THE USE OF HANDS-ON LEARNING TECHNIQUES TO PROMOTE
STUDENT ACHIEVEMENT IN SECONDARY MATHEMATICS CLASSROOMS
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisor. This thesis does not include
proprietary or classified information.
Samantha A. Bullard
Certificate of Approval
_____________________________
Donald R. Livingston, Ed.D.
Thesis Co-Chair
Education Department
______________________________
Sharon M. Livingston, Ph.D.
Thesis Co-Chair
Education Department
THE USE OF HANDS-ON LEARNING TECHNIQUES TO PROMOTE
STUDENT ACHIEVEMENT IN SECONDARY MATHEMATICS CLASSROOMS
A thesis submitted
by
Samantha A. Bullard
to
LaGrange College
in partial fulfillment of
the requirement for the
degree of
MASTER OF EDUCATION
In
Curriculum and Instruction
LaGrange, Georgia
May 10, 2011
Mathematical Hands-on Learning Techniques
iii
Abstract
This study aimed to determine whether the use of hands-on learning increased
achievement in secondary math classrooms. This study was completed using an action
research design in a suburban Georgia high school. A treatment and control group
comprised of Math 2 students was studied to determine differences in student attitudes
and test scores. Several different hands-on learning techniques were researched and
implemented in the treatment group. Pre and post test scores were analyzed
quantitatively while the lesson plan, student and teacher surveys, focus group and
reflective journal were coded for themes. The post-test scores from the treatment and
control groups failed to show significant gains in the treatment group over the control
group. However, student attitudes were more positive and several students expressed
more motivation in the treatment group.
Mathematical Hands-on Learning Techniques
iv
Table of Contents
Abstract…………………………………………………………..……………………….iii
Table of Contents…………………………………………………………………………iv
List of Tables …………..…………………………………………………………………v
Chapter 1: Introduction……………………………………………………………………1
Statement of the Problem………………………………………………………….1
Significance of the Problem……………………………………………………….1
Theoretical and Conceptual Framework…………………………………………..3
Focus Questions……………………………………………………………...……5
Overview of Methodology………………………………………………………...5
Human as Researcher……………………………………………………………...6
Chapter 2: Review of the Literature………………………………………………………7
Types of Hands-on Learning……………………………………………………...7
Computer Manipulatives………………………………………………………….8
Virtual Manipulatives……………………………………………………………..9
Direct Instruction………………………………………………………………...11
Student Attitudes…………………………………………………………………13
Negative Impact of Manipulatives……………………………………………….14
Summary...………………………………………………………………...……..15
Chapter 3: Methodology…………………………………………………………………16
Research Design…………………………………………………………………16
Setting……………………………………………………………………………16
Sample and Participants…….……………………………………………………17
Procedures and Data Collection Methods…………………………….………….18
Validity, Reliability, Dependability and Bias…………...……………………….20
Analysis of Data………………………………………………………………….22
Chapter 4: Results……………………………………………………………………..…26
Chapter 5: Analysis and Discussion of Results……………………………………….…37
Analysis………………………………………………………………………..…37
Discussion…………………………………………………………………….….41
Implications………………………………………………………………………43
Impact on Student Learning……………………………………………………...44
Mathematical Hands-on Learning Techniques
v
Recommendations for Future Research………………………………………….45
References………………………………………………………………………………..46
Appendices……………………………………………………………………………….48
Mathematical Hands-on Learning Techniques
List of Tables
Tables
Table 3.1
Data Shell………………………………………………….18
Table 4.1
Treatment and Control Group Prior to Treatment…………27
Table 4.2
Pre-Test/Post-Test Data Control Group…………….……..28
Table 4.3
Pre-Test/Post-Test Data Treatment Group………………...29
Table 4.4
Post-Test Scores-Treatment and Control Group…………..30
vi
Mathematical Hands-on Learning Techniques
1
CHAPTER ONE: INTRODUCTION
Statement of the Problem
Discipline issues are becoming more prevalent in secondary mathematics
classrooms. Students can be rude, defiant and disobedient. The number of students
displaying chronic behavior issues is growing. Students in the classroom who do not
exhibit these behavior problems are displaying frustration in the lack of time they are
being served in the classroom while teachers are occupied with disciplinary action.
The Georgia Performance Standards outline a performance task driven curriculum
for high school math students that is very difficult to implement while dealing with
discipline problems within the classroom. In conjunction with discipline problems, the
amount of time that students are off-task is also rising. Students are spending less time
engaged in instructional activities and are found to be unmotivated to attempt new
assignments. Deshler states, “It soon became clear that many classrooms were out of
control: large numbers of students were tardy for class, student behavior during class was
inappropriate, and the amount of time spent teaching the targeted interventions was
limited” as cited in Sprick, (2006). These discipline issues are often the result of
students being off task and not engaged in the current activity or instruction. This thesis
will investigate the effects incorporating more interactive teaching techniques has on the
behavior, achievement, and attitude of students in secondary mathematics classrooms.
Significance of the Problem
With the implementation of the state of Georgia’s staunch standards in secondary
mathematics classes, it is critical for disciplinary distractions to be limited. Teachers are
required to teach in-depth concepts to students that require attention and participation.
Mathematical Hands-on Learning Techniques
2
With the implementation of these new standards comes rigorous standardized testing.
Struggling students are becoming frustrated, resulting in off task behaviors and negative
attitudes. These off task behaviors result in lower test scores and poorer performance in
subsequent classes. With the implementation of spiraling curriculum, success in
following classes is critical to the understanding of concepts in previous classes.
Many students are also becoming frustrated with constant classroom interruptions.
Teachers are exhausting efforts to correct behavior problems rather than devoting needed
time to answering questions and providing intervention methods aimed to help struggling
students. Deshler also states, “When instruction did take place, it didn’t reach all of the
students and was often compromised because of the poor work environment; teachers
were frequently interrupting their lesson to regain control of their class.” (as cited in
Sprick, 2006, p.xv)
Students who are disengaged, off task and often engage in inappropriate
behaviors, distract other students who were previously on-task and require disciplinary
action. This results in lower grades for many students. In particular, the grading rubric
for math support classes is largely based on the participation of students during class.
Their behavior and ability to remain engaged is used to determine their grade for these
classes. Students with chronic behavior issues are compromising not only their grade but
the grades of those that they are distracting.
Students are frustrated and are losing interest in the study of mathematics. With
dropping scores on the End of Course Tests in Georgia, along with the rising drop out
rates in our school, many students are failing to meet the math requirements for a high
Mathematical Hands-on Learning Techniques
3
school diploma and are becoming more likely to drop out of high school (Georgia
Department of Education, 2010).
Theoretical and Conceptual Frameworks
When examining the underlying theories related to this thesis, one will find that it
is heavily influenced by constructivist thought. In 2001, Tomlinson asserts that
constructivist classrooms require teachers to be learning facilitators, rather than lecturers
(as cited by LaGrange College of Education, 2008). This thesis will investigate the
association of hands-on discovery tasks to enhance student learning and decrease student
behavior problems. When implementing hands-on activities, teachers play a facilitator
role, aiding students in their discovery of new knowledge. This thesis aligns with the
first tenet of the LaGrange College of Education’s Conceptual Framework, enthusiastic
engagement in learning, since it promotes a constructivist approach to teaching students
using hands-on discovery.
This thesis demonstrates a deep knowledge of curriculum as outlined by the first
tenet of the Conceptual Framework, as well as the first and second domains of the
Georgia Framework for teaching and the second Core Proposition for Experienced
Teachers (LaGrange College of Education, 2008). Teachers must use a wide variety of
methods and resources in order to implement the activities, techniques and behavior
plans. Teachers will also follow all national and state standards in the construction and
implementation of these lesson plans. Teachers must demonstrate knowledge of learners
as they implement these strategies. Educators must understand how to provide diverse
learning instruction to support different learning styles of students and promote active
engagement in these lessons.
Mathematical Hands-on Learning Techniques
4
On a national level, the second proposition of the National Board for Professional
Teaching Standards states, “Teachers know the subjects they teach and how to teach
those subjects to students” (as cited by the Lagrange College Education Department,
2008, p.12). To implement hands-on activities educators, must have a deep
understanding of their content in order to adapt it to different activities and modes of
instruction. The idea of hands on instruction also aligns with the second tenet in the
Conceptual Framework. The first competency cluster of this tenet outlines the planning
skills needed adequately facilitate a constructivist learning environment. The hands-on
activities outlined in this thesis require educators to plan appropriate lessons and prepare
accordingly. This planning and preparation also reinforces the third domain in the
Georgia Framework and the second and third Core Propositions for Experienced
Teachers. Teachers will support the personal development of all students by learning
their individual needs and learning styles.
