Sarah Glaz is a Professor of Mathematics at the University of Connecticut. She has been awarded a 2005 University of Connecticut Provost's General Education Course Development Grant for the development of the course featured in this article.
Briefly, the new course is designed to do the following:

  Cover all the topics from Intermediate and College Algebra  needed for lower division Q (quantitative) courses, including both traditional Intermediate Algebra  topics, and logarithmic and exponential functions.

   Cover all topics with a college oriented approach. That means that topics from Intermediate Algebra have to be approached with an emphasis on the elements that are necessary for more advanced mathematics courses and for courses from other disciplines.

   Engage students in elementary mathematical modeling projects on a regular (weekly) basis throughout the semester, as a way of training them to solve multi-step problems from other disciplines, a necessary skill for their next course employing mathematics.

   Ensure, through appropriate teaching techniques, that the level of difficulty of this course is not greater than the level of difficulty of Math 101; and engage students with the material and the course in ways that help alleviate math anxiety, build confidence in their ability to work with mathematical concepts, and (dare I say it) start liking mathematics.

To cover the additional material and mathematical modeling projects, the course meets for longer hours than a traditional course at this level. Precisely, the course meets three times a week for a time equivalent of a 5 credit course, that is 250 minutes a week. The course contains, in addition to the review material, sufficient college material to warrant some graduation credits. Students who successfully complete the course earn 3 credits that count toward graduation. This offsets the inconvenience of longer meeting times, and is an additional incentive for students who need to reinforce their high school algebra, to actually enroll in the course.

I have taught an experimental version of the course during Fall and Spring semesters of academic year 2005/2006. In Fall semester 2006 the course becomes a permanent part of the mathematics department's curriculum, under the title: Math 104Q: Introductory college Algebra and Mathematical Modeling [4]. It will be offered simultaneously on five of  UConn's six campuses, and coordinated electronically and online. In what follows, I share the results of my two years involvement in the development of this course, as a way of offering suggestions and soliciting feedback from colleagues at other universities who face similar issues.

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After an extensive search, I adopted a book which is compatible with my aims for the course: Intermediate Algebra, by K. Elayn Martin-Gay [11]. This book is both pedagogically sound, and charmingly written. Moreover, the Instructor's Resource Manual with Tests [12] associated with the book provides a large number of imaginative practice and test exercises, and a small, but attractive, collection of group projects. More group projects were developed by me specifically for this course. The course covers 2 to 4 sections of material, and one mathematical modeling group project every week.

I now invite you to visit the course's web site of  Math 104Q: Introductory College Algebra and Mathematical Modeling [4], and scroll down to the Syllabus table. The topics, for most part, are traditional Intermediate Algebra material, including: linear equations, lines, systems of equations, polynomials, rational and radical expressions. As mentioned in the Introduction, the syllabus also includes Chapter 9: Logarithmic and Exponential Functions, which is not usually part of a remedial mathematics course. Note that the semester does not end with Chapter 9, as one would expect, but rather with three possible topics from earlier chapters. This was done in response to students' comment that the material of Chapter 9 is new to them, and they need more time to absorb it before being tested on it in the Final Exam.

The Intermediate Algebra topics  are covered using a College Algebra approach. I will illustrate this point with two, out of the many possible, examples:.

  First example: Throughout the semester there is an emphasis, at all times, on the notion of a function, rather than on that of an algebraic expression. This includes making sure students become familiar and are comfortable working with function notation, domain, range, graphs (by hand), and interpretation of graphs. The last is  particularly important for science courses, where the formula itself is secondary to the graph, when interpreting behavior of phenomena. This emphasis will be further elaborated when we discuss the mathematical modeling group projects.

