Mathematics Skill Development Framework

Quick Overview

The main objective of the the first three phases is to provide skills and practices of actual or potential use for adult and daily life in street, including trades and professions mastered outside of colleges. Each year of instruction in the first three phases may optimize the actual or apparent take-home value of skill and concept mastery. Skill development may be motivated further by providing games and activities which make skill mastery fun. The second objective in the first three phases and the explicit objective in the fourth is to prepare for calculus and calculus-based college programs.

Primary Level: The first phase essentially corresponds to existing practices in the elementary or primary school level provision of basic skills in counting, measuring and figuring, most of clear value for adult and daily life. Students may easily take or accept early skill mastery as part of growing up - preparing for adulthood.

Some small refinements to speed and enrich matters are implied by site lessons on decimal subtraction, prime recognition and factorization, andon justifying not only addition and subtraction of fractions by raising terms, but also comparison, multiplication and division. The site algebraic account of how raising terms leads from simple cases to the general case will sooner or later be translated into words and numerical examples which students can follow without a mastery of algebra.

Late Primary or Early Secondary Level: This phase begins with prealgebraic skills and practices with actual or potential value for adult and daily life clear to students, teachers and parents in a practice first manner. For take-home value an operational command of skills and practices - how to do to or apply, and when - is more important than any explanation or theory. Albeit, the latter may and should be given where it helps a student alone or in a group to obtain that operational mastery in a repeatable, reproducible and observable manner. In this skill mastery is seen. Writing is an iterative affair. There is one site to do: Provide a clear description of the counting, measuring and figuring skills and practices with numbers, maps-plans-diagrams drawn to scale, and formula evaluation, that students should master by the end of this first phase. The description will provide a checklist for skill development.

The second phase will continue with more practical skills and work habits of actual or potential value for adult and daily life. Mathematical methods will be limited to the counting, measuring and figuring that can be done with numbers; with drawing and measuring instruments; with maps, plans and diagrams drawn to scale; and with algebra limited to formula evaluation. In the evaluation of arithmetic expressions and formulas, the domino effect of errors will be emphasized, so that students may value the care and diligence needed to avoid such errors not only in mathematics but also in daily life.

Euclidean logic favoured in mathematics may be introduced in this phase or the next in a mathematics free manner and perhaps in a reading and writing course, to develop greater precision in reading and writing needed to ease or avoid learning difficulties. Since the general algebraic way of writing and reasoning with letters and symbols has been a great source of difficulty and mathematics anxiety in the past, this phase employs algebra only in form of formula evaluation.

The aim here is to provide and leave the greatest possible operational command of skills and methods of value for adult or adult life. For example in money matters, computations with compound growth or interest and computations with geometric sums forward and backwards may serve daily or adult life in understanding and calculating loan, mortgage and pension payments. Formulas may be introduced and confirmed numerically, and justified in the next phass. In modern times, students may enter the adult world responsibilities between 16 and 24 years of age. All depends on how long they study. In consequence, the material here is given 3 to 12 years before the adult lives of learners begin. I suspect more good is done in giving the skills early than delaying because we do not know when studies will end. The phrase better late than never is modified here to say better early than never. We do not when the schooling of a child or teenager will end.

Mid-Secondary Level: The third phase is based on the premise that smaller and extra steps in site material for talking about numbers and quantities, for talking about calculations and for introducing algebra will ease or avoid algebra troubles and anxiety associated with the shorthand roles of letters and further symbols in and beyond algebra. In particular, site algebra chapters and steps show how to systematically introduce algebraic skills and concept. With that, the actual or potential practical use of algebra forwards and backwards in adult or daily life can be emphasized. Besides that, an elementary account of Euclidean geometry based on the direct use of logic provides a model for reason with implication rules alone and in sequence, apart from the conflicts of daily life

At the high-end of this level, or the initial end of the next, the algebraic and set theory development of probability along with recursive or inductive development of formulas may be presented as an island and body of knowledge with some take home value, some intellectual value, and value for building the algebraic abilities required in the next phase.

This phase continues by introducing algebra-based skills and practices needed to prepare students for calculus-based studies. But those skills and practices, with actual or potential value for adult or daily life easily recognized and illustrated are put first. Thus two ends for mathematics education are served together, with at least one, the take-home value, providing context if not motivation. Site strength in building and clarifying algebra skills and concepts provides a base for this. The olde problem of the shorthand roles of letters and symbols being a partial or full mystery to students and some of their teachers is directly and effectively eased or banished. In addition, logic skills and Euclidean Geometry are lightly introduced, the first for their take-home value for easing or avoiding difficulties at home, work and school, and the second to introduce the general mathematical role of logic in a small, compact island of mathematical thought.

