Culturally Relevant Homework 1

TITLE

Except where reference is made to the work of others, the work described in this project is my own or was done in collaboration with my Advisor. This project does not include

proprietary or classified information.

________________________________________________

Leah Larkin Selbie

Certificate of Approval:

________________________________________________________________________

Donald R. Livingston, Ed.D. Sharon M. Livingston, Ph.D.Associate Professor and Project Advisor Assistant Professor and Project AdvisorEducation Department Education Department

Culturally Relevant Homework 2

Can Culturally Relevant Homework be an Effective

Learning Strategy for Middle School Math Students

A project submitted

by

Leah Larkin Selbie

to

LaGrange College

in partial fulfillment of

the requirement for the

degree of

SPECIALIST IN EDUCATION

in

Curriculum and Instruction

LaGrange, Georgia

December 9, 2010

Culturally Relevant Homework 3

Abstract

This research project is focused on the topic of homework because homework is regularly used as a learning strategy in American schools despite multiple studies that have yielded mixed results on its effectiveness. The subjects were given culturally relevant homework, where topics were of interest to interested the students and studentsthey had a choice in the assignment, over the course of one math unit to see if this type of homework would be able to increase achievement. Subjects were divided into three subgroups: low level performers, middle level performers, and high level performers. Results were analyzed with anAn ANOVA was analyzed to determine if the groups had significant gains and if some groups had gainswere more significant than others.

Culturally Relevant Homework 4

Table of Contents

Abstract……………………………………………………………………………………3

Table of Contents………………………………………………………………………….4

List of Tables and Figures…………………………………………………………………..

Chapter 1: Introduction…………………………………………………………………….Statement of the Problem………………………………………………………….5Significance of the Problem……………………………………………………….6Theoretical and Conceptual Frameworks…………………………………………7Focus Questions…………………………………………………………………..9Overview of Methodology…………………………………………………….….9Human as Researcher……………………………………………………………11

Chapter 2: Review of the Literature………………………………………………….….12

Chapter 3: Methodology…………………………………………………………………17Research Design…………………………………………………………….……17Setting……………………………………………………………………………19Sample/Subjects/Participants…………………………………………….………19Procedures and Data Collection Methods………………………………….……20Validity and Reliability Measures………………………………………………….Analysis of Data…………………………………………………………………….

Chapter 4: Results…………………………………………………………………………..

Chapter 5: Analysis and Discussion of Results…………………………………………….Analysis……………………………………………………………………………..Discussion…………………………………………………………………………..Implications…………………………………………………………………………Impact on Student Learning………………………………………………………...Recommendations for Future Research…………………………………………….

References…………………………………………………………………………………..

Appendixes…………………………………………………………………………………

Culturally Relevant Homework 5

CHAPTER 1: INTRODUCTION

Statement of the Problem

Homework has always been a topic of concern in American education.

Questions abound over how much homework is acceptable for each grade level, what

type of homework is best, and can homework be an effective education strategy that

helps students learn. As the debate over homework continues, teachers continue doing

what they have always done; they assign homework. According to Baines(2007),

America “leads the world in assigning homework a whopping 140 minutes per week in

mathematics for secondary students” (p. 24). Yet, even with all of this homework

American students are doing, the achievement scores of American students are slightly

above the international average as seen in the 2003 TIMSS study (Mullis, Martin,

Gonzalez & Chrostowski, 2004).

This study will seek to show that culturally relevant homework that is

differentiated for American students can be just as beneficial as traditional methods that

are currently being used in most of America’s classrooms. Baines (2007) additionally

states, “Because most American teachers assign worksheets and exercises from textbooks

for homework, a student’s level of engagement working at home may be less than

optimal” (p.24). Research shows that engaged students are more likely to obtain

knowledge and skills in an assignment and more likely to complete the assignment

(McCoy, 2008, p.14). This study seeks to answer the question if American schools

continue to assign homework, can culturally responsive homework with constructivist

strategies be effective for learners in seventh grade math.

Culturally Relevant Homework 6

Significance of the Problem

Educators in American schools continue to assign homework assuming it is good

for students. Supporters of homework state that homework can yield significant

increases in academic achievement levels of students (Hong & Milgrim, 2000). In

opposition, another study finds that homework may undermine academic achievement

(Baines, 2008). Despite the disparity over the effectiveness of homework, it is still a

widely used educational strategy in American schools.