  Second example: Emphasis on a more mature, college oriented approach, to the topic of factorization of trinomials. All  Intermediate Algebra books I have looked at, approach factorization of trinomials via the shortcut  possible only when the roots are integers. It often stays at this stage, and students arrive to a Pre Calculus course without realizing the relation between the roots of a polynomial and its factorization, and occasionally, without even knowing the formula for roots of a quadratic equation. As an illustration of how this material may be taught with a more mathematically mature approach, without making it more difficult for the students, click on: Roots and Factorization of Trinomials (this handout can also be downloaded from the course's web page by clicking on the Pink Panther folder: Student's Handouts). This handout contains a summary of  lessons in factorization of trinomials using the quadratic formula for roots. To fill in the details for complete lesson plans, one needs to add blackboard examples,  student class work on examples where the Instructor circulates and makes individual comments, and one good group project that trains and reinforces students' factoring ability. Consider making the formula for roots of quadratics memorable to your students by introducing them to the Quadratic Formula Rap . This rhyme originally appeared in the 1885 Proceedings of the Edinburgh Mathematical Society (unknown author) [21], and was suggested as rap lyrics by Lawrence Lesser [10].

I will end this section by mentioning my approach to the controversial issue: Should one or shouldn't one allow use of calculators? On one hand, we want students to develop a "number sense", and correct bad habits in simple arithmetic, which can have consequences on their later use of mathematics. This can only be done by practicing arithmetic without using a calculator. On the other hand, "real life" mathematics-- the reason we want everyone to be mathematically literate, seldom deals with "nice" numbers, and a calculator is necessary in order to avoid interminable, routine calculations. To accomplish both aims, I do not allow the use of calculators on exams and on individual homework exercises, but ask students to own and use a scientific calculator for the mathematical modeling group projects.

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The third link on the top right-hand corner of  the course's web page Math 104Q: Introductory College Algebra and Mathematical Modeling [4]  is a Pink Panther folder titled Student's Handouts. It contains nine handouts summarizing the knowledge students are supposed to acquire during the semester. The handout: Conversion Rectangle Trick reviews material from Algebra I, while the other eight handouts cover the material of the course one chapter at a time. I distribute these handouts in class as we reach the appropriate material in the syllabus, and encourage their use during individual  and group homework assignments. To emphasize their importance, I single out these particular handouts by xeroxing them on paper of various colors. The Pink Panther folder is there for easy access in the future when the students are enrolled in their next science course.

Distributing handouts in class at appropriate times accomplishes several things:

   Handing each student every handout individually, as a gift, is one way of showing my students I care, and establishing a relationship in which students feel comfortable asking questions and a warm classroom atmosphere conducive to learning.

   The summary of material for each topic, teaches by example, a correct approach to the material for effective leaning. That is, it shows students that one should study carefully the multitude of
information received on each topic, and then recognize and summarize what is important and merits additional attention.

   A short, one page, summary of each topic, lists concisely and precisely what students need to absorb on each topic. These concrete summaries present each topic as a finite, small, number of concepts, rather than as a potentially limitless and nebulous amount of information. This takes the edge off the math anxiety associated with the perceived vastness of mathematical knowledge, and makes students more comfortable with the material and more confident in using it.

In addition to the handouts in the Pink Panther folder, I distribute other handouts during the semester. These include: course information, solutions to particular homework assignments and exams, coverage and suggested practice exercises for exams, and various humorous handouts designed to relieve tedium, increase comfort, or boost  students' confidence in their ability to work with mathematical concepts. Consider making your students aware of the phenomenal power of their minds by sharing with them a handout that was given to me by one of my students: The Paomennehal Pweor of the Hmuan Mnid.

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In the second  week of the semester I take the class for a visit to the Mathematics Department's Computer Lab. This one-time visit has two purposes: providing information, and generating enchantment. Both information and enchantment can be found on the course's web page Math 104Q: Introductory College Algebra and Mathematical Modeling [4].

   The information is derived by accessing the course's web page, where students can find general information about the course, exam dates, syllabus, office hours, information about online and on-campus tutoring, and more. Note that precise information on homework assignments is not included on the course's web page. Appropriate use of  web pages can provide a wonderful opportunity for enhancing students' learning experience, but it is not an adequate replacement for classroom instruction. To encourage class attendance, a very important component for successful completion of this course, I never include on the course's web page all the information students receive in class.