A great deal of reflection and subsequent action is required when implementing
behavior plans and hands-on activities. The third tenet in the Conceptual Framework
describes how teachers must reflect on the effects of actions on students and others and
should seek to grow professionally based on these reflections to improve their own
practice. On a state level, the fifth and sixth domains of the Georgia Framework for
Teaching require educators to implement instructional experiences based on their
knowledge of students, and contribute to teaching as a profession. Nationally, the fourth
domain of the National Board for Professional Teaching Standards states, “Teachers
think systematically about their practice and learn from experience” (as cited by the
LaGrange College Education Department, 2008, p.12). Educators must constantly be
Mathematical Hands-on Learning Techniques
5
learning from experiences and adapting to better serve students. Focusing on behavior
plans and hands-on activities, teachers must constantly be reflecting in order to
continually be implementing plans and activities that work for a specific group of
students. Educators must also remember that plans that work with a specific group of
students must sometimes be altered to serve the needs of a different group of students.
Focus Questions
The main research question of this study was analyzed by investigating three
direct focus areas. The first focus question centers on the pedagogy of the problem. The
second aims to answer questions directed at student outcomes and the third focus
question aims to reflect holistically on the implementation of the study.
1.) What is the process of implementing hands-on learning techniques?
2.) How do math grades differ in classrooms utilizing hands-on learning techniques
versus traditional, direct instruction?
3.) What are the attitudes of teachers and students about hands-on learning
techniques?
Overview of Methodology
This study was modeled using a classroom action research design with both
quantitative and qualitative components. The study was conducted using two Math II
classes in a suburban area high school. Two samples of students were chosen based on
pre-test scores, one sample from a treatment class that will employ hands-on learning
techniques and another sample from a control class that will continue to utilize direct
instruction. An instructional coach also critiqued the unit lesson plan of the treatment
group, and several teachers took surveys regarding their opinions on the implementation
Mathematical Hands-on Learning Techniques
6
of hand-on learning techniques. In addition to the qualitative data gathered with the
lesson plan rubric, interviews and a reflective journal kept by the researcher, quantitative
data was gathered in the form of pre and post test scores of both the treatment and control
group. Issues of validity, reliability and bias were considered and addressed to ensure
that the study was valid, reliable, and free of bias. All data gathered in this study was
organized and analyzed by focus question as well as holistically.
Human as Researcher
With less than three full years of experience, I am fairly new to the classroom.
However, I have studied both mathematics and education extensively. I teach at a Title
One low income high school with more than 2,400 students. I have also interacted with
students outside of the classroom by coaching Junior Varsity Volleyball. I have
previously taught sections of Algebra 2, Mathematics I, Mathematics II, as well
Mathematics II Support classes. I believe that by utilizing more hands-on activities,
disciplinary issues in classrooms will decrease, thus strengthening students’ scores and
understanding in mathematics. I believe that these plans will increase student
engagement and will encourage them to behave in order to maximize time in the
classroom.
Mathematical Hands-on Learning Techniques
7
CHAPTER TWO: REVIEW OF THE LITERATURE
Hands-on learning techniques are slowly becoming more and more prevalent in
secondary mathematics classrooms. Teachers are finding the students need to become
more engaged in the material to stay focused and gain an interest in the field. The goal of
mathematics teachers is to help all students understand math in a dynamic and coherent
way. While most people view mathematics as a collection of unrelated ideas, it is the
responsibility of the teacher to show these as interconnected concepts. It is very
important for students to create and understand their own visual representations of
mathematical concepts. These goals are achieved through the use of five critical best
practice strategies. Small group discussion, the use of manipulatives, physical
representations, visual representations, and symbolic representations are used to promote
student achievement (Zemelman, Daniels, & Hyde, 2005).
Types of Hands-on Learning
There are several different types of hands-on techniques that are being utilized in
math classrooms today. The most prevalent, especially in geometry classrooms, is the
use of physical manipulatives. James Heddens (1997) defines manipulatives as concrete
models that involve mathematics concepts, appealing to several senses that can be
touched, and moved around by the students. He also goes on to state that the
manipulative can be any material or object from the real world that children can move
around to show a mathematics concept (Heddens, 1997). These physical objects can
range from 3-dimensional shapes such as spheres and cylinders to tiles used to
demonstrate fractional pieces in terms of a whole unit.
Mathematical Hands-on Learning Techniques
8
Cooperative learning is also a technique that can be utilized in many of today’s
classrooms. Cooperative learning involves students working in groups to collaborate to
solve a problem. Groups are given a task in which the input of all members is essential to
completing the task (Schul, 2011).
Manipulatives and cooperative learning are also often intertwined in lesson plans
together. With the use of graphing calculator software students can now use programs
geared toward the discovery of new concepts in math. Students can cooperatively use
these programs to answer specific questions in the hopes that they will discover
overarching mathematical concepts. While the technology has changed drastically in the
past decade, Ernest (1994) found that after a review of the manipulatives, calculators
were identified as one of the most useful items. In 1994, Ernest observed teachers using
TI-81 graphing calculators with fewer capabilities of the TI-84 and TI-89 graphing
calculators we have teachers have today.
Also, for manipulatives to be effective tools Heddens (1997) suggests that each
student have materials to manipulate independently; with students actively involved with
manipulatives, interest in the content will be aroused. Good mathematics manipulatives
are durable, simple, attractive and manageable (Heddens, 1997). With small budgets,
some teachers may have trouble finding appropriate math manipulatives for each student
in her class.
Computer Manipulatives
Concrete manipulatives can no longer only be equated with physical
manipulatives. Today, computers can supply representations that can be just as
meaningful as physical manipulatives. Computers also have several key attributes that
Mathematical Hands-on Learning Techniques
9
can make them quite attractive to educators. They offer manageable and clean
manipulatives. They can limit distractions that are often present when physical
manipulatives are used, and can mirror desired mental actions closer than some physical
manipulatives (Clements & McMillen, 1996).
Computers also offer flexibility and allow for changing the arrangement of a
representation. For example, most database software will sort and arrange data in
different ways just with the push of a few buttons. Computers can also store and retrieve
configurations from previous work. Students and teachers can save and retrieve work to
review, grade, or finish. Computers can also dynamically link multiple representations,
such as pictures, tables, graphs, and equations (Clements & McMillen, 1996).
The use of computer manipulatives can also encourage students to pose new
questions and make conjectures. This kind of digital manipulative also allows students to
explore their own conjectures while decreasing psychological cost of making wrong
conjectures. Students also move easily from naïve to empirical and ultimately to logical,
problem solving thinking as they test and study their conjectures. Computers also allow
for scaffolding of problem solving. These scaffolding techniques allow students to build
on their initial ideas and construct more analytical approaches (Clements & McMillen,
1996).
Virtual Manipulatives
In 2002, Moyer, Bolyard and Spikell defined virtual manipulatives as “an
interactive, web-based visual representation of a dynamic object that presents
opportunities for constructing mathematical knowledge” (Moyer et al., 2002, p.373).
These virtual manipulatives provide students an interactive environment to seek answers
Mathematical Hands-on Learning Techniques
10
to their questions and can pose and solve their own problems. Students can gain
immediate feedback about their actions and can reflect and conceptualize their results
(Moyer et al., 2002).
The National Council of Teachers of Mathematics’ Principles and Standards of
School Mathematics states that visualization is an important tool in problem solving
(cited by Niess & Walker, 2009). The NCTM also states that students need multiple
visualization opportunities to fully develop this skill. Technology is becoming a key tool
in classrooms as we try to convey meaning to abstract mathematical concepts. The
Principles and Standards for School Mathematics states, “Technology is essential in
teaching and learning mathematics; it influences the mathematics that is taught and
enhances student learning.” (Niess & Walker, 2009, p.37) Niess and Walker (2009) state
“Analysis is at the heart of reasoning in mathematics. Students need experiences that
guide them as they learn to reason mathematically”(p.37).
Virtual manipulatives can also be classified as static or dynamic visual
representations. Static representations are essentially pictures. They can be used to
visualize concrete manipulatives, but cannot be manipulated in the ways that dynamic
representations can (Moyer et al., 2002). Dynamic visual representations are objects.
They can be manipulated much in the same way that concrete manipulatives can be. Just
as a student could slide, flip or rotate on object, dynamic representations allow students to
slide, flip, and rotate objects on a computer using the touch of their mouse (Moyer et al.,
2002).
It is important to realize that these virtual manipulatives are not really hands-on
activities. They are more abstract than many of the touchable manipulatives that can be
Mathematical Hands-on Learning Techniques
11
utilized. However, in some situations these virtual manipulatives may be found more
useful.
According to Clements and McMillen (1996), teachers should consider the
following when choosing computer manipulatives:

Guide students to alter and reflect on their actions, always predicting and
explaining.

Have students work cooperatively in pairs.

If possible, use one computer and a large-screen display to focus and extend
follow-up discussions with the class.

Recognize that information may need to be introduced before working on
computers, such as the purpose of the software and how to use the software and
the hardware (p.279).
These are a few of the suggestions that can aid teachers in choosing the best computer
manipulatives for their use.
