**5. How such a program relates to the seven features of successful college calculus programs**

Based on our site visits to five doctoral-granting mathematics departments with college calculus programs which we identified as more successful than other programs, we identified seven features of college calculus programs that we hypothesize are related to these programs' successes [13]. In [14], I discuss how each of these features can be thought of while implementing diversity, equity, and inclusion practices. Here, I consider how the above articulation of a forward-thinking calculus program would relate to the seven characteristics.

By my definition above, a forward-thinking calculus program is designed so that all components of the course support a diverse population of students to thrive. A diverse student population will include a diversity of mathematical backgrounds and experiences, cultural diversity, language diversity, as well as diversity of genders, ages, races and ethnicities, sexual orientations, and physical and mental abilities. While each of these types of diversity can influence the design of a forward thinking calculus program, here I foreground the role of diversity in mathematical backgrounds and cultural diversity.

*A rich and engaging calculus curriculum* designed to support a diverse population of students to thrive would acknowledge the needs of the students taking the course, including what additional mathematical preparation they need to thrive in the course and what components of calculus are needed in their future courses and careers. At my own institution, the calculus coordinator is often surprised and disappointed by calculus students' algebraic knowledge – one example is how persistent many students' belief that

$$a^2 + b^2 = \{a \, \star b\}^2. \tag{1}$$

One way to respond to this realization is to blame the students for not being prepared enough, and to continue assessing their calculus learning by inherently relying on their lacking algebraic understanding, resulting in believing that the students also lack calculus understanding. A different way to respond to this realization is to blame the system responsible for educating these students, and either infuse algebraic lessons in to the calculus lessons or to not rely on students' algebraic skills for them to demonstrate their calculus understanding (for example, by not assigning algebraically messy functions and by delegating algebraic manipulations to technology). A forward-thinking calculus program would additionally learn what majors the students are pursuing and what calculus content students need to thrive in those majors – while STEM is constantly developing and growing, as should the mathematics we teach students to support them in STEM.

A mathematics department engaging in bringing their calculus program into the modern day should use *local data* to inform these changes. The types of local data collected can include quantitative outcome measures (such as grades and persistence) and qualitative measures of experience (such as focus groups with students who have persisted and those who have not). The value in the quantitative data is that it can identify trends and patterns and can be used to examine the prevalence of an observation. One downside is that the experiences of the majority can overshadow the experiences of the minority, and when designing a calculus program to support a diverse population of students to thrive, it is the voices of the minority that become especially important. Qualitative data can complement the quantitative data by illuminating experiences of a smaller number of students. One way to gather such data is to hold a number of focus group interviews with students (as done in the bates College example previously discussed), especially students from demographic groups and with experiences not held by the majority of the population. This could be holding a focus group of students of color in calculus to identify how they are experiencing the calculus program, and specifically how they are experiencing the calculus program as students of color. A similar focus could be taken by speaking to transfer students, first generation students, "nontraditional" students (typically older than traditional students), and students who have not taken calculus before. The quantitative and qualitative data gathered can be used together to inform curricular decisions (what do our students need from this course?), pedagogical decisions (what have students been experiencing in our courses, and what needs to change?), and programmatic decisions (is this calculus program achieving the goals that we want it to?).

*Coordination* of a calculus program designed to support a diverse population of students to thrive raises questions about what is fair. A primary goal of coordination is to ensure that all students (including those being taught by different instructors) experience a similar course and that their grades reflect this objectively. This need for similarity and objectivity speaks to a desire for the course to be fair for all students, though this inherently assumes that all students are coming in with the same preparation and resources. By acknowledging that this is not the case, the role of coordination becomes not to ensure fairness but to ensure justice for all students. A fair coordination system will seek to ensure that students are graded as objectively as possible and that this grade is only based on their knowledge. A just coordination system will seek to ensure that all students are given an opportunity to communicate what they have learned – which may entail acting in ways that do not seem fair to other students.

