Thank you for coming to visit my blog! I have started this blog as a pedagogical experiment into teaching mathematics, and presented it in the form of a textbook.
Mathematics has many real life applications. In fact, people always have their reasons for studying mathematics, such as getting college credit, getting job skills, interest, solving mysteries, and even investigating the beauties of the universe. I hope this blog will help you achieve your aim.
Pedagogical Introduction
I have purposely introduced material in such a way that people can build up from the foundation. Many a time mathematics teachers or lecturers ask of students to accept results without introducing a proper structure to mathematical learning and proofwriting. Also, they tend to do this rigidly. This results in a poor mathematical memory, an undesirable result.
This implies that the basis to memory is the true understanding of the concept. If we forget a formula, we could always rederive it from what we can still remember. In fact, we can do it both ways. An example to illustrate is as below:
"There are 2 geometrical principles, the Pythagoras Theorem and the Cosine Rule. Let's say you forget one of them, and we remember the other. The Cosine Rule was historically derived from the Pythagoras Theorem, for the proof, refer to the section "Sine and Cosine Rules". The idea of dropping perpendiculars and examining the height of the triangle should come naturally (instinct). Then comes setting up of the proof. To rederive (NOT prove, or we would encounter circular reasoning) the Pythagoras Theorem from the Cosine Rule, we set A=90 degrees."
Often, then, mental links between topics matter. I believe that this is something not often encouraged of students. There is also another dimension to mathematical thinking, and that is one that involves elegance. A common example would be as follows:
"In a m by n rectangular grid, where m and n are natural numbers at least 2, we start off at the bottomleft most square. Our aim is to move to the topright most square. The rule is that we can only move to the right 1 step at a time, or upwards 1 step at a time. How many ways are there to do this?
A person may think of assigning numbers to each square, such as the number in each square is the sum of numbers in the square to its left and the square below it. The bottomleft most square starts with 1, and we proceed on to find the numbers in all squares, until we reach the last square. The number represents the number of ways to get to that square.
However, an elegant approach is to consider each move. There are m+n steps to get there. We can either go right or up in each step, but we can only move right m times and up n times. Hence, the question translates to 'How many ways are there to arrange m R's and n U's in a row?' The answer is then (m+n)!/m!n!."
How do we know to come up with an elegant method for the above problem? Again, it comes with practice and intuition, and one needs to investigate each problem into detail. There is no one size fit all method (and I wished there actually is.), and this is something not often encouraged in learning mathematics. People often look for the only way to solve the problem, but I encourage spending time learning about the structure of each problem, and learning everything it has to teach you. More examples will be included later.
Structure of this blog and how to use it to your advantage
In this blog, I have arranged the materials in a topical manner. In each topic, I subdivided them into smaller useful principles, instead of just introducing theorems. For each post, I have also included what the pre-requisites and related topics are, and link them by stating which posts to also take note of. This helps to encourage interdisiplinary thinking and improves mathematical thinking and sensitivity. Do not just read through the material, understand the rationale and think of why people present information in this manner historically.
Also, after each topic, I would include problems of different difficulty levels. I have not segmented them, but compiled a list of more helpful problems. Each problem has something to teach all of us, so try all problems and take the problems and solution writing seriously. Solutions are not posted, and need to be requested via email.
I have also included sections of proper logical reasoning and writing proofs. I believe this will help the student in the solid mastery of the subject.
Aims of writing this blog
There are several aims of creating this blog. I list them as follows:
1) To provide a comprehensive resource in learning mathematics. I believe there are very few resources out there that can give people a gentle introduction to advanced topics or developing the thinking skills of a student. I also hope that any reader who is only looking for reference can find easily what they want in here.
2) To encourage solid progressive learning. This is what has been lacking in many students, and they tend to remember results and apply them. After 2 years, no one really remembers offhand.
3) To encourage mathematical thinking in solving problems. In fact, you do not need this in real life. However, for those who are keen on really learning mathematics well, I suggest looking through the content and try out the problems. Write out the complete proof too, and keep them neatly in a folder or exercise book. You can review your mathematical maturity as you go along.
4) To encourage thinking across the different areas in mathematics. In fact, there are multidisciplinary problems, and I have also listed them out in this blog as a separate section.
As of now, this blog is still incomplete. I have only started it recently, and hope to finish it as soon as I can. Over time, I will try to expand on its coverage, and will add in new problems now and then.
Due to my lack of knowledge of blogging, I may not be able to bring out the full advantage of my untested pedagogy. If possible, I would humbly ask for help for the following:
1) Contents Page with hyperlinks to my other posts.
2) Setting up internal links between related posts.
3) How to write and display proper mathematical expressions (not too tedious)
4) Searching the blog for relevant references.
Every person has limits, and I admit this work may not be good. I welcome all suggestions, questions or criticisms. Please do email them to me at shinato2010@gmail.com, and I will try to reply as soon as I can. Pardon me for my slow replies sometimes.
Shinato, King of a Higher Plane
