INSTRUCTOR’S GUIDE

AND

INSTRUCTOR’S RESOURCE WEBSITE

FOR A STUDENT-CENTERED, ACTIVITY-BASED GENERAL EDUCATION MATHEMATICS COURSE

I. INTRODUCTION

Excursions in Modern Mathematics can serve as the text for mathematics courses with a variety of different goals, and for courses that are organized in a variety of different manners. This Instructor’s Guide is designed specifically to support faculty and departments offering courses with goals and organization similar to those described below.

Goals for Students. The Instructor’s Guide and supporting materials were developed for use in a course using Excursions in Modern Mathematics with the following goals for students:

· Students will be better able to think logically about situations with quantitative components;

· Students will be better able to make use of their mathematical, graphing, and computational skills in real situations;

· Students will be better able to independently read, study and understand quantitative topics that are new to the students;

· Students will be better able to explain and describe quantitative topics orally and to discuss quantitative topics with others;

· Students will be better able to explain quantitative ideas in written form;

· Students will improve their “number sense”, learn some details of a variety of situations where mathematics is used, and become engaged and have fun doing mathematics.

In short, students should be able to deal with situations involving quantitative components that they will encounter: as students in other college level courses, particularly including statistics and the social sciences; as citizens confronted with a wide array of public policy issues; as members of the work force, teaching elementary school, reporting the news, or managing an office; and as parents, participating in the education of their children.

Use of Classroom Time:

The materials and approaches described in the Instructor’s Guide were developed for use in a course with the goals described above. A basic assumption has been made that the best way for students to learn to communicate with others about quantitative issues is to require that students communicate with others. Similarly, it is assumed that the optimum way to enable students to learn to write about quantitative matters is to require them to write about quantitative matters.

A guiding principle for a course with these type of goals is that approximately one third of class time should be devoted to directed instructional activities with the instructor lecturing, introducing topics, helping students review specific computational skills that will be needed, and summarizing key ideas. The other two thirds of the class time can then be devoted to actively involving students with other students completing worksheets, discussing mathematics with others, engaged in group activities, and beginning long-term projects. The materials included in the Instructor’s Guide are designed to fully engage students in activities designed to meet the stated goals.

Nature of Materials – a low tech course:

Most of the supplemental materials provided on the Instructor’s Resource Website support paper-based activities. Basically the materials are designed to be printed by the instructor, and then duplicated for distribution to students. Some of the materials do lead to student work on interactive Websites and many assignments could be distributed electronically, but the materials basically support a paper-based course with a minimum of technology required.

By the same token, very few of the student activities require the use of programmable graphing calculators. Students are expected to make regular use of ($10.00) scientific calculators.

Experience at Virginia Commonwealth University:

The materials in the Instructor’s Guide were developed by VCU and other colleges within the Virginia Collaborative for Excellence in the Preparation of Teachers (VCEPT)[1], with support from the Division of Undergraduate Education of the National Science Foundation. At

Virginia Commonwealth University,

· Approximately 1400 students complete the semester course each year;

· Approximately 25 different individuals teach sections of the course; during a typical year this includes five full-time faculty, six graduate teaching assistants and 14 part-time faculty;

· Most students enroll in self-contained sections of 30-35 students. Others enroll in lecture/recitation sections. Under the lecture/recitation format, the 150-student lecture meets weekly and the 25-student recitation sections meet twice each week. Under both formats, each section is assigned an undergraduate teaching assistant to support in-class activities and help with grading;

· Many instructors arrange class time so that students are engaged in activities 2/3 of the time; in other sections, the faculty member is lecturing or leading class discussion as much as 2/3 of the time;

· Students in all sections complete the equivalent of the following:

§ Make weekly entries in a learning log in response to specific writing prompts;

§ Work as a member of a team to develop and offer an oral presentation on a mathematical topic;

§ Write two three-page typed mathematics papers;

§ Make a poster presentation;

§ Participate in a large number of in-class activities, usually in collaboration with other class members;

§ Complete four quizzes and three exams.

· All instructors participate in a one-day workshop prior to each semester; graduate assistants and those teaching the course for the first time participate in a weekly seminar/planning session.

Organization of Instructor’s Guide

One section of the Guide is devoted to each of a number of different types of course activities. The section contains an explanation of how the type of activity is designed to help students meet the goals of the course. Then some experiences with using the approach are described and various grading alternatives are discussed.

The supplemental materials are available on an Instructor’s Resource Website www.prenhall.com/tannenbaum. The materials are organized by the chapters of the text. Within each chapter there are many more activities and assignments than could possibly be used in one semester.

These materials were developed in MS Word. Permission is hereby granted to all instructors to print any materials and duplicate for use in any course and/or to modify materials before distributing to students.

The electronic materials are regularly revised. We would appreciate any suggestions for improving the materials. In addition, we would very much appreciate any materials developed by other faculty to be considered for inclusion in the Guide. Please send suggestions or materials to whaver@atlas.vcu.edu. Attachments in MS Word are preferred.