   The enchantment is provided through a collection of math links: Math Links for Information and Fun [5]. This collection can also be reached from a number of other places on the Mathematics Department's web pages, including the second link on the top right-hand corner of the course's web page. I invite you to browse through this collection, and become enchanted again with the way mathematics connects to every field of human interest. Favorites among the students are the links connecting art and mathematics, and the geometry links scattered through several categories.

The first link on the top right-hand corner of the course's web page is in a category of its own. It is a very informative and well written article on Coping With Math Anxiety [14] from the Platonic Realms  web site. I recommend that anyone suffering from Math Anxiety or teaching students who suffer from it, read this article.

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One of the most effective strategies for teaching mathematics to students who are not  mathematically inclined is the technique called cooperative learning (or group work). Cooperative learning is a teaching strategy in which the class is split into small groups. Each group works as a team on a mathematical project, with minimal interference from the Instructor. This technique has the power to engage the students with the material in ways that no other method of teaching does. Other benefits include: students learn to work as part of a team and use their peers as resources,  they learn how to communicate mathematics, they are actively involved in the teaching and learning processes which enhances both their ability to do mathematics, and their confidence in this ability. In addition, the teacher benefits from having a more engaged, interested and energetic class, less absenteeism, an opportunity to observe how their students approach problems, and less homework to grade---  each group submits only one report per project. There are a number of  written and online sources discussing the theoretical foundations of cooperative learning, and guiding teachers who wish to introduce this technique in their classes. A few written examples are: A Practical Guide to Cooperative Learning in Collegiate Mathematics [8], Reading in Cooperative Learning for Undergraduate Mathematics [2], and Cooperative Learning in Undergraduate Mathematics [16]. Online sources can be easily found through a Google [7] search. But, ultimately, the place where a teacher truly learns how to use the technique is in his or her own classes, where by trial and error, one can modify the basic principles to suit the personality of the teacher and the mathematical level of the students.

Cooperative learning plays an important role in the pedagogy of this course. The technique is employed in ways that alleviates students' math anxiety, increases the comfort level of interaction among students and between students and teacher, and maximizes students' engagement with mathematics. Here are a few general suggestions for introducing cooperative learning in a course of such mathematical level:

  Start friendly:  To get started give the class an opportunity to form their own groups by asking students to pair up for group work with a friend or with the person sitting next to them. This results in small, two people, groups for the first and second group projects. Occasionally larger groups get formed this way. Monitor that the initial groups have no more then four students each, the optimal group size. The first assignment should give them time in class to get started on the group work, and make sure they need to be in touch after class to complete the assignment. After the second group activity, ask each member of the group to submit to you in writing how well their group functioned, and if they wish to remain with the same group members. Next class period assign formal groups of three or four students each, taking their opinion into account.

  Minimize potential conflicts with a Team Performance Agreement (TPA): Once you assign official groups, give class time for the groups to discuss and prepare a Team Performance Agreement (TPA). Click on Team Performance Agreement  to open a handout designed to guide the groups through the discussion of issues necessary for TPA preparation. The idea for a TPA originated in Thomas DeFranco and Charles Vinsonhaler's book: PProblem SSSolving [1]. The TPA page provided in this section is a modified version of theirs. Request that each group submits to you a typed "legal"  Team Performance Agreement (TPA), signed by all its members.

  Monitor group interactions: During group activities circulate, peek over their shoulders, eavesdrop on their conversations, and smile encouragingly. Tell them to call you if they need a little help to get started or to get unstuck. If many groups are slow to get started on an activity, or get stuck in the same place, ask them all to stop work for a minute and give a hint to the entire class to help them get going. It should be a little noisy in class during group activity. That means they are talking with each other, which is what you want. Listen to the noise they make. If you hear a hum like a hive of busy bees, it means all goes well. If it is very quiet in the room, this is not a good sign. Encourage interaction, even if it means asking them to share with each other what exactly they do not understand in the problem. It is a way of getting each of them engaged with the problem, which will eventually lead them towards a solution.

