Why Bother - Context
and Motivations for Mathematics Education - which
ones are convincing?
Page Sections: [What Will You Like?] [Why Bother -
Context and Motivation for Mathematics Education] [Common Benefits
versus Technical Needs] [Yet Another New ]
[Learning by
Rote or With Understanding] [Horrible
(Pointless) Course Design/Instruction] [Horrible
(Unreliable) Mathematics and Teachers Certification
Practices] [End Notes]
Key Questions: What observable skills, if any,
do you want to see in the mathematics education of yourself or others?
How should they learnt or taught, why and when?
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"Would you tell me, please, which way I ought
to go from here?"
"That depends a good deal on where you want to get to," said
the Cat.
"I don’t much care where--" said Alice.
"Then it doesn’t matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an
explanation.
"Oh, you’re sure to do that," said the Cat, "if you only walk
long enough."
(Alice's Adventures in Wonderland, Chapter
6)
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Site lessons and directions for learning and teaching
provides direct, very detailed answers to Alice's question at least
in the subject of mathematics education. The motivation for that stems
from observation of common fears and difficulties, and from course
design and delivery in school systems which may give students a decade
or more of mathematics lessons with no observable result except for an
absence of skill and confidence. That is absurd.
Site books and topics span many topics seen in
mathematics and logic. The presentation has two motivations. The
first is make the hard easier to learn and teach. As a student and
teacher, I sensed or saw some gaps in skill and concept development.
Here are my remedies. The second motivation is to provide a coherent
thought-based or logical development of skills and topics. All is done
at least in part via the presentation of starter and further lessons,
fresh or recycled, in site sections. The thought-based development of
skills and concepts given or outlined in site pages may help those who
want to understand as well as do.
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Volume 1B, Mathematic
Curriculum Notes, begins with inductive or progressive
principles for observable skill development, continues with a
discussion of barriers to skill and concept development, and
reflects on possible remedies. But the question of goals and
objectives, or ends and values for mathematics education, was not
addressed.
People may keep their thoughts and conclusion
private. However, the ability to write and draw on paper or on
screen allows people to develop and share their thoughts and
conclusions, step by step, in an observable manner for the sake
of communication and verification or correction, in the process
develop common knowledge. Or, dreams may be located in the mind
in a private manner, apart from reality, but rational ideas
located in the mind are those which can be discussed and refined
on paper or media which serve to extend and record our minds and
memories.
The ability to write and draw steps in a manner that
peers in the form of co-workers, fellow students and supervisors
may see and judge in terms of content and completeness implies
skill and knowledge into public form that turns instruction into an
observable and verifiable affair for better and worse.
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Primary and Preschool Mathematics - the
beginning: Eighteen short and inexpensive booklets
available in bookstores provide parents and teachers, skill and
concept pathways at the
preschool to grade 3 and at the grade
4 to 8 levels. Booklet content give exercises and short
explanations that parents may give children or preteens to
check and develop skills and concepts. See if the grade
4 to 8 booklets can be completed before grade 7 or 8
begins. Learning how to do and apply arithmetic carefully
and fully with decimals, fractions and even signs is needed is
needed in daily life, so much so, that learning how by rote is
justified where explanations overwhelm. The same may be said
of map and plan usage, money matters, time and date matters and
measurement matters - those involving length, time, amount
alone or in the description of rates and proportions.
Selecting those booklets, reading them from end to end,
provides a standard and lower bound for primary and preschool
mathematics. It further provides a rational base for site
junior high school mathematics guides.
Towards the end of primary school and during secondary
mathematics, these ends,
values and habits for skilful and observable work and
learning need to be emphasized. Thoughts cannot be read. They
need to be expressed and recorded.
-
Junior High School Mathematics - the
middle: Three guides for arithmetic,
algebra and
geometry
identify skills to master and say how to to do so, one at a
time, one after another, with the aid of site
material. Logic mastery in
seeing the difference between one and two implications,
using implication rules one at time, one after another in
chains of reason could be part of this step or the next -
the earlier the better as long as that does not overwhelm
students. - the earlier the better because logic mastery by
testing and improving precision in reading and write is known
to ease or avoid learning difficulties. The skills
emphasized in the guides reflect the twin objectives of serving
common needs and (ii) preparing students for calculus in light
to full-strength forms. The guides include methods, old
and new, to give a full and firm base for (i) and (ii).
