MASTERING MATHEMATICS
We argue that students need to supplement school math classes with
afterschool tutoring. This is already done in all academically
highachieving countries where parents recognize that high school and
college are the gateways to desirable, uppermiddleincome professions.
And they realize that students will need extra help to enter the better
colleges and to do well there. The best private and public schools
in America already offer this as part of their program. Sadly, most
public schools do not offer such help.
Experience and cognitive
science teaches that the most costeffective math teaching (or
afterschool tutoring) promotes individualized "mastery learning". This
implies a very small class size to include "onetoone" tutoring. The
teaching should be spread out over the entire school year to take
advantage of the spacing effect. Simple class review, isolated homework
help or quick prepping to pass entrance exams can be helpful but are not
enough to build the needed foundations for serious collegelevel work.
Our recommended process
for "mastery learning" is very competitively priced and uses learning
materials based on research in cognitive science.
If you are an early middle school student
we invite you to contact us.
To learn more please
review and explore the Q&A Set below.
+ How do we learn mathematics?

The mathematics we need for most professions today has
been under development for over 2500 years. This cultural achievement
can only be taught and learned by students in a formal
stepbystep manner.

Formal human learning of anything  particularly
mathematics  requires the teaching of three critical student skills:

Motivating
his/her focus to direct the effort of a very limited working memory

Actively
connecting and integrating any new with relevant prior learned knowledge

Effectively practicing recall and use of
concepts, skills and reasoning to "learn" (i.e. long term retention).

All knowledge is
contextual, hierarchical and can be represented in the form of questions and
answers. We use speciallydesigned Q&A Study Notes to help us focus
attention and support recall practice.

Simple study skills should be
learned early and used.

Learning and problem solving calls for
thinking and reasoning skills. Such skills are best taught using schemas
in the form of selfdirected questionanswer sequences.
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+ How well are
California K12 students doing  particularly in mathematics?

Not
well. In 20102011
only 76% of California students graduated from high school.

Most of the
failing students
couldn't pass the CAHSEE  the
California High School Exit Exam. In mathematics this is a
Grade 8 level  not a Grade 12 level  test.
A student passes if he can answer about 65% of the test
questions. In earlier times that was called an "F".

Concerning
the "proficient students" that did pass: more than 60 percent of the nearly
40,000 firsttime freshmen admitted to the California State Universities
require remedial education in English, mathematics or both.

The 25,000
freshmen needing these remedial classes all had taken the required college
preparatory curriculum and earned at least a B grade point average in high
school.

California ranks below average in states with Massachusetts leading, .

Internationally the United States ranks poorly  with Finland leading.

In sum: California
public education is not meeting the needs of students seeking a higher
education.

To meet
student needs math education for all
students needs to be publicly  and privately  supplemented as is routinely done in all highachieving
countries.
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+ Why is mathematics so difficult for so many?

School policies on curriculum, funding and class
size
means that students are only repeatedly exposed to the mathematics
they are meant to learn. Schools do not teach
to mastery. Ask: how will they reteach students who have either
never grasped earlier lessons or who have already forgotten them?

Despite curriculum standards educators
are in conflict on the levels of math content & mastery needed.
Teachers, students & parents may believe that certain math content won't
be needed for their chosen careers. They are then not serious in their
math studies. Later they may decide to choose a career that does need
more advanced math content. Who will then provide the needed foundation
skills in time?

Students start at different levels, learn at different rates and often have
very different levels of family support. Can schools individualize supplementing student
work needed for mastery learning?

Many students become lost and discouraged because they don't have the needed
foundation skills. This results in "math anxiety". Can schools
really deal
with this "math anxiety" and help them "catchup"?

Rarely does a teacher have time to individually guide proper study & review.

Many schools cannot afford to have students keep their textbooks. How can
students best use textbookbased materials for restudy or review?

Often students, parents and teachers expect too little and give up too early.
How do schools help students who falter but don't qualify for "Special Ed"?

Historically only a small
elite were taught mathematics. Although mathematics is a difficult subject
we now know that almost everyone can learn it. Do schools encourage new teaching strategies
to help students to master the more complex mathematics today?
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+ How are
highachieving schools different?

Parents work with teachers to help students
create a context for nextclass learning.

Parents proactively guide students in homework,
reading ahead, previewing video lessons  and use tutors if they
can't do it themselves.

Teachers are welltrained, seek constantly
to improve and work with students on problem solving skills and become
independent learners.