II. IN-CLASS ACTIVITIES

A large collection of worksheets and other materials to support in-class student activities is provided on the Instructor’s Resource Website. These materials are designed to help instructors create a situation where students are actively involved doing and learning mathematics. However, the importance of the lecture (faculty-centered) portion of the course should not be underestimated.

Centrality of Lecture Component

Good faculty lectures can greatly assist student learning. Indeed, it is recommended that 1/3 of course time be devoted to lecturing, more than any other activity.

· Organization/Expectations. Students need a clear understanding of what will be happening in the course, what topics will be studied, and what they will be expected to do (assignments to turn in, presentations, papers, quizzes and tests). In addition, students need to be told why they are expected to do certain things. This especially true for students who have experienced mathematics classes in which they spent most of their time listening to an instructor lecturing and then working problems individually. So, if students are told to work with another student completing a worksheet they also need to be told why: so that they will have the experience of talking about mathematics with some one else; so that they will have the experience of figuring out a quantitative idea without being told by an instructor; because they will learn the mathematics more thoroughly through this activity.

· Introducing Topics/Big Picture. At some point, near the beginning of each topic, students need to gain an understanding of the “big picture”. Connections with other topics, or other disciplines, that are obvious to the instructor often go unnoticed by students unless specifically described. In addition, a broad overview of the topic is very useful to the students as they put the various aspects of the topic in context.

· Most motivated students can concentrate on a good lecture for 10 minutes and a well-prepared 10 minute overview can provide a good prelude to the study of a topic in more depth.

· Confirmation/Consolidation/Validation. Students need help from their instructor in consolidating what they have learned. Students can be actively and intellectually involved for an entire class period, but gain nothing if they don’t summarize what they have learned. A short review by the instructor at the end of the period is very helpful.

Student Worksheets

Student worksheets are provided on the Instructor’s Resource Website. They are in Word, so they can be easily modified for use in different courses. A class period can be easily structured around a worksheet. The instructor can:

· begin by talking about the general topic and what you will want students to do;

· tell students that they will work in pairs and that they need to find someone to work with (tell students to introduce themselves - lots of people are shy);

· tell students to put both of their names on their paper (in the beginning of the semester this is better, since it forces students to work together - later in the semester they can each have their own sheet);

· have students begin to complete worksheet;

· walk around and help individual pairs of students; having an undergraduate assistant is very helpful here, particularly for classes with more than 25 students;

· give short explanations and clarification to the entire class as students progress through the worksheet (go out in the hall and get a drink of water, if you find yourself interrupting too often);

· spend the last five minutes helping students consolidate what they have learned;

· assign homework;

· collect worksheets or assign completion of worksheets as part of homework (if students have their own worksheets) or explain that class will continue with worksheets during the next meeting.

Other In-Class Student Activities

At the beginning of the semester most classes can be centered around student worksheets, with students working in pairs. As the semester progresses, and students are assigned papers, presentations and longer projects class time can be spent productively in other ways:

· group project meetings. On many campuses students find it difficult to find time to meet outside of class. Particularly in this case it is helpful to provide students a little face-to-face time to work and plan together. They can continue to work together out of class by telephone, email, or instant messaging. In most classes, students make very good use of this time. If they don’t there is an easy solution stop providing this time.

· paired discussion of homework assignment or learning log entry. At any appropriate time during the class you can ask students to discuss a particular assignment with one or two other students. This is a good way for the student and the instructor to find out if the students understand what is being studied.

· activity stations. For activities that require supplies (dice, spinners, etc. for probability and sample spaces or scissors and rulers for the golden rectangle) the instructor can set up learning stations around the room and have student rotate through the stations.

Active Student Engagement in Large Classes

Of course, actively engaging students in large lecture sections is a lot more difficult than in small sections. But, after gaining experience with smaller sections, many faculty successfully break up their large lectures with the same type of student activities.

In summary, there are many ways to fully engage students during class time. The need for this engagement becomes clear when students are given a problem to do immediately following a lecture. Very often the instructor will have introduced a problem and discussed some subtleties associated with it. When the students begin their work, some students have no clue as to the problem, let alone the subtle points. Only when they begin to work themselves do they gain an understanding of the nature of what was discussed in the lecture.

III. LEARNING LOG/JOURNAL

One tool that many instructors use to improve communication skills is the Learning Log or Journal.

As described in the student handout, General Learning Log, which is included in the Instructor’s Resource Website, “a learning log is a running account of what’s going on in our class. It allows you to keep track of what you are learning, how much you understand, and how much more you may need to know in order to understand the content. A learning log also helps to create a dialogue between the student and the content as well as the student and the teacher. For the former, the student can question the material, how it works, or the practicality of it. It’s a means to think about and digest the material covered in class. For the later, the student has a private means of asking questions, or expressing opinions about what’s going on in the class. As a result, the professor is more in tune with the understanding and the concerns of the individuals in the course.”

