Our Philosophy

Why Math Tutoring?

At Cardinal, we believe that the highest calibre of teaching and learning is found only in individual instruction.  Since ancient times, the tradition of one-on-one engagement through Socratic dialogue has been the most effective and efficient mode of learning.

Academic lectures in a classroom or auditorium can be engaging and dynamic, and a good orator can stir an audience of a dozen to several hundred, inspiring students' curiosity, and giving them the impetus to explore ideas on their own.  However, as often as not, formal lectures in a classroom environment can be a waste of time and energy, and utterly demoralising to a student whose particular interests are not addressed.  This is where private tutoring can be a real saviour.

When a good teacher pays careful attention to the threshold of a student's comprehension and the frontier of his or her learning progression, not a second is wasted in advancing to a higher level of understanding and proficiency.  Through constant effort on the part of the student and teacher, via incremental steps, there is nothing to prevent even the most frustrated and forsaken student from returning to a level of competence, and then accelerating beyond to a position of excellence.

In just a couple of hours of math tutoring per week, any student committed to improvement in his or her studies can within weeks develop new strength and become comfortable with mathematics, and, within months, dramatically advance to a state of mastery.

The personal value of this new confidence, and the economic value of this skill for entrance into desired programmes of study and professional careers, is inestimable.

What is Mathematics?

Certainly, it is the bane of many students' careers. Nevertheless, it has been called the queen of the sciences and the language of God.  But what is it?

Mathematics is a discipline of thought which attempts to give insight and order to our complex world by describing aspects of it in a simple and precise way.  Its purview is wide, and it has applicationsto virtually every field of study.  But generally, it attempts to describe the nature of quantity, space and form, structure, change, reason and inference, and even randomness.

To claim that mathematics is simple would of course be far from the truth.  However, knowledge of the language and concepts of the field can yield remarkable insights into speciously complex problems.

What good is it?  Without mathematics, we would not have electricity, motorised transportation, computers, skyscrapers, astronomy and space travel, money, most forms of energy, or modern medicine.  So clearly it is important.  But why is it so difficult?

There are a number of challenges in achieving mastery of this field.  In fact, complete mastery is impossible, as the field is now far too vast for a single mind to learn, and ever expanding.  But fortunately there are a few basic subjects which are highly useful in everyday life, and a few more advanced topics which have excellent applications in many professions.

The first challenge in learning math is conceptual.  Courses in mathematics are concept-dense.  They are not simply learned by repetition and experience, but require deep thought.  This requires intense concentration, and great respect.  Even then, the frustration of thinking hard and seeing no progress in understanding can be devastating.  Fortunately, with good instruction and proper discipline, in time everything is possible.  The "Eureka!" moment of understanding is a joyous catharsis which cures all ills.  And nothing is more satisfying that wielding a mathematical tool to solve a hard problem.  Bringing math to a tough problem is like bringing a ring of keys to a vault.  All you have to do is select the right key.

The second challenge is abstraction.  Mathematics is a model for the world, but not one that sits on an architect's table; rather, it is one which can be held only in the mind.  Thus it requires focus and imagination to behold, and to relate to reality.

The third is technical.  Mathematics is also a language, and one whose words have intuitive analogues, but whose definitions are at once precise and absolute, and also completely self-contained.  The language must be learned, and this requires memory and reason.  In applying established ideas,memory becomes a substitute for thought.  As such, formulas are committed to memory and applied when needed.  But they must be applied according to strict rules of logic.  This takes great care and practice.

Why should I learn math?  For all this trouble, there must be some reward.  Indeed there is.  Firstly, mathematics is powerful.  It has applications in physics, chemistry,biology, as well as computer science.  But it is also essential to business, accounting, economics and finance, as well as engineering, medicine, law, and any quantitative social science where data is collected and analysed to formulate a theory.  In fact, math can even be used to create beautiful art and music.

Secondly, math is beautiful.  If you can appreciate the beauty of a sunflower, an ancient tree, an ocean wave, a human eye, a symphony, a galaxy, or the Pyramid of Cheops, then you can appreciate mathematics.  And, perhaps ironically, it is the precision and simplicity of the concepts which makes the subject a joy to learn and know.  And its timeless nature makes it truly precious.

Lastly, math is fun!  Many problems in life are complex but straightforward to solve.  The fun of math problems is that they are often quite simple to state, but their solution is not at all obvious.  For example, how long is a binary elimination tennis tournament of 128 players, with one match time per day, where each round is a round robin?  Or, if you meet a new person each day of a year, what is the probability that you meet one of them on his or her birthday?  With a humble spirit and a little patience, the answer will be found.  And unlike in nearly all other fields,the right answer is always indisputably correct!

How can I learn mathematics?  First, you must have desire.  Desire for knowledge, desire for truth, desire for understanding.  Then, you must have devotion.  Math does not come easily, even to a mathematician; it takes discipline and time.  Then you must have a good teacher, and good books.  And then the sky is the limit.