**Theresa Anderson**

Development and evaluation of supplementary geometry tasks### DEVELOPMENT AND EVALUATION OF SUPPLEMENTARY GEOMETRY TASKS

By Theresa Anderson

Spring 1999

The purpose of this project was to develop and evaluate a set of supplementary high school geometry tasks, with a focus on navigation. The tasks were written in the context of real-world situations and structured to integrate previously learned concepts while introducing new ideas.

The tasks were field tested in a high school class during the fall of 1998. The tasks were evaluated to determine appropriateness for geometry students, level of student engagement, and whether instructions for the students were clear and complete.

There is a wealth of material concerned with the motivation of students and the link between motivation and student achievement. A brief summary of some of these writings is included to emphasize the important role that the affective domain has on student engagement in mathematics. One method for motivating students is the use of projects in a mathematics class. Literature related to the support for and guidelines for projects is reviewed. Current textbooks were examined to determine if, and how often, projects were integrated into the curriculum.

A unit of curriculum developed for this project, entitled

*Navigating a Course through Geometry*, is included as an appendix. This material consists of directions for the teacher, student worksheets, and sample student work.**Leslie Leah Anderson-Mills**

A change in mathematics graduation requirements### A CHANGE IN MATHEMATICS GRADUATION REQUIREMENTS: A CASE STUDY

By Leslie Leah Anderson-Mills

Spring 2004

This thesis examines the effects of increasing the mathematics requirements for graduating seniors at a small rural high school. This policy, essentially a detracking system, requires all graduates to complete 3 years of mathematics with 2 of those years being algebra and geometry. A quasi-experiment with a nonequivalent control group was used with longitudinal data collected from students’ Stanford Achievement Test ninth edition scaled scores; eighth and eleventh grades. Achievement gains were measured using a two-sample T-test. Categorical differences were analyzed using a Chi-square. Significance lever of .05 was used throughout. There was a substantial improvement between the treatment and control group, 21%. This was no statistically significant, perhaps due to variability issues. Additionally, 30% of the control group verses only 10% of the treatment group had a drop in achievement. The treatment group have 30% of students move into high achiever status while the control group had a 25% shift. Upon further examination of subgroup data, Hispanic, low SES and high achieving students were not harmed by the policy. Not only did any of these subgroups not suffer declines in achievement, but also graduation rates remained unchanged. This implies that students can and should be expected to meet reasonable educational goals.

**Ray Atara**

The integration of chaos theory into the precalculus curriculum### THE INTEGRATION OF CHAOS THEORY INTO THE PRECALCULUS CURRICULUM

By Raymond J. Arata

Summer 2000

This study was designed to develop and field test a set of materials integrating chaos theory and fractal analysis into the precalculus curriculum. Because chaos theory, dynamic systems, and fractal analysis are related topics, these topics will be referred to collectively as chaos theory.

Chaos theory is contemporary field of mathematics that has grown rapidly in popularity and scope during the last two decades.

However, mathematics curricula in this country do not typically include chaos theory as a major topic, especially at the high school level. It is this author’s belief that chaos theory can be taught at the high school level and integrated into the precalculus curriculum.

A set of lessons was developed which integrated the presentation of complex numbers into the presentation of Julia sets and the Mandlebrot set. There were two matched pair groups of students within on classroom, the blue group and gold group. The gold group was presented with the Mandlebrot and Julia sets along with complex numbers. The blue group was presented with complex numbers only. At the end of the unit, a quiz testing only complex numbers was given to all students. Another set of lessons was developed which integrated the presentation of sequences and series into the presentation of the Cantor middle thirds set, the Koch snowflake, and the Sierpinski triangle. The gold group was presented with sequences and series only. The blue group was presented with both sequences and series and the Koch snowflake, the Sierpinski triangle, and the Cantor middle thirds set. At the end of the unit, a quiz testing only sequences and series was given to all students. All students also completed an attitude survey.

The results of the field study were very interesting. In the complex numbers unit, the gold group scored significantly higher on the quiz than the blue group. In the sequences and series unit, there was no significant difference between the quiz scores of the gold group and the blue group.

The results of the attitude survey indicate that the students felt that the inclusion of chaos theory topics enhanced the learning of the precalculus topics. More significantly, they felt that the learning of other precalculus topics would be more enjoyable if chaos theory were included.

The results of the study suggest that integrating contemporary and interesting mathematical topics such as chaos theory into the traditional precalculus curriculum can enhance the learning of precalculus and the enjoyment of mathematics.

**Kim Battaglia**

Challenges facing students transferring into a reform algebra curriculum### CHALLENGES FACING STUDENTS TRANSFERRING INTO A REFORM ALGEBRA CURRICULUM

By Kimberly L. Battaglia

Spring 2004

The objective of this thesis was to identify challenges to transfer students entering a College Preparatory Mathematics (CPM) algebra curriculum. The case study was conducted in an eighth grade algebra classroom over the course of one academic year. The subjects were five students who transferred into the class during the year.

Initial interviews were designed to provide data concerning their background. In subsequent interviews, students worked through written assessments of math content from the units that they had missed. During each interview, students were observed to see how they did each problem, whether they were able to complete the problem on their own, or if they needed help.

The transfer students found making guess and check tables, coordinate graphing, geometry and algebraic symbol manipulation to be challenges. Other challenges not specifically related to the CPM algebra curriculum included working in collaborative groups and motivation.

