Love stories in a differential equations classroom

We believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (2009) and Gutiérrez (2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.


Introduction
Though there are a growing number of mathematics classroom resources that discuss culture (e.g.Algebra Project by Bob Moses, Rethinking Mathematics by Rico Gustein and Mathematics for Social Justice by Gizem Karaali and Lily Khadjavi), the undergraduate mathematics curricula are still largely devoid of cultural relevance (Brown et al., 2019).This paper expands on our previously published work on how to more broadly engage students in the classroom.In Ma et al. (2023), we mentioned a specially designed bundle of in-class activities that teach ideas about mathematical modeling while simultaneously providing 'a place where students both learn about their own and others' cultures and also develop pride in their own and others' cultures' (Aronson & Laughter, 2016, p. 167).Fostering such culturally inclusive activities and creating a learning environment where students can explore mathematical modeling while engaging with their own and others' cultures is a clear indication of delivering a culturally relevant education.
We believe that cultural competence can help students learn mathematical content and also that learning mathematical content can help students learn about themselves and others.That is, the mathematics classroom can be a space where students 'recognise and honor their own cultural beliefs and practises while acquiring access to the wider culture' (Ladson-Billings, 2006, p. 36).In particular, asking students to share their personal and cultural experience on a specific topic, such as love, can help to 'validate each student's culture, and bridge gaps between school and home through diversified instructional strategies and multicultural curricula' (Gay, 2010 as cited in Aronson & Laughter, 2016, p. 165).
Love is an eternal topic.People use various methods to explore and praise love in literature, music, film, art, and life, as an expression of their feelings and yearning for life.Mathematicians from various areas of expertise have dedicated significant time to the study of love and have reached valuable conclusions.For example, Backus (2011) may claim that love is a numbers game, but relying solely on fate, the odds of him meeting the right woman on any given night are approximately 1 in 285,000.By scoring a recorded conversation on specific categories, the specific affect coding system (SPAFF) can predict if there is a 'happily ever after' for a particular couple (Coan & Gottman, 2007).The optimal stopping theory suggests that in online dating apps, it is advisable to reject the initial 37% of matches and then choose the next best option (Swanson, 2016).Some experts use ordinary differential equations (ODEs) to study love.For example, Strogatz (1988) modeled Romeo and Juliet; Rinaldi (1998aRinaldi ( , 1998b) ) and Rinaldi et al. (2014Rinaldi et al. ( , 2013) ) simulated relationships in Laura and Petrarch, Pride and Prejudice, Beauty and the Beast, respectively.Other mathematicians followed suit and wrote additional papers that studied love using differential equations (Elishakoff, 2019;Jafari et al., 2016).All of these examples are Eurocentric examples of cis-gender, heterosexual romantic relationships.In the teaching package described in this paper, we broaden the definition of love to be more inclusive to those who may not identify with the characters above.
Students will actively engage in collaborative work, starting with constructing a minimal relationship model.Through model revisions, they will iteratively expand the model twice, ultimately developing a comprehensive relationship model using a system of ODEs.Moreover, students will gain valuable experience in solving these ODE models using various methods, such as the characteristic polynomial method, the matrix form, or the Laplace transform method.
In this paper, we present the theoretical underpinnings to design the pre-class assignment, three in-class activities, and a group final project-a package that focuses on the study of love through mathematical modeling in an ordinary differential equations course.In Section 2, we provide the theoretical frameworks for such a design.Section 3 summarises the activities and project, highlighting their key features and intended learning outcomes.Section 4 explains how we apply the concepts of agreements and dimensions from the theoretical framework to design the projects.In Section 5, we give suggestions about ways the designed materials can be implemented in a classroom setting.Section 6 concludes the paper's main contributions and suggests directions for designing similar course materials according to the agreements and dimensions described in the theoretical framework.
Finally, in the Supplemental Material, we provide detailed descriptions of the activities and project.

