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Apple Classrooms of Tomorrow Research
Report Number 11

High School Mathematics: Development of Teacher Knowledge and Implementation of a Problem-Based Mathematics Curriculum Using Multirepresentational Software


Authors

Jere Confrey, Ph.D.
Susan C. Piliero
Jan M. Rizzuti
Erick Smith
Cornell University


Apple Computer, Inc.
1 Infinite Loop
Cupertino, CA 95014


Preface

Begun in 1985, Apple Classrooms of Tomorrow (ACOT)SM is a research and development collaboration among public schools, universities, research agencies and Apple Computer, Inc. ACOT explores, develops and demonstrates the powerful uses of technologies in teaching and learning. In all ACOT endeavors, instruction and assessment are as integral to learning as technology.

Supporting a constructivist approach to learning, technology is used as knowledge-building tools. As students collaborate, create media-rich compositions and use simulations and models, researchers investigate four aspects of learning: tasks, interactions, situations and tools. The research is formative. The findings guide ACOT staff and teachers as they refine their approach to learning, teaching and professional development. ACOT teachers and students often use the most advanced technologies available, including experimental technologies, to help us envision the future and improve the educational process.

ACOT views technology as a necessary and catalytic part of the effort required to fundamental restructure America's education system. We hope that by sharing our results with parents, educators, policy makers, and technology developers the lessons of ACOT will contribute to the advancement of educational reform.

The original research summarized here took place during the 1989-90 school year and was reported in a symposium, "Multiple Perspectives on the Implementation of Multi-Representational Software in a Secondary Classroom," at the 1990 meeting of the American Education Research Association.

Introduction
A Call for Action

Recent publications of the National Council of Teachers of Mathematics (e.g., NCTM, 1989; Wagner & Kieran, 1989) call for major changes in the curriculum, assessment, learning activities, and use of computers and calculators in mathematics instruction. The proposed changes reflect a contemporary view of what it means to be mathematically literate in an information society. This math literacy requires new competencies, including the ability to apply mathematical ideas to problem situations and work with others to set up and solve problems. Such abilities are not acquired by individual drill and practice. Problem-solving experiences must be central to the school experience, providing opportunities for students to work through problems embedded in complex contexts, verify and interpret their results, communicate mathematically, and reason logically. These suggested reforms are based on the premise that mathematics is much broader than a static set of concepts, definitions, and skills to be mastered.

Educators maintain that the introduction of computers into math classrooms can dramatically improve instruction beyond basic skills, but research documenting these predictions is relatively scarce. Still, it seems that appropriate technical tools, linked to a competent curriculum, and accompanied by active teacher support, could offer a genuine alternative to existing practices for teaching mathematics. Thus, the goals of this project are to build a problem-based mathematics curriculum and multirepresentational software to support students' problem-solving activities, provide teacher development in this setting, and conduct research on the implementation process.

Project Design: A Constructivist Approach*

Based on a constructivist framework, the instructional goal of this project is to create a learning environment that promotes student construction of mathematical concepts through repeated cycles of developing a problematic, acting to resolve the problematic, and reflecting on these actions (Confrey, 1989b). The term problematic refers to a disturbance, or roadblock, between where the student is, and where s/he wants to be. It is the problematic that calls the solver to action and to subsequent reflection on the action. As a result, the learner develops new understandings and this completes the process of knowledge construction.

Instruction . The constructivist teacher seeks opportunities for students to engage in tasks with potential for promoting conceptual development. Frequently, the teacher uses problems for exploring student conceptions and supporting their extension and development. In individual, small group, or whole class settings, the constructivist teacher listens to students and tries to understand their mathematical interpretations, which may be different from his/her own. The success of a constructivist instructional model depends on how well the teacher is able to recognize the student's problematic, hear legitimate alternative solutions, and promote reflection (Confrey, 1989b). Ultimately, the student must decide on the adequacy of his/her construction.

Curriculum. The curriculum developed for this project consists of contextual problems designed to be used in conjunction with problems selected from the course textbook or developed by the teacher. The research team developed two units for the precalculus curriculum, the first on linear functions and the second on geometric sequences, exponential and logarithmic functions. (See example problem on page 16.)

Software. In a constructivist learning environment that encourages students to express and defend alternative solutions, students need multiple ways to represent their ideas. Symbolic representations, formal proofs, calculator keystrokes, graphs, figures, statements of contextual problems, and spoken language are common forms used to represent mathematical ideas. Based on the notion that students learn more effectively when they can represent mathematical ideas in multiple forms, the Cornell research team developed Function Probe (Confrey, 1989a). Function Probe is an interactive software tool that encourages learners to explore the concept of function by enabling them to represent the function through tables, graphs, calculator keystrokes, and algebra, and to move between and coordinate actions among these multiple representations.

