https://wiki.preterhuman.net/index.php?title=High_School_Mathematics&feed=atom&action=historyHigh School Mathematics - Revision history2021-05-15T04:25:55ZRevision history for this page on the wikiMediaWiki 1.35.0https://wiki.preterhuman.net/index.php?title=High_School_Mathematics&diff=20202&oldid=prevNetfreak at 18:01, 24 September 20202020-09-24T18:01:15Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:01, 24 September 2020</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <p>Option I: Pay 35 for up to, but not including, the first hour. Pay an</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <p>Option I: Pay 35 for up to, but not including, the first hour. Pay an</div></td></tr>
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</table>Netfreakhttps://wiki.preterhuman.net/index.php?title=High_School_Mathematics&diff=20201&oldid=prevNetfreak: Created page with "<strong>Apple Classrooms of Tomorrow Research<br> Report Number 11 <br> <br> High School Mathematics: Development of Teacher Knowledge and Implementation of a Problem-Based Ma..."2020-09-24T18:00:43Z<p>Created page with "<strong>Apple Classrooms of Tomorrow Research<br> Report Number 11 <br> <br> High School Mathematics: Development of Teacher Knowledge and Implementation of a Problem-Based Ma..."</p>
<p><b>New page</b></p><div><strong>Apple Classrooms of Tomorrow Research<br><br />
Report Number 11 <br><br />
<br><br />
High School Mathematics: Development of Teacher Knowledge and Implementation<br />
of a Problem-Based Mathematics Curriculum Using Multirepresentational Software</strong><br />
<br />
<p><br><br />
</p><br />
<br />
<h3>Authors</h3><br />
<br />
<p>Jere Confrey, Ph.D.<br><br />
Susan C. Piliero<br><br />
Jan M. Rizzuti<br><br />
Erick Smith<br><br />
<cite>Cornell University</cite> <br><br />
<br><br />
<br><br />
Apple Computer, Inc.<br><br />
1 Infinite Loop<br><br />
Cupertino, CA 95014<br><br />
<br><br />
<hr align="LEFT"></p><br />
<br />
<h3>Preface</h3><br />
<br />
<p>Begun in 1985, Apple Classrooms of Tomorrow (ACOT)SM is a research and<br />
development collaboration among public schools, universities, research agencies<br />
and Apple Computer, Inc. ACOT explores, develops and demonstrates the powerful<br />
uses of technologies in teaching and learning. In all ACOT endeavors, instruction<br />
and assessment are as integral to learning as technology.<br><br />
<br><br />
Supporting a constructivist approach to learning, technology is used as<br />
knowledge-building tools. As students collaborate, create media-rich compositions<br />
and use simulations and models, researchers investigate four aspects of<br />
learning: tasks, interactions, situations and tools. The research is formative.<br />
The findings guide ACOT staff and teachers as they refine their approach<br />
to learning, teaching and professional development. ACOT teachers and students<br />
often use the most advanced technologies available, including experimental<br />
technologies, to help us envision the future and improve the educational<br />
process.<br><br />
<br><br />
ACOT views technology as a necessary and catalytic part of the effort required<br />
to fundamental restructure America's education system. We hope that by sharing<br />
our results with parents, educators, policy makers, and technology developers<br />
the lessons of ACOT will contribute to the advancement of educational reform.<br><br />
<br><br />
The original research summarized here took place during the 1989-90 school<br />
year and was reported in a symposium, &quot;Multiple Perspectives on the<br />
Implementation of Multi-Representational Software in a Secondary Classroom,&quot;<br />
at the 1990 meeting of the American Education Research Association.<br><br />
</p><br />
<br />
<h3>Introduction <br><br />
A Call for Action</h3><br />
<br />
<p>Recent publications of the National Council of Teachers of Mathematics<br />
(e.g., NCTM, 1989; Wagner &amp; Kieran, 1989) call for major changes in<br />
the curriculum, assessment, learning activities, and use of computers and<br />
calculators in mathematics instruction. The proposed changes reflect a contemporary<br />
view of what it means to be mathematically literate in an information society.<br />
This math literacy requires new competencies, including the ability to apply<br />
mathematical ideas to problem situations and work with others to set up<br />
and solve problems. Such abilities are not acquired by individual drill<br />
and practice. Problem-solving experiences must be central to the school<br />
experience, providing opportunities for students to work through problems<br />
