Mathematics and Logic - Skill and Concept Development

Three Skills For Algebra begins with logic to test and sharpen reading and writing, continues with a high school level arithmetic review exercises to find gaps in solution writing skills, and continues again with algebra review chapters to strengthen algebraic abilities. The chapters and exercises were written to fill gaps in their senior high schoool command of mathematics

The leading elements of Why Slopes and More Math gives a light technical context for the senior high school study of slopes, of factored polynomials and of end-point and interior maxima and minima of functions through sign analysis of formulas for slopes and derivatives of functions, formulas given in factored form. Thus leading elements of online book Why Slopes and More Math begins with the algebraically easy part of a first course in calculus, the part that comes after algebraically harder decussion of limits, continuity and convergence.

The leading elements of both books together show how different starting points make calculus and senior highs chool mathematics easier.

Top Level Links

Which Way to Go, Why and How

"Would you tell me, please, which way I ought to go from here?"
"That depends a good deal on where you want to get to," said the Cat.
"I don't much care where--" said Alice.
"Then it doesn't matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an explanation.
"Oh, you're sure to do that," said the Cat, "if you only walk long enough."
(Alice's Adventures in Wonderland, Chapter 6) --

Ends and Values for Learning

The question why learn or study a subject or a topic appears is a sign of intelligence. Some students have parents who say mathematics mastery is important. But many have parents who in recalling their experience express a dislike for mathematics after primary school. But if we combine ends and values from earlier times, we may arrive at overlapping sets of ends and values for learning and teaching primary and high school mathematics. The first ends reflect the actual or potential needs of adult or daily life, and in trades and activities that do not require common studies. The third end reflect the needs of calculus-based college programs and of advanced, senior high school science courses. The first two ends are more immediate than the third end. For the first two ends, if not the third, over-preparation is better than under-preparation to students and their families earn their livelihoods and to rationally defend their interests in a world where daily behaviour, and contracts involving money matters or income have huge consequences for individuals and their families.

• Early mathematics skill development in primary school and junior high school may provide common arithmetic and geometric needs in daily and adult life. That may include say the common needs of precollege trades and professions. Preparation for daily or adult life at home, at work and in travel requires us to count, measure and calculate with money, time, length, area, volume, speed and rates of change on paper or with the geometric help of maps, plans and diagrams carefully drawn to scale. Arithmetic mastery may include formula evaluation. Early skill development should make us want to avoid the domino effects of errors. That has value for all multistep methods in- and out-side of mathematics. Early skill development, well done, may make mastery of routine skills and concept common, while providing a partial base for college studies. Focus mostly on method and ideas with actual and then take-home value may lead students and their families to value and want mathematics and logic in early instruction.

  The scout motto "be prepared" for what may come applies. For better and worse, numerical and logical skills and concepts are needed in daily and adult life to understand others, to read and write instructions precisely, and to correct yourself or others. There is a great risk of making incorrect decisions if you do not fully understand the numerical and logic reasoning used in arguments and agreements between yourself and others. Mastery of logic and basic mathematics, the more the better, will help you quietly recognize faulty decision making, yours or that of others.

• In early or besides mastery ofmathematics, logic mastery leads to more or full precision in reading, writing, speaking and listening. More or full precision will ease or avoid confusion in following and giving instructions in many arts and disciplines at home, in school and in the workplace. Logic mastery sooner rather than later is best for its take-home value for studies and then wokk. But when may depend on each student. Before or beside logic mastery, early skill mathematics or quantitative skill development may emphasize how to do and record measurement and arithmetic steps precisely, so that the steps can be seen and verified, and so that students become aware of the need to avoid the falling domino effect of errors. In this falling domino effect, a mistake in one step leads the following steps to being in error, except in the lucky case where a second or further mistake cancels the effect of the earlier ones. For that, there should be no credit. Plug: The leading math-free chapters of online Volume Three Skills for Algebra on implication rules and their use in deductive reason may lead the not too young to logic mastery.

