Developing Meaningful Learning with Grade Four Students:
Using PEEL Procedures to Improve My Teaching Practice

by
David R. Turner

 

A thesis submitted to the Faculty of Education
in conformity with the requirements for
the degree of Master of Education

 

Queen's University
Kingston, Ontario, Canada
November, 1999

Copyright © David R. Turner

 

Abstract

This study has its roots in my desire to investigate and incorporate new teaching practices that would help develop my students’ skill and ability to make conscious personal connections to their learning. Teaching procedures that were integrated into my practice originated with the Project to Enhance Effective Learning (PEEL), and address specific concerns about student learning behaviours. Venn diagrams were used to help my students build understanding of place value in mathematics. Concept maps were used to help my students build comprehension and understanding of material they read, and to develop deeper personal connections to story characters and events. Topic and task questions were used to help students develop reflective thinking skills by creating connections between lessons and topics in mathematics. This study demonstrates that making changes to my practice can help my students develop meaningful learning.

This study took place during the 1997-1998 school year with my Grade 4 class at Percy Centennial Public School in Warkworth, Ontario. Twenty-one of my students participated in this study. The data include students’ Venn diagrams, concept maps and “Thinking Linking Logs,” in which students recorded their responses to the topic and task questions. The student data together with the procedures I incorporated into my practice are analyzed within the context of recent literature on teacher change (Briscoe, 1994; Guskey, 1986; Richardson, 1990), reflective practice (Osterman, 1998) and constructivist learning theory (Grennon Brooks and Brooks, 1993; Edwards, 1994).

The study suggests that (a) teachers can incorporate new procedures into their practice if these changes are perceived to address teachers’ specific concerns about observed student learning behaviours and (b) students can develop meaningful learning if teachers integrate procedures into their practice that encourage students to develop good learning behaviours.

 

Acknowledgments

I would like to thank my thesis supervisor, Dr. Tom Russell, for his guidance and support through both the data collection and writing phases of this thesis. I appreciate all of his suggestions, encouragement and patience throughout the project. I also thank Dr. Howard Smith, a valued member of my thesis committee, for all of his insights, comments and suggestions that went into the final version of the thesis.

Thanks are extended to Dr. Ian Mitchell, of Monash University, for spending a cold February afternoon with me to discuss PEEL and my research. He contributed greatly to my understanding of PEEL and data collection techniques in the action research cycle.

I am grateful to my principal, Peter Chrisomalis, for supporting and encouraging me in my quest to improve my practice. Thanks also to the students and families of the participating students from my Grade 4 class at Percy Centennial Public School during 1997-1998. Their co‑operation in sharing their data with me and their interest in my work are greatly appreciated.

Without the support of my family, this thesis would not have been possible. Thank you, Linda, for your ever-present support, patience and encouragement.

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

 

CHAPTER 1: INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Purpose of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Outline of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Research Setting and Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

 

CHAPTER 2: THEORETICAL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Controlling Change in Teachers’ Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Reflective Practice and Teacher Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Origins and Foundations of PEEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Research Design: Why PEEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Research Design: Action Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13

Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

The PEEL Procedures Used in this Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Concept Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Topic and Task Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Analyzing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

 

CHAPTER 3: USING VENN DIAGRAMS TO BUILD UNDERSTANDING

OF PLACE VALUE IN MATHEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Place Value and Grade 4 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Defining and Assessing the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Re-Conceptualization of My Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Implementing the Procedures: Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Working with Tens and Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Working with Hundreds and Tens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Working with Hundreds and Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

 

CHAPTER 4: USING CONCEPT MAPS AS READING RESPONSE

TO BUILD UNDERSTANDING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Summaries of the Short Stories and Novels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Short Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

The Tiger Skin Rug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Dinner at Alberta’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Mountain Rose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

The Case of the Mysterious Tramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chocolate Fever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Read Aloud Story: The Chocolate Touch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Novels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Titanic Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A Friend Like Zilla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Devil’s Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Concept Maps and Reading Response . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Short Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Tiger Skin Rug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Dinner at Alberta’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Mountain Rose. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 37

Novel Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Titanic Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A Friend Like Zilla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Devil’s Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Summary: Concept Maps as Reading Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

 

CHAPTER 5: USING VENN DIAGRAMS AS READING RESPONSE

TO BUILD UNDERSTANDING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Comparing Characters Using Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

Comparing Point of View using Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Making Personal Connections to Characters Using Venn Diagrams . . . . . . . . . . . . . . . . . 46

Summary: Using Venn Diagrams for Reading Response . . . . . . . . . . . . . . . . . . . . . . . . . 47

Using Concept Maps and Venn Diagrams Together as Reading Response . . . . . . . . . . . . 49

Progressing Toward Creating Meaningful Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

 

 

CHAPTER 6: USING TOPIC AND TASK QUESTIONS IN MATHEMATICS

TO DEVELOP REFLECTIVE THINKING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Topic and Task Questions in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Using Topic and Task Questions to help Students Make Personal Connections to Lessons .54

Dividing Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Using Topic and Task Questions to help Students Make Connections Between Lessons . . 62

Using Topic and Task Questions to Help Students Make Connections Between Topics . . 63

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

 

CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . 68

PEEL Procedures as Agents of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

PEEL Procedures as Improvements to my Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

PEEL Procedures and the Development of Meaningful Learning . . . . . . . . . . . . . . . . . . . 73

Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

 

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

 

APPENDICES

A: Using Venn Diagrams to Build Understanding of Place Value In Mathematics . . . . . 82

Selected Samples of Caroline’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83

Selected Samples of Ian’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Selected Samples of Olivia’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Selected Samples of Tania’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B: Using Concept Maps as Reading Response to Build Understanding . . . . . . . . . . . . . 101

Selected Samples of Anne’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Selected Samples of David’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Selected Samples of Neil’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Selected Samples of Olivia’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Selected Samples of Peter’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Selected Samples of Caroline’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

C: Using Venn Diagrams as Reading Response to Build Understanding . . . . . . . . . . . . 123

Selected Samples of Evan’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Selected Samples of Kevin’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Selected Samples of Neil’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Selected Samples of Peter’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Selected Samples of Caroline’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136

D: Topic and Task Questions in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Selected Samples of Caroline’s “Thinking Linking Log”. . . . . . . . . . . . . . . . . 140

Selected Samples of Fay’s “Thinking Linking Log”. . . . . . . . . . . . . . . . . . . . . . 150

Selected Samples of Neil’s “Thinking Linking Log”. . . . . . . . . . . . . . . . . . . . . . 160

Selected Samples of Ian’s “Thinking Linking Log”. . . . . . . . . . . . . . . . . . . . . . 171

 

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

List of Figures

Figure 1 Venn Diagram Procedure Using Tens and Ones . . . . . . . . . . . . . . . . . . . . . . . 22

 

 

List of Tables

Table 1 Overview of Procedures Used, Time Lines and Data Collected . . . . . . . . . . . . 3

Table 2 Some Poor Learning Tendencies Identified in PEEL . . . . . . . . . . . . . . . . . . . 11

Table 3 Good Learning Behaviours Identified in PEEL . . . . . . . . . . . . . . . . . . . . . . . . 12

Table 4 Connecting Specific Teaching Concerns with Poor Learning

Tendencies, Good Learning Behaviours and PEEL Procedures . . . . . . . . . . . 15

 

Table 5 Assessing Concept Maps and Venn Diagrams

as Indicators of Understanding in Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

 

Table 6 Assessing Topic and Task Question Responses

as Indicators of Personal Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

 

Table 7 Concept Map Connections Matrix for The Tiger Skin Rug . . . . . . . . . . . . . . . 35

Table 8 Concept Map Connections Matrix for Dinner at Alberta’s . . . . . . . . . . . . . . . 36

Table 9 Concept Map Connections Matrix for Mountain Rose . . . . . . . . . . . . . . . . . . 38

Table 10 Topic and Task Questions Used with the Thinking Linking Log . . . . . . . . . . 54

 

CHAPTER 1

INTRODUCTION

This study has its roots in my desire to develop and implement new teaching practices and procedures that would help develop my students' skill and ability to make conscious connections between new learning and their existing knowledge. This study was undertaken to improve my teaching practice by including specific procedures and strategies that will help build a deeper understanding of school knowledge by my students. My goal was to increase my students' skill in reflective thinking about what they are learning and their understanding of how this new learning can connect with what they already know. Encouraging students to consciously identify the internal connections they make between lessons could lead to a greater understanding of those lessons and to more meaningful learning.

Grennon Brooks and Brooks (1993) claim that understanding is constructed when students combine new information or learning with the knowledge they already have. This is an internal process of finding connecting points between the existing and the new knowledge. The extent to which this process of assimilating new information occurs for individual learners can vary widely.

Teachers have observed in their classrooms students who excel at learning new skills sitting next to students who struggle to grasp a basic understanding of the same skill. The success or failure of a student to learn something new depends upon that student's ability to make the connections and to integrate the new learning with his or her existing knowledge. Students who seem to learn easily appear to have developed specific skills that help them learn. The decisions a student makes during a lesson have some influence over that student's success in learning during that lesson. When learners make poor or inadequate decisions, consciously or unconsciously, then incomplete or inadequate learning is the likely result.

Teachers may develop procedures for their practice that not only encourage students to make connections between new learning and their existing knowledge, but also help students recognize, develop and consciously choose good learning behaviours. Thus students may develop the understanding that learning is an individual and internal process, and by developing good learning behaviours, their ability to learn may be improved.

 

Purpose of the Study

This is a study of my own teaching practices. My aim is to seek out and implement specific teaching procedures that will help my students develop good learning behaviours during lessons and that will help my students make conscious connections between new learning and their existing knowledge.

 

Outline of the Study

The procedures that were selected as practice improvements were chosen from the Project for Enhancing Effective Learning (PEEL) (Baird & Northfield, 1992). This project, begun in Australia in 1985 at Monash University, undertook to explore ways to change teaching practice in a collaborative forum so that students would understand more clearly the learning process and be able to connect more deeply with their school work.

To help my students build their understanding of school knowledge, I introduced two PEEL procedures early in September, 1997: Venn diagrams and concept maps. In February, the students began recording the links they were making between mathematics lessons in a "Thinking Linking Log." Students' "Thinking Linking Logs," concept maps and Venn diagrams were collected as part of the regular activities of mathematics and reading classes. A summary of the procedures used as well as the time lines and data collected is presented in Table 1.

 

Table 1

Overview of the Procedures Used, Time Lines and Data Collection in this Study

 

Chapter Two continues the discussion of various models to improve teaching practice, and PEEL in particular. Chapter Three describes in detail the introduction of Venn diagrams into my practice of teaching place value of hundreds, tens and ones in mathematics. This part of the study occurred early in the school year (September), and was the first PEEL procedure attempted by me and the students. The Venn diagram procedure helped channel the students’ thinking about place value, allowing them to focus on specific components of the concept as they completed the tasks each day. The student work also acted as a formative-type evaluation, providing me with specific information about each student’s understanding of each place holder in the place value system.

Chapter Four describes the use of concept maps and Venn diagrams as reading response organizers. These procedures were introduced to the students with several pieces of short fiction in the first term (September). During the third term, several students participated in novel studies that also used these two procedures. Chapter Four provides a summary of each story used for this study to familiarize the reader with the content to which the students were responding. Data collected from students’ reading responses indicate the ability of each of these procedures to allow students to demonstrate their understanding of the connections they have made between the characters and the story events, as well as between the characters and themselves.

Chapter Five describes how topic and task questions were used with the “Thinking Linking Log” in mathematics to help students develop deeper connections to their work. During three units of study (division, fractions and decimals, and quadrilaterals), the students were directed through questions to consider the connections they could make to their learning. Students were making personal links to these lessons, developing connections between lessons, and making links between mathematics topics.

Conclusions are presented in Chapter Six. The usefulness of the PEEL procedures to address teacher change in general is discussed. Also summarized are the improvements these procedures have made both to my practice and to the development of meaningful learning with my students.

Research Setting and Participants

Percy Centennial Public School is located in a predominantly rural setting in a small village in eastern Ontario. There are about 360 students attending this Junior Kindergarten to Grade 8 school. About 90% of the students travel to and from school by bus. My grade four class consisted of 31 students, 17 male and 14 female.

I teach all subjects to this class with the exception of French, which is taught 40 minutes every day by another teacher. In March of the school year of this study (1997-1998), I was appointed Acting Vice-Principal at my school, and this reduced my teaching time with the class to 50%. I continued to teach mathematics, reading and music, while another teacher taught the rest of the curriculum for the remainder of the school year.

Permission to undertake this project was obtained from my principal early in September 1997. I informed my students’ parents of the project informally through my September newsletter, and again during Open House in mid-September. An Official Letter of Consent was sent home for parents to sign and return if they were willing to grant permission for their child's work to be collected to become part of the study. Twenty-one families gave consent for the use of their children's work in this study.

All of the 31 children in my class participated in the PEEL procedures used for this study. All of the students' work was collected and assessed for school reporting purposes. Only the work from the 21 students for whom consent was given to participate in this study was collected and examined as part of this research. The work of the 10 children whose parents did not give consent to be part of the study or who did not return the consent form was not collected for use in this project. Children whose work did not become part of this project suffered no academic penalty for not participating. Names of the children involved have been changed to ensure the anonymity of the participants.

 

CHAPTER 2

THEORETICAL FRAMEWORK

This chapter reviews a number of studies on the general nature of teacher change. The Project for Enhancing Effective Learning (PEEL) is also reviewed in the context of the research on teacher change. This study is discussed in the context of teacher change and PEEL.

