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From LtoJ : the effects of an assessment approach on student From LtoJ : the effects of an assessment approach on student

achievement in mathematics at the middle school level achievement in mathematics at the middle school level

Toni Andersen University of Northern Iowa

Copyright ©2004 Toni Andersen

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From LtoJ : the effects of an assessment approach on student achievement in From LtoJ : the effects of an assessment approach on student achievement in mathematics at the middle school level mathematics at the middle school level

Abstract Abstract This study investigated the effectiveness of the assessment approach From LtoJ as applied to the development of procedural skills and conceptual understandings in mathematics at the eighth grade level. The focal question of this study was: Does the literature support the assessment process used in the From LtoJ program? The information was gathered from approximately one hundred students taking pre-algebra in a three-year span. The curriculum database focused on the essential information a student must acquire in the areas of curriculum vocabulary and concepts, as well as their computational ability with fractions. Each week, random items were selected from the curriculum and given to students in the form of a question. Three separate quizzes were given in the areas of vocabulary, concepts, and computation of fractions.

Using the assessment approach From LtoJ, student achievement and success was increased and student failure was reduced in mathematics when compared to previous test results. Students were assessed through weekly quizzes. Analyzing the results enabled the teacher to make immediate adjustments in instruction to insure continuous improvement, both for individual students and for the class sections as a whole. The data collection indicated when the curriculum was presented in a continuous manner through the weekly assessments, students learned to internalize and retain knowledge of essential information. Thus student achievement was impacted in a positive manner using the assessment approach of From LtoJ.

This open access graduate research paper is available at UNI ScholarWorks: https://scholarworks.uni.edu/grp/275

https://scholarworks.uni.edu/grp/275

From LtoJ: THE EFFECTS OF AN ASSESSMENT APPROACH ON STUDENT ACHIEVEMENT IN MA THEMATICS

AT THE MIDDLE SCHOOL LEVEL

A Graduate Literature Review

Submitted to the

Division of Middle Level Education

Department of Curriculum and Instruction

In Partial Fulfillment

Of the Requirements for the Degree

Masters of Arts in Education

UNIVERSITY OF NORTHERN IOWA

By Toni Andersen January, 2004

ABSTRACT

This study investigated the effectiveness of the assessment approach From LtoJ as

applied to the development of procedural skills and conceptual understandings in

mathematics at the eighth grade level. The focal question of this study was: Does the

literature support the assessment process used in the From LtoJ program? The

information was gathered from approximately one hundred students taking pre-algebra at

the eighth grade level in a three-year span. The curriculum database focused on the

essential information a student must acquire in the areas of curriculum vocabulary and

concepts as well as the computational ability of fractions. The curriculum databases were

developed by the teacher using the standards and benchmarks of the coursework. Each

week, random items were selected from the curriculum and given to students in the form

of a verbal and/or written question. Three separate quizzes were given in the areas of

vocabulary, concepts and computation of fractions.

Using the assessment approach, From LtoJ, student achievement and success was

increased and student failure was reduced in mathematics when compared to previous

test results. A collection of essential curricular information was distributed to students at

the beginning of the school year. Students were assessed through weekly quizzes

consisting of randomly selected information from the three separate databases of

assessment questions. Analyzing the results of the individual quizzes and class total

correct of questions enabled the teacher to make immediate adjustments in instruction to

insure continuous improvement in student learning, both for individual students and for

11

the class sections as a whole. The data results were informative in nature for both

students and educators to determine the progression of student learning. The data

collection indicated when the curriculum was presented in a continuous manner through

the weekly assessments, students learned to internalize and retain knowledge of essential

information. Thus student achievement was impacted in a positive manner using the

assessment approach of From LtoJ.

lll

This Graduate Literature Review by: Toni Andersen

Titled: From LtoJ: The Effects of a Assessment Approach on Student Achievement in Mathematics at the Middle School Level

Has been approved as meeting the research requirement for the

Degree of Masters of Arts in Education

I~ I ;) rt/i Date Approved

I ~ I J --o+ Date Approved

I / ,. ),,,_},/

~ Curricu::nd lnstructi

lV

John E. Henning

Donna Schumacher-Douglas

Rick Traw

DEDICATION

This project is dedicated to those who helped me realize my dreams with support, inspiration and the magic of love.

All teaching involves the transmission of knowledge. To my students who have taught me to be curious. You have shared your thoughts and questions. You have shown me that I could learn more about teaching by listening and thinking. It is my ambition that you will be more knowledgeable, more open to life, and more understanding of the world than when you first entered my classroom. It is my hope that you remember our motto: Each person builds his own destiny.

To my husband, Bob, I thank you again and again. You gave me reassurance and understanding when I most needed it. When there just weren't enough hours in the day and you were "Mr. Mom"-Thank you! You are the motivation to do all the positive things I do in life. Thanks for believing in me.

To our children, Jacob and Caitlin, your patience while I worked on my "homework" taught me more about life than any education could have ever taught me. Thanks for being great kids and for making me laugh. To our son, Ryan, thank you for the late-night MSN pep talks. Your quest for higher learning is inspiring. Once again, each of you have taught me that being a parent is the highest calling an individual can obtain in this life.

I'll love you forever, I'll like you for always, As long as I'm living my baby you'll be. - Robert N Munsch

V

ACKNOWLEDGEMENTS

This project became a group effort and I want to thank all whose support has been

invaluable. First, my advisor and professor, Dr. Donna Schumacher-Douglas, who

provided encouragement and humor when perspective was needed as well as a wealth of

information and knowledge. Second, Teresa Coenen, "principal extraordinaire", who

provided valuable insights about the "bigger picture" as I began this project. For her

answers to those "quick questions" I am grateful. To my colleagues who expressed

interest and offered support during this lengthy process, I truly believe we have the best

middle school due to the fact that we have the best middle school staff. Finally, Sandy

Leinen, who as we sat in class that first night and said, "If you're in, so am I." I_ truly

appreciate all you have given me during this adventure called a masters program -

encouragement, support, and most of all, your friendship. Thank you one and all.

Vl

TABLE OF CONTENTS

Page DEDICATION ......................................................................................... V

ACKNOWLEDMENTS ............................................................................. vi LIST OF CHARTS ................................................................................. viii LIST OF TABLES .................................................................................. ix LIST OF APPENDICES ............................................................................. x

CHAPTER 1. INTRODUCTION ................................................................... l Purpose of Study .............................................................................. 4 Research Questions ........................................................................... 4 Importance of Literature Review and of Existing Database .......................... 5 Descriptions ofTerillS .............................................. · ......................... 6

CHAPTER 2. METHODOLOGY AND PROCEDURES .................................. : ... 7 Method of Identifying And Locating Sources ......................................... . 7 Sources of Student Achievement Information .......................................... 8

CHAPTER 3. REVIEW OF LITERATURE AND EXISTING DATABASE ............ 11 Middle School Philosophy ............................................................... .. 11 Assessment at the Middle Level ............................................ ." ............. 13 Mathematics And Achievement At The Middle Level. .............................. 15 From Total Quality Management To DataNotGuesswork To From LtoJ ......... . 17 Review OfDatabase .... ... ................................................................ 21

CHAPTER 4. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .......... .48 Summary Of Study ..................................................................... 48 Implications of Data Results ............................................................. 51 Recommendations ......................................................................... 54 Conclusion .................................................................................. 55

REFERENCES ....................................................................................... 57

Vll

LIST OF CHARTS

Page Chart 3 .1: 2000 Essential Vocabulary Class Scatter Matrix ................................. 26

Chart 3.2: 2000 Essential Vocabulary Class Run Chart ....................................... 27

Chart 3.8: 2001 Essential Vocabulary Class Scatter Matrix .................................. 31

Chart 3.9: 2001 Essential Vocabulary Class Run Chart ...................................... .32

Chart 3.15: 2001 Computation of Fractions Class Scatter Matrix ........................... 33

Chart 3.16: 2001 Computation of Fractions Class Run Chart ................................ 34

Chart 3.22: 2001 Essential Concepts Class Scatter Matrix .................................... 35

Chart 3.23: 2001 Essential Concepts Class Run Chart .................................. : ..... .36

Chart 3.29: 2002 Essential Vocabulary Class Scatter Matrix ................................ .40

Chart 3.30: 2002 Essential Vocabulary Class Run Chart .................................. 41

Chart 3.36: 2002 Computation of Fractions Class Scatter Matrix ........... _. .............. .42

Chart 3.37: 2002 Computation of Fractions Class Run Chart ............................ .43

Chart 3.43: 2002 Essential Concepts Class Scatter Matrix ................................... .44

Chart 3.44: 2002 Essential Concepts Class Run Chart ......................................... .45

Vlll

LIST OF TABLES

Page

TABLE 1. ITBS Math Total-Grade 8 ................................................ 22

TABLE 2. ITED Math Total-Grade 9 ................................................. 22

TABLE 3. Local Achievement Data 02-03 ............................................. 23

TABLE 4. ICAM Mathematics 02-03 ................................................... 23

TABLE 5. ICAM Mathematics 01-02 ............................................... .. .. 23

TABLE 6. Math Achievement. ............................. · ............................. 24

lX

LIST OF APPENDICES

Page Appendix A Letter of Permission .............................................................. 60

Appendix B Turning Points: Preparing American Youth for the 21 st Century Recommendations for Transforming Middle Grade Schools ............... 61

Appendix C "From LtoJ Concepts" ........................................................... 63

Appendix D Essential Information - Essential Vocabulary ................................. 65

Appendix E Essential Information- Computation of Fractions ........................... 69

Appendix F Essential Information - Essential Concepts .................................... 70

Appendix G Chart 3.3: 2000 Essential Vocabulary Student Run Chart 1st Period ...... 71

Chart 3.4: 2000 Essential Vocabulary Student Run Chart 2nd Period ...... 72

Chart 3.5: 2000 Essential Vocabulary Student Run Chart 3rd Period ...... 73

Chart 3.6: 2000 Essential Vocabulary Student Run Chart 4th Period ...... 74

Chart 3.7: 2000 Essential Vocabulary Student Run Chart 5th Period ...... 75

Appendix H Chart 3.10: 2001 Essential Vocabulary Student Run Chart 1st Period .... 76

