# 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**

- Primary Instructor Responsibilities

- Sample Course Documents

- Teaching Team Member Experience

- Math in Modern Society Curriculum
- Introduction to Global Climate Change Curriculum
- High School Workshops

- Community Involvement

- High School Workshops & Visits
- Mentoring & Teaching Conferences

- Department Involvement

- Entry Level Committee
- Algebra ConcepTests
Working Group
- First Year TA Mentoring
- Freshman Advising

- Awards
- Statistical Summaries & Student Comments

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.

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.

**Math 110: College Algebra**(Fall 2005 - 4 units)

Section Web Page; Section Calendar; Sample
Exam, ConcepTest Examples

College Algebra is a prerequisite course for Calculus, Business Mathematics,
Statistics, and several science courses. Each semester there are 30 to 60
sections offered, and a common final is administered to all sections. I
strive to maintain enough flexibility to respond to my students
needs, while staying on schedule and preparing them for their future classes
and the final exam. While many of the topics in this course have been
introduced to students in high school, we stress a deep understanding of the
subject rather than an algorithmic approach. To this end, I often use ConcepTests during the course lecture. These are
multiple choice questions that require a deep understanding of the subject to
answer correctly. Students hold up colored flash cards to signify their
answer. This allows me to get a quick and accurate assessment of how many
students have grasped the subject. I then have students discuss their
answers in small groups, and after a few minutes I ask them to answer the
question again. Students have responded remarkably well to this
process. Circulating through the class and listening to their
discussions, I am impressed by the level of their debates and mathematical
reasoning. It gives students valuable practice expressing mathematical
concepts verbally. Furthermore, there is no better way to learn a concept than
to try and teach it to someone else. As students share their answers and
ideas, their understanding is strengthened.

In response
to poor pass rates amongst incoming Native American students, the university
decided to enroll all incoming Native students taking College Algebra into the
same section. Teaching this section has been a very rewarding
experience. Because these students work closely together in many of their
classes, they have brought a sense of community and collaborative learning to
the class. Even my non-Native students seem to be working more in study
groups and participating more in class. Several of the Native American
students are enrolled in a program which promotes going to instructor office
hours and tutoring lab hours. As a result I've had the opportunity to
work extensively with these students one-on-one. I recruited two Native
American math majors to work as preceptors for the course. They hold
regular office hours and participate in class discussion sessions. These
preceptors are an excellent resource for all my students, and provide valuable
role models. For the most part, these Native American students are doing
very well in this course. I fully expect a very high pass rate among
these students.

**Math 115A: Business Math I**(Summer 2005, Spring 2004, and Spring 2002 - 3 units each)

Section Web Page, Section Policies, Outline of Project #2, Guidelines for Project #2 Write-Up

Developed here at the University of Arizona, the materials for this course have
been published and distributed by the MAA. This is an in-depth
interdisciplinary course coalescing topics from computer science, mathematics,
and business. The electronic course textbook consists of interlinked
PowerPoint presentations. Students spend the semester working in groups
on two, involved, term projects. In the
first project, students perform an in-depth statistical analysis of historical
bank loan records in order to determine whether or not they should foreclose on
a particular borrower. In the second project, they determine a fair
market price for a European call option. Mathematical concepts and
computer skills are developed and presented in the context of these
projects. Topics covered in this course include: basic probability and
statistics; probability density functions;
cumulative distribution functions; and Bayes' Theorem. In addition to
teaching students the mathematical content of the course, I taught them the
business applications of that content, and how to use advanced Excel and
PowerPoint features. These classes are taught in the Integrated Learning
Center where all classrooms are equipped with computer projectors, document
cameras, and lap tops for the students. Thus, I was able to demonstrate
the computer skills in class and have students practice them on their
laptops. For each project, students submitted a written and oral
report. All student work was word-processed and all student presentations
were delivered using PowerPoint. Samples of student work can be provided
upon request.

The first
time I taught this class, I worked hard to prepare and present the lectures in
PowerPoint in order to provide students with a model for their own
presentations. After having taught the course once, I felt that
presenting the lectures in PowerPoint had distanced the students. I found
they were far too content to sit back and watch, rather than take notes, question,
and become actively involved in the class. To promote active learning
instead of passive listening, I decided to use board work to present the
mathematical content of the course. This gave me greater
flexibility. I was able to lead students to the relevant mathematical
concepts through examples and discussion. I continue to use the computer
projection system to demonstrate various Excel, Word, and PowerPoint
features. Using StarBoard software,
I am able to make annotations directly on the
screen.

**Math 115B: Business Math II**(Fall 2002 - 3 units)

Section Web Page, Sample Lecture Notes

This is the second course in the two semester Business Math sequence.
Like Business Math I, it uses an electronic text and encompasses topics from
business, computer science, and mathematics. The mathematical and
computer content of the course is developed around two term projects. In
the first project students use marketing data to determine the demand function
for a new product and find the price which will maximize profit. In the
second project, students determine a bidding strategy to use in an oil lease
auction. In this course I taught students to perform more advanced
simulations using Excel. I also built on the presentation and report
writing skills they gained in Business Math I. The mathematical content
of the course includes basic differentiation and integration.