 

Controlling Change in Teachers’ Practice

A common thread throughout the literature on teacher change is the source of the change that the researcher intends to see implemented. Much of the literature examines the degree to which innovations have been successfully implemented (Guskey, 1986; Richardson, 1990). Researchers involved in assessing teacher change with these innovations have expressed their frustration at teachers’ unwillingness or inability to implement their suggestions (Briscoe, 1994). Teachers often reject or discount research findings that support intended changes in practice, claiming them to be impractical or inapplicable in their situations. In the teacher’s experience of the classroom, research holds little validity. Most of these studies report failure to create the changes in practice that they had intended (Guskey, 1986). The most useful consideration in predicting or determining successful implementation of a change in practice is the origin of the change. The research indicates that teachers have not implemented innovations that originate from outside the classroom, nor have innovations external to teachers’ experience become integrated into teachers’ practice (Guskey, 1986; Richardson, 1990).

In contrast to these findings, recent studies show that teachers actually are changing their practice, and that the changes are being made almost continuously (Briscoe, 1994). Accommodations for student needs, new curricula and changes in school administration all contribute to the need teachers see to develop new procedures and strategies for their practice. Indeed, this research indicates that for changes in teacher practice to be lasting and meaningful, the teacher must initiate the change (Briscoe, 1994; Guskey, 1986; Richardson, 1990).

 

Reflective Practice and Teacher Change

Reflective practice as a strategy to improve professional practice was developed by Schön (Osterman, 1998). Reflective practice is the reorganization and restructuring of knowledge based on experience, leading to new understandings of teaching. Osterman (1998) identified four separate stages that together constitute a cycle of reflective practice: experience, assessment, re-conceptualization and experimentation. A specific teaching experience begins the cycle. As the teacher reflects on the experience, he or she considers the intended or expected outcomes, what actions were taken by the teacher and the students to achieve the outcomes, and what the actual outcomes were. If, through this reflective analysis, the teacher discovers a discrepancy between what was intended and what actually occurred, the teacher must consider how to change his or her actions to align more closely the intended and the actual outcomes. The search for explanations to and resolution of the discrepancy brings the teacher in contact with new ideas and new strategies. The teacher then experiments with the new strategies. Successful experiences are likely to become integrated into the teacher’s practice. Teachers construct meaning from their experiences by reflecting on those experiences.

Edwards (1994) suggests that problems arising from practice create intellectual dissonance. Teachers’ efforts to correct the dissonance lead to the construction of new knowledge or practice. Changes in practice result from teachers reflecting on the nature of the dissonance and then restructuring their understanding and knowledge as solutions are sought.

Both Osterman’s reflective practice model and Edwards’ constructivist model are based on Cobb’s (1988) idea that learning is rooted in the active construction of knowledge by learners. Knowledge and beliefs are formed within the learner and require the active involvement of the learner. Each learner constructs new knowledge and understanding as he or she makes sense of personal experiences. Through reflection, the learner compares new knowledge to existing knowledge. Learning occurs when the learner attempts to integrate the new with the current, creating new understanding.

While much of the literature about teacher change identifies an intransigence on the part of teachers asked to implement change, this intransigence has been attributed to the intended changes being initiated by parties from outside the school or classroom. More recent studies of teacher change indicate that teaching practice is fluid, changing frequently according to the decisions made by individual teachers to meet the needs of their students, school and community. Successfully implemented innovations that become integrated into teachers’ practice must originate with the teacher and must be viewed as a necessary improvement to practice to increase the effectiveness of student learning.

Teachers have generally viewed the results of education research as impractical and inapplicable to their situations and experience. Traditional research has examined and measured teaching practice usingtheory as the guide. Reflective practice is a strategy that teachers themselves can use to examine and improve their own practices. From this perspective, teachers’ personal theories about learning are examined and measured usingpractice as the guide.

 

 

 

Origins and Foundations of PEEL

The Project to Enhance Effective Learning (PEEL) seeks to improve the way teachers teach and students learn. PEEL is a response of teachers and researchers to three general concerns in education: schools do not meet the needs of society, schools are not as effective as was originally thought, and theories of learning and ability have changed (White, 1997).

Schools teach students facts and skills, yet student understanding of these facts and skills is often inadequate (White, 1997). Students may know how to add, multiply or divide, for example, but they may not recognize which particular operation to apply to specific problems. PEEL was begun to help develop new teaching practices that would address the concerns and problems that teachers recognized through their experiences in their classrooms. The aim of PEEL is to improve practice as a means to improve the effectiveness of student learning.

The mission of schools is to create life-long, independent learners. The Ontario Ministry of Education and Training released a new elementary school curriculum in 1997 and 1998 that seeks to “prepare students for a lifetime of learning” (1997a, p. 3). Baird (1997) suggests that the process for creating life-long learners is not well defined, and that although the difference between good and poor achievement is measurable, the difference between good and poor learning is not quite so clear. PEEL endeavours to identify specific student behaviours that act as impediments to learning. Teaching procedures were developed to teach students how to improve their learning by using behaviours that would affect learning.

In the past, a student’s school performance was linked closely to his or her assumed ability. Traditionally, ability was considered an innate trait, a fixed trait from birth. The amount of ability a person had could not be changed. Ability was equated with intelligence, something a person was born with. Variations in school achievement were attributed to variations in the amount of ability each student possessed. Consequently, schools and teachers did not try to teach students to be more able or more effective learners. More recent theories view ability as a combination of a number of specific information-processing skills that can be taught and learned (White, 1997). PEEL teachers sought teaching procedures that would help students learn information-processing skills that would lead to improved ability to learn.

PEEL began as a joint project between researchers at Monash University and a group of teachers at Laverton High School in Melbourne, Australia. Unlike traditional staff development programs, PEEL began with teachers’ concerns about student learning in their classrooms. Using the teachers as researchers, and using a reflective practice model of collaborative action research, the teachers examined their classroom experiences. The findings of PEEL studies are consistent with a constructivist view of learning. Students who were taught to relate new knowledge to existing beliefs generated new personal meanings, and they achieved better understanding of the material than did a second group of students who completed question-and-answer tasks with the same material. Poor learning was found to be related more closely to poor information-processing skills and habits than to intellectual ability (Baird, 1992). Poor learners were ones who were not actively engaged in the learning process. Through observation by teachers, nine poor processing habits were identified, called poor learning tendencies. These are listed in Table 2. This list of poor learning tendencies led to the creation by PEEL teachers of a corresponding list of good learning behaviours that teachers could encourage.

Learning, the integration of new knowledge with existing knowledge, cannot be directly observed. Learning is a process that occurs in the mind of the learner. However, Mitchell (1992a) recognized that students who were actively engaged with their learning would display a number of behaviours that could act as indicators that meaningful learning was happening.

 

 

1. Superficial AttentionNo processing of information to generate personal meaning.

2. Impulsive Attention Some parts of the learning are attended to, others are ignored.

3. Premature Closure Finishing work, believing it to be finished, when it is not finished.

4. Inappropriate Application Application of a memorized action where it does not apply.

5. Staying Stuck No strategy to cope with being stuck except to call the teacher.
6. Non-Retrieval No attempt to connect current lesson or new knowledge to previous lessons, existing knowledge, or own views.

7. Ineffective Eradication Persistent use of seemingly changed misconceptions.

8. Lack of Internal Reflective Thinking Learner is not thinking reflectively about the current topic.

9. Lack of External Reflective ThinkingDoes not make links among subjects or with the outside world.

(adapted from Mitchell, 1992a, p. 179)

 

Table 2

Some Poor Learning Tendencies Identified in PEEL

 

Some good learning behaviours are listed in Table 3. The teacher-researchers developed specific procedures to integrate into their practice to help students learn and practice good learning behaviours.

 

 

 

A MONITORING BEHAVIOURS 1. Seeks Assistance 2. Checks Personal Progress 3. Plans and AnticipatesB CONSTRUCTING AND RECONSTRUCTING BEHAVIOURS 4. Reflects on the Work 5. Links to Beliefs and Experiences 6. Assumes a Position(adapted from Mitchell, 1992b, p. 63)

 

Table 3

Some Good Learning Behaviours Identified in PEEL

 

Research Design: Why PEEL?

PEEL incorporates all the criteria identified in the literature to encourage lasting, effective and meaningful change in teaching practice. The need for a change in practice is identified by teachers from their observations and experiences in the classroom. The change does not originate from outside the classroom. Teachers construct new meaning and new understandings of their practice and of student learning. New procedures, developed in response to the recognition of problems in student learning (poor learning tendencies), are tested in the classroom and measured against improvements in student achievement. Procedures that successfully generate good learning behaviours and improved learning in students become integrated into practice.

 

Research Design: Action Research

Action research is one strategy available to teachers who want to study their own practice and its effects on student learning (Burch, 1993; Halsall & Hossack, no date; Henry & McTaggart, 1998; McNiff, Lomax & Whitehead, 1996; Torres, 1996). One action research model involves five steps: problem formation, planning an intervention, gathering data, analyzing and reflecting on the data, and reporting the findings of the project (McNiff, Lomax and Whitehead, 1996). At this point in the cycle, the teacher-researcher may reflect on the outcomes of the intervention and start re-examining practice to develop a new research project, beginning a new cycle of research. As teachers review the successes and failures of each lesson and activity and modify lessons to overcome perceived failures, they engage in a cycle of revision. It is the small revisions with each activity and lesson that together constitute improvement of practice. Teachers actually are continually revising and changing their practice. Within a formal action research project such as the present study, teaching experiences and their interpretation are documented, analyzed, and reported to other teachers.

 

Context

The central objective of this study is to modify my own teaching strategies to encourage students to develop deeper understanding of the work and to link new knowledge to previous knowledge and experience. Two of Baird’s (cited in Mitchell, 1992a) poor learning tendencies summarize my concerns in my classroom: the non-retrieval of one’s existing beliefs and understandings that are related to the current lesson, and the lack of reflective thinking (an inability to make connections between and among lessons, topics, ideas and personal experience). Table 4 illustrates the relationships among the targeted poor learning tendencies, good learning behaviours, and the specific procedures chosen for this study.

Mitchell and Mitchell (1992) describe in detail a wide range of classroom procedures that are consistent with PEEL objectives and contribute to the enhancement of learning. The procedures have been classified according to each of the poor learning tendencies as well as more general concerns about learning. I incorporated Venn diagrams, concept maps and topic and task questions in my practice in an attempt to improve the ability of my students to seek links between lessons and activities within three topics in mathematics, and to seek links between schoolwork and their personal experiences in reading. Documenting these changes to my classroom practice to address these concerns and collecting and analyzing data as students respond to my changes is the central focus of this study.

Neither concept maps nor Venn diagrams are new procedures to teaching. Concept mapping has been the focus of several studies of teaching practice (Hyerle, 1995-1996; Prater & Terry, 1988; Rafferty & Fleschner, 1993). Marino (1996), Rivers (1996) and Sullivan (1996) report their individual experiences in incorporating concept mapping into their practices as a result of their participation in PEEL Similarly, studies of teaching and learning in mathematics have included Venn diagrams as important procedures (Dodridge, 1973; Gray & Sharp, 1996; McGinty & Van Beynen, 1985; Van Dyke, 1995).

 

The PEEL Procedures Used in this Study

Concept Mapping

In creating a concept map, students arrange a series of keywords on a sheet of paper, drawing lines to connect words together. Students also indicate how any two connected words are related. This activity is designed to encourage students to make links in their knowledge. Concept maps can be used to link ideas in a single lesson, in a number of lessons, or across an entire unit of study, or concept maps may link a lesson or unit to the "real" world.

Venn Diagrams

Venn diagrams display relationships between two or more different ideas, objects, characters, or concepts. The diagram is of two or more interlocked and overlapping circles. Each circle is assigned a specific descriptor. The students fill in the circles with examples that fit each descriptor. Examples that fit two or more of the descriptors would be placed inside the overlapping areas of the circles. Developing Venn diagrams requires the learner to become

precise in his or her understanding of the descriptors.

 

Table 4

Connecting Specific Teaching Concerns with Poor Learning Tendencies, Good Learning Behaviours and PEEL Procedures

 

 

Topic and Task Questions

This procedure is the basis of the Thinking Linking Log used with mathematics. The procedure helps students link lessons and topics together. At the close of a lesson, students are asked to respond to questions such as, "What do you think the main point of the lesson was?" or "What did today's lesson have to do with yesterday's lesson?" Task questions require students to relate to what they did during the lesson: "What was the main thing you had to do, and why do you think we had to do it?"

 

Data Collection

Data collection for this study centres on reading and mathematics. Concept maps and Venn diagrams were used as reading response with novel study projects in April, May and June. In mathematics, our first topic was number sense and numeration. Students constructed Venn diagrams to demonstrate their understanding of place value. These Venn diagrams were collected in the first term. During the second and third terms, students used a reflective journal in math to record their responses to topic and task questions designed to create links between lessons in the topics of division, fractions and quadrilaterals. All materials collected were dated to provide a chronology of the students' demonstrations of the development of their skills and connections.

 

Analyzing the Data

Participating students were assigned a student number to help with computerized data organization. In reading, all possible links were identified for the concept map for each story. The links each student made between terms were entered into a computer database file. Mathematics Thinking log entries were also entered into a computer database. These entries were linked to a description of the specific lesson or activity for that day and the specific "linking question" for that day's work. These entries were then sorted by date and by student to achieve a linear time line for each student.

Concept maps and Venn diagrams were sorted into categories using a four-point scale designed to assess each student's level of understanding of each story read. The scale, shown in Table 5, is modeled after the Ontario Ministry of Education and Training Achievement Level chart for Language (1997a, p. 9).

The students' Thinking Linking Log entries were sorted into categories based on the type and nature of the connections they made for each lesson. A four-point scale developed for this purpose was modeled after the Ontario Ministry of Education and Training Achievement Levels chart for Mathematics (1997b, p. 9). This scale is shown in Table 6.