Chart 3.11: 2001 Essential Vocabulary Student Run Chart 2nd Period ... 77

Chart 3.12: 2001 Essential Vocabulary Student Run Chart 3rd Period ... 78

Chart 3.13: 2001 Essential Vocabulary Student Run Chart 4th Period ..... 79

Chart 3.14: 2001 Essential Vocabulary Student Run Chart 5th Period ..... 80

Appendix I Chart 3.17: 2001 Computation-Fractions Student Run Chart 1st Period ... 81

Chart 3.18 2001 Computation-Fractions Student Run Chart 2nd Period .. 82

Chart 3.19: 2001 Computation-Fractions Student Run Chart 3rd Period .. 83

Chart 3.20: 2001 Computation-Fractions Student Run Chart 4th Period .. 84

Chart 3.21: 200 l Computation-Fractions Student Run Chart 5th Period .. 85

Appendix J Chart 3.24: 2001 Essential Concepts Student Run Chart 1st Period ...... 86

Chart 3.25: 2001 Essential Concepts Student Run Chart 2nd Period ..... 87

Chart 3.26: 2001 Essential Concepts Student Run Chart 3rd Period ...... 88

Chart 3.27: 2001 Essential Concepts Student Run Chart 4th Period ...... 89

Chart 3.28: 2001 Essential Concepts Student Run Chart 5th Period ....... 90

Appendix K Chart 3.31: 2002 Essential Vocabulary Student Run Chart 1st Period ... 91

X

Chart 3.32: 2002 Essential Vocabulary Student Run Chart 2nd Period ..... 92

Chart 3.33: 2002 Essential Vocabulary Student Run Chart 3rd Period ..... 93

Chart 3.34: 2002 Essential Vocabulary Student Run Chart 4th Period .... 94

Chart 3.35: 2002 Essential Vocabulary Student Run Chart 5th Period .... 95

Appendix L Chart 3.38: 2002 Computation-Fractions Student Run Chart 1st Period .. 96

Chart 3.39: 2002 Computation-Fractions Student Run Chart 2nd Period . 97

Chart 3.40: 2002 Computation-Fractions Student Run Chart 3rd Period .. 98

Chart 3.41: 2002 Computation-Fractions Student Run Chart 4th Period .. 99

Chart 3.42: 2002 Computation-Fractions Student Run Chart 5th Period 100

Appendix M Chart 3.45: 2002 Essential Concepts Student Run Chart 1st Period ..... 101

Chart 3.46: 2002 Essential Concepts Student Run Chart 2nd Period .... 102

Chart 3.47: 2002 Essential Concepts Student Run Chart 3rd Period ..... 103

Chart 3.48: 2002 Essential Concepts Student Run Chart 4th Period ..... 104

Chart 3.49: 2002 Essential Concepts Student Run chart 5th Period ..... 105

Appendix N Heartland Middle School Pre-Algebra Standards and Benchmarks ..... 106

Xl

From LtoJ: The Effects of an Assessment Approach on Student Achievement in Mathematics at the Middle School Level

CHAPTER 1

INTRODUCTION

Mathematics instruction has been particularly challenging in recent years. The

National Council of Teachers of Mathematics (NCTM) states the following:

Evidence from a variety of sources makes it clear that many students are not

learning the mathematics they need or are expected to learn. The reasons for the

deficiency are many: in some instances, students have not had the opportunity to

learn important mathematics. In other instances, the curriculum offered to

students does not engage them. Sometimes students lack a commitment to

learning. (NCTM, 2000, pg. 4)

In current educational practices, teachers use assessment methods to obtain results

data when analyzing student learning and skill level. When using results data, the

information does not clearly illustrate where the students are in regards to the learning of

essential knowledge. It is often collected too late to adjust instructional strategies to meet

the needs of the students. Deming (1993) outlined the changes necessary in education for

teachers to have ''up-to-the-minute" data to clearly show where students are in

comparison to the curricula being taught in the classroom. Though Deming's (1993)

Total Quality Management (TQM) is a business model, it is often applied in educational

settings because it is student-centered as well as data-driven. The collection of process

data gives the teacher more responsibility for student learning as well as the flexibility to

alter instruction to meet the learning needs of students. The collected data can provide a

wealth of information regarding student achievement and teacher instruction. Without

theory and data, the teacher and students move through the school year with no

improvement in the classroom (Jenkins, 1997).

The management theory, Total Quality Management, (Deming, 1993) offered the

following aim for education: increase the positives and decrease the negatives so that all

students keep their yearning for learning. From this basic premise, the framework for

DataNotGuesswork (Jenkins, 1997) was developed and applied to the educational setting

and Leddick and Jenkins (2001) further refined the concept of quality in the classroom ..

DataNotGuesswork (DNG) emphasizes the attainment of curriculum benchmarks by

analyzing student data (Jenkins, personal communication, September, 2001) Jenkins

(2003) continued to focus on the improvement of student learning through the assessment

approach, From LtoJ. "From LtoJ is used to measure learning of information, student

performance, student enthusiasm, and classroom performance. "(Jenkins, 2003, p. 14)

From LtoJ utilizes graphs to display information regarding student achievement. In

gathering and analyzing data, the histograms consider the "L" and the "r' curves of the

graphs. At the beginning of the school year, when students are provided the essential

information to be learned and the performance standards to be demonstrated, the curve of

the students' assessments should be in the shape of an "L" where students have much to

learn about the curriculum. Many students do not know the information thus there are

many who receive zeros on the assessments. The "J" curve represents the end of the year

progress. Many students have met the year's expectations of learning the curriculum thus

there are many who receive high marks on the assessments.

2

Using the From LtoJ assessment approach, the goal was to increase student

achievement and reduce student failure in middle school mathematics. Databases of

essential concepts, vocabulary and fractional operations were written by the teacher for

grade level eight based on the curricular standards and benchmarks following the

conceptual guidelines provided by Jenkins who developed the assessment theory of From

LtoJ. Database questions were deemed appropriate if they matched a particular

benchmark ofrequired knowledge for the students by the end of the eighth grade year.

Student learning was assessed weekly through separate quizzes consisting of ten

vocabulary items, five concept problems, and five fraction problems randomly selected

from the each of the three databases. The teacher entered student results into a software

program and generated graphs that displayed class progress. Analyzing the results data

enables the teacher to make immediate adjustments in instruction to insure continuous

improvement in learning.

In 2000, the learning process, From LtoJ was implemented in the eighth grade

mathematics classroom for Essential Vocabulary. The scatter matrix shows the class

period together as a single unit. With the use of a transparent overlay of the student run

charts for each class period, comparisons can be made from class to class. In 2001 - 2002,

the.implementation of the From LtoJwas expanded to include the Computation of

Fractions and Essential Concepts as well as Essential Vocabulary. Students can take more

responsibility when given the opportunity to improve their own learning. When it is

tangible, the students know what they are to be learning, and they know at any point how

they are doing. In the eighth grade mathematics classroom, large graphs posted in the

3

classroom display class progress in mastering the essential information. In addition, each

student keeps an individual graph to record personal progress towards mastery.

It should be noted that the students in all three years employed various strategies

to increase their retention of the essential information as the school year progressed.

Intentional memorization begins with not just having students memorize words or

processes, but with the instructional process itself. Vygotsky (1978) calls this type of

memory deliberate memory, which is a higher mental process. It refers to the use of

memory strategies and processes that are employed to retain and internalize knowledge.

Many students employed the use of flash cards as a study aid to review words or skills

already taught and to preview information yet to come, while other students worked on

practice problems.

Purpose of Study

This review of literature and of an established database was designed to provide

information on the use of the assessment approach, From LtoJ, in the middle level

mathematics classroom. The purpose of the study was to examine the effects at the

student level of participating in the From LtoJ program. This study explored how the data

gathering and analysis process of From LtoJ can positively effect student achievement in

the pre-algebra classroom.

Research Questions

The focal question of this study is: Does the literature support the assessment process

used in the From LtoJ program? Other questions flowed from this primary question:

• How does the From LtoJ process prepare students to construct their own

mathematics knowledge?

4

• What are the students' perceptions of learning using From LtoJ in the

mathematics classroom?

• How does From LtoJ meet the unique developmental needs of middle school

students?

Importance of Review of Literature and of an Existing Database

The quality of the educational system in the United States has been closely

scrutinized by the public during the past decade. (NCTM, 2000, pg.20) With this scrutiny

comes the need for school systems to implement assessments and accurately measure

student achievement that will provide timely feedback to the students and to the general

public. Unless a school employs assessments that are specific to the curriculum actually

taught, it cannot accurately determine how well its students are learning (Marzano, 2003).

If one of the main purposes of assessment is to inform teachers about their

teaching, it can then be stated that teachers view assessment data mainly as an indication

of the effectiveness of their teaching. Teaching mathematics in the world of high-stakes

assessments requires results data to clearly inform all stakeholders that students are

receiving a quality education. Teachers need to be clear about the reasons for using

assessment techniques and be able to convey that message to students and parents alike

(Wilkins, 2002).

This study can serve as a model for mathematics te~chers and othet ~d\icators

interested in using the assessment approach From LtoJ to enhance stud~nt lean\it\g and

achievement.

5

Descriptions of Terms

In order for readers to have a common understanding of the terms used in this

paper, the following terms are defined:

Assessment: A set of strategies for discovering what students know or can do.

Evaluation: A value assigned to student performances and a judgment about the quality

of a student's performance or product ofleaming (Educators in Connecticut's

Pomperaug Regional School District 15, 1996). The typical end products of

evaluation are grades.

Process Data: Data collected continuously over time. (Jenkins, 1993)

Quality in Learning: Integration of philosophy, theories, processes and tools of quality

and learning to identify, analyze and remove barriers to learning (Bergman,

2002)

Quality: Meeting and exceeding existing and future student wants and needs. (Bergman,

2002)

Results Data: Data collected at the end of a learning cycle. (Jenkins, 1993)

Test: The presentation of a standard set of questions to be answered. Used to

determine what a person knows in various formats such as multiple-choice,

true/false, or essay examinations.

Total Quality Management (TQM): A management philosophy, supported by a set

of processes and tools, aimed at achieving a culture of continuous improvement,

driven by all the people within an organization. (Garfield, 1992)

6

CHAPTER2

METHODOLOGY AND PROCEDURES

Education reform calls for improvement in student achievement through varied

assessment (Jackson, 2000; NCTM, 2000; United States Department of Education, 2001).

