At the national joint meetings in San Diego in January, 1997, and in Baltimore in January, 1998, there were contributed paper sessions on Developmental Programs That Work, sponsored by the Developmental Mathematics Subcommittee.  The programs discussed below were chosen from those presentations.  They appear in alphabetical order, by speaker's name. 

Suzanne Doree, Augsberg College, The Log-Divides Formula or An Example of Content We Should
(and Should Not) Be Teaching in Developmental Mathematics Courses

Rick Gillman, Valparaiso University, Quantitative Problem Solving at Valparaiso University

Laurie Hopkins and Amelia Kinard, Columbia College, Combining Experiential Learning with the TI-92 in Developmental Algebra Courses

Frances B. Lichtman, Alma College, A Foundations Course for Underprepared College Students

Mercedes McGowen, William Rainey Harper College, Implementation of a  Developmental Algebra Curriculum Project: An Examination of the Role of Technology and Its Impact on the Nature of Knowledge Acquisition

Teri Jo Murphy, College Mathematics Instruction in Transition

Lisa  S. Yocco, Georgia Southern University, Tailoring the Transition from Developmental to College Mathematics

The curriculum group of the NKATE (NSF Kentucky Advanced Technological Education project) has developed a reform intermediate algebra level course, which it has piloted for five semesters, since Spring, 1995.  The group is comprised of eighteen University of Kentucky Community College System faculty members.  Most of the group are mathematics faculty, but the group also includes faculty from physics, industrial engineering technology and computer information systems.

The group considers reform of the intermediate algebra course to be the most crucial to community college mathematics programs.  It has very high failure rates, and students who do succeed have low success rates in subsequent courses; the traditional intermediate algebra course clearly does not perform its mission.

The course, An Introduction to Functions Through Applications, has maintained a core of the drills-and-skills found in the traditional course, but has drastically altered the approach.  Topics are motivated by real-world applications or student faculty experiments that generate data.  The course relies heavily on collaborative activities and uses technology, but takes care not to let the technology determine the curriculum.  The course was designed in the spirit of the MAA Guidelines and the AMATYC Standards for problem solving, connecting with other disciplines, and technology.

The program is being evaluated externally; preliminary follow-up data indicate results even better than project participants expected.

 Lillie R.F. Crowley, lillie@pop.uky.edu
Lexington Community College
Cooper Drive
Lexington, KY 40506-0235
(606) 257-2797

Darrell H. Abney, dha@pop.uky.edu
Maysville Community College
1755 U.S. 68
Maysville, KY 41056
(606) 759-7176

There is much exciting work being done these days on how we teach math.  At the session on Developmental Mathematics last January I heard many interesting ideas about pedagogical choices and overall construction of curricula.  (I talked about the latter myself, in fact.)  What was missing, I thought upon reflection, was an equally thoughtful discussion of what we teach:  what “works.”

I teach a highly successful reform algebra course designed for students who have taken some high school algebra but are not ready to take a college level mathematics course or a quantitative reasoning course in chemistry, physics, or statistics.  There are many aspects of the Applied Algebra course that I believe contributes to its success.  One important aspect is keeping the course from getting over-packed with content.  The “less-is-more” philosophy is certainly not new but it is also not easy to implement.  There are times when I am still tempted to teach students everything I know about a subject, or at least everything I “usually” teach about the subject.

There are many examples of concepts that a student in calculus or pre-calculus needs to know but that a student in an “alternative” college math course or its pre-courses does not.  It seems to be the case at many schools, such as mine, that many of the students in intermediate algebra courses do not continue to pre calculus and the majority of students in intermediate algebra courses do not take calculus.  Although intermediate algebra courses are typically designed as a “pre pre-calculus” course, they function, for many students, as a pre-course to an “alternative” college math course such as liberal arts math, finite math, statistics, etc.

The particular area I will focus on is logarithms.  As an example, I think that students in intermediate algebra courses, or other pre-courses to “alternative” college math courses, ought to learn how to solve exponential equations using logarithms (hopefully preceded by using a numeric method).  I think, however, that only students in calculus and its pre-courses need to study the laws of the logarithm.