Writing in the Learning Log is in response to Writing Prompts. Some generic writing prompts are included in the general section. In addition, samples of possible writing prompts are included for each chapter.

The primary purpose of the Learning Log is to encourage students to think about the mathematical topic being studied. Students often focus on determining an algorithm to follow instead of on understanding the key ideas. The prompts ask the student to explain the mathematics in his or her own words, and the intent is that, in doing so, the student will be forced to think more deeply about what is happening. The process of expressing mathematics in the English language typically requires a clarity of understanding beyond that necessary to complete a set of calculations. This process is described by some people as “writing-to-learn.”

The secondary purpose is to develop communication skills. However, it should be noted that, as the Learning Log is described in this section, the student will not usually receive immediate feedback. In addition, the student is well aware that the instructor already understands the mathematics. Therefore, the student is not really explaining the situation to someone else. However, the assignment is still very valuable, since the very act of writing is the beginning of the communication process.

(Of course, there are large numbers of ways to structure Journal/Learning Logs that address the two drawbacks described in the previous paragraph. In some courses, instructors design the Journal/ Learning Log to be at the center of the learning experience, with students expected to spend large amounts of time on this activity, and with a proportionate amount of the grade determined by this work. Possible features include: expectations that students write regularly in the Journal without specific prompts; requirement that students read the Journal/Learning Log of other students; “two-minute paper” entry in log at end of each class period.)

However, as presented in this section, the Learning Log plays a less central role in the course. It is described as “one-of-many” ways that students are enticed to become engaged in the course.

In-class and Out-of-class Entries

Most instructors assign most of the entries as out-of-class exercises. But there are a number of situations in which asking students to respond in-class to a writing prompt can be very effective:

· End of class “two-minute paper.” Students can be asked to summarize the key idea that they learned in class. Or students can be asked to write down what they don’t understand. Entries of this nature are probably most effective on days when Learning Logs are to be collected;

· Middle of class prompt to be shared and discussed with another student. This use of the Learning Log can be particularly effective in large lecture sections. After the instructor has introduced an important/difficult idea, students can be asked to explain the idea in their own words in their Learning Log. Then each student can exchange the response with another student and discuss the topic with this student. This exercise actively involves all students in the class, helps students improve their written and oral communication skills, and reinforces the mathematics that is being studied;

· Prompt asking students to think of examples of situations to be introduced. For example, before introducing Fair Division, students can be asked to write down a number of situations in which goods need to be divided fairly. After writing in response to the prompt for a few minutes, students then share their responses with the rest of the class.

Including some in-class prompts can provide the instructor with an opportunity to explain to the students why writing is important. Writing is a means to achieve two goals of the course: learning the mathematical ideas and learning how to communicate mathematical ideas. By the very act of devoting precious class time to student writing, the instructor emphasizes the importance of this writing.

However, as stated earlier, in most cases students are asked to respond to writing prompts outside of class. Given additional time, students can be asked to reply in more depth. Types of prompts include:

· Request that students contrast and compare. For example, after studying Euler circuits and then Hamilton circuits, many students profit from writing down ways that these circuits are similar and ways that they are different;

· Assignment to redo portions of a quiz or test that the student answered incorrectly. An example of such a prompt is included in the General section. This tends to work well two or more times near the beginning of the semester and then gets “old” for the students;

· Progress report on long-term project. This type of prompt helps get students started on the project in a timely manner and gives them a chance to ask questions;

· Homework problem from text. Many of the “Running” exercises at the end of each chapter of Excursions in Modern Mathematics make excellent Learning Log prompts.

Relative Importance of Reading/Responding/Critiquing Student Entries

Virtually all instructors who assign Learning Logs/Journals believe it is important that some one read what the students write. Many students have very limited experience writing mathematics and need to be motivated to write. In addition, instructors can learn about what students are learning and which topics they find difficult.

Similarly, most instructors think it is important to respond to any student questions, any concerns expressed by the student, and any suggestions, criticism or praise about the course.

On the other hand, many instructors do not believe that it is necessary, or even beneficial, to critique student entries in the Learning Log. This is particularly the case with corrections of student writing. If students know that their grammar, spelling and sentence structure is to be graded they will tend to write less and to think about the mathematics less. In addition, most experienced writing instructors report that students don’t pay much attention to red marks all over their papers.

Some instructors are uncomfortable with not correcting student errors. If they feel strongly, of course they will make the corrections. But time does provide some constraints. Even those faculty who believe that it would be desirable to critique all student writing sometimes reach the conclusion that it is better to have students write, and not have their writing be critiqued, than it is to not have them write at all.

The approach advocated in this Instructor’s Guide is to focus on increasing the engagement and learning of students, not on identifying and correcting each student error.