The success of a mathematics program depends on the extent to which it accomplishes its

goals and objectives (NCTM, 2000). A classroom assessment program must give the

teacher information about each student's status and progress toward goals to determine if

learning has taken place (NCTM, 2000; Stiggens, 1988). All assessment should have as

its primary purpose the improvement of student learning (Jenkins, 1993). A variety of

procedures are used to assess student progress in today's classrooms. One such

assessment approach is From LtoJ.

Method of Identifying and Locating Literary Sources

According to Gall, Borg, and Gall (1996), there are four types of knowledge that

research contributes to: description, prediction, improvement, and explanation. This study

placed emphasis in the area of description due to the fact that there is little known about

the educational outcomes when From LtoJ is implemented.

Multiple sources of information were used in order to ensure validity as well as

increase the reliability of the data. The process of the literature review included searching

and reviewing a wide range of information resources. After an initial online search of the

resources located in the Educational Resources Information Center (ERIC) databases,

7

visits to the University of Nebraska at Omaha's University Library were utilized to

collect copies of original articles.

The literature search yielded sources of varying relevance and quality. The

following criteria to include research sources was considered:

1. relevance and/or historical value to topic,

2. reputation of author(s),

3. reputation of publishing house, and

4. date of publication.

(personal communication, D. H. Schumacher-Douglas, September, 2003)

In order to retrieve accurate information of the topic at hand, keywords were

utilized to provide a focus for the search. Keywords and terms used included middle

level, achievement, assessment, W. Edwards Deming, Total Quality Management, Data

Not Guesswork, From LtoJ, and other less successful terms. Different levels of

examination were employed to determine relevancy. A superficial scan of abstracts and

recordings of different searches provided a basis for the collection of original works.

Critical reading of the collected literature provided an overview of the research, identified

gaps or limitations in the research and fostered future research ideas. Information was

also gathered through workshop attendance on the topic of the assessment approach,

From LtoJ as well as personal conversations regarding the implementation of From LtoJ

and student achievement in the mathematics classroom.

Sources of Student Achievement Information

The study examined the proficiency rates of the Heartland Middle School (HMS)

District Math Assessment, the Iowa Test of Basic Skills (ITBS), the Iowa Test of

8

Educational Development (ITEDs) and the Iowa Comprehensive Assessment of

Mathematics (ICAMs). Proficiency is defined as the combined percentage of students in

the intermediate and high achievement levels on the ITBS, ITEDs, and ICAMs. This data

will be used to explain and clarify student achievement as related to the assessment

approach of From LtoJ.

Student achievement information was analyzed using an established database of

essential information. This essential information was gathered from the curriculum of the

eighth grade mathematics- classroom. There were specific content items required to be

retained in the students' knowledge base by the completion of the eighth grade year.

Utilization of the HMS eighth grade mathematics standards and benchmarks as well as

the text provided the basis for the ·questions asked of students in the assessment process.

These content items were formulated into test questions and used in weekly

quizzes to assess student comprehension. The assessments were given once a week to the

mathematics students. The essential vocabulary (Top Vocab) quiz is given first. Students

are asked a random selection of ten vocabulary terms. The teacher provided the students

with the definition and the students wrote down the correct term. The computation of

fractions quiz was given next with students calculating answers to five problems seen

individually on the overhead. Students were then asked to solve five questions relating to

the essential concepts of the mathematics course. All quizzes utilized the random

selection of assessment items. This provided a comprehensive approach to the questions

asked of students. These assessment results were tabulated using the individual and class

scores from the weekly quizzes. Student achievement was assessed using individual and

class graphs to track progress. Individual scores are based on a nmneric score; the

9

number correct out of the possible correct. The class graphs are based on a percentile

score to allow for absences or students transferring out of the school district.

The assessment results for this study were gathered at Heartland Middle School

which, in 2003, was comprised of approximately four hundred twenty middle school

students in grade levels six through eight. The student population was 51 % male and 48%

female. In eighth grade, the student population was 58% male and 42% female. The

percentage of students receiving free-or-reduced lunches was 3 8% of the student

population. The students of Heartland Middle School were predominately Caucasian with

minority students making up less than 2% of the population. Minority students at

Heartland Middle School were representative of the Asian and Hispanic ethnic groups.

This population make up was also representative of the community at large.

This was a rural school district with total enrollment of approximately 1,550

students with approximately 82.9% of the graduating class of2002 pursuing higher

education. The number of dropout students was less than 0.5% for grades 11-12. The

rural community was dependent on agriculture for its economic stability with light

industry also contributing to the economy.

10

CHAPTER3

REVIEW OF LITERATURE AND OF EXISTING DATABASE

Education cannot be limited to the confines of a textbook used in a particular

grade level. The learning from one year must be carried over and applied in subsequent

years. (Jenkins, personal communication, 2001). With this premise, a review of literature

pertaining to the issues of middle school philosophy, assessment and achievement in

mathematics, and the assessment approach From LtoJ will follow. First a review of

middle school will be presented. This will be followed by a discussion of assessment and

achievement in mathematics: Finally, based on the mathematics assessment data gathered

at HMS, the development of the process of the assessment approach of From LtoJwill be

explored.

Middle School Philosophy

The middle school concept has been part of American education since the early

1960s. It is unique in that it responds to the academic needs of young adolescents, but in

a developmental framework. It is the blending of academics, pedagogy, and develop

mental needs of young adolescents.

A 1989 task force established by the Carnegie Council on Adolescent

Development (CCAD) presented a report on its findings regarding middle grades

research and effective educational practices (see Appendix B). The task force's report,

entitled Turning Points: Preparing American Youth for the 21 st Century, established eight

essential principles for improving middle level education:

11

• Large middle schools should be divided into smaller learning communities.

• Middle schools should teach a core of common knowledge to all students.

• Middle schools should be organized to ensure success for all students.

• Educators and principal are to have the major responsibility and power to

make decisions about adolescent schooling.

• Middle schools are to be staffed by teachers who are experts at teaching

young adolescents.

• Schools should promote good health; education and health are linked.

• Families should be allied with school staff through mutual respect, trust, and

communication.

• Schools and communities need to be partners in educating young adolescents.

Exemplary middle schools develop programs and practices around the needs of

young adolescents (CCAD, 1989; NCTM, 1997; National Middle School Association

[NMSA], 1995). The primary agenda of middle school education is to promote young

adolescents' intellectual development (Jackson, 2000; NCTM, 2000). The middle grades

are a powerful place where adolescents can develop a mature sense of self-esteem, a

constructive method for expressing themselves regarding their interests, and an avenue to

direct their endless energy (CCAD, 1989). An academically rigorous learning

environment addresses these critical developmental stages. Middle school education is

organized around teams of teachers from diversified subject areas providing instruction

on a common set of subject-matter. The inter-disciplinary approach is emphasized,

encouragtQg students to draw broader connections, think analytically rather than merely

memorize, and engage their persdnal creativity as a resource in the educational process

12

(CCAD, 1989). Research indicates a positive correlation between student achievement

and the degree to which the middle school model is implemented: the more complete the

implementation, the better the result. (CCAD, 1989; Jackson & Davis, 2000).

Assessment at the Middle Level

The National Council of Teachers of Mathematics (1987) states that public

expectations for middle school mathematics are as follows:

• Set up problems with the appropriate operations

• Approach and work on problems using a variety of techniques

• Understand the underlying mathematical features of a problem

• Work with others to solve problems

• See the applicability of mathematical ideas to common and complex problems

• Solve open problem situations typical of most real problems formulated

• See and value the beauty and utility of mathematics.

Middle schools across the United States have implemented aggressive initiatives

to increase expectations for students. Many different kinds of assessments should be used

to determine a student's understanding and progress (NCTM, 2000). The purpose of

assessment should be to improve learning and teaching. It is gathering information about

student learning that can be used to provide feedback to students about their learning as

well as feedback to the instructor about students' achievement of course goals (Angelo &

Cross, 1993).

Assessment involves the processes of gathering data to make educational

decisions about students, giving feedback to students about their progress, and judging

13

instructional effectiveness (American Federation of Teaching, National Education

Association [AFT-NEA], 1990).

As high-stakes, state-mandated performance assessments become a matter of

public record, educators are more accountable for student achievement and the use of

"best practices" in the mathematics classroom is mandatory (Vogler, 2002). Assessment

has been discussed by Warren and Nisbet (2001) stating:

In the area of informing the teacher, assessment was most often used to help plan

the next phase of instruction and to evaluate one's own effectiveness in teaching.

In the area of informing the learner, assessment was often used to provide a

positive experience for students regarding their strengths not weaknesses (pg.

350).

In discussing assessment, Jenkins (2003) states:

Ideally a measurement system has two components: One begins at the state level,

moves to the district and then to the school level. The other system begins with

the classroom, combines classrooms to create school data, and then schools are

combined for district data. The first is results data; the second is data on the

learning process. Both are necessary to adequately monitor learning for all

students (pg. 31 ).

Kohn ( 1999) describes the assessment of learning in much the same manner:

Learning doesn't take place at a district or state level; it takes place in a

classroom. Therefore, the assessment should be focused on students' learning

over time by the person in the best position to judge the quality of that learning.

14

There's an inherent problem with any one-shot test that's designed and then

scored by somebody far away.

The current reporting practices of standardized testing limit the use of alternative

classroom assessment practices (Jenkins, personal communication, 2003). Methods of

reporting students' achievement in mathematics need to change so that various types of

assessments can be incorporated in the reporting methods used to communicate with

students and parents (Schmidt & Brosnan, 1996, pg.17). The uses of traditional methods

of assessments such as tests, quizzes, homework, are easy to quantify and calculate.

However, "educators and school districts need to review grading and reporting policies to

best convey what is valued in mathematics (attainment of concepts, knowledge and skill,

problem solving, and a positive attitude toward mathematics) to the parents and students"

(Schmidt & Brosnan, 1996, pg.19). If knowledge is viewed as more than a collection of

facts and ideas implemented in rote procedures, active individual internalization of

knowledge can be thought of as a goal of education (Albert, 2000, pg. 25).