In conclusion, depending on when a student transfers into the class, teachers should be aware that the transfer students could have some of the challenges reported. However, findings suggest that most challenges students faced can be minimized with brief, individual teacher interventions.

**Mike Buzbee**

A guide to portfolio assessment in secondary mathematics classes### A GUIDE TO PORTFOLIO ASSESSMENT IN SECONDARY MATHEMATICS CLASSES

By Michael Buzbee

Summer 2004

The purpose of the project was to develop of clear, useful, and helpful teacher handbook for incorporating portfolios in secondary mathematics classes.

By providing an assortment of classroom ideas, activities, and suggestions that complement portfolios, the design of the handbook was intended to help minimize the frustrations that seem to arise when teachers attempt something “new and different.” The strategies for developing portfolio assessment in the handbook support constructivist philosophy and curriculum, and are designed to most mathematics education settings. Activity templates with suggestions are provided throughout the handbook.

Questionnaire data was collected from eight teachers with experience teaching secondary mathematics. The teachers were asked to rate, and comment on the helpfulness and clarity of the handbook. Responses to the questionnaire also provided information as to ways in which the handbook could be improved. The overall response from teachers was that the handbook was clear and helpful.

**Ted Cluver**

A study of the mathematics education master's degree program at CSU, Chico### A STUDY OF THE MATHEMATICS EDUCATION MASTER’S DEGREE PROGRAM AT CALIFORNIA STATE UNIVERSITY, CHICO

By Theodore Mathew Cluver

Spring 2004

California State University, Chico currently offers MA/MS degrees in mathematics education in Interdisciplinary Studies. This study explores opinions and perspectives of those who have inquired or participated in these master’s degree programs by use of a survey. Two different surveys were developed according to the level of involvement of the participant. That is, there was a survey for those who inquired but never applied for admission and a survey for those who are currently enrolled or have completed the program.

The purpose of this study is to understand how well the elements of the CSU, Chico’s mathematics education advising pattern in Interdisciplinary Studies meets the needs of participating K-12 teachers. The survey revealed that the participants like the format of the summer-only program, but would like more course offerings. It also revealed that the faculty is considered the most positive aspect of the program. For those that had inquired, the major obstacles for not enrolling were personal reasons, time conflicts, and distance to travel to the University.

**Janet Cowan**

Sacred geometry### SACRED GEOMETRY

By Janet Cowan

Spring 1999

Sacred geometry is customarily regarded as having one of the following meanings: (a) describing certain shapes that were often used in the design of sacred buildings, such as temples and cathedrals; or (b) having to do with geometric forms that possess elegance or mathematical beauty. This project brings a different focus to these traditional definitions by including a special understanding of the process of creation. The project is a computer-generated color animation video, created using Lightwave 5.6 software, which illustrates the relationships among different geometric patterns regarded as sacred in both ancient and modern teachings. The video shows the evolution of these forms, beginning with the void of formlessness and continuing through the creation of a single point, a sphere, a circle, and a 19-circle pattern called the flower of life. Four of the five Platonic solids are then shown as evolving from the flower of life pattern. The accompanying script describes the metaphysical meanings of the symbols from the viewpoints of several philosophical and religious traditions, focusing mainly on Judeo-Christian and ancient Egyptian teachings. The review of related literature includes the three components of visualization, evolution, and the objects of sacred geometry.

**Carol Cumming**

Handbook for teaching addition of fractions in the middle school### HANDBOOK FOR TEACHING ADDITION OF FRACTIONS IN THE MIDDLE SCHOOL

By Carol Ann Cumming

Summer 1999

This master’s project was to provide a set of lessons for the middle school teacher to use to supplement the mathematics curriculum that encompasses the area of fractions and in particular the addition of fractions. The review of literature suggests the necessity for a variety of techniques to teach the fraction curriculum to middle school students. It indicates the need for the middle school teacher to use hands-on materials to explain and illustrate fraction concepts.

The format for the Handbook is simple and concise with the ten lessons. The handbook lessons present basic fraction concepts of equivalency, common denominators, and addition of fractions using different manipulative devices including tangrams, geoboards, paper strips, money, and pattern blocks. The handbook includes instructions for the teacher, demonstration suggestions, diagrams of the problems, worksheets for the students, questions for the teacher to use to lead students to discover connections and concepts, and answers for the problems on the worksheets.

**Phyllis Cummis**

Performance standards and tasks for seventh grade### PERFORMANCE STANDARDS AND TASKS FOR SEVENTH GRADE: ALGEBRA AND FUNCTIONS

By Phyllis Ann Cummins

Summer 1999

New mathematics standards were adopted by the State of California in 1997. These standards are quite different from what has been previous practice in California’s public schools. One major change is that algebra is recommended as a required class for all eighth grade students. The standards for seventh grade algebra and functions are also more intense than what has been the practice in my school district.

As a seventh and eighth grade teacher, I have a concern about the ramifications of these new standards. My goal is to prepare my students for future mathematics classes in the best manner possible.

The purpose of this project was to develop and evaluate a focused, coherent set of performance tasks for the seventh grade algebra and functions standards adopted by the State of California. Tasks were developed using “real world situations”. Ideas for the tasks came from various sources.

These tasks were used in my classroom with seventh and eighth grade students as an end of the year assessment. Each set of finished tasks was evaluated using a four point rubric, and the results were tabulated.

My sample was confined to my small group of seventh and eighth graders in a rural school in northern California. The tasks were given to grade levels only if the particular skill had been taught in class. Some tasks were given to both seventh and eighth graders and some were given to only one grade or the other.

