Tuesday, 20 December 2011

Beta, Alpha, Theta, Delta - Our Four Brain States

It would be safe to say that the average person knows very little about brain states. Indeed, it is doubtful whether many people know that there are four brain states and that all our brains function according to these states - Beta, Alpha, Theta or Delta.

Simply put, Beta refers to that state when our brains are most alert; concentration and our ability to think are heightened and in the ideal condition to write an examination, present a paper, synthesize information or tackle situations where we need an intensified mental sharpness. The second state, known as Alpha, refers to a relaxation of the brain that allows our thoughts to run freely and for creativity to blossom.

Theta is a very different state of mind. Theta is the mode in which the brain is neither highly active, nor completely relaxed but slows down the activity of brain waves to the point of sleep. This state exists on the edge of sub-consciousness and is where we are drawn into memories and expansive thoughts, meditation and, sometimes, true inspiration.

The actual state of sleep is the Delta state. Our brainwave frequency is slow and undulating and we drift into deep restorative sleep that allows our brain to operate optimally in other states.

When we speak of a brainwave, we may not realize that we literally have an electrical power in our brains that causes waves to course through our brains at varying speeds. Scientists have been able to measure the frequency of these waves, and to determine the effect of the waves on the way in which our brains work. The high frequency waves cause arousal of the brain and occur when we are intensely busy with an activity, particularly involving speech. This is the beta state of mind.

The Alpha brainwaves occur less frequently than beta waves, are slower and higher in scale or amplitude and so your brain would be working, but far less actively in this state. You may be reflecting on your activities undertaken in the more alert state of mind, or working on more mundane tasks. The slowing of the brain waves continues as we reach the Theta state. The slow frequency of these brainwaves can put us in a mellow mood, daydreaming as we perform routine, simple tasks. We free our minds and allow ideas to flow as we drive slowly along a common route, take a shower or a bath. We usually enjoy this unhurried state of mind.

Our brainwaves slow down to no more than a couple of cycles per second when we fall into a deep sleep. As we move through the Rapid Eye Movement period of our sleep, so our brains will have sped up and we be in the theta state of mind, closer to being awake. As we awake, our brainwaves speed up until we reach the beta state. Some of us have brains that get up to speed very quickly and so we can jump out of bed the minute the alarm rings; for others it is a much slower process.

This knowledge of how the brain works has allowed religious groups to utilize the frequencies. For example, the Buddhist chant echoes the rhythm of the brain in theta mode and transports them into deep meditation. In the western world, too, experts in the field have discovered that they can help people suffering from illness such as depression and post-traumatic stress by guiding them into the various stages of the brain, especially the theta stage, to deal with their distress.

Don't Worry Be Happy If you Aren't Happy Then What Are You?

Live life to the fullest, spread your arms and breathe, don't be afraid of change, the world is for you to have not the other way around. Use the power of your mind to free yourself from the shackles of negativity.

Life is to short to be unhappy stop wasting time on negative impulse. Take negativity and use it as fuel for positive thought.

Saturday, 10 December 2011

5th Grade Math

This year my son entered the 5th grade and had a little trouble in math. Not anything that threatened him failing, but problem enough that it could affect his performance in the future if he did not get the concepts this year. I found with him that what really works is making him do problems, and do them correctly on concepts that he is having difficulty with. He's smart and understands it, he's just a boy and flies through things at warp speed so he can go play basketball with his friends.

This was the case the other day when he was struggling with fractions. I wrote down a whole bunch of practice problems and made them a little tougher than the ones I knew he would get in class. For each problem he got wrong he would do two more problems. It worked great because he wanted so desperately to not have to do more that he got them all right.

If you are a concerned parent with a 5th grader, I have included a basic curriculum of 5th grade math:

Place value and number sense-determining the value of different digits and knowing what numbers mean.

Fractions and mixed numbers-understanding fractions in the form of mixed numbers and improper fractions.

Geometry-shapes and their measurements.

Add and subtract fractions-adding and subtraction improper fractions and mixed numbers-understand.

Decimals-understanding what decimals mean.

Multiply fractions-muliplying fractions as mixed numbers and improper fractions.

Addition and subtraction-adding and subtracting large numbers.

Divide fractions-understanding how to divide fractions.

Mixed operations-add, subtract, multiply, and divide whole numbers, fractions, and decimals.

Add and subtract decimals-using decimals to add and subtract.

Algebra-elementary algebraic concepts.

Multiplication-multiplying decimal, fractions, and whole numbers.

Coordinate Graphs-understanding the basics of graphs.

