Thursday, March 29, 2012

Differential & Integral Calculus, Lec 1, Math 31A, UCLA

In this video of a math course at the university of California at Los Angeles are simply defined the notions of differential and integral calculus, slope and limit. Simply put, the differential calculus is the calculation of the rate of change and the integral calculus defines the accumulation of change. The equation of the derivative of a function tells you how this function changes meaning when it increases and when it decreases. The integral calculus tells how much changes happen. The velocity of a car can be represented by the equation of a derivative. This function derivative tells you when the velocity increases, decreases and when the car comes to a stop. The integral gives the distance traveled by the car in absolute value of course.

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Evaluating integrals using area interpretation

Since the integral is an area under curve its calculation can be done without calculating the antiderivative but rather by calculating the areas under curve. This is what this video shows

Wednesday, March 28, 2012

AP Calculus AB: Revolving Solids Known Cross Sections

This is a problem concerning the volume of a solid with known cross section. The calculation of the volume is found in three steps:

1) Express the area of the cross section as a function of a variable
2) Find the limits of integration
3) Integrate the function

The cross section can be a triangle, a square, rectangle, washer, etc. A washer has the shape of a disk with a circular region cut out from its center. Its area is found by subtracting the area of the inner circle from the area of the bigger circle.

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Saturday, March 24, 2012

Definition of some basic math words and expressions

In every body of knowledge the definition of words is important. Here is the definition of some basic words and expressions used in math.

Mathematics (math) is the study of numbers, quantities, shapes, and space using mathematical processes, rules, and symbols. There are many branches of mathematics and a large vocabulary associated with this subject.
Here are some math words and terms you will likely come across but may not know their precise meanings.
Algorithm A step-by-step mathematical procedure used to find an answer.
Coefficient A number that multiplies a variable. For example, 9 is the coefficient in 9x.
Denominator The bottom number in a fraction. The denominator represents the number of parts into which the whole is divided. For example, 6 is the denominator in the fraction 5/6.
Equation A mathematical statement used to show that two expressions are equal. It contains an equals sign. For example, 16 - 9 = 7 (the expression 16 - 9 and the expression 7 are equal).
Greatest Common Factor (greatest common divisor) The largest number that will divide two or more other numbers equally. For example, the greatest common factor of 32 and 48 is 16.
Improper Fraction (mixed fractions) A fraction that has a larger numerator than denominator. For example, 5/6 is an improper fraction.
Inverse Operations Opposite or reverse operations. Addition and subtraction are inverse operations, as are multiplication and division.
Negative Number A number that is less than zero. A minus sign is used to show that a number is negative. For example, -12 is a negative number.
Numerator The top number in a fraction. The numerator represents the number of parts of the whole. For example, 5 is the numerator in the fraction 5/6.
Ordinal Number A number that shows place or position, as in 2nd place.
Positive Number A number that is greater than zero. While a minus sign is used to signify a negative number, the absence of a minus sign signifies a positive number.
Prime Number A number that can be divided evenly only by itself and 1. For example, 7 is a prime number.
Square Number A number that results from multiplying another number by itself. For example, 49 is the square of 7 (7 x 7 = 49).
Square Root of a Number A number that is multiplied by itself to produce a square number. For example, 7 is the square root of 49. It is designated by the symbol √.
Variable A quantity that can change or vary, taking on different values. It is typically represented by a letter of the alphabet. For example x is the variable in 9x (x can be any number that is being multiplied by 9).
These are just some of the many words and terms found in mathematics. It is important to know the meanings of words and terms you will encounter as you progress through the study of math.

Wednesday, March 14, 2012

New technologies can help physically challenged and diverse learners learn math

Every mathematics teacher wants to be able to help their students learn more math and learn math better.  The typical mathematics classroom contains a diverse range of students who differ in their readiness to learn.  Quality mathematics teachers seek new strategies to reach their students and help them grow.
Differences in learning occur for a variety of reasons.  Some students may have academically encouraging homes.  Some students may have academic learning disabilities. Other students may have physical differences.  And just as with physical growth, some students may simply grow in their mathematical abilities at different rates.
Regardless of the reason, mathematics educators often strive to find tools and resources to help meet individual student needs and differentiate instruction.  Handheld, mobile technologies may offer just the means to do that.  As this recent article from SmartPlanet, details there are new opportunities for the visually impaired learner using “haptic” technology.
Haptic means relating to the sense of touch.  Through a research project at Vanderbilt University, an android app is being developed to help learners who have difficulties with their vision to learn mathematics – a subject where visual data such as graphs, charts, and symbols are relied upon for communication.
Many learn better through doing rather than speaking or hearing. Mathematics can be difficult to teach to these learners.  In addition to assisting the visually impaired, such technology may open the door for the kinesthetic learner.  With handheld devices becoming downright commonplace, this seems like an opportunity with a lot of promise.