 

Table 5

Assessing Concept Maps and Venn Diagrams as Indicators of Understanding in Reading

1	2	3	4
- limited understanding of concepts or knowledge   - reasoning is inconsistent   - partially complete, but inappropriate explanations   - uses a few simple ideas     - communication is unclear, imprecise	- limited understanding of concepts or knowledge   - consistent reasoning   - incomplete but appropriate explanations     - uses several simple related ideas   - communication shows some clarity and precision	- general understanding of concepts or knowledge   - consistent reasoning   - complete and appropriate explanations     - uses ideas of some complexity     - communication is clear and precise	- thorough understanding of concepts or knowledge   - consistent reasoning   - complete and appropriate explanations     - uses complex ideas     - communication is clear, precise and confident

 

 

 

Table 6

Assessing Topic and Task Question Responses as Indicators of Personal Connections

1	2	3	4
- limited links   - inconsistent reasoning   - explanations incomplete, not related to topic     -uses a few simple ideas	- limited links   - consistent reasoning   - explanations incomplete, related to topic     - uses several simple ideas	- links are general   - consistent reasoning   - explanations are complete, connected to topic     - uses ideas of some complexity	- thorough links   - consistent reasoning   - explanations are complete, connected to topic   - uses complex ideas

 

 

Summary

The literature indicates that meaningful and enduring change in practice is more likely to occur when teachers consider the change to be practical and applicable in their classrooms (Briscoe, 1994). Teachers themselves, in their role as classroom leaders, are in a position to recognize the type of changes needed in practice to satisfy the needs of the students and the curriculum. Reflective practice and action research are models of inquiry (McKernan, 1991;

McNiff, Lomax and Whitehead, 1996) that provide a framework for teachers wanting to initiate and monitor change in their practice.

PEEL is a collaborative action research project that seeks to improve the quality of student learning. By identifying poor learning tendencies that are barriers to learning, PEEL has attempted to show teachers what poor learning looks like. Integrating new teaching procedures into practice not only discourages poor learning tendencies but also teaches students good learning behaviours that can improve learning. Teachers can train students in information-processing skills to improve their ability to learn.

While I engage in reflective practice to improve my teaching, I am teaching my students to be more reflective in their learning. I am also teaching my students ways to construct new knowledge. Both teacher and students are constructing new knowledge in the same manner. I have identified specific poor learning tendencies and good learning behaviours that are the focus of this study. I have also listed and described specific procedures I incorporated into my practice.

 

CHAPTER 3

USING VENN DIAGRAMS TO BUILD UNDERSTANDING

OF PLACE VALUE IN MATHEMATICS

 

Place Value and Grade Four Mathematics

Place value is the foundation of our number system. Relationships between and among numbers are understood using place value. Recognizing how and why numbers are represented using place value is essential to achieving greater understanding of number operations and relationships. The National Council of Teachers of Mathematics developed a series of standards for the teaching of mathematics in elementary schools. The Council stated that “understanding place value is a critical step in the development of children’s comprehension of number concepts” (NCTM, 1989, p. 39), and that “children must understand numbers if they are to make sense of the ways numbers are used in their everyday world” (NCTM, 1989, p. 38).

The Ontario Ministry of Education and Training curriculum guide,The Common Curriculum (1995), which specified outcomes for student learning with respect to place value, was in effect at the time of data collection. This curriculum specified that by the end of Grade 3 students would be able to “apply the concept of place value and use whole numbers and simple fractions” (Ontario Ministry of Education and Training, 1995, p. 73). In the newOntario Curriculum (1997), Grade 4 students are expected to know how to represent the place value of whole numbers to ten thousand and of decimal numbers to hundredths (Ontario Ministry of Education and Training, 1997, p. 22).

 

 

Defining and Assessing the Problem

Place value tends to be a difficult concept for children to grasp. In my classroom, several students struggled with the construction of three-digit numbers. When asked to write a number with seven tens, nine ones and two hundreds, students recorded the numerals in the order they were given in the problem (792) rather than reorganizing them according to their named place value (279). There was a need to address the difficulty these students had in demonstrating an understanding of place value.

 

Re-Conceptualization of my Practice

Students having difficulty understanding place value exhibited a poor learning tendency. They were not actively processing the lessons and concepts to generate personal meaning. This is the tendency of superficial attention. My goal was to change my practice so that the students would learn how to focus their attention and to generate meaning with place value.

One PEEL procedure well-suited to this situation involves Venn diagrams. Given two specific descriptors (one for each circle), the students would direct their attention to numbers that fit the descriptors and fill in each section of the diagram. The Venn diagram acts as a lens for a student to focus thinking on one specific task: presenting numbers that fit the descriptors.

 

Implementing the Procedure: Observations

Mathematics class occurred in the first period of the day. The Venn diagram exercises were presented at the beginning of the period, to be completed upon arrival in the class. Parallel to these activities, the students worked through other place value activities using manipulatives and textbook exercises. The students were given a Venn diagram of two linked circles each day. Each circle was labeled with a specific place value descriptor. Selected samples of student work with Venn diagrams and place value concepts are included in Appendix A.

 

Working with Tens and Ones

During the first week one circle was labeled "Numbers with x tens" and the other circle was labeled, "Numbers with y ones." (Figure 1). The values of x and y changed each day. The students’ task was to complete the outer sections of the diagram with at least 5 numbers, and to complete the centre section with at least 3 numbers.

 

Figure 1

Venn Diagram Procedure Using Tens and Ones [TABLE UNAVAILABLE IN WEB FORMAT]

The first Venn diagram in this series used nine ones and seven tens. One circle was labeled “Write 5 numbers with 9 ones.” The other circle was labeled “Write 5 numbers with 7 tens.” Nineteen Venn diagrams were collected. All students filled the “9 ones” section with correct numbers (9, 19, 29, etc.). Seventeen students recorded numbers with seven tens in the outer section of the “tens” circle (70, 71, 170, etc.). Two students recorded incorrect numbers in the “tens” section. Julie identified numbers with seven "ones," rather than numbers with seven "tens" (17, 27, 37, etc.). Tania recorded the number "7," then continued with a list of numbers in the seventies. The centre section required students to identify and record numbers with both nine ones and seven tens. Seventeen students accurately completed this section of the diagram. Two students had difficulty with this task. Both Caroline and Tania realized that numbers in the centre of the diagram must have both a nine and a seven, but did not position these digits correctly. Caroline identified five numbers using the digits 7 and 9. Two were correct (179 and 279). Three numbers were incorrect (197, 917, and 719). It appears that Caroline took the first number in her list, 179, and rearranged the digits in random order to create other numbers. Neither Caroline nor Tania recognized their errors until they were guided to correct responses.

The next Venn diagram activity asked the students to focus on numbers with 2 tens and numbers with 5 ones. Caroline's performance improved with this diagram. She wrote three numbers in the centre section, two of which met the criteria: 25, 225, (correct) and 145 (incorrect). Caroline preserved the ones digit correctly (5) in creating numbers for this section. In completing the first activity, Caroline used the two identified digits (7 tens, 9 ones) in random order to produce numbers for the centre section (197, 917 and 719). In the second activity (2 tens, 5 ones), the numbers that did not fit the centre section were not produced by rearranging digits. Caroline demonstrated a stronger connection to place value concepts with the second activity.

Tania's performance also showed improvement with the second activity. She recorded four numbers in the centre section, two of which were correct: 25, 225, (correct), and 250 and 25000 (incorrect). Her response to this diagram appears similar to her response to the first diagram. In both, Tania used zeros to generate new numbers. In the first diagram, the zeros appear to be placed at random. In one instance, she inserted a zero in the tens place of her original number (79) to create a three-digit number (709). In a second instance, Tania added zeros to the right side of her original number (7900). In the second diagram, Tania’s numbers reveal an attempt to use a pattern to help her find more than one correct number. She did not insert zeros into the middle of her numbers this time. The first number on her list was 25, a correct response. The next number on the list was 225, created by adding a hundreds digit to 25. The next two numbers on her list were created by adding zeros to the right side of her first number (25). This produced numbers that did not match the descriptions, but Tania’s use of a patterning strategy appears to be an important step forward in her construction of new understandings of place value and number.

Of the 17 students who completed the diagram accurately, Julie demonstrated confidence in her understanding of place value. She filled her diagram with 38 numbers, all in their correct sections of the Venn diagram. Eleven students wrote numbers in the margins of the page. These numbers did not fit descriptions for either circle. When asked about these “extra” numbers, Neil explained that “you can use all the numbers. Some numbers go in one circle, some go in the other circle, some go in the middle. All the rest can go outside the circles.”

 

Working with Hundreds and Tens

The next series of Venn diagrams directed the students to think about the hundreds place and the tens place. The first diagram of this series used "numbers with 8 hundreds" and "numbers with 2 tens." I collected 17 responses to this exercise and found that 9 students successfully completed the diagram. Confident students filled the diagram with many numbers. Margot recorded 17 numbers, Olivia recorded 27 numbers, and Glenda entered 66 numbers in the diagram. Bill used a repeating pattern to ensure he was writing different numbers. For numbers with 8 hundreds, he wrote 10,800, 11,800, 12,800, 13,800 and 14,800. Similarly, he wrote 10,020, 10,120, 10,220, 10,320 and 10,420 for numbers with two tens. For numbers that fit both descriptors, he wrote 10,820, 11,820, and 12,820.

Anne, Caroline, David and Tania were able to identify numbers for the outer sections of the diagram, but they did not correctly identify numbers for the centre section. Fay was able to correctly fill in the hundreds section only. Caroline and Tania successfully completed the outer sections of the diagram. Neither student successfully identified any numbers for the centre section. Caroline wrote 802, 803 and 804 in the centre section. Tania left the centre section blank.

More students had difficulty completing this task than the previous diagram using tens and ones. One student, Fay, successfully completed only one section of this diagram. All of the other students were able to fill in both outer sections of this diagram. Twelve of the 17 students were also able to accurately complete the centre section.

 

Working with Hundreds and Ones

With the third series of Venn diagrams, I used the hundreds place and the ones place. Even though the tens place is not mentioned as part of this activity, the students would have to think about its role as a placeholder as they generate three-digit numbers to complete the diagram. This, I felt, created a more difficult problem. Students were asked to identify numbers with three hundreds and numbers with six ones.

Caroline, David, Peter and Olivia completed all sections of the diagram accurately. Identifying numbers for the centre section was problematic for Ian, who had successfully completed all previous diagrams. He listed 326, a correct response, but then added 36 and 116 to his list. While both of these numbers are incorrect, all three numbers have six ones, one of the required descriptions.

Caroline correctly filled in all sections of the diagram. For numbers with three hundreds, she wrote 305, 307, 306, 304, and 309. For numbers with six ones, she wrote 66, 76, 86, 96 and 36. In the centre section, she wrote 316, 306 and 356. I noted that Caroline wrote 306 in both the hundreds circle and the centre section. This number should only appear in the centre section, because it has both three hundreds and six ones.

Tania also demonstrated an improvement in her understanding with this diagram. She correctly identified numbers to fill both outer sections. While Tania did not list any numbers in the centre section of the previous diagram, in the centre section of this diagram she recorded three numbers: 706, 716, and 726. Although none of these numbers has the digit three in the hundreds place, they all have six ones. Tania also used a pattern of counting by tens to generate similar numbers. This strategy preserved the digit in the both the hundreds place and the ones place. Even though the hundreds digit is incorrect, there is evidence of an increase in Tania’s processing of the problem information as she searches for correct solutions. Tania has demonstrated an increase in her ability to construct meaning of numbers and place value.

 

Summary

This was the first PEEL procedure I attempted and the first PEEL procedure the students encountered. At this early stage in the project, I was working on faith in the evidence from the PEEL teachers in Australia. Using Venn diagrams to build understanding of place value was a new approach to a concept sometimes confusing to students. Students were directed to focus their thinking on two specific values at a time (tens and ones, for example). Much of my prior teaching relied on textbook exercises. Students were able to complete the textbook work, but when confronted with place value in other contexts (regrouping with addition and subtraction, for example), students could not call upon a clear understanding of how to use place value. The information the Venn diagrams provided to me about the areas of strength and weakness of the students in my class was immediate and compelling. By beginning the school year with this procedure, I was able to identify quickly the students who displayed poor learning tendencies and who had weak information-processing skills in mathematics.

 

 

CHAPTER 4

USING CONCEPT MAPS

AS READING RESPONSE TO BUILD UNDERSTANDING

My reading program began with the students reading and responding to short stories found in the Grade 4 reading anthology, Cross the Golden River (Booth, 1986). Five stories were presented and read during September and October. I introduced two PEEL procedures as reading response activities with these stories: the concept map and Venn diagram. The objective was to help students build their understanding of characters, settings and story events from their reading by constructing meaningful connections. During the third term I introduced novel studies to small groups of students. As part of the written component for this work, the students worked with either Venn diagrams or concept maps or, with some novels, both procedures. Three groups of students participated in the research component of this novel study during which the students were not sorted into reading groups according to reading ability. Instead, the students were free to choose the story that interested them most, and reading groups developed based on topical interest.

 

Summaries of the Short Stories and Novels

Background information about the stories is helpful for the reader to understand the student responses to the follow-up procedures. In this section I provide short summaries of the stories the students read. It is my intent that this information will allow the reader to place the student responses in context, providing a common starting point for the discussion that follows.

 

Short Stories

 

The Tiger Skin Rug (Rose, 1986). The main character of this story is a hungry tiger who traded places with the Rajah’s tiger skin rug when Rajah’s servant was cleaning it outdoors. In this way, the tiger got into the Rajah's palace, so that he could eat the table scraps after the Rajah's meals and never be hungry again. No one in the palace realized that the switch had occurred. One night three robbers broke into the palace to steal the Rajah's gold and silver. The tiger, still posing as the rug, faced a dilemma: keep still and quiet and not reveal himself, or scare the robbers away. The tiger decided he had to protect the Rajah and scare the robbers away. The Rajah, after getting over his own scare from finding a live tiger in his palace, befriended the tiger to repay him for saving his riches. The tiger lived a long life in the palace and was never hungry again.

