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Running head: EFFECTIVENESS OF ALTERNATIVE INSTRUCTIONAL METHODS IN ALGEBRA

Assessing the Effectiveness of Alternative Instructional Methods versus Traditional Techniques in the Algebra Classroom

J. Michael Dillon

Walden University

Research Seminar III: Quantitative Research in EducationEDUC-8438-003

Dr. Gerald Giraud, Facilitator

8-17-08

Week 11

Alternative Methods 1


Part I

Introduction and Study Design

A stigma surrounds mathematics education in American Schools. According to Ashcraft

(2002), the culture of the United States tends to assume that math is too difficult, requires too

much effort, and is unimportant for the average person. Much of this attitude stems from

previous experiences with mathematics. Marilyn Burns has suggested that two-thirds of

American adults hate and/or fear mathematics (Furner & Berman, 2003). In addition, it has been

estimated that only 7% of Americans have had positive experiences in mathematics from

kindergarten through college (Furner & Duffy, 2002). In the end, many students do not like to

take math courses. They have a difficult time expressing mathematical thought verbally and in

writing, and they are not used to having an active role in learning mathematics (Walmsley,

2003). The expectation by many individuals is that mathematical knowledge can be handed to

them without the need for effort, study, or struggle. This general perspective regarding

mathematics creates a desperate need to gather research about ways to turn the tide of

mathematical difficulties for students. Since mathematics instruction and learning includes a

broad range of issues, the topic of interest addressed in this study will focus primarily on

learning in the Algebra I classroom.

Review of Current Literature

The difficultly with the stigma around mathematics education is that a paradox in the

world exists. At the same time that many students struggle with and give up on mathematics, the

need for mathematical knowledge is becoming increasingly important worldwide as socieity

becomes data and technology oriented (Ashcraft, 2002). Wilkins and Ma (2003) stated, “In the


present quantitatively complex society, a person needs a functional knowledge of mathematical

content to make informed decisions as a citizen and as a worker” (¶5). According to Furner and

Berman (2003), the demands of today’s high-tech, increasingly interconnected world make it

imperative that children are given opportunities to build confidence in their ability to do

mathematics. Therefore, a balance between the difficulties that students face in mathematics and

the needs of a fast-paced, technology-driven world must be found.

Unfortunately, the balance has not yet been struck. Currently, American students are not

performing well in mathematics. This lack of achievement cuts across gender, racial and ethnic

lines, and with respect to other students around the world. Ultimately, a decline in mathematical

achievement can be found among many students. Furner and Berman (2003) suggested that the

math scores of students generally tend to decline from 4th grade through 12th grade, and most

students graduate from high school with an understanding of mathematics that is well below the

minimum expectations suggested by teachers, the National Council for Teachers of Mathematics

(NCTM), and other organizations. This results in an inability for students to successfully apply

mathematics concepts to problems faced in the real world (Franz & Pope, 2005). A majority of

students are not performing well, yet they need the information in order to open doors to many of

the opportunities that exist in the real world.

It is hard to deny that there are difficult issues that must be faced in the world of math

education. Student motivation, achievement, and success are important; and the lack of these

things can be attributed to a variety of things. School districts and teachers have a significant

influence on the math education of the students in their care. In many instances, the school is the

primary purveyor of mathematical knowledge. Teachers and administrators play a critical role in

creating a school environment that fosters a desire to learn mathematics and fuels students’


aspirations to be successful. Every characteristic of the school setting can have an impact on

student learning. School policies, resources (or lack thereof), teacher quality, teacher

expectations, and tracking can affect student performance (Bol & Berry, 2005). If schools and

individual teachers create positive policies and routines, model life-long learning, set high

expectations, provide proper support for students, and locate the necessary materials and

resources for a quality education, then students have a greater chance of achieving. However,

poor instruction, lack of resources, insufficient number of math courses, and misinformation

about math can negatively affect student performance (Furner & Duffy, 2002).

