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"A simulator for drilling operations is composed of a set of nonlinear coupled partial differential equations that describe the simultaneous flow of hydrocarbons in a well. This mathematical model represents a 'virtual laboratory' where the finer mechanisms related to a number of different physical effects can be studied in detail," Evje goes on to explain.

In order to compute reliable solutions, it is crucial to have a model that is well defined mathematically. Mathematical methods are applied in order to derive upper and lower limits for various quantities like masses and fluid velocities, which provide insight into the parameters that are important for the control of these quantities. In addition, they allow proof of the existence of solutions for the model in a strict mathematical sense. In this paper, the author demonstrates that under certain assumptions, a solution exists.

At the time, the technology we used (in math class) was a graphing calculator,” he said. “Today, these kids have the privilege to learn math in a new, innovative way

Only the Fuse algebra application and a few key tools are loaded onto the devices. At school, students are blocked from inappropriate sites via firewalls. The App Store, where iPad users can purchase games and other applications, is locked on the device, but school officials are looking at opening the online store in the future. Freshman Catherine Rodriguez, 14, flashed a big smile as she received her new iPad. Math isn’t her strongest subject; she hopes the new technology will help her, she said. Passing math is a big concern for Rodriguez’s mother, who took three years to pass pre-algebra, she said.

Charles Cook, vice chancellor for instruction at Houston Community College. “I think the Statway project really has it right, where we’re looking at mathematical reasoning and problem solving and how math can serve you as a tool to further understanding, further investigation and further knowledge.”

Carnegie’s $14 million Quantway/Statway initiative is developing two mathematics programs: one-year courses that build quantitative literacy (Quantway) and statistical proficiency (Statway). They’re infusing both with teaching strategies that build students’ confidence as math learners.

The graduation rate is 68% for Latinos, 59% for African American students and 56% for students who are learning English. This compares with 83.4% for whites and 89.4% for Asians.

Among eighth-graders statewide, about 4,200 dropped out during the academic year; more than 13,000 finished eighth grade but didn't show up for ninth, the traditional beginning of high school.

L.A. Unified's estimated graduation rate for the four-year period is 55%. However, the state's new system places the district's rate at 64.2%.

About two-thirds of the school's students are low income and 18% are learning English.

Test score gains are highest in elementary schools and drop off precipitously in middle and high school.

Across L.A. Unified, fewer than 20% of high school students scored proficient or better in general mathematics, algebra 1 and 2 and geometry. In fourth grade, by contrast, 67% of students tested as proficient or better in math.

In math, the numbers are 76% for Asians, 61% for whites, 41% for Latinos and 34% for African Americans.

Mazzocco’s group studied 71 ninth-graders whose math abilities had been tested annually since kindergarten. Students completed two quantity estimation tasks. In one series of trials, participants saw an array of blue and yellow dots flash for a fraction of a second on a computer screen and indicated whether more blue or yellow dots had appeared. In other trials, students saw nine to 15 yellow dots flash on a screen and estimated how many dots were shown.

Theoretically, you could have performed the calculations with a pencil and paper, but no one actually did so. Yet the Supreme Court still held them to be unpatentable mental processes.

The Federal Circuit displays a similar confusion about a more recent case, this one involving an algorithm for digital image half-toning. The court upheld the patent despite the fact that mathematical formulas were "admittedly a significant part" of the patent's claims. In Tuesday's opinion, the court argued that the half-toning patent was different from the fraud-detection patent because the half-toning algorithm "required the manipulation of computer data structures (e.g. the pixels of a digital image and a two-dimensional array known as a mask)."

Of course, a "computer data structure" is just a way of organizing numbers and symbols. When I served as a teacher's assistant (TA) for computer science courses in grad school, I would regularly draw diagrams of data structures on the whiteboard and perform example calculations on them. Similarly, many branches of math involve manipulating "data structures" like ordered pairs and matrices—presumably the Federal Circuit doesn't think you can patent new results in linear algebra because "data structures" are involved.

