Tag: Calculus

The Common High-School Tool That’s Banned In College

By William Pang

Step inside any high-school math class in the United States, and chances are youll find students staring down at their Texas Instrument calculators, nimbly typing commands into those $100 pocket computers. Calculators are so commonplace in modern American education that a TI-84 or -89 can be found stashed away in many homes, mementos from taking the SAT or computing integrals on the Advanced Placement calculus exam.

Still, college professors remain divided on the use of calculators in their classes. When I took my freshman math courses at McGill University in Montreal last school year, I had to revert back to pencil and paper, clumsily lining up columns to do addition and long-multiplication problems at my professors request. This isnt an unusual predicament: According to a 2010 national survey by the Mathematical Association of America, nearly half of Calculus 1 college instructors prohibit students from using graphing calculators on exams.

The debate over the use of calculators in math classrooms has ensued for more than four decades nearly as long as the contemporary calculator has been around. Although the abacus has been in use since the time of the Sumerians and Ancient Egyptians, it wasnt until 1958 when the Texas Instrument engineer Jack Kilby invented the integrated circuit, which paved the way for the cheap and compact computer chips used in most electronic devices today. (Kilby later won the Nobel Prize in physics.) A decade later, calculators would no longer be stored in gigantic cabinets with a price tag of over $700,000; they would substantially diminish in size and gradually become more affordable. Today, prices for graphing calculators hover around $80.

The debate over the use of calculators in math classrooms has ensued for more than four decades nearly as long as the contemporary calculator has been around.

By the mid 1970s, 11 percent of Americans owned a calculator. Four-function calculators those that only perform addition, subtraction, multiplication, and division gradually entered the classrooms, dividing educators and parents alike. Debates over the role of calculators in the classroom quickly emerged, and arguments for and against their use have hardly changed since then. Proponents of the calculator argued that machines could help students make sense of abstract mathematical notations through real-life problems, making math fun and interesting. Opponents worried that students would become over-dependent on calculators, losing the ability to do simple arithmetic operations and exercise a solid sense of numbers.

In 1986, Connecticut became the first state to mandate the use of calculators on state tests, signaling the beginning of a calculator-dependent generation. But the most consequential move came three years later from the National Council of Teachers of Mathematics (NCTM), which advocated for the use of calculators from kindergarten through grade 12. The guidelines set by the NCTM were soon adopted into many local and state curricula. In 1994, the College Board made substantial changes to the SATs math section to allow the use of calculators. The 1995 Advanced Placement calculus exams were the first to require the use of graphing calculators, a powerful electronic aid that is still used in most high-school math classes today.

For W. Stephen Wilson, a math and education professor at Johns Hopkins University, using a calculator is akin to relying on a crutch when one doesnt have a bad leg. I have not yet encountered a mathematics concept that required technology to either teach it or assess it. The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them. There might be teachers who can figure out a way to enhance learning with the use of technology, but its absolutely unnecessary, Wilson wrote in the journal Education Studies in Mathematics.

Proponents of using technology in classrooms argue that graphing calculators, particularly those equipped with programs that can compute algebraic symbols, would reduce the need for students to memorize formulas and perform time-consuming computations. But Wilson fears that students who depend on technology will fail to understand the importance of mathematical algorithms.

Yes, a calculator could effortlessly churn out the derivative of an equation, but would students understand how to find the answer using the fundamental theorem of calculus or the definition of the derivative? The idea, Wilson says, is not to have students mindlessly perform mechanical operations, but for students at all levels to apply linear thinking in understanding the beauty of efficient algorithms. If a student cant master long division, how can she grasp derivatives and integrals?

Wilson says he has some evidence for his claims. He gave his Calculus 3 college students a 10-question calculator-free arithmetic test (can you multiply 5.78 by 0.39 without pulling out your smartphone?) and divided the them into two groups: those who scored an eight or above on the test and those who didnt. By the end of the course, Wilson compared the two groups with their performance on the final exam. Most students who scored in the top 25th percentile on the final also received an eight or above on the arithmetic test. Students at the bottom 25th percentile were twice as likely to score less than eight points on the arithmetic test, demonstrating much weaker computation skills when compared to other quartiles.

Its worth noting that calculators are also more likely to be barred in math exams at research universities than at two-year colleges and regional public universities. Out of the 50 national universities ranked at the top by U.S. News and World Report, only four schools had policies allowing electronic devices on Calculus 1 exams. One explanation is that selective institutions are less likely to offer remedial math courses and generally accept students who possess a strong math background.

