Advice and Suggestions for Course Design and Delivery.
What to tell Students
1. Aim for Care and Precision
in reading, writing, reason and arithmetic.
Understanding that carefulness and precision are needed through will ease
or avoid confusion and difficulty in work and studies.
Online logic chapters
2 to 5 from site book Three Skills for
Algebra will test or develop care and precision in reading and
writing.
Testing or developing precision reading and writing
provides a first step in building skills and confidence for work and
studies in many arts and disciplines. It should also the first
step for the greatest benefit from site appetizers and
lessons.
While logic chapter 2
to 5 in the online book Three Skills for
Algebra may lead to precision in reading, writing,
reasoning, these online arithmetic review
exercises could lead to precision in arithmetic. Many
students are not aware of the need for results in arithmetic to
repeatable and reproducible, and hence verifiable - that, is they can
be checked.
The arithmetic review
exercises need not be done now. When you try them,
exercises involving operations or calculator buttons you have yet to
meet can be skipped until later.
The ability to figure well, without errors, that is in a repeatable,
reproducible and hence verifiable manner, was once considered to be a
sign of intelligence, a sign of skill and competence. It is
because a person who does arithmetic well has learnt to follow
multi-step methods with decimals, fractions or both, and has learnt
that lack of care in one step of a method usually leads to wrong or
inaccurate results. So person is aware of the need for care and
precision while following instructions or writing and giving
them. Whence mastery of logic and learning to figure well all
imply or demonstrate greater care and precision for work and studies,
and so point to fewer difficulties for both work and studies.
Take note.
Tell Students: When you were young, to read,
write and spell, you learnt all the alphabet not just some.
With regrets, algebra and higher mathematics demands mastery of times
tables, arithmetic operations with decimals (addition, subtraction,
multiplication, long division) and with fractions. During arithmetic
mastery, you learnt or will learn that an error in one step of a
multi-step calculation usually leads to wrong results. So each step,
the work, has to be done with care and precision. The same or
similar care is needed in algebra, logic and beyond.
2. Aim for Fraction and Algebra Skills and Sense
The careful and precise use of whole numbers and fractions in solving
linear equations will provide a firmer base for further mathematics
instruction. Some steps can be followed or designed to avoid
integers and signed numbers. Emphasizing that solutions of linear
equations can be checked before being submitted for correction or marking
will eventually lead to greater student independence. When a check
fails, the mistake or mistakes fall between the start of the solution and
the end of the check.
For all high school and older students, the unique site area solving linear equations
introduces and illustrate fractional ops on line segments (sticks) to
geometrically develop algebra and fraction sense and skills. That with
solving triangular and essentially-one-unknown simultaneous equations
begins a new, yet tried and tested path to make solving word
problems, substitution, the distributive law, and solving
simultaneous equations much easier learn and teach. The geometric
approach is intended to promote the algebra approach, and it usually
does. Where it does not, students will need further help.
Fractional operations on stick diagrams to build algebra and
fraction skills and sense is a site invention - a co-invention
with an offline author.
While some students will want to leap into the algebraic way of
solving linear equations, staying a little longer with or leaving and
returning to the stick diagram approach will reinforce fractions skills
and sense. The fraction parts of the site area Fractions,
Ratios, Rates, Proportions & Units offers
fraction skills by rote, automation
is important, and also fraction skill development with more operations
on line segments to provide and perfect a thought-based mastery.
The latter is recommended only for (i) students who want to learn, and
enrich or perfect their skills and knowledge, and (ii) students
who dislike or want to go beyond rote learning of methods. That
being said, drill, practice and correction with fractions and solving
linear equations may build skills and confidence through the mastery of
methods with repeatable, reproducible and hence verifiable results.
3. Aim for a Greater Use of Words in Mathematics
Many people have difficulty with algebra. Even gifted or
advanced students will have gaps in their understanding of the
shorthand role of letters and symbols in mathematics. After all,
arithmetic and algebraic expressions, the longer ones, are difficult or
impossible to read aloud, symbol by symbol, bracket by bracket, and so
on. And when expressions are read aloud, there is the frustrating
possibility of confusion in the order of operations. The reader
may mean one thing, and the hearer may write something else. That is
not good for precision in reading and writing in arithmetic and
algebra. This difficulty in reading arithmetic and algebraic
expressions aloud, including those appear in the algebraic described
properties of arithmetic with numbers, leads to silence, a dearth of
words, in skill development and communication. We look at and
digest mathematical expression in a glance instead of reading them
aloud. Mathematics in and beyond arithmetic and algebra has been a
silent discipline where written letters and symbols, together or
separately, are used for communication. That makes learning and
teaching harder.