Remedial college and remedial secondary level education may
follow these guides.
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Senior High School or First Year College
Mathematics - three ends or three bases for further
instruction
A first
common, base part gives
-
a natural stopping point for students who
would like to would end their mathematics, with some topics
and skill that have take-home value - serve common need -
while a quick view of the role of logic in mathematics.
There is more to mathematics than being given a method and
data to use in it; and
-
a base for further studies for students who
plan to pursue intermediate or advance studies in
mathematics, science, engineering and commerce at an
intermediate or advanced level.
This second
middle part gives
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preparation for a light form of
calculus.
-
a light form of calculus sufficient as end in
itself, or as an
appetizer for those going on to the strong form
This third
and last part (35% done) describes
-
Calculus with proofs
-
preparation for calculus with proofs.
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Posing the question of what observable skills should met
and mastered posits a viewpoint of education in which the ability to
obtain results and to express ideas in a visible form for peer or
teacher review and interaction has great value - is an end for
instruction. While education may lead to private thoughts with great
freedom, the material world demands skilful mastery of rules and
practices from application to, if possible, the ability to develop that
observable mastery in others.
Site pages not only provide goals for education, site
pages also say how to meet the goals with the aid of appetizers and
lessons, fresh or recycled. Starter lessons and alternative routes
may make skill and concept easier to learn and teach. The net result is
an alternative curriculum for secondary mathematics education from
arithmetic to calculus plus answers and questions about what be met and
mastered in the service of common needs of daily life at home, at work
and study, and after that in the service of technical needs of trades
and professions, or mathematics itself.
.
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Primary School Level: Practical and common needs
are served by learning to count, do arithmetic carefully, master time
and date matters, master money matters for buying and selling goods,
work and calculate with measures, use maps and plans to find or
estimate lengths, areas, angles and location. Learning about the
domino effect of errors in calculations leads to incorrect results
should imply better work and study habits in situations where rules and
patterns need to be applied carefully, one at a time, one after
another, to arrive at good results. The care required to figure well
is an observable sign of diligence or wits of the practical
kind.
-
Current Secondary School level - Preparation for
Final Examination: After primary school, many students and
teachers do not know why skills and topics are covered, except that
their mastery is likely to be required by final examinations. There is
something rotten in that. (The site author as a teacher has had to
teach course which contained material not of service to students, but
still required for graduation.
-
University level Science and Engineering
Instruction; Courses are demanding. Students in being admitted are
given the chance to succeed and to prove that they are able, but
success is not guaranteed and indeed about half the students in
university level calculus will fail or drop-out. Science and
Engineering programs use mathematics course to select among the able,
those willing to sit down and study.
The initial educational aim of developing reading, writing and arithmetic
skills prepares students for adult life. In well-off societies, five to
12 years of schooling is required by laws for the sake of child and their
futures. In poorer societies, going to school is a privilege and not a
right nor an obligation. That is unfortunate. Reading, writing and
arithmetic was the first aim of primary level schooling. Further
education of young teens, or apprenticeships in the workplace, has had
the tasks of preparing students for trades or for further studies or the
task of polishing social skills manners or the task of keeping people in
school instead of being idle on the streets.
According to the 21st
year book of the National Council of Teachers of Mathematics, 1953,
instructors and course designers should be learning engineers presumably
for skill and concept mastery. The viewpoint of education that says true
knowledge is a private affair, located in the mind, apart from observable
and verifiable skill development shifts education from the material to
the immaterial and does not favour observable and verifiable skill
mastery. Oops.
Aiming for observable and hence verifiable & correctable mastery of
skills and methods gives a tangible, material, concrete goals and
pathways for instruction and self-instruction - lean, critical or just in
time, as you like. A do-this, do-that approach for instruction
from elementary to advance levels with the focus on skill development,
one small step at a time, one small step after another, could build
confidence and give a viable, accessible, operational command of
mathematics and logic at many levels.
Competence, communication, reason and problem solving
can all be described concrete in terms of observable. And in terms of
skills, student centered education would mean providing skills and ends
with take home value that serve common needs first, and in building
abilities for work and study provide confidence and self-esteem.
