An Exploratory Study of Preschool Teachers

AN EXPLORATORY STUDY OF PRESCHOOL TEACHERS’ PERCEIVED KNOWLEDGE, BEHAVIORS AND ATTITUDES/BELIEFS REGARDING THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS (NCTM) PROCESS STANDARDS

A dissertation submitted to the Kent State University College of Education, Health, and Human Services in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Julia A. Stoll

May 2015

A dissertation written by

Julia A. Stoll

B.S., Bowling Green State University, 1993

M.A., Kent State University, 2000

Ph.D., Kent State University, 2015

Approved by

______, Director, Doctoral Dissertation Committee Anne Reynolds

______, Member, Doctoral Dissertation Committee Karl W. Kosko

______, Member, Doctoral Dissertation Committee Karen L. Kritzer

Accepted by

______, Director, School of Teaching, Learning and Curriculum Alexa L. Sandmann Studies

______, Dean, College of Education, Health and Human Services Daniel F. Mahony

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STOLL, JULIA A., Ph.D., May 2015 Teaching, Learning and Curriculum Studies

AN EXPLORATORY STUDY OF PRESCHOOL TEACHERS’ PERCEIVED KNOWLEDGE, BEHAVIORS AND ATTITUDES/BELIEFS REGARDING THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS (NCTM) PROCESS STANDARDS (249 pp.)

Director of Dissertation: Anne Reynolds, Ph.D.

The purpose of this study was to find out what preschool teachers know about the

NCTM Process Standards. A 56 item survey instrument was designed to assess, a) teachers’ perceived content and pedagogical content knowledge, b) teachers’ reported behaviors, and c) teachers’ attitudes and beliefs, all regarding the value and importance of the mathematics processes in preschool settings. The recommendation in the literature is that quality mathematics instruction is important in preschool and should include a focus on these processes. There were 217 preschool teachers in the state of Ohio that completed the online survey.

Data analysis revealed areas where teachers would benefit from more professional development opportunities. Teachers have some knowledge of the mathematics processes but not a strong knowledge base. They had the most knowledge of mathematical connections and least knowledge of problem solving. A look at reported behaviors revealed that teachers promote representation of mathematical ideas the least, and for attitudes/beliefs teachers are least confident with reasoning & proof and representation. There was a meaningfully significant correlation between teachers’

reported behaviors and attitudes/beliefs. Further analyses indicated that teachers with a state teaching license had more knowledge of the processes, particularly for reasoning & proof and representation. Also, work setting did have an affect on teacher behaviors with significant differences found between teachers who work in Head Start programs and teachers in public school settings, with Head Start teachers reporting more frequent behaviors that promote the mathematics processes.

ACKNOWLEDGMENTS

I would like to thank my family, friends, professors, and especially the wonderful people on my dissertation committee. With the extra few years it took me to complete this work I can’t thank my dissertation committee director, Anne Reynolds, Ph.D., enough for hanging in there with me. Also, on my committee I want to thank Karen

Kritzer, Ph.D. and Karl Kosko, Ph.D. for their time, feedback, and words of encouragement. Thank you Dr. Kosko for the personal knowledge and resources that helped in my analysis of the survey data.

I thank my parents for ongoing “check-ins” and for the many times they took care of my darling, four-year old son, Finley, so I had time to write and my sister Marcia for the statistics talks that were ever so helpful in my understanding of the data. Also, I never would have completed this work without the entire Hutchins family: Pam, Tom,

Emily, Eva, and Trey. Since Finley was born they have graciously provided him a home away from home where he feels like one of their family.

I wish to thank Debbie Shama-Davis, Ph.D. from the KSU Research & Evaluation

Bureau for the initial input provided in the design of my survey instrument. Also, I would like to thank Eddie Bolden, evaluator for the bureau, who provided an abundance of support with the SPSS program analyses and reports.

Who I was as both a teacher and a learner was transformed during my master’s program at KSU where I had the opportunity to engage in critical conversations with my professors, Carol Bersani, Genevieve Davis, Nancy Barbour, Beverly Timmons, Rich

Ambrose. Thank you to all of them, especially Genevieve Davis, for inspiring me to continue my graduate work. Last but certainly not least, I want to thank all of my colleagues at the KSU Child Development Center. I would have never finished the program without their support, encouragement, and help in many ways from offering feedback on the survey to numerous cheers of “You can do it!” They are my family and have supported me as any family would by going above and beyond what anyone would have expected.

TABLE OF CONTENTS

ACKNOWLEDGMENTS ...... iv

LIST OF TABLES ...... x

CHAPTER Page

I INTRODUCTION ...... 1 Statement of the Problem ...... 1 Significance of the Study ...... 11 Research Goals and Objectives ...... 15

II LITERATURE REVIEW ...... 18 Introduction ...... 18 Reform Mathematics Curriculum ...... 20 Mathematics Education: An Historical Lens ...... 20 Learning with Understanding ...... 23 NCTM Mathematics Process Standards Defined ...... 26 Problem solving ...... 26 Reasoning and proof ...... 28 Communication ...... 30 Connections...... 31 Representation...... 33 The State of Mathematics in Early Education ...... 34 Children’s Informal and Formal Mathematics Knowledge ...... 40 The Vital Role of the Preschool Teacher ...... 44 Measures of Mathematics Instruction in Early Education ...... 47 Developmentally Appropriate Practice in Early Education ...... 53 Teacher Knowledge, Behaviors, and Attitudes/Beliefs ...... 56 Teacher Knowledge Defined ...... 57 Teacher Attitudes and Beliefs Defined ...... 61 Relationships Between Teacher Knowledge, Attitudes/Beliefs and Behaviors...... 62 Conclusion ...... 65

III METHOD AND PROCEDURES ...... 67 Introduction ...... 67 Research Objectives ...... 67 Research Questions ...... 68

Research Design...... 71 Dependent Variables ...... 71 Independent Variables ...... 72 Population and Sampling Plan ...... 73 Validity and Reliability ...... 74 Face validity and content validity ...... 75 Reliability ...... 76 Development of the Instrument: Teaching Mathematics Processes in Preschool Survey (TMPPS) ...... 76 Survey Design ...... 79 Perceived knowledge of NCTM mathematics process standards ...... 83 Self-reported behaviors of mathematics teaching of the NCTM process standards ...... 86 Attitudes/beliefs of teachers on the value or importance of the NCTM process standards ...... 89 Demographic information of survey respondents ...... 91 Phase 1 Development: Cognitive Interviews and Content Reviewer Data...... 92 Content reviewers ...... 92 Cognitive interviews ...... 95 Participants for Cognitive Interviews ...... 95 Procedures for cognitive interviews...... 96 Results from cognitive interviews ...... 98 Phase 2 Development: Pilot Study ...... 99 Participants ...... 99 Procedures ...... 101 Reliability of survey ...... 102 Data collection procedures for research study ...... 106 Data Analysis Procedures ...... 108 Research Questions 1-3...... 108 Research Questions 4-6...... 108 Research Question 7 ...... 109 Research Question 8 ...... 110 Research Questions 7-8 Additional Information ...... 111 Conclusions ...... 112

IV RESULTS ...... 113 Overview ...... 113 Observed Reliability of Instrument ...... 113

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Operational Definitions of the Variables ...... 114 Mathematics Content Knowledge (MCK) and Mathematics Pedagogical Content Knowledge (MPCK) ...... 114 Teacher Reported Behaviors ...... 115 Teacher Attitude/Beliefs ...... 115 Descriptive and Demographic Data ...... 116 Research Questions 1 – 3: Preschool Teachers’ Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Regarding the NCTM Process Standards Teachers’ Perceptions of their Knowledge of the NCTM Process Standards ...... 121 Teachers’ Reported Behaviors that Promote the NCTM Process Standards ...... 123 Teachers’ Attitudes/Beliefs toward the NCTM Process Standards ...... 124 Research Questions 4, 5 and 6 ...... 125 Research Question 4 ...... 126 Research Question 5 ...... 127 Research Question 6 ...... 128 Research Question 7 ...... 129 Research Question 8 ...... 132

V DISCUSSION AND CONCLUSIONS ...... 136 Introduction ...... 136 Teacher Background Information ...... 136 Research Questions 1 – 3 ...... 138 Teachers’ perceived knowledge scale ...... 138 Teachers’ reported behaviors scale ...... 140 Teacher attitudes/beliefs scale ...... 142 Research Question 4 ...... 144 Research Question 5 ...... 146 Research Question 6 ...... 147 Research Question 7 ...... 149 Research Question 8 ...... 150 Summary and Implications for Practice ...... 153 Future Research ...... 155 Conclusion ...... 159

APPENDICES ...... 161 APPENDIX A DRAFT SURVEY SCALES...... 162 APPENDIX B CONTENT REVIEWER AND COGNITIVE INTERVIEW DATA ..171 APPENDIX C PILOT STUDY SURVEY ...... 182 APPENDIX D LARGE-SCALE RESEARCH STUDY SURVEY ...... 197

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APPENDIX E SUMMARY OF DAP EXPECTATIONS FOR THREE TO FIVE YEAR-OLDS ...... 213 APPENDIX F SUPPLEMENTAL RESEARCH QUESTIONS ...... 217 APPENDIX G RESEARCH QUESTION 7 ADDITIONAL DATA ...... 220

REFERENCES ...... 223

LIST OF TABLES

Table Page

1. List of Research Objectives ...... 16

2. Comparison of the CLASS and NCTM ...... 50

3. MPCK Definition in Relation to Knowledge of NCTM Processes ...... 59

4. Type of Early Education Setting ...... 74

5. Highest Level of Education ...... 74

6. Survey Items for Each Scale and Each NCTM Mathematics Process ...... 80

7. Items of Developmentally Appropriate Practices for Self-Reported Behaviors of Mathematics Teaching of the Process Standards Scale ...... 81

8. Definitions of the NCTM Process Standards ...... 84

9. Examples of Best Practice in Early Education Settings...... 88

10. Item Criteria/Rubric for Content Reviewers – Example Using the First Four Questions ...... 93

11. Cognitive Interviews Summary ...... 100

12. Descriptive Statistics for the Pilot Study ...... 103

13. Cronbach’s Alpha for Pilot Study ...... 104

14. Reliability Analyses ...... 114

15. Distribution of Participants by Type of Early Education Setting Where One Works ...... 117

16. Frequency of Response on Years of Preschool Teaching Experience ...... 118

17. Distribution of Participants by Highest Level of Education ...... 119

18. Frequency of Response on Participation in Preschool Mathematics Professional Development Opportunities ...... 120

19. Teacher Education and Familiarity with NCTM Process Standards ...... 121

20. Descriptive Statistics for Perceived Knowledge of Preschool Teachers ...... 122

21. Descriptive Statistics for Reported Behaviors of Preschool Teachers ...... 123

22. Descriptive Statistics for the Attitudes/Beliefs of Preschool Teachers ...... 125

23. Correlation of Teacher Perceived Knowledge and Teacher Reported Behaviors ...... 126

24. Correlation of Teacher Perceived Knowledge and Teacher Attitudes/Beliefs ...... 127

25. Correlation of Teacher Reported Behaviors and Teacher Attitudes/Beliefs ...... 129

26. Test of Mean Teacher Licensure Differences between Yes or No Respondents on Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Regarding NCTM Process Standards ...... 131

27. Distribution of Participants by Type of Early Education Setting Where One Works after Re-classification ...... 133

28. Item Criteria/Rubric for Content Reviewers ...... 172

29. Content Reviewer One Results ...... 174

30. Content Reviewer Two Results ...... 176

31. Developmentally Appropriate Expectations For Three To Five Year Olds ...... 214

32. Mean Scores and Correlation of Feelings of Adequacy and Reported Behaviors ...... 218

33. Means and Standard Deviations Comparing Teacher’s Level of Education and Total Scores on Perceived Knowledge, Reported Behaviors, and Attitude/Beliefs ...... 219

34. One-Way Analysis of Variance Summary Table Comparing Teacher’s Level of Education on Teacher’s Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Total Scores...... 219 xi

35. Independent T-Test Means Tested for Sections I, II, and III of the Survey ...... 221

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CHAPTER I

INTRODUCTION

Statement of the Problem

It is unclear what preschool teachers know about the National Council of

Teachers of Mathematics (NCTM) Process Standards and what they are currently doing to promote children’s development and use of these Standards in the classroom. While research concludes that young children are capable of higher level thinking, it is unclear to what extent teachers promote higher level thinking such as problem solving and reasoning, if they plan ways for children to make connections between mathematical ideas and personal experience, and if they encourage children to communicate and represent their mathematical ideas.

Mathematics reform efforts since 1989, when NCTM published the first nationally recognized set of mathematics standards for K-12, are informed by theory and research in both curriculum and teaching that emphasize teaching mathematics for understanding. In 2000, NCTM published a revised set of Standards for grades Pre-K-12.

These Standards include five mathematics processes that are vital to the learning of mathematics content with understanding: problem solving, reasoning and proof, communication, connections, and representation. Additionally, not only is teaching mathematic content such as number concepts, geometry, and measurement emphasized, but, more importantly, there is an emphasis on teaching these concepts for understanding by developing children’s knowledge and use of mathematics processes.

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In 2010, the Common Core State Standards (CCSS) for mathematics were developed in order to contribute to the knowledge base of educators across the country.

These standards consist of Standards for Mathematics Content and Standards for

Mathematical Practice. The Standards for Mathematical Practice are based on NCTM

Process Standards and on other components of mathematics proficiency outlined in the

National Research Council’s report, Adding it Up in 2001

(www.corestandards.org/assets/CCSSI_Math%20Standards.pdf). The mathematics processes are practices children learn and engage in to understand better the mathematics content. The CCSS are designed to inform the teaching of mathematics in grades K-12, not preschool. Individual states have standards for preschool age children (e.g. Ohio’s

Early Learning and Development Standards.) These individualized preschool standards contain varying degrees of focus on mathematics processes.

This study aims to focus on the foundation of the CCSS, which is an emphasis on learning mathematics content with understanding. For teachers to implement the CCSS effectively, they must understand the NCTM Process Standards (Wenrick, Behrend, &

Mohs, 2013). Since most mathematics educators should already be familiar with the

NCTM Standards documents that have informed the development of both the CCSS and individual state standards for preschool children, these Standards will be used as the framework for the development of the survey questions in this study. The five NCTM

Process Standards were specifically chosen as the focus because promoting these in preschool curricula can effectively provide challenging mathematics environments for children and an increase in higher-level thinking that can lead to greater achievement.

The problem is that these processes are typically not a focus of instruction in traditional preschool curricula (Greenes, Ginsburg, & Balfanz, 2004; Clements & Sarama, 2009;

Ginsburg, Lee & Boyd, 2008).

Even with research on mathematics education reform indicating that standards-based teaching improves student achievement (Ross, McDougall, &

Hogaboam-Gray, 2002), few attempts have been made to examine the standards-based teaching practices of preschool teachers, particularly those behaviors related to the five process standards defined by NCTM. There are many studies of teaching for understanding using mathematics processes for elementary, middle and secondary school-aged children (Blanton & Kaput, 2005; Carpenter, Moser & Bebout, 1988;

Carpenter, Ansell, Franke, Fennema & Weisbeck, 1993; Farmer, Hauk & Neumann,

2005; Kosko, & Norton, 2012; Nicol & Crespo, 2005; Russell, 1999). There is also research on how young children develop mathematical knowledge from a psychological perspective, particularly in the area of number concepts (Baroody, 1987; Baroody, 2000;

Fuson, 1992; Gelman & Gallistel, 1978; Pepper & Hunting, 1998; Hunting, 2003). While these studies may also look at processes of mathematics, they are typically focused upon how children use a process to construct meaning for a particular mathematics concept

(e.g., studying children’s reasoning about part-whole number relationships [Hunting,

2003]) and are not the main point of study (Sophian, 1995).

Studies that focus on the mathematics development of preschool-aged children also report mostly on the quantity of mathematics experiences children receive or focus on the mathematics content that children are taught (Doucet & Tudge, 2004; Early et al.,

2005; Feiler, 2004; Graham, Nash & Paul, 1997). Fewer studies look at the teaching methods of preschool teachers, specifically methods that focus on mathematics processes or higher-level thinking (Feiler, 2004; Ginsburg et al., 2008). For example, Kilday and

Kinzie (2009) state that there is less research done to identify high-quality mathematics teaching in preschool and much more research on language, literacy, and social-emotional development. More research is needed that looks at how teachers are engaging children in complex mathematical thinking during activities such as problem solving, reasoning, communicating and using the language of mathematics, making connections to informal knowledge and other concepts, and the representation of their thinking in multiple ways.

Overall, there seems to be a lack of attention by teachers on planning specific process-oriented mathematical experiences for young children. In a large-scale study of state-funded preschool programs by the National Center for Early Development and

Learning (NCEDL) the researchers found that about 66 percent of a child’s time in a full-day program is spent engaged in curriculum activities with 48 to 49 percent whole group instruction, 29 to 31 percent free choice/centers, and only 11 to 12 percent small group instruction (Early et al., 2005). The NCEDL studies concluded that the majority of mathematical experiences that were observed were integrated or embedded in experiences rather than activities with a primary focus on mathematics (Early et al.

2005). Most teachers offered mathematics experiences in play with puzzles, blocks, games, songs, and fingerplays (Clements & Sarama, 2007; Graham et al., 1997). Cross,

Woods and Schweingruber (2009) suggest that these integrated experiences may not be

5 enough to support children’s development of mathematical knowledge. It is suggested that activities are needed where mathematics is the primary focus (Clements & Sarama,

2007). The NCEDL studies also noted that in the six percent of observations where mathematics was observed (compared to 14 percent in literacy), the focus was on skills and factual knowledge with little opportunity for conversations, integration, understanding, and problem solving (Cross et al., 2009). In general, studies have found that “children’s abilities extend beyond what is introduced in most programs” (Clements

& Sarama, 2007, p. 463).

In another study of six preschool classrooms, data showed that higher- level mathematics concepts were not observed often which suggests that more attention should be given to this in preschool settings. Over 70 percent of the math-mediated language that was observed was either “number” or “spatial sense” and was lower level thinking skills (e.g., labeling numbers or locations versus operations and patterns). Some examples of the mathematics observed were also low level such as commenting on the size or quantity of objects (Rudd, L. C., Lambert, M. C., Satterwhite, M. & Zaier, A.

2008).

