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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

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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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.
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- 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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## Introduction to Section Content + General Advice

See too the general advice on helping your child or teenager in mathematics below.

1. Speaking Skills suggests how to improve the speaking and listening skills of your child.

2. Reading & Writing offers ideas for the development of these skills.

3. Preparing for science -Teaching a boy or girl to cook or to follow any multi-step method precisely, in a repeatable and reproducible manner, will help in science and all area of work and study.

4. Learning Takes Time and Effort: Four Things for a Student to Know. Quote in full of an article from Speaking of Learning that refers back to words at this site, no longer online.

5. Patience Please. Reflects the inductive idea that learning takes time. If you see a difficulty, you need to identify the source and retreat before it in order to practice skills that restore confidence and then to practice skills that remove the source of the difficulty. Teaching, tutoring or parenting takes time and patience. Good luck. Nothing is certain.

6. Who is in Charge? For better or worse, you the parent or guardian may be the first and longest term instructor of your child. Do your best

Parents and teachers need to say no for small things of little consequence to build and maintain authority to say no for larger matters. Parental authority: ; use it or lose it.

7. Student Motivation Here a discussion of the challenge. Not the solution. ;

Students with parents who say mathematics mastery is important, or education in general is important, will ; often have more goals, more will and more staying power in school and college - no guarantees here -but is part of the solution.

8. Talk to Your Child or Teen. For many, those without learning difficulties, the will to learn is often more important than ability. Encourage the will. ; That is part of the solution.

9. Discipline in Schools: Streaming by Student Cooperation. Societies that want quality education will preserve the authority of teachers in the classroom, while providing safeguards so that the that teachers do not abuse that authority. In the first instance, students may be streamed by their willingness to cooperate with teachers, and then by their academic destination. People with good education will not want to go into classrooms if [A.] student disrespect is a constant danger; and [B] present-day course design in their discipline areas is inconsistent and irrational according to their previous training in the discipline. Uniform standards in education are mixed blessing. While in the first instance they raise standards, over time central planning or bureaucratization may lead to those standards being lowered. In large enough regions, different school systems should develop their own standards, with multiple independent centers for course and curriculum design, each trying to offer a different design, each staff in a way that rejuvenation is continuous or mass retirement in one is not simultaneous with mass retirerment in another.

10. Work & Study Ends, Values and Methods. These appear to be missing in schools.

11. Parents Need to Follow & Supervise the Education of their Child or Teen.

If your child falls behind, provide extra help during the school year or during summer vacations. Ask your school for a list of observable skills that it and you should be verify. If there is no list, form one alone or with other parents. If ; there is no list of observable skills, your child school system has no idea where it is heading. It is lost. ; ;

## General Advice For Mathematics Skill Development

In general, explain to your child or teenager that skills in mathematics and other subjects need to be seen to believed. In arithmetic and beyond, talk about the need to do and record work step by step, so that each step can be checked as done or later. Emphasize too the domino effects of errors, that is, how an error in one step leads all or most following steps to be in error as well. If local textbooks are too hard for you to follow then your child or teenager will have difficulty as well.

For children or preteens ages 3+ to 12 years of aged learning mathematics, use parent friendly, Work Booklets to build or rebuild skills and confidence. Emphasize mastery arithmetic and geometry has value in adult and daily life. say this mastering is part of growing up, like learning to read exactly what is meant, and to write exactly what they mean.

For children or teenagers, ages 12 and up, site material may help you check and develop skills in stronger ways that seen in school. Mastery of site material unfortunately requires adult level patience and reading abilities that most students, except for the gifted and talented do not have. So you or a tutor will have to give less gifted learners ages 13 to 19, skills and concepts to master in parallel to the skills and concepts that teachers provide in school.

Site material was written to provide alternative and clearer paths for development of key skills and concept in mathematics and logic because of gaps and steps too large in the secondary level course design and delivery. The tutor or school system able follows site steps in arithmetic, geometry and algebra offers your child a firmer base for calculus and senior high school mathematics. But there is no 100% guarantee your child or will be able to follow the site program alone or with help - that remains to be seen.

In the early years of secondary school, your teenager may need to be tutored to prepare them for course test and finals, besides following site steps as site material and your local school may follow two different paths. But within a year or two of effort, site steps, ends and values for skill development should help your child do well in school. But again there is no guarantee your child will be able to follow the site program - that remains to be seen. So you have explore site material or rely on the judgement of others to decide what route to follow. Good luck.

If you are lucky, your local school does not follow the new-age, constructivist approach to education. In it,

• Children are told the best way to learn a subject is from discovery and from their personal experience.
• Teachers are also told that true knowledge for each student is a private matter, a product of their reflection on their personal experience, a product that can not be reliably tested nor corrected in class and on final examinations. in class or in final examinations.
• Teachers are told to introduce idea indirectly, so that students may discover them for themselves. In this, various kinds of problems and challenges are to be given to set the stage for this self-instruction.

Teachers are further told that giving students skills and concepts to master is a substandard form of instruction. At the same time, school authorities and parents like you want their children and teenages to do well on final examinations. Such final examinations test student mastery of skills and concepts that used to be given directly instead of through hints. There is a contradiction between theories that hold students should be taught indirectly, and the practice of wanting students to do well on tests and final examinations of skills and concepts that previous educational theories and practices tried to develop directly. That being said, site material stems from the observation of gaps and steps too large or missing in the direct explanation of key skills and concepts. In particular, site material aims to remedy the foregoing by providing teachers and tutors, and the gifted or talented, clear and direct explanations likely to ease, avoid and even explain for common fears and difficulties.

Clear How-TOs, ends and values for the direct explanation of given skills and concepts need to be well-known and well-documented before instructors, well-versed or not in mathematics and logic, are called upon to develop skills and concepts indirectly. Otherwise, indirect instruction will simple compound the shortcomings of direct instruction in skill and concepts development. The problem with constructivism is that moves the focus of education away from worldly concerns.

This viewpoint of constructivist-led indirect instruction stems from the perspective that arts and disciplines at home, in school and the workplace require an observable and verifiable mastery of visible and hence verifiable skills. Skills and abilities in many arts and discipline need to be seen to be shared, to be well-defined and believed. In particular, skill in cooking at home and in a restaurant may involve some thought or reflection that we cannot see, but that skill needs to be seen to believe. In general, we do not want cooks who think, we want cooks that can do. Likewise, we want cloths designers and makers, plumbers, electricians, interior decorators, construction workers, factory workers, engineers, book-keepers, accountants, scientists and mathematicians who not only think about what needs to be done in the privacy of their own minds, but also display skill in ways that can be seen and judged. The format of that display may vary from discipline to discipline.

In mathematics itself, practical concerns suggest the following ends and values. Instruction should first serve on the development of basic skills and concepts with of service in daily and adult life. Second, it should serve the needs of pre-college or precalculus occupuations. Finally, it should serve the needs of calculus and precollege courses in science and technology. Calculus is required for college level studies in accounting, commerce, engineering, science, technology, mathematics and mathematics education. Here serving the first end may serve the next two, and serving the second may serve the last. Further, there is no harm in college bound students seeing skill and concepts for adult and daily life, and for precollege work destinations as a parachute or safety net as there is no guarantee that their college studies will succeeed. Mastery of logic, awareness of the domino effect of errors, and awareness of how many later skills and concepts depend on earliers could provide further ends and VALUES to guide both learning and teaching.

Finally, in reaction to my description of these worldly ends and values for instruction or skill and concept development, a colleague kindly observed the absence of higher level cultural end and values here. So more thought may be required.

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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.
... 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.