Mastery-Based Grading Through the Lens of Ignatian Pedagogy

Eric Bucher
Mentor: Bin Yu (Department of Political Science)

Acknowledgment. I would like to thank my mentor Dr. Bin Yu for her guidance, support, and insight
throughout this project.

1 What is Mastery-Based Grading?
Our goal as educators is simple. But often simple to state goals are the hardest to accomplish. We want our students to learn. It is what pushed us into the careers that we are in and it is what drives us to pursue self-improvement every semester.

Often I find in my classroom that something gets in the way of true teaching, and that is evaluation. How do we
know when our students have truly learned something? What does that look like? In this article I will speak from
the perspective of teaching in a mathematics course. Traditional evaluation involves exams, quizzes, homework,
and in general percentages. Each problem is assigned a point value and students receive points for correctly
answered questions. But what does a 83% mean? How much did my students really learn? Does this mean that a
student understands 83% of the concepts on the exam? Does it mean they are 83% likely to correctly answer a
calculus problem? And does a student with an 83% deserve a better grade than a student with an 82% on that exam?

It is questions like these that made me seek a better way to evaluate student learning. Mastery-based grading, or
sometimes Standards-based grading, is a system that only awards a student when they have fully mastered a
concept. Students are allowed multiple opportunities to give evidence of their mastery of core concepts throughout
the semester and as long as the concepts are eventually mastered. They receive no credit for topics that are not
eventually mastered. Typically there are standards presented for students to master and they are graded based on
how many of the standards the students are able to master by the end of the semester. The key component of the
methodology is to encourage learning from their mistakes, growth, and rewarding students for their knowledge
base at the end of the semester.

I chose to implement a mastery-based grading system in MATH 225 – Foundations of Mathematics course during
the Spring 2020 semester. My goal was to utilize this course structure and the principals of Ignatian pedagogy to
develop a course framework that would have students naturally engage in discernment and solidarity and kinship
on their journey through the course.

2 How does Mastery-Based Grading relate to Ignatian Pedagogy?
The five primary Ignatian values are reflection, discernment, solidarity and kinship, service rooted in
justice and love, and cura personalis. When I first came across the notion of Mastery-Based grading it struck me
as being deeply connected to the Ignatian values of discernment and solidarity and kinship. Essentially the practices that make Mastery-Based Grading effective for education, already tap into these Ignatian values and this seems like an opportunity to help develop discernment and solidarity and kinship in our students both with and without addressing them directly. If we can help our students live these Ignatian values in our classroom, this can help them experience them more readily outside of it.

2.1 Discernment

“Discernment invites us to be open to God’s spirit as we consider our feelings and rational thought in order to
make decisions and take action that will contribute good to our lives and the world around us” [Xavier.edu/missionidentity].

There is an opportunity to promote discernment in a Mastery-Based classroom by having students spend time
reevaluating their own learning to see where they were lacking mastery. One of the benefits of Mastery-Based
grading is that students can always receive perfect grades if they are willing to learn from their mistakes and
reengage with the material that they initially did not fully grasp. This process involves students having a stronger
understanding of what it is they know. This encourages students to not be satisfied with what they currently know
and always be striving for knowing more. And this starts with self-evaluation from the students, which is at the
heart of discernment. By giving the students numerous opportunities to demonstrate mastery of a given topic and
providing no penalty for mistakes made along the journey to mastery, the process encourages and rewards students for engaging in thoughtful and meaningful discernment. The goal is to show them that discernment truly provides an opportunity for them to grow and learn, leading to a greater understanding of the Ignatian value.

2.2 Solidarity and Kinship
“Solidarity and kinship invites us to walk alongside and learn from our companions, both local and afar, as we
journey through life” [Xavier.edu/mission-identity].

Mastery-Based classrooms are designed to present students with a deeper understanding of their own
strengths and limitations. Equipped with this deeper self-understanding it seemed opportune to give the students
opportunities to bring their understanding and lack of understanding to their peers. When designing the course I
had the students work in groups as a form of reflection over which standards they were able to achieve and which
ones they weren’t. They were encouraged to help those who haven’t mastered standards they fully grasped. One
way of encouraging this was to reward students for teaching each other. One way to achieve mastery of a standard
was to help teach other students various topics. The goal of adding this feedback process to the course was to help
students engage with solidarity and kinship by giving tangible rewards (in this case grades) for learning from and
teaching our companions.

3 Course Design

3.1 The standards structure.
The course was designed with 19 standards, which the students set out to master. Their grade was computed as a
percentage of these 19 standards that they completed. An opportunity to master each standard would be offered at
least 3 times for each standard in what I called reviews. Students are allowed to attempt any standards that are
offered that they have not yet mastered each opportunity. In addition to the written opportunities, students are able to schedule a “discussion review” which allows them to complete two standards by talking me through standards they have not yet mastered. Finally there is opportunity to demonstrate mastery in non-conventional ways. For instance, students that are found helping other students in class with a difficult topic that they themselves have not yet mastered can receive mastery. After every attempt students receive either an M for mastery, a J for journey person, or an A for apprentice, which indicates how they did. An M means that a student has complete this standard and doesn’t need to show any additional proficiency. A J means that the student is showing great progress but has not demonstrated mastery yet. An A means that the student has work to do on the journey to mastery. Any grade other than an M will not give any credit, and in order to achieve an M you must have a clear and well communicated understanding of the nuances of the topic. The students fill out a chart throughout the semester to track their progress in each of the standards. Below is an example of what the charts that are presented in each syllabus might look like throughout the semester, keeping in mind that a student who achieves an M doesn’t need to demonstrate future work on the standard:

The goal of the standards structure is to help students take deep ownership over their understanding of the material.

A student that engages deeply with the material and the process and truly learns something will receive the same
results as a student who found the concept easy to grasp the first time.

3.1 The feedback loop.
In section 2 we discussed how Mastery-Based Grading intrinsically promotes Ignatian Values, but I wanted to have
a more deliberate implementation of Ignatian pedagogy in this course as well. This is in the feedback loop that was
built around the Reviews in the course. After taking the review, the students were broken into groups where I
paired students with a variety of success in each standard. For instance if Review 1 covered standards 1-6, I would
make sure a student with an M on standard 1 was in each group, but also that a student with an A on standard 1 was also in that group. This was not always possible, but as the semester went on we had enough variety in grades that this was easily doable. The students were asked to engage in the act of discernment as homework. They wrote a reflection on which standards they found difficult and which they thought they were strong in. On this reflection
homework they also wanted to outline one standard, which they felt they could help a student master, and one
standard they wanted someone else to help them with. The goal was to create an open, accepting environment in
which the students were prepared to both learn from their peers and to teach their peers. This process was called the Ignatian Feedback Loop in our class. The students would arrive in class, get with their groups, and then help each other master various skills so that on the next review they would have a strong chance of mastery. If a student did a particularly good job of teaching and supporting their fellow student(s) then it was possible that I would award mastery during these feedback days.

Students were asked during exit interviews to evaluate how the different Ignatian values helped their understanding of the material and how the feedback loop in general helped or hindered their conceptual understanding. Below is a diagram that I think helps understand how this feedback loop is designed to involve students in Discernment and Solidarity and Kinship:

circular triangle chart with review pointing to discernment homework pointing to classroom activity, pointing back to review

4 Student feedback.
At the conclusion of the semester I conducted oral exit interviews with each student via ZOOM. During
the interviews, the students were asked 3 questions regarding this project and I will provide some select
comments made by the students, a brief summary of their overall response, and my primary take-aways
from their responses. I intentionally designed the questions to address the process both in reference to the
Ignatian Values and without.

4.1 How did the standards method help or hinder your understanding of the material?
• “The standards were honestly the best thing I have seen at Xavier this semester.”
• “I really liked the standards approach, it gave me opportunities to learn from what I missed.”
• “I loved the standards. I think I always felt like I could get my grade back on track… I didn’t, but that’s
more on me.”
• “At first it really scared me. After the first standard I only had like 1 out of 7 or something like that, but like
you said in the beginning you eventually get them in the end.”

Generally speaking the students really seemed to enjoy the standards based model of grading. They seemed to feel
as if it allowed them to learn from their mistakes and they were not being punished for learning. They also seemed
to like how it instilled in them a sense of control over their final grade. The forgiving nature seemed to be
particularly appreciated during the remote learning format of the class.

