Application to NSF Sponsored Workshop

Improving and Assessing the Impact of Programs to Encourage High School Girls to Pursue Science, Engineering, and Mathematics

Santa Clara University

Santa Clara, CA

August 5 - 7, 1999

From:

Charlene Morrow

Director, SummerMath

Mount Holyoke College

50 College Street

South Hadley, MA 01075-1441

413-538-2069 (phone); 413-538-2002 (fax)

cmorrow@mtholyoke.edu

http://www.mtholyoke.edu/proj/summermath

Below I have outlined the theoretical underpinnings of our program, given a brief program overview, described many of our outcome measures, and given some sense of our outcomes in very general terms. SummerMath has been in existence since 1982, and I have been directing the program since 1986. I have mailed to you an envelope of materials, including two published articles about SummerMath, additional program information, and a flyer about my recent book, Notable Women in Mathematics (containing 59 biographies of women mathematicians, many of them contemporary figures). I would very much look forward to the opportunity to share and plan with other program directors in furthering the agenda of women’s participation in SMET fields.

The theoretical framework for presented in Women’s Ways of Knowing. provides a good basis for understanding our work at SummerMath. This framework is an important tool in understanding the educational needs of women and for analyzing educational reform efforts for gender equity. This particular need arose for us at SummerMath, Mount Holyoke College, because we offer a mathematics program for young women in high school (described in more detail later in this paper). Using tools such as the Curriculum and Evaluation Standards (NCTM, 1989) and the Professional and Teaching Standards (NCTM, 1991), it is becoming easier for us to determine the ways that our program meets the emerging standards produced by the current reform effort. What seems more difficult, however, is to find a tool of analysis that will help us understand how we are or are not meeting the educational needs of our students as females. The framework presented in Women's Ways of Knowing gives a way for us and other mathematics educators to pay careful attention to the needs of our female students.

What does a mathematics program look like when it strives to incorporate principals of both constructivism and women's education and development? Below is a very brief overview of one such program structure developed at Mount Holyoke College: SummerMath.

SummerMath is an intensive, four-week program for 60 young women in high school that provides new perspectives and new experiences of mathematics, computing, and science. The student body is very diverse, academically, geographically, and racially (usually about 70 percent minority student enrollment). Some of our students have achieved good grades in math, some have not, but all desire a new and deeper relationship with mathematics. The prerequisites for entering are the wish to have a different experience with mathematics and the willingness to engage oneself actively in the educational enterprise. No particular grades, courses, or test scores are required.

Based on constructivist and connected models of learning described above, SummerMath instructional methods are designed to remedy the blind memorization and rule-following behaviors that limit so many students' understanding of the underlying concepts of mathematics. SummerMath helps students replace such unproductive methods with more flexible problem-solving approaches. The program provides the challenge of rigorous study and difficult problems with the support of a community of teachers, residential staff, and peers.

A great strength of SummerMath lies in the diversity of its students. They vary in age, mathematical experience, race, ethnicity, and place of home, coming from cities, suburbs, and rural areas all over the US They come with different expectations and experiences for living together, learning, and communicating. Learning to live together with respect for all is not easy. A key element linking the teaching staff to students and providing support for students in both the classroom and dormitory is the group of residence and teaching assistants (RA's). These RA's are college-age women who live in the dormitory with students and help students both adjust to group living and meet the academic challenges of SummerMath. In this way, RA's are the primary mentors for students as they participate in SummerMath. Each RA has charge of about ten students in the dormitory and assists in the classroom or in recreation. As top college students interested in mathematics, science, and/or education, the RA's have an understanding of both the program and students that has made an invaluable contribution to SummerMath. Each RA takes her RA group on such outings as roller skating, shopping, and ice cream forays.

For an entire week before the students arrive, the staff of about thirty people meets to prepare for the program. Activities fall into three categories: Doing mathematical and computer activities; engaging in community-building exercises; and raising our consciousness about issues such as racial and gender equity. It also includes experience and reflection on mathematical activities in order to build understanding and commitment to constructivist principles of active learning. The focus of staff preparation is to look within for potential points of connection that can foster the growth and development of our students.

The primary classroom strategies used at SummerMath are as follows:

• Students work together to determine solution paths for problems. Teachers do not usually give answers to problems.
• The teacher serves as facilitator and guide rather than lecturer or expert.
• Students interact mainly in pairs and small groups.
• Students represent solutions in a variety of modes: Pictorially, graphically, numerically, verbally, and algebraically.
• Teachers ask students to explain in detail all solutions to problems and to validate their solutions.
• Students make mathematical connections to areas outside of mathematics.

The academic portion of the program for all SummerMath students is composed of three ninety minute classes five days per week. Each student takes Fundamental Mathematical Concepts and Computers/Logo Programming for the entire four weeks. In addition she takes three different workshops, each two weeks long. Classes are small and we maintain a student/teacher ratio of approximately 12:1. A number of extracurricular, evening, and weekend activities are offered as well. Each part of the program is described below.

