In many arts and disciplines, it is absurd to  ask an  apprentice to solve open or difficult problems before mastery of standard methods and routines.  Most arts, trades, professions and disciplines in the work place and in college or university studies demand and prize the careful mastery of routine rules and patterns, one at a time, one after another, alone or in combination, all in a repeatable, reproducible, observable and correctable manner. Within each discipline, critical thinking or maximum benefit and least harm demands a knowledge of the origins and limitations of rules and patterns, practices, steps and methods. Mathematics is a discipline in which the ability to  carefully use and combine of rules and patterns, one at a time and one after another, alone or in combination needs to be written carefully on paper to demonstrate and record skill and comprehension. 

In general for pleasure and for applications, we may record, develop and read our thoughts and reasons by writing and drawing words, symbols and diagrams on paper or alternative media.  Spelling and writing stems from our ability to draw. The detailed and deliberate record of our thoughts and reasons may be read or seen or heard later to restore or share them. There-in an extension of our memories and reasoning faculties, singly and collectively. While mind reading is not possible, we may project our thoughts and visions into words and diagrams to communicate and reason alone or in company, and thus to generate or solve problems. That projection may be seen and corrected by ourselves and peers.  Education needs to develop and maintain projection habits and provide it food for thought and, for better and  methods for decision making, for arriving at conclusion, if that possible, along with a knowledge of their benefits, origins and limitations of reason or decision making.  See Volume 1A.

Page Sections: [Top] [Notes for Teachers and Instructors] [Steps & advice to improve marks, performance and comprehension]  [Key Appetizers and Lessons] 

Ends, Values and Means: 

• The will and ability to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause  escapes  attention is an end, values and means for mathematics education in the training of students and in course design and delivery. 
• Skill and confidence in  mathematics, a written art or discipline,  may follow from care and precision in following and recording the steps in methods and routines in a well-formatted, readable, reproducible and repeatable manner for verification or correction. Here care and precision is another end, value and means for skill and concept development. 
• Mastery of  thought-based paths for understanding and developing skills, concepts and comprehension is a further end, value and means for mathematics education, and may be considered an extension of the care and precision, required so that no subtlety, no clause and no detail escape attention..

Ideas that cannot be expressed on paper with diagrams, words and symbols are not part of observable skill & comprehension.  Compare and contrast that view with the Allegory of the Cave in Plato's work The Republic where knowledge is based on shadow interpretation. Compare and contrast that view with the dominant constructivist theory of skill and concept  learning, in which mastery is a subjective affair, not for observation nor correction in an objective manner; and in which changes in delivery style in a shadowy manner was suppose to lead to a subjective (anarchistic) view of knowledge, one that in retrospect resembles the state of knowledge before striving for objectivity was the norm in science and  technology, if not law. 

Key Appetizers and Lessons

Page Sections: [Top] [Steps & advice to improve marks, performance and comprehension] [Notes for Teachers and Instructors] [Key Appetizers and Lessons] 

The following appetizers and lessons, online chapters included,  can be read or seen separately. Each one is different. Altogether, they provide  technical themes and content-oriented standards for senior high school and college mathematics studies and instruction to meet or exceed.

1. If you are able to read logic chapters 1 to 5  in online volume Three Skills for Algebra, you are not too young nor to old for site material and directions, a good fit is expected. Logic - Chapters 1 to 5 develop greater precision in reading and writing for work and studies in and apart from mathematics. Improve your skills and confidence. Some chapters are easier than others. Chapters 2 is hardest. Chapters 3, 4 and 5 are easier.
   
   Online math & logic  jigsaw puzzles. Each appetizer and lesson, each site page, gives a piece of a math and logic education jigsaw puzzle. Look at the pieces, and try to fit them together by trial and error, one at a time and one after another, in pairs and in larger groups.  Putting the pieces together takes time. If a piece does not fit, try another and another. Each art and discipline, and each problem in daily life, is like a jigsaw puzzle - one or more. You need to find  the pieces and check that  they all present, and put them together by trial and error, with time and labour, starting with the easier parts - the straight edges.  In mathematics, the straight edges are provided by  mastery  of linear chains of reason in Logic - See chapters 1 to 5.
   