Another form of a virtual manipulative is the video clip. Video clips can be short
clips from movies, television shows, or professionally prepared educational videos. The
clips can be used to introduce new concepts and engage students in a real-world
application or use of what they are learning. These video explore mathematics in nature,
art, and other real-world contexts (Niess & Walker, 2009).
Direct Instruction
Direct instruction is known to many educators as the more traditional teaching
technique. Direct instruction is an alternative to more hands-on teaching techniques.
Years of advocacy of hands-on discovery techniques have tarnished the reputation of
Mathematical Hands-on Learning Techniques
12
direct, teacher-driven forms of instruction (Silver, Strong, & Perini, 2008). However,
teachers do not necessarily only utilize one teaching technique. Silver et al. (2009) states,
“Just as certain topics lend themselves to inquiry-based learning, so do particular skills
need to be mastered and used on command by students” (p. 35). While direct instruction
can differ from teacher to teacher, many direct instruction techniques utilize a simple,
four phase process. The first step in direct instruction is modeling. During this stage, the
skill is modeled by the teacher who thinks aloud while performing the skill or working
through the problem (Silver, et al., 2009). Next, directed practice allows the teacher to
use questions to lead students through the steps. The next step, guided practice, allows
students to generate their own leading questions while working through the problem. The
teacher observes and coaches while providing feedback as students work. Finally,
students are prepared to practice more examples independently.
Silver (2009) found that teachers who spent more time demonstrating and
explaining procedures and skills are more effective than teachers who spend less time
explaining procedures and skills (as cited by Silver et al., 2009). Similarly, Silver states,
“A large body of research supports the effectiveness of Direct Instruction as an ideal
technique for teaching new skills to both general-education and learning-disabled
students” (Silver et al, 2009, p.38).
Direct instruction lessons are founded on four basic principles (Silver et al, 2009).
The first principle is effective modeling. An ideal direct instruction lesson begins with an
effective modeling session. This session demonstrates every step in the concept and
demonstrates how each step is performed. Another important principle is emerging
independence. The ultimate goal in direct instruction is to move students from
Mathematical Hands-on Learning Techniques
13
dependence on the teacher to self-directed application of the skill. Questioning is another
integral skill to direct instruction. Questions force students to analyze steps to solve a
problem or master a skill. Finally, ongoing assessment is crucial to effective direct
instruction. Direct instruction should incorporate multiple practice opportunities to
provide feedback to students (Silver et al, 2009).
Student Attitudes
Patricia Ernest (1994), a researcher with the University of Montevallo, conducted
a three year study in which she studied the use of manipulatives in the classroom. She
reported, “students seemed to comprehend tasks with accuracy, employed discovery and
problem solving strategies, were anxious to share their discoveries and solutions, engaged
in lively student/student interaction related to the content, and exhibited an excitement
about learning” (Ernest, 1994, p.8). Ernest also reports that students are more willing to
participate in math projects when manipulatives are used. Observers found that students
enjoyed working with the manipulatives as expressed through their verbal and nonverbal
actions. Observers also noted a very high “on-task” involvement rate of 100% in most
cases. Students were also more willing to respond to questions and several students
extended their discovery learning beyond the assignment. Observers also noted students
were anxious to share their discoveries with their classmates and were engaged in lively
peer conversations related to the content. Overall, students exhibited excitement about
their learning (Ernest, 1994).
A study by Sowell, (as cited by Ernest, 1994), concluded that mathematics
achievement is increased by the long term use of manipulatives and that students’
Mathematical Hands-on Learning Techniques
14
attitudes toward mathematics are improved when they are instructed with manipulatives
(Ernest, 1994).
Computer manipulatives have been found to increase student motivation and
focus their attention. These manipulatives have also been found to be especially
motivating to students who have previously been unsuccessful in mathematics (Clements
& McMillen, 1996). Hoyles, Healy and Sutherland, studied pairs of students as they
worked with computer manipulatives. They found that the computer can draw the
attention of pupils and become a focus for discussion, thus resulting in very little off-task
talk (as cited by Clements & McMillen, 1996).
Negative Impact of Manipulatives
It is important to note that with any teaching technique there are also downfalls.
Heddens (1997) believes “Manipulatives that are improperly used will convince students
that two mathematical worlds exist-manipulative and symbolic. All mathematics comes
from the real world. Then the real situation must be translated into the symbolism of
mathematics for calculating.” (p.2) Heddens gives the following example, if you put
three goats with five goats, and you get eight goats is a real world situation. On the
mathematical level, we would say 3+5=8. While these are not two different worlds, they
are in the same world expressing the concept in two different ways (Heddens, 1997).
The use of manipulatives do not guarantee student achievement. Fennema (as
cited by Clements & McMillen, 1996) found that classes not using manipulatives
outperformed other classes that were utilizing manipulatives. Using manipulatives can
also lead students to begin working and solving problems systematically without
understanding. For example, many students learn counting, adding, and subtracting
Mathematical Hands-on Learning Techniques
15
using base-ten blocks or beans and bean sticks. However, Thompson discovered students
often fail to properly relate their manipulations of the base-ten blocks with the notations
system we use today to represent their actions (as cited by Clements & McMillen, 1996).
Students may also derive mental conceptions different from the intended lesson
(Clements & McMillen, 1996).
Summary
There are many different types of hands-on learning including manipulatives and
cooperative learning techniques. Manipulatives can be classified as concrete or computer
based, virtual manipulatives (Clements & McMillen, 1996). Direct instruction is the
more traditional teaching method in secondary mathematics classrooms. A common
misconception is that educators must follow one method or the other. However, while
some concepts lend themselves easily to use of manipulatives, many lend themselves to
direct instruction techniques (Silver et al, 2009). As with all teaching strategies there are
downfalls to each one. While the use of manipulatives seem to excite students and
promote achievement, they can sometimes lead to misunderstanding of concepts or the
misconception that two different worlds exist- the manipulative and the symbolic.
Mathematical Hands-on Learning Techniques
16
CHAPTER THREE: METHODOLOGY
Research Design
This study was conducted using action research, using both qualitative and
quantitative components. According to Hendricks (2009), “The purpose of action
research is for practitioners to investigate and improve their practices.” This is especially
important for educators seeking to improve their teaching techniques (p.3). In action
research, practitioners, focus systematically on different ways to deal with issues that
they face. These issues may range from instructional practices, social issues of
schooling, collaboration with colleagues, or supervision of staff (Hendricks, 2009).
More specifically, this study was modeled using a classroom action research
design. Classroom action research is conducted by teachers in their classroom. While
this is normally a solo research study, collaboration among classroom teachers can occur.
The primary goal of researchers using classroom action research is to improve their
practices. This type of research values interpretations made by teachers based on the data
they collected from their students (Hendricks, 2009). According to Johnson (2009),
“action research as the potential to change education; to keep our teaching practices
evolving” (p.29).
Setting
This study took place in a high school math classroom. The school is located in a
suburban area in mid-west Georgia. The high school has grades 9 through 12 and is a
federal Title I school, based on the percentage of students receiving free and reduced
lunch. The school is widely supported by many local businesses and the community
Mathematical Hands-on Learning Techniques
17
throughout the year. This setting was chosen because the researcher is employed by the
school and gained access to conduct the study there. Permission to conduct the study was
received from the high school principal and from the county in which the school is
located. The Institutional Review Board also approved the study.
Sample and Participants
The study utilized a sample of students taken from two different rosters. Samples
of 15 to 17 students were chosen from the treatment class and control class rosters.
Samples were chosen to form comparable groups to study. Samples were chosen based
on scores on a pre-test administered before the study started. Grades were ordered and
the sample was chosen that the control and treatment groups would not exhibit significant
difference before treatment. The sample was largely made of fifteen to sixteen year old
10th graders. The samples were comprised of a majority of white and African American
students, while very few students were of Hispanic origin. Boys and girls were also
equally represented in the sample. While SES level was not a factor in choosing samples,
the sample was drawn from a Title I school, indicating a high population of low
socioeconomic students.
An instructional coach was also utilized to gain feedback. She was a willing
participant who knew the purpose of the study. She has been a teacher for more than
twenty years at several different schools and has been serving as an instructional coach at
the school of the study for two years. She was chosen based on her knowledge of the
material and of the sample. Several teachers were also interviewed using a survey. They
were all willing participants in the study.
Mathematical Hands-on Learning Techniques
18
Procedures and Data Collection Methods
When conducting this study, both qualitative and quantitative data were gathered.
Table 3.1 shows how these data relate to each of the outlined focus questions of this
study.
Table 3.1 Data Shell
Focus
Question
Literature
sources
Type: Method,
Data , Validity
How are
data
analyzed
Rationale
FQ1
What is the
process of
implementing
hands-on
learning
techniques?
Clements
&
McMillen
(1996)
Type of Method:
Instructional Plan,
rubric and
interview
Type of Data:
Qualitative
Type of Validity:
Content
Coded for
themes
Recurring
Dominant
Emerging
Looking for categorical and
repeating data that form
patterns of behaviors
FQ 2
How do math
scores differ
in classrooms
using handson learning
techniques
and
classrooms
using a more
traditional
teaching
technique?
