ASU College Algebra
COLLEGE ALGEBRA REDESIGN: INFUSING CONCEPTUAL ACTIVITIES AND
TECHNOLOGY IN THE CLASSROOM TO PROMOTE STUDENT LEARNING
PI: Dieter Armbruster
Co-PIs: Marilyn Carlson, April Strom, Lance Ward, Kacie Koch
Abstract
This project aims to redesign ASU’s large-enrollment College Algebra course in ways
that research suggests will improve students’ understanding, pass rates, and motivation to take
additional mathematics. It will also address Arizona’s shortage of mathematics teachers through
a novel training and mentoring of mathematics majors to serve as apprentice course instructors.
The traditional ASU College Algebra course serves approximately 1700 students with over 80
course sections each year. Approximately 20% of ASU’s freshman class enrolls in College
Algebra annually. First Year Mathematics (FYM) faculty, graduate teaching assistants, and
competitively selected undergraduate students currently teach these sections. The pedagogy in
this three-credit course consists of lecture-based instruction with minimal collaboration among
students during class meetings. Most traditional sections do not infuse computer technology into
classroom activities; however, teachers have the option of using online homework assignments.
These assignments focus on students’ procedural learning, rather than conceptual understanding.
These methods of instruction are showing poor results. Over the past seven years, the
percentage of students who fail or withdraw from ASU's College Algebra program has ranged
from 35% to almost 50%. This is remarkable given that College Algebra repeats material that
students have already taken in high school. Indeed, repetition is a large part of the problem. In a
traditional College Algebra course, students receive a repeat dose of a curriculum and
instructional methods that did not succeed for them in high school and are not succeeding for
them at the university.
The course redesign we propose will incorporate curriculum and teaching techniques that
research has shown to be more effective. Research has demonstrated that developing students’
conceptual understanding through instruction focused on student learning and sense-making
provides students with powerful tools for success in mathematics, for example (Carlson, 1998;
Monk, 1992; A. Thompson, Philipp, Thompson, & Boyd, 1994). National calls for redesigning
College Algebra courses have also supported efforts to infuse active learning, problem solving,
and data analysis into course instruction (Mathematical Association of America, 2007). Our
course redesign plan incorporates two main components: (a) improving course curriculum and
instruction and (b) developing a secondary certification program for competitively selected, well
prepared, and carefully mentored mathematics majors who will tutor and teach the redesigned
College Algebra course. We will improve curriculum and instruction by incorporating active
learning in each class session with a focus on research-based conceptual activities designed to
promote student interaction and understanding. The newly designed secondary mathematics
certification program will prepare undergraduate mathematics majors to tutor in College Algebra
during their junior year and to teach the course during their senior year. Experienced faculty
from the redesign team will mentor and supervise these undergraduate teachers during their
tutoring and teaching experiences.
ABOR’s investment in this redesign project will reap a double return: Students in the
redesigned courses will be better prepared for subsequent mathematics courses, and our
mathematics majors will be better prepared to teach high school mathematics. In addition, by
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ASU College Algebra
providing future mathematics teachers with meaningful training in the most advanced teaching
methods and technologies, our project will support Arizona’s school districts in their struggle to
meet the quality teacher goals of federal No Child Left Behind regulations.
Our research plan will utilize multiple methods for data collection and data analysis.
Using pre- and post-tests, we will compare students' success in the redesign pilot with student
success in the traditional College Algebra course. In addition, we will interview students from
both the pilot group and the traditional group. These videotaped interviews will focus on
students' understanding of key College Algebra concepts. We will also track all College Algebra
students' subsequent enrollment and success in future mathematics classes.
Cost savings from the redesign will primarily come from using undergraduates to teach
and tutor the course in lieu of expensive graduate and FYM faculty. They will receive college
credit, instead of salary, toward secondary certification. Faculty mentors will work closely with
undergraduate teachers to help develop their knowledge and skills. Previous interventions to use
carefully selected and trained undergraduates to teach College Algebra have been highly
successful (Carlson, 1994). This cost savings will ultimately translate to fewer faculty being paid
to teach College Algebra. Funds saved as a result of the redesign will be allocated to redesigning
other mathematics courses and providing leadership training opportunities for faculty teaching
redesigned courses.