My overall take away is that the Mastery-Based component of this project was very successful. The students felt
empowered, took pride in their learning, and really had a deep understanding of what they knew and what they
didn’t know by the end of the semester. A longevity study would be the only way to understand if they truly
learned more from this process than had I done a traditional grading model, but from the initial feedback and the
quality of the work presented I believe the students had a better grasp of the course material than in previous
semesters.

4.2 How did the group work after standards help or hinder your understanding of the
material?
• “ The feedback loop was helpful, it helps to see that other students are stuck too.”
• “ I liked talking to the other students. I thought I knew things but talking to them made me realize that I
didn’t know them as well.”
• “I thought teaching others things really helped.”

In general, the students seemed to gain from teaching each other. Most students found it beneficial to see that other students were in similar positions as them. Also they seemed to learn a lot from playing the role as teacher
Overall I think the feedback loop helped students engage with each other instead of just the material. Very often in
mathematics it can feel like an isolated experience so I believe this helped the students discuss and communicate
mathematics in a way that deepened their understanding. I also believe it helped make students feel more open to
communicate openly throughout the semester about what they don’t understand, making student engagement very strong throughout the later parts of the semester.

4.2 How did this course impact your feelings towards the Ignatian values of discernment
and solidarity and kinship?
• “ I think the reflections we had to write helped me be successful. It was nice to see exactly where I was not
understanding things.”
• “ I felt a responsibility to help others in class, which I realized really helped me. I think I might try and do
that in other classes too.”
• “ Maybe discernment. I think that one was the one that I had to think about the most, because I was always
thinking about how to fix my reviews. One thing that was nice is since I could always get full points for a
standard it really forced me to keep thinking about them until I got them.”
The students in general didn’t seem to have as deep a connection to the Ignatian values as I would have hoped. I
think something very subtle happened in this course, in that the students were engaging with discernment and
solidarity and kinship without realizing that they were. Because the course and activities were designed to have
these values as the underlying structure a lot of students seemed to really appreciate it without having a name for it.

I was surprised how many students gave answers like “I really liked helping other students, it gave me a sense of
pride and joy” while still saying, “I don’t really think I engaged with kinship”.

Overall my one take away from this is that it would be beneficial to have more activities that take them through the
two Ignatian values prior to the first review. Possibly we could have a writing exercise or simply discuss the values
for a day leading up to this. I think this might help tie the course to the larger Ignatian values of the university.

Back to Top

Statistical Inference

Max Buot, Ph.D.Max Buot, Ph.D.
Mentor: Nancy Bertaux, Ph.D. (Economics)

I attempted to incorporate the Ignatian mission into the Statistical Inference (MATH 312) class I taught in the Spring 2008 semester. This course is typically offered every two years, and is aimed at advanced majors in mathematics, especially those undergraduates who are interested in pursuing advanced degrees in statistics. Although the list of topics in MATH 312 is typical for such a course, my effort to include Jesuit values made it a unique pedagogical experience.

To be honest, I found the task of explicitly demonstrating the Ignatian mission in an upper-level statistics course to be challenging. In my view, fulfilling this responsibility would require creative and careful planning to ensure that the course remains true to its objectives: introduce abstract statistical theory, prove the main results rigorously, and apply the results to solve a wide array of data analysis problems. As a significant number of students in the class may continue their mathematical education in graduate school, a solid foundation on the course topics is of utmost importance. Assignments dealing with Jesuit principles may not appear relevant or appeal to a student in a math class.

In order to create an avenue through which Ignatian ideals could be integrated in MATH 312, I interpreted the Ignatian phrase Finding God in All Things as a general invitation to "see the big picture". In particular, students are encouraged to appreciate the wonder of God in their daily lives, search for God's presence in past events, and contemplate the future path that God's will has in store for them. Specific to the Statistical Inference course, this means providing a sense of perspective that elevates their understanding of the ideas presented in the class. To this end, I created supplementary assignments involving journal article readings. The purpose for each article reading can be classified into one of three categories which correspond to my basic interpretation of Finding God in All Things. For classification into the first category, the article should demonstrate the wide applicability of MATH 312 topics in the present-day. For the second category, the article should place the statistical methods discussed in the course in historical context: Why was the statistical inference method needed? What problems motivated the development of the theory? And for the third category, the article should illustrate some questions and problems which are at the forefront of statistics research today.

The following is an example of how an article reading was incorporated into MATH 312. The theory of maximum likelihood estimation is based on the assumption that the parameter of a probability distribution is a fixed constant, whose value is unknown, but is to be estimated using a random sample of data. Within the statistics community, methods developed in this context are called frequentist methodology. However, if the parameter itself is modeled with a probability distribution, then the statistician is said to be employing Bayesian inferential methods. To better understand the differences between the frequentist and Bayesian paradigms, I assigned article readings from various sources, such as statistics journals and mainstream periodicals such as The Economist and Scientific American. This activity extended the students' perspective on the historical implications of these two statistical philosophies, since scholarly arguments between frequentists and Bayesians played a key role in the development of statistics as a discipline in and of itself in the 20th century.

To help encourage dialogue among the students, I set aside time during the semester for discussion of the articles. I also required that each student write down a brief summary of their readings. Although there were only a few occasions in which article readings were assigned, the students found this activity to be a valuable component to the course and a welcomed excursion from the usual lecture dynamic.

The phrase Finding God in All Things captures the spirit of Ignatian values. With thoughtful consideration, each one of us can discover evidence of God's wonder, inspiration, and love every day. As a second-year faculty member at Xavier, my involvement in the Ignatian Mentoring Program has given me an opportunity to reflect on how God is revealed in my own life, on and off campus. It has truly been a worthwhile endeavor. Eight years prior to my arrival at Xavier, I was a busy graduate student and post-doc who was consumed with scholarly activities in the field of statistics. As a participant in the Ignatian Mentoring Program, my perspective on my academic career path has been positively broadened.

Although I have described a special approach I implemented in MATH 312, I have also developed a clearer sense of my role as a faculty member at Xavier University. As a professor, it's important for me to view Jesus Christ as the ideal teacher. One specific event, the washing of the disciples' feet on the night of the Last Supper, is a valuable reminder for those who are called to lead: The act of leading requires the willingness to be of service. A teacher, who in fact is a "leader of students", needs to anticipate situations in which students seek guidance and direction. This means being organized and prepared for each lecture, and being able to provide examples to elucidate abstract mathematical concepts to the student. As Jesus often made use of parables to illustrate religious lessons, moral truths, and apparent paradoxes, the use of examples, demonstrations, and discussion can serve a similar purpose in my own classroom. With respect to scholarship, an active research program is an exemplary way for a faculty member to contribute to the advancement of knowledge in his academic specialty. Engagement in research is not only an exercise of discovery in which the scholar strives to uncover new and exciting ideas, but it also supplies the scholar a deeper perspective on the discipline he teaches, which in turn is reflected in a student's educational experience.

Back to Top

Mathematics and Democracy - Student Reflections on the Case for Quantitative Literacy


Minerva Catral, Ph.D
Mentor: Thomas Wagner, Ph.D. (Communications)


Acknowledgement. I am grateful to David Burns for his encouragement and guidance, and much thanks and appreciation are due to my mentor, Thomas Wagner, from whom I have learned so much this past year.

1. Hopes and Goals
One of the wonderful pluses in being at Xavier is that there is no lack of opportunity for growth--expansion outside one's discipline and comfort zone is a process that continues on. I came into the Ignatian Mentoring Program (IMP) bringing with me my own (limited) expertise in my chosen field of study, with the goal of getting a broader general view from interactions and discussions with colleagues who bring with them their own expertise in their respective disciplines. The hope then is that this widened perspective might translate to a more informed instructorship and a more effective teaching and learning environment in the classroom.

I had always been curious about what output I might have at the end of participation in one of the mission-centered programs of the university. My goal in this IMP participation was to achieve greater awareness of the Jesuit mission of the university, and indeed to find out how I might contribute to this mission as I run my mathematics classes. I have learned a lot from the readings and other resources on Jesuit education and Ignatian pedagogy, but the most learning came from the sharing and discussion with others of their own experiences in incorporating the mission in their classes.

My meetings and discussions with my mentor Thomas Wagner have been invaluable. As a mathematician my classroom goals have mostly focused on "The Math". However Thomas has given me a lot of good insight on how I might be able to reach students who may not be as excited as I am about the exquisiteness(!) of mathematics. For example, during his visit in one of my MATH 150 classes he noted how my illustration of a quadratic function as a model of the number of vehicles parked in a parking lot as a function of the time of day might be improved to something more tangible (or maybe more realistic?).

