Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

Online Volumes
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

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More Site Areas 
1. Help Your Child or Teen Learn 
2. Solving Linear Equations
3. Fractions Ratios Rates Proportions & Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. More Calculus
More Site Areas 
8. Complex Numbers 
9. Qc Maths  Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
14.  Français
15. Algebra, Odds & Ends, Etc
More Site Areas 
16. Math Education Essays
17. Telling & Working with Time
18. Maps, Plans & Drawings
19. Quantitative Skills for  home, shopping and work 
20. Statistics Useful, or Not.
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YOU are better than YOU think. Show yourself  how:  

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Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.


After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.


Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Mathematics From Primary School to College

Ages 5 or 6 ] Ages 6 or 7 ] Ages 7 or 8 ] Ages 8 to 9 ] Ages 10 to 13 ] Age 14 ] Where is it going ] (These links also appear in the left margin)

Setting clear goals, identifying what should be mastered and testing knowledge of those goals, is the key to good  mathematics education.  Parents, teachers and students need logical, tested guides to what should be learnt and why.  Instruction needs to provides steps small enough so that most (preferably all) can follow alone or with help.  

Warning: If the text and workbooks provided by school authorities are not clear to parents, they will not be clear to teachers - the mathematics background of most primary school teaches and many high school mathematics teachers is often equal or inferior to that of parents. . Unfortunately, modern theories of education in English lands appear to be based on subjective psychological or cognitive theories of knowledge which contradict the essentially objective rule- and pattern- based view of mathematics held by university mathematics professors. So there is a division of labour  in which professors of mathematics in leading universities are excluded from math teaching training and course content decisions at the  primary and secondary school level. So mathematics lessons in secondary school and primary school has become diluted with subject matter and standards not endorsed or not supervised by mathematicians. At the primary and secondary level, mathematics lessons may be given by people with a weak knowledge of mathematics. At the college level, math-teacher training programs may be led by professors of education whose technical knowledge of mathematics is less than that recommended by mathematics societies for high school teachers. 

Primary and secondary  teachers could buy large-print math  workbooks for grades 1 to 6 (Primary School Mathematics) in duplicate to  post their pages on walls under the title tutor training program for students to see and  review what they should know from present or earlier studies.  Students in difficulty can be asked to explore the posted material to regain confidence and fill gaps in their knowledge. The JumpMath books described next are not big-print. 

Elementary Mathematics: To learn to read, write and spell, students need to master the alphabet - learn it and not forget it. Anything less would lead to difficulties or fear in or of reading, writing and spelling. Likewise,  to learn high school and college mathematics, and to avoid fears and difficulties,, algebra, geometry, trig and even calculus, students need to master the following efficiently and fully to the point of automation, the how with and if necessary without comprehension of why: addition and times tables, decimal methods for arithmetic; angle, length and time telling or measurement;  fraction skills and sense besides calculator usage skills. Alone or with help, parents, teachers and older students, those taking charge of their own education,  need to check mastery,  develop the missing ones, or  verify the missing ones are being develop in school.

Toronto JumpMath Work Books (Grades 3 to 8) for home and school 

The jumpMath home and school mathematics program asserts the following: 

One feature distinguishes our workbooks from regular math textbooks, however: in the JUMP workbooks, teachers are consistently shown how to help students who are having trouble moving forward by breaking mathematical concepts and operations into the most basic elements of understanding and perception

in its Teacher Manual - Fractions, page 2.  The jumpmath publication page offers a downloadable fraction unit and describes workbooks for home and school (grades 3 to 8). Copies of the workbooks for grades 3 and 4 bought for inspection are well-done. The workbooks may cover more than necessary. 

  • The jumpmath program publication page offers a downloadable fraction unit and describes workbooks for home and school (grades 3 to 8) - Schools should see the bulk order prices. Parents should consider chaperoning their children, grades 3 to 8, through the home version of the workbooks.

The jumpMath program (created by a mathematician) appears to cover the middle years of primary and high school instruction well. So I recommend its consideration besides any other program parents and teachers choose for students in grades 3 to 8 in school or for remedial instruction. See what is what is best.  

Older Site Mateial Questions and activities in the webpages

Ages 5 or 6 ] Ages 6 or 7 ] Ages 7 or 8 ] Ages 8 to 9 ] Ages 10 to 13 ] Age 14 ] Where is it going ]

will allow you to judge the mathematics and logic skills of your charges in primary school and high school instruction. A simpler route may be to acquire the jumpmath workbooks for home schooling (I have seen those for grades 3 and 4) and chaperone your offspring through them - time and patience required.

Secondary School Mathematics

The page Where is it going gives or suggest reasons for  high school that seem to have been forgotten. 

Teens and pre-teens need to obtain fraction sense and an efficient command of operations with fractions and whole number without a calculator.  If that is not done, all further mathematics instruction is sabotage or suboptimal.  The teacher who finds his or her students do not have the prerequisite for the current course needs to review and consolidate those prerequisites - not doing so may turn instruction into a formality.  The new site area  Solving Linear Equations with Stick Diagrams and Fractions introduces algebra while illustrating and reinforcing fraction sense and skills. As parent, you could take your son or daughter through it carefully. 