Methods of Collection

A variety of different methods have been used successfully. For some reason, instructors tend to decide which method works best for them and to feel quite strongly about this decision. Time-management issues are crucial (see section VIII).

One method is to require students to write in a small notebook or to turn in the responses in a pocket folder. Using this method, the notebooks are collected three or four times during the semester. All of the entries since the last collection are read and the instructor provides appropriate response. The student is told, in the notebook, whether or not his work is satisfactory and whether additional response to some prompts is necessary. Instructors using this method can usually read and respond to a month’s worth of Learning Log entries for a class of 25 students in not much more than one hour. The student has the responsibility for keeping track of the materials. Alternately, the instructor may note in a grade book the date and degree of completeness of the writing as directed by the writing prompts for that set of prompts.

Another method is to collect each response the day after it is assigned and to record that the response has been received in a central grade book. The student writing is returned to the student along with the instructor’s comment. This method takes less time for each reading, encourages students to complete work in a timely fashion and provides immediate feedback. Also, the volume of paper to be carried back and forth to class is smaller. The total amount of time spent grading, collecting, and recording grades may be greater but many instructors think it takes no more time than the first method.

Another method is to have the students respond electronically. Various web-based materials exist for faculty who wish to handle the Journal electronically.

Use of Learning Log in Determination of Course Grade

Again, a number of different methods may be used, and different instructors tend to have different views concerning the best method. Possible methods include:

All students required to maintain adequate Learning Log. Logs are graded as satisfactory/unsatisfactory and if the grade is not satisfactory, the student does not pass course. Not surprisingly, students do not like this approach and tend to respond negatively.

A grade assigned to Learning Log. The grading is averaged along with other grades to determine a final course grade. This method is often used when there are no other writing assignments, and the course grade is determined primarily by quiz and test grades. Obviously student work must be graded very carefully, and the grading takes a substantial amount of time.

Learning Log graded as satisfactory or unsatisfactory; if grade is satisfactory, low test or quiz grades are not counted in determining course grade. Instructors who use this method usually permit make-up tests only under very special circumstances. Students like this method, and virtually all students complete their Learning Logs.

Learning Logs graded as satisfactory or unsatisfactory and a predetermined (perhaps 3%) number is added to the student’s average. Students like this method and perceive it as fair. Some instructors do not think it is appropriate to discount a poor test grade entirely, and therefore prefer this method to the previous one. A negative consequence is that a higher percentage of students think that they will not need the extra points and stop writing in their Log at some point during the semester.

IV. TEAM ORAL PRESENTATIONS

Team or group oral presentations are often an effective way to fully engage students; to increase the depth of student understanding of particular mathematics topics; to give students experience working as a member of a team; and to require students to develop the ability to communicate quantitative information.

The approach taken in this Instructor’s Guide is that the oral presentation is one-of-many activities in which the student will engage. The goal is to fully engage the students for a short period of time. Therefore, it is important that: the project is very well defined; the mathematics to be mastered is explicitly stated; the nature of the oral presentation is clearly explained; and the grading criteria is provided at the time the assignment is made.

Value of Team Assignments:

When employers are asked to describe what they seek in college graduates, they invariably include “the ability to work as a member of a team”. While a general education mathematics course cannot assume the responsibility for the total education of a college graduate, the course can provide students the opportunity to work together on a well-defined task. This Instructor’s Guide takes the view that the primary purposes of the team oral presentation are to learn mathematics and to develop communication skills, but that the experience of working as a member of a team is a valuable secondary purpose.

Grading:

Typically each student on the team receives the same grade. The grade is based (in order of importance) on: 1) the correctness and specificity of the particular mathematics assigned to the team; 2) the context within which the mathematics is presented; 3) the oral conventions (clarity of presentation, eye contact, visuals). A sample of a grading rubric appears at the end of the section.

Tips for Oral Presentations:

· Size of teams. Many instructors think that teams of four members are optimal. This number is small enough so that each member can actively participate and large enough to require a collaborative effort by students

· Formation of teams. There are many ways to form teams. The nature of the institution and its student body (commuter vs. residential; part-time vs. full-time; age distribution) perhaps should be considered. Virtually any method of forming groups will work if the instructor proceeds confidently. For example, teams can be formed by students; instructors may form teams; or team membership can be determined by a random process.

· Team Meetings. It is recommended that some class time be provided for team meetings (perhaps twice for 15 minutes). Instructors should emphasize that it is not necessary for the teams to meet in person. Good means of communication include telephone and instant messaging (it is easy for students with any type of email to set up an instant messaging group).

· Conflicts Among Team Members. It is not uncommon for conflicts to develop in one or two of the groups. Indeed, one of the reasons employers want colleges to provide team assignments for future employees is so that the students will have the experience of working through such conflicts. If conflicts are reported to the instructor, it usually is sufficient for the instructor to express sympathy and to explain that part of the purpose of the assignment is to give students experience with dealing with such situations. Sometimes it makes sense to assure the students that in any future group assignment, the conflicting students will not be expected to work together.