Mathematics and Achievement at the Middle Level

The research literature on student achievement reveals that students learn in

different ways (Tomlinson, 1995). Mathematics achievement among eighth graders was

influenced by motivation, attitude, and academic engagement. Singh and Granville

(2002) found "school-related motivation and attitude can be affected by more positive

school experiences and better instructional approaches. Educators must work to

strengthen positive attitudes and modify negative attitudes by promoting better classroom

practices and providing positive experiences in mathematics" (pg. 330).

15

The Professional Standards for Teaching Mathematics (1991) stated a need for the

development of instructional practices that would support mathematics teachers to

become more proficient in:

• Selecting mathematical tasks to engage students interest and intellect

• Providing opportunities to deepen their understanding of the mathematics

being studied and its applications

• Orchestrating classroom discourse in ways that promote the investigations and

growth of mathematical ideas

• Using and helping student use, technology and other tools to pursue

mathematical investigation

• Seeking and, helping students seek, connections to previous and developing

knowledge

• Guiding individual, small-group and whole class work (NCTM, 1991).

These practices represented a considerable change from the customary

mathematics classroom where the teacher typically answered homework questions,

provided a brief explanation of new materials, and then assigned the homework with

students spending the remaining time doing their homework. (NCTM, 1991). The NCTM

Standards ( 1995) call for the use of many new teaching practices ( such as orchestrating

group work, facilitating classroom discussions, nurturing reasoning and conjecturing, and

employing multiple forms of assessment) most of which are not currently used by many

math teachers. Educators "need to reject the notion that knowledge is a collection of facts

and ideas implemented in rote procedures with students functioning as passive learners"

(Albert, 2002, pg. 147).

16

The language of mathematics can sometimes be confusing for students to

comprehend and apply appropriately. Many mathematical vocabulary words are unique

to the subject area, some terms have different meanings in everyday usage than in

technical mathematics, and some are used in mathematics in more than one way.

However, terms, phrases, and symbols are essential when communicating mathematical

ideas; and becoming fluent with them is vital for children's mathematical learning

(Rubenstein & Thompson, 2002). The goal is for mathematical language to become fluid,

natural, and "a powerful tool. .. to foster the learning of mathematics" (NCTM, 2000).

It is widely assumed that an improvement in self-concept leads to an improvement

in academic achievement. Research has examined the relationship between academic

achievement and self-concept. According to Skaalvik and Valas (1992):

The perspective holds that the achievement--self-concept relation changes in early

adolescence (at the end of elementary and the beginning of middle school). As

self-concept becomes better established and more stable, it may increasingly

affect motivation and study behavior, which in turn may affect academic

achievement. Performance is affected through student motivation and study

behavior. Motivation is affected directly by achievements (pg. 29).

Educators should have an awareness of the perception: If elementary students think they

are able to do the mathematics task, the belief of the students at the middle level will be

positive and, as a result, increase student achievement.

From Total Quality Management to DataNotGuesswork to From LtoJ

The assessment approach From LtoJ has its basis in the work of Dr. W. Edwards

Deming, Dr. Susan Leddick, and Dr. Lee Jenkins. An American statistician, W. Edwards

17

Deming, developed a management theory for business that is being implemented in the

educational practices of public schools. The theory, Total Quality Management (TQM),

focuses on customer satisfaction, employee empowerment, and product quality. TQM

provides a method of managing and structuring learning that promotes challenge and

internalization (Deming, 1993). Weaver (1992) suggests:

Students become involved in their own education by questioning the learning

process. Teachers view education through students' eyes. Instead of using

standardized tests and grades to measure students' progress, assessment of

student progress is done regularly throughout the school year. By doing so,

schools avoid bringing problems to students' attention at the end of the year,

when it is too late to do anything about them (pg. 1 ).

The Total Quality Management concepts relevant to this study included:

• Focus on the learning process, not the rating process.

• Objectives are posted or handed out to help students keep a sense of direction.

• Students are taught strategies for learning, remembering, and applying what

they have learned in the classroom.

• Teachers communicate interest to students.

• The teacher expresses expectations for improvement.

• No one is complacent about student achievement; there is an expectation that

educational programs will be changed so that they work better (Cotton, 1994,

p 35).

The ideas ofTQM appear to be relevant to improving teaching and learning. The

fundamental idea is that feedback from students can help provide direction for

18

improvement. A number of studies have examined the value of providing the results from

frequent assessment to students. One approach that has been studied provided teachers

with weekly performance graphs on individual students. Children in classrooms in which

teachers received this feedback performed at significantly higher levels than students in

classrooms in which the performance graphs were not available (Whitehurst, 2003).

Assessment feedback should be returned to the student quickly so that adjustments and

improvements can be made in a timely way. The combination ofTQM and assessment

can be viewed as one way to build a partnership in learning between teachers and

students (Angelo, 1993).

The Quality Teacher by William Glasser (1993) provides information on six

conditions for quality schoolwork:

• There must be a warm, supportive learning environment.

• Students should be asked to do only useful work.

• Students should be asked to do the best they can do.

• Students should be asked to evaluate their own work and improve it.

• Quality work always feels good.

• Quality work should never be destructive (p. 42).

From the Total Quality Management System, the DataNotGuesswork metho·d was

suggested by Deming and developed by Jenkins while a superintendent of schools in the

Enterprise School District (Redding, CA) and the Antioch School District (Antioch, CA).

The method was then refined and systematized by Leddick, a consultant in organization

design and continuous improvement. Leddick assisted with Deming's public seminars and

served as his personal aide during the last ten years of his life.

19

The benefits ofDataNotGuesswork according to Jenkins and Leddick, were:

• Focus: It makes the learning explicit. It defines for the teacher and student

what is supposed to be learned in a given amount of time.

• Feedback: The teacher can observe the data and respond with appropriate

changes in instruction.

• Engagement: Students can take more responsibility for their own learning.

• Steadfast performance: The preview/review component ofDNG ensures that

students retain what they've learned. They are-not given permission to forget.

(Jenkins, 2001, p.15).

From DataNotGuesswork, the assessment approach, From LtoJ, was developed

by Jenkins, a former classroom teacher and math/science coordinator. Most recently a

California school superintendent, Jenkins assists teachers and administrators with the

improvement of student learning through the implementation of the assessment approach,

From LtoJ. The theory fosters student independence by developing self-discipline that

will serve the student throughout his or her educational career. The implementation of

frequent assessments reinforces skills, while students practice the concepts of the

required curriculum.

The basic conceptual framework of From LtoJ according to Jenkins (2003) is

outlined below:

• Students are told at the beginning of the year exactly what information is to be

learned that year. Many other interesting topics might be discussed, but this is

the essential information.

• The work from prior grades is connected to current grade-level expectations.

20

• From the beginning, students are quizzed on a sample of end-of-the-year

expectations. These items are randomly selected at the time of the quiz to

eliminate cramming and "psyching-out" the teacher to ascertain what will be

quizzed.

• Students graph their progress toward meeting the end-of-the-year standards.

This is called a student run chart. [Note: Class periods as a whole may be

graphed together using the student run chart.]

• The teacher graphs the progress of the class-as a whole. The teacher not only

needs to know the contribution of each student, also where the class as a

whole is going. This is called the class run chart.

• In order to "leave no child behind", teachers not only need to be able to look

at the progress of each child on their individual student run chart, but the

progress of the class as a whole. The teacher must be able to see the progress

of each student on one sheet of paper. The statistical tool that accomplishes

this is the scatter diagram (Jenkins, 2003).

Review of Database

Data collection took place for the school years 1999 - 2000, 2000- 2001, and

2001 - 2002. A variety of assessment tools were utilized to determine student knowledge.

Assessment instruments included the Harlan Community School District Math

Assessment, the Iowa Test of Basic Skills (ITBS), the Iowa Test of Educational

Development (ITED) and the Iowa Comprehensive Assessment of Mathematics

(ICAMs). Three purposes for assessment at the Harlan Community School District were

(1) to determine each student's developmental level within a content area, (2) to identify a

21

student's strengths and weaknesses in subject areas, and (3) to monitor year-to-year

growth in the basic skills. For the purposes of this study, student achievement data

focused on proficiency level of students. Proficiency level is defined as the combined

percentage of students in the intermediate and high achievement levels on the Iowa Test

of Basic Skills, Iowa Test of Educational Development, and the Iowa Comprehensive

Assessment of Mathematics. The Iowa Test of Basic Skills was utilized at the eighth

grade level and the Iowa Test of Educational Development was utilized at the 9th grade

level.

Table 1. ITBS Math Total - Grade 8

Percentage of Percentage of Percentage of Math Total Students Proficient Students Proficient Students Proficient Grade 8 1999-2000 2000-2001 2001-2002 !TBS All Students 79.6 75.8 81.5 Male 81.5 81.7 77.7 Female 78.2 69.8 85.2 Race/Ethnicity Data is not disaggregated because group size is less than 10. Free/Reduced 60.0 55.6 69.2 Non-Free/Reduced NA* NA* 84.7 *Not Available

Table 2. ITED Math Total - Grade 9

Percentage of Percentage of Percentage of Math Total Students Proficient Students Proficient Students Proficient Grade 9 2000-2001 2001-2002 2002-2003 !TED All Students 87.5 85.7 82.7 Male 89.7 82.9 78.8 Female 85.1 88.0 86.0 Race/Ethnicity Data is not disaggregated because group size is less than 10. Free/Reduced 79.2 87.0 82.7 Non-Free/Reduced NA* 76.2 82.7 *Not Available

22

Table 3. Local Achievement Data 02-03

Local Student Achievement Data Compared with State and Nation Percentage of Students Proficient

ITBS and ITED 2002-2003

Math Total HCS* Iowa Nation Grade 8 (!TBS) 83.2 73.1 60.0 Grade 9 (ITED) 82.7 81.3 60.0 • A pseudonym is used.

Table 4. !CAM Mathematics 2002-2003

!CAM Math Module: Patterns, Functions & Algebra Eighth Grade Students-2002-2003

141 Students Tested* All Students Total Students Levell Level 2 Level 3

(Below 25%) (Above 85%) N % N % N %

Students Included 140 8 5.7% 99 70.7% 33 23.6%

*Includes students identified as excluded from testing and students with invalid answer documents.