Introduction
Development of students’ conceptual understanding through instruction focused on
inquiry methods that promote sense making of key ideas provides students with powerful tools
for success in mathematics (Carlson, 1998; Monk, 1992; A. Thompson et al., 1994). National
calls for redesigning current College Algebra courses have also supported efforts to infuse active
learning, problem solving, and data analysis into course instruction as a way of building stronger
conceptions of College Algebra topics (American Mathematical Association of Two-Year
Colleges, 2006; Mathematical Association of America, 2007). Our proposed redesign of ASU’s
College Algebra program adopts research-based recommendations for teaching College Algebra
conceptually with meaningful activities designed to promote critical thinking, collaborations and
communication among students.
Over the past seven years, the percentage of students who fail or withdraw from ASU's
College Algebra program has been unacceptably high, ranging from 35% to almost 50%. This
failure affects the lives of hundreds of ASU students, who often drop out of the university after
failing their initial mathematics course. Their lack of success can be traced in part to the students'
own lack of preparation, coupled with their guaranteed acceptance into the College Algebra
course. But the problem is magnified when the course concepts, which these students have
already encountered in their high school coursework, are re-taught to them using the techniques
and teaching styles that did not work the first time. That is the situation currently in ASU College
Algebra; under-prepared students are receiving a repeat dose of a curriculum and instructional
methods that did not succeed for them in high school and are currently not succeeding for them
at the university. Our course redesign thus intends to take a fresh approach by adopting methods,
technology and curricula that mathematics education research points to as effective. Using the
research to guide us, we intend to align the instructional delivery, curriculum and tools of ASU’s
College Algebra program with the best knowledge and most current tools that have proven
successful for College Algebra students in this nation and internationally. By improving the
experiences of large numbers of students in freshman mathematics at ASU we will contribute to
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ASU College Algebra
improving the retention rate of ASU students. This effort will also open doors to subsequent
courses for students who have never had the opportunity to pass College Algebra before. By
making the course more relevant we anticipate that we will also improve these students’ views
about the importance and usefulness of mathematics in a broad range of decisions they will be
confronting during their lifetimes.
The current ASU College Algebra course consists largely of skill-based tasks, which
mostly require memorization of the procedures essential for completing these tasks. Yet research
strongly suggests that memorizing mathematical procedures leaves students with little
understanding of the important concepts necessary for success in future mathematics courses.
Our redesign project will focus more on the key ideas of College Algebra than on repetitive
calculation exercises. Among the key ideas we will explore are variation, covariation, rate of
change, function, additive reasoning, and multiplicative reasoning. These ideas are central to
College Algebra and foundational for students’ success in calculus (Oehrtman, Carlson, &
Thompson, in press).
Additionally, mathematics education research has illuminated the importance of
instruction that focuses on student inquiry and applying mathematical concepts in real-world
contexts. In contrast to the traditional College Algebra course, our redesign project will infuse
contextual applications in every class session. These applications will require students to grapple
with the mathematics and make sense of the concepts through a process of inquiry and reflection.
This engagement of students in class activities will shift instruction in the course from a more
teacher-centered to a more student-centered experience. They too will help students see the
relevance and usefulness of mathematics in science explorations. Our project will produce a
comprehensively redesigned model for College Algebra, incorporating training workshops,
enhanced curriculum, computer-based instructional supports, and revised assessments. Over the
course of several years, we hope to create a national model for instruction not only in College
Algebra, but in the entire range of entry-level college mathematics courses, with the knowledge
and tools acquired from this project laying the foundation for improvements we plan to make in
both precalculus and calculus.