· Practice Sessions: It is highly recommended that you ask each group to conduct a practice session with you (or with a student assistant); the groups need to know their time limit. If you have a student assistant it is often very effective to have each group meet with the assistant, perhaps in the hall, while you lead the rest of the students in in-class activities.

· Class Size: Team oral presentations are most effective when the class size is 24 or less (this allows for six groups with four members). With more than five or six groups, students become restless (and instructors bored).

· Instructor Summary: Even with well-posed assignments and practice sessions, students will describe some mathematical ideas with less than ideal clarity and also make some incorrect statements. Following each student presentation with an instructor summary can provide an opportunity to clarify key ideas and correct any misstatements. Tell the students ahead of time that you will provide such a summary, so that they do not think that you are being critical of their presentation. Alternately, a class session, directed by the instructor, may be devoted to summarized key ideas from multiple group presentations.

Conducting the Presentations:

· Scheduling Presentations. It helps to schedule the presentations ahead of time. If you have six presentations you may wish to devote a week of class time to presentations. If your class meets three days a week for 50 minutes each, this would make it possible to have two presentations a day (each 15 minutes, with 4 minutes for questions and 6 minutes for you to summarize and highlight key points).

· The Student Presentation. Students should know ahead of time how time limits will be enforced and the basis for their grading. If possible, let the students complete their presentation without interruption from the instructor.

· Questions From Rest of Class. If your class is quiet, emphasize that if they don’t ask questions, you will. If you have a student assistant, ask the assistant to “plant” some questions.

· Homework. It is important to assign homework (one possibility is for groups to assign homework as part of their presentation). If you collect homework, it should be due the next class day.

· Positive Feedback. If possible, as will usually be the case, provide positive feedback to each group.

Assigning Different Grades to Different Members of the Same Team:

Again, different instructors have different opinions about this. If you feel it is important to preserve the option of different members receiving grades, by all means do so. If you don’t believe that different grades for a team effort (except for an individual who does not participate at all) are appropriate, don’t assign them. Students will almost universally accept either method with no questions asked, as long as the instructor describes the grading procedure up front, and as long as the instructor does not display uncertainty about this issue.

Rationale for preserving option of different grades: some students will do more work than others; some students will do virtually no work if instructor does not assure that they do; students should be rewarded for what they do, not what others do or the luck of the draw concerning team members; having students grade the performance of team members is a valuable experience; it’s the only “fair method”.

Rationale for not assigning different grades: the emphasis should be on the overall product, not on who made what contribution; peer pressure is more effective than the instructor in getting full participation; students should be rewarded for what is accomplished, not for input; realistically, on the job if you are a member of a team and your team fails, you will not be rewarded even if you tried very hard; “who said life was fair.”

Mechanisms for obtaining feedback concerning performance of individual team members. Of course, instructors have the option of providing a great deal of grade differential or very little. Some forms that are useful in obtaining feedback from group members are contained in the ‘General’ section of the Instructor’s Resource Website.

Have Fun: If students know that you place a high value on their presentation, they will do outstanding work. In many classes, at your suggestion, students will dress professionally, and their presentation will also be professional in nature! Team oral presentations are hard work, but usually productive and fun for both faculty and students.

EVALUATION OF PRESENTATION

IDEAS/CONTENT OF SPECIFIC ASSIGNMENT	weak strong
overall situation described correctly	_________________
correctly tells the "story"; this portion of the grading sheet is modified depending upon the particular assignment	_________________
complete and correct mathematical details	_________________
understood by students in class	_________________
STYLE/ORGANIZATION
unique/ individual introduction that makes you want more	_________________
interesting presentation	_________________
addresses all parts of the assignment	_________________
all members of team contributed	_________________
actively involved rest of class	_________________
CONVENTIONS
speaking style and eye contact	_________________
class could hear speakers and see visuals	_________________

Comments:

V. POSTER SESSION

What is it?

A student poster session can be described to a faculty member as an analogue of a poster session at a professional meeting in which mathematicians describe their results on a poster and then stand by the poster, prepared to describe the work in more detail to anyone who is interested. It can be described to students as being similar to a Science Fair. In either case, the work is described repeatedly to individuals or to small groups of interested people.

In a typical poster session that is a part of a general education mathematics course, students choose (or are assigned) one of four selected topics. They are then provided with a few specific questions associated with their topic. They then, on their own or with another student, study the mathematics involved. They prepare a poster (and associated backup sheets or other props) answering the questions and then, during the poster session, stand by their poster and explain their topic to other students in the class.

