Appetizers and Lessons for Mathematics and Reason  ( Français)  
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 Logic mastery is key to easing or avoiding learning difficulties in work & studies. 

Online Volumes (Book Orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3.
Why Slopes & More Math 1995

Site  Folders for Instructors & Adults
A. Public Policy Essays
B.  Mathematics  Education Essays  2006-7
C -Logic & Applied Math Program  
    for education,  June 22, 2008 
D. Quebec English Math Ed -  Standards to
 avoid  in course design & teacher education 
E. Help your child or teen
How TOs/ Ref.-08- 2008
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Logics in Maths
Before LAMP   Back ] Next ]


LAMP  Prequel - Previous Instruction

In the first years of education, primary school and even before, children in well-protected environments may be enthused, initial reluctant aside, about going to school to learn and growing up to follow in the footsteps of their parents or guardians.  For students with sight, learning to read, write and do arithmetic is based on our ability to recognize shapes as different or similar, and to draw or write those shapes on paper.  Elementary or primary education may or should introduce students to decimal notation; to associated place-value methods for the arithmetic operations of addition, subtraction, multiplication and short & long division;  to the meaning of fractions and arithmetic operations with them; to measures and calculations - more addition, subtraction, multiplication and division - with whole and fractional units of time, length, area, volume, money and mass (or weight) present in our cultures. As time permits, parents and teachers of children may expose students to planning and making meals; to following daily, weekly and longer schedules; to buying and selling goods and services in many situations - travel, work and restaurants. 

Applications: Book that teaches another language, say French, may describe people or a family in situations at home, at work, in town, on the land, at seas or in air,   course designs and materials and in that description introduce to associated vocabulary.  Likewise, course materials in mathematics may describe the calculation present in common place situations and activities with simplest ones first.  Examples in primary and secondary school years could place students in the mathematical shoes of buyers and sellers, carpenters, painters, construction workers, health workers, taxpayers, wage earners, business operators, investors, bankers, planners, navigators, surveyors, and so on. The description of mathematical reasoning  would provide a context and motivation for mathematics learning and teaching whose extent depends on local culture and conditions.  The mathematics of activities that are not in local favour - a favor that depends on time, place and culture - should be presented so  that students can recognize what to avoid.  

Operation viewpoint of arithmetic operations:  Whole numbers and fractions may describe how many units or objects are present. They may also describe position in terms of placement (1st, 2nd, 3rd, etc) or in terms of coordinates ( 10, 0, -10.5).  Whole numbers and fractions may be classified as numerical adjectives.  Then physical addition, subtraction, multiplication and division may be described in terms of and imply corresponding operations on whole numbers and fractions.  That implies our numerical adjectives are closed under these numerical operations when the corresponding physical operations are feasible. Before the introduction of signs, a larger quantities cannot be subtracted from smaller ones.  Division by a whole number (answer to the question how many whole times one count goes into another and what is the remainder, if any)  may lead to a quotient and remainder calculations.  But also, division by a whole number or fraction (how many times a shorter length goes into another length)  may also be described by a whole number plus a fraction.  

Signed Numbers:  That being said, if signs are introduced as prefixes on whole numbers and fractions, and these signed numbers are used as coordinates along a line and to describe displacement, arithmetic operations on signed numbers can be implied and thus defined by physical operation on displacements. Displacements to the left or right are easily added in a head to tail many. Unsigned or positive multiples of a displacement are easily defined through repeated addition, While negative multiples are defined by a change of direction.  The latter and the adjectival description of displacements as signed multiplies of a unit displacement (vector) implies the definition of products of signed numbers. Whence signed numbers are adjectives on which arithmetic operations are implied by their roles as adjectives. 
 

Put Performance First. That is to say or suggest, skills and confidence in arithmetic may follow in the first instance from a rote or near rote mastery of calculation methods, so that results are obtained in a repeatable, reproducible and hence verifiable manner. While thought based development and comprehension of operations on whole numbers and fractions, including place value methods for operations on decimals, is preferable. It may aid mastery of arithmetic operations and should be emphasized for that, but any further comprehension of why operations work need not be required. In the first instance, skill and confidence may be produced by emphasizing drill and practices, so that students figuring skills becomes automatic.  However, for students who refuse to apply methods before understanding the origins or justifications, the course materials should provide explanations in an appendix for the sake of completeness.  

 

LAMP (first draft, June 2008) a program for adult  and teen mathematics education

Mathematics education standards implied by calculus should be a factor, not the only one, yet not a forgotten nor hidden one in course design 

Area Intro
Introduction
Arithmetic
Geometry
Algebra
Logic
Calculus

Musings - More Ideas

More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education

For further musings or thoughts see site books.




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