Although this article focuses on describing the outcomes of a paper requirement for my MATH 150
Honors class in Fall 2012, the broader outcome of my own IMP mentoring this past year goes far and beyond this particular class project.


2. Paper Requirement for MATH 150
"Jesuit Education and Ignatian Pedagogy values the five educational principles comprising the Ignatian pedagogical paradigm: context [understanding student life and culture], experience [providing intellectual and affective learning opportunities], reflection of meaning for self and others, action [the external expression of learned content] and evaluation of student growth." http://www.xavier.edu/jesuitresource/

The goal is to embody these principles in every course that I teach. In particular, I put focus on my MATH 150 course, and as a class project I had my students perform a close look and reflection on a topic that is intimately connected to mathematics--the problem of quantitative illiteracy in today's society.

MATH 150 is an informal introduction to Calculus, the mathematics used to describe the process of change. This course is designed for a general audience and fulfills a core mathematics requirement. In this course, a three-fold development of calculus (numerical, graphical and algebraic) replaces the traditional (purely algebraic) approach. Strong emphasis is placed on clearly communicating questions and interpretations of the results obtained, and to provide logical and convincing arguments for the results.

The students in my MATH 150 Honors class in Fall 2012 were given required readings on a collection of essays from Mathematics and Democracy: The Case for Quantitative Literacy. This volume, which came out more than a decade ago, consisted of writings and reflections by a variety of professionals both inside and outside of mathematics. These essays address the issue of the disconnect between the mathematics taught in schools and the mathematics that is needed for a citizen to be a functioning member of society. The students were asked to write a paper in response to a list of questions about quantitative literacy. The result was some very thoughtful and enlightening student essays on the subject. Excerpts from these essays are given in Section 3.

2.1 Paper Guidelines
Math 150-07H Fall 2012
Mathematics and Democracy: The Case for Quantitative Literacy

Source: Steen, Lynn Arthur, executive editor, 2001. Mathematics and Democracy: The Case for
Quantitative Literacy, Princeton, NJ: National Council on Education and the Disciplines. Available at
www.maa.org/ql/mathanddemocracy.html.

Read at least the first five essays in this collection, then formulate your paper as a response to the following questions.

1. What does it mean to be quantitatively literate?

2. Why is it important for society to strive towards a quantitatively literate population?

3. According to the authors, mastery of school mathematics concepts does not necessarily imply
quantitative literacy. Give some of the evidences that were cited to support this claim.

4. To conclude your paper, give some of your own reflections on quantitative literacy and give your own opinions about the arguments presented in these writings.


3 Student Reflections
The following are excerpts from the papers that the students have written.

3.1 On the meaning of quantitative literacy
"It is clear that "quantitative literacy" means different things to different people. However, a working definition of any type of literacy must evolve with the times. For example, literacy as it was defined in the 1800s would be very different from literacy as it is defined today because of new developments in almost every area of life. If changes in society are not accounted for, a standard for literacy would soon be out of date and inapplicable.

A better term for quantitative literacy may be "numeracy," which implies a comfort and confidence with communicating using numbers. Numeracy is described as a "habit of mind," or a way of looking at situations that come up in everyday life. Numeracy is different from mathematics in that it cannot be separated from context nor taught apart from its real-world applications. In fact, the word numeracy itself refers to an understanding of the most commonly encountered real-world applications of quantitative reasoning. Quantitative literacy, or numeracy, is "rooted in the connection between mathematics and reason." "

3.2 On the importance of a quantitatively literate population
"In a democracy that stresses independence and free thought, a quantitatively literate population is essential. Today, data is easier to gather and more accessible than ever, and our society is flooded with quantitative information. From advertisements to politics to news stories, we are constantly bombarded with figures. In this type of environment, a quantitatively illiterate person is at a decided disadvantage.
For example, an American citizen could not make informed decisions on how to vote on tax policies, entitlement programs, or health care if he or she cannot interpret the graphs and statistics provided in support of and in opposition to these measures. Also, jobs are becoming increasingly competitive and employers are looking to hire candidates with a solid background in data analysis and interpretation. In this type of job market, the odds are clearly against the quantitatively illiterate person. Furthermore, quantitatively illiterate persons in America experience a profound lack of social power because they lack the skills to "think for themselves," "ask intelligent questions," and "confront authority confidently." In a democracy such as our own, it is important that no one group becomes marginalized, and any group that lacks the aforementioned quantitative skills would face a much greater risk of marginalization."

"Leaders--government officials, scientific experts, health agencies, and media organizations--all use statistics to educate, persuade, and control the public. Without quantitative literacy and an understanding of how to analyze numbers and their context, people are left without the confidence or the ability to question or confront these authority figures... If we cannot understand numbers, we are relinquishing our control to those who can and allowing them to exercise that control over us."

3.3 On school mathematics versus quantitative literacy
"Quantitative literacy is much different than statistics and mathematics. It is an empowering tool that moves away from the uncertainty and abstract nature of mathematics into a certainty rooted in real data and real situations. While mathematics teaches skills that are typically associated with success in the educational world, quantitative literacy involves tools that allow success in life."

"Mastery of school mathematics does not necessarily mean quantitative literacy. Quantitative literacy is an extension of subjects in understanding. Numerate students are created by using mathematics in everything they do. Quantitative students are created by learning that "content is inseparable from pedagogy and context is inseparable from content," and applying it to their everyday life and professions... Quantitative literacy is not isolated to one subject; instead, it is about applying all the information learned from different subjects and applying them to one's life. Quantitative literacy expands from numeracy and reason."

3.4 Further reflections
"Reading these chapters really opened my eyes to the issues that the United States faces. I agree with the authors of the chapters that quantitative literacy is important and needs to be taught in schools. It was not until this class that I actually began to catch a glimpse of how math is actually important to everyday life and how it can be applied to certain situations... I have never been interested in numbers and have never understood how the math I was doing in high school would ever be used when I was older... Throughout high school and middle school I always excelled in math. I memorized the formulas and did well on the test. After the test the information left my head as quickly as I had learned it."

"Although I still do not see any clear-cut solution to the problem of quantitative illiteracy, I certainly can identify with feeling quantitatively illiterate. Even before reading "Math and Democracy," I knew that I am probably innumerate in quite a few ways. I can watch a math problem be solved in front of me and then spit it out back on a test, but when it comes to looking at a situation and figuring out how to use math to solve the problem, I end up confused and frustrated every time."

"I was never shown the importance of quantitative literacy, and thus should not be expected to recognize that importance on my own. I assume that many other students, like myself, also lacked education that stressed this literacy, which explains why there is a lack of public concern for innumeracy, unlike the large concern for illiteracy; we are taught the importance of reading and writing at an early age."

"I do not believe that quantitative literacy should be improved in the classroom despite the fact that countless acts of legislation has put priority on science and mathematics. It is unfair to subject the majority of the populace to advanced courses for the needs of a few. Furthermore, as technology continues to advance and quantitative literacy is needed, technology can act as a simplifying aspect, instead of becoming quantitatively literate, technology can act as a translator in order to understand the world around us, instead of individuals being forced to become quantitatively literate."

"As a student who was taught the same "school mathematics" that Schoenfeld experienced during high school, I am new to the concept of quantitative literacy. I have always seen mathematics as a discipline that did not apply to me... required mathematics were frustrating and seemed unnecessary. However, after reading these articles I am actually embarrassed by my lack of interest or concern with skills that affect my everyday life."

"While arguments could arise exponentially for the various sides of quantitative literacy, and I do see benefits of advocating for greater stress of it study, I think there is a broader idea to approach society's intellect from. As far as I have observed in my life thus far, as a student and American citizen, I believe our society is becoming lazy. One characteristic of laziness is exhibited by people's lack of analytic skills and hands-on problem solving with data, as discussed with this case study. But another aspect is people's seemingly lack of interest in anything that has to do with their minds instead of having a computer do it for them... For these reasons, I believe a greater emphasis needs to be put on "thinking" in general. Yes, much of that thought is numeracy, but I see its growth in conjunction with philosophical and scientific thinking growth."

