Undergraduate Research Initiatives

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1 Matthew A. Beauregard, PhD Teaching Statement Teaching is about inspiring, motivating, guiding, and transforming lives. Our profession comes with its natural hardships, blessings, and gratification. In pursuit of knowledge and understanding, mathematics is ubiquitous and central to scientific truth. Therefore, while teaching pedagogy may differ, we remain consistent in our goal to push student innovation, reasoning, and critical thinking. Here, I offer an overview of my teaching philosophy, student research and development initiatives, and brief, unsolicited, select comments from students. Teaching Philosophy Mathematics has been a dominant presence throughout my entire life. For as long as I can remember I have been enthusiastic about the study of mathematics and its applications. That enthusiasm led to my decision to pursue a career as a mathematician. Moreover, I realized early in my schooling that I thoroughly enjoyed communicating concepts, ideas, and thoughts; be it in a seminar, conference, among colleagues, or other students of mathematics. I realized I had an aptitude to communicate mathematical concepts clearly and effectively. Teaching has been an integral part of my academic career from high school to post doctoral. I thoroughly enjoy the challenge it brings, the lessons you learn, and the reward of watching students succeed. I feel I share an equal responsibility with the student in achieving their understanding of the material. On one hand, part of my responsibility is to create a positive learning environment that will keep the students engaged, interested, and comfortable to promote the free exchange of ideas. On the other hand, the teaching should promote ideas and tools that will make the students better thinkers, problem-solvers, and self-learners in and out of the classroom. When I consider the study of mathematics I believe there are two closely related reasons for studying the subject. One is that an individual may study mathematics due to an inherent interest in the beauty of the material itself, while another may study mathematics for the purpose to use it as a tool to explain some phenomena. Both reasons share a fundamental similarity: The study of mathematics develops critical thinking and problem solving skills which is highly desirable in any professional field. As a mathematics teacher I feel it is important that every one of my students develop an appreciation for mathematics in the real world as well as an understanding that through the study of mathematics their critical thinking skill set is strengthened and will help them in their future careers. Every interaction with a student in and out of the classroom is an opportunity to help keep them engaged, make them feel comfortable, and empowered. These are foundational to the development of a positive learning environment in the traditional or online classroom. In my experience, these are accomplished in a few concrete ways. First, it is important to be accessible to students so that there is never any hesitation on their part to come with problems and ask questions. I institute an open door policy in which anyone is welcome to talk mathematics with me throughout the day. It is not uncommon for students to ask mathematical questions from other subjects they are studying. In addition, I guarantee all correspondence will be returned within a 24 hour time period and frequently utilize video chat sessions through Skype or Google Chat. This openness and dedication lets the student realize I care about their success and am willing to help them achieve it. Secondly, I believe in the power of using frequent positive and constructive feedback. When giving feedback it is important to start with something positive about their performance, followed by the constructive feedback, and then finish off by reinforcing something positive they have done. This principle is employed in all forms of correspondence between students. I have found being genuine and frequent in this practice motivates students, improves morale, keeps them positive, and in turn empowers them. Thirdly, I consistently reach out to students I sense are struggling. In fact, for introductory courses I have instituted a preliminary exam, testing on prerequisite material, given in the first week of classes. This identifies students who may potentially struggle and freely opens a channel to communicate with them early in the term. This early interaction is an opportunity to empower and enable the student which will help them become more 1

2 successful. Lastly, it is important to allow students to discover things for themselves. As a teacher I act as a guide, giving a useful tip or suggestion to help students draw their own conclusions. Otherwise our own enthusiasm can dwarf a student s moment of triumph! In the classroom I employ various techniques to keep students actively engaged and to address different learning styles. In the classroom I engage the students through lecture, create discussions through online forums, use illustrative props, group work to help students learn how to communicate ideas and concepts with others, and utilize technologies such as Mathematica or Matlab c. These techniques are easily transferable to the online environment. It also keeps the classroom dynamic and exciting for the student. In addition, it provides numerous avenues to illustrate applications of the mathematical concepts, deepening their understanding and appreciation of mathematics. For instance, Mathematica s capacity for live animations can bring to life series solutions to partial differential equations. In this manner, students can make numerous observations, oftentimes leading to conjectures. As a result, the lecture can be aided and guided by student observation. The teaching of mathematics is quite organic. Employment of the above techniques provides flexibility and the ability to monitor student learning and performance, thus increasing student mastery. Of course, I also access a student s mastery of the material through standard methods such homework, quizzes, and exams. However, these are after the fact type of measurements. Hence, I routinely ask summary questions, that ask students to summarize a particular activity or lesson. In addition, I gauge their understanding through the employment of another simple exercise. I ll start by asking each student to close their eyes. I then ask them a question, often with a yes/no answer. I will then have each student place a closed fist against their forehead and then put up one finger if their answer is yes. This has been a quick and playful tool providing priceless information. As a mathematician I enjoy the challenge teaching brings. Each student is unique. Each class is unique. In each experience I have continued to become a better mathematician, teacher, and person, through the sanctification it offers. I am committed to creating the best atmosphere for learning so that each student has the opportunity to achieve their potential in their current or future careers. It is a blessing to be given the opportunity and talent to teach our next generation of mathematicians! Undergraduate Research Initiatives It was my second year at the university and I was making my typical morning walk over to the mathematics building. The cool breeze was befitting to New Hampshire s early autumn season. That fresh, clean, and crisp air always reminded me of the winter still to come. My differential equations professor had regular office hours prior to our 8am class. It was rarely full and I loved to take advantage of that. This day, when I came, his door was already open, music was playing from the computer. While the music was playing a full spectrum of data was being displayed on the computer screen. I asked him what he was doing and he began to explain his research. He was interested in the mathematics of dynamical systems and how they can be utilized in sound compression algorithms. I was aghast, you could research mathematics? Apparently so! It was shortly thereafter I successfully applied and received an undergraduate research grant. To a large part, that experience and mentoring motivated me to pursue advanced studies in applied mathematics. My story is not unique, institutions have realized that incorporating undergraduate research into curriculum helps attract and retain students in mathematics and the physical sciences. It gives students priceless experience in application of mathematical tools and ideas, critical thinking, collaboration, communication, and many other attractive skills. Of course, mathematical research can be extremely technical requiring vast training and experience. However, there are open research problems and areas that are readily available and attractive to undergraduate students. In the past, I have worked with undergraduates on traffic flow modeling and singular perturbation theory. In fact, one of my previous students studying traffic flow continued their work and has a paper currently under review! Here, I offer a few readily available research projects closely linked to my research expertise and provide a brief synopsis of each. 2

