Constructivism or Rote Learning
CONSTRUCTIVISM OR ROTE LEARNING: PROMOTING GREATER ACADEMIC ACHEIVEMENT
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.
Zontrina` C. Morton
Certificate of Approval
______
Donald R. Livingston, Ed.D. Sharon M. Livingston, Ph.D. Thesis Co-Chair Thesis Co-Chair Education Department Education Department Constructivism or Rote Learning
CONSTRUCTIVISIM OR ROTE LEARNING: PROMOTING GREATER ACADEMIC ACHEIVEMENT
A working thesis submitted by
Zontrina` Morton
To
LaGrange College
In partial fulfillment of
the requirement for the
degree of
MASTER OF EDUCATION
in
Curriculum and Instruction
LaGrange, Georgia
May 4, 2011 Constructivism or Rote Learning
Abstract
The purpose of the study was to identify if discovery learning or rote learning promote greater academic achievement for ninth grade math 1 students. The study compares the academic achievement of discovery learners to the academic achievement of rote learners of ninth grade REP students and teachers views on discovery learning and rote learning. After a broad assessment of the literature, data from test scores, student observations, and surveys were gathered throughout this study. The results from the study showed that there is no significant difference between the academic achievement of discovery learners and rote learners. The results also provided evidence that discovery learning promoted positive attitude changes towards mathematics. Survey results also showed that teachers were undecided about which method was best for students. Constructivism or Rote Learning ii
Table of Contents Abstract……………………………………………………………………………..iii
Table of Contents…………………………………………………………………...iv
List of Tables………………………………………………………………………..v
Chapter One: Introduction…………………………………………………………..1 Statement of the Problem……………………………………………………1
Significance of the Problem…………………………………………………2 Theoretical and Conceptual Frameworks……………………………………3 Focus Questions……………………………………………………………..5 Overview of the Methodology………………………………………………6 Human as Researcher………………………………………………………..7
Chapter Two: Review of the Literature……………………………………………..8 Cognitive Structures and Learning Mathematical Concepts………………..8 Discovery Learning and Rote Learning……………………………………11 Student Attitudes Towards Mathematics…………………………………..15
Chapter Three: Methodology………………………………………………………18 Research Design……………………………………………………………18 Setting……………………………………………………………………...18 Sample/ Subjects/ Participants……………………………………………..19 Procedures and Data collection Methods………………………………….19 Validity/Reliability/Dependability/Bias…………………………………....22 Constructivism or Rote Learning iii
Analysis of Data…………………………………………………………....26 Validation………………………………………………………………….27 Credibility………………………………………………………………….27 Transferability……………………………………………………………...28 Transformational…………………………………………………………...28
Chapter Four: Results………………………………………………………………29
Chapter Five: Analysis and Discussion of Results…………………………………40 Analysis…………………………………………………………………….40 Discussion………………………………………………………………….44 Implications………………………………………………………………...46 Impact on Student Learning………………………………………………..47 Recommendations for Future Research……………………………………48 References………………………………………………………………………….49 Appendices………………………………………………………………………....52
List of Tables
Table 3.1 Data Shell…………………………………………………………….…..20 Table 4.1 Pre Test Results for Experimental and Control Group…………………..32 Table 4.2 Pre Test and Post Test Results for Experimental Group………………..33 Table 4.3 Pre Test and Post Test Results for Control Group…………………...... 33 Table 4.4 Post Test Results for Experimental and Control Groups………………...34 Table 4.5 Cronbach’s Alpha Student Pre and Post Test Survey…………………...35 Table 4.6 Cronbach’s Alpha Teacher Survey………………………………………35 Table 4.7 Chi Square Student Pre Survey…………………………………………..37 Constructivism or Rote Learning iv Constructivism or Rote Learning 1
CHAPTER ONE: INTRODUCTION
Statement of the Problem
“When I was in elementary school I loved math, but now I hate math?” Several students say this as they come into class. In elementary school the students are taught to be rote learners and memorize all math facts without understanding what they are doing.
However, most adults today learned math from the “master,” being the teacher. The teacher was writing on the board explaining how to do methodical problems, and then we were supposed to emulate what she/he had done. The teacher would always do a sample problem first giving all of the steps to solve the problem, then give us one just like it with the numbers and variables being the only difference from the problem she/he just did.
We knew how to follow the steps for the problem, but for us the concept of mathematics stated by Dreyfus and Eisenberg, is a “digestive process rather than a creative one” (as cited by Mann, 2006).
The traditional styles of teaching mathematic solutions are either right or wrong with only one way to solve the problem. The problem with rote learning, or the step by step process of learning, is that the students can follow the steps, but they do not have a full understanding of the mathematical concept being taught. Students are able to replicate what someone has shown them, but are often unable to think through a problem.
Students should be able to solve problems requiring independence, judgment, originality, and creativity. When using the discovery method, students have an opportunity to get the right answer using many different methods. Creativity is a process of becoming responsive to problems, deficiencies, gaps in knowledge, and being able to explore Constructivism or Rote Learning 2 solutions by guessing, or formulating their own assumption. In this study I will explore, identify, and test how students can retain mathematical concepts through discovery learning compared to the traditional style of teaching.
Significance of the Problem
Research stated that the essence of mathematics is to be able to think creatively, not simply arriving at the right answer (Mann, 2006). If students are forced or taught to become rote learners, they are stuck seeing math as black and white, right or wrong. The only thing they will remember are the basic multiplication facts, counting money, and some telling time. When they graduate and go into the “real world,” they will not be able to relate the material they had learned in school to anything in life. This may ruin their self esteem due to their declining grades if they go to college or, God forbid, someone asks them to think of a solution to a problem. They will not be creative thinkers, they will be dependent upon colleagues or managers, but they will be able to follow steps that are given to them. If a student is a discovery learner, they will be able to relate real world problems, not only in class, but in life. They will be critical thinkers, independent thinkers, problem solvers, and they will not have to memorize basic steps in order to solve a problem. Discovery learners are able to think through problems themselves.
Dreyfus and Eisenberg suggested that for a transfer of learning, students need more than drill and practice, they also need to understand the mathematical concepts beyond the practice exercises (as cited by Mann, 2006). Constructivism or Rote Learning 3
Theoretical and Conceptual Frameworks
When studying the different teaching methods for this thesis, it is clear that discovery learning is closely related to the constructivism theory of teaching. The main focus of this thesis is to identify if discovery learning promotes greater academic achievement than the traditional style of teaching. Constructivism is the idea that learners construct their own knowledge (Hein, 1991). Piaget believed that human beings possess mental structures that assimilate external events, and convert them to fit their mental structures (Bhattacharya & Han, 2009). Students need to be given different tools of learning according to their learning style. Students in a constructivist classroom are actively engaged and participating in activities based on their particular style of learning.
Theory and research support the proposition that all four of the learning profile categories
— learning style, gender, culture and intelligence preference — impact learning, indicating that learning is enhanced when a mode of learning or approach to learning matches an individual's learning preferences (Tomlinson, 2009).
This thesis relates to the first tenet, knowledge of content and the second tenet, exemplary professional teaching practices of the LaGrange College Education
Department’s (2008) Conceptual Framework. The instructor, which serves as a facilitator, must have a great level of knowledge of the content before using a constructivist’s ideas. According to LaGrange College Education Department’s
Conceptual Framework, “While all knowledge begins with experience, not all knowledge can be adequately constructed without understanding the central concepts, tools of inquiry, and structures of various disciplines” (LaGrange College Education Department, p. 3). This means that all students have some sort of prior knowledge, so they should be Constructivism or Rote Learning 4 able to relate the new concepts with their prior knowledge and be able to apply, explore, and develop ways of solving real world problems independently or collaboratively. In a constructivist’s classroom, the teacher allows the students to use their own style of learning that is conducive for them.
Constructivism is differentiated instructions where a students’ prior knowledge and experience offer them multiple approaches to learning (LaGrange College Education
Department, p. 5). As stated in LaGrange College’s Education Departments Conceptual
Frameworks, A challenge that teachers face in a constructivist classroom is to apply the constructivist principles while simultaneously meeting the content and testing requirements of the state departments of education and local school boards. The idea of constructivism and differentiated instructions is an important part of discovery learning.
It can be implied that the discovery method of teaching is having the students start from scratch with them not having any prior knowledge, and having them basically “recreate the wheel.” As suggested by LaGrange College Education Department Conceptual
Frameworks, the process of differentiated instruction starts with teaching the concept, present new knowledge, and gives students a work period where they can apply the new information in active, cooperative, and meaningful ways (p. 7).
On the national level, Proposition three of the National Board for Professional
Teaching Standards Core Propositions for Experienced Teachers states, “teachers know the subjects they teach and how to teach those subjects to students” (as cited by
LaGrange College Education Department, p. 12). The awareness that teachers know the subjects that they teach play a very important role in constructivism, due to the fact that teachers have to know how to strategically plan lessons for all types of learning styles. Constructivism or Rote Learning 5
Constructivism is about having the students become active learners and for them to use their own creative minds while solving real life problems in a classroom environment.