Approximately one half of the performance tasks I developed were successful with the seventh grade class I worked with in this study. All the tasks were successful with the eighth grade class. This might imply that the seventh graders did not have enough background preparation for these tasks. It is possible that with proper preparation, these tasks could be successful with seventh grade students.

**Laurie DeMaranville**

Construction of polygons by tying knots with ribbons### A COURSE IN MATHEMATICS FOR SECONDARY STUDENTS

By Laurie DeMaranville

Fall 1999

This thesis explores construction of regular polygons by tying torus knots with ribbons. A model is developed that starts with a particular knot diagram of a torus knot, converts that diagram to a straight-line knots diagram and then widens the straight-line diagram into a ribbon.

Predictions can be made regarding the polygons that can be constructed. The torus knots are broken into different classes—some yield polygons, others do not and some appear to form polygons, but it is not proven. Conclusions are primarily based upon the relationship between the number of longitudinal and meridional cycles of the torus knot, the arrangements of the crossings of the knot diagram, and the direction of the fold of the ribbon. A pentagon and all regular polygons with seven or more sides can be formed by tying specific torus knots with ribbons.**Katy Early**

Mental mathematics and the development of place value understanding### MENTAL MATHEMATICS AND THE DEVELOPMENT OF PLACE VALUE UNDERSTANDING

By Katy Early

Spring 2003

The purpose of this study was to examine the development of understanding of place value among fifth and sixth grade students in a classroom where mental mathematics was a regular part of the curriculum. At the beginning and the end of the study, students explained in writing the meaning of each of the digits in the number 25. During the six-month study, children engaged in mental calculation practice twice a week. The solved the problems independently, and then explained and defended their mathematical reasoning to their peers. On several occasions, students also wrote about their mental processes.

The results of the study show that the members of the class derived a variety of useful mental processes for subtraction and multiplication, and that individuals developed preferences for particular methods. Conversations among students led to a classroom culture that valued invented procedures and accurate use of mathematical language in the explanation of any method, whether standard or invented. Children revealed ideas about place value during class discussions by identifying quantity value for digits as they shared their thinking, and by explaining how their own and peers’ inattention to place value caused mental calculation errors. On the pre-test, 55% of the students wrote explanations of place value which were considered successful. On the post-test, this figure rose to 91%.

Mental mathematics was found to be of value for developing place value ideas for students at all levels of ability in the class. The teacher reported that other aspects of mathematical learning were also enhanced, and particularly that the class needed less instructional time to learn new procedures and to become efficient in their use. The regular inclusion of mental mathematics in classroom instruction for intermediate grade students is an accessible means of strengthening fundamental concepts in mathematics while simultaneously simplifying the learning of new concepts.

**Ronna Eaton**

Eighth grade open-ended mathematics problems to enhance the teaching of problem-solving and writing in mathematics### EIGHTH GRADE OPEN-ENDED MATHEMATICS PROBLEMS TO ENHANCE THE TEACHING OF PROBLEM-SOLVING AND WRITING IN MATHEMATICS

By Ronna Ewing Eaton

Fall 1995

Students are being challenged to become mathematically powerful. In order to become powerful students need to gain the ability to do purposeful and worthwhile mathematics as it relates to usefulness in life. Research shows that problem-solving increases this type of mathematical competency, and increasing the number of experiences in solving divergent problems will improve students’ problem-solving abilities. The interest a student has in the problem assigned has a direct and positive correlation as to whether the student will successfully solve the problem. The relationships between problems, problem-solving behavior, classroom environment, and students affect the outcome for successful problem-solving.

The purpose of this project was to create a series of eighth grade open-ended mathematics problems. Twelve problems were designed to help eighth grade students improve their problem-solving skills. The problems address all eight strands of the

**Mathematics Framework for California Public Schools**, (California State Department of Education, 1992). The problems were developed to provide students with writing as a tool and technique in mathematical problem-solving and to provide teachers experience with holistic scoring (rubrics).Scoring rubrics were discussed with the students and they were taught how to score problems to help improve with own abilities to respond to problem-solving in developing written communication in mathematics. Revision was encouraged to improve understanding and the overall product.

The results of the field testing revealed that students profited from their experiences in writing in mathematics using constructed response answers to open-ended problem-solving experiences.

**Patricia Evans**

Lessons in functions for fourth grade students### LESSONS IN FUNCTIONS FOR FOURTH GRADE STUDENTS

By Patricia Evans

Fall 1995

This study was designed to develop and field test a set of lessons to introduce the notion of function to fourth grade students by classroom teachers. Few lessons are available at the elementary concept level in function. Yet, research has shown that understanding is achievable by young children.

Seven problems were developed and field tested in three classrooms in 1995. The teachers were chosen for their interest in mathematics curriculum development. The field testing was an action research model.

The teaching model used small work groups with minimal teacher intervention followed by whole class discussions or presentations to the class by each group. The teacher role was as guide, not director or leader.

The appendix includes the lessons, classroom notes, and samples of student work. Masters are included for the lessons as needed.

The results of the field test were positive, based on the responses of the participating teachers. Limitations do exist, including size of the sample, socio-economic and cultural make up of the sample, length of the study, and the narrow focus of the topic.

**James Friedrich**

Does the use of geometer's sketchpad improve student achievement?### DOES THE USE OF GEOMETER’S SKETCHPAD IMPROVE STUDENT ACHIEVEMENT?