The acknowledgment that not all students are entering college calculus with the same mathematical experiences, preparation, and resources has a significant affect on the role of *placement* into mathematics. During our site visits to the more successful college calculus programs, we observed placement systems designed to place students into the highest course in which students could be successful. A key component of a placement system that is able to place students in this way is to have multiple options for courses that acknowledge the differences in student experiences. In our

**147**

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

more recent work, we have seen examples of a broadened variety of college calculus courses that acknowledge that students come into college calculus with different prior experiences. The majority of these courses focused on supporting students on the lower cusp of placing into calculus (as determined by a placement exam, standardized test scores, or high school grades), such as calculus infused with precalculus and co-calculus (see [27] for details about these course structures), though we did observe courses designed to support students at the higher end of the placement, such as in the accelerated calculus course developed to support students who had already been exposed to calculus in high school. Such course variations enable a placement system

Through CSPCC and PtC, we observed growing support of *active learning* in the calculus sequence. While the specific implementations of active learning vary (including partner talk, group work, whole class discussions, and student presentations at the board), the common underlying element is that these classes engage students in mathematical activity during class. To engage students in rich mathematics during class time in a way that supports all students to thrive involves deep attention to and care of the mathematics of the students rather than only of the textbook or the instructor [23]. Another way to say this is that instead of describing the classes as "student-centered" I would describe them as "student-thinking centered." Such classes assume that students make sense of mathematics differently from one another and differently from the textbook or the instructor, and that such differences do not make their meanings incorrect; rather, drawing out multiple mathematical meanings for one problem leads to richer discussion and a richer understanding of the content. Forward-thinking calculus programs value and leverage the diversity of ideas present in a mathematics class composed of a diverse student population by engaging students in rich mathematics, eliciting their meanings of the mathematics,

What I describe here as a forward-thinking calculus program is far different from my own experiences as a college calculus student, and likely far different from the experiences of the majority of novice college calculus instructors (including graduate students, post-doctoral fellows, and new faculty). With this in mind, it becomes even more critical to provide teaching preparation to novice instructors involved in the teaching of calculus. One critical need for such preparation is purely pedagogical – while secondary teachers go through years of pedagogical preparation and apprenticeship, new college instructors are often expected to learn on the job. An additional need, that becomes pronounced when teaching to a more diverse population of students, is to help novice instructors understand that their students are not all like them (and are not all on their way to an advanced degree in mathematics) and to value what these students bring to their class. One professional development experience that can support this is to look at student work in a non-evaluative way; by looking at student work to understand what the students do understand and how they are making sense to come to their solution, rather than evaluating how many points a solution earns, instructors can learn to appreciate the

The final component of a forward-thinking calculus program to consider is the *supports that exist outside the classroom* that are designed to support a diverse population of students to thrive. Through the CSPCC and PtC work, we have observed tutoring centers specific to calculus content and shared workspaces in the mathematics department for students to informally gather to work on calculus together. Through the sites we have visited, we have seen much value in these supports, with many students sharing how impactful they were to their learning. We have also seen a number of rich supports for students that reside outside the mathematics department; for example, a mentoring program for students of color in STEM and

to give students the course options in which they can be successful.

and engaging with the students' meanings of the mathematics.

richness of their students' mathematical thinking.

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

*Theorizing STEM Education in the 21st Century*

relying on their lacking algebraic understanding, resulting in believing that the students also lack calculus understanding. A different way to respond to this realization is to blame the system responsible for educating these students, and either infuse algebraic lessons in to the calculus lessons or to not rely on students' algebraic skills for them to demonstrate their calculus understanding (for example, by not assigning algebraically messy functions and by delegating algebraic manipulations to technology). A forward-thinking calculus program would additionally learn what majors the students are pursuing and what calculus content students need to thrive in those majors – while STEM is constantly developing and growing, as

A mathematics department engaging in bringing their calculus program into the modern day should use *local data* to inform these changes. The types of local data collected can include quantitative outcome measures (such as grades and persistence) and qualitative measures of experience (such as focus groups with students who have persisted and those who have not). The value in the quantitative data is that it can identify trends and patterns and can be used to examine the prevalence of an observation. One downside is that the experiences of the majority can overshadow the experiences of the minority, and when designing a calculus program to support a diverse population of students to thrive, it is the voices of the minority that become especially important. Qualitative data can complement the quantitative data by illuminating experiences of a smaller number of students. One way to gather such data is to hold a number of focus group interviews with students (as done in the bates College example previously discussed), especially students from demographic groups and with experiences not held by the majority of the population. This could be holding a focus group of students of color in calculus to identify how they are experiencing the calculus program, and specifically how they are experiencing the calculus program as students of color. A similar focus could be taken by speaking to transfer students, first generation students, "nontraditional" students (typically older than traditional students), and students who have not taken calculus before. The quantitative and qualitative data gathered can be used together to inform curricular decisions (what do our students need from this course?), pedagogical decisions (what have students been experiencing in our courses, and what needs to change?), and programmatic decisions (is this calculus