• Mid- and senior- high school mathematics and logic skill development may build on earlier development to serve the needs of senior high school science and technology courses, and the needs of calculus-based college programs in commerce, science, engineering and technology. Calculus in the first instance consists of calculation of slopes for linear and nonlinear curvers y =f(x). The key role of slopes in calculus explains why slopes and rates of change need to be introduced earlier studies. Hint, Hint Site volume 3 with its light calculus previews offer a context for the study of slopes, factored polynomials and function maxima and minima may amuse and inform students in courses leading to calculus and in the first weeks of calculus.

Students heading for calculus-based, college programs in business, if they avoid demanding high school science courses, will not see senior high school mathematics used before arriving in college. To compensate, long-term value needs to be emphasized. Emphasize how the calculations and logic of college level programs requiring calculus will be more difficult to follow or use and bend to our future requirements with a weak mastery of mathematics. Site volumes 2 and 3 in forming and reforming the views of students and teachers in senior high school mathematics as indicated above may inform and amuse, and in the process provide some context and motivation for the study of slopes, factored polynomials, function maxima and minima, and calculus too.

Oral Development - Words have been missing

All the ideas described briefly below are explained in more detail in site algebra starter lessons and in site Volume 2, Three Skills and Algebra. The arithmetic related ideas could have been placed with site arithmetic lessons instead.

Before and besides the role of letters and symbols in algebra, we may use words and numerical examples to talk about about and show how to calculate totals and products by adding and multipling subtotals and subproducts. We may also talk informally but precisely about counts and measures as being known or not, constant or not, forgotten or not, and variable or not. Many technical terms may be introduced and understood before and besides the letters and symbols. Moreover, to gossip or talk about people, places and activities, we need names, labels and phrases to identify them. In mathematics, names and descriptive phrases such as the compound growth formula, the rectangular area calculation, the distributive law and the Chinese Square Proof of the Pythagorean Theorem allow us to gossip and talk about calculations and further ideas in situations where symbols and diagrams cannot be formed nor read. Most formulas, methods and practices in mathematics and logic are named. For people wanting and able to talk about what they learning with others, learning the names becomes an asset and not a burden.

Using rules and formulas forwards and backward, and talking about it may end a further silence. Talking and writing about the forward and backward use of rules and formulas provides a unifying verbal theme for the study of logic, mathematics and science in school and college studies. Most if not all rules and formulas are not only used directly in a forward sense but also indirectly or backwards. Determing the constant in a proportionality relation uses the relationship, an equation, backwards. Once it is found, the proportionality relations may then be used or rewritten forwards and backwards to compute or express the value of one number or quantity in terms of others. The example here may not be familar to you if you have not seen them, but by talking about the forward and backward use of rules, formulas and proportionality relations, the backward use will be expected and not be another surprise for students weak and strong of mathematics, logic and science. This forwards and backwards use is common pattern previously met and mastered case by case in silence. Talking and writing about it introduces or extends the oral dimension of skill and concept development.

Visual Development of Algebra and Calculus

Site algebra starter lessons and the online chapters of Volume 2, Three Skills for Algebra, material, show how to learn and teach skills and concepts with words, forwards and backwards. Algebra starter lessons include a geometric, stick diagram introduction for solving linear equations in a way that visually proves or improves fraction skills and sense. Here fractional operations on stick diagrams are suppose to make the algebraic solution of linear equations easier to grasp. However, in entertaining a group of students during a one hour, substitute teaching assignment, one keen student could not make the transition from solving with stick diagrams to solving algebraically. It was not my place to give him extra instruction. He may have been better served by more stick diagram examples, or by a leap to the algebraic method. I cannot say.

Geometry too can help with the introduction of calculus and in providing motivation or context for the study of slopes (remember the domain name is whyslopes.com) and the study of factored polynomials alone and in ratios (rational expressions). See site Volume 3, Why Slopes and More Mathematics, online in full with a fall 1983 why slopes prequel. Volume 3 in a preview of calculus provide geometric motivation for the study of slopes and factored polynomial to the location of maxima and minima of functions.

