Original Site Title: Appetizers and Lessons for Mathematics and Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept Development with How-TOs Français: 26 pages
for college students, gifted teens, home-tutoring and K1-12 schooling. Avid readers in school and out may like Site Volumes.

Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons. See Site Map

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits
Ages 12+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5. Fraction Operations by Raising Terms Solving Linear Equations: Take I Take II


Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math
, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles

Welcome: Site content may develop critical thinking, improve reading and writing, and build mathematics skills. See online chapters on on logic and pattern based reason.

Teachers: This December 2011, 5-phase framework offers a context for mathematics & logic instruction. Phases 1 to 3 focus on skills with actual or potential value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus.

Site Review: Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation for high school and college ... mathematics. Read more.

Home < Archives < Mathematics Education Essays << teacher certification

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In deciding which teacher certification program to follow in a school or faculty of education, or in a teacher training college, you need to know if the certification process is well-balanced and well-defined.  In a teacher certification program there will be invariably be practicums or field experiences course in which you interact with a host teacher and field experience supervisors Those personnel and the structure of the practicum or field experience, along with  the preparation and support you will get for the field experience are critical.  Your success in the teacher certification program, its evaluation of your suitability for becoming a primary or secondary school instructor, will be based on those experiences.

Question - pointing to or calling for best practices

  1. Are host teachers screened for their class-room management and subject expertise?  Ideally, the host will be a master teacher, familiar with the nuances and quirks of the local curriculum and course material.

    If host teachers are not screened, the field experience is become a lottery. In the worse case,  an unscreened host teacher may base your direction in the field experience and the judgment of your suitability for becoming a high school instructor on your knowledge of  substandard  practices. The use of  untrained or unguided  host teachers with unscreened people and content to make pass-or-fail judgments on student teachers contradicts assessment practices taught in schools and faculty of education, and in the large, university codes for fairness and due support in assessment of student teachers.  .
     
  2. Are there any guidelines or training for the host teachers?

    If there are no guidelines, if any thing the host teachers wants is satisfactory in the certification program,   the degree of support and direction is not governed, and what is provided and sanctioned may depend on the host teacher inclination. Since host teacher varies, the ease or difficulty of your certification journey will depend on chance.
  3. How are the field experiences structured? Is there any advanced notice to students of who will be their host teacher(s) and a course to be met.  

    In some teacher certification programs, a field experience is a random event. You are assigned a host  teacher at random, and two or three days after arrival in the host teacher classroom, you are expected to deliver lesson plans and lessons. And in that randomness, the subjects meet may be one of 5 plus, and you will not have time to prepare.  In other field experiences, you will meet the host teacher before the start of the school year and collaboratively plan or anticipate your interaction while building up trust and identifying far in advance the subject or topics in which you will have to plan and deliver lessons. The latter represents a best practice.

  4. What is the certification program field experience policy with respect to students knowledgeable in teachable subject without being familiar with the  local course materials and practices? Often there are several courses in a high school mathematics program, and if you have not met them before as a student in them, advance warning would be useful. 

    If there is no policy that indicates the teacher education program in question may fail to give you the support you need and so let you fail. So If you are an older student starting a teacher certification program,  go the program office which monitors  student teaching and ask what will happened when a student teacher, familiar in general with a subject  is not familiar with the current course materials and practice?  That question is important. If there is no special consideration for that circumstance, there may be tacit assumption that you are familiar with the current course materials and practices

    Key Questions: Doe the teacher certification program describe in sufficient detail the high school program in your field? Is local course material clear and sufficient for course planning and delivery. Are their undocumented practices? Is there nonsense or bureaucratic bullshit in course materials and course definitions?  Do you have to ask for dozen examples of past finals to reverse engineer course content.?
  5. Are field experience supervisors expert, master teachers,  in the course material in which your practicum or field experience will occur? Will they be able to provide check and balance for host teacher quirks? The question is important where professor of mathematics education are not the field supervisors, or where professors of education are expert in delivery style but not content.  There is no guarantee in North American that your professor of mathematics education will know mathematics. He or she may have become a professor of mathematics education without a good undergraduate knowledge of the subject in schools and faculties where "style" is more important than substance.

    The certification program needs to provide access to a master teachers familiar with the course material and its nuances or quirks to provide a check or balance on the host teacher's expectations practices. The assumption that a certified or qualified teacher is course material and delivery expert often fails in secondary schools where compulsory subjects like mathematics have to be given regardless of whether or not the subject is in the zone of comfort of the host teacher. I have met cases where substandard  textbooks or course material are approved and required.  If your first encounter with such substandard texts is in a field experience, and master teachers are not available to provide aid, to supervise your field experience, and to check or balance nonsense in course materials and your unscreened host teacher behavior.  your do-or-fail field experience may end in a disaster.