Why arent high schools taking their cue from math professors at Harvard and MIT? Because most college students wont major in STEM subjects and wont need advanced math knowledge for much of their work. Dan Kennedy, a high-school teacher at Baylor School, argues that to set a reasonable expectation for all students, calculators should be used because many real-world problems cannot be solved without technology. Students, he says, would be better served by learning probability, statistics, computer literacy, financial mathematics, and matrix algebrathe kind of math that requires the use of graphing calculatorsnot the kind of theoretical math that dominates math competitions.

David Bressoud, a math professor at Macalester College in Minnesota, has a different theory: He thinks that large research universities typically ban calculators because the devices are essentially obsolete there. The larger universities have traditionally had computer-lab resources, [and now] it is easier to expect that all students have access to a computer, Bressoud said. Computers, Bressoud says, are a much better tool for teaching calculus because they are more flexible and faster than calculators.

At Macalester, first-year calculus is known as Applied Multivariable Calculus 1. Computers are heavily encouraged in class, and professors arent slowly chalking away proofs and theorems on the blackboard. Unlike those at traditional college math classes, Macalester professors take the word applied seriously: A lecture on functions, for example, is demonstrated using the Body Mass Index, a function of height and weight used to determine whether a person is obese. Students in their first year of calculus also learn differential equations, a topic that is generally covered only when students have three semesters worth of calculus under their belts. The aim is that, by introducing differential equations early on, students understand how mathematical models are generated. Why? Because these models are used in many fields, including, but not limited to, economics, environmental science, psychology, and medicine.

The calculator debate also plays into a larger discussion of whether colleges should be less theoretical and more practical. For technology advocates, an increased emphasis on technology is often seen as a way to prepare students for the real world. Calculator opponents tend to see it differently. The goal of university education, they contend, is for students to get a good grasp of the theoretical foundation of a subject, not to master calculators or computers. After all, todays technology might become drastically different 20 years from now, but a good foundation will always last.

Even Socrates once quipped that a reliance on writing would lead to the deterioration of memory. And many of the best practices in pedagogy teach that memorization does have its merits when it comes to education, despite the invention of the internet and search engines. Drawing the line between the use of and barring of calculators could prove difficult, says Jon Star, an education professor at Harvard University. That line is also moving as time goes on, at different levels. There might be students who bring different skills who put the line in different places, Star noted. Im very wary of anyone who says it should be at one extreme or the other.

Many schools opt for the middle way. While calculators might not be allowed on tests and exams, colleges know that tech-savvy students will utilize programs such as Wolfram Alpha, a powerful web-based computational tool, to aid with calculus assignments. Homework problems often require calculator use, asking for solutions that involve cumbersome values or algebraic symbols that are too tedious to compute by hand. Will future professors who were born in a generation of smartphones and tablets change the course of this discussion? Only time will tell.

This story originally appeared on TheAtlantic.com.

Read more: http://www.huffingtonpost.com/entry/the-common-high-school-tool-thats-banned-in-college_us_586ff9f8e4b0eb9e49bfbb2c

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Trigonometry and Calculus

Why do those two words bring fear to the minds of most people?

Have you ever had a trigonometry course or a calculus course? If you were a science or math major in college then you almost certainly have taken those courses! Did you like learning that type of math? Was it fun for you or does it evoke bad memories?

Hopefully if you’ve had those courses, they were fun for you. I was thinking about that as I was reading this article I found below.  It briefly describes the history of trigonometry and calculus. Hope you enjoy it!

 

Discovering Trigonometry
Trigonometry is a category of mathematics that studies triangles, as well as the spatial relationships between triangle sides and degree angles between these sides. Trigonometry is used to define trigonometric functions.

Trigonometric functions describe the relationships between the angles and sides and are also applied to cyclical phenomena, such as waves.

Trigonometry itself is very similar to geometry, but is slightly more complex. It utilizes functions such as sine, cosine and tangent to analyze areas of angles. These and other functions of trigonometry are used in a variety of career fields including but not limited to: acoustics, architecture, astronomy, biology, chemistry, civil engineering, computer graphics, metrology, medical imaging, music theory and several other fields.

