THE EFFECTS OF STUDENT-CENTERED ACTIVITIES
ON STUDENT MATH FACT AUTOMATICITY
Linda M. Drawbert
B.A., California State University, Sacramento, 1975
THESIS
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF ARTS
in
EDUCATION
(Curriculum and Instruction)
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
SUMMER
2010
THE EFFECTS OF STUDENT-CENTERED ACTIVITIES
ON STUDENT MATH FACT AUTOMATICITY
A Thesis
by
Linda M. Drawbert
Approved by:
__________________________________, Committee Chair
Rita M. Johnson, Ed.D.
__________________________________, Second Reader
Julita G. Lambating, Ph.D.
____________________________
Date
ii
Student: Linda M. Drawbert
I certify that this student has met the requirements for format contained in the
University format manual, and that this thesis is suitable for shelving in the Library
and credit is to be awarded for the thesis.
_____________________________________
Robert H. Pritchard, Ph.D., Department Chair
Department of Teacher Education
iii
___________________
Date
Abstract
of
THE EFFECTS OF STUDENT-CENTERED ACTIVITIES
ON STUDENT MATH FACT AUTOMATICITY
by
Linda M. Drawbert
Statement of Problem
California’s accelerated math standards state that first graders know their
addition math facts and corresponding subtraction facts to 20. Research shows that
many students do not meet this standard. The demands of NCLB’s accountability and
high stakes testing make for a fast paced math program that neglects the needs of
students who do not have strong math intelligence or have math disabilities. Learning
stages involving hand-on activities are being given short shrift in lieu of rote
memorization. With a review of literature highlighting the importance of learning
modalities, multiple intelligences and brain-based learning, could student-centered
activities that combine all these elements produce gains in math fact fluency leading to
math fact automaticity?
iv
Sources of Data
Information was obtained through research on the topics of high stakes
accountability, learning theories, modalities of learning, multiple intelligences, and
brain-based learning. Data was collected from informal observations of students
involved in the study and from assessments and tests administered to the researcher’s
second grade students and students in another second grade class that acted as a
control. Students attended a suburban school in Sacramento’s San Juan Unified School
District.
Conclusions Reached
Students in the experimental group continued to improve their math fact fluency
even after the study ended. The mean difference between the pretest and the postposttest was significant. Males in the experimental group also had a mean difference
between the posttest and the post-posttest that was significant. Since improved
performance in math fact fluency leads to math fact automaticity, the effects of
student-centered activities on student math fact automaticity were shown to be
positive. An additional benefit of student-centered activities was a more positive,
relaxed attitude by students toward learning math facts.
, Committee Chair
Rita M. Johnson, Ed.D.
_______________________
Date
v
ACKNOWLEDGMENTS
I would like to take this opportunity to thank the people who have offered their
time, expertise and guidance throughout the formation of this thesis.
Dr. Rita M. Johnson
Dr. Julita G. Lambating
Dr. Frank R. Lilly
I would also like to thank my colleagues Nancy Sinnwell, Ann Munsee, and Shana
Walters for their unwavering support and encouragement during the writing of this
thesis.
vi
TABLE OF CONTENTS
Page
Acknowledgments ....................................................................................................... vi
List of Tables ............................................................................................................... ix
List of Figures............................................................................................................... x
Chapter
1. INTRODUCTION .................................................................................................. 1
Purpose of the Thesis........................................................................................ 1
Statement of the Problem ................................................................................. 1
Significance of the Thesis ................................................................................ 2
Limitations ........................................................................................................ 4
Definition of Terms .......................................................................................... 6
Organization of the Thesis................................................................................ 7
2. LITERATURE REVIEW ....................................................................................... 9
High-stakes Accountability .............................................................................. 9
Learning Theories ........................................................................................... 12
Multiple Intelligences ..................................................................................... 16
Modalities of Learning ................................................................................... 17
Brain-based Learning ..................................................................................... 19
Implications for Teaching............................................................................... 21
Summary......................................................................................................... 25
3. METHODOLOGY ............................................................................................... 27
Methodology................................................................................................... 33
Activities......................................................................................................... 35
Research Methods .......................................................................................... 37
Expectations ................................................................................................... 40
vii
4. DATA ANALYSIS .............................................................................................. 42
Informal Observations .................................................................................... 42
Daily Assessments .......................................................................................... 44
Math Fact Fluency Tests ................................................................................ 46
5. CONCLUSIONS AND RECOMMENDATIONS ............................................... 55
Conclusions .................................................................................................... 55
Recommendations .......................................................................................... 61
Appendix A. Performance Levels Graphs on Sums Assessments .......................... 62
Appendix B. Graphs of Individual Performance on Daily Assessments .................. 65
Appendix C. Performance Graphs on Math Facts Fluency Tests ............................ 69
Appendix D. Make a Ten Column Addition ............................................................ 74
Appendix E. Domino Card Activities ...................................................................... 76
Appendix F. Number Line Floor Mat....................................................................... 87
Appendix G. Math Fact Fluency Test and Daily Assessments ................................ 92
References ................................................................................................................. 108
viii
LIST OF TABLES
Page
1.
Percentage of Del Paso Manor Students Enrollment by Ethnicity ................. 29
2.
2009-2010 Math Grade 2 Classroom Benchmark Reports,
Experimental Group .................................................................................. 30
3.
2009-2010 ELA Grade 2 Classroom Benchmark Reports,
Experimental Group .................................................................................. 31
4.
2009-2010 ELA Grade 2 Fluency Classroom Benchmark Reports ............... 31
5.
2009-2010 Math Grade 2 Benchmark 3 (EOY) District Exam
Report ........................................................................................................ 32
6.
2009-2010 ELA Grade 2 Benchmark 3 (EOY) District Exam
Report ........................................................................................................ 32
7.
2009-2010 ELA Fluency Grade 2 Benchmark 3 (EOY) District
Exam Report .............................................................................................. 32
8.
Performance Levels for Addition Math Fact Fluency, as
Determined by Researcher ........................................................................ 45
9.
Data for All Pre and Post Tests ...................................................................... 47
10.
Comparison of the Means for the Experimental and Control
Groups, All Tests ....................................................................................... 52
ix
LIST OF FIGURES
Page
1.
Comparison of Experimental and Control Group Performance
Levels on Pre-Test ..................................................................................... 49
2.
Comparison of Experimental and Control Group Performance
Levels on Post-Test ................................................................................... 50
3.
Comparison of Experimental and Control Group Performance
Levels on Post-Post-Test ........................................................................... 51
4.
Comparison of the Means for the Experimental and Control
Groups, All Tests ....................................................................................... 52
x
1
Chapter 1
INTRODUCTION
Purpose of the Thesis
The purpose of this teacher action research study is to investigate the effects of
supplementing the current math curriculum with activities aimed at musical,
kinesthetic/tactile, visual, linguistic, and mathematical intelligences. Through the use
of assessment and observations, the researcher of this thesis will document and
analyze the effect of integrating activities that engage multiple intelligences in the
teaching of basic addition math facts to 18. This three-week study takes place in the
researcher’s self-contained second grade classroom.
Statement of the Problem
California, in 1997, adopted accelerated standards for math, K-12. As
delineated in Mathematics Content Standards for California Public Schools (1999), the
primary standards for learning addition and subtraction math facts are: Kindergarten:
2.1 Use concrete objects to determine the answers to addition and subtraction
problems (for two numbers that are each less than 10); Grade 1: 2.1 Know the addition
facts (sums to 20) and the corresponding subtraction facts and commit them to
memory; and Grade 2: 2.1 Understand and use the inverse relationship between
addition and subtraction (e.g., an opposite number sentence for 8 + 6 = 14 is 14 - 6 =
8) to solve problems and check solutions.
2
In the article “First-Grade Basic Facts: An Investigation into Teaching and
Learning of an Accelerated, High-Demand Memorization Standard” (Henry & Brown.
2008), the authors found that only 32% of the students demonstrated Basic Facts
Competence on even half of the facts. This left them with the question whether
students are unable to achieve this standard in first grade or does the teaching method
need to change.
As a state adopted text, the Scott Foresman math textbook, second grade
edition, includes twelve chapters. It covers so many math concepts under California’s
accelerated math program that teachers must tightly pace their lessons. The strategies
for learning addition facts in the second grade Scott Foresman text, the current
adoption used by San Juan Unified School District, include Lesson 1-2: Using
Doubles and Near Doubles and Lesson 1-3: Using 10 to Add 7, 8, and 9. Students are
expected to practice math facts at home with flash cards. Computer tutorials, games,
and board games are utilized at school. This researcher found that fewer than 25% of
the students in her class scored above 80% on this study’s initial test of 45 random
addition facts to 18 in four minutes.
Significance of the Thesis
Math fluency is the number of correct answers divided by the number of
minutes it takes to answer the questions. The standards are set at a young age so that
multi-step problem solving can be undertaken. This automaticity of basic facts serves
to free sufficient mental resources for a learner to focus their attention on the more
3
complex aspects of a numerical task (Gray, 2004). Automaticity is when memory
recall of facts or information is instantaneous. In Brain-based Learning, Jensen (2008)
states “the current understanding is that multiple memory locations and systems are
responsible for our learning and recall” (p. 155). The more senses teachers use to
engage students while learning new material, the more connections and more locations
for memory to be stored. Recall is stimulated by trigger information associated with
mental, emotional, and physical states while learning. Previous studies (Brackney,
2007; Miller, 2006; Montgomery, 2007) have used multiple intelligence applications
in their research. In these studies, treatments focused on one multiple intelligence at a
time even when more than one intelligence was tested. This action research studies the
effects of concurrent activities based on multiple intelligences. It is this researcher’s
expectation that a higher percentage of students who learn math facts through a variety
of activities that stimulate the senses will have better recall than those who learn math
facts through standard methods taught in the state adopted text and through rote
memorization. It is also the researcher’s expectation that the engaging nature of the
activities will motivate the students to learn the facts.
Previous math fact fluency studies focused on the learning of disparate facts
(Poncy, Skinner, & Jaspers, 2007) and fact families (Brackney, 2007). This action
research study focuses on learning the sums of numbers to 18.
4
Limitations
There were a few limitations to this action research study. One limitation was
the time needed to implement each lesson. The lessons built upon each other, with
each succeeding lesson reviewing all the previous facts learned. Beginning lessons
required approximately 15 minutes while later lessons took up to 30 minutes. This
impacted the pacing of the standard curriculum for the three-week period of the study.
A second limitation was the small sample size. Twenty second-graders were
the participants in this research. During the three-week study period, there were 37
absences. Data was analyzed for the 17 students who took all three assessments: the
pretest, the posttest right after the completion of all the lessons and the post-posttest
administered three weeks later.
