Dr. Strangekidney, or How I Learned to Love Renal Pathophysiology

The year was 2006.  The class was renal pathophysiology, the bane of medical students everywhere.  Our text was from Harvard Medical School, and even Harvard admitted on the back cover that this is a “difficult subject for even the most advanced students.”  I don’t remember such a depressing statement–before or since–on any other textbook in my 30 years of studying all kinds of difficult subjects, including organic chemistry I and II, physical chemistry I and II, differential equations, formal languages and automata, advanced calculus/physics for engineers, and everything I’ve studied in medicine!

Certainly not the most advanced student in my class, I nevertheless resolved to somehow conquer this untamed beast of medical knowledge.   I don’t know what I was thinking.  Nephrology is, perhaps, the most difficult subject taught during the basic science years of medical school. The material is extremely complex, the time allotted to learn it too short, and the student finds it difficult to retain the numerous relationships between facts long-enough to place them into a top-level or conceptual context.  What made it more difficult was that there were several other subjects to learn during that block of basic science.

After seeing a fellow classmate use a story mnemonic for a set of facts in another class, I suddenly realized how I can use a giant story mnemonic to learn all the renal pathophysiology we had to learn.  By imposing a mnemonic system onto individual facts, and onto relations between them, a story arises, a concrete story that significantly facilitates retention of this abstract subject. Retention, in turn, facilitates reasoning about the subject on a higher level.

Even though I’ve never used any illicit drugs, nor even tried a cigarette, I began to spend bizarre, psychedelic afternoons and evenings learning renal pathophysiology while writing a very strange, nonlinear story in the margins of our textbook and printed PowerPoints.

For example, I represented sodium (Na) as a Morton salt container.  Since sodium moves into the renal tubular cell from the luminal membrane (the side that faces the tubule) and out of the cell from the basolateral membrane (faces away from the tubule), I visualized Morton moving toward the light (lumen ~ light) inside the cell. Once inside, Morton runs down a ladder to first base (“base-o-ladderal”).

We learned that charged particles can’t diffuse freely across lipid bilayers; they must use channels or transporters to move across cell membranes (which are lipid bilayers). Morton was charged with a crime, so he can’t move across a cell membrane without personal transporters or without showing up on a TV channel.

The loop of Henle is the portion of the nephron between the proximal and distal convoluted tubules.  I represented the loop of Henle as Don Henley, the famous singer-songwriter.  Na is reabsorbed in the loop of Henle via Na+-K+-2Cl- cotransporters. In my story, this became:  Don Henley eats salted (Na+) bananas (K+), then drinks swimming pool water (Cl-) to quench his thirst. (Mr. Henley himself never did this, of course, to my knowledge.  This was only a memorizational tool.)  The loop of Henle is regulated by the tubular flow rate: Henley flows, baby!

The distal tubule became the Orient.  Na is reabsorbed there via Na+-Cl- cotransporters.  This became: in the Orient, Morton salt containers float in pool water which empties into the Ganges river.  The distal tubule is regulated by the tubular fluid flow rate: the Ganges ebbs and flows, regulating life in the Orient.

I represented the collecting tubules of the nephron as museum collections.  Na is reabsorbed there via Na+ channels. The museum collections display televisions that show Morton salt containers 24/7!  The collecting tubules are regulated by aldosterone and atrial natiuretic peptide.  Aldosterone became Aldoster, the Spanish playboy, a collector of Morton salt containers, while atrial natriuretic peptide became an atrium filled with nature art: the collections include an atrium filled with nature paintings that pep you up.

And so on for hundreds of associations (including renal biopsies and other images, equations, etc.) in an ever-enlarging story involving the same “characters.”  I studied this mnemonic system twice before the exam:  once while building it, and once as a review.

The exam was a multiple-choice, Scantron-based affair.  We were under a lot of time pressure.  Halfway through, I suddenly realized the exam was misnumbered–it didn’t correspond to the Scantron!  I had misbubbled at least fifteen questions and felt I had no choice but to erase everything and start over.  Trying not to panic, I carefully erased everything, went back to the beginning, and quickly filled everything in again, question after question.  By the time I reached the point where I’d left off, I didn’t have much time left.  I picked up the pace significantly and still managed to finish.

Walking out of the examination room that day, one of my classmates (who was a bona fide “most advanced student”) told me she thought this was the hardest test she’d ever taken.