Theoretical framework
We leverage the works of Gutiérrez (2018) and Rendón (2009) to build a classroom experience in which students can not only learn mathematics but can also have opportunities to develop as socially responsible global citizens.Rendón (2009) posits seven unspoken academic agreements in which we have unwittingly adopted merely as a matter of academic membership.We have agreed to these norms as much as we have agreed to our given names, language, morals, and culture.Rendón (2009) also suggests a revised agreement with each unspoken agreement.While these agreements are for any general education, Gutiérrez (2018) posits eight dimensions for specifically rehumanizing mathematics for Latinx, Black, and Indigenous students and teachers.We begin with a description of each agreement from Rendón (2009) accompanied with the associated dimension(s) by Gutiérrez (2018), if applicable.It is important to note that the relationship between the agreements and the dimensions is not necessarily in a one-to-one correspondence.Rather, they support and complement each other like two performers in a dance.
The first agreement posited by Rendón (2009) is the agreement to privilege intellectual/rational knowing.We can see this enacted in the education system through standardised testing such as the SAT, which tests students' verbal and mathematical skills as an indication of their college aptitude.Rendón (2009) suggests including more opportunities for students to demonstrate their knowledge.In particular, the work of Gardner (2000) on multiple intelligences is leveraged to create the new agreement, the agreement to work with diverse ways of knowing in the classroom.The idea is to encourage and celebrate visual/spatial, interpersonal/social, intrapersonal/introspective, or kinesthetic intelligences, in addition to language and logical intelligence.
The next agreement by Rendón (2009), the agreement of separation, discusses how academia has traditionally seen teaching/learning, and faculty/students as separated entities that are not relational.For example, information flows from teacher to student, as faculty are the sole experts.Additionally, faculty and students should keep their distance; any outreach to students is seen as coddling.In this agreement, Rendón (2009) says that students learn content from a distance and through the lens of only one discipline.This is opposed to the new agreement proposed by Rendón (2009), the agreement to embrace connectedness, collaboration, and transdisciplinarity.In this agreement, teaching and learning is seen as democratic, participatory, and relational (Rendón, 2009).The lines between learning and teaching are muddled so that teachers and students are both producers and beneficiaries of knowledge (Rendón, 2009).This is not unlike the participation/positioning dimension of Gutiérrez (2018).In this dimension, Gutiérrez (2018) proposes recognizing artificial hierarchies in the classroom and shifting authority and the positions of power from the teacher (or artifacts like a textbook) to the students.
Within this same agreement, Rendón (2009) puts forth validation theory in which faculty and students are encouraged to know each other, and faculty are encouraged to offer students assistance, even in non-content areas of social and emotional health.Similarly, Gutiérrez (2018) calls for bringing in a human element into doing mathematics by 'recognizing mathematics as a living practise' (Gutiérrez, 2018, p. 5).That is, that mathematics is impacted by social and political forces, as well as cultural and historical influences.With the new agreement and this dimension, students learn knowledge that cuts across disciplinary boundaries.
The agreement of competition pits and sorts students using measures that may be biased.Indeed, Kidder and Rosner (2020) analyzed available SAT data provided by the ETS (October 1998 sampling) and found that 'the test development process unintentionally, but consistently and predictably, increases the disparate impact of the SAT' (Kidder & Rosner, 2020, p. 49).Kidder and Rosner (2020) showed that in the experimental section 1 of the SAT, questions where Black and Chicanx students outperformed White students were often rejected or omitted from future tests, whereas questions White students performed better on were retained.In Rendón's new agreement, the agreement to engage diverse learning strategies (i.e. competitive and collaborative; individual and community-based), the emphasis is on fostering collaborative learning environments and inclusive practises, where students and faculty work together on community-and problem-based activities, building relationships and engaging in meaningful interactions.
For Rendón ( 2009), there is little room for errors and imperfection in the traditional classroom and curriculum.Instead, Rendón ( 2009) posits that education is about acquiring facts rather than the process of learning.Typically students are trained to act as people who know information rather than seekers of knowledge.This falls into the the agreement of perfection.On the contrary, the agreement to be open and flexible about being grounded in knowing and unknowing provides space for students to explore and 'stay comfortably afraid' (Obama, 2022, 2:17) of the unknown.This is a sort of precursor to the dimension of creation (Gutiérrez, 2018).Gutiérrez (2018) calls for giving students experiences beyond reproducing mathematics (such as those required of a textbook) and instead encourages students to invent new procedures or new mathematics.
The fifth agreement is the agreement of monoculturalism.The curriculum, texts, and even the physical space of the classroom often validate Western structures of knowledge while subjugating knowledge of indigenous people and people of color (Rendón, 2009).Rendón puts forth the agreement of multiculturalism and respect for diverse cultures.We should not only expand the theoretical frameworks we use regarding knowledge construction and production but also celebrate diverse ways of knowing.This particular agreement is connected to two rehumanizing mathematics dimensions: cultures/histories and windows and mirrors (Gutiérrez, 2018).In the former, Gutiérrez (2018) calls for learning mathematics from cultural and historical perspectives so that students can have an opportunity to reconnect with their roots.The latter calls for opportunities for other students to see themselves (mirrors) in the curriculum as well as opportunities for others to see them (windows) in the curriculum.In this way, students are 'seeking to understand themselves and others in a relationship' (Gutiérrez, 2018, p. 5).Note that these two dimensions share a thread with the second agreement in which students can see learning as relational.
The agreement to privilege outer work is one with which many academics may be familiar.It is the struggle to balance work with personal goals, often prioritizing our outer lives and neglecting our inner lives by not reflecting or resting enough.The agreement to balance our personal and professional lives with work, rest, and replenishment helps us to reframe our overall health.
Lastly, the agreement to avoid self-examination is an overspending of time and energy looking outward to the institution without spending enough time examining our internal views and assumptions.Rendón proposes that those in academia (faculty, staff, and administrators) examine how our own positionalities impact the student experience.We should shift our focus on what we can change in terms of the curriculum, how we engage with students, as well as how we share power and the physical classroom.The revised agreement is the agreement to take time for self-reflexivity.
Underlying all these agreements is a call for broadening mathematics (Gutiérrez, 2018).This dimension summons us to shift from seeing mathematics as merely a logical discipline to one that is rich with culture, history, and creativity, where students can 'draw upon other parts of themselves (e.g.voice, vision, touch, intuition)' (Gutiérrez, 2018, p. 5).It is the hope that by engaging in these agreements students will take ownership-the last dimension of rehumanizing mathematics according to Gutiérrez (2018).That is, students will do the mathematics for themselves rather than fulfill a school assignment.