Homework assignments are shown to be most effective if students are interested

in completing the work and if the homework that is assigned does not stimulate negative

thoughts or emotions (Trautwein, 2009). When a student is assigned homework that does

not interest them or is so time consuming that they have negative feelings, then the

homework is not effective. Therefore, students need to have homework that builds on

their interests. Using differentiated homework assignments that are culturally relevant

can keep students engaged and be a factor in whether the student completes the

homework (McCoy, 2008). Additionally, differentiating assignments allows students to

have a choice in their assignment. This helps students feel an element of control in what

they do and increases motivation (McCoy, 2009). Culturally relevant homework that

engages and differentiates can be used in the classroom and be a very effective tool for

student achievement.

Culturally Relevant Homework 7

Theoretical and Conceptual Frameworks

This homework study will be completed with social constructivist methods.

Social constructivism is “a theoretical base from which teacher education candidates

learn how to be critical educators who can create learning environments in which

learning is both enjoyable and rigorous” (LaGrange College Education Department,

2008, p. 3). Social constructivism relies on the idea that learners construct knowledge

based upon external factors, such as the environment and the society they live in.

Students enter a classroom with their own learning styles, their own set of experiences,

their own ideas and understanding of how things work, their own backgrounds, and their

own failures and successes. All of these attributes translate into how well a student will

be able to learn new information. Scott Inch(1995) is an applied mathematician and

professor of mathematics who discovered the effectiveness of constructivism after many

years of teaching. In his definition:

Constructivism states that children (and adults) make sense of their world by

relating new experiences to those they already understand. They evaluate new

concepts and ideas to see how they fit together with existing information. Often

children encounter “discrepant” concepts; physical objects, ideas and phenomena

that don’t quite make sense. When faced with this seemingly contradictory

information, the child must either interpret what he sees to conform to his present

set of rules, or generate a new set of rules to account for the new information.

(p.111)

Culturally Relevant Homework 8

Using assignments that have cultural relevance to student’s everyday lives is a teaching

method that is centered around social constructivist ideas.

The research completed in this study adheres to all tenets of the Conceptual

Framework of the LaGrange College Education Department [LCED](2008). The first

tenet of the Conceptual Framework is based on enthusiastic engagement in learning. In

this tenet, social constructivism is evident. Using differentiated homework that is

culturally relevant to student’s lives is a form social constructivism. Students will be

“actively involved in the teaching and learning process” (LCED, 2008, p. 3).

The research in this paper will also adhere to the second tenet of the Conceptual

Framework by using exemplary professional teaching practices. The teaching methods

used in the project will link the student’s backgrounds and lives with classroom content

in order for students to make personal connections on a deep and meaningful level

(LCED, 2008, p.5). The goal of the study is for information that students learn in the

classroom to be applied to real life situations outside the classroom (LCED, 2008, p.5).

Finally, this study will include caring and supportive classrooms and learning

communities, which is the third tenet in the Conceptual Framework. This study will be

reliant upon relationships between teachers and students to enhance learning. Because a

diverse population of learners with even more diverse lives will be subjects in the project,

thoughtful and considerate actions will be used in the project.

Culturally Relevant Homework 9

Focus Questions

The purpose of this project is threefold. First, the research will seek to determine

whether culturally relevant homework can be just as beneficial to student as traditional

drill and practice homework. After that determination is made, the project will center on

implementing change within the school. Information will be collected that will determine

the attitudes students and teachers about the changes in homework methods. Also, the

project will focus on the approach used to implement changes in the school and whether

the implementation was successful.

The research and literature involved in this project will be centered on three focus

questions. These questions will guide the research and the methods that are used in this

project. They are as follows:

1. Can student’s achievement increase from homework that is differentiated and

culturally relevant to their lives?

2. How do learners and teachers feel about differentiated and culturally relevant

homework?

3. How was the process of change assessed within the school?

Overview of Methodology

This study will use comparative education to implement a differentiated,

culturally relevant method of assigning math homework to middle school students. The

study will be set in middle school in western Georgia and will specifically focus on

seventh grade math. Subjects of this project will be seventh grade math students and

participants will be seventh grade math teachers. To answer focus question one, students

Culturally Relevant Homework 10

will receive culturally relevant homework that is differentiated throughout a geometry

unit. Students will be divided into three groups based on their 2010 math CRCT scores.