Subgroups Total Students Level 1 Level2 Level 3 (Below 25%) (Above 85%)

Gender Male 75 3 4.0% 51 68.0% 21 28.0% Female 65 5 7.7% 48 73.8% 12 18.5%

Table 5. !CAM Mathematics 2001-2002

!CAM Math Module: Patterns, Functions & Algebra 2001-2002

125 Students Tested Test Group Levell Level2 Level3

Low High All Eighth Grade 9.6% 60.8% 29.6%

Male 12.5% 59.4% 28.1% Female 6.6% 62.3% 28.1%

23

Table 6. Math Achievement

1992 2000 Norms Norms

Math High High High Grade 8 Performance Performance Performance

ITBS 99-00 00-01 01-02 All Students 16.9 11.8 33.0

Male 18.6 12.7 31.8 Female 13.0 9.5 28.0

Intermediate Intermediate Intermediate Performance Performance Performance

99-00 00-01 01-02 All Students 62.7 64.0 53.5

Male 62.9 69.0 46.0 Female 65.2 60.3 57.2

Low Low Low Performance Performance Performance

99-00 00-01 01-02 All Students 20.3 24.3 13.5

Male 18.6 18.3 22.2 Female 21.7 30.2 14.8

2000 Essential Information

Chart 3.1 is the scatter diagram for 2000 Essential Vocabulary. This is an eighth

grade scatter matrix with each dot representing a class period. Continuous improvement

is noted by the upward trend of the data points. With a class of 20 students, the possible

number of words correct is 200 points. The number of the words correct range from 20 to

40 in Week l. In Week 30, the number of words correct range from 120 to 140.

Chart 3.2 is the class run chart for 2000 Essential Vocabulary. Each point

represents the total number of corrected answers as a normalized score or percentage. The

eighth grade class of2000 for Essential Vocabulary attained 15% correct in Week 1 and

74% correct in Week 30. The graph showed an upward trend of the data points with 10

data points showing all-time high points to that point in the school year. For example, in

24

Week 21, the class attained a percentage of 54% which was the highest percentage they

achieved.

Charts 3.3 - 3.7 (See Appendix G) are student run charts for 2000 Essential

Vocabulary for class periods one through five. With class sizes of 20 students in each

class period, the total possible number of items correct is 200 words. It should be noted

that for the purposes of data collection at the district level, each class size was set at

twenty to maintain consistency. First period scores range from 36 to 121 words in Weeks

1 through 30 respectively. Second period scores ranged from 30 in Week 1 to 158 in

Week 30. Third period scores ranged from 28 in Week 1 to 152 in Week 30. Fourth

period scores ranged from 25 in Week 1 to 155 in Week 30. Finally, fifth period scores

ranged from 35 to 153 in Week 1 through Week 30 respectively.

25

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data points showing all-time high points to that point in the school year.

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Chart 3.15 is the scatter diagram for 2001 Fractions. This is an eighth grade

scatter matrix with each dot representing a class period. Continuous improvement is

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70 to 80.

28

Chart 3.16 is the class run chart for 2001 Fractions. Each point represents the total

number of corrected answers as a normalized score or percentage. The eighth grade class

of2001 for Fractions attained 32% correct in Week 1 and 77% correct in Week 30. The

graph showed an upward trend of the data points with 15 data points showing all-time

high points to that point in the school year.

Charts 3.17 - 3.21 (See Appendix I) are student run charts for 2001 Fractions for

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to 75 in Week 1 through Week 30 respectively.

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is noted by the upward trend of the data points. With a class of20 students, the possible

number of concept questions correct is 100 points. The number of the questions correct

range from 5 to 20 in Week l. In Week 30, the number of questions correct range from

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Chart 3.23 is the class run chart for 2001 Essential Concepts. Each point


eighth grade class of2001 for Essential Concepts attained 22% correct in Week l and



29

Charts 3.24 - 3.28 (See Appendix J) are student run charts for 2001 Essential

Concepts for class periods one through five. First period scores range from 34 to 67

words in Weeks 1 through 30 respectively. Second period scores ranged from 24 in Week




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is noted by the upward trend of the data points. With a class of20 students, the possible


80 in Week 1. In Week 30, the number of words correct range from 140 to 180.






Charts 3.31 - 3.35 (See Appendix K) are student run charts for 2002 Essential

Vocabulary for class periods one through five. First period scores range from 82 to 183

words in Weeks l through 30 respectively. Second period scores ranged from 50 in Week




Chart 3.36 is the scatter diagram for 2002 Fractions. This is an eighth grade

scatter matrix with each dot representing a class period. Continuous improvement is

noted by the upward trend of the data points. With a class of 20 students, the possible

number of fractions problems correct is 100 points. The number of the problems correct

range from 10 to 40 in Week l. In Week 30, the number of problems correct range from

80 to 95.

37

Chart 3.37 is the class run chart for 2002 Fractions. Each point represents the total

number of corrected answers as a normalized score or percentage. The eighth grade class

of 2002 for Fractions attained 24% correct in Week 1 and 89% correct in Week 30. The

graph showed an upward trend of the data points with 11 data points showing all-time

high points to that point in the school year.

Charts 3.38 - 3.42 (see Appendix L) are student run charts for 2002 Fractions for

class periods one through five. First period scores range from 40 to 96 words in Weeks 1

through 30 respectively. Second period scores ranged from 27 in Week 1 to 92 in Week

30. Third period scores ranged from 16 in Week 1 to 90 in Week 30. Fourth period scores

ranged from 23 in Week 1 to 82 in Week 30. Finally, fifth period scores ranged from 12

to 75 in Week 1 through Week 30 respectively.

Chart 3.43 is the scatter diagram for 2002 Essential Concepts. This is an eighth



number of concept questions correct is 100 points. The number of the questions correct

range from 20 to 40 in Week 1. In Week 30, the number of questions correct range from

70 to 90.

Chart 3.44 is the class run chart for 2002 Essential Concepts. Each point


eighth grade class of2002 for Essential Concepts attained 32% correct in Week l and



38

Charts 3.45 - 3.49 (See Appendix M) are student run charts for 2002 Essential

Concepts for class periods one through five. First period scores range from 44 to 90

words in Weeks 1 through 30 respectively. Second period scores ranged from 29 in Week




39

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Improved test scores are a by-product of improved learning (Jenkins, 2003). The

use of standardized tests to determine student achievement is commonplace in many

schools. In the eighth grade mathematics classroom, students are assessed using multiple

assessment tools. Using three years of data collected after the implementation of the

assessment approach From LtoJ in the area of Essential Vocabulary, comparisons show

an improvement in percentage of students proficient on the Iowa Test of Basic Skills. In

comparing the students as eighth graders in 1999 - 2000 and the same students as

freshman in 2000 - 2001, the number of students proficient increased 7.9%. Similarly,

the eighth graders of 2000 - 2001 as freshman in 2001 - 2002 increased the proficiency

rate by 9.9%. The eighth graders of2001-2002 increased the proficiency rate by 1.2% as

freshman in 2002 - 2003.

The use of the Iowa Comprehensive Assessment of Mathematics at the eighth

grade level is given to the students in the spring of each school year. Of most interest in

analyzing the data for the ICAM is the low number of students who are considered at the

low level. Eight students out of 140 students, or 5. 7%, in 2002 - 2003 were considered to

be a Level 1. In comparison, 11 students out of 125, or 9.6% in 2001 - 2002 were

considered to be Level 1.

Through the use of the class period student run graphs, students saw the

progression of learning. Students no longer had permission to forget essential information

previously taught and had the essential information databases from which to preview

upcoming essential information. In all the data collected and reviewed, learning was

constant throughout the year. Perhaps the best indicator of success, however, cannot be

46

graphed. In speaking to educators at the high school levels, it was remarked that the

retention of essential information from the eighth grade year to the freshman and

sophomore years was evident in the high school classrooms (T. Peterson & B. Custer,

personal communication, 2003). Less review was needed and less time was wasted

providing time for more in-depth instruction of the curriculum (T. Peterson, personal

communication, 2003).

Students were interviewed informally regarding their achievement and continuous

improvement in the mathematics classroom and meticulous field notes were recorded as

the assessment theory of From LtoJ was implemented and utilized during the process of

data collection. Students were also asked to summarize their individual learning

processes in their math journals. Middle school students seem to have valuable insights

into their own thought processes. When presented with the question: Why does From

LtoJ work in our classroom?" students' replies carried the same general thoughts.

Comments made included: " I think this works because it identifies what we need to

learn, so I can focus on those things and track how I'm doing." and "I have to know this

stuff to do well."

47

CHAPTER4

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Summary of Study

The importance of being mathematically literate in order to function in today' s

society has been well documented by educators and employers alike. Student learning is

related to many factors and the most important may be different depending on the

educational level of the student. Factors that influence learning may change as the goals

and interests of students change. It is important for educators as well as parents to know

that middle level students who have the opportunity to study concepts associated with

algebra, geometry and statistics are more likely to develop their knowledge of these

subjects and will then exhibit continued growth in those content areas (NCTM, 2000).

Likewise a lack of emphasis on concepts in mathematics curriculum will impede student

learning.

All middle school students should be engaged in learning curriculum that

provides a balance of knowledge and skill acquisition (NCTM, 2000). From LtoJ

provides a varied approach to instructional practices as well as a means for adolescents to

develop a lasting understanding of mathematical concepts and skills. With From LtoJ,

students are actively engaged in their own learning as they demonstrate knowledge of

mathematics.

Mathematics assessment must reflect what is important for student to learn, rather

than what is easy to assess. Rather than serving as tool to measure learning, the From

48

LtoJ assessments served as a tool to organize the learning of essential information. The

primary academic and conceptual needs identified were:

• Basic skill (essential concepts, vocabulary, and fractions) development and

retention,

• Building student understanding of the curriculum; and

• Building confidence in the knowledge of the curriculum (Jenkins, 2003).

In using the From LtoJ process, the focus of assessment became improvement

rather than grades and performance relative to peers, progress rather than fixed

achievement and a student's next steps to learning rather than past accomplishments.

Students can succeed in this program even if their basic skills are deficient.

Even though From LtoJ is more difficult than the traditional drill or rote method

students formerly used to demonstrate their learning, it caused less stress for the students.

In documented field notes, students agreed that From LtoJhelped them in some way with

basic understanding of important mathematical concepts. Students responses echo this

process:

• The program was fun, way better than just doing assignments.

• From LtoJ taught us new ways to understand mathematics.

• From LtoJhelped us learn to solve problems and made us think.

