A. Before You Stop
In school before you stop studying mathematics, please do the
following.

Look at the chapters Implication Rules
and Chains of Reason. These two
chapters may help you to use and understand precisely rules,
instructions, patterns, definitions and recipes in every
subject and every area of skill or specialization including
mathematics.

Read the first chapters on algebra. The description in them of
the three key skills for algebra and the algebraic examples
will, I hope, help you step from arithmetic to algebraic way of
writing and thinking plus a little beyond. These chapters try
to explain and describe with everyday words, how (algebraic)
shorthand notation is used to describe and do mathematics after
arithmetic. There is more to mathematics than just doing
arithmetic well.
B. Why Study
Mathematics courses are preparation for business calculations, for
handling your money sensibly and for courses in sciences,
engineering and technology. You should view mathematics as an
opportunity to strengthen your thinking skills.
In mathematics courses you should not only meet calculations to do
but also the chains and threads of reason and persuasion which
justify them and links them together. Understanding and following
the rules and patterns of mathematics, practices and nurtures an
ability to think and reason well. Mathematics provides a neutral
territory for the practice of rule and patternbased reason and
logic. The opinions and views you meet in daily life say and care
little about what mathematical conclusions should be.
If you find yourself in a course which gives formulas and numbers
to use in them but does not expect you to use algebra, you are
wasting your time. Your time would be better spent studying
algebra, and then taking a more advanced course that respects your
intelligence. Similarly in college, if you find a course which
gives you formulas and numbers to use in them and also talks at
length about rates of change without expecting you to understand
calculus,^{6} then a calculus course would be of
better use of your time.
^{6} Calculus in the first instance
provides formulas for the slopes of (nonlinear) curves and for
the rates of changes of numbers or quantities.
C. When to Study
Look at the description of courses you will take in and outside
mathematics. From their description, see what mathematics course(s)
you are expected to take with them (corequisites) or before them
(prerequisites). Methods taught in a corequisite mathematics
course are too often covered after they are required in another
course. So take a mathematics course before there is any
possibility the methods in it will be needed in another course.
Then you may master the methods before they are required and not
after.
D. More Advice
If you follow the advice and the cautions below, you should have a
mathematical foundation for any subject requiring calculation. In
mathematics do the following.

Master arithmetic. Also master weights and measures. After you
have mastered the rules of arithmetic, learn to use a
calculator.

Master the algebraic way of writing and thinking. Also master
the use of rules and patterns to arrive at conclusions.
Mathematics after arithmetic builds on our abilities to talk
about numbers and quantities, to describe calculations and to
change the way calculations are done. Mathematics after
arithmetic also depends on our ability to follow and understand
rules and patterns. See the first chapters on reason and
algebra in this book.

After algebra, take trigonometry and geometry.

Learn about money matters. Take a course on money calculations,
preferably after a course in basic algebra. Most of us handle
money for credit or investment. In your last year of studies
before starting work, take a course on the mathematics and
arithmetic of personal finances. This course should include
balancing of budgets, description of typical household expenses
for individuals or families in rented or mortgaged properties,
problems involving simple and compound interest, and
mortgage/pension calculations. Traces of such calculations
appear in elementary and high school mathematics, but they are
forgotten years before you need them. A course like the one
described should be offered in schools and colleges for
students in any art or science. Ask for one to be given, if it
is not already offered. All this is practical mathematics.
It should be more widely known.

If you go to college, take a year or two of the mathematics
subject called calculus, a year of probability and statistics,
and a year of matrix computations (or linear algebra). Calculus
courses usually have trigonometry and algebra as prerequisites.
Calculations in many trades, including business, engineering,
computer technology, physics and health science, require
calculus.
Mastering the rules and patterns of mathematics and reason
(there is a connection) is good practice for mastering the rule and
patterns of all disciplines. To master mathematics you need to read
your course notes or course textbook carefully. Examples, solutions
and proofs show you patterns to follow or imitate. Here every step
not understood hides an idea from you.
The problems you find easy to solve should be done to restore and
build your confidence and to reassure yourself that you have
understood what they require. But after you have done a few such
problems, you should look at the ones which appear harder. The
problems which appear to be too hard should be noted and
remembered. You can return to them later by yourself or with help
from another. What is hard for you to solve may be easy for
another, and viceversa.
E. Cautions
When I taught a remedial algebra course, one of my students was a
high school gym teacher. One of his past assignments was to teach
algebra.^{7}
^{7}This should not occur, but in many
school systems it does. When it does, it shows a lack of respect
to students and a lack of purpose for education. It also suggests
circumstances beyond the control of students and teachers.
In some schools due to circumstance beyond their immediate control,
some instructors are required to explain ideas outside their own
specialties. When or if you meet such an instructor, be polite and
do not become a troublemaker. If a teacher sees you as a threat or
troublemaker, you may suffer. When you meet a misplaced instructor,
politely and diplomatically try to transfer to another class in the
same subject or read the course textbook yourself and get a tutor.
F. More Keys to Better
Learning
Here are several more comments on learning mathematics or
another subject.