Most of us are used to solving exponential equations by explicitly using the laws of the logarithm but that is not the only way to teach the concept and skill. It has been my experience that for some students it’s not even a successful way; a student fails to understand the laws of the logarithm and concludes, incorrectly, that s/he cannot solve exponential equations using logarithms and, consequently, that  “logs are difficult.”

I have been successful in teaching students to solve exponential equations using logarithms without necessarily understanding, or even having seen, the laws.  My students report that “logs are convenient and easy.”

This partially answers the following questions.  How do we decide which specific concepts (such as solving exponential equations or the laws of the logarithm) a student in a particular course needs to learn?  Once we have identified a concept as important for students to learn in a particular course, how to do we teach that concept (such as solving exponential equations) without requiring students to learn concepts (such as the laws of the logarithm) that we have identified as not as important for them to learn?  How do we cut topics (like the laws of the logarithm) while still being mathematically responsible?

While I focused on particular examples about logarithms, I hope that it will inspire us to revisit the choices we have made about a wide range of content.

Suzanne Doree, Ph. D., doree@augsburg.edu
Department of Mathematics
Augsburg College # 61
2211 Riverside Drive
Minneapolis, MN  55105
612-330-1059 

In reviewing the current literature on developmental mathematics and the admissions standards of the University, the departmental committee doing this work identified the problem as one of incoming students lacking problem solving confidence and ability, rather than one of a lack of mechanical skills.  (All admitted students have “successfully” completed three years of college prepatory mathematics.)  The committee established the following criteria for the design of the course:

• Emphasis should be placed on problem solving skills.
• As much work as possible should be done in a collaborative setting.
• There should be some focused review of algebra skills, particularly in the context of solving problems.
• The course should address the different perspectives that students might bring to it as non-traditional students or international students.
• The course should not be viewed as a burden to either the students enrolled in the course or the faculty teaching it.

The committee decided to use the COMPASS software published by ACT as a placement test, ADVENTURES IN ALGEBRA by Quant Systems as an online algebra tutorial system, and to devote in class time to collaboratively solving problems and discussions of the problem solving process.  The course, which consists of multiple sections with controlled enrollment, is collaboratively taught by several faculty in the department in rotation so that experienced faculty can  introduce the course to other faculty, and everyone has an opportunity to teach the course.

Among our insights from the first year of teaching the course are the impact of guess and check being an “efficient” way to solve a problem, and some understanding of student separation of meaning from mechanics.

Rick Gillman, rgillman@orion.valpo.edu
Department of Mathematics and Computer Science
Valparaiso University
Valparaiso IN, 46383
(219)-464-5067
 

The availability of computer algebra systems in handheld technology raises new questions about the applicability of such technology in introductory algebra courses.  This program takes advantage of the new handheld computer algebra systems and new information about the way in which students learn to transform developmental mathematics at this college.  The pilot program, which was implemented last fall, was so successful that we have incorporated this new program into all developmental algebra courses.   The students in the program work with algebraic manipulatives and realistic problems to understand the concepts of algebra and then use the TI-92 graphics calculator to facilitate the symbolic manipulations.  Students were expected to understand the concepts; test and homework questions frequently included discussions of how and why.  Paper and pencil manipulations were taught, but students were allowed to use their calculators on all tests and homework.  For this pilot study, all students were required to rent a TI-92 from the mathematics department or to purchase their own.

The data we’ve gathered includes both anecdotal information and a comparison between students in traditional courses and in the new course in the dimensions of performance, attitude and persistence.  Preliminary longitudinal results indicate the effect this kind of preparation may have on performance and attitude in subsequent mathematics courses.

Laurie Hopkins (lhopkins@colacoll.edu) and Amelia Kinard
Mathematics Department
Columbia College
Columbia, SC 29203
Phone:  (803)-786-3669
Fax:    (803)-786-3809
 

At Alma College, we offer a foundations course for underprepared college students.  Students are carefully placed in this course based on mathematics ACT scores, previous coursework in mathematics, and scores on the College mathematics placement examination. The previous mathematics experience of these students has been generally negative.  The goal of this course is to help students learn a variety of problem solving techniques, to prepare students for real world quantitative experiences, and to provide access to higher-level mathematics courses.

The content of the course includes arithmetic of real numbers, simple equations, applications of percent, geometry, probability and odds, descriptive statistics, and graphing (lines and exponential equations).