The software was designed in response to research on student methods and, thus, has features which allow students to, for example, build calculator buttons as generalizations of numeric procedures, or to fill tables to organize data and discern patterns. As a dynamic medium, the representations can be easily manipulated. For example, graphs can be translated, stretched, and reflected by actions carried out with the mouse. Finally, all windows keep an easily accessible history to assist students and teachers in reflecting on student work. (Screen images illustrating the multirepresentational character of the software are on pages 16 17.)

Teacher Development. Teacher development in this project included a week of intensive preparation during the summer and continued support during the school year. In addition, the reflection process the teacher experienced after planning and debriefing interviews contributed to her development.

Implementation of the curriculum and software tool was closely linked to teacher development and the teacher's contribution to the project, which was recognized as critical and transformative. The teacher was creating new knowledge as she worked with the resources provided. Since she was compelled to use the software and curriculum within the constraints and possibilities of everyday instruction, the teacher and the research team developed a partnership enabling each member to apply his or her own expertise to make the innovation work.

The Study

This report summarizes work that took place during the 1989-90 implementation year, after Function Probe software and two curriculum units had been developed. The study involved one high school mathematics teacher in the Apple Classrooms of Tomorrow (ACOT) project and her precalculus class of ACOT students.

The class being studied was located in a midwestern urban high school with a student body of approximately 1300 (33% black, 62% white, 3% Asian). There were four ACOT classrooms in this participating high school, with nine teachers and approximately 120 students from ninth through twelfth grades. All incoming ninth graders scoring above 37% on mathematics and reading tests were eligible to apply to the ACOT program and selections were made at random from this applicant pool, controlling only for race and gender. The precalculus class participating in this research consisted of the ACOT senior class of 22 students currently in the fourth year of the program.

The teacher was a veteran teacher with sixteen years of experience and three years as part of ACOT. Prior to the project, she described her teaching as traditional with little use of applications or multiple representations.

Data Collection

Data collection took place during a year-long period from March 1989 to March 1990. Research focused on implementation of the innovation and included five weeks of data collection in September-October 1989 on linear functions and five weeks in January-February 1990 on exponential functions. Data collection of the teacher's practices prior to the implementation phase took place in March 1989 and July 1989 (during the teacher development week), and then continued through the 1989-90 school year.
The research team used the method of triangulation (Lincoln & Guba, 1985) in data gathering. This approach enabled the team to collect data from a variety of perspectives and then to examine the interrelationships between them. Jan Rizzuti studied the development of individual students; Erick Smith concentrated on small group interactions; Susan Piliero studied the teacher's development of pedagogical and content knowledge, including her beliefs and practices. Project director Jere Confrey led the software and curriculum development efforts, provided guidance and support for the teacher, and oversaw the conduct of the research.

The research team gathered data from the following sources: 1) printed records of student work on Function Probe; 2) videotapes of all class sessions; 3) audio and videotapes of individual students and teacher interviews; 4) unit pretests and posttests; 5) observation notes taken during classes; 6) lesson plans, worksheets, tests, and other instructional artifacts generated by the teacher; 7) copies of student assignments and tests collected by the teacher.

Findings and Interpretations

Individual Students

Research findings indicated that students of average and above average abilities were able to construct strong conceptualizations of linear functions and exponential functions by working with multirepresentational software. Using Function Probe, students were able to represent relationships and operations algebraically, graphically, and in tabular form, and their understandings of functional relationships were strengthened and broadened by coordinating different representations. In addition to the use of multiple representations to understand functions, the use of contextual problems also contributed to their understanding.

Results also showed that although the software and curriculum played important roles, they were not enough to ensure strong conceptual development in most students. The teacher's role was critical in the development process.

Student use of Function Probe's table, graph, and calculator windows advanced learning in several areas, including the following:

Table Window. Students used the table window to organize the information of a problem and generate information such as differences or ratios between entries, although they often had difficulty identifying the variables clearly. They could fill in two columns with the sense of co-varying rates of change, but often had difficulty coding the relationship between the two columns algebraically.

Students also used Function Probe to test guesses and hypotheses. For example, one student used the RATIO command to verify his guess that a sequence was geometric, and another built an algebraic equation, checking the computer equation values against her expectations.

