Introduction to Mathematical Thinking 1#:Thinking Outside The Box📦

Photo by Thomas T on Unsplash

Mathematical thinking is essential if you want to make the transition from high school math to university level mathematics.Even I, who is a graduate student of Mathematics, is still trying to gain this competence. I thought that my university was not enough in this regard. I do not want to accept a system with millions of new theorems and exams at the end. And I wanted to create my own education.

School math corresponds to learning to drive.A key feature of mathematical thinking is thinking outside the box. But how ?

They won’t teach this in university.

I think Critical and Analytical Thinking maybe the solution for thinking outside the box.Thinking outside the box is such a valuable ability in today’s world.At high school, the focus is primarily on mastering procedures to solve various kinds of problems.

Photo by Chinh Le Duc on Unsplash

That gives the subject very much the flavor of a cookbook, full of mathematical recipes, thinking inside boxes. At university, the focus is on learning to think a different way, to think like a mathematician, thinking outside.I want to learn this skills for my personal growth. So I started this course from University of Stanford.

At university we have to learn how to approach a new problem, one that doesn’t quite fit any template we’re familiar with. It comes down to learning how to think about a problem in a certain way. The first key step is learn to stop looking for a formula to apply or a procedure to follow.

If you can’t solve a problem by looking for a template to follow or a formula to plug some numbers into or a procedure to apply, what do you do? The answer is you think about the problem a certain way. Most of the time I won’t feel as though I am making any progress. So I understood that my goal has to be understanding, not doing. Before this course I took too much notes but now I understood that taking too much note is not efficient.

What is mathematics?

• That might seem strange for the people like me probably spent several years being taught math.If all you want to do is learn new mathematical techniques to apply in different circumstances, then you can probably get by without knowing what math is really about.

In the 19th century, there was a major change in the nature of mathematics.First, it became much more abstract.

Second, the primary focus shifted from calculation and following procedures to one of analyzing relationships. The change in emphasis wasn’t arbitrary. It came about through the increasing complexity of what became the world we are familiar with.

The Science of Patterns, according to that description of Keith Devlin, the mathematician identifies and analyzes abstract patterns. There can be numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. They can be either real or imagined patterns, visual or mental, static or dynamic, qualitative or quantitative, utilitarian or recreational. They can arise from the world around us, from the pursuit of science or from the inner workings of the human mind.Different kinds of pattern give rise to different branches of mathematics.

Anyone said branches of Mathematics?..what are those?

For example, arithmetic and number theory study the patterns of counting and number. Geometry studies the patterns of shape.

Photo by Shubham Sharan on Unsplash

Calculus allows us to handle patterns of motion. Logic studies patterns of reasoning. Probability Theory deals with patterns of chance. Topology studies patterns of closeness and position.(I think one of the most compelling course in my B.S😅)

Photo by Photo Boards on Unsplash

Fractal Geometry studies the cell similarity found in the natural world.

Major consequence of the increasing abstraction and complexity of the mathematics in the 19th century was that methods developed to solve important real world problems had consequences that were counter-intuitive. For example, Banach-Tarski Paradox.

As a summary ;in theory, take a sphere and cut it up in such a way that you can reassemble it to form two identical spheres, each the same size the original one. And that wasn’t the only surprise,mathematicians had to learn to trust the mathematics above our intuitions. Just as physicists did with the discovery of relativity theory and quantum mechanics.

What was the first use of numbers?

Photo by Volkan Olmez on Unsplash

People certainly measured land and they used various kinds of yardstick but they didn’t use numbers. And they certainly counted seasons, but you can count without numbers. You can count with notches in sticks and you can count with pebbles and so forth. Our ancestors only invented abstract numbers in order to get money💵.

At least that’s the best available evidence that we have and that we think happened about 10,000 years ago.The main focus in the 19th century became concepts and relationships. That was a revolution in mathematics which took place in Germany.

Guess…According to Keith Devlin, the most valuable mathematical ability in today’s advanced nations is?

• Adapt old methods or develop new ones.But the crucial ability in today’s world is adapting old methods or developing new ones.And Keith Devlin adds real Math takes time( Yeah…I guess I need to realize that I need more time on this too.👩🏻‍🌾)

In mathematics, things are different. Precision is crucial, and it cannot be assumed that all parties have the same contextual and background knowledge in order to remove ambiguities. Moreover, since mathematical results are regularly used in science and engineering, the cost of miscommunication through an ambiguity can be high, possibly fatal.

Mathematical statements use language in a highly restricted way.

So this allowed us to create the language of mathematics.

• For example “!” means factorial in math,we use ”∩” to denote intersection of sets.

Ancient Greek mathematicians seemed to be the first to notice that all mathematical statements can be expressed using one of these simple forms.

When that study is carried out in a mathematically formal way, it’s known as formal logic or mathematical logic. The study of mathematical logic is a well-established branch of mathematics studied, and used to this day in University Departments of Mathematics, Computer Science, Philosophy, and Linguistics. It gets a lot more complicated then the original work carried out in ancient Greece by Aristotle and his followers, and by the stoic logicians.

The subject known as formal logic was first studied by which ancient civilization?

• The answer is the Greeks.

To sum up,university mathematics is not focused on learning procedures to solved problems, it’s about thinking a sand away. And mastering a new way of thinking is best learned by working in groups.

I am looking for a group to learn Math together.If you want to join me write a comment underneath this article.

Love

Şevval👩🏻‍🌾