Our Philosophy on Teaching
Dr. Margo Alexander, Senior Lecturer:
As a Senior Lecturer in mathematics, my ultimate goal is to prepare students to function confidently and knowledgeably in the area of mathematics. Effective teaching requires understanding what students know prior to coming to college and what they need to learn while here and then challenge them, support them and help them learn. Subsequently the students’ curriculum should focus on the big picture of mathematics, with instruction that enables students to build new knowledge actively from their experiences in and outside the classroom. To contribute to these teaching and curriculum goals, I employ many strategies, with my own professional focus on teaching technologies and their effective use in classrooms and online environments; collaborative and co-teaching approaches that expand student experiences via multiple teaching styles and instructor areas of expertise simultaneously in the same classroom; and the teach-the-teachers model through which my engagement with pre-service or in-service teachers has exponentially-increased reach, as my students will eventually teacher their own students for years to come. Highlights of my approach in each of these areas are summarized in the following statement of my instructional interests, goals and qualifications.
The struggle is real… and necessary
During my graduate studies at the University of New Hampshire, my office was located next to that of a lecturer named Yitang Zhang who went by “Tom.” Tom was a quiet, kind, and fairly eccentric faculty member who could regularly be found walking up and down the halls with his hands behind his back or holding a cup of tea in one hand. On these walks, he’d be thinking – constantly thinking. On most days you could find Tom pacing the halls of the Mathematics Department like this. On such a walk, he might stop next to a classroom for a moment, look inside to see what was happening, and then continue his pacing… and thinking. Thinking about what? I didn’t ask, so I suppose I didn’t really know at the time.
For seven years after finishing his Ph.D., Tom struggled to find an academic position. To make a living, he worked various jobs for friends including a job at a Subway restaurant. But Tom didn’t give up. In 1999, he caught a break. A friend helped him secure a teaching position at the University of New Hampshire; there he lectured calculus and the occasional number theory course.
In 2014, the mathematics world exploded with news that someone had proved the Weak Twin Prime Conjecture. This foundational conjecture in number theory had remained unproven for hundreds of years. Surely to solve such a problem, that person must already be a well-established and well-respected number theorist, right? But it turned out that the solution actually came from an unheard of mathematician: Tom Zhang. I suppose I now know what Tom was pondering as he took his regular walks. Apparently, Tom had been working on and struggling with this problem for several years. No one ever expected him to solve it. Yet despite the difficulty and the many failures along the way, he never gave up.
Certainly, Tom Zhang is talented. There is no denying that. Natural talent, however was not the main reason he went from being an unheard of mathematician to appearing in news stories around the world. The key was perseverance. Even in my own research, I struggle with problems for weeks, months, and sometimes years. The successes I have are due to my willingness to fight, to struggle, and to be frustrated sometimes to the point of anger. The struggle is real and it is necessary. One of the most important components of my philosophy of teaching is to regularly emphasize this type of self-patience. Students often come to me frustrated or even crying because they can’t understand something despite great time and effort. I view it as my job to tell them to be kind and patient with themselves. Learning new mathematics at any level, even at the research level, can be a painful battle. Shortcuts are in rare supply. Those who can persevere through the struggles are the ones who make significant progress and contributions to their respective fields. It’s easy to forget this if the content you are teaching is old news to you. Good mathematics teaching requires that you recall the struggle and apply empathy.
The value of independent mathematical thinking
As an instructor, teaching a student how to think and grow independently is the greatest achievement I can possibly hope for. In teaching mathematics, it is tempting and all too easy to let the content and course structure become routine. From an instructional standpoint, very little about the mathematics changes from semester to semester. Science will forever improve on itself and the humanities change with the times, but the rigidity of mathematics is enduring. Despite this constancy, the students in my classes come and go. Every person that sits down apprehensively on the first day and hands in their final exam on the last has their own personal interests, dreams, and struggles. To maintain an awareness and sensitivity to this reality is crucial for successful teaching, especially in courses as foundational and difficult as calculus.
In my classroom, I seek to impart upon my students the same passion for the beauty and creativity inherent to mathematics that I developed during my own studies. This forms the framework for my dynamic and sometimes animated lectures. Students who see an instructor excited by the mathematics are much more likely to actually become interested in the content. Additionally, when students actually value mathematical thinking, they will be much more likely to struggle through challenging concepts and are more likely to apply formal reasoning to other aspects of their lives. I want students to walk away from my class with critical thinking experience that is applicable beyond the scope of the course content. To this end, I regularly pose challenging problems and get students to think about the problem solving process itself in addition to learning traditional techniques.