The site introduction of complex numbers is geometric instead of algebraic. It follows or re-invents a path in a 1951 book on Secondary Mathematics (possibilities) by Howard Fuhr, a mathematician who masqueraded as a mathematics education professor at Columbia University and who as part of the NCTM leadership in the 1960s help develop and implement the college-oriented Modern Mathematics Programs for skill and concept development in primary and high school mathematics from counting to calculus. The level of rigour in this geometric introduction of complex numbers is not less than that in the geometric introduction of trigonometry using triangles and/or the unit circles drawn in a Cartesian plane.

More Concept and Skill Development Notes

Site composition was driven by a search to remedy the skill and concept development difficulties stemming from steps too big and steps without clear value for students and their families in earlier programs in mathematics and logic education - programs which aimed for student mastery of selected skills and concepts. In consequence, site lessons and lesson ideas include many expositional innovations to aid skill and concept development. Most, if not all, are mathematically correct, with a few small departures from earlier views to make instruction simpler.

In calculus and secondary mathematics, late primary mathematics too, there are many different starting points for instruction.

The choice of starting point need not reflect the conventions of higher mathematics, conventions that may be arbitrary despited being widely accepted. Instead, the choice of starting point may reflect the objective of making skill and concept development simpler for students and their teachers. The harder starting points may be left to advanced studies involving fewer students and teachers.

• Mathematics Literacy: Since students may leave school early, we need to show them and give them mastery of reading, writing, arithmetic and geometry with actual or potential take-home value for their daily and adult life in local and distant communities. While learning mathematics with comprehension is best, the take-home value of basic and routine skills needed for daily and adult to important to insist upon mastery with comprehension. In this course design and delivery should emphasize the domino effect of errors in multistep methods, numerical or geometric. And in arithmetic, students should be shown how to do and record steps in a manner that their skills can be seen and checked as done or later. In practical, skill-based arts and disciplines from cooking to mathematics, skills needs to be demonstrated to be believed, and indeed to be both taught and mastered. In general, there are too many skills for a student to find them or their refined form by discovery. The challenge for early mathematics instruction is to identify and provide observable and thus verifiable skills with take-home value that serves common or routine needs while seamlessly preparing students for late instruction.

• Geometry with Proportionality First: To quickly support the common, actual or potential, geometric use of maps, plans and diagrams drawn to scale in daily and adult life, and in precollege trades and professions, the site webvideo exposition of geometry may include SAS, ASA and SSS methods or practices for the construction of similar or proportional triangles, and in general assume that in maps, plans and diagrams drawn to different scales that corresponding angles are equal and corresponding lengths are proportional. So the similarity or proportionality present in maps, plans and diagrams drawn to scale may be exploited to indirectly measure angles and lengths, and quantities computed from them. Trigonometry may then be introduced as a way to calculate angles and lengths instead of obtaining them direct from measurements, actual or of the corresponding angles and lengths on maps, plans and diagrams drawn to scale. The early mastery of common, and easily understood and repreated practices with maps, plans and diagrams drawn to scale provides a context for and even implies the assumptions and axioms of Euclidean Geometry.

• Geometry with Congruence or Isometery Second: For simpler or more accessible account of Euclidean geometry, the site account does not include a proof of the Pythagorean thereom. The Chinese Square Dissection proof provides a more accessible alternative. The latter is presented online in Volume 2, Three Skills for Algebra. Without the Pythagorean thereom, Euclidean geometry may be easy enough to return to the North American classroom in a way that shows high school or college students how logic in the form of implication rules alone and in direct deductive chains of reason appears in mathematics.

• Counting and Arithmetic with Decimals: Decimal place value is the key to counting. We assume every set of fewer than 10, 100, 1000 and 10000, etc, can be divided into a group of upto 9 units, a group upto 9 groups of ten, upto 9 groups of 100 and upto 9 groups of 1000 in manner that the count between 0 and 9 of units, 10s, 100s and 1000s are unique, albeit the division of set elements into groups of units, 10s, 100s and 1000s is not unique. The foregoing division or partition gives a unique, multidigit decimal way to write and record the count or number of set elements in which each unit has a place value. The concept of place value leads to and easily justifies arithmetic counting shortcuts involving the addition, comparision, subtraction and multiplication and even division of decimals. The details are given in the site arithmetics section along with North American, metric (or SI) and UK-German methods for writing and reading aloud with words multidigit decimals without and then with a decimal point. Comprehension of operations with decimals enriches early instruction and may help some master these operations. Others, most others perhaps, may find full explanation of why some operations work too complicated for their liking. For them skill and confidence in decimal methods may follow learning how to use the methods to obtain repeatable and reproducible results via steps observable and, if need-be correctable.

