Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
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Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics.
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
Mathematical Induction - a light romantic view that becomes serious.
Responsibility Arguments - his, hers or no one's
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6.
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals.
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this.
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign.
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function?
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms.
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
Early High School Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
Use them not only for locating points but also for rotating and translating in the plane.
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. Many human-made objects
are similar by design.
Complex Numbers Appetizer. What is or where is
the square root of -1. With rectangular and polar coordinates, see how to
add, multiply and reflect points or arrows in the plane. The visual or geometric approach here
known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of
"imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails
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www.whyslopes.com >> Parent Center
Introduction to Section Content + General Advice
See too the general advice on helping your child or teenager in mathematics
Speaking Skills suggests how
to improve the speaking and listening skills of your child.
Writing offers ideas for the development of these skills.
science -Teaching a boy or girl to cook or to follow any
multi-step method precisely, in a repeatable and reproducible manner,
will help in science and all area of work and study.
Time and Effort: Four Things for a Student to Know. Quote in full
of an article from Speaking of Learning that refers back to words at
this site, no longer online.
Please. Reflects the inductive idea that learning takes time. If
you see a difficulty, you need to identify the source and retreat
before it in order to practice skills that restore confidence and
then to practice skills that remove the source of the difficulty.
Teaching, tutoring or parenting takes time and patience. Good luck.
Nothing is certain.
in Charge? For better or worse, you the parent or guardian may be
the first and longest term instructor of your child. Do your best
Parents and teachers need to say no for small things of little
consequence to build and maintain authority to say no for larger
matters. Parental authority: ; use it or lose it.
Student Motivation Here a
discussion of the challenge. Not the solution. ;
Students with parents who
say mathematics mastery is important, or education in general is
important, will ; often have more goals, more will and more staying
power in school and college - no guarantees here -but is part of the
Talk to Your
Child or Teen. For many, those without learning difficulties,
the will to learn is often more important than ability.
Encourage the will. ; That is part of the solution.
Schools: Streaming by Student Cooperation. Societies that want
quality education will preserve the authority of teachers in the
classroom, while providing safeguards so that the that teachers do
not abuse that authority. In the first instance, students may be
streamed by their willingness to cooperate with teachers, and then by
their academic destination. People with good education will not want
to go into classrooms if [A.] student disrespect is a constant
danger; and [B] present-day course design in their discipline areas
is inconsistent and irrational according to their previous training
in the discipline. Uniform standards in education are mixed blessing.
While in the first instance they raise standards, over time central
planning or bureaucratization may lead to those standards being
lowered. In large enough regions, different school systems should
develop their own standards, with multiple independent centers for
course and curriculum design, each trying to offer a different
design, each staff in a way that rejuvenation is continuous or mass
retirement in one is not simultaneous with mass retirerment in
& Study Ends, Values and Methods. These appear to be missing
Need to Follow & Supervise the Education of their Child or
If your child falls behind, provide extra help during the school year
or during summer vacations. Ask your school for a list of observable
skills that it and you should be verify. If there is no list, form
one alone or with other parents. If ; there is no list of observable
skills, your child school system has no idea where it is heading. It
is lost. ; ;
General Advice For Mathematics Skill Development
In general, explain to your child or teenager that skills in mathematics
and other subjects need to be seen to believed. In arithmetic and beyond,
talk about the need to do and record work step by step, so that each step
can be checked as done or later. Emphasize too the domino effects of
errors, that is, how an error in one step leads all or most following
steps to be in error as well. If local textbooks are too hard for you to
follow then your child or teenager will have difficulty as well.
For children or preteens ages 3+ to 12 years of aged learning
mathematics, use parent friendly,
Work Booklets to build or rebuild skills and confidence. Emphasize
mastery arithmetic and geometry has value in adult and daily life. say
this mastering is part of growing up, like learning to read exactly what
is meant, and to write exactly what they mean.
For children or teenagers, ages 12 and up, site material may help you
check and develop skills in stronger ways that seen in school. Mastery of
site material unfortunately requires adult level patience and reading
abilities that most students, except for the gifted and talented do not
have. So you or a tutor will have to give less gifted learners ages 13 to
19, skills and concepts to master in parallel to the skills and concepts
that teachers provide in school.