 

Dinner at Alberta’s (Hoban, 1986). In this story, a young crocodile named Arthur needs to improve his table manners to impress a young female crocodile named Alberta. Arthur’s parents and his younger sister Emma work with him to improve his manners, with limited success. When Arthur's family has dinner with Alberta's family, Alberta's younger brother, Sidney, mimics Arthur's attempt to use his new manners. As the dinner progresses, the antagonism escalates. After dinner, Sidney invites Arthur outside on the pretense of showing him his tree fort. When they come back inside, both boys’ clothes are torn, Arthur has a bump on his head and Sidney has a fat lip. This part of the story is written to imply the boys were fighting. The reader must infer that these two characters were fighting outside based on this evidence.

 

Mountain Rose (Stren, 1986). When Rose was a baby, her parents left her with a kindly aunt and sent her twin brother to live with another relative. During a high school wrestling match, Paddy Flanagan, a famous wrestling coach, offers to coach Rose in her quest to become the Ladies Wrestling Champion of the World. Rose ultimately became a prize-winning wrestler, winning the world title. As champion, Rose is challenged to a wrestling match by Gardenia Gus, the Men's Wrestling Champion of the World. During this match, Rose discovers that Gus has an elephant-shaped birthmark that looks just like her own. Rose realizes that Gus is her long-lost brother, separated from her at birth.

 

The Case of the Mysterious Tramp (Sobol, 1986). This is an Encyclopedia Brown mystery. All Encyclopedia Brown stories are written with enough clues for the reader to be able to solve the mystery. In this story, Mr. Clancy was hit on the head with a pipe and robbed as he was working on the engine of his stalled truck. The main suspect, Mr. Clancy's assistant, John Morgan, was in the truck at the time of the attack and provided an eye witness account of the event to Encyclopedia Brown. According to John Morgan, while Mr. Clancy was attending to his truck, a tramp came out of the woods, attacked and robbed Mr. Clancy before Morgan could get out of the truck to stop him. Encyclopedia Brown did not believe Morgan's story and accused Morgan himself of committing the crime. The student's responsibility was to determine whether Encyclopedia Brown was correct, and how it could be proved.

The actual solution to this mystery is related to the engine trouble Mr. Clancy was trying to fix at the time of the robbery. John Morgan was in the cab of the truck when he said he saw the tramp attack Mr. Clancy. However, because the hood of the truck was up and Mr. Clancy was in front of the truck, John Morgan could not have seen anyone attack Mr. Clancy. This was the flaw in John Morgan's story that the students needed to discover.

 

Chocolate Fever (Smith, 1986). Chocolate Fever is about Henry Green, who came down with an unknown illness that covered his body with brown spots. After being inspected by the school nurse and an odd doctor, he was diagnosed with "Chocolate Fever." When the brown spots began popping, it became clear that the spots were full of chocolate. The excerpt ended without finding a resolution for this boy's illness and dilemma.

 

Read Aloud Story: The Chocolate Touch (Catling, 1952). In September I began reading this story aloud to the students. This is a variation of the story of King Midas and the golden touch. The Chocolate Touch featured a young boy called John Midas who liked to eat chocolate. He found a magical candy store and purchased a piece of special chocolate with a strange coin he found on the sidewalk. When he ate this special chocolate, he found that everything that he put to his lips turned to chocolate. Eventually, he only had to touch an object for it to turn to chocolate. At first, he thought this was a good gift, but soon came to regret having this "touch". His lesson learned, especially after he kissed his mother and turned her into chocolate, John found the magical candy store again and was able to have the "touch" and its effects reversed.

 

Novels

The three novels that these students read were Titanic Crossing (Williams, 1995), A Friend Like Zilla (Gilmore, 1995), and Devil's Bridge (DeFelice, 1992).

 

Titanic Crossing (Williams, 1995). This novel combines factual information about the sinking of the Titanic with a fictional story about two families as they cross the Atlantic on the doomed ship. The Trask family was returning to America after the death of the father, accompanied by an uncle. The family had lost its wealth and was reliant upon Grandmother Trask in America for support. Grandmother was not fond of her daughter-in-law and wanted the children to live near or with her. She used her power to force her daughter-in-law to comply with her wishes. Albert Trask, the son, faced a conflict with his mother regarding his schooling. On arrival in America, Albert was supposed to go to military school. He wanted to go to art school, an idea which his mother, uncle and grandmother opposed.

On board the ship, Albert befriends Emily Brewer, a girl his age who is also travelling with her family to America. On the evening of the sinking, Albert's mother disappears in third class, visiting a friend she met earlier in the day. When the ship struck the iceberg, Albert's mother did not make it out of third class, leaving Albert to look after his younger sister, Virginia. Albert and Virginia survived the sinking due to Albert's quick thinking and maturity in the absence of their mother. On board the rescue ship, Albert and Virginia were reunited with Emily, who had lost her entire family in the disaster.

Once in America, Albert had to face his grandmother over the issue of his schooling. His experience on the Titanic had caused Albert to grow up quickly, giving him the strength to stand up to his grandmother, who finally allowed him to attend art school.

 

A Friend Like Zilla (Gilmore, 1995). In this story, a young girl (Nobby) and her family travel to Prince Edward Island for a vacation and family reunion. The family that owns the rental cottages where Nobby and her family are staying has a 17 year-old daughter, Zilla. Zilla is mentally disabled, making her different from other 17-year-olds, in Nobby’s eyes. Nobby and Zilla soon become good friends. Nobby's Uncle Chad and Aunt Audrey arrive for the reunion. Nobby and her Uncle Chad do not get along because he continually corrects and belittles her. When Uncle Chad met Zilla and realized that Zilla is mentally disabled, he teased her and made fun of her. One morning Uncle Chad went out for a run and did not return. The families organized a search party to look for him. Zilla and Nobby went out looking, even though Nobby was not keen to find him. Zilla followed the sound of seagulls to a drop-off near the cliffs by the sea and found Uncle Chad in a gully with a broken ankle. After he was rescued, Uncle Chad softened his attitude toward both Nobby and Zilla. The story ends with Nobby and Zilla becoming better friends, and Uncle Chad offering to start a new friendship with both girls.

 

Devil’s Bridge ( DeFelice, 1992). This is a story about a young boy, Ben, who enrolled in the town's annual striped bass fishing derby. Ben's father, known as Pop, holds the Derby record for catching the biggest striped bass. Pop is dead, and Ben lives with his mother, who is dating a man called Barry Lester. Ben does not like Barry because he feels that Barry is trying to step into his father’s place. While scouting out good fishing locations, Ben overhears that Freddy Cobb and another man plan to cheat to win the Derby and beat Pop’s record. The rest of the story is about Ben trying to stop these men from getting away with their plan. On the day of the Derby, Ben himself caught a fish that would beat his father's record and win the Derby, so long as Freddy Cobb was disqualified. However, Ben decided that it was more important to let the fish live free in the ocean than to win the Derby, so he let the fish go. Barry caught Freddy cheating and so preserved Pop’s record. Ben started to see Barry differently after this and thought that they may become friends after all.

 

Concept Maps and Reading Response

Reading response activities are meant to help students consolidate their understanding of what they have read. These activities may also be used by the teacher to assess, in either a formative or a summative way, student comprehension skills. I introduced concept mapping into my practice for reading response to help my students improve their comprehension by increasing their ability to create and identify connections among characters, events and settings of the stories they read.

Completing a concept map requires students to consider a number of relationships that may exist among key elements or concepts taken from a piece of writing. Students create as many connections as possible among the terms, and then physically connect pairs of terms by drawing lines between them on paper. Each student then writes the explanation or reason for the connection on the line. With careful selection of the terms used in the map, students are directed to examine, identify and explain key relationships within a story. Samples of the students’ concept maps are included in Appendix B.

 

Short Stories

 

Tiger Skin Rug. This was the first story for which students were presented with a concept map as a response activity. I used six terms with the map - four characters (Rajah, Robbers, Tiger, Servant) and two concepts (friendship and being afraid). Nineteen maps were collected. The data in Table 7 show how many students made connections for each pair of terms. The students' explanations for their connections indicated the strength of each connection, and to what extent each student understood the story.

The students were able to create connections among the major characters. All of the students were able to create a connection between the tiger and the robbers. Sixteen students created connections between the tiger and the Rajah. Eleven students created connections between the Rajah and the servant. Eight students created connections between the Rajah and the robbers. Six students created connections between the servant and the tiger. The students used story events to explain the connections they made among the characters. The tiger saved the Rajah from the robbers. Seventeen students responded that either the tiger scared the robbers away or the robbers were afraid of the tiger. Julie and Evan recognized the dilemma the tiger faced. Julie wrote, “the Rajah would find out that he [the tiger] was not a real tiger skin rug.” Evan explained that “the tiger was afraid when the robbers broke in.” The students used the concept map procedure to create and explain the relationships among the main characters in the story.

 

Table 7

Concept Map Connections Matrix for The Tiger Skin Rug. (19 Respondents)

	Rajah	Robbers	Tiger	Servant	friendship	afraid
Rajah	X	8	16	11	12	4
Robbers	X	X	19	0	2	19
Tiger	X	X	X	6	11	2
Servant	X	X	X	X	5	0
friendship	X	X	X	X	X	0
afraid	X	X	X	X	X	X

 

The mapping exercise focussed the students’ thinking on these relationships and provided a way for them to express the connections they made while reading the story. The students’ responses gave me specific information about each student’s comprehension of the events of the story. By examining the links and the written explanations, I was able to determine the extent of each student’s comprehension of the interactions among the characters as well as their understanding of the story events. The concept mapping activity stimulated the students to think about the story as a whole rather than as a series of disconnected events.

Dinner at Alberta’s. This story revolved around a single theme: the effect of good table manners. There were more characters in this story than there were in The Tiger Skin Rug, and the characters interacted in different ways at different points in the story. This was the second concept map exercise for both my students and me. I used the six characters for this concept map: Arthur, Alberta, Sidney, Emma, Mr. Crocodile and Mrs. Crocodile. I was pleased with the students’ work with the map from the first story, and I looked forward to seeing what the students would do with this one. I expected the students to expand upon the relationships and interactions among these characters as Arthur attempted to improve his manners to impress Alberta. Seventeen concept maps were collected from this exercise. Table 8 shows the possible combinations of terms together with the number of student responses for each pair of terms.

 

Table 8

Concept Map Connections Matrix for Dinner at Alberta's (17 Respondents)

	Arthur	Sidney	Alberta	Emma	Mrs. Crocodile	Mr. Crocodile
Arthur	X	10	16	15	9	10
Sidney	X	X	9	1	0	0
Alberta	X	X	X	10	0	0
Emma	X	X	X	X	11	10
Mrs. Crocodile	X	X	X	X	X	10
Mr. Crocodile	X	X	X	X	X	X

 

All of the students created connections among these characters. Links were created between Arthur and Alberta by all of the students, explaining that they were friends, or that Arthur liked Alberta. Fifteen students created connections among the characters that were based solely on familial relationships. The students recognized that Arthur and Mrs. Crocodile were son and mother. Similarly, Arthur and Emily were connected because they were "brother and sister." Mr. Crocodile and Arthur were connected because they were father and son. Sidney and Alberta were connected because they were brother and sister. Mrs. Crocodile and Emma were connected because they were mother and daughter. Mr. Crocodile and Emma were connected because they were father and daughter. None of these students used the story events to explain the relationships among these characters.

David and Neil were the only two students to use supporting evidence from the plot to explain the connections they created among the characters. David wrote that Mrs. Crocodile, Mr. Crocodile and Emma were trying to teach Arthur how to eat. Neil also realized that Emma and Mrs. Crocodile were “helping Arthur with his manners.”

While the students correctly identified the family relationships among these characters, many did not explain their links using story events. From this experience, I discovered that some of the terms I must include on the concept map must tie the characters to their feelings or to other story elements such as main plot points or settings.

 

Mountain Rose. This was the first story for which the students responded with both a concept map and a Venn diagram. After my experience with the mapping activity forDinner at Alberta’s, I chose four characters and two concepts to include on the concept map: Rose, Gus, Paddy, Desdemona, family and famous.

I collected five concept maps. The possible combinations and links among these terms are identified in Table 9. The students were able to make strong connections with the main character, Rose. All of the students were able to create connections between Rose and Paddy, explaining that Paddy was Rose’s coach. The students also created connections between Rose and Desdemona, indicating that Rose wrestled Desdemona for the championship title. All of the students were able to create connections between Rose and Gus, explaining that they were both famous wrestlers. David and Olivia made the additional connection that Rose and Gus were sister and brother. There were fewer connections among the other terms. None of the students created connections between Paddy, Gus and Desdemona. There was no interaction between these characters in the story, making links among these terms difficult for the students to create.

 

Table 9

Concept Map Connections Matrix for Mountain Rose (5 Respondents)

	Rose	Gus	Paddy	Desdemona	family	famous
Rose	X	3	5	5	5	4
Gus	X	X	0	0	1	4
Paddy	X	X	X	0	0	1
Desdemona	X	X	X	X	0	0
family	X	X	X	X	X	0
famous	X	X	X	X	X	X

 

Novel Studies

The longer stories provided the students with more and varied information about the characters and their actions. When the relationship and interaction among the characters were strong, the students were better able to identify how characters could be linked. Examples of events and actions from the story were plentiful, and students used these to justify the links they made.

 

 

Titanic Crossing. The concept map for this story was assigned after the students had read the entire story. I chose three characters and three concepts to include on this map: Emily, Albert, Virginia, brave, scared and spoiled. Four students (Holly, Julie, Caroline and Scott) read the novel, and I collected four concept maps.