Grade for collegiality: Give a grade on the group report submitted. Every member of the group receives the same grade. This encourages collegiality, not competitiveness.

Optimize group work timing: I favor 15 to 30 minutes group work time at the end of a class period. This requires a single transition from one type of learning activity (lecture) to another (group work), rather then two or more transitions. In general, it is advisable to avoid the transition from group work to lecture during a class period, as students find it difficult to become relatively passive listeners after being actively engaged in group work. Assign sufficient class time for students to get a good head start on the group project. The intention is to have groups finish the assignment as homework. Still, if the group work occurs at the end of a class period, it provides you with a natural alternative activity for groups who finish the project early. Have them start cooperatively on their regular homework assignment. This encourages out of classroom cooperation on all homework assignments, and, by extension, for exam preparation. Both activities are extremely beneficial for strong and weak students alike.

I will elaborate further, on the cooperative leaning aspect of this course, in the next section, where I give specific examples of  group projects.

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The syllabus table on the course's web page  Math 104Q: Introductory College Algebra and Mathematical Modeling [4], lists a selection of group projects following each chapter. Some of these projects are taken from the Instructor Manual that comes with the textbook, and other are original group projects I developed specifically for this course. Students engage in one  group project every week. This means that the class works on a total of 14 group projects during the semester, one or two for each chapter on the syllabus. All group projects have some relation to mathematical modeling. At least one group project for every chapter is a mathematical modeling project with real data. The rest are group projects designed to fulfill a variety of other pedagogical needs, including reinforcement of mathematical techniques such as factoring polynomials, or the rules of exponential and logarithmic functions. In addition to the benefits outlined in the previous section, working on these group projects, makes students aware of the place of mathematical concepts in a broad cultural setting. This often results in an appreciation of, previously disliked, abstract mathematical notions, and a new motivation to master them.

The three group projects discussed below give a flavor of the nature of the group project I developed for this course.

Algebraic Poetry:  Click on How Many Bees in Lilavati's Swarm?  to open a handout of a group project on solving linear equations (with solution). This project falls into the category of uses of history in the teaching of mathematics. Bhaskara [18] was a twelve century Indian mathematician who authored the book Lilavati, written in verse, for the mathematical education of his daughter. The citation comparing Bhaskara to "the crest on a peacock" comes from  [9], and the translation of this charming poem comes from [17].  As a group project , it seemed particularly appropriate to introduce it just after finishing the "word problem" section, as a replacement for a number of more conventional homework exercises. It reinforces the four customary steps students are advised to follow to understand and solve "word problems".  The naming of these steps: Understand, Translate, Solve, and Interpret is K. Elayn Martin-Gay's [11]. This choice of words brings home the fact that mathematical modeling is just "a translation" of real-life phenomena into mathematical language, and that after obtaining the mathematical solution, one needs to interpret the solution into the real-life language of the original problem. The Trial and Error technique advocated in the Understand step, is my suggestion. It develops students' number intuition, and their grasp of  the trial and error aspect of the mathematical modeling process. It also bridges the gap between the concreteness of the numerical solution, and the abstractness of the equation. Finally, students are delighted to explore a connection between the language of poetry, which many of them love, and the language of mathematics. They even forgive me and Bhaskara, for introducing so many "hateful" fractions into the equation.

Other poems from Lilavati appear on various web sites, and can be used as basis for additional group projects. Here is one, Mathematical Problem [20], which is not quite appropriate for that purpose.
There are more modern sources of algebraic poems, for example How Old Is the Rose-Red City? [19] by Martin Gardner. For the written source, complete with the story connected to this rhyme, see [3].