Although external factors greatly influence mathematics education, the most important

factor that must be considered is the student. Learning begins with the individual. Academic

self-concepts, attitudes toward success in mathematics, confidence in one’s ability to learn

mathematics, mathematics anxiety, self-efficacy, and locus of control are all things that affect

student performance, achievement, and goals (Hall & Ponton, 2005). Bol and Berry (2005) also

added that attitudes toward mathematics, readiness, and motivation are student characteristics

that can account for achievement or lack thereof. Affective factors within students in addition to

cognitive abilities can also have a significant impact on student success in math (Ma & Xu,

2004). Ultimately, teachers, parents, and other adult figures must work hard to guide children as

they learn. However, a child must also contribute to the process and take an active role in his or

her own education.

There are a variety of things that contribute to the dynamics of math education.

Individual characteristics, influences from teachers, parents, and communities, previous

experiences, national and global pressures for success, resources and funding, and so forth all

contribute to how and what students learn in the math classroom. The goal of this study is to


focus on a specific instructional setting in order to assess how different teaching techniques can

help students find success in mathematics.

Research Problem

During units of most Algebra I programs associated with simplifying algebraic fractions,

students generally tend to perform below average on assessments and earn lower overall grades

for the grading period. The problem lies in determining which instructional strategies (i.e.

traditional, alternative, technology-based, and so forth) will allow students to maximize

achievement in this type of unit. In addition, many students have negative attitudes associated

with learning mathematics. Therefore, a secondary problem lies in assessing how instructional

strategies influence student motivation and attitude while learning algebraic content

Purpose of Research

The purpose of this study will be to determine if the use of alternative teaching/learning

strategies that supplant (or at least supplement) the traditional classroom routines used during the

unit covering algebraic fractions will produce better results on assessments and with respect to

the overall grades for the unit.

Research Question

The primary research question is: Do students who experience and participate in

alternative teaching/learning strategies for the unit on simplifying algebraic fractions as part of

the Algebra I curriculum earn higher overall grades for the unit as compared to students who

experience and participate in the traditional classroom routine? An additional research question

will include: Does the use of alternative instructional strategies affect students’ attitudes

regarding mathematics?

Hypotheses


The null hypothesis is that there is no significant difference in the overall grades earned

by students who experience alternative instructional methods as compared to those that

experience the traditional classroom routine. The alternative hypothesis is that there is a

significant difference in the overall grades earned by students who experience alternative

instructional methods as compared to those that experience the traditional classroom routine.

Variables

The independent variable is the classroom setting (alternative or traditional) the student is

in. The setting is defined by the instructional strategies employed by the instructor. The

dependent variable is the overall grade of the student.

Methodology

Population Definition

The population being studied includes students in high schools in northeastern Iowa. The

state is divided into AEAs (Area Education Agencies). These organizations are part of the public

education system and are designed to assist school districts in many different supportive roles.

In the designated AEA, there are 24 school districts (each with varying numbers of buildings)

and 19 non-public schools. There are approximately 38,000 K-12 students and 4300 educators

that receive support from the agency. A majority of the schools in the AEA are located in small,

rural communities. The average size of the high school populations in the AEA is about 200.

However, there is one urban district with several large high schools that serves a city of

approximately 60,000 people. The total high school population for this district is about 3500

students.

In general, the student population across the AEA is primarily White. However, there are

several schools district that do have more diverse (as compared to others schools in the AEA)


populations. For example, one small, agricultural-based community has the unique characteristic

of being relatively diverse. A meat processing plant in that town has drawn many families of

Hispanic (particularly Mexican and Guatemalan) descent to the community to work. There are

also several families that have emigrated from Eastern European countries. Thus, the diverse

nature of the community is reflected in that school’s student population. In these districts, about

15% of the student body is of minority descent. Many of these students receive ELL (English

Language Learner) services until they are considered proficient in English (up to a maximum of

three years). In addition to ELL services, a large number of these students attend mainstream

classes.