Site book Three
Skills for Algebra with its online postscripts point to a
greater and clearer use of words in mathematics in two different, but
mutually supporting ways,
-
Greater Use of Words in Describing Numbers, Amounts and
Quantities: Algebra chapters 8 and 9
identifies our ability to describe numbers, amounts and quantities
with written or spoken words before or besides symbols. The
online postscripts use words to explain what is a
variable, constant, or parameter, and do so without the use of
symbols in a manner that students can grasp years before
functions, a senior high school topic, are met. The online
postscript brings mathematics in closer alignment with physics where
numbers or quantities that may vary in one sense or another, are
called variables.
-
Greater Use of Words in Common Operations on Equations and
Formulas: Chapter 14 on
compound interest or growth formula introduces the direct and
indirect use of formulas or equations, and shows the difference
between algebraic and numerical solutions, and point to the
ability of the algebraic solution to give the numerical
solution in full, or just the end results.
For all formulas in high school and college mathematics, we may now
identify (A) direct and indirect use, and the (B) numerical
and algebraic solutions that may be possible in the indirect
use. The repeated use of two phrases in dealing with
formulas, one at a time and one after another, gives voice to a
previously unnamed and hence hidden operations and themes in
mathematics learning and teaching. Remembering the phrases (A) and
(B) while you study or teach will make the methods of algebraic
ways of reasoning clearer and provide a focus or two for their
study.
Words and names are powerful. Once an mathematical
object or operation is named, and clearly described with words,
we can use names and words, again and again to point out
recurring patterns. The foregoing lessen the silence that accompanies
arithmetic and algebraic expressions, formulas included, because they
are so awkward to read aloud term by term, parentheses by parentheses.
The foregoing introduces a new avenue for mathematical learning and
teaching. Formula, operations and properties of real numbers,
etc, known and named can be mentioned and discussed without being
present in written form. The result is or will be more written or
spoken communication in mathematics based on words to supplement or go
beyond expressions better seen and read silently at glance, than read
aloud in a way that communicates order of operations clearly and
precisely.
4. Develop More Algebraic Thinking Skills, Those Needed for
Calculus
Preparation for the full strength use of algebraic ways of writing and
thinking in Calculus. The algebraic way of writing and reasoning is
required at full strength in calculus. This full strength requirement,
and algebra shock, is too sudden and terminal for many students.
This aim (and the previous ones) point to a remedy, one that many, not
all students have enjoyed.
The following methods or path for easing or avoiding algebra shock in
calculus may be seen at the start of calculus and prior to that, in
course in analytic geometry after or as part of the discussion of slopes
to straight lines and the factorization of polynomials alone or in the
numerators and denominators of quotients (rational functions).
Calculus gives the best framework for understanding
calculations met in business, science and engineering. Describing the
same calculation without calculus is long and shallower process.
Shortcomings in the development o algebraic skills and concepts, those
needed for calculus, led schools and course design to fill student with
topics not needed for calculus and supposedly simpler. Good preparation
for college mathematics (or calculus) requires mastery of most,
but not all the topics, you meet in high school mathematics: exact
arithmetic with whole numbers and fractions, algebra, geometry without
and with coordinates, and trig.
Calculus in the first instance is a subject of slope and rate related
calculations, as is or reversed, with applications.
The online version of site Volume 3, Why Slopes and More
Mathematics, includes a geometric calculus
preview before a more algebraic perspective in chapters 2 to 6 . The
geometric
calculus preview explain how slope related calculations, forward,
not reversed, appear in calculus. That gives context or explains
why slopes appear repeatedly in earlier high school and college
mathematics. In an courses where slopes and then polynomials and
rational functions are met, the geometric calculus
preview and chapters 2
to 6 could be used to (i) understand and explain
extreme points and identify where factored or easily factored
polynomials and rational functions are increasing or decreasing; and
(ii) to develop students algebraic reasoning concepts and skills.