3 1. Analysis of nonuniform finite differences - adaptive finite difference methods rely heavily on nonuniform grids. In consideration of boundary-value problems in ordinary and partial differential equations, how does their inclusion affect the underlying matrices? Moreover, in what cases is the truncation error and/or convergence rate improved as the grid size goes to zero? 2. Traffic flow modeling - cellular automata with its collection of rules can be utilized to model traffic flow at a microscopic level. The set of rules can be as complicated as you like. Can additional rules be established in order to better represent reality? Furthermore, what effect does the inclusion of particular rules have on observable statistical quantities? Can we determine optimal paths through a network of roads? 3. Swarm intelligence - a collective or swarm may exhibit intelligent behaviour as a byproduct of simple rules that each individual may follow. A discrete dynamical system can then be developed that, in a macro-scale, can be modeled by a system of coupled reaction diffusion equations. How does the established rules correlate to the reactive terms in the continuous case? Can a particular outcome be inverted to establish a set of microscopic rules? If so, can this process be optimized? How does the macroscopic model predict long-term behaviour as the collective size changes? How best to incorporate a multi-scale model that captures the micro and macro scales effectively? 4. Linear stability analysis - many mathematical models in biology, chemistry, and engineering exhibit known steady state solutions, oftentimes homogenous. For instance, a straight elastic filament under tension is a steady state solution to the static Kirchhoff equations. Are these solutions stable? For unstable solutions, a weakly nonlinear analysis can be undertaken to derive amplitude equations. A theoretical or numerical study can then provide the student with useful insights into the physical phenomena being studied. Student Development Initiatives In the past, it has often been assumed that in order to improve the stature and recognition of an institution s undergraduate or graduate programs they must attract the brightest students and supplant them with their highest performing faculty. This strategy is valid but it is incomplete. Nationally, there has been a movement to include a broad array of student development programs supplementing the traditional approach to education. Student research initiatives are a natural part of these programs. From matriculation to graduation, our institutions must provide students with such development. What does this mean for mathematics? To a large part, there should be an institutional push toward the improvement of oral and written communication skills. To help accomplish this goal I have often implemented a writing component to my courses. For undergraduate students, this has included expository writing on historical mathematical figures, current scientific discoveries, as well as summaries of undergraduate lecture events. For more mature audiences I have included research term papers. These papers study recent research articles relevant to our current course being studied. The students then present pieces of their papers to their classmates throughout the term. A side benefit of this experience is that students are guided toward potential mentors within the institution. For instance a recent undergraduate student studying differential equations, selected a paper on boundary layer theory. The student s interest was perked enough that they continued throughout the duration of the semester in an undergraduate research project. Although my development initiatives are self-employed, I find that they have been effective at challenging our students into deeper thinking and rational reasoning. Moreover, these activities have been directly addressed in student evaluations and the overwhelming majority have indicated that it was a gratifying part of the course. 3

