Standards and Principles for Mathematics Instruction as Skill Development
Rigour Modularized
From arithmetic to calculus in mathematics, skill development depends on
showing and explaining what to do in an observable, repeatable and
reproducible manner. In that, mathematics programs for instruction may
set observable standards for the presentation and communication of work
in visible steps that can be corrected or confirmed. In mathematics as in
other skill-based arts and disciplines, some steps may performed by rote
or automatically while others steps may be done with full or partial
comprehension. In the latter case, reasons to justify steps may included
in the steps or not. Thus skill development is possible at many levels.
Which way to go depends on the ends, values and abilities of students and
teachers. The latter may vary between individuals and shift over time.
At the level of algebra and geometry, where one student has mastered
decimals and fractions by rote and another with more comprehension, there
may be no observable difference between their algebraic and geometric
works.
In my own case while studying what might be possible for the
thought-based development of mathematics from counting and arithmetic in
primary school to calculus in college or the end of high school, I
observed gaps in my comprehension of decimal arithmetic - my advance
studies in mathematic despite their algebraic-deductive bent had taken
arithmetic methods for granted.
Skill development in arithmetic, algebra, geometry, calculus and
statistics may involve varying mixes of explanation, comprehension and
practices between different classrooms in the same or different schools
while still leading to repeatable and reproducible results. In the
foregoing, the ability to do is a must while the ability to understand
and justify steps is optional, a question of depth and style.
Mathematics skill development is robust and flexible enough for practice
or learning to do to be accompanied by great variation in the depth and
extent of theory and comprehension. It appears that many aspects or steps
of arithmetic, algebra, geometry and calculus may be met and mastered
with varying degrees of compartementized logical or thought-based
development of skills and practices. In general, by showing students how
to do and record work in steps, steps that may be done automatically or
include a degree of self-validation - reasons for them, instruction sets
an observable and verifiable standards, standards that may vary between
modules or topics. Variation is to be expected because of variation
between different programs and within each program, because of variation
between different classrooms. The latter variation may be lessened by
course material in which common formats for doing and recording work in
visible steps are clearly put. No matter what format is selected, doing
and recording steps in sequence will illustrate the domino effects of
mistakes. Emphasizing that domino effect provides a general end, value
and tool for building skills and confidence in all disciplines where
multistep methods appear.
Reflections
The high school mathematics I met in the first two years of a UK grammar
school introduced practices algebra and geometry which I saw and
partially mastered, but for which the context was unclear. In a practice
first, theory yet to appear approach, the logical foundations of
mathematics was not emphasized. As a student, I saw the algebraic
shorthand role of letters and symbols appear in the statement and use of
the quadratic formula. But dogmatically, as a matter of principle, I
would not use the formula until I had understood its derivation. With
some struggle, I did. The struggled required a rationalization of the
role of letters and symbols in algebra. On moving to Montreal, I met a
high school program with text books that emphasized the axiomatic
development and justification of algebra and geometry. I was hooked on
the logical development of mathematics. That being said, I sensed again
that the exposition of mathematics was incomplete - course material and
textbooks employed the algebraic way of writing and reasoning with
letters and symbol, but did not speak of and did rationalized their
shorthand role in this algebra way. That was disconcerting.
While I continued to study mathematics for the sake of understanding
whatever I might be doing in future, I sensed that the axioms for algebra
and axioms for geometry were somewhat disjointed. I further sensed that
the axioms did not sanction the decimal arithmetic I had met, the
trigonometric practices I was then meeting, and the use in chemistry and
physics of units in calculations. However, I did not then have the skill
and confidence to express my discomfort to my instructors. Instead, I
waited for my discomforts to pass. It did not. In further studies at the
university undergraduate and graduate level in mathematics, the
discomforts continued. The full set-theory algebraic-logical
development of mathematics provided an edificed for pure mathematics, one
that did sanction common calculation skills and practices with decimals
and units; one that assumed the algebraic way of writing and reasoning
without offering any rationalization of it. Over the last four decades,
the explanation of my discomforts shifted from assuming lack of ability
on my part to recognizing gaps in the exposition of school and university
level mathematics.