College Algebra Redesign: A Model for Success
Central to our mission in this redesign effort is the idea of shifting instruction in the
classroom to be more student-centered as opposed to traditional courses in which the center of
instruction lies with the teacher. This active role of learning will require students to engage in
meaningful collaborative activities inside and outside of the classroom environment. Our vision
of this redesign incorporates classroom activities that provide opportunities for students to collect
their own data using calculator-based technology and to work with hands-on manipulatives to
make sense of the mathematics. In this effort we are guided by our team’s deep familiarity with
the literature in mathematics education research. Indeed, teaming mathematics faculty with
mathematics education researchers is a particular strength of our project, one that will expand the
university’s human capacity to deliver quality instructional experiences.
In the language of ABOR’s RFP, our College Algebra redesign project is best described
as a supplemental model since the basic structure of the traditional course will remain while
being augmented with instruction that focuses on understanding and using foundational ideas
through active learning during class. All class sessions will continue to be conducted in the faceto-face format and students will meet in class three hours per week. However, in-class instruction
will be student-centered with less emphasis on the traditional lecture format currently used in
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College Algebra at ASU. During class students will be engaged in activities that are scaffolded to
promote inquiry. Similar to the traditional course, enrollment will remain capped at 20 students
per section.
Relative to the five principles of successful course redesign cited by NCAT, this project
has been designed to maximize success in College Algebra by reforming the curriculum and
instruction. The project will also contribute to the development of a new program for secondary
mathematics teacher certification.
 Redesigning the whole course: Our project will redesign the entire College Algebra
curriculum at ASU using innovative instructional activities based on research. In an
effort to maintain consistency and improve quality of instruction among the course
sections, we plan to implement the use of PowerPoint presentations for instructors to
use in scaffolding classroom activities and the major conceptual issues that should
emerge from the classroom discussions. These presentations will be made available
for students to download and use as class notes. The use of structured lessons for all
sections will decrease the amount of “course-drift” that many past projects
experienced. Consistent with Florida Gulf Coast University’s redesign model, our
course will use a common syllabus, textbook, and course website for scheduling
assignments and sharing course materials. Faculty experts will design course activities
into approximately seven modules, which we describe later in this proposal. Faculty
will also revise existing assessments (chapter tests and final exam) to align with the
new course objectives.
 Encouraging active learning: Class activities will engage students in meaningful
collaboration to discuss solution approaches for tasks designed to develop students’
understanding of key concepts of the course. Students will consistently be required to
present their ways of thinking about problem situations to each other and to the entire
class. Students will focus on learning key ideas through inquiry and making sense of
data and contextual situations, both inside and outside the classroom. Furthermore,
online homework assignments will require students to take an active role in learning
procedural and skill-based tasks in order to prepare them for upcoming classroom
activities. These online assignments will also provide students with tutorial support
and reinforce concepts previously learned in the course.
 Provide students with individualized assistance: The college algebra redesign project
will encompass learner-centered classrooms that allow instructors to facilitate
individual student learning during class sessions. (This is in contrast to the current
situation in which instructors lecture about the content and illustrate procedures for
obtaining answers to problems, while paying little attention to individual learning.)
Offering individualized assistance during class can help instructors learn what areas
are most difficult for students. This allows instructors to help students, either
individually or collectively, at the time the difficulty is known rather than requiring
students to seek help outside of class (which they often do not). This improves quality
of the course instruction and reduces costs by replacing expensive labor (full-time
faculty and graduate students) with inexpensive labor (undergraduates) to aid in
outside tutoring. For our project, we plan to leverage pre-service mathematics teachers
by offering them credit to tutor in the ASU Mathematics Tutor Center and/or the
CRESMET Tutor Center. Institutions such as the University of Alabama and Virginia
Tech have also successfully incorporated undergraduate assistants and peer tutors in
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ASU College Algebra
their mathematics tutoring centers. Our project will further utilize undergraduate
mathematics majors who wish to obtain secondary certification by teaching College
Algebra as part of their field experience. In addition, the online tutorial support
provided by the software will be available to students 24 hours a day, every day. Such
automated tutorial support will free instructors from spending class time on low level
skills and procedures, thus allowing for greater attention to individual student
development of key concepts.
