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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.

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

www.whyslopes.com >> Parent Center >> 8 The Effect of Negative Remarks Next: [9 Streaming by Student Cooperation.] Previous: [7 Student Motivation.]   [1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

# Negative Talk May Harm Motivation

In a 1988 junior college mathematics course, I found myself facing a student whose aim in my class to obtain the borderline mark of 60% to pass. In challenging his lack of effort directly and trying to explain why he needed to study mathematics for his future plans to become a police officer, I found that I was the first to confront him on his lack of will and effort and that his mother had said mathematics after arithmetic was a waste of time.

The latter remark is understandable. In course design and delivery reaps what it has sown. For too many years, secondary mathematics in covering mathematics after arithmetic has had technical difficulties in introducing algebra. Secondary mathematics has immersed students in skills and concepts whose presentation when done well may be algebraically and logically correct, but whose presentation is unclear because of a lack of systematic development of algebraic and logical skills. For many students and many of their mathematics teachers, many unversed in mathematics, the sole apparent reason for skill and concept mastery is preparation for the next final examination. Year after year, decade after decade, The latter has left a poor impression of mathematics education on high school leavers and graduates - the parents of today's high school and college students included. So aversion to mathematics may be inherited. Site ideas on emphasizing skills and practices with clearer value for adult and daily life may ease or remove that aversion and lack of context for mathematics education.

## For Positive and Active Learnings

Talk to your children and teens. Ask them about their school days. See if they are passive learners - working to rule - doing the least amount possible, or opposing their own instruction. Regardless, encourage your offspring to be positive, independent, self-directed learners, learners actively and deliberately seeking to obtain and perfect skills and knowledge - alone or with help from readings, tutors and teachers. Teachers may meet dozens or hundreds of students each week, and so have not the time nor inclination to be personal tutors. So active cooperation with attention to details, the will to listen and try instead of passive or overt opposition or resistance to education, will all help your charges in school and out.

Your children and teens have decisions to make. Whether to try or not, to cooperate in their own education is one of them. The proper choice may be harder when a teacher is not charming but the decision still has to be made. So talk to your children and teens. Ask them about their school days, and guide them if you can. In raising kids, if you are unlucky, there may be two difficult even stormy periods during the horrible two's and the horrible teen years. Wait for the storms to pass, if you can. Giving your child or teen the will to learn counts more for his or her education than being well-educated yourself. Students with limited gifts or abilities with the will to learn and to keeping trying may go further than student with many gifts. Much depends on will, effort and the chance to study, preferably with clear material.

## Do not be Negative - Find a silver lineing.

A teenager planning to become a policeman said he did not work at learning mathematics, that he was doing the least possible (in fact less) to pass, since his mother had told him, mathematics after arithmetic was a waste of time. My late or too late objection to that came as a surprise - no earlier teacher had done that. For students to do better in school, they need parents who value education and say so repeatedly for the sake of motivation, especially when and where immediate take-home value is missing - not self-evident. While I disliked writing essays and book reports as a teen, had difficulty deciding what to writing, I would not today tell any child of mine that reading and writing skills, and understanding the characters in book or play, is unimportant. With the passage of time, I might have acquired the maturity to do well and avoid the misery of my earlier high school days in expressive and creative subjects.

Help your child with homework without always doing it for them. With school work, your objective should be for the child or teen to write as much as possible alone, or with a few kind hints, while you encourage the greatest possible independence. In mathematics, the vertical alignment of equal signs in evaluatio of formulas and arithmetic expressions provides a pattern to follow and encourage. For good work habits, it will help your teenager or pre-teen do and record work in steps that can be seen and checked, and make the domino effects of care and mistakes easier to see and correct.

When you child or teens writes an essay, point out mistakes in spelling and sentence structure. Say when sentences are too long, run on and on, or include too many ideas. While writing that, I fear that the advice is not always not followed in site material. So do as I say, not as I do. Writing essays will be easier for teen or pre-teen if there is a story to tell. In the case of a writing block, ask your son or daughter leading questions: Do they have a story, a chain of reason, a chain of instructional steps to present, or a collection of examples, to present in an organized manner. In writing as in packing a suitcase, the order of packing is not important as long as all is packed. Other times, sequence is important. Further say, if the son or daughter cannot answer the question posed by their teacher, than they should improvise by picking a topic, close or far, to demonstrate and practice their writing and story-telling skills, fictional or not, in a technical or non-techical area. In that the aim is to produce a work of art, fiction being easier and requiring less rigour than non-fiction. If you teen is lucky, his or her teacher may accept a good essay, on topic or close. As a rule of thumb, producing one is better than producing none.

### Learning from and Correcting Mistakes - Before Submission

How to limit your help. For the sake of your sons or daughters mark, you or a tutor may help more initially. But eventually, your son or daughter should do more and more. In doing writing for mathematics or for producing an essay, the objective is for your son or daughter to eventually learn how to independently produce so well in correct, well-formated manner. You or tutor may help by identifying errors in form, content and logic well-ahead of the homework or assignment due date. You son or daughter may have to learn that not corrrecting the errors will lead to lower mark.

Remember on tests and final examinations in school, your son or daughter will be alone. So the aim of help from a parent, sibling or tutor should be independence - show how to mathematics and write essays alone, and identify errors small and large. The aim is for your son or daughter to learn from them in order for errors to be caught and corrected before submission and in order for like errors to be avoided in future written work. Good luck.

## Postscript: Advice-Summary for Young Writers

Teachers are not interested in what you say or think. They are interest in developing and/or observing skill. You do not have to put your heart into your words - use your head instead: Teachers are interested in how you write: Do your words tell a story or present a theory which others can follow in steps or pieces that other may follow? Are your words presented carefully without spelling and grammatical errors?

You are presenting written work to teachers not only for grades or marks, but also for corrections, to learn where you are making mistakes in content and format. But that learning from mistakes could take place before you hand in your work if you ask others to identify errors, small or large, in your work that you may correct before giving your work to your teacher, and mistake or lack of quality penalized. The reward for this extra attention to detail may be higher marks, justified we hope by the fact that have learnt from correction or feedback of others.

And in essay writing, if you have a mental block for an assigned topic, you have nothing clear to say about it, talk to others to see if you can unblock, and failing that write about another topic to exercise and demonstrate your story or argument telling or development skills, and to exercise demonstrate your spelling and grammatical skills, all in the hope that your instructor kind - preface the essay with words to say you had a writing block. The foregoing advice is more easily followed in writing essays than it is in answer given mathematics questions. That's life.

www.whyslopes.com >> Parent Center >> 8 The Effect of Negative Remarks Next: [9 Streaming by Student Cooperation.] Previous: [7 Student Motivation.]   [1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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.