By James Friedrich

Summer 02

Many studies exist for LOGO and BASIC programming that analyze student achievement, but few studies exist that measure the effectiveness of the Geometer’s Sketchpad. The purpose of this study was to measure the effectiveness of the Geometer’s Sketchpad at Laguna Creek High School, in Elk Grove, California. Pre-existing data acquired from normal practice was used to measure student achievement. A pretest, the Geometry Readiness exam by the Mathematics diagnostic Testing Program (MDTP) of UC/CSU, was used to determine each student’s level of performance at the beginning of the year. A district level posttest was used to measure any gain or loss of ability. The treatment group consisted of 95 students who used the Geometer’s Sketchpad for 12 labs and instruction of geometry lessons while using Sketchpad as an instructional tool. The teacher also reviewed homework problems that could be done on the Sketchpad. The control group consisted of 137 students, 31 from Laguna Creek High School and 106 students from Elk Grove High School. The control group did not have any access to the Sketchpad. A t-test and an Analysis of Covariance compared the MDTP scores of the treatment group to the control group and CSRS scores of the treatment group to the control group.

Results of this study showed that students using the Geometer’s Sketchpad did significantly better than the control group on the End of Course Exam giving by the Elk Grove School District. The End of Course Exam was a 45 question multiple-choice exam that measured comprehensive topics of the geometry course. The topics ranged from identifying figures, applying properties of known shapes, to solving algebraic equations using properties of circles. An analysis of covariance further concludes that the treatment group did significantly better than the control group.

**Bradley Fulton**

The development and evaluation of a middle school replacement unit on exponentiation### THE DEVELOPMENT AND EVALUATION OF A MIDDLE SCHOOL REPLACEMENTUNIT ON EXPONENTIATION

By Brad S. Fulton

Summer 1999

This project involved the development of a replacement unit to facilitate the teaching of exponents to middle school students. After noting the lack of conceptual understanding among students when a traditional algebraic approach was used, the author designed activities that would foster conceptual development. Physical and manipulative models as well as real-world applications of exponents were used. In addition natural number exponents, physical models of zero and negative powers were designed. Fractional values of exponents were also explored in the unit.

The literature review for the study focuses on three areas. The first section deals with how multiplicative reasoning develops. The next section explores pedagogy for effective teaching of such a unit. This section includes research on cooperative grouping, Multiple Intelligences Theory, and brain-based learning. The last section examines existing curricula and evaluates them in relation to the literature.

The activities were evaluated through a field test on a group of 56 eighth graders heterogeneously grouped into two classes in California. Of the total students, 36 were present for all three phases of the assessment. The classes included regular education, Resource Specialist, Special Day, and English Language Deficient students. The field test included of an assessment given at three stages during the teaching of the unit. It was first given as a pretest, and again at the culmination of the unit of instruction. It was given a third time five weeks after the unit of instruction was finished to test for retention. The assessment instrument consisted of three portions: computational fluency, conceptual understanding, and application for problem-solving. Students showed strong gains in conceptual understanding and also showed improvement in application. Lesser gains were posted in computational fluency. Assessment data is included in the paper.

The appendices include the assessment instrument and the replacement unit activities.

**Stephen Gonzales**

Technology-enhanced hands-on mathematics curriculum using authentic assessment for designated at-risk secondary students### TECHNOLOGY-ENHANCED HANDS-ON MATHEMATICS CURRICULUM USING AUTHENTIC ASSESSMENT FOR DESIGNATED AT-RICK SECONDARY STUDENTS

By Stephen Gonzales

Summer 1999

This project was designed primarily to develop and evaluate curriculum that motivates at-risk students to become interested in mathematics, realize the value of mathematical skills, and ultimately become successful in applying their knowledge. Although other projects have effectively used hands-on learning, problem-based learning, authentic assessment, and technology, to my knowledge, none have used them simultaneously with at-risk students. Utilizing knowledge gained from the literature review, I wrote a unit of instruction that ties together other successful models, while also integrating innovative lessons to teach mathematics to at-risk students.

To achieve the objectives stated in the previous paragraph, I began by writing a unit of instruction entitled the Architect Unit. The unit consists of 11 lessons that employ methods such as CAI (Computer Assisted Instruction), hands-on learning, thematic teaching, and authentic assessment. Teamed in groups of two, students designed their own floor plans, built their own quarter-inch scale model homes, and presented their work at a public exhibition. The exhibition was three fold with an oral component, a career component, and a rubric scoring component. Scoring was completed by outside community members, one of which was a professional architect.

The most significant result of this study appears to be that the Architect Unit of instruction can be used to help at-risk students successfully learn mathematics. This is verified by the rubric scores given by the community judges involved in the project. Nine out of ten teams of students achieved successful scores. In addition, the overall class mean on the pre-test improved from 17.22 to a class mean of 21.75 on the post-test, showing a gain in mathematical ability. It should be noted that the post-test was not given directly after instruction, but four weeks after CAI. The overall success of the students in this study shows a major change in their performance, especially considering that the students were selected because of their record of academic failure and inability to complete junior high school.

With less money being allocated for at-risk programs, like continuation schools, researchers need to continue the development of curriculum for secondary at-risk students.