• Counting and Arithmetic with Fractions: The fraction three quarters when written or read aloud means three times a quarter. A quarter œ is a unit fraction. Proper and improper fractions with the same denominator all give a number or count of a unit fraction, that associated with the same denominator. With the aid of decimal representations forms of numerators, it is an easy matter to count, add, compare, subtract and even divide multiples of a single unit fraction. It also an easy matter to multiply a multiple of a single fraction by a whole number - to form a multiple of a multiple. By long division and regrouping, each improper fractions is equivalent to a mixed numbers. In primary and secondary school, students may be shown how to add and subtract fractions with unlike denominators by raising terms to convert each fraction to another equivalent fraction, so after conversion, each has a common denominator and so is a multiple of a common unit fraction. Following this, students may be shown how to compare, multiple and divide fractions by rote. Site fraction lessons in contrast show how raising terms to obtain like denominators explains and justifies methods to compare and divide fractions while raising terms to ensure the numerator of the multiplicand is a multiple of the denominator of the multiplier explains and justify methods for fraction multiplication. The justification of arithmetic with fractions sets the stage for the justification of arithmetic with decimal fractions (multiples of one-tenth, one hundredth, one thousandths) that usually denoted by multidigit decimals with digits after and even before a decimal point.

  Comprehension of operations with fractions agains enriches early instruction and may help some master these operations. Others, most others perhaps, may find full explanation of why some operations work too complicated for their liking. For them skill and confidence in decimal methods may follow learning how to use the methods to obtain repeatable and reproducible results via steps observable and, if need-be correctable.

• Prime Numbers and Fractions: For algebra alone or as part of calculus, and for operations with complex numbers, students need an efficient command of arithmetic with fractions where the denominators are say less than 200. Prime factorization of whole numbers less than 200 is useful here. The development of prime number factorization methods in the site arithmetic section shows how to use time tables to recognize small primes, and how to use an olde square rule method to quickly and efficiently obtain prime number factorization of whole number less than 289 = 17², and to recognize primes less than 289 as well. The foregoing path as demostrated in site arithmetic section may be easier for people to learn and teach. Prime factorization is also useful for a "simplification" of roots involving whole numbers or their fractions, a simplification often seen in trig and calculus. Mastery of exact arithmetic in high mathematics requires mastery of some cosmetic standards or conventions for the expression of fractions, roots and radicals.

• Arithmetic with units and denominate Numbers - missing. Units of measure and counting appear directly in daily and adult life, and also in science and technology. Units of measure also appear in the description of speed, acceleration and other first and second order rates - rates that may be described as derivatives in calculus. Modern mathematics programs did not mention nor sanction the use of units and their multiples (denominate numbers) in high school and college studies, albeit this use appear in science courses and in some practical examples met in mathematics courses in trigonometry and before. The site account of arithmetic and fractions with units compensates for this. Albeit, the compensation is given in a do this, do that manner, because of a lack of words on my part to provide greater comprehension. Readers are invited to provide remedies. Early algebra courses today may introduce monomials (products of letters or "variables" to various powers) and operations on them alone and in fractions before students understand the computational significance of monomials and operations on them. Site algebra starter lessons explanation of equivalent computation rules may provide a remedy for that. But before or besides algebra, The same exercises with monomials given by numerical multiple of products of units to various powers may be more meaningful to students, while be a prerequisite to the numerical description of rates and proportionality constants.