Some time later, we got the results.  I got one of the highest scores (if not the highest) on the exam–nearly a perfect score.  So, this crazy technique works.  Or it worked in that case.  I think it worked by imposing a concrete story onto abstract facts that are otherwise poorly associated (e.g., not intuitively associated) with other facts.

Years later, I discovered that Scott Young, the celebrated learner/self-improver, used an analogous strategy for learning new things.  He recommended practicing mnemonic techniques over and over until they become second-nature.  (I don’t know if he still recommends this because I haven’t kept up with his work.)

However, I didn’t ever use a giant story mnemonic again.  It was just too unwieldy for me.  I found that it’s more efficient to use a variety of learning techniques.

We moved on to the next block of material, I didn’t have any reason to practice my mnemonic system, and by the time I took a nephrology rotation during our clinical years of medical school, I found I had forgotten much of what I had learned and had to spend time relearning it.

How about you?  Do you use mnemonics on the fly while studying?  What’s your approach to learning subjects that are extremely rich in detail and in relationships between details?

My next essay was going to be on the following topics:  how I prepared for the SAT by myself and scored at the 99.9th percentile; scored 5 (highest) on multiple AP exams (Calculus AB, Literature and Composition, Biology, etc.); used the same pattern of preparation to study for the GRE in a month and got a similarly high score as on the SAT; and studied for the MCAT in less than two months and scored high-enough to impress any med school admissions committee.  However, I found this excellent article today that says much of what I wanted to say:  How to Get a Perfect SAT Score, by Allen Cheng.  Pair it with A Mind for Numbers and with How We Studied for the Boards and you have a complete top-level strategy for all of these exams.

If you’d like for me to publish my essay, let me know, and I’ll see if I can add to the excellent advice presented by Mr. Cheng.

Update 2/15/16: a friend read this essay and recommended that I check out Joshua Foer’s Moonwalking with Einstein.

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How We Studied for the Boards

I love learning new ways to learn almost as much as I love learning new stuff.  A few weeks ago, I read and summarized A Mind for Numbers by Barbara Oakley. A quick read, it’s the companion book to “the #1 most popular MOOC (massive open online course) of all time,” Learning How to Learn, taught by Oakley, an engineering professor, and by Terrence Sejnowski, a computational neuroscientist.  I signed up for the course a long time ago but didn’t check it out until this month.

Why did I summarize it, you may ask?  Well, I got into the habit of distilling books into high-yield summaries when I observed my brilliant medical school classmates summarizing our textbooks, back in the day.  (When a bunch of smart kids study the same hard stuff over four years, they learn some things from each other.)  I still summarize practical books whenever I can, both to improve retention and to have a quick personal reference on hand for later use.  It’s great.

If you’re interested in learning strategies or in math, engineering, the mathematical sciences, or chess, I recommend you check out A Mind for Numbers.  It’s packed with research-supported insights about how best to learn such material.  A lot of it rings true with me from my computer science days.  I had to figure out many of these strategies on my own back then.  I wish there had been books like this!  It would have saved a lot of time.

However, for medical students, residents, and fellows, it’s not sufficient.  Medicine is less concept-heavy and more detail-heavy–at least until the most important details are memorized and understood–than math-related subjects are. In this article, I’ll discuss a medicine-specific strategy that many of my physician colleagues and I have used over the years to learn dense, complicated information quickly and effectively, and to do well on board exams.  I’ll follow this with specific, high-yield examples that worked for us in studying for Step 1–arguably the most important exam of a physician’s career besides his specialty-specific boards–and for the internal medicine and rheumatology boards.  Finally, I’ll end the article with general study tips that have worked for me.  Here we go:

The Two Most Important Tenets of Studying for a Medical Board Examination

1. Find the highest-yield information and memorize it cold–with active methods such as mnemonics, teaching others, lecturing “to the wall,” writing notes out from memory, writing your own multiple-choice questions, etc.–before moving on to information that’s less likely to be tested.

2. Identify the highest-yield question banks–the ones that best resemble the exam–and test yourself frequently while studying.  Look up every unfamiliar answer choice (even the wrong ones) following each question.  As Oakley explains in her book, testing practices retrieval of information–a critical aspect of learning.  Try to buy or obtain practice exams that closely resemble the actual test and take them after every few weeks of studying to gauge your progress. (You hit two birds with one stone by mastering question banks because board exams supposedly test your mastery of the subset of knowledge that’s most important to your clinical practice.  Whether or not board exams actually test this subset is a controversial topic that we won’t discuss here.)