4 Comments from students I just wanted to comment and say thanks for the time and attention you gave my son. You definitely had a positive impact on him 1. Thank you so much! I will remember this semester for the rest of my life, and I really appreciate you taking the extra time to help me with all of my questions after class time. This semester was most definitely a struggle academically and extracurricularly. You helped me forget the extracurricular struggles by sparking my interest in the subject, and suggesting that I create a schedule for my life. I am excited to use my knowledge I ve learned this semester for real world applications in the future. I do plan to stay in touch, as you have been one of my favorite professors. Once again, thank you for all you ve done for me this semester. Having you was an honor, sir. Your words of wisdom and encouragement will stick with me forever, thanks. I think you re a great professor, one who actually cares about his students. It s usually just a name to a face, and nothing more. I don t know if you remember me, but you were my Calc 1 teacher at the University of Arizona 3 years ago. I think you are possibly teaching at Baylor now? Well, whether you remember me or not, I am now beginning my senior year in Aerospace Engineering and am ing you to briefly say THANK YOU! Over the past couple semesters, I ve realized that most students attribute poor grades to the fact that they had a bad teacher or don t know the basics for the course. Whereas, I have always said the reason I do so well in school is because I had a great Calc 1 teacher who provided me with a solid mathematical foundation. Your class was actually my very first college class (if you can remember we had class at 9 am so it literally was my first class), but you pushed us to learn to understand the material, not just achieve the best grade, which is a mindset I have maintained throughout all of my classes. I know several students who are very bright, but failed Calc 1 and ultimately dropped out of engineering because of it, so I really do think having you as my first professor made a world of difference for my success in college. I went on to complete a math minor (which I have a 4.0 GPA in), tutor math on campus for the Think Tank, and have just accepted a job offer for after graduation at Honeywell Aerospace working on their Control Momentum Gyroscopes. Of course, several people and events have encouraged me and influenced a great amount of my achievements, but when I think back to where it all began, it is in your math class. I am sure there are plenty of your students who would say the same thing as me, and I hope you are continuing to encourage and inspire your students to succeed. I wish all freshman could have at least one teacher like you because it is extremely difficult to maintain an excitement for school if you do not have that good initial experience. So again, thank you, and I hope all is well with you and your family! Dear Professor Beauregard, I am beyond excited to tell you that I have submitted all of my graduate applications! I am looking forward to our semester break. I wanted to thank for your help and advice throughout this process. I don t think I could have made it through without the encouragement I received from you. I [have] appreciated your counsel so much. Dear Dr. Beauregard, Thank you so much for being an excellent professor this semester. This class, although always a challenge, taught me so much and gave me a better idea of how mathematics is applied, especially in the engineering world. Dr. B. is one of the best teachers I have ever had. He always comes to class excited to teach and ready for his students to learn. He goes after office hours if we need help and is always concerned for how the student is doing. He has good examples and clearly knows the applications. 1 This was received from a parent through . 4

5 If every professor could be like this everyone would be majoring in mathematics. I am very thankful for the immeasurable support Dr. B. provided me in this class. I wish I could have him again! Thanks for the great semester Dr. B. I enjoyed the class immensely and you are the best teacher that I ve ever had. I ve been blessed w/ great ones all my life, but you were the best. You were always helpful, especially with your virtual [office] hours, open door policy, and [through] . Dr. B. is fantastic, I wish the class [was] longer! Dr. Beauregard was such a refreshing teacher to have. Always quick to respond to ask questions, willing to go the extra mile to help you understand the material. He makes a point at making physical connections from the mathematics we are learning to the outside world. Its the first time I have really had to think about things in a physical sense in math! Fantastic. Dr. B. is still my favorite professor despite the fact that he left U of A. It was an absolutely privilege to have him as a teacher. He challenges you on problem sets, but the applications [were] great. When I think of the archetypal professor I always think of Dr. B. I am so happy that I got the chance to work with him before he left. Dr. B. is by far the best prof I have encountered. His passion for math makes his students want to learn. I learned an exceptional amount in this class. He is always available and willing to answer questions and clarify material. I recommend if given the opportunity to take a class by Dr. B. it will be an experience you will not regret. He was really concerned that the class understood the material. He is very bright and always helpful through . Very respectful! Dr. B. it is clear that you love what you are teaching, it made it easier to understand and go along with you. I thank you for being so personable, which made it easier for me to approach you with questions. Dr. B. enthusiasm and knowledge really showed and help excite and motivate me to learn more. He made me rise to the occasion (he s not easy!) and not just do the bare minimum. He just made the class so fun that I had to! Thank you! Well, he kept me awake. I didn t think that was possible 2. Dr. B. is quite funny and has a way of capturing your attention. I wish the class was longer! Incredible teacher. He s genuinely honest and caring, not to mention hilarious. He explains the material well and is always there to help. Plus his mathematical stories are great! Dr. B. is an amazing teacher, his enthusiasm and energy for the class is truly inspiring and refreshing. It truly makes the class worthwhile and exciting. His knowledge of the material and their applications helps makes the class more interesting. Dr. B. I just want to send a quick note. How did you keep my attention all year!? Thank you for dedication to us! Dr. B. I loved your goofy style and your passion for the material and us. Thank for such a wonderful semester. Your constant encouragement has [meant] a lot! Truthfully, Dr. B., this was a really tough and enjoyable course. Although, I should have studied more to get a higher grade, I still found myself more and more interested and engaged in PDE s every class we had. Thank you! Dr. B. sense of humor is great, he loves telling the story behind the mathematics. Always engaging! I even changed my major! 2 I still find this to be the most entertaining comment I have received thus far! 5