• Algebra Starter Lessons. Showing students how to do and record numerical and algebraic steps in ways that can be seen and checked when done or later makes their mastery of multistep methods observable, and hence verifiable or correctable. Showing should also make students aware of the domino effects of mistakes, and the care needed to avoid or correct such errors. The introduction and assumption of methods to compute totals and products using subtotals and subproducts employs practices that are too complicated in high school instruction to derive from the usual axioms for arithmetic with real numbers. But the assumption of these methods or practices extends the usual axioms and from the perspective of advance mathematics gives a very redundant set of axioms. But the same redundancy is justified as it makes early instruction easier and more effective, and the extra assumptions have immediate take-home value for daily and adult life not present in the usual axioms. Now the usual axioms are best understood besides or after a math-free mastery of logic. The usual axioms for the distributive, commutative and associative law are algebraically described. Many students find the algebraic description too remote or abstract. But if we introduce the concept that each algebraic expression give a unique computation rule, one that that be evaluated on paper or with the aid of a program on a calculator or computer, we may observe from numerical examples that different computation rules appear to be equivalent in the sense that they give the same result. This small step of introducing the concept of equivalent computation rules provides another context, a different starting point, for understanding and explaining distributive, commutative and associative laws in arithmetic with many kinds of numbers, and eventually with numbers being replaced by computation rules - those with numerical values.

• Arithmetic without Calculators: To be over-prepared is better and less risky than being under-prepared. A written, calculator-free mastery of arithmetic with signs, decimals, fractions; with units of measures; and with number theory practices is needed for a full, traditional, mastery of algebra, trigonometry, complex numbers and calculus. A full mastery of arithmetic with units of counting and measures also has value for adult and daily life, and for further studies in commerce, science, engineering and even mathematics itself.

  In modern urban life we depend on machines to simplify our daily life. But calculators usage both simplifies and weakens mathematics mastery, or that needed to understand decimals, fractions, algebra, trigonometry and calculus. As a master of my subject with standards for skill and concept development, I see the student who can only do arithmetic with the aid of a calculator as being handicapped from being too spoilt in earlier instruction. Any expectation that quantitative skills and disciplines can be well-taught without a written mastery of arithmetic with decimals and fractions is false.

  Again, manually learning how to do and record work in steps that can be seen and corrected as done or later may begin with evaluation of arithmetic expressions and algebraic formulas. While calculators are useful, failure to require manual student mastery of arithmetic removes a starting point for observable skill and concept development. In particular, mastery of observable steps that can be seen and confirmed or corrected as done or later is also is key part of showing and demonstrating abilities in problem solving, in writing proofs and employing multistep methods at home, at work and in studies in many arts and disciplines.

Addressing Students and Family Alienation -

More on Ends and Values

Mathematics after primary school has been difficult and without immediate value for one generation after another. While some students have parents who did well or who encourage skill and concept development, other students have parents who may say mathematics after arithmetic is a waste of time. High school and college students may attend courses because those courses are required. In high school and college, students who base their efforts only on whether or not their teachers are pleasing have a shallow context and motivation for learning. Students for whom doing well in tests and finals is the only motivation also have shallow reason for learning. Cultural values for learning may appeal to some. But practical ends and values may appeal to more.

In primary school, students and their families may see the 4Rs (reading, writing, reasoning and arithmetic) as being useful in adult and daily life. There-in lies content and motivation. But at the junior- and mid-high school level, some mathematics and logic lessons are of actual or potential service to daily and adult life for decision-making and money-matters at home and the work place. Other lessons only have long-term value for college programs that some students may never enter or complete. Instruction may lean to the first kind of lessons initially to provide ends and values easily understood and appreciated by students and their families. Emphasizing the more useful methods and concepts first may help retain student motivation and also help those who have leave school early. But eventually, high school and college mathematics has less and less take-home value besides more and more value for future studies or courses that students may not see. Here again, instruction may focus on the take-home value, when present to provide motivation.

At the precalculus level, instruction should focus on two kinds of skills and concepts, those that have actual or potential take-home value for daily & adult life, and for precollege trades and activities; and those that prepare students for a light and then deeper command of calculus. In the former, I would include a set-based development of probability theory. In both streams, I might include matrix operations but not linear programming. The latter can be left to college programs in commerce, science, engineering and technology. I would restrict high school mathematics to computations and proofs that are lead to repeatable and reproducible results, and to the computation of averages useful in small business for estimating demand for products and services being sold. Further elements of descriptive statistics, I would leave to college studies, or to high school courses on critical thinking.