A third limitation was the absence of continued data from a second control
group comprised of 21 second grade students who were practicing math facts by
taking biweekly 35-problem quizzes. The first control group of second graders who
were not practicing math facts during class time had seventeen students take all three
assessments. Having more data on control groups would have made analysis of results
more meaningful. Was improvement due to the intensity of the practice or the
activities?
A fourth limitation was the willingness of the learners to be actively involved
in the experience. Initially, the study was to focus on students who were not proficient
with math fact fluency. It was the assumption of the researcher of the study that
5
students already proficient would not find the activities worthwhile. However,
observations by the researcher found that all students actively participated and no
negative comments were heard. Taking a personal interest in doing well on the
assessments proved a challenge to at least two students who had days when they did
not complete half their normal number of problems. One other student, wanting to
finish in the allotted time, randomly wrote answers that resulted in a very low
percentage of problems being correct.
A limitation for anyone wanting to replicate this study is that the materials
used for the activities, domino cards and number line floor mats, are not available
commercially. The domino cards must be run off on card stock to provide a set of 54
cards for each student (see Appendix E). Felt number line floor mats must be made.
(see Appendix F).
Assessments were created from the website Free Math Tests: Create Math
Tests/Custom Math Tests at http://www.rbechtold.com/math.html. A limitation in
using this web site was that although parameters for the sum of numbers were allowed,
the researcher could not eliminate two-digit numbers as addends in the math facts
randomly generated for the sums greater than 10. Would these two-digit addition
problems have a negative influence the test results?
6
Definition of Terms
The following terms have been defined as follows for the purpose of this study.
Basic addition math facts: the 100 addition facts of two single-digit numbers,
from 0+0 to 9+9.
Benchmark assessments: district tests in math, English language arts, writing
and reading fluency administered at the end of each marking period to measure
students’ progress and provide teachers with data to modify instruction.
Domino cards: playing cards representing the 54 basic addition math facts with
domino pips and corresponding numbers in the corners developed by the author of this
study to provide students with a visual aid to learning math facts.
Doubles: the addition math facts where both addends are the same (Example:
3+3, 8+8, etc.)
Fact Family: the three numbers related by addition and subtraction. (Example:
7+8=15, 8+7=15, 15-7=8, and 15-8=7)
Family of facts: facts with the same addend. (Example: 5+1, 5+2, 5+3, 5+4,
etc.)
Make a ten: using partners for the sum of 10 to add columns of numbers; also a
strategy used to solve facts with addends of 7, 8, or 9. (Example: 8+7=(8+2) +5=15)
Near Doubles: the addition math facts where one addend is 1 or 2 more than
the other addend. (Example: 1+2, 4 +5, 5+6, 6+8, etc.)
7
Number line floor mat: a 9” X 162” felt strip with the numbers 1 through 18,
spaced vertically, used to provide students with a kinesthetic experience to learning
addition facts by hopping out or stepping off the addition facts, counting on from the
first addend to arrive at the answer. (Example: What number + 5=9? The students start
at 5, then counts the number of hops to reach 9. Upon reaching the number, the student
states the math fact, 5+4=9)
Retrieval: the act of accessing information in long-term memory
Sums of numbers: addition facts that add up to a certain number. (Example:
Sum of 10 includes the facts 0+10, 10+0, 1+9, 9+1, 2+8, 8+2. 3+7, 7+3, 4+6, 6+4, and
5+5)
Organization of the Thesis
This thesis is comprised of five chapters. Chapter 1 is the introduction, which
includes the purpose of the study, the statement of the problem, the significance of the
thesis, limitations, and the definition of terms used in the study. Chapter 2 is the
review of literature. Topics covered in this chapter include high stakes accountability,
learning theories, multiple intelligences, modalities of learning, and brain-based
learning. Chapter 3 is the methodology. It provides a description of the participants,
the setting, an explanation of the design of the study, the activities, and assessments
used to collect data. Chapter 4 presents an analysis of the data collected. Chapter 5
contains a summary of the findings and recommendations for future research. People
who would have an interest in this research include teachers, school administrators,
8
and curriculum developers. The Appendices include copies of the domino cards of
addition math facts, activity sheets and workmat developed by the researcher,
directions for making the number line floor mat and templates for numbers, copies of
the daily assessments and tests and graphs of individual and group performance levels
on the assessments and tests and an explanation of the activity for adding columns of
numbers, Make a Ten.
9
Chapter 2
LITERATURE REVIEW
I hear and I forget. I see and I remember. I do and I understand. (Confucius, 500 BC).
How do students learn? What can teachers do to facilitate this learning for all
students in an era of high-stakes accountability? This review looks at the everchanging landscape of education in America, the theories of learning expounded over
the years and implications for instruction based on the peer reviewed articles and
research.
High-stakes Accountability
The changing demands on school reform during the last two decades have
changed and intensified the roles of teachers. This intensification, enacted by policymakers through regulations and procedures for a fast-paced, rigorous curriculum that
have created a climate of high-stakes accountability, finds teachers at odds with their
vision of best practices and the pedagogies they’re directed to implement (Ballet,
Kelchtermans, & Loughran, 2006; Valli & Buese, 2007).
The No Child Left Behind Act is designed to help all students meet high
academic standards by requiring that states create annual assessments that
measure what children know and they can do in reading and math in grades 3
through 8. Districts and schools that do not make sufficient yearly progress
toward state proficiency goals for their students first will be targeted for
assistance and then be subject to corrective action and ultimately restructuring.
10
Schools that meet or exceed objectives will be eligible for ’academic
achievement awards.’ (U.S. Department of Education, 2001, Introduction)
The data derived from all the testing is to allow parents, administrators,
policymakers and the general public to track the performance of every school in the
nation (U.S. Department of Education, 2001). Corporate America views schools as
data-driven institutions and sees data as a way to strengthen the future workforce
(Emery & Ohanian, 2004). President Obama has enacted his administration’s plan for
school reform, “Race to the Top.” With $315 million from the Statewide Longitudinal
Data Systems program, states will expand their data systems to track students'
achievement from preschool through college and link their achievement to teachers
and principals (U.S. Department of Education, 2009). Tying teacher evaluation to
student’s achievement on tests presupposes the student’s mind is a blank slate ready to
be filled with knowledge Steven Pinker (2002) refers to the philosophy of The Blank
Slate as “the idea that the human mind has no inherent structure and can be inscribed
at will by society or ourselves” (p. 2). This metaphor has gained prominence in
popular culture. Walt Disney said, “I view a child’s mind as a blank book. During the
first years of his life, much will be written on the pages. The quality of the writing will
affect his life profoundly” (Tk, 2001, p. 130). But this idea is not recent. John Locke
(writing in 1690) used the metaphor of the mind as “white paper void of all character,
without ideas” (1959, p. 121) on which is written by experience.
11
William Godwin (1797) postulated “children are a sort of raw material put into
our hands” (para. 37). But, by the time a child enters school, they have already
experienced informal education. His mind can no longer be considered blank (Jensen,
2008). More recently, Susan Ohanian (2001) observed,
People who aren’t familiar with junior high students-or any students in any
other grade, for that matter, have the naïve notion that kids learn what teachers
teach. The truth of the matter is that few schools have ever stopped teaching
the right stuff; somewhere along the way a lot of kids stopped agreeing to learn
it. (p. 21)
NCLB’s goal of having 100% of America’s public school students at the
proficiency level by the year 2014 is changing how instruction is delivered. Teachers
are now required to move through the curriculum on the district’s schedule because
benchmark tests have to be given within a prescribed time period. Curriculum content
must match state standards and be aligned with the state tests. With districts
mandating the pace at which units are covered and when assessments are given,
teachers are supposed to differentiate instruction to reach all students. The narrow
focus on teaching to a state test may produce inflated gains in scores. But the
fundamental concern is with improved achievement, not just higher test scores
(Amrein & Berliner, 2002; Koretz, 2005; Stecher & Hamilton, 2002).
Is 100% proficiency an unreasonable goal? Rosenberg (2004) finds that
different tests yield different percentages of proficiency, since “proficient” is a
12
judgment about how students ought to perform. However, under NCLB, proficiency
defined by NAEP standards, is so high that it is completely out of reach. Having a goal
that is unobtainable, no matter how hard teachers try, can do more to demoralize than
to motivate greater effort. Goals need to provide a challenge, but not be set so high
that they are unachievable (Linn, 2003). Gardner (1997) wrote,
What to make of all of this? I say, pay one’s respect to school and to IQ tests,
but do not let them dictate one’s judgment about an individual’s worth or
potential. In the end, what is important is an individual’s actual achievements
in the realms of work and personal life. (p. 42)
Learning Theories
Each child comes to school with a cultural background and an intelligence all
their own. Teachers can help students learn by using teaching strategies that utilize
ideas based on cognitive development theory, the different types of intelligences, and
modalities of learning and knowledge of the brain’s pathways to memory for retrieval.
Over the years, through observation and analysis, behavioral psychologists
have developed theories to explain how people learn. The Theories of Learning in
Educational Psychology (n.d.) homepage presents a comprehensive list of theorists,
gives an overview of their theories, and organizes the theories into four perspectives.
This review is meant as a background of basic theories, therefore only a sampling of
theorists for each perspective has been selected.
13
Behaviorist Perspective
Schools owe much of their educational practices to behaviorists. These
practices include “systematic design of instruction, behavioral and performance
objectives, programmed instruction, competency-based instruction, and instructor
accountability” (Merriam & Caffarella, 1991). Skinner (Public Broadcasting System
[PBS], 2009) is closely associated with operant conditioning, the pairing of responses
with particular consequences, believing that learning is developed through imitation
and reinforcement. A behavior that gets a positive consequence is more likely to be
repeated. The theory that a behavior can be conditioned is part of a teacher’s
objectives when planning a lesson. Teachers often use positive reinforcement for
behavior management in the classroom. NCLB subscribes to this theory in its
rewarding of schools that meet their API and the sanctioning of schools that fail to
meet their API.
Skinner devised a technique known as shaping. He believed that complicated
tasks can be broken down into small segments and learned through rewarding
successive approximations of desired behavior. These incremental rewards
strengthened the learning. Computer games and tutorials use many of the principles of
Skinner's shaping technique (PBS, 2009).
14
Cognitive Perspective
Piaget (1973) defined three periods of intellectual development in children:
1. the period of sensorimotor intelligence from birth to age 2;
2. the period of preparation and of organization of concrete operations of
categories, relations and numbers from age 2 to age 11, which is further
subdivided into the preoperational stage from ages 2-7 and the concrete
stage from ages 7-11; and
3. the period of formal operations from ages 11 to 14. Although Piaget based
his three periods on age levels, understanding of concepts builds on
previous learning.