The idea that teachers know how to teach those subjects to students relate because a teacher must know their students and how they learn to be effective in the classroom.
Therefore, teachers must use a variety of instructional strategies to encourage students to become critical thinkers.
The National Board for Professional Teachers Standards, LaGrange College
Education Department’s (2008) Conceptual Framework, and the Georgia Framework for
Teaching are all on one accord when it comes to professional teaching practices. On the state level, Domain three of the Georgia Framework for Teachers states, “Learning
Environments: Teachers create learning environments that encourage positive social interaction, active engagement in learning, and self-motivation” (as cited by LaGrange
College Education Department, p. 11). Teachers need to use various teaching strategies to keep their students engaged in learning. Teachers must focus on student performance while using the constructivist ideas in the classroom to ensure that the lessons they are creating are meaningful. By using constructivist ideas in the classroom teachers should notice the positive reactions, self-motivation, and the active and engaging learning environment they have created.
Focus Questions
Students are not fully understanding and retaining mathematical concepts; they are memorizing the process. In order to provide a deeper understanding of mathematical concepts, I had to determine if discovery learning or rote learning promote greater Constructivism or Rote Learning 6 academic achievement. The following focus questions guided this study in order to determine the effectiveness of discovery learning.
1. Why do cognitive structures matter when teaching mathematical concepts?
2. What effects does rote learning have compared to discovery learning?
3. What were the students and teachers attitudes about discovery learning and rote
learning?
Overview of Methodology
Research was conducted using action research, employing both quantitative and qualitative data. Action research was used given that the research was conducted in the classroom. Two classes were selected to participate in this study. The research time for this study was two weeks. The quantitative data analyzed with statistics that completed the academic achievement data were recorded and evaluated using the qualitative ethnographic method due to a small sample size. The ethnographic method displays the data as a description of academic achievement. This method helped determine if there was a difference in test scores between students using discovery learning and rote learning. Surveys were given to the teachers to get their opinions on discovery learning versus rote learning. The survey questions were evaluated using chi square analysis.
Student’s academic achievement for the discovery learners and rote learners were documented by student observations, using the ethnographic method, and using student surveys. Constructivism or Rote Learning 7
Human as Researcher
As a fourth year teacher of ninth and tenth grade mathematics I have come to realize that student's are not retaining mathematical concepts they are memorizing the process. Mathematics teachers try to teach their students using different teaching strategies; however, discovery learning is supposed to be a better strategy than any other.
As a child I was taught with the drill and practice method. I knew how to do the problems, but did not have a full understanding of why I was doing what I was doing. I knew enough to get through the courses but did not retain many of the concepts that were taught. Now, I want to know if discovery learning provides a deeper understanding to promote academic achievement than rote learning. So I can be sure when my students take the End of Course Test they would have retained most of the mathematical concepts learned. Constructivism or Rote Learning 8
CHAPTER TWO: REVIEW OF THE LITERATURE
The research for this study is aligned with the focus questions above to create a starting point for how this study is conducted and to speculate the results from the data collected. The research used was collected with the intention of identifying if discovery learning or rote learning promote greater academic achievement. The literature was examined to find studies pertaining to cognitive structures, discovery learning, rote learning, and student attitudes towards mathematics.
Cognitive Structures and Learning Mathematical Concepts
Discovery learning and rote learning are intended to promote academic success.
Learning mathematics requires that children create and re-create mathematical relationships in their own minds (Burns, 1992). Learning should take place by doing not mimicking.
Cognitive structures are the mental process people use to make sense of information (Garner, 2007). Mentioned in Getting to Got It: Helping Struggling Kids
Learn How to Learn, (Garner, 2007), cognitive structures are broken down into three interdependent categories as follows: The first category is comparative thinking structures which process information by identifying how bits of data are alike and different. They include recognition, and memorization to name a few. The second category is symbolic representation structures which transform information into culturally acceptable coding systems. They include verbal and nonverbal language, mathematics, music and rhythms. Lastly, the third category is logical reasoning Constructivism or Rote Learning 9 structures which use abstract thinking strategies to systematically process and generate information. They include deductive and inductive reasoning, analogical and hypothetical thinking (Garner, 2007). Research found on cognitive structures suggests that the more educators learn about how cognitive structures affect learning the more optimistic they can be.
Cognitive theory focuses on how people develop structures of knowledge. In cognitive theory, knowing means more that the accumulation of factual information and routine procedures for reasoning and solving problems. It focuses on how knowledge is encoded, stored, organized, and retrieved (Pellegrino, Chudowsky, & Glasser, 2001).
Research also suggests that cognitive theory emphasizes what type of knowledge someone has. The purpose of assessment is not to determine what people know, but how, when, and whether they use what they know. This type of information is hard to capture in traditional tests, which focus on how many someone answers correctly or incorrectly with no additional information given as to how they derived those answers. Cognitive structures require more complex tasks that reveal information about thinking patterns reasoning strategies, and growth over time (Pellegrino, et al, 2001). Students should be able not only to understand concepts that have been presented and learned, but must also have the ability to know when to use these concepts. This is the only the beginning of becoming proficient in the use of cognitive thinking/reasoning in math. This proficiency is accomplished in steps or strands which are touched upon in the following description.
Mathematical proficiency has five strands: 1. Understanding: Comprehending mathematical concepts, operations, and relations- knowing what mathematical symbols, diagrams, and procedures mean. 2. Computing: Carrying out mathematical procedures, Constructivism or Rote Learning 10 such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. 3. Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. 4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something not yet known. 5. Engaging: Seeing mathematics as sensible, useful, and doable- if you work at it- and being willing to do the work (Kilpatrick & Swafford, 2002). Knowledge learned with understanding provides a foundation for remembering or reconstructing mathematical facts and methods, for solving new and unfamiliar problems, and for generating new knowledge (Kilpatrick &
Swafford, 2002).
In the 1980s and 1990s computational skills were appearing to be less important.
However, the notion that students should understand and be able to use math was becoming more prevalent. Students were expected to invent math with little or no assistance (Kilpatrick & Swafford, 2002). These efforts led to increased attention to memorization and computational skill, with students expected to internalize procedures presented by teachers or textbooks. The clash of these contrasting positions is called the
“math wars” (Kilpatrick & Swafford, 2002). When people advocate only one strand of proficiency, they lose sight of the overall goal. Such a narrow treatment of math may well be one reason for the poor performance of U.S. students in national and international assessments (Kilpatrick & Swafford, 2002).
Students become more proficient when they understand the underlying concepts of math, and they understand the concepts more easily if they are skilled at computational procedures. U.S. students need more skill and more understanding along with the ability Constructivism or Rote Learning 11 to apply concepts to solve problems, to reason logically, and to see math as sensible, useful, and doable (Kilpatrick & Swafford, 2002).
Proficiency is much more likely to develop when a mathematics classroom is a community of learners rather than a collection of isolated individuals (Kilpatrick &
Swafford, 2002).
One of the strongest findings from research is that time and opportunity to learn are essential for successful learning (Kilpatrick & Swafford, 2002).
Discovery Learning and Rote Learning
Discovery learning is designed to promote academic triumph for students with different learning styles. Rote learning is designed to promote academic success through repetition or rehearsal. Some educators favor discovery learning while others believe rote learning is more effective. However, the one doing the work is the one who is learning. Teaching students to formulate, understand and solve problems is vital; therefore, the National Council of Teachers has called for more engaging, meaningful mathematics instruction for all students (Lubiensky, 2007). The mathematical knowledge and skills that were traditionally being taught are no longer required. However, students need to learn problem solving, critical thinking and mathematical confidence to be able to apply learned knowledge in various situations and acquire new knowledge on their own.
Classroom exploration and discovery learning are believed to improve retention and deepen understanding. However, students discover greater purpose of their learning through their cognitive engagement (Simonet, 2008). This is, in essence, how students Constructivism or Rote Learning 12 think about what they are learning and formulate a deeper understanding of the material’s significance (Simonet, 2008).
Research suggests that rote learning causes students to have difficulty working independently without an example in front of them. Research also suggested that in order for students to learn, they need to use their cognitive structure to process the information instead of memorizing the process (Gardner, 2007). Math evokes memories of failure in parents. Students fear and lack self confidence when it comes to math (Tilton, 2003).
Students are not all self regulated learners. They may not know what they understand and what they don’t understand. Sometimes they think they understand when they really do not (Douglas & Frey, 2008). Research suggests that students need hands on instruction in order to understand math. So many students need to be physically engaged in a task in order to learn (Tilton, 2003). The background knowledge that students bring into the classroom affects how they understand the concepts being taught. Therefore, identifying and confronting misconceptions that interfere with learning is an important part of the learning process (Douglas & Frey, 2008). Most math classes in elementary grades use more manipulatives. The higher a student moves up in grade level they use less manipulatives and more paper and pencil methods.