2. Fraction Starter Lessons:  point to an efficient, operational command of exact arithmetic with whole numbers and fractions. There-in lies a first standard and a must for all of secondary school mathematics.
   

3. Solving Linear Equations - begins with a geometric way to visualize and solve linear equations, and then introduce ideas to make word problems and simultaneous easier to understand and explain. Older students can read the examples here in sequence to review and understand how to solve linear equations, how to present solutions (appearance is everything after content). Teachers can use the examples or similar one here to introduce the topic and to reinforce fraction skills and sense. (If I was to redo this, I would choose coefficients to ensure whole number solutions while working with the stick diagrams here.) Pay attention to the format of the solutions here. Errors in format lead to errors in finding or calculating solutions. Here-in lies a second standard and goal for all of secondary mathematics.
   

4. Three Skills for Algebra,  See how to use words before and besides symbols. See how to talk about numbers and quantities, and how to describe calculations, and see a hint of the 4th skill. Here we are filling some gaps in your education - indeed in the education of all who have learnt and taught algebra, or written books about it. There is a missing link here. We are providing words that been missing not in the doing, but  in the discussion and explanation of algebra. Read all about. BREAKING THE SlLENCE with a Better and Greater Use of Words in learning and teaching mathematics.  Here-in lies a goal for secondary II mathematics.
   

5. Using the Compound Interest Formula forwards and Backwards - there-in lies a 4th skill for algebra, The backward use of equations and fomulas has been a silent part of high school and college courses. The first innovation here is to break the silence by describing that practive with words, that is the phrase "Forwards and Backwards" or "Directly and Indirectly". BREAKING THE SlLENCE  Continued.  The second innovation here is to name ,  illustrate and contrast the concetp of numerical and algebraic solutions of equations. Be satisfied if you can solve the backward use problems numerically - the arithmetic approach. Be estatic if you understand the algebraic approach and its greater power. Here in lies a connecting theme, goal and standard for mid to senior high school mathematics and science. 
   

6. Arithmetic,  Watch these videos to perfect skills and comprehension of whole numbers and fractions, etc, etc.  You may think that arithmetic mastery is for primary school students, and further studies in mathematics should not demand skill in computations with a calculator.  That may be true when you go shopping, but the ability to do arithmetic in an efficient, repeatable and reproducible manner, no errors please, is a must for senior high school mathematics and calculus.  As student, you have master the basics - learn to walk, before you run. That being said, if you are adverse to arithmetic, most of the topics can be understood without a great command of arithmetic
   

7.  Euclidean Geometry,  Here a lean treatment that will connect construction and duplication of triangles with isometry, parallel lines and how to recognize parallelograms. The ability to follow short Chains of reason developed or asked for in  site logic chapters apart from mathematics appear in connection with geometry. 
   

8. Complex Numbers - a simple geometric approach If you have mastered polar coordinates, this visual and geometric approach will complete your earlier understanding. Students of electricity, engineering and physics knowlingly or not, employ complex number ideas in the basic or advanced concepts and calculations. If high schools mathematics introduced simple geometric approach to saying how to add and multiply points in the plane, many difficulties would disappear. This geometric approach or its easy consequences simplifies learning and teaching of the law of signs, of unit circle trigonometry and of vectors.
   
   Remark: The site area on   Euclidean Geometry shows how the latter implies the distributive law for complex numbers
   

9. Calculus - Geometric Preview  and  Calculus - Algebraic Preview. The first preview explains why slopes or rates of change are studied in senior high school mathematics. The second one goes further into slope interpretation and shows how factored polynomials and sign analysis thereof helps in saying where a function y  = f(x) is increasing or decreasing. Calculus is the door-opener for studies in science, engineering, nursing (for some reason), medicine, accounting and on on. Calculus is the subject which requires skill and concept development and mastery in arithmetic, algebra, logic,  geometry and trigonometry. The full strength requirement for the latter is a shock for student who did not know about those requirements and also for those who know.  The calculus previews develop and motivate geometric and algebraic skills and understanding before or during calculus.

