My mathematical past

1.) When I understand the principles of some formulas, I would feel so satisfied and try to go further. I do not need to memorize the formulas, just deriving them when I use them. Also, when I can solve a mathematical question that my friends cannot, I would feel I have made an achievement.

2.) When I can't get the ideas (e.g.,"visualizing" the formulas) of some formulas and just apply them without any thoughts, I feel frustrated.

3.) At this moment I do not have any role models of a math "teacher" but I want to have a math teacher who can visualize the mathematical theories and use daily examples to explain the theories.

4.) I don't like a math teacher who just asks students to memorize the formulas, without explaining the principles of these formula.

Response about Skemp on two approaches to teaching and learning mathematics

As I read the article, the three things that made me "stop":

1.) ".., there are two kinds of mathematical mis-matches which can occur." When I read this paragraph, I totally had the same feeling. When I was a mathematics student, I also thought I just needed to know the calculation procedures (i.e., understand instrumentally), without any intention to understand the concepts behind. However, when I became a private tutor of mathematics, I understood it's important to understand how the calculation procedures came from since this can help students understand other similar concepts and they do not need to tediously memorize the formulas.

2.) "It is easier to remember." I also feel the same way if students can have relational understanding of mathematics. When I taught students about the area of a triangle, I explained why the formula is 1/2 x base x height. Then they can apply the same idea to calculate the areas of parallelograms and trapeziums, and this interesting inter-relation of area calculation between different shapes can motivate students to learn mathematics. It can also help students develop logical thinking which is useful in other subjects. 

3.) The final "stop" is when the writer applied the analogy of finding routes in a new town to explain the relational understanding of mathematics. Instead of remembering a number of particular routes to certain locations, the writer constructed a cognitive map of the town in his mind. This map helped him produce an infinite number of routes to go from one place to another. He would not be lost even if he made a wrong turn. This analogy can help readers easily understand the benefits of relational understanding of mathematics.

I agree with Skemp's point of view about the benefits of relational understanding of mathematics because as a private tutor of mathematics, I totally believe if students can have relational understanding of mathematics, they can handle mathematical problems more easily and develop logical thinking to solve other problems.