Mathematics Curriculum Shifts

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Mathematics Course Design (Curriculum) Shifts

Site innovations for mathematics and logic education were initially developed to fill skill and concept gaps and flaws sensed in the high school exposition of modern mathematics curricula prevalent from mid-1950s to the 1980s in schools and colleges. However, exploration and refinement of ideas for learning and teaching points to an alternative thought-based development of high school mathematics (algebra, geometry, trig and functions) needed for calculus. The net result may be fewer but more effectives hours in high school mathematics.

These curriculum shifts could be the basis for a leaner and more effective mathematics instruction.

~~Two~~ Three Shifts - clearer and effective ways to develop algebra and fraction skills and sense: The puzzle of how to introduce the algebraic way of writing and reasoning clearly and directly was first met by in high school days 1965-70. Difficulties of fellow students and instructor in understanding and explaining algebra slowed the site author's education. The first algebra chapters in the 1995-6 Volume 2, Three Skills for Algebra, point to a solution - a greater verbalization in mathematics in which the overlooked ability of describing or talking about numbers and quantities is recognized and emphasized. That is before and then besides the introduction of letters and symbols in algebra as placeholders for numbers and quantities in calculations or their description. The spring 2005 site area Solving Linear Equations with fractional operations on stick diagrams also introduces algebra in a parallel approach to the foregoing, which comes first is a matter of taste, while consolidating fraction sense and skills. The two approaches together provide a solid base for algebra for students starting their teenage years, or later remedial instruction. Algebra self-instruction alone or with help allows student to benefit immediately. For self-instruction, the algebra chapters in Volume 2 are recommended first.

There is a fourth skill for algebra in Three Skills for Algebra, namely a development of the ability to talk about or describe the numerical and algebraic use of formulas and equations with short descriptive phrases: (i) forward and backward use (or direct and indirect use) and (ii) algebraic and arithmetic (numerical) solutions. These phrases appear in Chapter 14. can be used through out high school mathematics to identify recurring themes - key objectives - and to provide another fresh perspective on the algebraic way of writing and reasoning.

In mathematics, I would like to see the first two years of secondary school consolidate arithmetic and introduce algebra skills. Then I would like the third year to be given as a reward. That is, I would like it to provide applications, one at a time, and one after another, to develop a favorable impression for students who have begun to dislike the subject and may drop out, but an impression that need not be terminal as it includes motivation for further studies.

~~Third~~ Fourth Shift - Complex Numbers & Easy Consequences: Vectors & coordinates, polar & rectangular, are used in a very simple, logical development of complex numbers., one that implies a quick, logic-based development of senior high school mathematics (and the use of complex number methods with eⁱ in technical and engineering schools.)

Technical note: Assumption that the head to tail addition of vector described displacements in the line or plane is independent of our choice of rectangular coordinate systems implies the distributive law for real and complex numbers. In other words the geometric assumption that the coordinate description of sum of displacements gives a new logical development of the properties of real and complex numbers in ways that simplify and provide a base for analytic geometry and trigonometry - that favored in university program without explanation. This logical development based on geometry covariance, an idea that appears in relativity, provides an axiomatic shift for mathematics education with consequence for high school and college studies. See the logic chapter Islands and Divisions of Knowledge for thoughts on multiple starting or entry points in the deductive arrangement of ideas. Self-instruction in complex numbers alone or with help allows student to benefit immediately At the college level in engineering and physics, the properties of complex numbers and benefits for trig via the cis function were often presented as efficient shortcuts without proof. Here is a justification that may accelerate college and high school instruction.

A further shift - calculus re-arranged.: Calculus demands full mastery of logic, fraction skills and sense, algebra, analytic geometry, trig and functions. That demand provide a standard and goal for high school mathematics instruction which needs to be emphasized as the coverage of more and more topics in high school may distracts learning and teaching from the full mastery.. Even with that full mastery, calculus employs the algebraic way of writing and reasoning at full strength. The site calculus introduction employs geometric and algebraic previews, and decimal view of error control in computations, to develop the multiple full strength uses of the algebraic way of writing and reasoning gradually and systematically in ways that should eliminate or avoid some calculus perils, and allow more to go further. Calculus self-instruction alone or with help allows student to benefit immediately. Note in a recently seen discussion of the modern mathematics curricula of the 1960's, there is mention of a slope-oriented analysis which site geometric and algebraic previews may duplicate. If that is the case, site previews are re-inventions and not new.

Expert Instruction (Mastery Learning): In classes, grades of 50%, 65% or 80% in a sequence of assignments and tests say how well you are doing, but do not say what you have missed. If the teacher or marker identifies and correct all mistakes in your answers, you can learn from your mistakes, and you know what you missed. In my classes, I intend to make a checklist of skills and topics, so that I can record which ones have been mastered to report to student a grade - the percentage of skills and topics which appear to be mastered, and to track and report what remains to be reviewed by the student or re-taught. Efficient learning (more gain for less effort) might follow. But I am advocating here what I have yet to do in class, an expert approach to learning and teaching. Tutors too can be hired to follow this approach instead of being hired to improve marks.