Some critics have found Piaget’s stages to be too rigid. On his Learning and Teaching
website, James Atherton (2009) states, “many children manage concrete operations
earlier than he thought, and some people never attain formal operations (or at least are
not called upon to use them)” (Atherton, stages of cognitive development, para. 1).
Bruner was a constructionist theorist. His stages of development are enactive,
iconic and symbolic (Bruner, 1996). In math, students often understand concepts first
by manipulating objects, the enactive stage; then by seeing the concepts in picture
form, the iconic stage; and ultimately understand the symbols used to represent
concepts, the symbolic stage. He believed that children must go through all of the
stages in successive order to create understanding. With the mandate of No Child Left
Behind and standards based curriculum, where the pace of curriculum is guided by
15
textbook manufacturers (Emery & Ohanian, 2004), the early stages of learning are
given insufficient instructional time and students are introduced to the more advanced
stage before strong connections can be made.
Humanistic Perspective
Carl Rodgers felt that the emphasis on cognitivism took the joy out of learning.
He believed that the highest level of learning took place through experiential learning.
Personal involvement allowed feelings and emotion to be part of the learning
experience. He also believed that teachers should be genuine with students and value
them as individuals (Rogers, 1983). “High stakes testing is having the effect of
eliminating whatever there has been of learning for the joy of it, learning to develop
higher-order thinking skills or learning something because it is what one is interest in”
(Emery, 2004, p. 202). The fast pace of curriculum delivery due to testing schedules
and the intensification of teacher roles due to accountability have also had a negative
impact on the relationships teachers form with their students (Valli & Buese, 2007).
Social Perspective
Vygotsky, a constructivist, believed that learning precedes development (Moll,
1991). He did not label the progression children make in understanding as stages. He
saw the social and cultural influence very important to a child’s development. He
believed learning took place in social environments where the knowledge of others
helps shape an individual’s leaning. His More Knowledgeable Other (MKO) theory,
suggests a scaffolding design of assistance provided by the more knowledgeable peer
16
or adult. His Zone of Proximal Development (ZPD) concept about the difference of
what a child can learn on his own and what a child can learn with guidance allows the
child to develop skills with less frustration and greater growth (Vygotsky, 1978). This,
too, is in contrast to the one-size-fits-all view of No Child Left Behind.
Multiple Intelligences
In light of these theories, where children create their own understanding
through discovery and guidance by more knowledgeable others, it is important to
understand that children have multiple intelligences (Gardner, 1993). Gardner defined
the nine intelligences as logical/mathematical, linguistic, musical, bodily/kinesthetic,
spatial, interpersonal, and intrapersonal, naturalist, and existentialist (Gardner, 1997).
The logical/mathematical learners recognize abstract patterns and have strong problem
solving and reasoning skills. Linguistic learners like to read, discuss, and listen.
Musical learners easily remember melodies and rhythms. Bodily-kinesthetic learners
like to touch and gesture and learn through their senses. Spatial learners like to draw,
design things, and build. Interpersonal learners understand and care about people,
learning through group interaction. Intrapersonal learners are self-motivated and enjoy
working alone. The naturalistic learner is in touch with nature. The existentialist
learners are reflective and prefer learning in context. Although everyone has all these
intelligences, some are stronger in some people and weaker in others. With practice
these intelligences can be strengthened. Traditional texts and teaching focus on the
learners with strong logical/mathematical and linguistic intelligences. These are
17
easiest to assess with standardized tests. Most standardized tests questions measure
cognitive skills on the three lowest levels of Bloom’s taxonomy: knowledge,
comprehension, and application (Anderson, 1999; Bloom, Englehar, Furst, Hill, &
Krathwohl, 1956).
Modalities of Learning
Learning style is the way in which each learner begins to concentrate on,
process, absorb, and retain new and difficult information (Dunn & Dunn, 1978). Three
learning styles are visual, auditory, and tactile/kinesthetic. Visual learners need to see
information in graphs, charts, and diagrams. Auditory learners learn best through
lecture, focusing on tone of voice, speech, and verbal cues. Tactile/kinesthetic learners
learn best through manipulation, hands-on activities, and movement.
Taking a more in-depth look at preferences for learning, The Dunn and Dunn
Learning-Style Model (1978) encompasses both developmental and biological
elements. Its defined learning styles encompass twenty-one elements in five strands:
environmental, emotional, sociological, physiological, and psychological.
Environmental elements involve lighting, sound, and location. Emotional elements
include motivation, persistence, responsibility, and structure. Sociological elements
differ: group work or independent projects, teacher directed or project-based.
Physiological elements are visual, auditory, and kinesthetic. Psychological elements
correspond to right brain/left brain, impulsive or reflective, and global versus analytic
18
tendencies (Dunn, 2000). On her website, Dr. Teresa Dybvig and Church (2009)
emphasize,
Since we tend to teach the way we learn best ourselves, we often find it
mysteriously difficult to reach students who learn differently. When we don't
teach in the way the student learns best, lessons can be a struggle, even though
we may like the student and believe in their talent and intelligence (para. 2)
A meta-analysis of Dunn and Dunn model by Lovelace (2005) concluded that
teaching with learning styles in mind would produce higher achievement and improve
student attitude toward learning. However, Kavale and LeFever (2007) were not
convinced of Lovelace’s findings because it focused on a single model of learning
styles.
Kolb (1984) defined four styles of learners: accommodators, convergers,
divergers, and assimilators. Accomodators work concretely and actively through the
question, “What would happen if I do this?” Assimilators conceptualize abstractly and
reflectively observe on the question, “What is there to know?” Convergers
conceptualize abstractly and actively experiment to answer the question, “How does
that work?” Divergers work concretely and reflexively think on the question, “Why?”
Kolb believed the most effective problem solving and learning occurred when people
used all four types of learning styles. Assimilators and convergers would score higher
on standardized tests than divirgers, and accommodators would find the test difficult
(McCarthy, 1997). The 4Mat Learning System developed by McCarthy (1997) has
19
teachers deliver lessons through each of the four styles in sequence: diverger,
assimilator, converger, and accomodator. This system allows students to work in their
preferred style and to develop skills in the other three styles. Since standardized
testing does not make allowances for learning styles, this system of cycling through
learning styles would seem to be beneficial to all types of learners (Wilkerson &
White, 1988).
Brain-based Learning
With the advances in technology over the last 20 years, neuroscientists have
been able to discover many aspects of the brain’s structure and how it works. The
brain contains nerve cells (neurons) that receive and process signals and relay through
synaptic connections to other neurons of the brain.
As a nerve cell is stimulated by new experiences and exposure to incoming
information from the senses, it grows branches called dendrites. Dendrites are
the major receptive surface of the nerve cell. One nerve cell can receive input
from as many as 20,000 other nerve cells. If you have 100 billion cells in your
brain, think of the complexity!” (Weiss, 2000, p. 28)
says Marian Diamond, a neuroscientist and professor of neuroanatomy at University
of California, Berkeley. With this magnitude of information, very little gets stored in
long term memory.
For information to have meaning, it must be relevant, involve emotion, and be
in context. Rote memorization of unrelated facts has little to do with true learning
20
(Jensen, 2008). When we can connect rote memory to experience, we understand and
remember it more easily (Caine & Caine, 1997). Memories associated with the five
senses create the strongest associations in long-term memory. Long before science
made it possible to study the working of the brain, John Locke (writing in 1690)
regarded perception and the internal reflections on them to be the most important
aspects of understanding. He wrote in An Essay Concerning Human Understanding,
Our observation employed either, about external sensible objects or about the
internal operations of our minds perceived and reflected on by ourselves, is
that which supplies our understandings with all the materials of thinking. This
great source of most of the ideas we have, depending wholly upon our senses,
and derived by them to the understanding, I call SENSATION. (Locke, 1959,
para. 1)
The MRI and fMir allow researchers, through visual imagery, to observe the
workings of the brain and learn which areas of the brain are active when engaged in a
variety of tasks. In turn, cognitive theorists use this information to connect good
teaching practices to maximize learning in the classroom. “Brain-based education is
the engagement of strategies based on principles derived from understanding of the
brain” (Jensen, 2008, p. 4). These strategies are based on how the brain learns best.
Information can be stored in multiple areas of the brain-the visual cortex, the audio
cortex, the motor cortex and the prefrontal cortex-and is joined by pathways of
neurons. Multiple memory locations and systems are responsible for our learning and
21
recall (Jensen, 2008). Having multiple pathways to memory available would suggest
that recall of information would be more accessible to learners with different learning
styles and intelligences.
Implications for Teaching
Robert Marzano (2003) defines several of the factors that are the primary
determinants of student achievement. Of the three teacher-level factors: instructional
strategies, classroom management and classroom curriculum design, teachers have
control over only the first two. The California State Board of Education, in 1999,
determined the scope and sequence of the mathematics curriculum for public schools
with the adoption of content standards. With these standards came state adopted
textbooks. Scott Foresman California Math textbook was specifically written to meet
the Mathematics Content Standards for California Public Schools: Kindergarten
Through Grade Twelve. Its first and second grade texts teach addition facts through
several lessons of mnemonic strategies such as counting on doubles, near doubles,
making tens, and fact families. With the dense curriculum of California’s accelerated
math standards, textbooks provide few lessons for each strategy.
Most instruction jumps directly from the characterization of addition and
subtraction through simple physical models to the memorization of number
facts without acknowledging that there is an extended period during which
children count-on and count back to solve addition and subtraction problems.
(Carpenter & Moser, 1984, p. 200)
22
Henry and Brown (2008), in their study “First-grade Basic Facts: An
Investigation into Teaching and Learning of an Accelerated, High-demand
Memorization Standard,” found that only 32% of the students demonstrated Basic
Facts Competence on even half of the facts. This left them with the question whether
students are unable to achieve this standard in first grade or does the teaching method
need to change. Research studies of educational practices in several high-performing
countries (Fuson & Kwon, 1992; Fuson, Stigler, & Bartsch, 1988) found that students
were not simply drilled on basic facts using rote memorization techniques, but
received explicit and sustained instruction on redistributed derived-facts strategies in
first grade. Henry and Brown concluded, “At least for the present, it makes sense for
teachers and professional developers to develop a healthy skepticism for the pacing
and instructional strategies recommended in at least some state approved textbooks”
(p. 181).
Cooper (2005) postulates a hierarchy of eight number fact skills students use to
solve math problems:

8) counting with errors,

7) guessing the facts,

6) counting or sequencing correctly,

5) referring to mnemonic clues,

4) using number relationships,

3) converting to reverse operations,
23

2) recalling number facts (delayed) and

1) automatic recall of facts.
In teaching strategies that use multiple intelligences and modalities of learning, all
students may not achieve automatic recall of all number facts, but they may move up
the hierarchy to increase their speed and accuracy in their computations.