How you find a solution to a problem is not important provided
you understand fully the solution. (Some teachers may
disagree.)

If you have to copy solutions blindly then you will not
understand ideas well enough to pass tests and the final
examination.

You should ask another to check that your written responses or
solutions are both understandable and wellwritten. Mistakes
brought to your attention in any manner improve your
understanding. If such checking improves your ability to avoid
mistakes in the future, then such checking should I believe be
encouraged. Again, some teachers may disagree.

Students who know and identify in their solution those steps
which are doubtful deserve more respect than students who
don't. Knowing exactly where one is sure and where one is not
is the sign of an alert mind.

Correct answers obtained accidentally, for instance by
canceling errors in a solution should not be given full marks.
Errors in a solution show that the subject is not carefully
mastered.

Learning is better done in a cooperative atmosphere where
students help each other to understand instead of a competitive
one, where the success of one student is at the expense of
others. (But you can not always choose your environment.)

Seeing two or more approaches to a subject can be better than
one. What appears hard in one approach may appear easier in
another.
G. Calculators and Computers
Calculators lessen the need for us to do arithmetic but, in using
them, mistakes can be made. Here you need to know in advance what
kind of answer a computation will yield. If you think you have made
a mistake in entering numbers or instructions, you need to reenter
them again. If a different result appears from before, at least one
of your efforts, the original or the check, will be in error.
(Logic Question: What can you say for sure if the results agree?)
Suggestion: remember or learn how to do arithmetic by hand and how
to estimate the expected size of results for addition, subtraction,
multiplication and division.
Computer programs can perform arithmetic and algebraic or symbolic
operations. They can also draw graphs and solve some equations
rapidly. These programs do not provide solutions to all possible
problems. For the solutions they can provide, you have to
understand the statement of the initial problem. Beyond this, a
computer (or another student) cannot understand the chains of
reasoning for you. Understanding is an personal affair. No computer
and no other person can do this for you. But if you know what to
expect from a calculation, calculators and computer programs can
help you check your expectations and explore mathematical ideas.
Here you can learn from your mistakes. In some cases, computer
software can tutor you. They can tell what to expect in various
circumstances. Today there are computer programs and online
computer books which may help you master mathematics and other
subjects. More are appearing everyday. I know of them, but I have
no experience with them.