Graph Window. Many students used graphs as secondary representations after using the table or calculator to find relationships within the problem. They were also able to link the numerical data from tables and calculators to the graphical representations in order to better understand functional concepts such as slope and rate of change.

Many students developed a strong qualitative sense of functional relationships that is not as easily developed when examining hand-drawn graphs. By rapidly graphing functions and presenting them as objects for examination, Function Probe allowed students to concentrate on global features of graphs, such as shape, direction, and location, leading them to a deeper understanding of different types of functions.

Other features of the graph window proved to be important learning tools for the students. They changed the scale of the graph window to situate their graphs on appropriate axes and examine the extreme points of graphs. They used the translating, stretching, and reflecting features to coordinate the algebraic and graphical forms of functions. They used the point-locator to find points of intersection between graphs. Students quickly became facile with these features and were able to spend more time solving problems and discussing their solutions with others.

Calculator Window. Students used the calculator window extensively to build operational and numerical forms of functional relationships, as well as carry out basic calculations. For instance, they used the repeated multiplication inherent in an exponential function to model the actions of a computer virus problem; then they built algebraic equations in the form of calculator buttons to model the functional relationship of this situation. Textbooks typically present full-blown algebraic representations of functions, which do not support students' development of this important ability.

Small Group Interactions

The research on small group interactions yielded two major results: 1) collaborative group work can play a critical role in students' development of diverse approaches and solutions to problems as well as their conceptions of the problem-solving process; 2) effective collaborative learning cannot be implemented rapidly; it must grow in conjunction with changes in other classroom practices, including the nature of teacher student interactions, assessment practices, and ways of reporting outcomes of group interactions.

Analysis of small group interactions focused on the evolution of the problem-solving process from the development of individual problematics to the achievement of a solution through negotiated consensus. Results indicated that three processes are necessary for successful collaborative learning: 1) individuals need to identify their own problematics (the issues they believe must be resolved to solve the problem), which evolve throughout the problem-solving process; 2) the group must create a social framework for developing a negotiated consensus that allows group actions to lead towards resolution of individual problematics; 3) individuals must have opportunities to reflect on group solutions in relation to their evolving problematics.

Changing from an individualistic to a cooperative approach in mathematics learning was well-received by students in form, but making it become an effective means for learning was more difficult. Initially students' conceptions of mathematics and mathematics learning caused them to be sceptical of the value of seeing other methods. They focused exclusively on getting the answer, completing the task, and moving quickly to the next task. However, this changed over the year. Students learned to read a problem and begin a discussion on how to proceed. They began to view problems as puzzles to be worked out over time, allowing for conflict resolution and exploration of alternative solutions. This change can be attributed to several factors: the teacher's active solicitation of alternative viewpoints; the software's ability to support rapid explorations, which facilitated development of negotiated consensus; and group presentations that provided a forum for arguing and discussing alternative solutions.

The students' collaborative skills improved, despite a gradual decrease in group structure and class time spent on group work. In the beginning of the year, group activity filled most class periods, groups were composed heterogeneously, and group roles were assigned. By the end of the year, no roles were assigned, little attention was paid to group composition, and group work often occurred intermittently over the class period. This suggests that a mathematics learning environment that promotes student methods and allows for diversity may be at least as important as the use of structured group work in promoting collaborative learning.

One of the major difficulties for teacher and students was finding a balance between closure and exploration. In a classroom where alternative viewpoints were encouraged and valued, it was difficult to maintain a balance between providing time to reflect on presented approaches and exploring further alternatives. Although reaching satisfactory closure remained an issue throughout the project, the teacher undertook activities that aided this process. These included: 1) placing greater importance on her own understanding and exploration of the subject matter; 2) experimenting with assessment for collaborative work through activities such as group presentations; 3) visiting groups unobtrusively and remaining for a longer time, thus encouraging group interactions and increasing her understanding of group processes and solutions.

The Teacher

Conceptions of Mathematics. Initially, the teacher described mathematics as two separate subgroups: school mathematics and the mathematics used in real life. Later, she began to see mathematics as having infinite actions and processes that extend beyond mere computation and manipulation of algebraic symbols and numbers. The teacher also developed a more positive appraisal of her own mathematical abilities and this increased confidence was evidenced in the classroom by her willingness to devote more instructional time to problem-solving activities and to student-initiated departures from her own instructional script.

