Lesson Plans and Web-Based Tutorials for College Factoring Workshops

Lesson Plans and Web-Based Tutorials for College Factoring Workshops

by

JOHN N BRUNNER

December 2010

A Project Submitted

in Partial Fulfillment of

the Requirements for Degree of

MASTER OF SCIENCE

The Graduate Mathematics Program

Curriculum Content Option

Department of Mathematics and Statistics

Texas A&M University-Corpus Christi

APPROVED: ______Date: ______

Dr. George Tintera, Chair

______

Dr. Elaine Young, Member

______

Dr. Sherrye Garrett, Member

Style: APA

Abstract

The process of factoring polynomials can be a difficult skill to master, especially for beginning algebra students at the college level. One of the many basic skills that a developmental mathematics student is exposed to and must comprehend is the process of factoring. This process involves finding an equivalent expression that is a product (Bittinger, Ellenhogen, & Johnson, 2006). Without a strong background in basic mathematical skills, students can easily develop frustration and disillusionment, hindering their potential to be successful college graduates.

A local community college provides support for students enrolled in developmental mathematics courses, including workshops, supplemental instruction, and one-on-one tutoring. Factoring workshops are taught twice a semester, providing support for students enrolled in remedial level mathematics classes. At the present time, no written lesson plans or videos are provided by the mathematics department for facilitators of the workshops.

The purpose of this project was to develop eleven detailed lesson plans on topics presented in the factoring workshops as well as extensive web-based tutorials on factoring polynomials by use of video recordings. These lesson plans will be integrated into future workshops for use by presenters or student tutors. The video tutorials will allow students unlimited access for viewing on the college’s web site if additional instruction is needed.

Dedication

For my inspiration and understanding I give full thanks to Dr. Nadina Hutchings. Before you entered my life, I used to wonder what I might accomplish. You will always be in my heart, my thoughts, and I miss you so very dearly.

Acknowledgements

I would like to thank my committee chair Dr. George Tintera for all his patience, assistance and caring, and Dr. Elaine Young for her professional guidance and the wonderful aspects of mathematics you have taught me so that I can become a better teacher in the future. I would also like to thank Dr. Sherrye Garrett; through your course of 2008 I received inspiration to read again and ponder how a child learns.

I would finally like to thank my family and friends, especially Faydale Curtice for the many hours of assistance you have provided me throughout this project. I am always truly grateful for your patience and caring. To my dear friends that I met while attending graduate school, Nimisha Bhakta, Sonia Carrera and Conchita Marshall, I am truly blessed with such wonderful people.

Table of Contents

Abstract ii

Dedication iii

Acknowledgements iii

Introduction 5

Literature Review 7

Results 11

Summary 15

References 16

Appendix A: Complete Lesson Plans 20

Appendix B: Student Packet 133

Appendix C: Student Packet Answer Key 155

Appendix D: Hyperlinks for Videos 161

Appendix E: Additional Problems and Solutions Available for Students 162

iv

Introduction

Some students enter the higher education environment without the fundamental mathematical proficiency and problem solving skills that are needed to succeed in a college level mathematics course (Hammerman, 2003). Without a strong background in the basic mathematical skills, students do not succeed in higher level courses such as calculus or linear algebra, nor will they be eligible for many high-technology and high-paying fields after graduation (Paglin & Rufolo, 1990). A local community college provides one-on-one tutoring, access to use of computers for online homework assignments, extended operating hours six days a week, and implementation of factoring workshops, which are held twice a semester.

Attending the workshops is a diverse array of the school population, ranging from elementary algebra to college algebra level students. The workshops are presented on Saturdays during the first two weeks of the fall and spring semesters, to coincide with the start of classes. These workshops usually last about five hours, with an hour lunch and intermittent breaks for students and instructors. Workshops are held again during the last two weeks of the semester to assist students with their first exposure to factoring techniques while taking elementary algebra classes.

One area of concern is the lack of structured lesson plans for use by the instructors or fellow tutors. Previous workshops have been somewhat disorganized with a “shoot from the hip” attitude. This conveys to the students a diminished level of professional development, as well as may confuse the remedial level students even further in their quest to understand the factoring process used in mathematics.