Senior Secondary Level - Mostly for Calculus: The fourth phase consists of all algebraic and geometric skills and concept of technical value for calculus or beyond in mathematics and for calculus-based college programs whose take-home and immediate practical value is nought or too little to be worth mentioning earlier. This topic will include polynomials, rational functions and functions in general. That being said, critical path analysis may imply a treatment of function lighter than that given in site material.

Calculus in the first instance is the subject of slope related computations. Done forward and backwards, those slope related computation may explain formulas given earlier for areas and volumes of geometric objects. Calculus further providesa language for the description and elaboration of ideas and methods in the money matters and in the science, technology, engineering and mathematical arts and disciplines of modern life. Like another language, it may be used for fiction and non-fiction. But language mastery is a pre-requisite to understanding which is which, and shades of gray inbetween.

For context or motivation in this phase, the site fall 1983, why slopes calculus appetizer and chapters 2 to 6 located in the online version of Volume 3, Why Slopes and More Mathematics, will help. For students heading for commerce or business subjects, most precalculus and even calculus will lack take-home value and any cross-curricula value. But students in senior high school biology, chemistry and physics may see the cross-curricular use of probability skills and methods; the use of arithmetic computation with units and denominate numbers; the use of proportionality relations and other formulas forwards and backwards; numerically and algebraically; the use of quadratics in describing falling objects close to the earth; the forward and backward use of logarithms and exponentials in describing exponential or compound growth and decay; and the use of conics in describing parabolic dishes and in describing orbital motions around a planet or star. So the cross-curricula value of precalculus topics depend on the studies and academic destination of students. All the foregoing examples may complement the most obvious reason for most topics at this level, preparation for calculus.

In precalculus, the full technical coverage of functions, given in site advanced algebra steps for the sake of completeness, might be delayed and presented in a just in time and reduced manner, as needed during calculus instead of fully and ahead of calculus. What logical option works best may be determined by experience. So the framework here may be implemented in several different ways, some more lean than others. In that site chapters and steps include many lessons and ideas to make skill building more direct, simpler and less demanding in the zone of natural talent.

Senior Secondary or College Level - Calculus: The fifth phase covers calculus lightly to deeply in the final years of secondary school or the first year of college. Algebraic ways of writing and reasoning are employed at full strength in calculus. Site calculus previews and chapters show how to ease or avoid algebra schock. Volumes 2 and 3 show how to check, consolidate and expand arithmetic and algebraic skills in ways that make the hard easier. The site coverage of differentiation methods may also help.

Students in this phase or the previous may also be given a light course on calculus which only shows how to differentiate and integrate polynomial and rational functions. Doing that concretely, would help the study of constant accelleration motion in physics and justify formulas for areas and volumes mastered earlier without explanations.Doing that concretely, see the site version to come, may aid algebraic skill development more effectively and concretely than starting with a comprehensive study of a zoo of functions y = f(x) and how the introduction of parameters y =a f(b(x-c))+ d shifts and dilates their graphs. Which way works best may be established empirically alone or with the aid of some critical path analysis.

The exposition of calculus includes many topics to prepare for further skill and practices in mathematics. Calculus may learnt and taught not only to prepare students for more courses, but also to consolidate earlier skills and concepts. The exposition of calculus may also look backwards and complete earlier comprehension. A decimal error control perspectives of limits and continuity may set the stage for a decimal free epsilonic view, while also providing a framework to explain the role of finite decimals in approximating computation with real numbers. The earlier approximation of areas by covering with finer and finer grids of smaller and smaller squares may be connected to integration. That is, formulas for areas and volume given earlier may be explicitly justified.

End Notes

Different communities and school systems with their different life styles may have differing views of skill and practices are locally the most important. That may imply variation in three or more phases. However, the five phase approach may be robust to meet the needs of communities with varying degrees and length of exposure to the money matters and technology of Charlie Chaplin's modern times. Where children or teenagers are expected to leave school, Each year of early instruction could maximize the local take-home value of skill and concept for the sake of students and their families.

Some school system may include statistics in a greater or less depth. The statistics I have seen in high school seems to be a distraction from the development of skills with take-home value and the thought-based development of skills and concepts needed for calculus. Not talking about statistic above makes the five phaser easier and quicker to describe and implement.

Secondary mathematics and calculus do not need to stand on a minimal set of axioms - the choice of which being somewhat arbitrary, but on a consistent set of axioms that makes the operational command of mathematics and logic easier to obtain in a repeatable, reproducible manner which can be seen and verified, either empirically in an it works basis, or deductively. In the five phases above, we depart from the modern mathematics curricula or extend them by covering and empirically sanctioning skills and practices in common use at home, at work and other subjects.