative affair. 

5. More on Which Way to Go:  In retrospect, primary and some secondary school mathematics serves common needs - develops logic and quantitative skills of obvious value in daily life at home and at work:  Applications include date and time matters, geometry in the form of map and plan usage based on drawing objects to scale;  money matters in the form of saving and in the form of buying and selling goods and services - working for a living included; chance and probability in the form of recognizing and avoiding risk, or making  decisions in risky, not all is certain, circumstances;   and the use of numbers and units to count, measure and locate objects.   Arithmetic to helps in counting and measuring. In all the foregoing,  ends, values and methods for work and study  may be met and mastered. But much secondary school mathematics - the trade and pre-university streams - serve technical needs in the form of preparation for trades that required a few  quantitative skills that are not common, and preparation for college studies and training in mathematics, science and commerce.  Here mathematics has been put on a bureaucratic pedestal. Education authorities or parents consider it unfair for some students not to be taught pre-university mathematics even while observable skills and values,  those serving common needs may be missing. But common needs should be served first so that a five year or even a decade of study in mathematics has take-home value. That being said, when a student asks why a topic must be learnt in secondary mathematics, the answer is likely to be one of two: (a) - preparation for a final examination, no more, no less; that is why; and (b) the topic is required by and has long-term value for college studies, all in a manner beyond the comprehension of teachers, student advisors and most parents. In practice, there is bureaucratic social contract in high school mathematics - teachers prepare student for yearly final examinations, with students and parent expecting and demanding that, no more and no less. Students will tell teachers who try to explain why a method works that explanation of why is not needed as teachers are hired to present only correct methods. For better or worse, here are three roles for mathematics and language instruction: (i) serve common needs and as part of that provide ends, values and methods for work and study;  (ii) aid reading, writing and reasoning skills by showing students the difference between the  situation A if B and the situation A if and only B, and by exposing students to direct and even indirect deductive chains of reason with both math-free and mathematical examples - that serves common needs because contract and instruction at work, home and school need to be understood and written carefully; and (iii) last prepare students for the technical requirements of trades and college disciplines with the unfortunate, but explicit understanding, that many may try, and many of them will fail in the preparation for college disciplines, those which rest on calculus.  In that, we need to explain that mathematics in the form of calculus provide necessary  tools and a logical base and language  for studies in commerce, science, engineering, technology. In that, we need to explain that calculus taught at the college or senior high school level demands a full strength mastery of the skills and topics in earlier mathematics that serve common needs and/or are present simply because calculus and beyond requires them. While there may be greater motivation for skills and topics that may serve common needs, sooner or later, the technical topics have to be covered as well. They should be explicitly identified as preparation for calculus and beyond,  and/or as exercises which support and require ends, values and methods for work and study; and/or as exercises in which the thought-based development of skill and ideas in all or part provides a model for reason.  The foregoing circumstances (many may try, but not all will succeed) and the foregoing motivation for elements that prepare for technical needs may not please everyone. Yet providing them (and further ones) honestly is we hope, better than none, or better that the motivation, this topic appears on the final examination context for learning and teaching. Hopefully, present day discussion and documentation  of rich learning environments and problems, routine and open, will provide more context for secondary mathematics education.  Here as in other matters, site material provides another lower bound for skill development and motivation for it.  Not all is certain.

Page Sections: [Page Top] [Mathematics Starter Lessons for Ages 14+] [Forewords to Site Books] [Two Paths for Learning and Teaching ] [Why Bother - Context and Motivation for Mathematics Education] [Common Needs with or  versus Technical Needs] [Elements of New Mathematics Education Program ] [More on Learning by Rote or With Understanding] [Horrible (Pointless) Course Design/Instruction] [Unreliable Teachers Certification Practices] [Two Gaps] [End Notes]

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