**3. An example of a college calculus program affords an anti-deficit perspective**

Through the PtC work, I hoped to find a mathematics department where the calculus program was thoughtfully crafted to best support today's college students – a more diverse population of students, that includes more students of color, and more first-generation and low-income students than before [20]. We did not find a program that had an explicit focus on supporting a diverse population of students to thrive in mathematics, but we did see a calculus program developed to support every student in their construction of mathematical meanings in calculus. This program was developed based on research rooted in radical constructivism, and not with an explicit attention to equity. However, I believe this program affords an anti-deficit approach to mathematics by viewing every student's mathematical understanding as valuable and part of the construction of richer mathematical meanings. This calculus program illustrates that by sincerely valuing every student's mathematical understandings, and leveraging research to support each student's rich construction of mathematical meaning, a diverse population of college calculus students can mathematically thrive.

#### **3.1 Background on DIRACC**

Project DIRACC (*Developing and Investigating a Rigorous Approach to Conceptual Calculus)* is an NSF-funded college calculus curriculum developed by Pat Thompson and his colleagues based on years of research on student understandings of calculus (see [21] for a description). This curriculum is self-described as "Newton meets Technology", focusing on developing meaning for infinitesimals (while utilizing animations and interactive apps) rather than emphasizing the notation and formality of Leibniz. This curriculum is shared online for free, and is currently being implemented in at least two large, public, doctoral granting mathematics departments, including one involved in Progress through Calculus.

In this chapter, I will draw on my experience at the one university involved in PtC (referred to as Large State University; LSU), where DIRACC is the curriculum used for all calculus courses for science, computer science, and mathematics majors. The undergraduate population of LSU is approximately 50% white students, 20% Hispanic and Latinx students, 7% Asian, and 5% Black and African American. In the DIRACC calculus courses I observed, I estimated that approximately 30% of students were Black, Latinx, and/or Native (based on appearance). At LSU, there is a separate (and more procedurally oriented) college calculus course for engineering majors. The DIRACC courses are taught by instructors, mathematics education faculty, and doctoral students pursuing degrees in pure and applied mathematics and

**143**

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

mathematics more than students at other institutions.

**3.2 Shift in curriculum**

problems being set in other contexts.

and students of color's majors).

**3.3 Shift in pedagogy**

mathematics education. This course is coordinated by a full-time instructor, and this coordination includes weekly meetings for all instructors, where the topics of discussion during the meetings include understanding the mathematics and student thinking related to the mathematics for the upcoming section. Preliminary results from PtC indicate that students in DIRACC outperform students at comparable universities on a calculus content assessment and maintain positive beliefs towards

To best serve the students in our calculus classes, we need to learn what is motivating them to pursue degrees requiring calculus – whether future career goals or general interest in learning – and rethink our calculus curriculum to be in line with these interests. It is well established that in today's economy, STEM jobs pay significantly more, on average, than non-STEM jobs [22]. Given this widespread knowledge, we cannot ignore that one contributing motivation for students to pursue STEM is future job and wage prospects. When sitting in Calculus I classes across the country, it often seems that everyone knows the students are there not to learn deep and interesting mathematics, but to get a grade in the course that allows them to continue pursing whatever STEM degree they are hoping for in order to get a good job. I believe that we are missing a big opportunity in our calculus classes to inspire these STEM-intending students about the magic and beauty of calculus. The great majority of calculus courses I have visited have been "mainstream" courses, meaning to serve all STEM students, although in actuality the great majority of the content is driven by the needs of the engineering students, with occasional word

In a forward-thinking calculus system, there would be a meaningful connection

Through PtC, I observed three DIRACC calculus courses at LSU, and though the three courses looked different, in each I witnessed a racially diverse group of students equitably engaging in rich mathematics, contributing to constructing mathematical meaning as a class. In one class, the instructor stood in front of a 40-person class, while he randomly selected students to answer questions related to a context problem they worked on. The questions he asked were substantive and open ended,

between the content taught in calculus, the needs of the majors whose students are taking calculus, and the interests and motivations of the students enrolled in our courses. It would be these latter two driving the content, rather than historical precedents. The DIRACC curriculum achieves this by forgoing Leibniz's precise notation in favor of Newton's more intuitive ideas – skipping the formalities of ideas such as limit to spend more time supporting students to understand the ideas of infinitesimals and how this can support meaningful understanding of rate of change functions and accumulation functions. This curriculum was designed explicitly to support students in developing rich mathematical meanings, and is thus inherently responsive to how students think about calculus and what todays' students should be learning in a calculus course. As currently taught, I witnessed this curriculum equitably engaging a racially diverse student population in rich mathematics. This curriculum could go further in the future by engaging the diverse learners as whole people, by situating the mathematical content in contexts that are especially interesting and relevant for them (where these contexts could be identified by talking to students and using local data to identify trends in women

mathematics education. This course is coordinated by a full-time instructor, and this coordination includes weekly meetings for all instructors, where the topics of discussion during the meetings include understanding the mathematics and student thinking related to the mathematics for the upcoming section. Preliminary results from PtC indicate that students in DIRACC outperform students at comparable universities on a calculus content assessment and maintain positive beliefs towards mathematics more than students at other institutions.

### **3.2 Shift in curriculum**

*Theorizing STEM Education in the 21st Century*

few programs at the department level [19].

**an anti-deficit perspective**

students can mathematically thrive.

