TEACHING PORTFOLIO


Winter 2002

Table of Contents

  1. Teaching Philosophy
  2. Teaching Toolbox
  3. Evaluation of Teaching
  4. Development at UCSB
    • Teaching Assistant Training
    • STIA
    • Technology
    • Teaching Associate Experience
  5. Future Plans

Appendices:

1. Teaching Philosophy


A college education develops the critical thinking skills and the moral behavior of people in order for them to become better citizens. As professors, we develop students' critical thinking skills by having them take a wide variety of classes that require them to think outside the box and to develop a pattern of learning that will last a lifetime. We develop their moral behavior by teaching them more about the world around them and their contributing role in its existence.

As a mathematics professor, I have the special opportunity to accomplish these goals through a subject area that most students do not enjoy. It is my role as a teacher to guide the students in their struggle to understand and appreciate the beauty of mathematics using the many different tools in my Teaching Toolbox . As a result of this struggle, students develop a better understanding of the world they live in, they can communicate more clearly in our technical society, and they have developed the tools necessary to continue their learning process throughout their lifetimes.

2. Teaching Toolbox

Word Problems:
The centerpiece of my Teaching Toolbox is the use of word problems. Since my primary goals involve how the students are going to use mathematics outside the classroom, word problems are essential. Word problems help students to see how mathematics can solve problems that they might encounter outside the classroom. They also require students to transfer ideas from English into mathematical terms in order to develop the idea and solve the problem and then back to English to communicate the solution. Thus they are the ideal method for accomplishing the goals outlined in my Teaching Philosophy.
Homework:
I assign homework for many different reasons. The primary reason is to promote constant feedback and interaction. By having the homework turned in as often as every class period, I am able to discover how the class is doing with the material so that I can change the pace of the class or review if appropriate. I am also able to give the students constant feedback so that they can make choices about the type of material on which they need to focus.
Another reason for homework is to motivate students. Unfortunately most college freshmen are not self motivated. What I have learned through my teaching experience is that I have to make homework count toward students' final grades in order to motivate them to complete the homework assignments. Because of this as one can see from my most recent syllabus (see Appendix C), I now make homework count as a large part of the student's grade.
The final reason for assigning homework is to give the students an opportunity to discover how to tackle difficult problems on their own. In order to accomplish this task, as part of every homework, I assign problems which cannot be done without referring to another part of the text or another text altogether. By doing so I am teaching them how to read a math textbook by making them use items such as the Table of Contents and the Index which most of them never use.
Projects:
One of the most rewarding teaching techniques that I have incorporated into my teaching toolbox has been special projects. (For some examples of these projects, the reader is referred to Appendix D .) The goal of the projects is two-fold.
First, the projects are all designed to show the connection of mathematics with the world around them. For example, one of the projects had the students study the effects of credit card debt. It was very rewarding when, as the students handed in the assignment, they commented on how credit cards can really cause a lot of problems if they get out of hand. At that point I realized that I had accomplished one of the goals from my Philosophy of Teaching by helping the students to develop their moral behavior.
Second, projects teach students how to communicate using mathematics. Each of the projects requires the students to type out their discoveries in a professional way. For example on the credit card project they had to type a letter explaining their circumstances and how they plan to get out of debt.
Exams:
The use of exams is one area that most instructors feel is not important in teaching. In fact, I have heard several people say that they do not like exams because it takes away from classroom time where the students could learn something. If exams are used properly, they become one of the most useful tools in our Teaching Toolbox. The best exams do more than just evaluate a student's understanding of the material in order to assign the student a grade. Exams are also useful in helping the students pace themselves, giving the students feedback to help them know what they need to spend more time learning, helping the students see the connections between the different topics covered, and helping the instructor know what material needs to be reviewed in the future.
In order to accomplish these goals, exams must be given frequently and need to be written in a way to discover where the weaknesses are in the students' understanding. One way that I have been able to minimize the consumption of class time is to create multiple choice exams that test on conceptual concepts rather than computational concepts. (See the sample exams in Appendix E .) Another way that I try to accomplish these goals is by grading all of the exams in a timely manner, preferably the day the exam is given, so that I can change the class work to fill in the gaps in understanding. Also, by passing back the exam shortly after the exam is taken, the students are able to make decisions about how they can improve their learning methods so that they can earn a better grade on the next exam.
Group Work:
One of the primary purposes of group work is to get the students to interact with one another and to communicate mathematical concepts. I have also found that teaching is the best way to learn material and working in groups encourages the students to teach one another. As a result of this, students are being taught using a wider variety of methods and are able to understand the same material from different perspectives. Finally, I have found that having the students work in groups helps to maintain the energy level of a classroom. No matter what the classroom size, from 15 to 150, when I sense that people are getting bored or are not quite understanding the concepts I have them break up into groups and give them a problem to work on. I then move around to the different groups and help them to get past their roadblocks. It is at these times that I have been able to be most effective with the students who are having the most difficulty with the subject matter.
Quizzes:
Quizzes, whether scheduled or not, are one of the best ways to get students to take the material seriously and to give them instant feedback. One of my special tools is the bonus quiz. I pull out this tool when I feel that students are starting to become apathetic toward mathematics because they are struggling with the concepts. What I do is I give the students four or five problems at the beginning of class and tell them that they need to turn in their results by the end of the period. They are allowed to use anything to solve the problems, especially myself. Since I call it a quiz and make it extra credit to their homework scores, the students take it seriously and work hard, even though it has yet to really change a grade directly.
Internet:
The internet, when used properly, is a very powerful tool in the teaching of mathematics. Unfortunately, the proper use of the web has not been completely developed. An option that I use the internet for is to refer students to some of the tutorials available on-line. One example is the Visual Calculus program from the University of Tennessee. On this site students are able to supplement the material learned in class with practice quizzes and tutorials. It is also able to give the students immediate feedback on how they are doing through practice quizzes.
Another use of the internet is for distance learning programs such as the Calculus Quest program at Oregon State. These programs work well for the self motivated student who is able to learn from a textbook. However, for the average student there is still a need for personal interaction to help them through the process. Therefore, the only way that I currently can see a program succeeding for the average students is to make sure that each student has access to a tutor as well as the on-line course.
A third use of the internet is to help with real life projects. With the flow of information over the web, it is becoming easier to create real life projects for the students to work on. One project that I have done for my precalculus class is to have them model the tides with trigonometric functions. They retrieve the tidal information from the National Oceanic and Atmospheric Administration web site at http://www.noaa.gov and create a model of the tidal flow over a period of a week.
When used properly, the internet can be very useful. It is up to us as teachers to make sure that we use it in the most beneficial manner for our students.
Computer Algebra Systems:
With the advent of the hand held computer algebra systems from Texas Instruments, math teachers are having to face new and challenging questions. I have seen the benefit of using a CAS in my Differential equations classes to help the students to find the eigenvalues of a matrix. However, I first have them do the process without the technology, and then teach them how to use the technology. This way I can help them to become too dependent upon their calculators. For more information about Computer Algebra Systems and my opinions on their use, the reader is referred to my paper in the appendix .