The recommended focus may mean fewer topics are taught. For students not heading for calculus-based studies, less with a focus of skills and concepts with take-home value may be best. In the preparation of students for calculus and senior high school mathematics, multiple topics with no short-term value may be met. That short-term value will vary between students. Students in courses required to prepare for calculus who do take mathematically demanding, senior high school courses will see more short-term value. In general, calculus and preparation for calculus is a long demanding path which many find difficult or hard to complete. But, here is a plug, site Volumes 2 and 3, make the path easier and throught calculus preview make calculus and precalculus easier and more appealing.

Course Design Balancing Act

Still More on Ends and Values

To serve the skill and concept needs of the common person in the street, we need to put first those skills and concepts with actual or potential value for daily and adult life. Then students may attend school and go home with methods that help themselves or their families in money and other matters. Near the end of school coverage of arithmetic, geometric and logic (or reading and writing) skills and concepts with actual or potential service for daily and adult life, more algebra and higher level geometry skills may be introduced to revisit and reinforce the foregoing service while being of service to more trades and activities at the precalculus level, and also being of service or preparation for senior high school science courses and perhaps later studies in calculus. The multiple ends and values in the foregoing need to be balanced. The balance may depend on the local or immediate needs of students and their families, that is, how long students are likely to remain in school; on whether or not, they are likely to see all all ends and values served; and on whether or not, the students are quick or slow learners.

The concept and skill development standards and principles for instruction in results-oriented arts and disciplines, as espoused in site material, seek to provide students with an observable and verifiable know-how of the ideas and methods currently forming and characterizing each art or discipline. The latter presents a moving targets as best practices in each may vary over time and place. But in a moving target, concept and skill mastery may be seen or empirically measured by student response to questions. In each such art and discipline, students are expected to retain know-how and build on it in a progressive manner, with regression being a sign of weakness, or absence too long from practice in an art or discipline. Each art or discipline comes with different cultural and practical values, some more important than others in ways that may justify its instruction or not in each school or school system.

Morover, course design and delivery needs to acknowledge that there are multiple intelligences in learning and teaching styles. A style that is suitable for instruction in the humanities where conclusions are highly subjective is not suitable for instruction in mathematics and science where the benefits, origins and limitations of ideas and methods should be shared.

A Stronger Base for Modern Mathematics

Site lessons and lessons ideas from counting to calculus provide a foundation for college level studies of modern mathematics. Site lessons and lesson ideas offer student and their teachers a mastery of concepts and skills with comprehension, based on a redundant set of practices and axioms, whose redundancy can be explained and removed in college course in or leading to modern mathematics. The ends and value further offer reasons for mathematics and logic mastery that students and their families are more likely to appreciate before preparation for calculus becomes the main focus of instruction at the senior high school level. For calculus, Chapter 14 of site Volume 3, Why Slopes and More Mathematics, offers a decimal, error control development of limit and continuity concepts that may stand alone, or be used to make the epsilon-delta development much easier to understand and explain. Site departures in early instruction from modern mathematics are intended to provide TCPITS an more accessible view, but they are also intended to develop the logical and algebraic maturity needed for college and senior high school students to study modern mathematics if they choose or where it appears in their programs of study.