Silver et al
(2008)
Dependent T
Independent T
Effect size
-To determine significant
difference between same
group tested twice.
-To determine if significant
differences between two
groups exist.
-Measure the magnitude of
the treatment effect
Heddens, J.
(1997)
Ernest, P.
(1994)
Type of
Method:
Teacher made Tests, quizzes
and essays
Type of data:
Interval
Type of
Validity:
Content
Mathematical Hands-on Learning Techniques
FQ 3
What are the
attitudes of
teachers and
students on
hands-on
learning
techniques?
Clements &
McMillen
(1996)
Ernest, P.
(1994)
Zemelman et
al
(2005)
Type of
Method:Reflective
Journal Surveys,
focus groups,
interviews etc.
Type of Data:
Qualitative
Type of Validity:
Construct
Coded for
themes
Recurring
Dominant
Emerging
19
Looking for categorical
and repeating data that
form patterns of
behaviors
This study employed a classroom action research design and collected and
analyzed multiple forms of data. This form of research is conducted with the purpose of
improving ones practice (Hendricks, 2009). A lesson plan and rubric for treatment was
developed and graded by a curriculum specialist to address the pedagogical content of the
unit being studied. The lesson plan encompassed a two week circle geometry unit. It
outlined several hands-on learning activities that should be implemented. The plan was
evaluated by a mathematics instructional coach using the rubric, then amended to utilize
the most appropriate forms of hands-on learning for each concept. The lesson plan and
scoring rubric are available in Appendix A. A pre and post test was used to collect data
on both the treatment and the control group. The pre test was given to students before
any material was addressed and the post test was given as a summative evaluation. The
pre/post test is available in Appendix B. Student and teacher attitudes were chronicled
and analyzed using a reflective journal kept by the teacher using daily prompts, see
Appendix C. An open ended survey given to a math curriculum specialist and several
teachers of varying experience was also used. This survey is located in Appendix D.
Student attitudes are evidenced by surveys given to the students in both the treatment and
Mathematical Hands-on Learning Techniques
20
control group before the treatment was employed (see Appendix E). The survey was
given to students in the treatment group at the end of the unit to note any changes that
were evidenced after the treatment. Finally, a focus group of students in an independent
class was created to gain an understanding of their thoughts. The focus group prompts
are located in Appendix F.
Validity, Reliability, Dependability and Bias
The first focus question outlined in this study seeks to discover how hands-on
lessons are implemented in the secondary classroom. Qualitative data were gathered.
Content validity is defined as “The extent to which an assessment procedure adequately
represents the content of the curricular aim being measured” (Popham, 2008, p.53). To
ensure content validity, a detailed lesson plan was followed throughout the
implementation. This lesson plan was formulated by the researcher, but to ensure
construct validity, it was critiqued by an instructional coach that is well versed in the use
of hands-on learning in the classroom.
To ensure dependability, the data collection and treatment were kept consistent.
The lesson plan was followed as written. Any changes to the original plan were
documented and accounted for. The length of time for the study was also adequate. It
encompassed several days and allowed for many uses of hands-on learning to be
implemented. The researcher kept control of the study setting to ensure dependability of
the research.
Bias is also an important factor in ensuring an accurate study. Bias is present
when elements of the assessment distort a student’s performance merely based on the
student’s gender, ethnicity, etc. (Popham, 2008). The lesson plan was actively reviewed
Mathematical Hands-on Learning Techniques
21
by both the researcher and the instructional coach to ensure that activities would not give
a group of students an unfair advantage over another group. Disparate impact was also
considered when reviewing test items. Disparate impact is evident when a subgroup
performs lower than the population (Popham, 2008). Since some activities involved the
use of a computer, instructions and teacher guided instruction were utilized to ensure that
students who are more familiar with computers would not have an unfair advantage over
students who are not familiar with common computer functions.
The second focus question aims to discover any differences in the achievement of
the treatment and control group. Interval data were collected and analyzed. Pre and post
test scores were collected before and following the study. To ensure content validity, the
pre and post test were assessed by a colleague teaching the same standards to ensure that
the questions were adequately assessing the standards that were to be addressed and
covered through the unit.
A test/retest correlation will also be utilized to ensure that the assessment was
reliable. The pre and post test were identical to prevent any irregularities in reliability.
The test was also reviewed by a fellow teacher and checked for several different forms of
bias before the assessment was given. Offensiveness, unfair penalization and disparate
impact are all forms of bias that can be present in assessment (Popham, 2008). Data
collection was also conducted only by the researcher to reduce the risk of incompetent
“helpers” sabotaging the data collection (Salkind, 2010).
The third focus question seeks to understand the attitudes of both students and
teachers concerning the use of hands-on learning in the classroom. Qualitative data were
gathered in the form of a reflective journal and interviews with students and fellow
Mathematical Hands-on Learning Techniques
22
teachers. To ensure construct validity, questions asked to teachers were kept constant
from one teacher to another. The interviews conducted with teachers of different
experience levels. Questions were formulated to accurately assess the teacher’s attitudes.
Students were interviewed in a focus group type setting. Talking points were formulated
before the meeting and were read by a fellow teacher for construct validity.
To ensure dependability, data were accurately recorded by the researcher during
interviews and by a note taker during focus groups sessions to ensure that accurate,
unbiased notes were kept. An adequate number of participants were also chosen to
participate in the interview and focus group. Questions and focus group prompts were
also checked for bias before being asked. The questions were not unfair to one group of
another and did not elicit a particular response.
Bias was also strongly considered throughout the development of this study.
According to Popham (2008), “assessment bias applies to all forms of testing, not merely
paper and pencil tests” (p. 83). Therefore, all lessons and activities were reviewed to
ensure that they did not provide advantage or appeal to one subgroup more than others.
The pre and post test were also reviewed by the researcher and another teacher on an item
by item basis to ensure that the questions were absent of bias.
Analysis of Data
Data collection for this study were collected and analyzed specifically by focus
question and also holistically. Qualitative data were gathered to answer focus question
one, “How are hands-on lesson plans implemented in the classroom?” A lesson plan was
developed and drafted by the researcher to cover an entire unit in a secondary math 2
classroom. This lesson plan included specific instructions on how to implement several
Mathematical Hands-on Learning Techniques
23
hands-on lessons throughout the unit, along with resources that would be needed. This
lesson plan was analyzed using a rubric by an instructional mathematics coach at the
researcher’s school. The rubric graded the standards and learning objectives, technology
use, cognitive tasks, assessment and preparation. Comments were given by the coach to
address all of these important topics. These comments were coded for themes. Themes
were organized by recurring themes, dominant themes, and emerging themes-themes that
were not previously thought of by the researcher. The comments and suggestions of the
instructional coach were recorded to ensure that accurate detail was kept.
The third focus question focused on the student and teacher attitudes towards the
implementation of hands-on learning techniques. Interviews with several teachers were
conducted to gain insight into their experiences. Interviews with both new teachers to the
field and experienced teachers were conducted. An interview with a mathematics
instructional coach, who is in charge of mathematics technology at our school, was also
conducted.
A survey was given to Math 2 students in both the treatment and control group.
This was a quantitative survey using a Likert-type scale for responses . The survey was
given before treatment and post treatment to the treatment group to examine any
significant changes in responses. A focus group discussion of Accelerated Math 2
students was also held to gain insight into their attitudes towards the use of hands-on
learning. The focus group was transcribed with the aid of another teacher to ensure that
accurate notes were kept. Students were asked open ended questions and their thoughts
were transcribed. Student names were not used in the transcript.
Mathematical Hands-on Learning Techniques
24
The interviews and focus group transcripts were also coded for themes.
Recurring and dominant themes were noted as well emerging themes that came to light in
the sessions.
Focus question two aimed to assess whether hands-on learning promoted
achievement in secondary math classrooms. Pre and post tests were conducted in both a
treatment and a control group of Math 2 classes. The treatment group utilized hands-on
learning throughout a math unit, while the control group continued to use a direct
instruction technique. Both groups took the same test and the pre and post test were
identical. An independent t test was run using pre test scores to show that the groups had
no significant difference before treatment was given. The decision to reject the null
statement, there is no significant difference in the beginning groups, has been set at p<
.05. Then, a dependent t test was used on each class to show that significant gains were
made by both groups. Finally, another independent t test was run on both classes post
treatment. The decision to reject the null statement, there is no significant difference in
the two groups post treatment, has been set at p< .05. An effect size calculation was also
utilized to measure the magnitude of the treatment effect. Unlike the t tests, the effect
size will calculate the magnitude of the effect regardless of the sample size.
The data of this study were also studied holistically. Emerging and recurrent
themes were analyzed using all data collection methods. Great care was taken to ensure
that the study was valid. Consensual validation was achieved by study approval attained
not only by the faculty at the research institution of the researcher, but also at the school
where the researcher teaches. Results of this study were also compared to the results of
similar studies to ensure epistemological validity.