Parents, teachers and students expect students
to excel  students work hard in and after school.

Students spend more hours on daily practice
to gain & maintain mastery of skills. Class work is
supplemented after hours if parents cannot do this.

Parents & teachers diagnose student difficulties
early  with intervention to correct problems occurring quickly.

Teachers & tutors motivate students and
teach until students learn  they teach use of schemas to make transfer
of learning to new situations easier.

Teachers & tutors guide students to become
independent learners  comprehensive testing & review is the norm.

Student success is measured by their degree
of effort & improvement  they are explicitly motivated to be the best they can be.
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+
Is "math anxiety" real? Can it be overcome?

YES  "Math anxiety" is very real because students don't "learn topics
to mastery" and so get lost.

Learning difficulties must be diagnosed
carefully as these should drive intervention remedies.

A comfortable starting point for reteaching
& relearning should be found. These become anchor points for conceptual
connections to all future new learning.

The reteaching approach should use special
baby steps to assure learning mastery for each lesson.

Students must learn and apply disciplined
study skills needed for catchup and beyond.
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+ What math is needed for responsible citizenship  a
highly technical career?

The needed lessons are derived
directly from the California Mathematics Standards  this includes both
the 1997 set and the new California Core Mathematics Standards. Both
curriculum standards are bestinworldclass. They are clear as to what
needs to be mastered by when.

We recommend students purchase very affordable
USED math textbooks  ideally teacher edition texts keyed directly to the
California 1997 standards.

When teaching to the newer Core Standards
we carefully crossreference our lessons to these earlier very valuable texts.

Math lessons should include explanations
with worked examples. Ideally lessons should include the following elements:
vocabulary; concept definition exploration; mental integration with prior
knowledge; worked examples for all skills to be mastered; and schemas to
foreshadow the transfer of concepts to newer domains.
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+ Are there shortcuts to learning math  is there an
easier "Royal Road"?

Learning math to mastery
requires strong motivation, persistent focused effort and repeated mental
integration of new with prior knowledge.
It is the
only “Royal
Road” to learning mathematics  it works  but it isn't for everyone.

In
The Talent Code we learn
that talent is not born – it is won through daily personal effort over
many years. As in other fields it takes almost 10 years to become
proficient to worldclass level as is needed for commerce and the
professions.

Learning must be to mastery
because math concepts & skills rigorously build on each other. This means
developing automating fluent recall
of vocabulary, math facts, problem solving schema, use of standard
algorithms, symbol manipulations and basic math concepts & skills.

In elementary math it is particularly
important to master the arithmetic of whole numbers, fractions and signed numbers.
This should lead to setting up equations using elementary algebra
and tackling math word problems with confidence by middle school.
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+ What does cognitive science say about learning
mathematics and best classroom practice?

Cognitive science teaches that
without mental integration and repeated review we typically
forget 40% of what we are exposed
to within 37 days

Class room learning is best accomplished if the student prepares for the
lesson, takes good notes, does all homework exercises and reviews class
notes repeatedly. Class preparation for a class should be in the form
of tutored, advance reading of the textbook or viewing related videos

To achieve & maintain fluent
recall of any newlylearned material (i.e. “to learn”) we must first connect
it with what we already know; mentally integrate it, review it within hours
of first exposure, and then practice daily for retention on a
special schedule shown in the figure.

We truly "learn" in the
repeated effort
to correctly recall what we may have forgotten.
We practice efficiently when we repeatedly try to retrieve what we have the most trouble recalling.

Accomplished learners develop
a practice of
justifying and
explaining each step taken. This
is most easily done when our learning is encouraged and personalized by
a teacher/tutor as in music or sports instruction.

All this requires patient guidance
and a focused effort – with teaching not finished until the learner has
learned. This is called teaching
to mastery.

Progress towards mastery depends on a persistent
motivation, how much time is spent on the right kind of daily practice and
on dailyweekly access to an expert teacher for continued guided practice,
corrective feedback and encouragement.

To repeat: Learning means long
term recall based on a permanent change in long term memory. We have
learned music when we can consistently perform well  either in practice
or in public. We learn sports when we can meet goals or execute plays that
respond creatively to an opponent’s best effort. We have learned mathematics
when we can provide clear answers to wellformulated questions and can skillfully
solve math problems.
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+ What is at stake financially? What can parents do?