Checking for understanding gives the students a model of good study skills. With teachers consistently checking for understanding students become aware of how to monitor their own understanding (Douglas & Frey, 2008). There are different methods of checking for understanding such as, oral language, writing, projects and performances, and test. Constructivism or Rote Learning 13
Student-centered education, which is also called discovery learning, has a more integrated curriculum that bases learning more on student interests, small-groups individual instruction, and prefers individual diagnostic evaluation. A relatively new call for reform comes from those who wish to have schools pay more attention to developing student’s practical thinking abilities. They share a common concern for focusing less on facts, knowledge, and rote skills and more on thinking, problem solving, understanding, and creating (Chall, 2000).
Student-centered schools follow student interests and integrate materials across subject areas. The teacher serves as a facilitator of learning: providing resources to help students plan and follow their own interests and to keep records of learners’ activities and use of time. A rich variety of materials, including manipulatives are used; as well as, a wide range of activities based on individual interests. Students work in small groups individually with teacher guidance based on their own initiative and the teaching target is the individual child. The students are permitted to move around freely and cooperate with other learners. The use of time is flexible, often permitting uninterrupted work sessions determined by the learner. They are evaluated based on comparisons of learners with themselves rather than with their classmates or grade standards. There is a preference for diagnostic rather than a normal referenced evaluation and a de-emphasis on formal testing because learners proceed at different rates (Chall, 2000).
Teacher-centered schools standards are established for each grade level with specific subject areas being taught separately. The teacher serves as a leader of the class and is responsible for the content, leading lessons, recitation, skills, seatwork, and assigning homework. The teacher-centered pattern is more formal, with a curriculum Constructivism or Rote Learning 14 divided by grade levels and different subjects, ant that textbooks and tests are more widely used in that setting (Chall, 2000). The teachers work with the commercial textbook and a smaller range of activities are teacher-prescribed. The whole class is moved through the same curriculum at roughly the same pace. The teacher may occasionally teach small groups and may provided a degree of individualized instruction, but the teaching target is the whole class. Students are not allowed to move around freely; meaning child-child interactions are restricted. The day is divided into distinct periods for teaching different subjects. They are evaluated using the norm-referenced tests and grade standards, and the usual informal and formal testing. Students are assigned to grades by age (Chall, 2000).
Students who experience this reform tradition are encouraged to explore, develop conjectures, prove, and solve problems. The assumption is that students learn best by resolving problematic situations that challenge them through conceptual understanding
(Akisola & Olowojaiye, 2008). Several studies in the area of mathematics have shown that instruction, especially at the secondary school level remains overwhelmingly teacher-centered, with greater emphasis being placed on lecturing and textbook than helping students to think critical across subject area and applying their knowledge to real world situations. However, there is a need to adopt some of the recent reform based instructional strategies, along with some traditional practices that have been overlooked and underutilized in secondary mathematics (Akisola & Olowojaiye, 2008).
Building student capacity for mathematics thinking and reasoning concluded that students must first be provided with opportunities, encouragement, and assistance before they can engage in thinking, reasoning, and sense-making in the mathematics classroom. Constructivism or Rote Learning 15
Consistent engagement in such thinking practices; they maintained, should lead students to a deeper understanding of mathematics as well as increased ability to demonstrate complex problem solving, reasoning, and communication skill upon assessment of learning (Akisola & Olowojaiye, 2008). Learning mathematics requires that children create and recreate mathematical relationships in their own minds (Burns, 1992).
Students’ Attitudes Towards Mathematics
Allport (1935) states that an attitude is a mental and neural state of readiness, organized through experience, exerting a directive a directive or dynamic influence upon the individual’s response to all objects and situations with which it is related. In order for students to be successful in math, negative attitudes need to change to can do attitudes
(Tilton, 2003). Attitudes are psychological constructs theorized to be composed of emotional, cognitive and behavioral components. Attitudes serve as functions including social expressions, value expressive, utilitarian, and defensive functions, for the people who hold them (Akinsola & Olowojaiye, 2008.) Attitudes are connected to social cognitive learning theory as one of the personal factors that affect learning (Akinsola &
Olowojaiye, 2008.) It is generally believed that students’ attitude towards a subject determines their success in that subject. A favorable attitude results in good achievement in a subject. Research suggests that a student’s attitude towards mathematics could be enhanced through effective teaching strategies. It has been confirmed that effective teaching strategies can create positive attitudes on the students towards school subjects
(Akinsola & Olowojaiye, 2008.) Since attitudes refer to those actions that result from and are influenced by emotion, consequently, the affective domain relates to emotion, attitudes, appreciations, and values. In the mathematics classroom the affective domain Constructivism or Rote Learning 16 is thus concerned with students’ perceptions of mathematics, their feelings towards solving problems, and their attitudes about school and education in general (Akinsola &
Olowojaiye, 2008.)
Attitudes of students can be influenced by the attitude of the teacher and his method of teaching. Studies carried out have shown that the teacher’s method of mathematics teaching and his personality greatly accounted for the students’ positive attitude towards mathematics and that, without interest and personal effort in learning mathematics by the students, they can hardly perform well in the subject (Yara, 2009.)
The Organization for Economic Co-Operation and Development suggests that it can be argued that students ‘attitudes should relate to academic achievement; however, it is not obvious if success in learning engenders a better attitude or if a positive attitude toward school is a factor for motivating students to learn (OECE, 2010.) In the field of mathematics education, research on attitude has been motivated by the belief that attitude plays a crucial role in learning mathematics (DiMartino & Zan, 2003), but the goal of highlighting a connection between a positive attitude and achievement has not been reached. PISA also suggests that students’ attitudes towards school are weakly correlated with achievement (OECE, 2010.)
Cheung (1998) wrote about the attitude toward mathematics and the ages of 11-
13 year olds. He states that these ages are particularly important in the development of a mathematical attitude. This is the time when negative attitudes become most noticeable.
Although he goes to say that the reason behind this is unclear. Possible reasons behind this are the greater prevalence of abstractions in mathematics material. In his research, using he found a positive correlation between attitude and mathematics achievement. Constructivism or Rote Learning 17
The correlation showed that the more positive the attitude, the higher the level of achievement in the student.
Evidence abounds that the conventional teaching method which is the traditional method commonly used in schools, is inadequate for improved students attitude towards mathematics. If learners are not assisted or encouraged to perceive positively most of the things they learned in mathematics classes, their performance will be affected. It depends entirely on the teacher to help learners develop positive attitudes towards the learning of mathematics (Akinsola & Olowojaiye, 2008). Constructivism or Rote Learning 18
CHAPTER 3: METHODOLOGY
Research Design
The research design for this study was action research. A classroom action research design was implemented because the research will take place in the classroom with the teacher as the researcher. Action research allows teachers to investigate ways to deal with issues related to student achievement, classroom management, students with special needs, and motivation (Hendricks, 2006). Action research is for practitioners to investigate and improve their practices (Hendricks, 2006). Action research is a process where participants examine their own educational practice systematically and carefully, using the techniques of research (Ferrance, 2000.) The term action research is the idea that teachers will start a cycle of posing questions, gather data, write reflections, and decide on a course of action (Ferrance, 2000.)
My research focus of this study is to examine ninth grade students’ achievement level and attitudes towards mathematics using the constructivist approach and the traditional approach. The methods that will be used to collect data are assessments, meaning pre and post tests, student surveys measuring their attitude before and after the treatment, and reflective journaling.
Setting
The research study took place at a Title I high school in an urban area of Georgia; which is a predominately middle to lower class school. The school population consists of
1118 students. There are 466 ninth grade students, 373 tenth grade students, and 279 eleventh grade students. The race and ethnic percentages are as follows: 3% Asian, 84% Constructivism or Rote Learning 19
Black, 10.4% Hispanic, 0.3% Native American, 1.9% White, and 0.4% Multi-Ethnic.
The location of my classroom is on the second floor on the designated ninth grade hall.
Subjects
The subjects are the students of two Math I REP classes. REP students are extremely low level learners. The subjects in the REP class have been socially promoted to ninth grade due to their age or their parents signing a waiver. Therefore, students who were considered to be retained are included in this study. The average age of a ninth grade student is fourteen; however, there are a few students that are sixteen and seventeen in the classes as well. There are sixteen students in the first period class and twelve students in the second period class. The first period class consists of one teacher, 7 females, 9 males, 12 Black, 3 Hispanic, 0 White, and 1 Asian student. The second period class consists of one teacher, 4 females, 8 males, 8 Black, 4 Hispanic, 0 White, and 0
Asian students. These classes were chosen because they take place at the same time of day and they were conveniently accessible.
Procedures and Data Collection
There are three methods that were used to collect data for my study. The types of data used were pre and post assessments, student attitudinal surveys, and a reflective journal. Focus Question One collected data using reflective journaling, an instructional plan rubric, and an instructional plan interview during the spring of the 2010-2011 school year. Reflective journaling (see Appendix D) was used to reflect on the daily lessons, student behaviors, and to analyze the students’ skill level. The instructional plan rubric
(see Appendix B) was used to lay out research plan. The instructional plan was critiqued Constructivism or Rote Learning 20 by two fellow colleagues. After the critique the colleagues were interviewed to discuss their suggestions of what can be added or done differently according to the rubric.