Students who scored below an 800 will be labeled as low performers. Students who

scored between 800 and 825 will be labeled middle level performers. Students who

scored higher than 825 will be labeled as high level performers. A dependent t-test will

be used for each group to determine whether differences between the subgroups exist

based on the pre-test data. The same test will be used to deteremin differences between

the subgroups based on their post-test data. An ANOVA will analyze pre/post test scores

to determine if there was a significant difference in student learning within the subgroups.

An effect size calculator will determine which group showed the most increase.

Quantitative data collection methods will show the attitudes and opinions of the

subjects and the participants involved in the study. These data will be collected through

student surveys and teacher interviews. The purpose of the qualitative data will be to

answer focus question two and determine how well the idea and implantation of

differentiated and culturally relevant homework is received at the school. The survey

will use a four point Likert scale and a chi square test will measure the questions in the

survey to determine if any single question is significant. Cronbach’s Alpha will also be

used to measure whether the answers of all of the subjects and participants are trending in

the same direction for each question.

Another aspect of this project will show whether demographics play a role in the

effectiveness of new methods and the implementation of new methods. This study will

also be completed in another 7th grade classroom at another school. The study will be

exactly the same, but the demographics of the school are different. An independent t test

Culturally Relevant Homework 11

will determine if there is a significant difference in the two schools before the study

begins and at the close of the study. This portion of the study will provide additional

answers to focus questions one and three.

After quantitative and qualitative data are collected and analyzed, the project will

move towards organizational change. A systematic method of implementing new

strategies involving culturally relevant homework strategies that differentiate for all types

of learners will be implemented. This portion of the project will answer focus question

three. Additional qualitative data will be collected and analyzed for this portion of the

study through interviews with administrators within the school. The interviews will be

used to determine the impact and process of organizational change.

Human as Researcher

I obtained my teaching certification at LaGrange College in 2009 where I

received a Master’s degree in the Art of Teaching. I have been teaching in a 7th grade

math classroom for three years and am currently working on my Specialist in Education

degree. In this short amount of time, I have noticed that students question the need for

math outside the classroom and that students question the need for homework. However,

math continues to be a very important subject in school, especially with the increase of

high stakes testing and homework continues to be used in most classrooms as an

instructional tool, even though students place little value on homework. In doing this

study, I believe that students will benefit from homework that is differentiated where they

have a choice in the assignment and that is designed around their interests. The goal is

for students to relate to the assignments so they will be motivated to complete the

Culturally Relevant Homework 12

assignment and inspired to learn more. I think when students see that they can apply

what they learn in school to topics that interest them outside the classroom, then they will

become better students and their performance will increase all around.

Culturally Relevant Homework 13

CHAPTER 2: REVIEW OF THE LITERATURE

Homework is widely used as a learning strategy in schools throughout our

country. Many researchers have studied the impact that homework can have on a child’s

education and the results vary between studies. One study may claim that homework is

beneficial while another study may state that homework does not help a student in any

way (Cooper, Lindsay, Greathouse, & Nye, 1998). Regardless of the effect homework

has on learning, the 2003 Trends in International Math and Science Study (TIMSS)

results show that 91% of 8th grade American students report a medium to high level of

time spent on math homework each week (Mullis et.al, 2003). Other studies that focus

on the topic of homework discuss the beliefs and ideas that surround homework.

According to Xu (2006), parents and teachers alike believe that homework is used to

reinforce classroom learning, to help students improve their study skills and to promote

personal responsibility. Homework is viewed as a valuable education tool by students,

parents and teachers (Trautwein, Nigglie, Schnyder, & Ludtke, 2009). Since homework

seems to be here to stay, the following review will report ON two different homework

approaches, whether cultuarally relevant homework can improve student’s educational

achievement, and how teachers and students feel about the use of it in a 7th grade math

classroom.

Homework as a learning tool

The main reason that teachers assign homework is to improve academic

achievement. Teachers want homework to help students retain the lesson and increase

their understanding of the material covered in class (Cooper, et.al, 1998). If homework is

Culturally Relevant Homework 14

truly beneficial to students, then an increase in achievement should happen after

homework is assigned to students. The TIMSS study shows that a vast majority of 8th

grade math students are receiving homework on a regular basis, but the study also shows

that American students are falling behind leading countries in the area of mathematics

(Baines, 2008). An average American student received more homework per week than

the average Korean student, but the average Karen student scored 82 points higher than

the average American in the TIMSS study (Baines, 2008).