The foregoing also provides a way to ease or avoid difficulties in
the first and further weeks of calculus.
Calculus is also the subject in which the many facets of what is a
variable, constant and parameters appear as well. That being said,
chapters 2 to 6 in
Why Slopes and More
Mathematics provide a slow and effective, induction, into the
algebraic way of writing and reasoning required.
The chapters with the aid of slope interpretation
identify interior and end-point extreme points (maximums and minimums).
Polynomial and rational function formulas given for slopes (they are
not computed in these chapters), and given in factor formed, are used
in slope sign analysis. The sign analysis of these factored
polynomials and rational functions indicate the intervals where a
function y = fix) is increasing or decreasing, and thus indicates the
interior or end-point location of extreme points. By skipping
over lengthy discussion of limits and derivative calculations to the
sign analysis of derivatives or slopes, these chapters provide a
context for the skipped material while developing the algebraic
maturirty needed to understand the skipped material.
5. Make Limits, Convergence and Continuity Easier
Bring Back, O Bring Back, the Bonny Decimals.
Calculus and mathematics instruction became more complicated in the
second half of the 20th Century due to the course design which employed
decimals to represent numbers and to do calculations, but which also
did not favor nor mention them in presenting theory. In
particular axioms for real numbers (algebraically described properties
of arithmetic with real numbers represented by decimals) do not
mention decimals in high school mathematics despite coverage of
scientific notation and decimal-baed accuracy or estimates in
arithmetic. Then in calculus, hard to grasp, extremely algebraic,
decimal free explanations of limits, continuity and convergence are
inaccessible. As a mathematics students, the site author
Professor Whyslopes, struggled repeatedly to understand the
decimal approach in technical and intuitive manner. For many
students, that decimal free approach is a very large barrier to
understanding. So today, calculus courses may teach students the
properties of limits, continuity and convergence by example (rote
learning) and not attempt to provide any theoretical framework.
The modern mathematics description of limits, continuity
and convergence is often decimal free. The modern mathematics
curricula of the 1950's and 1960s, echoes of which still appear in some
or all classrooms today, did not mention nor sanction the decimal
representation of real numbers and so did not sanction methods of
decimal arithmetic, while still employing in numerical results.
Moreover the decimal perspective of limits, continuity and convergence
was not mentioned or sanctioned as well.
Limits, Continuity and Convergence in Calculus.
A decimal viewpoint of limits, continuity, and convergence, and the
associated question of limited or unlimited error control in function
evaluation or computations, is sufficient for most students and
its provide an model which also makes the decimal -free viewpoints
easier to understand and grasp - provides a context for the
latter. Therefore chapter 14 in Why Slopes and More
Mathematics introduce the decimal viewpoint while the appendices to this volume push
(or review) the decimal into advanced calculus or real analysis. That
provides the proofs of theorems often given without in first and
further courses in calculus.
6. Aim for a Greater Use of Complex Numbers
This aim points to a change in course design and delivery at the
secondary and tertiary (college level). Senior high school and
college students may use these underlying ideas in their
self-instruction. Course design changes indicated here will most likely
not occur in their school days.
A simple and clear way to understand and explain complex numbers (site starter lesson,
pre-calculus level) is to introduce addition of points in the plane
using rectangular coordinates; to introduce their multiplication via
polar coordinates; and then to assume or geometrically imply the
arithmetic properties of complex numbers. Implicit here is the
assumption, that every point in the plane has both rectangular and
polar coordinates. From the numerical properties of complex
numbers, algebraically described, we can obtain several easy
consequences: a new proof or confirmation of the Pythagorean theorem,
the properties of trigonometric functions; and a geometric, complex
number development of trigonometry. Details are given in the site area
on complex numbers. All the
foregoing suggests simpler path for high school trigonometry and
simpler, complex number developments of trig expressions for dot and
cross-products of vectors in the coordinate plane. University
level schools of engineering and science will appreciate the shortcuts.
They can be also be used in senior high school mathematics before
calculus if time permits besides the other curriculum obligations.