embedded in complex contexts, verify and interpret their results, communicate<br />
mathematically, and reason logically. These suggested reforms are based<br />
on the premise that mathematics is much broader than a static set of concepts,<br />
definitions, and skills to be mastered.<br><br />
<br><br />
Educators maintain that the introduction of computers into math classrooms<br />
can dramatically improve instruction beyond basic skills, but research documenting<br />
these predictions is relatively scarce. Still, it seems that appropriate<br />
technical tools, linked to a competent curriculum, and accompanied by active<br />
teacher support, could offer a genuine alternative to existing practices<br />
for teaching mathematics. Thus, the goals of this project are to build a<br />
problem-based mathematics curriculum and multirepresentational software<br />
to support students' problem-solving activities, provide teacher development<br />
in this setting, and conduct research on the implementation process.<br><br />
</p><br />
<br />
<h3>Project Design: A Constructivist Approach*</h3><br />
<br />
<p>Based on a constructivist framework, the instructional goal of this project<br />
is to create a learning environment that promotes student construction of<br />
mathematical concepts through repeated cycles of developing a problematic,<br />
acting to resolve the problematic, and reflecting on these actions (Confrey,<br />
1989b). The term problematic refers to a disturbance, or roadblock, between<br />
where the student is, and where s/he wants to be. It is the problematic<br />
that calls the solver to action and to subsequent reflection on the action.<br />
As a result, the learner develops new understandings and this completes<br />
the process of knowledge construction.<br><br />
<br><br />
<cite>Instruction</cite> . The constructivist teacher seeks opportunities<br />
for students to engage in tasks with potential for promoting conceptual<br />
development. Frequently, the teacher uses problems for exploring student<br />
conceptions and supporting their extension and development. In individual,<br />
small group, or whole class settings, the constructivist teacher listens<br />
to students and tries to understand their mathematical interpretations,<br />
which may be different from his/her own. The success of a constructivist<br />
instructional model depends on how well the teacher is able to recognize<br />
the student's problematic, hear legitimate alternative solutions, and promote<br />
reflection (Confrey, 1989b). Ultimately, the student must decide on the<br />
adequacy of his/her construction.<br><br />
<br><br />
<cite>Curriculum. </cite>The curriculum developed for this project consists<br />
of contextual problems designed to be used in conjunction with problems<br />
selected from the course textbook or developed by the teacher. The research<br />
team developed two units for the precalculus curriculum, the first on linear<br />
functions and the second on geometric sequences, exponential and logarithmic<br />
functions. (See example problem on page 16.)<br><br />
<br><br />
<cite>Software.</cite> In a constructivist learning environment that encourages<br />
students to express and defend alternative solutions, students need multiple<br />
ways to represent their ideas. Symbolic representations, formal proofs,<br />
calculator keystrokes, graphs, figures, statements of contextual problems,<br />
and spoken language are common forms used to represent mathematical ideas.<br />
Based on the notion that students learn more effectively when they can represent<br />
mathematical ideas in multiple forms, the Cornell research team developed<br />
Function Probe (Confrey, 1989a). Function Probe is an interactive software<br />
tool that encourages learners to explore the concept of function by enabling<br />
them to represent the function through tables, graphs, calculator keystrokes,<br />
and algebra, and to move between and coordinate actions among these multiple<br />
representations.<br><br />
<br><br />
The software was designed in response to research on student methods and,<br />
thus, has features which allow students to, for example, build calculator<br />
buttons as generalizations of numeric procedures, or to fill tables to organize<br />
data and discern patterns. As a dynamic medium, the representations can<br />
be easily manipulated. For example, graphs can be translated, stretched,<br />
and reflected by actions carried out with the mouse. Finally, all windows<br />
keep an easily accessible history to assist students and teachers in reflecting<br />
