Posted in Educational Reform, Math

Math Is A Creative Endeavor

Many of us come away from our compulsory math education with the impression that math and creativity have nothing to do with each other, that math is the epitome of convergent thinking: there is just one right answer, and just one way to find it. At best, this is only half true.

Mathematical statements are precise, and have binary true/false values (if they are well-defined), but those of us who enjoy math see it as a creative exploration of logical relationships. The answers in math may be convergent, but the ways of arriving at them are infinitely divergent.

For example, consider the question “what is 5 + 5?” Ostensibly, it has just one answer, 10. But how many ways are there to arrive at this answer? Some people might do it by counting. Others might do it by multiplying 5 times 2. Others might do it by looking at their hands. Many probably know it by rote memorization. In terms of the inner cognitive process of computation, there is literally no limit to the variations of thought involved even with such a simple calculation.

Now consider something a bit more sophisticated (yet still relatively simple), like a proof of the Pythagorean theorem. It has been known at least since the Babylonians, definitively proved at least since Euclid, yet over the centuries hundreds of proofs of the Pythagorean theorem have been recorded, including an original one by president James Garfield. The question “how many ways are there to prove the Pythagorean theorem” is a classic example of divergent thinking in action.

If the idea of math as a creative endeavor seems surprising to you, don’t worry, it’s not your fault. It’s simply a result of outdated teaching methods. When I teach math, I do it in a way that engages both sides of your brain, so that it is actually engaging, interesting, satisfying, and yes, creative.

Posted in Books, Inspiration, Math

Schneider On Nature’s Patterns

Excerpted from A Beginner’s Guide to Constructing the Universe, by Michael S. Schneider:

Nature’s patterns and those of our inner life are familiar to everyone and always available to us. The  power which we seek is the power with which we seek. When we feel separate from the archetypes of nature, number, and shape we make them mystical, but this only keeps our selves in a mist. There is no need for secrecy and “occultism” any longer. These are everyone’s life-facts. We can apply them to better appreciate the world, and we’ll need them once we realize the urgency of cooperating with the way the world works.

Posted in Books, Inspiration, Math

Schneider On Sacred Mathematics

Excerpted from A Beginner’s Guide to Constructing the Universe, by Michael S. Schneider:

When the lessons of symbolic or philosophical mathematics seen in nature, which were designed into religious architecture or art, are applied functionally (not just intellectually) to facilitate the growth and transformation of consciousness, then mathematics may rightly be called “sacred.”

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 Books, Inspiration, Math

Schneider On Our Mathematical Nature

Excerpted from A Beginner’s Guide to Constructing the Universe, by Michael S. Schneider:

This quantitative approach keeps us dull to the potential wisdom that the familiar counting numbers can teach us. When imaginatively taught to people beginning at an early age, mathematics can delight, inspire, and refine us. It can make us aware of the patterns with which the world and we are made. Instead, math is taught as a servant of commerce, without regard for its basis in nature. It is viewed as a distant subject that instills much more anxiety than wonder and inspiration. Mathematics is seen as outside us to be occasionally called upon, rather than woven into the fiber of our existence.

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.

Posted in Educational Reform, Math

A Mathematician’s Take On Standardized Exams

Theoretically, if a test is designed to measure what a student has learned, then there should be no such thing as “preparing” for the test. This is why I believe that the best “test prep” method is to focus on learning what is taught in school, and ignoring the test itself until the day of. This philosophy may go against the grain of the conventional wisdom perpetuated by test prep companies and anxious parents, but it got me a full-ride scholarship to the college of my choice based on my ACT score, so I know there is something to it.

In general, with the prevalence of test-prep courses, software, and services, what standardized tests really measure is not how effective a student’s overall education has been, but rather how well the student has prepared for the test. This goes to show that sometimes what you focus on doesn’t expand, it actually shrinks.

One of the most common complaints students have about learning math is, “I’ll never use this”, and, when it comes to math as it is usually taught and measured on standardized exams, they are right. Math professor Sanjoy Mahajan writes on the Freakonomics blog about how the math questions on standardized exams are unrealistic, and how they could be written to reflect the ways that people actually use math in real life. I especially enjoyed reading this article because Sanjoy’s methods reflect the ways that I teach students to think about computations: using their brains in natural, intuitive ways, rather than like a fleshy digital calculator.