Remark 1: Teachers will find several approaches to the
derivation of the arithmetic field properties of complex numbers in
site pages. All assume points in the planes drawn on paper can be
represented by ordered pairs of rectangular and polar
coordinates. All stem from different assumptions about Euclidean
geometry, the use of coordinates and/or real numbers. The most extreme
route is to (i) assume decimals can be used as coordinates, (ii)
assume the completeness of this representation by accepting infinite
decimals expansions as coordinates, and (iii) imply or derive the
properties of real numbers from assumptions about geometry and the use
of unit lengths and directions to define coordinates systems. The
latter route employs geometry instead of set theory to represent and
obtain the properties of real numbers. The latter route met first
could provide a context for modern mathematics even though modern is
context-free in another sense.
Remark 2 : Pure mathematics from the algebraic statement of
axioms (assumed patterns) provides a thought-based development or
codification of concepts and statement in mathematics with no logical
dependence on suggestive drawings nor physical argument. So pure
mathematics is said to be context-free , but there is a context for
this context-free development. In contrast to pure mathematics,
applied or mixed mathematics employ suggestive drawings and physical
arguments to introduce and employ coordinates or other device from pure
mathematics to model objects in space and time. That steps beyond
pure mathematics, while assuming those calculations in the model which
can be done in a mathematics or context free (formally or purely) way
in order not to accidental introduce further physical assumption into
the calculations or the results. Finally, primary,
secondary and tertiary mathematics from learning to count and
recognizing geometric shapes to the definition of trig functions using
right triangles and unit circles in a coordinate plane is not part of
pure mathematics. Before pure mathematics can begin, mathematics
education has to inductively (use of manipulatives and suggestive
drawings appears here) develop the necessary numerical,
algebraic and deductive skills and sense. The set theory axioms
of pure mathematics imply the axioms about real numbers which appeared
in modern mathematics course design. But there was a mistake.
Modern mathematics course design did not mention decimals and the
emphasis on context-free development was inconsistent with the high
school and college development of Euclidean Geometry, Trig and Calculus
with the aid of suggestive drawings.
Modern mathematics course design inserted set language into course
design - that description awkward in parts can be still be retained
and refined as it provides a finer or more precise language for the
development of some skills and concepts. How will be explained
While the logical development of pure mathematics is and should be
context-free to avoid dependence on suggestive drawings or physical
assumptions, both of which can be useful but misleading, pure
mathematics itself codifies the properties of real numbers used as a
coordinates in mixed or applied mathematics. Comprehension of how
and why provides a context for the avoidance of suggestive drawings and
physical arguments in pure mathematics.
7. Understand Three Kinds of Reason in Mathematics
There are three kinds of rule-based intelligence in mathematics, logic
and most pattern-based subjects.
-
The first kind met in primary school arithmetic consists of
skills with repeatable, reproducible and therefore verifiable results
- results that are then considered right or wrong.
-
The second kind also met in primary school consists of pattern or
rule recognition. The development or exploitation of the ability to
recognize or suggest simply patterns in order to predict the next
element in a sequence. If the prediction fails, another pattern is
required.
-
The third kind, assumption-based, deductive reason, appears after
inductive mastery of logic, that is mastery of implication rules If A
then B and their use. The third kind follows the use of implication
rules and definitions and assumptions, one at a time and one after
another, to arrive at logical conclusions. Here chains of reason how
to be posed in a readable, repeatable, reproducible and
therefore verifiable manner.
For third kind of thinking in mathematics, there was a search for
secure assumptions, so that deductive reason could proceed in a
consistent and reliable manner. Unfortunately, uncertainty
results in mathematical logic imply more can suggested than proven in
mathematical theories which are not finite. So the assumptions made for
the third kind of reason stem from experience or trial and error over
time. That identifies modern pure mathematics as another empirical
art. But mathematics by providing a format for measurement
and calculations remains the queen of science, a queen in the
hierarchy of empirical arts.
Pre-coordinate Euclidean geometry, the original model for pure
reason in mathematics, with its assumptions and deductive chains of
reason is still worth presenting in part if not in full in high school
mathematics in a selective manner to build algebraic-deductive skills
and geometric skills and sense. However, the empirical nature of
pre-coordinate and hence coordinate-free Euclidean Geometry is implied
by diagrams with subtle faults that imply incorrect conclusions -
subtleties detected with the use of coordinates in advance mathematics
courses.
|
|
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.
|
|