**Lori Holcombe**

The development of interactive software for the introduction of trigonometry### THE DEVELOPMENT OF INTERACTIVE SOFTWARE FOR THE INTRODUCTION OF TRIGONOMETRY

By Lori Holcombe

Spring 1995

Computer-Assisted Instruction (CAI) uses state-of-the-art technology to allow a more complete understanding of basic concepts for the aspiring mathematician. This project applies findings of current educational research to the introduction of the basic elements of trigonometry. The computer gives the student the freedom to explore interesting areas without the tedium of calculating and graphing by hand. This will enable more thorough learning of elemental trigonometry than has been evident using traditional modes of instruction in a basic course.

This original computer program is a creative introduction to trigonometry. The program is arranged in 38 screen presentations in HyperCard® format for Macintosh computers. Each screen interactive and allows students to manipulate graphics and calculations.

Dynamic graphics animate each lesson. Prerequisite algebra and geometry skills are reviewed. The unit circle is employed in the definitions of degree, radian, sine, cosine, and tangent. The original arc-based definitions are used to link geometric concepts to the trigonometric basics. Cartesian graphs are thoroughly explored.

Three test screens provide a means of assessment for the user. Several screens offer discovery environments which allow the investigation of key ideas. It is recommended that this program be used by students interested in an alternative approach to independent learning and by teachers as an enhancement to classroom instruction.

**Shelly Hollenbeck**

An evaluation of an intervention program for middle-school mathematics: connecting math concepts### AN EVALUATION OF AN INTERVENTION PROGRAM FOR MIDDLE-SCHOOL MATHEMATICS: CONNECTING MATH CONCEPTS

By Shelly Hollenbeck

Summer 2005

Many efforts have been made to develop programs and curriculums that are successful in helping at risk and low achieving students increase their knowledge and understanding of mathematics. The purpose of this study was to evaluate the effectiveness of one such program, Connecting Math Concepts (CMC), in the San Juan Unified School District in Carmichael, California. Pre-existing data acquired from normal practice in education was used to measure student achievement.

An experimental group of 47 students was chosen from one specific middle school in the San Juan Unified School District that adopted the CMC curriculum and used it as a mathematics intervention program in the 1999/2000 school year. A control group of 96 students having similar characteristics (gender, ethnicity, socioeconomic status) and comparable Procedures Scale Score and Total Math Scale Score on their sixth grade Stanford Achievement Test-9 was chosen from all ten middle schools in the same school district. The two groups were then compared for the next five academic years using state and national standardized testing scores, California High School Exit Exam Scores, and class schedules to determine when the students successfully completed an Algebra 1 course.

Results of the study showed no overall significant difference between the study groups. However, two other findings of interest were exhibited. First of all, the experimental group demonstrated less proficiency on the California Standards Test for General Math, but slightly more knowledge and understanding on the California Standards Test for Algebra, and significantly more knowledge and understanding on the California Standards Test for Geometry compared to the control group. Secondly, only 27% of the experimental group who had already passed an Algebra 1 class failed the California High School Exit Examination, whereas 42% of the control group who had already passed Algebra 1 failed the exam.

**Veronica Johnson**

Computer assisted instruction using accelerated math program and student achievement### COMPUTER ASSISTED INSTRUCTION USING ACCELERATED MATH PROGRAM AND STUDENT ACHIEVEMENT

By Veronica Johnson

Spring 2003

A study was undertaken to evaluate whether the use of computer-assisted instruction utilizing the Accelerated Math (AM) program would improve student achievement in mathematics. The study compared student achievement gains in math between those students who used AM material and those who used teacher-prepared materials. The Math Lab classroom instruction for both groups was a combination method of direct instruction and individualized instruction based on the progress and pace of the student learning ability. The contrast group students were from the Math Lab classes in a high school in northern California during the school year of 2000-2001, and the experimental group students were from the Math Lab classes in the same high school during the 2001-2002 school year. The teacher for both groups was the author. The tool chosen to measure student math performance was the Algebra Readiness (AR50/90) Test developed by the Mathematics Diagnostic Testing Project. Students in both groups took the same test three times during the course. The data analysis indicated that there were significant gains for both the experimental and contrast groups from pre-test to post-test as measured by the AR50/90 Test. However, there was no statistically significant difference in achievement mean gains between the two groups. The Accelerated Math program was no better than the teacher developed curriculum as measured by student gains in mathematics achievement according to this limited study.

**Jennifer Johnston**

Doctoral programs in mathematics education### DOCTORAL PROGRAMS IN MATHEMATICS EDUCATION: WHO GETS IN?

By Jennifer Johnston

Spring 2004

This study was designed to collect information about the types of courses and experiences a master’s degree student in mathematics or in mathematics education would need prior to entering a doctoral program in mathematics education. Faculty members who oversee dissertation research and/or evaluate applications for doctoral programs in mathematics education answered a 6 question survey concerning a variety of areas such as the focus of master’s degrees and bachelor’s degrees, experience writing a thesis or taking comprehensive exams, teaching experience requirements, prerequisites and characteristics for success. Responses were categorized and tallied for each topic. Data were then analyzed (a) collectively and (b) by sorting the data according to the type of department where the program is housed.

Results, conclusions, and recommendations indicate that doctoral program candidates should have: (a) a master’s degree completed (in either mathematics or mathematics education), (b) a bachelor’s degree in mathematics, (c) written a master’s thesis, (d) taken a wide range of courses in mathematics, education, and mathematics educations (e) taught public school for at least three years, (f) at least a GPA of 3.0 and GRE scores of 1000, and (g) strong work ethics.

There is a shortage of qualified individuals prepared to work in the field of mathematics education. Future research in the area is recommended. Current doctoral students could be used in a new study to evaluate the master’s degree courses and experiences that were the most helpful in preparing them for their doctoral studies.