Site advice and directions for learning and teaching mathematics will take time to understand and follow. Follow  closely, but not too closely - site advice and directions are approximately correct, for some circumstances, not all.

Where is Mathematics? Keep your ears and eyes open. At school, at home, in going out, watch for the occurrence of measurements and calculations, and how they are done, and why.  The result may be some questions to motivate your mathematics studies. The assignment here is to collect questions.  The forthcoming site areas in preparation (preparation postponed)

17**. Telling & Working with Time
18**. Maps, Plans & Drawings
19**. Quantitative Skills for  home and  work, etc, etc 
** Means Planned - Here are descriptions for teachers, not students.

aim to develop skills and concepts in context. Give  methods to use and apply in a repeatable and  reproducible manner in common situations involving mathematics. One of the textbooks I am reading for learning French organizes its lessons around themes: going to a restaurant or theatre, riding on a bus or train or plane, visiting a shop, working as a carpenter, and so on. The lesson then provides the words or vocabulary useful in each setting or activity. Mathematics courses may describe similar visits or activities,  and connect the latter to mathematics.  There-in lies a value-providing ends and context for meeting rules and methods of mathematics.

Students and teachers do not have to see a thought-based development of all rules and patterns,  if the rules and practices lead to repeatable, reproducible and hence verifiable (right or wrong) results. But a thought-based development helps in understanding the benefits, origins and limitations of rules and patterns, customs and practices, so that the latter can be applied carefully and not mis-applied.

The thinking part of an art or discipline:

The  thinking part of an art or discipline comes after the assumption & careful  mastery of some rules and patterns, steps and methods, practices and conventions.  Careful mastery means you can use the latter to arrive at results in a repeatable, reproducible and hence verifiably right or wrong manner.  The thinking part of a subject begins when you start to combine rules and patterns, steps and methods, practices and conventions, to obtain new ones in a repeatable, reproducible and hence verifiable manner. Thinking or critical thinking within an art or discipline continues through recognizes the benefits, origins and limitations of rules and patterns, steps and methods, practices and conventions, so that the approximations in the application of the latter are known or avoided.  The combination of rule and patterns, customs and practices, steps and methods, one after another, may lead to short parallel strands of reason and hence a thought-based development of an art or discipline beside the empirical mastery of rules and patterns etc with confidence building results that should be  repeatable,  reproducible and hence verifiable. Once the ability to form or follow strands of reasons within an art or discipline is present and respected or appreciated, fuller and fuller thought-based developments can be offered, if not in class, then in print. The first phase of education could be based on rote - here are the facts and methods - learn to use them in a repeatable, reproducible and hence verifiable right or wrong manner.  Later phases may then build on that via a mix of deductive and rote mastery of further rules and patterns.

Page Sections: [Top] [Steps & advice to improve marks, performance and comprehension]  [Key Appetizers and Lessons] [Notes for Teachers and Instructors]

Notes & Remarks for Teachers and Instructors  

For Methods and a focus Pre-Secondary, Junior Secondary and Remedial  College Instruction, see (i) First Year High School Math - Lesson Plans with Fraction Focus (ii) Second Year High School Math - Lesson Plans with an algebra focus (iii) Algebra Lesson Plans.  Upper primary school instructor may read site lesson plans for secondary I and II to know what their students will meet, and to prepare for it.

In too many schools or college, it is impolite to suggest that better methods for instruction exist. Any suggestion implicitly criticizes the classroom habits and formation of fellow instructors. But an environment in which sharing and offering fellow instructors ideas for instruction is impolite slows effective reform in instruction, reform based on a free sharing and refinement of what works or could work better.  Allow yourself the freedom to consider and discuss site suggestions for better instruction without being offended.  Bon Appetite.

Logic: The site introduction of logic is math-free. It is aimed at improving work and study skills while hinting at the role of logic (implication rules) in mathematical proof and definitions.  The site introduction of logic showing the need for greater precision in reading and writing may lead readers to cultivate that precision. The math-free aspect may allow development of logic skills and concepts in parallel in and outside of maths. 