Tuesday, March 6, 2012

Instruction for masses knocks down campus walls

The pitch for the online course sounds like a late-night television ad, or maybe a subway poster: “Learn programming in seven weeks starting Feb. 20. We’ll teach you enough about computer science that you can build a Web search engine like Google or Yahoo.”

But this course, Building a Search Engine, is taught by two prominent computer scientists, Sebastian Thrun, a Stanford research professor and Google fellow, and David Evans, a professor on leave from the University of Virginia.
The big names have been a big draw. Since Udacity, the for-profit startup running the course, opened registration on Jan. 23, more than 90,000 students have enrolled in the search-engine course and another taught by Mr. Thrun, who led the development of Google’s self-driving car.
Welcome to the brave new world of Massive Open Online Courses — known as MOOCs — a tool for democratizing higher education. While the vast potential of free online courses has excited theoretical interest for decades, in the past few months hundreds of thousands of motivated students around the world who lack access to elite universities have been embracing them as a path toward sophisticated skills and high-paying jobs, without paying tuition or collecting a college degree. And in what some see as a threat to traditional institutions, several of these courses now come with an informal credential (though that, in most cases, will not be free).
Consider Stanford’s experience: Last fall, 160,000 students in 190 countries enrolled in an Artificial Intelligence course taught by Mr. Thrun and Peter Norvig, a Google colleague. An additional 200 registered for the course on campus, but a few weeks into the semester, attendance at Stanford dwindled to about 30, as those who had the option of seeing their professors in person decided they preferred the online videos, with their simple views of a hand holding a pen, working through the problems.
Mr. Thrun was enraptured by the scale of the course, and how it spawned its own culture, including a Facebook group, online discussions and an army of volunteer translators who made it available in 44 languages.
“Having done this, I can’t teach at Stanford again,” he said at a digital conference in Germany in January. “I feel like there’s a red pill and a blue pill, and you can take the blue pill and go back to your classroom and lecture your 20 students. But I’ve taken the red pill, and I’ve seen Wonderland.”
Besides the Artificial Intelligence course, Stanford offered two other MOOCs last semester — Machine Learning (104,000 registered, and 13,000 completed the course), and Introduction to Databases (92,000 registered, 7,000 completed). And this spring, the university will have 13 courses open to the world, including Anatomy, Cryptography, Game Theory and Natural Language Processing.
“We’re considering this still completely experimental, and we’re trying to figure out the right way to go down this road,” said John Etchemendy, the Stanford provost. “Our business is education, and I’m all in favor of supporting anything that can help educate more people around the world. But there are issues to consider, from copyright questions to what it might mean for our accreditation if we provide some official credential for these courses, branded as Stanford.”
Mr. Thrun sent the 23,000 students who completed the Artificial Intelligence course a PDF file (suitable for framing) by e-mail showing their percentile score, but not the Stanford name; 248 students, none from Stanford, earned grades of 100 percent.
For many of the early partisans, the professed goal is more about changing the world than about making money. But Udemy, a startup with backing from the founders of Groupon, is hoping that wide use of its site could ultimately generate profits. And Mr. Thrun’s new company, Udacity, which is supported by Charles River Ventures, plans to, essentially, monetize its students’ skills — and help them get jobs — by getting their permission to sell leads to recruiters.
“We’re going to have detailed records on thousands of students who have learned these skills, many of whom will want to make those skills available to employers,” said Mr. Evans, the Virginia professor. “So if a recruiter is looking for the hundred best people in some geographic area that know about machine learning, that’s something we could provide, for a fee. I think it’s the cusp of a revolution.”
On Feb. 13, the Massachusetts Institute of Technology, which has been posting course materials online for 10 years, opened registration for its first MOOC, a circuits and electronics course. The course will serve as the prototype for its MITx project, which will eventually offer a wide range of courses and some sort of credential for those who complete them.
The Georgia Institute of Technology is running an experimental two-semester MOOC, known as Change 11, a free-floating forum that exists more in the online postings and response of the students — only two of whom are getting Georgia Tech credit — than in the formal materials assigned by a rotation of professors. Next year, Richard DeMillo, director of Georgia Tech’s Center for 21st Century Universities, hopes to put together a MOOSe, or massive open online seminar, through a network of universities that will offer credit.