Forces of Nature: Click on Hurricane Season,  and Hurricane Tracking Chart, to open the handouts that make up a group project on the coordinate system and lines (with solution). This is a group project in environmental mathematical modeling using real data. Students are asked to play the role of meteorologists working at the National Hurricane Center (NHC) [13] in Miami, Florida, and are engaged in  hurricane tracking and modeling of hurricane damages. The NHC site is the source of all the data for this group project, and the enclosed Saffir-Simpson Hurricane Scale, and Hurricane Tracking Chart. The Hurricane Tracking Chart is used by the students to track the 1992 hurricane Andrew, which was one of the most ferocious hurricanes in US history. I developed this group project the year before hurricane Katrina hit US, but in any event, even today, it is not possible to prepare a group project on hurricane Katrina, since not all relevant data is yet in. This is a good group project to introduce following the chapter on the coordinate system and lines during a Fall semester. In the Fall semester  the class reaches that point in the curriculum during the hurricane season, and this project generates considerable excitement. Employing role playing is designed to further engage the students with the drama of hurricane tracking. Hurricane tracking essentially employs coordinate system plotting, and makes this activity, and its related issues of coordinate scaling, very natural and manageable. Further graphing of the Wind Speed Curve gets students used to interpreting graphs, and the predictions regarding Andrew's strength on its second landing, and the magnitude of Andrew's damages, show them the limits of our ability to model and predict nature. The second part of the project, assessing hurricane damages employing a linear model, is designed to illustrate the natural meaning of slope, y-intercept, and graph of a line. According to NHC, the relation between hurricane wind speed and damages is not linear, but exponential. This may be used to prepare a continuation of this group project to be introduced after the sections on exponential functions. But, by the time the class reaches that point in the syllabus, hurricane season is over, and I opt  for a group project on modeling the growth of the black bear population, which is on the rise in Connecticut.

The possibilities for developing environmental mathematical modeling group projects using real data are endless. All phenomena in which forces of nature show their strength involve quantitative aspects. Much of the data can be easily found online, and the mathematics involved ranges from the most basic to the most sophisticated. Another source of environmental mathematical modeling consists in the various damages human beings inflict on the environment. Those vary from water and air pollution, to accidents involving hazardous materials, to nuclear disasters and global warming. The reader can find links to various environmental sites that can serve as sources of data and inspiration for group projects, by checking the links page on Math 108QC: Mathematical Modeling in the Environment [6], a course I have developed  a few years ago.

  Hands-on Group Projects: Click on The Largest Box  to open a handout of a group project on polynomials (with solution). This project is a modification of an exercise that is appearing for years in various Calculus, Pre Calculus and College Algebra books. The purpose of the original exercise is to find the maximum of a function using either the derivative or a calculator-drawn graph. In The Largest Box the exercise is converted into a mathematical modeling project at Intermediate Algebra level, which, without mentioning more advanced mathematical concepts, prepares students for these concepts. I introduce the project in the middle of the chapter on polynomials, after an interval of time in which the notion of a function is not in use. The Largest Box, in addition to practicing basic polynomial manipulations, reintroduces and reinforces the notion of a function, including domain and range calculations and visualization. But, the real impact of this project comes from its hands-on approach,  which transform abstract mathematical notions into concrete and familiar entities. Each group is given the cardboard backing of a notepad and asked to construct an actual box. The box has its practical uses, and also provides an opportunity to discuss in class some mathematical and ethical aspects of modeling container boxes  for cereal manufacturers. Students' reaction to this group project made it very special to me. In students' own words: Math Rocks!  Clicking on this link opens a photograph of the box constructed by one of the groups in my Spring 2006 experimental class (whose number was Math 195Q).

It is difficult to come up with hands-on mathematical modeling group projects, at this mathematical level. Another type of hands-on group projects consists of certain puzzles involving cut outs. These can be  useful as entertaining group activities for reinforcing mathematical techniques. An interesting example is the Logarithmic Equations Cut-outs [15] on The Handley Math Page of David Pleacher.

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This course is meant to effect a complete change in the way remedial mathematics is delivered at  the University of Connecticut. As such, the course affects many people, across all six campuses of the university, including: students, Instructors, Student Advisors, and administrators. In order to ensure that the resulting course responds to the needs of all those affected by it, it was important to get feedback on the contemplated changes, throughout the entire period of the course's development. I set up a feedback system, which evolved along with the course, and eventually became the system through which the course is and will continue to be coordinated across all university's campuses.