An honest teacher will own up to the same thing
The creative process of communicating significant mathematical concepts to others is something that is developed over time. When I first began teaching, I thought I was pretty hot stuff – a natural. Years later, of course, I see that I was sorely mistaken. An honest teacher will own up to the same thing. Just as with the process of learning mathematics itself, there are no shortcuts to becoming a good teacher. If you are just looking to get the job done and you are not genuinely and actively looking to improve and grow, you probably aren’t as good as you think you are. Our students deserve better! For me, it took years of lecturing and interacting with students to hone my oratory skills, my ability to know ahead of time what students will struggle with, and to know what concepts require extra explanation and visualization. I still seek to improve by finding new ways to capture student interest and apply new computational software.
If you are a young teacher who wants to improve, find a passionate and experienced teacher of a lower-level course who respects and is respected by their students. Attend their lectures: sit, listen, and watch. Ask other instructors how they deal with tricky classroom management situations. Take it all in and make it your own.
One does not create mathematical thinkers by doing mathematics
Conducting mathematics research (for me, in topology) is something I will do for many years simply because I love it. Research keeps the mind actively engaged in high level mathematics. It informs my teaching and offers students a glimpse into higher mathematical worlds. While research is rewarding in the sense that it allows one to push the boundaries of what is known a tiny bit further, it is almost always unclear (even in applied mathematics) when certain mathematical advancements will affect society if they do at all. Mathematics needs to be done, but one does not create mathematical thinkers by doing mathematics. The greatest impact factor a mathematician can achieve is through his or her students. The pedagogy we choose and the way we act in the classroom can have an enormous impact on the lives of hundreds or thousands of real people.
The practical stuff
Teaching in a classroom setting gives me the opportunity to make mathematics accessible and meaningful, encourage both academic and non-academic maturity in students, and to design effective curriculum. Over the past several years, my preferred method of instruction has evolved into a combination of traditional lecture and cooperative learning. Students learn best when they receive scaffolding for problem solving and can make sense of concepts on their own. I have often implemented this philosophy at Georgia State by allowing students in-class time to practice solving problems while I am personally available to them (in-class, office hours, and extra review sessions). Building a strong rapport with my students has allowed me to regularly implement this highly interactive approach to teaching.
When teaching a class of fewer than 20 students (e.g. honors calculus), I like to give students the opportunity to learn in a cooperative workshop environment. This is a setting in which students, with guidance, can make computations, derive results, and explore new ideas in groups. One of the advantages of this approach is that students leave with a sense of ownership of the content. Additionally, students are more likely to master concepts when they are given the chance to communicate what they understand.
In a large lecture course, where group-work is less practical, I like to create a forum for students to solve problems creatively by introducing a metacognitive perspective into lectures. Providing definitions, theorems, and working through computations is important, but will not take students to the highest levels of learning. When introducing a new concept or working through a problem in lecture, I describe aloud the type of thinking that might lead one to a definition or to the next step in a solution. Encouraging students to model this behavior gets them thinking about how they might apply their experiences to solve new problems. This approach can make class more enjoyable for students since it is a reminder that mathematics is an evolving human enterprise, a fact often hidden by the formal nature of mathematics curriculum.
In any classroom setting, creating a rapport with students is crucial. I naturally maintain a high level of energy in-class and work hard to make sure students feel comfortable approaching me with questions and comments. Most importantly, my students know I sincerely wish for them to succeed in their goals. Exhibiting consistent encouragement and patience allows students to see I understand the struggles and frustrations that arise in learning mathematics and am there to support them. After all, learning new mathematics is challenging even at the research level. This part of my philosophy of teaching is particularly important for reaching students that lack strong backgrounds and are quickly turned off to mathematical challenges.
Mathematics is an important area of study for any college student, regardless of their major or career choice. Not only are the mathematical skills themselves useful, but studying math offers a unique opportunity to greatly boost creative thinking and problem solving skills. In my teaching experience, I have tried to emphasize this aspect of learning as much as possible so that the student gains the maximum benefit from their time both inside of and outside of the classroom. Additionally, the student is more likely to engage in class if they know the applications will be far reaching outside the classroom.
As a graduate student at Columbia University, I had my first exposure to teaching. I was the instructor of a few of my own classes and I had a substantial amount of TA work as well. During this time, especially through my work as a TA, I experienced many students asking me for help with homework or exercises. After addressing these sorts of questions several times, I realized quickly that it is the wrong approach to simply give students the answer to their questions. Rather, I would try to get them to work through the problem with me so that we can see exactly where they were getting confused and so that they could address the problem themselves. This way, the student knows exactly what areas they need to focus on for the future and they also learn how to approach other similar types of problems so that they are less likely to need help the next time. In this way, they greatly improve their problem solving skills in general as well, not just in their math class.