• It was a challenge every week, but I became confident in my abilities. I

always wanted to do better each week.

The assessment approach, From LtoJ, provided:

• continual feedback for both the students and teacher,

• a team approach to learning and instruction,

49

• a means to improve the working and learning processes, and

• increased cooperation and communication as students and teacher work

toward continuous improvement.

The nature and purpose of this study was to gather descriptive information

concerning the assessment approach From LtoJ and student achievement. By becoming

actively engaged in this process, the benefits included: the immediate monitoring of

student progress on the acquisition of the essential information; increased student

motivation to learn and retain essential information; and increased student perception as a

community of learners (Jenkins, personal communication, 2003).

Through the implementation of this long-term improvement strategy, students

displayed greater achievement on the passing rates. The curriculum was presented in a

continuous manner. Students learn to internalize and retain knowledge. Weekly quizzes

are given to emphasize understanding. The quizzes which are based on the standards and

benchmarks of the curriculum develop both basic and new skills as well as retention of

prior learning. One of Deming's principles for management is to "drive out fear." This

process of continuous improvement built confidence and the instant feedback helped

boost that confidence while quickly identifying where weaknesses in individual and class

learning lie.

Rather than a teaching system, From LtoJ can be considered a learning system by

providing a focus that makes the learning of the essential information explicit. As the

authors of Principles and Standards for School Mathematics suggest, "Classrooms in

which students are encouraged to present their thinking and in which everyone

50

contributes by evaluating one another's thinking provide rich environments for learning

mathematical reasoning" (NCTM, 2000, pg. 57).

The primary aim of the From LtoJ implementation focused on increased student

success and the reduction of failure. Students had time to create their own knowledge, to

learn concepts and skills and were given opportunities to see material over and over until

learning and retention took place. Students' problem-solving and higher level thinking

skills improved but mathematics communication improved as well through the From LtoJ

implementation.

Implications of Data Results

The ultimate goals of education are performance and achievement. The goal in the

eighth grade mathematics classroom was to implement an assessment approach where

continuous improvement could take place. The essential element of continuous

improvement is to score higher than the individual student, class period or grade level,

ever have before.

Students understand from the data collected and posted in the classroom that

essential information assessments are providing feedback about their learning as well as

continuous improvement in the area of mathematics. While students in the classroom

kept individual student run charts, the class scatter matrices, the class run charts, and the

class period run charts were the focus for the instructor. When students saw that they

were moving together toward improvement as a team, they expressed more confidence

about themselves and their abilities. (Various students, personal communication, 2001-

2003).

51

Continuous improvement was not about group totals versus individual effort; it

was about both combined. Class run charts were valuable in that they gave a view of

learning over time. The use of the scatter matrix provided an overview of how all class

periods were performing as a grade level.

At the beginning of the school year, scores are low, in part, to the learning process

required to introduce the procedures to the students. Lack of retention of prior instruction

was also a factor in the low scores. However, the students' ability to do the vocabulary,

fractions, and concepts quizzes increased over time and an upward scoring trend was

established in the graphs.

As the process continued in 2002 - 2003, the class run chart and the individual

class student run charts, students made gains in all areas of essential information.

Through the continued use of the From LtoJ learning process, learning was improved and

students experienced a growth in knowledge.

Educators have an opportunity to alter the negative attitudes and strengthen more

positive attitudes toward mathematics by promoting better classroom practices and by

providing positive experiences in mathematics. In regards to mathematics instruction in

the middle level classroom, the NCTM (1987) recommended:

• basic skills in mathematics be defined to encompass more than computational

facility,

• the success of mathematics programs and student learning should be evaluated

by a wider range of measures than conventional testing, and

52

• more mathematics study should be required for all students and a flexible

curriculum with a greater range of options be designed to accommodate the

diverse needs of the student population.

The From LtoJ process addresses each of these points by providing a deeper

understanding of mathematics through the use of frequent assessments. These frequent

assessments, whether done weekly or otherwise, evaluate student learning. The

presentation of the essential information throughout the curriculum provides flexibility to

meet diverse learner needs and gives direction for instructional improvement. The teacher

can constantly monitor the students as individuals and as a whole. The collection of

assessment data in an on-going manner provides the teacher with feedback during

instruction as to the areas or concepts that need continued focus for increased

improvement.

The use of From LtoJ meets all three mathematics recommendations through the

implementation of vocabulary, conceptual and computational assessments. This

assessment approach implementation provides a flexible curriculum and evaluation

method to meet the needs of all learners, not just the high achievers. Because one of the

few places students must ''talk mathematics" is in the classroom, teachers must give

attention to mathematical language learning. Identification must be made of the

difficulties in learning mathematical language and some strategies to deal with these

difficulties. Teachers must increase their sensitivity to issues in language learning and

create strategies to help students become fluent in mathematical communication.

53

At a time when middle school students' enthusiasm toward learning decreases

incrementally, granting autonomy to a student appears to be less of a motivating factor

than increasing feedback from their graphed assessment data. This feedback enhances a

student's sense of competence through involvement.

Recommendations

The following recommendations should be considered in the middle school

mathematics classroom:

1. Educators must take advantage of every opportunity to use strategies that

reinforce intrinsic motivation to sustain effort throughout a student's

education.

2. Students should see their assessment data frequently graphed and displayed.

3. Interaction with assessment data by students should be frequent, in order for

them to see and value progress being made.

Of all the reform recommendations made by the NCTM, making mathematical

connections is among the more difficult to achieve. At the middle school level, students

are beginning to appreciate the real power of mathematics. Mathematical connections can

relate mathematical topics to students' daily lives and to other mathematical topics but

are probably most helpful in relating mathematics to other curriculum areas. These

connections help students understand mathematics better and see it as a useful and

interesting subject to study.

Based on data collected, the following actions were deemed necessary:

• Revision of the middle school math essential information to narrow the focus

and limit the amount of mastery information items per grade level.

54

• Revision of problem and test item databases to make them more concise.

• Establishment of essential information should include "real-world" problem

assessment, such as time, money, and measurement to provide an appreciation

of mathematics is used outside the school setting.

• Establishment of essential information databases at sixth and seventh grade

levels to provide continuous improvement at all middle levels. Student

learning can be improved with "current grade (to learn), prior grade (to

review), and future grades (for information)" (Jenkins, 2003).

• The development of a specific assessment instrument to be administered at

beginning and end of the year assessing the retention of essential information.

• Disaggregation of collected data for specific assessment instrument into

various sub-groups to gain insight regarding instruction for those sub-groups.

• Further data needs to be collected regarding the frequency of assessment data

displayed and how viewing the results in graphic form influences the students'

motivation to learn.

Continuous improvement is needed to improve student learning. The

implementation of the stated recommendations may further improve student

learning and increase student achievement.

Conclusion

"What we assess we must value" (Jenkins, 2003, personal communication). Many

studies have been done in regards to both student achievement and math assessment as

separate issues. From LtoJ provides a means for ongoing data collection to adjust

teaching strategies throughout the year to improve the quality oflearning by end of the

55

year. Middle level teachers "must do everything possible to enhance the probability of

every student being successful" (Stevenson, 1992, p. 215). The assessment approach

From LtoJ offers students the means to construct their own knowledge while retaining

essential information. Higher order thinking skills emerge from basic skills to improve

student achievement. Continuous improvement is the focus of the teaching and the

learning when From LtoJ is implemented in the middle school mathematics classroom.

56

REFERENCES

American Federation of Teaching, (1990). National Council on Measurement in Education, National Education Association.

Angelo, T. & Cross, P. (1993). Classroom assessment techniques: A handbook. Hoboken, NJ: Wiley & Sons.

Albert., L. and McKee, K. (2002). In their own words: achieving intersubjectivity through complex instruction. Learning and Instruction. In V. Spiridonov, I. Bezmenova, 0. Kuoleva, E. Shurukht, & S. Lifanova, (Eds.). The summer psychology conference 2000, the zone of proximal development. Moscow, Russia: Institute of Psychology of the Russian State University for the Humanities.

Carnegie Council on Ado le scent Development ( 1990). Turning points: Preparing American youth for the 21st century. New York: Carnegie Corporation of New York.

Deming, W. E. (1993). The new economics for industry, government and education. Cambridge, MA: Massachusetts Institute of Technology.

Gall, M., Borg, W. & Gall, J. (1996). Educational research. White Plains, NY: Longman Publishers.

George, P. (2002). No child left behind: Implications for middle level leaders. Westerville, OH: National Middle School Association.

Jackson, A., & Davis, G. (2000). Turning Points: Educating Adolescents in the 21st Century. New York: Teachers College Press.

Jenkins, L. (1997). Improving student learning: Applying Deming's quality principles in classrooms. Milwaukee, WI: ASQ Quality Press.

Jenkins, L. (200 I). DNG: DataNotGuesswork handbook. Scottsdale, AZ: From LtoJ Consulting Group, Inc.

Jenkins, L. (2003). Improving student learning: Applying Deming's quality principles in classrooms. (second edition.) Milwaukee, WI: American Society for Quality.

Jenkins, L. (2003). From LtoJ handbook. Scottsdale, AZ: From LtoJ Consulting Group, Inc.

Kohn, A. ( 1999). Why students lose when tougher standards win. Educational Leadership, 57(1), 18-22.

57

Marzano, R (2003). What works in schools: Translating research into action. Alexandria, VA: Association for Supervision and Curriculum Development.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation methods for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

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

Rubenstein, R & Thompson, D. (2002). Understanding and supporting children's mathematical vocabulary development. Teaching Children Mathematics, 9(2), 107-113.

Schmidt, M. & Brosnan, P. (1996). Mathematics assessment: practices and reporting methods. School Science & Mathematics, 96(1), 17-22.

Singh, K., Granville, M., and Dika, S. (2002) Mathematics and science achievement: effects of motivation, interest, and academic engagement. Journal of Educational Research, 95(6), 323-333.

Skaalvik, E. & Valas, H. (1999). Relations among achievement, self-concept, and motivation in mathematics and language arts: a longitudinal study. Journal of Experimental Education, 67(2), 135-150.

Stiggens, R. (1988). Make sure your teachers understand student assessment. Executive Educator, 10(8), 24-30.

Stevenson, C. (1992). Teaching 10 to 14 year olds. New York: Longman.