Subject-Matter Expertise. Subject matter was no longer the traditional material of school mathematics, but the construction of new understandings of mathematics through the process of exploring, reviewing, analyzing, integrating, and applying multiple representations of functions and transformations on them. The teacher showed evidence of substantial growth in her own subject-matter knowledge.

Conceptions of Mathematics Instruction. The teacher moved from the view that she should develop mastery in specific teaching techniques to the view that she needed to develop awareness of her own mathematical processes. In September, interviews focused on issues of grouping, class management, and grading. By January, however, interviews focused on subject matter. She identified the need to approach subject matter as both a teacher and a learner. As a learner, she had to construct her own meanings for the mathematics in light of her current understandings. As a teacher, she had to understand ways of analyzing and approaching the problems rather than merely arriving at a satisfactory solution.

Reflection and Point-of-View. There was significant change in the teacher's ability and desire to reflect on the events of the class through the students' perspectives of subject matter. She described her past practice of "shutting down" her thought processes after class as a survival technique she developed in response to years of teaching six classes a day and not having the inner resources to look ahead or reflect back. That practice changed, as she later reported a strong need to go back and view lessons from the students' points-of-view. She began to take notes on student solutions and comments and used these notes to organize her own thinking about their interpretations and strategies, which affected her planning for subsequent lectures and discussion.

It became increasingly important for the teacher to allow students opportunities to share their insights and solutions with the rest of the class. She allotted class time for group presentations during which she not only asked students to share their solutions, but to field questions from her and their classmates about their thought processes, choice of representations, and conclusions. By acknowledging alternative ways of approaching and solving problems, and allowing students the opportunity to share these approaches, the teacher felt students had developed a healthy autonomy and taken on a greater responsibility for their own learning.

Assessment. In September, much of the teacher's discussion on assessment revolved around how to assess the group work, but by January, her grading encompassed more individual work. By the year's end, group work accounted for 15% of a student's grade, with the remainder based on tests, quizzes, handouts, class notes, and homework. Her tests and quizzes showed an increased ability to design nonroutine questions that required students to coordinate multiple methods and/or representations. In addition, the problems no longer needed to conform to the exact format of previously assigned homework problems or examples worked out in class.

Curriculum. Prior to participating in the research project, the teacher tended to avoid word problems, focusing on basic skills acquisition and keeping applications to a minimum. Problem solving later became the heart of her instructional program, and daily use of Function Probe software helped the computer become more integrated into the mathematics classroom. The textbook was no longer the sole guide for how instruction should proceed, but one of several sources for instruction.

Class Time. The efficient use of time, always a concern for teachers, became especially problematic in a classroom environment that nurtured diversity of methods and multiple representations of concepts. There was often little or no time spent on helping students tie together key themes and conclusions from a lesson before turning to new activities. And there was little reflective discourse about the broader mathematical concepts that help students develop a more cogent world-view of mathematics, as well as a more integrated understanding of the specific mathematics content. However, the teacher's development over the year enabled her to recognize the need for closure and reflective discourse, which she intends to include in subsequent years.

Conclusions

By examining this classroom from the perspectives of the teacher, individual students, and small groups, the research team was able to document, examine, and critique thoughtfully the process of implementation while serving as a resource for the teacher. Each team member became closely identified with her or his perspective. During joint sessions of video data analysis, team members noted the impact of classroom activities on their perspectives and advocated interpretation from those points-of-view. This prompted the team to view the classroom as an interactive, negotiated system, and to understand the forces that support and impede change.

Conclusions that can be drawn from this work include the following:

  1. Creating sustainable changes in classrooms requires a systematic, but incremental approach. New curriculum and software must be recognizably connected to the existing curriculum. Computer software has the potential to alter the curriculum dramatically but to change it incrementally within the constraints of a conservative standardized testing system and the demands of a mandated curriculum.

  2. The teacher is a critical participant in the process. Change can be very demanding and requires significant teacher support. This teacher was already competent in her use of software and hardware applications and thus did not carry this added pressure. Nonetheless, the pressures of learning new ways of teaching and learning, and of developing mathematical knowledge and performing well under constant scrutiny made her year stressful while exhilarating.

  3. The philosophy governing teacher development must be consistent with the philosophy supporting student learning. Teachers will need adequate opportunities to develop and change their subject-matter knowledge and to learn how to listen to students.

  4. Institutional changes such as longer classroom periods and partnerships among mathematics teachers which allow adequate opportunities for reflection are necessary, especially during the initial phases.

  5. Students also need support in the change process. In the absence of a textbook as the primary resource, students cannot gauge their progress using traditional benchmarks such as pages covered or chapters completed. New forms of assessment must be devised within technological environments to assist students in evaluating and gaining satisfaction from their own progress.