The solution of jigsaw puzzles where pieces are inspected and fitted together in a persistent trial and error fashion until a picture emerges and puzzled is solved provides a model for combinatorial, opportunistic and thinking-out-of the box problem solving in and outside mathematics - whatever works.  Calls for problem solving in mathematics and other disciplines may be shaped and refined by showing showing students how build comprehension and how to reason by combing rules and patterns, reliable and ethical ones preferred, in a repeatable, reproducible, verifiable and hopefully ethical manner. 

The site coverage of logic ends with thoughts, not definitive, on indirect methods of proof and reason.  See the last logic chapters and postscripts in Volume 1A, Pattern Based Reason.  Volume 2, Three Skills for algebra ends with duplicates of the same chapters but omits the postscripts. Indirect reason could be illustrated in detective and mystery stories. But the few stories (fiction) I have read end in sudden revelations and  of how the main character solved the problems, revelations involving clues and evidence not previously available to the reader. Remedies would be welcome.

Algebra: The site introduction of  algebra, what is a variable, solving linear equations and operations on polynomials clarifies nuances and subtleties while providing a clearer and greater role for words and geometry in its mastery and exposition. Thus, it builds and sets new and higher standards. There is one pre- or co-requisite to the mastery of algebra, namely efficient, calculator-free arithmetic skills with whole numbers and fractions.  The lack of drill and practice to develop and maintain the latter undermines high school instruction from algebra to calculus. Mastery of figuring skills with whole numbers and fractions should be kept and polished in K5-12. Parents may hope for a sound development of figuring skills, but should trust verify as well.  Centralized and bureaucratized design and implementation of mathematics education may eventually lower standards. 

Arithmetic and algebraic expressions where order or operations are implied by position and/or parenthesis are best seen and understood in silences, non-verbally, like pictures and diagrams. Site words on three or four skills for algebra and on what is a variable compensate for that silent aspect of mathematics.  Learning to talk about numbers and quantities, easily and precisely provides a striking advance for the development and comprehension of mathematical disciplines.

The site use of fractional operations on stick diagrams to visually and geometrically introduce the algebraic methods for  solving linear equations employs letters to denote unknown lengths - a concept easier to grasp, more concrete, that asking students to let a letter or symbol denote an unknown number (or a variable). Starting algebra with letters x to represent lengths may be an accidental return to the role of geometry in algebra, a role hinted at by reading x² as x-squared and x³ as x-cubed.  Fractional operations on sticks (line segments0  by themselves may consolidate comprehension of fractions and illustrate the exact role of fractions in algebra. That being said, with hindsight, I would start with equations that have whole number solutions instead of fractional ones.  

Remark: For example, the site stick diagram method is a geometric path, effective for most,  not all, invented in spring 2005 to introduce algebraic solution methods for linear equations.. One pupil had difficulty with the transition from geometry  to algebra.  In applying the method, you need to control or handle the risk.  The latter student need special treatment. With him, if I had had him for more one hour, I would have taught the algebraic form by rote, so he could obtain and check solutions, even if he did not understand the attempted geometric development. 

After the introduction of algebraic methods for solving linear equations, the site area on solving linear equations introduces (i) triangular or permuted triangular systems of equations; and (ii)  system of equations in essentially one unknown - the solution of the latter requires use of  (a) the associative law for multiplication and (b) the distributive law.  The verification of answers (an important part) forces students to look for mistakes between the start of their solution and the end of their check. An in all the foregoing, require students to format their work so that the sequence of steps in their figuring or reasoning process is recorded in a clear, legible and sequential manner.  

Suggestion: Have students avoid in place operations in which the sequence of those operation is unclear. In place of quantity, seek quality and for that require or give marks for clarity and format in student presentation of their solution steps and solution verification steps. In general, written solutions should be and become stand-alone and self-sufficient units in the notes of a student which record and communicate the use of mathematical methods. Implement this standard when and where there are calls for the development of communication and reasoning skills in mathematics. 

