Mathematics Education Essays

Welcome to the mathematics education revisited (essays, rants and proposals) site section. In the past essays and rants used to be spread though out site sections. Now they are all located in this site area. Section material represent reflections and reactions to what has been done in the past. When I first heard of constructivism in the 1990s, I read its then and revised principles and standards (NCTM emitted) to see what proposed in the hope of supporting local trends with a few methods (lessons that had worked) because I did not have an academic position, and prospect for one were dim. Instead I found myself in a counter-trend position with ends and values for mathematics education strikingly different from dominant position and more closely aligned what is called the mathematically correct. At the same time, constructivist calls to engage students, to provide rich learning problems and environments and to empower students through such environments had a strong appeal. They represented an ideal to strive.

My exploration and expression of ideas and methods for instruction stemmed from an observation of steps too large or gaps and eventually inconsistencies in skill and concept course design that the constructivist movement inherited from earlier efforts. I had some remedies which I felt compelled to explore and report. While the latter was motivated by the difficulties of my students 1983-89 while I taught in college posts, the remedies posted online 199502010 are too late for past students, but may be timely for their grand-children. That is another case of better late than never.

That being said, the constructivist principles or standard that true knowledge is a private affair, located in the mind, apart from reliable observation and testing, and not to be challenged because everyone's ideas and reflection should be respected, appears to be in contradiction with the old-fashion ends and values that called for observable skill development in a tangible or observable manner. Each of us in a daily lives from cooking to driving has and follow skills or routines to obtain repeatable and reproducible results. In that, there is an element of rote learning. In rules and practices are given and followed carefully (we hope) to avoid mistakes. Performance, know-how, observable skill development, does not require comprehension. They may be learnt by rote. For material ends and subjects in education, some practices may be learnt and combined by rote. But there may be intelligence or observable skill in learning to how apply implications, rules and practices carefully (avoid mistakes) and in mastering the further super-practice of how to combine implications, rules and practices to obtain more, all in an observable and recorded manner for peer- or self-correction.

The site two level framework POMME for Progressive (meaning step by step) or Practical Observable Motivated Mathematics Education is based on teachers and students being given or offered clear ends, values and methods for skill development via small and alternative step likely to work and likely to ease or avoid common fears and difficulties. The focus of instruction is on credible and reliable skill development. True skill development has to been seen and reliable to be credible. And this student confidence and self-esteem comes learning to do. The rich learning environment and problems currently sought to support constructivist may also lead to projects and paths that enrich and motivate observable and verifiable know-how development. POMME tangible ends and values provide a context and motivation for its methods, while it methods support the ends and values. In that, awareness of the domino effect of approximations and errors in figuring and reasoning may be both a method and also an end and value for instruction. So ends, values and methods for instruction may overlap.

• 2010-02-20 Mathematics Ed. References (Material that influenced POMME)
• 2010-02-21Words For Teachers, Mathematicians & Maths Education Professors
• 2010-08-26. The Big Question Which Way to Go, How and Why, hanging over mathematics (and science) education There is a big difference between what may be possible in the thought-based development of each - or what can taught in a hands-on and observable manner.

Before the advent of POMME, many ideas were explored in formative essays. The essay still include reflections yet to be included in the implementation of POMME. Just for the record, this site has existed for a three to four years or not more.

Mathematics Education Rants and Essays
Section Content.

1. Site History and Content - through site reviews 1995 onward.
2. Site Eurekas - Site Highlights, an old view
3. Site Origins
4. About Site Lesson Plans - Another tour of Site Content

Lesson Plans, Aims and Goals (Ends, Values and Means?)

1. Three Aims for Students - Ends and Values
2. Three Goals for Mathematics Education, etc - Ends, Values, Unifying Themes
3. Lessening or Avoiding Algebra Difficulties
4. Algebra Lesson Plans
5. Algebra, Geometrically
6. Mathematics Curriculum Shifts
7. Advice and Suggestions for Course Design and Delivery
8. Teaching Tips - from fractions to Calculus
9. Math Education Perils (Arithmetic, Algebra, Calculus)
10. Talk the algebra talk
11. First Year High School Math - Lesson Plans with Fraction Focus
12. Second Year High School Math - Lesson Plans with an algebra focus
13. Third Year High School Math - Lesson Plans with a Focus on Slopes
14. How Letters Appear in Mathematics
15. Map, Plans and Drawings, a multi-year project

Links

1. Links - Just a few.
2. Activities to Engage Students - links to explore

Ideas and Principles For Instruction and Educational Reform

1. Inductive Principles For Instruction - systematic skill and concept development.
2. Fairness in Education - requires systematic development of all skills and concepts.
   