Below in Table 3.1, the research design, literature sources, data sources, and the analysis of the data for this study are current and prearranged by focus question.
Table 3.1: Data Shell
Focus Question Literature Sources Type of Method How these data Rationale and Data are analyzed?
Why do cognitive (Dreyfus & Method: Coded for Qualitative: structures matter Eisenberg, 1966) themes There are when teaching Instructional Plan categorical mathematical (Garner, 2007) Rubric Recurring or repeating concepts? data (Kilpatrick & Instructional Plan Dominate Swafford, 2002) Rubric Interview Emerging Focus Learning Task Reflective Question Journaling
Student Two observation
Data: collected test Qualitative score data What effects does (Mathcounts, Method: Descriptive Quantitative: rote learning have 1998) and inferential there are no from the compared to Assessments Statistics significant discovery (Chall, 2000) differences learning on test Quantitative Data Pre/Post Test between rote spring scores? (Akisola & learning and Olowojaiye, 2008) Data: Dependent T- discovery semester of Test learning. Quantitative Independent T- the 2010- Test 2011 school
What are students (Akinsola & Method: Descriptive Qualitative: year. The and teachers Olowojaiye, 2008) and inferential There are attitudes about Student surveys Statistics categorical discovery (Yara, 2009) or repeating teacher made learning and rote Teacher surveys data learning? (DiMartino & Quantitative: Zan, 2003) Focus Group there are no pretest results significant Reflective differences Journal between rote learning and Data: discovery learning Qualitative Constructivism or Rote Learning 21 were used to determine how much students have retained using the discovery method; as well as, the traditional method of learning throughout the semester. These scores will be used to determine how many students in each subgroup were able to improve their test scores by 10 percent. Classroom assessments and the teacher made post tests will also be measures of improvement in how much a student has retained. These scores will also determine how many students in each subgroup whom is on target to meet or exceed the goal of 70% on the End of Course Test. Permission was granted to use the test scores of the subjects if the information was kept nameless. In studies of growth or development, equivalent forms of the same test may be administered at the beginning and end of the experimental instruction to measure the comparative effects of different methods or materials of instruction (Engelhart, 1972.) Observations are also used in focus question two to watch the behavioral patterns of people in certain situations to obtain information about the phenomenon of interest. Observation is an important way to collect information about people because people do not always do what they say they do
(Christensen & Johnson, 2008.)
Focus Question Three used the method of teacher and student surveys (see
Appendixes E and F). The surveys were given to the educators in the school once throughout the study. The goal of the survey was to determine how educators taught students’ in their own classes and to establish if majority used the discovery method of teaching or the traditional approach. Surveys or questionnaires were appropriate for this study because they provide a proficient way to collect and analyze data (Patten, 1998.)
Researchers use questionnaires to obtain information about the thoughts, feelings, attitudes, beliefs, values, perceptions and other characteristics of the research participants Constructivism or Rote Learning 22
(Christensen & Johnson, 2008.) Focus Question Three also used focus groups (See
Appendix C) as a method to obtain information from students and educators. A focus group is a group interview in which a moderator leads a discussion with a small group of individuals to examine, in detail, how the group members think and fell about a topic
(Christensen & Johnson, 2008.) Focus groups were used to determine how students learned and retain information taught in mathematics class. Teachers and students were given prompts that were answered in a descriptive dialogue. The dialogue consisted of how the teacher related the concepts to the students; for example, did the teacher use manipulatives or lecture.
Validity/Reliability/Dependability/Bias
Focus Question One states: Why do cognitive structures matter when teaching mathematical concepts? In Focus Question One the data was collected using performance learning tasks, an instructional plan rubric, and an instructional plan interview. The validity for Focus Question One was content related because the learning task, instructional plan rubric and interview accurately reflect the Georgia Performance
Standards for ninth grade students (Popham, 2008). Focus Question One was valid research because the performance tasks are widely used by mathematics educators in
Georgia. The tasks and rubric were designed to assess Math skills aligned with the
Georgia Performance Standards for ninth graders. The performance tasks have been constructed by math experts for internal consistency and bias to ensure the questions can be measured in a consistent manner which makes the task dependable (Popham, 2008).
The instructional plan rubric was reviewed by math experts for internal consistency and bias to ensure the students progress can be measured in a consistent manner which makes Constructivism or Rote Learning 23 the rubric dependable (Popham, 2008). All instruments will be checked for unfairness, offensiveness, and disparate impact. Popham (2008) states that assessment bias refers to the qualities of an assessment instrument that offends or unfairly penalizes a group of students based on their gender, race, ethnicity, socioeconomic status, religion, or other such group defining characteristics.
Focus Question Two states: What effects does rote learning have compared to discovery learning on test scores? Focus Question Two collected data from assessments.
These assessments consisted of Pre and Post Test results. The validity for Focus
Question Two was content related because the test results accurately reflect the Georgia
Performance Standards for ninth grade (Popham, 2008). Focus Question Two was valid research because the pre and post tests are designed to assess ninth grade math skills that are aligned with the Georgia Performance Standards. The tests were benchmark test that are used amongst all math teachers in the county; therefore, it is considered interval data.
The tests are designed by experts for internal consistency and bias, and that makes these test dependable (Salkind 2007). The data collected by the pre and post test were analyzed using a parallel test for reliability correlation. All instruments will be checked for unfairness, offensiveness, and disparate impact. Popham (2008) states that assessment bias refers to the qualities of an assessment instrument that offends or unfairly penalize a group of students based on their gender, race, ethnicity, socioeconomic status, religion, or other such group defining characteristics.
Focus Question Three asks: What are students and teachers attitudes about discovery learning and rote learning? Focus Question Three consists of surveys which were measured using nominal data on a five point Likert scale. The survey measures Constructivism or Rote Learning 24 teachers’ attitudes towards the discovery method of learning and rote learning. It also measures students’ attitudes towards mathematics as a whole. The Cronbach’s Alpha determined if there is a strong internal consistency among the responses to the questions
(Salkind, 2007). The internal consistency reliability of the data collected by the surveys was analyzed using the Cronbach’s Alpha method.
The survey data measured teachers’ attitudes towards discovery learning and rote learning. Popham (2008) states construct validity is a procedure in measuring the inferred construct accurately; therefore construct validity was confirmed by hypothesizing that most math teachers may favor the hands on approach while others may not favor hands on as strong. The surveys prompted teachers to specify how their students learn in their classroom in order for the data to be investigated.
All instruments will be checked for unfairness, offensiveness, and disparate impact. Popham (2008) states that assessment bias refers to the qualities of an assessment instrument that offends or unfairly penalizes a group of students based on their gender, race, ethnicity, socioeconomic status, religion, or other such group defining characteristics.
The survey has also been used by other researchers in a similar study and confirmed it would return accurate results. The researchers did not mention that the survey held any type of bias.
Focus Question Three was answered by collecting data from reflective journaling and Focus Groups. The validity sought was construct validity because it deals with student attitudes. Journal prompts were used in order to make the data more reliable. The Constructivism or Rote Learning 25 validity was obtained by hypothesizing that teachers who prefer constructivism over rote learning would have some type of bias when answering the Focus Group questions. This will also be true for teachers who prefer rote learning over constructivism. There were a total of five respondents. Three of the respondents stated that students who use the discovery method will have a deeper understanding of the content. The other two taught using the traditional method and didn’t think there was a difference between hands on and constructivism. They thought it was up to the student to get what they need out of the lesson by listening to the lecture, taking notes, and doing the assignments. In order to make the data reliable, prompts were used when recording the Focus Group answers .
Analysis of Data
The qualitative data from Focus Question One is analyzed using coded for themes to look for categorical and repeating data that form patterns of behaviors. The data in
Focus Question One were written in the form of a narrative by the teacher using reflective journaling to document student behavior. A learning task was also used to determine the mastery of the content. An instructional plan rubric followed by an interview was used as well to determine if any methods needed to be added to or taken away from the actual lesson. The interview was done with math experts to see what they thought needed to be done in the instructional process to ensure student engagement and mastery of the content.
The journaling was divided into three categories that included; attitude, engagement, and mastery. Journals of the student observations were analyzed by looking for patterns of positive or negative attitudes, student engagement, and how well they Constructivism or Rote Learning 26 master the concept. It was also analyzed to see if there was a difference between the two groups’ attitudes, engagement, and mastery. The instructional plan interview was analyzed by looking for patterns or repetition amongst the experts that were interviewed.
It was analyzed to see if there was a difference in the interviewee’s comments about the instructional plan rubric.
Focus Question Two uses quantitative data. The assessment data used for Focus
Question Two is analyzed using the Independent t- test and a Dependent t- test to determine that there is no significant differences between the control group and the treatment group. The decision to reject the null hypothesis has been set at p < .05. The results came from teacher made pre and post tests. The information that the test scores provided was to distinguish if there is a significant difference between the control group and the treatment groups’ mastery of the Georgia Performance Standards. The test scores also measured if there is a significant difference in academic achievement amongst the groups. The teacher gave a pretest and a post test to both groups. These scores were coded by student and analyzed by looking for significant differences in scores between the two groups. There is no significant difference between discovery learning and constructivism. The decision to reject the null hypothesis has been set at p < .05.
