Posted in Educational Reform, Math

Research Shows Timed Testing Causes Math Anxiety – DUH!

I recently came across an article in Education Week where Stanford mathematics education professor Jo Boaler highlights a number of research studies that establish something those of us who pay attention have known for a long time: timed testing causes math anxiety. Can you say “Duh!”

The article and its sources describe the results of various studies of the cognitive, behavioral, emotional, and neurological effects of timed testing, which I will summarize below. Not to scare you or anything, but

  • Timed testing creates math anxiety that disproportionately affects the highest and lowest performing students
  • Math anxiety tends to persist and grow over time with repeated exposure to negative stimuli, leading to lasting consequences, including limitation of career options
  • Math anxiety actually has measurable neurological effects that inhibit the recall of known facts as well as the acquisition of new knowledge – that’s right, sending your kids to school can actually prevent them from learning and lead to lasting brain damage
  • Math anxiety causes emotional distress that can contribute to self-image issues that persist throughout adult life
  • Math anxiety is on the rise and is directly correlated with common public school teaching policies
  • Timed testing kills curiosity and enthusiasm and leads students to see math as a matter of performance and competition rather than as a fascinating subject with intrinsic value, which corresponds to am immense loss of value for society
  • Timed testing leads students to equate effectiveness and achievement with rapidity, which is so far from the truth it’s not even funny

This article just goes to highlight two things I have always said, that more fear = less learning, and that the public school system is accomplishing the exact opposite of what it should be doing, at an alarming rate. It just adds more evidence to the pile that for many students, school actually does more harm than good, and sane alternatives are needed, like, yesterday.

So what is the solution? The article and its references also highlight the positive changes that need to happen, specifically that learning needs to take place in an emotionally uplifting, stress-free environment that uses positive reinforcement and encourages exploration, imagination, and creativity, and that develops divergent thinking skills alongside convergent thinking skills.

For kids who are suffering under the yoke of public schooling, I always do my best to control and counteract the psychological damage caused by such practices as timed testing, and empower them to discover and use their innate mental superpowers.

Posted in Educational Reform, Math

Does Being Good At Math Make You “Smart”?

It continues to confound and amaze me how often I have the following type of exchange:

Person: “What do you do?”

Me: “I teach math.”

Person: “Oh my god, you must be so smart! I hated/was terrible at math!”

My experience learning and teaching math has shown me that not only is math just another subject that anyone who desires can learn and become skilled at, what’s more it is an innate human ability that everyone has, just like the ability to use language, recognize faces, run, swim, or climb trees. Along with math, all of these are activities that we recognize as skills that can be developed and improved, yet still innately human and instinctive for us as a species. All humans can do math, just like all birds can fly and all cats can hunt.

So, why is math considered to be a measure of “smartness” rather than an innate ability that anybody can develop? I speculate there are at least two factors feeding into the persistence of this pattern:

One is that the Newtonian-Cartesian scientific paradigm, which emphasizes the supremacy and superiority of rational, quantitative, and convergent modes of thought over and above the workings of insight, intuition, divergent thought, and artistic perception, has been supremely successful in the areas of science to which it readily applies, which has led to a kind of arrogant intellectual monism in which rational, linear, convergent thinking is seen as superior to nonrational, nonlinear, or divergent thinking.

The other is that math is generally taught in primary school in ways that are demonstrably inefficient and counterproductive, and that lead to disempowerment and sap intellectual curiosity, so only people with exceptional talent or exceptional immunity to cultural conditioning tend to thrive intellectually in such an environment. In an ideal learning environment, different people would pursue different subjects to different degrees determined by their interest and motivation, but nobody would come away with a phobia of any particular subject, or of learning in general, as is far too often the case in traditional public schools.  This is why much of the work I do as a tutor and academic coach consists of what is essentially PTSD rehabilitation therapy for math-related social anxiety.

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 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 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.