Holly did not write explanations for her connections between the terms. Instead, she indicated the strength of the connection by the number of lines she drew between the terms. For example, she drew two lines between Albert and brave, which may indicate a stronger connection than between Albert and scared, which she connected with only one line. She connected Emily and brave with 17 lines, and Virginia and spoiled with seven lines.

Julie and Caroline recognized Albert's bravery because he did not board the first available life boat. Scott also recognized Albert's bravery, although he did not specify an explanation. None of the students linked bravery with Virginia. This is consistent with the story, in which Virginia did not display specific acts of bravery.

The students reported the connections between scared and the characters in the story. Julie and Caroline noted that Albert was scared when he discovered he did not know where his mother was. Holly indicated a connection between Albert and scared without explaining how these terms are connected. Scott did not make a link between Albert and scared. Julie also identified the point when Virginia did not know where her mother was as a point of fear. Caroline noted that Emily was scared when the boat was going down. Holly did not explain her connections, but she made a connection between Emily and scared.

 

A Friend Like Zilla. Two students read the novel A Friend Like Zilla. The students completed a concept map using three characters (Zilla, Nobby, and Uncle Chad) and three concepts from the story (judge, mean and forgive). Uma created links among the six terms but did not write explanations for the links. Anne included her explanations for the links she created among the terms.

The connections the students made between Chad, Zilla and Nobby are not clear. Both students created a link between Nobby and Zilla. Anne explained that these girls are friends. Uma created a link between Nobby and Uncle Chad, but Anne did not. Anne did not create a link between Zilla and Uncle Chad while Uma created a connection without explanation.

Similarly, the students did not make strong connections to the concept terms. Anne did not create a link between Nobby and being mean, while Uma made this connection without explanation. Uma also created a link between Zilla and being mean. Anne did not make this link. Both students linked Uncle Chad with being mean. Anne wrote that "he was yelling at Nobby and Zilla." The connections with forgiveness were also mixed. Both students linked forgiveness with Nobby, and forgiveness with Zilla, but it is unclear why the students found Nobby and Zilla forgiving. Anne did not link Uncle Chad and forgiveness. Uma made this connection, but did not explain her thinking.

 

Devil’s Bridge. Four students read the novel Devil's Bridge. The students completed a concept map using two characters from the story (Ben and Freddy), one character who was not present in the story but who had influence over the other characters (Pop - Ben’s father), and three key story events (Ben's big striper, Pop's record, and the Derby). I left the students to reflect on the circumstances that linked these characters to these specific events.

To guide the students to consider Ben's conflict with his image of his father, I included the terms Ben, Pop, Pop's record, and Ben's big striper. Both students recognized that Pop and Ben are related as father and son. Peter and Neil reported that Ben caught a bigger fish that would beat Pop's record. Neil also noted that Ben did not enter his fish in the Derby. Both students also created connections among Ben, Freddy, Pop's record and the Derby. While Neil noted that Ben had overheard Freddy talking, both Peter and Neil explained that Freddy cheated to try to win the Derby. Both students recognized Freddy's illegal attempt to break Pop's record.

 

Summary: Concept Maps as Reading Response

Completing a concept map exercise directs student thinking about characters, their actions and story events in specific ways. While narrowing students’ thinking to a specific concept, the activity broadens student thinking about the story as a whole as the students considers each character’s relationship to the selected concept. In completing the concept map, the student creates a complex visual rendering of the interconnections within the story. The activity allows students to record any rationale they can use from the story to support the connections they make. The nature of each explanation becomes the measure of the depth of understanding of the story. For example, with The Tiger Skin Rug, all students who connected the terms “afraid” and “tiger” noted that the robbers were afraid of the tiger and that the tiger scared the robbers away. These are obvious, yet important observations in developing an understanding the story. Two students also considered being afraid from the tiger’s perspective, noting that the tiger was afraid of being discovered while posing as the rug. This connection is equally important in understanding the story, although it was treated more subtly in the text.

The responses recorded on a concept map show each student’s understanding of the relationships in the story. Traditional question-and-answer responses may also provide the teacher with an accurate measure of a student’s understanding of a specific point in the story, but answers to comprehension questions may not provide the student with a view of the complex and varied ways the characters and themes interact in a story. The completed concept map provides a student with a graphic representation of the complexity of the story. Although students construct the map by considering one pair of terms at a time, as the map grows the student is able to visualize how the characters and concepts fit together to create the whole story.

Teachers need to set the terms for each concept map carefully. Using the characters alone, as I did with Dinner at Alberta’s, produced unsatisfactory results. Few students used story events to report relationships between pairs of characters beyond family or friend. Carefully selected terms may lead students to report connections and relationships that would be difficult to obtain from question-and-answer responses. The concept map procedure directs the students to reflect and report on the story rather than reflecting and reporting on questions about the story.

Some of the parameters that I set up for this procedure may have limited the student responses. I prepared a standardized “concept map response” form, consisting of two columns of three boxes for the students to write the terms for each map. Generally, students drew more connections between terms in adjacent boxes than they did between terms that were in boxes that were farther away from each other. A more open-ended approach may encourage students to consider all pairs of terms. By providing students with a list of the terms and a blank piece of paper, the students would be able to locate the terms in the map so that each could visualize the connections between each pair of words.

Concept mapping as reading response was introduced into my practice as an alternative to a more traditional question-and-answer strategy. Concept mapping activities helped students build connections among main story elements - characters, events and settings. By practising the construction of such connections, students may improve their comprehension and understanding of what they read. The evidence presented in this chapter suggests that not only did my students’ ability to create connections improve over time, the nature of the connections they made also improved.

 

CHAPTER 5

USING VENN DIAGRAMS

AS READING RESPONSE TO BUILD UNDERSTANDING

I introduced Venn diagrams into my practice with reading as an aid to help students build understanding of what they read. Students listed similarities and differences between characters, settings, or events from their stories. The Venn diagram procedure provides a vehicle different from the concept map that helps students build understanding of what they have read. The activity directs students to reflect on two specific story elements. While creating a concept map helps students to create links among several terms or concepts, completing a Venn diagram guides students to examine the similarities and differences between only two terms.

Students completed Venn diagrams for three different types of reading response. At the beginning of the year, to coincide with our Venn diagram work with place value in mathematics, students used Venn diagrams to compare characters from the same story. Students also compared characters from two different stories. Students completed Venn diagrams to examine point of view. The viewpoints of both Mr. Clancy and Mr. Morgan from The Case of the Mysterious Tramp were examined. Later in the year, as part of our novel studies, Venn diagrams were used to help students make personal connections to the characters in their stories. All of these activities encouraged students to sort information from their reading and to use the information to improve their understanding of the characters in their stories. Samples of the students’ Venn diagrams used as reading response are included in Appendix C.

 

 

 

Comparing Characters Using Venn Diagrams

A Venn diagram was first used to help students compare Rose and Gus, two characters from Mountain Rose. Rose and Gus, siblings separated at birth and reunited at the end of the story, provided a rich supply of similarities and differences. I collected 16 Venn diagrams from this exercise. The exercise was designed to help students increase their understanding of the sibling relationship between Rose and Gus, who shared many similarities that the students noted: they both liked wrestling and both were wrestling champs, both their parents had died, and they both liked to eat. All students recognized that a major difference between Rose and Gus was their training diets. Gus's training diet consisted mostly of junk food while Rose ate only healthy foods. Although David, Julie, Bill and Scott were the only students to record that Rose and Gus were family, Uma, Holly, Olivia, Fay and Margot noted the birthmark that both Rose and Gus shared. As a first attempt using Venn diagrams for reading response, most students were able to recognize major similarities and differences between these two characters.

The next reading response Venn diagram activity was to compare two characters from two different stories: Henry Green fromChocolate Fever and John Midas fromThe Chocolate Touch. Both stories were about boys who loved chocolate. The students were asked to record at least five characteristics that made the two boys different, and at least three characteristics that made the boys similar. I collected responses from 14 students.

All of the students noted that the diseases were different, that John's disease was affecting his touch and Henry's disease was more measles-like with the spots. Julie and Richard both noted that while Henry did not know why or how he got his disease, John's disease developed from him eating a magic candy. Bill indicated that both boys had spots. Glenda wrote that John had spots just on his nose, while Henry had spots all over his body. Nine students indicated that both boys went to see a doctor. Richard also noted that John went with his dad and Henry went with his mom. Four students recognized that the doctors were different. Neil went further with his response, indicating that the doctors in both stories were "nut cases.”

Part of the written work forDevil’s Bridge involved completing a Venn diagram to help students explore Ben's feelings for his mother's boyfriend, Barry Lester, and his father, Pop. The two descriptors were Barry and Pop. Neil wrote some obvious differences between the two characters: Barry was living and Pop was dead, Pop won a record and Barry did not, Pop was a great fisher and Barry knew little about fishing, and Pop had a boat while Barry did not. The only similarities that Neil described were that both were "boys" and that they both sat in Pop’s chair. Ben thought that, by sitting in Pop's chair, Barry was trying to replace Pop. Peter only identified that Barry was not a good fisher, and that Pop was a great fisher. Peter provided more information about their similarities: both Pop and Barry liked Ben and his sister Kate, both men were heroes, and both men were smart.

 

Comparing Point of View Using Venn Diagrams

The Case of the Mysterious Tramp presented a different opportunity to use a Venn diagram to help students build the understanding of the story. This was a mystery story. Several different versions of the crime story were told, each told by a different character. Each character had a specific point of view regarding the crime, and John Morgan in particular tried to hide his involvement in the crime by altering his story. The students used a Venn diagram to examine both Mr. Clancy’s and John Morgan’s versions of the crime. There were several similarities in the two versions of the crime, but the differences led the students to understand what really happened.

This was the students' first attempt to discern different points of view. The students were able to identify several facts common to both characters’ stories. Julie and Kevin recorded that in both stories the motor of the truck overheated. Anne wrote that both men acknowledged that Mr. Clancy was hurt. Richard and Neil recorded that in both stories somebody hit and robbed Mr. Clancy. Glenda wrote that both stories related that Mr. Clancy was working on his truck and that he was hit on the head with a piece of pipe. Tania wrote that both men knew each other. Richard and Neil wrote that Mr. Clancy did not see a tramp. Tania wrote, "Mr. Clancy got hit by John Morgan." Although this actually happened, Mr. Clancy would not know this, since he was hit from behind. In this exercise, Tania demonstrated her understanding of what happened in the story, but did not report the events from Clancy's point of view. Tania has shown understanding of the story, but not of point of view. Anne, Richard, Glenda and Neil wrote that Morgan said he saw the tramp come out of the woods with a pipe.

Many students demonstrated their understanding of the solution. Julie knew that John Morgan "stole his walet [sic]." Kevin wrote that John Morgan "stels [sic] mony. [sic]" Glenda wrote that, "John Morgan hit him (Clancy) on the head" and that Mr. Clancy "had a peice [sic] of pipe out of the truck." Tania wrote that John Morgan "robbed Mr. Clancy."

 

Making Personal Connections to Characters Using Venn Diagrams

In the third term, when the reading program focussed on longer chapter book stories, the students prepared Venn diagrams to compare a character to themselves. These activities helped students create personal connections to the characters and events of their stories.

The students who read A Friend Like Zilla completed a Venn diagram comparing themselves to Nobby. Uma noted that she had "nice uncles" while Nobby had a "mean uncle." The other obvious difference between them was their age: Nobby was eight and Uma was ten. Both Uma and Nobby had friends, brown hair, brown eyes and nice aunts. Anne noted that both she and Nobby were girls, kind and nice. Anne wrote that she was nine and small, while Nobby was a teenager and big. In fact, Zilla was the teenager, not Nobby.

Students who read Devil’s Bridge completed a Venn diagram designed to explore their personal connections with Ben. Neil found that he was quite different from Ben. There were two similarities between Ben and him: they both have a bike, and their hair is brown. There were more differences than similarities: Ben loved fishing while Neil "feels so-so about fishing," Ben's mom worries too much while Neil's mon doesn't worry too much, Ben's dad was dead while Neil's dad is alive, and Neil hasn't had poison ivy while Ben had.

Peter also made direct connections between himself and Ben. He wrote that Ben's father was a great fisher, and that Ben had Barry in his life. Peter noted that he was the same as Ben, because both had adults other than their fathers living with them. He noted that he was similar to Ben because they both like fishing, they both have someone living with them they don't like, and both their fathers had died.

 

Summary: Using Venn Diagrams for Reading Response

The Venn diagram exercises support the students’ work with concept maps. Both procedures were introduced to help students create more meaningful connections to the material they read. The procedures appear to help students improve their understanding and reading comprehension by directing their thinking and reflection to identifying relationships among the characters, events and settings of the stories they read.

Venn diagrams as reading response narrow student thinking to consider relationships that exist between two story elements. I chose this procedure to help students improve their understanding in three ways. First, students used Venn diagrams to compare the actions of the characters from their stories. Second, students completed a Venn diagram to help them identify point of view. Third, students used Venn diagrams to help them make personal connections to the characters in their novels.

The students showed little trouble identifying similarities and differences between characters. They recognized major similarities and differences between Rose and Gus from Mountain Rose. They were able to differentiate between the two characters from Chocolate Fever and The Chocolate Touch, and their separate illnesses. The students who completed the Venn diagram for Devil’s Bridge were able to compare Pop and Barry, noting the differences and similarities between these two men in Ben’s life.

Students were presented with a Venn diagram to describe an event from The Case of the Mysterious Tramp from two points of view. Most students were able to identify the elements common to the two sides of the tale. Most students were able to identify the parts of the story that were told differently from contrasting points of view, although some were not able to report the differences from the different viewpoints.