3. Evaluation of Teaching


Throughout my time at UC Santa Barbara, my teaching philosophy and abilities have been greatly affected by the feedback of students.

One thing that I have realized from my evaluations is that students enjoy someone who is high energy. When I had my class of 200 students break up into groups and I went from group to group to help them, evaluations were wonderful. (See the ESCI Form for Spring 2000 ) Because of this feedback I have tried to come into my classes with more energy. It also has the added benefit of making teaching even more rewarding.

Another part of my teaching that has been developed by reading student evaluations has been the way I communicate with the students. I have had problems coming across as condescending to the students when they ask a question. As one can see from my evaluations for Fall 1996 many of my students felt that I did not allow for questions to be asked in class. Since then I have made changes in the way I interact with students and when one looks at my most recent evaluations this area has greatly improved.

I have also discovered that many students struggle with working in groups. This has supported my understanding that it is not beneficial to have the class work entirely in groups. When I did this as part of a special program for at risk students, many of the students did not feel like they were getting enough instruction without a traditional lecture. (See the evaluations for Fall 1998.) In response to these comments, I have tried to find a good balance between telling the students how to do the problems and guiding them through the discovery process themselves. It is a fine balance and one that I am continuing to improve and evaluate with each group of unique students.

A final topic that seems to come up over and over on evaluations is that students are very happy when they know exactly what will be on the exams. It has been a challenge to balance having the exams too predictable and making sure that they test understanding and not just the ability to regurgitate information. I was definitely able to do this during the summer of 2001 with my differential equations class. I was able to create exams that tested just conceptual understanding and thus I was able to tell them exactly what the exams would look like.

Overall, the feedback of students has been very helpful in developing my abilities as a teacher. No matter where I go and what type of teacher evaluations that the school has I will continue to get feedback from students in every possible way.

