What is Math? Math can be interpreted many different ways. For me math is about discovering new ideas and processes in which help you solve a problem. It is also about being able to use logic with math such as thinking about real world examples and if measurements make sense. There are many methods to solving problems which I believe students should discover on their own. Some people think math is procedural while others believe it is about discovery. I think math is both a procedure and also a discovery. Since I want to be a teacher math is the process of students creating a strategy that works for them. I could teach a procedure to my students, however the chances of them actually retaining knowledge from that lesson will be slim to none, whereas if they discover a method of their own they will be able to recall it more easily in the future.
As far as discoveries go, I have not done a lot of research. However they main discovery I have learned about it Euclid's Postulates. As well as what separates Euclidean and Non-Euclidean geometry. Another big idea in math that I have studied is the Pythagorean Theorem. This discovery was introduced to me in middle school and was continually brought up throughout my education. Next, is the Quadratic Equation, this has also been brought up several times throughout my education starting in high school. There are several methods that I have used, however these are the most common and the ones I have actually spent time discovering and learning myself.