**Amber Laherty**

Integrating literature into the middle school mathematics classroom### INTEGRATING LITERATURE INTO THE MIDDLE SCHOOL MATHEMATICS CLASSROOM

By Amber Laherty

Summer 2003

The intent of this project was to develop a set of lessons designed to integrate literature into the middle school mathematics classroom. Five different lessons were field tested with students in grades six through eight. Six different teachers tested the lessons. Each lesson was introduced by the teacher reading a book out loud to a class. After completing the lessons, students and teachers completed an opinion survey. The purposes for the survey were to study the appeal of the literature book, and to study the interest of the students in integrating literature into the mathematics classroom. The outcome of this project, for the most part, found each literature book appealing. The most important conclusion of this study appeared to be that integrating literature into the middle grades mathematics classroom is a positive experience for students.

**Melanie Liotta**

Problem solving prompts for the intermediate grades: assessing students through communication in problem solving### PROBLEM SOLVING PROMTPS FOR THE INTERMEDIATE GRADES:

### ASSESSING STUDENTS THROUGH COMMUNICATION IN PROBLEM SOLVING

By Melanie Close Liotta

Spring 2002

The objective of this project was to develop and evaluate a series of prompts that would improve both student problem solving skills and attitude towards mathematics. Each prompt consisted of a series of several problems, ranging from rudimentary to challenging, and also included a writing component where students needed to communicate their mathematical reasoning. A 4-point rubric was developed to accompany each prompt to assess the students’ understanding. The first prompt, together with the attitude survey, was administered at the beginning and end of the year for comparison.

The study was conducted in a sixth grade classroom over the course of one academic year. The students’ scores only showed improvement in prompts within the same standard, and not overall. The pre- and post- of the first prompt showed significant improvement, particularly in communication, with the number of students showing at least substantial understanding doubling. The attitude survey also showed significant improvement in the students’ attitude towards mathematics, with the number of students showing a ‘good’ attitude tripling.

In conclusion, problem solving abilities may not be transferable across content areas, or may take longer than one academic year to show significant improvement, but rich problem solving experiences can improve attitude towards mathematics.

**Mark Marvis**

Development and evaluation of a statistics unit for beginning algebra### DEVELOPMENT AND EVALUATION OF A STATISTICS UNIT FOR BEGINNING ALGEBRA

By Mark Mavis

Summer 2001

The purpose of this project was to develop and evaluate a three-week unit on statistics for a high school algebra course. The unit would be a supplement to an existing algebra course that did not already include statistics as a content objective.

The unit was field tested in junior high and high school classrooms during the spring of 2001. The unit was evaluated to determine appropriateness for a beginning algebra course and to determine if the statistics content was relevant.

There is a challenge from national and state (California) standards to teach statistics to all high school students. Most high schools teach a traditional mathematics sequence that ignores statistics. There are also nationwide and statewide emphases on establishing algebraic thinking throughout all grade levels. Proportional reasoning is an essential skill that is part of every algebra curriculum. A review of the research on proportional reasoning is included to show both the complexity of this area of mathematics and the need to develop this kind of thinking throughout the middle grades (6-8) and into high school.

The unit of curriculum developed for this project, entitled

*Two-Way Tables*, is included as an appendix. It is a unit that combines statistical ideas with proportional reasoning activities yet still has students engaged in algebraic thinking. This material consists of directions for the teacher as well as student worksheets, quizzes, and a test.**Stuart Moskowitz**

The integration of the graphing calculator into the algebra classroom at the community college### THE INTEGRATION OF THE GRAPHING CALCULARTOR INTO THE ALGEBRA CLASSROOM AT THE COMMUINTY COLLEGE

By Stuart Moskowitz

Spring 1994

This study was designed primarily to develop and field test a set of supplementary activities integrating graphing calculators into the Intermediate Algebra curriculum at the community college level.

Graphing calculators first became economically feasible in 1986. The advantages of this new technology over computers were realized immediately. An entire class set of calculators could be purchased for less that the cost of a desktop computer, placing the power of technology and visualization into the hands of every student.

Initial research and curriculum material was directed towards precalculus and calculus course. In 1989, the National Council of Teachers of Mathematics (NCTM) recommended in

**Curriculum and Evaluation Standards for School Mathematics**that all students beginning in ninth grade have access to graphing calculators (NCTM, 1989, p.124). This includes prealgebra and algebra as well as precalculus and calculus.At the time this study was begun in 1991, little or no research had been conducted in community college algebra classrooms. This project was designed to develop curriculum material for use at that level.

The author began writing a series of activities to supplement the existing traditional course material. These activities were then field tested in the author’s Intermediate Algebra classes at Butte College during the 1991/1992 and 1992/1993 school years.

A set of TI81 graphing calculators was made available by the mathematics department for classroom use only. Student reactions were determined by overall performance and an optional daily journal.

The results of the field study were mixed. Of the students who submitted journal entries, 32% made only positive comments, while 41% made only negative comments. The remaining 27% made both positive and negative comments. Some students commented that since they were unable to use the calculators outside of the classroom, many of the calculator’s potential benefits were not realized fully. With only limited access, some students were frustrated because they did not have time to become acquainted completely with the calculator. There did not appear to be enough classroom time for the students to become comfortable with the calculator as well as learn the algebra.

The results of the study suggest that supplementing the existing curriculum is not the best method for integrating graphing calculators into the curriculum; more research is needed to determine better methods. The entire course content needs major changes in order to realize fully the potential of this new technology.