Another common characteristic among the schools in the AEA deals with students that

have been identified as having special learning needs and/or coming from poor families. There

are a large number of students that have been identified as coming from low-income homes

based on the statistics for those that qualify for free and reduced lunch. Based upon the

scheduling and special-service logistics for the schools, most of the students with special needs

attend mainstream classes and receive special support at other times of the day. Thus, in any

given classroom at any given time, teachers can expect to have diverse students with diverse

learning needs. These needs may include (but are not limited to): ELL (English Language

Learner) support, special education services and IEPs (Individualized Education Plans), TAG

(Talented and Gifted) resources, and so forth.

Since the focus of the study involves improving general algebra proficiency, the Algebra

I course is an ideal setting for conducting the research. Almost every student has to take this

course at some point in his or her mathematics coursework. In addition, the algebra classes are

likely to be composed of diverse groups of learners with diverse learning needs


Sampling Process

Procedures. The sample will consist of 20 Algebra I classes (out of approximately 100

Algebra I classes throughout the AEA). The sample will be drawn using a multi-stage, stratified

random sampling procedure. In order to ensure that the sample will reflect the various types of

school settings in the AEA, three strata will be identified based on population size: 1) small

school; 2) medium-sized school; and 3) large school. Based on the percentages of schools that

fit into each stratum, individual Algebra classes will be randomly selected to be part of the

sample. In order to maintain the integrity of the sample with respect to its reflection of the larger

population, every effort will be made to utilize support services to ensure that every student,

including those with special needs have a clear understanding of the nature, benefits, and risks of

the study. Once the sample has been identified and all necessary consents from administrators,

participating teachers, students, and parents have been obtained, the twenty sample classes will

be divided equally between two groups: the control group will consist of classrooms where the

normal instructional strategies are employed; and the test group will consist of classrooms that

will utilize the alternative instructional strategies. The assignment of classes to each group will

be based on ensuring that the two independent groups are as similar as possible with respect to

student make-up, school type (i.e. small, medium, etc.), teacher characteristics, and so forth.

Rationale. The sampling procedures outlined here are intended to develop a sample that

is representative of the larger population. If the sample is truly representative, then the hope is it

will be possible to generalize the results of the study to the population as a whole. The use of a

stratified random sampling procedure increases the likelihood of generating a random sample. A

sample size of 20 Algebra I classes was determined based on the feasibility of gathering data


from the locations and on creating a sample size that would be large enough to make the

detection of treatment effects possible.

Potential Solutions Based on Current Research

During the course of the review of the literature, a variety of potential solutions were

presented. First, students need to be challenged by the material. They also need to be given

opportunities to make relevant connections to the material. According to Turner and Meyer

(2004), teachers need to find the appropriate balance of challenging materials and activities with

positive, emotional support for the students. Second, real-world applications need to be

incorporated into the curricular content in order to create opportunities for students to make

relevant connections to the material. Third, students need to be given opportunities in the

classroom to explore their attitudes regarding math and need positive support from the teacher in

the process of changing negative attitudes. Fourth, the teacher needs to implement varied and

dynamic teaching strategies that meet the learning needs of the students. In many ways, the

teaching strategies implemented by classroom teachers need to nurture a student’s desire to learn.

Finally, a learning environment needs to be created that encourages communication, writing,

questioning, self-reflection among all students, and a desire to learn.

The solutions uncovered in the review of the literature are also reflected by the NCTM.

In the area of student learning, the NCTM suggested that teachers need to observe and assess

several characteristics of math students. These student characteristics include: their confidence

in using math to solve problems; the abilities to communicate ideas and reason; flexibility in

exploring mathematical ideas; willingness to try a variety of methods when problem-solving;

willingness to keep trying when working on mathematical problems; curiosity and inventiveness

when doing math; the ability to reflect on and self-monitor their own thinking and performance;


their focus on the value of math in its application to real life and other subject areas; and their

appreciation for math in regard to its existence in other cultures and characteristics as a language

(Furner & Berman, 2003). The characteristics suggested by the NCTM provide a framework for

the types of things that teachers should look for in how their students respond and behave in the

math classroom. These characteristics can provide clues into how well a student is truly doing in

the process of learning mathematics.