Teaching bundle summary
The five-component teaching bundle introduced in this paper is suitable for an undergraduate course on ODEs.This bundle can be used after the characteristic polynomial method has been covered, usually at the beginning of discussing systems of ODEs.The overall mathematical goal of the bundle is to engage students in the modeling and model revision process and solve a system of ODEs with different types of techniques, such as converting it back to a higher-order linear ODE through the process of modeling, using the matrix form, or using the Laplace transform method.The overall cultural goal of the bundle is to include students' perspectives in the classroom for a more personal learning experience.This section will provide a summary of the bundle and then the next section details how it connects to each revised agreement/dimension.
The general structure has 5 parts: Pre-Activity Assignment, Activity 1, Activity 2, Activity 3, culminating with the Final Project.The Pre-Activity Assignment asks students to think about relationships between two subjects (could be between two people, a beloved pet and a person, two companies, or two countries, for example.)They are asked to draw on their own personal experiences and/or cultural experiences.This is the first step towards inviting students to participate in the process of developing a mathematical model.Students should get a sense that the project is personalised rather than objective as in a textbook assignment.
The Pre-Activity Assignment helps to set up Activity 1 and may be extended to the final project.Activity 1 introduces students to a system of two first-order linear ODEs that model the feelings of one subject solely based on their interaction with a second subject.They will convert the system of ODEs into a second-order linear ODE and solve it by the characteristic polynomial method.Finally, they will discuss the limitations of the model, which helps students start to think about models not as final outcomes but as works in progress.
With the limitations identified in Activity 1, Activity 2 begins the revision process of the mathematical modeling.The students will expand the model to include one's feelings about themselves (or their personalities) and their interactions with the other subject.In other words, students rebuild the model from Activity 1 to now include a change of feelings about one subject based on their reaction to the other subject, as well as their reaction to themselves, then solve it with the help of matrices.Again, we repeat the process by asking students to re-examine the limitations of the new model.
In Activity 3, students will continue to build a more sophisticated model to get closer to real-life scenarios.In this activity, we add terms to model the boredom, for the lack of a better word, between the two individuals.Students will practise using the Laplace transform method to solve ODEs during this activity.As always, there is still room to improve the model.
Finally, the students apply what they have learned from Activities 1-3 about modeling relationships in one of many different scenarios they propose, such as (1) a fictitious romantic relationship where one person is secure while the other centers oneself and retreats from others; (2) a relationship depicted in the student's culture; (3) a personal familial or interpersonal relationship; and (4) the competitive and collaborative relationships between two companies or two countries.Students are given various choices, from the type of scenario they wish to model and explore to the format of their final project.For details on possible scenarios and any of the five parts of this teaching bundle, see the Supplemental Material.
An overarching goal of using this five-part bundle is to actively engage students in the revision process and emphasise the evolution of model-building.This process of iterative model revision holds significance in the classroom as it mirrors the real-life practise of mathematical modeling, going beyond the presentation of a few popular techniques and their 'final outcome.'Instead of simply training students to mimic known problem-solving approaches, this approach prepares students to develop new models and enhance existing ones, nurturing their ability to think creatively and critically.