Automaticity of addition and subtraction facts enables students to focus their
working memory on more complex aspects of problem solving (Gray, 2004). Students
who are able to recall number facts quickly and who are fast in using mnemonic
strategies perform better on arithmetic skills tests (Imbo & Vandierendonk, 2007).
Researchers (Ackerman, Anhalt, & Dykman, 1986; Geary, Hoard, Byrd-Craven,
Nugent, & Numtee, 2007) investigating the relationship between automaticity and
learning disabilities including attention deficit hyperactivity disorder and dyscalculia
have found that students employing counting strategies fail to put the basic facts into
long term memory because their working memory is too short to make the
connections, which in turn leads to poor performance in math. Other research studies
(Bielsker, Napoli, Sandino, & Waishwell, 2001; Haught, Kunce, Pratt, Werneske, &
Zemel, 2002; Miller, 2006; Montgomery, 2007; Poncy, Skinner, & Jaspers, 2007)
have shown achievement in math fluency and automaticity through the use of art,
music, games, touch math, mnemonics and regimented programs as higher than that of
rote memorization. In providing instructional strategies that engage many learning
styles, teachers provide multi-sensory experiences. Gray states, “An environment that
24
promotes practice with numbers in a variety of settings can be used to encourage the
development of basic numeracy skills without the need for mindless repetition and
rote” (p. 43).
Rote memorization puts isolated facts into long-term memory without making
connection to meaning since it is devoid of context, emotion and relevance (Jensen,
2008). Investigating memory and cognitive function, Weaver (2006) examined the
actor’s immersion in the emotional, physical, and mental senses of the experience to
communicate meaning when learning lines. Bielsker et al. (2001) suggest, “The key to
successful teaching is to access as many of these memory lanes as possible. The
students will be able to recall information faster if they have a variety of ways to
connect the information to learning” (p. 43).
When learning is relevant, involves active processing and is set in an
environment that is relaxed and safe (Wilson, 2007), students will develop the
emotional connections that lead to long-term memory. Teaching with learning styles
in mind will increase achievement and motivation in students (Kritsonis, 1997-1998).
Robert Marzano (2003) postulated three student-level factors that influence
achievement: home atmosphere, learned intelligence and background knowledge, and
students’ motivation. Of the three, student motivation is the one teachers have some
control over. “Constructivist learning, where children make active sense of the formal
mathematics they are adding to their repertoires, decreases the stress and increases the
25
depth of thought, the joy, and the creativity in mathematics study” (Bank Street
Corner, 2009, para. 11).
In 1946, Edgar Dale developed the Cone of Experience model to emphasize
the progression of conceptual understanding from concrete experience to abstract
expression (though not necessarily in an order or hierarchy). Dale did not put numbers
to his model, but later versions with the levels labeled with percentages of what people
generally remember appeared without his consent (Dale, 2008). His model delineates
the beliefs of Piaget, Vygotsky, Bruner and Confucius: the best learning takes place
through active participation.
Summary
In summary the NCLB’s mandate that all students score at the proficient level
on standardized tests by 2012 presents a conundrum for teachers. How do teachers
balance the need to cover the prescribed curriculum while providing engaging learning
experiences to students with diverse learning styles and multiple intelligence
strengths?
Strictly adhering to the rigorous fast-paced curriculum guidelines adopted to
meet state standards precludes lessons that provide in-depth experiential learning that
students at various stages of cognitive development need for understanding. Textbooks
are geared toward linguistic and logical/mathematical learners with visual and
auditory learning styles. Students who have strength in musical, bodily/kinesthetic,
spatial, interpersonal and intrapersonal, naturalist or existentialist intelligences need
26
differentiated instruction. Brain-based research suggests that experiences that engage
all the senses produce strong memory connections to multiple areas of the brain. These
stored memories are then made available for retrieval through multiple pathways.
Using information from this literature review about multiple intelligences and
learning styles, my hypothesis is that students who learn math facts through
instructional strategies that combine songs, movement, and games will produce higher
gains in achievement on posttests than students who learn math facts only through
textbook mnemonic strategies and rote memorization.
27
Chapter 3
METHODOLOGY
Del Paso Manor Elementary School (DPM) is nestled in a residential area
bounded by the business corridors of Watt, Eastern, Marconi and El Camino Avenues.
The neighborhood is comprised of single-family homes, duplexes and apartment
complexes. Students walk, ride their bikes, and come by car. Bus transportation was
one of the budget cuts made by San Juan Unified School District. The school
playground, with its primary and intermediate play structures, blacktop, playing field,
grass areas, basketball courts and gazebo with tables and benches, shares the
boundaries with a public park maintained by Mission Oaks Recreation and Park
District. A 50 year old school that has benefited from modernization money, DPM has
a large multipurpose room/cafeteria with a curtained stage for assemblies and class
performances, a computer lab of 34 iMac computers established through a technology
grant and maintained through PTA budgeting, and a library that houses reference
books and a 1700-book collection for the Accelerated Reader Program used by the
students. PTA funds have established individual classroom and school-wide gardens.
The positive climate of the school is not just in its surroundings and amenities.
Del Paso Manor Elementary is a dynamic K-6 school of approximately 504
students that offers many resources to its students and parents. A strong PTA raises
money to provide funds for field trips, classroom supplies, computers, library books
and assemblies. The Bridges After-School Program provides free academic
28
intervention and social activities to 80 students. Band and choir offer after school
programs. There are 22 fully credentialed classroom teachers, a librarian, a resource
teacher, a resource aide, a language specialist, a Spanish bilingual instructional
assistant, a half-time computer tech, a speech therapist, a cafeteria service worker, a
secretary, a clerk, a principal, three noon aides and two custodians employed at the
Del Paso Manor. DPM is a magnet school for students identified as gifted and talented
(GATE) students throughout the district. The Rapid Learner (RL) program is grades 26. Approximately 27% of the student body is in the RL program. Gatepost and a
GATE advisory committee support the RL program with a Spanish program and
additional field trips.
As a teacher who has taught students in grades 1, 2, 5, and 6 at Del Paso
Manor School during the past 23 years, the researcher has been witness to many
changes at the school. During the first years at DPM, neighborhood and RL programs
were separate entities with SIP funds being split by enrollment in each programs.
Later, a technology grant spearheaded by an RL parent scrutinized this practice. For
grant approval change was made to ensure that all students and parents felt they
belonged to Del Paso Manor Elementary not just to the program in which they were
enrolled. Teachers created activities that could be shared across programs and grade
levels. PTA sponsored whole school projects. Today, Del Paso Manor Elementary is a
2010 California Distinguished School. Its two signature practices are collaboration and
differentiation.
29
Del Paso Manor is a culturally diverse community (see Table 1). The transfer
of students caused by a nearby school closure has created changes in the ethnology of
the school. An increasing number of students identified as English Language Learners
(ELL) are adding to the diversity. This year DPM had 82 identified ELL students.
Students receive 30 minutes daily instruction with a language specialist or with a
Spanish bilingual instructional assistant.
Table 1
Percentage of Del Paso Manor Students Enrollment by Ethnicity
Racial/Ethnic Category
African-American
American Indian
Asian
Caucasian
Hispanic or Latino
2007-2008
8%
2%
11%
60%
18%
2008-2009
8.7%
2.3%
10%
54.6%
22.4%
Del Paso Manor is academically diverse as well. Its school report card reports
a 2008-2009 API of 851. But this score is not disaggregated between RL and
neighborhood programs. With the RL scores taken out, the overall API would be
considerably lower. Over the last 10 years as a second grade teacher, the researcher
has found successive second grade classes have come in with less preparation to meet
state standards.
The researcher’s second grade students were the participants in the
experimental group in this study. They were a mix of cultures and ability levels. Thirty
30
percent of the students were ELL, with proficiency levels ranging from beginning for
the Japanese student, early intermediate for two Spanish students and intermediate for
two Spanish and one Russian student. Sixty percent belonged to two parent
households, 25% percent belonged to single parent families and 15% belonged to
blended families.
San Juan requires trimester benchmark tests for all elementary students with
scores reported to DataDirector (see Tables 2, 3, and 4). Test scores are for trimesters
2 and 3 to encompass only those students who took both Benchmark tests in math and
English Language Arts. District reading fluency test results are also charted to show
the academic level of students in the experimental group. Performance levels were set
for the benchmarks by the district. Tables show the number of students scoring at each
level.
Table 2
2009-2010 Math Grade 2 Classroom Benchmark Reports, Experimental Group
Math
Benchmark 2 (03/10/10)
Performance Level
Proficient: 70.01-100%
Basic: 44.01-70%
Below Basic: 24.01-44%
Far Below Basic: 0-22%
# Students
10
8
2
0
Benchmark 3
(06/02/10)
# Students
10
8
2
0
31
Table 3
2009-2010 ELA Grade 2 Classroom Benchmark Reports, Experimental Group
English Language Arts
Performance Level
Proficient: 70.01-100%
Basic: 44.01-70%
Below Basic: 24.01-44%
Far Below Basic: 0-22%
Benchmark 2 (03/09/10
# Students
7
11
0
0
Benchmark 3 (06/01/10)
# Students
9
6
5
0
Table 4
2009-2010 ELA Grade 2 Fluency Classroom Benchmark Reports
Performance Level
Advanced: 75.01-100%
Proficient: 50.01-75%
Approaching: 25.01-50%
Below: 0-25%
Benchmark 2
March 11, 2010
3
9
4
4
Benchmark 3
June 3, 2010
4
8
3
5
A comparison of district scores, experimental group scores and control group
scores on the End of Year Benchmarks shows the performance levels of students in
the experimental group relative to the control group and to the district’s total second
grade students for which scores were recorded in DataDirector (see Tables 5, 6, and
7).
32
Table 5
2009-2010 Math Grade 2 Benchmark 3 (EOY) District Exam Report
Performance Level
Proficient: 70.01-100%
Basic: 44.01-70%
Below Basic: 24.01-44%
Far Below Basic: 0-22%
District
(2127 students)
75%
20%
5%
0%
Experimental Group
(20 students)
50%
40%
10%
0%
Control Group
(21 students)
52%
33%
14%
0%
Table 6
2009-2010 ELA Grade 2 Benchmark 3 (EOY) District Exam Report
Performance Level
Proficient: 70.01-100%
Basic: 44.01-70%
Below Basic: 24.01-44%
Far Below Basic: 0-22%
District
(2204 students)
67%
24%
8%
1%
Experimental Group
(20 students)
45%
20%
35%
0%
Control Group
(21 students)
48%
29%
24%
0%
Table 7
2009-2010 ELA Fluency Grade 2 Benchmark 3 (EOY) District Exam Report
Performance Level
Advanced: 75.01-100%
Proficient: 50.01-75%
Approaching: 25.01-50%
Below: 0-25%
District
(2530 students)
41%
34%
13%
10%
Experimental Group
(20 students)
20%
40%
15%
10%
Control Group
(21 Students)
N/A
N/A
N/A
N/A
Students in the experimental group had significantly more students in the
performance levels of basic and below basic than the district in math and English
Language Arts but comparable numbers to the control group. This research action
33
study with the participation of this class of students with only half of them proficient
in math, language arts and reading fluency should make data significant. The results
should prove useful to any teacher with students who struggle academically. It was the
expectation of the researcher that analysis of the data would provide clues to which
level of students benefit most from the integration of student centered activities for
math automaticity.