Multiply decimals-decimal multiplication.

Data, charts, and graphs-understanding various displays of data.

Patterns-identifying patterns.

Division-using the division algorithm

Consumer math-math of percentages, sales, and other life applications.

Ratios, proportions, and percents-understanding ratios, proportions, and percents.

Division with decimals-understanding how to do division with decimals.

Problem solving-using various skills to calculate problems.

Measurement-identifying, converting between, and using measurements.

Number theory-prime numbers and LCM

Time-understanding units of time and using calculations with time.

Probability and statistics-calculating basic probability and statistics and making predictions.

Friday, 9 December 2011

What Math eTextbooks of the Future Desperately Need

Have you ever read a research paper and found a mathematical mistaken in it? Indeed I have, and perhaps you are not skilled or knowledgeable on the topic, but there are mistakes, and they do exist. Often these peer-reviewed papers do not get the adequate time necessary to hash out all the issues or find the mistakes. Further, often research papers have a lot of math in them, but they are solving the wrong problem, or attacking the problem the wrong way, and yet they publish the paper anyway.

Of course, I've also seen mistakes in college textbooks, supposedly written by the professor, and a group of grad students. Yes, it happens, and sometimes the professor points it out to the students along the way, or the correction will come about the next year. This seems unfortunate when you are paying $225 that college textbook in the first place, and so perhaps I might shed some more light on this problem, I'd like to tell you about something I recently read.

There was an interesting post on SlashDot on March 4, 2012 titled; "Math Textbooks a Textbook Example of Bad Textbooks," by Samzenpus, where the words of Keeghan were reiterated, namely that;

"There may be a reason you can't figure out some of those math problems in your son or daughter's math text and it might have nothing at all to do with you. That math homework you're trying to help your child muddle through might include problems with no possible solution. It could be that key information or steps are missing, that the problem involves a concept your child hasn't yet been introduced to, or that the math problem is structurally unsound for a host of other reasons."

Now then, in the future I suspect that math textbooks will be holographic, give it five more years and they will project holograms of the shapes, and images which you are trying to figure out. Putting things in three and four dimensions with a projected hologram makes a lot of sense. Imagine the advantages for a student who can visualize the math problem using holographic imagery. You see, in solving these problems in this way they will be using the spatial part of the brain, and not the language part of the brain were they are trying to determine what all the symbols mean.

Indeed, I bet that the students learn math better, quicker, and go on to enjoy it more, therefore do better at the subject and get better grades. Perhaps, we are just a few technologies away right now from having all the math and science engineers, scientists, and future generation of mathematical intellectual superstars graduating from our high schools and colleges. I wouldn't be surprised, and I hope that if you are involved in any of these types of technologies, that you will be thinking here, please consider it.

Thursday, 8 December 2011

Calculus - Derivatives

Derivative is the central concept of Calculus and is known for its numerous applications to higher Mathematics. The derivative of a function at a point can be described in two different ways: geometrical and physical. Geometrically, the derivative of a function at a certain value of its input variable is the slope of the line tangent to its graph through the given point. It can be found by using the slope formula or if given a graph, by drawing horizontal lines toward the input value under inquiry. If the graph has no break or jump at that point, then it is simply the y value corresponding to the given x-value. In Physics, the derivative is described as a physical change. It refers to the instantaneous rate of change in the velocity of an object with respect to the shortest possible time it takes to travel a certain distance. In relation thereof, the derivative of a function at a point in a Mathematical view refers to the rate of change of the value of the output variables as the values of its corresponding input variables get close to zero. In other words, if two carefully chosen values are very close to the given point under question, then the derivative of the function at the point of inquiry is the quotient of the difference between the output values and their corresponding input values, as denominator gets close to zero (0).

Precisely, the derivative of a function is a measurement of how a function transforms with respect to a change of values in its input (independent) variable. To find the derivative of a function at a certain point, do the following steps:

1. Choose two values, very close to the given point, one from its left and the other from its right.

2. Solve for the corresponding output values or y values.

3. Compare the two values.

4. If the two values are the same or will approximately equal to the same number, then it is the derivative of the function at that certain value of x (input variable).

5. Using a table of values, if the values of y for those points to the right of the x value under question is approximately equal to the y value being approached by the y values corresponding to the chosen input values to the left of x. The value being approached is the derivative of the function at x.

6. Algebraically we can look for the derivative function first by taking the limit of the difference quotient formula as the denominator approaches zero. Use the derived function to look for the derivative by replacing the input variable with the given value of x.