The same large-scale study of state funded pre-k programs by the NCEDL, (Early et al., 2005), and two others, the Study of State-Wide Early Education Programs

(SWEEP) and the Multi-State Study of Pre-Kindergarten, utilized two observation instruments to analyze overall program quality and classroom instruction by observing teacher behaviors; however, the focus was not specifically on the teaching of mathematics. These multi-state studies took place between 2000 and 2004 in 11 states

6 and collected data using the Early Childhood Environment Rating Scale, Revised

(ECERS-R; Harms, Clifford & Cryer, 2005) and the Classroom Assessment Scoring

System (CLASS; Pianta, La Paro & Hamre, 2008). The results of these instruments and teacher questionnaires showed that improvement is needed in the areas of curriculum and pedagogy. Findings indicated that much of children’s time in the classroom is spent doing routine activities and waiting in line – “children are not engaged in constructive learning or play a large portion of the day” and “children have relatively few meaningful interactions with adults during the pre-k day” (p. 31). The results also indicate that classroom quality does not match what research says is needed for optimal learning and that “instructional quality, in terms of helping children learn new concepts and providing useful feedback, is especially problematic” (p. 31). Results from the ECERS-R and the

CLASS reported that what they call “process quality” was lower than expected (i.e.,

“instructional quality was low and learning interactions between teachers and children were infrequent”) (p. 32). The instruction lacked the quality indicators mentioned above

(constructive learning and play, meaningful interactions with adults, integration, problem solving) related to basic curriculum and pedagogical practices, which are foundational to more complex, higher-level thinking in mathematics (represented by the five mathematics processes). For example, if there were few meaningful interactions between teachers and children, then opportunities for engaging in the processes of communication and reasoning about mathematical ideas would also be infrequent.

Other educators agree that the mathematics processes are not typically evident in teaching practices (Copple, 2004; Feiler, 2004; Ginsburg et al., 2008). Feiler (2004)

7 points out that more research is needed on mathematics instruction in preschool classrooms. She suggests that research is needed that could document what it might look like to put into practice (in a preschool context) the recommendations from the

Conference on Standards for Prekindergarten and Kindergarten Mathematics Education

(Clements, Sarama, & DiBiase, 2004). One recommendation pertinent to this study is that, “mathematical processes such as problem solving, reasoning and proof, communication, connections, and representation; specific mathematical processes such as organizing information, patterning, and composing; and habits of mind such as curiosity, imagination, inventiveness, persistence, willingness to experiment, and sensitivity to patterns should all be involved in a high quality early childhood mathematics program”

(DiBiase, 2004, p. 3). Fieler (2004) describes a study of how a teacher took her values of listening and supporting children’s thinking in contexts outside of mathematics and applied these to a mathematics lesson on problem solving, something she never thought her 4-year olds were capable of understanding. The result was that “the processes of solving problems, reasoning, and communicating about mathematics served as a bridge between her well-established practices and teaching mathematics” (p. 398). She states that examples like these may help teachers build on their strengths and that beginning with what is already valued by many teachers, such as problem solving, critical thinking and communication, can help them “develop their understanding of mathematical content” (p. 399).

Carol Copple (2004), affiliated with NAEYC, also noted that the majority of preschool teachers in varied early education settings need more knowledge of materials

8 and experiences that are valuable for mathematics learning. Preschool teachers need

“sustained interactions” with children “related to math ideas.” They need

“developmental knowledge of children’s math learning,” and they need to “follow up on children’s comments and actions to deepen and extend their awareness of mathematics”

(p. 85). Recommendations for teaching, professional development, and research in the area of early mathematics have been made repeatedly in recent years in order to capitalize on young children’s mathematics capabilities (Clements et al., 2004; Cross et al., 2009;

Ginsburg et al., 2008; The National Mathematics Advisory Panel, 2008). Yet, implementing research-based recommendations (important for the improvement of the mathematical development of young children) is ultimately the teacher’s responsibility.

Therefore, it is important to learn more about what teachers are doing in preschool classrooms relevant to children’s mathematical thinking and to use this data to make improvements to both pre-service and in-service teacher professional development.

With the rising number of children in preschool programs and the early years of schooling being so important for future mathematics success, it is important to address what NCTM demands as the need for high quality programs where all children have opportunities for a challenging mathematics curriculum taught by knowledgeable and effective mathematics teachers. How preschool programs interpret the meaning of “high quality”, however, is very different. There are substantial differences in program philosophy, curriculum goals, and teaching methods in early education settings. What mathematics content is taught, and how it is taught is just the beginning. The quality of these experiences, the mathematical content knowledge and pedagogical content

9 knowledge of the classroom teacher, the teacher’s beliefs about mathematics instruction, and the culture of the classroom, all vary substantially from one program to the next

(Copple, 2004; Cross et al., 2009; Stipek, 1991). Many preschool programs use teacher- directed approaches where basic skills are stressed; other programs are more child- centered where social-emotional development is emphasized (Cross et al., 2009; DeVries,

Zan, Hildebrandt, Edmiaston, & Sales, 2002; National Research Council, 2001b; Stipek,

1991). Skipper and Collins (2003) describe the mistaken use of either a basic skills approach or an approach that focuses only on social/emotional development. With the former there is often a highly scripted mathematics curriculum that is not developmentally appropriate. With the latter, there is play with no teacher interaction and, therefore, only incidental learning. An important determination would be to assess teacher knowledge and practices in a variety of early education settings including public school preschool, Head Start, and for-profit childcare centers. Large-scale studies mentioned earlier such as the SWEEP and the study by NCEDL, examine teaching practices in only state funded pre-K programs. It is unclear how this data would compare with observations of other types of settings such as for-profit childcare centers.

Case studies, where observations and interviews are the primary means of gathering data, are most effective. While observations of teachers in the classroom are ideal, they are very costly and time-consuming to implement on a large scale; logistically, they are not practical for gathering data from a large sample of teachers. There are two observation instruments in particular that have been found to be valid and reliable measures of teacher effectiveness in preschool settings, the Classroom Observation of

Early Mathematics – Environment and Teaching (COEMET) (Sarama & Clements, 2007) and the Classroom Assessment Scoring System (CLASS) (Pianta et al., 2008). The

CLASS is not focused on mathematics specifically but looks at the overall quality of a preschool program. The COEMET is designed specifically to study mathematics and does include dimensions of both math content and math processes. Ross, McDougall,

Hogaboam-Gray and LeSage (2003) agree that “small-scale observational studies provide the most convincing evidence of implementation” of reform or standards-based teaching.

However, they make the point that this method is too expensive to be used for larger samples such as large-scale school improvement projects.

In 2009, Kilday and Kinzie completed a comprehensive comparison of measures designed to assess the quality of mathematics instruction in early childhood settings. Of the nine they studied, only three were viewed as quality measures (one being the

COEMET). Therefore, they suggest, “more instrument development is needed in this field to yield a measure for Pre-Kindergarten that offers supports for administration so that accurate ratings can be obtained by individuals with less context-specific knowledge” (p. 371). They state that self-report surveys are less expensive and in some cases provide a “relatively accurate picture of classroom practice” (p. 344). Currently, there are no empirically valid and reliable survey instruments available that specifically measure preschool teachers’ knowledge, behaviors, and attitudes regarding the NCTM

Process Standards. Therefore, one will be developed for the purposes of this study. Since the validity of self-report surveys has been challenged (Ball & Rowan, 2004), it is

11 necessary, when developing this type of instrument, that both reliability and validity issues are addressed thoroughly.

Significance of the Study

Both the National Research Council in Paths Toward Excellence and Equity

(Cross et al., 2009) and the report of the National Mathematics Advisory Panel (2008) agree that quality mathematics instruction is important in preschool. Teachers of young children should not be exempt from using the NCTM Process Standards and the theoretical base from which they were generated to teach mathematics. Research indicates that very young children are capable of understanding and developing the informal knowledge of mathematics that prepares them for more formal schooling

(Baroody & Ginsburg, 1990; Clements et al., 2004; Clements & Sarama, 2007; Cross et al., 2009). Children are capable of “complex mathematical thought” (Rudd et al., 2008;

NCTM, 2000), “possess mathematical concepts and skills long before they enter formal schooling,” and “engage in some form of arithmetic problem solving” (Ginsburg, 1989).

However, early childhood programs have been criticized for curriculum that lacks depth and intellectual rigor (Copple, 2004; National Research Council, 2001b).

In response to a national policy agenda where both early childhood education and mathematics education have become a priority, the Committee on Early Childhood

Mathematics was formed by the Mathematical Sciences Education Board of the Center for Education at the National Research Council to capitalize on the efforts to improve preschool education. The view that mathematical competence is necessary for success in today’s society is recognized by policy makers (Cross et al., 2009) and is widely

12 supported by The National Council of Teachers of Mathematics. NCTM (2000) emphasizes how the workplace requires the need for higher levels of mathematics, including problem solving. They stress the need for more students to pursue careers that are mathematics intensive such as engineering, science, and statisticians. To prepare children for these careers, all children need access to high quality programs beginning at an early age. The National Association for the Education of Young Children (NAEYC) supports this view as well in their joint position statement with NCTM (NAEYC/NCTM,

2002). They state, “high-quality, challenging, and accessible mathematics education for

3-to 6-year old children is a vital foundation for future mathematics learning” (p. 1). It is widely accepted that children should be exposed to mathematics in the early years

(Baroody, 1987; Clements et al., 2004; Cross et al., 2009; Gifford, 2003; Ginsburg, 2009;

NAEYC/NCTM, 2002; NCTM, 2000; National Research Council, 2001a; Perry &

Dockett, 2002). Proponents of curriculum standards suggest that “agreed-upon, empirically based goals” would provide teachers with a framework for planning curriculum that is both appropriate and challenging, and would “raise the bar on the learning experiences and outcomes for children” (Bredekamp, 2004, p. 78-79).

With a steady increase in the number of children attending one or two years of preschool before formal schooling, it is important to study the early education context in which mathematical development occurs. Between 1970 and 2008, children ages 3 to 4 years experienced the largest increase in enrollment rates in school settings with an increase from 20 to 53 percent (The Condition of Education, 2010). Between 1990 and

2012, the enrollment rate increased from 59 to 64 percent. Full day attendance increased

13 from 39 to 60 percent (The Condition of Education, 2014). According to the Early

Childhood Longitudinal Study, Birth Cohort of 2001, 57 percent of the approximately 4 million children born in 2001 were primarily in center-based settings. Of this 57 percent,

13 percent were in Head Start programs and 45 percent were in other center-based settings such as For-Profit childcare centers (Early et al., 2005). While there are more children in preschool, mathematics has typically not been a focus of curriculum planning and implementation (Early et al, 2005).

The development of mathematics knowledge and language at a young age has been shown to improve overall academic performance into elementary school (Duncan,

Dowsett, Magnuson, Huston, Klebanov, et al. 2007; Jordan, Kaplan, Ramineni, &

Locuniak, 2009; Klibanoff, Huttenlocher, Levine, Vasilyeva, & Hedges, 2006). In one particular study, early mathematics skills predicted later reading achievement even more so than early reading skills predicted later math achievement (Duncan et al., 2007).

Knowing the impact that early mathematics instruction can have on future development, it becomes very alarming when studies over the past ten years show children in third through fifth grade still do not understand the mathematics but rely on memorizing procedures to solve problems. Moreover, in 2010 the U. S. Department of Health and

Human Services, Administration for Children and Families, looked at the mathematics achievement of children in Head Start programs and found no significant gains for preschoolers and no impact on mathematics achievement in kindergarten or first grade.

Klein and Starkey (2004) also suggest that differences in mathematics achievement in elementary school are the result of early development during the preschool years where

“these differences are either increased or reduced by instructional practices in school” (p.

344). Therefore, young children must be exposed to meaningful opportunities for mathematical learning, to continue to build on their informal mathematical knowledge that “provides the basis for later mathematical learning” (Clements, 2004, p. x). With the majority of children ages 3 to 5 years in center-based programs, it is important to study the opportunities young children have for quality mathematics experiences in the early education context.

Young children in preschool have the same rights as elementary, middle, and high school students to have knowledgeable and effective teachers. It is evident that mathematics instruction is important in preschool and that young children are capable of higher-level thinking but it is unclear what knowledge preschool teachers have of mathematics processes that promote learning mathematics with understanding, and if they are promoting children’s development and use of these processes in their classrooms. The data gathered from a valid and reliable self-report questionnaire is a first step in assessing what type of mathematics experiences children in preschool are currently offered in a particular school district, region or state and will be valuable in determining what type of professional development opportunities to provide teachers. In a subsequent study, follow up interviews and classroom observations will be conducted with a sample of the participants from this study in order to make judgments about the reliability of participant responses on the questionnaire.

Research Goals and Objectives

The goal of this study is to assess preschool teachers’ knowledge, behaviors and attitudes regarding the NCTM Process Standards to inform teacher education programs, and improve the quality of professional development practices and opportunities. Data from this study is used to look for differences between licensed and non-licensed preschool teachers, and between teachers who work in different types of early education settings. This information can be used to determine if licensed preschool teachers know more about the mathematics processes than do non-licensed teachers. The data can also be used to determine what professional development programs are needed for teachers in different preschool settings. For example, do preschool teachers in public school settings know more about the mathematics processes than preschool teachers in For-Profit settings do? Another goal is to look for correlations between teacher knowledge, behaviors, and attitudes/beliefs regarding the mathematics processes to determine a focus for professional development programs. For example, if teacher attitudes are found to affect teacher behaviors in the classroom, then professional development opportunities should focus on promoting the value of the mathematics processes for children’s understanding of mathematics content. See Table 1 for the complete list of objectives for this study.

In order to gather data from a large sample of preschool teachers a valid and reliable, self-report survey instrument will be developed and emailed to participants.

Items for the survey instrument will be based on the five NCTM Process Standards: problem solving, reasoning & proof, communication, connections, and representation.

Table 1

List of Research Objectives

1. To determine what preschool teachers believe they know about the NCTM Process Standards, (i.e., how the Process Standards are defined by NCTM and what NCTM suggests for implementing them when teaching young children).

2. To determine the frequency with which preschool teachers believe they engage in behaviors that promote children’s development and use of the mathematics processes.

3. To determine preschool teachers’ attitudes toward the value and importance of incorporating the mathematics processes in the classroom.

4. To look for possible correlations between teacher knowledge, teacher behaviors, and teacher attitudes/beliefs regarding the NCTM Process Standards. 5. To compare the knowledge, behaviors and attitudes/beliefs of licensed vs. non-licensed teachers and of teachers who work in various types of early education settings.

Chapter II will review the literature related to the teaching and learning of mathematics at the preschool level and will discuss the theoretical framework for the self- report questionnaire, including definitions of NCTM’s five Process Standards. It will also include research regarding reform and recommendations for improvement in mathematics education at the preschool level along with valid measures that are currently being used to assess the quality of instruction in preschool settings. The review of the literature will also provide information on relationships found between teachers’ mathematics knowledge, classroom behaviors, and attitudes/beliefs toward mathematics.

Chapter III will discuss the rationale and construction of the self-report questionnaire. It will address issues of modality, social desirability bias (SDB), item development, participants and sampling procedures, and content validity and reliability.

Statistical methods for each research question will be described.

CHAPTER II

LITERATURE REVIEW

Introduction

The problem this study will address is that we do not know enough about teachers’ perceived knowledge, attitudes/beliefs, and behaviors in preschool classrooms that result in children’s development and use of higher order thinking skills in mathematics. These higher order thinking skills, or components of mathematical thinking, will be defined in terms of the NCTM Process Standards: problem solving, reasoning & proof, communication, connections, and representation. The classroom teacher is the most important factor in creating quality early childhood mathematics experiences for young children in preschool. However, research indicates that the lack of mathematics content knowledge (MCK), mathematics pedagogical content knowledge

(MPCK) (Ball, Thames, & Phelps, 2008; Fennema & Franke, 1992; Hill et al., 2008b;

Ma, 1999), and lack of knowledge about Developmentally Appropriate Practice (DAP) in teachers can be obstacles to ensuring classroom experiences where children are engaged in mathematical thinking or the use of the mathematics processes to better understand mathematics content. Researchers have studied mathematics content in preschool

(Baroody, 2000, 2004; Clements, 2004; Kamii, 1982; Ginsburg, 1983; Steffe, Cobb, von

Glasersfeld, Richards, 1983; Steffe, 1991), the quantity of mathematics experiences in preschool (Doucet & Tudge, 2004; Early et al., 2005; Feiler, 2004; Graham et al., 1997), and in general, what constitutes a quality preschool experience (Copple & Bredekamp,

18 19

2009; Harms et al., 2005; Piante et al., 2008). However, there is less research on the teaching and learning of mathematics processes in preschool settings (Stipek 2006;

Resnick 2001). Most have been studies of primary grade classrooms, particularly in the area of problem solving (Carpenter, Fennema, Franke, Levi & Empson, 1999; Cobb et al., 1989; Hiebert et al., 1996).

Large-scale studies typically collect data from one type of early education setting such as state funded programs but many children attend For-profit and family childcare programs where teacher credentials and curricula vary widely. Therefore, it is important to collect data from teachers in a variety of settings, on a large scale, so the most effective curricular decisions can be made and professional development opportunities can be provided at the county and state level for teachers in all types of settings. In order to shed light on teachers’ perceived knowledge, attitudes/beliefs, and behaviors in all types of preschool classrooms that do or do not promote children’s use of mathematical processes, a self-report survey instrument will be developed to gather data on what teachers perceive to know about the five NCTM Process Standards and what they report they are or are not doing to implement them.

The first section of the literature review describes the theoretical framework in which the NCTM Process Standards are grounded, how NCTM defines the five processes, and how understanding and using the mathematics processes can influence teachers’ effectiveness and children’s achievement in mathematics. Research regarding the state of mathematics education in preschool follows in the second section along with valid measures that are currently being used to assess quality instruction in preschool

20 settings. The third section describes research findings on the relationships between teacher knowledge, behaviors, and attitudes/beliefs, considering both a teacher’s

Mathematics Content Knowledge (MCK) and Mathematics Pedagogical Content

Knowledge (MPCK). It also describes how a self-report survey that focuses on mathematics processes can contribute to teacher effectiveness in early education settings.

Reform Mathematics Curriculum

The focus on quality mathematics for all children is a product of a reform movement in mathematics education that spanned the last four decades. The most visible product of reform was the publication of NCTM’s, Curriculum and Evaluation Standards for School Mathematics in 1989. The development of these standards was an effort to promote the learning of mathematics content with understanding. This NCTM publication was in response to traditional drill and practice methods and students’ lack of problem solving abilities necessary in an ever -changing work force. The reform movement is best understood by looking at the history of the change in beliefs about how children learn and the resulting changes in mathematics education over time.

Mathematics Education: An Historical Lens

For much of the twentieth century, the accepted approach to children’s learning was influenced by behaviorist theory that learning was a passive activity and children could be filled with knowledge simply by listening to others (NCTM, 1970). In the

1960s and 1970s, a philosophical shift took place in educators’ beliefs about how children learn mathematics. Forms of cognitivism became increasingly more accepted with the most interest in constructivism and the work of Piaget (Noddings, 1990; Battista,

1999). Constructivism is based in the belief that knowledge is “actively built up by the cognizing subject” rather than passively received (vonGlasersfeld, 1990, p. 22). This

“cognitive revolution” as it was called, laid the foundation for the current reform movement in mathematics education (Battista, 1999). While today there are still differences of opinion among mathematics educators and researchers regarding what children can or should learn, there is consensus that knowledge is not gained in a passive manner through transmission (Noddings, 1990). We are not to be “filled up” with knowledge by listening to others.