**Martha Ogness**

Reinforcing Fundamental Math Skills in High School Math Classes### REINFORCING FUNDAMENTAL SKILLS IN HIGH SCHOOL MATH CLASSES

By Martha Ogness

Spring 2005

With the new California State high school graduation requirements, it has become increasing important that students attain mastery of basic computational skills while continuing on in an Algebra 1 course, our Fundamental Testing Program, the FTP, was designed. The purpose of this study was to evaluate the FTP. The FTP is a series of short low-stakes tests designed to review and reinforce basic math skills that have been identified as necessary to successfully complete Algebra 1.

Three contrast groups were used to test the effectiveness of the FTP. One class implemented the standard FTP; one class implemented a modified FTP using calculators on the tests, while a third control class did not take part in the FTP. A pretest and posttest given to all three groups measured the growth of each student. The mean gain score of each class was calculated. The mean gain score of the group doing the standard FTP was 6.00. The mean gain score of the group doing the modified FTP using calculators was 4.05. The control group not participating in the FTP had a mean gain score of 1.07. The mean gain scores of the FTP groups were significantly higher than those of the control group (alpha = 0.05). The group doing the standard FTP had a significantly higher mean gain score than those of the FTP class using calculators.

The results of this study provide evidence that the FTP is a worthwhile intervention for use in high school math classes to assist students in mastering and maintaining basic computational skills while continuing on in a college preparatory program.

**Joseph Palmer**

Investigating polygons using LOGO### INVESTIGATING POLYGONS USING LOGO

By Joseph Michael Palmer

Spring 2003

POLY is a procedure written in the computer language LOGO. It primarily consists of commands to move forward and turn right, in the same prescribed amounts. Investigating the figures drawn by this process is rich in mathematics, primarily geometry. Altering the POLY procedure to encompass compounded angles re-affirms the mathematics of POLY for all such variations, as well as leading to further mathematical investigations.

**Alma Quesnel**

Attitudes of elementary school Latino children towards mathematics### ATTITUDES OF ELEMENTARY SCHOOL LATINO CHILDREN TOWARDS MATHEMATICS

By Alma D. Quesnel

Spring 2002

This investigation is a qualitative case study designed to examine whether and when the Latino population in elementary school develops negative attitudes toward mathematics. Thirty eight randomly selected Latino students in two California elementary schools, on urban and one rural, were interviewed to collect information about their perception of what mathematics is, their attitude towards mathematics, their perception of their own performance in this discipline, and their perception of the utilization of mathematics outside of school.

The subjects’ answers were transcribed and then arranged in a matrix in order to identify, interpret, and summarize the patterns in each categorical question. Subjects in this study tended to conceptualize mathematics almost exclusively as arithmetic. The provided few examples of how mathematics is used outside of school. The research found that the Latino population studied tends to like mathematics until about grade four; by grades five and six, negative attitudes emerge with a sense of inadequacy especially evident in the female subjects.

**Beryl Scarbrough**

How fifth grade students make sense of decimals### HOW FIFTH GRADE STUDENTS MAKE SENSE OF DECIMALS

By Beryl G. Scarbrough

Fall 2003

Rational numbers represented in decimal form present various difficulties for middle school children. The purpose of this study was to examine how fifth grade students made sense of decimals in order to provide teachers with a lens through which they may view their own classroom and make informed decisions regarding instruction of decimals. Research shows that symbol manipulation without understanding is ineffective. The review of research focused on how students understand mathematical concepts in general and then looked specifically at the complexity of learning decimal notation. The research review suggested that knowledge of quantity determined understanding of decimal numerals. This investigation then, focused on how ten fifth grade students determined quantity of a decimal numeral. Data was collected from a written test and the student’s explanations of answers given on that test. The results revealed: 1) The underachieving students interpreted decimal numerals using whole number rules and 2) The higher achieving students who revealed knowledge of quantity appeared to interpret decimal quantity by connecting them with their fraction equivalent.

**Nikki Smith**

Density: an applied conceptual mathematics and science curriculum for middle school grades### DENSITY: AN APPLIED CONCEPTUAL MATHEMATICS AND SCIENCE CURRICULUM FOR MIDDLE SCHOOL GRADES

By Nickoleen Margaret Smith

Spring 1999

The purpose of this project was to develop a middle school curricular unit which utilized integration of conceptual content and constructive teaching strategies to significantly improve student comprehension of identified mathematics and scientific concepts and subconcepts. The unit developed for this purpose focused on the concept of density. The project utilized 12 hands-on laboratory investigations which maximized student involvement.

The unit introduced mathematical and scientific concepts and subconcepts progressively, spiraling previous data to aid in the sense making of new and more challenging information. The integration of a mathematical strand of measurement and calculation, necessary for the interpretation of the scientific data generated by the laboratory experiments, also led students to realize the essential role mathematics plays in conceptual understanding. Embedded, often informal, daily assessment allowed early identification of misconceptions, giving rise in turn to extensions and opportunities to remediation.

A field test of the unit was conducted with three eighth grade classes and one seventh grade class. Prior to, and immediately following, the field test, pretests and posttests of mathematical and scientific concepts and subconcepts, were administered. Of the two science tests administered, one was an informal evaluation which consisted of group definitions and illustrations of density and its subconcepts, as well as volume, mass, and measurement. The second science test consisted of a 10-question essay evaluation which required students to define and explain terms and concepts. A 30-question math test, based upon measurement and the utilization of scientific measurement tools, density calculations, and the interpretation of graphic data, was administered and evaluated.