Site material was written to provide alternative and clearer paths for
development of key skills and concept in mathematics and logic because
of gaps and steps too large in the secondary level course design and
delivery. The tutor or school system able follows site steps in
arithmetic, geometry and algebra offers your child a firmer base for
calculus and senior high school mathematics. But there is no 100%
guarantee your child or will be able to follow the site program alone
or with help - that remains to be seen.
In the early years of secondary school, your teenager may need to be
tutored to prepare them for course test and finals, besides following
site steps as site material and your local school may follow two
different paths. But within a year or two of effort, site steps, ends and
values for skill development should help your child do well in school.
But again there is no guarantee your child will be able to follow the
site program - that remains to be seen. So you have explore site material
or rely on the judgement of others to decide what route to follow. Good
If you are lucky, your local school does not follow the new-age,
constructivist approach to education. In it,
- Children are told the best way to learn a subject is from discovery
and from their personal experience.
- Teachers are also told that true knowledge for each student is a
private matter, a product of their reflection on their personal
experience, a product that can not be reliably tested nor corrected in
class and on final examinations. in class or in final examinations.
- Teachers are told to introduce idea indirectly, so that students may
discover them for themselves. In this, various kinds of problems and
challenges are to be given to set the stage for this self-instruction.
Teachers are further told that giving students skills and concepts to
master is a substandard form of instruction. At the same time, school
authorities and parents like you want their children and teenages to do
well on final examinations. Such final examinations test student mastery
of skills and concepts that used to be given directly instead of through
hints. There is a contradiction between theories that hold students
should be taught indirectly, and the practice of wanting students to do
well on tests and final examinations of skills and concepts that previous
educational theories and practices tried to develop directly. That being
said, site material stems from the observation of gaps and steps too
large or missing in the direct explanation of key skills and concepts. In
particular, site material aims to remedy the foregoing by providing
teachers and tutors, and the gifted or talented, clear and direct
explanations likely to ease, avoid and even explain for common fears and
Clear How-TOs, ends and values for the direct explanation of given skills
and concepts need to be well-known and well-documented before
instructors, well-versed or not in mathematics and logic, are called upon
to develop skills and concepts indirectly. Otherwise, indirect
instruction will simple compound the shortcomings of direct instruction
in skill and concepts development. The problem with constructivism is
that moves the focus of education away from worldly concerns.
This viewpoint of constructivist-led indirect instruction stems from the
perspective that arts and disciplines at home, in school and the
workplace require an observable and verifiable mastery of visible and
hence verifiable skills. Skills and abilities in many arts and discipline
need to be seen to be shared, to be well-defined and believed. In
particular, skill in cooking at home and in a restaurant may involve some
thought or reflection that we cannot see, but that skill needs to be seen
to believe. In general, we do not want cooks who think, we want cooks
that can do. Likewise, we want cloths designers and makers, plumbers,
electricians, interior decorators, construction workers, factory workers,
engineers, book-keepers, accountants, scientists and mathematicians who
not only think about what needs to be done in the privacy of their own
minds, but also display skill in ways that can be seen and judged. The
format of that display may vary from discipline to discipline.
In mathematics itself, practical concerns suggest the following ends and
values. Instruction should first serve on the development of basic skills
and concepts with of service in daily and adult life. Second, it should
serve the needs of pre-college or precalculus occupuations. Finally, it
should serve the needs of calculus and precollege courses in science and
technology. Calculus is required for college level studies in accounting,
commerce, engineering, science, technology, mathematics and mathematics education.
Here serving the first end may serve the next two, and
serving the second may serve the last. Further, there is no harm in
college bound students seeing skill and concepts for adult and daily
life, and for precollege work destinations as a parachute or safety net
as there is no guarantee that their college studies will succeeed.
Mastery of logic, awareness of the domino effect of errors, and awareness
of how many later skills and concepts depend on earliers could provide
further ends and VALUES to guide both learning and teaching.
Finally, in reaction to my description of these worldly ends and values
for instruction or skill and concept development, a colleague kindly
observed the absence of higher level cultural end and values here. So
more thought may be required.
www.whyslopes.com >> Parent Center
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based Reason
Online Volume 1A,
Pattern Based Reason, describes
origins, benefits and limits of rule- and pattern-based reason and decisions
in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not
reach it. Online postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.
1996 - Magellan, the McKinley
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology
-- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The
NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4,
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
Lines-Slopes [I] - Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.
Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x)
Arithmetic Exercises with hints of algebra. - Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.
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