Methodology
Students learn the words to songs they like automatically by hearing them over
and over. A study by Wallace (1994) found that “music, when repeated, simple, and
easily learned, can make a text more easily learned and better recalled than when the
same text is learned without melody” (p. 1473). The songs for addition facts by Ron
Brown have catchy tunes and are unique thematically, making them fun for students to
sing and move to. With daily repetition, the author expected increased motivation to
learn and readily recall the math facts presented in the songs.
One strategy for learning addition math facts presented in the Scott Foresman
text adopted by San Juan unified School District was “Make a ten.” Since our math
system is base 10, place value is a fundamental concept. The first step in the strategy
was to use counters to represent the numbers. The first number of counters was placed
on a ten-frame mat, a 5 X 2 grid. The second number of counters was added to the grid
until it was full, thus making a 10. The remaining counters added to 10 to get the sum.
(Example: 7+5=(7+3)+2=12.) The next phase of this process was to make the tens
34
without counters. Therefore finding partners of 10 is an essential component of
addition. Since students have 10 fingers, this is a less abstract concept for them. The
researcher found students adept at finding the partners of 10. Practice was done when
students were asked to add the number of laps the class had run during their biweekly
Walk/Run Across America activity. Each student’s number of laps was recorded on
the board and on graph paper with column heading of hundreds, tens and ones. The
column of numbers was then added by making tens. Two numbers (partners) that
added to 10 were circled, then a 1 (ten) was placed in the tens column. More partners
were recognized and circled with the resulting 1 (ten) placed in the tens column.
Students became proficient at renaming numbers so that partners could be made
adding more than two numbers to make the 10. When all numbers were circled except
those that did not add up to 10, they were added to get the ones digit. The numbers in
the tens column were added to get the tens digit (see example in Appendix D).
In observing the effectiveness of this strategy for learning the addition facts for
the sum of 10, the researcher chose to focus this action research on the teaching of
addition math facts as the sum of numbers instead of the traditional family of facts.
The assessments were generated with the sum of addends as the determining factor
from Free Math Tests: Create Math Tests/Custom Math Tests at
http://www.rbechtold.com/math.html.
35
Activities
Songs
The CD Addition by Ron Brown and produced by Intelli-tunes provided the
songs for this study. The songs teach addition facts as the sum of numbers. Song titles
include: “Five”, “Six Rap”, “It’s a Seven”, “Eight”, “Nine Alive”, “Partners for Ten”,
“11 Party”, “Twelve Rap”, “Thirteen!”, “Highway 14”, “Fifteen March”, “S-S-SSixteen”, and “Seventeen.” A new song was introduced each day of the study with
song lyrics, copied from the booklet that came with the CD, distributed each day.
Students listened to the song, singing along while the song replayed three times. The
following day, all the previous songs were sung before the new song was introduced.
Domino Addition Fact Cards
The author created a set of addition fact playing cards that resemble domino
tiles (see Appendix E). Each card has a line dividing its face into two square ends.
Each end is marked with a number of pips or is blank. Each card represents two
related addition facts. An addition problem is printed in the top right hand corner to
represent the number of pips in each half of the card. The related addition fact is
printed upside down in the bottom left hand corner so that it corresponds to the
number of pips in each half when the card is turned around. A class set of the 54 cards
was printed on card stock. Cards were put into sets of sums. After a sum was
introduced through song, students received the addition fact domino cards and a Sum
Card for that day. They would then mix them up with the other domino cards they’d
36
received previously. It was then their task to find the cards that matched the day’s
sum, place the cards on a workmat and record the addition facts that related to that
sum on a worksheet. Students recorded all related sums by turning their workmats half
way round to see the addition problems that had been upside down. Students who
needed the visual cues of the pips counted the pips to find the correct sum. Each
student wrote his/her name on the back of the cards. After the lesson, each student’s
cards were placed in a plastic bag with his/her name. The bags were collected and kept
in a storage box until the next lesson.
Number Line Floor Mats and Missing Addend Game
The author created vertical number lines so students could physically move
their bodies along it (See instructions in Appendix F). Felt strips 9” X 162” were cut
from bulk felt. This size was chosen to keep costs down and to allow them to fit on the
classroom floor. The strips were divided into 18, 9” X 9” squares on which 4” sticky
back felt numbers were pressed. Inexpensive commercial numbers were not available,
so the author drew the block numbers, made templates to create stencils, traced the
numbers on 9” X 12” sticky back felt sheets and cut them out. Ten felt mats were
made for a class of 20 students. Students played in pairs. Finding a space on the
classroom floor, they folded the mat so that the sum of the day was at the end.
Students pooled their domino addition fact cards of the day. The first player asked the
other, “What number plus (one of the addends) equals (sum)?” The partner stepped on
the mat at the first addend, then hopped and counted the number of spaces it took to
37
reach the sum. To complete their turn, the student had to say the complete number
sentence, (addend) + (addend) = (sum). The student then chose a new addition fact
card and asked the partner to find the missing addend. Play continued until all cards
were played. Students who completed their card stacks early were allowed to use all
the cards in their bags to play the missing addend game. Cards were then sorted by
name and returned to the plastic bags at the end of play.
Research Methods
The first day of the study a 45-problem test of basic math facts to 18 with
single digit numbers was administered to students. Students were asked to complete as
many problems as they could in four minutes. Having observed students adding twodigit numbers in the course of math lessons over the year, the researcher anticipated a
low rate of proficiency. In order to alleviate frustration and build confidence, the
researcher chose to begin this study of math facts that were familiar to students, the
sum of 5.
After tests were collected, the students were given a copy of the song lyrics for
“Five.” Students heard and sang the song through three repetitions. The author used
finger cues to reinforce the number facts, displaying a clenched fist for zero, full hand
for five, pointer finger for one, four fingers minus the thumb for four, pointer and
middle fingers for two and pointer, middle and ring finger for three. Students were
encouraged to display the finger signals for numbers during the song. With the fast
pace of the song, many miscues with the hand gestures enlivened the lesson. Once the
38
song ended the third time, students were given a folder in which to put the song sheet.
They were each then given a set of domino cards with the facts to the sum of 5 and a
Sum of 5 card. Students were instructed to place these cards on a workmat with spaces
marked for six cards. Students then received a worksheet on which to write each
number sentence with the sum of 5, i.e. 0+5=5, 1+4=5, 2+3=5. Students were
instructed to turn the workmat half way round to view the cards in the opposite
position and write the resulting number equations, i.e. 5+0=5, 4+1=5. 3+2=5. This
worksheet was then put into their folder. The Number Line/Missing Addend game was
then explained to the students. Students picked partners, combined their sets of sum
cards, unfolded the mats to end with the number five and laid them on the floor.
Partners took turns asking missing addend questions and hopping on the number line
floor mat to find the answers, then stating the number sentences until both partners’
sets of domino cards had been played. When sufficient time had passed for this to be
accomplished, students were asked to fold up the mats, sort their cards and place them
into plastic Ziploc bags. Bags were collected and stored in the classroom.
The second day of the study, students got out their folder and song sheet for
the song, “Five”. After singing it through three times, the song sheets were put away
and an assessment of 45-math facts with sums to 5 was administered with a time limit
of 4 minutes. Following the assessment, the students received the song lyrics for “Six
Rap”. After singing the song through three times with appropriate hand gestures,
students were given their Ziploc bag of domino cards, a Sum of 6 card and a packet of
39
cards with facts with the sum of six. Students were instructed this time to mix all the
cards up, find the sum of six cards and place them on their workmat and write the
number sentences for all the sums of six. The Number Line/Missing Addend game
was played this time with sums of six cards.
Students chose their partners from the group who finished writing the number
sentences when they did. Thus the students were always active. There was no set place
in the room students had to set up their number line floor mat, so choice was involved
in this as well. It was the researcher’s expectation that having a choice of partners and
location would engage the students and promote ownership of the game.
The third day of the study, students got out the folder and song sheets for the
songs, “Five” and “Six Rap.” Both songs were reviewed before a 45-problem
assessment on addition facts with sums to six was administered. Following the
assessment, a song sheet with the song, “It’s a Seven” on one side and “Eight” on the
other was handed out. “It’s a Seven” song was introduced and sung three times. A
Sum of 7 card and a set of domino addition fact cards with the sum of seven were
handed out. Students mixed these cards with the sum of 5 cards and the sum of 6 cards
before finding all the sum of 7 cards. After writing all the number sentences created by
these cards, students played the Number Line/Missing Addend game.
Successive days’ activities repeated this procedure. Students reviewed the
previously taught songs, took an assessment test of the previously studied sums,
learned a new sum song, received the Sum card and set of domino cards related to the
40
day’s sum, shuffled and sorted domino cards for the sum of each day, wrote the
number sentences for each fact of the day’s sum, and played the Number Line/Missing
Addend game.
During the Number Line/Missing Addend game, if students used all their
domino cards for the day’s sum, they got the rest of their cards, mixed them together
and reviewed the other addition facts. This was especially true after the sum of 14
because there were so few new facts to learn for the higher sums.
After all math facts of sums to 18 were taught, the same instruments used as
the pretest was administered as a posttest. All the songs were reviewed just prior to the
posttest. Three weeks later the same instrument was used as a post-posttest with no
review of the addition facts to 18. Would they have better recall of facts for having
participated in the musical, kinesthetic and visual activities? Informal observations
were made of students while engaged in the activities and while taking the
assessments. What was the attitude toward the activities? What strategy did they use to
get the answers during testing? Would students who performed at the below level of
math fluency on the pretest show more improvement on the posttest than the students
who performed at the approaching level? Would math fact fluency be maintained after
the study ended and intense practice of math facts stopped?
Expectations
The expectations were that many students who were struggling with math fact
automaticity would improve their math fluency on the addition facts to 18. Students
41
who performed at the below level on the initial test would perform at the approaching
level or better on the posttest and exhibit less anxiety when taking a timed test of math
facts. Students who performed at the approaching level in the pretest would experience
success and would perform at the proficient level on the posttest. Students who were
already proficient or advanced would continue to succeed. Another expectation was
that students would have fun learning the addition math facts.