Goals of Assignment

The goals of the poster session assignment are to provide students with the experiences of: mastering a topic with a quantitative component based on independent reading and inquiry; writing about a mathematical topic; explaining a mathematical topic to others. These goals are achieved with most students. A further goal is to provide some students with the experience of delving deeply into a particular mathematical topic and understanding it at a level beyond what they previously experienced in any area of mathematics. In a large enrollment, multi-section general education mathematics course in which the poster session assignment forms only a small portion (perhaps 10%) of a student’s course grade, it is unreasonable to expect all students to become immersed in their topic. However, a positive feature of this assignment is that it often does fully engage some students, and not only those with strong manipulative skills:

A Unit Based Around the Poster Session

The assignment will typically consist of four specific topics chosen from the text questions on the other topics. Together these topics will form the bulk of a unit in the course. In this unit:

· the instructor will introduce the four topics to be studied and “set up” the interesting questions to be answered;

· each student will select (or be assigned) his or her topic and begin independent study;

· homework assignments will be given for each topic; students are expected to complete the assignments concerning their own topic before the session and questions on the other topics after the session;

· instructor and students will review/study required computational skills;

· to provide time for independent study, another related topic in the unit may be studied by class as a whole;

· students will write about topic(s) in learning log, meet with other students or instructor, and/or turn in completed assignments in order to demonstrate mastery of their topic;

· the poster session will take place;

· instructor will review each topic after the session;

· students will write about topics in learning log or as one of the major papers in the course;

· students will be tested on the mathematics studied. (Be sure the students understand this beforehand!)

The Particular Topic

Typically a particular topic consists of a topic or a partial chapter in Excursions in Modern Mathematics. The assignment should clearly state what questions the students need to answer, what ideas they need to understand and be able to explain, and which exercises in the text they should be able to do. It should also be clear that all of these questions should be answered on the poster or on sheets of paper available to those who visit the posters. There are a number of specific assignments included on the Instructor’s Resource Website. They are included with the material for various chapters and identified as poster/presentations.

The Poster Session

The session itself can take place over one or two days. The students should know ahead of time when they and their poster will be on display.

· A program. If you have access to a copier, it is very easy to make up a program. The program can list the presenters and their topics by time slots. This adds a degree of importance to the activity. But its major purpose is to keep things organized and moving smoothly.

· Introduction of poster presenters. It is usually useful to have each presenter introduce him/her self and the topic of the poster at the beginning of each time slot (this way no one can hide).

· Instructor visiting each poster. It goes without saying that it is very desirable that an instructor visit each poster and ask a few questions to the student (however, see grading below).

· Invite a visitor(s). Visitors serve a useful purpose. Primarily they give students the opportunity to explain their work to more “outsiders.” But, they also provide a means to enable more members of the university/college community to learn about what is going on in your course. The first year, you will probably want to invite a friend. After you have some experience, you may wish to invite the department chair, a Dean or faculty members from other departments. Visitors are usually very favorably impressed by the quality of student work, and by the depth and intrinsic interest of the mathematics being studied in a general interest mathematics course.

· Ensuring that students take seriously visiting other student posters. Students often tend to be “polite” and “nice” and don’t ask the poster presenters to explain the details of their work. In most classes the requirement to write about the topics described in the other posters and to be tested on this material will take care of this potential problem. It also helps to tell the students what they are expected to do.

Grading

Have the students turn in their posters (and backup work) to you. The grade will need to be primarily based on this work, since the instructor will often only have a very short time to hear each student describing his or her work.

Alternate Sources of Evaluation:

· If you have visitors, you may ask each visitor to identify the three students who made the most impressive oral description.

· You may ask students in the class to evaluate other presentations.

If at all possible complete your evaluation (with the help of the visitor) immediately after the session is completed. It is very easy to forget details.

Of course, the oral presentations have intrinsic value, whether or not they can be formally

evaluated. They usually elicit a great deal of student work and students often report that the poster session, and the activities leading up to the session, was the high point of the course.

VI. STUDENT PAPER

One method of helping students develop their mathematics is the use of formal papers. As described in this Instructor’s Guide, a student paper is a 3-4 page typed paper on a mathematics topic. The paper will require the students to describe in their own words the mathematical ideas that are under consideration. Typically the student is assigned to describe the mathematics within a particular context: a report on a classroom activity, a letter home, a description of the college mathematics course for a high school senior, or an article for the New York Times. Writing within an assigned context encourages students to be creative and to describe the mathematics in their own words.

Purpose:

The major purpose of the assignment is for students to learn mathematics. This purpose should be kept in mind while writing the assignment, describing the assignment to students and grading the work.

Writing of this nature is often called writing-to-learn, with the understanding that the emphasis is on learning mathematics. The idea is that if students are required to understand the mathematics well enough to put the key ideas in their own words, within a specific context, they will develop an understanding at a level beyond which they normally achieve.

A secondary purpose of the assignment is to give the student some additional experience in communicating mathematics. Again, the emphasis here is on communicating mathematics, not on generic communication. It is not realistic to expect that the writing in one mathematics class will significantly improve a student’s overall writing (after all, students have been studying writing for years in their English classes). It is realistic to expect that students can increase their ability and confidence to write about mathematics, bringing to bear their existing writing and thinking abilities to mathematical reasoning.