   Can education be fair if students are tested on natural talents instead of developed ones? Mastery of a skill, say the algebraic way of writing and reasoning, is regarded as a natural talent when and only when we do not know how to systematically develop that skill or concept. Site material reduces the number of natural talents required in the mastery of mathematics. Find the four skills for algebra in chapters 8 to 14 of Volume 2, Three Skills for Algebra, to see how to develop the algebraic way of writing and reasoning, and thus make mathematics fairer.
   
3. Apprenticeship in art, trades and disciplines, a classical view.
4. Education is an Empirical Art
5. Key Notes and Themes
6. For a Leaner Mathematics Curriculum
7. Need for a Mixed Mathematics Curricula
8. Extent and Need for Quantitative Skills depends on your society
9. Ways to be a Better Instructor - Ideas and Methods - try with caution
10. Four Ways to Improve Education Reform, and avoid disaster.

Logic and Reason in Mathematics

Mixing Rote & Thought-Based Development

1. Cultivating Intelligence - Why value careful mastery of rules and patterns, steps and methods, practices, in a repeatable and reproducible manner.
2. Multiply Kinds of Reason in mathematics - Essay I
3. Multiply Kinds of Reason in Mathematic- Essay IIs - On the hierarchical development of rules and patterns, steps and methods, and practices in pure and applied mathematics (mixed mathematics). What is proof? What options are there for a thought-based development and verification of college and pre-college mathematics?
4. Theory of Knowledge - Stories, Longer and longer
5. Formal or Informal Peer Review
6. Education in Mathematics, Science and Technology - All based on empirical verification and empirical skill development and verification. But in mathematics we can offer a full thought-based development while in science and technology, we can introduce the scientific method and introduce lab equipment, but can only provide a full-thought based development through visits to the lab and library. The lab alone is insufficient.
7. Maths Instruction in General - Three Goals A B and C to Set for Student, Supporting those goals and why rewrite the curriculum
8. Operational Viewpoint - Aim for an Operational Command of Mathematics First.- For students with no immediate interest in the know-why, a focus on the practice, an operational command of key skills and concepts may make comprehension later of the know-why easier and more appealing. The calculus teacher may says to students - learn to do now and to understand later.
9. How to Set Standards for textbooks and course materials - Need for Inspection by University Domain experts outside of Education Faculties to ensure bureaucratic course design and textbook composition does not lead to nonsense in mathematics education.

Teacher Training

1. Teacher Certification Issues and Cautions
2. Math Ed. Professors - Mathematics Background of, Trust but verify - see transcripts. Knowledge should be above or beyond calculus. If that is not the case, explain why not.

Archives:

1. About this site - old version
2. Old Site Entrance - 2010-05-1
3. Yet Another Old Site Entrance
4. Maths Ed Stopping Rule - 2010-04-25
5. A New Mathematics Curriculum - 2010-04-29

Challenges for Education Reform

1. Five Decades make a difference
2. Managing Reform - Assigning Responsibilities. (Should anyone be responsible? Should anyone be in charge? Is reform headless?)
3. Mathematics in Context - What Context?
4. What Should be Learnt and When?
5. Grouping Students - Streaming?
6. Learning Takes Time and Effort
7. Making the Hard Easier but Ignoring how and so missing the Point
8. Hook, Line and Sinker - Mathematics Education Inconsistencies - Reform in North America
9. More on Mathematics Education: Covers: For a leaner curriculum, Education an empirical art, More on testing, Constructivism versus Empirical Methods.
10. Four Skeptical Essays on Constructivism Revisited - Incompleteness
11. Euclidean Model for Development. Damage Reversal
12. Educational Follies - Learning By Discovery incomplete, cannot work, compound difficulties.
13. An Educational Inconsistency.
14. Modern Education

But teaching by indirect instruction requires not only a knowledge of what can be taught directly, but also a knowledge of how to explain all elements indirectly. Anything less invites or compound difficulties. Ouch.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science