Mathematical Hands-on Learning Techniques
25
Credibility of the study was also achieved. Multiple data sources were used
structurally corroborate the study. Both qualitative and quantitative data sources were
gathered and analyzed. Throughout the study, opposing points of view were also
researched and discussed to guarantee fairness. Throughout this study, great care was
taken to be precise and accurate. The researcher worked to present a tight argument and
a coherent case to exhibit rightness of fit.
Transferability of the study was also considered. Great care was taken to allow
for the replication of the study by others for future research. This study was
transformational for several parties involved. Catalytic validity was attained as the study
caused a positive change for participants of the study. The results of this study prompted
positive change within the researcher and others involved.
Mathematical Hands-on Learning Techniques
26
CHAPTER FOUR: RESULTS
Focus question one aimed to answer the methods of implementing hands-on
learning activities in secondary math classrooms. A detailed lesson plan was constructed
by the researcher (see Appendix A) and was critiqued using a rubric by a mathematics
content specialist at the school of the researcher. The content specialist is also in charge
of technology use in the mathematics department. The first objective of the lesson plan
was to clarify the learning objectives of the unit. The specialist agreed that the standards
and learning objectives were “very specific and focused.” The lesson plan also aimed to
implement adequate technology. According to the rubric, the lesson plan linked the
curriculum to technology use and content learning is extended. The specialist asserted,
“The technology activities accelerate the learning by removing the tedious calculations
that tend to bog the students down-while still effectively demonstrating the content.”
The rubric also considered the cognitive tasks outlined in the lesson plan. The
lesson plan requires synthesis and evaluation of the information. The tasks go beyond
existing understanding to create own original position or product. The specialist stated,
“The tasks provide experience with the theorems. The content becomes dynamic and
personal through student involvement.”
According to the rubric, the lesson plan failed to allow students to design their
own assessment tools. The lesson plan merely informed students of an assessment tool
created by the teacher. The content specialist suggested having small groups of students
devise their own review materials prior to the test. However, due to time constraints
during the study, this suggestion was not implemented.
Mathematical Hands-on Learning Techniques
27
The preparation for learning tasks was also considered when evaluating the lesson
plan. The specialist determined that adequate preparation would be expected and that
multiple resources would be organized for student use. The specialist asserted that the
variety of the tasks, “provide opportunities for engagement of body and mind.” Finally,
the overall focus of technology use was considered. The technology use was determined
to be primarily transforming. The specialist was confident in the problem solving
activities. The specialist stated, “Students must use tools to apply learning and create
solutions to real-world dilemmas…authentic learning.”
The second focus question aimed to determine whether test scores differed in
classrooms that used hands-on learning techniques as opposed to a more traditional,
direct instruction approach. An independent t-test was conducted on both the treatment
and control group prior to any treatment to ensure that both groups showed no significant
difference at the α=.05 significance level. Table 4.1 displays the data on the groups prior
to any treatment.
Table 4.1-Independent t-test for Treatment and Control Group Prior to Treatment
t-Test: Two-Sample Assuming Unequal Variances
Mean
Variance
Observations
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Treatment
Control
38.23529412 46.26667
247.3161765 196.4952
17
15
0
30
-1.527426508
0.068565376
1.697260359
0.137130751
2.042270353
Mathematical Hands-on Learning Techniques
28
The null statement is there is no significant difference in both groups prior to
treatment. According to Table 4.1, t(30)  1.5274, p  .05 , therefore, since the obtained
value fails to exceed the critical value, the null hypothesis fails to be rejected. The two
groups did not have any significant differences before treatment.
After treatment, two dependent t-tests were conducted on each groups’ pre-test
and corresponding post-test. The null hypothesis, there is no significant difference in the
pre-test scores and post-test scores was rejected if p< .05. Table 4.2 displays the data on
the pre-test/post-test for both the control group.
Table 4.2 Dependent t-test of Pre-Test/Post-Test Data Control Group
t-Test: Paired Two Sample for Means
Pre-Test
Post-Test
Mean
46.266667
69.46666667
Variance
196.49524
653.552381
15
15
Observations
Pearson Correlation
Hypothesized Mean Difference
Df
0.4502955
0
14
t Stat
-3.9129009
P(T<=t) one-tail
0.0007808
t Critical one-tail
1.7613092
P(T<=t) two-tail
0.0015616
t Critical two-tail
2.1447886
As evidenced by table 4.2 the obtained t-value exceeded the corresponding critical
value at the α=.05 confidence level for both groups. For the control group:
t(14)  3.9129, p  .05 , therefore, the null hypothesis is rejected, meaning significant
differences were present from the scores of the pre-test and corresponding post-test.
Mathematical Hands-on Learning Techniques
29
In table 4.3, the results show the treatment group performed similarly. For the
treatment group: t(16)  8.0829, p  .05 , thus, the null hypothesis is rejected, meaning
there is significant difference in the pre-treatment and post-treatment scores.
Table 4.3 Pre-Test/Post-Test Data Treatment Group
t-Test: Paired Two Sample for Means
Mean
Variance
Observations
Pearson Correlation
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
Pre-Test
38.23529
247.3162
17
0.130097
0
16
-8.08287
2.43E-07
1.745884
P(T<=t) two-tail
t Critical two-tail
4.85E-07
2.119905
Post-Test
79.4706
261.265
17
The reliability coefficient of the treatment group was r(16)  0.45, p  .05 ,
correlating the pre-test and post-test scores. The reliability coefficient of the control
group was r(14)  0.13, p  .05 . An effect size calculation was also calculated on both
groups. The effect size of the treatment group was calculated at r  .79 , representing a
high effect size while the effect size of the control group was r  .490 representing a
more moderate effect size.
Finally, an independent t-test was conducted to compare the post-test scores of the
treatment and control groups. This test was conducted at α=.05 confidence level. The
null statement, there is no significant differences in the control group post-test scores and
the treatment group post-test scores was rejected if p<.05. Table 4.3 displays the results
of this independent t-test.
Mathematical Hands-on Learning Techniques
30
Table 4.4 Independent t-test for Post-test scores for Treatment and Control Group
t-Test: Two-Sample Assuming Unequal Variances
Mean
Variance
Observations
Hypothesized Mean Difference
Df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
Treatment
Control
79.47058824 69.46667
261.2647059 653.5524
17
15
0
23
1.30307709
0.10272186
1.713870006
0.205443719
t Critical two-tail
2.068654794
According to Table 4.3, t(23)  1.3031, p  .05 , therefore, since the obtained tvalue fails to exceed the corresponding critical value, the null hypothesis fails to be
rejected. Thus, there is no significant difference in the post-treatment scores of the
control group and the treatment group. Using the means and standard deviations of each
of the groups post-treatment the Cohen’s d value was calculated at d  .4677 . The
calculated percent of non-overlap was about 33% .
The third focus question aimed to explore teacher and student attitudes regarding
the use of hands-on learning in the classroom. A survey was given to students in the
treatment and control group to access their attitudes towards hands-on learning.
However, after noting that the students did not take the survey seriously and that several
students did not answer the survey, the researcher chose to disregard the results.
Survey type interviews were also conducted with several teachers at the school of
the researcher. Teachers interviewed were asked their initial feelings towards the
implementation of hands-on learning activities. Teacher 1, with 32 years of teaching
experience, responded that she hates them, while several other teachers with varying
Mathematical Hands-on Learning Techniques
31
levels of experience responded with similar responses. Teacher 2, a math specialist with
18 years of experience, stated that she loved to implement these activities and it is what
she has always tried to do in her classroom. Teacher 3, a special education teacher that is
in the math collaborative setting, responded that her initial feeling when considering
these activities is the sizes of her collaborative classes. She hopes that she has a very
small class when trying to implement these activities.
Teachers were also asked about their current use of hands-on learning activities.
The math content teachers all responded that they occasionally implement these activities
with two stating that they typically implement them 2-3 times per semester. Teacher 3
responded that since she is collaborative teacher in the classroom she only conducts these
activities when they are planned by the content teacher. Teacher 2 responded that she
tries to implement these activities as often as possible. She stated, “But not for every
lesson, some concepts lend themselves more naturally to hands-on learning.” Teacher 4,
with seven years teaching experience, stated that while she has tried implementing
Versatiles in her lower level math classes, she typically does more hands-on activities in
higher level math classes, such as trigonometry. Versatiles are tiled flip boards that
follow twelve questions on a given topic. If students answer all questions correctly, when
their boards are flipped they display a specific pattern. If students answer a question
incorrectly, they have immediate feedback on which question was incorrect and they
have the opportunity to correct those problems.
All teachers agreed that these activities did require extra planning and preparation.
