Postscript - February 2011

In two years of UK grammar school 1965-7, mathematics lessons consisted of given rules and patterns in algebra, trig and geometry, all given in what I presume was a mathematically correct manner. I was too young to know otherwise. In then next three years of English Quebec secondary schooling, rules and patterns were also given but in axiomatic structure. That is, rule and pattern mastery started from axioms - assumed patterns algebraically or geometrically put. The fine print in my Quebec high school textbooks emphasized or valued starting from a minimal set of axioms. The development was essentially logical. But there were four flaws in the Quebec portion of my secondary school and junior college education - nuances small and large.

1. The arithmetic mastery of decimals and fractions met in my UK primary and secondary school days was required, but not explicitly sanctioned. That departed from the ideal in textbook fine print of building mathematical knowledge on explicitly given axioms

2. The algebraic way of writing and reason was required to understand the shorthand role of letters and symbols in the axioms and in the further development of algebra, geometry and science in my UK and Quebec studies. But no course and the fine print in all of my textbooks did not provide any sanction for this shorthand role. While I found my own rationalization, self-constructed, I saw the instruction of myself and fellow students slowed by the silent assumption use of algebraic skill. Course design and delivery assumed it without clearly or explicitly discussing it. Talking about three skills for algebra, a lesson given in fall 1983, represented my second effort to address this flaw. The first was in a 1975 handout at a McGill University open house.

3. The use of order pairs as coordinates in the plane ias in the analytic approach to geometry was mixed with synthetic Euclidean Geometry, the line, circle and triangle drawing approach. Having two approaches unreconciled departed from the fine-print promise in algebra of a minimal set of axioms.

4. The use of drawings and diagrams, disowned in the algebraic course view of mathematics, was present in both high school level trigonometry and in later college level calculus. And geometric drawings were employed along side mathematically and algebraically deep utilization of epsilons and delta views of continuity and convergence, with the underlying theory avoiding decimals, while examples and illustrations employed or required decimals.

My strength and weakness as a student and a human being was and may still be a reflex to take everything literally. So I was disappointed with my high school and college education because the algebraic way of writing and reasoning was not introduced in a clear step by step manner. In all the courses I took and in all the textbooks I saw, this shorthand way of writing and reasoning was required while the effort to explain it was absent or, when present, insufficient. I was also disappointed by the espousal of an ideal, the consistent and full logical development of mathematics from axioms - assumed patterns. Course design and delivery failed to deliver. In retrospect, following graduate studies and doctoral degree in mathematics, and further thought, the ideals espoused represented the hopes and motivation of modern mathematics. But mathematicians if not mathematics educators since Godel in the 1930s were aware that the hopes were not feasible. None the less, the modern mathematics curricula echoed those hopes and emphasized a rigour in ways that many tried to take literally.

In retrospect, mathematics is an empirical subject. Its development is a mix of practice and theory. Given that, I first recommend K3-9 observable skills and practices with take home values be learnt and taught in manner that emphasizes their value, with explanations to aid mastery without overwhelming it, with the end of making students aware of the domino effect of errors in calculations and reasoning, and with the end of showing students how to do and record steps in manner they and others can do or check. With skills that have take home value, students may expect instructors to teach correct methods - methods that can be learnt by rote if the student wishes, methods for which explanations why are available for reading by students when or if they want.

For skills that have high take home value for life in the street or work, rigourous mastery is more important than comprehension. But in the preparation of students for college programs, those that may employ one variable calculus, or more. skill development needs to show students how to use and combine rules and patterns in applications, and in the development of further rules and patterns. The ability to apply and the ability to reproduce in all or part what has been shown is in itself an observable skill. Skills that can be seen can be described, confirmed or corrected. But the thought-based development of skills and concepts does not require a minimal set of axioms. Minimality here represent a value of higher mathematics education. The development requires a convenient and consistent set. This set may provides the opportunity for students to see how rules and patterns may be applied to obtain results and combined to obtain further rules and patterns. The set develops and sanctions decimal, algebraic, geometric and logic skills and practices, so that the flaws indicated above are avoided while the stage is set for further studies in college programs. That represents the current or last objective of site material.

The initial objective when writing began was not so large. Writing began with the inductive criteria for course design and delivery, with the question of how to motivate skill development, and with the hope of making the modern mathematics curricula of the years 1965-90 more accessible - less challenging to learn and teach. The initial aim was to report the inductive principles and a few appetizers and starter lessons to educational authorities, and then leave further work in this matter to others. However, I was not formally qualified to present ideas to educational authorities, or academic committees, more ideas followed, More over, in 1989, before I started writing, education had shifted from valuing skill development in reading, writing and arithmetic, mathematics included, to saying true knowledge is a personal affair, located in the mind, apart from observation and correction of teachers, and not associated with perfomance, that is, observable rule and pattern mastery. That subjective movement in education, led in US and Canadian mathematic education by the NCTM rtoday is the reverse of that espoused by the NCTM in the 1950s. Site material provides a rational alternative.

A K1-9 emphasize of skills and practices with current or potential take home value for work or life in the street represent student-centered skill development. The K7-12 parallel or subsequent emphasis on skills for one-variable calculus and college programs in technical fields represents college oriented skill development for careers - not guaranteed - that may benefit the student or society. Such instruction has intellectual and/or take-home value for some, not all. That is not ideal. But recognizing this situation represent a step forward from the situation in which secondary mathematics instruction is clouded in mystery, with the question why learn or teach this has the bureaucratic answer: preparation for final examinations. Moreover, this college oriented instruction can be offered, does not have to be taken nor required, once most skills with take-home value have been covered. The latter needs to be done first. It will be useful to all, including those students aiming for college studies who might other wise miss it. As a high school teacher, I once had to give a mathematics course required for graduation to a group of students who have benefited from a review and consolidation of skills with take-home value. Instead, their time was wasted because of government standards for education that forced the learning and teaching of topics with no academic nor take home value for the students in question. That situation needs to be addressed.

END OF POSTSCRIPT

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