on student work. (Screen images illustrating the multirepresentational character<br />
of the software are on pages 16 17.)<br><br />
<br><br />
<cite>Teacher Development.</cite> Teacher development in this project included<br />
a week of intensive preparation during the summer and continued support<br />
during the school year. In addition, the reflection process the teacher<br />
experienced after planning and debriefing interviews contributed to her<br />
development.<br><br />
<br><br />
Implementation of the curriculum and software tool was closely linked to<br />
teacher development and the teacher's contribution to the project, which<br />
was recognized as critical and transformative. The teacher was creating<br />
new knowledge as she worked with the resources provided. Since she was compelled<br />
to use the software and curriculum within the constraints and possibilities<br />
of everyday instruction, the teacher and the research team developed a partnership<br />
enabling each member to apply his or her own expertise to make the innovation<br />
work.<br><br />
</p><br />
<br />
<h3>The Study</h3><br />
<br />
<p>This report summarizes work that took place during the 1989-90 implementation<br />
year, after Function Probe software and two curriculum units had been developed.<br />
The study involved one high school mathematics teacher in the Apple Classrooms<br />
of Tomorrow (ACOT) project and her precalculus class of ACOT students.<br><br />
<br><br />
The class being studied was located in a midwestern urban high school with<br />
a student body of approximately 1300 (33% black, 62% white, 3% Asian). There<br />
were four ACOT classrooms in this participating high school, with nine teachers<br />
and approximately 120 students from ninth through twelfth grades. All incoming<br />
ninth graders scoring above 37% on mathematics and reading tests were eligible<br />
to apply to the ACOT program and selections were made at random from this<br />
applicant pool, controlling only for race and gender. The precalculus class<br />
participating in this research consisted of the ACOT senior class of 22<br />
students currently in the fourth year of the program.<br><br />
<br><br />
The teacher was a veteran teacher with sixteen years of experience and three<br />
years as part of ACOT. Prior to the project, she described her teaching<br />
as traditional with little use of applications or multiple representations.<br><br />
</p><br />
<br />
<h3>Data Collection</h3><br />
<br />
<p>Data collection took place during a year-long period from March 1989<br />
to March 1990. Research focused on implementation of the innovation and<br />
included five weeks of data collection in September-October 1989 on linear<br />
functions and five weeks in January-February 1990 on exponential functions.<br />
Data collection of the teacher's practices prior to the implementation phase<br />
took place in March 1989 and July 1989 (during the teacher development week),<br />
and then continued through the 1989-90 school year.<br><br />
The research team used the method of triangulation (Lincoln &amp; Guba,<br />
1985) in data gathering. This approach enabled the team to collect data<br />
from a variety of perspectives and then to examine the interrelationships<br />
between them. Jan Rizzuti studied the development of individual students;<br />
Erick Smith concentrated on small group interactions; Susan Piliero studied<br />
the teacher's development of pedagogical and content knowledge, including<br />
her beliefs and practices. Project director Jere Confrey led the software<br />
and curriculum development efforts, provided guidance and support for the<br />
teacher, and oversaw the conduct of the research.<br><br />
<br><br />
The research team gathered data from the following sources: 1) printed records<br />
of student work on Function Probe; 2) videotapes of all class sessions;<br />
3) audio and videotapes of individual students and teacher interviews; 4)<br />
unit pretests and posttests; 5) observation notes taken during classes;<br />
6) lesson plans, worksheets, tests, and other instructional artifacts generated<br />
by the teacher; 7) copies of student assignments and tests collected by<br />
the teacher.<br><br />
</p><br />
<br />
<h3>Findings and Interpretations <br><br />
<br><br />
Individual Students</h3><br />
<br />
<p>Research findings indicated that students of average and above average<br />
abilities were able to construct strong conceptualizations of linear functions<br />
and exponential functions by working with multirepresentational software.<br />