Teacher 4 also noted that it required more time to gather and organize materials and that
she typically make changes to an activity based on past experience. Teachers were also
Mathematical Hands-on Learning Techniques
32
asked about difficulties when implementing these activities. Three teachers, including
the math specialist, expressed concern for the cost of materials. They stated that
purchasing some needed materials are often paid for with their personal investment. An
emerging idea in the survey interviews were the destruction of some of the materials and
students becoming easily off task. Teacher 3 stated that in her past experience, she has
found that high school students will complain that they are being treated like elementary
children. Several teachers noted that hands-on activities take up too much instructional
time. Another recurring idea was the amount of understanding gained from these
activities. Several teachers worried that they did not learn the intended concepts properly
using the activity and questioned if the goals of the activities were achieved.
Gains of using hands-on activities were also discussed. The Teacher 2 stated that
students exhibit a better understanding of concepts after completing these activities. She
also finds that students tend to like math more and exhibit higher engagement. She also
finds that hands-on activities provide teachers the opportunity to get to know their
students better. Teacher 1 found that students can exhibit better recall of information if
they have done a hands-on activity with the information, while Teacher 5, with five years
of experience, found that students sometimes enjoy doing something different from the
normal routine. Teacher 4 stated, “Students pay more attention when we do hands-on
activities. They want to know what to do and are pleased with a final product.”
However, she also stated, “It is hard to gauge how much time an activity will take.”
Teacher 2 has found very few gains in implementing hands-on activities.
Student attitudes in class during hands-on activities were also surveyed. One
teacher replied that a few students will have a better understanding due to the activity.
Mathematical Hands-on Learning Techniques
33
Teacher 4 stated, “I think that students enjoy the activity while they are doing it. For
some students, the hands-on activity is what is needed to make a concept click. I’ve seen
the light bulb shine bright after an activity for some students and that makes the work to
prepare the activity worth it.”
Finally, teachers were surveyed regarding their use of technology. Most teachers
responded that they typically use calculators, both scientific and graphing, in their
classrooms. Teacher 5 added that he also uses an overhead projector and an LCD
projector for lesson presentations and video clips. Teacher 4 summed up her push to
implement technology in the last year, “I have made a great effort this year to implement
technology in the classroom. I have a SmartBoard this year and I have used it almost
every day to deliver notes to students. I have also used answer responders on many
occasions to take quizzes. I have also used graphing calculators in the past.” She further
states, “Students like the technology. They are more engaged and willing to participate in
class when technology is used. I have found that some students do not like the answer
responders, but I’m not sure that I’ve implemented them in the best way possible.” She
also finds that students seem to be more engaged when technology is used. They
especially enjoy interacting with the technology individually. Teacher 1 stated that she
does not use technology as “she is old and decrepit.”
A focus group with 22 Accelerated Math 2 students was also conducted. This
group was separate from the control and treatment group. A recurring theme throughout
the session was that they enjoyed doing activities that involves building or producing a
product. An emergent idea from the focus group session was the use of food in activities.
Several students in the focus group all mentioned that they enjoyed activities that
Mathematical Hands-on Learning Techniques
34
involved candy or food. Several students also mentioned that they did not usually enjoy
hands-on type activities. They stated that they would rather be given instruction and then
practice problems rather than attempting to discover a concept.
Students in the focus group were also polled about the types of technology that
they use in their other classes, and what types of technology they enjoy. They stated that
in other classes, including previous math classes, they have used calculators, laptops,
SmartBoards, and hand held responders. Students stated that they enjoyed using the
graphing calculators in their math classes. Laptops were also mentioned several times by
several different students. Laptops are currently being used in various classes such as
history and English, and the students enjoy completing projects on them. They do varied
activities utilizing the laptops such as research and creating PowerPoint presentations.
Students are also currently using hand held responder devices in chemistry and history
classes. While some students asserted that they enjoy using the hand held responders,
about one-third of the students stated that they did not have preference of the responders
over paper and pencil tests.
Finally, a detailed journal was kept by the researcher outlining the thoughts of
each activity implemented and any student responses that were exhibited. The first day
of the study, students created flip charts out of construction paper to record thoughts of
each concept we studied throughout the unit. After we created these charts, we labeled
each page with an important concept that would be studied. The students seemed to
really enjoy these flip charts. They mentioned that it was something different, rather than
constantly taking lecture notes in their notebooks. Students also seemed to take
ownership of the charts. They were very consistent about recording information they had
Mathematical Hands-on Learning Techniques
35
learned in these charts and many students personalized them to reflect their personality.
Overall, students seemed to enjoy this idea and all of the students in the treatment group,
except one, kept their chart throughout the end of the unit.
The next day, after learning new vocabulary and a few new concepts, students
were given VersaTiles to supplement their assigned problems. While students worked
problems, they would match their answer to an answer located in an answer box and
move a corresponding tile to a matching letter. After, each set of twelve problems were
completed, the tiles would be turned over to reveal whether the students matched all of
their answers correctly. Students really seemed to enjoy this activity as well. It was also
very quick and easy to assess which students had correctly mastered the concepts and
which students were still struggling. The visual and kinesthetic learners seemed to
achieve the most out of this activity. One student refused to participate in any of the
activities and therefore was not included in the data collection sample.
Students enjoyed the human circle activity. Students commented on how they
enjoyed the room being arranged in a different configuration. Desks were placed around
the outside walls and students faced the middle of the room. Students asked if the desks
could stay in this configuration. Students stayed attentive and followed along using a
worksheet provided by the researcher. There was slight trouble getting students to
volunteer to participate in the activity. However, many students who are normally off
task during instructional time were attentive and volunteered to participate.
Closer to the end of the unit, students used graphing calculators to discover
several theorems related to angles and arcs. The software was pre-loaded onto the
calculator and students followed an instructional worksheet that would lead them through
Mathematical Hands-on Learning Techniques
36
the activity. This hands-on learning technique was much more difficult to monitor in the
classroom setting. Most students were not familiar with the graphing calculators and
took the majority of the allotted time trying to figure out how to operate them. It was
also difficult to monitor their progress while navigating the room and focusing on
individual calculator screens. The students commented several times how they enjoyed
using the calculators and would love to use them more often.
The final activity utilized a mobile laptop lab. Students were all assigned a laptop
and given instructions to a particular website. The website allowed them to move points
around a circle and automatically displayed segment lengths. Students made conjectures
using simple calculations relating the segment lengths to known theorems. This activity
was very hard to manage due to the technology involved. Students liked using the
computers and appreciated having their own individual computer rather than sharing
computers in a group. However, the battery life of the laptops used was not sufficient to
last an entire class. Laptops were constantly being shuffled to allow for chargers to reach
electrical outlets. Although, the expected battery life of the laptops was supposed to be
longer, they did not prove to be long enough. Some students also had trouble logging on
to the network and with a class set of laptops all using wireless internet, the internet
capabilities would sometimes be very slow. However, students did seem to be rather
attentive and concerned about their work. Even with technological frustration levels
high, the students seemed to enjoy the activity.
Mathematical Hands-on Learning Techniques
37
CHAPTER FIVE: ANALYSIS AND DISCUSSION OF RESULTS
Analysis
For focus question one, how are hands-on activities implemented in the secondary
classroom, data were collected using a unit lesson plan and a scoring rubric. The lesson
plan, outlining all hands-on activities to be implemented, was scored using a rubric by a
mathematics instructional specialist. The rubric focused on several areas including
technology, content knowledge, and assessment. The rubric was analyzed qualitatively
using comments and suggestions of the specialist.
The data collected from the instructional specialist accurately measured the
strengths and weaknesses of the lesson plan and its components. Direct quotes from the
specialist were reported whenever possible. The lesson plan and rubric comments
produced significant results.
The researcher considered suggestions mentioned by Clements and McMillen
(1996) and Ernest (1994) when formulating the unit lesson plan. Clements and McMillen
suggested several key ideas, mentioned previously in Chapter 2, that were followed, such
as the idea students working cooperatively in pairs as well as the teacher guidance
throughout each hands-on activity. Throughout the study, teacher guidance during each
activity was required for the students to produce the intended results. Ernest (1994) also
found that calculators can be the most useful manipulative in the mathematics classroom.
Throughout the study, the use of graphing and scientific calculators seemed to be
essential to understanding the material, and allowed the students to complete the designed
activities in the amount of time needed. Many of the activities in the lesson plan would
not have been possible without the aid of calculators.
Mathematical Hands-on Learning Techniques
38
Focus question two, how do math grades differ in classrooms using hands-on
activities, was analyzed using pre and post test scores obtained from both a treatment and
a control group. Data were coded and analyzed using several statistical tests including
independent and dependent t tests, reliability correlation coefficients, and an effect size
calculation. Pre-test scores from both groups were analyzed using an independent t test
with the following null statement: there is no significant difference between groups pretreatment. The pre-test data produced an obtained value that failed to exceed the critical
value. Therefore, at the   .05 confidence level, the null statement fails to be rejected.
Students who had previously taken the class were not used in the sample since their pretest scores proved to be outliers in the data set.
