Calculus in society

Summary: Since its introduction in the seventeenth century, calculus has been applied to more and more practical endeavors, from engineering and manufacturing to finance.

Since its initial development in the seventeenth century, calculus has emerged as a principal tool for solving problems in the physical sciences, engineering, and technologies. Applications of calculus have expanded to architecture, aeronautics, life sciences, statistics, economics, commerce, and medicine. Contemporary society is impacted continually by the applications of calculus. Many bridges, high-rise buildings, airlines, ships, televisions, cellular phones, cars, computers, and numerous other amenities of life were designed using calculus.

Since the 1970s, calculus in conjunction with computer technology has resulted in the emergence of new areas of study such as dynamical systems and chaos theory. Such vast applications have established the study of calculus as essential in preparation for numerous careers. Indeed, calculus is considered among the greatest achievements of humankind, making it worthy of study in its own right in a society that places rational thought and innovation in highest esteem. Recent curricular and pedagogical reforms in calculus have made it more academically accessible to the school population.

What Is Calculus?

Calculus originated from studying the physical motions of the universe, such as the movement of planets in the solar system and physical forces on Earth. It involves both algebra and geometry, in combination with the concepts of infinity and limits. In contrast to algebra and geometry, which focus on properties of static structures, calculus centers on objects in motion. There are two principal forms of calculus, differential calculus and integral calculus, which are inversely related. At its most basic level, differential calculus is used in determining instantaneous rates of change of a dependent variable with respect to one or more independent variables; integral calculus is used for computing areas and volumes of nonstandard shapes.

Who Invented Calculus?

In the late seventeenth century, Isaac Newton (1646–1727) of England and Gottfried Wilhelm Leibniz (1646–1716) of Germany independently invented calculus. Isaac Newton began his development of calculus in 1666 but did not arrange for its publication. He presented his derivations of calculus in his book, The Method of Fluxions, written in 1671. This book remained unpublished until 1736, nine years after his death. Gottfried Leibniz began his work in calculus in 1674. His first paper on the subject was published in 1684, 50 years earlier than Newton’s publication. Because of these circumstances and fueled by the eighteenth-century nationalism of England and Germany, a bitter controversy erupted over who first invented calculus. Was it Isaac Newton or Gottfried Leibniz?

Investigators found that Leibniz had made a brief visit to London in 1676. Supporters of Newton argued that during that trip, Leibniz may have gained access to some of Newton’s unpublished work on the subject from mutual acquaintances within the mathematics community. However, these two prominent and outstanding mathematicians used their own unique derivations and symbolic notations for calculus, with Newton developing differential calculus first and Leibniz developing integral calculus first. For many decades, the calculus feud divided British mathematicians and continental mathematicians, and it remains a historical mystery into the twenty-first century. It was an unusual controversy in that it erupted rather late in the development of calculus and was ignited by the respective followers of Newton and Leibniz. In the twenty-first century, the general consensus is that both Newton and Leibniz invented calculus, simultaneously and independently.

Isaac Newton (1646–1727): The Man

Isaac Newton was revered in England during his lifetime and is recognized as one of the foremost mathematicians and physicists of all time. In addition to his invention of calculus, Newton is famous for designing and building the first reflecting telescope, formulating the laws of motion, and discovering the white light spectrum. He held many prestigious positions, including Fellow of Trinity College, Lucasian Professor of Mathematics, Member of Parliament for the University of Cambridge, Master of the Royal Mint of England, and many others. Even though Newton was extremely productive and admired universally for his work, on a personal level he was humble, cautious of others, and angered by criticism. His modest nature is embodied in his famous statement, “If I have seen farther than others, it is because I stood on the shoulders of giants.” His works in mathematics and physics were recognized throughout Europe when he was honored as Fellow of the Royal Society of London in 1672. He subsequently served as the Society’s president from 1705 until his death. In 1705, Newton was knighted in Cambridge by Queen Anne of England for his contributions to the Royal Mint. In 1727, Newton’s name was immortalized in English history by his burial in London’s Westminster Abbey and by the accompanying monument honoring his contributions to mathematics and science.