A solid mathematics education is
needed in the typical professions of the upper middle class. These include lawyers, physicians, optometrists, dentists, engineers,
professors, architects, economists, political scientists, pharmacists, school
principals, civil service executives and civilian contractors.

Financially, here is what is at
stake.

Students need to be taught mathematics until it is learned to mastery.

Public school choice & reform is in the future  current school class
work needs to be supplemented today.

Quick fix tutoring and most homework help may be
valuable  but only if it is directed towards student mastery.

Parents need to get more personally involved in coaching and/or tutoring
their children. They need guidance in coaching  unless public schools
provide it.

The best class supplementing model is a "class laboratory" shadowing
the current school lessons to be learned. In that setting tools
should be made available to students that are consistent with accepted
principles of cognitive science.

There really is no other "Royal Road". The sooner there is supplemental
tutoring the less costly it will be.
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+
What are the costs of private tutoring? Are our costs
competitive?

Professional tutoring companies
charge $40+ per hour. Professional university trained tutors charge
$60+ per hour. Some peer students can offer valuable quick help at
perhaps $15 per hour  these may help temporarily with homework and quick
fixes.

Our inperson hourly rates are comparable. However, because "mastery
learning" involves more than a quick fix, we offer lower hourly rates for
longer internet tutoring sessions,

We use SkypeScreenSharing.sessions using our special materials. They are
effective, convenient and less costly at $60 for twohour sessions. Weekly
1/2 hour guitar lessons cost that much.

There are no contract obligations  but payment is expected monthly in advance
 conveniently by check or PayPal using your credit card.

Please be aware: The "catchup learning program phase" may take some time
and may be costly; we can advise you on this.

After "catchup" mature students may need only ½ hour weekly advisory lessons
plus ½ hour of parentmonitored daily mastery practice.

We encourage & help prepare parents to tutor their own children  using
our materials to become "independent learners".
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+
When should mathematics "learning to mastery" begin?

Learning to mastery should begin in Kindergarten and certainly no
later than Grade 3 to build good foundations for Grade 4 arithmetic and
reasoning skills.

We "teach to mastery" beginning at ANY level  including
beginning at the college remedial level. At higher levels it will of
course be necessary to relearn to mastery from lower levels of student
skill. We use diagnostic testing to determine that level of prior
knowledge that has been mastered. Then we outline a plan for "catch up".

We guide motivated students to excel in Advanced
Placement Calculus in High School Grade 12. The legendary Master
Teacher Jaime Escalante learned that students must have a good
foundation early in middle school  this to include developing good study
habits.

We follow Escalante in preparing for the AP Calculus Exams – except
that we offer our program privately, after school, and at greater student
convenience.

We actively seek "qualified"
Grade 68 students who wish to pursue yearround preparation for AP calculus
classes in high school.

A student is "qualified"
if he/she is enthusiastic about learning, willing to work hard and whose
parents are financially able and willing to support a minimum 100 hour annual program.
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+
What are our qualifications to tutor or teach?

Alfred (“Mister Al”) Peschel
has a 1961 B.A. in Mathematics from Columbia
University,
with one year of study at the famed
Göttingen
University.

In his 32 year career in
California
aerospace he used many advanced mathematical techniques – orbital mechanics,
trajectory simulation, operations research, multivariate statistical
analysis and computer software engineering metrics.

He served on national technical
boards and represented both TRW (Grumman) and the USA in ISO standards development
for software management metrics.

Upon retiring, he followed a
dream  enrolling in the thenlocal LA Harbor College to refresh courses
from Basic Arithmetic through Advanced Calculus.
While there he was hired to tutor high school mathematics for the
college’s
Upward Bound Program. That is where he was given the name "Mister
Al".

Afterwards he researched how mathematics is best taught for learning.
He applies the principles of the
Unified Learning Model . Look for
more description of it at our
"Royal Road" Blog.