Moving toward that goal, a more comprehensive mathematical model of human metabolism was used recently to design an NIH clinical trial that is comparing the effects of reducing fats versus carbohydrates in obese adults.

Working with the Dana Center, the community college association was able to get all 50 two-year college presidents to sign on.

The Texas project is inspired by the Carnegie Foundation’s work, developed with the Charles A. Dana Center at the University of Texas at Austin, in which students needing math remediation weren’t all put on a path toward calculus.

Kay McClenney is a University of Texas at Austin project director who helped develop a recent report showing the scope of the remediation problem at the state’s community colleges.

Some adaptive-learning software, which tailors lessons and exercises to individual student progress, also uses visual representation to demonstrate relationships.

online drawing programs that allow students and instructors to draw and manipulate shapes and graphs—like the Geometer’s Sketchpad, made by Emeryville, Calif.-based Key Curriculum Press, or the independently run GeoGebra, which has established dozens of institutes across the globe—immediately give users a sense of the relationships that govern geometry, algebra, and even calculus.

While software from Bellevue, Wash.-based DreamBox Learning also uses visual approaches, it differs from MIND’s software because it lets students in grades K-3 choose their own visual representations. After completing a problem one way, students will often be prompted to solve the same problem by choosing a different visualization to reinforce the concept.

its ability to import data from thrice-yearly Washington state standardized testing.

content services like Learn360, from Woodbury, N.Y.-based AIM Education Inc., offer the ability to combine resources into playlists of media set specifically for the needs of individual students, to help give some of that multidimensional understanding of content.

Institutions are equally affected. A recent report by the Community College Research Center put the annual cost of remediation at $1.9-billion to $2.3-billion at community colleges and $500-million more at four-year colleges.

There is growing recognition that the traditional semester-long course sequence used by many community colleges doesn't work, says Michael Lawrence Collins, associate vice president for postsecondary state policy at Jobs for the Future, a Boston-based nonprofit that studies education and work-force issues. It's inefficient to have students take up to a year and a half in remediation when many need just pieces of what they're being taught.

The Virginia system is betting on that promise. Its colleges will soon replace their semester-long developmental-math courses with nine units, which can be taken as one-credit classes or Web-based lessons with variable credit hours that allow students to complete more than one unit in a self-paced computer lab and classroom. The number of units that students are required to complete will depend on their placement-test scores and intended program of study. Students focused on the liberal arts will have to show competence in only five units, for example, mastering basic algebra concepts such as linear equations. Students who plan to pursue careers in science, technology, engineering, math, or business administration must complete all nine units.

Although he has help from a tutor in the class, it's a lot to juggle, he says. Mr. Wyrick does like how the Web-based class allows him to peek at students' quizzes in real time and track their progress a lot faster than by sorting through homework and test papers himself. "It allows me to intervene even before they ask for help," he says.

There is one certainty. The Virginia Community College system is not the only one anticipating the outcome. Equally curious are researchers and other colleges searching for successful ideas. "There is a risk," Mr. Collins says, "but there is also power in being that bold."

The Virginia Community College system is poised to find out. Starting in 2012, it will adopt a new systemwide developmental-math curriculum that will allow students to focus only on those math concepts they haven't already mastered rather than taking a series of semester-long math courses.

Half of all incoming students in the system need developmental education—and three-fourths of those students fail to graduate or transfer within four years.

the advent of new technologies means that all adults now need to be able to reason mathematically in order to work and live in today's society.

if we do not get our elementary students motivated to study these subjects, then we are facing an uphill battle as we continue our goal that ALL students feel confident and become successful in mathematics and science.

That's particularly true of Williams, who began working as a treatment plant intern while still a teenager. Now 22, Williams has created several YouTube videos on wastewater math. He ended his presentation last week by showing his students one of the questions from the plant operator examination — a complex equation whose answer required at least a page of careful analysis — and assuring them that they had the skills to complete it."All the algebra you've learned? You know all these formulas," Williams said. "And this problem is just one formula stacked on top of another."