Critics of homework offer insight as to the negative effects of homework. They

state that homework causes too much of a burden to students, and that is creates negative

emotions in parents and students. These critics also point out that there is a scarce

evidence of the positive effects that homework has on students (Trautwein, et.al, 2009).

Opposite views maintain that homework has positive effects on students.

Supporters of homework believe that, “perhaps the most important advantage of

homework is that it can enhance achievement by extending learning beyond the school

day,” (Marzano & Pickering, 2007). A 2007 study found that achievement in

mathematics was better when a teachers assigned homework often, but did not

necessarily use lengthy homework assignments (Trautwein, et.al, 2009). Lengthy

assignments result in boredom with the assignment, which decreases achievement

(Cooper, et.al, 1998). Teachers must keep in mind the age of the student and their ability

level to work with direct instruction when assigning tasks in order for homework to have

a positive impact. An opportunity for additional practice in what a student has already

learned should increase student performance. Because homework allows extra practice

Culturally Relevant Homework 15

for students, the achievement gap between high and low performing students can be

decreased by homework (Trautwein, et.al, 2009).

Traditional homework vs. culturally relevant homework

This study will focus on culturally relevant homework, which is less prevalent

than traditional homework. The most popular type of homework assignment is a drill and

practice assignment. (Trautwein, et.al, 2009). This method, hereinafter referred to as

traditional homework, allows students to repeatedly practice tasks learned in a previous

lesson. This method allows for additional practice, which can result in increased

understanding of the material.

However, traditional homework is the same for all students and does not take

student preferences into account. Homework is beneficial for student’s achievement

when the individuality of each student is taken into account (McCoy &, Tysinger, 2009).

A traditional homework assignment may decrease a student’s engagement level and will

not have a positive impact on a student’s achievement. Often teachers assign worksheets

or problems from a textbook and the level of engagement may not be the best for learning

(Baines, 2008). When students are not actively engaged, they become bored easily and

do not care about the task at hand.

Culturally relevant homework could be a more beneficial type of homework that

will CAN help improve student’s achievement. Culturally relevant homework is

homework that allows practice what they learn, but allows students to have a choice

based on their individual preference and includes material that is relevant to the students.

Glasser’s choice theory states that students want to be in control and have a sense of

Culturally Relevant Homework 16

power when they are learning. It THIS THEORY also states that personal choice will

have a substantial, positive impact on student’s behavior, engagement and performance

(McCoy &, Pool, 2009). One research study, CITED BY MCCOY AND POOL (2009)

of four different classrooms found that student engagement decreases as the number of

choices in an assignment increase. The researcher concluded that students sometimes do

not understand what is expected of them or have no experience with assignment choice,

so teachers must clearly present the expectations to the student (McCoy, Pool, 2009). In

a culturally relevant homework assignment, students are allowed to choose the problem

they want to complete based on their own personal preferences and individual

differences.

Effective learning environments also include connections across subjects and

connections to real world examples (McCoy, Sellers, 2009). Culturally relevant

homework is based on real life topics that teenagers relate to easily. By relating

homework material to real life topics, students can apply what they learn inside the

classroom to situations outside the classroom. Using topics that are meaningful to

teenagers will keep their interest in an assignment, which will, in turn, increase student

involvement.

Another positive effect that culturally relevant homework can have on

achievement comes through increased motivation. Students will benefit most from

homework when they are motivated to spend their time on an assignment and when

completing the assignment does not evoke a negative emotion within the student

(Trautwein, et.al, 2009). Student choice is also shown to increase self-motivation

(McCoy, Pool, 2009).

Culturally Relevant Homework 17

Attitudes about culturally relevant homework

The attitude a student has on ABOUT homework will playS a big role on the

effectiveness of the homework. Research states that as students age, their attitude about

homework plays a big part in their completion of the homework and in their class average

(Xu, 2006). Many students feel that homework will not help them learn and are bored by

the material that is being taught. Culturally relevant homework can help change students

attitudes about homework and can keep students interested in the lesson. It is important

to improve the attitudes students have about homework, so they are more likely to

complete given assignments.