The Assignment:

As always, it is important that students understand what they are expected to do, and why. If the instructor does not explain the purpose of the assignment, many students will view the assignment as “busy work” and will not fully engage themselves with the mathematics. A sample assignment is provided at the end of this section.

Grading:

A sample grading plan is also provided at the end of this section. Of course, there are many different ways to grade papers. However, in most areas of the country, college freshmen will not have previously been assigned mathematics papers, and will not know how this mathematics writing will be graded. Therefore, it is very useful for the students to be told how the paper will be graded at the time the assignment is made.

On the sample grading plan, three aspects are emphasized: the mathematics; the context; the conventions:

Mathematics: Clearly the most important; you do need to be sure that the students understand that the mathematics is most important to the grader.

Context: Context is important because it forces the students to put the ideas in their own words. This internalization and re-description is central to the learning process. To aid in this process, tell the students that you are looking for papers that are fun to read.

Conventions: Be clear that this is the least important area, but that you expect correct spelling, grammar, and punctuation. There is no excuse for not running a paper through spell check. Many instructors also encourage students to have friends read drafts of papers and for students to seek assistance from writing centers. (Be clear on your academic integrity policy. Some instructors might view such help as inappropriate; therefore, you need to make your policy clear).

Framework for Writing Assignment:

As stated in the introduction to this section, it is recommended that students be given a framework or context for their writing assignment. Frameworks that have proven to be effective include:

· Description/critique of specific oral presentation or poster session presentation. Students are expected to choose a topic other than the one of their own presentation, and then critique a specific presentation on this topic. Their paper should include the names of the students making the presentation. The paper should focus on mathematics – the same mathematics (answering the same questions) as the original assignment. However, the student author should explain how the author came to understand the mathematics through the presentation – or, alternatively, how the presentation was incorrect or incomplete;

· Article for the New York Times. Students should first state which type of article they are writing: news, feature, column, editorial, obituary, etc. Naming a particular publication helps establish the audience for the author. For example, you could tell the students that Times readers are reasonably well-educated, interested in public policy, generally well-informed, but have very limited mathematical background. Again, it is important to let students know that the structure for the article is important, but the bulk of the assignment (and the primary basis of grade) is to explain the mathematics;

· Memo to your boss. Students should be told they are in a position where they need to make a recommendation to their boss. Their recommendation needs to include a justification and the boss has not studied Excursions in Modern Mathematics. Examples might include a plan for preparing, daily, a route to deliver packages to a particular set of locations. The employee needs to explain to the boss why the schedule for each day is not guaranteed to be optimal. Whatever the particular area of the recommendation, students will be told that the memo should compare and contrast the benefits of various approaches.

· Letter home. Have the students state whether they are writing to their parents, a high school friend or a high school mathematics teacher. The assignment could be to write a letter describing their Excursions in Modern Mathematics course. While the bulk of the letter should be explaining a particular mathematics topic, information about the instructor, how class time is spent and particular classmates should be interspersed throughout the paper. Don’t worry; most of your students will humor you and report on how much they love the course and their instructor.

Paper: NEWSPAPER ARTICLE

Purpose: To explain in detail one of the mathematical topics studied in Chapters 1, 3 or 5.

Assignment: To write a 3-4 page article that could appear in the New York Times. The article may be in the form of a news article, a feature article, an op-ed piece, an obituary, or any other form. It probably will be about an event that you have imagined, but it could be based upon a real situation should you choose to do so.

Suggested Prewriting: Try to talk out the concepts (with anyone who is willing to listen) in order to check your understanding. If you can talk about them, you can write about them.

Length: Probably 3-4 pages, be sure it’s typed and double spaced. Whatever space you need to describe all of the features of the mathematical situation that you are describing as well as the format of whatever newspaper format you have chosen.

Audience: 1 math professor and the readership of the New York Times (a well educated readership; however, most readers have not studied the topics of this course).

Grade: Your grade will constitute 10% of your course grade. Be sure you credit any sources that you use and if you use a sentence or phrase from another source, be sure to place that phrase in quotation marks.

Good Luck!

EVALUATION OF GENERIC PAPER

IDEAS/CONTENT OF SPECIFIC ASSIGNMENT	weak strong
overall situation described correctly	_________________
correctly tells the "story" of the particular assignment; this portion of grading sheet varies depending upon assignment	_________________
complete and correct mathematical details	_________________
equations/constructions correct	_________________
STYLE/ORGANIZATION
unique/ individual introduction that makes you want more	_________________
interesting presentation	_________________
addresses all parts of the assignment; in particular addresses each issue raised under the topic chosen	_________________
includes enough details and examples	_________________
CONVENTIONS
generally free of grammatical errors	_________________
generally free of spelling and punctuation mistakes	_________________

Comments:

VII. LONG-TERM STUDENT ACTIVITIES

The group oral presentation, poster session, and student paper are all examples of long-term student activities (or at least activities that extend beyond a class period or typical homework assignment). Clearly, there are variants that can be made among these various activities: any one of these assignments could be extended beyond that described here (the paper could be 8-10 pages or the oral presentation 50 minutes); the poster session could be assigned to a pair of students instead of to individual students; the entire class could visit the poster sessions sequentially. Different approaches work better for different faculty members and at different institutions. This Instructor’s Manual offers some models as a starting point. Following are two issues that relate to long-term student assignments.