Academic achievement is the main factor that helped determine if there is a significant difference in the way students are learning. Analyzing the Independent t- test score data for a difference between the treatment group and the control group helped determine which method (if any) was more appropriate for student achievement.
Survey data from Focus Question Three was analyzed by using the Chi Square method (Salkind, 2007). The Chi Square method tested to see which of the survey Constructivism or Rote Learning 27 questions were significant and which questions were not significant. Chi Square was used as a desire to find what questions (items) are significant (and which ones are not). The significance level is reported at the p < .05, p < .01, and the p < .001 levels. This data was useful in determining students feel about math and how teachers are instructing their students.
Validation
This study was dependable in that ‘consensual validation’ was gained because it was approved by LaGrange College (Eisner, 1991). Research experts reviewed the study and agreed that it was well researched and presented. Epistemological validation of the results was achieved by comparing the results of this study to those found in the literature
(Denzin & Lincoln, 1998).
Credibility
Credibility was established in this study because the three focus questions called for gathering and analyzing data in several ways. Eisner (1991) calls this process
‘structural corroboration, or triangulation, where a confluence of evidence comes together to form a compelling whole. Within Eisner’s definition are embedded the concepts of fairness and precision. Fairness and precision were achieved throughout the literature review because many different views of the way students should learn were presented.
Also, the data collected was from a diverse population. The subjects came from different backgrounds and their situations were different as well. The teachers had different views on the way students should learn and this will be evident in the results. Eisner refers to precision as the ‘rightness of fit’ in that the study was extensively researched on a valid Constructivism or Rote Learning 28 topic. This topic being constructivism or rote learning is an issue in many schools because everyone has their own view of how a student should and can learn. The test score data, surveys, journals, and observations provided evidence to strongly support the results of the study.
Transferability
Since the method of student learning being; the constructivist method and rote learning is a widespread issue in most schools, this study provided useful information to researchers and educators. Eisner (1991) calls this process ‘referential adequacy’ where perception and understanding by others will increase because of the research.
Transformational
Catalytic validity was strong in this study seeing that the results better the readers understanding of the influence that constructivism and rote learning have on students’ academic achievement (Lather as cited by Kinchloe & McLaren, 1998). Educators can take this knowledge and make appropriate choices for students in the future.
Chapter 4: RESULTS
In a mathematics classroom many students struggle; however, with the new rigorous curriculum that covers Algebra 1, Geometry, and Statistics students seem to struggle at a greater capacity. In general, students entering ninth grade encounter challenges in mathematics. There were a total of 22 students who participated in this study. A constructivist approach in the classroom was used on the experimental group to Constructivism or Rote Learning 29 determine if it would significantly impact student achievement compared to the control group.
Focus Question 1 (Why do cognitive structures matter when teaching mathematical concepts?) was measured by using an instructional plan, instructional plan rubric, learning task, student observations, and reflective journaling. The instructional plan and rubric interview were used to determine if the study was conducive for the students, and planned for accurate results. After the instructional plan interviews some changes were made based on suggestive commentary from veteran teachers. Teacher one stated that “The essential questions addressed the enduring understanding of the students because it encourages them to use higher order thinking and helps them identify ways to relate probability to real life.” Teacher one suggested to “Instruct students on how to use the probability function using a graphing calculator.” Teacher two suggested that
“Interactive websites should be used” throughout the study. When asked how would other assessments yield better results: Teacher one responded “Standardized questions should be used throughout the lesson. Teacher two suggested “EOCT verbiage should be used at all times.” Student observations were used to determine the understanding the students showed and the learning task was used for the students to demonstrate the mastery of the concepts. Reflective journaling is another method used to collect data about the necessity of cognitive structures in a mathematical classroom. Information was gathered about how students derived their answers, visible attitude changes and understanding when and how to use the mathematical concept. The reflective journal recorded strategies that were used in the classroom using the constructivist approach and student attitudes towards mathematics. Constructivism or Rote Learning 30
Throughout the reflective journaling data was recorded about the instructional process and student attitudes. Looking back at the journals the control group was given notes, sample problems, and practice problems for them to do. In the journal the best part of the lesson was that the students were actively engaged and participating during instruction. When they were given their assignment they felt good about doing and completing the work. One student stated “without my notes, I wouldn’t know what to do.” Shortly after that comment I heard a few students say “this is easy!!” The students in the control group were closer to reaching their goal because they had notes to refer back to, samples of how to work the problems out, and they practiced in class. The experimental group; however, was a little different. They were not given any notes or sample problems. When the experimental group started class they were placed in randomly selected groups. They were all given a learning task along with a manipulative.
The task was not explained prior to their reception; however, a time for completion was given. Student one raised her hand and asked “Can you explain this to us, because we don’t understand?” Another student said “We don’t understand either!” Shortly after that other members of the class were saying “I don’t get it, you didn’t teach us this!” At that point students started to sit in their chairs and talk about everything except the assignment. The students felt confused and they just did not even try to figure out what was being asked. The students were also upset because the lesson was not taught and they had nothing to refer to as a reference like they were used to. The class was then pulled together as a whole group. The task was read and explained in minor details without examples. Then a lot of “Oh’s” were heard throughout the entire class. The students had a minor setback because they did not understand the lesson they were given. Constructivism or Rote Learning 31
Therefore, the lesson started off not going well, but ended with the students enjoying using the manipulative and understanding what the lesson asked them to do.
Focus Question 2 (What effect does rote learning have compared to constructivism on test scores?) Pre and post tests were used to show student progress.
There were 15 questions on the pre and post test. The Math 1 concepts used on the pre and post test were probability, summary statistics, permutations, combinations, and expected value.
An examination of the pre test scores of the control group showed that the mean scores were 32 percent. The pre test of the experimental group showed the mean score as
39 percent. Table 4.1 indicates that the pretest shows no significant difference at t (19)=
1.7, p > .05 for all pre test performed by the constructivist approach and the traditional approach. Therefore, the null hypothesis must be accepted.
Table 4.1 Pre Test Results for Experimental and Control Groups t-Test: Two-Sample Assuming Unequal Variances
Pre Test Experimental Pre Test Control Mean 38.72727273 32.72727273 Variance 51.61818182 85.01818182 Observations 11 11 Hypothesized Mean Difference 0 Constructivism or Rote Learning 32 df 19 t Stat 1.702411112 P(T<=t) one-tail 0.052491656 t Critical one-tail 1.729132792 P(T<=t) two-tail 0.104983312 t Critical two-tail 2.09302405
T(19) = 1.70, p > .05
The pre test of the control group showed that the mean scores were 28 percent and the mean scores of the post test were 73 percent. The pre test of the experimental group showed that the mean scores were 39 percent and the mean scores of the post test were 72 percent. One hundred percent of the students in the control group and the experimental group increased their scores. Neither group suffered losses and no changes in their scores.
This indicates that the control group performed the same as the experimental group. The results showed that all of the students showed an increase of knowledge in probability, summary statistics, permutations, combinations, and expected value. The post test scores ranged from 46% - 91% in the constructivist group and 40% - 95% in the traditional drill and skill group. In table 4.2 the t-tests shows the differences in the pre test and post test at p < .05 for all tests given to the control group and the experimental group. The effect size for the pre-posttest of the experimental group is r = -0.8486, the control groups effect size is r = -0.85632, and for the study is r = -0.0259. This data concludes that there is no significant difference in scores of the students in the control group versus the students in the experimental group. Therefore, the null hypothesis must be rejected.
Table 4.2 Pre Test and Post Test Results for Experimental Group t-Test: Paired Two Sample for Means
Pre Test Experimental Post Test Experimental Constructivism or Rote Learning 33
Mean 38.72727273 72.45454545 Variance 51.61818182 169.8727273 Observations 11 11 Pearson Correlation -0.001747498 Hypothesized Mean Difference 0 df 10 t Stat -7.510675462 P(T<=t) one-tail 1.01884E-05 t Critical one-tail 1.812461102 P(T<=t) two-tail 2.03767E-05 t Critical two-tail 2.228138842
T(10)= 7.51, p < .05
Table 4.3 Pre Test and Post Test Results for Control Group t-Test: Paired Two Sample for Means
Pre Test Control Post Test Control Mean 28.45454545 73.18181818 Variance 137.4727273 226.5636364 Observations 11 11 Pearson Correlation -0.204500533 Hypothesized Mean Difference 0 df 10 t Stat -7.102592175 P(T<=t) one-tail 1.6426E-05 t Critical one-tail 1.812461102 P(T<=t) two-tail 3.2852E-05 t Critical two-tail 2.228138842
T(10)= 7.10, p < .05 In Table 4.4 the t-test results showed no differences in the post tests at p > .05 for all tests performed by the groups. The data shows that the students in both the control group and the experimental group showed significant increases in their student learning.
