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 It’s simply a result of outdated teaching methods.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 A Better World Through Mathematics

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

In this time of rapid change and transition of the roles of traditions and institutions, we have the opportunity to restructure education and teach children differently, to expose them to harmony in all its forms, in nature, music, art, and mathematical beauty.  Perhaps children steeped in harmony will become a generations of adults who will strive to achieve harmony in the world.  And perhaps they will transform our relationships with our environmental matrix to treat the soil, water, air, plants, and creatures differently, cooperatively, in ways born of understanding of the whole, respect for its parts, compassion, and common purpose.  Comprehending nature’s speech will let us listen to what she is telling us in her own native language, which is also our own.  If we can see and understand nature as a harmony in which there is room for diversity and in which we participate, we’ll want to transform ourselves and our relationships to align with that harmony.

We often act as if inner human nature was unconnected with outer nature, and we judge the outer world by one standard, ourselves by another.  Familiarity with the principles of geometry can help reconcile this artificial division.  The geometry outside us shows us the principles within ourselves.  It’s time we, as a global whole, relinquish old models of looking and learning and begin to cooperate.  Literacy in nature’s script dispels the stereotype of nature as disorganized, unintelligible, and hostile.  This book is about reshaping our vision and constructing a new perspective aligned with life-facts.  Learning nature’s language and reading its message helps abolish the attitude of separateness and encourages us to appreciate diversity.  It will lead to nothing less than our own transformation as we find all nature’s principles within ourselves.

To learn to view the world in terms of its patterns requires a shift within us.  But once this shift occurs and we see the familiar world in terms of its shapes and principles, a light turns on and the world brightens, comes into sharper relief.  Everything speaks its purpose through its patterns.  Even without knowing it we use the same designs found in nature.  Look at a microscopic diatom and see a cathedral rose window.  Ultimately, the same energy that motivates and guides the natural world does the same for us.  All universal designs are found in human body proportions, which we have seen can be repackaged to produce the proportions of a crystal, plant, animal, solar system, and galaxy.  It is as if the universe is one single organism, motivated by a single power, developing in many ways to gradually become aware of itself through the awareness of the creatures and forces it produces.

Posted in Books, Inspiration, Math

Schneider On Harmony

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

Symbolic and sacred mathematics encode subtle experiences whose purpose is different from that of secular mathematics. They can invigorate, refine, and elevate us. Our role as geometers is to discover the inherent proportion, balance, and harmony that exist in any situation. The study and experience of numeric and geometric proportion infuses in us an appreciation of proportion everywhere. The study of balance teaches us to recognize and seek a sense of balance in our lives. The study of harmony develops our sense of harmony in all relationships. Actually to see and work with unity and wholeness in geometry and natural forms, rather than just read about them, can help abolish our false notion of separateness from nature and from each other. It is this notion that ultimately fuels competition for the “goods of the earth” and contributes to environmental crises.

Posted in Books, Inspiration, Math

Schneider On Universal Principles

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

Studying, contemplating, and living in agreement with universal principles is a social responsibility and can be a spiritual path. it is becoming clear that when we cooperate with nature’s ways we succeed; when we resist, we struggle. Implications for our environmental crises are obvious. Rather than an antagonist, nature can be our teacher to learn from and cooperate with to mutual benefit. To understand nature better, we first need to recognize the roles of its basic patterns.

Posted in Books, Educational Reform, Math

Schneider On The Personalities Of Numbers

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

It’s a shame that children are exposed to numbers merely as quantities instead of qualities and characters with distinct personalities relating to each other in various patterns.  If only they could see numbers and shapes as the ancients did, as symbols of principles available to teach us about the natural structure and processes of the universe and to give us perspective on human nature.  Instead, “math education” for children demands rote memorization of procedures to get one “right answer” and pass innumerable “skill tests” to prove superficial mastery before moving on to the next isolated topic.  Teachers call this the “drill and kill” method.  Even its terminology informs us that this approach to math is full of problems.  It’s no wonder countless people are innumerate.  We’ve lost sight of the spiritual qualities of number and shape by emphasizing brute quantity.

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.

Posted in Math, Online Tutoring

Practice Doesn’t Necessarily Make Perfect

The saying “practice makes perfect” requires some qualification: “Perfect practice makes perfect.” It is not mere mindless repetition that improves skill, but deliberate practice with the aim of improvement. Deliberate practice, the kind that requires “careful reflection on what worked and what didn’t work,” is the subject of this Freakonomics article by math professor Sanjoy Mahajan, particularly as it relates to math and science education. In it, he proposes the question, “What would an educational system look like that took seriously the principles of deliberate practice?”

This is another one of the ways that the role of a tutor is invaluable for creating true expertise. In math, students typically get graded on their results only, with little, if any, attention given to the process of arriving at the results. But math is not just about arriving at the right answer; that’s something a computer can do. For a human, it is also about arriving at the result in the most artful and efficient way possible. As a tutor, I have the opportunity to directly observe and shape the student’s thought process in the way that a teacher rarely does. This way, the student not only gets feedback about whether their answer is correct, but also about how to improve their thought process. A good tutor can make the difference between struggling for hours to come up with an answer and being able to arrive effortlessly and artfully at a result. A teacher can teach, but a tutor can make the learning process fun, changing the work from a grind into a delight.