42
Chapter 4
DATA ANALYSIS
Chapter 3 detailed the methodology used to conduct the study. A pretest,
posttests and post-posttest were administered to ascertain the effectiveness of the study
on the class as a whole and on individual students. Data on daily assessments of
previous addition math facts studied was collected for each student. Math fact
automaticity, the instantaneous recall from memory, is the goal of learning math facts,
but first students must increase their math fact fluency, the number of problem
answered correctly in a given time period. The expectations were that many of the
students who were struggling with math fact fluency would succeed in improving their
performance level on a 45-problem, 4-minute test of basic addition math facts to the
sum of 18. It was expected that fewer students would rely on finger counting to obtain
answers during the posttests. Another expectation was that all students would actively
participate in the activities as observed through informal observations.
Informal Observations
In the interest of not limiting the data, the author chose to informally observe
the class as a whole. Seven students relied on counting their fingers during the pretest.
These students completed 6 to 35 of the 45 problems on the test. On the post-posttest,
four of these same students were observed using their fingers, but the number of
problems completed ranged from 17 to 43. Although the use of fingers was still a
strategy for some students, the number of facts needing the strategy had decreased.
43
Students had positive comments on the songs and eagerly sang along with each
successive song. “Can we listen to the math songs?” was a suggestion made by
students during free choice time. Students would often be out of their seats dancing
along with the songs. “This is fun,” was a comment about the Number Line/Missing
Addend game made by a student who was adept at math fact automaticity. Most
students were eager to finish the daily math facts tests within the time limit. As time
was kept on the board, students wrote their ending time on their papers when they
finished each test. The decision to begin with sums to 5 allowed all students to feel
successful in taking and completing the tests of sums up to 10. One student was so
eager to complete the test on time, the student guessed at many problems resulting in
many incorrect answers. When the researcher explained that the importance of the test
was to get as many right as possible, the student put more effort into getting problems
correct instead of finishing all 45 problems in 4 minutes. The student’s percentage of
correctly completed problems rose steadily from a low of 53% on the sums to 13 to a
high of 84% on the sums to 16. Attitude toward the tests had an effect on test results.
Two students had difficulty staying focused on the sums to 17 assessment, completing
just 9/45 and 12/45 problems compared to their scores on the sums of 16 test of 24/45
and 45/45 respectively (see graphs in Appendix B). “This is hard,” was one negative
comment made by a student trying to find all the facts for the sum of 17 in the stack of
45 domino addition fact cards. The student used the strategy of counting the pips to
find the correct cards. The student had expected to find cards to put in all six boxes on
44
the workmat. When the researcher explained that not all the boxes would be needed
for each sum, the student was able to complete the task. The student’s attitude become
more positive, quickly writing the number sentences and moving on to the Number
Line/Missing Addend game.
Daily Assessments
Math facts automaticity is defined as the instantaneous recall of addition facts
from memory. Math fluency is the number of correct answers divided by the number
of minutes it takes to answer the questions. For this study, a 45-problem test format
with a 4-minute time limit was established. Similar timed math fact tests in use at Del
Paso Manor School, known as “Busy Bees,” had a format of 35 problems to be
completed in 3 minutes. This kept the two tests within the same performance range.
The researcher set the performance levels for the assessments and the tests used in this
study (see Table 8). With math fluency as the goal, the performance levels were set
higher than the district performance levels on the benchmark tests described in
Chapter 3. San Juan Unified School District uses a scale of 4 performance levels on
their report cards for K-6 schools: exceeds standards (4), meets standards (3),
approaches standards (2) and below standards (1). Above and below in each level is
represented with a + or -. For the purpose of this study, intermediate levels were used
to delineate small improvements in student test scores.
45
Each day, after reviewing the previously learned math fact sums by singing the
corresponding songs from Addition by Ron Brown, a 45-problem assessment of math
facts of sums up to the number learned was administered with a 4-minute time limit.
Table 8
Performance Levels for Addition Math Fact Fluency, as Determined by Researcher
Performance Level
4, Ad, Advanced, Exceeds standards
3+, Pr +, Proficient +, Meets standards
3, Pr, Proficient, Meets standards
3-, Pr -, Proficient -, Meets standards
2+, Ap +, Approaching standards
2, Ap, Approaching standards
2-, Ap -, Approaching standards
1+, Be +, Below standards
1, Be, Below standards
Lowest
Percentage
100%
96%
88%
80%
72%
64&
56%
48%
0%
Ave. score out
of 45
45
43.2
39.6
36
32.4
28.8
25.2
21.6
0
A statistical analysis using Grade Pro was done on the results of the daily
assessments using the experimental group. Assessments of sums to 5, sums to 6, sums
to 7, sums to 8, sums to 9, and sums to 10 had an average proficiency rate of 97.4%
with a standard deviation 4.25. The high proficiency rate can be attributed to previous
retention of these facts from first grade (see Appendix A). They were also the easiest
for students who still count on their fingers to visually get answers. Sums to 11 had an
average of 96.7% with a standard deviation of 4.4. Sums to 12 had an average of
91.9% with a standard deviation of 11.1. Sums to 13 had an average of 79.6% with a
standard deviation of 17.3. Sums to 14 had an average of 79.9% with a standard
46
deviation of 14.6. Sums to 15 had an average to 82.2% with a standard deviation of
20.4. Sums to 16 had an average of 83.5% with a standard deviation of 21.7 the first
time it was administered and 93.7% with a standard deviation of 11.9 when it was
administer to the same students the next day. Sums to 17 had an average of 82.0%
with a standard deviation of 25.8. Sums to 18 had an average of 82.1% with a standard
deviation of 16.4. The greater standard deviation for the tests of the sums to 13 and
higher, suggest that the students who use finger counting had more difficulty with
these math facts. The previously mentioned abnormally low scores of two students on
the assessment of sums to 17 account for the highest standard deviation of 25. The
data suggest that more practice on the sums to 13 and above is needed. This
conclusion is supported by the data for the two tests on the sums to 16. The average on
the second test increased from 83.5% to 93.7% with a decease in standard deviation
from 21.7 to 11.9. The data was taken for students who took both tests on the sums to
16 (See Appendix B).
Math Fact Fluency Tests
Three tests were administered to the experimental group and the control group.
The pretest was administered to both groups before any review of addition math facts
was undertaken. This test consisted of 45 facts to the sum of 18 randomly sorted with
a time limit of 4 minutes. The same instrument was used for the posttest administered
to both groups at the end of the study. Students in the experimental group reviewed all
the math facts by singing along with Addition by Ron Brown prior to taking the test.
47
The control group had no review. Three weeks after the end of the study the same
instrument was administered as a post-posttest to both groups with no review of
addition facts to 18 by either group prior to the test.
Data for all pre and posttests are compiled in Table 9. Students in both the
experimental group and the control group were listed alphabetically by first name then
given a student number. 19 females and 15 males participated in the study.
Table 9
Data For All Pre and Post Tests
Student
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Group
Experimental (1)
Control (2)
1
2
1
2
1
2
2
1
2
2
1
1
1
2
1
1
1
2
1
1
2
2
1
1
Score/45
Pretest
21
45
35
35
45
37
29
23
44
43
6
25
23
22
20
18
25
19
18
35
42
15
45
19
Score/45
Posttest
23
45
38
36
45
33
39
31
45
45
31
30
21
35
24
15
20
22
26
40
45
15
45
28
Score/45
Post-Posttest
23
45
41
44
44
31
29
25
45
45
27
32
30
24
31
31
17
30
21
43
45
14
45
36
Gender
Male (1)
Female (2)
1
2
1
2
2
2
1
2
1
2
2
1
1
1
1
2
2
2
2
1
2
1
2
2
48
Table 9 (Continued)
25
26
27
28
29
30
31
32
33
34
2
1
1
2
2
2
2
2
1
2
13
10
41
25
33
5
29
8
34
42
13
21
30
33
44
2
34
17
24
45
17
16
40
34
37
7
21
19
32
44
2
1
1
1
2
2
2
1
2
1
Grade Pro was used to ascertain performance levels using results on the three
sets of test results of the experimental group and the control groups (see Figures 1, 2,
and 3). Each group was comprised of 17 students who had complete data for all tests.
Data was used to make individual performance graphs using Excel (see Appendix C).
Groups graphs comparing scores on each test was also made using Excel (see
Appendix C).
49
Pretest Math Fact Fluency
7
Number of Students
6
5
4
3
2
1
0
Be
1
Be+
1+
Ap- 2-
Ap
2
Ap+
2+
Pr3-
Pr
3
Pr +
3+
Ad
4
Performance Level
Experimental Group
Control Group
Figure 1. Comparison of Experimental and Control Group Performance Levels on
Pre-Test
The Experimental Group had more students performing at the performance
levels of Below and Below+ than the Control Group. The Control Group had more
students at the performance levels of Proficient and Advanced than the Experimental
Group.
50
Posttest Math Fact Fluency
7
Number of Students
6
5
4
3
2
1
0
Be
1
Be+
1+
Ap- 2-
Ap
2
Ap+
2+
Pr3-
Pr
3
Pr +
3+
Ad
4
Performance Level
Experimental Group
Control Group
Figure 2. Comparison of Experimental and Control Group Performance Levels on
Post-Test.
The gains made from the pretest to the posttest by the Experimental Group
may be attributed to the intensity of practice. The gains made by the Control Group
could possibly be attributed to in-class practice for California’s STAR testing that
concluded the week before the posttest. However, the data also shows a marked
decrease in the standard deviation for the Experimental Group between the pretest and
the posttest. This signifies achievement by the lowest performers. Eleven students
performed at the below level on the pretest. Four of those same students performed at
the approaching level on the posttest. The standard deviation for the Control Group
51
had a slight gain between the pretest and the posttest. Two below performers moved to
the approaching performance level; four proficient performers moved to the advanced
performance level.
Post-Posttest Math Fact Fluency
7
Number of Students
6
5
4
3
2
1
0
Be
1
Be+
1+
Ap2-
Ap
2
Ap+
2+
Pr3-
Pr
3
Pr +
3+
Ad
4
Performance Level
Experimental Group
Control Group
Figure 3. Comparison of Experimental and Control Group Performance Levels on
Post-Post-Test.
Students in the Experimental Group continued to improve performance from
the posttest to the post-posttest. Some students in the Control Group lost ground.
Data analysis of the three tests was made using Excel (see Table 10). The data
show both groups made mean gains during the study period (see Figure 4).
52
Table 10
Comparison of the Means for the Experimental and Control Groups, All Tests
Test
Pretest
Posttest
Post-posttest
Experimental Group
Mean
SD
26.06
11.421
28.64
8.750
31.41
9.193
Control Group
Mean
SD
28.59
13.139
32.24
13.562
31.24
12.562
Mean
Comparison of the Means of All Tests
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Pretest
Posttest
Post-posttest
Test
Experimental
Control
Figure 4. Comparison of the Means for the Experimental and Control Groups, All
Tests.
