Here’s something tutors hear all the time:
“I understand the math when it’s taught in class, and I do the homework, but I have a really hard time with tests.”
That innocent statement embodies one of the fundamental misunderstandings many students have: the difference between understanding and mastery.
Understanding means “getting it,” making sense of the details and the steps, seeing how the puzzle pieces fit together. Mastery, on the other hand, means reaching a level of skill with the content such that we can perform the learned task quickly, accurately, independently, and under pressure. Mastery is where we want to be on test day.
There are many things I understand but have not mastered. I understand how to throw a curveball, but I can’t consistently hit the strike zone with a good curve. I understand how to juggle, but no one’s going to hire me to perform in a circus because I can only juggle with three balls, I don’t know any tricks, and I often mess up and drop a ball.
I am not this guy:
You have not truly learned something until you’ve moved beyond understanding. To truly, deeply learn something, such that you won’t soon forget it, means to walk the path toward mastery.
The Mastery Path
The mastery path often begins with the unpleasant feeling of ignorance and confusion. We’re exposed to a brand-new idea, and it makes no sense to us. But the teacher teaches, we examine the diagrams, we listen to analogies, we wrestle with the puzzle pieces, and hopefully, soon enough, we understand the idea.
When we have this moment of insight, things click, light bulbs go off, it’s dramatic, it’s emotional, and it feels like we’ve finally arrived at our destination. But we haven’t. Understanding is a critical step along the way, but it isn’t the end of the road. Far from it.
Understanding is a key milestone on the road to true learning. It is a necessary prerequisite for mastery because we can’t master something we don’t understand. So the first and most urgent goal for any student who is trying to learn new content is to reach understanding.
The two most common errors students make in the learning process are: 1) not urgently pursuing understanding, and 2) not continuing to practice after reaching the point of understanding.
The first error is caused by the incorrect belief that the “I get it!” moment is the end of the road. After all, if that’s the end of the road, what’s the rush? Students who don’t grasp that learning is about more than understanding see no reason to prioritize reaching the “I get it!” milestone. They don’t recognize that waiting until the last minute to figure something out means they cannot truly learn it.
The second error is sometimes caused by the same incorrect belief, while at other times it is caused by overconfidence. In fact, at the “I get it!” stage, most of us – myself included – have a tendency to slip into overconfidence. While looking at a list of topics that will be on the test, a student who has reached this stage might cross a topic off the list, saying, “Oh, I know that.” If I then offer a practice problem or ask the student to explain the topic to me, we very often discover that he either doesn’t fully understand it or doesn’t fully remember it. Either way, he doesn’t really know it because he never truly learned it.
It is critical for students to know that long-term memories are formed by spaced repetition: spread out practice that occurs after reaching the “I get it!” milestone. It is also critical for students to know that they’re still liable to be slow and make lots of mistakes unless they practice what they know. Approaching mastery almost always means doing extra work beyond what the teacher asks for. It means practicing on your own, even though no one is making you.
I don’t always choose to walk the mastery path, and results are, well, unsurprising. Here’s an example:
I figured out how to multiply matrices last summer, but I haven’t practiced this math operation enough times to be able to perform it quickly and without errors. In fact, it has been nearly a year since I reached an understanding of matrix multiplication, and during that year I’ve only practiced it twice, so I would probably need to refresh my memory of how it works before I could even begin.
This highlights another feature of the mastery path: We’re never standing still.
By not practicing matrix multiplication, I didn’t just stay at the beginner-level with that skill, I went backward. On the mastery path, there is no standing still; there is only motion, forward and backward. In order to maintain your current position on the path, you have put in a little effort, the equivalent of marching in place. If you don’t practice what you know, you’ll forget it. This should come as no surprise because one of the fundamental principles of neuroplasticity is “Use it or lose it.”1
Now, I must give an important caveat about this diagram. The mastery path serves as a quick, visual reminder of the principles I’m describing in this article, but it is greatly oversimplified.
First of all, the path from ignorance to mastery is rarely a straight line. There are often false-starts and back-slips and wrong turns along the way. Mistakes and failures are the norm. Frustration is common, especially at the beginning. For this reason, you can’t very well walk the mastery path without a growth mindset.
Secondly, although I deliberately made the second arrow longer, it is not at all to scale. In reality, the second arrow should be thousands of times as long the first. It takes much, much longer to reach mastery than it does to reach understanding.