With a shift in thinking about how knowledge is generated came the need to change how mathematics was taught in school. As educators have redefined what is meant by “learning” from a theoretical perspective, instructional practices have also changed. Traditionally, mathematics instruction has reflected Thorndike’s bond theory

(Jones & Coxford, 1970), that arithmetic is taught best by breaking it into isolated facts and skills to be taught and tested separately. Since the early 1900s the primary form of mathematics instruction in school was to state a rule, give examples, and then provide problems for students to solve. These drill techniques were used with little thought given to student understanding. Even though other theories emerged (Jones & Coxford, 1970),

Thorndike’s “connectionism” overshadowed them. One such theory in the 1930s was

William Brownell’s meaning theory (1944). He believed that mathematics should be taught for understanding of mathematical relationships by emphasizing children’s ability to think. Another theory in the 1820s was Warren Colburn’s inductive, discovery

22 approach (Jones & Coxford, 1970). He believed that teachers should use a specific series of questions that would lead a child to his own rules.

However, it was not until the 1960s and 1970s that work of psychologists, especially Piaget and Bruner, began to influence beliefs about how children learn and mathematics education practices. Traditional practices, rooted in passive learning, were not supported by new scientific research, leading to questions about how mathematics should be taught. In response, proponents of constructivist theory put forth the concept that teaching “must focus on, guide, and support their [children’s] personal construction of ideas” and that “such instruction encourages children to invent, test, and refine their own ideas rather than to blindly follow procedures given to them by others” (Battista,

1999, p. 430). Noddings (1990) describes this as “pedagogical constructivism,” that, “the acceptance of constructivist premises about knowledge and knowers implies a way of teaching that acknowledges learners as active knowers” (p. 10). The goal for teachers then is to facilitate students’ sense making, that is, to understand the mathematics versus just doing mathematics.

Battista (1997) goes on to clarify what such instruction requires. One is for teachers “to help students build mathematical structures that are more complex, abstract, and powerful than they currently possess” (p. 5). He suggests teachers do this by posing thoughtfully chosen problem solving tasks to students and encouraging them to reflect on their actions. This leads to cognitive restructurings followed by the accommodation of new information. Teachers must also, “foster meaningful learning in the classroom,” and in collaboration with their students, “establish a culture of inquiry in which members

23 pose questions, solve problems, share ideas, and think critically” (p. 6). As a result of these constructivist ideas, the focus of mathematics in schools has shifted away from rules, procedures and rote memorization to reasoning, problem solving, and meaning making.

Learning with Understanding

In history, there were times when it was assumed that we understood what was meant by understanding, meaning, or thinking (Hiebert et al., 1997, p. xiii). However, the current view is that it is very complex and difficult to define. Wiggins and McTighe

(2005) describe it as “a mental construct, an abstraction made by the human mind to make sense of many distinct pieces of knowledge” (p. 37). Hiebert et al. (1997) describes understanding as something that is always changing and growing and a result of personal connections where the more relationships one makes, the better one understands. Going back further in history, the idea of transfer was considered important to the ability to understand something conceptually. In thinking about understanding as both conceptualizing and transferability, Dewey (1933) explains that conceptualizing is necessary for information to be transferred which results in a more complex understanding of new experiences. Whitehead (1929), too, had similar ideas about the importance of transferability to understanding. “Education with inert ideas, …or those that are merely received into the mind without being utilized or tested, or thrown into fresh combinations is not only useless: it is above all things, harmful…Let the main ideas which are introduced be few and important, and let them be thrown into every combination possible” (p. 1-2). The phrase sense-making is used often by educators

24 when referring to understanding as the result of facts acquiring meaning for the learner when put into relationships with other things. Wiggins and McTighe (2005) view the construction of relationships as the heart of all understanding.

Bruner (1960) stated years ago that learning should serve us in the future through

“nonspecific transfer”, the transfer of principles and attitudes rather than skills (p. 17).

Bruner refers to a focus on the bigger idea or principle and then defining the more specific parts as needed to solve a particular type of problem, a contrast to Thorndike’s behaviorist approach that you must learn all of the parts to understand better the whole.

For Bruner, instead of just learning skills, the focus should be on learning a general idea that can be used to solve novel problems. Bruner describes this type of transfer as the

“continual broadening and deepening of knowledge in terms of basic and general ideas”

(p. 17). We may describe this today as gaining a better understanding of a concept. To do this a teacher must have mastery of the structure of subject matter. Bruner presents a case for the importance of structure by emphasizing that fundamental ideas one learns have greater applicability to new problems. These big ideas or principles are still viewed today as essential for the basis of transfer (Wiggins & McTighe, 2005). For example, the

“big ideas” of mathematics content that young children should learn, as described by

Clements and Sarama (2007), are number and operations, geometry and spatial sense, and geometric measurement along with processes such as problem solving, reasoning, and classification.

Even Bloom, in his influential Taxonomy of Educational Objectives: Cognitive

Domain (1956), stresses the need for application. He states that processes such as

25 application, synthesis, analysis, and evaluation contribute to new understanding. Wiggins and McTighe (2005) clarify this further stating that doing something correctly does not provide evidence of understanding because it might have been an accident or done by rote. They believe understanding is achieved when a student can give an appropriate explanation for something done correctly. Wiggins and McTighe clarify that “transfer is the essence of what Bloom meant by application” (p. 41).

Knowledge and skill are necessary elements of understanding, but not sufficient

in themselves. Understanding requires more: the ability to thoughtfully and

actively “do” the work with discernment, as well as the ability to self-assess,

justify, and critique such “doings.” Transfer involves figuring out which

knowledge and skill matters here and often adapting what we know to address the

challenge at hand. (Wiggins & McTighe, 2005, p. 41)

Brownell stressed in his meaning theory that transfer, and thus understanding, comes naturally while students solve mathematical problems rather than as something we should teach directly (Hiebert, 1997, p. 22). Dewey (1900/1990) stressed that teaching that focuses on imitation or repetition without reason does not lead to reflection, or the

“power of reflective attention” and ultimately, understanding (p. 147). These approaches are reflected in NCTM’s standards documents, which state that the focus should be on a child’s depth of understanding, how well they can apply what they have learned, rather than on “regurgitation of isolated facts” and this learning “emerges from experiences or problem solving activities” (Hiebert, 1997, p. ix). The other NCTM Process Standards also play a role in children’s understanding. According to Hiebert et al. (1997),

26 understanding develops as we make connections, especially through communication when we talk, listen, write, and interact with others. The process of representation is critical in promoting understanding because children can use personally meaningful representations as a means for reflection (NCTM, 2000). NAEYC/NCTM (2002) describes the NCTM processes as a necessary means to learning content knowledge and that the processes “develop over time and when supported by well-designed opportunities to learn” (p. 7).

NCTM Mathematics Process Standards Defined

Problem solving, reasoning, communication, connections, and representation are described by NCTM and mathematics educators as processes necessary for learning mathematics content with understanding. NAEYC/NCTM (2002) adds that, “children’s development and use of these processes are among the most long-lasting and important achievements of mathematics education” (p. 7). Each of the processes is defined according to NCTM in Principles and Standards for School Mathematics (2000).

Problem solving. NCTM defines problem solving as “engaging in a task for which the solution method is not known in advance” where “solving problems is not only a goal of learning mathematics but also a major means of doing so” (p. 52). Lester and

Kehle (2003) describe problem solving as coordinating prior experience and knowledge to “generate new representations and related patterns of inference that resolve the tension of ambiguity that prompted the original problem solving activity” (p. 510). Teachers must see the importance of ambiguity in planned activities and embed problem solving naturally in planned experiences for children. In the preschool years problems should be

27 presented in a variety of contexts, they should be related to daily routines, and often times arise from stories. Traditionally, problem solving was taught after children had learned the basic skills however, reform research stresses that children learn “basic skills, higher- order-thinking skills, and problem-solving strategies” while engaging in problem solving

(p. 121). Therefore, teachers must choose and pose problems with the intent that children will develop new understandings in the process of solving them.

According to NCTM, teachers must know the mathematical concepts they want their students to learn so they know how to effectively intervene to support children’s thinking (e.g. when to probe, give feedback, withhold comments, plan similar tasks).

NCTM (2000) states that “problem solving is natural to young children because the world is new to them, and they exhibit curiosity, intelligence, and flexibility as they face new situations;” therefore, we must “build on children’s innate problem solving inclinations and preserve and encourage a disposition that values problem solving” (p. 116).

In NCTM’s description of problem solving, it is clear that teachers must contribute to the development of student dispositions and a classroom culture of curiosity, persistence, reflection, and self-assessment. Teachers must promote children’s reflective habits by continually asking questions that prompt them to check their understanding. To build these habits and dispositions in their students, teachers must believe that children are capable, give children time for thinking, listen carefully to children’s explanations, and structure an environment that values the work of children (p.

119). An essential part of this environment is encouraging children to work together so

28 in the process of explaining their thinking and hearing new ideas suggested by their peers, they may change their strategies as warranted (p. 121).

Problem-solving opportunities are, therefore, an essential component of classroom curriculum that the teacher must seize when they are meaningful to her students. Dewey

(1900/1990) cautions, however, that teachers should capitalize on children’s own questions where they would naturally use “reflective attention”, such as judging, reasoning, and deliberation, rather than the use of external means to make an activity interesting by “offering a bribe” for completing a task or threats of punishment. Dewey speaks of the importance of providing more than “ready-made materials” where children may, only by accident, have a “motive for reflective attention” (pp. 148-149). He stresses the fundamental necessity of teachers to “lead a child to realize a problem as his own, so that he is self-induced to attend in order to find out its answer” (p. 149). In order for teachers to offer children these opportunities, they must have a well-developed understanding of what is meant by problem solving and developmentally appropriate practice in preschool.

Reasoning and proof. NCTM describes reasoning as using evidence to draw conclusions (Martin & Kasmer, 2009/2010) and Russell (1999) explains reasoning as the

“development, justification, and use of mathematical generalizations” (p. 1). NCTM

(2000) states that, “systematic reasoning is a defining feature of mathematics that is found in all content areas” (p. 56-57). NCTM describes characteristics of reasoning and proof as noting patterns or regularities in real world situations and stresses the importance that students expect mathematics to make sense; to realize that sense making is a

29 developing understanding of mathematical ideas that occurs when we make connections to prior knowledge through reasoning (Martin & Kasmer, 2009/2010). In the early grades, reasoning includes informal observations as well as explanations and pattern- recognition and classification are important elements of reasoning.

Teachers can encourage students’ reasoning by asking specific types of questions.

NCTM (2000) gives the following examples: By asking, “Why do you think it is true?” they are helping children learn that “all assertions should have reasons.” By asking,

“What will happen next?” or “Is this always true?” they help children learn to make and test conjectures. By asking, “Why does this work?” they help children develop an argument, and, by asking if something works “always, sometimes or never,” they are helping children learn to generalize (pp. 56-58). Teachers should plan many regular opportunities for children to “reason from what they know” and use the “language of logic” themselves as a model for children (NCTM, 2000, p. 58-59).

Young children are capable of mathematical reasoning (Ginsburg, Klein &

Starkey, 1998). With this belief, the classroom environment should be one where reasoning becomes a habit of mind (p. 56). Many opportunities should be provided for children to work together, learn to listen to the ideas of others and make comparisons to their own ideas. Children should be expected to explain their thinking and become the authority in determining right and wrong (Wenrick et al., 2013). It should be an environment where children’s ideas are respected and nurtured, and where children can expect mathematics to make sense.

Communication. NCTM (2000) defines communication as essential for understanding in mathematics. It is a way for children to share their ideas and clarify their understanding – “the ideas become objects of reflection, refinement and discussion”

(p. 60). Listening to the viewpoints of others can reveal to students different aspects of a problem and help students learn to be critical thinkers. Communication also means using objects, drawings, and charts to share thinking with others, and using symbols only after children have made meaning of mathematical ideas from these other forms of communication.

Teachers can help students understand mathematics by choosing worthwhile topics for discussion, plan time for students to reflect, model and pose questions to clarify expectations for student work, and accept and use multiple forms of communication.

Teachers should help students learn to talk about mathematics. “As students articulate their mathematical understanding… they begin using everyday, familiar language” that teachers can build upon to add more formal mathematics language (p. 63). NCTM says that teachers should present challenging problems, expect students to explain their thinking, give them time to talk with and listen to their peers, guide students to rephrase when others do not understand and model appropriate vocabulary (p. 130).

Research suggests that, for classroom discourse to be effective, there must be opportunities for students to share their ideas and solutions and respond to the solutions of their classmates (Sherin, 2002), be opportunities to work in small groups where different points of view are expressed (Marzano, Pickering, & Pollock, 2001), and be a sense of security and comfortableness with sharing their thinking with others, ask

31 questions, and make mistakes (Ball & Friel, 1991; Sowder et al., 1988). Therefore, teachers have a huge responsibility in shaping the discussion based on what they know about their students to maximize the learning potential of these interactions among peers and teachers.

Connections. NCTM (2000) describes connections in mathematics instruction as the “interrelatedness of mathematical ideas” (p. 64). Connections should be made among mathematical topics and to other subject areas. Mathematics should be embedded in many activities throughout the day including those for physical education, music, science, and literacy. Conceptual understanding is NCTM’s goal and children’s understanding is both deeper and longer lasting when they make connections among mathematical ideas

(p.64). NCTM emphasizes that for children to see the utility of mathematics and not view mathematics as an arbitrary set of rules the conceptual understanding must be linked to procedures.

Teachers should make explicit the connections to be made. Since preschoolers learn mathematics when connections are made to the real world, teachers should capitalize on spontaneous moments that arise where mathematics learning can be emphasized. Teachers can also make children aware of connections when they ask questions or comment on the mathematical aspects of children’s play or routine activities.

It is important for teachers to know what previous knowledge students bring to a situation so they can plan appropriate experiences including tasks or problems in new contexts that revisit previously explored concepts.

The classroom environment should be developed where children are expected to use what they already know to make connections to mathematics concepts. Young children often do this by using concrete objects such as fingers; the use of these objects should be encouraged. Experiences for children should be purposeful and authentic such as encouraging mathematical thinking within long-term projects and investigations with encouragement by teachers to use their own strategies to make connections among mathematical ideas, vocabulary, and representations (p. 134).

NCTM states that understanding of mathematical ideas occurs when one makes connections to prior knowledge. This refers to children’s assimilation and accommodation processes. Since assimilation is the end product that we hope to gain with cognitive development, as teachers we need to provide children opportunities for this to occur. Teachers are often told today that they need to build on children’s prior experiences. What is being recommended is that teachers need to be aware of what children already know or what they have some experience with so that the teachers can provide the appropriate stimulus that the children need for assimilation and accommodation to occur.

Not only has Piaget’s theory on assimilation and accommodation affected the practice of teaching but also assessment. With assessment, we are now more cognizant of what it really means to “understand” something. If we compare understanding to

Piaget’s theory, we can see that understanding really is about how much we have assimilated and/or accommodated the stimulus. The more we assimilate the more our understanding grows quantitatively, and, the more we accommodate, the more our

33 understanding grows qualitatively. As Wadsworth (1977) points out in describing

Piaget’s work, if a child only assimilates, then he/she know a lot about one thing and similarities, but cannot see differences. If a child only accommodates, then he/she can differentiate among many things but cannot make generalizations. The implication for teachers, then, is to know more about each child, his/her mathematical knowledge and his/her experiences so interesting, meaningful and challenging experiences can be offered to each child.

Representation. NCTM (2000) defines representation as “the act of capturing a mathematics concept or relationship in some form and to the form itself” so representation refers to a process or a product (p. 67). Goldin and Kaput, (1996) distinguish between internal and external representations. Internal representations are the intellectual pictures or mental abstractions one acquires through experience that provide meaning to external representations such as symbols, diagrams, and graphs. NCTM

(2000) explains that having access to a variety of mathematics representations and the ideas they represent gives children a means to expand their capacity to think mathematically (p. 67). Therefore, NCTM says that representations are essential for children’s understanding, and for communicating effectively, for making connections, and as models for problem solving. They are “powerful tools for thinking” (p. 136).

According to NCTM (2000), teachers should promote children’s use of multiple forms of representation, especially by providing technological tools. Teachers should encourage children’s sense making by promoting the use of a variety of ways to represent ideas then help children begin to translate these to representations that are more

34 conventional. NCTM stresses that when children represent their ideas it helps teachers gain insights into the ways children interpret and think about mathematics. Encouraging children to share representations helps them explain their thinking and appreciate the diverse thinking of their peers.

The classroom environment should include models of conventional ways of representing mathematics situations. It should also be one where multiple forms of representation are encouraged and where teachers and peers listen to one another’s ideas.

For young children objects should be provided that they can manipulate to represent an idea, which is foundational to the use of abstract symbols.

While the five NCTM Process Standards are defined separately, it is clear that they are all connected and overlap one another. For example, reasoning is essential when developing the logic for problem solving, and representations are invaluable for strengthening children’s communication skills. Reasoning occurs as a result of making connections to prior knowledge, and representations aid children in the problem solving process.

The State of Mathematics in Early Education

In the past 15 years, national attention has been given to improving the mathematics education of preschool age children. Major publications by NCTM,

NAEYC, the National Research Council, and findings from the Conference on Standards for Prekindergarten and Kindergarten Mathematics Education specifically address the mathematics education of children age three to six. Highlighted in this section is the fact that, with each new publication or conference during this time period, research on the

35 value of teaching the mathematics processes for student understanding of mathematics has emphasized recommendations that teachers embed the processes in their classroom practices. It is not clear, however, if these recommendations have been fully realized or implemented in the majority of early education settings.

When in 2000, NCTM published Principles and Standards for School

Mathematics for children pre-K to grade 12 it was the first time preschool was included in the Standards documents. These Standards include five Content and five Process

Standards. The five Content Standards are: number and operations, algebra, geometry, measurement, and data analysis & probability. The five Process Standards are: problem solving, reasoning and proof, communication, connections, and representation. NCTM states that these five processes should be integrated within the learning of all mathematics content. Critics note that these Standards, however, do not provide teachers clarity on what the appropriate expectations for preschoolers should be because the grade band

(pre-K-2) is too broad, and, therefore, are not as useful for preschool teachers as they might be for 1st and 2nd grade teachers (Bredekamp, 2004; Cross et al., 2009; NAEYC/

NCTM, 2002).