Table 4.4 Post Test Results for Experimental and Control Groups t-Test: Two-Sample Assuming Unequal Variances Constructivism or Rote Learning 34
Post Test Experimental Post Test Control Mean 72.45454545 73.18181818 Variance 169.8727273 226.5636364 Observations 11 11 Hypothesized Mean Difference 0 df 20 t Stat -0.121145393 P(T<=t) one-tail 0.452392375 t Critical one-tail 1.724718218 P(T<=t) two-tail 0.90478475 t Critical two-tail 2.085963441 T(20) = -0.12, p > .05
Table 4.2 and 4.3 shows the results of the pre test and post test scores of students in the traditional group and the constructivist group. There is no significant difference shown between the two tests for the traditional method and the constructivist method.
For the control group the significance level is T(10)= 7.10, p < .05, and the experimental group the significance level is T(10)= 7.51, p < .05. The Cohen’s d effect size for the post to post test was d = -0.05186.
Focus Question 3 (What are students and teachers attitudes about discovery learning and rote learning?) was examined by the use of an attitudinal survey completed by all students before and after the treatment. The survey consists of 20 questions about students’ attitudes and the way they learn mathematics. Crobach’s Alpha reliably determined that the survey measured the total score accurately. Cronbach’s alpha was computed for each survey question and the results are in table 4.5. Since the Cronbach alpha is low this means that there was a wide variation in the responses. The responses showed little to no consistency. Cronbach’s alpha was used because internal consistency reliability is used to see if the items on a test are consistent with one another and that they represent one area of interest (Salkind, 2009). Constructivism or Rote Learning 35
Table 4.5 Cronbach’s Alpha Student Pre and Post Survey
Pre-Survey α = .07 Post Survey α = .06 A survey was used to evaluate the teachers attitudes about discovery learning
compared to rote learning. The survey consists of 10 questions about hands on teaching
versus the traditional style of teaching. Crobach’s Alpha reliably determined that the
survey measured the total score accurately. Cronbach’s alpha was computed for each
survey question and the results are in table 4.6. The Survey for the group of teachers is
65% reliable. Cronbach’s alpha was used because internal consistency reliability is used
to see if the items on a test are consistent with one another and that they represent one
area of interest (Salkind, 2009).
Table 4.6 Cronbach’s Alpha Teacher Survey
Teacher Survey α = .65
The survey was completed by each student involved in the study. The survey was
used in order to determine whether the constructivism or the traditional approach affects
student’s attitudes in mathematics. In order to determine if constructivism or the
traditional method affected student attitudes a chi square test was used. Chi Square was
computed for each survey question and then tested for a significance level. Table 4.7
arranges the chi square value for each survey question. Table 4.7 shows the chi square
significance value at or below the .001, .01, and the .05 level for nine survey questions.
As seen in table 4.7 there were eleven questions whose chi square value showed
significance at p < .05. The data shows six post survey questions ( 2, 4, 10, 14, 17, and Constructivism or Rote Learning 36
18) had a significant increase in the way students felt about math and a significant decrease in three survey questions. Question numbers 13, 19, and20 showed a significant decrease in students attitudes. The data shown in post survey question 13 showed a significant change. The data shows that the students feel that they can be creative and discovery things in math. The data in question 19 showed a significant change because in the pre survey students seemed to worry that they would get poor grades in math.
However, the data in the post survey show that the students no longer worry about getting poor grades in mathematics. Lastly question 20 had a significant decrease because the students don’t just do math because they have to. When students have the ability to apply concepts to solve problems, reason logically, and see math as useful and doable, their attitudes will transform (Kilpatrick & Swafford, 2002).
Table 4.7 Chi Square Student Pre Survey
Survey Questions n = 21 Pre χ Post χ
1. I have always hated math 10.19** 12.09**
2. I stop working as hard after I do badly 10.19** 20.19***
3. I rarely ask questions during math class 5.42 7.19
4. Knowing math will help me in my future 16.14** 25.42***
5.Males are naturally better at math than 7.33 5.42 females
6. Math is a fun subject 7.33 5.42
7. Math has been my best subject 3.04 3.04
8. Math has been my worst subject 1.61 1.76
9. Females can do just as well in math as 19.71*** 19.71*** Constructivism or Rote Learning 37
males can
10. I like to do hands on activities in math 23.04 23.04***
11. Math is just memorizing formulas 4.47 5.90
12. Math is not important in everyday life 12.09** 12.09**
13. In math, you can be creative and discover 45.79*** 18.76** things by yourself
14. I try to learn math because it helps me 6.85 15.90** develop my mind and helps me think more clearly
15. Using the internet (or a computer) is a 7.80 7.80 good way for me to learn math
16. Working with other students helps me 7.80 7.80 have a better attitude towards math
17. I keep trying even if the work is hard for 11.14* 28.76*** me to do
18. I work as hard as I can in math 2.57 14.47**
19. I worry that I will get poor grades in 17.80*** 4.47 math
20. I only do math because I have to 18.28*** 13.52**
*p < .05, ** p < .01, ***p < .001
The last method used for focus question 3 was a focus group. I conducted a focus group with 6 math teachers to determine if there was any categorical or repeating data.
Appendix A will disclose the focus group questions. During the focus group, 8 questions were asked. The teachers were divided in their styles of teaching. Teachers 1, 4, and 5 were constructivist, teachers 2 and 6 are traditionalist, and teacher 3 does both. The next question is for the teachers to describe how they evaluate their students. Teachers 2 and
6 agreed “I assess my students by giving tests, quizzes, classroom practice, bell ringers, and tickets out the door.” However; teacher 3 added “I don’t always give my students the traditional style test. I also give them culminating task so they are able to use Constructivism or Rote Learning 38 manipulatives.” Teachers 1, 4, and 5 added “We do computer assessments, and on class work assignments, test, and quizzes are from learning task that involve manipulatives.”
The following question is how does the traditional style and constructivist style of teaching affect academic achievement. Teacher 1 stated “Hands on gives the students a deeper meaning of what is being asked and the students are allowed to figure the solution out on their own.” Teacher 4 stated “I just like the fact that when students are discovering, the information is sticking.” Teacher 3 stated “Students need to be able to discover, but not all of the time. Students need the repetition in order to grasp what to do and when to do in math.” Teacher 1 “When students get home they don’t have any manipulatives to discover, so I thing that is a waste of time. And on the EOCT they are unable to use the manipulatives as well, so waste of time!” Teacher 2 added “ Traditional learning is how we learned in school so I think that is best because the students have notes to refer back to if they don’t understand what they are doing. They also need to practice in class and at home to understand the concepts fully anyway.” The last two questions were referred to teachers 1, 3, 4, and 5. What (if any) barriers have you encountered in trying to use the discovery method and how often is the discovery method used in your classroom? Teacher 3 stated “The only barriers are that the students don’t know how to use the manipulatives and they don’t always know how to interpret what the questions are asking, so they sometimes need extra guidance. In my classroom the discovery method is used maybe 1-2 days a week. “Maybe.” Teachers 4 and 5 stated
“At least 2-3 times a week.” No one had anything further to share. Constructivism or Rote Learning 39
Chapter 5: ANALYSIS AND DISCUSSION OF RESULTS
Analysis
The purpose of this study was to determine if discovery learning or rote learning promoted greater academic achievement in mathematics. To explore Focus Question 1:
Why do cognitive structures matter when teaching mathematical concepts? Focus
Question 1 was measured by using an instructional plan, instructional plan rubric, learning task, student observations, and reflective journaling. The qualitative data from
Focus Question One was coded for themes to look for categorical and repeating data that form patterns of behaviors. The data in Focus Question One were written in the form of a Constructivism or Rote Learning 40 narrative by the teacher using reflective journaling to document student behavior. A learning task was also used to determine the mastery of the content. An instructional plan rubric followed by an interview was used as well to determine if any methods needed to be added to or taken away from the actual lesson. The interview was done with math experts to see what they thought needed to be done in the instructional process to ensure student engagement and mastery of the content.
Journals of the student observations were analyzed by looking for patterns of positive or negative attitudes, student engagement, and how well they master the concept.
It was also analyzed to see if there was a difference between the two groups’ attitudes, engagement, and mastery. The instructional plan interview was analyzed by looking for patterns or repetition amongst the experts that were interviewed. The data showed that there were repetitive comments made about the instructional plan rubric as well as the attitudes that were observed in the classroom.
In chapter 2 the literature states that students become more proficient when they understand the underlying concepts of math, and they understand the concepts more easily if they are skilled at computational procedures. U.S. students need more skill and more understanding along with the ability to apply concepts to solve problems, to reason logically, and to see math as sensible, useful, and doable (Kilpatrick & Swafford, 2002).