The data generated by the pretests and posttests were analyzed statistically. Academic growth and understanding, as evidenced in the analysis, were significant. This information supports the efficacy of constructive methods of education coupled with integrated, conceptual, curriculum. The application and generalization of the mathematical and scientific concepts covered in this curriculum show evidence of lateral transfer in laboratory experiences dealing with convection of the Earth’s crust and dispersal of airborne pollutants.

**Dan Sours**

Integrating advanced placement multiple choice and free response tests into an advanced placement calculus course### INTEGRATING ADVANCED PLACEMENT MULTIPLE CHOICE AND FREE RESPONSE TESTS INTO AN ADVANCED PLACEMENT CALCULUS COURSE

By Daniel Mark Sours

Fall 2004

This project used interviews, observations and student work to examine the development of six “traditionally” taught high school calculus students’ knowledge of the derivative in the context of a teaching experiment that integrated Advanced Placement Calculus test problems into the curriculum. The students’ knowledge is defined as understanding the ability to represent the derivative using ratio, limit and function in the following contexts: (1) verbally as a rate of chance, (2) graphically as the slope of a tangent line, (3) symbolically as a difference quotient, and (4) in paradigmatic contexts such as velocity. All students showed increased knowledge of the limit of the difference quotient definition of the derivative that could be attributed to the multiple choice problems. All students showed their strongest understanding of the derivative in a graphical sense in terms of the slope of a tangent line. Those students who took a more active role in the free-response problems showed a greater knowledge of the derivative than those who did not. Among those who took an active role, students who recognized the value of these problems showed even greater knowledge

**Gail Standiford**

Developing facility with algebraic concepts through concrete activities### DEVELOPING FACILITY WITH ALGEBRAIC CONCEPTS THROUGH CONCRETE ACTIVITIES

By Gail Helen Standiford

Summer 2005

The focus of this project was to develop a series of concrete lessons and to evaluate the impact of these materials on student performance in preparing them for their first year algebra course. The project materials consisted of a problem-solving unit taught at the beginning of the school year, several Problems of the Week to assess algebraic thinking, and six focused lessons to develop understanding of variables, expressions and equations. Two pretests and posttests were administered to evaluate the effectiveness of the materials. The data analysis showed these materials were effective in the setting of this project, thus they will be made available to other middle school teachers.

**Elisa Sunflower**

Lessons in place-value for third, fourth and fifth grade students### LESSONS IN PLACE-VALUE FOR THIRD, FOURTH AND FIFTH GRADE STUDENTS

By Elisa Sunflower

Summer 1994

This study was designed to develop and field test a set of lessons in 1s, 10s, and 100s place-value for third, fourth and fifth grade students for use by classroom teachers. Most lessons at the elementary concept level in place-value are part of second and third grade curriculum. Yet research has shown that a high percentage of fourth and fifth grade students do not understand the basic concepts of place-value (Ross, 1986; U.S. Department of Education, 1992).

Five problems with imbedded assessment and a pre-post assessment task were developed and field tested in five classrooms in the spring of 1994. The teachers were chosen for their “constructivist” classroom practices. The field testing was a collaborative action research model with the field test teachers contributing to the development of the lessons.

The teaching model used small group work with minimal teacher intervention followed by presentations to the class by each group. Teachers guided but did not direct the instructions. Included in the appendix are the lessons, a set of field test teacher beliefs, sample teacher dialogue, and the results of a field test teacher questionnaire. Each lesson includes class vignettes and/or samples of individual verbatim. Masters are included for the lessons as needed.

The results of the field test were positive based on the responses of the participating teachers. However, limitations of the study include lack of a control group, homogeneous cultural make-up of the classrooms, high dependence of the lessons on language, the length of the study, and the narrow focus of the place-value topics.

The research team consisted of the author and Sharon H. Ross, Associate Professor of Mathematics. Dr. Ross is preparing a research paper to report the student outcomes in understanding place-value.

**Thomas Vohs**

Assessing the effect of implementing mathematics history lessons with algebra I student### ASSESSING THE EFFECT OF IMPLEMENTING MATHEMATICS HISTORY LESSONS WITH ALGEBRA I STUDENTS

By Thomas M. Vohs

Summer 2005

According to California mathematics standards, the use of history in mathematics classes has been de-emphasized. At the same time, however, research shows that student interest in mathematics is an indicator of student achievement. Internationally, the use of history to enhance mathematics instruction has gained in popularity.

The challenge to have all students successfully comprehend Algebra I has placed an increased value on raising student interest in this subject. A review of literature has been included to show that algebra has evolved through an interesting, interconnected history of people and ideas. Some researchers maintain that the stages of this history correlate with the way students learn. Other educators who have implemented history in the classroom found positive influences on student attitude.

The purpose of this research was to assess the effect that four lessons involving the history of mathematics would have on motivating Algebra I students. Data from this project could be used to support math instruction that integrates math history into the standard curriculum. The tested lessons provide direction or ideas for how to add math history to the curriculum.

The four lessons, which included personal stories and ideas about four famous mathematicians were tested in three high school Algebra I classrooms during the 2004-2005 school year. The pre/post student surveys were given to measure any change in attitudes and to give specific feedback on student feelings about using history in Algebra I. Statistical analyses indicate that the students who were exposed to the history had a more positive opinion of mathematics than those who were not given the history lessons.