Using Function Probe, students were able to represent relationships and<br />
operations algebraically, graphically, and in tabular form, and their understandings<br />
of functional relationships were strengthened and broadened by coordinating<br />
different representations. In addition to the use of multiple representations<br />
to understand functions, the use of contextual problems also contributed<br />
to their understanding.<br><br />
<br><br />
Results also showed that although the software and curriculum played important<br />
roles, they were not enough to ensure strong conceptual development in most<br />
students. The teacher's role was critical in the development process.<br><br />
<br><br />
Student use of Function Probe's table, graph, and calculator windows advanced<br />
learning in several areas, including the following:<br><br />
<br><br />
<cite>Table Window</cite>. Students used the table window to organize the<br />
information of a problem and generate information such as differences or<br />
ratios between entries, although they often had difficulty identifying the<br />
variables clearly. They could fill in two columns with the sense of co-varying<br />
rates of change, but often had difficulty coding the relationship between<br />
the two columns algebraically.<br><br />
<br><br />
Students also used Function Probe to test guesses and hypotheses. For example,<br />
one student used the RATIO command to verify his guess that a sequence was<br />
geometric, and another built an algebraic equation, checking the computer<br />
equation values against her expectations.<br><br />
<br><br />
<cite>Graph Window</cite>. Many students used graphs as secondary representations<br />
after using the table or calculator to find relationships within the problem.<br />
They were also able to link the numerical data from tables and calculators<br />
to the graphical representations in order to better understand functional<br />
concepts such as slope and rate of change.<br><br />
<br><br />
Many students developed a strong qualitative sense of functional relationships<br />
that is not as easily developed when examining hand-drawn graphs. By rapidly<br />
graphing functions and presenting them as objects for examination, Function<br />
Probe allowed students to concentrate on global features of graphs, such<br />
as shape, direction, and location, leading them to a deeper understanding<br />
of different types of functions. <br><br />
<br><br />
Other features of the graph window proved to be important learning tools<br />
for the students. They changed the scale of the graph window to situate<br />
their graphs on appropriate axes and examine the extreme points of graphs.<br />
They used the translating, stretching, and reflecting features to coordinate<br />
the algebraic and graphical forms of functions. They used the point-locator<br />
to find points of intersection between graphs. Students quickly became facile<br />
with these features and were able to spend more time solving problems and<br />
discussing their solutions with others.<br><br />
<br><br />
<cite>Calculator Window</cite>. Students used the calculator window extensively<br />
to build operational and numerical forms of functional relationships, as<br />
well as carry out basic calculations. For instance, they used the repeated<br />
multiplication inherent in an exponential function to model the actions<br />
of a computer virus problem; then they built algebraic equations in the<br />
form of calculator buttons to model the functional relationship of this<br />
situation. Textbooks typically present full-blown algebraic representations<br />
of functions, which do not support students' development of this important<br />
ability.<br><br />
</p><br />
<br />
<h3>Small Group Interactions</h3><br />
<br />
<p>The research on small group interactions yielded two major results: 1)<br />
collaborative group work can play a critical role in students' development<br />
of diverse approaches and solutions to problems as well as their conceptions<br />
of the problem-solving process; 2) effective collaborative learning cannot<br />
be implemented rapidly; it must grow in conjunction with changes in other<br />
classroom practices, including the nature of teacher student interactions,<br />
assessment practices, and ways of reporting outcomes of group interactions.<br><br />
<br><br />
Analysis of small group interactions focused on the evolution of the problem-solving<br />
process from the development of individual problematics to the achievement<br />
of a solution through negotiated consensus. Results indicated that three<br />