Since both the treatment and control group exhibited no significant differences,
the use of hands-on learning activities were utilized in the treatment group over the next
two weeks. The control group continued to follow a more traditional, direct instruction
approach. At the end of two weeks, a post test was given. To ensure dependability, the
same test was used for both the pre-test and post-test. Post-test scores were analyzed
using a dependent t-test to ensure that both groups showed significant gains using either
instructional technique. The null statement was rejected at the   .05 confidence level,
using the null statement: there is no significant difference in the group from pre-test to
post-test. Therefore, both the treatment and control group showed significant gains
throughout the unit. The means for both groups were much higher on the post-test than
on the pre-test showing great gains throughout the unit. Finally, another independent ttest was conducted on the post-test scores of the treatment and control group using the
null statement: there is no significant difference in the groups post-treatment. At the
Mathematical Hands-on Learning Techniques
39
  .05 confidence level, the obtained value failed to exceed the obtained value.
Therefore, the null statement fails to be rejected. Even though the t-test failed to show
significant difference in the treatment and control group post-treatment, it is important to
mention that the mean test score for the treatment group was much higher than the mean
of the control group. The mean of the treatment group was 79.47 while the mean of the
control group was 69.47. The treatment group’s post-test scores also exhibited a much
lower standard deviation than the control group. An effect size calculation was also
conducted on both groups using correlating pre and post-test data. The effect size of the
treatment group was also much higher than the effect size of the control group. The
calculated effect size asserts that the treatment group did show more gains in the post-test
scores than that of the control group. Although the effect size calculation cannot be used
to generalize to a larger population, the reported effect size of this study had a significant
magnitude.
These results confirmed what was previously stated in the literature. According to
some sources, student achievement would be improved with the use of hands-on learning
techniques (Zemelman et al., 2005).
Focus question three “What are student and teacher attitudes about the use of
hands-on learning” was answered using qualitative data. Teacher surveys, a student
focus group and a reflective journal were all recorded and analyzed. The teacher surveys
produced valuable insight into teacher’s thoughts and experiences with the
implementation of more hands-on learning techniques. A majority of teacher expressed
frustration with the implementation of hands-on activities. Many have had poor past
Mathematical Hands-on Learning Techniques
40
experiences with students vandalizing manipulatives and materials as well as an
exhausting number of hours preparing such activities.
The gains of use of hands-on learning were also discussed. While about half of
the teachers interviewed expressed that they had seen great gains with the use of such
learning, the other half expressed frustration seeing very little gains in this instruction
technique compared to direct instruction. A special education teacher even noted that
students sometimes felt babied when asked to complete activities rather than taking notes.
Overall, the idea of using technology in the classroom was a little more positive.
Teachers mentioned that they were already using calculators, overheads and other forms
of technology. Many expressed openness to implementing new forms of technology
when it becomes available. Some mentioned the amount of time needed to incorporate
more technology, but felt that gains in student achievement would be evident.
The math curriculum coach was a huge supporter of hands-on learning and
technology in the classroom. She expressed that that she has seen significant gains in
student achievement and student attitudes throughout her teaching years of teaching when
implementing technology and hands-on learning. She mentioned how much more time
was required when planning of these activities, but felt that the students were more
motivated to learn the material when they played an active role in their learning.
A focus group of Accelerated Math 2 students was conducted to gain insight into
their thoughts and experiences of hands-on learning in their high school classes. Many of
the students expressed how they enjoyed hands-on activities that allowed them to build or
make a final project. The students seemed to feel ownership and pride from putting
together something that showcased their work.
Mathematical Hands-on Learning Techniques
41
Finally, a reflective journal was kept by the researcher throughout the study and
recorded in after each hands-on activity. The reflective journal provided qualitative data
that was coded for themes. Overall, the researcher found positive reaction from students
when implementing the hands-on learning activities. Many students commented on how
much they enjoyed doing something different. However, some students did have some
complaints about doing different activities, some of which required them to leave the
classroom. Surprisingly, many of the complaints were made by lower achieving students.
Many of the higher achieving students, who normally do not seem to mind taking notes
and listening to direct instruction, seemed to really enjoy doing the hands-on lessons.
The qualitative data collected from both students and teachers also modified
current research. Ernest (1994) found that student attitudes improved and that students
seemed more motivated when using hands-on learning. A study by Sowell, (as cited by
Ernest, 1994), also found that student attitudes improved when manipulatives were used
in the classroom. However, this study found that while some attitudes improved and
some students seemed more motivated, other students did not exhibit these behaviors.
Some students showed the same attitudes and motivation during and after treatment as
they had before treatment started.
Discussion
Overall, this study found that while hands-on activities can sometimes motivate
students and increase their achievement, some students seem to resist the change and
prefer direct instruction. This study also found that while student achievement can be
slightly higher in classroom using hands-on learning, the gains were not significant
Mathematical Hands-on Learning Techniques
42
enough to draw precise conclusions about one method producing higher achievement
than the other.
It is also important to realize that this study was conducted using a Mathematics II
Geometry unit, which lends itself towards hands-on learning. Other topics in high school
math courses may not lend themselves so well with the use of hands-on learning
methods. This study also found that students may or may not prefer this type of learning.
An emerging theme that was found when conducting this study was the lower achieving
students resisting the change to try hands-on learning activities. Many of them
complained when the class left the room to complete activities in another room. Some of
the also did not want to work through frustrations with new technology to complete an
activity. Many of the lower achieving students in the class would become frustrated or
irritated in the process of the activity and would want to go back to a more traditional,
direct instruction technique.
Teachers typically try to implement many hands-on learning activities in
classrooms with low achieving students under the impression that it will increase their
achievement. However, this study was inconclusive in proving higher achievement for a
diverse group of students.
This study focused primarily on the practice of teaching mathematics in the high
school setting. This study found that while hands-on learning can have positive results
for learners, it may not be the best method for all topics. It also may not be the best
method for teaching every student. With the influx of new technology in high school,
this study was relevant in determining if the mere implementation of hands-on learning
would necessarily guarantee increased achievement.
Mathematical Hands-on Learning Techniques
43
Throughout this study many steps were taken to ensure that the study remained
credible. Structural corroboration was achieved by gathering data from several different
sources focused on three specific focus questions. Both quantitative and qualitative data
were gathered and recorded as accurately as possible. The review of literature also
presented opposing views to ensure fairness. Research was presented in an unbiased
fashion and all opposing views were adequately researched and considered.
Efforts were taken to ensure that this study was conducted methodically in order
to produce a coherent, tight case. Validity and reliability were considered when
determining data collection methods for each focus question. While the case and data
collection methods were strong, the sample size for this study was very small. The
evidence of this study are not strong enough to assert judgments concerning a large
population.
Implications
This study found that while the use of hands-on learning can be used to motivate
students and increase their achievement, it is not going to produce significantly different
results than students in a traditional, direct instruction setting. The quantitative results of
this study showed that both the treatment and the control groups made significant gains
throughout the unit, but in the end, one failed to produce significantly higher results than
the other. This study was conducted using a small sample, therefore, it would be hard to
generalize these results for a much larger population.
The qualitative data gathered throughout this study proved to be very valuable.
Student opinions were uncovered that were previously not considered. While most
teachers believe that students like doing hands-on activities, this did not prove to be true
Mathematical Hands-on Learning Techniques
44
for all students. As with all concepts, there were different opinions and complaints from
each student. This study shaped all participants in a positive way. Whether they
preferred it or not, the students experienced a new method of learning that they may or
may not have been exposed to be before. The researcher was positively impacted by
learning new methods of teaching and learned new ways to try to motivate students. The
other teachers who volunteered to be surveyed also were impacted by the study. Many
neighboring teachers expressed interest in the activities being conducted and tried several
of them in their classrooms as well.
This study can easily be replicated by any teacher in a high school math
classroom. The lesson plan used is referenced in appendix A and can be altered to fit any
unit or concept being taught.
Impact on Student Learning
This learning positively impacted student learning. Students were given the
opportunity to expand their ideas by trying new ways of learning that they previously
may not have used in a mathematics classroom. Students explored new technology that
they may use to expand their mathematical background. While some students did not
enjoy the hands-on learning activities, all students were given the opportunity to try new
ways of learning that may benefit them at some time in their educational career. Most
students also appreciated that their teacher was investing interest in their learning and that
new activities were being considered. This improved the classroom atmosphere not only
from student to teacher but also from peer to peer.
Mathematical Hands-on Learning Techniques
45
Most teachers involved in this study also mentioned that they would try some of
the new ideas in their classrooms to expose their students to the different hands-on
learning techniques as well as new technology that are available.
Recommendations for Future Research
This study had positive impacts on all participants involved. However, there were
several key components of this study that can be strengthened when considering future
research. First, this study took place over the course of one unit, which elapsed two
weeks. In the future, a longer study that encompassed an entire course would produce
much tighter argument. Also, the sample size used in this study was relatively small.
Two classes of fewer than 20 students each were used. The results would be better
generalized if a much larger sample was used. Since this study, many teachers at the
school of the researcher have expressed positive interest in the idea of incorporating
hands-on learning and technology in the classroom. Therefore, with the support of other
teachers, it may be possible to conduct a much larger study encompassing more students,
and even different levels of mathematics courses.
