|
|
|
Before the introduction of signs, that is negative and positive numbers, finite decimal expansions extend this idea of greater than or more than. A finite decimal expansion in particular counts the number of units, tenths, thousandths and so on that the number it represents can be divided into. Beyond this, students may be shown or pointed to the comparison of (unsigned) numbers with infinite decimal expansions, albeit such comparisons may be rare due to the prevalence in everyday computations of finite decimal representations and expansions. When dealing with unsigned numbers, the ideas of greater than and more than imply that the larger number can be obtained from the lessor number through the addition of a (unsigned) number.
www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,Foreword + Chapters 1 to 10 + 12
Area Map Inductive Principles 1 Introduction 2 For & Against Math 3 Algebraic Thought 4 Why Slopes & SQRT of -1 5 Books & Articles to Read 6 Unruly Origins of Algebra 6. Axiomatic Civilization 7 Geometry, 2 Ways 8 Modern Instruction 9 The Two Ends 10 The Transition 10 Explaining Logic 10 Explaining Algebra 10 Why Sets in Math. 12 Four Phases Links
Chapter 11: Primary School Mathematics
11 Primary Math 11 Cue Cards 11 Counting 11 Decimals - Addition 11 Decimals -Times 11 Decimals & Subtraction 11 Fractions and Division 11 Notational Conflict 11 Reciprocals Etc 11 Decimals - Ratios 11 Size Comparison 11 Numbers, +ve or -ve 11 Rename < Sign 11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's Area: Help Your Child or Teen LearnMost students in high school are not heading for calculus, but most topics in high school mathematics are present due to calculus. Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math Education Essays. Site areas and pages provide pieces of the a Mathematics Education, Jigsaw Puzzle, in formation.
-Inductive principles for course design & delivery require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for what was missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.
Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.
Modern mathematics curricula introduced an inconsistency into course design and delivery. They did not sanction the use of decimals nor the use of diagrams in skill and concept development but decimal arithmetic and diagrams are needed for student comprehension and for an operational mastery of quantitative skills. That implies the need for an mixed-math curricula based on a systematic development of operational skills, sufficient for applications and sufficient to provide a base & context for the very optional study of pure mathematis.
|