 Build in ongoing assessment and prompt (automated) feedback: Our team will adopt
course management software, called WileyPLUS, to facilitate online homework
assignments and assessments. Students will use the software to complete assignments
designed to prepare them for an upcoming week of classwork. Instructors will use
these assignments to assess students’ procedural knowledge prior to beginning the
conceptual lessons each week. The software provides immediate feedback on each
student’s progress and also graphs the progress of the entire class. Instructors can
gauge students’ level of engagement and understanding to further prepare class
instruction to meet the needs of the class. This component of our redesign project is
similar to the University of Massachusetts Amherst’s plan of requiring students to
complete low-stakes assessment prior to class to reduce the amount of class time spent
on tasks understood by students, thus utilizing class time more efficiently.
 Ensure sufficient time-on-task and monitor student progress: Through the use of
computerized assessment, instructors can monitor students’ progress on online
assessments by investigating students’ time on assignment and number of task retakes.
The course management software allows instructors to oversee individual student
progress while also assessing class progress outside of the classroom. Similar to the
universities of Alabama and Idaho, attendance will be mandatory for College Algebra
students. Study groups will be encouraged outside of class.
Our redesign team will align the new curriculum, instructional methods, and assessments
with the most recent research knowledge of the experiences students need to: (a) continue taking
and succeeding in undergraduate mathematics courses; (b) develop an understanding of
fundamental concepts (such as variable, rate of change, function) needed for understanding
calculus; (c) become better mathematical thinkers and problem solvers, acquire improved
communication abilities, and (d) acquire improved confidence in their ability to persevere in
solving complex problems. This redesign project will require the retraining of all ASU faculty
and graduate students who teach College Algebra. These faculty and graduate students currently
use mostly traditional lecture format and have no knowledge of the body of research that has
revealed instructional sequences that lead to students’ understanding the key ideas of the course.
The workshop models and tools that are designed and used to retrain College Algebra instructors
in realizing the goals stated above will be institutionalized to ensure that the course redesign and
improvement is sustained. The broad dissemination of what will be a research-tested model for
College Algebra will set in motion a new approach for revising undergraduate mathematics
courses at ASU.
Learning Materials
All faculty in the course redesign are committed to employing research-based materials
that have been successfully used to prepare students for beginning calculus. Additional materials
supported by research to enhance secondary mathematics and science teachers’ understanding of
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ASU College Algebra
the function concept will be implemented in this course. These materials will be revised and
scaffolded to meet the needs of College Algebra students. The faculty are also committed to
enhancing these materials by including computer-based supports and improved training for ASU
instructors and graduate students who teach College Algebra. Since the faculty, graduate and
undergraduate students who teach College Algebra at ASU and elsewhere tend to change from
year to year, we are committed to developing tools and workshops that will ensure that as new
teachers are assigned to the course, they too are trained to provide quality, coherent, and
meaningful instruction.
The focus for this project will be on redesigning instructor training and instructional
formats rather than authoring new curricular materials. The curricular groundwork has been laid
for the College Algebra pilot materials. Our efforts will concentrate on adapting these materials
to the College Algebra audience.
The College Algebra redesign course will be organized into seven modules: Proportional
Reasoning, Covariation I, Linear Functions, Composition, Quadratic Functions, Covariation II,
and Exponential/Logarithmic Functions. Each module will consist of the following components:
 Lesson Logic: This in-depth lesson narrative provides instructors with the important
ideas contained in the lesson along with helpful hints on the type of conceptual
questions to address during the activity. The lesson logic is for College Algebra
instructors only, not students.