Calculus: The site introduction of calculus shows why slopes and factored polynomials are studied in high school. This two part introduction eases or avoids algebra shock in calculus begins by providing geometric and algebraic calculus previews which are understandable, skill and confidence building, presentable even before calculus begins. There lies a further advance for the development of algebraic skills in and before calculus. Putting these previews at the start of a calculus in essence gives a simple, easy to understand, preview of the derivative-based max-min analysis in differential calculus, one that develops algebraic skills slowly and systematically and gently instead of sudden  The site introduction of calculus goes on by adopting and sanctioning a  decimal,  error control viewpoint of limits, convergence and continuity. The site coverage of real analysis   goes further in providing decimal-based proofs of the theorems of calculus, usually given without proof in first courses on calculus.  Calling for the return of decimals and their explicit sanction in course design and delivery contradicts the 1950`s and 1960`s modern mathematics curricula, curricula continuing today in diluted form, but it should ease or avoid algebra shock in both calculus and university level courses on real analysis. 

Complex Numbers: The site introduction of  complex numbers provides a simple, visual, geometric introduction of the addition and multiplication of points, arrows and vectors in the plane in a manner that might be enjoyed in college level instruction today in science, engineering and mathematics, in the present-day training of electricians; and in junior high school courses where rectangular and polar coordinates are mentioned.  Easy consequence senior high school and college level consequences include trig formulas for dot and cross-products, and yet another proof of the Pythagorean Theorem.  The site introduction revamped could also be a basis for a leaner high school curriculum in which the role of signed numbers as coordinates appears before the law of signs.  The late physicist R. Feynman described his subject as a the addition and multiplication of arrows in the plane. Secondary mathematics too could be described in the same way.

A Word about Mathematics Education. Reason, communication and problem solving need to be based on a skill and concept development and some perfection in reading, writing and arithmetic. In elementary school or before, children may learn the or an alphabet, say 26 letters, and the digits 0 to 9. Some children may object to meeting and mastering the alphabet - too many letters, why bother.  If we said to these children, let make reading and writing simpler. You only have learn 20 letters, not all 26.  That would cause problems in reading and writing, and understanding words and their meaning. In mathematics students need to master the use of decimals to efficiently represent and efficiently do arithmetic on whole numbers and fractions. That mastery is useful or should be useful with weights, measures and calculation in daily life, albeit education that goes on and on beyond the age of 14 delays and hides  that usefulness, that is a problem to address.  The situation today where students are taught mathematics from primary school to college without knowledge or efficient mastery of arithmetic is similar to students study language without being equipped with a knowledge of the alphabet and a mastery of spelling and grammar. In particular, we would not tell a child or teen that a complete knowledge of the alphabet is optional.  Schools, colleges, teachers and parents should not be telling and should not be allowing students to think that mathematics can learnt or taught without efficient figuring skills with whole numbers and fractions.

In mathematics and logic, rules and methods  for solving routine problems need to be mastered well. When rules and methods for solving routine problems are approximate, for some circumstances, not all, students need to learn that as well.   Meeting and mastering rules and methods, exact or approximate, for solving routine problem provides students a model for tackling non-routine or authentic problems.  

Site  lessons and lesson plans focus on the technical development of skills and concepts with what may be a repeatable, reproducible and verifiable methods for building skills, comprehension and confidence.  

This technical development may be dry and the hints of applications in that development to abstract or remote. 

Thus there is a need for detailed examples of mathematics in practice and context, culturally dependent, say in buying and selling, in construction and production at home and at work, in paying taxes (ouch), in keeping records of income and expenses, and in further common trades and practices of a society or culture. The examples might show or emphasis how mathematical operations seen in or learnt in one context help in another.  For example, in teaching people to speak and write French, the textbooks I am reading offer scenes or scenarios (eating at home or in a restaurant, buying and preparing food,  traveling on a train, driving an car, visiting a hospital) to introduce and develop vocabulary and language skills in context and in a comprehensive manner for that context. Mathematics education would benefit not only from site technical innovation, fresh or recycled, standing on the work of others, but also from a series of spiraling and expanding vignettes introducing or developing the mathematics employed in common place activities, trades,  professions and school subjects, elementary to advanced. 