4. Development at UCSB

Teaching Assistant Training
When I arrived at UCSB I had already taken some courses in a teaching credential program and so the portion of the training designed to help us learn to teach was mostly review. However the TA Training program did not just focus on how to teach. I also learned how to interact with other graduate students as well as the faculty at the university. I realized that teaching at a university like UCSB involves more than just what occurs in the classroom. It also involves helping students deal with the university around them. As a TA the students (especially the freshmen) looked to me for advise in how to work with the system of such a large school as UCSB. Even though I was also new to the campus, because I had already been on campus for several weeks with my department orientation and the campus wide orientation I was able to know where to direct the students when they had questions.
STIA
During the summer when I participated with STIA I was able to develop my teaching in new and unique ways. Throughout my time at UCSB, a majority of my interaction came from within my own department. What STIA allowed me to do was to learn what is happening in other departments and how they go about teaching their subjects. This was particularly useful for myself since as a mathematics teacher I have been charged with the task of training the students at UCSB to be ready to go into their major classes. For instance I was able to learn from a Psychology grad student about how impacted the psychology major is and why the psychology students are required to take calculus and what they are supposed to learn during the class. I was also able to learn from Physicists how similar the physics classes are to the mathematics classes. I was able to take some of their ideas, such as using physical models, that they use in their classes and apply them to my own classes. (See the exams from my Differential Equations course.) Overall, I found the STIA experience very useful and I look forward to working as a STIA fellow this upcoming summer.
Technology
During my time at UCSB I have learned that the use of technology in the mathematics classroom is one of the most controversial subjects in the mathematics community. Within our own mathematics department we have both extremes on the subject. We have those who believe that using technology in the classroom is essential in the teaching process and we have those who believe that it is the worst possible way for the students to be taught be cause they learn no mathematics. Because of these politics, as a grad student I have learned to stay out of the debate. However, I have taken advantage of the opportunities to learn more about the technology that is available for the teaching of mathematics. One example was the conference that I attended at Loyola Marymount where one of the main talks was on the use of technology in the mathematics classroom. It was at that conference that I discovered the concept of using the technological tools that the students would be using outside the classroom such as spreadsheets. One of the speakers gave a talk about what the other disciplines wanted from the mathematics curriculum and almost every one of the disciplines wanted the students to be able to use spreadsheets to work with mathematics. Not one of the disciplines mentioned a graphing calculator or a computer algebra system. (For more information about computer algebra systems click here .) Because of this, as well as some of my personal experiences teaching with graphing calculators, I have decided to look into how to use spreadsheets in the mathematics classroom as well as internet based teaching tools such as java applets and internet tutorials. (See the Archives Site for some examples of these.)
Teaching Associate Experience
One of the best parts about my time at UCSB has been the opportunity to teach many of my own classes. The experience that I have gained in doing so has changed the way that I teach as well as given me a better idea of what I would like to teach in the future. As a teaching associate I have been able to experiment with different teaching styles and techniques. I have been able to use small groups, lectures, projects, homework, and many other techniques in order to develop my teaching toolbox. I have found through this experience that the best way to learn how to teach more effectively is to teach and to see what works for the different types of students. It is something that cannot be learned in a classroom and changes with each new group of students.
Also through the teaching associate experience I have found that I really enjoy teaching the lower division math courses that most people would like to avoid. I have found that I am able to really help these students to move beyond their fear of mathematics and to see the subject for what it is. Because of this I plan to focus on these areas of instruction in my future faculty positions.

CCUT Portfolio
By compiling the CCUT portfolio I was able to reflect on my experiences at UCSB and to learn from them.  Without this requirement, I probably would have missed out on the reflection experience and the immense learning that goes with it.  This reflection has helped me to see how my teaching has developed through the years by reviewing my teaching evaluations.  I have also been able to develop my teaching philosophy and to verify that how I am teaching genuinely reflects my philosophy.  It is all too easy to fall into a rut and to not teach the way that you intend.
Besides the evaluative benefits of the CCUT portfolio the portfolio ended up being very useful for my job search process.  I was able to make CD's of my porfolio to give to each of the schools that I applied to.  While the job that I decided to accept did not see this portfolio since it is primarily a research postdoc in my next job search in a couple of years I will update my portfolio and use it again. 

5. Future Plans

This coming fall I will be a visiting assistant professor at the University of Tennessee at Knoxville (UTK).  While the position is primarily research oriented, I look forward to developing my teaching techniques.  One thing that I am planning to do is to participate in the Project NExT program.  This is a program established by the Mathematical Association of America to develop the future of mathematics university professors.  Another thing that I look forward to is the opportunity to teach upper division undergraduate classes.  This is a new opportunity and should be a great learning experience that will be very beneficial for my future teaching.

I would also like to get involved in the training of teachers for the next generation of students. I would like to show them how all of the different math subjects are interrelated. For instance how Abstract Algebra relates to high school Algebra. I would also like to become involved in programs that develop math students writing ability. Because I have found as a teacher that the communication of mathematics is often more important than the computation. This is particularly important for future teachers.

Most importantly, I am looking forward to becoming involved in the entire life of the university. As a member of the faculty, I see it as my responsibility to do more that teach mathematics. Our goal is to develop students' critical thinking skills by requiring them to think outside the box and to develop a pattern of learning that will last a lifetime and to develop their moral behavior by teaching them more about the world around them and their contributing role in its existence. Whether this would be through committee assignments, student mentorship projects, or community activities I am excited to be a part of a growing and developing university community.



Appendices

Appendix A: STIA information


STIA Certificate

Appendix B: Technology research paper


"The Use of a Computer Algebra System in First Year College Mathematics Classes "

Appendix C: Sample syllabi


Precalculus Summer 2001
Linear Algebra and Differential Equations Summer 2001

Appendix D: Sample project


Credit Card Project for Precalculus

Appendix E: Sample Exams


Differential Equations Exam #1
Differential Equations Exam #2
Differential Equations Exam #3

Appendix F: Student evaluations and comments