Gottfried Leibniz (1646–1716): The Man

Gottfried Leibniz is recognized as one of Germany’s greatest scholars of philosophy, history, and mathematics. He was the son of a philosophy professor and a leader in the philosophy of metaphysics. His optimism is reflected in his words, “We live in the best of all possible worlds.” On a personal level, Leibniz was considered likeable, friendly, and somewhat boisterous. Professionally, Leibniz was employed by a succession of German princes in the capacities of diplomat and librarian. He planned and founded several academies throughout Europe. For his knowledge of law, he was appointed Councilor of Justice for the Germanic regions of Brandenburg and Hanover. Similarly, Russian Tsar Peter the Great appointed Leibniz as Court Councilor of Justice for the Habsburgs. For his work in mathematics (derivations in calculus and invention of the binary number system), in 1673, Leibniz was appointed Fellow of the Royal Society of London, a society honoring outstanding mathematicians and scientists throughout Europe. By 1706, however, Leibniz’s stellar reputation had begun to disintegrate. Accusations of plagiarism regarding the invention of calculus were unrelenting until Leibniz’s death in 1716. In contrast to Newton, the only mourner at Leibniz’s funeral was his secretary. Eventually, more than a century after his death, Leibniz’s outstanding contributions to mathematics were recognized in Germany when a statue was erected in his honor at Leipzig, one of Germany’s major centers of learning and culture.

Interestingly, it is Leibniz’s symbolic notations for calculus, namely dy/dx and ʃy dx, that have stood the test of time. These notations are most prevalent in calculus classrooms in the twenty-first century because of their consistency with the operations of differential equations and dimensional analysis. The most significant contribution to mathematics by Newton and Leibniz was their derivations of the Fundamental Theorem of Calculus, a theorem that unites both differential and integral calculus.

Building on Newton’s and Leibniz’s Work

Following the invention of calculus, additional contributions to calculus were made by John Wallis (1616–1703), Michel Rolle (1652–1719), Jacob Bernoulli (1654–1705), Guillaume de l’Hôpital (1661–1704), Brook Taylor (1685–1731), Colin Maclaurin (1698–1746), Joseph-Louis Lagrange (1736–1813), Bernard Bolzano (1781–1848), Augustin-Louis Cauchy (1789–1857), Karl Weinerstrasse (1815–1897), and Bernhard Riemann (1826–1866).

The Power of Calculus

The power of calculus in contemporary society rests primarily in its applications in the physical sciences, engineering, optimization theory, economics, geometrical measurement, probability, and mathematical modeling.

The following is a sampling of basic applications using the two major branches of calculus.

Applications of Differential Calculus

• Environmental science: An oil tanker runs aground and begins to leak oil into the ocean and surrounding land areas, resulting in potentially devastating consequences. Differential calculus can be used to supply information essential for assessing the leakage and resolving the problem. For example, the rate and volume at which the oil is leaking can by determined using calculus.
• Business and economics: Important applications of calculus in business and economics involve marginal analysis (known as the first derivative). Marginal costs, revenues, and profits represent rates of change that result from a unit increase in product production. This information is valuable in developing production levels and pricing strategies for maximizing profits.
• Medicine: Calculus can be used for evaluating the effectiveness of medications and dosage levels. For example, calculus can be used in determining the time required for a specific drug in a patient’s bloodstream to reach its maximum concentration and effectiveness.
• Biology and chemistry: Assessments of chemical treatments for reducing concentration levels of biological contaminants (such as insects or bacteria) can be determined by calculus. For instance, calculus can be used in measuring the concentration levels, effectiveness, and time necessary for a chemical treatment supplied to a body of water to reduce its bacterial count to desired minimal levels.
• Physics (velocity and acceleration): For moving objects (such as rolling balls or hot-air balloons), their maximum velocities, accelerations, and elevations can be determined using calculus.
• Politics: The number of years required in a city for the rate of increase in its voting population to reach its maximum can be determined using calculus.
• Manufacturing: The design of containers, meeting specific constraints, can be determined using calculus. For example, calculus will supply the dimensions of a container that will maximize its volume or minimize its surface area.