The research in this study proved this statement. The subjects understood the math concepts and were able to apply the concepts to solve the problems. This is demonstrated in table 4.1 showing the differences in the scoring of the constructivist learner and the rote learner. Constructivism or Rote Learning 41
The results of this research did not support cognitive theory’s focuses on how people develop structures of knowledge. In cognitive theory, knowing means more that the accumulation of factual information and routine procedures for reasoning and solving problems (Pellegrino, Chudowsky, & Glasser, 2001). Focus Question 2 (What effect does rote learning have compared to constructivism on test scores?) Students were given a pre test prior to any instruction or discovery learning task and a post test after the end of the unit. Drill and skill assignments were used for the control group and discovery learning task were used for the experimental group. The data from the pre and post test were analyzed to determine if the two groups would show significant differences in the pre and post test scores. The results from the t-test show that there is no significant difference in the pre test and post test scores. Therefore, table 4.2 shows that neither the constructivist approach nor the rote approach showed any major academic achievement in the mathematics classroom.
As stated in the review of the literature, research by Gardner (2007) suggested that rote learning causes students to have difficulty working independently without an example in front of them. The study proved this research to be correct. However, it wasn’t proved just for the rote learners, but the constructivist learners needed guidance as well. As indicated in chapter 4, the reflective journal the constructivist learners could not start or complete the task without an explanation. When they were given the task they read it, but were unable to understand what was being asked. The students were asking several questions until the task was read aloud and explained in minor detail. After the explanation they completely understood and worked cooperatively from there. The research in the study did not fully support what research also suggested. The research Constructivism or Rote Learning 42 suggested that in order for students to learn, they need to use their cognitive structure to process the information instead of memorizing the process (Gardner, 2007). The kind of student Gardner is addressing is the constructivist learner. However, the traditional drill and skill learners scores were nominally higher than the students who used the discovery method, but not statistically higher. Therefore, memorization is a technique that students used in order to learn and process the information needed to promote their own academic achievement in mathematics.
Research also suggested that discovery learning is designed to promote academic triumph for students with different learning styles. However, rote learning is designed to promote academic success through repetition or rehearsal. This statement is proved true for both groups. As seen in table 4.1 the constructivist group experienced academic success when learning by working in groups and using manipulatives to discover the answer.
Tilton (2003) stated that so many students need to be physically engaged in a task in order to learn. The research in this study showed that students do not necessarily have to be physically engaged at all times in order to learn. The students in the control group did not partake in any physical engagement and 100% of the students achieved academic success in mathematics.
Focus Question 3 (What are students and teachers attitudes about discovery learning and rote learning?) was examined by the use of an attitudinal survey completed by all students before and after the treatment, a teaching method survey completed by 6 math teachers, and a focus group that included 6 math teachers. Constructivism or Rote Learning 43
The student survey was completed before and after the treatment. Students had higher positive responses after completing the unit. For the control group they had no treatment however, they felt more confident; therefore, this created a more positive experience. The experimental group which received the treatment of hands on learning had positive attitudes on learning math but neither group showed any more of an attitude gain than the other.
Research states that in order for students to be successful in math, negative attitudes need to change to can do attitudes (Tilton, 2003). Success depends entirely on the teacher to help learners develop positive attitudes towards the learning of mathematics (Akinsola & Olowojaiye, 2008). This part of the research was proved to be true. As the experimental students were observed in the beginning they would look at the tasks and not even attempt to solve the problems. However, after some encouragement from the teacher trying to get them to have a deeper understanding of what was being asked that is when the light bulbs went off. After the encouragement they were on their own.
Tilton states that in order for students to be successful in math, negative attitudes need to change to can do attitudes (Tilton, 2003). If learners are not assisted or encouraged to perceive positively most of the things they learned in mathematics classes, their performance will be affected. In the study when students were not given any instructions to assist them in solving math problems the students shut down. However, when students were encouraged and given assistance their behavior was positively affected. Moreover, the students that were not assisted or encouraged their performance suffered but after assistance was given they were successful. If students do not Constructivism or Rote Learning 44 understand what is being asked then they will not perform well. When students fully understand the material or concept they are working on they are encouraged to do their best because their confidence level is high. In a mathematics classroom the traditional student’s attitude can affect their performance due to the amount of assistance or encouragement given by the instructor.
Discussion
The results of this study indicate that in mathematics there is a great possibility that there is a link between the constructivist learner, rote learner and academic achievement. However, the findings in this study suggest that there is no significant statistical difference between constructivism and rote learning in this test of achievement.
During this study the reflective journaling indicated that for the experimental group certain questions on the tasks could have been worded differently because the students did not always understand what the questions were asking. For the control group they understood the material after a lesson and some guided practice problems. All of the lessons went well due to the fact all of the students had a general understanding of probability prior to starting the learning sessions. However, the control group was given notes and plenty of practice and the experimental group had hands on activities and maniupulatives that were used to determine the mastery. The students in the experimental group enjoyed the hands on activities. The students in the control group were uninterested in taking notes and completing all of the practice problems they had to do. I learned that the students enjoy hands on activities, but they need guidance as well.
If the students have too much of one strategy they will eventually get exhausted of doing the same type of activities. This study was done in a short period of time (2 weeks) Constructivism or Rote Learning 45 which was not enough time to get a true picture of retention. This was only enough time to unfold which style had a greater affect on student attitudes and academic achievement.
This was only a step towards validating the different learning methods, and to see which one showed a significant difference.
Credibility was guaranteed throughout the study by the use of multiple data sources. The outcomes were measured by the internal level of measurement. The pre test and post test showed test- retest reliability for the control group and the experimental group. Content validity was increased by having other math teachers look over and then discuss the instructional plan rubric to see if the items accurately reflect the content and the ideas that the student should understand. Bias was decreased by checking the tests for offensiveness and fairness. This research does not suggest support for either constructivism or rote learning in mathematics. On the other hand, this study suggests through observations and reflective journaling that the students enjoyed the constructivist methods more than the traditional group. During the process of reflective journaling many students in the experimental group enjoyed playing with the manipulatives, working in groups, and being able to actually see and touch what they were learning.
Hence, the control group did not like taking notes. I often heard “we have to take notes again?” However, one student stated “without my notes, I wouldn’t know what to do.”
The traditional students did not enjoy note taking, but appreciated having the notes to refer back to. Therefore, this research should be regarded as a step toward validating that there is no significant statistical difference between the constructivist approach and the traditional approach; however, there is a difference in student attitudes.
Implications Constructivism or Rote Learning 46
The implications of this study support the literature. Swafford (2002) suggests that students become more proficient when they understand the underlying concepts of math, and they understand the concepts more easily if they are skilled at computational procedures. The constructivist approach impacts student achievement differently because the constructivist group enjoyed their activities while the traditional group was uninterested with taking notes and doing practice problems. Therefore, this study discovered that there is no significant difference between the discovery learners and the rote learners with regard to achievement in one unit of mathematics. The combination of the approaches had a strong impact on the students’ level of mathematical understanding.
In this study all of the students made academic gains and none of them stayed the same or fell below. The learning method did not matter when it came to academic achievement.
However, the learning method mattered when the students were using manipulatives and working in groups. The students in the experimental group enjoyed their learning method while the control group despised the traditional drill and skill method. When students deepen their understanding in mathematics they gain confidence and positive attitude change towards mathematics.
Impact on Student Learning
This study impacted student achievement by using the constructivist approach and the traditional approach. The constructivist approach impacted the students learning because they were introduced to new ways of learning mathematics. The constructivist learners were allowed to discover their own ways to solve different types of problems dealing with statistics and always actively engaged. These learners had different types of manipulatives to use throughout the classroom; as well as, technology. The students Constructivism or Rote Learning 47 were more apt to do their class work when they found that they had the actual material in hand that matched the learning tasks. The students on the traditional track were bored; however, the notes, drill and practice, and more practice gave them a way to be successful. The traditional students had an opportunity to look at notes and examples when they did not understand. Therefore, when they understood what they were doing all of the students were actively engaged during every lesson.
Even though the findings in the research showed no significant statistical difference, student learning was improved due to the fact that there is a 100% gain in the control group and the experimental group. The results of the study show that there was a difference in student attitudes; therefore, math teachers should use and create enjoyable lessons for their classes.
Recommendations for Future Research
According to the results of the study, there are several recommendations for future research. The first recommendation for this research is to increase the length of time. The length of time should be at least until the final assessment of the course, or the teachers could move throughout the grade levels with the same students to track their long term progress. More time would allow the validity to be stronger and may give a better outlook on which method is best. The second recommendation is to increase the sample size. The third recommendation is to include the socioeconomic status, gender differences, and race in the study to see how that affects student attitudes and achievement. Students should receive both the constructivist approach and the traditional approach throughout their math classes. This will give them a greater sense of Constructivism or Rote Learning 48 accomplishment when they can see for themselves that no matter which method is taught, they can think and achieve on their own. Lastly, in order to promote lifelong learners, teachers should use enjoyable learning methods to teach mathematics and keep the students engaged.
References
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Appendix A
Instructional Plan
Lesson Title: Probability Subject Area: Math 1 Grade Level: 9th Duration: 1-2 Weeks Objectives: Students will: 1. learn what probability is, 2. learn different ways to express probability numerically: as a ratio, a decimal, and a percentage, and 3. learn how to solve problems based on probability.