Tomlinson, C. (1995). How to differentiate instruction in mixed-ability classrooms. Alexandria, VA: Association for Supervision and Curriculum Development.

United States Department of Education, (2001). Retrieved September 15, 2003, from http://www.ed.gov/nclb/landing.jhtml

Vogler, K. E. (2002). The impact of high-stakes, state-mandated student performance assessment on teacher's instructional practices. Education, 123( 1 ), 3 9-56.

Warren, E. & Nisbet, S. (200 l ). How grade 1-7 teachers assess mathematics and how they use the assessment data. School Science & Mathematics, 348-356.

Weaver, T. (2000). Total Quality Management. Retrieved September 15, 2003, from http://www.ericfacility.net/databases/ERIC Digests/ed347670.html

58

Whitehurst, G. (2003). Papers and presentations, mathematics and science initiative. Retrieved September 14, 2003, from http://www.ed.gov/print/rschstat/research/progs/mathsscience/whitehurst.html

Wilkins, J. (2002). Predicting student growth in mathematical content knowledge. Journal of Educational Research, 95(5), 288-299.

59

Appendix A - Letter of Permission

April 21, 2003

TO WHOM IT MAY CONCERN:

Toni Andersen, 8th grade math teacher at HCMS, has access and permission to use any data collected by the school district for the purpose of completing her master's degree in middle level education.

Sincerely,

Teresa Coenen Principal

60

Appendix B: Turning Points: Preparing American Youth for the 21 st Century

Recommendations for Transforming Middle Grade Schools

The Task Force calls for middle grade schools that:

• Create small communities for learning where stable, close, mutually respectful

relationships with adults and peers are considered fundamental for intellectual

development and personal growth. The key elements of these communities are

schools-within-schools or houses, students and teachers grouped together as

teams, and small group advisories that ensure that every student is known well

by at least one adult.

• Teach a core academic program that results in students who are literate,

including in the sciences, and who know how to think critically, lead a healthy

life, behave ethically, and assume the responsibilities of citizenship in a

pluralistic society. Youth service to promote values for citizenship is an

essential part of the core academic program.

• Ensure success for all students thought elimination of tracking by achievement

level and promotion of cooperative learning, flexibility in arranging

instructional time, and adequate resources (time, space, equipment and

materials) for teachers.

• Empower teachers and administrators to make decisions about the experiences

of middle grade students through creative control by teachers over the

instructional program linked to greater responsibilities for students'

performance, governance committees that assist the principal in designing and

coordinating school-wide programs, and autonomy and leadership within sub-

61

schools or houses to create environments tailored to enhance the intellectual

and emotional development of all youth.

• Staff middle grade schools with teachers who are expert at teaching young

adolescents and who have been specially prepared for assignment to the

middle grades.

• Improve academic performance through fostering the health and fitness of

young adolescents, by providing a health coordinator in every middle grade

school, access to health care and counseling services, and a health-promoting

school environment.

• Reengage families in the education of young adolescents by giving families

meaningful roles in school governance, communicating with families about

the school program and student's progress, and offering families opportunities

to support the learning process at home and at the school.

• Connect schools with communities, which together share responsibility for

each middle grade student's success, through identifying service opportunities

in the community, establishing partnerships and collaborations to ensure

students' access to health and social service, and using community resources

to enrich the instructional program and opportunities for constructive after

school activities.

Carnegie Council on Adolescent Development. ( 1989). Turning Points: Preparing American Youth for the 21st Century.

62

Appendix C: From LtoJ Concepts

Variation and Classroom Graphing

l. From LtoJ is used to measure learning of information, student performance, student enthusiasm, classroom management (i.e. attendance, behavior), and to meet special education requirements.

2. Process measurement provides feedback during the course. Results measurements are the final data at year's end or data from standardized assessments.

3. The three basic graphs are the class run chart, student run chart, and the scatter diagram.

4. The student overlay combines the student run chart and scatter diagram to precisely answer the question, "How is my child doing compared to other students?"

5. Effective use of data requires the study of patterns and trends, not merely two data points.

6. Random sampling of end-of the year items provides students a constant review of what has been taught and a constant preview of what is yet to come.

7. The square root of "n" is an ample sample size for accurate data, if collected weekly or bi-weekly.

8. The classroom ( or department/school) histogram's shape should progress from an "L" to a bell and finally to a ''r.

9. There are two kinds of variation: special and common.

l 0. Seven points in a row are needed to rule out good and bad luck.

11. The four generations of management are (1) I'll do it myself, (2) do it the way I tell you, (3) management by objective, and (4) agree on an aim and work together to accomplish it.

12. Data can be disaggregated to see how sub-sets of students are performing and aggregated to study whole grade levels and departments.

13. Ranks data should be replaced with control charts.

63

System 14. The seven elements of a system are: aim, input, supply, process, feedback,

customers, and output.

15. The primary aim of schools is to increase student success and reduce failure.

16. Dr. Deming estimated that problems are the fault of the system of96% of the time and of the people 4% of the time.

17. There's no shortage of good people -- unless we choose to create one.

18. The improve system results, work on system processes.

19. A clear aim unifies people and practices.

Epistemology 20. The learning sequence is data, graph the data, gain insight, test hypotheses,

and gain knowledge.

21 . Experience is not the best teacher; continually testing theories is.

22. It takes one example contrary to a theory to require that you revisit the theory.

23. Ask why at least five times to search out root causes.

24. Information is about the past; performance is learned to create a better future.

25. the # l requirement of leaders is to create more leaders.

26. Dr. Deming's learning cycle has four parts: plan, do, study, and act.

Psychology 27. Dr. Deming called the Western Society practices of destroying people's self

worth, "The Forces of Destruction." These forces include stars, grades, incentives, management by numbers alone, and management by objectives.

28. It is not the responsibility of educators to motivate students but to determine what is causing them to lose their motivation and stop such practices.

29. A major responsibility of all leaders is to drive out fear from their organizations.

30. Leaders help people remove system barriers.

64

Appendix D - Essential Information - Essential Vocabulary Database

The Top Vocab! -Mathematical Vocabulary

There are words in the real world that are math related. We should know these words so that we can read, write, and, most importantly, think mathematically. This list contains math vocabulary used in 8th grade math. You are familiar with some of the words, while others will be new. Each week, there will be a quiz consisting of ten vocabulary words chosen randomly from this list. They may be repeated from one week to the next week or they may not. That's random! Your responsibility is to learn these words and retain the definitions as we progress through the school year.

1. Algebra - a study of arithmetic in which symbols, usually letters, represent numbers 2. Order of operations-the rules to follow when more than one operation is used 3. Variable- a symbol, usually a letter, used to represent a number in mathematical

expressions or sentences 4. Exponent-in a power, the number oftimes the base is used as factor. 5. Factor-when two or more numbers are multiplies, each number is a factor of the

product. 6. Power - a number that can be written using an exponent. 7. Base - in a power, the number used as a factor, or the base of a parallelogram or a

triangle is any side of the figure, or in a percent proportion, the number to which the percentage is compared, or the bases of a prism are the two parallel congruent faces.

8. Associative property of equality - ( a + b) + c = a + (b + c) or ( ab )c = a(bc) 9. Commutative property of equality - a+ b = b + a or ab= ba 10. Evaluate - to find the value of an expression by replacing the variables with

numerals. 11. Addition property of equality - If you add the same number to each side of an

equation, the two sides remain equal. If a = b, then a + c = b + c. 12. Subtraction property of equality - If you subtract the same number to each side of an

equation, the two sides remain equal. If a = b, then a - c = b - c. 13. Inverse operation - pairs of operations that undo each other. Addition and subtraction

are inverse operations. Multiplication and division are inverse operations. 14. Division property of equality - if the same nonzero number diyides each side of an

equation, then the two sides remain equal. If a = b, then a + c = b + c. 15. Multiplication property of equality- If each side of an equation is multiplied by the

same number, then the two sides remain equal. If a= b, then ac = be. 16. Distributive property - the sum of two addends multiplied by a number is the sum of

the product of each addend and the number. a(bc) = 17. Rectangle - a quadrilateral with four congruent angles but not four congruent sides 18. Perimeter - the distance around a geometric figure 19. Square - a parallelogram with all sides and all angles congruent 20. Parallelogram - a quadrilateral with two pairs of parallel sides and congruent angles 21. Area - the number of square units needed to cover a surface enclosed by a geometric

figure

65

22. Altitude - a segment in a quadrilateral that is perpendicular to both bases, with endpoints on the base lines or the segment that goes from the vertex of a cone to its base and is perpendicular to the base

23. Height - the length of the altitude of a triangle or a quadrilateral 24. Inequality- a mathematical sentence that contains one of the following signs: <, >,

*, ~. ;::, or¢. 25. Integer-positive and negative whole numbers 26. Graph - to draw or plot the points named by those numbers on a number line or

coordinate plane 27. Coordinate - the number associated with a point on the number line or coordinate

plane 28. Absolute value-the number of units a number is from zero on the number line 29. Opposite - two integers are opposites if they are represented on the number line by

points that are the same distance from zero, but on opposite sides of zero. The sum of opposites is zero.

30. Additive inverse - two integers that are opposites of each other 31. Matrix - a rectangular arrangement of elements in rows and columns 32. Row - numbers side-by-side horizontally 33. Column - numbers stack on top of each other in a vertical arrangement 34. Element- each number in a matrix is called an element 35. Coordinate system- a plane in which a horizontal number line and a vertical number

line intersect at their zero points 36. Origin - the point of intersection of the x-axis and y-axis in a coordinate system 37. X-axis- the horizontal number line which helps to form the coordinate system 38. Y-axis-the vertical number line of the two perpendicular number lines in the

coordinate system · 39. Quadrant- one of the four regions into which two perpendicular number lines

separate the coordinate plane 40. Ordered pair- a pair of numbers used to locate a point in the coordinate system (x,y) 41. Ratio - a comparison of two numbers by division 42. Rate - in a percent proportion, the ratio of a number to 100 or a ratio of two

measurements having different units 43. Unit rate - a rate with a denominator of one 44. Proportion - a statement of equality of two or more ratios 45. Percent - a ratio that compares a number to 100 46. Frequency table- a table for organizing a set of data that shows the number of times

each item or number appears 4 7. Statistics - the branch of mathematics that deals with collecting, organizing, and

analyzing data 48. Bar graph- a graphic form using bars to make comparisons of statistics 49. Histogram- a special type of bar graph that displays the frequency of data that has

been organized into equal intervals. The intervals cover all possible values of data; therefore there are no spaces between the bars of the graph.