Cultural Note: What examples are appropriate or their selection will need to reflect and expand upon the cultural and economic history or origin of students and their parents, and which activities are common place or dreamt of.  In particular, the expansion pollution age  industrial, agricultural and resource based societies provides a context, common place examples of arithmetic and geometry, that may be absent from others societies coming into contact with or surrounded by that expansion. That for better or worse raises the question of how the other societies may adapt or react to that contact and the changes it wrought. For examples first nation societies, indigenous people,  may express and describe numbers and quantities differently  and not have the language, the words, to directly describe  examples of arithmetic and geometry present in the  industrial, agricultural and resource based societies which today, for better or worse, dominate the planet, Malthusian style. That clash of cultures is least demanding on larger societies  (save for the advent of change due technology or resource exhaustion) with their greater inertia and is most on smaller societies with less inertia.  

Ends and Values:  In many arts and disciplines, there are practices, customs and values  to be met and mastered one at a time, one after another, alone or in combination.  Customs have developed over many years, decades and even centuries in ways a student cannot fully anticipate. So explanations, clear and direct, not confused, are needed to communicate the evolved and often less than obvious, customs and practices. As part of the teaching process, students may be given situations or puzzles to extend or challenge their skills and knowledge, and to set the stage for mastery of a custom or practice. But at the end of the day, the instructor should describe the custom or practice, and encourage its mastery in a repeatable and reproducible manner. And in that customs and practices may be learnt by rote or with some explanation, preferable full and as much as the student can grasp. There are some arts, trades and disciplines in which  mastery of rules and patterns in a repeatable and reproducible manner is more important than and may serve in place of a thought-based development. That being said,  critical thinking within an art, trade disciplines about when to apply or follow a step or method depends on an knowledge of the benefits, origins and limitations of rules and patterns to avoid errors or mis-application. There-in lies a justification for a thought-based development and a general discussion of rule- and pattern-based reason.

 High school mathematics beside calculus implied content should include methods and routines useful to student and their close ones in the task and problems of daily life, including money matters, weights, amounts and measures.  Such methods and routines could be taught in way that would engage students and also aid the basic math and logic skills needed for calculus. Basic skills include the ability to write and thus show the steps in a calculation for verification or correction.

Mathematics education reform in secondary and primary school needs to strike a balance between the standards implied by calculus for mathematics mastery and the standards implied by social theories of learning. Diluting the primary and secondary school  mastery of mathematics needed for calculus,  to make mathematics more engaging for social reasons,  is similar to inviting students to swim in the deep-end of a swimming hole while limiting their instruction to learning how to wade.  Yet insisting that all students be strongly prepared for calculus is also impractical,  even with site innovations for making that preparation easier to learn and teach.  How to strike the balance is a question left to another day.   Course design in  secondary and/or primary school needs to explicitly identify the mathematics needed for calculus and the full-strength mastery of that material, while also preparing students for the frequent appearance in of elementary mathematics (arithmetic, geometry and algebra) - the question of what would disappear from daily life if there were no knowledge of numbers,  geometry and algebra might guide the formation of students in an engaging manner while emphasizing an operational command of every day mathematics in ways that appear to be reliable in a repeatable, reproducible, observable manner.  Balance in mathematics education might provide students with all the foregoing while striving to do so gently in a manner that serves the needs of calculus.    

 Education reform in secondary and primary schools appears to be in  rebellion against the demands of calculus for a full strength mastery of key skills and concepts.  Course design cannot properly cover nor prepare for the part of secondary school mathematics needed for calculus, while moving to an more engaging manner that mathematics that does not meet the dryer and more technical demands of calculus.  Diluting coverage of the part needed for calculus in secondary school, and the preparation for that part in primary school, appears to compound the fears and difficulties that stem from formally covering that part or formally preparing for it.   

Mathematics is an art and discipline in which the apprentice has to learn how to follow routines and methods in standard,  repeatable and reproducible well-formatted, manner to provide a base for the communication and record of reasoning in a manner the apprentice & others may check and verify. The notes and written work of the apprentice must show mastery of routines and methods in each topic, before tackling any further dependent topic. While solving difficult problems may be part of research in many arts and disciplines, students need to see the benefits, origins and limits of routine problem solving methods to recognize when non-routine methods are required.  The apprentice needs to see the benefits and limitations of the master methods before transcending them.