Do both steps in parallel, not in sequence.  The better you execute these steps, the higher your score on a medical board examination.  That’s all there is to it.  The devil, however, is in the detail.

Example #1: USMLE Step 1

I attended Baylor College of Medicine (BCM) in the late 00s.  BCM’s average Step 1 score, compared with other medical schools, was rumored to be among the top few in the nation, if I remember correctly (a quick search failed to verify this, though).  And it has continued to rise.  There were several indirect reasons for our high scores.  We studied for and took Step 1 after we had completed several key clinical rotations (such as internal medicine, pediatrics, general surgery, etc.).  Also, BCM seemed to have a bias for admitting good test-takers (our average MCAT score was unusually high).  There were probably other indirect reasons, too.  However, these advantages aside, it remains the case that some study strategies are more effective than others.  The following was a popular Step 1 study strategy at our school (warning to current med students: this is from the late ’00s):

*Total study time: no longer than 8-10 hours daily for 8 weeks or less.  Many of us took weekends off.  Treat studying like a job:  get up at the same time, start studying at the same time, and finish studying at the same time each day.  Plan to stop studying a couple days before the exam so that you can “recharge” before you take it.

1. Memorize First Aid for the USMLE Step 1 cold.  This book was incredibly dense, disjointed, and painful to study.  But it was pure gold when it came to getting a high score on Step 1, especially when memorized as well as possible.  Some of us sort of memorized it by reading through it 4-5 times, each time more quickly than the last (e.g., 3 weeks for the 1st pass, 2 weeks for the 2nd pass, 1 week for the 3rd pass, etc.).  The pharmacology section was most critical.  First Aid required a lot of supplemental studying:

a. Rapid Review or Lippincott Biochemistry: I used RR Biochemistry as a reference, looking up any concepts I didn’t fully understand in the relevant section of First Aid and expanding that section with my own notes.

b. BRS or Rapid Review Pathology: read it twice.

c. BRS Physiology: read it twice, taking notes in First Aid.

d. BRS Behavioral Science: for those who didn’t take a psychiatry rotation before studying for Step 1.

e. Dr. Edward Goljan’s pathology audio lectures: best to take notes on these lectures while listening to them during the first month of studying.

f. Goljan’s High-Yield Review (~100 pages): best studied the week before the exam.

g. High-Yield Neuroanatomy: I remember really enjoying this book during med school, but I don’t recall how much I studied it for Step 1.

h. High-Yield Biostatistics: read it very quickly, taking notes in First Aid.

i. Skim the images in Robbins & Cotran Pathologic Basis of Disease.

2a. Kaplan Qbank or USMLEWorld.  The latter had recently debuted, back when I was studying.  With better explanations, diagrams, and a lower price point, it eventually became the “gold standard” question bank.  I don’t know if this is still the case.

2b. NBME practice exams.  These exams felt very different from Step 1, but word on the street was that one’s score supposedly correlated well with one’s actual Step 1 score.  Form #3 was considered most representative.

I first learned about the above strategy from an upperclassman who went into emergency medicine.  Later, one of my classmates who did very well on Step 1 told me that he used the same strategy.

In my own case, things didn’t unfold so neatly:  I read First Aid five times, took notes in First Aid while reading relevant parts of Rapid Review Biochemistry, then read BRS Physiology once and skimmed BRS Pathology and High-Yield Biostatistics quickly.  Also, and this scares me in retrospect, I only completed 36% of the Kaplan Qbank.  However, I took two NBME practice exams a few weeks apart during the second month of studying and did very well on them, predicting the great score that I got on the actual Step 1.  I think that in my particular case, knowing First Aid backward and forward was the key to doing well.

I shared this strategy with a friend who carried it out perfectly, then studied supplemental material–both question banks, other review books–and got an even higher score than I did.  We shared the strategy with other friends.  They all did well, too.  I told our preclinical directors about it.  The study strategy for Steps 2 and 3 is analogous.