4 Conclusion
Overall I am happy with the papers that the students have produced. It is encouraging to see from the writings (and subsequent class discussions) how the students' viewpoints about their own mathematics education (and education in general) have been affected by the readings from Mathematics and Democracy. It is interesting how the subject of grades has come up in these papers, and that good grades, even excellent grades, do not ultimately translate to true learning. I found the student essays to be very enlightening and these will certainly guide me in the years to come in my capacity as mathematics educator. My hope is that these too have contributed to the students' growth and that these have truly reflected the five principles of Jesuit education and Ignatian pedagogy mentioned at the beginning of section 2: context, experience, reflection of meaning for self and others, action, and evaluation of student growth.

Back to top

Mathematics and Social Justice-Creating Awareness of Social Issues for Pre-service Teachers

Carla Gerberry, PhD (Mathematics and Computer Science)
Mentor: Laney Bender-Slack, EdD (Education)

Goals of the project:

For this project, I had two goals. The first was to infuse my class with the Ignatian pedagogy. The second was to use this pedagogy in order to raise awareness, in my pre-service teachers, about the issues facing those who live in economically poor areas.

The Ignatian heritage highlights reflection, discernment and solidarity and kinship. These are key elements of teaching for social justice. At Xavier, we strive to help students think about others in new ways. As future teachers, it is important for my students to have a heightened awareness of issues they will face besides just teaching their content. Their students will come to them with an outside life and that will influence who they are in class. For those who choose to teach in a high-needs school district, this will be even more important. Issues of homelessness and home life will be more prevalent. Being aware of this will enable my students to be more sensitive and understanding of their student's lives and allows them to stand in solidarity with them.

The project:

The problem

The course in which this project was implemented was a mathematics problem solving content course for middle school teachers. Generally, the course covered a wide range of topics such as logic, probability, geometry and algebra. This allowed a lot of flexibility when conceptualizing an activity that could raise awareness of social issues while still being mathematically rich. I wanted this project to focus on the interweaving of Ignatian pedagogy and the content of the course. More specifically, I wanted my students to critically examine other areas of the country besides where they were from.

The project was based on a problem from "Rethinking Mathematics: Teaching Social Justice by Numbers" edited by Eric Gutstein and Bob Peterson. Specifically, it was a modification of "The Geometry of Inequality" by Andrew Brantlinger. This problem requires students to examine certain aspects of the South Central Los Angeles area. For example, the number of movie theaters, community centers and liquor stores within a given area. The purpose of this project was to raise awareness about the disparities in different regions of the country.

The modification

For this project, students were divided in to groups and given different cities: South Central Los Angeles, Chicago, New York City, and Cincinnati. The point of this derivation from the original was to have a whole class discussion about the differences in large cities across the US. Students gave presentations at the end of the semester pertaining to their findings, thoughts, connections and mathematics.

Assignment given to students:

Students were given the following assignment for the project.

For your final presentation and write-up, please make sure to include the following:
1. A print off of the map/area you will be covering.
2. A summary of what the math is for the project (e.g. proportions and how you will use them).
3. A paragraph on the Ignatian themes we have covered in class and how these will tie in to your project.
4. A discussion of how these themes may or may not influence your teaching or speak to you as a teacher/human/member of society.

Bi-weekly discussions and presentations:

From the dialogues in class and student's final presentations, it was obvious that the students had learned from this assignment and had an increased awareness of the Ignatian pedagogy. In class discussions brought rich ideas such as the meaning of diversity and how this was different in different areas. Every other week, students were assigned an Ignatian theme of the week. The themes we discussed were discernment, solidarity, kinship service rooted in justice and love, and reflection. For each theme, students came to class and had to provide evidence of their reflection on the topic. Students discussed how they could scaffold ideas of solidarity in their classroom extensively.

Student Reflections:

"As influential and impactful as each Ignatian quality is, I feel as though the two that I would teach/incorporate the most are solidarity and kinship. Students, people for that matter, cannot go through life without having relationships and interaction with others. It is essential that children grow up learning what a solid relationship is and how to build one. Treating others as brothers and sisters and working together is something that every person needs to learn how to do and is something that they will have to do; I think that it is the most beneficial of the Ignatian qualities to teach students."

"A couple of the values that I feel are key to understand when teaching a classroom are solidarity and kinship. Being able to realize that there are other people in this world that are not as fortunate as you are is important, because it gives you the opportunity to see where others are coming from, and what perspective they hold. As a teacher, it means that you must be there for your students, and provide help for them even though it may not be directly related to academics. Sometimes a student just needs someone to understand them, listen to them, and be there. Moreover, if a teacher was to explain these values to their students, then the students can be more aware of how they treat each other in the classroom as well. This would create an atmosphere of love, and unity because the students would put themselves into shoes, and become more willing to stick together as a class."

"Making these kinds of comparisons and observations are great ways to make math relevant to the real world, and possible get down to the root issue of some of the social injustices of this world."

"I think that many of these lessons I have learned about Ignatian pedagogy throughout the semester will greatly influence my teaching. I hope to implement what I have learned during my Jesuit education here at Xavier into my classroom someday, especially in relation to the values of solidarity, kinship, and service. I would love for my students to receive an education that is grounded in values much more important than the Pythagorean theorem or the Gettysburg Address. Teaching students how to relate to others and make decisions with others in mind, especially people who are most vulnerable in our society, will be crucial in developing mindful citizens that are conscious of the need to be an active advocate for social justice."

Self-reflection:

Reading through my student's reflections has shown me how they benefited from our class discussions and project. Anecdotally, at the end of our last class session, my students told me that they were nervous about the project because it was so open-ended. In the end, they found it fascinating to learn about different communities, some of which they have been to and some not. Looking at census data alone shed light on topics and issues of fairness and diversity. We had a great discussion about how diversity can mean a lot of things and that when considering what a community looks like, you must consider multiple aspects such as access to positive and negative attributes of the given community.

For example, my students found it very interesting that the amount of liquor stores in Cincinnati nearly matched the number of grocery stores. This created a discussion about the consequences of having this happen in a population and how it may hinder real growth in a community.

For me, this project has increased my interest in the Ignatian pedagogy and I am anxious to incorporate more aspects into my future courses. This semester has given me ideas about what to do with my Number Sense and Geometry courses. In the future, I hope to have parallel assignments that will make my students think about how real life can and will influence their students participation and learning in school. With an Ignatian lens, they will have tools to help all of their students in a rich and meaningful way.

Back to top

Allocation of Charitable Resources: A Practical Illustration of
Quantitative Literacy's Role in Ignatian Discernment

David J. Gerberry, PhD
Mentor: Richard Mullins, PhD (Chemistry)

As the Jesuit values of Xavier University were a significant factor in my decision to join the
Xavier community, I was excited to participate in the Ignatian Mentoring Program (IMP) during
the 2014-2015 academic year. In my first years at Xavier, I felt that I had imparted the Ignatian
principles of discernment, compassion and care for the whole person primarily through one-on-one
interactions with students. While I plan to continue to do so, the IMP has shed light on ways
that I can incorporate Jesuit values directly into the classroom setting in a way that is genuinely
connected to the mathematical aspects of a course. I would like to acknowledge my mentor, Richard
Mullins, as I have benefited greatly from his guidance and experience.

1. Project Setting

For my project, I attempted to incorporate Ignatian ideals into my Spring 2015, MATH 120:
Elementary Functions course through an assignment and subsequent class discussion regarding
charitable giving. MATH 120 is a freshman-level, non-major mathematics course that revisits
the general idea of a function from high school and aims to give a deeper understanding of each
of the major mathematical function types (e.g. linear, exponential, logarithmic, logistic, power
and polynomials). Depending on mathematics placement scores, many Xavier students must take
MATH 120 before progressing to Calculus or Statistics courses required for their major. The project
was carried out in both of my MATH 120 sections, giving a total of 65 students.

2. Project Assignment

The project is best-described through the actual assignment description that was given to the
students (included here between horizontal lines).

MATH 120 D. Gerberry Charitable Giving Project
As students (and a professor) at Xavier University, the Ignatian Tradition is a central theme of
our environment. While Ignatian ideals are more prominent in some courses and less so in others,
it is my belief that all courses make a contribution to our spiritual development. Specifically,
mathematical ideas can even make important contributions. This project is my idea of how to
ensure that my students come out of MATH 120 knowing that this is so.

Charitable Giving

A practical, yet simple question that illustrates the relevance of the Gifts of Our Ignatian Heritage
is the following:

"Which charities will I decide to support?"
"How much money will I give to each of these charities?"

The Gift of Discernment invites us to be open to God's spirit as we consider our feelings and
rational thought in order to make decisions and take action that will contribute good to our lives
and the world around us.

