Posted in Teaching & Learning, Tips for Students

How To Learn Math (Or Anything Else) Faster

I always say it’s better to play keep-up than catch-up, which means staying a little ahead of the progression of concepts in a class, rather than a little (or a lot) behind.  This means that you have to proceed at the rate the class progresses.  But what can you do if you feel like a class is progressing faster than your ability keep up?

Logically, there are only three things you can do to increase your rate of learning:

  1. Spend more time studying than you already do. If you are currently spending two hours per day, spend three, for example. Note that this approach has diminishing returns.
  2. Improve your process by learning how to study more efficiently. This requires that you invest time in learning how to learn faster. There are various resources and approaches for this, such as Scott Young’s Ultralearning.
  3. Hire a guide. Working with a tutor can increase the overall rate at which you learn by roughly a factor of three while reducing the stress you experience, and a tutor who is also a good academic coach will help you improve your process so that you can learn more efficiently on your own as well.
Posted in Math, Teaching & Learning, Tips for Students

How To Learn Math (Or Anything Else)

As the title of this post indicates, the process I am about to describe is actually the natural learning process for anything, not just math.

First, Get Curious

Learning has to start with curiosity. You can be curious about math as a means to an end (say, if you want to be a financial analyst or engineer, for example), or you can be curious about math for its own sake (if you want to be a mathematician, this will probably apply to you).

Remember, curiosity is an emotion, and emotions are generated in part by what we focus on and what we tell ourselves. So, if you want to learn something but don’t have any curiosity about it, generate some! Curiosity is the glue that makes new knowledge stick.


Once you get started, let your curiosity lead you into new and interesting territory. Venture forth into the material with an agenda of pure discovery. Let this be an open-ended, non-directive process, with no particular goal in mind, the same way that you would read a novel or watch a movie.

Think of the last new movie that you saw. Can you remember the setting? The plot? The characters? Were you trying to memorize any of those things? That should be proof enough to you that this method works.


In math, practice could take the form of performing calculations, solving problems, or writing proofs. In other subjects, it could take the form of answering practice questions or re-communicating what you have learned, either by writing or speaking. It is a Law of Learning that the harder you work for a particular piece of knowledge the better you will retain it, so don’t shirk on effort here. Just like with working out, the more you sweat, the more you get!

Get Help

Once you have put in sufficient practice, you are in an ideal position to ask for and receive help. Asking for help from a teacher, tutor, or mentor at this point will help you fill in any gaps that you have identified in your understanding, uncover any blind spots you may have and improve your process. This is how you go from proficient to efficient. Having first put in the effort to understand the material yourself will prepare you to appreciate and receive what is being offered.

Posted in Books, Educational Reform, Teaching & Learning, Tips for Students, Tips for Teachers

The Role of Neural Associations in Learning

Excerpted from Triggers, by Stanley Mann:

“Multisensory imagination is the world’s finest teaching machine, and we all possess it.  We simply need to learn how to use it.”

I just came across this great quote in a book I’m reading. The book is about using triggers, or neural associations, to direct one’s attention in constructive ways. The quote comes from a chapter on using triggers to enhance learning ability. It resonates with me on a number of levels.

First of all, the idea of multisensory imagination is a little-known but extremely powerful learning enhancement tool, and is the sort of thing that we should be teaching kids about in school instead of pumping them full of facts and figures. To truly learn anything you have to make it real in your mind, fully engaging with it in your imagination with as many sensory channels as possible; both sight and sound at the very least, and ideally touch, smell, and taste if possible. Conversely, teaching can’t work unless it engages the imagination and the emotions through multiple sensory channels. Educators and curriculum designers need to be mindful of this principle in order to be effective.