*Coordination* of a calculus program designed to support a diverse population of students to thrive raises questions about what is fair. A primary goal of coordination is to ensure that all students (including those being taught by different instructors) experience a similar course and that their grades reflect this objectively. This need for similarity and objectivity speaks to a desire for the course to be fair for all students, though this inherently assumes that all students are coming in with the same preparation and resources. By acknowledging that this is not the case, the role of coordination becomes not to ensure fairness but to ensure justice for all students. A fair coordination system will seek to ensure that students are graded as objectively as possible and that this grade is only based on their knowledge. A just coordination system will seek to ensure that all students are given an opportunity to communicate what they have learned – which may entail acting in ways that do not seem fair to other students. The acknowledgment that not all students are entering college calculus with the same mathematical experiences, preparation, and resources has a significant affect on the role of *placement* into mathematics. During our site visits to the more successful college calculus programs, we observed placement systems designed to place students into the highest course in which students could be successful. A key component of a placement system that is able to place students in this way is to have multiple options for courses that acknowledge the differences in student experiences. In our

should the mathematics we teach students to support them in STEM.

program achieving the goals that we want it to?).

**146**

more recent work, we have seen examples of a broadened variety of college calculus courses that acknowledge that students come into college calculus with different prior experiences. The majority of these courses focused on supporting students on the lower cusp of placing into calculus (as determined by a placement exam, standardized test scores, or high school grades), such as calculus infused with precalculus and co-calculus (see [27] for details about these course structures), though we did observe courses designed to support students at the higher end of the placement, such as in the accelerated calculus course developed to support students who had already been exposed to calculus in high school. Such course variations enable a placement system to give students the course options in which they can be successful.

Through CSPCC and PtC, we observed growing support of *active learning* in the calculus sequence. While the specific implementations of active learning vary (including partner talk, group work, whole class discussions, and student presentations at the board), the common underlying element is that these classes engage students in mathematical activity during class. To engage students in rich mathematics during class time in a way that supports all students to thrive involves deep attention to and care of the mathematics of the students rather than only of the textbook or the instructor [23]. Another way to say this is that instead of describing the classes as "student-centered" I would describe them as "student-thinking centered." Such classes assume that students make sense of mathematics differently from one another and differently from the textbook or the instructor, and that such differences do not make their meanings incorrect; rather, drawing out multiple mathematical meanings for one problem leads to richer discussion and a richer understanding of the content. Forward-thinking calculus programs value and leverage the diversity of ideas present in a mathematics class composed of a diverse student population by engaging students in rich mathematics, eliciting their meanings of the mathematics, and engaging with the students' meanings of the mathematics.

What I describe here as a forward-thinking calculus program is far different from my own experiences as a college calculus student, and likely far different from the experiences of the majority of novice college calculus instructors (including graduate students, post-doctoral fellows, and new faculty). With this in mind, it becomes even more critical to provide teaching preparation to novice instructors involved in the teaching of calculus. One critical need for such preparation is purely pedagogical – while secondary teachers go through years of pedagogical preparation and apprenticeship, new college instructors are often expected to learn on the job. An additional need, that becomes pronounced when teaching to a more diverse population of students, is to help novice instructors understand that their students are not all like them (and are not all on their way to an advanced degree in mathematics) and to value what these students bring to their class. One professional development experience that can support this is to look at student work in a non-evaluative way; by looking at student work to understand what the students do understand and how they are making sense to come to their solution, rather than evaluating how many points a solution earns, instructors can learn to appreciate the richness of their students' mathematical thinking.

The final component of a forward-thinking calculus program to consider is the *supports that exist outside the classroom* that are designed to support a diverse population of students to thrive. Through the CSPCC and PtC work, we have observed tutoring centers specific to calculus content and shared workspaces in the mathematics department for students to informally gather to work on calculus together. Through the sites we have visited, we have seen much value in these supports, with many students sharing how impactful they were to their learning. We have also seen a number of rich supports for students that reside outside the mathematics department; for example, a mentoring program for students of color in STEM and

a tutoring center and gathering space for Native students in STEM. Such programs could be made richer with more of a partnership with the calculus programs. While the mathematics departments' main focus is on supporting students mathematically, there is an opportunity for calculus programs to acknowledge calculus students as multifaceted people, and identify existing supports on campus that the calculus program could integrate into.