Understand the complex motives that go into your charity decisions
- Am I just going with the crowd?
- Is this charitable contribution actually self-serving?
- Am I afraid to be embarrassed for not giving?
- What aspects of this decision are emotional and which are reasoned, scientific even?

Do my decisions actually serve the "Greater Good"?
- Most, if not all, charities are worthy causes, but how do we choose the greatest good?

The Gift of Reflection invites us to pause and consider the world around us and our place within
it. It calls us to infuse a culture of attention, reflection and reverence throughout the University.
Can we recognize new opportunities to do more good?
- Will we simply continue a pattern of charitable giving indefinitely?
- Will we embrace the causes that our parents have?
What is really important to you and what is important to the world as a whole?

The Gift of Service Rooted in Justice and Love invites us to invest our lives into the well-being
of our neighbors, particularly those who suffer injustice. This encourages and develops a culture of
mutually beneficial community engagement as an expression of faith that promotes justice.

Who do my charitable donations actually serve?

- People like me (similar views and circumstances)?
- People in my area, my state, my country, etc.?
- Do they serve the people that are in the greatest need?

Assignment

Consider the following two scenarios:
Scenario 1: You graduate from Xavier, get a good job and have a reasonable amount of money per year
that you can donate to charity. Let's assume you have $5,000 per year to donate.

Scenario 2: You graduate from Xavier, become the CEO of a company and have lots of money per year
to donate to charity. Let's assume you have $10,000,000 per year to donate.

For both of the scenarios above, decide how you would allocate your pool of money to charities.
Specifically, tell me the following:

1. Names of the charities you will donate to.
2. How much money you will give to each charity?
3. Explain why you have chosen this charity.

Some Guidance
You should include anything that you give money to:
Sum up donations you might give to church for the year
Donations to political campaigns
Donations to organizations: Humane Society, Red Cross, Cancer Society, etc.
Things you buy to donate (e.g. buy coats, blankets, food to give to homeless)

Don't include
Fundraisers where you are buying a product (e.g. Girl Scout cookies, Boy Scout popcorn)
Donations of old clothes or items to Goodwill or something similar

Directions
Write this up in Word or something
Turn in electronically through Canvas
Due on Monday, March 30
Email me if you have questions
You just need to answer the 3 questions on this page, not everything I mentioned along with
Ignatian Gifts. Those are simply there to get us thinking.

Grades
Obviously, I can't tell you that you picked bad charities and give you a bad grade. You are free to
pick whatever you want. Basically, I see grades only going three ways:
Bad grade: Obviously didn't take this project seriously and didn't put any time or thought
into it.
100%: Did take it seriously. Put in a decent amount of time and thought and it shows.
Above 100%: Took it very seriously. Obviously did some research, went above and beyond.

3. Charities chosen by students

In Tables 1-4, I present the results of the students' allocations of charitable resources. Tables 1 and
2 give the number of donations made to specific charities whereas Tables 3 and 4 give the total
dollar amount given to specific charities. My initial reactions to the students allocations were:

I was impressed with Xavier students' experience in volunteering. Many students had first-
hand experience working with the organizations that they would donate to.

All student donations were to very worthy causes. Given my own immaturity at their age, I
fully expected to see some answers that were nearly indefensible. To my surprise, not a single
donation went to the Organization for Depressed Cats or something similar.

Wounded Warrior Projects and other veteran-related causes were very popular with the students.
Specifically, this helped me realize how much less impact the recent wars in Afghanistan
and Iraq have had on me (someone in my mid-thirties) than on my students. This also led
to classroom discussion about the role of government in particular causes. More specifically,
how do we respond when we believe a cause should be funded by the government instead of
charitable organizations? Should we not donate to this cause? Should we donate more?

I was also surprised at how many students personally know someone that has benefited from
the Make-A-Wish Foundation.

One very important aspect of quantitative literacy is being able to recognize when someone
is using quantitative data in a deceptive way. When presenting the data in Tables 1-2 to the
class, I labeled them simply as \Most Important Charities." In doing so, I implicitly defined
\most important" as \receiving the greatest number of donations." We then moved on to consider
\receiving the largest amount of money" as the definition as "most important" and to discuss
other quantitative measures that we could use to measure importance. The takeaway message was
something we discuss frequently in MATH 120:

The definitions that you choose and the assumptions that you make are incredibly
important when using mathematics to model real-life issues,


and to be weary of quantitative data that is presented without precise definitions or descriptions
of the data.

Two tables of the number of donations made to particular charities

4 Categories of charities
We then moved on to examine how the resource allocations changed when the amount of charitable
money went from $5,000 to $10,000,000.

To do so, I categorized each charity according to its primary cause and primary geographic
scope. Of course, significant overlap exists for several charities so assumptions needed to be made
in order to categorize the data. As a class, we discussed that again, this is a point where someone
with ill-intent could be deceptive in their presentation of the data. The categories we used are shown
in Table 5. Charities were also grouped according to primary geographic scope: local, national or
international.

With the data categorized, we were able to examine general trends in the students' charitable
allocations. Specifically, we investigated if the allocations changed significantly when the amount of
money to donate increased from $5,000 to $10,000,000 (see Figure 1) and came up with explanations
why this should or shouldn't be the case.

We also used the data from our charity allocations to determine if differences existed between
the giving patterns of male and female students (see Figure 2). We noted that male students seemed
to donate a larger percentage of money to religious organization and the poor while females donated
more to military-related causes. I used this opportunity to briefly describe theeld of statistics
(which many students will go on to after MATH 120) and that statistics could be used to decide if
our data is sufficient to conclude that female and male students actually do have different patterns
of charitable giving.

Pie charts of resource allocations by charity type and geographic scope for the scenarios of $5000 and $10,000,000 available to donate

Tables showing number of donations made to particular charities

5. Connections to MATH 120 course material

At this point, I attempted to make connections between our charitable giving project and the
material we had been covering in MATH 120. Specifically, for each charity we realized that the
amount of \good" (e.g. meals delivered, lives saved, DALY's, etc.) that an organization can do will
depend on the amount of money available.

Therefore, we could imagine a function where
Input of function is amount of money available to a charity
Output of function is amount of \good" (e.g. meals delivered, lives saved, DALY's, wishes
granted, etc.) that can be done with that money.

Pie chart of the percentage of money donated by female students to each category of charity

Figure 2: Comparison between resource allocations of female and male students for the scenario of
$5,000 available to donate.

We realized that any such function should be increasing (i.e. more money should imply more
"good"). We considered the multiple function types that we have studied through the semester
(see Figure 3) and discussed which would make the most realistic mathematical model. We also
discussed how the eciency of the organization could be measured by the rate of change of the
function and that the best places on the graph to donate would be at the points where the slope is
largest (i.e. at inflection points).

Linear regression, another important course topic for MATH 120, was used to discuss whether
or not it is a possible for a worthwhile charity to actually be over-funded? Specifically, we examined
the data in Figure 4 that shows the levels of National Cancer Institute research funding for different
forms of cancer as a function of the percentage Disability-adjusted Life Years (DALYs) lost to each
form of cancer. The quantitative data seems to indicate that certain forms are overfunded (e.g.
breast, prostate, leukemia) while others are underfunded (e.g. lung, colorectal). We discussed
reasons why this might be the case.

6. Further questions for in-class discussion

Our charitable giving project concluded with a discussion of several more general questions that
could be a part of our discernment process for choosing charities.

Are all lives worth the same? Is a life in Cincinnati worth more than one in Florida? Is an
American life worth more than others? Is a child's life worth more than a 95 year-old's?
For the most part, the students agreed that making such statements sounds and feels terrible.
Looking at our data though, we realized that our allocations do reflect this in some ways.
Moreover, we discussed that governments, NGO's and public health policymakers make these
judgments everyday and that quantitative metrics even exist for these decisions such as
Disability-Adjusted Life Year (DALY), Quality-Adjusted Life Year (QALY).

Graphs of different function types studied in math 120

Graph of NCI research funding relative to societal health cost as measured by DALYs

Should our charitable donations go to organizations with specific goals or to more general
causes?

Again, students agreed that specific goals certainly sound better on the surface. In our data
we noticed that a large portion of students' donations went to cancer research. While many
donated to fight specific forms of cancer, there were also many donations to the American
Cancer Society in general. We discussed the benefit of allowing experts at the American
Cancer Society allocate funds to different cancers to achieve the greater good but also realized
that our personal connection to specific forms of cancer is an important factor.