Furthermore, neither getting it nor mastering it can truly be thought of as black-and-white checkpoints. Both exist on a spectrum. As we approach the “I get it!” milestone, we may experience several glimmers of insight as our understanding gradually builds. And we may continue to refine our understanding long after that milestone.
Along the path from understanding to mastery, we steadily increase our prowess, progressing from beginner to intermediate to expert. But even after reaching world-class status in any particular skill, there remains room to grow. The diagram vastly oversimplifies the long continuum from utter cluelessness to the non-existent ideal of complete mastery.
In the real world, mastery doesn’t mean perfection. Masters still make mistakes; masters can still improve. And, as a competition like the Olympics makes clear, there are differences in ability even among the greatest performers in the world. The pursuit of mastery is really a lifelong endeavor, as athletes and artists often demonstrate.
World-renowned cellist Yo-Yo Ma is one of the best examples of the unending pursuit of mastery. Although he was a child prodigy who reached proficiency on the cello in 1960 at the tender age of five, Ma has never ceased working on becoming better. Even with over 90 albums and 18 Grammy awards, his career and his growth as a musician appear to be far from over.2
Mastery of this sort is a lofty goal and can be very intimidating. You may have heard that it takes 10,000 hours of deliberate practice to reach a level that might be called mastery. For most school subjects and micro-topics within those subjects, true mastery isn’t really the goal. Instead, “excellence” or even “good enough” is the appropriate goal. And rest assured, it takes much, much less time to reach good enough. That is a manageable, doable task that can be accomplished in the normal time-frames that students have to work with. The main point of this article is that “I get it!” is not good enough.
Because we never really reach true mastery but only approach it, author Daniel H. Pink says in his book Drive: The Surprising Truth About What Motivates Us that mastery is like an asymptote:3
This graph shows, quite correctly, that we make rapid, large gains in the beginning–the most important work–and then we make incremental gains thereafter–the hard work of consolidating memories and refining skills. While the graph shows that you cannot go from zero to hero without putting in a great deal of time, it also shows that you can go from zero to decent rather quickly.
One way to know if you’ve reached the level of “good enough” is to notice how much mental effort it takes to recall, talk about, or use the information. If it still feels slow or strained, you’re not there yet. If manipulating the ideas is fast, easy, and automatic, you’re there. For example, when someone says, what’s 2+2? Do you have to think about the answer? No, you reflexively know the answer because you have that concept deeply memorized.
So far, we’ve described the benefits of approaching mastery as the following: making fewer mistakes, becoming faster, and forming stronger long-term memories. But the 2+2 example I just gave demonstrates yet another benefit of deep learning: the ability to utilize knowledge without adding to our cognitive load.
When we know something so well that we can use the information automatically and without effort, we’re left with an unburdened mind. An unburdened mind has energy to spare, which is precisely what we’ll need to learn new ideas, plan ahead, strategize, and solve complicated problems.
Walking the Path
I’m now going to explain how to walk the mastery path. I will break up the instructions into the two main phases of the mastery path. Although there is generally a difference between the way you pursue understanding and the way you pursue mastery, some of the same tactics, such as spaced repetition and mental recall, are helpful during both stages.
The Pursuit of Understanding
On the path from ignorance to understanding, there will be a fair amount of time spend not getting it before arriving at the “I get it!” milestone. It’s common to feel frustrated and disheartened at this stage. But fear not! The following strategies can shorten the journey to understanding.
Whether you’re learning new material from a teacher, a book, or a YouTube video, the first step is always the same: Pay attention! Now, this may shock you, but paying attention is easier when there are fewer distractions, so do everything you can to eliminate distractions. If you’re at home, perhaps that means studying in a dedicated, quiet space, away from your siblings, your pets, and the television. Wherever you are, the ultimate distraction–your phone–should be turned off, set to airplane mode, hidden out of sight, placed in another room, or at least silenced.
One way to improve your focus as well as your retention of the content is taking notes. But this can be uncomfortable when you’re brand new to an idea. When you’re in the “I don’t get it.” stage, it’s normal to be confused by what you’re learning. You can’t tell which details are important and which ones aren’t because you don’t know what’s going on. That’s okay. Take notes even though it doesn’t make sense yet. You’ll put the puzzle together later. Right now, you’re just collecting the pieces.
