**2.1 Brief history of calculus**

The invention of calculus is traditionally given shared credit to Isaac Newton and Gottfried Wilhelm Leibniz, who each independently developed the theories around infinitesimal calculus in the late seventeenth century. Bressoud points out that their contributions to calculus lie in connecting theories related to differentiation and integration, rather than in developing the theories of the individual ideas [2]. Newton, primarily a physicist, was motivated to pursue calculus in order to provide a scientific description of motion and magnitude. When documenting his ideas related to calculus, he did so primarily for himself, using a mixture of notations that made sense to him. Leibniz, described as a polymath or someone with a wide array of knowledge akin to a renaissance man (specifically, his interests included metaphysics, law, economics, politics, logic, and mathematics), was motivated to pursue calculus in order to provide a metaphysical explanation of change. He purposefully developed a clear and consistent notation system to document his work that we still essentially use today. While these two men came to be interested in calculus for very different reasons (one from more applied motivations and one coming from more pure mathematical interests), today they continue to share the honor of being credited with this field.

Although Newton and Leibniz are credited with the invention (or discovery, depending on your scientific philosophy), many mathematicians came before them, to develop the ideas they built on, and after them to refine their ideas [3]. Even as recently as 2014, researchers are challenging the credit given to Newton and Leibniz by finding evidence that other mathematicians developed the formal ideas of calculus long before the 1670's. For example, a group of mathematicians in Southern India from the Kerala School developed and published work on many fundamental ideas of infinitesimal calculus 300 years prior to Leibniz and Newton [4]. With this note aside, college calculus today is a direct descendent of the work of Newton and Leibniz, using Leibniz's notation, and so I consider these as the birthplace of the college calculus we still see today.

### **2.2 Evolution of calculus education**

So how did calculus come to hold the place as the integral (pun intended) component of so many students' college educations? To answer this question, I draw significantly from Alan Tucker's "History of the undergraduate program in mathematics in the United States" [5]. In the 1700s and 1800s, mathematics was studied as one of the main topics (along with Latin, Greek, and Hebrew) in college following the English college model. The goal of mathematics in such an education was as a "classical training of the mind instead of the language of science and engineering it is today" ([5], p. 689). The students attending these colleges were mostly male and mostly from the upper-class. Although Newton's and Leibniz's work developing calculus into a more systematized and valued field occurred in the late 1600s, calculus wasn't taught widely in college until the late 1800s. It was during this time period that colleges shifted from delivering a classical curriculum to a more practical curriculum. This is largely due to the fact that land-grant public universities were established in 1862 by the Morrill Act, and calculus became more standard for technically-oriented students.

By the early 1900s, most colleges allowed students to choose the courses for their study, which led to an increase in college enrollment and a decrease in mathematics study. It was during this time that mathematics was no longer viewed as part of a classic education, and instead a tool useful for engineers and scientists (as it continues to be seen today). During this time, calculus became an elective in US college preparatory high-schools and a mandatory subject for college preparatory high schools in Europe.

**139**

*Towards a Forward-Thinking College Calculus Program DOI: http://dx.doi.org/10.5772/intechopen.87940*

course in high school to be successful in college [8].

**2.3 Current state of college calculus education**

Also around this time, the presentation of calculus changed from following the order of topics in which the theory was developed (first integration, then differentiation, then series, and lastly limits), to the order most beneficial for rigorously proving theorems of calculus (first limits, then differentiation, then limits, and lastly series) [3]. Little has changed since this time. The undergraduate mathematics curriculum for mathematics majors has become more and more solidified, aided by guidance from the Mathematical

Association of America and their Committee on the Undergraduate Program in Mathematics (CUPM) guide. Tucker notes that in the more recent history, though the curriculum has not changed much, "the greatest area of change and concern in the past 40 years has been the articulation between high school and college mathematics." By the early 1980s, "mathematics faculty were dealing with large numbers of students in freshman courses who showed limited knowledge of needed algebra skills" ([5], p. 702). He attributes these changes to states increasing their high school mathematics requirements and watering down the material to meet these higher expectations, as well as the rise of high stakes testing. Near this time, AP Calculus became an increasingly expected course in high school. During the late 1990s (during which time I was in high school and also the earliest data recorded online of AP participation by subject), approximately 100,000 high school students took the AP Calculus AB exam [6]. From my experience, AP Calculus AB was viewed as a course that only students extremely interested in pursuing mathematics or physics as their college major would take. By 2018 that number had tripled [7]. As part of a large, national study on college calculus conducted in 2010, my research team identified that two-thirds of all students in a college calculus class had already taken a course in high school called calculus (many of these being AP Calculus AB or BC), and half of the students we surveyed believed they needed to take a calculus

Over the past nearly 50 years, there has been tremendous attention paid to reforming college calculus, which has resulted in more attention to problem solving in applied contexts, an increased focus on supporting students' development of conceptual understanding rather than only procedural fluency, and often more active learning techniques employed in the classroom, including student-centered instruction and more technology use [1, 9, 10]. These changes have certainly resulted in more variation in the college calculus instruction across the country, with some programs very much still rooted in these reforms and others holding on to a pre-reform calculus model. That said, the basic content being taught in all of these programs is still essentially a course on Newton & Leibniz's ideas, taught in the order best suited for proving calculus, regardless of the presentation, the students being taught, the pedagogy, or the contexts for the word problems.