 PowerPoint Presentation: The lesson will be formatted into a PowerPoint presentation,
which will pose various problems for students to work on within their respective
groups. Students will be provided opportunities to justify their reasoning on
whiteboards for other students in the class to view. Activities will also involve a
hands-on component where students are required to collect and analyze their own data
using calculator-based technology (e.g., Texas Instruments Calculator-Based
Laboratory). In addition, visual simulations of the mathematics will be included in the
activities. Instructors will demonstrate various mathematical ideas using instructional
software such as Geometer’s Sketchpad, Graphing Calculator, and Fathom. The
PowerPoint presentation will be made available for students to download as class
notes.
 Reflective Homework: These assignments will be designed to promote student
reflection on the topics learned in class. This writing-intensive homework will serve as
a method of formative assessment to gather information about whether students were
able to make important connections during in-class activities.
 Online Homework: These homework assignments will prepare students for the
upcoming class activities, thus freeing time for in-class instruction to focus more on
the conceptual aspects of algebra. Online assessments will gauge students’ learning of
the mathematical procedures and concepts and will provide the instructors with
immediate feedback on class progress.
Sample Activity: The Box Problem
One example of a problem often taught in traditional College Algebra courses is the Box
Problem. This problem involves key ideas about variation and covariation. Understanding these
ideas helps students to grasp functional relationships as they relate to contextual situations. Yet
traditional courses typically overlook these powerful ideas and simply focus on more procedural
components of the problem. An example of the traditional Box Problem is illustrated below.
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ASU College Algebra

The Box Problem (Larson & Hostetler, 2004)
An open box is to be made from a piece of paper (8.5” by 11”) by cutting equal squares of length
x from the corners and turning up the sides. Find the following:
1. Verify that the volume of the box is given by the function V (x)  x 8.5  2x 11 2x .
2. Determine the domain of the function V .
3. Sketch the graph of the function and estimate the value of x for which V (x) is maximum.

In this traditional format of the problem,
students are provided the function first with additional

follow-up questions about domain of the function, as well as the function’s graph. With this task,

volume changes
students are not required to think deeply about how the
with respect to changes
in the size of the cutout. Connections among the size of cutout, respective changes in length,
respective changes in width and respective changes in volume are largely ignored. The focus of
the traditional version of this task is on the procedural and abstract algebraic components of the
situation with little to no emphasis on the meaning of the variables.
In contrast, the redesign course will utilize the Box Problem in a very different way to
illuminate the important components of variation and covariation. As students come to grasp the
ideas of variation and covariation, they are building conceptions of the algebraic representation
of the function. In addition, the redesigned Box Problem will engage the students in hands-on
and active learning. They will first create physical boxes with various cutout sizes to help bridge
the conceptual gap between the size of cutout x and the volume of the box, V (x) . This
knowledge will be used to create the algebraic function. The sample slides below illustrate the
introduction to the Box Problem that we will use in the redesigned course.


The next portion of the discussion for the Box Problem in the redesigned course will be
geared to simulating the changing box with the changing size of cutout. This computer-aided
simulation provides a powerful opportunity for students to visualize the size of the box as the
cutout size changes dynamically. Sample slides below illustrate this portion of the activity.
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ASU College Algebra

The redesigned Box Problem will focus on the meanings of the formula and variables,
such as the meaning of x , the meaning of 11 2x and the meaning of 8.5  2x as they relate to
the physical dimension of the box. The lesson will engage students in generalizing the arithmetic
processes and building the algebraic formula from concrete models of the boxes to more abstract
forms such as the algebraic representation
of V (x)  x 11 2x

 8.5  2x . In this activity,

students will also discover how to conceptualize how two variables change in tandem with one
another by using a teaching strategy known as the Finger Tool. Developed by ASU Professor
Patrick Thompson, the Finger Tool involves having students move their index fingers—each of
 one independent and one dependent—along imagined x
which represents a changing quantity,
and y axes. This helps students to grasp the coordination of the changes in the dependent
variable with respect to changes in the independent variable. As when responding to the
traditionally taught Box Problem, students will also discuss the maximum volume of the box.

However, the explorations leading up to this question should assure that students understand
the
meaning of both the algebraic procedures and graph as they relate to the real context of the box.