In the area of teaching, the NCTM also provided suggestions for teachers. Furner and

Berman (2003) stated:

In the NCTM vision, mathematics teachers would encourage students, probe for ideas, and carefully judge the maturity of a student’s thoughts and expressions. By contrast, most current teaching practices in mathematics classrooms do not provide sufficient time for critical thought. (¶3)

Furner and Berman also noted that math teachers need to encourage independent thinking,

promote group work, utilize technology, use manipulatives and other concrete learning tools, and

balance drill and practice problems with real-world problem-solving exercises. Teachers also

need to emphasize and promote equity among students and provide all students with

opportunities to learn mathematics (Furner & Duffy, 2002). Ultimately, the NCTM (2000) stated

that “effective mathematics teaching requires understanding what students know and need to

learn and then challenging and supporting them to learn it well.” (p. 16).

Instrumentation Tools and Materials

The goal of the study is to determine if a difference exists in the grades and overall

attitude between algebra students that experience traditional teaching/learning techniques and

those that receive instruction that capitalize on alternative methods. In this study, traditional

techniques will include: teacher-centered lecture and demonstration, teacher-led board-work and

examples, textbook-based homework assignments, and paper-and-pencil assessments.


Alternative methods will include (but are not limited to): technology-based instruction;

alternative presentation techniques such as PowerPoint slideshows; Internet activities; small

group collaboration and problem-solving; writing and journaling activities; activities based on

multiple intelligence models; and assessments that reflect the nature of the instructional

strategies. These strategies will be used either in place of or in addition to traditional techniques

at the discretion of the participating teachers and researchers.

The primary, non-survey source of data will be the overall grades earned by the students

during the unit on simplifying algebraic fractions. For the control and test groups, overall grades

(in percentage form) from the completed unit will be obtained for the students that comprise the

classes. The use of multiple forms of data also helps to improve the validity of tests results.

Therefore, a standardized test will also be administered to students in all classes in the sample to

assess their understanding of the key concepts addressed in the unit. See the appendix for a

sample of the data collection sheets.

In order to ensure that the data collected is high-quality and will provide accurate results,

the reliability and validity of the data must be established. Trochim (2007) defined reliability as

the level of how consistent a test or other measurement is. There are many different techniques

for establishing the reliability of a measurement. Internal consistency will be used as the

reliability estimate for the overall grades the students earn for the algebra unit. Since the overall

grades are based on several different types of activities and assignments (i.e. tests, homework,

etc.), each of the different activities should yield grades within a similar range. For the

assessment test, alternate forms of the test will be administered to different groups of students to

establish the reliability of the test. The assessment test will have several different types of


questions that address the general concepts of the unit; therefore, a split-half technique may also

be used as secondary estimate of the reliability.

Validity is an estimate of how well a test or measurement provides accurate results that

represent the true nature of the phenomenon in question (Trochim, 2007). The validity of the

overall grades earned by the participants as well as the assessment test will be established by

addressing the convergent validity of the measurements. Convergent validity is the degree to

which a test yields results that are consistent with other measures of the same phenomenon. In

this study, overall grades (which include grades from various activities), the assessment test, and

general observations made by the researcher and teachers will all provide information about the

success of students in the unit. These measures provide distinct ways to assess student progress

and should align if they are indeed valid sources of data.

A survey (see Appendix) will also be utilized in order to gather data about the secondary

research questions. The survey is designed to assess student attitudes regarding mathematics,

perceived strengths and weaknesses in mathematics, and preferences for different instructional

strategies. The survey incorporates a variety of questioning strategies in order to gather both

quantitative and qualitative data. The use of qualitative data can be used to provide additional

meaning to the results of the statistical analyses.