  6. Does the teacher certification program provide instruction on classroom management?

    With the advent of compulsory school attendance in authoritarian societies, instruction was teacher and subject centered, in a way that manners and good behaviors was required. Otherwise, there might be stiff penalties. In that environment, students were afraid to misbehave. But in societies which are granting and recognizing more and more freedom for their youth, school attendance may be required while the consequences of not cooperating with the schools and their instructors become less and less. In many societies today, school attendance is compelled while poor  manners and behaviors are not enforced. So teachers have to artfully employ the limited means at their disposal for classroom and hallway order and management. That being said, your participation in school activities and your teaching a course in a way that maintains or develops students will to cooperate and participate in their own education minimizes the need for display of fairness and firmness.  Repeatedly providing students opportunities for success is one way to encourage  the will to learn and work in each. Tat being said, you will sooner or latter give a course in which the motivation of most if not all, is success in the final examination.  Good luck. You will be meeting students, students likely to be well-mannered later on, who are in your face rude. In all interactions, look for a path or intervention in which the student or students decide that cooperation is in their best interest.
  7. For the subjects you plan to teach, will the teacher certification program besides describing or exhorting best classroom practice, describe what  is done as well? There again appears the question of style being touted but not supported by education professors who are style experts, not content experts.  The latter is required for teaching experience supervision. If the latter is missing, run or take your chances. 

    If a teacher training program focuses on what should be rather than what is, you will missing some necessary  preparation for the field experiences and lesson planning.

    Teacher education programs may exhort you to act in a certain manner and say what should be without focusing on what is?  Besides course on pedagogy, you may need a course from a master teacher familiar with the current  course materials in a primary and secondary subject program, mathematics, science, history or a language such as English or French or the language of instruction. For example in mathematics or science, a master teacher at the elementary level or secondary level, would describe the ends and means of each year of instruction, course by course, to identify how later skills and concepts are erected on earlier ones in a linear or spiral (repetitive) manner. A master teacher would also describe the benefits and limitations of current course materials, and identify nonsense in them. Nonsense can appear and continue in course materials if the latter are generated, approved and required in the absent of a master teacher or subject master to object.

  8. For the subjects you plan to teach, does the college education library keep on it shelves or computers, full set of course materials (textbooks and governmental course definitions and final examinations) for student-teachers and Professors of Education to consult?

    The presence of course materials - what you will meet in the classroom or in your field experiences - will allow you to get a headstart. And if the course materials or government documentation are not clearer complete, access to several, preferably a dozen, recent final examinations may define course content for you.  That is important for subjects or school courses which you have not followed in your school days due to their newness or because you followed another path in school days, while or before current materials were in use.

  9. What are your withdrawal rights for the practicum's or field experiences? Do they exist? What are the penalties?

    In many college or university courses, there is a penalty-free withdrawal period. So if you do not like the courese or its instructors, you can withdraw. However, withdrawal in teacher certification may be deemed a failure. But the right to withdraw without penalty may be very precocious in field experiences where host teachers and/or field supervisors and everyoneelse in the chain of command are not master teachers in the subject at hand.
  10. If  you fail a field experience due to unanticipated difficulties with course practices and  materials in subject will you be allowed to continue the program at later date after studying the subject in question in greater depth?

    This may be important if you discover you subject knowledge is not a great as thought, or if you meet undocumented expectation in the field experiences, or if you meet nonsense and bureaucratic bullshit in the course material and course definitions that leaves you in a garbage-in, and hence a garbage out situation. 
  11. If you fail a field experience due to classroom management difficulties, will the certification program allow you to re-enter in a few years time to try again?

     With the passage of time, maturity and out-of-school experience in the workforce may provide the pose and greater maturity needed for classroom management?
  12. If you fail and make an appeal, in the appeal process itself, will there be a master teacher in your teachable subjects, so that any subject related aspects of your appeal are heard?
  13. Does the teacher certification program or host institution provide insurance cover for the possible hazards met in primary and secondary schools, gentle to tough?

Teachers & Tutors: Site pages offer better or best practices for providing skills - simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do. Others are welcome to refine or exceed it. Please do.

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

The Logic of Injustice: How Texas sent an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate is not compensation for years or decade of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.


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Home < Archives < Mathematics Education Essays << teacher certification

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Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

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

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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