August 2011 Postscript  Context
In mathematics programs for ages 4 to adult, the main objective
appears to be the preparation of students for college level studies
and careers based on calculus or statistics. But many or most
students do not complete secondary school. While mathematics
programs for ages 5 to 12 may develop numerical and geometric
skills and practices of actual or potential value for daily and
adult life, programs for ages 13 to 17 focus on a preparation for
calculus or statisticbased college studies, and they do so
imperfectly.
Most secondary school mathematics  in North America at
least  is taught by instructors who have not seen how or why
calculus nor statistics is used in college studies and beyond. Most
instructors started to teach mathematics not because they were
trained in it, but because no one else was available to teach it.
Besides that, year after year, the only immediate reason for
understanding or explaining a skill or practice is its likely
appearance on the next final examination. In a discipline whose
mastery is taken to be a sign of good figuring and reasoning
abilities, how or why the skill or practice will be needed in
college will not be clear to learners and their teachers. Learners
taking advanced courses covering biology, chemistry, physics and
money computations will see high school mathematics in use  better
late than never. Students in trade oriented courses may also see
some applications. But almost everyone else will not. While
mathematical preparation for college studies may be wanted and
respected for its long term value, for most, the shortterm value
of doing well on test and finals is not inclusive and not
appealing.
Present day students and teachers will have to do their best with
local course design and materials. In site material, online books
and further site material will help. See the advice in the other
column. Here the description of alternative ends, values and paths
for mathematics and logiclanguage skill and practice development
may provide some immediate context and motivation.
Doing Your Best with local course design and materials
If you find studying hard or not, then you should try to master
logic  see site logic chapters  and you should be aware of and
try to avoid the domino effects of errors in multimethods at home
in cooking, at school in mathematics etc, and at work in following
instructions. Logic mastery with the greater precision in reading
and writing it provides is one way to ease or avoid troubles in
following instruction at school today and in work tomorrow.
For reading and writing, all the letters of your alphabet have to
be met and remembered. Anything less will lead to difficulties in
spelling and understanding words. The child who complains there are
too many letters too remember will soon be corrected. For
mathematics mastery, skill in counting, measuring and calculating
with decimals, fractions, percentages and signs is a must. But too
many students are not shown fully how to do so in the last years of
primary school and the first years of secondary school. But such
skill is a must for all further mathematics in secondary school and
college. Site arithmetic steps will help. Use them to check for
gaps in your command of fractions, primes and arithmetic with units
of measure.
In my school days, the introduction and use of algebra included
steps too large for most to follow. Site algebra chapters and steps
provide starter and advanced lessons to give a remedy. In it,
smaller and extra steps provide a more gradual, less steep, paths
to try. See what works. Looking for a remedy began in the author's
high school days. The first breakthrough came with the fall 1983
invention of three lessons: Three skills for algebra, why slopes
and two logic puzzles. Talking about the three skills adds words to
the exposition of mathematics. The lesson on why slopes appear in
secondary mathematics lesson was an algebraically light calculus
preview. Calculus is the subject of slope related computations
forwards and backward for lines or linear functions y = ax+b and
more generally for curves or nonlinear functions y = f(x).
Preparation for calculus is the main reason for earlier mathematics
in college or secondary school. Calculus and many other subjects
require precision reading and writing. That was one reason for the
two logic puzzles. While lower level mathematics may be taught by
rote in a practice first, theory second or not all at all manner,
that is easier, students heading for calculusbased college
programs should meet and be able to master some of the theory  too
much would be overwhelming. Site logic chapters provide a worky
introduction, almost mathfree, to the kind of logic used in
advanced mathematics, logic whose mastery in all or part has take
home value in sharpening skills and in easing or avoiding learning
difficulties at home, work and school.
In the bottom margin of each page there are links to key lessons.
Explore them. If one or a few are not to your liking, try the
others. Site content indicated above should fill a gap or two in
your education. That should be reason to explore more site
material. Good luck.
Alternative Paths for Mathematics Education
The question of how and why learn and teach mathematical and
logical skills and practices has bothered me for decades. As a
student, I wanted to understand why I should study mathematics 
its topics and in general. My fathers story that he was handicapped
in college level chemistry and physics since he had not studied
mathematics extensively provided general motivation. That general
aim did not compensate for the lack of immediate motivation for
some topics  the teacher reply that they were required, or the
expectation that I bring my own motivation, were not helpful. And
when I taught mathematics, I wanted to be able to tell my students
why they should learn this or that  to provide a context. I
remember teaching a final year high school courses to students,
they needed for graduation, but I did not see how its topics would
help them with their academic and career aims. Teaching calculus or
upper level high school mathematics is more rewarding for the
instructor and students when there is a chance they are aiming for
college programs in commerce, science, technology, engineering or
mathematical subjects. At least then, there is a context for
teaching and learning. But fourfifths of secondary students are
mathematics education orphans. Apart from meeting graduation
requirements, mathematics courses cover skills, practices and
concepts with little or no takehome value for them, short or
longterm, while other skills and practices that could serve their
daily or adult lives go untaught. That is annoying. Ideas follow to
suggest how mathematics education could be more helpful in
developing skills and practices with takehome value for daily and
adult life, and also for college programs which require advanced
mathematics. The ideas may not help your secondary school
education, but they might help the secondary education of your
children and if you go to college, the mathematics education there
of yourself and others  Colleges have more freedom than secondary
schools to change course design and materials.
Imagine mathematic and logic education may come in overlapping
layers or levels. The first level, usually for ages 3 to 14, would
cover all the counting, measuring and figuring skills and practices
that employ numbers and/or mapsplansdiagrams drawn to scale with
actual or potential takehome value for daily and adultlife. In
that time, date and calendar matters; money matters; logic and
decision making matters; and elementary knowledge of chance,
probability and odds may be useful. The first level would describe
some calculations with words  adding by adding subtotals would be
one example; and other calculations with formulas. Calculation
methods would be given for routine or common situations in daily or
adult life, all in a scoutlike, be prepared for what is likely,
context. Instructors would show how to do and record work in steps
that can be seen and checked as done or later. Avoiding the domino
effect of mistakes in counting, measuring, figuring and in general
would be emphasized as an end, value and tool for skill development
and mastery. This first level might cover methods that are easily
understood and verified in class until just before doing so becomes
too repetitive. And if mathematics education was to stop or not be
appreciated beyond this first level, at least the first level would
provide skills and concepts that would or could be useful sooner or
later.
The second level of mathematics would introduce the use of algebra
for solving for unknowns or getting formulas for them, and the use
of algebra to say when different computations give the same result.
In the past, the algebraic shorthand role of letters and symbols
has not been clearly introduced. I saw that in my school days. In
my school days, I was able to rationalize the shorthand roles of
letters in finding formulas for solutions, in solving equations and
in describing when different calculations would give the same
result. But many other students, more gifted than I in their
reading and writing abilities could not. And one of my physic
teachers did not understand algebra. The foregoing combination of
students and teachers in my physic course made my instruction
slower than needbe. Site algebra steps, smaller and extra, provide
a more gradual path, less steep, for developing algebra skills and
practices. A natural context for commutative, associated and
distributive laws in arithmetic is provided by the new concept of
equivalent computation rules. The latter has little or no takehome
value for daily or adult life, but it makes preparation for
mathematicsbased college programs more accessible. Learning how to
use formulas forwards and backward allows several related formulas
with takehome value given earlier to be replaced by one. The
foregoing may lead to a fuller and stronger mastery of money
related calculations, there be takehome value in that for daily or
adult life. Stopping here might leave a favourable impression of
mathematics,
Three further topics, study in any order, in mathematics have some
actual or potential takehome or intellectual value for daily and
adult life.