Furthermore, students will be prompted to continue reflecting on related ideas. We will
prompt them to find different-sized boxes with the same volume and values of the cutout for
which the volume increases. This will help illuminate ideas about the physical nature of the box
by providing meaning of how two different sized cutouts can produce the same volume but
different orientations of the box. The activity will also include discussion about amounts of
change relative to ideas of covariation and multiple representations of function.
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ASU College Algebra
In summary, the redesigned College Algebra course will extend traditional problems into
classroom activities with rich, probing questions embedded in contextualized and meaningful
problem situations. The redesigned Box Problem illuminates the kinds of questions students will
tackle during class where the center of instruction is shifted from the teacher to the student. Our
redesign team will continue modeling such activities to provide more meaningful and productive
experiences for students in College Algebra.
Cost Reduction Strategy
Student enrollment at ASU is currently growing and all indicators suggest this enrollment
will continue as the university strives to accommodate an unprecedented number of students
enrolled. As a result, the number of students who need to take College Algebra will undoubtedly
parallel the growth of the university. To accommodate this growth without increasing the number
of faculty hired to teach College Algebra, the redesign project utilizes undergraduate
mathematics majors as teachers of College Algebra for their field experience toward secondary
mathematics certification. The cost savings of this project is calculated to be $243,744 per year,
given we project that half of all College Algebra sections will be taught by undergraduate
teachers. The change in teaching personnel will allow the enrollment of College Algebra to
increase without increasing the number of expensive faculty.
Additionally, the team will develop training materials and workshops to provide new
teachers will the necessary knowledge for teaching College Algebra using the redesigned
materials. These materials will be developed by a team of experienced researchers and faculty
who will share instructional materials to eliminate duplication of effort in course curriculum
construction. These materials will be provided through a course management website for
instructors to use. Not only will this save the university money in funding teachers to create new
course materials, it will also increase new-instructor support of the project, since they will not be
required to develop their own set of materials independently.
Funds saved as a result of the redesign project will be used toward continued redesign of
other first year mathematics courses at ASU (i.e., Precalculus, Brief Calculus). The cost savings
will also provide training opportunities for future College Algebra teachers (FYM faculty,
faculty mentors, and undergraduate mathematics majors).
Redesign Project Timeline
The timeline for the College Algebra redesign project can be categorized into four
phases. Phase 1 is the planning stage during which the leadership team, composed of ASU
faculty and mathematics education researchers, will work to revise current curricular activities
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and research tasks for the College Algebra audience. We plan to organize the course material
into seven modules and create online assignments to align with these modules (using course
management software which accompanies the course textbook). Furthermore, the leadership
team will begin training instructors in the new teaching strategies and use of conceptually
focused curricular materials to support the instructors in providing effective instruction for the
redesigned course. The leadership and instructional team will work together throughout the
summer and fall sessions to continue refining the course materials.
Phase 2 will involve piloting the new College Algebra course with 10 sections during the
spring 2008 semester. The leadership team will conduct data collection through pre- and post-test
assessments and clinical interviews to ascertain the effectiveness of the redesigned course as
compared to the traditional course. The leadership team will continue training new instructors
during this phase.
The leadership team will continue course revision during Phase 3 of summer 2008. The
team will analyze the data collected during the pilot study, and use that data to inform the
refinement of both the curriculum and instructor training as they prepare for full implementation
in Phase 4. The project is planned for full implementation of all College Algebra sections at ASU
to be taught using the redesigned model developed during the 2008-2009 academic year. The
Department of Mathematics and Statistics has committed to support a full implementation, which
will involve their providing the necessary support to carry out the redesigned College Algebra
beginning in Fall 2008.