Late in the year, as part of the novel study, some students completed Venn diagrams that compared themselves to characters in their novels. This exercise was designed to help students make personal connections to their reading. Anne and Uma, who read A Friend Like Zilla, were able to identify basic differences. Peter and Neil, who read Devil’s Bridge, made strong personal connections to Ben.

As with the concept mapping exercise, the responses the students wrote to complete the Venn diagrams were open-ended. Each student was given the flexibility to report the connections or relationships that they could generate. Some students were able to generate basic connections, while others developed skill in identifying deeper personal connections. Venn diagrams were useful to me as diagnostic tools. I was able to assess the student responses to determine the degree to which the students attended to the story and how much of the story they understood.

Venn diagrams focused student thinking on the relationships between two story elements. The internal questioning and reflecting that occurred as students considered how two characters were the same or different helped each student build a connection between the two characters. The students used the Venn diagrams to explore and report the relationships between characters.

 

Using Concept Maps and Venn Diagrams Together as Reading Response

Mountain Rose was the only story for which the students responded with both a concept map and a Venn diagram. The students used a Venn diagram to compare Rose and Gus. All of the students were able to report major similarities and differences between these two characters. Interestingly, while only one student identified the brother-sister relationship in the concept map, nine students noted it on the Venn diagram. Uma, Holly, Olivia, Fay and Margot noted that both Rose and Gus shared the same birthmark, a connection none recorded on the concept map. David, Julie, Bill and Scott noted that Rose and Gus were family. This information was not included on their concept maps. David was the only student to record this fact on both the concept map and the Venn diagram.

If the concept map was examined in isolation, it could be concluded that these nine students did not make this important connection while reading the story and that they missed this key piece of understanding. The Venn diagram provided a second opportunity for these nine students to report what they knew about the story. The Venn diagram activity served as a cross-reference for the concept map connections.

 

 

Progressing Toward Creating Meaningful Learning

Producing concept maps and Venn diagrams helped my students improve their information-processing skills by directing them to reflect upon the relationships that exist among the characters and events. These exercises focused student attention on establishing relationships, rather than on identifying a sequence of events. This illustrates the power of these procedures. Rather than considering a story in a linear fashion as a sequence of events, students who create relationships within the story start to recognize the complexity of the story.

As my students began to work with concept maps and Venn diagrams in September, they started to create connections and visualize the relationships within the stories. Characters were being connected to their actions, thoughts and feelings. For example, with Tiger Skin Rug, all students recognized how the characters were connected to each other. The tiger became part of the Rajah’s life. The tiger worked to protect the Rajah by scaring away the thieves. The tiger was afraid he might lose his preferred life, while the Rajah was afraid of the thieves. The story became more than a series of events as the characters were linked to each other in more complex ways. Similarly, students were able to examine the detailed similarities and differences between Rose and Gus with a Venn diagram.

The exercise with Dinner at Alberta’s helped me to understand that concept maps must contain terms that relate to story themes or concepts for my Grade 4 students. Whether because of their age or because this type of response work was new to them, most students had difficulty connecting characters to story events without the presence of terms that identified plot themes. Yet two students were able to create connections to the story. David and Neil used information from the story to explain the connections they made among Arthur, Emma and Mrs. Crocodile.

Creating connections using concept maps required the students to focus their thinking on specific characters, events, times, and places. When completing the map, the students had to reflect on each pair of terms and find a way to link them together. With the exercise for Titanic Crossing, for example, the students had to consider how and when Albert displayed bravery. The concept map gave them the criteria with which to assess each episode involving Albert. Was he brave at the beginning of the story? Was he brave at the end of the story? When in the story did Albert display bravery? What did Albert do to demonstrate his bravery?

Creating connections with Venn diagrams required students to practice similar skills. The exercise focussed their thinking to consider the relationship between two characters or, as with the novel studies, the relationship between themselves and a character. The students who read A Friend Like Zilla focussed their attention on the ways Nobby was similar to them. Those who read Devil’s Bridge considered their connections to Ben. The internal questioning as students completed these activities focussed their thinking.

At the same time, these activities allowed for open-ended responses. Students were able to record any rationale they could find to support each connection. The explanations the students gave to support the connections they created are the key to assessing the improvement of comprehension. As in the case of Dinner at Alberta’s, most students explained their connections in a superficial way: Arthur was Mrs. Crocodile’s son. Emma was Arthur’s sister. These basic explanations do not link the characters to the story but rather to each other. David and Neil created links that were supported with evidence from the story, indicating with their explanations that Emma and Mrs. Crocodile were helping Arthur with his manners. These students made (or at least reported) links that were more relevant and contextual to the story. David and Neil demonstrated a deeper connection to the story than did the other students. They also demonstrated deeper reflection about the connections they were making. The explanations the students provide to support the connections they create are indicators of the development of meaningful learning.

 

CHAPTER 6

USING TOPIC AND TASK QUESTIONS IN MATHEMATICS

TO DEVELOP REFLECTIVE THINKING

I developed a general strategy with the class that required the students to be specific in telling me what they didn't understand. Many usually approached me for help, stating, “I don't get this,” or “I don't know what to do.” I found myself automatically going over the entire lesson, paraphrasing and using different examples. I soon realized that I could not do this individually for each student in one period. I then began to ask them to tell me specifically which part of the problem they did not understand. Initially, they would immediately reply, "All of it" or "the whole thing." My instruction to them was to read the question or problem over again carefully and find the part that was giving them trouble. They were to come back to me when they could tell me exactly their problem so I would be better able to help them better. Interestingly, many students did not reappear with the requested specific question. What was happening, I discovered, was that upon careful rereading of the problem, the students were generally able to find a solution to the problem that was troubling them. This gave me more time to help the students who genuinely needed additional help.

 

Topic and Task Questions in Mathematics

The topic and task procedure was developed to help students reflect on and become more aware of their own learning. The procedure requires students to reflect on the day’s lessons and their work, as well as the previous day’s lesson and work. The goal is to create a link between the current and the previous lesson. To begin working with this procedure, the students started a math journal that we called the "Thinking Linking Log,” the purpose of which was to record responses to questions that were designed to help the students make connections between mathematics lessons. This journal was used exclusively with mathematics. Selected samples of student entries in the Thinking Linking Log are included in Appendix D.

The questions for the procedure were chosen to help students create three different types of connections. The first set of questions helped students make personal connections to each day’s lessons and activities. Students were asked to reflect upon the main purpose of the day’s lesson, what they had to do, and why they thought they had to do it. The second set of questions helped students create connections between the day’s lesson and previous lessons. The questions were designed to help students recognize the continuity between adjacent lessons and ideas, and recognize the relationship between the lessons. The third set of questions helped the students recognize and develop an understanding of how the current topic related to other topics in mathematics.

The Thinking Linking Logs were introduced in March 1998 and were used with three separate mathematics topics: division, fractions and quadrilaterals. The questions used for each of the three types of connections intended are shown in Table 10.

Student responses were assigned a value from one to four, based on the quality of the link established by the student and of the reasoning the student used to create the link. Responses that used complex ideas and complete explanations scored higher than those that used simple ideas or incomplete explanations. All of the student responses were entered into a computer database and linked to a description of the lesson of the day, the date, and the specific question for which the student was to write a response.

These records were sorted by the lesson topic, the question asked and the response rank for that particular question.

 

Table 10

Topic and Task Questions Used with the Thinking Linking Log

Using Topic and Task Questions

to Help Students Make Personal Connections to Lessons

The construction of knowledge requires learners to connect new information and experiences to the knowledge they already have. I chose the topic and task questions procedure from PEEL because it provided a framework for my practice to help my students generate awareness of the connections they made to their math lessons and activities. The questions directed the students to reflect on their work and its relevance to them.

The procedure was first introduced during the division unit in March. The students were briefed on how to think about responding to the questions, since they were not accustomed to replying to this type of question. Their skill in thinking and reflecting developed as they continued to write in their logs during our work with fractions and quadrilaterals.

 

Dividing Large Numbers

Grade 4 is the year that students are introduced to dividing larger, two- and three-digit numbers by a single-digit divisor. In the past, I have noted that children generally have more difficulty learning the division algorithm than learning how to add, subtract or multiply. One strategy for helping students construct meaningful understanding of division is to encourage them to think and reflect on what they are doing and why they are doing it. These topic and task questions were used throughout our lessons on division.

The first entry in the log was after an introductory lesson about remainders. The activities involved sorting various numbers of photographs into groups of either eight or six so they could be placed into photo album pages. The students needed to determine how many photographs were left over after all pages have been filled, and how many pages were required to include all the photos. At the end of the lesson students were asked to identify the main point of the lesson. I had stated directly five times that this lesson was meant to help the students consider the meaning of the remainder.

Eighteen responses were collected. Seven students replied that the main point of the lesson was to sort photos. Five students stated that the lesson was about dividing by eight and by six. Three students reported that the main point was to work with the hundreds chart to help with division. One student claimed the lesson was to "make us smart" and another stated that the purpose was to write a sentence for the answer. One student reported that what to do with a remainder was the main purpose of the lesson. This was the first time the logs were used, and I was hopeful that responses would improve with time.

The second lesson involved estimating when dividing numbers less than 100. There is a large emphasis on estimating in the Grade 4 program, and we had much practice estimating sums, differences, products and measurements prior to this unit. The students used Valentine materials to estimate the number of craft items that could be produced from a known quantity of raw materials. The students wrote to explain the main thing they had to do and why they thought they had to do it. The student log entries reflect their connection with their prior experience with estimating. Ten of the 19 students responded that they were estimating. Six of these 10 stated that the lesson would help them learn how to estimate with division. Two stated that the work was for practice. One student said that estimating could be useful.

Long division, or dividing two- and three-digit numbers, was introduced as repeated subtraction. The demonstration activity involved distributing books either four at a time, or by multiples of four. Each time a group of books was distributed, the students subtracted that number of books from the total, and made note of which multiple of four we used to take away from the total. An integral part of this lesson was the introduction of the use of the "long division" symbol. At the end of the exercise, the students were asked to write a response to what they thought was the main point of the lesson. Julie, Olivia, Ian and Glenda all claimed the lesson was about putting books away. Holly, Margot, Neil, Quentin, and Tania stated that the lesson was about dividing. Holly, Margot and Neil explained that the lesson was about dividing big numbers. Evan indicated that this lesson presented a new way to divide.

The next lesson also involved dividing numbers in the hundreds, but this lesson stressed using the traditional long division approach. After this lesson students again reflected on the main thing they had to do in the lesson. Olivia, Fay, Margot and Tania all related they were doing long division with numbers in the hundreds. Tania also added that this work was very hard for her.

The final division entry in the Thinking Linking Log culminated a week of review after the March Break. We had practised long division of numbers in the tens and the hundreds using three different methods. We also examined remainders and what a remainder might mean in specific problems. This entry was the precursor to the summative test for this unit on division. The students were asked to explain how to divide as if they were telling someone who knew nothing about division.

Fourteen students responded to this task. Eight students indicated that division was about sharing or putting things into equal groups. Two students did not actually explain division, but stated that they would explain it step-by-step. They did not identify the steps they would use. One student reported that division is the opposite of multiplication and another student used the long division symbol to help explain his thinking.

 

Fractions and Decimals

The next topic in which the linking logs were used in mathematics was fractions and decimals, in April. The first lesson involved using four parallel number lines, all divided into hundredths. We labelled the first number line in tenths using fractions, the second number line in tenths using decimals, the third number line in fourths using fractions, and the fourth number line in fourths using decimals. The objective of the lesson was to demonstrate equivalence between specific fraction and decimal numbers (1/10 = 0.1 = 0.10). At the end of the period, students were asked to respond to the question, "What was the main point of today's lesson?"

Sixteen students responded to this question. All indicated that the lesson was about fractions and decimals. Neil, Holly and Quentin linked this fractions and decimals exercise to working with number lines. Glenda stated that the lesson helped her connect fractions and decimals together. Olivia stated that the lesson helped her compare fractions to decimals. David, Ian and Margot reported that the lesson helped them to see how fractions and decimals can mean the same amount.

During the second lesson, the students compared two fractions with different denominators to find which is larger. The exercise compared a specific number of pieces of one pizza with a specific number of pieces of another pizza. Each pizza was cut into a different number of pieces (e.g.., Which is more: three pieces of a pizza cut into five pieces, or five pieces of a pizza cut into eight pieces?)

After the lesson, the students were asked to describe the main thing they had to do, and why they thought they had to do it. Ian wrote that his work was about who got more pieces of pizza, and that this work would help him learn to compare fractions and decimals. Evan and Peter both identified the concept of comparing the number of pieces of pizza, but the purpose of this work to Peter was to help him pass the grade. Evan's purpose was to become smarter so he could be able to do fractions and decimals in high school.

One of the lessons was designed for students to order and compare decimals. Much of this work involved comparing distances measured in metres to determine which distance was either the farthest or the shortest. Students were invited to respond to the same questions that were posed when they were comparing fractions (pizza activity): What was the main thing you had to do today, and why do you think you had to do it?

Seven students responded. All indicated that the activity was about comparing decimals. David, Uma, Margot and Olivia stated that comparing decimals was the focus of the work. Bill stated that he was finding out who got the farthest distance. Neil indicated that he was comparing numbers. Quentin declared that he was measuring people kicking Kleenexes, which was one of the activities of this exercise. When asked why they thought they had to do this work, David and Bill stated that they were trying to see who got the highest number or the "most of decimals." Quentin's purpose was to learn how to measure decimals. Neil and Olivia wrote that they were learning to compare decimals, or working to get better at comparing decimals. Scott wrote that the main thing he was doing during this activity was picking the highest decimal. His purpose in doing this was learning to do it in his head. Both Caroline and Tania indicated that the main work they were doing was measuring so they could get better at measuring.