Indirect Instruction Benefits and Limits

Indirect instruction in mathematics has the advantage of enriching skills and concept mastery in classes where there is time for individual and group creativity, and where teachers not all trained in mathematics are shown how to provide problems and circumstances which studentsmay investigate to discover or build their own ccmprehension. But with or where teachers are not fully versed in mathematics or course content, it appears far simpler to provide instructors with lessons easily understood and repeated in class, which avoid the affects o steps too large, not for all, but for most to follow. For example, the verbal introduction of algebra in Volume 2, Three Skills For Algebra, and the first six or seven chapters of Volume 3, Why Slopes and More Mathematics, provide lessons and lesson idess, easily understood and repeated in class, and in the process may introduce learning difficulties of students with instructors at many levels. Here advocates of indirect instruction while declaring student mastery of given skills and concepts in direct instruction to be a substandard objective for mathematics education essentially kept the course design and content which direct instruction employed in its identification of skills and concepts for student mastery. Moreover, educational authorities at the precollege level retain final examinations for the yearly end of most high school mathematics courses. Final examination by their very nature test student mastery of chosen skill and concepts. Due to the continued presence of mathematics final examinations which tests student mastery of given skills and concept, fairness in student evaluation requires all the given skills and concepts be clearly explain, illustrated and checked first in class. So direct instruction is still required. For fairness, student mastery of given skills and concepts requires both teacher and student awareness of how later skills and concepts stand on earlier ones. Otherwise, students will be promoted with the necessary background to succeed.

Each program of instruction aim at mastery of given ideas and methods has varying degrees of success and failure, and of motivation and alienation for students and their families. In the case of modern mathematics programs for secondary mathematics and calculus, the step by step development was clear to some and due to the presence of steps too big, not for but for some, confusing for others. Site material in providing smaller steps allows steps too big to be recognized and gives remedies - full or not - to be tried and tested. Smaller steps should allow more to go further.

Page Contents: Which Way to Go, How and Why || Practical Ends and Values - the why || Oral Development - words have been missing || Visual Introductions of Algebra and Calculus || More Concept and Skill Development Notes || Addressing Students and Family Alienation || Course Design Balancing Act || A Stronger Base for Modern Mathematics ||

Appendix - Closing Words

Still More on Ends and Values

In Canada and the USA five decades ago, that is, in the 1960s, modern mathematics programs set forth the substance or content of high school mathematics and calculus in a way that served the technical needs of college bound students, those heading for university programs in commerce, science and technology. Mathematical discussion leading to those modern mathematics programs acknowledged the college-orientation and mentioned that the needs of student not college bound were not being addressed. However, for the sake of inclusivity, modern mathematics programs were adopted for all instead of the just college bound. The tradition of mathematics course design incomprehensible to parents - different from what they had seen in school or college may have begun then. From my perspective as a student and then as an instructor, the modern and pre-modern mathematics course design in algebra at least included steps too large not for all, but for most to follow. In primary and secondary schools, the college-oriented selection of content in modern mathematics programs helped move course design and delivery away from serving the actual or potential needs of daily and adult life, and in doing so offered less context and motivation for mastery of given ideas and methods.

For senior high school mathematics, preparation for college programs of the technical or mathematical kind provides one context and motivation for college-bound students. Senior high school mathematics is of clear service to students headed for science or engineering take senior high school courses in physics, chemistry and biology. But students headed for college commerce programs - those best studied with the help of calculus - may or should be encourage to take senior high school courses in science. Otherwise, physics-bsaed examples in senior high school mathematics will have less value for them.

To provide context and motivation for more students and their families, primary school and the leading years of high school may deliberately develop mathematical and logical skills, concepts and work habits with clear or identifiable actual or potential benefit in daily or adult life. In particular, Primary school and the leading years of secondary school may focus on basic skills in counting, figuring and measuring of actual or potential service to daily and adult life. Students and their families may value this focus. The leading years of secondary school may further introduce and emphasize skill, concepts and work habits of service in senior high school and college mathematics. In this standards for mastery of given skills and concepts may be maintained and supported by lesson and lesson ideas deliberately chosen and continually reviewed to make the hard less so, without loss of substance; and guided by the setting of final examinations set not by amateurs but by subject experts. Here multilevel course design and delivery requires knowledge of and respect of how later ideas and methods depend on earlier ones. In this, while high school mathmatics as a whole may prepare student for calculus-based college studies, the inclusion of skills and concepts of service, actual or potential, to daily and adult life; money handling included, or to pre-college trades and professions could provide and sweeten the context of secondary mathematics for all. And in the preparation for calculus, set formalism may appear to set the stage for the later study perhaps of pure mathematics in college, but in a minimal manner, one that does not overwhelm.

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