Several methods will be used to establish the reliability of the survey instrument. The

survey incorporates various types of questions to elicit information about student attitudes

regarding mathematics. Therefore, internal consistency will be used to estimate the reliability of

the survey. The survey also incorporates several open-ended questions. Teachers participating

in the study will be asked to interpret the data collected from these questions. By having both


the researcher and the teachers assess the data form the survey, the reliability of the instrument

can also be estimated using an inter-rater strategy.

The validity of the survey will be established by examining the convergent nature of the

data. In order to confirm assessments of the student attitudes regarding mathematics, the results

of the survey will be compared to observations of classroom behaviors. These behaviors might

include lack of participation, disciplinary concerns, low (math) self-esteem, incomplete

assignments, apathy, and so forth. The opposites of these behaviors could also indicate positive

attitudes about mathematics. Due to the fact that different classes have unique characteristics

and that different teachers incorporate various management techniques, general criteria will be

established for assessing classroom behaviors and their link to math attitude.

Data Collection Process

In order to collect the non-survey data, classroom teachers at each classroom site will be

responsible for leading the instruction, implementing assignments, completing the unit, and

calculating and recording the grades for the unit. In order to ensure some level of uniformity

among the different classes (which will be instructed by diverse teachers), a rubric will be

developed in conjunction with the teachers involved in the study for assigning grades based on

student performance during the unit. Classroom teachers will also be responsible for

administering and collecting the standardized tests for assessing content understanding; however,

the researcher will be responsible for grading these tests. At the completion of the unit, the

researcher will collect, compile, and analyze the non-survey data.

The survey will be administered in two phases. The survey will be administered before

instruction for the unit begins. The same survey will also be administered at the conclusion of

the unit. The purpose of the pre-/post-unit survey will be to assess if the use of alternative


instructional techniques has any impact on the student attitudes or motivation regarding

mathematics. In both cases, the classroom teachers will be responsible for distributing the

surveys to all of the students in their classes. In order to ensure a high rate of return, the surveys

will be completed anonymously. If students are absent on the days that surveys are taken, the

classroom teacher will have a small window of time to administer and collect the instruments.

Although a 100% rate of return would be ideal, communication with the classroom teachers will

be essential in the event that some students are unable to complete the survey. As always, ethical

considerations will a play vital role in protecting the participants in the study.

Analysis Plan for Non-survey Data

Type of relationship and test. In order to the test the claim that students who receive

alternative instruction for the algebra unit will earn different grades than those students whose

receive traditional instruction, the grades earned by the students will be analyzed using an

independent-samples t-test. This test will assess the difference in the mean grades earned in the

both the treatment and control groups. Using a significance level of 0.05, the goal will be to

determine if a statistically significant difference is shown. Although the test will be formulated

as two-tailed test, the underlying assumption is that students receiving alternative instruction will

perform at a higher level than those in the control group.

Rationale. The rationale for the choice of an independent-samples t-test is based on

several factors. First, a hypothesis test will serve as the vehicle for addressing claims about

samples that undergo different treatments. Second, it will be possible to calculate the mean and

standard deviation for each sample. However, the mean and standard deviation for the

population will be unknown and an assumption of equal variances will not be made. Therefore,

a t-test will allow the use of the sample mean and standard deviation as an estimate of the


population mean and standard deviation. Third, the following conditions for an independent-

samples t-test are met: 1) standard deviations for the populations are not known (and are not

assumed to be equal); 2) the samples are independent and are simple random samples; and 3) the

samples come from a normally (or very close to normally) distributed population (Triola, 2004).

Finally, the use this type of analysis procedure will provide a strong basis for establishing

conclusions about the effectiveness of various types of instructional methods in the algebra

classroom and can be used to complement the data collected from the survey instrument.