Trigonometry is taught starting in middle and high school. It can be taught as a separate course, but is also taught as a preliminary course for calculus. Trigonometry develops student’s knowledge of both pure and applied mathematics. College level trigonometry is required for several different career majors to help students develop a further understanding of angles and spatial relationships.

Trigonometry was first developed in the 2nd century BC by the Greek mathematician Hipparchus. Hipparchus developed what is known as the first trigonometric table. He used trigonometry, and other mathematical functions, to develop lunar and solar theories. He also used trigonometry to study the motion and orbit of the sun and moon. Though trigonometry was developed by Hipparchus, the study of triangles can be traced all the way back to Egyptian mathematics and Babylonian mathematics.

The Egyptians and Babylonians used trigonometry to develop theorems on ratios of triangle sides. The Babylonian astronomers used early trigonometry to measure the angular distances on the celestial sphere. They used this to detail records of rising and setting stars, planet motions, and solar and lunar eclipses.

During the Hellenistic period, the Greeks took the early Egyptian and Babylonian trigonometry and developed the chord, which developed the use of arcs. A chord of a circle is the geometric line segment whose endpoints are on the circle’s circumference. The chord joins two points on any curve, but it is not limited to an ellipse. The chord that passes through the center point of the circle also functions as the circle’s diameter.

The trigonometry that is currently used today was developed by European mathematicians, Sir Isaac Newton and James Stirling in the 17th century. Newton and Stirling created the general formulas that are currently used to solve trigonometric functions.
By: Dennis McLynn
Article Directory: http://www.articledashboard.com

Dennis McLynn is the Vice President of Strategic Marketing & Business Development for High Points Learning. HighPoints Learning (HPL) is a leader in Web-based math education and instruction. HPL offers an online math tutoring program that helps raise participants’ math scores an average of 15 points in pre and post testing. HighPoints Learning services the 3-12 grade market. For more information visit: www.ehighpoints.com

So with all of this history under your belt, you’re ready to go hit the trig books right?  😉

Have a great day!

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Does A Math Tutor Really Help?

In order to help some students learn math more easily, it’s a good idea to hire a math tutor. Here’s a nice article which explains why a student might need a math tutor and the benefits of having one.

When And How A Math Tutor Can Really Make A Difference

If you’re having trouble understanding pre-algebra, calculus, or anything in between, you could greatly benefit from a math tutor. For those who need to study up for a high school exam, the GRE, or just a simple college course quiz, a math tutor can substantially increase your leverage in understanding the logic behind an array of problems. There are tons of live, online tutors that are available for immediate pay. Others might be available in your local community or even University or school. Many who freelance offer their services online who will charge per-hour for various forms of assistance.

Those who advertise their services online can help you with immediate assistance if they charge for an instant chat program or Skype call, for example. For less immediate needs, some may simply ask for an email to be sent once a payment via PayPal is sent. Others will grant you access to message boards that operate round the clock so that you can ask a multitude of users available for questions. This may be one route to take, but there are also many free website s and message boards that you can visit, though answers to your questions and math problems may not be as speedy or even correct.

A math tutor may be accessible directly from your high school, college, university, or other educational institution. Asking the administration of your school or learning facility is the best to go about finding out if there are available resources for your math needs. On large university campuses, many bulletin boards dawn print out ads from students who specialize in various subject areas. If you can find one of these and do a quick search, perhaps finding a fellow student for help may be the best bet. Prices are likely to be low and you would be helping a statistically poor college student!

Because your educational career is of the utmost importance, finding math help is essential. This will help you gain leverage in future courses while helping pass the ones that are required. It may also come in handy once graduated, as extensive tutoring can truly open the eyes of those seeking help, revealing to them a new perspective on the subjects at hand. If you need some numerical assistance, you can easily find some help with a little research and perseverance.

Your education is nothing to toy around with, so it is important to not be afraid to seek and ask for help. This is particularly true with math, the #1 ranked response by college level students as the most difficult subjects to grasp both in high school and undergraduate. No matter which grade or educational level you may be at, if you need math assistance, it is important to seek this help out. With the right help, these problems can cease to exist, making your future endeavors stress free and easier to grasp.

By: aayana

Article Directory: http://www.articledashboard.com

Looking for a Middletown NJ math tutor to help you or your child grasp this essential subject? Visit us today at www.1on1tutorsnj.com to learn more about how we can help.

Here at Cherry Hill Mathnasium we offer a great math tutoring program. Please stop by at our website or give us a call at 1-(856) 874-0050.

 

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