In 2000 the Conference on Standards for Prekindergarten and Kindergarten

Mathematics Education took place to develop standards and teaching methods for mathematics education that were more specific to young children. As a result of this conference, Clements, Sarama, and DiBiase (Eds.) (2004), published several recommendations for promoting high quality mathematics education for young children.

They include guidelines for what children should know, what teacher expectations for

36 children should be, what environment/context best supports children’s learning, the role of the teacher, and what forms of professional development would be most effective for teachers.

In 2006, NCTM responded to the need for more focus in the curriculum at each grade level by publishing the Curriculum Focal Points for Prekindergarten through

Grade 8 Mathematics: A Quest for Coherence. This document outlined the three most important concepts to be taught at each grade level. Just as was recommended by the

2000 Conference, the focal points for prekindergarten are number and operations, geometry, and measurement. The focal points emphasized the Process Standards that are described by NCTM as necessary for understanding of the content.

Guidelines for what to teach young children have been elaborated on in a publication for the Second Handbook of Research on Mathematic Teaching and Learning

(2007), where Clements and Sarama identify five big ideas in early childhood mathematics (pre-k/k): (a) number and quantitative thinking, (b) geometry and spatial thinking, (c) geometric measurement, (d) patterns and algebraic thinking, and (e) data analysis (in order of prevalence) (p. 466). In agreement with others, they note that the big ideas of mathematics are what guidelines should be based on (Bruner, 1960; Fuson,

2004; Griffin, S., Case, R., & Capodilupo, A. 1995; National Research Council, 2001b;

Tibbals, 2000; Weiss, 2002) and are defined as “those that are mathematically central and coherent, consistent with children’s thinking, and generative of future learning” (p. 463).

They also emphasize processes such as reasoning and problem solving, and classification and seriation.

In a publication by the National Research Council (Cross et al., 2009), the

Committee on Early Childhood Mathematics provided the necessary detail for what preschool children should know and what they can do that is not present in the NCTM documents. They reported their summary of the existing research and made several recommendations for improving the mathematical competence of young children.

Recommendations from the National Research Council in 2009 and from the Conference on Standards for Prekindergarten and Kindergarten Mathematics Education in 2004 send a similar message about what is quality mathematics in preschool:

 Mathematics guidelines for curriculum planning should focus on and

elaborate the big ideas of mathematics. (Clements et al., 2004, Cross et al.,

2009)

 “Connections – between topics, between mathematics and other subjects, and

between mathematics and everyday life – should permeate children’s

mathematical experiences.” (Clements et al., 2004, p. 3)

 “…mathematical processes such as problem solving, reasoning and proof,

communication, connections, and representation; specific mathematical

processes such as organizing information, patterning, and composing; and

habits of mind such as curiosity, imagination, inventiveness, persistence,

willingness to experiment, and sensitivity to patterns should all be involved in

a high-quality early childhood mathematics program” (Clements et al., 2004,

p. 3).

They have also made several recommendations specific to the role of the teacher in creating high quality mathematical experiences for young children. Selected recommendations regarding teacher practices from Clements et al. (2004) are:

 Recommendation 6: “Mathematical experiences for very young children

should build largely upon their play and the natural relationships between

learning and life in their daily activities, interests, and questions.” (p. 4)

 Recommendation 7: “Teachers’ most important role with respect to

mathematics should be finding frequent opportunities to help children reflect

on and extend the mathematics that arises in their everyday activities,

conversations, and play, as well as structuring environments that support such

activities. Teachers should be proactive as well in introducing mathematical

concepts, methods, and vocabulary.” (p. 4)

 Recommendation 8: “Teachers should purposefully use a variety of teaching

strategies to promote children’s learning. Children benefit from a thoughtful

combination of carefully planned sequences of activities and of integrated

approaches that occur throughout the day. Successful early childhood

teachers build on children’s informal knowledge and everyday activities,

considering children’s cultural background, language, and mathematical ideas

and strategies.” (p. 4)

 Recommendation 10: “Teachers should endeavor to understand each child’s

own mathematical ideas and strategies. Teachers should use those

understandings to plan and adapt instruction and curriculum.” (p. 4).

 Recommendation 11: “Teachers should help children develop strong

relationships between concepts and skills. Skill development is promoted by a

strong conceptual foundation.” (p. 4)

In 2002 (updated in 2010), NCTM and NAEYC published their joint position statement on early childhood mathematics. In this document, Early Childhood

Mathematics: Promoting Good Beginnings, similar recommendations are made as those listed above to help ensure that children have access to effective and challenging, research based curricula and teaching practices. For example, specific to the NCTM

Process Standards they state that teachers should, “use curriculum and teaching practices that strengthen children’s problem-solving and reasoning processes as well as representing, communicating, and connecting mathematical ideas” (p. 3). They expand on this by making a general suggestion for teachers related to DAP teaching practices and that is to, “provide ample time, materials, and teacher support for children to engage in play, a context in which they explore and manipulate mathematics ideas with keen interest” (p. 3). Other recommendations from NAEYC/NCTM that match those from the previous publications include the need to focus on the most important mathematical ideas, the importance of planning for the needs and abilities of individual children, building on children’s informal mathematics knowledge and interest, and providing time for student reflection and planned extensions to explore mathematical ideas. It is clear that there is a consensus from educators in the field of early childhood education and early mathematics education on what is needed to improve the quality of early education experiences for young children.

NAEYC and NCTM agree that in order to see continued improvement of mathematics in this country an overall commitment to education in the early years is necessary. While NCTM and state departments of education have developed standards to improve student learning, this can result in the teaching of isolated skills that are not meaningful to young children and teachers’ lower expectations of children’s mathematical learning potential (Stipek, 2006). Therefore, it is important for teachers to have a thorough understanding of mathematics and mathematics teaching, child development, and developmentally appropriate teaching practices for young children.

NAEYC/NCTM (2002) stress that, while a child’s earliest experiences can have long- lasting outcomes, knowledge of mathematics is “not yet in the hands of most early childhood teachers in a form to effectively guide their teaching” (p. 2).

Children’s Informal and Formal Mathematics Knowledge

Young children bring much informal mathematical knowledge with them to school (Baroody, 1987; Clements & Sarama, 2007; Cross et al., 2009; National Research

Council, 2001a), and the teacher makes planning and instructional decisions that help children develop formal mathematical knowledge. Children develop informal mathematical knowledge during play and daily routines or interactions and, with teacher guidance, these informal notions can be built upon to develop more formal mathematics concepts through “mathematizing.” The term mathematizing describes what children do when they begin with an informal notion of a mathematics concept developed from everyday experiences and then think about them in mathematical terms (Cross et al.,

2009). Clements and Sarama (2007) provide a more formal definition where

41 mathematization includes “redescribing, reorganizing, abstracting, generalizing, reflecting upon, and giving language to that which is first understood on an intuitive and informal level (p. 534-535). Baroody (1987) states that informal knowledge is the crucial step between children’s intuitive knowledge, based on their perceptions, and the formal, abstract knowledge that is the focus of elementary school curriculum. Informal knowledge provides a necessary foundation for learning the formal school mathematics.

Cross et al. (2009) state, “mathematics processes are a means for making sense of abstract mathematics and for formulating real situations in mathematical terms – that is, for mathematizing the situations they encounter” (p. 43).

According to Piaget, children develop through the process of assimilation where new knowledge is constructed when connections are made to previous knowledge.

Therefore, teachers must build on children’s prior knowledge. This is a key idea of the

Connections Process Standard. Baroody (1987) points out how counting plays a crucial role in the development of this informal knowledge and teachers should draw on children’s counting knowledge to learn new skills or concepts. “It is counting that puts abstract number and simple arithmetic within the reach of the young child” (p. 33).

Without these connections, children have difficulty with the formal, abstract mathematics and resort to memorization and rote learning without understanding. Eleanor Duckworth

(2006) explains it this way:

Intelligence cannot develop without matter to think about. Making new

connections depends on knowing enough about something in the first place to

provide a basis for thinking of other things to do – of other questions to ask – that

demand more complex connections in order to make sense. The more ideas about

something people already have at their disposal, the more new ideas occur and the

more they can coordinate to build still more complicated schemes. (Duckworth,

2006, p. 14)

Some educators question whether or not teachers of preschool children should let them “develop spontaneously” or if they should “guide” children in the “process of getting to know the world around them” (Steffe, 1991, p. 21). Some teachers have the belief that everything a child needs to know in mathematics before formal schooling they can learn on their own through daily interactions and play without a teacher’s deliberate or planned instruction or guidance (Skipper & Collins, 2003). Others believe that teachers need to also plan specific mathematics activities (Clements & Sarama, 2007;

Ginsburg et al., 2008). Either way, teachers “need an extensive knowledge base, one that includes a deep understanding of mathematical psychology” (Baroody, 2004, p. 157). A lack of understanding about appropriate curriculum and instruction in preschool often means that efforts are not made by teachers to relate mathematics to what individual children know, to provide meaningful opportunities during inquiry-based investigations for the use of mathematical thinking, to use play and daily activities as opportunities for

“mathematizing,” and to view children as capable of using and developing mathematical processes or decision making which develops intellectual autonomy (Clements et al.,

2004; Rudd et al., 2008).

One example cited by Clements and Sarama (2007) is research that shows there are no differences in the amount of mathematics that both low-and middle-income

43 children demonstrate in free play (Ginsburg, Ness, & Seo, 2003; Seo & Ginsburg, 2004).

This is significant because these studies discuss how low-income children are typically behind their middle-income peers in academic achievement in mathematics. One explanation, offered by Clements and Sarama (2007), is that, for low-income children, mathematizing has not been learned, and, in order for this to occur, children need opportunities to connect informal mathematics in play to the formal mathematics of school; “to bring the ideas to an explicit level of awareness” by reflecting on and discussing their pre-mathematical activities (p. 534). Therefore, engagement in language, especially the language of mathematics becomes a “foundational ability” rather than a

“basic mathematical ability” that can help to bridge the gap between the achievement of low-and middle-income children (Clements & Sarama, 2007; Ginsburg & Russell, 1981).

This explanation supports the notion that teachers need to have a planned mathematics curriculum and include the process of communication, which promotes a focus on language development in mathematics instruction. Rudd et al. (2009) also view the mathematics processes of communication and connections as important in preschool. In one study they focused on math-mediated language, in other words, how teachers talk with children. They emphasized the importance of language as a tool for teaching mathematics and concluded that there is a need for teachers to help children make connections between new mathematical terms and what the children already know.

The Vital Role of the Preschool Teacher

Reform efforts to improve student understanding in mathematics education at all grade levels are dependent upon the effectiveness of the teacher. Effective teaching leads

44 to mathematics improvement for students (NCTM 2000). According to NCTM, effective teaching requires (a) knowing / understanding mathematics content, pedagogical strategies, and students as learners, (b) maintaining a challenging and supportive learning environment, and (c) continually seeking improvement of one’s own teaching (p. 17).

Regarding the first requirement, knowing/understanding mathematics, pedagogical strategies, and students as learners, there are many studies of both preschool and elementary teachers that show evidence that a teacher’s mathematical knowledge affects teacher behaviors, attitudes, and student achievement (Appalachian Rural, 2000;

Botha, Maree, & de Witt, 2005; Brown, 2005; Hill et al., 2008b; National Advisory

Panel, 2008; Perrin, 2008; Weiss, Pasley, Smith, Banilower & Heck, 2003). Most of these studies refer to mathematics content knowledge, with only a few studies that focus on preschool teachers’ knowledge of mathematics processes. For example, Botha, et al.

(2005) asked teachers in South Africa how often they used the five processes and Sarama and Clements (2007) developed an observation instrument to examine teacher practices related to math processes. Smith (2000) found that teachers’ high pedagogical content knowledge correlates with students’ positive attitudes toward mathematics. No study, however, has directly assessed teacher knowledge of the mathematics processes in order to determine the effect this knowledge may have on teaching practices and student achievement. The National Advisory Panel (2008) states,

Research on the relationship between teachers’ mathematical knowledge and

students’ achievement confirms the importance of teachers’ content knowledge. It

is self-evident that teachers cannot teach what they do not know. However,

because most studies have relied on proxies for teachers’ mathematical

knowledge (such as teacher certification or courses taken), existing research does

not reveal the specific mathematical knowledge and instructional skill needed for

effective teaching, especially at the elementary and middle school level. Direct

assessments of teachers’ actual mathematical knowledge provide the strongest

indication of a relation between teachers’ content knowledge and their students’

achievement. (National Advisory Panel, 2008, xxi)

Research suggests that a majority of teachers may not have the necessary mathematical knowledge (Ball & Bass, 2000; Ball & Friel, 1991; Ball, Hill & Bass, 2005;

Clements, 2004; Ma, 1999, Moseley, 2005; Thompson, 1992). Mathematical knowledge has two components, mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK). Content knowledge refers to understanding the structure of mathematics in the areas of number, geometry, and measurement.

Pedagogical content knowledge refers to understanding child development, connections among math concepts, student misconceptions, and appropriate teaching strategies. Both are essential because while teachers do make planning decisions in advance, more importantly, they make judgments and decisions during unplanned opportunities for mathematical learning that arise from the students’ daily work and discussions. Simon

(1995) describes teaching as a “process of continually adapting and adjusting while co- participating in mathematical activity with students” (p. 238) and therefore, it is believed that these unanticipated events are an essential aspect of teaching (McClain & Cobb,

2001). Teachers decide when and how to intervene and support children’s discussions,

46 what resources to use, and “how to guide the development of the classroom microculture”

(Simon, 1995, p. 237).

Regarding the second requirement, having a challenging and supportive learning environment, research of preschool teachers’ practices suggests that children are exposed to few mathematics experiences in the early childhood classroom (Early et al., 2005).

Teachers’ lack of MPCK could be the reason for teachers’ missed use of opportunities for supporting mathematical thinking (Copple, 2004; Cross et al., 2009; Stoll, 2000) and why teachers plan isolated tasks not connected to children’s prior knowledge, experience or current investigations. Teachers view mathematics as only counting, sorting and identifying shapes, and they limit the mathematics curriculum to only basic skills

(Copple, 2004; Cross et al., 2009; Graham et al., 1997; Lee & Ginsburg, 2007; National

Research Council, 2001b; Pianta et al., 2005; Rudd et al., 2008). Much of what is taught is social knowledge or social conventions (Kamii, 1982) that children can only learn from others through transmission such as the names of shapes, colors and numerals. A focus by teachers on basic skills results in less emphasis on cultivating children’s higher-level thinking such as the mathematical processes of problem solving, reasoning, and communication (Thompson, 1992; Fennema et al., 1996; Stipek, 2006). Preschool, children need more opportunities to construct relationships - to think logically, reason, and solve problems. Several studies show that teaching isolated skills may result in short-term positive effects on achievement, however, in the long term, children demonstrate lower achievement scores, less confidence, and less pleasure in learning

(Miller & Bizzell, 1983; Schweinhart, Weikart, and Larner, 1986).

The third requirement is continually seeking improvement of one’s own teaching.

In NCTM’s Professional Standards for Teaching Mathematics (1991), they state that,

“teachers are key figures in changing the ways in which mathematics is taught and learned in schools” (p. 1). Graham and Fennell (2001) agree with NCTM that, “teacher development is increasingly viewed as a career-long process” (p. 320) so if we want to improve teacher effectiveness then appropriate professional development opportunities must be provided. This is challenging in that, as NCTM (2000) states, “teaching mathematics well is a complex endeavor and there are no easy recipes…for helping all teachers become effective” (p. 17). Just as effective teachers get to know their students as learners in order to vary their teaching practices to meet individual needs, professional development offerings must be based on what we know about the needs of teachers.

Therefore, we need to find effective and efficient ways to find out what teachers know or do not know about the NCTM Process Standards, in order to extend or enhance their knowledge appropriately.

Measures of Mathematics Instruction in Early Education

Cross et al. (2009) state that while there is research on the effectiveness of mathematics curriculum in early education settings (Clements & Sarama, 2007b; Greenes et al., 2004) this doesn’t always shed light on effective instructional practices. Several instruments are available that are designed to evaluate the quality of mathematics instruction in early and elementary classrooms that are “derived from and aligned with the theoretical bases for the NCTM Standards” (Kilday & Kinzie, 2009, p. 366). Those discussed here are the Early Childhood Environment Rating Scale, Revised (ECERS-R)

(Harms et al., 2005), the Classroom Assessment Scoring System (CLASS) (Pianta et al.,

2008), and the Classroom Observation of Early Mathematics Environment and Teaching

(COEMET) (Sarama & Clements, 2007).

The ECERS-R (Harms et al., 2005) is a research-based quality assessment tool used in preschool settings all over the world. The focus is not specifically mathematics but rather a look at the overall quality of early education environments for young children. There are seven subscales related to space and furnishings, program structure, personal care routines, activities (including math/number), language and reasoning, and interactions among teachers, children, and families. The section on mathematics is too general to influence major improvement in mathematics knowledge or practice as the focus is mainly on if activities to teach basic content exist (e.g. counting, shapes, size), if the materials are accessible and well organized, and if there are occasional planned mathematical experience by teachers. No examples are provided as to what these materials and planned experiences should be. The ECERS-R does include a few scales that assess the frequency and quality of teacher / child interactions: providing children many avenues for communication, including talk about logical relationships, and reasoning about their problem solving experiences. This emphasis on communication, reasoning and problem solving, while not specific to only mathematics, illustrates how important these processes are to quality education programs for young children.

One of the most widely used quality assessment instruments for preschool settings is the

CLASS (Pianta et al., 2008). A preschool classroom is scored on 10 different dimensions related to items such as classroom climate, teacher interactions with children, and

49 teaching methods. The items are based on comprehensive research on effective instruction in the field of early childhood education. While the instrument does not specifically focus on one subject area such as science, math or literacy, the same type of critical thinking that the NCTM Process Standards encompass is infused within the 10 dimensions. For example, Table 2 shows that under the dimension of Concept

Development, Pianta et al. (2008) include Analysis and Reasoning which is defined as

“why and/or how questions, problem solving, prediction/experimentation, classification/comparison, evaluation” and is rated high quality if “the teacher often uses discussions and activities that encourage analysis and reasoning.” NCTM also defines reasoning in mathematics in similar terms. The Reasoning and Proof Process Standard emphasizes that children should have opportunities to respond to “why” questions and be encouraged to describe and justify their thinking to others (see Table 2). Other examples, under the dimension of Concept Development, are two items that refer to what the

NCTM would define in mathematics as the process of making connections. These are

Integration and Connections to the Real World. Integration is defined as “connected concepts” and “integrates with previous knowledge.” Connections to the Real World is defined as “real-world applications” and “related to students’ lives.” Again, NCTM defines connections in mathematics in similar terms. The Connections Process Standard emphasizes the need to connect the mathematics that children learn to their prior knowledge, other subject areas, and to children’s real world experiences [Table 2].