Of course, other methods were also effective.  I’ve heard that at Caribbean medical schools, students are drilled with endless multiple-choice questions and mock Step exams for months, becoming expert at taking the test.  (There was, and may still be, a bias against Caribbean med students and foreign medical graduates when I was in residency.  They had to obtain high scores on the Step exams to have a fighting chance for an accredited residency slot in the US.)

There’s more than one way to eat a pomegranate.  And I’m sure the game has changed since 2008.  Spaced-repetition software is more popular now than it was back then.  There are medicine-specific, computer-based spaced-repetition systems like Firecracker (see below) that weren’t around when I was in medical school.  Optimal learning strategies are better-understood.  Every generation of students is savvier about learning and has better learning tools available to them.

Example #2: American Board of Internal Medicine

My internal medicine residency program had a very high ABIM pass rate.  Again, they preferred to accept medical students with high Step scores (the thought–whether or not it’s correct–was that this predicts internal medicine board pass rates).  However, there were other critical things about our program that set us up for easily passing the ABIM:

Continuous, Active Learning of High-Yield Material

Residents don’t have much time to study.  While seeing patients, we tested ourselves and learned actively by constantly questioning our treatment strategies.  We constantly looked up important points, asked ourselves why we were ordering, say, enoxaparin for one patient and not for another, or an echocardiogram for one patient with a certain condition and not for another with the same condition, etc.

Studying High-Yield Information and Practicing with a High-Yield Question Bank

We had already studied in-line with residency training, but for the ABIM, we had to round out our knowledge, which was done by studying the latest version of the ACP’s Medical Knowledge Self-Assessment Program (MKSAP), which our program director purchased for each of us.  MKSAP is a series of review books for every subject in internal medicine:  cardiology, pulmonology, nephrology, general medicine, etc.  It’s also a large question bank.  The wisdom passed down to us from prior generations was that knowing the question bank was enough to do well on the exam, and that knowing all of MKSAP by heart was sufficient for getting a high score.

This is all concordant with what we know now about learning any body of knowledge.

(Once again, there was more than one way to study. Some residents preferred other review series.)

Example #3: American Board of Internal Medicine – Rheumatology

Rheumatology is, in some ways, a difficult branch of medicine to study.  Many of the diseases we encounter (or must be able to recognize and treat but might never encounter) are at the case report level and could show up on, say, the popular television show, House, M.D.  Unlike the internal medicine boards, which mostly tests management of commonly-encountered diseases on the wards and in clinic, the rheumatology board exam has a large proportion of infrequently- (or never-) seen conditions, even by rheumatologists who trained in large, diverse cities.  It also has some basic science questions that never come up in clinical practice.  This is somewhat understandable because the field changed radically in prior decades by sophisticated new medications capable of putting previously-debilitating diseases into remission.

The general consensus is that the highest-yield review book is Rheumatology Secrets.  Thankfully, the third edition of Rheumatology Secrets came out just last year.  Until this edition was published, it was difficult to find an up-to-date, high-quality, concentrated review of rheumatology.  The second edition was published in 2002–not long ago–but rheumatology has exploded since then with new treatment options, better understanding of disease processes, new guidelines, etc.  I’m glad our program director bought each of us a copy last year.

The Rheumatology Image Library is also a critically high-yield source of information.  Memorize it cold, ideally using spaced-repetition.  It takes at least two weeks of part-time studying to get through this image bank the first time.

In rheumatology, the question banks to study are the CARE modules.  Use spaced-repetition to learn (at least) the most recent five years’ worth of CARE questions completely, looking up anything you don’t understand.

Some people swore by the UCSF Rheumatology Board Review.  Others didn’t find it useful.  My co-fellows and I didn’t go to it, so I can’t comment on its efficacy.

Following the advice of fellows who graduated before us and of young attendings who had recently taken the exam, I read Rheumatology Secrets twice, studied five years’ worth of CARE questions in the months leading up to the exam, went through the Image Library twice the month before the exam, and did very well on my rheumatology boards.

Here are other, more general tips for studying medicine that I’ve found particularly helpful:

*Distill bloated texts into high-yield summaries, then study those summaries.

*Concept-mapping might be even more effective than summarizing.  (Summaries were easy to type and store, so that’s what I got into the habit of doing, but in the age of tablet computers, concept maps might be digitalization-friendly, too.)