The students respond well to a highly interactive, participatory approach, combining collaborative activities, board work, and discussion, with little lecture.  Topics are presented in a way that is intertwined with life experience.  Genuine data is used along with reading selections from newspapers and magazines.  Students write about their math experience in journals that are turned in to the instructor every other week.  Although arithmetic skill acquisition is part of the course, calculators are essential for problems involving extensive computation.

A glimpse of the classroom environment:
• To review homework problems, students are paired at the board, one who understands a problem with one who does not.  After working together, the student who did not initially understand the problem will explain it to the class.  Often the entire class works at the board.
• Students work in small groups interpreting graphs selected from current news articles.   Afterwards, they read the accompanying article and observe the quantity of information conveyed by the graph alone.
• Working in small groups, students examine spatial relationships. They decide how much paint will be needed for their dorm room, the amount of materials needed to build a bookshelf, and compare the height of a can of three tennis balls to the circumference of the can.

Approximately half of the students in this course have subsequently completed a course in beginning algebra.  Many students have also completed a course in elementary statistics or a course in mathematics for elementary teachers.

Frances B. Lichtman, Lichtman@alma.edu
Department of Mathematics and Computer Science
Alma College
Alma, MI 48801

The large numbers of students enrolled in developmental math programs at colleges and universities and the historically low rate of success in subsequent college mathematics courses reinforce the need to understand how these students develop and/or reconstruct their mathematical understanding and skills to provide a foundation for successful college work.  Many students in developmental algebra courses have learned mathematics that does not make sense to them. These students have great difficulty interpreting function notation and other symbolic expressions.  They have not learned to distinguish the subtle differences symbols play in the context of various mathematical expressions.  The ability to flexibly interpret and use ambiguous mathematical notation is necessary for successful mathematical thinking, yet many students enrolled in this course are unaware of this ambiguity, or of the need to develop flexible strategies and to consider the context of the problem.
 
The reform algebra curriculum used in the study was developed to meet the needs of returning students and adult learners enrolled in the community college and in four-year developmental programs.  The curriculum promotes the view that learning mathematics is a matter of mastering a sizeable, interrelated body of knowledge and acquiring mathematical power, with the concept of function used as an organizing lens.  Skills are taught in context and the use of the graphing calculator, integrated throughout the materials, facilitates the development of the function concept and incorporates data analysis.  The materials are designed to support students actively engaged in their own learning and teachers as they attempt to improve pedagogy and increase effectiveness while coping with substantive changes in what they teach and how they teach it.  Two goals of the curriculum are to have students make sense of mathematics and to develop a positive attitude towards mathematics and its power for problem solving.
 
We have studied the ways in which undergraduate developmental algebra students construct and organize new knowledge into their existing knowledge structures and examines the role of technology in this development. The nature of the processes and competencies needed by students who use graphing calculator technolgy are a major focus.  Pre- and post-test questionaires, student work, concept maps, and task-based interviews were used to profile student understanding.  Results indicate that successful students do different kinds of work and demonstrate their ability to use a diverse, flexible set of strategies and algorithms for solving mathematical problems. The divergence between the more successful and least successful is greater when students attempt to answer questions which are viewed as “pencil and paper” questions by the students, such as solving an equation or factoring a quadratic expression.  Though still evident, the divergence between the more successful and least successful is less when students answer questions utilizing technology and various representations, such as tables and/or graphs, or when solving systems of equations.

Mercedes McGowen, mmcgowen@harper.cc.il.us
William Rainey Harper College
Palatine, IL. 60067-7398
 

Using classroom observations and videotapes, and a journal the course instructor kept, we followed an instructor through her first semester of attempting to implement these strategies.  Results from these data sugggested that (a) although the course employed active learning and student collaboration, the content presented remained at lower cognitive levels and (b) the instructor experienced frustration in trying to balance content coverage with student involvement, in learning to release control to the students, and in discarding traditional notions of remediation.

These changes described are a lot for instructors to handle all at once.  Instructors tend to teach the way they were taught -- and to return to those strategies by default.  Watching videotapes of the classes and maintaining a journal helped, but were short-term and not intensive.  Recommendations include (a) increased ongoing professional development opportunities (consultations with professional development staff, support to attend local and national meetings that address instructional and content issues) and (b) departmental support (tangible and intangible) for reform efforts.  In light of the instructor’s frustrations and dissappointments, and essentially isolated without an ongoing support structure for her efforts, it was too easy to slip back into traditional teaching patterns.