After receiving my Ph.D. in 2013, I taught for a year at Morehouse College. Morehouse is a smaller school with lower admission standards than Columbia University, so my time at Morehouse gave me a very different perspective on teaching. At Morehouse, I learned that sometimes it is beneficial to be very hands on with your students. I had very lively discussions with my students during my office hours, sometimes even several students from the same class. Again, it was more about working together with the students to get them to understand not just the right answer, but the right reasoning and the right process. I also had the opportunity to teach some precalculus level classes at Morehouse. This experience really helped me realize that in order to be a truly effective teacher you have to understand why a particular problem is confusing to the students in the first place. It’s not enough to simply show them the right way to do it, it is also important to understand the issue from their point of view in order to make the lectures as accessible and useful as possible. To this end, I encouraged frequent questions and tried to make sure it was a stress-free environment as much as possible. All these concepts were reinforced outside the classroom with office hours and frequent homeworks and inside the classroom with midterm exams and the final exam.
Lastly, during my time at Georgia State, I have had a lot of time to hone my teaching techniques even further. Now that I have more experience teaching, I have gotten a lot more skilled at detecting some of the more common points of confusion and some of the more common mistakes. During my lectures, I always try to emphasize the areas I know are common sources of confusion, and I take extra care to point out commonly made mistakes to try to get the students to avoid them. Additionally, I try to make sure they understand the process, not just the result. For example, in my Calculus I classes when we are learning derivative rules, I try to make sure the students don’t just memorize the formulas, but also understand how to recognize when each rule applies, always emphasizing the question “what step do you do first”. A student who is able to take any problem and break it down into a first step has taken big progress towards internalizing both the material of Calculus I and learning a valuable life lesson about how to approach unfamiliar problems.
All in all, I am very passionate about teaching, and about teaching math specifically. I greatly value the opportunity I have to help students grow both in and out of the classroom. I feel that I have really developed a style that the students can latch onto and relate to, as evidenced by the fact that many times each semester I have students asking me if I am teaching the next course in sequence the following semester. Even though I feel I have made great progress in my first few years of teaching, I know there is always room for improvement, and I will never stop looking for new ways to pull in the students and get them engaged in the material of the class.
When I am asked, “What do you teach?” the reply is always, “People.”
My teaching philosophy centers about the principle of motivating students to learn/engage/understand the mathematical concepts for application and the advancement of civilization.
I believe that anyone can learn the formalities of mathematics and everyone deserves to have a quality mathematics education by a content expert with applications to real life. My Ph.D. provides this expertise, and my education background (BS degree) fuels my curiosity about how people learn and what motivates their imagination and interest in the material.
While constructing lessons I have flipped the traditional lecture pedagogy. I have thrown out the traditional “What- How-Why” approach of the mathematic topic. We know this very well, for example: “Today, we’ll cover section 4.1, definition of a derivative today. Here’s how it works and then if there’s time, I’ll show you why you’d need it in real life”. My new philosophy is the “Why-How-What” approach. I start with a (seemingly nonmathematical) problem that needs a solution (picture, power point slide, etc.) and has real-world consequences/application and ask them to solve it. Students are emotionally engaged and motivated to solve the problem (that’s the ‘Why’ – and they are curious!). After they try to solve the problem and have exhausted their previous mathematical knowledge, they are responsive to ‘How’. They attempt to understand how to extend their knowledge to build a new/improved solution method. The entire time, they are connecting the new material with the limitations of the old material and working the solution because the answer means something to them. The final phase is naming the method/topic. This way, the student has the correct vocabulary going forward into more advanced mathematics/science classes.
In response to the success of this new pedagogy, I needed an assessment rubric that was more meaningful to the student – just like the material. Thus, I found a ‘mastery’ method of assessment that was already in practice at certain schools in Georgia. This assessment method requires that I assign a number of mastery to students’ test/quiz problems (0 = zero knowledge, 1 = minimal knowledge; 2 = developing knowledge; 3 = moderate knowledge; 4 = complete knowledge). Then students receive feedback on what they know rather than ‘subtraction’ of points for “incorrect answers”. From the mastery percentage, the students can learn how many points they have earned on a problem. Thus, for a 10 point problem, if they earned 3 mastery (3 out of 4 possible), they earn 75% of the points, thus 7.5 points go along with their knowledge.
Through this process, students are actively taking responsibility for their learning and are willing to master the material because they are motivated by real world solutions. In addition, they realize that they are in control of their grade/points by the mastery level they achieve. We work on in-class quizzes to get practice on what mastery looks like and we talk a lot about it in class. The students learn to be ‘self reflective’ on their work by reworking previous exams to analyze their mistakes and are moving towards an overall conceptual learning approach rather than using ‘practice tests’ as a guide for instrumental mathematics understanding. I am pleased since I can see an 80% (average) ABC rate since implementation.
My ultimate goal is for students to actively engage in the material during class, become passionate about lifelong learning using their mathematical ‘6th sense’, and to use their curiosity and imagination to drive future solutions to make a global impact on civilization.