*(For residents, fellows, and attendings) Every day, study a bit of something you have not yet mastered or that you don’t know.  As with any other daily practice, it’ll amount to a lot over time.  Many physicians continue this habit–to the benefit of their patients–throughout their working lives.  (I picked up this tip from the chief of cardiology during my residency.)

*Don’t expect to outsmart medical exams.  They’re designed to prevent cleverness.  They often test how solid your knowledge base is by asking questions about “corner cases,” so you must have a strong working knowledge of the material.

*Master the high-yield information you’ve already learned, but don’t spend too much time on it before moving on to new material.  It’s comfortable to study familiar material, but focusing on important stuff you don’t know will make you much stronger.  (I adapted this tip from learning how to play guitar.)

*Study purposefully, with good, concrete reasons for why, how, and what you’re studying.  Don’t study on autopilot.  Don’t just reread stuff passively.  Change your study strategy if self-testing shows you it isn’t working well.  (I adapted this from a learning strategy that my guitar instructor emphasizes.  It worked well for me while studying for the rheumatology boards, too!)

*Cut out Facebook, Twitter, and other social media distractions in the months leading up to the exam. This will give you an immediate advantage because your attention will be less fragmented.

*Technology is a double-edged sword.  When used properly, it can help you learn more efficiently.  A friend of mine who recently reached the USCF National Master level in chess says that technology has helped today’s kids reach levels of chess skill unheard of before computer-assisted training became possible.  However, technology can also be a crutch that prevents you from learning what you need to know.  Don’t offload everything to your “other brain” (e.g., smartphone).

*Study mindfully–this is more subtle than it sounds and is another idea from learning a musical instrument.  I find that I can be stuck, for months, on a mediocre level of performance of a tricky piece for guitar if I don’t mindfully break the problems down into small pieces and find elegant solutions.  (This usually involves finding a different fingering sequence or pattern that eliminates the prior problems.)  As soon as I approach the issues mindfully, I move out of my plateau and begin to improve again.

*Interleave studying with seeing patients, when possible.  This gave us a large advantage to students at many other medical schools when it came to studying for Step 1.  Seeing patients will give your studying traction.  It solidifies what you’re learning by giving it context and making it practical.  It even helped, in a different way, while I was studying for my rheumatology board exam.  Since I was building my practice in an underserved area while studying for the boards, I saw patients with serious conditions who had been waiting for months or longer to be seen by a rheumatologist.  Seeing brand-new patients is very different from inheriting patients from other rheumatologists.  One has to think more carefully and catch on to more subtle clues.  This unique situation and my studying each informed the other.

*On practice tests and then on the actual exam, stay coldbloodedly focused on the relevant parts of questions.  Board exam questions are filled with distractors.  A technique that worked for me was to scroll down to the answer choices after reading the first couple sentences.  That helped me figure out what to look for as I read the rest of the question.

*Test yourself frequently in conditions as similar to the actual exam conditions as possible, to take advantage of unconscious cues.  If this isn’t possible, then study in multiple different settings so that you don’t set up unconscious cues tied to recall.  (E.g., studying in pink rooms but taking the test in a white room might really mess you up.)

*Keep your studying short and intense.  Approach it like a full-time job and then enjoy your afternoons and evenings.  Many of us studied for Step 1 full-time for no longer than 8 weeks.  The accepted wisdom was that one would begin to forget memorized material if one studied longer than 8 weeks, that it was more psychologically daunting to study hard for so long, and that it would be tempting to not study at full intensity.

*Spaced repetition:  there are general spaced-repetition programs like Anki and Mnemosyne that anyone can use and also dedicated spaced-repetition learning software, such as Firecracker, for pre-meds and med students.  (Firecracker wasn’t around when I was in med school.)  You can also just use your own spaced-repetition algorithm with paper and pencil.

*Mnemonics and mnemonic systems are useful both for short-term retention (e.g., long-enough to ace a test) and, in combination with long-term spaced repetition, for permanent retention.  There are entire books/websites on mnemonics, so I won’t go into detail here.  My favorite kind of mnemonic is the story mnemonic.  Read my next essay on how I used a giant story mnemonic to ace one of the hardest courses I’ve ever taken.

*Become a savvier strategist.  Consider learning chess or another strategy game.  Better yet, learn how to play many games.  This will increase your flexibility and effectiveness in everything you do.

*Try to get at least eight hours of sleep each night.  Try to exercise most days.