When the class used the pattern blocks (triangles, trapezoids and rhombuses) to cover hexagons to demonstrate improper fractions and mixed numbers, they were again asked to identify the main point of the activity. Nineteen responses were collected, and 16 students indicated that the main point was working with pattern blocks to form either improper fractions or mixed numbers. Uma wrote that the focus was how many ways one hexagon could be made with the other pattern blocks. Holly wrote that the main point was to determine how many ways one whole could be made using 1/6, ½ and 1/3.

 

Quadrilaterals

This was the last unit of the year. Begun in May, it followed our work with fractions and decimals. The first quadrilateral lesson had the students create four-sided shapes using elastics and geoboards to meet specific criteria: two sides longer that the other two sides; all sides the same length; all sides different lengths; all square corners; two square corners; no square corners; two sides parallel; and no sides parallel. When the figures had been created on the geoboards, the students drew the shapes on dot paper and labelled each shape with its name: square, rectangle, parallelogram, rhombus, irregular quadrilateral and kite. The thinking-linking question for this exercise was "What the main point of today's lesson?"

Fifteen students wrote responses to this question, and all of them indicated that the lesson had to do with making shapes. Nine students either used the term "quadrilateral” or indicated the shapes were four-sided in their responses.

 

Summary

I chose to introduce topic and task questions into my practice to help make the students more aware of the connections they were making during mathematics lessons and activities. The students had not had much experience thinking about their learning. Initially, they had difficulty identifying the main ideas or concepts in our math lessons and articulating their ideas and thoughts clearly. Throughout the division unit students expressed their ideas of what they were to do and what the lessons were about in general terms. The connections they reported were also general, noting that the lessons were simply about division. Some students recorded even more generally that they thought the lessons and activities were to make them smart.

With practice and experience, the students responded with greater attention to specific detail. During the work with fractions and decimals, students wrote with more confidence. Entries included notations that the work was to help them compare fractions to decimals and to understand the meaning of a decimal number and a fraction. Mathematical language also began to appear in several of their responses. Students used terms such as “improper fraction” and “mixed number” in their log writing.

As we began to study fractions and decimals, the students demonstrated improved ability in thinking and writing reflectively. When asked to comment on the main idea behind the number line activity, 8 of 16 students related their responses directly to the activity of using number lines to identify decimal and fraction equivalents. One student claimed that the lesson had helped her compare fractions and decimals, and three students reported that the lesson helped them see how fractions and decimals can mean the same amount. These responses align directly with the objective and the expectation of the lessons. Compared to the earliest responses in the division unit, these students had started to develop their critical reflective thinking skills, and were making better connections to their lessons. The students' ability to articulate their ideas about what they were learning was also improving. Some students were starting to think about the context of the lesson as well as its content.

By modifying my practice to include the opportunity for the students to write about what they were doing and learning, the students improved their ability to communicate their thoughts and reflections about the meaning of mathematics. Fay demonstrated improvement in both her information-processing ability and her awareness of her learning. Math had been a particularly difficult subject for Fay, and she needed much practice with manipulative materials before she could begin to grasp mathematical concepts. She had particular difficulty understanding fractions, especially equivalent fractions and how whole numbers could be represented by fractions. Fay worked with several activities using pattern block shapes to help her visualize fraction equivalence. With one activity, she was to cover the hexagon blocks with trapezoid, rhombus and triangle blocks to show that one whole could also be written as a fraction: 2/2, 3/3 or 6/6. With another activity Fay explored the relationships between one trapezoid (½ a hexagon) and three triangles (3/6 of a hexagon) to help her understand equivalence between different fractions that mean the same amount. This work was later extended to help her improve her understanding of improper fractions and mixed numbers. Two and three hexagon blocks were used in these activities, in which she would cover two hexagon blocks with trapezoids, rhombuses or triangles to indicate that two wholes could also be written as 4/2, 6/3, and 12/6. In a similar way, Fay modelled mixed numbers with the pattern blocks to show that three trapezoid blocks could also mean one and one-half hexagon blocks.

Fay wrote daily to explain what she was doing and why she was doing it. In her first entry she wrote, “the main thing I had to do was fractions with blocks. I did this so that I could learn how to work with fractions and pattern blocks.” After completing an activity that helped her explore the relationship between fractions and whole numbers larger than one, Fay wrote, “Today we are learning about fractions, with shapes like 12/6, 2 ½, 3 1/3." In this entry Fay wrote both an improper fraction and two mixed numbers, the exact concepts with which she had been working. This was considerably more specific that her earlier entry about working with fractions and pattern blocks.

 

Using Topic and Task Questions

to Help Students Make Connections Between Lessons

One objective in teaching division is to introduce the students to different methods to “do long division.” I used topic and task questions in my practice to help students make conscious connections between the day’s work and the previous day’s work. By recalling last day’s work, students would be able to identify the similarities and differences between different methods and approaches to “long division.” As the students progressed through the unit, they regularly wrote in their logs, building connections between previous knowledge and the current work.

The unit on dividing larger numbers began with the demonstration of division of a two-digit number using books, described earlier. The next lesson also focussed on dividing two-digit numbers. These lessons introduced the students to the same concept (long division) from two different approaches. This provided an opportunity to direct the students to think about how each lesson was linked to the previous lesson. Fifteen of the 17 responses noted only that both lessons involved division. David’s entry, while still quite general, revealed that David recognized that both lessons were about the same kind of dividing. Bill, Holly, Fay Margot and Neil wrote more specifically that both lessons involved dividing larger numbers. Uma and Glenda recognized that both lessons used the long division symbol. Similarly, Tania, Anne, and Ian wrote that both days they did long division.

The next two lessons focussed on developing the traditional approach to long division using the closest multiple of the divisor to divide three-digit numbers. After the first lesson, students wrote about what they thought the lesson had to do with last day's lesson. The previous day's lesson involved dividing two-digit numbers using the repeated subtraction method. Eleven of 13 students recognized the general link that both lessons were about division. David, Uma, Olivia, Ian and Tania wrote more specifically that both lessons were about long division, and Fay and Margot wrote that both lessons were about dividing with hundreds. Similarly, Holly wrote that both lessons were about dividing with big numbers.

 

Using Topic and Task Questions

to Help Students Make Connections Between Topics

The last mathematics unit featured quadrilaterals and was begun in May. The students had been using their logs since March and were familiar with the type of questions they were being asked to reflect upon. One specific activity in this unit required the students to recall and use symmetry, a topic that was taught and practiced during the fall term as part of a unit about transformational geometry (slides, flips and turns). This presented me with an opportunity to direct the students’ thinking toward making connections between topics.

The first connection to be explored was between quadrilaterals and symmetry. In one activity, the students followed oral directions to create an origami frog from a square piece of paper. When the task was complete, the students were asked to reflect on their work and respond to two questions. The first question was about symmetry. I was interested to see which students could recognize connections to previous work with symmetry and our work with quadrilaterals. Seventeen students wrote to explain how the frog showed symmetry. Fourteen students identified at least one line of symmetry on the frog, and explained that the frog was the same on both sides of the line of symmetry. Eight supported their written response with a diagram to show the line of symmetry.

The second question asked the students to describe how this activity could be linked to our work with quadrilaterals. The same 17 students wrote responses to this second question. Seven students connected the shape of the starting paper (a square) to quadrilaterals. Five students identified quadrilateral shapes in the frog. Peter, Margot and Ian wrote that both the frog and quadrilaterals have lines of symmetry. Evan and Fay connected both lessons by the amount of difficulty they had with both topics. Fay wrote that both the frog and the quadrilateral work were easy, while Evan claimed that making the frog was hard, just like our "other math."

Measuring the perimeter and the area of various quadrilaterals was the final topic of this unit. The first activity involved the students measuring the perimeter of quadrilateral shapes using centimetre rulers. When the students had completed the perimeter activities, they responded in their logs to the question, "What is perimeter?" Nineteen responses were collected, with 16 students noting that perimeter is a measure of the distance around an object. The three remaining students wrote inaccurate descriptions that indicated incomplete understanding of perimeter. Bill wrote that "perimeter is something big or small." Holly replied that "perimeter is something you can measure with." Tania did not complete her sentence: "perimeter is a."

The second group of activities focussed on measuring the area of quadrilaterals. The lesson involved measuring the area of quadrilateral shapes using square centimetres and centimetre graph paper, as well as geoboards and elastics. For the first exercise, the students used centimetre grid paper and counted the number of squares found inside each quadrilateral shape. They paid special attention to counting and estimating the number of part-squares and half-squares so they could calculate an accurate measure of the number of squares in each shape. The following day, the students were introduced to the concept of the square centimetre. On this day, the areas of shapes were measured using squares that were exactly one square centimetre. Final activities were designed for students to recognize relationships between area and perimeter. The students created quadrilateral shapes that conformed to specific perimeter and area criteria. For example, students were challenged to create three different rectangles so that each rectangle had an area of 12 square centimetres. Similarly, students endeavoured to create pairs of quadrilaterals that either shared the same perimeter but had different areas, or shared the same area but had different perimeters.

At the end of the perimeter and area activities, the students wrote responses to two related questions. My objective for these questions was to determine the degree to which students were able to make connections between perimeter and area. The first question asked the students, "How are perimeter and area different?" Eighteen students responded to this question, and 8 were able to make a solid connection between the perimeter and the area of an object. Neil wrote, "Area and perimeter are different because perimeter is the space around an object while area is the space an object takes up." Ian wrote, "Area and perimeter are similar because perimeter is outside area is inside. If there is perimeter there is area. Perimeter is outside area inside." Glenda wrote, "They are different because area is the inside and perimeter is the outside." Margot wrote, "They are different because area is space and perimeter is the distance around."

Ten students indicated poor understanding of the connection between the two measurements. Caroline wrote, "They are different because perimeter means square, area means round." Tania wrote, "Perimeter has to do with shapes and area doesn't." Uma, Bill, Holly, Olivia, Caroline, Evan, Peter and Tania recognized generally that area and perimeter are both related to measurement. Tania wrote that “Area and perimeter are similar because both you have to measure." Caroline wrote, "Area and perimeter are the same because they both mean length." These students were better able to describe how perimeter and area are similar than they were in describing how they were different. By asking both questions, I was able to gain a more complete picture of the level of understanding of these students.

Neil and Ian made the strongest connections between the two concepts. Neil wrote, "Area and perimeter are similar because area is the space inside the perimeter." Ian wrote, "Area and perimeter are similar because perimeter is outside area is inside. If there is perimeter there is area. Perimeter is outside, area inside." Kevin was also able to make a good connection between the two measurements: "Perimeter is the outside of a shape. The other one is the inside."

 

Summary

The students had been recording their reflections in their logs for three months when they began to think about identifying the connections between topics. When asked to relate the nature of quadrilaterals to symmetry, the students accurately remembered the concept of symmetry and successfully applied it to the current activity, the origami frog. They were able to make a successful connection between these two topics.

After working with perimeter calculations, 16 of 19 students were able to write an accurate explanation of perimeter. After working through several activities that explored the relationship between area and perimeter, the students responded to two questions: How are area and perimeter different? and How are area and perimeter similar? Eight students were able to provide a clear understanding of the two concepts in their logs. More interesting, though, are the responses from the students who did not have a clear idea. In previous entries for previous units and lessons, students who did not fully understand a topic wrote generally or vaguely in their journals. With the perimeter and area questions, however, the students who did not completely understand the relationship wrote clear descriptions of how they understood these concepts. Caroline wrote, "They are different because perimeter means square and area means round." Tania wrote, "Perimeter has to do with shapes, area doesn't."

While both students show an inaccurate understanding of the relationship between perimeter and area, they show increased ability to explain their thinking in writing. I was more interested in these students' ability to articulate their understanding and their thinking about perimeter and area than in whether they had the correct idea or not. The students who wrote these ideas clearly were students who struggled with mathematics all year. Even the students who did not grasp the concepts clearly were able to explain their thinking in a clearer fashion than they could during the division lessons.

 

This chapter and the previous three chapters describe three different ways in which I extended my teaching practices with a view to improving the quality of my students’ learning. Each of these teaching procedures is taken from the PEEL project. By introducing significant new elements into the ways I teach language arts and mathematics, I created significant challenges for my students and for myself. It was one thing to read a range of PEEL procedures and to select those that seemed most appropriate to my Grade 4 classroom. It was quite another to experience these personally and to be challenged myself as I also sought to offer my students greater challenge in their classroom learning experiences. I had to make PEEL my own, and so did the children.

 

 

CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS

Much literature on changing teacher practice suggests that meaningful change can occur and endure only when it develops within the classroom situation of individual teachers. Each teacher recognizes specific and unique problems in the classroom, arising from the changing needs of students or curriculum. The identification of a need as a result of the teacher’s reflection on experiences with students in the classroom is the beginning of the learning cycle identified in both reflective practice and action research models. Teachers begin to re-conceptualize their practice as they work to create solutions to an identified problem. Solutions are tested in the classroom, and those that the teacher deems successful are incorporated into practice. The method used in this study is based on this reflective practice/action research learning cycle recognized in the literature. This study was initiated to examine my teaching practice as I implemented new procedures that seemed likely to help my students improve the quality of their learning. In reflecting on my own practice and my students’ behaviours and work, I felt that I needed to change the way I taught to help the students improve their information-processing skills. Further analysis, coupled with an introduction to PEEL during one of my courses at Queen’s University, helped me identify two poor learning tendencies that I used as targets for this study. Students paid only superficial attention to the work, with little demonstrated ability to generate personal meaning from their work. The students also demonstrated a lack of ability to reflect on and make conscious connections between lessons and topics.

The objective of this study, therefore, is to document and examine the impact of introducing into my practice teaching procedures that would help students develop good learning behaviours. My initial premise was that these procedures would help students improve their abilities to link new learning to their experience and knowledge and to reflect on and identify the connections that exist between lessons, concepts and topics in their studies. Implementing changes in my practice to improve the students’ ability in these two areas could lead to the development of more meaningful learning.

