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When students at Prescott High School sign up to take math as seniors, not all of them will be wading into precalculus or calculus, with in-depth explorations of derivatives and trigonometric functions.

Some will instead end up using mathematics to study the Electoral College, or the security of Internet passwords, or how delivery companies ship packages to different locations on time.

The northern Arizona high school is one of many that have created alternative math courses for 12th graders who are not interested in taking precalculus or calculus, or not academically ready for those classes. Across the country, interest in introducing new senior-year math options is rising as more states require four years of the subject for graduation and schools explore alternatives for both struggling and high-achieving students.

The most popular emerging courses include statistics and discrete math鈥攂oth of which are offered at Prescott High School鈥攁s well as classes in quantitative reasoning, math modeling, and math in business and finance.

The academic demands of those courses vary. Many target students who have completed Algebra 2, but who do not want to take precalculus or fear they would find it too much of a strain. Some are being drawn up for elite students who have flown through all the math classes available to them.

Supporters of these alternative courses say they have little, if anything, in common with the relatively low-level 鈥渃onsumer math鈥� courses that for years were a fixture in many schools, mostly as an option for struggling students.

Administrators at Prescott High, located about 90 miles north of Phoenix, launched their alternative courses in the late 1990s, when Arizona鈥檚 public, four-year universities raised their math admission requirements, recalled Joe Peters, a teacher who chairs the school鈥檚 math department.

School officials wanted options other than precalculus or calculus for college-bound students who did not expect to major in math, Mr. Peters said. They created two semester-long courses: Discrete Math, in which students perform problems with a heavy emphasis on real-world applications, in business and other areas; and Statistics and Probability, where they cover topics such independent and dependent events, game theory, and odds. Students typically take the classes in succession.

Difficult in a Different Way

About a quarter of Prescott High School鈥檚 seniors follow the discrete math and statistics path. A slightly higher percentage take either precalculus or calculus, Mr. Peters estimated; the remainder take either an advanced algebra or a less demanding business-math course.

Students must complete Algebra 2 to take the discrete math and statistics, Mr. Peters noted. Determining whether those two classes are easier than precalculus is difficult, he said, and dependent on the topic being covered any given day. The two classes keep seniors engaged in challenging math, and prepare them for the statistics they will encounter in college, even in non-math subjects, he explained.

鈥淎t the college level, statistics is pretty tough,鈥� Mr. Peters said. For many students, 鈥淚t鈥檚 like, 鈥榃hoa鈥攈ere鈥檚 a whole new math topic you haven鈥檛 had.鈥� This is a very good class to have in high school.鈥�

Many of the best alternative, senior-year math courses are 鈥渉arder in a different way,鈥� than precalculus, because they require more in-depth math reasoning, said William McCallum, a professor at the University of Arizona, in Tucson, who has studied fourth-year options.

Subject Appreciation

Last year, Arizona state officials decided to phase in a requirement that students take four years of math to graduate with a traditional high school diploma. Around that time, Mr. McCallum, the director of the university鈥檚 Institute for Mathematics & Education, began working with state, school, and university officials from Arizona and other states to evaluate senior-year math options.

Schools will need a greater array of fourth-year choices as state graduation requirements increase, Mr. McCallum said. He is convinced that many students who would struggle in precalculus or calculus鈥攁nd even those who don鈥檛 do well in Algebra 2鈥攃an thrive in demanding, alternative courses such as statistics and mathematical reasoning. Many high school seniors might re-engage in math enough to choose a major or career that relies heavily on math, he said.

鈥淲e want students to have an appreciation for the power of mathematics, and an inclination to use it鈥攏ot to want to skip over it,鈥� Mr. McCallum said. When students today begin to fear math, he said, many of them 鈥渘ever come back.鈥�

An obvious barrier to offering alternative courses, Mr. McCallum noted, is the likely need for districts to hire more math teachers鈥攁lready in short supply鈥攐r to decide which math classes receive priority. He and others suggest that districts can overcome those hurdles by offering courses over the Internet, with supervision from math teachers, and by arranging to share teachers among schools.

For years, students who struggled in later high school math, or avoided it, were channeled into relatively unchallenging consumer-math courses, said Zalman Usiskin, who directs the University of Chicago School Mathematics Project. Many of those low-level courses began to vanish in the 1980s, in the wake of rising state math requirements and mandates that students take specific courses, he said.

The College Board offers Advanced Placement tests in both calculus and statistics. Participation on the AP Statistics exam has risen by 69 percent since 2003; about 98,000 students took that test in 2007. But AP Calculus is still far more popular, with 212,000 students having taken the Calculus AB exam in 2007, a number that has also grown. That trend did not surprise Mr. Usiskin, who noted that many school officials and students have come to regard calculus as a prerequisite for majoring in math in college.

Many Options

Many of the high school course offerings that focus on building applied math skills have traditionally been 鈥渙f very mixed quality,鈥� said John Kraman, a senior policy analyst at Achieve Inc., a Washington organization that advocates for higher standards.

If high schools are going to offer alternative courses to precalculus and calculus, those classes should focus on preparing them for college and the workforce, Mr. Kraman said. Otherwise, they risk simply repeating previous math content, 鈥渃losing doors for kids,鈥� he said.

Yet Achieve officials believe alternative senior-year courses can benefit students, Mr. Kraman said. The organization, in partnership with the Charles A. Dana Center at the University of Texas at Austin, is promoting alternative, fourth-year high school courses that present demanding math in innovative ways, on a . The 鈥淔ourth Year Capstone Courses鈥� listed include AP and International Baccalaureate classes, as well as courses from states such as Arkansas, Indiana, and Virginia.

The Dana Center is devising its own alternative, senior-year math course, Advanced Mathematical Decision Making, which will emphasize statistics, financial and economic literacy, basic trigonometry, and the application of algebra and geometry to many types of problems. At least 50 high schools in Texas are planning to pilot the class next academic year.

For Kena Fedorchak, a senior at Prescott High School, the alternative to calculus鈥攄iscrete math鈥攈eld strong appeal.

The 17-year-old had fared well in precalculus as a junior, but he worried he would struggle in calculus the next year. Discrete math has forced him to use math creatively, he says. This week, he and three classmates are finishing a project in which they calculate the probability of rescuing people lost in the wilderness, factoring in several variables鈥攖he terrain, the size of the search area, and the number of searchers.

鈥淚t鈥檚 very simple, compared to precalculus, but you鈥檙e having to use math in different contexts,鈥� the senior explained. 鈥淚t鈥檚 opened my eyes to how often math is used in everyday life.鈥