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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

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Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

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Welcome. The leading elements of Online books Three Skills For Algebra, and Why Slopes and More Math. stem from early college lessons that amused and informed recent high school students, science, engineering and education students, and adults in remedial to advanced mathematics lessons.

    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.

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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) --

Site lesson and lesson ideas in implying step by step a stronger and more effective path for direct instruction may mute or lessen the claims in education physchology or constructivism 1990 onward that direct instruction does not provide an effective way to teach mathematics - the fears and troubles of too many or most being evidence of that.

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.

For mathematics and logic instruction, preparing children and teenagers to earn income as adults may meet employers needs but more importantly, it may meet the needs of students and their families for income from employment or self-employment, and defending their own interests while changing jobs or being fired. High school, trade school, undergraduate university programs and graduate university programs may open doors for worthwhile employment, but depending on economic times, education too long may also distract from gainful or worthwhile work. There are no simple answers. Where does education for the sake of students begin and where does it end? It may begin by showing students early how to handle money matters in daily and adult life from not going into debt while buying or selling to evaluating the immediate or long-term value of a mortgage, a pension plan or the income stream and benefits of a job with or without benefits may help them face or avoid common situations and difficulties.

  • 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.

Arithmetic and algebraic expressions are often to complicated to read aloud, term term by term. Diagrams too are better seen than "read aloud". Outside of mathematics, a picture is worth a thousand words. In mathematics, a symbol, an expression or a diagram better seen and grasped in silence may also be worth a hundred to a thousand words. There has been a great silence in arithmetic, algebra, geometry and calculus because mathematical ideas and methods are often better written and drawn in silence instead of being expressed and explained aloud. Yet we may deliberately use more words to introduce skills and concepts clearer, to talking unifying themes, and to improve communication in circumstances where writing or drawing is not possible. While demonstration how appears in site material, we will identify where the greater use of words is possible. There is more to mathematics than be given a formula, and numbers to use in it. But remember, pictures and diagrams too can be employed alone and besides words to make skills and concepts easier to learn and teach.

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.

In describing how to calculate averages and how to compute the perimeter of a polygon, word descriptions of how may be simpler or not to understand and explain than formulas. As a first example, the average of a set or sequence of numbers is given by their total divided divided by the number (count) of set or sequence elements. As a second example, the perimeter of a polygon is given by the sum of the lengths of it sides, or more briefly by the instruction: add the sides. As a third example, the total area of a region consisting of non-overlapping subregions is given by a sum of subareas. In early mathematics instruction, how to compute this or that may be easier to understand and explain with words with the use of letters or symbols being more complicated. But for the compound interest or growth formula, for the quadratic formulas and later for the chain rule - do not worry what computations these phrases name or identify, the the letters and symbols in them are worth a thousand words. The greater use of words advocated for earlier instruction here is not possible in later instruction. So the silence will return.

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.

The big steps in modern mathematics programs were too hard for many to follow. Site material offer smaller steps to compensate. Before modern mathematics programs, instruction had a greater focus on skills and concepts with value for daily and adult life - work, mortgages and investments included. The discussion of ends and values above suggests preparation for daily and adult life as much as possible first and foremost, and on preparation for college second while emphasing anything in the latter that could have take-home value.

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.

For example, the site development of prime numbers begins with a definition that is not the most general but with a definition that is likely the easiest for students to understand and apply. For a second example, the site essay on what is a variable, by talking or writing about numbers and quantities varying in one sense or another, we provide a prequel to the later, more formal and more algebraically advanced view of what is a variable, a prequel that is easily understood because it is wordy and pre-algebraic. For more examples, see the site geometric development of complex numbers before trig, and see light calculus preview in Volume 3, Why Slopes and More Math, and see, still in Volume 3, the decimal prequel to the epsilon-delta view or definitions of limits and continuity.

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 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

In modern mathematics programs for secondary mathematics education, direct instruction aimed at student mastery of given concepts and skills has been uncertain and unreliable due to steps too big or hard for most to follow, and due a college-oriented choice of concepts and skills with value too long-term for students and their families. Those steps too big undermined course design and delivery. However, direct instruction can address its own problems by serving short- and long-term ends and values in the selection and arrangement of course topics, and in offering smaller, more accessible and reliable steps for concept and skill mastery. The key question is whether or not remedies based on the smaller and alternative steps in site lessons and lesson ideas, alone or with the proposed ends and values above, will be effective..

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.

When and where direct instruction has clear steps or lessons to provide student mastery of important skills and concepts, teachers and course designers may provide circumstances and pose questions to indirectly lead student to formulate ideas and skills and gain the experience on which direct instruction may stand. But where direct instruction lacks those clear steps and lessons, it is doubtful that indirect instruction will provide a practical and clearer path to to student mastery of the given skills and concepts. The ability to explain matters directly is likely a prerequisites to the ability to provide skill and concept mastery indirectly.

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.

www.whyslopes.com - Home Page

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Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.

2000 - Waterboro Public Library, home schooling section:

CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as how (and why) to learn mathematics in school; pattern-based reason; finding a number; solving linear equations; painless theorem proving; algebra and beyond; and complex numbers, trigonometry, and vectors. Also section on helping your child learn ... . Lots more!

2001 - Math Forum News Letter 14,

... new sections on Complex Numbers and the Distributive Law for Complex Numbers offer a short way to reach and explain: trigonometry, the Pythagorean theorem,trig formulas for dot- and cross-products, the cosine law,a converse to the Pythagorean Theorem

2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8

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

2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4

... section Solving Linear Equations ... offers lesson ideas for teaching linear equations in high school or college. The approach uses stick diagrams to solve linear equations because they "provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence." The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

Why study slopes - this fall 1983 calculus appetizer shone in many classes at the start of calculus. It could also be given after the intro of slopes to introduce function maxima and minima at the ends of closed intervals.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.

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