In addition to training teachers, the leadership team will work during Phase 2 with
undergraduate mathematics majors seeking secondary certification. Since the secondary
certification program is new to ASU, the team will first focus on recruiting and then training
undergraduate assistants in tutoring College Algebra. This will serve as an important part of their
preparation for teaching the course. During Phase 3, undergraduates will be assigned to faculty
mentors to further prepare for teaching College Algebra during Phase 4 of the project and
beyond. The undergraduate mathematics majors will meet weekly with their mentor during their
teaching semesters to continuously reflect on their teaching practice. Figure 1 illustrates the
College Algebra redesign project timeline’s four phases.
Phase 1
Planning stage, revise curriculum, author online materials
curriculum, author online materials
Phase 2
online materials
Pilot 10 sections of revised college algebra, collect data (assessments, interviews)
revised college algebra, collect data (assessments, interviews)
collect data (assessments, interviews)
Phase 3
interviews)
Continue course revision based on preliminary findings
revision based on preliminary findings
preliminary findings
Phase 4
findings
Full implementation of revised co
of revised college algebra
algebra
Summer 2007 – Fall 2007
Spring 2008
Summer 2008
Figure 1: College Algebra redesign timeline.
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ASU College Algebra
Redesign Research Plan: Measuring Success
The effectiveness of the pilot course will be compared with the current instructional
approach during the first semester of its implementation, using valid tools and methods to
compare the redesigned sections with the current sections. Our research plan consists of two
components: (a) researching the College Algebra students and (b) researching the College
Algebra redesign teachers.
Research of Students
Our research plan is focused on the conceptual development and learning of College
Algebra students in the redesign course as compared to students in the traditional course. Among
our primary tools for measuring student achievement will be the Precalculus Concept
Assessment (PCA) instrument, developed by Marilyn Carlson and identified through more than a
decade of testing as a valid assessment tool for determining if students have acquired an
understanding of the major concepts of College Algebra and precalculus mathematics (Carlson,
Oehrtman, & Engelke, 2007). In their study, Carlson, Oehrtman, and Engelke reported that 85%
of students who received a score of 11 or higher on the PCA succeeded in first semester calculus,
while 80% of students who received a score lower than 11 on the PCA either withdrew or failed
first semester calculus. Therefore, the PCA instrument was integrated into this research to
provide insight on students’ understanding of precalculus concepts, such as the concept of
function, before and after their College Algebra experience. Students in both the pilot course and
the traditional course will be assessed using the PCA as a pre- and post-test assessment
instrument and their results will be compared to determine the effectiveness of the redesign
course materials. Finally, we will administer a previously developed and validated tool, the
Views About Mathematics Survey (VAMS), to assess shifts in students’ confidence and views
about the methods and nature of mathematics. Results of the pilot group will be compared to
those of the traditional group.
In addition to the quantitative measures listed above, the team will also collect qualitative
data to provide a more in-depth investigation of students’ understanding of College Algebra. We
will conduct clinical interviews for College Algebra students from both the pilot group and the
traditional group. A total of 10 students from each group will be selected to participate in this
study. Each student will be interviewed three times at various points in the semester to study
their conceptual growth on specific topics relative to College Algebra. The interview protocol
will include conceptual tasks designed to measure students’ ways of thinking about College
Algebra topics such as variable, rate of change, covariational reasoning, additive reasoning, and
multiplicative reasoning. Research has shown that these algebra concepts are foundational to
building a profound understanding of the function concept, which is central to the college
algebra and precalculus curriculum (Carlson, 1998; Carlson, Jacobs, Coe, Larsen, & Hsu, 2002;
Confrey & Smith, 1994, 1995; Monk, 1992; P. Thompson, 1994). This qualitative data will be
compared with the quantitative data to provide a more holistic picture of the effectiveness of the
redesigned College Algebra course. The qualitative data will also provide the instructional team
with evidence of student thinking throughout critical points of the semester to help refine course
instruction as necessary.
Research of Redesign Teachers
The team will also investigate the development of the teachers as they prepare for and
implement the redesigned course materials. This data will be used to enhance teacher training
materials and workshops as we prepare to expand our redesign efforts for improving course
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instruction in precalculus and brief calculus courses.