Reality Check: Economic conditions that provide employment at
the end of a short or prolonged stay in secondary and college systems
would help as well with self-esteem. It takes a village and a viable,
sustainable, economic prospects to raise a child with confidence and
hope.
While some people complain that too people are not doing
real work, the use of machines and energy in farming, fishing,
construction and mining implies fewer hands are needed to do perform
physical labor needed to provide food, shelter clothes, medicine and
construction. That leaves more time for idle time, office work and
goods and services, optional or essential. Many strive to be part of
the flow, control and design of goods and services, essential and
optional as individuals or as employees or employers in private and
public affaires. The main problem facing society is the provision or
lubrication of goods and services, all in a way that will not lead to a
population whose demands exceeds local or global resources, or to an
impoverished population largely apart from the flow of goods and
services. Failure to plan is planning for failure. While governments
should not be in full charge of economics due to the nature of
bureaucratic decision making, governments need to provide limits and to
provide safety nets or emergency rules and plans to provide rations and
basic necessities. Idle hands are avoided via people working in
service industries (education, health, government) in societies where
mechanization and greater productivity implies more can be produced
with fewer people gainfully employed.
Page Sections: [What Will You Like?] [Why Bother -
Context and Motivation for Mathematics Education] [Common Benefits
versus or with Technical Needs] [Yet Another New ]
[Learning by
Rote or With Understanding] [Horrible
(Pointless) Course Design/Instruction] [Horrible
(Unreliable) Mathematics and Teachers Certification
Practices] [End Notes]
Mathematics and logic education may serve both common and technical
needs. Common needs are presently served by the development (we hope) in
primary school mathematics.
As said above, practical and common needs are served by
learning to count, do arithmetic carefully, master time and date
matters, master money matters for buying and selling goods, work and
calculate with measures, use maps and plans to find or estimate
lengths, areas, angles and location. Learning about the domino effect
of errors in calculations leads to incorrect results should imply
better work and study habits in situations where rules and patterns
need to be applied carefully, one at a time, one after another, to
arrive at good results. The care required to figure well is an
observable sign of diligence or wits of the practical kind.
A practical knowledge of geometry would entail students
have experience with measuring actual distances and angles in their
environments and in construction projects, and also in doing geometric
figuring, surveying and navigation with maps and plans using land areas
or objects drawn to scale. That practical knowledge of drawing to scale
(similarity applications) may be provided by any formal discussion of
similarity and trigonometry.
Games of chance and risk present in daily life provide a
context for introducing probability theory. Money matters may range
from knowing the monetary value of paper and coins, their use in buying
and selling goods and services with percent mark-ups, discounts and
commissions. Money matters may also include cautions regarding the
balancing of personal, household and business income and
costs.
Money matters of the more technical kind would employ compound interest
and geometric sum formulas forwards and backwards. The full coverage of
probability and money matters falls in the category of serving common
needs that have technical prerequisites. The latter should be presented
in a lean or minimal manner, so that the technical support an a practical
or operational command, and all further details are absent or delayed
until later study.. For example, an geometric summation formula may be
mastered numerically with no general explanation of why the formula
works. Explanations may be left for later study, or provided as a
reference where covering in class would be overwhelming for the students
and/or instructors present. Skills and concepts with the greatest
immediate or likely take-home value or benefit should be put first and as
early as possible too, given the risk that students will drop out. Do
that may lessen the risk, or at least provide mathematics education with
take-home value to those students.
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In education that puts common needs firsts, the objective is to
develop a practical operational command of rules, patterns and
formulas sufficient in the first instance for solving common or
routine problems in a repeatable and reproducible manner. In
daily life, the ability to follow steps for handling routine
task is more important than theory or explanations why in the
first instance. Of course explanations why that do not
overwhelm and do not distract from the ability to do are of no
harm and may even provide an deeper understanding necessary for
the variation of methods. And explanations why may give or
imply mastery of a whole family of methods and patterns for
handling a family of like problems.
Explanations why numerical, geometric or algebraic methods work
are less important than the ability to apply them with steps
done and recorded in an observable and hence verifiable or
correctable manner to give results, intermediate to last. That
sets a standard for showing work for work and studies in
general, and so serves a common need - provides a first
benefit..
Reference: Ends,
Values, Methods for Work and Study.