Mathematical Hands-on Learning Techniques
46
References
Clements, D, & McMillen, S. (1996). Rethinking concrete manipulatives. Teaching
Children Mathematics, 2, 270-279.
Ernest, P. (1994). Evaluation of the effectiveness and implementation of a math
manipulatives project.Proceedings (Report No. SE-057 682). Nashville, TN:
Annual Meeting Mid-South Educational Research Association. (ERIC Document
Reproduction Service No. ED391 675).
Georgia Department of Education (2010). EOCT Statewide Scores. Retrieved
from:http://www.doe.k12.ga.us/ci_testing.aspx?folderID=244&m=links&ft=EOC
T%20Statewide%20Scores
Heddens, J.W. (1997). Improving Mathematics Teaching by Using Manipulatives.
Retrieved October 3, 2010, from Kent State University website:
http://www.fed.cuhk.edu.hk/~fllee/mathfor/edumath/9706/13hedden.html
Hendricks, C. (2009). Improving schools through action research: a comprehensive guide
for educators. Upper Saddle River, NJ: Allyn & Bacon.
Johnson, A. (2009). What every teacher should know about action research. Upper
Saddle River, NJ: Pearson.
Moyer, P, Bolyard, J, & Spikell, M. (2002). What are virtual manipulatives?. Teaching
Children Mathematics, 8(6), 372.
Niess, M, & Walker, J. (2009). This rock 'n' roll video teaches math. Learning and
Leading with Technology, June/July 2009
Popham, W. (2008). Classroom assessment: what teachers need to know. Boston, MA:
Allyn & Bacon.
Mathematical Hands-on Learning Techniques
47
Salkind, N. (2010). Statistics for people who (think they) hate statistics: excel 2007
edition. Thousand Oaks, CA: Sage Publications, Inc.
Schul, J. E.(2011) Revisiting an old Friend: the practice and promise of cooperative
learning for the twenty-first century. Social Studies, 102( 2), 88 - 93
Silver, F, Strong, R, & Perini, M. (2008). The strategic teacher: selecting the right
research-based strategy for every lesson. Upper Saddle River, NJ: Prentice Hall.
Sprick, R.S. (2006). Discipline in the secondary classroom. San Francisco, CA: John
Wiley & Sons.
Zemelman, S, Daniels, H, & Hyde, A. (2005). Best practice: today's standards for
teaching and learning in america's schools. Portsmouth, NH: Heinemann
Educational Books.
Mathematical Hands-on Learning Techniques
48
Appendix A
Lesson Plan: Math 2 Circle Geometry Unit
Stage 1 – Desired Results
GPS and/or Elements (use only the elements that you teach in THIS lesson!):
MM2G3. Students will understand the properties of circles.
a. Understand and use properties of chords, tangents, and secants as an
application of triangle similarity.
b. Understand and use properties of central, inscribed, and related angles.
c. Use the properties of circles to solve problems involving the length of an arc
and the area of a sector.
d. Justify measurements and relationships in circles using geometric and
algebraic properties.
MM2G4. Students will find and compare the measures of spheres.
a. Use and apply surface area and volume of a sphere.
b. Determine the effect on surface area and volume of changing the radius or
diameter of a sphere.
Enduring Understandings:
Essential Question(s):
Students will understand that…
Circles have many properties and will be able to
answer questions concerning arcs, angles, tangents,
secants and chords.
Real World Understandings (What might transfer
to their world?):
Students will answer real world questions
concerning circles, such as pizzas. They will use
formulas to answer questions about surface area
and arc sectors.
What are the properties of tangents?
How do you use angle measures to find arc
measures?
How do you use the relationships of arcs and chords
in a circle?
How do you find the measure of angles inside,
outside and on a circle?
How do you find segment lengths in a circle?
How do you find circumference, arc length, and areas
of sectors and circles?
How do you find surface area and volume of a
sphere?
Knowledge (NOUNS for the GPS):
Skills (VERBS from the GPS):
Students will know…
Students will be able to…
Properties of chords, tangents and secants
Properties of central, inscribed, and related angles
Arc length of an arc and Area of a sector
Surface area and volume of a sphere
Understand
Solve
Justify
Apply
Determine
Real World knowledge (Where do they use this
KNOWLEDGE in their real world):
Students will solve problems involving circles in
many different career areas, including construction.
Real World Applications (Where do they use these
SKILLS in their real world):
Use of these skills is evidenced by students’ ability to
Mathematical Hands-on Learning Techniques
49
solve problems and work through tasks effectively.
Stage 2 – Assessment Evidence
Performance Task(s) and Product(s) to be
assessed (What will they put in my hand to
be assessed that they created individually):
Formal Assessment Grading Format(s) (How will
I grade it, letting them know in advance how to
receive every point in my grading scale):
Vocabulary flip charts
Versatiles completion (graded on successful
completion)
Human Circle participation
TI-84 Activity Worksheets
Dig This! Activity group answer
Surface area and volume worksheet for spheres
2-3 Quizzes covering individual lessons
1 unit post test
Stage 3 – Learning Plan
Procedures/Sequence:
Day 1:
* Students will complete the pre-test before beginning the Circle unit
*Students will construct tangible flip charts to be used throughout the unit.
*Students will define terms theorems from 6.1 and make concentration cards using words or
pictures, depending on the preference of the students.
Day 2-3:
* Students will learn theorems associated with finding arc measures in a circle using central
angles and the properties of chords in a circle. After a brief lessons, students will use
VersaTiles to complete 12 problems. Answers will be checked by comparing the resulting
picture to the key picture. Students will correct missed problems until they have successful
mastery.
Day 4:
Human Circle
*Students will use rope and elastic in a life size circle to compare central and inscribed
angles. Students will answer questions throughout the activity to aid them in discovering
properties of these angles. Students will also have the opportunity to use rope to
demonstrate properties studied during day 2 and 3.
Day 5:
Students will use TI-84 graphing calculators and Cabri software to complete the Angles and
Arcs Activity. This will help solidify theorems regarding angle measures inside and outside of
a circle. Students will visualize the concepts with the use of individual graphing calculators.
Mathematical Hands-on Learning Techniques
50
Day 6:
Students will use laptops and Illuminations software to complete an activity regarding
segment lengths in circles.
Day 7: Post-test
Enrichment, Hands-On, Student–Centered Activity:
(outlined above)
Materials:
Flip charts, index cards, versatiles, Human Circle materials (rope, tape and elastic), TI-84
calculators with Cabri software (class set), laptops with internet capability (class set), pizza
pan manipulatives, sports balls and seamstress tape.
1. Student LD: (i.e. Process, Product, Content)
Students may come in before or after school or during their study hall to receive more individual
help from the teacher. Collaborative classes will utilize the collab teacher to assist in all activities,
instruction and smaller group activities.
2. Student ESL: (Process, Product, Content)
Students will receive an outline of the project in their native language.
Closure :
What did we learn today about tangents…secants…etc.? …….Sue………Joey…
Mathematical Hands-on Learning Techniques
Appendix B
Pre-test/Post-test
51
Mathematical Hands-on Learning Techniques
52
Mathematical Hands-on Learning Techniques
Appendix C
Reflective Journal Prompts
What did we do today to incorporate manipulatives or a hands-on activity?
What part of today’s activity went well?
What could have gone better?
How did the student’s react?
How do I feel?
53
Mathematical Hands-on Learning Techniques
54
Appendix D
Hands-on Learning Teacher Survey
How many years of teaching experience do you have?
What are your primary classes taught?
What are your initial feelings regarding the implementation of hands-on learning
activities in the classroom?
Do you try to implement hands-on learning activities? If yes, what kinds of activities and
how often do you implement them?
Do these activities require extra planning?
What are some of the difficulties you have found with implementing these activities?
What are some of the gains that you have found in using these activities?
Do you feel that these activities influence a student’s attitude toward the concept?
Do you try to implement technology in your classroom? What technology items have
you tried in your classroom? What works? What doesn’t?
Have you noticed a change in students’ attitudes when implementing technology?
Mathematical Hands-on Learning Techniques
55
Appendix E
Student Likert-type Attitude Survey
5
Very
Much
I like math.
I think math is often hard to
understand.
I think it’s important for me to study
math.
I feel that math is useful for me right
now.
I am looking forward to studying
Geometry.
I think Geometry is going to be hard.
I have trouble visualizing three
dimensional objects.
I like to do hands-on work.
I like to work with a partner.
I like to work in cooperative groups.
My math homework is usually
interesting.
I generally get good grades in math.
I have been introduced to geometry in
the past.
4
3
Indifferent
2
1
Not
at all
Mathematical Hands-on Learning Techniques
Appendix F
Student Focus Group Prompts
What do you think of when your teacher says “hands-on” activity?
What are some of the hands-on activities you do in your other classes?
What kind of technology do you normally use?
Do you enjoy these activities?
Do you think you understand better after doing these activities?
Do these activities sometimes confuse you?
Would you prefer to do more hands-on learning activities in class?
56
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1