The improvement I was seeking in my practice mirrors the improvement I was seeking in the learning of my students. Its foundation lies in constructivist learning theory, in which new knowledge is created by the learner as new information or experience is incorporated into the learner’s existing knowledge and understandings. I was developing a new knowledge of teaching as I brought the PEEL philosophy and procedures into my practice. Simultaneously my students were developing new knowledge on two levels. They were increasing their awareness of their own learning, while also increasing their comprehension of what they were reading and improving their understanding of the content of their lessons.

I chose to concentrate my study on two core subjects of the elementary curriculum, language arts and mathematics. I selected three PEEL procedures that I thought I could incorporate into my practice in these subjects, that also addressed the poor learning tendencies I had identified. Concept maps were used in the reading program to help the students recognize connections between characters, story events and settings, in both short stories and chapter books. Venn diagrams were introduced for students to make connections between the similarities and differences of story characters, not only from each story, but also from different stories. Students also used Venn diagrams to create personal connections between themselves and story characters. In mathematics, students kept ajournal to record their reflections about the links they could make between their own knowledge and new lessons, as well as connections between lessons and concepts. Samples of student work were collected throughout the year to assess the success of each procedure in developing good learning behaviours and improve student learning.

 

PEEL Procedures as Agents of Change

PEEL began in Australia in 1985 with a group of teachers seeking solutions that would improve student learning. These teachers identified barriers to meaningful learning and then developed improvements to their practice to not only remove those barriers but to enhance effective learning. The students involved in the original PEEL project increased skill in learning and their demonstration of good learning behaviours led to corresponding improvements in their learning.

This study indicates that when I introduced three specific PEEL procedures into my practice, my students were led to develop good learning behaviours that indicate more effective understanding and deeper connections with lessons and concepts. Learning occurs when the student can reconcile new knowledge with existing knowledge. The PEEL procedures encouraged students to make these connections with lessons and reading material in a conscious way, contributing to the internal reconstruction of knowledge.

The goal of my practice is to develop independent learners who are able to derive meaning from all facets of the world around them. By adding the PEEL procedures to my practice, I am better able to teach my students how to make the connections necessary for them to construct meaning. The PEEL procedures began as innovations to my practice. Through practising the procedures regularly with lesson activities, the students’ learning began to improve. The noticeable improvement in student learning was the evidence that helped maintain the momentum in changing my practice. Over the course of this study, I have come to believe that it is possible to change students’ ability to learn by helping them become more aware of making the connections that are necessary to improve learning. PEEL principles and procedures acted as effective agents to change not only my practice but also the quality of my students’ learning.

 

PEEL Procedures as Improvements to my Practice

This study is about how I changed my practice to improve the quality of my students’ learning. There is much debate (Guskey, 1986; Richardson, 1990) about whether changing teacher practice involves changing teachers’ beliefs before introducing new practices, or changing teachers’ practices first, with the hope that the new practices will change teachers’ beliefs. This is a fundamental question, and one that I have considered during this study. I believe that teachers trying something new ask such basic questions as “Will it work?” or “Will this change produce the desired result?” The PEEL research indicates that the procedures are effective in improving specific aspects of student learning. But do teachers trust the research? The literature (Guskey, 1986; Richardson, 1990) suggests that they do not, or at least they do not consider the research valid for their situation or classroom.

This is the dilemma of heart and head. The head holds the knowledge of the procedures and the rationale, based on research, that justifies using them. The heart holds the belief that the procedures will produce the desired change in the students. I began this study by selecting and incorporating specific PEEL procedures into my practice in response to the needs I perceived in my students. The evidence provided by PEEL teachers who had already worked with the procedures was the basis of my faith that the procedures would lead my students to improved learning. As the school year progressed and as I began to see student responses that suggested improved connections to learning, my faith evolved as my belief in PEEL and the procedures strengthened.

One fundamental aspect of the PEEL research is that students everywhere exhibit some of the identified poor learning tendencies at some time during their school careers. This was one connection I was able to make between my situation and the PEEL research. I recognized that my students exhibited some poor learning tendencies. This common starting point in problem identification made it easier for me to consider applying PEEL to my practice. I was interested to see if the PEEL procedures would improve the quality of learning of my students.

I began this study by changing my practice to include just these three PEEL procedures. It was faith in the research, together with my curiosity about the results, that kept me going during the school year, especially at the outset. The early Venn diagram work with place value showed me that this procedure was effective in focusing student attention. It also allowed me to identify students who needed extra help with number sense concepts. Venn diagrams were perhaps the easiest of the three procedures to integrate into my practice, especially in mathematics. The place value activity became a “bell work” exercise, as I would have the Venn diagram for the day on everyone’s desk as they entered the room at nine o’clock.

Incorporating Venn diagrams into reading response was also not a difficult task. Choosing two characters, either from the same story or from different stories, and letting the students complete the diagram was relatively simple. Assessing these activities was also fairly straightforward, using the characteristics the students chose to include as a measure of their understanding. When the students started to include differences the characters had in their beliefs, or actions or motivations, that indicated to me that they were focussing on the characters and thinking in a deeper way.

Using Venn diagrams in reading was not as challenging as developing the concept maps. I devised a form with six boxes aligned in two columns of three. The students were good at connecting pairs of boxes that were across the column from each other and at connecting boxes that were either immediately above or below each other. There were characteristically fewer connections made between the terms in the boxes at the top and bottom of each column and in opposite corners. The students seemed to be locked visually by the placement of the boxes and by the placement of the terms in those boxes. It perhaps would have been better to give the students a list of the terms and have them create the map on a piece of plain paper, placing the terms in the map in places of their own choosing rather than using a pre-designed form that seemed to indicate some hierarchical relationship among the terms.

Another challenge for me was to devise the terms for the maps. I discovered that I must include terms that refer to the story themes or characteristics of the characters. When I used only characters’ names in the map forDinner at Alberta’s, the students responded with the family relationships among the characters and did not relate any information about how the characters interacted in the events of the story. For concept maps to work effectively with reading response, the teacher must be very careful in deciding which terms to include.

To summarize my personal sense of the beliefs vs. practices debate in relation to teacher change, I believe this study shows that it has been a combination of beliefs and practices in interaction with each other that offered me both specific procedures and supporting beliefs. I changed my actions as a teacher and I have documented here the results of those changed actions. Yet only by reflecting constantly on my own beliefs in relation to old and new practices could I sustain the changes and link them to other relevant elements of my teaching.

 

PEEL Procedures and the Development of Meaningful Learning

This study documents the development of specific good learning behaviours by my Grade 4 students. In reading, this study documents my students' development of understanding of what they have read. In addition, I have recorded the development of my students' skill in reporting the connections they could make between characters, story events and setting. In mathematics, I have demonstrated the development of my students' skill in reflecting on their personal connections to the lessons, as well as theirskill in reflecting on connections they could make between lessons and topics.

The data show that my Grade 4 students improved their information-processing skills to demonstrate meaningful learning in language arts and mathematics. These skills developed as a result of incorporating new teaching procedures into my practice. While the students’ initial responses to each procedure were vague and the connections they reported were general, the quality of student responses to lessons during the third term showed steady improvement. Responses were more articulate and connections were more definite. The study shows that, over time and with practice, not only did my skill and ability to embed the procedures into lessons and activities improve, but also the connections students were making between lessons and concepts became more articulate and specific to the topic or concept being taught. The data suggest that the students' thinking was redirected and refocused as they practiced the selected procedures throughout the school year and that the quality of the students' responses improved with practice.

This study demonstrates that by incorporating PEEL procedures into teaching practice, students can be taught to improve their ability to learn by exhibiting the good learning behaviours that encourage reflection upon learning andmaking personal connections. This study is the beginning of an awareness that as teachers we hold the key to developing and implementing procedures that can transform our practice. Before change can occur, teachers must recognize poor learning tendencies among their students and be willing to teach those students the good learning behaviours necessary to increase their success with learning.

 

 

Next Steps

The learning cycle of reflective practice for this project is complete. A problematic experience was identified; a solution was sought, found, and incorporated into my practice; and data were gathered to assess the effects of the new practices. In the process of analysing the findings, my practice has changed. I have become more of a risk-taker in exploring and attempting new procedures with my students. I have adjusted my teaching so that the students have more time to use the procedures to construct knowledge, and less time listening to directed lessons.

The PEEL procedures incorporated into my practice as a result of this study have been evaluated. Upon reflection on this project and its implications, one must consider the next steps. What will the reflective practitioner take from this study? What will the next steps be? The answer, in part, lies with PEEL.

PEEL began as a collaborative project in response to teachers' concerns about students' poor learning tendencies. Through collaborative work, PEEL teachers were able to identify, analyze and interpret their concerns about student learning. In addition, they were able to confer and share their successes and failures as they worked to incorporate into their practice procedures that would help students demonstrate behaviours necessary for good learning to occur. One of the strengths of PEEL is its collaborative component, which is not present in this study. Working independently to incorporate PEEL procedures into my practice was challenging. One course of future action and research would be to introduce PEEL to at least one of my colleagues at school so that collaboration between teachers can occur.

In this study I chose to address two poor learning tendencies in my classroom. I chose three PEEL procedures, and worked with them until they became a natural part of my practice. These procedures have become part of my regular daily routine of teaching. To build on the success of this project, it is sensible and appropriate to continue this work. In the next years, I shall continue to improve my practice by choosing other poor learning tendencies and integrating new learning procedures to continue to enhance the effective learning of my students.

 

 

REFERENCES

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Appendix A

Using Venn Diagrams to Build Understanding of Place Value in Mathematics

 

 

 

 

Using the Rating Rubric to Score Selected Venn Diagrams

 

 

Appendix A

Using Venn Diagrams to Build Understanding of Place Value in Mathematics

Selected Samples of Caroline’s Work

 

Appendix A

Using Venn Diagrams to Build Understanding of Place Value in Mathematics

Selected Samples of Ian’s Work

 

Appendix A

Using Venn Diagrams to Build Understanding of Place Value in Mathematics

Selected Samples of Olivia’s Work

 

Appendix A

Using Venn Diagrams to Build Understanding of Place Value in Mathematics

Selected Samples of Tania’s Work

Appendix B

Using Concept Maps as Reading Response to Build Understanding

 

 

 

 

Using the Rating Rubric to Score Selected Concept Maps

and Venn Diagrams as Reading Response

 

 

 

Appendix B

Using Concept Maps as Reading Response to Build Understanding

Selected Samples of Anne’s Work

 

Appendix B

Using Concept Maps as Reading Response to Build Understanding

Selected Samples of David’s Work

 

Appendix B

Using Concept Maps as Reading Response to Build Understanding

Selected Samples of Neil’s Work

 

Appendix B

Using Concept Maps as Reading Response to Build Understanding

Selected Samples of Olivia’s Work

 

Appendix B

Using Concept Maps as Reading Response to Build Understanding

Selected Samples of Peter’s Work

 

 

Appendix B

Using Concept Maps as Reading Response to Build Understanding

Selected Samples of Caroline’s Work

Appendix C

Using Venn Diagrams as Reading Response to Build Understanding

 

 

Using the Rating Rubric to Score Selected Venn Diagrams

 

 

 

Appendix C

Using Venn Diagrams as Reading Response to Build Understanding

Selected Samples of Evan’s Work

 

Appendix C

Using Venn Diagrams as Reading Response to Build Understanding

Selected Samples of Kevin’s Work

 

Appendix C

Using Venn Diagrams as Reading Response to Build Understanding

Selected Samples of Neil’s Work

 

Appendix C

Using Venn Diagrams as Reading Response to Build Understanding

Selected Samples of Peter’s Work

 

Appendix C

Using Venn Diagrams as Reading Response to Build Understanding

Selected Samples of Caroline’s Work

Appendix D

Topic and Task Questions in Mathematics

 

“Thinking Linking Logs”

 

Using the Rating Rubric to Score Selected Log Entries

 

Appendix D

Topic and Task Questions in Mathematics

Selected Samples of Caroline’s

“Thinking Linking Log”

 

 

 

Appendix D

Topic and Task Questions in Mathematics

Selected Samples of Fay’s

“Thinking Linking Log”

 

Appendix D

Topic and Task Questions in Mathematics

Selected Samples of Neil’s

“Thinking Linking Log”

 

Appendix D

Topic and Task Questions in Mathematics

Selected Samples of Ian’s

“Thinking Linking Log”

 

Vita

 

 

Name David Robert Turner

 

Place and Year of Birth Toronto, Ontario, 1957

 

 

EDUCATION

 

1971-1976 Stephen Leacock Collegiate Institute, Agincourt, Ontario

 

1976-1980 University of Waterloo, Bachelor of Environmental Studies

Honours Geography

 

1982-1983 Queen’s University, Bachelor of Education

 

1992-1999 Queen’s University, Master of Education

 

 

PROFESSIONAL EXPERIENCE

 

1983-1985 Field Studies Leader, Toronto Urban Studies Centre

Toronto Board of Education

 

1985-1986 Teacher-Demonstrator, Mobile Education Program

Ontario Natural Gas Association, Toronto

 

1986-1987 Historical Interpreter, Toronto Urban Studies Centre

Toronto Board of Education

 

1987-1990 Teacher, Grade 1, Smithfield Public School, Smithfield, Ontario

Northumberland Newcastle Board of Education

 

1990- Teacher, Grades 4-6, Percy Centennial Public School,

Warkworth, Ontario, Kawartha Pine Ridge District School Board

 

 

PROFESSIONAL DEVELOPMENT

 

1984 York University, Ontario Ministry of Education

Computers in the Classroom, Part 1

 

1987 York University, Ontario Ministry of Education

Junior Basic Qualification

 

1988 York University, Ontario Ministry of Education

Primary Basic Qualification