Showing work is a form of proof. In the later study of proof
in or outside of mathematics, steps involving the forward
and backward use of implication rules may done and recorded
as well for the sake of observable and thus verifiable or
correctable conclusions, one at a time, one after another.
Proof in mathematics and showing work in ways that imply
result is a mechanical affair, a mechanical method for
showing reason and communication results.
There is a progression. In doing and recording arithmetic,
geometric or formula evaluation steps in observable and
verifiable manner, people may learn to show work and value it
as proof of correctness for the work done. Doing work
carefully requires and encourages precision in the written and
drawn elements etc of each step. Seeing how to employ
implication rules If A then B directly to arrive at
conclusions, one at a time, one after another, is not more
complicated nor challenging than doing arithmetic steps one at
a time at time, one after another. But seeing the difference
between saying B if A and B if and only if A, and
seeing how how an implication rule holds if and only if
contra positive holds is more complicated to understand and
explain. Unlike arithmetic, the latter may be difficult for a
pre-teen and much easier for a student who 15 or so to
understand.
Another Benefit: Mastering the difference between saying
B if A and B if and only if A will sharpen
reading and writing abilities, and serve common needs because
sooner or later people will have instructions and contracts or
agreements to sign or avoid.
Reference: Logic Chapters 1 to
5 in Volume 2 give a mathematics free introduction to the
direct and indirect use of implication rules, alone and in
combination; to the difference between saying B if A
and B if and only if A; and to the codification or
axiomization of islands of pattern based knowledge into
deductive bodies. The foregoing provides a
mathematics-free introduction the geometric or Euclidean
style of reason in mathematics. See the site section
of Euclidean-Geometry
to informally see logic may appears in mathematics.
In preschool, the puzzle of fitting round, square and other
shapes into receptacles may introduce children to the concept
of trial and error. Jigsaw puzzles with few and then many
pieces further provide experience with the identification of
pieces likely to fit together due to like shapes or visual
clues, coupled with provides more trial and error experience in
mechanical and observable problem solving. Problem solving may
continue in mathematics with the identification of which
formula or method to employ to calculate or find an area,
perimeter, volume. Problem solving may be routine or not.
Mechanical experience in using rules and patterns carefully,
one at a time, one after another, alone or in combination
provides a base for pattern-based problem solving in general:
Given a problem, we try to remember or find a routine (in the
box) method to handle it and to do so in a show work manner
that implies to ourselves or others that method is applied
correctly. Thinking out of the box is only needed when we are
not familiar the problem at hand and its solution. Calls for
problem solving in mathematics and in general need to be
modified to emphasize that thinking out of the box should be
reserved for new problems and not well-know ones. The students
who has to think out of the box in order to solve a routine
problem may be demonstrating both great intelligence and
deficient training or self-application in the study of an art
or discipline. That being, giving students problems of an
unfamilar type to solve may require them to combine and extend
or go beyond what they have learnt previously. That can be good
exercise for developing opportunistic, trial and error, looking
at matters differently, problem solving.
Remark: Calls to provide students with
rich problems are fine. That works as long as students are
engaged in the problem solving process. However, asking
students to handle or run with rich problems may require more
out of the box thinking that necessary if students (i)
ability to solve routine problems; and (ii) their Ends,
Values, Methods for Work and Study. are both weak or
nonexistent.
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Students of art and design and computer games may learn
geometric views and projections. Surveyors and navigators may need a
knowledge of trigonometry functions of acute and obtuse angles.
Students heading into construction and electrical fields should know
about the use of maps and plans drawn to scale with solution of
triangles with trigonometry being a plus. Those heading into
electrical trades should know about phasors or complex numbers in the
description of alternating current.
Students heading for university studies in engineering, science, commerce
and itself need to master the mathematical subject of calculus,
elementary to advance level, in m The latter provide a technical language
for the expressions and development of computational skills and concepts.
Since students with different academic destination may be taught in the
same high school courses or program, the technical preparation of all
possible destinations needs to be covered by course design in mathematics
and/or science. And in that strength or weakness in the destination
subject and in mathematics may decide the academic destination or how far
each student continues in each.
Preparation for an operational command of calculus requires a mastery of
the following.
-
Arithmetic - exactly and efficiently without and with
calculators
- Arithmetic - Properties of Operations on Numbers
- Algebra (solving linear equations, forward and backward use of
geometric formulas etc.)