Analysis Plan for Survey Data

The survey instrument utilized in this study is intended to gather data that addresses the

secondary research question—how does the use of alternative teaching/learning techniques affect

the attitudes of students regarding algebra? This survey utilizes different questioning techniques;

therefore, the various questions will need to be analyzed in different ways. First, the open-ended

questions will be utilized to provide additional context for the conclusions reached in the

quantitative analysis. Descriptive statistics will be used to analyze the questions that require

participants to provide a rating for different classroom techniques and problems types. Finally, a

paired-samples t-test will be used to assess the responses of students on the Likert-style questions

regarding their feelings and attitudes about mathematics. The paired-samples approach will be

used since each student will complete the survey prior to and after the instructional period. The

data obtained from the survey instrument will hopefully provide insight regarding the affective

domains of mathematics and the experiences of students during the learning process.

Summary

Learning the content of the algebra curriculum is a critical step in a student’s

mathematics education. However, evidence suggests that large numbers of students struggle in


their math classes. Many teachers see this first hand in their classrooms, but the difficulties that

students face are also well documented in the literature. These struggles are also compounded by

the pressures of an ever-advancing, technology-based society, international competition, and

political mandates that stress success in mathematics. The goal of this research study is to

attempt to shed some light on instructional techniques that educators could potentially use to

enhance learning for their algebra students. A quantitative design will be utilized to compare the

grades of students that learn content regarding algebraic fractions taught with traditional and

alternative methodologies. Ultimately, the results of this study could provide support for

strategies that educators could use to improve learning and to help to change the attitudes that

students have regarding mathematics.


References

Ashcraft, M. H. (2002, October). Math anxiety: Personal, educational, and cognitive consequences. Current Directions in Psychological Science, 11(5), 181-185.

Bol, L., & Berry III, R. Q. (2005). Secondary mathematics teachers’ perceptions of the achievement gap. The High School Journal, 88(4), 32-45.

Franz, D. P., & Pope, M. (2005). Using children’s stories in secondary mathematics. American Secondary Education, 33(2), 20-28.

Furner, J. M., & Berman, B. T. (2003, Spring). Math anxiety: Overcoming a major obstacle tothe improvement of student math performance. Childhood Education, 79(3), 170.

Furner, J. M., & Duffy, M. L. (2002, November). Equity for all students in the new millennium:Disabling math anxiety. Intervention in School & Clinic, 38(2), 67.

Green, S. B., & Salkind, N. J. (2005). Using SPSS for Windows and Macintosh: Analyzing and understanding data (4th ed.). Upper Saddle River, NJ: Pearson Prentice-Hall.

Hall, J. M., & Ponton, M. K. (2005, Spring). Mathematics self-efficacy of college freshman. Journal of Developmental Education, 28(3), 26.

Ma, X., & Xu, J. (2004, May). Determining the causal ordering between attitude toward mathematics and achievement in mathematics. American Journal of Education, 110(3), 256.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author

Triola, M. (2006). Elementary statistics, Student Solutions Manual 10th edition, Mario F. Triola 2003. New York: Prentice Hall.

Trochim, W. M. K., & Donnelly, J. P. (2007). The research methods knowledge base (3rd ed.). Mason, OH: Thomson.

Turner, J. C., & Meyer, D. K. (2004, July/August). A classroom perspective on the principle of moderate challenge in mathematics. The Journal of Educational Research, 97(6), 311.

Walmsley, A. L. E., & Muniz, L. (2003, February). Cooperative learning and its effects in a high school geometry classroom. The Mathematics Teacher, 96(2), 112.

Wilkins, J. L. M., & Ma, X. (2003, September/October). Modeling change in student attitude toward and beliefs about mathematics. The Journal of Educational Research, 97(1), 52.


Appendix

The appendix includes samples for the non-survey data collection instrument and the survey tool.


Unit Grades Determined by Classroom Teachers– Data Sheet –

GROUP:(This will be the assignment to either the control or test group. “A” will the control group and “B” will be the test group.)

CLASS NUMBER:

(This will be a randomly assigned number between 1 and 20 to identify the classes.)

Dates for Instructions:

Date of Pre-Instruction Survey:

Date of Standardized Assessment Test:

Date of Post-Instruction Survey:

Student Number

Grade for Unit(Percentage)

Standardized Assessment Grade (Percentage)

1

2

3456

7

891011121314151617181920

Researcher will complete this section.