Many students are bored in class because material is not interesting or relevant to

them (McCoy, Punger, 2009). When teachers find a way to incorporate students

interests into assignments, for example by the inclusion of popular culture, then learning

is meaningful and memorable. This enhances a student’s view of the value of classroom

lessons and increases the engagement of the students (McCoy, Sellers, 2009).

The process of change

According to the No Child Left Behind law, students must meet the needs of all

students, particularly those students who are at risk, disadvantaged, or minority (Durden,

2008). In order for school improvement to be successful, the unique needs of a school

need to be determined (Beecher & Sweeny, 2008). In order to be able to assign

homework that is culturally relevant, a school must make sure that its teachers are aware

of the cultural backgrounds and the individual differences of their students. Without this

Culturally Relevant Homework 18

information, teachers will not be able to design assignments that peak student’s interests

and adhere to student’s learning needs.

Raising student achievement is a process that researchers believe requires

comprehensive school reform (Durden, 2008). In the implementation of culturally

relevant homework as the predominant type of homework that is assigned to 7th grade

math students, it must be assessed whether learning is facilitated through the cultural

experiences of the learner. Students create knowledge based on what they already know,

so new knowledge must be relevant to their lives. The process of change towards cultural

relevance must, “ allow learning to be channeled through the cultural reality and

experiences of the learner,” (Durden, 2008). The goal of implementing this change is to

decrease the achievement gap between low performers and high performers.

LITERATURE REVIEW NEEDS TO BE A MINIMUM OF 8-10 PAGES. EXPAND the lit review on change theory – also, the attitudes subsection is somewhat redundant of what was presented in the previous subsections.

Culturally Relevant Homework 19

CHAPTER THREE: METHODOLOGY

Research Design

An action research design has been used in this study. Charles and Mertler

(2002), authors of Introduction to Educational Research, found that action research is

completed by educators at a local school level and is intended to resolve an area of

concern within the school where the research is being completed. The authors also

define evaluation research as research that is used to apply to broad situations and to

analyze the quality of a program already in existence. This research project is a mixture

of action and evaluation research. Part of the research in this problems attempts to find a

solution to the problem of homework. Another part of this research is ato attempt to

make a change to the existing way of doing homework in a math class. With the action

research design, I was able to implement the use of culturally relevant homework, gather

and analyze data on the effectiveness of this type of homework. Through evaluation

research, I was able to implement a program of organizational change within the seventh

grade at my school.

Quantitative data was gathered to see the impact that culturally relevant

homework had on students of mixed ability level. This data was analyzed for low

performers, middle performers and high performers in the classroom. Each learner was

identified as low, middle or high with the use of a cutoff score from their 6th grade CRCT

data. I used a dependent t-test to determine if either of the groups were statistically

different based on their pre-test scores and based on their post-test scores. An ANOVA

was used to determine if there was significance among any of the subgroups and an effect

Culturally Relevant Homework 20

size calculator determined the size of the significance. Surveys were another source of

qualitative data. The surveys were used to measure the student’s attitudes towards

homework. A Chi-Square test and a Cronbach’s Alpha test were used to analyze the

surveys.

Qualitative data was gathered from students, parents, other math teachers and

school administrators in the form of focus groups and interviews. This information gave

insight into the attitudes of students and parents toward the use of culturally relevant

homework. The interviews were used to analyze the impact of organizational change that

was implemented because of this process.

Setting

This research study took place in a 7th grade math classroom in a rural school in

west Georgia. Four grade levels, 6th, 7th, 8th, and 8.5, are taught at this school. Students

range in age from 11-16 years of age and represent varying ability levels. White students

make up 55% of the school, black students 35%. THESE STATS ARE NOT RIGHT

and when do the 2010 report card stats come out?

The research took place in my classroom and was spread over 4 different classes

for a total of 110 students. I chose this as the setting for the research based on my

immediate access to the students. I have access to the students daily and to their school

records, which was used to put the students into sub groups. Permission for this research

study was obtained from the principal at my school, my Superintendent of Schools and

athe Institutional rReview bBoard at LaGrange College.