Does donating $1 million to Charity X and $3 million to Charity Y mean that Y is 3 times
as important to me?

Students resoundingly said \Of course not!" However, when looking at our data we see that
many students divided their charitable resources equally among causes. We discussed why
this might be the case only to conclude that psychological factors were at play.

What's better, donating all of our money to one charity or less to many charities?

Again, we concluded that psychological factors make donating to several charities feel more
satisfying. On the other hand, I presented an economic argument for donating larger amounts
to fewer charities based on overhead costs. Specfically, each credit card transaction or bank
transfer costs money and takes away from the amount of \good" that an organization can
accomplish. Moreover, an opportunity cost exists for the amount of time and eort a charity
spends attracting donors.

7 Conclusion
My original objective was to use charitable giving to illustrate the role that quantitative data and
objective aspects can play in the discernment process of a decision that is often dominated by
personal, emotional, moral and other subjective aspects. While certainly successful in doing so, the
project had unexpected benefits for me personally and my teaching. In the course of describing
why they had chosen particular charities, many students shared personal stories and experiences
with me that left me feeling much closer to the students and helped me to see them as more than
the single dimension that is \student in a math class." Later in the semester, I was able to use the
knowledge of my students' experiences to tailor example problems to topics that I knew aligned with
their interests. Moreover, the project introduced me to multiple excellent programs and charities
in our community. As Xavier students are already actively involved with many of these programs,
I see a natural path for my own expanded involvement in both our local community and the Xavier
community.

Back to Top

The Secrets to Peace and Joy: Change Myself by Love

Huizhen (Jean) Guo, Ph.D.
Mentor: Daniel Otero, Ph.D. (Mathematics/Computer Science)

I am a happy person. You may ask why I am happy.

Am I healthy? I am not very sick but I am not strong either. I always feel cold and wear much more than other people do. I need more sleep than other adults do. That is the reason I exercise on a regular basis.

Am I rich? I am certainly not poor; I have food to eat, clothes to wear, and a bed to sleep in. I am satisfied with my living conditions, but I am definitely not rich.

Do I have a happy family? My daughter and I often sing the song "I love you; you love me; we are happy family," and we do love each other and get along well, but my husband has not been living with us for many years, he is not eligible to work in the United States.

How about my work? I do enjoy teaching and my students, but my students do not seem to enjoy their classes that much and I have been working hard on that.

The reason I have joy and peace in mind is that I have faith in God's will, not my own will. God will lead my life. I love God and people as Jesus did. Because of my love, I am willing to change myself: changing my thoughts, perspectives, the way I look at the world, the way I look at other people. Just as the Bible says that all things work together to benefit those who love God, I would like to share what I have experienced spiritually with anybody who reads my article.

Parenting

Since I mentioned "change", you can guess I was not happy before I was willing to change. You are right. I was unhappy, actually anxious, for years. I was worried about too many things and did not realize the danger of my unhealthy mood. When my daughter reached the age of 15, the confusing and suffering age, the conflict between us triggered hidden problems in both of us. The resulting conflict made me even sadder. I was concerned and pondered, "If I can't get along with my daughter, how can I get along with other people? Who else can I get along with?" I have been walking in dark and struggling, seeking light since then, to find the path to happiness and peace. Human wisdom is limited. My friends' help didn't solve the problem. I turned to God, trying to find answers. I started reading the Bible, attending Bible study and Sunday worship more often. Then the turning point came. That was love, which God put in all creatures' hearts. After a particularly miserable night, the words came to me that "I love you, you should love others." When my heart is filled with love, I see hope and feel joy and strength while when I distrust others, the doubts hurt me first before they hurt others. Once I chose love, the remedy to any wounds, it began to work: changing me and letting me see the other side of me. As the phrase states, I too "focused on the stick in others' eyes, and didn't see the big log in my own eyes." Unconsciously, I set up two standards, the higher one for others and the lower one for myself. I put myself at the center, and expected others to run around me.

When I look at my daughter in a different way, the view is changed. I was upset because she paid too much attention to how she looked. But, do I want her to look unattractive? Certainly not. When I was in high school and college, though I was a top student in all subjects, I was not very confident because I didn't think I was pretty. In fact, I like to see people dressed beautifully, because they decorate the city just like trees, grass, and flowers do. I guess I didn't like the way my daughter dressed. But should a teenage girl have the same perspective on dressing as me? Actually, I have come to ask her opinion when I purchase dresses for myself, because she has a good fashion sense.

I was upset because my daughter didn't do well in math and science classes. The truth was, she thought she was not good at those subjects and gave up before trying. Now I ask myself: am I really good at everything when I was in high school? My handwriting was terrible, and I was often tardy for the first bell, in college too. Because I made good grades, my parents and teachers never criticized me. They spoiled me. Why don't I look at the subjects my daughter is good at, such as English, history, and journalism? She also excels at drawing and cooking. She is making progress everyday and even getting A's in math and chemistry now.

My daughter is in a vital stage of her life. She needs love, care, comfort, encouragement, and guidance. I should be the provider of her needs. God created her and has a plan for her. I don't need to worry about her future. I will love her no matter how much she achieves or accomplishes. I will help and support her as long as I can.

She is changing too, while I am changing. Now I am "the most caring mom in the world" and she is "the most lovely daughter." Sometimes I am wondering if what I said or did is correct or not, since it will influence her. So, I pray to God to give me wisdom to teach my daughter.

I thank God for putting love in our hearts. This love has saved us. The biggest lessen I have learned is that people are different. Every living life is created by God uniquely. We need to accept and respect the differences. People have different talents. Everyone is born to be useful.

Teaching

Better understanding my daughter helps me understand my students better. The experience of working out the relationship with my daughter helps me a lot when I am trying to work out the relationship with my students. I know all problems can be solved if I can love my students as I love my daughter.

Some students complained that my class was rigorous. My reaction was that math class is not like going to see a movie; it is supposed to be rigorous. College is the place where professors and students should work hard, not the place where they always have fun. Now I am thinking I should try to make the teaching process interesting by changing my teaching style. Maybe studying can be fun if we make it more interesting. That is part of my teaching responsibility.

Some students also complained about my English. I was very upset and thought: if other people could understand me, why didn't the students. It was just an excuse. Now I understand it from their point of view. An accent is a barrier when the material itself is hard to understand. I need to practice to continually improve my spoken English.

I was disappointed when some students could not solve linear equations or couldn't calculate the area of a triangle. I used to think that they had poor mathematics skills, and wondered what they did in high school and how they could learn college math if they were unprepared. Now I think this way: they are not math majors; it is understandable that they forget mathematics. While I took chemistry and physics in high school, I don't remember anything about those subjects now because I never use them. My students are here sitting in the class because they don't know, and I am here teaching because I do know. I used to complain that the students were lazy; they didn't want to attend classes and didn't want to do their work. Now I often remind myself that there were times in high school I skipped study hall and watched a movie. There were times in college I missed classes and went shopping. The students are young adults; they still need time to mature. Even adults make mistakes and delay work sometimes.

I began to spend more time preparing for classes. I give group quizzes, individual quizzes, group exercises and individual exercises, and hands-on activities in class to make it more interesting. For example, when I was driving to school one day an idea came to my mind. Random variable is a basic concept in statistics. I can let the students use either the miles away or times spent to measure "distance from home to school," and collect data on the two variables respectively. This helped them to understand the concept easily.

I used to get mad when I read students' evaluations. The good comments didn't make me happy while the negative comments hurt me at first then the hurt turned to anger. I still don't feel comfortable reading negative evaluations, but now I have learned I can, indeed, find something I can do to improve my teaching skills and to teach more effectively.

Statistics is used in research in a growing number of disciplines. When people say they don't like something, it may very well be because they don't like it. But, it may be sometimes they think they are not good at it. I hope, by teaching statistics, that I can help those who are not good at math/statistics realize that math/statistics is useful and not as hard as they thought. They can learn the material. I hope what they learn in school will benefit them in their work later.

I attended the Lilly Conference, a conference on college teaching, and the workshop prior to the conference in November 2005. The speaker at the workshop was Louis Schmier, author of "Random Thoughts". One thing he said impresses me, even now: you teach who you are. Students can tell if I am happy or sad, nervous or relaxed, prepared or unprepared. They can tell everything in my mind by my tone, the expression on my face, my attitude, my posture, etc. When I walk into the classroom with love, care in mind, they can feel that and can be affected by that.