Face to face feedback was provided by the students in my classes. Students were consulted on every aspect of the course including: group projects, choice of textbook, calculator policy, order of topics in the syllabus, length of class periods, pace of teaching, test related issues, and much more. I let the students know, at the beginning of each semester, that they are participating in an experimental class designed to improve the delivery of the preparation for sciences course at UConn, and that their input will affect the way the course will be delivered in the future. I solicited feedback frequently and  informally from the entire class and from individual students. In addition, many students gave me feedback on selected items on their own initiative. Their contributions, and their enthusiastic response to my request for input, shaped the final version of this course in ways I have not always anticipated. 

I have also received substantial feedback from the graduate student TAs who taught sections of Math 101 during the period of this course's development. I  met with the TAs once a week, to provide guidance and receive feedback. They provided input on several of the group projects, which I e-mailed them in advance, both before and after they tried them in their classes. They also helped me develop a write up for guiding Instructors through the introduction of cooperative learning in their classes (which is currently placed on the course's web page in the Instructor's Resources folder), and contributed to the fine tuning of other classroom experiments. I found the weekly, guidance and feedback, meetings to be a very satisfactory and effective way of coordinating the delivery of a course by TAs, and plan to continue this practice.

In order to communicate with colleagues I could not easily meet in person, I set up an e-mail list consisting of all instructors of Math 101 across university's six campuses, and other colleagues and Student Advisors who wished to be updated on the changes as they occur. This e-mail list turned out to be an efficient and pleasant way to keep in touch with colleagues with an interest in the new course, and to provide and receive useful information affecting its teaching. With an appropriate name change, I plan to continue using the e-mail list for exchanging information, and for coordinating the teaching of the course across all campuses.

A growing amount of material relevant to the teaching of this course was accumulating in my computer folder. To make this material easily available on all university's campuses, I placed it in a password protected file on the course's web page Math 104Q: Introductory College Algebra and Mathematical Modeling [4] , the link on the top left-hand corner: a Teacher folder titled Instructor's Resources. All Instructors teaching this course have access to the password. Readers of the present article have had a sample from most sections of the Instructor's Resources folder. For pedagogical and copyright reasons, this folder cannot be made accessible to students, or to the general public.

Fall 2006, is the first time that the course will be offered as a permanent part of the Mathematics Department's curriculum. At present, enrollments on all campuses seem healthy, and if students' response to the experimental versions is any indication, students will enjoy the course, and end up being better prepared for their next science course, as a result of enrolling in it.

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I am grateful to the University of Connecticut for awarding me a 2005 Provost's General Education Course Development Grant, which made the development of this course and the writing of this article possible. I also wish to thank for help and support many students and colleagues at the University of Connecticut without whom this project would not have gotten off the ground: my Department Head, Michael Neumann, for asking me to take this project on, and for his continuos support; all my colleagues in the Mathematics Department Undergraduate Program Committee, and the Math 101 e-mail list, especially David Gross, Judy Lewis, Gery Liebowitz, and Jeff Tollefson, for advise, input and encouragement; Associate Provost, Steven Jarvi, his team of Student Advisors, as well as other Student Advisors across the University, for the many ways in which they encouraged the development of this course; graduate students, Luke Hodges and Stephanie Hartman, who tried many of the group projects in their sections of Math 101, and whose input improved the group projects and many other pedagogical aspects of the course; my colleagues, Tom DeFranco and Charles Vinsonhaler, whose book  inspired me to experiment with cooperative learning; my colleagues, Jim and Cecile Hurley, who, both separately and together, provided valuable practical advise and moral support; our System Manager, Kevin Marinelli, for technical assistance and for photographing The Largest Box. Last, but not least, I wish to thank my students in Math 101 and the experimental versions of Math 104Q, without whose enthusiastic input and willingness to join me in experimentation, there would have been no such course.

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