Application of agreements and dimensions to the design of course materials
We aim to construct a culturally sensitive and inclusive classroom that welcomes students from all backgrounds while making the workload manageable for faculty members who are already overwhelmed with everyday obligations.In this section, we demonstrate how this project addresses each of the seven revised agreements and as such, the eight dimensions of rehumanizing mathematics.

Agreement to work with diverse ways of knowing in the classroom
Supplementing in-class, closed book, and proctored assessments with in-class modeling activities and projects, such as those given in the Supplemental Material, allows students to use diverse ways to demonstrate their understanding.For example, students may draw on other parts of themselves and present their final findings in various formats, such as poster presentation, oral presentation, song, dance, video, Instagram post, etc.This approach enables students to demonstrate a range of intelligences and their creativity.Indeed, in a study about engaging Calculus 1 undergraduates in tasks designed to foster creativity, students not only reported feeling more creative but also expressed greater enjoyment of mathematics and increased confidence in the subject when instructors 'prompt and encourage different approaches or divergent thinking' (Tang et al., 2022, p. 635).Conjuring feelings of joy and encouraging students to be creative while doing mathematics are critical aspects of rehumanizing mathematics Gutiérrez (2018).

Agreement to embrace connectedness, collaboration, and transdisciplinarity & dimension of participation/positioning & dimension of living practise
Since students are asked to reflect on their own experiences and cultures, the project in this paper gives the entire classroom community, including the instructor, opportunities to both produce and benefit from knowledge.
There is latitude for collaboration during the in-class activities and the final group project.For example, working on a group project together develops students' communication, problem-solving, and conflict-resolution skills.Students also gain exposure to diverse perspectives.
The final project also has the potential for transdisciplinarity.For example, students from economics may use a relationship model to help study the collaboration and competition among companies; students from physics may use a relationship model to study the exchange of heat based on positive or negative feedback; and students who love reading stories may study whether there is a happily ever after in a fairy tale of their choosing.