**3.1 Background on DIRACC**

**3. An example of a college calculus program affords** 

when selecting the 12 institutions involved in our study. This resulted in a number of institutions serving a more racially and ethnically diverse student population, and with a few of those institutions implementing programs specifically designed to support women and/or students of color and/or first-generation students to be successful in STEM. This work is ongoing, and we are in the process of learning more about these programs so that we can share more about them with others schools. One disappointing finding in our recent work has been the general lack of programs geared to increasing the diversity in STEM among research-oriented math departments across the country; while there were programs at the university and college level developed to foster diversity, equity, and inclusion in STEM, there were very

Through the PtC work, I hoped to find a mathematics department where the calculus program was thoughtfully crafted to best support today's college students – a more diverse population of students, that includes more students of color, and more first-generation and low-income students than before [20]. We did not find a program that had an explicit focus on supporting a diverse population of students to thrive in mathematics, but we did see a calculus program developed to support every student in their construction of mathematical meanings in calculus. This program was developed based on research rooted in radical constructivism, and not with an explicit attention to equity. However, I believe this program affords an anti-deficit approach to mathematics by viewing every student's mathematical understanding as valuable and part of the construction of richer mathematical meanings. This calculus program illustrates that by sincerely valuing every student's mathematical understandings, and leveraging research to support each student's rich construction of mathematical meaning, a diverse population of college calculus

Project DIRACC (*Developing and Investigating a Rigorous Approach to Conceptual* 

Thompson and his colleagues based on years of research on student understandings of calculus (see [21] for a description). This curriculum is self-described as "Newton meets Technology", focusing on developing meaning for infinitesimals (while utilizing animations and interactive apps) rather than emphasizing the notation and formality of Leibniz. This curriculum is shared online for free, and is currently being implemented in at least two large, public, doctoral granting mathematics

In this chapter, I will draw on my experience at the one university involved in PtC (referred to as Large State University; LSU), where DIRACC is the curriculum used for all calculus courses for science, computer science, and mathematics majors. The undergraduate population of LSU is approximately 50% white students, 20% Hispanic and Latinx students, 7% Asian, and 5% Black and African American. In the DIRACC calculus courses I observed, I estimated that approximately 30% of students were Black, Latinx, and/or Native (based on appearance). At LSU, there is a separate (and more procedurally oriented) college calculus course for engineering majors. The DIRACC courses are taught by instructors, mathematics education faculty, and doctoral students pursuing degrees in pure and applied mathematics and

*Calculus)* is an NSF-funded college calculus curriculum developed by Pat

departments, including one involved in Progress through Calculus.

**142**

To best serve the students in our calculus classes, we need to learn what is motivating them to pursue degrees requiring calculus – whether future career goals or general interest in learning – and rethink our calculus curriculum to be in line with these interests. It is well established that in today's economy, STEM jobs pay significantly more, on average, than non-STEM jobs [22]. Given this widespread knowledge, we cannot ignore that one contributing motivation for students to pursue STEM is future job and wage prospects. When sitting in Calculus I classes across the country, it often seems that everyone knows the students are there not to learn deep and interesting mathematics, but to get a grade in the course that allows them to continue pursing whatever STEM degree they are hoping for in order to get a good job. I believe that we are missing a big opportunity in our calculus classes to inspire these STEM-intending students about the magic and beauty of calculus. The great majority of calculus courses I have visited have been "mainstream" courses, meaning to serve all STEM students, although in actuality the great majority of the content is driven by the needs of the engineering students, with occasional word problems being set in other contexts.

In a forward-thinking calculus system, there would be a meaningful connection between the content taught in calculus, the needs of the majors whose students are taking calculus, and the interests and motivations of the students enrolled in our courses. It would be these latter two driving the content, rather than historical precedents. The DIRACC curriculum achieves this by forgoing Leibniz's precise notation in favor of Newton's more intuitive ideas – skipping the formalities of ideas such as limit to spend more time supporting students to understand the ideas of infinitesimals and how this can support meaningful understanding of rate of change functions and accumulation functions. This curriculum was designed explicitly to support students in developing rich mathematical meanings, and is thus inherently responsive to how students think about calculus and what todays' students should be learning in a calculus course. As currently taught, I witnessed this curriculum equitably engaging a racially diverse student population in rich mathematics. This curriculum could go further in the future by engaging the diverse learners as whole people, by situating the mathematical content in contexts that are especially interesting and relevant for them (where these contexts could be identified by talking to students and using local data to identify trends in women and students of color's majors).

#### **3.3 Shift in pedagogy**

Through PtC, I observed three DIRACC calculus courses at LSU, and though the three courses looked different, in each I witnessed a racially diverse group of students equitably engaging in rich mathematics, contributing to constructing mathematical meaning as a class. In one class, the instructor stood in front of a 40-person class, while he randomly selected students to answer questions related to a context problem they worked on. The questions he asked were substantive and open ended,

allowing every student to contribute thinking related to the question rather than simply answering correctly or incorrectly. The second class was a 120-person class where the instructor presented a slide presentation wearing a microphone, with three Learning Assistants circulating the room, and students discussing problems in small groups. The third class was a 30-person class where students spent the entire class working in groups of three-four on rich tasks while the instructor floated around the room, visiting with individual groups, and then bringing the class together for a whole-class discussion. The common element of these courses, in addition to the content being taught, was that the instructors authentically cared to understand what their students were thinking related to the mathematics, and that the instructors used this understanding of their students' thinking to connect the mathematics to the students' understanding of the mathematics – what Hackenberg has called exhibiting mathematical caring relations [23]. The DIRACC curriculum and its enactment at LSU illustrate a forward-thinking calculus program by centering the mathematics, and every individual student's construction of the mathematics, as the guiding forces.