A t-test to determine mean difference was done using SPSS. The mean
difference between the pretest and the posttest in the experimental group (M=2.88,
SD= 8.484) was not statistically significant, t(16) = 1.401, p = 0.180. The mean
53
difference between the pretest and the posttest in the control group (M=3.65,
SD=4.949) was significant, t(16) =3.038, p=0.008). So even though the both groups
had gains during the three-week study, the treatment was not the main reason for the
gain. However, more interesting data analysis results from the comparison of the
pretest and the post-posttest. The mean difference between the pretest and the postposttest in the experimental group (M=5.35, SD= 7.305) was statistically significant,
t(16) = 3.021, p = 0.008. The mean difference between the pretest and the post-posttest
in the control group (M=2.647, SD=5.267) was not significant, t(16)=2.072, p=0.55).
Was the gain of math fact fluency due to the student centered activities that allowed
for multiple ways of accessing information in the brain? Did the lessons aide students
in developing a schema for information storage and retrieval?
In order to examine whether this was equally significant for males and females,
further analysis of this data was done by gender. The mean gain from pretest to postpost test in the males of the experimental group (M=5.75, SD = 3.594) and the females
of the experimental group (M = 5.00, SD = 9.721) was not significant, t(15) = .205, p
= .840. The mean gain from the pretest to post-posttest in the males of the control
group (M = 3.43, SD = 4.650) and the females of the control group (M= 2.10, SD
5.840) was not significant, t(15) = .500, p = .625. For males, the mean gain from
pretest to post-posttest for the experimental group (M =5.75, SD = 3.694) and the
control group (M = 3.43, SD = 4.650) was not statistically significant, t(13) = 1.078, p
= .301. For females, the mean change from pretest to post-posttest for the
54
experimental group (M = 5.00, SD = 9.721) and the control group (M = 2.10, SD =
5.840) was not statistically significant, t(17) = .798, p = .436. Interesting results, since
the mean difference of the experimental group as a whole was significant but by
gender it is not. This may be due to small sample size. It would be worth pursuing
with a larger sample.
A post-posttest was administered to test the effects of student centered
activities on long term memory, The mean difference between posttest and postposttest in the experimental group (M = 2.47, SD = 6.3345) was not significant, t(16)
= 1.605, p = 1.28. The mean difference between posttest and post-posttest in the
control group (M = -1.00, SD = 6.134) was not significant, t(16) = -.672, p = .511.
Further analysis by gender produced some interesting results. The mean difference
between the posttest and the post-posttest for males in the experimental group (M =
3.63, 4.955) and the males in the control group (M = -2.86, SD 5.336) was significant
t(13) = 2.439, p = .030. The mean difference between the posttest and the post-posttest
for females in the experimental group (M = 1.44. SD = 7.518) and the females in the
control group (M = .30, SD = 6.584) was not significant, t(17) = .354, p = .728. Males
involved in the student centered activities increased their understanding more than
females.
55
Chapter 5
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
The purpose of this study was to discover the effects of student-centered
activities on math fact automaticity. During the author’s past 10 years of teaching
second grade at Del Paso Manor School in the San Juan Unified School District in
Sacramento, fewer than 25% of the students coming into second grade knew the
addition math facts by rote. Providing flash cards for home use and using computer
games did not lead to improved math fluency. Administering “Busy Bee” tests, a
series of progressive, timed math tests students must pass before going on the next
test, proved frustrating to the lower performing students. Would games and songs
provide more relevance and a reason for students to automatize the addition math
facts? Would student-centered activities help the lowest performers process
information in their own way allowing for growth in math fluency? Are gains made
during the study a result of the intense practice of math facts? Would gains in math
fact fluency remain stable after the study ends?
In observing students add two-digit numbers, I found many of them using
their fingers. When the sums were greater than 10, some students became confused. A
number line on their desks provided support for addition facts. As a second grade
teacher, I was concerned that students still needed to use their fingers with addition. In
my review of literature, I learned that this strategy is not uncommon. In the article
56
“Dyscalculia: Neuorscience and Education” Liane Kaufmann (2008) reviews research
findings that suggest a neurofunctional link between fingers and number processing.
Finger counting is natural for children because of their convenience and their
correspondence to our base-10 number system. The author relates behavioral research
that supports the notion that finger counting seems to be related to calculation
proficiency in elementary school children. With the research showing the importance
of finger use in developing number sense, the use of fingers for counting and
calculating increases in importance and should be encouraged rather than being
discouraged in favor of rote memorization in the first couple of years of school.
It is plausible to expect that the consistent use of fingers could positively affect
the formation of mental number representations (by facilitating the mapping
from concrete non-symbolic quantity knowledge to abstract symbolic number
processing) and thus also the acquisition of calculation skills. (Kauffman,
2008, p. 171)
Could a game improve math fluency for students who still needed to count on their
fingers? The Number Line/Missing Addend game was developed to explore this
question. It provides the kinesthetic movement and visual stimulus that some students
may need to make sense of abstract number concepts.
I began my teaching career in the late 1980s. The whole language approach to
reading was the educational paradigm. Thematic units were explored through all
avenues: projects, art, writing, and field trips. Math Their Way by Mary Barrata-
57
Lorton (1995) was a state adopted activity-based curriculum taught to assist primary
students in developing understanding and insight of math patterns through the use of
manipulatives. With this constructivist background and my review of brain-based
learning, multiple intelligences and modalities of learning, I now understand why this
approach is so powerful. Patricia Wolfe (2001) states in Brain Matters: Translating
Research Into Classroom Practice, “Information is not stored in a specific location in
the brain but in various locations-visual, auditory, and motor cortices-and is joined in
circuits or networks of neurons” (p. 72). Would activities that specifically target the
visual, auditory and motor cortices help students’ recall of math facts and lead to
automaticity? Would using them all within the same lesson enhance the neural
connections being made in the brain? With this question in mind, I set out to design
activities that would target those three areas. The domino cards with pips and
numerals provided the visual stimulation. Songs with interesting lyrics and rhythms
targeted the auditory cortex. The number line floor mat activity was designed to
stimulate the motor cortex.
In my review of literature, I found support for the use of student-centered
activities to aid in learning information. Jerome Bruner (1996) theorized that learners
transfer knowledge through three modes: enactive (action based), iconic (imagebased) and symbolic (language-based). Piaget (1973) postulated four modes of
learning: concrete, pictorial, semi-abstract and abstract. When teachers ask students to
memorize math facts without sufficient time to go through the learning stages or they
58
have a learning disability, their understanding is not solid and memory recall is
difficult. Current textbooks often start at the pictorial level and move to the abstract
level of cognition quickly. California’s current emphasis on standardized testing and
an accelerated curriculum preclude many hand-on activities because of pacing
demands.
The theories on modalities of learning and multiple intelligences give
compelling reasons for embarking on this methodology. The data suggest that the
basis for this study has merit. Students were beginning to make progress toward higher
math fluency leading to math fact automaticity. It is the recommendation of the author
that some changes to the methodology be made for further study. An added day of
practice for each of the fact sums to 13 and higher should be scheduled. An additional
activity to help students practice math facts should take the place of writing the facts
on the second day. The card game “War” can be played with the domino fact cards
already placed in the plastic bags from previous lessons. Students pick partners. They
shuffle their own cards and place them face-down. In unison, both players turn over
the top card in their own stack. Students determine the sum of each card placed in
“battle”. The student with the highest sum wins both cards, placing them on the
bottom of their stack. If the sum of the cards is equal, students each places three cards
from their stack face down on the first card, then place a fourth card face-up (a war).
The student with the highest sum on the fourth card wins and collects all the cards
played on that round. Play continues until one player wins all the cards, or time is
59
called. With the pips on the cards, low performing students would have the
opportunity to determine the sum without guessing.
As the saying goes, practice makes perfect. But type of practice, frequency and
intensity of the practice affect its effectiveness. Are we as teachers asking students to
learn lists of information? Eric Jensen (2008) states in Brain-Based Learning: A New
Paradigm of Teaching, “Factors involved in meaning making are relevance, emotion
and context” (p. 180). When students practice math facts as games, the information
becomes relevant and in a context they can understand. When students participate in
student-centered activities, they associate math facts with the positive emotion of
enjoyment and fun. These emotions trigger memory of the event and related
information forms a pathway in the brain that can be accessed when recall of facts is
needed. The use of the domino cards for the card game “War” and the weekly singing
of the math songs would provide continued practice throughout the year.
Motivation to learn the math facts is increased when students are engaging in
activities that are fun, involve talk with others, are visually stimulating, involve music
and allow movement. Student-centered activities accommodate the visual, auditory
and tactile/kinesthetic styles of learning (Dunn & Dunn, 1978). The data support the
expectation that student-centered activities give low performing math students who
may not be strong in mathematic intelligence or have dyscalculia, a math learning
disability, more pathways for storing information in memory and increase math fact
fluency. If they have stronger visual intelligence, the domino cards and the number
60
line floor mat can help them process the abstract concepts of numbers. If they have
stronger musical intelligence the lyrics and melodies of the songs can help them relate
the facts to music. If they have stronger kinesthetic intelligence, whole-body
movement along a number line floor mat can help them associate the hopping to
counting-on to find sums and missing addends. These activities strengthen math
fluency, and with practice lead to math fact automaticity.
The expectation of the study was that students would improve their
performance on math fact fluency during the study and that they would retain their
fluency performance levels after the study ended. The number of students who used
finger counting as a strategy dropped after the study, thus making retrieval of math
facts faster. Fifty-four percent (6/11) of the students at the below performance level on
the initial test increased their math fluency to approaching level on the second posttest. One hundred percent (3/3) of the students who were performing at the
approaching level on the initial test increased their math fluency to proficient level on
the second post-test. The question remains, “Why did two students fail to show any
progress?” An explanation for this may lie in the rigidity of a timed test. Both students
are compulsive about the correctness of their schoolwork. They often asked, “Is this
right? Like this?” They both scored in the proficient range of the district’s end of year
math benchmark, which is not timed. In their research study on strategy development
and working memory, Imbo and Vandierendonck (2007) found that “highly anxious
children used retrieval less often than did low-anxious children” (p. 303). Working out
61
the answers through strategies like finger counting takes more time than automatic
responses, resulting in lower performance levels.
The most compelling findings of the data analysis that support the hypothesis
that student centered activities will have a positive effect on math fact automaticity are
the experimental group’s statistically significant gains between the pretest and the
post-posttest and the statistically significant gains of males in the experimental group
between the post test and the post-posttest. Although the direct lessons stopped, math
facts continued to be part of classroom conversations about adding and subtracting
multi-digit numbers, measurement, and problem solving. It is heartening to find
evidence that providing student centered activities that combine different modalities of
learning and different intelligences can help students develop their own schema for
storing information in their brains.