2122232425


Thank you for taking the time to complete this survey. Please follow the instructions listed for each question or set of questions. Do not hesitate to ask your teacher if you have any questions or concerns. If you feel uncomfortable answering a question, feel free to skip that question.

Please respond to the following statements by placing an “X” under the one answer that best matches your feelings.

Question Strongly Agree Agree No

Opinion Disagree Strongly Disagree

1) The content we learn in Algebra is interesting to me.

2) I enjoy working on my math assignments.

3) Math is often difficult and frustrating for me.

4) Math would be much easier if I did not have to solve word problems.

5) I understand how to work with variables in most algebra problems.

6) Rate the following types of problems on a scale from 1 to 5. “1” is a type of problem that you do not like to do at all. “2” is a problem that you can do but do not care for. “3” is no opinion about the type of problem. “4” is a problem that you do not mind doing. “5” is a type of problem that you do like.

Rating Type of ProblemProblems with fractions (fractions have only numbers in them)Solving equationsSimplifying using “Order of Operations”Solving word problems (that do NOT use pictures, charts, graphs, etc.)


Solving word problems (that use pictures, charts, graphs, etc.)Evaluating problems (i.e. putting numbers in for the variables and getting an answer)Simplifying problems with variables (the answers still have variables)Problems with fractions AND variables

7) Rate the following types of classroom activities on a scale from 1 to 5. “1” is an activity that you do not like to do at all. “2” is an activity that you do but do not care for. “3” is no opinion about the activity. “4” is an activity that you do not mind doing. “5” is an activity that you do like.

Rating Type of Classroom ActivityWatching PowerPoint slideshowsWorking in small study groupsListening to the teacher go through problems and taking notesUsing the computer to learn new conceptsGoing to the board and working out problems with the whole classWorking on paper and pencil homework assignmentsPlaying review gamesWorking quietly by myself on assignments

Please answer the following questions. Feel free to write as much as you like.

8) What suggestions would you give to your teacher to make math class better?

9) Answer EITHER question a) or question b)…

a) If you like math or find it easy, what do you think are the reasons for why that is the case?

b) If you do NOT like math or think that it is hard, what are the things about it make it difficult?

10) This survey was about your feelings about math. In the space below, feel free to add any additional comments about your experiences with math.


Thank you again for taking the time to complete this survey.


Part II

According to Triola (2006), establishing a correlation involves demonstrating that some

type of relationship exists between two variables. Different types of correlations can exist, but

the simplest type of relationship is one that is linear. In a linear relationship, the variables are

related in a proportional manner. If a linear correlation is established, then either a positive or

negative relationship exists. In the positive case, both variables increase (or decrease) based on

the mathematical relationship. In the negative case, one variable increases as the other decreases

(and vice versa). The Pearson correlation coefficient is used to measure the strength of the linear

correlation and will yield values between negative one and positive one, inclusive (Green &

Salkind, 2005). The following exercises involve the use of SPSS software to analyze data for a

correlation between the teaching and research qualities among college professors.

Correlation (page 261)

The analysis is based on the following background information:

Betsy is interested in relating quality of teaching to quality of research by college professors. She has access to a sample of 50 social science professors who were teaching at the same university for a 10-year period. Over this 10-yeard period, the professors were evaluated on a 5-point scale on quality as instructors and on quality of their courses. Betsy has averaged these ratings to obtain an overall quality rating as an instructor (rating_1) and the overall quality of the course (rating_2) for each professor. In addition, Betsy also has the number of articles that each professor published during this time period (num_probs) and the number of times these articles were cited by other authors (cites) (Green & Salkind, 2005, p. 261).