The Euclidean model for reason is introduced by site logic
chapters in a mathfree way. Those chapters may improve reading
and writing skills, and help people see the difference between
one and twoway implications. Not seeing this easy difference
is a source of confusion in following and giving instructions,
in digesting information in and outside of mathematics; and in
agreements. The latter and their smallprint need to be read
and fully understood to avoid surprises. Remember in making an
agreement, all parties need an acceptable exit clause  an
option to use if things do not turn out as wanted. Good luck.
The site simplified coverage of Euclidean Geometry, errorfree
we hope, shows how implication rules can be used in sequence to
arrive at conclusions  often further implication rules.
Euclidean geometry provides a neutral territory for
illustrating the deductive use of implication rules, alone or
in sequence. All is simpler than you think.

Not all is certain. Earlier mathematics and life may provide
examples of chance, likelihood, probability and odds.
Probability theory in mathematics provides ways to estimate
what is likely to happend when not is certain. That help people
lessen or avoid risk. Mastery of equivalent computation rules,
how to use formulas forwards and backwards, and simple
operation with sets, should allow students to understand the
first elements of this theory and its notation. Learning about
probability, and above expected value of possible outcomes in
daily life or in playing games will help decision making in
matters of chance, when not all is certain, including the
methods for arriving at conclusions. The elementary study of
probability theory provides one model, more algebraic than
geometric, of mathematical reason.

The natural logarithm appears in the backward use of the
compound interest or growth formula. Secondary mathematics is
not the place to describe the origins of this logarithm
function and the antilog or inverse exponential function. But
a short, full theory of logarithms, roots and powers may be
develop from the algebraic description of properties of the
natural logarithm and its inverse. The elementary study of the
latter theory provides another model of mathematical reason.
Here it might include the forward and backward use of
exponential growth and decay models.

Site geometry steps cover and employ rectangular and polar
coordinates in the plane. The development of complex numbers
from the properties of these coordinates and their interaction
might provide another easy topic. In the prepartion of students
for geometrybased careers and for calculusbased college
programs, this topic would set the stage for the mastery of
trigonometry. Many geometric problems can be solved by drawing
to scale and then measuring. Some of these problems can be
solved by sketching and using trigonometric calculations in
place of drawing to scale precisely and measuring missing
angles or lengths. Trigonometry in the first instances allows
calculation guided by a sketch to be used in place of drawing
to scale. Therein lies a context for trigonometry.
Mathematics education may continue with further topics  see site
steps. The further ones also required by mathematicsbased college
programs. But those further topics would not have any actual or
potential takehome value for daily or adult life like the three or
four above. For students not heading for mathematics and statistic
based college studies, the three or four topics described above
might leave a favourable impression. Leaving a favourable
impression, one that includes multiple skills and practices with
actual or potential takehome value, and perhaps a thirst to learn
would be better than covering too much, and leaving a bad
impression or an alienated view of mathematics in the process. To
avoid the latter, less done to perfection would be better.
Mathematics education should continue beyond the first level only
while the underlying topics are easy for students to master, and
have some relevance for future studies or for life in the street.