Agreement to engage diverse learning strategies (i.e. competitive and collaborative; individual and community-based)
We intentionally provide opportunities for collaboration in the course materials, allowing students to build relationships through community-based or problem-based activities.For example, we design open-ended discussion questions with no right or wrong answers, encouraging students to engage in group discussions and brainstorm potential flaws in existing models while exploring methods for improvement.This also ensures that students with varying mathematical backgrounds can contribute to the discussion.Each activity set includes multiple parameter sets, allowing students to pair up and solve a subset of cases before comparing their results and discussing any discrepancies with their peers.We also guide students to attempt to solve the differential equation systems using two or more analytical methods, enabling them to naturally divide and conquer the task by focusing on one method at a time, and then comparing final solutions and the struggles encountered along the way.

Agreement to be open and flexible about being grounded in knowing and unknowing & dimension of creation
The project presented in this paper is particularly strong at addressing this agreement and dimension.The emphasis is on recognizing the limitations of a model and then revising it without a clear understanding of whether the model captures the scenario better or worse.It is only after taking a risk to adjust the model and testing that students will have an idea of its validation and accuracy.
Students come to understand the fact that no model is ever completely accurate.That is, '[a]ll models are wrong, but some are useful' (Box & Draper, 1987, p. 424).Rather than providing an exact answer, modeling requires the recognition of a pattern or structure which can be applied to real-world situations.By embracing the idea that there is no one 'correct' answer, students can approach future modeling problems with greater creativity and confidence.In the same Calculus 1 study mentioned above, students also reported that when instructors 'de-emphasise correctness in class' (Tang et al., 2022, p. 635), they felt creative, enjoyed doing mathematics, and felt confident in mathematics.Students can create models, ranging from simple to complex, without fear of producing 'wrong' answers rather than regurgitating processes objectively declared by textbooks or instructors.This approach to study modeling provides students with the freedom to explore the full potential of their mathematical abilities, encouraging them to aim as high as they can imagine.

Agreement of multiculturalism and respect for diverse cultures & dimension of cultures/histories & dimension of windows and mirrors
The power of this project lies within this agreement and these dimensions.This project gives space for students from diverse backgrounds to freely share their unique cultural celebrations in the classroom, adding more vibrancy and richness to our learning environment.For example, students may bring up the story about La Llorona, a woman indigenous to Central or South America depending on the culture, mourning the death of her children.Students can have the opportunity to share their versions with each other.In this way, students can see themselves in the curriculum while also learning about global identities.
In another example, White Day and Black Day are two unique holidays celebrated in some other East Asian countries and regions.White Day, observed on March 14 th , is a response to Valentine's Day, when people give reciprocal gifts to those who gave them gifts a month earlier (Wikipedia, 2023c).On the other hand, Black Day, celebrated on April 14 th , is a day for people who did not receive gifts on the previous two days and eat Jajangmyeon (black bean noodles) as a way to commiserate and celebrate their single status (Wikipedia, 2023a).Gift exchanges happen between romantic partners, friends, and coworkers.Both days are an important part of some modern Eastern Asian culture.White Day coincides with the International Marriage Day in Japan, celebrating the first official permission of marriage between a Japanese national and a non-Japanese national in 1873 (Wikipedia, 2023b).It would be an interesting practise to teach modeling about love relationships first on Valentine's Day and revisit the ideas, especially the revision of the model, on White Day and again on Black Day.
We would like to caution that deliberate facilitation of discussions is paramount to creating a meaningful environment that uplifts multiculturalism.Students arrive with various levels of cultural acceptance so the instructor must confront conflict in the classroom so as not to further marginalise students.

Agreement to balance our personal and professional lives with work, rest, and replenishment
The materials we design invite students to bring their own experiences into the classroom, so we can tap into a wider range of perspectives and enrich our teaching.Moreover, we avoid an endless pursuit of innovative models by revising one single model several times in one semester.As we strive to meet the demands of our work, this approach allows us to save time in designing course materials, which in turn affords us the opportunity to achieve a healthier balance between our personal and professional lives.