The first exercise involves conducting a correlational analysis among the four variables

and indentifying the: a) p-value for the correlation between rating_1 and rating_2; b) correlation

between cites and num-pubs; and c) correlation between cites and rating_1. The following

output was obtained using SPSS:


Correlations

1 .638** .232 .312*.000 .105 .028

50 50 50 50.638** 1 .347* .372**.000 .014 .008

50 50 50 50.232 .347* 1 .828**.105 .014 .000

50 50 50 50.312* .372** .828** 1.028 .008 .000

50 50 50 50

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)N

Overall quality as aninstructor

Overall course quality

Number of publications

Number of times citedby other authors

Overallquality as an

instructorOverall course

qualityNumber of

publications

Number oftimes cited byother authors

Correlation is significant at the 0.01 level (2-tailed).**.

Correlation is significant at the 0.05 level (2-tailed).*.

Figure 1. Correlations for quality ratings and publication data.

First, the p-value for the correlation between rating for the instructor (rating_1) and the rating for

the course (rating_2) was 0.000. This p-value gives an indication and a correlation is very

probable. Second, the correlation between the number of citations by other authors (cites) and

the number of publications (num_pubs) was 0.828, which indicates a strong positive correlation.

Finally, the correlation between the quality of the instructor (rating_1) and the number of

citations by other authors (cites) was 0.312. Although this value is significant for an level of

0.05, the final conclusion must be carefully considered since the coefficient is considered to be a

medium value.

The second exercise involves assessing the relationship between the number of articles

published (num_pubs) and the overall quality of the instructor (rating_1). The Pearson

correlation coefficient between the quality of an instructor and the number of articles published

p-value = 0.000 for correlation between rating_1 and rating_2

Correlation = 0.828 between cites and num_pubs

Correlation = 0.312 between rating_1 and cites


was 0.232 with a p-value of 0.028. In this case, there is not evidence of strong linear correlation

between the two variables. 0.232 is near zero, which indicates no correlation, and the p-value is

greater than 0.0083, which is required for significance based on the Bonferroni approach.

Therefore, there does not appear to be a correlation between the quality of an instructor and the

number of articles that he or she publishes.

The third exercise involves writing a “Results” section in APA format. For this data, the

“Results” section would appear as follows: Correlation coefficients were computed among the

variables: instructor quality, course quality, number of publications, and citations of research by

other authors. Using the Bonferroni approach to control for Type 1 error across the six

correlations, a p-value of less than 0.0083 (0.05/6) was required for significance. The results of

the correlational analyses presented in Figure 1 show that five out of the six correlations were

statistically significant and were greater than or equal to 0.312. The correlation between the

quality of the instructor and the quality of the course and the correlation between the number of

publications and the number of citations were greater and were significant for a lower value of .

The only correlation that was not statistically significant was that between the quality of the

instructor and the number of publications. In general, the results suggest that instructors who

have high quality ratings tend to teach courses that also have high ratings for quality. In

addition, instructors who have published more articles tend to be cited more by other authors.

On the other hand, there does not appear to be a relationship between the quality of an instructor

and the number of articles that he or she has published.

The final exercise involves creating a scatter-plot matrix of the relationships between the

four variables. The matrix is illustrated below in Figure 2:


Number of times cited by other authors

Number of publications

Overall course quality

Overall quality as an

instructor

Num

ber o

f tim

es c

ited

by

othe

r aut

hors

Num

ber o

f pu

blic

atio

nsO

vera

ll co

urse

qua

lity

Ove

rall

qual

ity a

s an

in

stru

ctor

Figure 2. Scatter-plot matrix of quality ratings and publication data.

Summary

The procedures used in these exercises demonstrated how to use SPSS to conduct a

correlational analysis among several variables. In addition to the numerical values associated

with the Pearson correlation coefficient, visual displays of the data can also be used to assess the

relationship among different variables. Based on the data, several strong linear relationships

were found among certain variables (i.e. the quality of the instructor is directly related to the

quality of the course). In other cases (i.e. instructor quality and the number of published

articles), the linearity among the variables was not strong. SPSS provides a fast way to generate

the statistical analysis, but it is ultimately important to understand the meaning of the various

tests in order to develop a proper interpretation of the results.


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