When writing first began in the last week of 1990, my objective was to
repair and extend the modern mathematics approach I had met in my
secondary and university days to make it more accessible and to provide a
thought-based path for building skills and confidence. That aim led to
site efforts to provide a thought based account of arithmetic. However,
those accounts are too complicated in full for most students and teachers
to follow alone or in class. One high school students explained to me
that my job as an instructor was to teach correct methods in class, and
the explanation of correct methods was not needed. Indeed, she may have
seen my explanations of methods that should be correct as a sign of
weakness or uncertainty on my parts as to whether or not what I was
teaching was correct. Since then my opinion of how arithmetic and further
mathematics should be taught has swung away from its full-thought based
development in class to the realization that the thought- and axiom-based
development of mathematics could be modularized. In that, the modules may
range from rote mastery of skills and practices to their axiomatic or
thought-based development. The key to that in each module appears to be
doing and recording work in visible steps that confirmed or corrected in
ways that emphasize the positive and negative domino effectd of diligence
and mistakes, respectively. Thus the need for a full logic- or
thought-based program is relaxed. The question of what should be included
in skill and practice development modules remains.
More
p>
Students who learn to figure need to understand place value. For each
arithmetic operation, some students will follow instructions without
needing nor wanting detailed explanation of how or why the operation
works. Others will be uncomfortable without greater comprehension. Thus
arithmetic steps may be mastered by rote learning, or with full but more
likely partical comprehension. The task of the teacher is to help
students to learn to figure with explanation of why full, partial or
absent in accordance with their needs. While instruction may and should
provide food for thought or reflection, instruction to be credible and
effective should develop skills, practices and abilities in an
observable, repeatable and reproducible manner. The latter implies
objective principles and standards In the instruction of students aged 4
to 13 years that appears to be true. Primary school logic, language and
quantitative skills and practice development has actual or potential
value for life at home and in the street easily understood and emphasized
by adults - teachers and parents included. But secondary school
mathematics with it focus on skills and practice needed for calculus- and
statistic-based college programs lacks immediate value for life at home
and in the street. Most people will not enter calculus- and
statistic-based college programs, and most who do will not complete them.
For two or more decades, most students and adults see no end and value
for secondary mathematics other than preparation for the next test or
final examinations
In reading, writing, reason and mathematics, learners 4 to 16
need skills and practices which might be useful for life at home, at
work, and on the street. Some will also need skills and practices for
calculus-based college programs.
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In and after arithmetic, students need to be shown how to do
and record work in steps that can be seen and checked. Students
also need to be given enough information for the work. Before
algebra begins in ernest, skills with decimals, primes, fractions
and signs are required. In that quality and accuracy is more
important than speed. In doing and writing work in clear steps, the
domino effect of errors will be seen. Avoiding this domino effect
is an end, value and tool for building skills and confidence at
home and on the street.
The child who says there are too many letters in the
alphabet is mistaken. The child will eventually learn that all have
to be learnt in order to read and write. Likewise in mathematics,
students wanting to learn algebra, geometry and calculus without a
proper command of arithmetic are mistaken. Later skills depend on
earlier ones. If skills are skipped or are not properly mastered,
difficulties will follow.
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Learning to think and act carefully is one reason for
arithmetic mastery. But careful thinking can also be learnt in
logic. It is a language skill. Seeing the difference between
saying A IF B and saying A IF AND ONLY IF B is a key step along
the path to careful thinking, reading and writing. The student
who learn to figure well and think, read and write with precision
may avoid some bad decisions - avoid actions or agreements in
which the numbers or words are not favourable. Learning to figure
and think careful is not just a matter of mastery useful skills,
it is a matter of self-defense.
Skills and practices may be mastered with full, partial or
no comprehension of why. In my pure mathematics education, the
axiomatic method provided a logical home for understanding and
eventually explaining university and late secondary mathematics.
In this home, the statement of axioms for sets, algebra or
geometry gave a base and framework for a deductive comprehension
for a large part of pure mathematics at the high school and
college mathematics. In some school districts, the axiomatic
approach may still be strong. In others, only a ghost may remain.
In retrospect, the axiomatic method I met - swallowed hook, line
and sinker - did not fully sanction all skills met in practice in
mathematics itself, in science and on the street. There were some
logical practical and instructional gaps. See Volume 1 and
1B.
