Growing up in Russia as a stereotypical intelligentsia child, I had a whole shelf of popular math books, including translations of Martin Gardner, Lewis Carroll, and Raymond Smullyan. I was effortlessly good in math throughout school, and in my first advanced math class in high school I had an amazing teacher who introduced us to “real” math, logically precise and beautiful. As a result, I got my undergraduate degree in applied math and, after moving to the U.S., received my PhD in pure math. Right afterwards I quit academia and became a web developer. I still loved math, but I wanted instant gratification as opposed to years of painstaking research. I’ve enjoyed my work but I’m still occasionally beating myself up over this. The PhD does add a certain pizzazz to the resume, as I discovered, so it wasn’t completely useless.
When I was in graduate school, I took an aikido class just for fun, and liked it enough to try to continue afterwards. Unfortunately I quickly discovered I was really bad at it. I was never an athletic person, didn’t have the natural ability other dojo students seemed to have, and just thought I would never be good enough, so I quit and thought that was it. However, something pushed me to try again, and after a few more years I realized I wasn’t really “getting it”. I was trying to go through the motions without understanding the concepts behind them — just like many students who study math. Once I had a grasp of the concepts, with the help of my awesome teachers, I still wasn’t as strong or quick or flexible as some others, but I could keep getting better.
During one of the seminars a teacher introduced to us the principles of kata and waza in Japanese martial arts, and something finally clicked for me.
What are Kata and Waza?
Kata is a “form” or prescribed pattern of movement used as a training tool in many traditional Japanese martial arts. Kata movements are practiced many times in order to become fluent and effortless, part of the student’s muscle memory. Examples range from one specific sword cut to several naturally linked movements.
Waza means “technique” in Japanese, but its meaning is actually wider. Waza is a complete response to the attack, that takes into account the attacker’s intent, speed, direction and other characteristics of the situation. To be successful, the nage (defender) not only has to execute the technique (which may include one or several katas) properly, but also have perfect timing, distance, power.
Similar distinction can be made in math. There are many formulas we can use, but the real mastery is in knowing which formula to apply, and how to apply it in a specific situation. Formulas and textbook examples are like katas — tools in a student’s tool belt, exercises they do over and over again to become familiar and comfortable using formulas. Waza is applying the right tools in the right way to solve complicated problems that come from real world or other sciences rather than a given textbook chapter.
Without waza, kata is pointless, but without kata, waza can’t be done right. In martial arts or math, you need both: formulas and complete solutions, exercises and complete techniques. This is true for other areas of life as well. Think about something you like to do and are good at — how would you apply kata and waza principles there?
Unfortunately, too often students think about math as kata only — a set of formulas and rules. Textbooks give them step-by-step methods for solving a problem, and students faithfully consult these steps as they’re doing their assigned homework. Naturally, it’s impossible to remember all the steps for all the methods they encounter, so in a few weeks or months students feel helpless when they encounter a similar problem because they forgot how to solve it. Or maybe they remember a formula that seems right, but it’s not really applicable in this problem. They are trying to use only kata where they need waza. That was my problem in aikido — I was trying to memorize the sequence of motions without understanding the reason behind them. Students need to develop mathematical intuition and deep understanding of principles behind formulas and methods.
Other students may have an opposite problem — they understand complicated concepts, but they don’t have a solid foundation in algebraic manipulation, or they don’t want to “waste” time doing simple exercises. They’re trying to do waza without kata, and as a result they spend too much time on a problem or make mistakes in calculations. That was also my problem in graduate-level math — I was content to understand the concepts and pass the tests, but I wasn’t persistent enough to spend more time working on problems, to go above and beyond the minimal effort in order to become truly proficient and ready for doing research.
Whatever your pursuit, don’t forget you need both waza and kata if you want to achieve a black belt!