Table 2

Comparison of the CLASS and NCTM

CLASS (2008) NCTM (2000) Problem Concept Development:  Ask questions that get them Solving 1) Analysis and Reasoning to check their  Why and/or how questions, problem understanding. solving, prediction/experimentation,  Engaging in a task for classification/comparison, evaluation which the solution method 2) Creating is not known in advance.  Brainstorming, planning, producing  Build on children’s innate problem solving inclinations. Reasoning & Concept Development: - Noting patterns & structure Proof 1) Analysis and Reasoning in real world situations  Why and/or how questions, problem - Classification solving, prediction/experimentation, - Learn to answer “why” classification/comparison, evaluation questions & describe thinking. Communication Regard for Student Perspectives:  Children share ideas 1) Student Expression  Children are expected to  Encourages student talk, elicits ideas explain or clarify their and/or perspectives understanding Quality of Feedback:  Children are given time to 1) Prompting Thought Processes talk with and listen to peers  Asks students to explain thinking, queries responses and actions Language Modeling: 1) Frequent Conversations  Back-and-forth exchanges, contingent responding, peer conversations 2) Open-Ended Questions  Questions require more than a one- word response Connections Concept Development: - Plan tasks in new contexts 1) Integration that revisit previously taught  Connect concepts, integrates with topics. previous knowledge - Connections should be made 2) Connections to the Real World among math topics, other  Real-world applications, related to subject areas, children’s students’ lives prior knowledge - Children learn math through connections to the real world.

Representation Instructional Learning Formats:  Promote children’s use of 1) Variety of Modalities and Materials multiple forms of  Range of auditory, visual, and movement representation. opportunities, interesting and creative materials, hands-on opportunities Note. From Principles and Standards for School Mathematics by the National Council of Teachers of Mathematics, 2000, Reston: VA, author. From Classroom assessment scoring system, manuel pre-K, by R. C. Pianta, K. M. LaParo, and B. K. Hamre, 2008, Baltimore, MD: Paul H. Brookes Publishing Co.

Table 2 shows the items from the CLASS that are comparable to the NCTM

Mathematics Process Standards.

As indicated in chapter one, research has shown that very little mathematical thinking occurs in the majority of preschool classrooms and what mathematics is evident typically consists of teacher-directed activities that focus on discrete skills or facts such as names of shapes and numerals, counting, simple patterning and sorting. Cross et al.

(2009), in describing findings from the NCEDL MS Study, state that, “children were less often exposed to instruction that was conversational, interactive and focused on understanding and problem solving” (p. 239). To formalize children’s intuitive notions and teach for understanding, teachers must also provide children opportunities for higher- level thinking by promoting problem solving, reasoning, communication, making connections, and representation. The CLASS also sees these processes as a necessity in quality preschool programming.

A quality assessment instrument that was developed specifically to focus on mathematics is the COEMET by Julie Sarama and Douglas Clements in 2007. It is a classroom observation tool that was designed to assess the quality of mathematics

52 teaching in preschool classrooms. This observation tool examines opportunities for mathematics during the routine activities of the school day and looks at qualities of specific mathematics activities planned by the teacher. It is evident when reviewing this tool that the NCTM processes are viewed as a necessary part of a quality mathematics curriculum in preschool. They are embedded in the majority of the 19 items that are rated under Specific Math Activity. Problem solving, for example, is emphasized under

Organization, Teaching Approaches, Interactions: “The teacher began by engaging and focusing children’s mathematical thinking – directed children’s attention to, invited them to consider, a mathematical question, problem, or idea” and under Expectations: “The teacher had high but realistic mathematical expectations of children” – “the teacher asked all children to try to solve problems and attempt various solution strategies.” (Sarama &

Clements, 2007)

The rest of the items also focus on the processes. Some excerpts include:

Eliciting Children’s Solution Methods: “The teacher asked children to share, clarify, and/or justify their ideas” (NCTM’s reasoning & proof). “The teacher encouraged children to listen to and evaluate others’ thinking/ideas” – “actively elicited communication between children, stated and reinforced the expectation that children would listen to each other” (NCTM’s communication). Extending Children’s

Mathematical Thinking: “The teacher encouraged mathematical reflection” – “drew out key ideas during and/or towards the end of the activity, helped children make connections to math ideas from other activities and/or real-life experiences” (NCTM’s connections).

From Kilday and Kinzie’s (2009) analysis of instruments designed to measure the quality of mathematics teaching in early childhood settings they conclude that the

COEMET is effective in measuring high quality mathematics teaching. One reason they state is that it uses the NCTM Standards to set goals and look at the lessons, teaching, and classroom environment, which all determine the quality of a program. It is evident that the NCTM mathematics processes are a vital part of a quality preschool classroom environment and curriculum.

Developmentally Appropriate Practice in Early Education

Quality preschool experiences can positively affect children’s later academic success (Barnett, Jung, Youn, & Frede, 2013; Duncan et al., 2007; Farran, 2000; Jordan et al., 2009). Studies show that children’s skills at entry to school predict future academic success (Stipek 2001, Phillips, Crouse, & Ralph 1998). Because children of low socio-economic backgrounds are typically behind their higher SES peers academically (Stipek & Ryan 1997, U.S. Department of Education, 2001), it is imperative that all children have access to high quality preschool programs. In the late

1980s (and revised in 1997 and 2009), the National Association for the Education of

Young Children (NAEYC), in an effort to improve the quality of preschool programs published guidelines for Developmentally Appropriate Practice (DAP) (Copple &

Bredekamp, 2009) that teachers and program directors can use to develop curriculum or plan appropriate experiences for children [Summary in Appendix E]. These practices provide a foundation for teaching that is based on research-based conclusions about how children learn. It is the individual teacher’s behaviors or practices in the classroom that

54 directly affect children’s cognitive development. The observation instruments described above all have elements that would be considered developmentally appropriate. As stated before, the COEMET describes behaviors that are DAP that directly related to mathematics but the others describe teaching practices more broadly. The following are examples of how DAP influence teachers’ instruction in mathematics:

- Play should be considered a learning context for mathematics, especially

block play, manipulatives, puzzles, and games. Children practice many

mathematics concepts during play. (Geist, 2009)

- Block play should be observed so that the teacher can engage children in

discussions of mathematics principles. Teachers should have planned,

systematic block building. (Kersh, Casey & Young, 2008)

- Board games should be offered to children because they increase numerical

knowledge. (DeVries, et al., 2002; Kamii, 1982)

Teachers also make planned decisions that influence children’s mathematical understanding regarding the appropriateness of the mathematical tasks, especially decisions that are part of daily routines or events, problems that arise naturally in daily living within a group context, and the use of mathematical tools, especially concrete objects. If tasks are meaningful to children, they are more motivated to think about particular concepts, make connections to other mathematical ideas, and apply these concepts to real-world contexts (NCTM, 2000). Researchers have found that children learn best when they have the intrinsic desire or motivation to learn (Dewey, Brownell,

Piaget). Bruner (1960 / 1977), for example, states that, “interest in the material to be

55 learned is the best stimulus to learning, rather than such external goals as grades or later competitive advantage” (p. 14). Dewey (1900 / 1990) agrees and emphasizes that teachers of young children need to recognize the “culture of imagination” inherent in how they learn where they “let their imaginations play about the current and familiar contacts and events of life” not just in myth and fairy tales. DAP support this notion of creating a classroom climate where children are encouraged to play off their interests, explore new materials, and use their imagination to discover new things.

Play is the primary vehicle for learning in the early years. However, teachers cannot rely on children spontaneously engaging in mathematics during play and should know that spontaneous mathematical thinking is not enough. Many materials and activities that promote mathematical thinking are essential such as books, puzzles, computer games, board games, and blocks, but it is the role of the teacher to formalize and extend these informal mathematics experiences of the children (Baroody, 1987).

NAEYC/NCTM (2002) states, “Significant benefits are more likely when teachers follow up by engaging children in reflecting on and representing the mathematical ideas that have emerged in their play. Teachers enhance children’s mathematics learning when they ask questions that provoke clarifications, extensions, and development of new understandings” (p. 11). However, teachers often focus on teaching mathematics content without attention to creating a classroom culture of inquiry due to pressures related to a lack of time during the school day to cover material and parent expectations for academic learning through traditional means such as worksheets. With the publication of the No

Child Left Behind Act (NCLB) (2001), Stipek (2006) explains how preschool teachers

56 are feeling the pressure to teach the academics included in this document. School districts are pushing up the cutoff date for children to begin kindergarten (Stipek 2002) so that the resulting older group of kindergarteners are ready to learn what used to be 1st grade concepts and achieve better scores on standardized tests. Preschool teachers are then pressured to make sure their students enter kindergarten knowing much of what used to be learned in kindergarten (Rudd & Lambert, 2009). Stipek (2006) warns that, “this could reduce the attention to things not tested such as critical, analytic, creative thinking and reasoning skills” (p. 456).

Teacher Knowledge, Behaviors and Attitudes/Beliefs

The quality of the learning environment has been found to influence children’s mathematical understanding. Davis, Maher, and Noddings (1990) stress the importance of teachers establishing a mathematical environment where children engage in mathematical activity that requires “constructive work with mathematical objects in a mathematical community” where a constructivist view is prevalent. The role of the community, which includes other learners and teachers, is to provide the setting, pose the challenges, and offer the support that will encourage mathematical construction (pp. 2-3).

According to Chapin and Eastman (1996), the learning environment consists of both external and internal characteristics. External characteristics relate to the physical environment, such as the room arrangement, and availability of materials. Internal characteristics relate to the personal qualities of teachers, including their attitudes and beliefs toward mathematics, attitude toward their students, and knowledge of mathematics. Chapin and Eastman emphasize that the internal characteristics, not

57 external characteristics, have the power to transform the classroom into a learning environment that promotes mathematics development.

Teacher Knowledge Defined

One problem that can affect the quality of teaching is differentiating between what it means to ‘know’ something and what it means to understand something. The idea of ‘knowing’ something can translate to remembering or memorizing something but does not mean there is a deep understanding of concepts. Linder, Costello and Stegelin

(2011), emphasize that an understanding of number sense goes beyond just naming numerals and counting but includes composing and decomposing sets, identifying relationships between numbers, and examining patterns in numbers. Without their own comprehensive understanding of mathematics content, teachers do not typically plan opportunities for children to move beyond counting in their development of number concepts.

John Dewey and contemporary researchers have emphasized the need for teachers to have a strong understanding of the subject matter (Ball, Hill, & Bass, 2005; Dewey,

1916; Ma, 1999; National Mathematics Advisory Panel, 2008). Steffe (1991) states, “A teacher in any particular discipline must be aware that in order to be a good teacher, one must above all have a thorough knowledge of the discipline. Of course, this condition is far from sufficient, but it is a necessary condition”(p. 32.) In a study by Thiel (2010), teachers were “open towards math” and saw its benefits for everyday living but

“perceived mathematical tasks as only those that involved the use of numbers and shapes.” In the same study, almost half of the teachers who felt reluctant toward math

58 and 12 percent of those who were not reluctant agreed that when “a child gives reasons for his opinion” that this does not further the development of mathematical thinking.

This illustrates the importance of teachers having knowledge of both mathematics content such as number and geometry and of processes such as problem solving and reasoning.

Mathematical knowledge for teachers includes both Mathematics Content

Knowledge (MCK) and Pedagogical Content Knowledge (PCK). It is not always clear what one refers to when using the term pedagogical content knowledge. Shulman (1986) first differentiated between the two in his research and since then PCK has been redefined or refined by many others, most notably Ball and Hill in the last ten years (Ball, Thames,

& Phelps, 2008; Hill, Ball, & Schilling, 2008a; Hill, Ball, Sleep & Lewis, 2007; Hill et al., 2008b; Ball & Bass, 2000). Grossman (1990) described it as knowledge of what students understand, knowledge of appropriate curriculum, and knowledge of appropriate instructional strategies. Manizade and Mason (2011) completed a synthesis of research on “teachers’ profession-specific knowledge”, a term used by Hill et al., (2007), which includes PCK. Manizade and Mason (2011) used this information to develop a definition of PCK as having four elements: (a) knowledge of connections among big ideas in math,

(b) knowledge of learning theories in child development, (c) knowledge of student misconceptions or areas of difficulty, and (d) knowledge of useful representations and appropriate teaching strategies (p. 188). This is the definition used for this study and is referred to as Mathematics Pedagogical Content Knowledge (MPCK). To highlight the usefulness of this definition for examining preschool teachers’ knowledge of the NCTM

Process Standards, the MPCK definition by Manizade and Mason was interpreted to

59 mean that teachers who are promoting children’s development and use of the mathematics processes must have knowledge of (a) how the five NCTM Process

Standards are intertwined and not viewed as five distinctly separate processes, (b) learning theory in mathematics education and child development, especially DAP, (c) students’ mathematical thinking, and (d) how NCTM envisions that teachers use the mathematics processes for children to learn mathematics content (see Table 3).

Table 3

MPCK Definition in Relation to Knowledge of NCTM Processes

Manizade & Mason (2011) This study –  Definition of PCK  Focus on NCTM Process Standards  Definition of MPCK

1. Teacher knowledge of connections 1. Teacher knowledge of how the five NCTM among big ideas in math Process Standards are intertwined and not to be viewed as five distinctly separate processes

2. Teacher knowledge of learning 2. Teacher knowledge of learning theory in theories in child development mathematics education and child development, especially DAP

3. Teacher knowledge of student 3. Teacher knowledge of students’ misconceptions or areas of mathematical thinking difficulty 4. Teacher knowledge of useful 4. Teacher knowledge of how NCTM envisions representations and appropriate teachers use the mathematics processes for teaching strategies children to learn mathematics content.

Note. Adapted from “Using Delphi methodology to design assessments of teachers’ pedagogical content knowledge,” by A. G. Manizade, and M. M. Mason, 2011, Education Studies in Mathematics, 76, 183-207. PCK = pedagogical content knowledge; MPCK = mathematics pedagogical content knowledge.

NCTM makes it clear that the mathematics knowledge a teacher must have consists of both content, such as number and geometry, and processes, such as problem solving and communication. The research surrounding child development and DAP also illustrates the importance of teacher knowledge of mathematics processes for understanding and overall quality of preschool programs.

Teachers’ MPCK has been found to influence student learning (Hill et al., 2005), and, therefore, it is an important part of a quality mathematics program in preschool.

Educators stress the importance of knowing how to teach mathematics content to a particular age group and that a lack of education by many preschool teachers results in less focus on mathematics language as a tool (Whitin & Whitin, 2003), fewer planned experiences that are connected to other math concepts or what children know (Rubenstein

& Thompson, 2002), and less attention to thinking and understanding (Stipek, 2006).

The research of Rudd et al. (2008) suggests that more attention on higher-level mathematics concepts is needed after finding a lack of mathematics activities planned in the preschool classrooms of degreed teachers at a quality university program. In a study by Howes (1997), a correlation was found between a teacher’s level of education and children’s language skills, cognitive skills, and complexity of their play. Botha, Maree and deWitt (2005) surveyed preschool teachers in South Africa during a workshop and found that teachers report they use communication the most (94.4%) then problem solving, reasoning, connections and representation the least (62.2%). He found that teachers will not plan for teaching math content in all areas or promote the math processes or use DAP strategies unless they know of these themselves (p. 713).

Therefore, a measure of teachers’ knowledge of mathematics, both MCK and MPCK, is important because mathematics instruction is heavily influenced by teacher knowledge of the subject.

Teacher Attitudes and Beliefs Defined

There is a positive relationship between attitudes and beliefs (Eagly & Chaiken,

1993; Fishbein & Ajzen, 1975) but not a strong consensus in the literature on how these are defined. In general, an attitude is considered a feeling and belief as what one considers as true (Maier, Greenfield, & Bulotsky-Shearer, 2013). Attitudes and beliefs have been described as multi-dimensional constructs where an attitude is composed of values, beliefs, and behaviors (Mueller, 1986; Maier et al., 2013). Oppenheim (1992) considers an attitude as the “tendency to respond in a given manner when confronted with certain stimuli” (p. 174) and Fishbein and Ajzen (1975), define attitude as “a learned predisposition to respond in a consistently favorable or unfavorable manner with respect to a given object” (p. 6). Teacher beliefs are what one considers to be true based on information that person has, whether or not that information is accurate (Atwater, 1994).

Hannula, Malmivouri and Pekhonen (1996) describe a teacher’s beliefs about mathematics teaching as determining a teacher’s attitudes and thus her behavior toward mathematics. According to these definitions, attitudes are feelings that affect how we react and respond to situations and beliefs are what a teacher perceives to know about subjects where both have influence over the other. Fishbein and Ajzen (1975) explain the relationship: “…once an attitude is established it may influence the formation of new beliefs” and “performance of a particular behavior may lead to new beliefs about the

62 object in turn influencing attitude.” For the purposes of this study, teacher attitudes and beliefs are considered one scale of measurement.

There have been surveys developed to measure teacher attitudes and beliefs in mathematics but they were designed for elementary teachers (Eagly & Chaiken, 1993;

Hannula et al., 1996; Sandman, 1980; Schifter & Simon, 1992; Wikins, 2008)) or the focus on teacher efficacy (Brown, 2005), or preservice teachers’ mathemtatics anxiety

(Gresham, 2007; Vinson, 2001). Bush and Kincer (1993) found that more research is needed specific to attitudes toward early childhood mathematics because most research is more general and can include even high school mathematics. There is consensus that both teacher attitudes and teacher beliefs influence teaching behaviors in the classroom, therefore, it would be beneficial to have a measure of preschool teachers’ attitudes and beliefs toward the NCTM Process Standards.

Relationships between Teacher Knowledge, Attitudes/Beliefs and Behaviors

In the area of elementary mathematics, much research has been conducted in the past 20 years on the relationship between changes in teacher knowledge and attitudes / beliefs and instructional practices. Some studies have shown that teacher knowledge of mathematics content (MCK) and mathematics pedagogy (MPCK) can change teaching practices, especially an increase in teacher knowledge of student thinking (Fennema &

Franke, 1992; Franke & Kazemi, 2001; Schifter & Fosnot, 1993; Simon, 1995; Simon &

Schifter, 1991). For example, in a study of 21 elementary teachers who participated in a

CGI (Cognitively Guided Instruction) workshop, 17 of these teachers changed their beliefs about children’s capabilities to solve problems without using traditional

63 algorithms. The change in beliefs influenced their teaching practices especially by focusing more on problem solving and the use of communication in mathematics.

However, the study also revealed that it was unclear if a change in beliefs impacted instruction or if changes in instruction impacted beliefs (Fennema et al. 1996).

In a 2008b article, Hill et al. note many past studies that they propose provides

“evidence for the proposition that stronger teacher knowledge yields benefits for classroom instruction and student achievement” (p. 431). However, they suggest that more research is needed on how this occurs. After a case study of ten elementary teachers that included an assessment of teacher mathematics content knowledge and classroom observations, Hill et al. conclude that “there is a powerful relationship between what a teacher knows, how she knows it, and what she can do in the context of instruction” (p. 496).

