Mathematics

Mathematics advances through several levels of abstraction, from arithmetic, through algrbra and geometry, to calculus and beyond.
At any stage, students may find that the study methods that worked for the previous step no longer work.
It's not that the work is harder (each class is just one step beyond the previous course).
But the type of work changes.

If you are finding that just working harder is not enough, it can be very frustrating.
It can also be counter-intuitive to let go of methods that have made you successful in previous courses.
A tutor may be able to help you adjust how you are studying to better suit the material.

What does this adjustment look like? It can include

• Understanding the context for the course
  (a map of where you are going)


• Understanding what will be different from previous courses
  (and why techniques that worked before may no longer work)


• Becoming consciously aware of one's study methods
  (Have you thought about the best use of homework?)


• Acquiring the skill of learning in different modes
  (moving beyond one's preferred methods)


• Understanding how to set one's own standards for performance
  (moving beyond grades).

For some students it may include adjusting the view of school in general.

Example

Real Analysis

There are several early milestones along the path of mathematical education.

In elementary school through high school:

• Arithmetic introduced manipulating numbers.
• Algebra introduced abstraction -- manipulating variables (letters) in place of numbers.
• Calculus introduced manipulating functions.

In the first two years of college:

• Linear algebra and vector calculus moved the manipulations into multiple dimensions.
• Differential equations combined aspects of algebra and calculus.
  Instead of finding numbers, x, that satisfy equations involving x, the student finds functions, f, that satisfy equations involving derivatives of f.
• Discrete math introduced methods for counting (premutations and combinations).

These courses form the common foundation of engineering and math majors.
They focus on the application of mathematics to solving problems, including calculating forces, accelerations, and strengths of magnetic fields, as well as describing heat transfer, population dynamics, planetary orbits, and compound interest.

Analysis of Functions of a Real Variable (Real Analysis, for short) is one of the next courses a math student would take.
The course is sometimes called Advanced Calculus or Theory of Calculus.
The "real" means that the function arguments and values will be real numbers (as opposed to complex numbers).
Calculus is sometimes described as consisting of limits, derivatives, integrals.
So "advanced calculus" suggests fancier techniques for calculating limits, derivatives and integrals.
But it might be more accurate to call the class "Advanced Approach to Calculus."
In this course, students learn to prove many of the results they learned in Calculus.

The change in focus to proofs suggests some changes to one's approach to studying.

• In a calculus course, a student can calculate a limit with a memorized technique, even if the definition of a limit is still a little fuzzy.
  In a Real Analysis course, knowing the precise defintion of a limit is required to use that definition properly in a proof.
  If a student doesn't know the vocabulary, it may be impossible to understand the questions.


• Traditionally, students learn to write proofs by studying proofs in lectures and textbooks.
  In previous courses, the student can succeed by focusing on only the conditions and conclusions of a theorem.
  In a real analysis course, the student must study the patterns and techniques used in the proofs.