Agreement to take time for self-reflexivity
On a personal level, students are witnesses to a changed classroom in which they are invited to bridge non-mathematical aspects of themselves with mathematical content.In learning about others' cultures, they can make connections to their own.We hope instructors can use this bundle as an opportunity to have discussions about history as well as culture so that students can critically examine ways in which they perpetuate or eliminate stereotypes about others.
On a content level, mathematical modeling is an iterative and cyclical process that involves several stages, including problem understanding, abstract representation, assumption setting, variable and parameter selection, mathematical formulation, problem-solving, analysis, validation with real-life data, model revision, and interpretation using plain language.A modeler's general approach is to report only the final explicit models, while the process of revision, even critical, is often neglected.Hence, 'the inner world of tacit knowledge and its impacts on mathematical modeling remain largely inaccessible to such approaches' (Brady et al., 2022).The discussion of revisions can provide valuable insights into the modeling process.Understanding the iterative nature of modeling and the importance of the revision process can help students develop a deeper understanding of modeling.Therefore, it is most beneficial for faculty members and students to teach and learn modeling by focusing on starting with a simple model and gradually deepening the discussion.The bundle we designed gradually demonstrates the modeling improvement and revision procedure, which helps students engage in authentic mathematical practises.

Structure of the bundle of worksheets and projects
To illustrate our ideas, we have prepared a package of course materials consisting of five components, comprising a pre-activity assignment (short essay), three in-class activities, and a final group project.While we expect detailed solutions for calculation problems, we encourage providing feedback rather than assigning grades for essays and responses in activities that involve interpreting model results and explaining model limitations.We believe grading the final project is the best way to evaluate the overall learning outcomes.
We also suggest that the timing of the activities be properly planned.One may consider incorporating the dates of these activities into various cultural or national holidays.For example, the Pre-Activity Assignment may be given before Valentine's Day (Feb.14th), and students can work on the first activity around if not on Valentine's Day.The White Day (Mar.14th) and Black Day (Apr.14th) from East Asian countries, introduced earlier, are all exactly one month apart.Students can revisit relationship models by working on Activities 2 and 3 on those days.It is good to explain the idea needed for take-home group projects between the first and second worksheets, if not sooner, so students have plenty of time to form groups and would be more engaged in those activities in class.
This five-component bundle illustrates the evolution of modeling ideas as well as analytical techniques in solving differential equations.Through these activities, students will have the opportunity to compare the advantages and disadvantages of using different methods to solve a linear system of ODEs, including the characteristic polynomial method, matrix form, and Laplace transform method.Additionally, as we explained earlier, they will engage in the challenging process of revising models and gain insight into the tactical aspects of modeling.These activities and the final project also prepare students to interpret simulation and modeling results using plain language and apply mathematical modeling skills to real-world problems.

Conclusion
This five-component bundle consisting of a pre-class assignment, three in-class activities, and a group project was designed following the seven agreements and eight dimensions.These activities expose students to valuable experiences discussing model revisions while guiding them to consider the advantages and disadvantages of three methods commonly used to solve the same type of problem.In the end, students have the freedom to model something they are personally connected to and may even choose to improve it by considering real interactions between multiple subjects, which extends the model to a nonlinear ODE system.This may lead to more interesting dynamical behaviors or a discussion of local linearization, both of which prepare students to use differential equation modeling in the study of the real world.
We highly recommend instructors use the full bundle instead of only part of it since it is a gestalt.If it is not possible due to constraints of in-class contact time, we suggest using, at minimum, the Pre-Activity Assignment, one in-class activity, and the final group project.That way, students can experience the evolution of modeling while working collaboratively to see themselves in the curriculum.
In addition to the relationship model discussed earlier, a plethora of other topics can be explored using a similar framework.For instance, infectious disease models, starting with the SI model and progressing through the SIR and SEIR models, could be introduced in class.A series of discussions on this topic could offer an excellent opportunity for students to gain a deeper understanding of the modeling process, and real data could be used to enrich the study.To add a cultural component, teachers may guide students to model the spread of a zombie apocalypse or gossip (or even misinformation) using infectious disease models.These activities could be introduced during special occasions such as Halloween, April Fools' Day, or other similar days, such as Dracula Day in Romania, the Hungry Ghost Festival in China, or the Majkat Day in Denmark.