You’re probably better at guessing what to write down than you think. Trust your instincts and write down whatever seems important. But here’s a simple rule of thumb to follow: If the teacher writes it on the board, you should write it in your notes.
Sometimes the teacher will even provide a handout that contains the most important information. This, it turns out, is poison disguised as dessert. Such handouts create the illusion of learning because you “have” the information. The problem is that you don’t have the information in your head. Although the ideas you’ll need for the test are on the handout, you probably won’t understand them or remember them unless you write them out yourself. When you re-write the information on the handout, make an effort to paraphrase and summarize in your own words. Change the order, add examples, or add drawings if that helps you grasp the ideas. Make it your own.
If the way the information is presented in class seems inadequate, remember that we live in the Information Age, so resources abound. If there’s a textbook for the class, that’s the best place to look. Start there and mine it for information. Again, take notes. If you still need more information, use the internet and type your topic into Google. Odds are good that many other students have looked for the same information, which means you’ll probably find helpful resources. Google results can be overwhelming, but there is very often a short YouTube video that can teach you the ideas. For math and science, check out Khan Academy. For history, science, and an increasing variety of other subjects, check out Crash Course.
As you’re engaging with these resources and as you’re attempting your homework, it’s likely that you’ll have questions. Write them down! Later, these written questions will serve as a guide when you engage with other resources. Sometimes typing your exact question into Google yields an answer. Most importantly, by writing down your questions, you’ll be much more prepared to engage with the greatest resource of all: other people.
The next day, ask your teacher whatever questions you wrote down. In most cases, your teacher is the best human resource at your disposal. But you don’t have to wait until the next day to ask someone. You may have a parent or sibling who could help. You probably have classmates whom you can contact. Perhaps one of them could explain the ideas to you.
Tutors, of course, love when a student shows up to a session with questions in hand. Students can get much more out of tutoring sessions when they have already struggled with the material, engaged with resources, taken notes, and discovered what their specific questions are.
The same rule for notes applies to homework: Struggle with the assignment even though you don’t get it. By doing so, you’re setting yourself up to understand it later.4 That struggle, it turns out, sends a very strong signal to the brain’s unconscious, which hears the signal and proceeds to work on figuring it out. Because you collected the pieces and made it clear that you’d like to be able to put them together, your unconscious will steadily work on the puzzle in the background.4
Unfortunately this superpower of the brain falters when we jump from one highly engaging activity to the next. A little bit of downtime in between studying and the next activity–or the next time you check your phone–gives the unconscious a moment to work its magic. So take a moment to pause and process after studying. Let it sink in. Allow some time for being bored and alone. At the very least, take a 30-second microbreak.
Figuring out anything complicated requires that we alternate back and forth between two brain modes: focused-mode and diffuse-mode.4 Focused-mode is active, energy-intensive studying, note-taking, and problem-solving. In focused-mode, we’re consciously engaged with the ideas. Diffuse-mode, on the other hand, is relaxed and passive. It occurs when we take real breaks, exercise, and sleep. In diffuse-mode, we’re unconsciously engaged with the ideas. To facilitate this back and forth that our brains use to sort things out, practice spaced repetition.4
I recommend working with the ideas three times during the first 24 hours and every day during the first week of learning. Working with the ideas could mean engaging with different resources, re-writing your notes, doing homework, or even just thinking about the ideas during downtime to see how much you can recall. These repetitions can be brief engagements of just a few minutes and still be beneficial. This pattern can help you quickly arrive at an “I get it!” moment, and can also help you solidify that understanding as you begin the pursuit of mastery.
The Pursuit of Mastery
As soon as you reach the “I get it!” milestone, you should restart the pattern of three engagements in the first 24 hours, followed by one per day for the next seven days. After the first week, you can taper off the repetitions and engage the ideas with decreasing frequency.
This may seem like a great deal of extra work, but it’s really important if you want to remember these ideas in the long run. The second round of spaced repetition is about securing your new knowledge. You just spent all that time and energy learning the content. Surely you don’t want to forget it and have to start over. You will forget unless you take steps to strengthen the memory.
The best ways to engage with the content to strengthen your memory all involve pencil and paper: writing the ideas out, doing practice problems, adding to your notes, or making some other form of written product. But remember that you can also mentally engage with the content when you’re nowhere near writing utensils. You’ll benefit from thinking about the ideas during downtime, such as walking, riding in the car, or doing the dishes.
