Postscript A. Three Remarks
Remark 1, Fall 2005
In the old view of mathematics education, the instructor may stand in
front of the classroom writing explanations and examples on the
blackboard, alone or with student help, to develop the skills and
concepts, one at a time and one after another. The instructor may also
collect written or walk-about the classroom to look at written work in
order to correct errors in notation and understanding. The teacher is in
charge of skill development and verification.
The newer view is that instructors should make mathematics attractive
with activities that are fun to do in which discovery and learning of
skills and concepts occurs, without the teacher standing in front of the
classroom and saying directly what should be learnt.
The old and new approaches should be combined so that students can learn
or discover through activities interesting to them while the instructor
states learning objectives clearly and beyond that verifies that the
desired skills and concepts have been mastered.
Remark 2, March 26, 2006.
Older education theory calls for course outlines and materials to set
forth performance and comprehension objectives - aiming for but not
always delivery, performance and understanding in a repeatable and
reproducible fashion. Marks were based on performance. Students learn
from course material (the theory) and from loss of marks due to the
identification of errors in performance.
Modern education theory calls for students to be engaged or hooked by
open-ended, course material and investigative, authentic, realistic
activities with performance. Drill and practice, mastery of skills and
concepts in a repeatable and reproducible manner not emphasized, not
demanded, and put aside. The latter de-emphasis appears to be empirically
unsound.
Remark 3. Critical Thinking
Site material here at www.whyslopes.com supports the development
of critical thinking and problem solving skills, and a discovery approach
to learning. Critical thinking requires the ability to follow multi-step
with care, see what is available and what works, before extraordinary or
out-of-the-box or lateral thinking is required.
Re-inventing the wheel is not efficient, but problem solving situations,
real or artificial, in which students have to go the limit or beyond of
their present body of knowledge can develop thinking skills. The extreme
constructivist view that knowledge is an individual affair, not for
correction, lies in contradiction with the growth and development of
technical knowledge in science, engineering and mathematics. The latter
seek and rely upon methods with repeatable and reproducible results. The
methods are learnt by trial and error, guided by existing or extended
empirical and theoretical patterns, in which nature in a behaviorist
manner may allow us to learn from mistakes - what does not work and what
recipes or methods do on a small if not a large scale.
While a teacher can not read the mind of a student, a teacher may see and
correct mistakes, minor to major, in the content and style of student
writings and further endeavors or products, so that the student may learn
from his or her mistakes, and possibly learn how to make fewer mistakes.
In the short span of education, several years or more, the student will
meet subjects in which individual construction or organization of skills
and concepts cannot in the first instance replace the early collective
and refined products of many minds.
Instruction is an empirical art with value judgments and decision
dependent on the subject at hand and what students produce - observable
behaviors or products only. Any else is subjective - not repeatable and
reproducible. In particular, the constructivist approach to instruction,
despite fine calls for authentic, realistic and engaging material and
practices in the classroom, calls that should be heeded and empirically
supported as much as possible, in its opposition to the testing and
measurement of skills and performance provide vacuous standards for
instruction and undermines the sequential nature of learning in which
skills and concepts at one level need to be learnt and verified before
the next level begins.
Next: Chapter 1, An introduction to the
problems of mathematics instruction.
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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
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
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
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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