Memorize or derive?

 What is the best way to learn mathematics? Is it to memorize formulas and algorithms, and remember when to apply what rule? Or maybe it is better to get to the point of each concept, make a visual aid, and let a kid derive a rule or a formula themselves? When I studied at the mathematics lyceum, we were taught not to memorize trigonometric formulas, but to understand the idea of formula derivation, so we were able to do it on the tests without any extra help. What I notice often, is that kids at secondary and high schools nowadays are given tons of rules, and formulas, but when 2-3 weeks pass after the test, they simply forget these algorithms, and their minds are already packed with new very intensive packs of mathematical information. As a result, the student gets credit, but the essence of the topic is lost. 

When I tutor a student, we spend some time making drawings and creating charts to understand the concept itself, then I ask them to explain it to me or imagine that they explain it to a friend, to give some new examples of how we can apply this concept. I believe in the interconnection of everything we learn in mathematics, there are no separate subjects in math. If you lack understanding of what is a variable is, or what is the root of the equation, we can not move further to the inverse or compound functions topic, for example. What I notice in US math curriculums for grade 10-11, is that they assume that if a student took an Algebra 1 course two years ago, he/she somehow remember it all, and they want them to dive into Algebra 2 without any review of the current skills. As a result, a student is overwhelmed with new material without having basic skills of working with variables or understanding the essence of argument and value for a function. 

There is no perfect curriculum, I personally like the International Baccalaureate (IB) Curriculum a lot because they have a lot of applied problems and projects, that help a student to get a deeper understanding of the concepts. I have been teaching this curriculum for 5 years during my work at the Pechersck School International, an IBO-accredited school in Ukraine. I work with students from grades 6-12 whose language of instruction is English and who follow any curriculum, given that mathematics knowledge is universal, and try to add my own perspective on learning in our classes. I love aha moments when something has clicked, and a student has connected their previous knowledge and a new piece, that didn't make sense before.