In his book, The Champion’s Comeback: How Great Athletes Recover, Reflect, and Re-Ignite, Jim Afremow distinguishes between two mindsets about practice. Some people just practice a skill until they’re able to get it right; others practice until they’re nearly unable to get it wrong.5 The second group of people are much more successful. So, once you’ve practiced enough that you can sometimes do it correctly, don’t make the mistake of thinking you’re done. Practice until you hardly ever get it wrong.
When you engage in practice, especially for a subject like math, try to get feedback about whether or not you’re getting the correct answers. That might mean choosing practice problems in the textbook for which the answers are in the back of the book, it could mean printing a worksheet from Google that has an answer key, or it might mean using an online tool like Khan Academy that checks your answers in real-time. Pay special attention to anything you get wrong. Use resources to figure out what to do, correct your mistakes, and come back to those problem types later.
As you pursue mastery, you may also discover that you have new questions. Because you now understand the basics, you can ask more profound questions. The program here is the same: Write down your questions, so you can ask someone later.
You may come to a point where you’ve enhanced your understanding enough that you would benefit from rewriting your notes. As you do so, make any changes informed by your new sense of mastery. Consolidate ideas that can be expressed more succinctly. Elaborate on ideas that you now have more to say about. Make special note of the best examples and most difficult concepts. This form of review is much more powerful than simply rereading or copying your old notes.
An excellent test of how far you are along the mastery path is your ability to teach the ideas to someone else. Being able to work with the content is one thing; being able to explain it clearly is another. Teaching is more challenging than learning, and it requires a much deeper understanding of the content. Not only is teaching a good way to review what you know, it is also a very quick way to discover any gaps that remain in your understanding. Plus, as they say, he who teaches learns. The person whom you explain the ideas to will probably ask some interesting questions or share new metaphors that will enhance your own understanding.
Teaching is just one version of self-testing. Taking actual practice tests on paper is another powerful way to see clearly what you know and don’t know. Sometimes practice tests are available in your textbook or online, and sometimes you have to invent them yourself. However you test yourself, check your answers. Then write your wrongs: On paper, correct or re-do anything you missed or forgot.
Any Progress is Good Progress
Remember that success and mastery exist on a spectrum. You don’t have to march along the mastery path all the way to the mountaintop for your effort to count. Every step you take in the right direction is beneficial. Every little bit of practice improves your skills. Every little bit of review strengthens your memory. Any progress is good progress.
1 Doidge, Norman. The Brain That Changes Itself. Penguin Books, 2006.
2 “Yo-Yo Ma.” Wikipedia. 10 June 2016. https://en.wikipedia.org/wiki/Yo-Yo_Ma#Awards_and_recognitions.
3 Pink, Daniel H. Drive: The Surprising Truth About What Motivates Us. Riverhead Books, 2011.
4 Oakley, Barbara. A Mind for Numbers: How to Excel at Math and Science (Even if you Flunked Algebra). Penguin, 2014.
5 Afremow, Jim. The Champion’s Comeback: How Great Athletes Recover, Reflect, and Reignite. Rodale Books, 2016.
Title Image: Loper, Chris. Original photograph. Text added.
Juggler: ryder, kevin. “Juggler.” https://www.flickr.com/. Creative Commons 2.0. Image cropped.
Mastery Path. Loper, Chris. “The Mastery Path.” 2016.
Mastery Asymptote: Lindquist, Rusty. “The pursuit of mastery, and the mastery asymptote.” Life-Engineering.com. April 15, 2010. http://life-engineering.com/the-pursuit-of-mastery-and-the-mastery-asymptote/.
Yo-Yo Ma: Daily, Ralph. “Yo-Yo Ma.” https://www.flickr.com/. Creative Commons 2.0.
Hiker: Creative Commons Public Domain. Courtesy of Pixabay. https://pixabay.com/en/wanderer-backpack-hike-away-path-455338/.
Teacher: Creative Commons Public Domain. Courtesy of Pixabay. https://pixabay.com/en/male-man-teacher-professor-213729/.
Football Player: Creative Commons Public Domain. Courtesy of Pixabay. https://pixabay.com/en/football-player-receiver-pass-858426/.