There are studies that have shown relationships between teacher attitudes toward teaching and learning and teacher behaviors in the classroom (Ajzen & Fishbein 1970 and 1980; Schuman & Johnson, 1976; Eagly & Chaiken, 1993, p. 157) and between teacher beliefs and behaviors (Schoenfeld, 1998). One problem regarding the beliefs – behaviors relationship is that some studies have found discrepancies between what preschool teachers believe about the importance of teaching mathematics and what they actually do in the classroom (Brown, 2005; Pianta & La Paro, 2003). In a book by

Carol Ann Wien (1995), she describes her observations of five preschool teachers over a five-month period to find out more about why developmentally appropriate practices were not evident in many early childhood classrooms. She reasoned that there were

64 competing ideologies that teachers were sometimes going back and forth between in terms of their practice. She describes these as frameworks of teacher dominion and developmentally appropriate practice. For example, one teacher explained that she did things a specific way because “…that was the way it was when I came, and I guess I think that’s the way it should be done, so that’s the way I do it” (p. 21). Another teacher, after being provided observations of her practice, was surprised to find herself portrayed as aspiring to a teacher dominion framework because “she thought she was much more child-centered than her practice indicated” (p. 37). A third teacher, who alternated between teaching styles, after reflecting on her teaching, articulated her true beliefs and made changes to her practice. These examples illustrate the need for professional development that offer teachers opportunities to increase their knowledge of what is appropriate and effective instruction and opportunities to reflect on their current behaviors to make changes in the classroom.

Several other studies illustrate the relationships between teacher attitudes, beliefs, knowledge, and behaviors. Graham et al. (1997) observed and interviewed four preschool teachers in two different settings. The authors found that there was very little informal or formal mathematics instruction for these classrooms even though the teachers stated that mathematics was important. In comparison, Trice and Ogden (1987) found that forty, first year teachers with positive attitudes spent more time during the day (an additional 20 minutes) teaching math. Thiel (2010), who also compared teacher attitudes and behaviors, found that teachers who were open toward math placed a higher emphasis on processes like problem solving in the classroom. Those who felt reluctant toward

65 math placed less emphasis on processes and application and tended to use more worksheets.

In two separate studies, Perrin (2008) and Appalachian Rural (2000), data showed that teachers who had more knowledge of the NCTM Mathematics Standards had more positive attitudes toward mathematics, attitudes aligned with the Standards, and more positive behaviors in terms of standards-based teaching. Weiss et al. (2003), found that the methods they chose to use in 9 out of 10 lessons and 28% of the content they taught were influenced by their beliefs and background knowledge.

In a study by Brown (2005), it was found that higher teacher efficacy and higher teacher beliefs in the importance of mathematics did not have a significant correlation with teachers’ use of standards-based methods of instruction. They concluded that this could be a result of teachers following district policies or teaching how they were taught when in school. Brown proposes there may be a lack of consensus among preschool teachers that math is important for preschool children and suggests that more research is needed on mathematics content beliefs of preschool teachers and the relationship to teacher practices and student outcomes.

Conclusion

The literature review highlights research in the field of early mathematics education that shows a need for more information on mathematics teaching at the preschool level. A historical look at mathematics reform illustrates the need for a focus on mathematics processes to promote student understanding of mathematics.

Recommendations from several research groups show the need for a focus on

66 mathematics processes at the preschool level. Research findings indicate three important reasons for this study: (a) time for mathematics during the preschool day is limited and focused mainly on basic skills with little attention to teacher-child interactions and higher-level thinking, (b) teachers of mathematics at all grade levels often lack content knowledge and pedagogical content knowledge that affects student achievement, and (c) positive relationships exist between teacher knowledge, behaviors, and attitudes so it is important to see if they also positively impact the teaching of mathematics in preschool.

Several valid observation instruments exist that examine some aspects of the mathematics processes at the preschool level but are not useful for a large-scale study.

Data collected in this study can inform professional development opportunities for in-service teachers and teacher education programs. The assumption is that an increase in teacher knowledge of the mathematics processes may influence teaching practices and ultimately improve children’s exposure to and understanding of mathematics concepts.

The next chapter will describe the methods and procedures for the study. This includes the rationale for the design of the self-report instrument and the procedures, results, and statistical analyses of each phase of development. Specifically, this includes information on the cognitive interviews, pilot study, and the research study. Validity and reliability measures are addressed along with details regarding the specialized sample of preschool teachers, and the administration of the survey.

CHAPTER III

METHODS AND PROCEDURES

Introduction

The goal of this study was to collect and analyze information on preschool teachers’ knowledge, behaviors and attitudes/beliefs regarding the NCTM Process

Standards. To gather this data a self-report survey instrument was developed. The instrument was named the Teaching Mathematics Processes in Preschool Survey

(TMPPS). The development of a survey is useful in order to collect district, regional, state or nation-wide data from teachers in early education settings that can be used to make recommendations to teacher education programs and meet the professional development needs of teachers.

The research objectives listed below provided the focus for data collection. The data collected from the survey was analyzed to answer eight research questions.

Research Objectives

1. To determine what preschool teachers perceive to know about the NCTM Process

Standards, (i.e., How the Process Standards are defined by NCTM and what

NCTM suggests for implementing them when teaching young children.)

2. To determine the frequency with which preschool teachers report they engage in

behaviors that promote children’s development and use of the mathematics

processes.

3. To determine preschool teachers’ attitudes/beliefs toward the value and

importance of incorporating the mathematics processes in the classroom. 67 68

4. To look for possible correlations between teachers’ perceived knowledge,

teachers’ reported behaviors, and teachers’ attitudes/beliefs regarding the NCTM

Process Standards.

5. To compare the perceived knowledge, reported behaviors and attitudes/beliefs of

licensed vs. non-licensed teachers and of teachers who work in various types of

early education settings.

Research Questions

1. What perceived Mathematics Content Knowledge and Mathematics Pedagogical

Content Knowledge do preschool teachers have of the five NCTM Process

Standards?

2. How often do teachers report they engage in behaviors that promote and develop

children’s use of the mathematics processes?

3. What are preschool teachers’ attitudes/beliefs toward the value of incorporating

NCTM Process Standards in the classroom?

4. How does teacher perceived knowledge of the Process Standards compare with

teacher reported behaviors that promote children’s development and use of the

mathematics processes?

5. How does teacher perceived knowledge of the Process Standards compare with

teacher attitudes/beliefs regarding the value and importance of the mathematics

processes in preschool?

6. How do teacher attitudes/beliefs regarding the value and importance of the

mathematics processes in preschool compare with teacher reported behaviors that

promote children’s development and use the mathematics processes?

7. Is there a difference between licensed and non-licensed preschool teachers in

terms of their reported behaviors, attitudes/beliefs and perceived knowledge in

regards to each of the five Process Standards?

8. Is there a difference between the perceived knowledge, reported behaviors and

attitudes/beliefs of preschool teachers who work in different types of settings in

regards to each of the five Process Standards?

As noted in Chapter II, there are few empirically validated measures available for studying the mathematics teaching of preschool teachers. There are several observation instruments such as the ECERS-R, COEMET, and CLASS for example, that address some aspects of mathematics teaching at the preschool level but these are not useful for gathering information from a large population of teachers. Current published surveys do not address the specialized objectives that this study targets regarding preschool teachers perceived knowledge of the NCTM Process Standards; therefore, a survey instrument specific to this line of inquiry, the TMPPS, was developed to address the research questions.

Rea and Parker (2005) suggest that surveys are advantageous because, (a) they can be used to make generalizations about all preschool teachers based on data collected from a small sample, and (b) survey data is standardized and therefore able to be replicated using statistical analyses. Teacher self-report surveys provide a relatively

70 accurate picture of classroom practice and are practical when collecting data from a large population of teachers in terms of time and expense compared to conducting classroom observations and teacher interviews (Rea & Parker, 2005). Because the information is personal and requires self-reporting by teachers, the survey approach is a good fit (Rea &

Parker, 2005).

The survey was used to describe information relevant to a group of preschool teachers at one point in time. Rea and Parker (2005) state that in order to understand the

“complexities of the population” descriptive information, behaviors of teachers, and teacher attitudes/beliefs are all important to address with a survey. Survey data from this study included demographic information of preschool teachers, teachers’ perceived knowledge of the NCTM mathematics process standards, teachers’ attitudes/beliefs regarding those standards, and teachers’ reported behaviors related to how they promote children’s development and use of the process standards. The survey data was also used to compare the perceived knowledge, attitudes/beliefs, and reported behaviors of licensed teachers and non-licensed teachers and of teachers who work in different early education settings.

In this chapter, I will provide an overview to address the validity and reliability of the instrument. Then I will describe the development of the survey in terms of context and design. This will include a description of the methods for survey development and the statistical measures that were used to assess reliability and validity and to answer the research questions.

Research Design

A self-report survey, developed specifically for this study, was used to investigate the perceived knowledge, self-reported behaviors, and attitudes/beliefs of preschool teachers regarding the NCTM Process Standards: Problem solving, reasoning & proof, communication, connections, and representation. The TMPPS was sent to preschool teachers in the state of Ohio through email using an online survey tool. The survey was made up of four sections. Sections one, two and three each focused on measures of the three dependent variables of the study: (a) teachers’ perceived knowledge, (b) teachers’ reported behaviors, and (c) teachers’ attitudes/beliefs, all regarding the five mathematics processes listed above. The fourth section collected demographic information that was used to answer Research Questions 7 and 8 regarding the affects of teacher licensure on the dependent variables and possible correlations between the dependent variables and the type of setting in which a teacher works.

Dependent Variables

Teachers’ Perceived Knowledge - Teacher knowledge is defined as both mathematics content knowledge (MCK) and mathematics pedagogical content knowledge

(MPCK). For this study, mathematics knowledge includes, (a) how the five NCTM

Process Standards are interconnected and not to be viewed as five distinctly separate processes, (b) learning theory in mathematics education and child development, especially DAP, (c) students’ mathematical thinking, and (d) how NCTM envisions teachers use the mathematics processes for children to learn mathematics content

(Manizade & Mason, 2011--see Table 3)

Teachers’ Reported Behaviors – Teacher behaviors are those that NCTM (2000) suggests in their standards document promote children’s use and development of problem solving, reasoning & proof, communication, connections, and representation. These behaviors include appropriate practices for children in preschool settings according to

NAEYC’s Developmentally Appropriate Practices (Copple & Bredekamp, 2009).

Teachers’ Attitudes/Beliefs – Teacher attitudes/beliefs are measured together due to the reciprocal relationship between the two attributes. Attitudes are defined as feelings that affect how teachers react and respond to situations and beliefs are what a teacher perceives to know about subjects where both have influence over the other (Atwater,

1994; Fishbein & Ajzen, 1975; Maier, et al. 2013; Oppenheim, 1992).

Independent Variables

There are two categories for the independent variable of teacher licensure status: licensed teacher and non-licensed teacher. A licensed teacher refers to someone with a state license to teach preschool age children. A non-licensed teacher refers to someone without a college degree or someone with a college degree, but not licensed by the state to teach preschool.

There are five categories for the independent variable of early education setting:

For-profit childcare center, public school program, university laboratory school, Head

Start program, and family/home provider. During the analysis of the data, the categories were revised to include a category labeled special education, and the for-profit childcare center category was renamed private childcare centers to include any center that was labeled by respondents as for-profit or private.

Population and Sampling Plan

The nature of this study was exploratory. The goal was to gather data from preschool teachers at a variety of settings that would provide a focus for future research regarding teachers’ mathematics knowledge and classroom practice. The research intent was mainly descriptive so broad coverage was more important than reducing error

(Henry, 1990). The target population was preschool teachers (children ages three to five years) in the state of Ohio, in settings that include public preschool, Head Start, for-profit childcare centers, family childcare, and laboratory schools.

A probability sampling procedure was not used as all preschool teachers in the state were targeted. To avoid selection bias, all preschool teachers were given equal opportunity to participate. This is considered a self-selected convenience sample. Using

GPower software, 176 participants were required for a medium effect size of 0.5, and alpha at 0.05 for an independent t-test.

A total of 217 participants completed the survey in its entirety. The survey results listed in Table 4 show that the majority of respondents teach in a public school setting and with children ages 3 to 5 years. The range of years of experience teaching the preschool age group is 0.5 years to 35 years. The majority of teachers have a master’s degree and only four reported not having a college degree of any kind (see Table 5). Of the 217 who responded, all but 13 have a degree in the field of education with 188 teachers having obtained a state teaching license to teach preschool.

Table 4

Type of Early Education Setting ______For-Profit Head Start Public School Other ______4.2% 18.5% 52.3% 24.9% n=9 n=40 n=113 n=54 ______

Table 5

Highest Level of Education ______High school diploma/ CDA Associate’s Bachelor’s Master’s Equivalency degree degree degree ______.9% .9% 10.6% 32.9% 54.6% n=2 n=2 n=23 n=71 n=118

Validity and Reliability

In order to answer the research questions a self-report survey instrument was developed, the Teaching Mathematics Processes in Preschool (TMPPS) survey.

Therefore, careful consideration was given to issues of validity and reliability. Wiersma and Jurs (2009) define validity as dealing with the “accurate interpretability of the results

(internal validity) and the generalizability of the results (external validity)” (pg. 9). They define reliability as the “replicability and consistency of the methods, conditions, and results” (pg. 10). The two are related in that the survey must be reliable for it also to be considered valid. To address issues of validity and reliability the survey development occurred in two phases:

a) Cognitive interviews & content reviewer data

b) Pilot study

In this section the participants, procedures and results for developing a valid and reliable survey instrument are described. Phase 1 consisted of a review of the survey items by practicing teachers and by early childhood mathematics educators to address both face and content validity. Phase 2 consisted of two groups of pre-service teachers completing the survey for estimates of reliability. The resulting revised instrument was then issued to in-service teachers to gather data regarding the research questions of the study.

Face validity and content validity. Phase one of the survey development was used to establish face validity and content validity. Content validity is defined by

Crocker and Algina (1986) as an assessment of whether the items represent the construct or performance domain they are intended to measure. The survey items each match a specific mathematics process. Additionally, the items for the Behaviors scale match a characteristic of what is Developmentally Appropriate Practice (DAP). The five NCTM

Process Standards and DAP in early childhood programs are defined in Table 8 and Table

7. The survey item and the mathematics process and DAP characteristic they match are listed in Table 6 and Table 7. Professors with expertise in mathematics education reviewed the survey items to assess content validity. The two individuals were chosen to be reviewers because they both have doctoral degrees in Curriculum and Instruction with a focus on mathematics and experience teaching mathematics methods courses at the university level, one of which teaches a course designed for the preschool years. Ten

76 current and former preschool teachers assessed the wording, order, interpretation, and clarity of the items to establish face validity of the survey. The procedures used to establish both content and face validity will be described later in this chapter.

Reliability. Besides the advantages of low cost and timely data when using internet surveys, Dillman (2000) found that when comparing mail and web survey methods, the reliability of the web survey was higher. Reliability measures the internal consistency of the items for the three constructs of the survey: knowledge, behaviors and attitudes/beliefs. It also measures the consistency of the questions in the subsections for those three constructs, the five mathematics processes. For example, does each question regarding participant perceived knowledge of problem solving correlate with all other questions meant to measure teacher perceived knowledge of problem solving?

Cronbach’s Alpha was used to test for internal consistency during the pilot study and the research study. Cronbach’s Alpha was computed for the three scales of the survey and for each reported factor or process standard within each section to provide an estimate of reliability coefficient.

Development of the Instrument: Teaching

Mathematics Processes in Preschool Survey (TMPPS)

After a review of the literature on early childhood mathematics reform efforts, the five NCTM Process Standards were chosen as the focus for this survey. They are problem solving, reasoning & proof, communication, connections, and representation.

Over the past thirteen years, the mathematics processes, identified as five standards by

NCTM, have consistently and prominently been identified as recommendations for improved teaching of mathematics in early education contexts.

Publications by NCTM (2000, 2006), the National Research Council’s Adding it

Up (2001), the Conference on Standards for Prekindergarten and Kindergarten

Mathematics Education (Clements et al., 2004), and NAEYC and NCTM (2002) have all identified the need for teachers in early education contexts to promote children’s development and the use of mathematics processes in order for children to learn mathematics content with understanding. For example, one recommendation is that,

“…mathematical processes such as problem solving, reasoning and proof, communication, connections, and representation; specific mathematical processes such as organizing information, patterning, and composing; and habits of mind such as curiosity, imagination, inventiveness, persistence, willingness to experiment, and sensitivity to patterns should all be involved in a high-quality early childhood mathematics program”

(Clements et al., 2004, p. 3). Another recommendation is “that teacher education programs provide more coursework and practicum experience that emphasizes mathematics for young children” (National Research Council, 2001a). Therefore, a survey of what teachers in early education contexts report they currently know, do, and believe about the NCTM Process Standards can inform teacher education programs and other professional development programs for in-service teachers. Constructs for the design of the survey instrument include:

a. Perceived Knowledge of NCTM Mathematics Process Standards

b. Self-reported behaviors of mathematics teaching of the Process Standards

c. Attitudes/beliefs of teachers on the value or importance of the Process Standards

The self-report survey instrument developed was used to assess three constructs: preschool teachers’ knowledge of the five NCTM Process Standards, teachers’ reported developmentally appropriate behaviors that promote children’s development and use of the NCTM mathematics processes, and teachers’ attitudes / beliefs on the value of the

NCTM mathematics processes in preschool. The three constructs are based on the review of the literature in early childhood mathematics (Clements et al., 2004; Cross et al., 2009;

NAEYC/NCTM, 2002) and early childhood education (Copple & Bredekamp, 2009).

Research suggests that children learn mathematics concepts such as geometry, measurement, and number operations with understanding when children are engaged in problem solving, reasoning, communication, making connections, and representation of mathematical thinking. Therefore, NCTM’s five Process Standards were used as the framework for the instrument’s development.

The first construct, teacher perceived knowledge, includes mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK). The second construct, teacher reported behaviors, is based on MCK and MPCK along with

NAEYC’s Developmentally Appropriate Practices for teaching preschool children

(DAP). The third construct, teacher attitudes/beliefs, is used to describe teacher attitudes/beliefs toward promoting children’s development and use of the mathematics processes in preschool classrooms. Teacher MCK, MPCK, and DAP are addressed in one of the recommendations from Cross et al. (2009):

“An essential component of a coordinated national early childhood

mathematics initiative is the provision of professional development to

early childhood in-service teachers that helps them a) to understand the

necessary mathematics, the crucial teaching-learning paths, and principles

of intentional teaching and curriculum and b) to learn how to implement

the curriculum” (p. 347).