The overall purpose of this project was to develop eleven lesson plans covering the topics that are discussed in the factoring workshops. In addition, sensing a need to “fill the gap” for those that are not able to attend the workshops in person due to work schedules, concurrent classes or family matters, the college’s audio visual facilities and the author of this project produced a series of videos mirroring the content of the lesson plans. Similar problems presented at the workshops were worked out, step-by-step, and were shown on video by use of document camera capturing. These tutorials will be available for the students to view on the college’s website for additional instructional support.

This project also developed additional problems, answers, and detailed solutions to five randomly selected problems from each of the ten topics. These problems will be posted on the college’s website, in print form only, for viewing by students, if they so choose.

This project was guided by two main principles:

1) Factoring in mathematics is extremely difficult and requires extended learning opportunities which can be provided by the workshops.

2) On line tutorials can provide additional support to students unable to attend the workshops.

Literature Review

At the local community college, the decentralized developmental mathematics program consists of the following three courses: Arithmetic and Geometry, Elementary Algebra, and Intermediate Algebra. These developmental mathematics courses normally serve multiple purposes. The primary goal is to strengthen student deficiencies in basic mathematical skills which are needed to succeed in other college math classes, as well as courses that require some form of math, such as chemistry, economics or engineering (National Association of Developmental Education, 2002 ). At many community colleges, developmental courses also serve a second purpose of strengthening students’ overall general learning skills before they enroll in regular college courses.

Remedial or developmental education classes, such as the ones offered at the local community college, are defined as coursework below college-level. Developmental education is a topic of much heated debate in higher education (National Council of Postsecondary Research, 2008). Colleges and states devote substantial resources to try and resolve the issue of remediation. In addition, students bear the cost of enrolling in the remedial level courses, with the possibility of lost wages and prolonged school time in order to graduate. In the U.S., almost a quarter of postsecondary students required remedial course work in mathematics in the fall of 2000, with almost 98% of public two-year colleges and 80% of four-year institutions offering at least one remedial mathematics course (National Council of Educational Statistics, 2003).

A national initiative on community college success studied over 46,000 students enrolled in 27 institutions reported over 70% of students were referred to developmental mathematics courses versus only 34% for developmental English (Biswas, 2007). The local community college in this project, from fall 2004 to fall 2008, reported 61% to 89% of first time students tested into a remedial level mathematics course, compared with only 30% to 59% for developmental English and only 27% to 46% for developmental reading courses (Institutional Research and Effectiveness, 2008).

In order for a person to be successful in any mathematics course, one must have “appropriate math knowledge” (Nolting, 2002, p. 42). To determine this mathematics knowledge, community colleges or universities require students to take placement exams in order to determine their current level of mathematics knowledge. These test scores are then used to place students into appropriate courses. Students that were successful in their recommended developmental courses tend to continue their enrollment in college for a greater number of terms (Stutz & McCarroll, 1993). The problem is that students disregard these placement suggestions and enroll in classes that are above the recommended level, or take the placement exam multiple times to avoid taking lower level mathematics courses (Nolting, 2002). This practice can lead to frustration and feelings of being overwhelmed when they enroll in the wrong class. Hollis (2009) recommends academic advisors become an integral part of the process to help developmental level students overcome their learning obstacles while attending college and to guide them in determining what level of classes should be taken.

Nolting (2002) asserts, “Learning math is different from learning many other subjects because it follows a sequential learning pattern” (p. 22). Sequential learning patterns are based on knowledge or skills acquired in certain mathematics classes will be used again in the future to build a stronger mathematical foundation. These sequential learning patterns can be strengthened if students enroll in appropriate classes using placement exams as well as scheduling mathematics courses sequentially and continuously in order to maintain continuity in the upper level educational learning process (Nolting, 2002).

The community college in this project incorporates remedial level assistance in the three following content areas: reading, writing, and mathematics. The student tutoring center is one of several key components in the overall process to assist students who are taking remedial level courses. It provides one-on-one tutoring, access to an adequate number of computers in order for students to work on assignments, extended operating hours six days a week, supplementary handouts, course content mastery (other than mathematics) and most importantly, factoring workshops.

