Ignatian Pedagogy in Collegiate Mathematics Education
Joy Moore, Ph.D.
Mentor: Leslie Prosak-Beres
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 http://www.ajcunet.edu/index.aspx?bid=526). 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 preservice 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, socioeconomical, 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.
Preservice 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.
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.
Ignatian Pedagogy: Connecting Biology Majors to Mathematics
Hem Raj Joshi, Ph.D.
Mentor: Lisa Close-Jacob, Ph.D. (Biology)
Super 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 & computation
- Appreciation and understanding of important phenomena: exponential growth and decay, limited growthâ?¦
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.
Statistical Inference
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.