Survey Design

A Likert scale is used for three sections of the survey instrument in order to collect a large set of data and provide more consistency in the responses across participants (Wiersma & Jurs, 2009). Using an open response format would be time inhibitive. The survey items were written as “I” statements. Participants were asked to report on their perceived knowledge, attitudes/beliefs, and behaviors regarding the mathematics processes. To address the issue of non-response rate for the survey, the

Qualtrics software program used to construct the survey was programmed to require that a participant respond to a question before moving on to the next question. Therefore, when the survey is completed, all items would be answered.

To reduce measurement error when developing a survey instrument, it is recommended in psychometric theory that multiple questions are used to measure a concept (Hox, 2008). Reducing measurement error increases the reliability of the scale or instrument. Therefore, multiple questions were developed for each of the three constructs: Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs. There were also multiple questions for each mathematics process (problem solving, reasoning &

80 proof, communication, connections, and representation) within each of the three constructs (Table 6). Table 7 identifies which items were designed to address specific aspects of what is meant by Developmentally Appropriate Practice for the Self-reported

Behaviors of Mathematics Teaching of the Process Standards scale.

Table 6

Survey Items for each Scale and each NCTM Mathematics Process.

NCTM Process Perceived Knowledge Self-Reported Attitude/Belief Standards items Behavior items items

Problem Solving 1, 21 26, 27, 29 41a, 42a, 43a, 44a, 45a 3, 10, 11, 18 Reasoning & 9, 16, 25 34 41c, 42c, 43c, Proof 44c, 45c 13 Communication 2, 12, 14, 17 32, 36, 38 41b, 42b, 43b, 44b, 45b 19 Connections 4, 23 28, 30, 39 41d, 42d, 43d, 44d, 45d 7, 20, 22 Representation 6, 8, 15 31, 33, 37 41e, 42e, 43e, 44e, 45e 24 All processes 35 46

5 Note. Negatively phrased items are in line 2 of each section and are italicized and bolded.

Table 7

Items of DAP for the Self-reported Behaviors of Mathematics Teaching of the Process Standards Scale

Developmentally Appropriate Practices Survey questions Teachers engage children in mathematical thinking in many ways.  Play: unit blocks, dramatic play, manipulatives, 29, 30, 31, 37, 38 puzzles, art, sand & water, woodworking, outdoors  Daily routines: snack, attendance, clean-up, transitions 27  Group games: card, dice & board games 31  Children’s literature & music 28  Projects / long term investigations 42

Teachers capitalize on spontaneous moments to introduce, revisit, examine and discuss mathematical ideas – model 35 mathematics language

Teachers provide children the following experiences regularly (classroom culture) :  Actively explore with materials 37, 38,  Interact with children and teachers 32, 33, 34, 36, 38, 39  Select their own activities from a variety of learning 29, 30, 35 areas  Work individually or in small groups 39  Engage in problem solving, reasoning and 26, 27, 29, 37 experimentation  Share ideas, ask questions, and negotiate with others 32, 34, 38 Note. Adapted from Developmentally appropriate practice in early childhood programs serving children birth through age 8: A position statement of the National Association for the Education of Young Children by S. Bredekamp and C. Copple (Eds.). (1997). Washington, DC: NAEYC.

There are also additional challenges cited in research that are specific to the

Reported Behaviors section of the survey. de Leeuw, Hox and Dillman (2008) state that behaviors are difficult to assess unless they are rare or very important, so the authors suggest that a short and recent reference period is used. The Likert scale provides options for the frequency of behaviors from as recent as several times a day to as infrequent as once a semester. This limits recall to a four or five month period of time.

Recall seems to improve with more time and repeated attempts so asking follow up questions in a separate extension to this study can provide the second attempt needed to get additional information (de Leeuw et al., 2008).

Another challenge is obtaining accurate information from participants, especially for the Reported Behaviors section and for the Attitudes/Beliefs section. Cannell, Miller and Oksenberg (1981) state that participants usually use inference strategies to arrive at an estimate of the frequency of behaviors, and we should be aware that “people commonly overestimate low frequencies but underestimate high frequencies” (p. 24).

One thought is to ask frequency questions in an open response format to avoid systematic biases associated with frequency scales even though answers may not be as accurate

(Cannell et al., 1981, p. 27). The open response format was not used with the hope that answers would be more accurate. Not using an open response format would also reduce the amount of time it takes respondents to complete the survey, which is important for increased response rates. One other suggestion by Cannell et al. (1981) is to encourage respondents to invest their time in giving thoughtful and accurate estimates for behaviors.

A carefully crafted cover letter was used to provide the encouragement needed for participants to provide the time and thought necessary in drafting their responses.

Perceived knowledge of NCTM mathematics process standards. NCTM’s

Principles and Standards for School Mathematics (2000) five Process Standards were used to define each construct, the five Processes, and develop the survey items. The

Teacher Perceived Knowledge section contains items that address a teacher’s mathematics content knowledge (MCK) and mathematics pedagogical content knowledge

(MPCK). For each Process Standard, NCTM describes what that process should look like during the early years (MCK) and what the role of the teacher should be in developing that process in the early years (MPCK). Using the Communication Process

Standard as an example, NCTM states that, during the early years, communication refers to ways for children “to share ideas and clarify understanding” (p. 60). This refers to teachers’ MCK and in the survey, and is represented by question #17 that says, “I feel it should be expected that children in preschool learn to question the mathematical strategies of others and/or ask for clarification.” NCTM, also in reference to the

Communication Process Standard, states that the teacher’s role is to provide time for children to pose questions, introduce mathematics language, and provide opportunities for them to exchange ideas with others. This refers to teachers’ MPCK and in the survey, is represented by question #14 that states, “I feel it is important to take the time during the school day – even if it means giving up time for other planned activities – to teach preschoolers how to work as a community of learners.” Table 8 provides the NCTM

84 definitions for both the MCK and MPCK of each Process Standard and the draft survey items can be found in Appendix A.

The descriptions of the five constructs, the five mathematics processes (Table 8), were used to inform the content validity of the survey items. For construct one, Teacher

Perceived Knowledge, the survey included 25 questions using a 5 point Likert scale format for respondents to answer with Agree, Tend to agree, Don’t know / Neutral, Tend to disagree, Disagree. [See Appendix A] Thirteen questions were written in the positive and twelve were written in the negative. [See Table 6] Since the original coding of the items had Agree as 1, Tend to agree as 2, Don’t know / Neutral as 3, Tend to disagree as

4, and Disagree as 5, the thirteen positively phrased items were reverse coded before analysis.

Table 8

Definitions of the NCTM Process Standards

NCTM Process Teachers’ Mathematics Teachers’ Mathematics Pedagogical Standards Content Knowledge Content Knowledge (MPCK) (2000) (MCK) Problem “Engaging in a task for which Teachers should: Solving the solution method is not  Present problems in a variety of known in advance” (p. 52). contexts and daily routines.  Know when to probe, give withhold “…children learn basic skills comments and plan similar tasks. along with higher-order-  Give children time for thinking. thinking skills and problem  Listen carefully to children’s solving strategies while explanations. engaging in problem solving  Encourage children to work together. (p. 121).  Encourage children to reflect on, explain, and justify their answers to others.

Reasoning and “Noticing patterns, structures Teachers should ask specific types of Proof or regularities in real world questions: situations and symbolic  “Why do you think it is true?” objects” (p. 56).  “What will happen next?”  “Is this always true?” Pattern recognition and  “Why does this work?” classification are important in These help children learn to make and test the early grades. conjectures, develop an argument, and generalize.

Children should be expected to explain their thinking and make comparisons to their own ideas. Communication A way to share ideas and Teachers should: clarify understanding. “The  Plan time for students to reflect, ideas become objects of model and pose questions. reflection, refinement and  Accept and use multiple forms of discussion” (p. 60). communication  Help students learn to talk about Children can communicate mathematics by introducing verbally and with other forms mathematics language. of representation.  Provide many opportunities for children to exchange ideas with teachers and peers. Connections “The interrelatedness of Teachers should: mathematical ideas” (p. 64).  “Plan tasks in new contexts that revisit previously taught topics” Connections should be made  “Ask questions that emphasize the among math topics, to other mathematical aspects of situations.” subject areas, and to real world  “Capitalize on unexpected situations. opportunities for learning.”  Embed mathematics in many activities throughout the day including those for physical education, music, science, and literacy.  “Challenge children to apply mathematics learning in extended projects and investigations” and use their own strategies (p. 133-135).

Representation “The act of capturing a Teachers should: mathematics concept or  Promote children’s use of multiple relationship in some form and forms of representation. to the form itself” (p. 67).  “Encourage students to represent their ideas in ways that make sense to Representation is both a them.” process and a product. It is  Help children begin to translate these used to help children organize to more conventional representation. their thinking.  Provide objects that they can manipulate to represent an idea.  Encourage children to share representations so it helps them “consider other perspectives and ways of explaining their thinking” (p. 139). Note. Principles and standards for school mathematics, by the National Council of Teachers of Mathematics, 2000, Reston, VA: Author.

Self-reported behaviors of mathematics teaching of the NCTM process standards. The Teacher Reported Behaviors section contains items that address a teacher’s MPCK and DAP for teaching young children. Along with the same information from NCTM identified above, NAEYC’s Developmentally Appropriate Practice in Early

Childhood Programs (Copple & Bredekamp, 2009) was used to develop the survey items related to teacher behaviors in early education contexts that are age appropriate and specifically promote children’s development and use of the five mathematics processes.

Table 7 provides the additional description of practices that are considered developmentally appropriate for young children by NAEYC that were used for item development. This information is based on current research in the field of early childhood education (Copple & Bredekamp, 2009) and informs the content validity of the survey items in this section. The items in this section of the survey should reflect both a) research on DAP in early education settings, and b) research on appropriate mathematics

87 for young children. Using the Communication Process Standard as an example, NCTM states that during the early years the teacher’s role is to provide time for children to pose questions, introduce mathematics language, and provide opportunities for them to exchange ideas with others. DAP suggests that children have opportunities to engage in mathematical thinking during play. Survey question #32 addresses these two definitions stating, “I facilitate children’s play to help children learn to communicate their mathematical ideas/thinking to others.” Table 9 provides characteristics of DAP and the survey items for the Reported Behaviors scale can be found in Appendix A.

The second construct, teacher reported behaviors, was included in the survey as

14 questions (25 for the pilot study) for respondents using a 6-point Likert scale to answer with "Never," "One or two times a semester," " One or two times a month," "One or two times a week," "Daily," or More than once a day." (See Appendix A) These were coded from zero to five respectively whereas the more frequent the desired behavior the higher the value it received.

There was also one additional question that allowed for an open-ended response regarding challenges participants perceived that they had encountered in making their own judgments about what and how to teach mathematics. There were seven options to choose from including “Following a prescribed curriculum," "not enough time for planning," "staffing issues," "length of school day," "access to materials or resources,"

"pressure from parents/families," or " no challenges.” They could also write in “other” perceived challenges that were not on the list of options.

Table 9

Examples of Best Practice in Early Education Settings

Developmentally Appropriate Practices for teaching mathematics in preschool Bredekamp & Copple (1997); Copple & Bredekamp (2009) Clements (2001) Kamii (1982) Teachers use the following ways to engage children in mathematical thinking (play being most important):  Play: unit blocks, dramatic play, manipulatives, puzzles  Daily routines: snack, attendance, clean-up, keeping records  Classroom happenings: voting, collection & distribution of materials  Projects / long term investigations  Group games- card games, board games  Children’s books, art, music, science, sand & water play, cooking, woodworking, outdoor experiences  Computer games Teachers capitalize on spontaneous moments to introduce, revisit, examine, discuss mathematical ideas  Teachers help children make connections among mathematics topics  Observe children during activities / play  Pose good questions for children to think about  Model the language of mathematics Teachers provide children opportunities to:  Actively explore with materials  Interact with children and teachers  Select their own activities from a variety of learning areas  Work individually or in small groups  Engage in problem solving, reasoning and experimentation  Share ideas, ask questions, negotiate with others

Note. Adapted from Developmentally appropriate practice in early childhood programs serving children birth through age 8: A position statement of the National Association for the Education of Young Children by S. Bredekamp and C. Copple, (Eds.). (1997) and Developmentally appropriate practice in early childhood programs serving children birth through age 8, 3rd Edition by C. Copple & S. Bredekamp (Eds.). (2009). Washington, DC: NAEYC. Number in preschool and kindergarten: Educational implications of Piaget’s theory, by C. Kamii, 1982, Washington, D. C.: NAEYC. Mathematics in the preschool by D. Clements, 2001, Teaching Children Mathematics, 7(5), 270-275.

This question regarding perceived challenges to making judgments about one's own teaching is included because one issue of data quality specific to classroom teachers is that they may have curricular or time constraints that inhibit their ability to act in ways they want to act regarding their promotion of children’s use and understanding of the

Process Standards in the classroom. The quality of the classroom environment alone can make a big difference in terms of opportunities provided for children, modeling by teachers that lend themselves to children’s comfort, confidence and ability in communicating mathematical ideas, solving problems as they arise in daily living, making connections and reasoning about mathematical ideas that arise in play, or representing their mathematical ideas in multiple ways (see Pianta et al., 2008 –

CLASS). Therefore, even if there are school constraints that teachers perceive to be affecting their actions or behaviors, knowledge of mathematics and knowledge of DAP in preschool settings can influence how teachers find a variety of ways to provide quality mathematics experiences within any constraints they may have. A statement at the beginning of this section indicates that there will be a question on the survey after the

Behavior items are answered where participants can share any perceived school constraints. This was done to encourage honest responses pertaining to their perceived classroom behaviors and to encourage them to complete the survey thoughtfully.

Attitudes/beliefs of teachers on the value or importance of the NCTM process

standards. As described in Chapter II, there is evidence that a teacher’s attitude and beliefs regarding a subject can influence her classroom practices when teaching that subject. Therefore, several items on the survey address teacher attitudes and beliefs

90 regarding the mathematics processes and their value or importance at the preschool level.

Attitudes are described as a predisposition to consistently respond in a certain way to stimuli (Fishbein & Ajzen (1975) and beliefs are described as what one believes to be true based on information that person has (Atwater, 1994). Beliefs determine attitude formation (Hannula et al., 1996) and attitudes are comprised of values and beliefs

(Mueller, 1986). The Attitudes/Beliefs scale contains items that look at both, (a) attitudes

- how teachers feel about their knowledge and teaching of the processes, and

(b) beliefs - what teachers consider to be accurate in regards to a focus on mathematics processes in preschool. For example, question #49 addresses teacher attitudes toward the math processes stating, “I feel confident in my knowledge of how NCTM defines each of the following process standards: Problem solving, communication, reasoning and proof, connections, and representation.” [Participants are asked to rate each of the five processes separately.] Question #41 addresses teacher beliefs regarding the math processes stating, “I feel the use of the following mathematics processes are necessary for preschool children’s understanding of mathematics content such as number concepts, measurement, and geometry: Problem solving, communication, reasoning and proof, connections, and representation.” [Participants are asked to rate each of the five processes separately.]

These items are general questions based on the NCTM mathematics processes and are designed to evaluate teacher confidence so as to determine the professional development needs of preschool teachers. Both the NCTM standards and the NAEYC documents were used to develop these items. This third construct, teacher

91 attitudes/beliefs toward promoting children’s development and use of the mathematics processes in preschool classrooms, was included in the survey as six questions for respondents to answer using a 4 point Likert scale format with Agree, Tend to agree,

Tend to Disagree, and Disagree. Questions #41 to #45 were coded as five distinct questions to address each of the five NCTM Process Standards where #41a addressed problem solving, #41b addressed communication, #41c addressed reasoning & proof,

#41d addressed connections, and #41e addressed representation. Question #46 addressed all processes in general (see Appendix A for survey items). Since the original coding of the items had "Agree" as 1, "Tend to agree" as 2, "Tend to disagree" as 3, and "Disagree" as 4, these 26 items were reverse coded before analysis.

Demographic information of survey respondents. The fourth section of the survey asks respondents to answer ten questions related to demographic information.

This is placed last because research suggests putting the questions that require the most thought first so that as respondents become mentally tired the easier questions are toward the end of the survey. The questions were chosen in order to not only provide data for the research questions regarding teacher licensure and type of work setting but also to provide data that could be used to answer research questions in future studies. Examples of demographic information in this section includes years teaching in general, years teaching preschool, educational qualifications, professional development training hours, how useful trainings have been for classroom teaching, and the type of setting in which they currently work (See Appendix A). Some of this data was described above under

“population and sampling plan” including the type of setting where they teach, ages they teach, years of teaching experience, and highest level of education.

Phase 1 Development: Cognitive Interviews & Content Reviewer Data

Phase one of the study is important for establishing content and face validity. The characteristics of the five Process Standards and what is considered Developmentally

Appropriate Practice in early childhood programs are listed in Tables 8 and 9 along with the items that have these characteristics (Tables 6 and 7).

Content reviewers. Two individuals with doctoral degrees in Curriculum and

Instruction with a focus on early childhood education and experience teaching mathematics methods courses at the university level were asked to review the survey items. They were given a draft of the items, the table that defines each of the processes and what is DAP, and pages in NCTM’s PSSM and in NAEYC’s DAP to review. At least two individuals had to agree that the item measured the area it was stated to measure from NCTM and NAEYC documents for “acceptable content validity to have been attained.” As suggested by Ross et al. (2003), four questions were asked of each of the two college professors (edited slightly to fit the purpose of this study):

a) Does the item involve an important dimension of mathematics for young

children?

b) What is the importance of this dimension of mathematics for teachers?

c) What is the meaning of this intended dimension of mathematics teaching?

d) Were any of the items unclear? In what way?

A rubric was used to analyze the responses of the content reviewers regarding how the constructs were defined and represented in the survey. For example, in Table 10, a sample rubric for the first four items of the survey is provided. The content reviewers were expected to agree with my interpretation of each item as indicated for the first three questions. If there was disagreement among the two content experts and/or myself then that question was discarded or revised. See Appendix B for Content Reviewer data.

Table 10

Item Criteria / Rubric for Content Reviewers - Example Using the First Four Questions

Q# Dimension Dimension Dimension meaning Best answer Items importance needing revision 1 Problem solving Math It is only a problem if 1 - agree knowledge you don’t know (MCK) something 2 Communication Math Time for children to 1 - agree teaching talk about math (MPCK) meaningful to them 3 Problem solving Math Capitalize on 4 – tend to teaching children’s interest and disagree (MPCK) motivation 4 Connections Math Accommodation and 1 - agree knowledge assimilation (MCK) Note. MCK = mathematics content knowledge; MPCK = mathematics pedagogical content knowledge.

