**Corrected**: In a previous version of this story, the group that is developing a linear algebra and geometry course of study was misidentified. The group is the Education Development Center.

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—both of which are offered at Prescott High School—as 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 “consumer math” 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’s public, four-year universities raised their math admission requirements, recalled Joe Peters, a teacher who chairs the school’s 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’s 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.

“At the college level, statistics is pretty tough,” Mr. Peters said. For many students, “It’s like, ‘Whoa—here’s a whole new math topic you haven’t had.’ This is a very good class to have in high school.”

Many of the best alternative, senior-year math courses are “harder in a different way,” 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’s Institute for Mathematics & Education, began working with state, school, and university officials from Arizona and other states to evaluate senior-year math options.

Some schools are experimenting with options for the fourth year of high school math, other than precalculus and calculus. Their content could include the following:

**One Typical Path:**

**9th:** Algebra 1

**10th:** Geometry

**11th:** Algebra 2

**12th:** Precalculus

**Senior-Year Alternatives**

**Statistics/Probability/Data Analysis**

Focus on having students collect and analyze data, conduct studies, measure probability, recognize patterns, and make estimates. (The College Board offers a test in AP Statistics.)

**Discrete Math**

Typically asks students to use math in “real world” applications for problem-solving. Students draw from different areas of math, such as algebra and geometry, using “discrete” rather than continuous variables.

**Advanced Math Reasoning/Quantitative Reasoning**

Under development by the Education Development Center, focus on study of the geometry and algebra of vectors and matrices, and its applications in technology, engineering, other areas.

**Engineering Math**

Examines how engineers use math in solving complex problems in industry, in areas such as computer, electrical, and aerospace engineering. Can incorporate trigonometry, functions, complex numbers, and vector analysis, among other types of math.

SOURCE: *Education Week*

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—and even those who don’t do well in Algebra 2—can 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.

“We want students to have an appreciation for the power of mathematics, and an inclination to use it—not to want to skip over it,” Mr. McCallum said. When students today begin to fear math, he said, many of them “never come back.”

An obvious barrier to offering alternative courses, Mr. McCallum noted, is the likely need for districts to hire more math teachers—already in short supply—or 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 “of very mixed quality,” 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, “closing doors for kids,” 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 Web site. The “Fourth Year Capstone Courses” 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—discrete math—held 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—the terrain, the size of the search area, and the number of searchers.

“It’s very simple, compared to precalculus, but you’re having to use math in different contexts,” the senior explained. “It’s opened my eyes to how often math is used in everyday life.”