With Peschel & Associates he
networks with other highly qualified teachers and tutors using special materials such as
his Q&A Study Notes.
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Q & A
The kinds of question we ask are as many as the kinds of things which we
know..... These then are the questions we ask, and it is in the answers to
these questions that our knowledge consists."
Greek Philosopher Aristotle  Posterior Analytics  Part 1 & 2  First
Sentences
Thales was a Greek philosopher who lived between 624546 BC in Miletus,
Ancient Ionia.
He is considered the father of Greek science, mathematics, and philosophy.
He is the first person to have asked questions about the nature of the
universe and considered the answers without thinking of gods, demons or
myths.
This was a crucial step in scientific reasoning and led to an intellectual
explosion which lasted hundreds of years in Greece.
He introduced Babylonian and Egyptian mathematics to Greece  and taught
mankind the concept of "demonstration"  of demonstrating why something
must be true.
This changed our world.
Here are three questions
he asked  and then answered:
How high is the pyramid?
How do I make a right angle at B to the line BC?
How far is the ship from the shore (without walking on water)?
+
History before Thales?
About 30000 BC : Paleolithic peoples in central Europe and France record
numbers on bones
About 25000 BC : Early geometric designs used
About 4000 BC : Babylonian and Egyptian
calendars in use
About 3400 BC : The first symbols for numbers
(simple straight lines) are used in Egypt
About 3000 BC : Babylonian begin to use a
sexagesimal number system for recording financial transactions . It is a
place value system without a zero place value
About 3000 BC : Hieroglyphic numerals in use in
Egypt
About 3000 BC : The abacus is developed in the Middle East and in areas
around the Mediterranean. A somewhat different type of abacus is used in
China
About 1950 BC : Babylonians solve quadratic
equations
About 1850 BC : Babylonians know Pythagoras's
Theorem
About 1800 BC : Babylonians use multiplication
table
About 1750 BC : The Babylonians solve linear and quadratic algebraic
equations , compile tables of square and cube roots . They use
Pythagoras's theorem and use mathematics to extend knowledge of astronomy
.
About 1700 BC : The Rhind papyrus (sometimes
called the Ahmes papyrus ) is written . It shows that Egyptian mathematics
has developed many techniques to solve problems . Multiplication is based
on repeated doubling, and division uses successive halving. 3.16 is the
value for " pi "
About 1000 BC : Chinese use counting boards
for calculation .
Thales, Miletus
624546 BC
About 530 BC : Pythagoras of Samos moves to Croton in Italy and teaches
mathematics , geometry , music and reincarnation
About 500 BC : Panini's work on Sanskrit grammar
is the forerunner of the modern formal language theory
About 500 BC : The Babylonian sexagesimal number
system is used to record and predict the positions of the Sun , Moon and
planets
About 465 BC : Hippasus writes of a " sphere of
12 pentagons" , which must refer to a dodecahedron
About 450 BC :Greeks begin to use written numerals
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LEARNING
TEACHING TO MASTERY
“Coach, I have taught my team to rebound, but they just don’t learn. … Swen, you say you taught them, but you have not taught until they have
learned.” Basketball
Coach John Wooden
LEARNING
"Not tomorrow, not the next day, but eventually
a big gain is made. Don‘t ask for the big, quick improvement. Seek the
small improvement one day at a time. That‘s the only way it happens – and
when it happens, it lasts.”
English Teacher & UCLA Basketball Coach Legend
John Wooden
PRACTICE
“If you want to learn to play an instrument or get better at playing an
instrument there is a guaranteed way that you can achieve this:
practice.
Like with anything we decide to learn, the more we do something the better
we get at doing it. This applies to everything we do in life. Its obvious
that practice is a requirement of becoming a better musician, so you want
to set up a good routine that is enjoyable and will keep you motivated to
want to continue practicing.”
Music Practice & Motivation
DAILY PRACTICE
"Musicians
advocate that practice is of utmost importance in the development of any
player. Jascha Heifetz, possibly the 20th Century's most amazing
violinist, said, "If I don't practice one day, I know it; two days, the
critics know it; three days, the public knows it."
Violinist
Jasha Heifetz quoted in 'Perfect Practice Makes Perfect: Inspiring Players
to Practice Their Art'
QUESTIONS & ANSWERS
“The kinds of question we ask are as many as the kinds of things which we
know..... These, then, are the … questions we ask, and it is in the
answers to these questions that our knowledge consists.”
Greek Philosopher Aristotle  Posterior Analytics  Part 1 & 2  First
Sentences
EXCELLENCE
“We are what we repeatedly do. Excellence, then, is not an act but a
habit.”
Aristotle quoted at Brainy Quotes
HARD
WORK
I have described the elements of my program. I believe that they can be
duplicated elsewhere with ease. The key, for the teacher as well as for
the student, is hard work. Hard work makes the future. When hard work is
combined with love, humor and a recognition of the "ganas" the desire to
learn, the ability to sacrifice, the wish to get ahead that burns in our
young people, the stereotypes and the barriers begin to crumble.
Master Mathematics Teacher Jaime Escalante 