Essay January 2007

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 (Optional Book Orders)	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. Test the Twiddla Whiteboard
[Site Entrance & Hub][ Back ] [ Area Entrance & Hub ] [ Next ][Site Exit] Essay January 2007

More on Mathematics Education, Etc.

Covers: For a leaner curriculum, Education an empirical art, More on testing, Constructivism versus Empirical Methods.

For a Leaner Curriculum

Where mathematics education reform is too bureaucratic or too rigid to consider ideas that should count, more generations of students will suffer from gaps in course design at the secondary & college level.

Education reform has led to more and more topics being included in secondary school mathematics. while old shortcomings linger and new ones born. A lean mathematics curriculum would focus on fraction and algebra skills and sense, 2D geometry with and without coordinates, trigonometric, and logic, all as preparation for calculus. A lean mathematics curriculum might include some application to demonstrate the usefulness of fractions, algebra and coordinates, and so invite the further studies. Preparation for calculus is key to college or university the thought-based as distinct from rote, study and comprehension of accounting, science and mathematics. In secondary school mathematics, statistics, 3d geometry, nets for 3D polyhedra, and transformation geometry are digressions for learning outside of mathematics, in say courses on social science, art or technical drawing, if need-be. Including these digression in core mathematics programs dilutes the preparation for calculus and calculus-based studies in mathematics Inclusions leads to a loss of focus in skill and knowledge development. Lean mathematics instruction could and should focus on mastery of fractions, algebra, 2d geometry with and without coordinates, logic, and trig.

Cut, cut, cut. Do the minimum well. Then enrich once the minimum is well-taught. Further cuts or shortening are possible by dropping artifacts in course design and delivery, topics not required for further skill and concept development. That being said, teachers still have to cover topics demanded by local school authorities. Site remedies may be woven into lessons to support and enrich local curricula, lean or not.

Education, An Empirical Art

In empirical arts, practices with repeatable and reproducible results come first, tested via trial and error, while theories and principles come later to summarize, to codify, to refine and even enlighten the practices. While practices or sequences of them in some empirical or hands-on arts in science, technology and business, assembly lines included, may comply with principles and standards, even be connected and organized and designed around said principles and standards, the forerunner to such organization consists of experience where principles and standards in formation and adaptation met reality - success and failure included.

Education is an empirical art. We may not read a student's mind, how a student thinks or links together skills and patterns, yet we can observe and test student performance, skill by skill, concept by concept, and encourage, but not guarantee, mastery of standard calculations and standard arguments or chains of reason in algebra, geometry and beyond. In some disciplines, not all, there are right and wrong answers due to methods that lead to repeatable and reproducible, and thus verifiable results independent of whom-ever applies the method. Learning how to apply and combine methods carefully to obtain reproducible and thus verifiable results is an old sign of intelligence in many old arts and disciplines in business, trades, science, engineering, technology and bureaucracy. The latter is subject to the limitations of rule and pattern based thought and practices, and the critical knowledge that not all is certain in empirical based thought and practice.

Critical thinking in science and technology begins with an awareness that what we hope for, dream of or construct in our minds remains speculation or faith IF or WHILE it or its consequence cannot be observe or tested directly, and thus corroborated if not confirmed. The foregoing is a rebuttal to the constructivist theory of learning, the part which opposes testing, the existence of questions with right or wrong answers, and which says student knowledge, if individually constructed, should not be contradicted. Empirically sound education must oppose wishful thinking. That being said, constructivist methods for engaging, authentic, genuine material and the development of critical thinking could be incorporated into education as an empirical art.

More on Testing. Knowledge empirically found or tested is relative and not absolute. Instruction which relies on testing skills and concepts can only identify errors in the mastery of the latter while correct responses only confirm, but do not guarantee mastery. But the level of student competence in a discipline defined by skills and concept mastery can be estimated from the degree of difficulty, the unlikelihood of correct responses if skills and concepts have not been mastered, and comprehensive of a test or series of test. Here individualized testing may be informative that mass testing. Empirical soundness of instruction and testing, the issue of lessons and associated tests with repeatable and reproducible results locally and beyond, should not be scrutinized in an absolute manner. Cognitive theory should look at education as an empirical art.

Constructivism versus Empirical Methods

After all is said, I found myself advocating an empirical approach to course design and delivery, an approach which may be combined with constructivist educational methods, those that work regardless of flaws in empirically unsound constructivist principles or theories, - principles and theories which imply subjectivity in mathematics and science, and beyond, which emphasize the empirical weakness of testing in education, if not in general, in place of the empirical merits. Constructivism with its advocacy of critical thinking in criticizing testing is contradicting the empirical basis of science and technology, the readiness to test in order to eliminate errors and so favor some success.

Managing or directing mathematics course design and delivery by insisting that pedagogical methods will work is a top-down approach to education reform. In the absence of testing, of clearly explicitly defined steps or building blocks which have worked, this top-down approach becomes an empirical gamble, like marketing and distributing a drug blindly in the hope that it work well and have no side effects. Besides hope in education reform, there needs to be verification - trust but verify. Otherwise, great leap forwards may do more harm than good.

While we cannot read a student mind to see what has been constructer or understood or not, or how, we can in good empirical form observe, correct and mark what is written or produced by students. Continuous testing, probing and observation of student performance is part of a continuous educational process. Through test feedback and/or direct explanations, students learn to avoid or discount wishful suppositions or constructs in contradiction with their environment in and out of school. Thus schooling can shape students minds rigidly. Or, schools can present rules and patterns of various arts and disciplines, and indicate the origins, benefits and limitations of rule and pattern based knowledge, the presence of uncertainty, the open ended nature of many situations or problems, a necessary disappointment for those of us nostalgic for certainty.

Spelling in a language requires knowledge of all the letters in its alphabet. We would oppose suggestions that students have to learn only part of alphabet. Some spellings are artificial. Students have to be given them. Students cannot discover them. Likewise in mathematics, we should oppose suggestions that students don't need fraction skills and sense, the prerequisite to algebra, or suggestions that pencils and paper calculation skills are not needed because of calculators and technology, or suggestions that students can discover mathematics by themselves. The structure of mathematics is inherited, handed-down and varying over time. Insistence on the discovery methods, insistence on cognitive dissonance, in learning mathematics leads to a loss of clarity and compounds existing confusions.

Putting constructivism subjective views of knowledge in charge of mathematics and science education is akin to rejecting the form of critical thinking in mathematics and science developed since the 14th century A.D. The placement invites cognitive dissonance (confusion) for all involved - students, teachers and parents. Bon Appetite.