I have been thinking about why I love my career. One reason is that teaching provides me with an opportunity to speak, to spread my thoughts and influence others. Yet, it is a big responsibility. I need to think carefully about what I say and what I do in classes.

I was nominated as a "professor of the year" recently. I know there is much to learn to become an effective teacher. I am confident and look forward to the challenge. God will help me and give me wisdom and strength to reach the goal.

Current Challenges

The biggest challenge I face now is that I am trying to accept the fact that, while my husband lives with us, he doesn't work. Even if he gets a work permit later, he might not work. However, he may change his attitude once he settles down and absorbs the US culture. In China, many people don't admire those who do certain types of jobs, such as farmers, and bus drivers. Instead, they admire those who make money or have power.

God has a plan for me and will guide me. I must listen and yield to God's will. God has prepared the best for me, just as parents always give the best to their children.

In Summary

When anger, hatred, and bitterness were controlling me, my brain was like a pot of glue and didn't function. That made me frightened. What was I going to do if I could not work? When joy and peace are comforting me, I can think clearly and my brain works well. I need to work. I need to support my daughter and myself, but I also enjoy working. Bad moods consume a lot of energy. Such a waste they are. I wished I could use all my energy in work. Now, I can, because I have learned to love and forgive others, to thank God for everything.

Everyone has three worlds, a spiritual one, an emotional one and a physical one. When my spiritual world is in right order, I am emotionally stable and physically energetic. Spiritual growth is a gradual process; it doesn't happen in one night.

I am still up and down sometimes. Every time when I am falling down I pray to God, and He lifts me up. I need to build up my spiritual world stronger and stronger by feeding myself with spiritual food just as I need to feed my physical body with earth-growing food. Some day, it will be strong enough and not be knocked down easily.

I grew up in a non-religious family. I was told there were no saviors in the world. Only people can save themselves. I would never have become a Christian if I had not come to the United States. There is an old Chinese saying, "It is easy for mountains to change to rivers and for rivers to change to mountains, but it is difficult for people to change." Look at me, I have changed; I am a new person, a happy person. The more I change, the happier I am. Nothing is impossible in God. I hope you can find joy and peace in God, too.

Back to Top

Ignatian Pedagogy: Connecting Biology Majors to Mathematics

Hem Raj Joshi, Ph.D.
Mentor: Lisa Close-Jacob, Ph.D. (Biology)

Photo of Dr. Hem Raj JoshiSuper goal:

  • Facilitate Understanding in Personally Relevant Manner
  • Magis - Novel Ways to Serve Students and University by Trying New Things
  • Cura Personalis - Considering Needs of Students
  • Challenging Students to Connect Topics

The discovery of the microscope in the late 17th century caused a revolution in biology by revealing otherwise invisible and previously unsuspected worlds. Mathematics broadly interpreted is a more general microscope. It can reveal otherwise invisible worlds in all kinds of data, not only optical (Cohen, PLOS Biology, 2004). For example, computed tomography can reveal a cross-section of a human head from the density of X-ray beams without ever opening the head, by using the Radon transform to infer the densities of materials at each location within the head (Hsieh, Computed Tomography, 2003).

The importance of mathematical and computational tools in every area of biological studies is well documented (Levin et. al, Science 275:334-343, 1997). Mathematical and computational challenges in population biology, ecosystems science, and epidemiology in particular have long been recognized. With new conceptual advances and technology, research initiatives that focus on integration of mathematics and biological issues are expanding very rapidly. There is a general and diffuse dissatisfaction with mathematics among the biologists (i.e. why I need math?). Today biology is becoming more mathematical, and all biologists need some mathematical skills to understand complex biological systems.

We would like to explore results from different biological experiments and connect them to relevant mathematics. To achieve this we will communicate with Biology faculty and develop a need based course (i.e. Teach mathematical skills that will be useful for biology majors).

Why should biology students study more math? There are two types of reasons:

Abstract reasons

  • Improve logical/rigorous reasoning ability
  • Ability to build models
  • Better appreciation of mathematics and computation
  • Appreciation and understanding of important phenomena: exponential growth and decay, limited growth...

Concrete reasons

  • Ability to perform important mathematical operations
  • Learn to interpret graphs
  • Ability to analyze data: Statistics

Designing a New BioMath Course for Xavier University

As a first step we will modify the existing calculus-based math course, and it will be offered for the first time in fall 2005. This course will help biology majors to understand the importance of mathematical models in biological sciences and use the knowledge in biology research projects. In the future, we will develop an entirely new math course for the biology major.

Back to Top

Ignatian Pedagogy in Collegiate Mathematics Education


Joy Moore, Ph.D.
Mentor: Leslie Prosak-Beres Joy Moore, Ph.D.

During my participation in a Manresa Experience in the Fall of 2007, I was introduced to the concept of Ignatian Pedagogy. Considering myself to be a practitioner of culturally relevant pedagogy, I was struck by the similarities between the two pedagogical approaches. I became interested in viewing my classroom practice through the lens of an Ignatian pedagogical framework. I have always maintained a reflective journal of my classroom practice and so I decided to use that as a place to begin accounting my observations. I made journal entries during the Fall and Spring semester of the 2008-2009 academic year, presented here in summarized form. The courses discussed include MATH 120 (Elementary Functions), MATH 150 (Elements of Calculus I), MATH 201 (Foundations of Arithmetic for Early Childhood Education) and MATH 211 (Foundations of Arithmetic for Middle Childhood Education).

Ignatian Pedagogy embodies five key teaching elements--Context, Experience, Reflection, Action, and Evaluation. Under each element, I provide a brief overview of the tenets of that element (referenced from Jesuit Education and Ignatian Pedagogy September 2005, Association of Jesuit Colleges and Universities, Rev. Peter-Hans Kolvenbach, S.J., Superior General of the Society of Jesus). I then provide my observations (italicized) regarding that particular element in my own pedagogical practice.

Context

What needs to be known about learners (their environment, background, community, and potential) to teach them well?

Cura personalis - personal care and concern for the individual--is a hallmark of Jesuit education, and requires that teachers become as conversant as possible with the context or life experience of the learner. Since human experience, always the starting point in a Jesuit education, never occurs in a vacuum, educators must know as much as possible about the actual context within which teaching and learning take place.

This is something I really try to instill in my pre-service teachers. The tenet itself is supported by the National Council of Teachers of Mathematics. In its description of a worthwhile mathematical task, two of the 11 points delineated are as follows:

The teacher of mathematics should pose tasks that are based on-
...
2. knowledge of students' understandings, interests, and experiences;
....
10. display sensitivity to, and draw on, students' diverse background experiences and dispositions;
....

At the beginning of the semester we explicitly discussed these points. Preservice teachers are asked to consider every assignment within this framework of context and to explicitly state how their homework addresses these tenets.

This tenet of Ignatian Pedagogy is very much in line with culturally relevant pedagogy, in that the culture of the learner is recognized, appreciated, and incorporated in the learning experience. Culture here is defined as the political, socio-economical, religious, racial, and moral background of the learner.

Experience

What is the best way to engage learners as whole persons in the teaching and learning process?
Teachers must create the conditions whereby learners gather and recollect the material of their own experience in order to distil what they understand already in terms of facts, feelings, values, insights and intuitions they bring to the subject matter at hand. Teachers later guide the learners in assimilating new information and further experience so that their knowledge will grow in completeness and truth.

I find I do this more (or better) in my calculus and precalculus classes. Using prior knowledge as a springboard for acquisition of new knowledge is the foundation of my pedagogical approach. In particular, I want my students to realize that they do know SOMETHING about mathematics. Overcoming issues of math anxiety is the greatest challenge in MATH 120 and MATH 150. The first thing students say on entering the classroom or my office hours is "I am not good at math", "I have never been good at math", "Math is not my best subject", "Math has always been my worst subject". Helping them realize they have a valid starting place in what they do know, empowers them, motivates, them, encourages them to try to learn more. So considerable time is spent gathering and recollecting "the material of their own experience in order to distil what they already understand" as a spring board for what knowledge I want them to acquire.

My approach is different with my preservice teachers. Particularly in MATH 201 and 211, students enter the classroom believing that they already know the content and are more than willing to "gather and recollect the material of their own experience". In fact, most preservice teachers enter the classroom with the intention of teaching the way they have experienced their own learning. So my intent here is to show many of them a different way of understanding what they know, so that they can create different learning opportunities for their future students. I want them to experience and consequently learn to facilitate learning opportunities that expand beyond rote memorization of facts, rules, and formulas and that delve into conceptual understanding of the mathematics at hand.