*Meditate daily for at least ten minutes.  Meditation is as old as the hills but research is only now beginning to discover its wonderful benefits.  A regular meditation practice will help you focus more easily when the big exam day arrives.  It will also help you avoid getting sidetracked when life (inevitably) tries to get in the way of your test preparation.

Do you use any of these approaches when learning new things?  If not, which learning strategies do you prefer?

Update 2/1/16: Dr. Oakley has graciously included this essay in the course material for Learning How to Learn!

“Cormorants” and Other Challenging Pencil Drawings

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Way back in January, 2011, I went on a whale-watching trip off the coast of Southern California with a friend–a Basque neurology resident–from Spain.  I had been drawing for only a few months, working through Drawing on the Right Side of the Brain, and wanted to challenge myself with something tricky, so I drew my hand drawing cormorants (I used a photo from the trip as a reference; the drawing was done with graphite on low-quality paper).  It was exciting to do something with a different visual perspective!

Since then, I’ve challenged myself in numerous other ways, all with the goal of gaining skill as rapidly as possible.

In November, 2011, I sketched this shell directly in five minutes…

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I wasn’t happy with it, because it wasn’t sufficient practice, so I drew the shell again, this time by drawing the negative space around it.  It took about thirty minutes and was drawn from a slightly different sighting angle than in the below photograph:

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My right hand drew itself drawing itself:

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That wasn’t easy.  Sketched it while flying back from a fellowship interview in September, 2012.  (It’s hard to draw on a plane because of the jitter and the shakes!)  The plane trip took a while, so I also drew this:

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Just as with any other skill, drawing requires deliberate practice for improvement.  Think of such challenges as “interval training” for your drawing ability.  (Interval training is one of the things runners do to gain speed and fitness.)

Learning Ruby with Codecademy

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I just finished the interactive Ruby course on Codecademy.  I highly recommend it!  It’s a free, interactive, engaging introduction to Ruby and is much more fun than reading a book.

I initially learned some Ruby (while learning Rails) back in fourth year of medical school and liked it a lot.  I then didn’t get to use it for years and sort of forgot it (but you don’t really forget…when you get back into it, it feels like reconstituting powdered milk or something: things come back to you pretty quickly).

I haven’t programmed seriously for a decade and was surprised at how much came back to me while taking this course.  I even remembered some esoteric stuff from C++ (I used to program in C++, Java, Lisp, and several other languages before I left computer science for medicine).

When I was a computer science undergraduate student, the emphasis was on theory, not on practical stuff like web programming (which was, quite frankly, rather looked down upon).  If you knew how to create websites, it was usually because you taught yourself.

Slow Down to Speed Up

Before I had a guitar instructor, I would learn a new piece or exercise by breaking it into chunks.  I would learn each chunk individually and then merge it with prior chunks until I could play the entire piece.

I still do this, but I’m also doing something else that’s at least as effective:  I practice each new piece at a tempo slow-enough to be able to play it in the correct time from the get-go.  At first, this tempo is very slow (like 30 beats per minute).  This is harder to implement than it seems, because our tendency is to hurry up and learn the piece yesterday.

I started using this practice strategy after my guitar instructor told me that the best guitarists he’s known have all started out by practicing very slowly.  He even suggested that the more slowly and correctly you’re willing to initially practice, the higher your eventual zenith of skill.

By implementing this strategy, I’ve witnessed a seeming paradox:  practicing very slowly helps me to progress more quickly.  Because I’ve always been learning piece X in the correct time, I no longer suffer from a major problem I had before:  awkward pauses “baked” into the song because I was faster in some sections than in others, thereby practicing these differences into the song itself even though they didn’t belong.

This strategy of slowing down to speed up is useful in other domains, too:

Running:  you want to start out slowly, both during each run (to warm up), and also when you’re just starting out as a runner (too many people make the mistake of doing too much too soon, thereby getting injured and not being able to run for weeks or months).

Strength training: lifting weights slowly (and lowering them even more slowly) may help you gain strength more quickly than otherwise, at least in the initial stages of a strength-training program.  It also reduces the risk of injury.

Learning to draw:  you start out very slowly, creating contour drawings, respecting negative space, assessing value, and carefully gauging perspective at every step of the way.