-
Analytic and Euclidean geometry
- Slopes, Lines and Direct Proportionality Relations
- Complex Numbers
-
Logic (Direct and Indirect Use of Implications, Difference
between one and two way implications, Recognizing when one hypotheses
is inconsistent with another.)
-
Right Triangle Trig Ratios for acute and even obtuse angles
- Periodic Trig Functions
- Quadratics in one and two variables
- Polynomials and their Ratios
-
Natural logarithms and exponentials, Powers and Radicals
-
Sets and Counting
-
Probability
-
Vectors
- Mathematical terminology and practices - Functions included
The topic emphasized in bold above serve common needs directly or
indirectly. For example providing the algebraic background for forward
and backward use of formulas in money calculations (loans, mortgages,
return on investment) and in refining or supplanting map and plan use
skills will help students handle common or likely problems in the daily
lives of themselves and immediate or future families. Thus skill and
concept with some take- home value is present.
If students heading for university studies are in a competitive
environment where continuation is based on academic performance due to
personal reaction to failure or due to elimination of students based on
their performance, we should mix and emphasize first those skills and
concepts with greatest take home-value with those skills and concepts
easiest for students to master.
Given that students may halt their studies or become discourage at any
time, there is no harm in attempting to put first or as early possible
those elements of the preparation for calculus providing skills with the
greatest take home value, that of greatest service to likely or common
needs. After that, students and teachers should be made aware that
further elements in the preparation are present because of the needs of
calculus or further studies. Those topics and how they are of service
should be labeled as such.
Students of probability may appreciate the role of sets (and even
functions) in formulating probability theories and in aiding the
identification and counting of possibilities. Remember a knowledge of
probability is needed for estimating odds or chances of success and
failure, and in that estimating expected returns or losses in cases
decisions are being made in the face of uncertainty. That uncertainty or
risk may be environmental and unavoidable. Or the uncertainty and risk
may be due to playing games of chance. The latter may range from throwing
dice, drawing a card, betting or making an investment. Thus a knowledge
of probability has take-home value in helping students or their families
avoid or minimize risks. That being said, expected value of choices or
outcome may justify some risk.
Students in physics, chemistry and biology courses may see
- the forward and backward use of linear equations, quadratics and
vectors in precalculus description and analysis of motions and forces.
- the description of gas law and gravitational laws as formulas and
proportionality relations that may used forwards and backward, along with
some probability and geometric formulas to imply or suggest or interpret
the laws (physical properties).
- the forward and backward use of logarithms, exponentials, powers and
radicals in the description of compound and continuous growth and decay
in biology, physics and commerce - geometric sums may be employed in the
description of the periodic stocking of ponds with fish or material that
may be diluted or flow out.
- the role of probability and counting methods in genetics.
The foregoing provides a further context for topics met in the
preparation for calculus and beyond.
Site coverage of complex numbers show how
the latter may be introduced with a high school level of rigor before
the introduction of periodic functions. The development is very simple.
The development is based on a technical item - a very simple geometric
proof of the distributive law for multiplication over addition. Before
the coverage of complex numbers, the properties of real numbers, those
given as axioms in modern mathematics curricula, can be derived or
implied by mathematical practices - those met and essential in the
earlier development of quantitative skills in the service of common
needs. The foregoing development implies the continuity between
earlier and later instruction - and a convergence with the modern
mathematics secondary curricula of the 1960s met in some American,
some European and some Asian schools.
"When I use a word," Humpty Dumpty said in rather a
scornful tone, "it means just what I choose it to mean -- neither
more nor less."
"The question is," said Alice, "whether you can make words mean so
many different things."
"The question is," said Humpty Dumpty, "which is to be master - -
that's all."
(Through the Looking Glass, Chapter
6)
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The inclusion in the preparation for calculus of modern mathematical
concepts and notation in the form of sets and in the set-based
description and codification needs to be lightly done, and to be
included in a just in time manner where it speed and aids skill and
concept development. Mathematics for the person in the street and the
for student outside of mathematics may a subject where notation is
employed where is helps instruction and not overwhelm students and
teachers. The underlying question: which is to be more important, that
is, the aim of making mathematics simpler and more accessible, and not
overwhelming, or the aim of introducing and emphasizing modern
mathematics notation and terms. The skill development program in the
parents area Help your Child/Teen Learn
gives a compromise - it propose a notational and conceptual light
introduction and development of calculus - an operational viewpoint -
be given first.