Fundamental Mathematical Concepts (FMC): In FMC students usually work on problems in pairs, occasionally joining in with other pairs to compare ideas or to listen to short presentations made by a teacher or by other students. Their work in pairs includes solving problems aloud to encourage reflection on their problem-solving strategies. Students learn to draw visual representations, to use multiple methods, and to critically assess their own understanding of ideas. As they go through such activities, they begin to replace their former strategies of memorization and blind imitation with more flexible problem-solving approaches. In each classroom, at least one instructor and one undergraduate assistant circulate, asking questions and guiding student explorations.

Because of the wide range of students, the mathematical content of the curriculum includes fractions, decimals, elementary algebra, geometry, and mensuration formulas; coordinate geometry of lines, circles, and other conic sections; quadratic polynomials and triangle trigonometry; the general notion of function; and the trigonometric and exponential functions. We designed the curriculum to come in packets of materials that allow a student to construct her own understanding of concepts and that encourage each student to discover mathematical principles on her own. This approach leads to deeper conceptual understanding and greater persistence in problem solving.

All classes are heterogeneous in relation to mathematics experience, confidence, and age. Within each class, students are paired so that the two students in each pair are working on the same level of material, but in each class students are at all FMC levels. The initial level of material for each student is determined by an extensive assessment done in the first week of classes.

Problem solving is pervasive throughout all levels of instruction, and a special structured approach to problem solving, known as pair-problem solving, is often used to facilitate the development of higher-order thinking skills. In this approach students work in pairs, with one student identified as the problem solver and the other as questioner. The problem solver solves the problem, vocalizing her thinking along the way, while the questioner asks questions in order to understand how the problem solver is approaching the problem. Students learn to reflect on their own thought processes, learn that there are reasons for the methods they use, and learn that they can express those reasons. In this way students also learn that there are many different effective approaches to solving a problem, that some problems have no solution while others have many solutions, and that a single problem may sometimes be validly interpreted in more than one way.

Computers/Logo Programming (Logo): The second major component of the academic part of SummerMath for pre-seniors is a course in problem solving in a computer environment using the computer language, Logo. This course has as its objectives: the generation of interest, confidence and competence in programming and the use of software, development of mathematical ideas through computing, and development of structured problem-solving skills in a computer environment. Explored using Logo are mathematical ideas ranging from elementary geometry: angle relationships, coordinate systems, and scaling, to recursive functions. Students learn to plan, organize, and revise their ideas by working on projects such as transformational geometry, tangram puzzles, patchwork quilt designs, and group murals. Work is generally done in pairs, but groups of students will often collaborate on a larger project. In four weeks, students are able to master the language to a level that allows them to initiate, develop, and complete a final computer project of considerable sophistication. At the end of the program, a computerized "slide show" containing samples of all the SummerMath students' Logo creations is prepared for presentation at the open house for the families and friends of SummerMath students.

Workshops: Mathematics workshops comprise the third portion of the academic program. Students choose two two-week workshops from a collection of possibilities. The choices this year will be:

• Biology of Growth and Genetics • Confidence Building

• LegoLogo: Engineering Design • Economics

• Unit Origami: Folding in Geometry • Architecture and Mathematics

• Medical Ethics • Kite Making and Aerodynamics

In addition to developing self-confidence and problem-solving skills, the primary objective of the workshops is to let students experience the excitement and power of using mathematics in the "real world." All workshops involve "hands-on" activities for students followed by "minds-on" reflection on those activities.

Recreation, Residential Life, and Other Activities: The recreational program is made up of various athletic choices: Dance, volleyball, tennis, racquetball, squash, running, walking, weight lifting, aerobic dance, Aikido, and swimming. Emphasis in afternoon recreation is on rejuvenation through emotional and physical expression, while at the same time providing opportunities to learn new sports. Creative writing is also offered.

Informal panels of professional women, graduate students, and undergraduates are invited to discuss their preparation for college while in high school, their undergraduate education, what it's like being a graduate student, their career(s) and changes in their career choices, what their jobs are like, and what it's like to balance their personal and professional lives.

The camaraderie and support provided by the students themselves and the residence staff is a vital and essential part of SummerMath. Group activities and cultural events serve to build a strong sense of community, one that cares about and has high expectations for each individual in the program. A respect for each student and an appreciation for the diversity of people is stressed. Students are able to take greater risks in mathematics learning as they become part of the strongly supportive community that SummerMath provides.

Assessment of learning is an on-going process at SummerMath. It begins, in all FMC classes, with an intensive week of problem solving, which accomplishes two purposes. First, it introduces a classroom style in which answers and methods are not given and in which the emphasis is on becoming aware of one's own thoughts and the process of solving a problem. Second, it assesses, for each student, an appropriate level of mathematics at which to begin.