50. Circle graph- a type of statistical graph used to compare parts of a whole 51. Line plot - a graph that uses an X above number on a number line each time that

number occurs in a set of data.

66

52. Data analysis - to study data and draw conclusions from the numbers observed 53. Cell- the basic unit of a spreadsheet; can contain data, labels, or formulas 54. Mean- the sum of the numbers in a set of data divided by the number of pieces of

data 55. Median- the middle number in a set of data when the data are arranged in numerical

order 56. Mode - the number or item that appears most often in a set of data 57. Range - the set of output values in a function or the difference between the greatest

number and the least number in a set of data 58. Scatter plot - a type of graph that shows the general relationship between two sets of

data 59. Parallel lines- lines in the same plane that do not intersect. The symbol II means

parallel. 60. Vertical angles - congruent angles formed by the intersection of two lines 61. Supplementary angles- if the sum of the measures of two angles is 180 degrees 62. Polygon - a simple closed figure in a plane formed by three or more line segments 63. Triangle - a polygon having three sides and three angles 64. Acute angle - an angle with a measure greater than O degrees and less than 90 degrees 65. Right angle- an angle that measures 90 degrees 66. Obtuse angle - an angle that measures greater than 90 degrees but less than 180

degrees 67. Scalene - a triangle with no congruent angles 68. Isosceles - a triangle that has two congruent sides 69. Equilateral - all sides of a figure are congruent 70. Perpendicular - two lines or line segments that intersect to form right angles 71. Complementary angles- if the sum of the measures of two angles is 90 degrees 72. Quadrilateral - a polygon having four sides 73. Rhombus- a parallelogram with four congruent sides and two pairs of congruent

angles 74. Trapezoid- a quadrilateral with exactly one pair of parallel sides. 75. Line symmetry- figures that match exactly when folded that divides a figure into two

halves that are reflections of each other. 76. Reflection- a type of transformation where a figure is flipped over a line of

symmetry. 77. Rotation- when a figure is turned around a central point. 78. Rotational symmetry - a figure has rotational symmetry if it can be turned less than

360 degrees about its center and still looks like the original 79. Congruent triangles -triangles that have the same size and shape. 80. Similar triangles - triangles that have the same shape but may not have the same size. 81. Tessellation- a repetitive pattern that fit together with no holes or gaps. 82. Transformation - movements of geometric figures to modify tessellations. 83. Translation- a method used to make changes in the polygons of tessellations by

sliding a figure horizontally, vertically, or both. 84. Prime number - a whole number greater than 1 that has exactly two factors, I and

itself 85. Composite number- any whole number greater than I that has more than two factors.

67

86. Prime factorization - expressing a composite number as the product of prime numbers.

87. Factor tree - a diagram showing the prime factorization of a number. 88. Fundamental theorem of arithmetic - a property of numbers that states every number

has a unique set of prime factors. 89. Greatest common factor -the greatest of the common factors of two or more

numbers. 90. Rational number- numbers of the form alb, when a and bare integers and b.cO. 91. Terminating decimal - a decimal whose digits end. Every terminating decimal can be

written as a fraction with a denominator of 10, 100, 1,000, and so on. 92. Repeating decimal- a decimal whose digits repeat in groups of one or more. 93. Bar notation- in repeating decimals the line or bar placed over the digits that repeat. 94. Probability- the chance that some event will happen. It is the ratio of the number of

ways a certain event can occur to the number of possible outcomes. 95. Multiple - the product of a number, a variable and any whole number. 28 is a

multiple of 4 and 96. Least common multiple - the least of nonzero common multiples of two or more

numbers. 97. Least common denominator- the least common multiple of the denominators of two

or more fractions. 98. Scientific notation - a way of expressing numbers as the product of a number that is

at least 1 butless than 10 and a power often 99. Mixed number - the sum of a whole number and a fraction (for example: 6 2/3) 100. Multiplicative inverse - a number times its multiplicative inverse is equal to 1. 10 l. Reciprocal - the multiplicative inverse of a number 102. Sequence - a list of numbers in a certain order, such as 2, 4, 6, 8, IO and so on. 103. Term- a number, variable, or a product of number and variables or a number in a

sequence 104. Arithmetic sequence - a sequence of numbers in which you can find the next term

by adding the same number to the previous term. 105. Geometric sequence -when multiplying by a constant factor, the sequence, forms

consecutive terms of a sequence 106. Circle - the set of all points in a plane that are the same distance from a given

point called a center 107. Center-the given point from which all points on a circle or sphere are the same

distance 108. Radius - the distance from the center of a circle to any point on the circle 109. Diameter - the distance across a circle through its center 110. Circumference- the distance around a circle 111. Percentage - in a percent proportion, a number (P) compared to another number

called the base (B) 112. Discount - the amount by which the regular price is reduced 113. Markup - the difference between the price paid by the merchant and the increased

selling price 114. Selling price - the amount the customer pays for an item 115. Interest - the amount charged or paid for the use of money

68

116. Principal - the amount of an investment or debt 117. Similar polygons - polygons are similar if the corresponding angles are congruent

and their corresponding sides are in proportion 118. Pentagon - a polygon having five sides 119. Scale drawing - a drawing that is similar but either larger or smaller than the

actual object 120. Scale model - a replica of an original object that is either too large or too small to

be built at actual size 121. Scale - the ratio of a given length on a drawing or model to its corresponding

length in reality 122. Dilation - the process of reducing or enlarging an image in mathematics 123. Scale factor - the ratio of a dilated image to the original image 124. Perfect square - a rational number whose square root is a whole number 125. Square root - one of the two equal factors of a number 126. Radical sign- the symbol used to indicate a nonnegative square root 127. Irrational number- a number that cannot be expressed as a fraction 128. Real number - the set of rational numbers together with the set of irrational

numbers 129. Hypotenuse - the side opposite the right angle in a right triangle 130. Legs - the two sides of a right triangle that form the right angle 131 . Pythagorean Theorem - in a right triangle, the square of the length of the

hypotenuse is equal to the sum of the squares of the lengths of the legs c2= a2 +b2

132. Function - a relation in which each element of the input is paired with exactly one element of the output according to specified rule

133. Function table - a table organizing the input, rule, and output of a function 134. Domain- the set of input values in a function 135. Linear function - an equation in which the graphs of the solutions form a line 136. Solid - a three-dimension figure 137. Prism- a three-dimension figure that has two parallel and congruent bases in the

shape of polygons 138. Edge - the intersection of faces of a three-dimensional figure 139. Vertex- the point of a prism where the faces intersect 140. Pyramid - a solid figure that has a polygon for a base and triangles for sides 141. Volume - the number of cubic units need to fill the space occupied by a solid 142. Circular cylinder- a cylinder with two bases that are paralle~ congruent circular

regions 143. Circular cone - a shape in space that has a circular base and one vertex 144. Surface area - the sum of the areas of all the faces of a three-dimensional figure 145. Tree diagram- a diagram used to show the total number of possible outcomes in

a probability experiment 146. Monomial - a number, variable, or a product of a number and one or more

variables 147. Polynomial - the sum or difference of two or more monomials 148. Binomial - a polynomial with exactly two terms 149. Factoring - finding the factors of a product 150. Trinomial - a polynomial with three terms

Appendix E - Essential Information - Computation of Fraction Examples

FABULOUS FRACTIONS

SKILL NUMBER SKILL EXAMPLES

1/8 + 1/10

ADDITION

1 OF 7/12 + 1/3 FRACTIONS

1/7 + 9/28

3/8-3/5

3/25 - 1/50

2 SUBTRACTION OF 1/3 - 1/4

FRACTIONS

3 ½ - 1 5/6

1 3/8 - 1/2

5/16 X 1/5

3 MULTIPLICATION OF 4/5 X 3/4

FRACTIONS

3 1/3 X 3/5

16/3 X 1/8

¾+½

DMSION

4 OF 3 + 6/7 FRACTIONS

1/8 + 1/3

1¼+3 ½

69

Appendix F - Essential Information-Essential Concepts Examples

Skill Skill Example Number

1 Add the negative integers. -4 + 7

2 Subtract the negative integers. 9 - -5

3 Multiply the negative integers. 4x-3

4 Divide the negative integers. -8 +4

5 Use order of operation. 7(6-4) + 3

6 Solving one step algebraic equations. 9 + n = 26

7 Solving two step algebraic equations. 3x-4 = 7

8 Solve and graph one step inequalities. n-9 < 26

9 Use a formula to solve a problem. A=l+w

10 Apply the commutative. property. Rewrite 6 + 4

11 Apply the associative property. Rewrite (9 + 3) + 7

12 Apply the distributive. Rewrite 7(3 + 8)

13 Combine like terms. 3(x + 4x)

14 Solving open addition problems. X + 7 = 29

15 Solving open subtraction problems. M-4 = 13

16 Solving open multiplication problems. 7w=35

17 Solving open division problems. Q + 14 = 6

18 Solving proportions. X= 10 3 15

19 Solve linear equations. 2x+y=10

20 Estimate to solve a problem. 203 X 19

21 Translate verbal phases into algebraic 7 less than twice a

expressions. number q.

70

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4 basic operations leaf plots, box-and- Uses tools and techniques

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with rational whisker plots) for determining area and numbers. • Identify and use volume.

• Simplifies powers, measures of central • Find the area of exponents, and tendencies (mean, various types of 2-square roots. median, mode). dimensional figures.

• Uses order of • Find the surf ace area operations to simplify Demonstrate an and volume of prisms expressions. understanding of • Find the volume of

• Find equivalent forms probability. cylinders . of decimals, • Find the probability of fractions, and one or more events. percents.

Algebra Geometry

Demonstrate the ability to Understands and applies solve linear equations. geometric properties in • Solve one-step order to solve problems.

equations using • Use proportions to rational numbers. solve problems

• Solve two-step involving similar equations with one figures. variable term (i.e., • Use the Pythagorean 3x-2=34). Theorem to find the

lengths of sides of Demonstrates the ability right triangles . to translate common language into algebraic form. Write simple algebraic expr~sions and equations from verbal phrases.

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