Skill development in mathematics and further disciplines to be student
centered needs to be put first those skills and concepts with greatest
take home value. In particular, primary and secondary school courses
should provide students with an operational command of skills likely to
be needed sooner or later at home or in the typical workplace. That
latter does not involve employment in science or engineering
Page Sections: [What Will You Like?] [Why Bother - Context
and Motivation for Mathematics Education] [Common Benefits
versus Technical Needs] [Yet Another New ]
[Learning by Rote or With Understanding] [Horrible
(Pointless) Course Design/Instruction] [Horrible
(Unreliable) Mathematics Teachers Certification Practices] [End Notes]
Many students want mathematical methods to be given in plug and play
manner. Many may think that schools hire instructors to present only
those methods which are correct. Thus explanations of why are not
needed. Indeed explanations of why and notation that may overwhelm may
overwhelm students or instructors not all full trained in mathematics.
Skills and concepts at the primary school and junior high school level
may be developed or given in a do this, do that manner with explanations
and notation include only when and where they aid the ability to do. The
aforementioned ends, values and methods for work and study. As indicated
above, learning to do and describe (record) steps, one at a time, one
after another, in a way that show work or allows the doer and others to
see what what is done makes the application of a method observable and
verifiable. Learning to show work in such a manner gives a model for
formal or deductive proofs. Opposition to explanations of why methods
work and to deductive proofs may decrease overtime as students become
more experience in doing and recording the steps of a method to obtain
and display results, intermediate to advanced.
At a practical level, mathematical and logical methods and routines may
be learnt and taught in a rote, do-this, do that manner. Then skill and
confidence is based on the ability to obtain with repeatable and
reproducible results in an visible and verifiable manner. Explanations
can be incidental and should be when and where the aim is to develop
quantitative skills with intermediate or long-term take-home value. But
once common needs have been met, the deductive nature of mathematics can
be introduced progressively but lightly.
For example, to learn and teach the addition and multiplication of
polynomials, associative and distributive laws are needed and employed
in a very general manner. To be mathematically rigorous in that
development, starting with the three number associative and
distributive laws for real numbers would require a long and detailed
argument one that would overwhelm students and distract from providing
them with an operational command of algebra. The site development is
informal but sufficient to provide an operational command of operations
with polynomials.
Secondary mathematics education cannot develop all properties of numbers
and numerical objects from a minimal set of axioms or practices -that
would be overwhelming for instructors and most students. Instead, more
feasible, secondary mathematics education has to provide an operational
command of an empirically consistent set of axioms or practices.
Technical concerns about rigor and minimal sets can be left to
undergraduate studies in mathematics, studies taken by the few rather
than than the many.
The gradual introduction and emphasis of proofs and the thought-based
development or origin of practices, and the deductive relations between
them may set the stage for an appreciation of the possibility and an
even an appreciation of mathematics as a subject in which most skills
and operation may be understood in a thought-based manner. That
provides an ideal or model for reason in the further study of
mathematics, science and law. The author of a story for the sake of
consistency will avoid "facts" or assumptions that lead to inconsistent
or contradictory events.
For keen or advance students, mathematics is very different from other
quantitative arts and disciplines. Measures of length, mass and time
in physics and chemistry may be made with simple instruments. Simple
formulas may be verified empirically. But after that, the study of
physics and chemistry becomes a plug-and-play matter. Chemical
substances arrive in containers identified by a label. Electronic and
further instruments are black boxes with inputs and outputs, with
innards and their operations unknown. In chemistry and physics,
students have no choice but to hope that their schools have hired
teachers who present correct and reliable methods. In chemistry and
physics, the physical properties of matter are given and tabulated
with the aid of numbers and formulas. But the thought-based development
of those formulas is largely absent and beyond empirical verification
or derivation in the high school science classroom despite the presence
of experiments and shallow hypotheses testing practices. Yet in
contrast, the thought- and logical development of mathematics skills
and concepts is possible, and can be provided as a reference (see site
pages) for students.