processes are necessary for successful collaborative learning: 1) individuals<br />
need to identify their own problematics (the issues they believe must be<br />
resolved to solve the problem), which evolve throughout the problem-solving<br />
process; 2) the group must create a social framework for developing a negotiated<br />
consensus that allows group actions to lead towards resolution of individual<br />
problematics; 3) individuals must have opportunities to reflect on group<br />
solutions in relation to their evolving problematics.<br><br />
<br><br />
Changing from an individualistic to a cooperative approach in mathematics<br />
learning was well-received by students in form, but making it become an<br />
effective means for learning was more difficult. Initially students' conceptions<br />
of mathematics and mathematics learning caused them to be sceptical of the<br />
value of seeing other methods. They focused exclusively on getting the answer,<br />
completing the task, and moving quickly to the next task. However, this<br />
changed over the year. Students learned to read a problem and begin a discussion<br />
on how to proceed. They began to view problems as puzzles to be worked out<br />
over time, allowing for conflict resolution and exploration of alternative<br />
solutions. This change can be attributed to several factors: the teacher's<br />
active solicitation of alternative viewpoints; the software's ability to<br />
support rapid explorations, which facilitated development of negotiated<br />
consensus; and group presentations that provided a forum for arguing and<br />
discussing alternative solutions.<br><br />
<br><br />
The students' collaborative skills improved, despite a gradual decrease<br />
in group structure and class time spent on group work. In the beginning<br />
of the year, group activity filled most class periods, groups were composed<br />
heterogeneously, and group roles were assigned. By the end of the year,<br />
no roles were assigned, little attention was paid to group composition,<br />
and group work often occurred intermittently over the class period. This<br />
suggests that a mathematics learning environment that promotes student methods<br />
and allows for diversity may be at least as important as the use of structured<br />
group work in promoting collaborative learning.<br><br />
<br><br />
One of the major difficulties for teacher and students was finding a balance<br />
between closure and exploration. In a classroom where alternative viewpoints<br />
were encouraged and valued, it was difficult to maintain a balance between<br />
providing time to reflect on presented approaches and exploring further<br />
alternatives. Although reaching satisfactory closure remained an issue throughout<br />
the project, the teacher undertook activities that aided this process. These<br />
included: 1) placing greater importance on her own understanding and exploration<br />
of the subject matter; 2) experimenting with assessment for collaborative<br />
work through activities such as group presentations; 3) visiting groups<br />
unobtrusively and remaining for a longer time, thus encouraging group interactions<br />
and increasing her understanding of group processes and solutions.<br><br />
</p><br />
<br />
<h3>The Teacher</h3><br />
<br />
<p><cite>Conceptions of Mathematics. </cite>Initially, the teacher described<br />
mathematics as two separate subgroups: school mathematics and the mathematics<br />
used in real life. Later, she began to see mathematics as having infinite<br />
actions and processes that extend beyond mere computation and manipulation<br />
of algebraic symbols and numbers. The teacher also developed a more positive<br />
appraisal of her own mathematical abilities and this increased confidence<br />
was evidenced in the classroom by her willingness to devote more instructional<br />
time to problem-solving activities and to student-initiated departures from<br />
her own instructional script.<br><br />
<br><br />
<cite>Subject-Matter Expertise</cite>. Subject matter was no longer the<br />
traditional material of school mathematics, but the construction of new<br />
understandings of mathematics through the process of exploring, reviewing,<br />
analyzing, integrating, and applying multiple representations of functions<br />
and transformations on them. The teacher showed evidence of substantial<br />
growth in her own subject-matter knowledge.<br><br />
<br><br />
<cite>Conceptions of Mathematics Instruction. </cite>The teacher moved from<br />
the view that she should develop mastery in specific teaching techniques<br />