  6. Methods for assisting students in working effectively in groups must be developed and supported with forms of assessment that promote students' own evaluation of their work as well as accountability in the larger instructional system.

Implications for the Future

This study has shown that when students and a teacher use multirepresentational software and contextual problems, they improve their abilities to explore ideas, investigate and solve mathematical problems, and construct meanings from their mathematical experiences. There is significant potential for a multirepresentational tool such as Function Probe to facilitate constructivist instruction; however, the teacher's role in supporting students' conceptual development is critical. Teacher development programs need to address the pedagogical issues of incorporating new curriculum and new technologies into the classroom in order to help students' conceptualizations develop to their full potential.

The larger vision driving this research, the development of an instructional model for mathematics based on a constructivist framework, will require a significant long-term effort. Software must be revised to respond to the issues emerging in the classroom practices. It must also be enhanced to make use of multimedia capabilities as a source for problem generation and to create new representational forms to link visual and audio images to the traditional representations on Function Probe.

Examples

The Parking Garage Problem

The mayor of our city, in an effort to encourage shopping at the new downtown City Center, has called together a task force to come up with a revised schedule of parking rates. Here are the four options of parking rates at the City Center parking garage. If you were on the city's governing board, which option would you choose and why?


Option I: Pay 35 for up to, but not including, the first hour. Pay an additional 50 for up to, but not including, the second hour. Pay an additional 50 for up to, but not including, the third hour, etc

Option II: Pay 10 for up to, but not including, the first hour. Pay an additional 50 for up to, but not including, each additional hour.

Option III: Pay 35 for up to, but not including, the first half hour. Pay an additional 25 for up to, but not including, each additional half hour.


Option IV: Up to the first hour is free. Pay 75 for each additional hour, etc.


Bibliography

Ball, Deborah L. (1990). "Research on teaching mathematics: Making subject matter knowledge part of the equation." In J. Trophy (Ed.), Advances in research on teaching: Vol. 2. Teachers' subject matter knowledge. Greenwich, CT: JAI Press.

Barnes, Douglas, & Todd, Frankie (1977). Communication and learning in small groups. London: Routledge.

Confrey, Jere (1990b). "An overview of the Function Probe project and its theoretical underpinnings." Paper written for the symposium, "Multiple perspectives on the implementation of multi-representational software in a secondary classroom", at the Annual Meeting of the American Educational Research Association, Boston, MA, April, 1990.

Confrey, Jere (1989a). Function Probe . Software for the Apple Macintosh Computer. Designed by J. Confrey, F. Carroll, S. Cato, P. Davis, and E. Smith.

Confrey, Jere (1984). "Towards a framework for constructivist instruction." A paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Madison, WI, October, 1984.

Confrey, Jere; Piliero, Susan; Rizzuti, Jan; & Smith, Erick (1990). Multiple perspectives on the implementation of multi-representational software in a secondary classroom. A symposium presented at the Annual Meeting of the American Educational Research Association, Boston, MA, April, 1990.

Davidson, Neil (1989). (Ed.), Cooperative learning in mathematics. Menlo Park, CA: Addison-Wesley.

Lincoln, Yvonna S., & Guba, Egon G. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage Publications, Inc.

Marcovits, Ziva; Eylon, Bat-Sheva; & Bruckheimer, Maxim (1986). Functions today and yesterday. For the Learning of Mathematics, 6 (2), 18-28.

Miles, Mathew B., & Huberman, A. Michael (1984). Qualitative data analysis. Beverly Hills, CA: Sage Publications.

NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston, Virginia: National Council of Teachers of Mathematics, Inc.

Shulman, Lee S. (1987). "Knowledge and teaching: Foundations of the new reform." Harvard Educational Review, 57(1), 1-22.

*The constructivist approach to learning asserts that learners "construct" their own meaning/knowledge from the information they acquire. This differs from the traditional approach which assumes a teacher can "deliver" knowledge to a learner.


Acknowledgments

Summarized by Linda Knapp

Research supported by grants from Apple Classrooms of Tomorrow (ACOT) and External Research of the Advanced Technology Group at Apple Computer, Inc., and The National Science Foundation (grant no. MDR8652160). All opinions and findings are those of the authors and not necessarily representative of the sponsoring agencies.

Assistance and support from Columbus Public Schools, Paula Fistick, Forrest Carroll, Simone Cato, and Paul Davis.

See Also