Culturally Relevant Homework 21

Subjects and Participants

Out of the 989 students in this school, Tthe subjects in this research study were

chosen from a group of 989 students. They were chosen because they werethe 110

students in my four math classes. A total of 110 subjects were used in this study. The

males made up 56% of the subjects while females made up 44% of the subjects. The

ethnicity breakdown of the subjects were 61% white, 29% African American, 3% multi-

racial, 3% American Indian, 2% Hispanic and 2% Asian/Pacific Islander. The ethnic

makeup of students in the research study is very similar to the ethnic makeup of the

school. Each subject in the study is in the seventh grade, but the subjects vary according

to age, socio-economic status, and ability level.

The subjects involved in this study were divided into subgroups according to their

ability level in mathematics. This process was done using 6th grade math CRCT scores

from the previous year. All students with a score of 800 or lower were identified as low

performers. Students with a score between 800 and 825 were identified as middle level

performers. Students who scored greater than 825 were identified as high level

performers. The cut scores used in the study are in line with those used in the state of

Georgia. Students who score less than 800 on the CRCT do not meet grade level

standards. Students who score between 800 and 825 are designated as basic performers.

Students who score higher than 825 are designated as proficient or exceeding the

standards.

Participants of this study included the school principal and assistant principal, 2

additional 7th grade math teachers and a number of parents whose children were subjects

Culturally Relevant Homework 22

in the study. The school officials and teachers were interviewed and the parents were

part of a focus group.

Procedures and Data Collection Methods

The students in the study are assigned at random by the registrar to one of my four

math classes. Pre-tests and post-tests were given to measure learning in the unit that was

being taught during the research study. During the research, students in 3 classes were

given culturally relevant math homework as opposed to traditional drill and practice

homework.

The data collected were analyzed according to subgroups. Each subgroup was

analyzed after the pre-test was administered using a dependent t-test. This was to ensure

that all groups were equal from the beginning of the study. After the post-test was

administered, a t-test was used to determine that each sub group improved. An ANOVA

was then used to determine if one group benefitted from the culturally relevant

homework more than another.

Another measure of quantitative data used in the study came from a survey using

a Likert scale. This survey, which can be seen in Appendix A, measured the student’s

attitudes toward culturally relevant homework. The Likert scale ranged from strongly

agree to strongly disagree. These data were analyzed using a Chi-Square test to

determine if any questions in the survey had significance. A Cronbach’s Alpha test was

also used to determine if the data from the survey were trending in the same direction.

Table 3.1 provides an outline of the data collected and the methods that were used in this

research study.

Culturally Relevant Homework 23

Table 3.1 Data Shell

Focus Question

Literature Sources

How data were gathered and what type of data

How these data are analyzed

Why these data provide valid data

Rationale

Strengths/Weaknesses

Can students benefit from homework that is differentiated and culturally relevant to their lives?

Trautwein, U., Niggli, A., Schnyder, I., & Ludtke, O. (2009);

McCoy, L., & Wake Forest University, D. (2009);

Hong, E., & Milgram, R. (1999).

Method:Assessment – pretest, posttest

Data:interval

Descriptive and inferential statistics – dependent and independent t-tests

Content validity

Determine if there are significant differences

Validity ReliabilityDependabilityBias

Culturally Relevant Homework 24

How do learners and teachers feel about this process?

Ravitch, D., & Cortese, A. (2009);

Durden, T. (2008);

Baines, L. (2008).

Method:SurveyInterview

Data:nominalqualitative

Chi-Square testChronbach’s Alpha

Construct validity

Look for categorical and repeating data

Validity ReliabilityDependabilityBias

How will change be implemented because of the research findings of this study?

Durden, T. (2008);

Baines, L. (2008).

Method: Interview

Data: qualitative

Coded for themes

Construct validity

Look for categorical and repeating data

Validity ReliabilityDependabilityBias

WHERE IS THE DISCUSSION OF THE REST OF YOUR STUDY – INTERVIEWS

AND FOCUS GROUPS – ALSO DIRECTING READERS TO THESE QUESTIONS

FOUND IN THE APPENDIXES.

Culturally Relevant Homework 25

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Culturally Relevant Homework 26

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Trautwein, U., Niggli, A., Schnyder, I., & Ludtke, O. (2009). Between-teacher

differences in homework assignments and the development of students'

homework effort, homework emotions, and achievement. Journal of Educational

Psychology, 101(1), 176-189. Retrieved from ERIC database.

Culturally Relevant Homework 27