to the view that she needed to develop awareness of her own mathematical<br />
processes. In September, interviews focused on issues of grouping, class<br />
management, and grading. By January, however, interviews focused on subject<br />
matter. She identified the need to approach subject matter as both a teacher<br />
and a learner. As a learner, she had to construct her own meanings for the<br />
mathematics in light of her current understandings. As a teacher, she had<br />
to understand ways of analyzing and approaching the problems rather than<br />
merely arriving at a satisfactory solution.<br><br />
<br><br />
<cite>Reflection and Point-of-View.</cite> There was significant change<br />
in the teacher's ability and desire to reflect on the events of the class<br />
through the students' perspectives of subject matter. She described her<br />
past practice of &quot;shutting down&quot; her thought processes after class<br />
as a survival technique she developed in response to years of teaching six<br />
classes a day and not having the inner resources to look ahead or reflect<br />
back. That practice changed, as she later reported a strong need to go back<br />
and view lessons from the students' points-of-view. She began to take notes<br />
on student solutions and comments and used these notes to organize her own<br />
thinking about their interpretations and strategies, which affected her<br />
planning for subsequent lectures and discussion.<br><br />
<br><br />
It became increasingly important for the teacher to allow students opportunities<br />
to share their insights and solutions with the rest of the class. She allotted<br />
class time for group presentations during which she not only asked students<br />
to share their solutions, but to field questions from her and their classmates<br />
about their thought processes, choice of representations, and conclusions.<br />
By acknowledging alternative ways of approaching and solving problems, and<br />
allowing students the opportunity to share these approaches, the teacher<br />
felt students had developed a healthy autonomy and taken on a greater responsibility<br />
for their own learning.<br><br />
<br><br />
<cite>Assessment. </cite>In September, much of the teacher's discussion<br />
on assessment revolved around how to assess the group work, but by January,<br />
her grading encompassed more individual work. By the year's end, group work<br />
accounted for 15% of a student's grade, with the remainder based on tests,<br />
quizzes, handouts, class notes, and homework. Her tests and quizzes showed<br />
an increased ability to design nonroutine questions that required students<br />
to coordinate multiple methods and/or representations. In addition, the<br />
problems no longer needed to conform to the exact format of previously assigned<br />
homework problems or examples worked out in class.<br><br />
<br><br />
<cite>Curriculum.</cite> Prior to participating in the research project,<br />
the teacher tended to avoid word problems, focusing on basic skills acquisition<br />
and keeping applications to a minimum. Problem solving later became the<br />
heart of her instructional program, and daily use of Function Probe software<br />
helped the computer become more integrated into the mathematics classroom.<br />
The textbook was no longer the sole guide for how instruction should proceed,<br />
but one of several sources for instruction.<br><br />
<br><br />
<cite>Class Time.</cite> The efficient use of time, always a concern for<br />
teachers, became especially problematic in a classroom environment that<br />
nurtured diversity of methods and multiple representations of concepts.<br />
There was often little or no time spent on helping students tie together<br />
key themes and conclusions from a lesson before turning to new activities.<br />
And there was little reflective discourse about the broader mathematical<br />
concepts that help students develop a more cogent world-view of mathematics,<br />
as well as a more integrated understanding of the specific mathematics content.<br />
However, the teacher's development over the year enabled her to recognize<br />
the need for closure and reflective discourse, which she intends to include<br />
in subsequent years.<br><br />
</p><br />
<br />
<h3>Conclusions</h3><br />
<br />
<p>By examining this classroom from the perspectives of the teacher, individual<br />
students, and small groups, the research team was able to document, examine,<br />
and critique thoughtfully the process of implementation while serving as<br />
