How To become Fluent in Math

Detail of Pythagoras with a tablet of ratios, numbers sacred to the Pythagoreans, from The School of Athens by Raphael. Vatican Palace, Rome, 1509

Detail of Pythagoras with a tablet of ratios, numbers sacred to the Pythagoreans, from The School of Athens by Raphael. Vatican Palace, Rome, 1509

Barbara Oakley shares how she rewired her brain to become fluent in math.

She talks about how schools focus too much on understanding and not enough on repetition and fluency, that students have to grasp the fundamental essence of an idea, even though that can quickly slip away without practice to consolidate them.

Students who have been reared in elementary school and high school to believe that understanding math through active discussion is the talisman of learning. If you can explain what you’ve learned to others, perhaps drawing them a picture, the thinking goes, you must understand it.

Japan has become seen as a much-admired and emulated exemplar of these active, “understanding-centered” teaching methods. But what’s often missing from the discussion is the rest of the story: Japan is also home of the Kumon method of teaching mathematics, which emphasizes memorization, repetition, and rote learning hand-in-hand with developing the child’s mastery over the material.

In the current educational climate, memorization and repetition in the STEM disciplines (as opposed to in the study of language or music), are often seen as demeaning and a waste of time for students and teachers alike [...] What this all means is that, despite the fact that procedural skills and fluency, along with application, are supposed to be given equal emphasis with conceptual understanding, all too often it doesn’t happen

The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition.

Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence

This echoes the ideas in this video by the Math Sorcerer: Stop Trying To Understand.

She explores the connection between learning math & science and learning sport. By using the procedure a lot, applying it in many situations, you will understand both the why and how. Stop focusing on understanding.

When you learn how to swing a golf club, you perfect that swing from lots of repetition over a period of years. Your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.

once you understand why you do something in math and science, you don’t have to keep re-explaining the how to yourself every time you do it

The greater understanding results from the fact that your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way.

When learning Russian, she focused on fluency, and not understanding of the language. She didn't want to simply understand Russian that is heard or read, she wanted "an internalized, deep-rooted fluency with teh words and language structure". How does she do this? By playing around with verbs, using them in sentences, learning when to (and not to) use them.

Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop.

I wouldn’t just be satisfied to know that понимать meant “to understand.” I’d practice with the verb—putting it through its paces by conjugating it repeatedly with all sorts of tenses, and then moving on to putting it into sentences, and then finally to understanding not only when to use this form of the verb, but also when not to use it.

This led her to the fundamental core of learning and development of expertise – chunking, which is how experts become experts, by storing thousands of chunks in their area of expertise in their long-term memory.

Chunking was originally conceptualized in the groundbreaking work of Herbert Simon in his analysis of chess

neuroscientists came to realize that experts such as chess grand masters are experts because they have stored thousands of chunks of knowledge about their area of expertise in their long-term memory

Whatever the discipline, experts can call up to consciousness one or several of these well-knit-together, chunked neural subroutines to analyze and react to a new learning situation. This level of true understanding, and ability to use that understanding in new situations, comes only with the kind of rigor and familiarity that repetition, memorization, and practice can foster

She applies the same strategy of language-learning into learning math and engineering. She played with the letters and alphabets in an equation, and build chunks around them.

I’d look at an equation, to take a very simple example, Newton’s second law of f = ma. I practiced feeling what each of the letters meant—f for force was a push, m for mass was a kind of weighty resistance to my push, and a was the exhilarating feeling of acceleration.

I memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side?

the truth was that to learn math and science well, I had to slowly, day by day, build solid neural “chunked” subroutines—such as surrounding the simple equation f = ma—that I could easily call to mind from long term memory

So, focus on building well-ingrained chunks of expertise through practice and repetition.

Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

gaining fluency through practice, repetition, and rote learning — but rote learning that emphasized the ability to think flexibly and quickly.

Fluency allows understanding to become embedded, emerging when needed.

Takeaways by Claude 3 Sonnet:

  • Practice > understanding for building true mastery
  • Classroom "understanding" can create an illusion of competence
  • Repetitive practice builds neural "chunks" that allow rapid, intuitive expertise
  • Experts draw on vast memories of chunked patterns/subroutines
  • Fluency is key - understanding follows fluency, not vice versa