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.

Explore

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.

Practice

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 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, Homeschooling, Independent Education

What’s Wrong With Compulsory Schooling?

If school is necessary and beneficial, why does it need to be mandatory? Things that are truly necessary, like eating and sleeping, don’t need to be enforced because people want to do them. Likewise, people, especially children, are gifted with an innate curiosity and industriousness that makes them want to explore, discover, and create. They want to help others, they want to do the things that adults do, and every child, at some point, dreams of saving the world. Children do not want to avoid meaningful work, they crave it. Passivity is not the natural state of the human organism, but it is a natural reaction to coercion.

The belief that people don’t want to help themselves is a self fulfilling prophecy. If we believe that children must be forced to learn, we rob them of their initiative and don’t allow their creativity and intellectual curiosity to flourish. Children forget most of the facts they are force-fed at school, but they learn the meta-lesson all too well: “You are not in control of your life.”

The reason kids don’t want to go to school isn’t that they don’t know what’s good for them, it’s that they do know what’s bad for them. They don’t want to spend their time on useless, unfulfilling tasks, they don’t want to be forced to compete, and they don’t want to be judged, ridiculed, or belittled. They want to flourish, and they seek out the things that help them do that whether we try to force them or not; in fact we can’t keep them from doing it. We don’t have to force children to learn any more than we have to force them to breathe.

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 Tips for Students

Your Own Reasons And Your Own Ways

My teaching philosophy is based on my observation that you can only enjoy success for your own reasons and in your own ways. This goes for all types of success, big or small, from learning math to enjoying a sense of happiness, fulfillment, and accomplishment in life.

When I begin working with a new student, my first question is always “Why do you want to learn this?” Motivation is the key to learning, and that is why this is the most important question to ask. You have to have your own reasons for learning, and the tutoring process always starts with getting the student in touch with theirs. “Because I should” or “because I have to” is an ineffective basis for generating the intellectual curiosity that learning requires.

Furthermore, I pay attention to the student’s cognitive process of perceiving, processing, and communicating about the subject. We all have our own ways of understanding, processing, and recalling, and I make the effort to communicate in a way that is compatible with the student’s way of thinking. My goal is not for the student to nod their head and say they understand; my goal is for the student to have an internal model detailed enough that they could impart their understanding to someone else. The aim is not for me to explain things to the student, the aim is for them to explain things to me.

However you are seeking success, seek it for your own reasons and in your own ways, because this is the only way you can find it. And if you’d like the help of an experienced guide to mastering math and excelling at academics, get in touch with me.

Posted in Tips for Students

Math Mistakes

A previous post looked at how to learn math, this post is about common mistakes that keep people from learning it.

1. Negative Self Talk

Telling yourself “This is hard”, “I hate this”, “I’m stupid”, or any of the many variations of these three main themes is wasted effort that both drains your mental resources and makes you miserable. You can just as easily talk yourself into learning math as you can talk yourself out of it, and increase your level of enjoyment at the same time, by training yourself to replace negative self talk with positive, or at least neutral, internal commentary.

2. Taking It Too Seriously

Allowing yourself to relax, go slow, be clumsy, and aimlessly explore is a crucial part of the learning process. Pressuring yourself to get it perfect right away will actually keep you from trying, leading you to fail before you even begin. Don’t worry about competing with anyone or solidifying your plans for the future. Just allow yourself to enjoy the process and learn at your own pace.

3. Not Taking It Seriously Enough

Learning math isn’t a life-or-death matter, but it does take practice and repetition, just like any other skill. You wouldn’t expect to sit out gym class by saying “I know how to do pushups.” Mere exposure to the material alone does not build mathematical skill; it takes practice and repetition to train your brain. The pay-off is that what seems hard at first becomes easy, and then automatic, allowing you to progress to greater levels of skill and understanding.

4. Cramming

Each day that you fall behind increases the proportional amount of work you have to do to get caught up, and decreases the chances that you ever will. In the extreme case, the chances that you will be able to fit several weeks or months worth of learning into a few days or hours are slim to none. And even if you do manage to successfully pass a test this way, the effort will be wasted in the long run because you will forget the material about as quickly as you learned it. Laying the foundation consistently is what provides the best short-term grades and long-term knowledge.

5. Not Getting Help

Not asking for help when you really need it can mean that you lose the opportunity to truly understand the material, or at the very least that you waste a lot of time following dead ends. You may waste three hours struggling to understand a concept that an experienced tutor or peer could explain to you in thirty minutes, allowing you to devote your time and energy to more productive pursuits. Don’t be afraid to ask for help when you need it, even if it is from an outside professional, because your time and the educational opportunities it represents is the most valuable thing you have.

Posted in Inspiration, Math, Tips for Students

How To Learn Math

1. Get Curious

Curiosity is one of the most powerful forces in existence, because you can’t learn anything unless you are curious about it. And the good news is that our innate sense of curiosity is insatiable; it is possible to get curious about almost anything if you engage it with your imagination. Trying to force yourself to learn without engaging your curiosity is tortuous, laborious, and ultimately ineffective. So whatever learning task you have in front of you, bring your curiosity to bear.

2. Explore

Exploration is the non-directed, interest-based satisfaction of curiosity. It is a process of trying things to see what happens, asking yourself questions and answering them, following interesting paths just to see where they lead. You’ll likely get a feel for an unfamiliar city better by taking a meandering walk through it than by following a guided tour. Take the time to see the material through your own eyes and get a feel for it before worrying about performance or setting agendas.

3. Practice

When you’re ready to learn specific techniques, give yourself plenty of time to practice. Like any new skill, it will be awkward at first and become easier and smoother the more you do it. Allow yourself to be clumsy and inefficient, and just keep going through the process. Before long it will start to make sense, next it will start to be intuitive, next it will start to become easy, and eventually it will become automatic. This lets you incorporate it into your regular thought process and proceed to build higher levels of skill.

4. Give It A Shot

Don’t let getting stumped throw you off track. When you’re exploring your edges you will run into a lot of roadblocks, but let that be a signal to try something new. The harder you try the greater the benefit, as well as the enjoyment. It wouldn’t be much fun to fill in a crossword puzzle with the answer key right in front of you would it? It’s the process of racking your brain and wrestling with the clues that makes it fun and even addictive. Unless you’re repeatedly getting stumped and unstumping yourself, you’re not learning or improving.

5. Get Help

Learning doesn’t need to be an entirely solitary activity, nor should it. We are social creatures and the interaction of communication and cooperation plays an important role in the learning process. Peers can help each other through a process of co-learning, and teachers, coaches, and guides can direct you along fruitful paths. The time to get help is when you are completely stuck and have reached a point of diminishing returns in your process of solo exploration. This is the point where you can benefit the most from peer discussion or from having an experienced guide to reveal tricks and shortcuts and direct your focus to the most fruitful avenues.