# High School Mathematics

**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:

- 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.

- 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.

- 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.

- 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.

- 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.

- 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?

<menu>

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.
</menu>

### 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.