Calculus: When and Why to Learn It
For over 30 years, calculus has been considered a pinnacle of high school mathematics, essential for careers in hard sciences and a prerequisite for top-tier colleges. However, math and science professionals are beginning to question the actual helpfulness of current high school calculus courses for advanced science fields. The increasing prevalence of data in various fields has spurred a movement toward emphasizing statistics and data literacy over calculus in high school math curricula.
The Rise of Data Literacy
Randy Kochevar, a senior research scientist at the Education Development Center, emphasizes that we understand the world through data in fields like gene expression and planet identification. Statistics and data analysis are fundamental not just for scientists, but for professionals in general.
The Common Core State Standards and the Next Generation Science Standards reflect this shift, calling for the teaching of data analysis and statistics. However, there is a disconnect; science standards expect students to have significant data skills by middle school, but it's unclear where those skills are being developed. Both sets of standards need more integration of data and statistics, as they were developed before the big data boom fully materialized.
Data production has exploded in recent years. In 1999, 1.5 exabytes of new data were produced, while by 2011, it had grown to over 14 exabytes a year. Today, we produce 2.5 exabytes of data every day, with the total doubling every two years.
Ironically, as big data and statistics become more prevalent, American students struggle with these concepts. From 2007 to 2017, 4th and 8th-grade students' scores on the National Assessment of Educational Progress in mathematics declined significantly in data analysis, statistics, and probability.
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Uri Treisman, a mathematics professor, notes that statistics is perceived as difficult because historically, it required calculus as a prerequisite. He is working to develop a statistics pathway as an alternative to calculus.
The Calculus Conundrum
Until about 1980, calculus was a higher education course for those interested in mathematics, physics, or other hard sciences. Only about 30,000 high school students took the course. School reformers then embraced calculus as a rigorous, college-preparatory course, leading to increased expectations from parents, district leaders, and colleges. David Bressoud, a mathematics professor, notes that calculus has become an accepted requirement for admission to top universities, even for students majoring in fields like French literature.
Today, approximately 800,000 students nationwide take calculus in high school, about 15 percent of all high schoolers, and nearly 150,000 take the course before 11th grade. Calculus classes are disproportionately white and Asian, with other student groups less likely to have access to the course.
Despite its status as a gold standard, university calculus experts argue that high school calculus is not a strong indicator of preparedness for postsecondary math in science fields. A report by the Mathematics Association of America and the National Council of Teachers of Mathematics found that many students who took Advanced Placement Calculus AB still retook calculus in college, and 250,000 students needed to take lower-level courses like precalculus or algebra. Taking calculus in high school was associated with only a 5 percentage point increase in calculus scores in college. The best predictor of success in college calculus was earning As in high school Algebra 1 and 2 and geometry.
Calculus and Socioeconomic Status
More than half of students who take calculus in high school come from families with a household income above $100,000 a year. In contrast, only 15 percent of middle-income students and 7 percent of those in the poorest 25 percent of families take the course.
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Statistics and data literacy advocates hope that diversifying math courses could broaden students' paths to STEM and other careers. Jobs requiring data literacy and statistics are among the fastest-growing occupations in the country.
Alternative Pathways and Integration
William Finzer, a senior scientist at the Concord Consortium, suggests two paths forward. One is to develop a separate statistics course, but this could concentrate opportunity among a small group of students. The other is a more holistic approach, where at least one class a year integrates data analysis into its curriculum.
EDC’s Oceans of Data Institute is building learning progressions for statistics and data literacy, based on the acronym CLIP:
- Complex, multi-variable data: Analyzing more than just simple relationships.
- Larger data sets: Sorting and understanding relevance from large datasets.
- Interactively accessed data: Working with data dynamically, not just static graphs.
- Professionally collected data: Considering the context and potential biases in data collection.
Finzer believes this approach would make data less foreign to high school graduates.
Ultralearning Calculus: A Different Approach
Instead of rote memorization, a more effective approach to learning calculus is intensive practice combined with forming deep intuitions about the principles involved. This strategy, known as ultralearning, involves going deep into the subject matter.
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The principles are less numerous than the superficial details. By really learning these, and then aggressively practicing their most common instantiations, that you have any chance of doing well on an exam that has novel problems.
This method can be applied to various fields, emphasizing the importance of understanding the fundamentals rather than just memorizing facts.
The Importance of Motivation
The motivation to learn is a critical factor in mastering math. Rather than focusing solely on the theoretical aspects, it's essential to show the practical applications and relevance of calculus.
Calculus in the Broader Educational Context
Calculus instruction in the United States is dysfunctional because it is considered a college-level course but taught to a significant portion of high school students in a way that favors those of higher economic status. In other countries, calculus is part of the curriculum for students on a STEM track, with a national curriculum and well-prepared teachers.
In the United States, there is often no explicit pre-service preparation for teaching calculus, and in-service support depends on the resources of the school district. Schools in wealthy districts often expect all students to take algebra by 8th grade and calculus by 12th grade, while many schools do not even offer calculus.
A study by Rosenstein and Ahluwahlia found that students who scored high on the AP Calculus exam did so because they liked mathematics, wanted to learn more, and hoped to get advanced placement in college. However, only about half of the students who study calculus in high school continue with mainstream calculus when they get to college.
Universities have been slow to address this problem, often using a curriculum that assumes no prior knowledge of calculus but moving at a pace that presumes familiarity. The Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) have promoted the policy that calculus in high school should be limited to those who are fully prepared and intend to seek advanced placement.
Re-evaluating Mathematics Instruction
The current system has exacerbated inequities in education. In 2019, 12.7% of all White graduating seniors had taken an AP Calculus exam, with two-thirds earning a three or higher. There is wide agreement that computational and data science need to be part of the mathematical preparation of students. The singular focus on calculus limits these efforts.
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