In FMC, the process continues with the daily interactions between student and teacher. Both teacher and learner take responsibility for assessing how well a concept is understood, based on the student's explanations of her thinking. Of course, this is almost always a new experience for the student, who generally has looked at her teacher as the authority in such matters. Midway through the Program a written evaluation of progress is made. Again, both teacher and student have responsibilities: the student completes a self-evaluation, the teacher completes an evaluation of each student, and the two compare and discuss results.

In Logo, the curriculum is designed so that each student can quickly find appropriate level projects on which to work. On-going assessment of progress, thoughts, and feelings in Logo classes are accomplished by weekly student writing, with oral follow-up done by the team of instructors. The two-week Workshops vary in their assessment methods, which are generally far less extensive than they are in FMC and Logo classes. Instructors of Workshops usually rely on oral interactions with students to guide their instruction and to inform students about their progress.

On Thursday in the final week of SummerMath, all students write extensively about their experiences at the program in response to an end-of-program questionnaire. They rate the program overall, and they respond to open-ended questions about their classes and the cultural and residential aspects of SummerMath. Instructors also complete written evaluations for each of their students at the end of the program. They comment about progress made, the ways in which the student participated, and suggestions to the student for her studies in the future.

We have also completed several years of attitude change research on our students showing the effectiveness of our approach in making positive changes in students' attitudes toward mathematics. Every year over 90% of our students show increases in confidence and persistence in mathematical problem solving. In 1998 a Mount Holyoke master’s thesis was written using interviews, attitude surveys, and evaluation data taken from the 1994 - 1997 programs.

Some of the most powerful statements about the results of the SummerMath program come from our students, as illustrated by the following quotes:

We have heard many other stories of young women taking charge of their learning, from simply asking the kinds of questions that will move learning ahead to having the courage to take honors and advanced level classes. The data we have collected on attitude changes also strongly supports these conclusions.

As illustrated by the above comments, students take home:

They become independent, but not isolated, problem solvers, retaining a sense of authority within themselves but able to effectively communicate with others about mathematics.

Since 1994 we have been holding a five day working conference for educators (including secondary mathematics teachers, principals, program developers, teacher educators, and college teachers). Their applications included reports of their previous contributions to gender equity and commitments to further endeavors to promote the mathematical education of young women. Teams of two or more persons from a single school system were encouraged to apply. The conference is held during the time that SummerMath is running.

Participants were introduced to a developmental model for women's learning and discussed its implications for mathematics education. They learned about and experienced our Pair Problem Solving strategy and worked many of the problems in the FMC curriculum. They attended SummerMath FMC and Logo classes and discussed the strategies they observed. Participants were introduced to the use of mathematical metaphors by Dr. Charlene Morrow, Past-President of Women and Mathematics Education, and professor of psychology and education at Mount Holyoke College. Dr. Maria Trumpler of Yale University and Acting Director of SummerMath also presented a session on women in the history of mathematics. Several participant-generated sessions were held where teachers presented to others materials and strategies based on their classroom experiences.

GenderWise sessions were very lively, and participants commented very positively on their experience, as illustrated in the comments below. Outcomes (more fully discussed in other papers) are indicated by the following sample of GenderWise participants’ comments:

• All the activities were important and I believe each session was integral and crucial to the conference but I especially liked the problem solving session. I discovered some things about the way I worked with students that may not be very helpful to them.
• Pair share (listener/problem solver) - Group as a whole discussions-- hearing others’ concerns and solutions/ideas - Talk on women in math history was excellent!
• The fact that the participants were such a diverse group, not all high school teachers, made the conference very stimulating. We came here with quite different goals and that made the conversations all the more interesting.
• Visiting the FMC class- seeing the approach in action was extremely powerful. It was a great opportunity to check my assumptions about the method, it provided an “up-close and personal” example of the theory in practice. I found that ninety minutes the most exciting of all. It was very energizing.
• Metaphor session-- I’m amazed how differently people feel about mathematics.
• Origami session-- Being forced to solve a problem with limited instructions is a humbling experience- similar to what our students go through.
• I’m excited to tell other teachers about pair-problem solving. I have a never seen this type of collaborative work environment before, and think that it could be extremely useful for some students. It really makes them question what they have done (or are doing) and gives a voice to all students.
• The idea of not providing answers to the students seems really new to me not in the sense that I don’t agree with it, but rather that it seems to contradict current practices.
• I learned/confirmed my belief that while ensuring that curricular material is presented in the classroom clearly and accurately is an important responsibility of the instructor, ensuring that barriers to learning and assimilating material by students are removed or diminished is also an important moral responsibility of the instructor.

A summary of strategies that we use at SummerMath, and which may help all educators address the needs of young women in the mathematics classroom, are as follows: