Erin McNicholas: Teaching Portfolio

Note: This document is best viewed on-line at:  http://math.arizona.edu/~emcnicho/School_Page/TeachingPort/teachingP.html

Table of Contents

  1. Introduction
  2. Teaching Experience
    • Primary Instructor Responsibilities
      • Sample Course Documents
    • Teaching Team Member Experience
  3. Curriculum Development
    • Math in Modern Society Curriculum
    • Introduction to Global Climate Change Curriculum
    • High School Workshops
  4. Department and Community Involvement
    • Community Involvement
      • High School Workshops & Visits
      • Mentoring & Teaching Conferences
    • Department Involvement
      • Entry Level Committee
      • Algebra ConcepTests Working Group
      • First Year TA Mentoring
      • Freshman Advising
  5. Course Evaluations and Student Comments
    • Awards
    • Statistical Summaries & Student Comments

Introduction

My goal as an instructor is not only to provide students with an understanding of the mathematical concepts covered in class, but a sense of how mathematics ties into the world around them, and an appreciation for the breadth and beauty of the subject.  To give students a solid understanding of the underlying theory, I stress the reasoning and logic behind the formulas and theorems covered in class.  I believe it is important that students have a deep understanding of the concepts and are not merely learning to plug numbers into formulas or perform mindless algorithms.  I believe that students must take an active role in their learning.  To promote this, I use class discussion and group activities to develop class topics.

In addition to helping my students master the mathematical content and develop their critical thinking skills, I have specific goals for each course I teach.  These goals usually depend on the course and the student demographic.  In my business math courses I try to impress upon students the importance of math to their future business careers.  I also try to improve their computer skills and professionalism.  In Math in Modern Society, a non-prerequisite course, my primary goal is to improve students' confidence and enjoyment of the subject.

For more information on my pedagogical views, please see my Teaching Philosophy.

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Teaching Experience

The following is a description of my teaching experience as a graduate student at the University of Arizona.  Over the past five and a half years I've had the opportunity to work for both the math department and the Graduate Interdisciplinary Program.  My responsibilities have ranged from primary instructor to teaching team member.  I've had experience in both large (over 100 students) and small (35 students or fewer) lecture settings. 

Primary Instructor

As primary instructor for several of the math department's undergraduate courses, my duties include: preparing and presenting all course lectures; assigning and grading student projects and homework; writing and grading all course exams; and assigning final grades.1  In addition, I had the opportunity to develop part of the curriculum for MATH 105: Math in Modern Society.  A description of this curriculum development can be found in the Curriculum Development section.  The enrollment in each of these math courses is capped at 35 students.  Links to sample course documents as well as descriptions of the course and the teaching methods I used, are provided below.  Student evaluations and feedback are provided in the Course Evaluations section.

Part of a Teaching Team

 1 For some of the large enrollment, multi-section courses like College Algebra,  there is a common final exam containing problems submitted by all section instructors.

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Curriculum Development

I've been fortunate to have the opportunity to develop curriculum for several courses and workshops.  I have always been interested in curriculum development, particularly in the development of interdisciplinary courses and projects.  I took the University's Teaching with Technology (TTE 501) course, where I developed projects and lesson plans aimed at the high school and undergraduate level.  These projects and lesson plans were based on NCTM Standards and incorporated technology and science applications.   Through this course, I learned how to perform class experiments using the TI View Screen and associated Calculator Based Laboratory accessories.  I also learned to use Geometer's sketchpad, Hyperstudio, Graphmaster, and several other math education software packages.  While I believe technology can be an effective tool for facilitating student understanding and interest, it needs to be implemented with care.  I try to use calculators and other tools to help students explore relationships and concepts covered in class.  At the same time, I stress the limitations of this technology and the importance of understanding the underlying theory when interpreting the output of these devices.  

Below is a description of the projects, course documents, and lessons I've developed for the Math in Modern Society course, the Introduction to Global Climate Change course, and the high school workshops on Cryptography and SETI. 

Math in Modern Society Curriculum

Since the majority of my students in this course were art or music majors, I wanted to choose topics which would stress the symmetry and beauty of mathematics, while simultaneously strengthening students' logic and reasoning skills.  I decided to do a unit on Graph Theory, and one on Groups, Fields, Symmetries, & Codes.  Since this second unit is not covered in the course text book, I obtained permission from the department to deviate from the text.  In addition to developing the content of this unit, I created supplemental course documents for student reference. 

My students responded extremely well to this unit on Groups, Fields, Symmetries, & Codes.  Their interest and involvement in the class increased.  For their second term project, I gave them a wide range of topic choices, but every student chose to do a project relating to this unit.   The quality of their Project 2 reports was noticeably higher than their Project 1 reports.  I've included a sample student report below.  The author of this report was a music major.  He researched the Fibonacci numbers and their appearance in music, and then applied what he had learned about modular groups to compose a piece of music using the Fibonacci sequence.  Using his background in music, he analyzed the musical properties of his composition.  

As part of this unit, I reserved a classroom for the evening and showed The Proof, a Nova episode describing Andrew Wiles proof of Fermat's last theorem.  Attendance at this optional showing was high, and several students remarked afterwards how surprised they were to understand some of the mathematics discussed in the show.   It was exciting and rewarding to see students working with and understanding advanced mathematical concepts such as modular groups and fields.  I think the inclusion of this unit gave students a much better understanding of what mathematicians do and what the subject of mathematics encompasses. 

Introduction to Global Climate Change Curriculum

During the spring semester of 2004, I received a special Graduate Interdisciplinary Program TA-ship to work on  curriculum development for the general education course, Introduction to Global Climate Change.  Having served as a graduate teaching assistant for this course during the fall, I was familiar with the course content and format.  

While serving as a TA for the course, I felt too much time was spent at the beginning of the semester covering the math and physics topics necessary to understand various processes of climate change.  It was my impression that students started the course eager to discuss and learn about climate change, and this beginning background material had diminished their interest.  I suggested we reorder the topics of the course, covering the math and physics concepts in the context of climate change, rather than as stand alone units.  I felt this reorganization would provide students with motivation and a clear understanding of how physics and mathematics tie into the study of climate change.  I reviewed each of Dr. Hirschboeck's PowerPoint lectures, determining which lectures would have to be altered to effect these changes.  I provided Dr. Hirschboeck with specific suggestions on how these alterations might be made.

In addition to proposing a reorganization of course topics, I reworked existing class assignments and developed new projects and activities.  I created a detailed syllabus outlining these changes.  This syllabus laid out the lecture schedule, in-class activities, homework assignments and due dates.  The changes I proposed, as well as the class activities and assignments I developed, were adopted and are currently being implemented.  

High School Workshops

 Each semester, local high school math instructors are invited to bring their classes to campus for a day-long mathematics workshop.  Each workshop is based on a particular mathematical subject or application, and has one to three high school classes in attendance.  This program is run and organized by the graduate students in the department.  Through a series of interactive lectures and hands-on activities, these workshops teach concepts not covered in high school courses.  Each workshop has a lead person, several graduate student volunteers, and a number of undergraduate volunteers.  The lead person develops the content of the workshop, organizes the volunteers, and leads various sections of the workshop.  The remaining sections are led by graduate volunteers.  Undergraduate volunteers help facilitate activities, and answer students' questions about college life.  The purpose of these workshops is to encourage high school students to take higher level math courses and consider careers in mathematics.  University recruitment is an added benefit.

 

During the spring 2003 and spring 2004 semesters, I organized and led workshops on Cryptography.  These workshops covered Caesar, Playfair, and the ADFGVX ciphers, as well as one-time pads and the Vigenere square.  Public Key Exchange methods and modern cryptographic practices were also discussed .  In the process, students were introduced to matrices and modular arithmetic.  Students had the opportunity to practice their newly acquired cryptographic skills through a series of activities and competitions. 

 

While the cryptography workshops were highly successful, they contained units involving exponential functions which made them too advanced for some high school classes.  During the spring 2005 semester I developed and led a workshop on SETI (the Search for Extraterrestrial Intelligence), which was designed to be accessible to Algebra I students.  This workshop demonstrated that even the most hypothetical situations can lead to interesting mathematics.  Students worked together to decode the Arecibo message which was broadcast into space in 1974.  The length of this binary message is the product of two prime numbers (73 and 23).  Arranging the message in an array of 73 rows and 23 columns reveals a pictogram conveying information about our planet, our genetic makeup, etc.  In the process of decoding the Arecibo message, students were introduced to various topics in chemistry, astronomy, and biology.  They reviewed prime and composite numbers, and learned ways of representing data.  In particular, they learned how to represent molecules numerically using the information contained in the periodic table.

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Department and Community Involvement

Community Involvement

I believe the role of an effective instructor extends beyond the classroom.  Universities serve not only their students, but the surrounding communities as well.  It is our responsibility to create ties with that community and take part in pedagogical debates occurring at local and national levels.

  • High School Workshops & Class Visits 

    During my time at the University of Arizona, I have organized and led three High School Workshops.  These day-long workshops are held on the University of Arizona Campus, and are open to local high school students.  These workshops emphasize the applications of mathematics and present topics not usually covered in the high school curriculum.     For more information on these High School Workshops, see the Curriculum Development section above.

    I've also taken part in the Calculus Class Visitation Project.  Each year the department sends a professor, a graduate student, and an undergraduate math major to visit every AP high school Calculus class in Tucson.  During these visits we talk to students about our experiences in mathematics, what mathematicians do, and how mathematics relates to various professions and careers.   Having participated in the Student Internship Program at Sandia National Laboratory, I talk in detail about research experiences and internships available to high school and undergraduate students.   The purpose of these visits is to encourage students to consider careers in mathematics, and to take more advanced mathematics courses when they come to the University.

     

  • Mentoring & Teaching Conferences    

    I was a member of the student panel at the 2004 AMS-NSF Conference on Mentoring in Mathematics.  As a panelist, I prepared and delivered a brief statement on what I believe makes someone an effective mentor.  I  also answered questions from the audience and had an opportunity to discuss undergraduate advising one on one with several of the conference attendees.

    During the 2006 AMS/MAA joint meetings in San Antonio, I took part in a panel discussion organized by the Young Mathematicians' Network (YMN) on making the transition from an undergraduate institution to graduate school.  

    Derek Habermass and I gave a talk at the 2006 MAA Southwest Sectional Meeting.  In out talk, ConcepTests: Implementations & Examples from a Wide Range of Courses, we provided examples of conceptests and discussed our experiences using them.

Department Involvement

I constantly strive to improve my teaching.  Involvement in departmental activities gives me the opportunity to learn from my colleagues.  Sharing classroom techniques and ideas, we are able to develop new and more effective ways of teaching our students.  Furthermore, involvement gives me the chance to influence department policies while gaining a better sense of how my teaching efforts contribute to the department's overall mission.  

  • Entry Level Committee  (2005-2006 Academic Year)

    I am currently the graduate teaching assistant representative on the Entry Level Committee.  This committee meets bi-monthly and works to improve the quality of entry level mathematics courses.  These courses include Math in Modern Society, Algebra, Trigonometry, Pre-Calculus, Calculus, and the Business Math sequence.  We examine the effectiveness of the currently adopted course text books and discuss other possible texts.  We look at student pass rates in these courses, and their corresponding placement grades to determine whether our current methods of student placement are effective.  We discuss ways of preparing new TAs, and discuss pedagogical issues which are common to all entry level courses (such as ways of improving students' basic algebra skills).

  • Algebra ConcepTests Working Group  (Fall 2005)

    I am currently taking part in a working group to develop ConcepTests for the department's College Algebra course.  ConcepTests were developed by Harvard professor Eric Mazur [1] to improve his students' conceptual understanding of  Physics concepts.  ConcepTests have since been developed for a variety of other courses, including Calculus [2].  ConcepTests consist of challenging multiple choice problems which require a deep understanding of course concepts to answer correctly.  Student are given a minute to think about their answer to the question, and then they respond using electronic responders, or by holding up flashcards.  This gives the instructor immediate feedback on how well students are following course topics.  After answering the question individually, students discuss their answers with their neighbors.  Through these discussions, students engage in peer instruction while improving their communication and reasoning skills.  After a few minutes of discussion, students answer the question again.  Studies have shown that implementing ConcepTests in the classroom improves students' performance on both conceptual and algorithmic problems [3].  

    As part of the College Algebra ConcepTest Working Group led by Dr. Lomen, I meet weekly with other College Algebra instructors to develop ConcepTests and discuss our experiences implementing these tests in the classroom.  I have been very impressed by my students' response.  They are clearly engaged and interested in discussing their answers with their neighbors.   I have noticed a marked improvement in my students' ability to express mathematical ideas verbally since implementing these tests.

  • First Year TA Mentoring  (Fall 2005)

    Many graduate students begin their first semester teaching College Algebra or Trigonometry.  While these graduate students undergo departmental and university TA training during their first year, it can be difficult to adjust to graduate courses while simultaneously teaching for the first time.  In an attempt to better help these students become effective teachers, I was asked to sit in on several classes throughout the semester, and then meet with each graduate TA to discuss their teaching techniques and possible areas of improvement.  After meeting with the TA, I meet with their faculty supervisor to discuss the TA's progress.  

  • Freshman Advising  (Summer 2001)

    Throughout the summer, incoming freshman come to campus for two day orientation and advising periods.  During this time they take placement exams and enroll in courses for the fall.  The department has two placement tests which are used to place incoming freshman who do not have advanced placement or college transfer credits in mathematics.  At each of these orientation sessions, I met with freshman to discussed their placement exam results, and help them determine which math classes would be most appropriate given their placement scores and expected major.  

[1] Eric Mazur, Peer Instruction - A User's Manual.  Prentice Hall, NJ, 1997
[2] S. Pilzer, M. Robinson, D. Lomen, et al., ConcepTests - to accompany Calculus, Hughes-Hallett et al., John Wiley & Sons Inc., NY, 2005
[3] A. Fagen, C. Crouch, E. Mazur, Phys. Teach., 40,  206-209 (2002)

 

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Course Evaluations and Student Comments

While student evaluations are by no means a perfect measure of teaching effectiveness, I value the feedback of my students and take their comments and suggestions seriously.  My student evaluations have been consistently high, even in the Business Math courses which traditionally receive lower evaluations.  Each semester students fill out an anonymous scantron form provided by the university, and a short answer questionnaire developed by the mathematics department.   Statistical summaries of these scantron results are provided below, as well as a sample of student comments from the department's questionnaire.  

Statistical Summaries & Student Comments

The scantron questionnaires for my Math in Modern Society course were lost.  Since there is no statistical summary of student feedback for this course, I have included a PDF file containing all the student responses to the department's short answer form.  These forms are unedited.  

The following link contains a sampling of student comments received in letters, e-mails, and on the math department's anonymous, short answer questionnaire form.

  • Sample of Student Comments (Coming Soon)

Awards

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