The ability to add, subtract, multiply and divide fractions whose denominators and numerators range from 1 to 100 say is sufficient provided students also have a good command of what is a fraction.  That be said, fraction sense and skill is needed to understand and do algebra beyond being given a formula and numbers to use in it.  The algebraic way of writing and reasoning needed the further or proper mastery of analytic geometry, trigonometry and calculus relies on fraction sense and skills. While some students and teachers unfamilar with the next level in mathematics may object to or not know the foregoing, the efficient mastery of fraction sense, what they are, and fraction skills is a must for going further in high school mathematics.  That mastery should be consolidated between ages 10 to 13 say. Anything less slows or stops learning.  All topics before students talk about solving equations may be used to emphasize fractions and allied concepts: ratios and proportions.  The aim of site areas in fractions, algebra,  geometry and so on is to provide students or their teachers and tutors (parents included) clear directions for understanding and explaining mathematics and its logic. 

Site books (online in full so you do not have to acquire them)  may help some students 14 plus to adult learn mathematics and logic, and parents guide their children.

Four Cautions

The advice offered here is approximately correct, for some circumstances not all. Pick and choose that which applies to yours. 

  1. Learning takes time and effort. Your child or teen should know that or be told, especially if you have succeeded in protecting them from worry. Do not assume he or she knows that learning takes time and effort. Marks in schools may be too generous  to the extent that students do not receive this message from teachers. I am a skeptic.
  2. Your charge also needs to be told that notes and work for doing problems, written on paper, needs to be written precisely. Ideas or work written incorrectly will be a source of error. For instance, methods for arithmetic (addition, multiplication, subtraction, and long divisions) rely on numbers being written in the proper place or column. Imprecision in location or alignment of numbers is a common source of error due to a change in meaning or interpretation between  writing and reading. Likewise, the algebraic way of writing and reasoning requires a proper and precise command of notation, otherwise what is meant or intended at time of writing will not be misinterpreted a moment or period later at the time of reading or further reasoning.
  3. Too many high school and college students not interested in mathematics believe arithmetic should be left to decimal computations with electronic  calculators. They forget how to do arithmetic by hand met if not learnt in earlier years.  But in algebra and beyond, operations with fractions appear and they need to be done exactly - the decimal approximations provided by electronic calculators cannot be used for the exact derivation of formulas. At the high school and college level, please check whether or not your son or daughter can add, multiply and simplify fractions efficiently. You may be surprised, sorry.
  4. Online help in reading, writing or mathematics has one limitation at the moment. The written work of students needs to be seen and corrected repeatedly for errors in presentation and notation. In mathematics especially, a student may master an idea (almost) without being to write arithmetic or algebraic calculations precisely and exactly on paper. The errors here in notation are dangerous.  A student may write one thing while meaning another, and then later read what is written inexactly. Imprecision in reading what is written is accompanied by an imprecision in writing mathematical thoughts or calculations on paper. Imprecision in one implies imprecision in the other.  The written work of students needs to be read and marked carefully, so that imprecision in writing (notation) is corrected. Here if there are few errors all should be identified, but if there are many, the most important ones should be identified and some left uncorrected in order not to discourage a student too much.
Ages 5 or 6 ] Ages 6 or 7 ] Ages 7 or 8 ] Ages 8 to 9 ] Ages 10 to 13 ] Age 14 ] Where is it going ] (These links also appear in the left margin)

www.whyslopes.com
Help your Child or Teen Learn:


Area Intro
1. Speaking Skills
2.  Reading & Writing
3. Preparing for Science
4. Learning Takes Time and Effort
5. Math Books: kids & teens
6. Math Books: teens & adults
7. Readings for  Parents
8. Patience Please
9. Who is in Charge
10. Motivation
11.  Will to Learn
12. Math K1-20
13. Links For Parents
14. JumpMath WorkBooks
15. Discipline in Schools

Maths for Ages 5+



Ages 5 or 6
Ages 6 or 7
Ages 7 or 8
Ages 8 to 9
Ages 10 to 13
Age 14
Where is it going

D What to do in School & Why  

E.How to Study Mathematics


To read, write and spell, your children need to learn and memorize the alphabet. Anything less would be absurd. That being said, learning and using mathematics demands that your children meet key skills and concepts, and not skip any. Where local schools do not provide the latter, you need to provide remedies.

Care and Precision: If your child  can learn to follow multi-step methods carefully and precisely in arithmetic, he or she may do so  in other subjects, as well. Get your child or teen, if you can, to sit down and study. Suggest he or she aim for skill and concept development and perfection for their own sake, not that of their teachers.

The will to learn is the key to success in school.  Parents do have to be educated to support or guide their children and teens. What matters more is support for the will to learn, for children and teens to be  told to try to learn and to ask teachers, their schools or classmates for help and more help, as needed. Teachers and parents need to push students, help them find the will to learn, teamwork helps.

The main reason and focus for high school mathematics is or should be preparation for calculus. That requires skill and knowledge perfection with fractions, algebra, geometry, trig and functions. Many high school programs do not provide this. Make sure alone or with help that your children and teens have a good command of fractions. 

 

 



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a 1983 McGill. Ph. D. in mathematics
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