Applications of Integral Calculus

• Inverse of differential calculus: In mathematics, most operations have inverse operations. In calculus, the inverse of differentiation is integration. Therefore, a fundamental application of integral calculus is to find functions that produce the answers to a problem in differential calculus.
• Measurement, area, and volume: Integral calculus can be used to find (1) the areas between the graphs of functions over specified intervals, (2) the surface areas of three-dimensional objects, and (3) the volumes of three-dimensional objects.
• Centroids: The centroid (or center of mass) of an object can be found using integral calculus. For two- and three-dimensional objects, the centroid is the balancing point of the object. Calculus can be used to locate the position of the centroid on the object.
• Fluid pressure: Integral calculus is essential in the design of ships, dams, submarines, and other submerged objects. It is used in determining the fluid pressure on the submerged object at various depths from the water’s surface. This information is essential in the design of submerged objects so they will not collapse.
• Physics (work): When a constant force is applied to an object that moves in the direction of the force, the work produced by the force is found by multiplying the force by the distance moved by the object. However, when the applied force is not constant or is variable, calculus can be used in determining the work produced by the variable force (for example, the variable force needed to pull a metal spring, or the force exerted by expanding gases on the piston in an engine).

The aforementioned applications are examples of the most elementary applications of calculus. In the technological world of the twenty-first century, applications of calculus continue to evolve. The consequences of calculus are ubiquitous in contemporary society and impact every walk of life.

Recommendations for Mathematics Curriculum Reform

In 1983, following a harsh report from the National Commission on Excellence in Education, U.S. society began to question seriously the effectiveness of its educational systems. The report, titled A Nation at Risk: the Imperative for Educational Reform, was commissioned by U.S. President Ronald Reagan. The report cited U.S. students for their poor academic performance in every subject area at every grade level and their underachievement on national and international scales. The Commission warned the United States that its education system was “being eroded by a rising tide of mediocrity.” In the years that followed, the Commission’s explicit call for educational reform in U.S. schools served to generate numerous curricular reform efforts at the pre-college and college levels.

In response to this call for reform, in 1987, the Mathematical Association of America (MAA) and the National Research Council (NRC) co-sponsored a conference held in Washington, D.C., titled Calculus for a New Century. The conference was attended by more than 600 college and pre-college calculus teachers. The conference focused on the nature and need for calculus reform in college and pre-college institutions throughout the nation. During that conference, the phrase “Calculus should be a pump, not a filter in the pipeline of American education” became a national mantra for calculus reform.

National educational assessments conducted in 1989 further supported initiatives for calculus reform. During the 1980s, approximately 300,000 U.S. college students were enrolled annually in science-based calculus courses. Of that number, only 140,000 students earned grades of D or higher. Thus, more than 50% of U.S. college students were failing the calculus courses required for their majors, which included mathematics, all of the natural and physical sciences, and computer science. These bleak statistics served to motivate concerned calculus teachers to examine the traditional calculus curriculum, as well as their own teaching methodologies, with the intention of increasing course enrollments, student achievement, and enthusiasm for the subject.

Their efforts resulted in major calculus reform initiatives as early as 1989. The first set of recommendations for reform in school mathematics (grades prekindergarten–12) came from the National Council of Teachers of Mathematics (NCTM). These recommendations were delineated in NCTM’s publication, Principles and Standards for School Mathematics (also known as NCTM Standards).

Four overarching standards (called Process Standards) were identified for improving mathematics instruction at all levels. These standards identified problem solving, reasoning and proof, connections, and communications as the four primary foci for mathematics instruction. During the 1990s, most U.S. states adopted this document as their curriculum framework for school mathematics. Decisions regarding the mathematics curriculum, textbook selections, and instructional strategies were revised in accordance with the recommendations of the NCTM Standards. Interestingly, the same document served to inspire pedagogical reform in mathematics at the college level, especially in calculus.

Traditional Calculus Versus Reformed Calculus

Until 1990, the calculus curriculum had remained basically the same for decades. The traditional calculus curriculum reflected formal mathematical language, mathematical rigor, and symbolic precision. Computations with limits, mathematical proofs, and elaborate mathematical computations were common practice in calculus classrooms. Students took careful notes, asked clarifying questions, and completed voluminous amounts of homework in preparation for test questions similar to those completed for homework. Instruction was teacher-centered and delivered through a lecture approach. Relevant applications were seldom considered, and graphing calculators and computers were rarely used in calculus instruction, and students were not allowed to use them for computations, graphing, or problem solving. Mathematics educators attributed the dismal performance of the majority of students in the nation’s calculus classes to this traditional calculus curriculum. Consequently, by the mid-1990s, calculus reform movements had been initiated in many of the colleges and pre-college classrooms throughout the nation.