Of the three developmental mathematics courses offered, the topics of factoring are covered in the last two courses. Factoring, in its simplest terms, poses the question, “What was multiplied together to obtain the given result?” (Bittinger, Ellenbogen, & Johnson, 2006, p. 304). In other words, to factor in mathematics is to find an equivalent expression that is a product (Hall, 2010). Terms are a single expression, called a monomial; a two-term expression is called a binomial; a three-term mathematical expression is considered a trinomial, and anything larger than three terms is labeled a polynomial (Bittinger, Ellenbogen, & Johnson, 2006).

However, the process of factoring in elementary and intermediate algebra can be a difficult skill to comprehend and master for students in remedial level classes (Kotsopoulos, 2007). This skill is understood quickly by some students, but others take a longer time to grasp the concept. One reason may be the diversity of learning styles and multiple intelligences that are present in the classroom setting (Gawlik, 2009). To assist students taking remedial level classes that are actively covering the topics of factoring, further allocation of resources should be provided by means of additional practice, extended study time, additional personnel, or a different teaching method (Hoda, 2006).

Computer use and all its encompassing technologies has become a normal part of a person’s daily routines. “Internet technologies make learning environments available without restrictions in time or place.” (Pahl, 2002, p. 1). Online courses and distance modules are growing rapidly in science, mathematics, engineering, and technology disciplines (Hauk, 2006; National Science Foundation, 1998). Technology has provided new ways of bringing instructional delivery to the student. The use of electronic tutorials will allow the student to review missed work and materials (Handal & Herrington, 2003). Each year more web-based tutorials are being used to deliver course work through distance learning, as well as to supplement traditional classroom settings. Evidence comparing the use of web-based tutorials show that academic achievement using web-based tutorials is equal to traditional classroom instruction (Singleton & Fernandez, 2005).

The use of technology, such as the web-based tutorials developed for this project, allowed students to learn the topics of factoring at their own pace, replay a tutorial as many times as needed to understand the concept, meet their own learning styles, and be exposed to a different teaching method. The additional resources available by use of web tutorials assist in engaging the student’s multiple intelligences (Roubides, 2004).

Results

The lesson plans developed for this project were initiated in July 2010 and concluded in August 2010. They include:

1) Tests for divisibility by the natural numbers 2 through 10

2) Greatest common factors

3) Factoring by grouping method

4) Sum of perfect squares

5) Difference of perfect squares

6) Sum of perfect cubes

7) Difference of perfect cubes

8) Monic trinomials: x2+bx+c

9) Non-monic trinomials: ax2+bx+c a≠1, 0

10) Techniques of factoring

11) Solving for specified variable

The lesson plans were developed in this order to maintain organization and create a level of sequential learning patterns (Nolting, 2002). Rules of divisibility will be used in later topics such as monic and non-monic trinomials, thus is presented first in the sequence.

A lesson plan template was chosen from www.docstoc.com, mainly due to simplistic layout and author’s membership. Some alteration of wording was done to fit the college level setting. Each lesson plan consisted of the following subcomponents:

1) Title

2) Student learning objective

3) Standards addressed

4) General goals

5) Required materials

6) Anticipatory set (Lead in)

7) Step-by-step procedures

8) Plan for independent practice

9) Closure (reflect anticipatory set)

10) Examples

For the step-by-step procedures section, problems used in the lesson plans were chosen from the author’s five personally owned textbooks. The numbers of problems chosen for each lesson plan ranged from 4 to 6 with detailed explanations being presented for the benefit of the instructor.

Incorporating from five personally owned textbooks, this author also randomly selected problems of varying degree of difficulty, developed listings of problems, and their respective answers for the purpose of additional support for students at the community college. After consultation, the author and an academic advisor initially agreed to provide ten additional problems from each of the major topics, excluding divisibility rules and techniques of factoring, with detailed step-by-step solutions. Upon implementing this strategy, it became quickly evident that the length of project would become too great. Therefore, the decision was made to curtail the number of detailed problems to only five. It is important to note that the problems worked out from each topic consist of detailed procedures, with each step scrutinized closely to maintain a logical flow for the sake of the remedial level students. This author feels strongly that deliberate concise steps shown on paper or on video will assist in the learning process.