Reflection

How may learners become more reflective so they more deeply understand what they have learned?
Teachers lay the foundations for learning how to learn by engaging students in skills and techniques of reflection. Here memory, understanding, imagination, and feelings are used to grasp the essential meaning and value of what is being studied, to discover its relationship to other facets of human knowledge and activity, and to appreciate its implications in the continuing search for truth.

Pre-service teachers in my courses are required to keep reflective journals. This practice may be considered unusual in a mathematics content course. But I want my students to reflect on their own learning experiences as a way to inform their future teaching practices. Thinking about how they felt about (affective domain) or how they understood (cognitive) a particular mathematical concept should inform their future practice. Student comments like, "I wish I had been taught this way. It makes more sense" or "I am frustrated by this method and my students will probably be too" are important aspects of the learning process for which I want my preservice teachers to make note. However, when I review their reflective journals, the content is not as reflective as I would like. Perhaps I need to rethink my direction for this requirement. Maybe check the journals more than twice a semester. Future goal: to be more specific about content of the reflective journal. Future requirement: it must be separate from class notes.

Action

How do we compel learners to move beyond knowledge to action?
Teachers provide opportunities that will challenge the imagination and exercise the will of the learners to choose the best possible course of action from what they have learned. What they do as a result under the teacher's direction, while it may not immediately transform the world into a global community of justice, peace and love, should at least be an educational step towards that goal even if it merely leads to new experiences, further reflections and consequent actions within the subject area under consideration.

I try to create opportunities for my preservice teachers that allow them to actually practice what they are experiencing. I require them to design lessons and make class presentations. Class discussions and peer feedback from these presentations add to their opportunity for reflection on actual practice. We have had very explicit conversations about issues of diversity in background, socioeconomic status, and types of learners. We have discussed issues of bias in standardized testing. Hopefully these types of conversations contribute to the development of this particular tenet. Though mathematics is often viewed as rote memorization of facts and figures, I want my students to experience it within various contexts so that they will teach it in like manner. Despite the fact that my courses are content courses and not method courses, I think it is important to give students an opportunity to do, practice, and teach mathematics, not just study it. Student feedback on my course evaluations indicates that this is a valuable part of the course for the majority of them.
Future goal: incorporate mathematics as social justice into my MATH 120 and Math 150 courses.

Evaluation

How do we assess learners' growth in mind, heart, and spirit?
Daily quizzes, weekly or monthly tests and semester examinations are familiar instruments to assess the degree of mastery of knowledge and skills achieved. Ignatian pedagogy, however, aims at evaluation which includes but goes beyond academic mastery to the learners' well-rounded growth as persons for others. Observant teachers will perceive indications of growth or lack of growth in class discussions and students' generosity in response to common needs much more frequently.

Open class discussions have been extremely revelatory for me this year, particularly in MATH 201 (Foundations of Arithmetic for Early Childhood Teachers). These preservice teachers have been very willing to be open in class discussions about their opinions, viewpoints, and understandings (or lack thereof as the case may be). This is has really serve to validate my requirement of class attendance and participation as 20% of the final grade.
It makes evaluation much more difficult for me if a student never says anything in class. Class participation also often differs from performance on standardized assessment methods. So students may exhibit understanding in their oral communication that does not translate to written assessment. Hence I include alternate means of assessment such as: board presentations, curriculum development projects, and worthwhile mathematical tasks.

In MATH 120 and MATH 150, the students who exhibit the most improvement in their academic performance are those who are willing to talk to me and others in class. Small group work really seems to help those students who are struggling with a concept, particularly when they are grouped with a student who exhibits strong academic performance. They may not be comfortable talking to me, but they seem to work well with their peers, if I form the groups appropriately.

Conclusion

Having identified the five aspects of Ignatian pedagogy in my own practice, I realized that I not only value the aspects of the pedagogy, but I believe all teachers should incorporate these aspects into their practice. Hence, my experience in the Ignatian Mentoring Program has led me to the development of a future research project (Fall 2009) that will investigate preservice teachers' beliefs about teaching mathematics within the framework of an Ignatian pedagogical approach. In an effort to provide preservice teachers with a learning experience that models the way I hope they will teach mathematics, my desire is to exemplify these tenets in my own classrooms, such that future teachers will exemplify like tenets in their classrooms; thereby perpetuating an endless cycle of Ignatian pedagogy within countless mathematics classrooms throughout the world. My desire, my prayer, is to fulfill Matthew 5:16:

Let your light so shine before men, that they may see your good works, and glorify
your Father which is in heaven.

Back to Top


Statistics and Privilege


Grigory Sokolov, Ph.D.
Mentor: Haider Raad, Ph.D. (Physics)

Brief Introduction and Motivation.
Being a mathematician (half statistician, half probabilist) myself, I find myself at the very beginning of the journey to expand my teaching style and to view my subject as something more than just a framework to interpret and understand the world we live in. The beauty I see in a theoretical---and often abstract---mathematics is a thing that is challenging to express and communicate to somebody else, especially if they are a student. And it is also quite hard---at least for me---to connect and relate to the Ignatian values.

Teaching at a Jesuit school, one should seek to expand their views and embrace the opportunity to provide more to their students than merely the subject they teach (however abstract and seemingly disconnected from “real” life). In that respect, I am quite lucky---especially with my core-level classes---as statistics is a topic that shows up in virtually every aspect of our everyday routine.

Idea and Implementation.
Driven by the ideas of encouraging deeper reflection, understanding of the injustices and disparities of the world outside, and striving for the common good, my mentor and I have decided that incorporating a project specifically tailored to promoting those values into my core-level statistics class seemed like a reasonable idea.

Having investigated various avenues for targeting the goals at hand, I decided to go with the data provided by the World Bank Group (an organization whose mission is to “end extreme poverty and promote shared prosperity in a sustainable way”). I compiled the data about two distinct characteristics of various low-income countries of the world, as well as that of the United States. Including the USA in the dataset was meant to contrast how people from privileged countries do not realize the full extent to which basic needs they take for granted aren't, in fact, universally accessible.

Specifically, I offered two datasets spanning the past several years---life expectancy at birth, and percentage of people using at least basic drinking water services---as the basis for a statistical project they were supposed to work on. Skipping a little bit ahead, I would like to mention that I seem to have not put enough care and guidance into how the project was structured. In retrospect, having a more explicit language and emphasizing the underlying motives of the project over its mathematical side would have probably achieved a better effect.

Reflection and Afterthoughts.
A month after the semester ended, I asked my students to reflect on the project, and provide some feedback regarding the lingering effects and impressions (if any) it had on them. Although the response rate was quite low---and this is something I definitely intend on working on in the future---some of the comments were quite encouraging (especially the ones regarding the project itself). Some of them are presented below.

Before I proceed, I would like to point the reader’s attention to the fact that response bias is obvious here. The students whom this project did not affect much probably did not bother to respond. The only statistical evidence one can get from this---I am a statistician after all---is that at least it reached some of my students. And this is a reasonable starting point.

-- I found it interesting to have a more real-world application of statistics. It helped me to better understand the concepts.

-- Outside of it being helpful practice of application for what we had learned it class, I found it interesting to see these statistics about other countries.

-- I really appreciated this project because it helped me to apply the concepts we had been learning in class. It gave me time to sit down and understand what we had been learning, and I feel that it really helped me boost my grade.

-- Personally I thought it was good for both real world application of statistic as well as a good way to use what we learned in class.

As is apparent from the comments above, I still have ways to go. Although the original purpose was to open the students’ eyes on the way the world as we know it is---with all the disparities, privileges, things we take for granted that others do not have access to---their takeaway was more of an “interesting real-life example”. Incorporating a more continuous process of reflection for the students seems to me at the moment to be the next logical move. Rather than relying on them to come up with a philosophical leap in an otherwise quite mathematical class, shifting the emphasis from theory (and even application) to the human aspect of the context of the data is my next step toward building a stronger sense of community and engagement among students, and becoming a better representative of Ignatian pedagogy.

Back to top



To provide feedback, please email: jesuitresource@xavier.edu

Jesuitresource.org is developed by The Center for Mission and Identity at Xavier University with support from the Conway Institute for Jesuit Education. Learn more about Jesuit Resource.