Recommendations
It is the recommendation of this researcher that student centered activities be
integrated into an intense four-week review of math facts at the start of the school year
in second grade if initial test data suggest that students do not have math fact
automaticity. It is also recommended that time for these student-centered activities
should be scheduled monthly, or even weekly, for maintenance of addition math fact
fluency. The Number Line/Missing Addend game should be reviewed as a precursor
to subtraction facts study. It is a further recommendation that student-centered
activities be integrated into the first grade curriculum when the facts are first taught.
62
APPENDIX A
Performance Levels Graphs on Sums Assessments
63
Performance Levels of the Experimental Group on Sums Assessments
Sums to 6
Sums to 7
12
Number of Students
Number of Students
12
10
8
6
4
2
0
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
6
4
2
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Pr -, 3-
Sums to 8
Sums to 9
Pr, 3
Pr +, 3+
Ad, 4
Pr, 3
Pr +, 3+
Ad, 4
Pr, 3
Pr +, 3+
Ad, 4
Pr, 3
Pr +, 3+
Ad, 4
Pr, 3
Pr +, 3+
Ad, 4
Number of Students
12
10
8
6
4
2
10
8
6
4
2
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
Performance Level
Sums to 10
Sums to 11
12
Number of Students
12
10
8
6
4
2
0
10
8
6
4
2
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Performance Level
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
Sums to 12
Sums to 13
12
Number of Students
12
Number of Students
Ap +, 2+
Performance Level
0
10
8
6
4
2
0
10
8
6
4
2
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Performance Level
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
Sums to 14
Sums to 15
12
Number of Students
12
Number of Students
Ap, 2
Performance Level
12
Number of Students
8
0
Be, 1
Number of Students
10
10
8
6
4
2
0
10
8
6
4
2
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
Pr, 3
Pr +, 3+
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
64
Performance Levels of the Experimental Group on Sums Assessments (continued)
Sums to 16, Assessment 2
Sums to 16
12
Number of Students
Number of Students
12
10
8
6
4
2
0
8
6
4
2
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
Performance Level
Sums to 17
Sums to 18
Pr, 3
Pr +, 3+
Ad, 4
Pr, 3
Pr +, 3+
Ad, 4
12
Number of Students
12
Number of Students
10
10
8
6
4
2
0
10
8
6
4
2
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
Pr, 3
Pr +, 3+
Ad, 4
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Performance Level
65
APPENDIX B
Graphs of Individual Student Performance on Daily Assessments
66
Experimental Group Student Performance on Daily Assessments
Addition Math Fact Fluency for Student 3
45
40
35
30
25
20
15
10
5
0
Number Correct
Number Correct
Addition Math Fact Fluency for Student 1
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
Addition Math Fact Fluency for Student 8
45
40
35
30
25
20
15
10
5
0
Number Correct
Number Correct
Addition Math Fact Fluency for Student 5
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
45
40
35
30
25
20
15
10
5
0
Addition Math Fact Fluency for Student 12
Number Correct
Number Correct
Addition Math Fact Fluency for Student 11
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
45
40
35
30
25
20
15
10
5
0
Addition Math Fact Fluency for Student 15
Number Correct
Number Correct
Addition Math Fact Fluency for Student 13
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
67
Experimental Group Student Performance on Daily Assessments (continued)
45
40
35
30
25
20
15
10
5
0
Addition Math Fact Fluency for Student 17
Number Correct
Number Correct
Addition Math Fact Fluency for Student 16
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
45
40
35
30
25
20
15
10
5
0
Addition Math Fact Fluency for Student 20
Number Correct
Number Correct
Addition Math Fact Fluency for Student 19
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
45
40
35
30
25
20
15
10
5
0
Addition Math Fact Fluency for Student 24
Number Correct
Number Correct
Addition Math Fact Fluency for Student 23
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
Addition Math Fact Fluency for Student 27
45
40
35
30
25
20
15
10
5
0
Number Correct
Number Correct
Addition Math Fact Fluency for Student 26
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
Daily Assessments
68
Experimental Group Student Performance on Daily Assessments (continued)
Number Correct
Addition Math Fact Fluency for Student 33
45
40
35
30
25
20
15
10
5
0
Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums Sums
to 5 to 6 to 7 to 8 to 9 to 10 to 11 to12 to 13 to 14 to 15 to 16 to to 17 to 18
16,2
Daily Assessments
69
APPENDIX C
Performance Graphs on Math Facts Fluency Tests
70
Experimental Group Performance on Addition Math Facts Fluency Tests
Addition Math Fact Fluency on Initial Test, April 14, 2010
Number of Students
7
6
5
4
3
2
1
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
Ad, 4
Pr +, 3+
Ad, 4
Pr +, 3+
Ad, 4
Performance Level
Addition Math Fact Fluency on Post-Test, May 17, 2010
Number of Students
7
6
5
4
3
2
1
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Performance Level
Addition Math Fact Fluency on Second Post-Test, June 11, 2010
Number of Students
7
6
5
4
3
2
1
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Performance Level
Pr -, 3-
Pr, 3
71
Control Group Performance on Addition Math Facts Fluency Tests
Addition Math Fact Flency on Initial Test, April 14, 2010
Number of Students
7
6
5
4
3
2
1
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Pr +, 3+
Ad, 4
Pr +, 3+
Ad, 4
Pr +, 3+
Ad, 4
Performance Level
Addition Math Fact Fluency on Post-Test, May 17, 2010
Number of Students
7
6
5
4
3
2
1
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Pr -, 3-
Pr, 3
Performance Level
Addition Math Fact Fluency on Second Post-Test, June 11, 2010
Number of Students
7
6
5
4
3
2
1
0
Be, 1
Be +, 1+
Ap -, 2-
Ap, 2
Ap +, 2+
Performance Level
Pr -, 3-
Pr, 3
72
Experimental Group Individual Performance Test Results
Addition Math Fact Fluency-Experimental Group
45
40
Proficient Level
35
Approaching Level
Number Correct
30
25
20
15
10
5
0
# 1 # 3 # 5 # 8 # 11 # 12 # 13 # 15 # 16 # 17 # 19 # 20 # 23 # 24 # 26 # 27 # 33
Student
Pretest
Posttest
Post-posttest
73
Control Group Individual Performance Test Results
Addition Math Fact Fluency-Control Group
45
40
Proficient Level
35
Number Correct
30
Approaching Level
25
20
15
10
5
0
# 2 # 4 # 6 # 7 # 9 # 10 # 14 # 18 # 21 # 22 # 25 # 28 # 29 # 30 # 31 # 32 # 34
Student
Pretest
Posttest
Post-posttest
74
APPENDIX D
Make a Ten Column Addition
75
Make a Ten Column Addition
Tens
Ones
When adding columns of single-digit numbers it is helpful
1
3
to make a ten by finding the number that add to ten. The addends in
1
5
these facts are partners, i.e. 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and
1
7
5. The partners are circled and the resulting ten is put as 1 ten in the
5
tens column. A ten may also be made by adding more than two
3
digits, i.e. 2 + 3+ 5. In that case all digits used to make a ten are
4
circled with the resulting 1 ten added to the tens column. When all
6
digits that can add to ten have been circled, the remaining non-
3
circled digits are added to become the answer in the ones place.
3
Then the digits in the tens column are added to become the answer in the tens place.
In the table, 5 and 5 are partners so they are both circled and a 1 (ten) is put in
the tens column. 7 and 3 are partners, so they are circled and another 1 (ten) is put in
the tens column. 4 and 6 are partners to make ten, so they are circled and a third 1
(ten) is put in the tens column. Three does not have a partner and is not circled. The
answer is 33.
Multi-digit columns of numbers can be added the same way beginning with the
ones column, then adding the digits in the tens column. The digits in the tens column
are added as tens so the resulting ten is actually ten tens and the ten is added as a 1
(hundred) to the hundreds column.
76
APPENDIX E
Domino Card Activities
77
Sum Family Worksheet
78
Sums Cards (Sheet 1 of 2)
79
Sums Cards (Sheet 2 of 2)
80
Domino Cards for Addition Facts to 18 (Sheet 1 of 6)
81
Domino cards (Sheet 2 of 6)
82
Domino Cards (Sheet 3 0f 6)
83
Domino Cards (Sheet 4 of 6)
84
Domino Cards (Sheet 5 of 6)
85
Domino Cards (Sheet 6 of 6)
86
Domino Math Facts Workmat
87
APPENDIX F
Number Line Floor Mat
88
Number Line Floor Mat
The number line floor mat is constructed of felt with felt sticky-backed numbers
applied vertically. It was constructed with 18 9” x 9” squares to save money and
space.
Materials needed (makes 9) :
4.5 yds felt
22-9” x 12” sticky-back felt sheets
4“ number stencils
Rotary cutter
Large rotary cutting mat
Stencil knife
Yardstick
White colored pencil
Directions:
1. Run off stencils and spacing
templates on heavy paper.
2. Cut out stencils with stencil knife.
3. Trace stenciled numbers onto sticky-back felt with white colored pencil,
spacing for as little waste as possible.
4. Cut out sticky-back felt numbers using stencil knife.
5. Cut felt into strips 9” x 162” using rotary mat, rotary cutter and yardstick.
6. Measure and draw lines on the felt strips every 9” to make 18 squares.
7. Use spacing template to position sticky-back numbers in each square.
Apply.
89
Number Templates of Stencils for Number Line Floor Mat (1 of 3)
90
Number Templates of Stencils for Number Line Floor Mat (2 of 3)
91
Number 1 and Positioning Templates of Stencils for Number Line Floor Mat (3 of 3)
Cut out. Use
for spacing
two-digit
numbers on
number line
floor mat.
Cut out.
Use for
spacing
single digit
numbers on
number
line floor
mat.
92
APPENDIX G
Math Fact Fluency Test and Daily Assessments
93
Math fact fluency pretest, posttest and post-posttest, Sums to 18
94
Math Fact Fluency Assessment, Sums to 5
95
Math Fact Fluency Assessment, Sums to 6
Math Fact Fluency Assessment, Sums to 7
96
Math Fact Fluency Assessment, Sums to 7
97
Math Fact Fluency Assessment, Sums to 8
98
Math Fact Fluency Assessment, Sums to 9
99
Math Fact Fluency Assessment, Sums to 10
100
Math Fact Fluency Assessment, Sums to 11
101
Math Fact Fluency Assessment, Sums to 12
102
Math Fact Fluency Assessment, Sums to 13
103
Math Fact Fluency Assessment, Sums to 14
104
Math Fact fluency Assessment, Sums to 15
105
Math Fact Fluency Assessment, Sums to 16
106
Math Fact Fluency Assessment, Sums to 17
107
Math Facts Fluency Assessment, Sums to 18
108
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