Of the 10 College Algebra redesign pilot sections, the leadership team will intensely
focus on at least two sections to investigate the effectiveness of the materials and instruction for
enhancing students’ learning of the concepts. These focus sections will be videotaped during
each class session with one camera on each student group and another on the teacher. This video
data will be reviewed and analyzed to further enhance the teacher training materials for future
implementations of the course. Video cases will be used in training workshops to showcase
exemplary models of conceptual learning inside the classroom. Moreover, all instructors for the
pilot redesign will be evaluated using the Reformed Teaching Observation Protocol (RTOP)
created by the Arizona Collaborative for Excellence in the Preparation of Teachers. The
measurement tool provides valuable information about the nature of teaching and will allow the
redesign team to facilitate in the development of the teachers from the redesign group.
In summary, the redesign project will incorporate multiple methods of data collection.
For investigating students, we will collect data from (a) PCA pre- and post-test for redesign and
traditional courses, (b) clinical interviews of students from both redesign and traditional sections,
and (c) VAMS assessment of students’ views of mathematics. For investigating the redesign
teachers, we will collect data from (a) RTOP pre- and post-evaluation of redesign instructors and
(b) videotape focus sections of College Algebra.
Timeline of Research Plan
Data collection for the project is planned throughout the semester during the pilot phase
of the project. Two focus sections of the redesign course will be videotaped throughout the
semester. Instructors will be evaluated using the RTOP during the first week of the semester and
again at the end of the course. Students from both the redesign group and the traditional group
will be assessed using the PCA and VAMS during the first and last week of the semester.
Student volunteers will be interviewed at the beginning of the course, during the middle of the
course and again at the end of the course. Figure 2 illustrates this timeline.
Videotape Focus Sections (2) of Redesign Course
RTOP Pre-evaluation
evaluation
RTOP Post-evaluation
evaluation
PCA Pre-test
PCA Post-test
VAMS
Pre-Survey
VAMS
Post-Survey
Student Interview 1
Interview 1
Week 1
Student Interview 2
Interview 2
Week 4
Week 7
Figure 2: Redesign research timeline.
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Student Interview 3
Interview 3
Week 10
Week 14
ASU College Algebra
REFERENCES
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Carlson, M. (1994). A successful transition to a calculator integrated precalculus curriculum:
Clues, surveys & trends. In Proceedings of the International Conference on Technology
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Carlson, M. (1998). A cross-sectional investigation of the development of the function concept.
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Education (pp. 114-162). Providence, RI: American Mathematical Society.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning
while modeling dynamic events: A framework and a study. Journal of Research in
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instrument: The development and results.
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exponential functions. Educational Studies in Mathematics, 26, 66-86.
Larson, R., & Hostetler, R. (2004). College Algebra (6th ed.). Boston, MA: Houghton Mifflin.
Mathematical Association of America. (2007). Curriculum Renewal Across the First Two Years:
Guidelines for College Algebra. Retrieved June 10, 2007, from
http://www.maa.org/cupm/crafty/CRAFTY-Coll-Alg-Guidelines.pdf
Monk, S. (1992). Students' understanding of a function given by a physical model. In G. Harel &
E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy,
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America.
Oehrtman, M., Carlson, M., & Thompson, P. W. (in press). Key aspects of knowing and learning
the concept of function. In M. Carlson & C. Rasmussen (Eds.), Making the connection:
Research and practice in undergraduate mathematics. Washington, DC: Mathematical
Association of America.
Thompson, A., Philipp, R., Thompson, P., & Boyd, B. (1994). Calculational and conceptual
orientations in teaching mathematics. In D. Aichele & A. Coxford (Eds.), Professional
development for teachers of mathematics (pp. 79-92). Reston, VA: National Council of
Teachers of Mathematics.
Thompson, P. (1994). Students, functions, and the undergraduate curriculum. Research in
Collegiate Mathematics Education. I. Issues in Mathematics Education, 4, 21-44.
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