Page Sections: [What Will You Like?] [Why Bother - Context
and Motivation for Mathematics Education] [Common Benefits
versus Technical Needs] [Yet Another New ]
[Learning by
Rote or With Understanding] [Horrible (Pointless) Course
Design/Instruction] [Horrible
(Unreliable) Mathematics Teachers Certification Practices] [End Notes]
Education that promotes students from one grade into another, year after
year, without providing basic skills - those with take home value in
urban and rural societies is not credible. Through this promotion
mechanism students complete primary and secondary school without ends and
values necessary for skill-based and skill-oriented work and study.
With mindless promotion, student may be enrolled in classes that cover
the technical topics required for calculus and beyond in mathematics
and science without the basic mathematics skill needed to serve common
needs and also to provide a base for the technical preparation for
calculus. Thus we have students trying to learn to add, subtract,
multiply and divide polynomials, meeting a technical need and employing
the time and energy they could have directed to the development of
skills and concepts with take-home value. The foregoing practices
leaves students with an incoherent view of mathematics. A teacher or
coach may take pleasure in preparing students for future studies or
careers. Mathematics education at the primary and secondary level
should focus on providing students with observable mastery of skill and
concepts, those with take-home value put first or as early as possible
in regular and remedial instruction. But teaching becomes a
bureaucratic profession when course design avoids or does not emphasize
skills with take-home and long-term value to the students in a given
class.
Page Sections: [What Will You Like?] [Why Bother -
Context and Motivation for Mathematics Education] [Common Benefits
versus Technical Needs] [Yet Another New ]
[Learning by
Rote or With Understanding] [Horrible
(Pointless) Course Design/Instruction] [Horrible (Unreliable)
Teachers Certification Practices] [End
Notes]
In modern times, schools and faculties of education may in accordance
with local regulations for teacher certification place student teachers
in pass or fail teaching practices in primary and secondary schools. But
there are or can be a few problems.
- Host Instructors (HIs) for mathematics and science teaching practices
are not necessarily screened for good classroom management practices nor
for subject knowledge. Without screening and without any training, host
instructors may be asked and given the power to say what is right or
wrong. That may lead to situations where older student teacher trainees
with more knowledge than the host instructor are expected to comply with
false or incorrect practices for lesson design and delivery. In some
North American states or provinces, about 50% of secondary mathematics
teachers are not be formally trained in mathematics or a quantitative
discipline, but having being allowed to teach secondary mathematics, are
permitted to serve as host instructors for teaching practices.
- Schools or Faculties of Education may appoint supervisors for
mathematics and science teaching practices, retired teachers or
principals, who are unversed in the subjects covered by the teaching
practice, and who may assume the host instructor being certified is
well-versed in the subject matter. Thus supervisors need not provide any
check nor balance for the unscreened and hence unpredictable skills and
expectations of host teachers for the subject matter at hand.
- Schools and Faculties of Education may require a mastery of calculus
for entrance into secondary mathematics teacher training programs, but
not provide evidence that their Professors of Mathematics Education have
mastered calculus. In secondary mathematics education, a knowledge of
calculus is necessary to see and understand why secondary mathematics
programs should aim for a full and mathematically correct mastery of
arithmetic, algebra, trigonometry, functions and all further
prerequisites to calculus. Secondary mathematics teacher programs are
substandard when given by Schools and Faculty of Education whose
Professors of Mathematics, Secondary and even Primary level, lack a
command of calculus.
- (A). Local government documentation of course content - what should
be taught at each level may be technically incoherent, incomplete or
incomprehensible. For example the Quebec 1990s documentation of its
secondary mathematics education program is too incoherent for the site
author to determine what was taught from its release, grade by grade,
from 1993 to 1996 say. The current Quebec secondary mathematics program
claims continuity with that earlier 1990s program. That is absurd. Thus
the local documentation, written at great cost in time and effort, stands
on thin ice. (B.) Local government education regulations may require the
use of textbooks too incoherent and incomprehensible for rational use by
anyone well-versed in mathematics. The G. Breton textbooks in use say
1995-2005 in Quebec (English translations) falls in that category. (C).
Local composition of mathematics final examination may vary in quality
and difficulty. But in an empirical manner determine what should be
taught or emphasized. (D) Local teacher training programs may focus on
the right way to teach (delivery style and classroom management), while
remaining silent on difficulties (A), (B) and (C).
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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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