Teri Jo Murphy, tjmurphy@AFTERMATH.math.ou.edu
Department of Mathematics
University of Oklahoma 

Many students are entering college with poor skills in mathematics.  The reasons for their math inadequacies are varied, but the fact remains that more and more students are enrolling in remedial algebra at the college- and university-level, and that College Algebra is the most dropped and/or failed course on many campuses.  Developmental teachers are steadfast in their commitment to the teaching of algebra, reminding students what, how, and when to study, and they employ any tool or trick of the trade to bring students up to the level that will enable them to pass their next course.  Despite all of the good teaching that takes place in the developmental classroom, there is, perhaps, a failure to prepare students for the potential rigidity or indifference that occurs in a College Algebra class.  Having been handled specially, these students will enter a college mathematics class with other students and will perhaps receive no special attention.  Because we cannot always change the future course, we must prepare them to survive in a more challenging environment.  How can we tailor the transition for these students from developmental courses to college-level courses?

To ease this transition, we introduced several techniques in our developmental math classes designed to help improve students’ ability to handle college-level mathematics courses.  One technique used was to integrate study skills as applied to a mathematics course.  Research has indicated that 50% of academic achievement results from a student’s IQ, 25% from the quality of instruction, and 25% from factors that can be modified, such as study habits. [Bloom, 1976]  Students have seen study skills before -- they know the SQ3R method for studying, the Cornell method for notetaking, time management techniques, and whether they are visual, audio, or kinesthetic learners.  What they perhaps have not seen is how these study skills relate to studying math.  For example, they may not have used a modified Cornell method for taking math notes, learned how to study for a math test, used tips for taking a math test (such as memory data dump), nor seen how to learn from mistakes made on tests.  Reading a math textbook is certainly different from reading a history textbook, and adjustments should be made to accommodate reading a math textbook, even if the student is simply skimming the section as a preview to the day’s lesson.

Secondly, students were given good organizational hints and encouraged to use them.  Good organization is essential in a math course so that the student does not fall behind and can quickly reference a topic if needed. Keeping an organized notebook consisting of  three sections (class notes and examples, homework, and problem log), transferring problems from the problem log to make a practice test, and making note cards with sample problems on one side and the solution on the other side are some of the methods that can help students’ success in a math course.  Study guides and test reviews are also helpful to (and popular with) the students, provided they do not count on the teachers’ preparation of review material as their sole resource for study materials.  Encouraging students to make their own “cheat sheet” is extremely beneficial to their understanding of important concepts and how these concepts relate to each other.  Reviewing notes or examples before students begin their assignment and learning to write every step of the problem without doing steps mentally can contribute to their success in later courses.

Additionally, instructors and students used precise mathematical language, to the extent that this was practical.  Instructors can endear themselves by offering shortcuts or “tricks,” but in the end the students may suffer.  Instructors should use mathematical terms correctly but avoid being pedantic.  As mathematics teachers, we have our own vocabulary -- words like simplify, evaluate, graph, and factor -- that often imply carrying out a list of operations to complete.  We often use the same words in different ways as applied in different contexts.  This can create confusion for the students.  A few examples follow that illustrate how students can easily become confused.

“Simplify” an exponential expression means rewrite the expression without using any negative exponents and carry out any exponential evaluations that are possible, while “simplify” a radical means rewrite the radical so that no perfect square factors appear under the radical.

“Factor” 40 means rewrite 40 as a product of prime numbers, while “factor” a polynomial means find two polynomials that multiply together to give the original polynomial.

“Graph” x = -3 might mean graph the point -3 on a number line or the vertical line x = -3 on a 2- dimensional coordinate system.

Consider the evils of words like “cancel.”  For example, when asked to “simplify [3xy(x+2)/3xz],” if the student is taught to “cancel the 3x” to get [y(x+z)/z] ,when asked to “simplify (3x+5y)/3x,” will the student “cancel the 3x” and get 5y?

Lisa  S. Yocco, lisay@gsvms2.cc.gasou.edu
Department of Mathematics
Georgia Southern University
Box 8121
Statesboro, Georgia 30460