I was
awarded the Eller Business and Public Administration Student Council
Certificate of Appreciation for my work teaching this course.

**Math 105: Math in Modern Society**(Fall 2003 - 3 units)

Section Web Page, Project 2 Options, Lecture Notes (Modular Arithmetic), Lecture Notes (Cryptography), Sample Student Report

While Math 105 satisfies the University's math requirement for graduation, it
is not a prerequisite for any additional math or science courses. Thus,
there is greater freedom in choosing the mathematical content. The course
is divided into four units: Surveys & Statistics; Consumer Finance; and two
units chosen from the topics covered in Peter Tannenbaum's
*Excursions in Modern Mathematics, 5th ed.*
Students in this course are predominately from the College of Fine
Arts. Many students started the semester with little to no
mathematical confidence, and some were openly hostile to the subject.
These are bright, creative people, but their mathematical background is often
weak. Because the majority of my students were Art or Music majors, I
selected topics for the two open units which would emphasize the beauty of pure
mathematics. I wanted topics that students had not been exposed to
before. Having struggled through math classes in high school, several
students felt behind before the semester even started. By choosing topics
that would be new to everyone, I wanted to instill in them the feeling that
they were all starting on the same footing, that their success in this course would
not be hindered by previous failures.

I worked
with Jill Newby, the mathematics librarian, to create a resource web page for
students. During the semester, students produced two term projects, and I was very impressed with their work. It was
rewarding to see students applying the concepts covered in class to topics
related to their major. For more information about student reports,
please see the Curriculum Development section.

**Math 111: Trigonometry**(Fall 2000, and Spring 2001 - 2 units each)

Teaching
Trigonometry was my first experience as a primary instructor. This class
is a prerequisite for Calculus, and a requirement for students in the College
of Architecture. For most lectures, I combined board work with class
discussion. I assigned small, in-class, group projects incorporating
calculator experiments and mathematical modeling. I then had
students present their results to the class. Using a TI graphing
calculator and view screen, I was able to instantly graph functions and point
out relationships to students.

**Part of a Teaching Team**

**NATS 101: Introduction to Global Climate Change**(Fall 2004)

Course Web Page

The general education requirements at the University of Arizona include various
interdisciplinary courses. Competitive TA-ships offered through the
Graduate Interdisciplinary Program (GIDP) provide funding for graduate students
wishing to assist in the teaching of these large courses. Working with
Dr. Hirschboeck from the tree ring department, I
gained first hand experience
developing interdisciplinary course activities and working with
undergraduate preceptors.

Each fall
there are approximately 150 students enrolled in Introduction to Global Climate
Change, which is taught in the Integrated Learning Center.
To promote student involvement we used wireless responders, allowing students
to answer questions posed during the lecture. The responses to these
multiple choice questions were displayed using PowerPoint. Incorporating
responders into the lecture made it possible for us to determine how well
students were grasping subjects covered in class. It also allowed us to
pose more philosophical questions to the class and get a sense of how the
students felt about various issues. The class was divided into 20
groups, each containing seven to ten students. Students discussed,
debated, and performed experiments with their teams. Undergraduate
preceptors were recruited from the class. Dr. Hirschboeck,
myself, and three other graduate TAs worked closely
with these preceptors outside of class, preparing them to lead group
discussions and facilitate lab experiments. In addition to working with
undergraduate preceptors, my duties included performing class demonstrations,
grading assigned papers and projects, and holding regular office
hours.

During the
spring of 2005, I was awarded a GIDP assistantship to work with Dr. Hirschboeck on improving the course content. I proposed a reorganization of course topics, and
developed new projects and activities. These changes were adopted and are
currently being implemented. For more information on this work, see the Curriculum Development section
below.

**Math 527: Principles of Analysis**(Fall 2003 & Spring 2004 - 1 unit each)

As a
teaching assistant for the Applied Mathematics Program's graduate analysis
course, I held weekly recitation sections and office hours. I prepared
and delivered mini-lectures on background material relevant to course topics,
and helped students work through challenging problems.

**Math 263: BioStatistics**(Spring 2005)

Assisting
Dr. Kim, I prepared and presented several lectures in his absence. I also
created homework keys and graded class assignments. These assignments
included a series of projects performed using Minitab.

^{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.

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.

**Sample Student Report****Examples of Supplemental Course Documents**

**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.

**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)

**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**

·
**Outstanding Graduate
Teaching Assistant Award, College of Science **(Spring 2004)

·
**Eller Business and Public
Administration Student Council Certificate of Appreciation **(Fall 2002)

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Last Updated October 1, 2006