Materials: The class will need the following: • Copies of Classroom Activity Sheet: Probability Problem Solving • Computers with Internet access (optional but very helpful) • Reference materials such as almanacs and encyclopedias • Copies of Take-Home Activity Sheet: Spin the Wheel! Constructivism or Rote Learning 52
Procedures: 1. Begin the lesson by asking students to define probability (the likelihood or chance that a given event will occur). Probability is usually expressed as a ratio of the number of likely outcomes compared with the total number of outcomes possible. Ask students if they can give an example of probability. 2. To help students understand probability, work on the following problem as a class: Imagine that you have boarded an airplane. The rows are numbered from 1 to 30, and there are six seats per row, three on each side of the isle. Seats in each row are labeled A through F. Using that information, work together as a class to solve the problems listed below. a. How many seats are in the airplane?180 seats b. What are your chances of sitting in row number 7?6/180, or 1/30 c. What are your chances of sitting in a window seat? There are two window seats per aisle, for a total of 60 window seats. Your chances of seating at a window would be 60/180, or 1/3. d. What are your chances of sitting in an “A” seat? There are 30 A seats, so your chances are 30/180, or 1/6. e. What are your chances of sitting in an even-numbered row? Of the 30 rows, 15 are even-numbered, so your chances are 15/30, or 1/2. 3. To figure out each problem, students must set up a ratio between the total number of outcomes—in these problems either the total number of seats or rows—and the specific question asked. Tell students that they will write their answer as a fraction, decimal, and percentage. Example: The chance of sitting in seat 7A is 1/180, .00555, or .555 percent. The ratio presented as a percentage helps make it clear if the probability of an event is great or small. 4. Distribute the Classroom Activity Sheet and tell students that they are going to work on several probability problems in class, expressing the answer as a fraction, decimal, and percentage. Students may work individually or with partners. The problems and the answers are listed below: a. Your sock drawer is a mess. Twelve black socks and six red socks are mixed together. What are the chances that, without looking, you pick out a red sock? What are the chances of picking a black sock? The total is 18 socks, and one-third of them are red (6/18, or 1/3, or .333, or 33.3 percent). The probability of picking a red sock is 1/3, or 33.3 percent. Because two-thirds of the socks are black (12/18, or 2/3, or—rounding up—66.7 percent), the probability of picking a black one is higher—2/3, or 66.7 percent, compared with 1/3, or 33.3 percent. b. You are rolling a regular die. What is the probability of rolling a 3?Of the total of six outcomes, 3 is one outcome. The probability is the ratio 1/6, .1666, or 16.66 percent. c. If you are rolling a regular die, what is the probability of rolling an even number? Of the six possible outcomes, half, or three outcomes, could be an even number. The probability is 3/6, 1/2, .5, or 50 percent. d. You are randomly choosing a card from a deck of 52 cards. What is the probability that the card you pick will be a king? Of the 52 possible outcomes, four outcomes are kings. The probability is 4/52, 1/13, .076, or 7.6 percent. e. You are visiting a kennel that has three German shepherds, four Labrador retrievers, two Chihuahuas, three poodles, and five West Highland terriers. When you arrive, the dogs are taking a walk. What is the probability of seeing a German Constructivism or Rote Learning 53
shepherd first? Out of a total of 17 dogs, 3 are German shepherds. The probability of seeing a German shepherd is 3/17, .176, or 17.6 percent. f. Two out of three students in Mr. Allen’s class prefer buying lunch to bringing it. Twenty students prefer buying lunch. How many students are in Mr. Allen’s class? Students can set up the following problem: 20/30, or 2/3, of the total number of students (X) buy lunch (20). To express that mathematically, 2/3 (X) = 20. Solve for X, which equals 30, so there are 30 students in Mr. Allen’s class. 5. After students have completed the Classroom Activity Sheet, go over their responses. Then assign the Take-Home Activity Sheet: Spin the Wheel! If time permits, review their answers during the next class period. Constructivism or Rote Learning 54
Appendix B
Instructional Plan Rubric
Criteria Describe & State Open-ended Questions
Essential How can you determine when to use the addition counting principle Does the essential question Question (s) and the multiplication counting principle? address the enduring What does probability mean? understanding? How does probability valid in real life? What are some different methods probabilities can be expressed? Instructional Computer, notebooks, hands-on projects In what ways can the use of Technology instructional technology be improved?
Materials Pencils, triangles, rectangles, spinners, dice, calculators, graphic organizers. Are there other materials that would be appropriate? Accommodation Student will be arranged in heterogeneous, small groups based on To what degree are the needs of s for special ability level. special learners being met? needs Directions will be read aloud. Extend time will be given. Standards MM1D1. Students will determine the number of outcomes related to Do the assessments align with covered a given event the standards stated? Apply the addition and multiplication principles of counting Calculate and use simple permutations and combinations MM1D2. Students will use the basic laws of probability. Find the probabilities of mutually exclusive events Find the probabilities of dependent events Calculate conditional probabilities Use expected value to predict outcomes Assessment Each group member should race their horses on their graphic Are there other assessments (formative) organizer in their notebook. that would yield better data? Each group member should be able to explain if the horse race was fair.. Each group member should be able to discuss the fairness of the dice and the spinner. Constructed spinner Each group member should be able to explain the connection between spinners and real life situations. Constructivism or Rote Learning 55
Appendix C
Focus Group Questions
Good Afternoon!
Thank you for taking the time out to meet with me. I will honor your time by making sure that we wrap up in the next 30-45 minutes.
Does anyone mind if I tape record this for my records? I won’t share the tape with administrators.
I am conducting an evaluation of how discovery learning vs. rote learning impacts the achievement of students.
My evaluation is formative and qualitative. This means that my primary point is to gather information that will help students and teachers improve, but right now I just want your opinions.
All information collected is confidential meaning I will not disclose the participants nor make quotes from this group. Please speak freely.
Are there any questions before we start?
1. Would you consider yourself a traditional teacher or the new age teacher (hands on)? Explain.
2. For the traditional teacher, describe the different ways you evaluate your students.
3. For the new age teacher, describe the different ways you evaluate your students.
4. Explain how you think discovery learning impacts student achievement.
5. Explain how you think traditional learning impacts student achievement.
6. What (if any) barriers have you encountered in trying to use the discovery method?
7. How often is the discovery method used in your classroom?
8. Is there anything else you would like to share?
Thank you for your time!! Constructivism or Rote Learning 56 Constructivism or Rote Learning 57
Appendix D
Journal Prompts
The activity went well because…
The activity did not go well because…
The biggest mistake made in the lesson was…
The best part of the lesson was…
The students moved closer to reaching the goals by…
The students had a set back because…
I could have done______differently.
The students enjoyed …
The students felt…
I learned that… Constructivism or Rote Learning 58
Appendix E
Student Survey
Mathematics I Attitude Survey
I am interested in your ideas about mathematics and this class. Your answers to the questions that follow will help me understand what you think mathematics is all about.
Course: Math I
Grade Level: 9th
Ethnicity Check only one.:
American Indian ____
Asian ____ African American/Black ____ Hispanic ______White ______Other ______(please specify) Gender:
Female Male
This questionnaire is not something to be graded and your answers are completely anonymous. Please tell us what you really think by putting an X in the box corresponding to Strongly Agree, Agree, Undecided, Disagree, or Strongly Disagree.
Thank you for your help! Constructivism or Rote Learning 59
Strongly Strongly Question Agree Undecided Disagree Agree Disagree l. I have always hated math. 2. I stop working as hard after I do badly on a math test. 3. I rarely ask questions during math class. 4 Knowing . mathematics will help me in my future.
5. Males are naturally better at math than females.
6. Math is a fun subject. 7. Math has been my best subject. 8. Math has been my worst subject.
9. Females can do just as well in math as
males can.
10. I like to do hands-on
activities in math. Constructivism or Rote Learning 60 Strongly Strongly Question Agree Undecided Disagree Agree Constructivism or Rote LearningDisagree 61 11. Math is just memorizing
formulas.
12. Math is not important in everyday life.
13. In math, you can be creative and discover things by yourself.
14. I try to learn math because it helps develop my mind and helps me think more clearly.
15. Using the internet (or a computer) is a good way for me to learn math.
16. Working with other students helps me have a better attitude towards math.
17. I keep trying in math
even if the work is
hard for me to do.
18. I work as hard as I
can in math.
19. I worry that I will get
poor grades in math.
20. I only do math
because I have to. Constructivism or Rote Learning 62
Appendix F
Teacher Survey Traditional vs. Discovery Question Strongly Agree Agree Undecided Disagree Strongly Disagree
I mostly lecture in my class.
I allow the student’s to discover math concepts.
I use the drill and kill method.
I always give notes.
I mostly use manipulatives in my class.
I relate what I am teaching to real life.
Students usually work individually.
Students usually work in groups/pairs.
I do most of the talking during class.
The students do most of the talking during class.
I serve as the facilitator.