For the past decade, I have been a part of a large research team studying college calculus. This research team has been led by David Bressoud, run under the auspices of the Mathematical Association of America, and funded by the National Science Foundation. Our research has come from two projects, the first begun in 2009 and focused on mainstream college differential calculus programs (typically called Calculus I) in all institution types, called *Characteristics of College Calculus (CSPCC);* the second begun in 2014 and focused on precalculus, differential and integral calculus programs at Masters and PhD-granting institutions, called *Progress through Calculus (PtC).* Our work has been generally focused on identifying aspects of college calculus programs that are more successful or innovative than comparative institutions, and supporting more mathematics departments to improve their programs based on these findings. For our purposes, success in college calculus is primarily marked by a large percentage of the students who plan to complete

#### *Towards a Forward-Thinking College Calculus Program DOI: http://dx.doi.org/10.5772/intechopen.87940*

*Theorizing STEM Education in the 21st Century*

**2.1 Brief history of calculus**

**2. How calculus established itself in STEM education**

they continue to share the honor of being credited with this field.

**2.2 Evolution of calculus education**

Although Newton and Leibniz are credited with the invention (or discovery, depending on your scientific philosophy), many mathematicians came before them, to develop the ideas they built on, and after them to refine their ideas [3]. Even as recently as 2014, researchers are challenging the credit given to Newton and Leibniz by finding evidence that other mathematicians developed the formal ideas of calculus long before the 1670's. For example, a group of mathematicians in Southern India from the Kerala School developed and published work on many fundamental ideas of infinitesimal calculus 300 years prior to Leibniz and Newton [4]. With this note aside, college calculus today is a direct descendent of the work of Newton and Leibniz, using Leibniz's notation, and so I consider these as the birthplace of the college calculus we still see today.

So how did calculus come to hold the place as the integral (pun intended) component of so many students' college educations? To answer this question, I draw significantly from Alan Tucker's "History of the undergraduate program in mathematics in the United States" [5]. In the 1700s and 1800s, mathematics was studied as one of the main topics (along with Latin, Greek, and Hebrew) in college following the English college model. The goal of mathematics in such an education was as a "classical training of the mind instead of the language of science and engineering it is today" ([5], p. 689). The students attending these colleges were mostly male and mostly from the upper-class. Although Newton's and Leibniz's work developing calculus into a more systematized and valued field occurred in the late 1600s, calculus wasn't taught widely in college until the late 1800s. It was during this time period that colleges shifted from delivering a classical curriculum to a more practical curriculum. This is largely due to the fact that land-grant public universities were established in 1862 by the Morrill Act, and calculus became more standard for technically-oriented students. By the early 1900s, most colleges allowed students to choose the courses for their study, which led to an increase in college enrollment and a decrease in mathematics study. It was during this time that mathematics was no longer viewed as part of a classic education, and instead a tool useful for engineers and scientists (as it continues to be seen today). During this time, calculus became an elective in US college preparatory high-schools and a mandatory subject for college preparatory high schools in Europe.

The invention of calculus is traditionally given shared credit to Isaac Newton and Gottfried Wilhelm Leibniz, who each independently developed the theories around infinitesimal calculus in the late seventeenth century. Bressoud points out that their contributions to calculus lie in connecting theories related to differentiation and integration, rather than in developing the theories of the individual ideas [2]. Newton, primarily a physicist, was motivated to pursue calculus in order to provide a scientific description of motion and magnitude. When documenting his ideas related to calculus, he did so primarily for himself, using a mixture of notations that made sense to him. Leibniz, described as a polymath or someone with a wide array of knowledge akin to a renaissance man (specifically, his interests included metaphysics, law, economics, politics, logic, and mathematics), was motivated to pursue calculus in order to provide a metaphysical explanation of change. He purposefully developed a clear and consistent notation system to document his work that we still essentially use today. While these two men came to be interested in calculus for very different reasons (one from more applied motivations and one coming from more pure mathematical interests), today

**138**

Also around this time, the presentation of calculus changed from following the order of topics in which the theory was developed (first integration, then differentiation, then series, and lastly limits), to the order most beneficial for rigorously proving theorems of calculus (first limits, then differentiation, then limits, and lastly series) [3]. Little has changed since this time. The undergraduate mathematics curriculum for mathematics majors has become more and more solidified, aided by guidance from the Mathematical Association of America and their Committee on the Undergraduate Program in Mathematics (CUPM) guide. Tucker notes that in the more recent history, though the curriculum has not changed much, "the greatest area of change and concern in the past 40 years has been the articulation between high school and college mathematics." By the early 1980s, "mathematics faculty were dealing with large numbers of students in freshman courses who showed limited knowledge of needed algebra skills" ([5], p. 702). He attributes these changes to states increasing their high school mathematics requirements and watering down the material to meet these higher expectations, as well as the rise of high stakes testing. Near this time, AP Calculus became an increasingly expected course in high school. During the late 1990s (during which time I was in high school and also the earliest data recorded online of AP participation by subject), approximately 100,000 high school students took the AP Calculus AB exam [6]. From my experience, AP Calculus AB was viewed as a course that only students extremely interested in pursuing mathematics or physics as their college major would take. By 2018 that number had tripled [7]. As part of a large, national study on college calculus conducted in 2010, my research team identified that two-thirds of all students in a college calculus class had already taken a course in high school called calculus (many of these being AP Calculus AB or BC), and half of the students we surveyed believed they needed to take a calculus course in high school to be successful in college [8].

Over the past nearly 50 years, there has been tremendous attention paid to reforming college calculus, which has resulted in more attention to problem solving in applied contexts, an increased focus on supporting students' development of conceptual understanding rather than only procedural fluency, and often more active learning techniques employed in the classroom, including student-centered instruction and more technology use [1, 9, 10]. These changes have certainly resulted in more variation in the college calculus instruction across the country, with some programs very much still rooted in these reforms and others holding on to a pre-reform calculus model. That said, the basic content being taught in all of these programs is still essentially a course on Newton & Leibniz's ideas, taught in the order best suited for proving calculus, regardless of the presentation, the students being taught, the pedagogy, or the contexts for the word problems.