Secondly, triggers in general, and learning triggers in particular, rely on the principle of association. Our brains are constantly making associations among various environments, stimuli, and emotional states based on our experience. Neurons that fire together wire together, so when two experiences are repeatedly juxtaposed, they tend to become linked. This means that to optimize your learning process, you need to create a positive physical and mental learning environment for yourself, so that you are comfortable and feeling good while you are learning. Conversely, if the learning environment is physically or mentally uncomfortable, such as by feeling coercive and emotionally unsafe, then the negative feelings it evokes will come to be associated with the process of learning itself, causing them to be re-experienced whenever future learning endeavors are undertaken. This is another principle of human psychology that educators and legislators need to take heed of.

Posted in Teaching & Learning, Tips for Students

The Feynman Technique

The Feynman Technique

This old standby is a favorite of students everywhere: “I understand it, I just can’t explain it…”

Those of us with experience learning and teaching know, of course, that this is a contradiction in terms. If you can’t explain something, you don’t understand it!

The contrapositive of this gives rise to the Feynman Technique for mastering any material:

1. Study the material you want to learn until you feel you have some grasp of it

2. Re-communicate it in some form to someone who doesn’t already understand it, making it as simple yet as complete and accurate as you can; you can do this by writing as if you were explaining it to someone, or by actually explaining it to someone.

3. If you don’t understand a particular point or detail well enough to explain it in simple, clear terms to someone who doesn’t understand it, return and review the material until you can.

4. Repeat as necessary/desired.

Where the real magic of this technique comes in is that in the process of explaining the ideas you want to learn about, you will be organizing and contextualizing your thoughts as you articulate them, so the process of communication itself generates comprehension!

There are many ways to put this principle into practice in your own learning process. Of course you can just write about what you are learning on a piece of paper and keep it to yourself, but this is likely to seem dry and lifeless. An even better approach is to engage others in your learning process, by actually explaining what you are learning to interested friends and relatives, or other students in the same class. You can write articles and blog posts, answer questions and provide homework help online, or even tutor other students.

Whatever you do to put yourself into a situation where you are re-communicating what you are learning for someone else’s benefit, making it as simple and clear as possible, will cement your own comprehension. Just remember, you can’t say you understand something until you can explain it!

Posted in Educational Reform, Math, Teaching & Learning

The Moore Method

“I hear, I forget. I see, I remember. I do, I understand.” — Chinese proverb

The Moore Method is a little-known method for teaching advanced math that gets great results. In essence, it sacrifices breadth of coverage for depth of understanding, i.e. it prioritizes quality over quantity when it comes to learning a subject.

In essence the Moore Method works by having the students present the course content themselves. In higher math, the semester starts with a list of definitions and theorems to prove from them, with new theorems being introduced as students progress through the material. However, I believe that this approach could (and should) be adapted to other topics and levels of study and scaled up. This type of participatory/active, rather than receptive/passive, classroom experience is a fundamental feature of the educational revolution that is sweeping the planet.

“That student is taught the best who is told the least.” Robert Lee Moore, inventor of The Moore Method

Posted in Independent Education, Inspiration, Teaching & Learning, Tips for Students

Albert Einstein’s Letter To His Son

I recently came across a letter from Albert Einstein to his son, aged 11, just before he became famous.  Historically, eccentric geniuses don’t tend to make the most reliable fathers, and Albert was absent for most of his kids’ lives while they were being raised, but this letter shows that this was clearly a matter of distraction rather than a lack of compassion.  My favorite thing about it is that Albert, never a stickler for academic conformity himself, tells his son that there are more important things than school, namely joy and learning:

“I am very pleased that you find joy with the piano. This [is] better even than school…Mainly play the things on the piano which please you, even if the teacher does not assign those. That is the way to learn the most…”

Posted in Inspiration, Teaching & Learning, Tips for Teachers

Bruce Lee On Teaching

“A good teacher can never be fixed in a routine. Each moment requires a sensitive mind that is constantly changing and constantly adapting. A teacher must never impose his student to fit his favorite pattern. A good teacher protects his pupils from his own influence. A teacher is never a giver of truth; he is a guide, a pointer to the truth that each student must find for himself. I am not teaching you anything. I just
help you to explore yourself.”

bruce lee