Mathematics/hard sciences:  you don’t rush over complex material; instead, you work through it carefully, taking more time to explore ideas you haven’t fully understood.

There is a time to speed up, but that time arrives only after you’ve worked slowly enough to gain significant skill.  For example, after you’ve mastered a song at a slower tempo, you can speed it up.  After you’ve gained some fitness, you can consider incorporating interval training in your runs.  After you’ve learned to draw with as little internal processing (stereotyping) as possible, you can start to incorporate gesture drawing into your practice sessions to increase your speed of drawing.  After you’ve learned, say, a section in your physics text and have solved some problems, you can consider timing yourself.  After you’ve learned a language, you can start to read more quickly to improve your overall comprehension (because questions that arise are often resolved by info you encounter later in the reading).

How I Fell in Love With Mathematics…and Only Then Became Good at It

A paper about the experience of mathematical beauty and its neural correlates was just published in Frontiers in Human Neuroscience.  It reminded me of how I fell in love with mathematics just before my senior year of high school.

Before that time, I found math a little frustrating.  I was decent at it and got good grades, but something critically important about math escaped me.  You could say that I had no trouble with math exercises, but I wasn’t skilled at solving math problems.  I didn’t know it at the time, but I lacked a sufficiently deep understanding of math and problem-solving in general.  All of this changed when I unexpectedly rekindled my childhood interest in logic puzzles.

Near the end of junior year of high school, I saw a curious little book on my English teacher’s bookshelf:  Fantastic Book of Logic Puzzles, by Muriel Mandell and Elise Chanowitz.  I pulled it off the shelf, started reading the back cover, and was immediately taken by such statements as the following:

“The puzzles you’ll meet inside this book are the world’s greatest and most baffling.” (This was not even nearly the case, of course.)

“To solve them, you need a plan of attack and logical reasoning…You’ll go on to solve tougher puzzles than you ever thought possible.” (True!)

It piqued my interest because it reminded me of how I used to enjoy solving logic grid puzzles as a seven- or eight-year-old.  My English teacher made a copy of it for me and I started working through it immediately.

Unlike the exercises I had encountered in math classes, which reinforced facts or blind techniques,  the logic puzzles in this book placed a heavy emphasis on deeply understanding the situations at hand.  You almost certainly wouldn’t solve most of them if you weren’t thinking carefully.

I finished the book, started studying for the SAT, and realized something very important:  I never learned geometry.  I mean, I took geometry in ninth grade (I was now about to start twelfth grade!), and I did “well”, but now that I knew how deep understanding felt, I realized I didn’t actually understand geometry on any significant level.

Nor did I know algebra.  Or trigonometry.  Or precalculus.

So, I sat down and gradually worked through all of these subjects from scratch while studying for the SAT.  I had learned, while solving logic puzzles, to constantly step back and question my assumptions.  I learned to hold only valid assumptions, and as few of them as possible.  I applied these habits of mind as I boldly attempted to prove theorems and other mathematical relationships by myself.  It was hard work, but enjoyable and addictive.

I became a mathematical explorer.  Math was no longer frustrating but fun.  I got into the habit of attempting to solve any test of ingenuity that came my way.  Paul Zeitz, in The Art and Craft of Problem Solving, asserts that this exploratory/investigative mindset is key to gaining skill at solving significant problems.  The problem-solver lays siege to the problem at hand, circling it until an opening is found–until the crux point of the problem is resolved–and then the problem surrenders its secrets to him, often in a trivial way, and is solved.

Gradually, I gained confidence and became skillful at math, doing very well in senior year calculus and in any other class (chemistry, physics) requiring accurate, logical thinking, as well as on standardized exams.  No longer intimidated by math, I switched from biochemistry to computer science after my freshman year of college and deliberately took the hardest math and science classes I could (much harder than entry-level “pre-med” classes), enjoying them greatly and doing very well in them, and, eventually, getting into one of the best computer science doctoral programs in the world.

They say the bumblebee doesn’t know it shouldn’t be able to fly.  I didn’t know I couldn’t become good at math, so I kept at it, and kept surprising myself, until I became pretty good.  I never did become great, but that’s okay.

(Who knows?  Maybe I would have become great if I had not switched careers and had continued to apply the principles of deliberate practice.  Maybe you could become great at some skill that currently escapes you, if you just drop the invisible script of “I don’t have the necessary talent” and get down to business!)