Need for Instructor to be clear about what constitutes original student work (avoiding plagiarism)

Students need to be clearly told what they are expected to do. By encouraging collaborative work and by assigning expository writing (as opposed to only having students work exercises) we are blurring the line between original work and plagiarism. The following clarifications are helpful to students:

· do not copy sentences or phrases from any source, including our text; put these ideas in your own words;

· even when you put ideas in your own words, you should reference your source.

Your assignments can be made in a way that encourages originality. For example,

· students can be told to include a specific current event as part of their paper (e.g., a current election or a divorce settlement that is in the news);

· If the paper is about a topic that was presented by other student in class, the assignment could require the writer to include the names of the presenter and details from the presentation.

Long-term Versus Moderate-Term Assignments

The approach described in this Instructor’s Guide is that of a large number of moderate length assignments rather than a smaller number of long-term assignments.

There are a number of reasons for this approach:

· Excursions in Modern Mathematics consists of a large number of different topics and, therefore, this approach fits well with the text;

· as non-math, non-science majors, the students in the course seem to be more willing to take on a large number of moderate term assignments;

· students are used to thinking of a mathematics assignment as ten discrete exercises to complete; therefore, the jump to the moderate term assignment by itself seems daunting enough.

VIII. INCORPORATING ALL OF THESE FEATURES IN A SINGLE COURSE

It would be undesirable, if not impossible, to introduce all of the features described above into a single offering of a one semester course. However, as described below, many features can be included if an instructor/department makes a decision to do so.

Advantages of Incremental Change

Unless an instructor is working as a member of a team under the leadership of individual(s) who have developed experience with the course or unless the instructor has experience offering this type of course in other situations, it probably is preferable to introduce new features incrementally:

· Instead of planning on making use of worksheets every class period, planning on using a worksheet once a week might be preferable;

· Or, a major change for a semester could be the use of the Student Learning Log. In this case more faculty time could be invested in reading the student responses and determining which types of prompts elicit the best student responses. If the activity is going well the instructor could: have students read each other’s responses and use this as a way to encourage oral communication; or, frequently make two minute entries at the end of class; or, ask for volunteers to describe their responses to the rest of the class. Alternatively, if the activity is not going well, the instructor could increase the time between assigning prompts.

· Or, the major change in the course could be the assigning of a term-paper. Again, this assignment could form a bigger part of the course requirement than if it were one of many special student activities.

Faculty Time Management

If, indeed, the general education mathematics course has many student-centered activities, there is a real danger that it will take more faculty time than is sustainable. A conscious effort needs to be made to be sure this doesn’t happen. A few suggestions:

· Adopt the point of view that having the students be actively engaged, learning and thinking about mathematics is more important than having every activity graded carefully. Basing a grade of a paper or assignment on completion of assignment, rather than accuracy of every detail, is entirely appropriate;

· If possible, obtain services of a student assistant and have some papers and tests graded by the assistant;

· Make use of the materials provided on the Instructor’s Resource Website. After you have used them once, you can easily modify them to better meet the needs of your course.

Possible Grading Scheme

The following grading scheme has been used in a course in which students were required to: take three exams and four quizzes; write two papers; participate in making a group presentation; make a poster session presentation; and turn in a dozen in-class/homework worksheets; and respond to weekly prompts in a Learning Log.

3 tests (10% each) 30%

4 quizzes (5% each) 20%

Group presentation 10%

Poster presentation 10%

2 papers (10% each) 20%

Small projects/worksheet 20 %

110%

The lowest grade (except the last test) was ignored for all students who successfully completed the Learning Log.

Of course, there are many ways to organize a course. The course whose grading scheme is described above calls for a large number of different student activities, each one of a limited duration. The various activities are linked together (for example, one paper, one quiz, and three writing prompts could be directly tied to the topics covered in the group presentations.) Another set of student activities would be tied to the unit studied through the poster presentations.

Of course, there are many other ways to organize a course and, as described in the section on incremental change, it would not be wise to introduce all of these student activities at the same time. Many faculty members would prefer only to have a few of these features in the course. However, it is possible to have a smooth, unhurried course employing all of these features.

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[1] College of William and Mary, Germanna Community College, J. Sargeant Reynolds Community College, Longwood College, Mary Washington College, Norfolk State University, Tidewater Community College,

University of Virginia, Virginia Commonwealth University, Virginia Union University.