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Counting and calculating with decimals, fractions and signs may
be mastered through a mix of rote learning and comprehension. For
learning to do and record work in steps that can be seen, students need
to be shown how with enough explanation to do - to understand how.
Rules and methods for doing and recording work in steps that be seened
skill and practice observable, repeatable and reproducible.
Mechanical rigour in logic, arithmetic, algebra, geometry and
calculus may be seen in the first instance as the ability to do, record
and present work in steps that can be seen as done or later for
confirmation correction. This mechanical rigour requires adequate
explanation and comprehension of how work may be done in visible steps.
Deductive rigour starts to appear in mathematics instruction when the
recorded steps are accompanied or extended by the statement of reasons
for them. The statement visible as written provide can also be
confirmed or corrected.
From pure mathematics, we have the notion that mastery of
skills and practices should be part of an axiomatic framework, with
reasons to justify each step fully understood if not written, in
practice steps may be learnt by rote or done automatically in ways
that require avoidance of the domino effect of mistakes. Rigourous
mastery of arithmetic, counting and measuring practices only requires
the ability to do in an observable manner for the sake of
confirmation and correction. In this, rigour consists of knowing how
to do in an observable, repeatable and reproducible manner. Higher
level rigour as in a deeper comprehension of the origin and a
thought-based justification of methods may come later in an optional
manner.
Rigour in algebra and geometry may vary from learning to do by
rote or automatically to understanding the nuances as to why each
step and substep is justified. In the latter, learning to record the
nuances or reason for each steps and substep could be part of skill
development. When the latter is emphasized, theory or greater
deductive rigour becomes within the reach of observable and
verifiable development and mastery of skills and practices.
Standards and Principles for Mathematics Instruction as Skill Development
Primary and secondary mathematics may cover several different modules.
Basic Skills. Modules with take-home value. Independent as possible for accessibility.
Dependent of some modules and submodules.
Primary School Modules with Context and Motivation
Presently, pre-school and primary school instruction of children 3+ to 13 develops skills, practices
and abilities with clear, likely or historical value for life at home; on the street, work includedl and for
further instruction. The work value of some skills may be more remote today due to higher school leaving
ages. The work- and take-home value of skills may also vary between actual and likely, and less likely,
due to large variations in home life in rural, urban and further communities.
To varying degrees, most of the following topics represents overlapping and mutually independent
sets of skills and practices with actual or potential value in family and community life.
- Counting Wholes and Arithmetic
- Counting elements of disjoint sets.
- Counting elements of overlapping sets by forming disjoint sets - a pratical alternative to the law of inclusion-exclusion
First example: Counting people when each has bought a different number of tickets. Here each ticket corresponds to a membership in a set.
- Counting Fractions and Arithmetic
- Whole and Fractional Multiples of Units
- Measurement and Calculations with units and fractions of units - TLM alone and in fractions as part of or besides physical circumstances.
- Equivalent Counts and Measures - Different ways to describe how many or much.
- Time and Date Matters - Basic to Advanced.
- Money Matters - Counting and investing money
- Money Matters - Buying and Selling Goods and Services
- Fractions with Units + Forward and Backward use of Rates
- Matters of Chance - Introduction to Risk Taking/Avoidance
- Spatial Sense - Position, above, below, besides, behind, in front of, nearer, farther.
- Geometry with Rulers, Tape Measures and Protractors
- Geometry with maps, plans and drawings
- Geometry with formulas, Counting and Arithmetic
- Logic Matters - Chains of Reason, jigsaw puzzles and logic puzzles
- Evaluation of Formulas and Arithmetic Expressions
- Activities at home, at school and on the street involving numbers and geometry
- Domino effect of care and diligence
- Graphs, Tables, Statistics, Variation and Dependence.
- > Signed Numbers for Coordinates; for debts and assets; for movements; for multiplication.
Secondary school mathematics will cover the algebra and geometry required for calculus- and probability-based
college programs in STEM, and some of the statistics employed in college programs outside of STEM. The coverage of
statistics will avoid needless complexity - delicate question of no service to STEM but only STEM prepared students
can answer.Keep the good parts of statistics - my adverse reaction to it lies in coverage too complex while
more basic or fundamental or useful skills go uncovered - a question of priorities.
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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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