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Saturday, March 3, 2012

Girl's verbal skills make them better at math study

A study published in the journal Psychological Science  suggests that the learning of mathematical concepts or mathematical terms is very important in learning math. The misconception about girls and boys learning math is that girls are less proficient in math than boys because of their verbal skills. In fact it might happen that certain girls might not have the ability to reflect enough in order to apply their verbal skills in the understanding of math. Moreover it is false to state that boys are better than girls in math. Is this statement based on preconceptions or studies?. What studies ever prove that? Even if these studies might exist it would be limited because of the worldviews or perceptions of those who did them. It is also important to know that a study never proves something true at a very high percentage. It might happen that certain conditions produce such or such results and if those conditions change the results change also. It has been repeated over and over for eons that boys are better than girls in math. Is that certain conditions common to the human race everywhere in the planet make that to happen and if those conditions reverse boys and girls equally learn math or the girls learn better than boys. One of the conditions is that educators think that learning is a conditioning process. This conditioning is rather artificial than natural. In artificial conditioning also called operant conditioning the conditions are created to produce certain actions. you are forced to learn something because of the results that this learning might produce. You might get punished if you don't learn or you might be rewarded if you learn. In these conditions you never have a natural motivation to learn the subject. These conditions deprive the learner of the joy of learning. He is forced to learn because of the consequences of not learning that might happen. In this anxious situation it happens that learning doesn't occur fully. Several scenarios might happen. One doesn't understand anything. One learns some parts of the subject. One learns all the subject just to avoid the consequences and forgets what was learned later. It is also interesting to know that educators don't teach students how to learn and why they have to learn a subject. The importance of learning the terms or concepts  is necessary in any subject. The ability to represent mentally the concepts and to progress gradually in the process of learning is also important. It is quite evident that the girls mastering this process had the ability to perform better than boys in this study,

While boys generally do better than girls in science and math, some studies have found that girls do better in arithmetic. A new study published in Psychological Science, a journal of the Association for Psychological Science, finds that the advantage comes from girls’ superior verbal skills.
“People have always thought that males’ advantage is in math and spatial skills, and girls’ advantage is in language,” says Xinlin Zhou of Beijing Normal University, who cowrote the study with Wei Wei, Hao Lu, Hui Zhao, and Qi Dong of Beijing Normal University and Chuansheng Chen of the University of California-Irvine. “However, some parents and teachers in China say girls do arithmetic better than boys in primary school.”
Zhou and his colleagues did a series of tests with children ages 8 to 11 at 12 primary schools in and around Beijing. Indeed, girls outperformed boys in many math skills. They were better at arithmetic, including tasks like simple subtraction and complex multiplication. Girls were also better at numerosity comparison — making a quick estimate of which of two arrays had more dots in it. Girls outperformed boys at quickly recognizing the larger of two numbers and at completing a series of numbers (like “2 4 6 8″). Boys performed better at mentally rotating three-dimensional images.  
Girls were also better at judging whether two words rhymed, and Zhou and his colleagues think this is the key to their better math performance. “Arithmetic and even advanced math needs verbal processing,” Zhou says. Counting is verbal; the multiplication table is memorized verbally, and when people are doing multiple-digit calculations, they hold the intermediate results in their memory as words.
“Better language skills could lead to more efficient verbal processing in arithmetic,” Zhou says. He thinks it might be possible to use these results to help both boys and girls learn math better. Boys could use more help with verbal strategies for learning math terms, while girls might benefit from more practice with spatial skills.
I would add to the conclusion of this study that educators apply the three steps of the learning process: learning of terms, pictorial representation of concepts and step by step mastery of the elements to be learned. In this way verbal and spatial skills will be developed both in girls and boys. Then with certain variations in the mastery of these skills there will not be polarization of this set of skills by both genders.