A New Kind of Calculus Class

The idea for this paper stemmed from a desire to construct a concept of feminism. As a math major taking an English class on feminism I wanted some way to realize what I have learned in this class. Having come to the realization that I have only practiced critical thinking - a form of de-construction - I wanted to try problem-solving - a form of construction. I read Peggy McIntosh' s article about an ideal feminist education as a challenge to create a classroom where every student can succeed (by success I mean feel included and learn from the class).

She gave examples in Biology, Philosophy, History of Art and English; I want to construct an example in Math. Since I am applying to graduate school for a PhD in mathematics, within the next two years I expect to have to teach an introductory calculus class at the undergraduate level. Thus I want to design a calculus class that embodies as much as possible McIntosh's ideal classroom. I set it in a classroom at University of Washington, Seattle, one of the schools I may attend. In order to design this I also read Eleanor Duckworth's piece on "The Having of Wonderful Ideas." She emphasizes encouragement of creative thinking in students in the classroom. The one other piece of information I kept in mind was due to John Dewey: "We must base our conception upon societies that actually exist, in order to have any assurance that our ideal is a practicable one. [T]he ideal cannot simply repeat the traits which are actually found [already; it must include change as will improve it]. The problem is to extract the desirable traits of forms of community life which actually exist,and employ them to criticize undesirable features and suggest improvement" (47). From this I gather that I need to draw on my experiences in teaching mathematics, my experience in watching others teach mathematics and my experience in inclusive classrooms such as Introduction to Critical Feminist Studies. I organize this paper by first giving the curriculum I planned and them explaining the choices I made for this curriculum, and end with a discussion about the possible experiences both I and the students of the class might have.

Syllabus:

Introduction:

"Calculus with Analytic Geometry (MATH 124) is a three-quarter introduction to calculus for students in the physical sciences and engineering. It covers differentiation and integration of functions of one variable, techniques for solving simple first order differential equations, an introduction to sequences and series, and an introduction to several variable calculus.

Special emphasis is placed on the process of translating from words into mathematics. Thus, story problems are at the heart of the course. Mastery of computational fundamentals is essential, but the ability to do routine exercises is not sufficient for success in this sequence.

Many students who have had some calculus in high school and done reasonably well on an Advanced Placement exam nevertheless choose to start out at the University of Washington with Math 124, in order to develop a deeper understanding of how calculus is used. We encourage such a decision, because we feel that it is better to build upon a solid foundation than to rush ahead into a more advanced course without a sure footing in the basic problem-solving skills of calculus.

To do well in Math 124 it is not necessary for students to have had a previous exposure to calculus. What is helpful for them is to have had practice solving word problems in their precalculus courses.

Topics covered in 124: Differentiation and integration, including such topics as maxima and minima, asymptotes, the mean value theorem, related rates, indeterminate forms, linear approximations and error estimates, the fundamental theorems of calculus, the substitution method, and differential equations, vector and curves in the plane and space, sequences and series, functions of several variables, partial derivatives, linear approximation, and tangent planes" (Entry Level Course Information).

Homework:

This course emphasizes problem solving and as such, completing problem sets are crucial to your understanding of the material. For each section I list problems that will not only test basic skills of the section but also push the limits a little further. Alongside each problem I list a point value. For each problem set you need to turn in 50 points worth of homework. This allows you some freedom to choose which problems you complete. I encourage you to work together on the homework, but the final write-ups must be completed on your own. There are not enough graders to grade all of the homework, so only several questions will be graded out of each assignment, but the grade with also reflect the completion of the assignment. If you want feedback about any of the homework, graded or not, come visit me in my office hours (listed below). Homework is due once a week. For help with the homework, there are your classmates (listed online), my office hours and TA office hours. The homework makes up 20% of your grade.

Additional Requirements:

• Weekly attendance to your quiz section to take a quiz.

• Attending two math events outside of class and writing a one page reflection on each experience. Examples of math events are the Math Across Campus Colloquium Series, UW-PIMS Colloquia, and also the Math Club offers a few events such as a movie night or lectures.

• A written mid-semester self-evaluation that talks about your development in the class in the terms laid out in the final self-evaluation below.

• A written final self-evaluation due at the end of the course discussing the change or lack thereof in the way you approach mathematics, what you have contributed to the math experience of your classmates and what you have gained from this class, in particular your classmates.

Combined, the quizzes and reflection papers are worth 15% of your grade.

Exams:

There are two midterm exams and a final exam. The exams total 65% of your grade: each midterm is worth 20% and the final is worth 25%.

Relevent Resources:

<Include a list of my office hours, all TA office hours, a list of websites including the one for the course, the one for tutoring in math and the one with all of the people in the class>

The syllabus described above is the one I would use as a graduate student at University of Washington, Seattle teaching an introductory calculus class of forty students in three one hour classes per week (Hacking and Frequently Asked Questions: TA Jobs and Financial Support). I used the university's description of the class as the introduction of the syllabus to acknowledge the fact that I am required to adhere to the standards of the university and that my students will be expected to have this course as background for further math course work. One problem with the way I have to run the course is that I have little to no control over the students entering my class - the university has placement exams and to a certain extent the students are self-selected because college attendance is not required in the United States and because this course is not required of all students (Overview of the Requirements for an Undergraduate Degree). To this extent, the diversity of my classroom is already (potentially) limited. Given this limitation, the class attempts to diversify the learning experience of the undergraduates in the class by requiring the community building exercise of attending two math events and also by requiring three written reports. By including in the grade students' math community efforts, I am trying to send two messages to students. The first is that there is a math community - people do meet and talk about mathematics. The second is that this community is valuable. They are being graded on their performance - as determined by them - within the community. What I hope the students will gain from these exercises is a sense that I notice their contribution to the class and that this is worth something. Since not all students are good test-takers or good at solving math problems, I want to give them the chance to talk about the skills they do bring to the class.