a resource for the teacher. Each team member became closely identified with<br />
her or his perspective. During joint sessions of video data analysis, team<br />
members noted the impact of classroom activities on their perspectives and<br />
advocated interpretation from those points-of-view. This prompted the team<br />
to view the classroom as an interactive, negotiated system, and to understand<br />
the forces that support and impede change.<br><br />
<br><br />
Conclusions that can be drawn from this work include the following:</p><br />
<br />
<ol><br />
<li>Creating sustainable changes in classrooms requires a systematic, but<br />
incremental approach. New curriculum and software must be recognizably<br />
connected to the existing curriculum. Computer software has the potential<br />
to alter the curriculum dramatically but to change it incrementally within<br />
the constraints of a conservative standardized testing system and the demands<br />
of a mandated curriculum. <br><br />
<br><br />
<br />
<li>The teacher is a critical participant in the process. Change can be<br />
very demanding and requires significant teacher support. This teacher was<br />
already competent in her use of software and hardware applications and<br />
thus did not carry this added pressure. Nonetheless, the pressures of learning<br />
new ways of teaching and learning, and of developing mathematical knowledge<br />
and performing well under constant scrutiny made her year stressful while<br />
exhilarating. <br><br />
<br><br />
<br />
<li>The philosophy governing teacher development must be consistent with<br />
the philosophy supporting student learning. Teachers will need adequate<br />
opportunities to develop and change their subject-matter knowledge and<br />
to learn how to listen to students. <br><br />
<br><br />
<br />
<li>Institutional changes such as longer classroom periods and partnerships<br />
among mathematics teachers which allow adequate opportunities for reflection<br />
are necessary, especially during the initial phases. <br><br />
<br><br />
<br />
<li>Students also need support in the change process. In the absence of<br />
a textbook as the primary resource, students cannot gauge their progress<br />
using traditional benchmarks such as pages covered or chapters completed.<br />
New forms of assessment must be devised within technological environments<br />
to assist students in evaluating and gaining satisfaction from their own<br />
progress. <br><br />
<br><br />
<br />
<li>Methods for assisting students in working effectively in groups must<br />
be developed and supported with forms of assessment that promote students'<br />
own evaluation of their work as well as accountability in the larger instructional<br />
system.<br />
</ol><br />
<br />
<h3>Implications for the Future</h3><br />
<br />
<p>This study has shown that when students and a teacher use multirepresentational<br />
software and contextual problems, they improve their abilities to explore<br />
ideas, investigate and solve mathematical problems, and construct meanings<br />
from their mathematical experiences. There is significant potential for<br />
a multirepresentational tool such as Function Probe to facilitate constructivist<br />
instruction; however, the teacher's role in supporting students' conceptual<br />
development is critical. Teacher development programs need to address the<br />
pedagogical issues of incorporating new curriculum and new technologies<br />
into the classroom in order to help students' conceptualizations develop<br />
to their full potential.<br><br />
<br><br />
The larger vision driving this research, the development of an instructional<br />
model for mathematics based on a constructivist framework, will require<br />
a significant long-term effort. Software must be revised to respond to the<br />
issues emerging in the classroom practices. It must also be enhanced to<br />
make use of multimedia capabilities as a source for problem generation and<br />
to create new representational forms to link visual and audio images to<br />
the traditional representations on Function Probe.<br><br />
</p><br />
<br />
<h3>Examples</h3><br />
<br />
<h4>The Parking Garage Problem</h4><br />
<br />
<p>The mayor of our city, in an effort to encourage shopping at the new<br />
downtown City Center, has called together a task force to come up with a<br />
revised schedule of parking rates. Here are the four options of parking<br />
rates at the City Center parking garage. If you were on the city's governing<br />
board, which option would you choose and why?</p><br />
<br />
<menu><br />
