Notes & Remarks for Teachers and Instructors

Teachers:  LAMP is a curriculum framework, a proposal, for  secondary, adult and college mathematics and logic instruction.

Teachers: 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.

Teachers:  Calculus requires high school math at full strength. If you have not taken calculus, weave site advice & that full-strength standard into your instruction. Site standards for technical skill and concept development, site advice and directions for instruction,  are implied by lessons and lesson plans, methods,  for meeting them and by a critical path analysis of what calculus or college level mathematics requires in earlier secondary and primary instruction. Standards are effective when and only when they are supported and implied by effective lessons or lesson plans for meeting or exceeding them.   

The anti-calculus rebellion in secondary and primary schooling:  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 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.   See LAMP. 

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 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.  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.  

Standard: 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. 

More on Standards: Calls for the use of technology in mathematics should not be seen as a call to encourage students to use calculators in place of mastering addition and times table, in place of mastering methods for arithmetic with decimals and in place of mastering exact arithmetic with whole numbers and fractions.  Prior to the use of calculators in mathematics,  calculus and algebraic skills employed in calculus  built on and so tacitly assumed and required the mastery of exact arithmetic with whole numbers and fractions. As said above, most topics in high school mathematics appear to be present due to the requirements of calculus - as preparation for calculus.  Their mastery in the style required by calculus is undermined when and where exact arithmetic skills are not developed nor kept in  primary and secondary school mathematics lessons. What primary and secondary mathematics education needs is a critical path analysis of its means and multiple ends to identify and develop a lean curriculum in which digressions are immediately useful and in which simplification  do no undermine the ends. Course design and delivery may continue as calculus preparation while explicitly supporting other ends for students (the grand majority?) in high school who do not take calculus. Otherwise course design will serve the needs of the few instead of the many. See spirals to come.

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.

Rule and Pattern-Based Knowledge. The site coverage of pattern based reason points learners in school and out to the role of rules and patterns in providing skills and comprehension in all arts,  disciplines and trades, all in contradiction with dominant UK, US and Canadian theories of instruction of knowledge. 

Mathematics Education: The site introduction of inductive principles for instruction and identification of nuances, subtleties and past shortcomings in course design and delivery provide practical methods and standards for instruction, present or future, all in an empirical, content oriented contradiction with present-day theories of instruction. If leading mathematicians had a view and a say regarding elementary and high school instruction, the past and present in education would be different. But dominant US, Canadian and (?) UK theories of education that govern math and science education education are likely inconsistent with university professor viewpoints of their disciplines.

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.  

Spirals to Come: 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.  Yet 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.  

Remark: Mathematics course design and delivery is a part of applied mathematics, not pure, in which old traditions need to be reviewed and refined, with identification and removal of subtle deficiencies. Time and a few iterations will be required to strike a balance between leanness, preparation for calculus, inclusion of motivating uses or applications. Should mathematics be based on (i) logic and formally stated patterns (axioms)  or on (ii)  that appear to provide repeatable and reproducible, so observable and verifiable, results. Option (ii) with hints of (i) may be best for the most accessible form of a mathematics curriculum, while inclusion of all logic, formal or informal, that explains why the patterns hold would provide the most complete or comprehensive form.  Each instance of a curriculum or its delivery might vary between the inclusive and comprehensive forms.  

Inductive Principles: Swimming may be a natural talent. But wading in from the shallow end of a pool and taking longer and longer glides in water, to practice swimming strokes, exposes that natural talent with more success and less fear than jumping in the deep-end alone or with a push. Site material provide college and senior high school students a chance to restart their mathematics education in ways that avoid fears and difficulties and enrich knowledge.   A concrete,  operational and practical command of mathematics and logic demands drill and practice, not too much, not to little, in using and combining methods, rules and patterns from arithmetic to calculus, in ways that can be seen and verified or corrected.  

Tutors and Teachers:  Require exact arithmetic with whole numbers and fractions. Show students how to format their work in the evaluation of arithmetic & algebraic expressions. Show how to solve linear equations. Introduce three skills for algebra. A 4-th skill, namely, the forward & backward use of formulas, equations and proportionality relations in arithmetic and algebraic style, is a unifying theme for high school and college mathematics.  Learn and teach proper format for work. Math is art form in which  arithmetic and algebraic steps should lead to repeatable,  reproducible & verifiable results. 

In critical path course design:  There should be a pre-test to see whether or where learning or teaching is needed, and a post-test to check for mastery and whether or not further learning or teaching is required.  In critical path course design,  there should be well-described, well-documented  methods logically designed and likely to work  with parachutes (alternative methods), given the mastery of earlier required elements in the  path.  That being said, critical path analysis also needs to take into account the abilities of students and the age at which those abilities may be expected.  The foregoing sets the stage for a mathematics education handbook.