Calculus reform efforts at the college level in the 1990s often applied the pedagogical recommendations found in NCTM Standards. These pedagogical recommendations were also reflected in the revised Advanced Placement Calculus (AP Calculus) and International Baccalaureate Calculus (IB Calculus) courses offered in the nation’s high schools. A measure of the subsequent success of the calculus reform movement at the pre-college level can be seen in the dramatic increase in numbers of students who took these courses from the 1980s into the twenty-first century. Specifically, the National Center for Education Statistics reported that the percentage of students completing calculus in high school had risen from 6% to 14% in the years from 1982 to 2004. The number of students completing calculus in high school continues to grow exponentially, at an estimated rate of 6.5% per year.

Several reform calculus curricula originated in the 1990s, and continue into the twenty-first century. The following examples are prominent reform calculus projects: Calculus, Concepts, Computers and Cooperative Learning (C⁴L) conducted at Purdue University; the Calculus Consortium at Harvard (CCH) conducted at Harvard University; and Calculus and Mathematica (C&M) conducted at the University of Illinois at Urbana-Champaign and at Ohio State University.

While these three reform calculus projects differ from each other in significant ways, they share the following characteristics:

• They use graphing calculators, computers, and computer algebra systems (CAS) extensively for instruction, exploration, and visual representations. Supporters argue that technology serves to alleviate the huge burden of algebraic computation so characteristic of traditional calculus. The rationale for this reform is that technology facilitates instructional processes that focus on the principles of calculus rather than on computational procedures. Moreover, the graphical and visual representations provided by these technologies offer alternative modalities for learning that accommodate students’ different learning styles. The curricula for CCH and C⁴L focus heavily on graphing calculators, whereas the curriculum for C&M relies heavily on the computer software, Mathematica.
• The teacher serves as a facilitator of learning rather than the main conveyor of knowledge. While the teacher continues to initiate instruction and answer questions, mathematical situations are often explored by groups of students, using cooperative learning strategies. Using the principles of constructivist learning, students are guided to discover mathematical properties for themselves in a laboratory setting.
• A major focus is placed on real applications from multiple disciplines. The intention is to raise students’ interest in the subject and motivate them with relevant applications.
• Mathematical rigor and formal language are de-emphasized. The abstractions of mathematical proof and rigor are postponed for several semesters to provide sufficient time for students to gain practical and intuitive knowledge of the subject.
• Assessment focuses heavily on students’ writing, explanations of problem solutions, and open-ended projects. Sometimes students’ explanations are valued as highly as the accuracy of their answers.

Whereas all of the above instructional practices have shown varying degrees of success in reform calculus classrooms, some areas of concern have been identified by those involved in the projects. Specifically:

• Focusing heavily on relevant applications sometimes results in the omission of important calculus content that cannot always be motivated by applications.
• The use of everyday language sometimes results in imprecise and incorrect mathematical definitions.
• Overuse of technologies for computation and graphing can weaken the development of students’ quantitative reasoning and computational skills in calculus.
• Real-world problems are sometimes too complex and frustrating to students because of the extraneous and irrelevant information they usually contain.
• Short-answer problems for assessment are often easier for students than describing their problem-solving procedures in writing.
• Constructivist approaches are often too time consuming, allowing insufficient time for covering the entire calculus curriculum during class time.

Resolution of these concerns will surely be addressed in future curriculum revisions, and changes or modifications will be made accordingly. However, these accommodations are consistent with the historical evolution of calculus, which is the study of change and systems in perpetual motion.

Summary

In the past, calculus was taught in ways that made it accessible to only a small proportion of the population. However, recent curricular and pedagogical reforms in calculus, both at the college and pre-college levels, have served to increase student success, include twenty-first-century-technologies, and triple course enrollments. Statistics indicate that calculus enrollments will continue to increase exponentially. These findings suggest that calculus instruction in the United States is responding positively to the academic needs of society.

Indeed, by combining the power of technology with calculus, new areas of mathematics are emerging (for example, fractals, dynamical systems, and chaos theory). These new branches of mathematics have allowed humans to mimic nature’s designs of mountain ranges, oceans, and plant growth patterns—which once were considered random acts of nature. In conclusion, calculus as a subject is still growing, and its applications are continually expanding to meet the needs of a dynamic, diverse, and technologically driven society.

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