<p>Option I: Pay 35 for up to, but not including, the first hour. Pay an<br />
additional 50 for up to, but not including, the second hour. Pay an additional<br />
50 for up to, but not including, the third hour, etc<br><br />
<br><br />
Option II: Pay 10 for up to, but not including, the first hour. Pay an<br />
additional 50 for up to, but not including, each additional hour.<br><br />
<br><br />
Option III: Pay 35 for up to, but not including, the first half hour. Pay<br />
an additional 25 for up to, but not including, each additional half hour.<br><br />
<br><br />
<br><br />
Option IV: Up to the first hour is free. Pay 75 for each additional hour,<br />
etc.<br />
</menu><br />
<br />
<p><hr align="LEFT"></p><br />
<br />
<h3>Bibliography</h3><br />
<br />
<p><b>Ball, Deborah L.</b> (1990). &quot;Research on teaching mathematics:<br />
Making subject matter knowledge part of the equation.&quot; In J. Trophy<br />
(Ed.), <cite>Advances in research on teaching: Vol. 2. Teachers' subject<br />
matter knowledge.</cite> Greenwich, CT: JAI Press.<br><br />
<br><br />
<b>Barnes, Douglas, &amp; Todd, Frankie</b> (1977). <cite>Communication<br />
and learning in small groups.</cite> London: Routledge.<br><br />
<br><br />
<b>Confrey, Jere</b> (1990b). &quot;An overview of the Function Probe project<br />
and its theoretical underpinnings.&quot; Paper written for the symposium,<br />
<cite>&quot;Multiple perspectives on the implementation of multi-representational<br />
software in a secondary classroom&quot;</cite>, at the Annual Meeting of<br />
the American Educational Research Association, Boston, MA, April, 1990.<br><br />
<br><br />
<b>Confrey, Jere</b> (1989a). <cite>Function Probe . Software for the Apple<br />
Macintosh Computer. Designed by J. Confrey, F. Carroll, S. Cato, P. Davis,<br />
and E. Smith.<br><br />
<br><br />
</cite><b>Confrey, Jere</b> (1984). &quot;Towards a framework for constructivist<br />
instruction.&quot; A paper presented at the annual meeting of the North<br />
American Chapter of the International Group for the Psychology of Mathematics<br />
Education. Madison, WI, October, 1984.<br><br />
<br><br />
<b>Confrey, Jere; Piliero, Susan; Rizzuti, Jan; &amp; Smith, Erick</b> (1990).<br />
<cite>Multiple perspectives on the implementation of multi-representational<br />
software in a secondary classroom</cite>. A symposium presented at the Annual<br />
Meeting of the American Educational Research Association, Boston, MA, April,<br />
1990.<br><br />
<br><br />
<b>Davidson, Neil</b> (1989). (Ed.), <cite>Cooperative learning in mathematics.</cite><br />
Menlo Park, CA: Addison-Wesley.<br><br />
<br><br />
<b>Lincoln, Yvonna S., &amp; Guba, Egon G</b>. (1985). <cite>Naturalistic<br />
inquiry.</cite> Beverly Hills, CA: Sage Publications, Inc.<br><br />
<br><br />
<b>Marcovits, Ziva; Eylon, Bat-Sheva; &amp; Bruckheimer, Maxim</b> (1986).<br />
<cite>Functions today and yesterday. For the Learning of Mathematics</cite>,<br />
6 (2), 18-28.<br><br />
<br><br />
<b>Miles, Mathew B., &amp; Huberman, A. Michael</b> (1984). <cite>Qualitative<br />
data analysis.</cite> Beverly Hills, CA: Sage Publications.<br><br />
<br><br />
<b>NCTM</b>. (1989). <cite>Curriculum and evaluation standards for school<br />
mathematics.</cite> Reston, Virginia: National Council of Teachers of Mathematics,<br />
Inc.<br><br />
<br><br />
<b>Shulman, Lee S.</b> (1987). &quot;Knowledge and teaching: Foundations<br />
of the new reform.&quot; <cite>Harvard Educational Review</cite>, 57(1),<br />
1-22.<br><br />
<br><br />
<sup>*</sup><font size="-1">The constructivist approach to learning asserts<br />
that learners &quot;construct&quot; their own meaning/knowledge from the<br />
information they acquire. This differs from the traditional approach which<br />
assumes a teacher can &quot;deliver&quot; knowledge to a learner. </font><br><br />
<hr align="LEFT"></p><br />
<br />
<h3>Acknowledgments</h3><br />
<br />
<p>Summarized by Linda Knapp<br><br />
<br><br />
Research supported by grants from Apple Classrooms of Tomorrow (ACOT) and<br />
External Research of the Advanced Technology Group at Apple Computer, Inc.,<br />
and The National Science Foundation (grant no. MDR8652160). All opinions<br />
and findings are those of the authors and not necessarily representative<br />
of the sponsoring agencies.<br><br />
<br><br />
Assistance and support from Columbus Public Schools, Paula Fistick, Forrest<br />
Carroll, Simone Cato, and Paul Davis.<br><br />
<br />
=See Also=<br />
* [[Apple Classrooms of Tomorrow]]<br />
* [[Apple Computer]]<br />
<br />
[[Category:Apple Research]]</div>Netfreak