An important aspect of the syllabus is the homework. I have made homework worth twice as much as Professor Paul Hacking did in his Calculus class for a few reasons. The main reason I do this is to place an emphasis on problem solving. Since the class is so large and the assignments are due weekly there is no way to provide feedback on each problem. So I dealt with this as Professor Hacking did by only grading two problems from each assignment, but also by making part of the grade reflect completion of the assignment. I think it is valuable for students to try a range of problems. The idea for this type of assignments came from Professor Josh Sabloff's course on Topology (I did not take this course, but heard about it). The idea behind letting the students choose problems stems from Duckworth's idea of students having an interest in the problems they ask themselves. In a smaller class, I would consider having the students ask their own questions, but in a class this large this is not feasible. I also wanted to encourage a diversity of ability levels of mathematics. In this respect it is important to know how I was going to design each problem set. The calculus text used at University of Washington, Seattle is by James Stewart. The problems in each section of this book are arranged from easiest to hardest, a fact every student realizes on doing just one problem set. Also, in the back of the book are answers to the odd numbered problems. So, I would make problem sets have eight of the more elementary problems, each worth five points, so a total of forty possible points and offer four of the harder problems, each worth ten points. In this way a student would have to complete at least two of the easier problems and one of the harder problems. In this way each student would be able to tailor her or his problems sets to her interest and ability level, without losing the rigor needed in an introductory calculus course. This would also be a step in the direction of letting the students control their own education. Ideally I would meet with each student to design homework that reflected her or his ability and interest level - at the end of the course I could advertise such a thing for smaller higher level courses.

The self evaluations would encourage students to examine what they are contributing to and gaining from the class and also to reflect on the environment created by the class. I got this idea from Professor Traynor in Senior Conference. These evaluations really encouraged me to make the most out of a class that I at first considered to easy. I think that self-reflection is important in an education, especially in a math class because in doing math problems it is easy to forget the point of why you do them.

One area that I did not talk about in the syllabus was my lectures. In a class of forty students in the one hour time frame, I would not be able to conduct the group work detailing the exploratory nature of mathematics that ideally I would run in a smaller class. However, in my lectures I could include snippets of the history that led to the math they learn in the class as well as ask them questions to consider on their own. For example, I could tell them about open problems in the field or ask them what they would have done in certain historical scenes (there was much argument surrouding the beginning of calculus involving Newton and Lebesgue). I would also mention controversies in math where applicable and ask them to reflect on their opinions. In short, I would encourage them as much as possible to think about math on their own terms and in the context of history. The idea for including history in class I took from Professor Paul Melvin's Abstract Algebra and also his Topology course. The value I see in including the history surrounding a subject came from learning about the importance of social location in Introduction to Critical Feminist Studies. What I hope students would take away from this is that there is no absolute right way to do mathematics. Especially in a course that teaches how to solve problems that have many methods to solving them, I think it is important for students to recognize the variability in doing math. I would hope to impart on them why we are studying Calculus. This would help them think about why they study what they study and if they actually want to study that.

In conclusion I think that implementing McIntosh's ideals for a classroom impossible in a forty student class in introductory calculus. However, I do think it is possible to move away from the regulatory style often found in classes of this nature to one that is more inclusive of the diversity of ability in students and also that conveys to the students a recognition that the nature of mathematics is so exploratory, inexact and grounded in history. While I would not be able to incorporate fully Duckworth's ideal of student directed classes, I think that this could perhaps evolve over time. As I teach more calculus classes, I could develop my own set of lecture notes for the course so that I would not have to rely so heavily of a text. With practice, I could develop a style of teaching and problems sets that include more student input. I am not sure if it is possible to reach either McIntosh's or Duckworth's ideal in a class of 40, but if it is, it would have to be something that I develop with time.

Works Cited

Dewey, John. "The Democratic Conception in Education." Philosophy of Education: An Anthology. Ed. Randall Curren. Malden, MA, USA: Blackwell Publishing, 2007. 47-54.

Duckworth, Eleanor. "The Having of Wonderful Ideas." The Having of Wonderful Ideas and Other Essays on Teaching and Learning. New York: Teachers College Press, 1987. 1-14

"Entry Level Course Information." <http://www.math.washington.edu/Undergrad/entry-level-courses/calctext.php>, 17 Dec. 2008

Frequently Asked Questions: TA Jobs and Financial Support

<http://www.math.washington.edu/Grads/FAQ/ta.html#teaching>, 17 Dec. 2008.

Hacking, Paul. MATH 124 Sections C and D, Autumn 2008
Calculus and analytic geometry I. Last Modified: 10/23/200 <http://faculty.washington.edu/hacking/124/>, 17 Dec. 2008.

McIntosh, Peggy. "Interactive Phases of Curricular Re-vision: A Feminist Perspective." Wellesley, MA: Wellesley College Center for Research on Women, 1983.

Overview of the Requirements for an Undergraduate Degree

<http://www.washington.edu/uaa/gateway/advising/degreeplanning/oasudr.php> 17 Dec. 2008.

Stewart, James. Calculus 5ed. Belmont, CA: Thomson Learning, 2003.