Results of the content reviewer rubrics for the first three questions are summarized. Neither content expert suggested a question be deleted. They both related each question to at least one process standard. There were only two questions where both reviewers disagreed with what I considered to be the dimension addressed by the

94 question, #29, “I pose mathematics problems to children who are engaged in block play” and question #31, “I provide and support children’s play with card and / or board games.”

The intent of question #29 was to address the value of integrating mathematics with children’s natural play interests – to formalize the mathematics they naturally when engaged in block play. Therefore, the dimension listed for this question was the

Connections Process Standard. The reviewers listed the dimension as Reasoning & Proof and Representation. The intent of question #31 was to address the value of representing number in a variety of ways such as those found in card or board games: dots or numerals on dice, pictures and numerals on cards, spinners or spaces on the board, and adding up numerals or counting objects. Therefore, the dimension listed for this question was the

Representation Process Standard. The reviewers listed the dimension as Problem Solving and Connections. However, because the five Process Standards are designed to be inter- related, it is easy to see how block play and board games can be used by teachers to support children’s development of all five standards. For this reason, and because blocks and games are very common materials in preschool classrooms, I have kept both of these items in the survey.

In response to question four on the rubric, “Were any of the items unclear? In what way?” one reviewer indicated that the wording of eight survey items was unclear so several revisions were made to these items. For items 7, 24, 25, 26, changes were made to one or two words in each item in order to clarify these for the reader. For items 14, 15,

41 the original wording was maintained and item 39 was removed from the final version of the survey.

Cognitive interviews. In an article by Karabenick et al. (2007) cognitive validity is described as one aspect of determining the validity of a survey that focuses on whether or not the interpretation of survey questions by respondents is the same as what was intended by the researcher. Cognitive interviews are described as a form of cognitive pretesting that can be used in the development of an instrument and are considered valuable when the data is collected in a systematic and quantifiable way for the purposes of determining the validity (cognitive validity) of the survey. Karabenick, et al. (2007) suggest that when using cognitive pretesting such as cognitive interviews that the interviews are structured, coding criteria is applied to the interview data, and the items with low cognitive validity are then modified “to improve their performance given the validity criteria adopted” (p. 142). The validity criteria used for item development were described previously and outlined in Tables 8 and 9 – the research based NCTM definitions of the five Process Standards and the research based definition of what are considered Developmentally Appropriate Practices in an early childhood classroom.

Appendix B contains notes from the cognitive interviews.

Participants for Cognitive Interviews

The survey was given to ten faculty members at the university laboratory school to complete in order to provide feedback on the wording and interpretation of the questions and to gain an estimate of how long the survey would take to complete.

Revisions were then made to the questions based on this feedback. Five are current preschool teachers, one is a current teacher of two and three-year-olds, one is a current kindergarten teacher, and three are former preschool teachers who are now in

96 administrative or supervisory roles at the center. The three former preschool teachers were provided modified directions for the behavior section of the survey. Rather than indicate the frequency of behaviors they currently exhibit in the classroom, they were asked to indicate what they imagine they would do if they were in a classroom at this point in time.

 Five teachers currently teach children ages 3 to 5 (Four with master’s degrees in

early childhood education and one working toward a master’s degree in family

and consumer studies / all with current Pre-K-3 certification). Total years of

experience teaching preschool age children ranges from 4 years to 30 years.

 One teacher currently teaching children ages 5 to 6 (Master’s degree in early

childhood education / current Pre-K-3 certification). Total years of teaching

experience is 32 years.

 One teacher currently teaching children ages 1 to 3 (Master’s degree in early

childhood education / certification Pre-K-3). Total years of teaching experience

is 29 years.

 Three former preschool teachers who currently work in other capacities at the

center (all with Master’s degrees in early childhood education, certification Pre-

K-3). Total years of teaching experience ranges from 5 years to 30 years.

Procedures for cognitive interviews. The directions provided to participants were to complete the survey all the way through without edits and to mark how long it took to complete. Then, they were to read through each question again and make any notes or edits regarding the wording, clarity, order, or interpretation of the question.

After the participant completed the survey, twice we met briefly to discuss their perceptions of what the questions intended to measure. Using a protocol suggested by

Karabenick (2007) three questions (adapted for this study) were asked, (a) “What question were you even slightly unsure of how to answer?” (b) “What do you feel this question is trying to find out from you?” (or, “What is your interpretation of what this question is asking?”) (c) “For this question, why did you choose that answer?” Then, for any questions that were misinterpreted by the teachers I asked them additional questions regarding wording or possible additions, (d) “Was it clearly stated or are their changes you would make?” (e) Do you feel any other questions should be added?” As the teachers responded to the questions, I made notes about their responses on a checklist that contained a space for notes on each survey item. [Appendix B: Cognitive Interview

Notes]

The participants were not provided with the definitions of the NCTM Process

Standards, nor an official description of Developmentally Appropriate Practices, which were the constructs for the development of the items. This was done because it was necessary to assess their knowledge of the five processes with the Perceived Knowledge items on the survey and I wanted to get their feedback on their interpretation of what was meant by these processes in the questions. For example, how did they interpret the word

“representation” when it was used in a question?

The participants were selected for this phase as a convenience sample; they are colleagues that work at the laboratory school. They are ideal participants for this phase because of their long-term experience in the field as preschool teachers and also because

98 the practice of offering and receiving critique regarding teacher work is typical at the school. They completed the survey at home at their convenience then were interviewed individually at the school when they had completed the survey and made their edits and notes.

Results from cognitive interviews. Results from the development phase, particularly the cognitive interviews, indicated that the survey took 15-20 minutes to complete. There were a total of 62 items in the first draft of the survey given to the 10 participants. There were three suggestions regarding construction of the survey not specific to a survey item. The first suggestion was to change the order of the Likert scale from starting with Strongly Agree to starting with Strongly Disagree. I decided not to make this change after seeking the opinions of others. There was not a general consensus either way. The second suggestion made by two teachers was to add definitions of each

Process Standard before the Attitudes/Beliefs section in order for respondents to respond honestly to how they feel about them and the need for more knowledge of them.

Therefore, I made this change. The third suggestion made by one teacher initially, and then agreed upon by others when asked, was to change the “Don’t Know” column (#3) in the Likert scale of the Knowledge section to a “Don’t Know/Neutral” response option.

For example, item #3 asks about a teacher’s thoughts on the value of planned lessons compared with capitalizing on spontaneous opportunities for mathematics learning. One teacher said she responded with a three on the Likert scale not because she does not know if one is more valuable but because she feels that both have value, which was a neutral opinion. Therefore, I made this change.

In terms of “Wording / grammar suggestions,” the teachers suggested edits for 15 questions. Most of these edits were made. Regarding the “Clarity / Interpretation suggestions,” changes were made to 14 questions. For example, item #4 states, “I feel the most important connection teachers can help preschoolers make is that between children’s intuitive mathematics and the mathematics they learn in preschool.” There was confusion by teachers about the meaning of intuitive versus what is learned in school. Therefore, I added the description of comparing the formal and informal mathematics children learn – “…the connection between children’s informal or intuitive mathematics knowledge and the formal mathematics learned in school.” Another change, for example, was made to item #8 where there was confusion about what was meant by representation. Therefore, I added a few examples in the question – “…to use drawings, objects, photos or other forms of representation.”

The list of all items with suggested changes and those that were edited is in Table

11 and provides a summary of the quality of the survey items. Of the items listed, those that are bolded and underlined are those that were revised or deleted. The resulting revised survey that was used for the pilot study is in Appendix C.

Phase 2 Development: Pilot Study

Participants. The survey was administered to two groups of pre-service teachers in an early childhood education program. This was a convenience sample in that the students were in the early childhood program at the university where I work. The first group of respondents included 48 students in their first semester of advanced study who

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Table 11

Cognitive Interviews Summary

Teacher Wording / Clarity / Deletion of Matched Perceived grammar edits Interpretation edits item response to Knowledge suggested suggested suggested NCTM Problem Solving #18 #10, #21 0 85%

Reasoning & #9 #13, #16, #25 0 68% Proof Communication #2, #14 #14, #17 0 94%

Connections #20 #4, #22 0 84%

Representation 0 #15, #6, #8, #24 0 83%

All Five #5 0 0 100% Processes Teacher Wording / Clarity / Deletion of Acceptable Reported grammar edits Interpretation edits item response Behaviors suggested suggested suggested Problem Solving 0 0 0 95%

Reasoning & 0 0 0 70% Proof Communication #38 #33 0 98%

Connections 0 #30, #42 0 90%

Representation #40 0 0 100%

All Five #37 #37, #43, #45, #46 0 100% Processes Teacher Wording / Clarity / Deletion of Attitudes/Beliefs grammar edits Interpretation edits item suggested suggested suggested All Five #50 0 Processes Additional Edits #57, #59 #55, #57 0 (Demographics) (Demographics) Note. Revised or deleted items are in bold and underlined.

101 had not yet had a mathematics methods course that emphasized the NCTM Process

Standards. The second group of respondents included 26 students in their fifth semester of advanced study who had three semesters of mathematics methods courses where the

NCTM Process Standards were emphasized as integral to quality curriculum planning and instruction. A comparison of the responses of these two groups was made to check the reliability of the survey items. It was expected that the fifth semester students would answer more of the Teacher Perceived Knowledge section items correctly, that they would have more positive attitudes/beliefs toward the NCTM Process Standards, and that they would indicate a higher frequency of appropriate behaviors related to integration of the mathematics processes. Since these are not practicing teachers, they would not be able to respond to the behavior items with a frequency indicator in the same way as the in-service teachers. Therefore, the directions were altered a bit. For the behavior sectionof the survey, they were asked to imagine they were teaching in a preschool classroom and respond with how often they thought they would engage in the described behaviors.

Procedures. I sought permission by the college instructors for the first and fifth semester students to administer the survey at the end of their seminar class time. One section, with 24 students in the first semester group, was a class that I taught. The fifth semester student group instructor and the other first semester student group instructor and

I emailed the students the week before asking if they would be willing to complete the survey and if so, that they should read the attached consent form that they would be given to sign the following week when they completed the survey. The following week the

102 students who agreed to stay at the end of class signed the consent form and completed a paper/pencil form of the survey that I collected. The instructions given to them were to first complete the survey and then they had the option of leaving any edits or comments on the survey.

In order to summarize data from the pilot study of pre-service teachers, descriptive statistics were used. Only respondents with complete data for each scale are included. [See Table 12] Mode effect can occur if there are differences in the format of the web survey and the format of the paper/pencil survey. In this study, the exact same format, wording and procedures were used for both the web and paper/pencil versions of the surveys to reduce the possibility of mode effect when comparing results of the pilot survey and the research study survey.

Reliability of survey. The purpose of the phase two pilot study was to provide an estimate of reliability – to test the internal consistency of the survey in terms of what it is supposed to measure and what it actually measures. With all three sections of the survey having items scored as continuous variables, Cronbach’s alpha was used to test the internal consistency of each mathematics process (problem solving, reasoning & proof, communication, connections, and representation) for each of the three constructs of knowledge, behaviors and attitudes/beliefs. The results, as indicated in Table 13 show

Cronbach’s alpha of .91 for the complete survey. The generally accepted minimum for reliability estimates in educational research is .70 or greater (Wiersma & Jurs, 2009).

When analyzed separately, each of the three scales also had favorable estimates of reliability with Cronbach’s alpha coefficients of .75 (Teacher Perceived Knowledge), .90

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(Teacher Reported Behaviors) and .91 (Teacher Attitudes/Beliefs). Therefore, the instrument is assumed to be a reliable measure of these constructs.

Table 12

Descriptive Statistics for the Pilot Study ______1st semester pre-service 5th semester pre-service teachers teachers ______Teacher Perceived Knowledge scale n 33 24 mean 86.64 96.21 SD 8.33 8.31

Teacher Reported Behaviors scale n 33 24 mean 53.61 67.54 SD 14.84 11.99

Teacher Attitudes/ Beliefs scale n 33 24 mean 91.39 95.33 SD 9.85 8.17

Total survey n 33 24 mean 231.64 259.08 SD 24.27 20.67 ______

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Table 13

Cronbach’s Alpha for Pilot Study

The Problem Reasoning Communica- Connections Representa complete Solving & Proof tion -tion survey - All 3 scales

14 11 18 19 16 questions questions questions questions questions n 56 63 64 60 61 63 alpha .917 .717 .566 .765 .816 .731

Teacher Perceived Knowledge Problem Reasoning Communica- Connections Representa (MCK / Solving & Proof tion -tion MPCK) Scale All 25 7 6 6 6 5 questions questions questions questions questions questions n 68 70 70 72 71 71 mean 89.88 23.56 20.86 22.79 20.51 17.99 sd 9.486 3.686 2.866 3.166 3.549 2.369 alpha .750 .439 .322 .447 .567 .163

Teacher Reported Problem Reasoning Communica- Connections Representa- Behaviors Solving & Proof tion tion Scale All 20 7 5 7 8 6 questions questions questions questions questions questions n 62 64 65 63 64 65 mean 55.29 20.31 15.05 24.09 17.52 sd 16.365 6.052 4.470 6.458 7.567 5.365 alpha .905 .786 .675 .809 .851 .737

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Teacher Attitudes / Problem Reasoning Communica- Connections Representa- Beliefs Solving & Proof tion tion Scale All 26 5 5 5 5 5 questions questions questions questions questions questions n 63 67 65 64 66 66 mean 36.70 7.01 7.85 6.70 7.23 7.27 sd 9.110 1.911 2.495 1.933 2.175 2.094 alpha .918 .587 .757 .667 .674 .639

After reviewing the results of the pilot study and notes from the cognitive interviews and content reviewers, there were several items that were deleted and not part of the final version of the survey. Item numbers 35 and 39 were the only ones negatively coded and when analyzed for reliability reflected something different from all other items. One referred to teachers’ use of worksheets and the other referred to the appropriateness of providing children accurate information when their reasoning is faulty.

While deleting these items did not change alpha significantly for this scale (from alpha of

.801 to .839), they were removed from the final version of the survey. In order to reduce the amount of time it took participants to respond, several other questions were deleted.

Items from the Teacher Reported Behaviors scale were chosen for removal because they had the most overlap and took the most time to answer so in addition to the two negatively worded items mentioned above, four additional items were also deleted. Item

#32 regarding teacher facilitation of play to promote communication was chosen for deletion because other questions related to the communication Process Standard were sufficient. Item #43 regarding the use of computers for problem solving was chosen

106 because not all teachers have computers for children’s use in the classrooms. Item #44 on making mathematical connections by using voting in the classroom was deleted because the topic of voting is narrow for the construct of Connections. Finally, item #45 regarding the rotation of materials to meet student needs was deleted because it was considered too vague or broad for recording frequency. With these changes, the Teacher

Behaviors section was reduced to a total of 14 items from 20 items.

Data collection procedures for research study. A web survey was used for this study. However, one concern with using the internet for distribution of the survey is that you may not have a representative sample because not everyone has internet access.

Teachers are considered a specialized population and typically have access to the internet through the school where they work. Das, Ester & Kaczmirek (2011) agree that a web survey is just as appropriate as other modes when a target population is used such as certain professional groups (p. 18) which, in this case, is preschool teachers.

Some studies show that respondents often choose the most socially desirable alternatives on survey items (Nunnally & Bernstein, 1994). This is known as social desirability bias (SDB). To reduce SDB or editing of answers Bradburn, Sudman, and

Wansink (2004) suggest using less threatening questions or using self-administered (e.g. web survey) vs. face to face because they provide more confidentiality. In a study by

Rossiter (2009) 15 years later, no SDB differences were found in the method of the survey and may have decreased even more due to the normalcy of web-based surveys in this country. Most research also shows that SDB is more prominent when personal,

107 embarrassing questions are asked. In this study, that is not the case so it can be assumed that SDB would be less significant.

The survey instrument developed for this study, Teaching Mathematics Processes in Preschool Survey (TMPPS), was administered via email. Two types of contacts were initially used to solicit participants. The Logistics Manager from OSU’s College of

Education and Human Ecology was contacted and agreed to send out emails regarding the survey to Ohio’s Early Childhood Quality Network. Also, Regional Early Childhood

Consultants of the State Support Team in Ohio were also contacted, and fourteen of the fifteen regional consultants agreed to email the survey to the teachers and directors in their region. Participants who completed the survey reported working in settings that included public school, special education classrooms or sites, private or for-profit childcare centers, and Head Start programs. The introduction to the survey included a statement about who the survey was for, why it was important for future professional development opportunities, and how long it would take. The introductory statement was followed by the university IRB approved consent form. By proceeding with the survey, participants gave their informed consent for the research study. The survey was sent to preschool teachers during the second week of February. There were a total of 130 respondents who completed the survey after the first email. A reminder was sent the last week of February via the Regional Consultants and when the survey closed on March 5,

217 respondents had completed the survey. Another 97 teachers had opened or started the survey but did not complete the survey.

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Data Analysis Procedures

Before data was analyzed, all surveys returned with missing or incomplete data were removed and all reverse coding was completed. As a result of this screening process, data from 217 preschool teachers was used to answer eight research questions.

The questions and how they were analyzed are included below.

Research Questions 1 - 3

1. What MCK and MPCK do preschool teachers perceive to have of the five NCTM

Process Standards?

2. How often do teachers report they engage in behaviors that promote and develop

children’s use of the mathematics processes?

3. What are preschool teachers’ attitudes/beliefs toward the value of incorporating

NCTM Process Standards in the classroom?

In order to address the first three research questions, descriptive statistics were used: mean, standard deviation, and range.

Research Questions 4 – 6

4. How does teacher perceived knowledge of the Process Standards compare with

teacher reported behaviors that promote children’s development and use of the

mathematics processes?

5. How does teacher perceived knowledge of the Process Standards compare with

teacher attitudes/beliefs regarding the value and importance of the mathematics

processes in preschool?

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6. How do teacher attitudes/beliefs regarding the value and importance of the

mathematics processes in preschool compare with teacher reported behaviors that

promote children’s development and use the mathematics processes?

To address research questions 4, 5, and 6, correlation coefficients, Pearson's r, were obtained to make comparisons between the three variables of teacher perceived knowledge, reported behaviors, and attitudes/beliefs. Pearson's r was used to compare teacher attitudes/beliefs and reported behaviors (#4), teacher perceived knowledge and reported behaviors (#2), and teacher perceived knowledge and attitudes/beliefs (#3).

Research Question 7

7. Is there a difference between licensed and non-licensed preschool teachers in

terms of their behaviors, attitudes/beliefs and knowledge in regards to each of the

five Process Standards?

Question 7 was analyzed using inferential statistics. The independent variable is licensure status: licensed or non-licensed teachers (2 levels). A licensed teacher refers to someone with a state license to teach preschool age children. A non-licensed teacher refers to someone without a college degree or someone with a college degree, but not licensed by the state to teach preschool. The dependent variables are Teacher Perceived

Knowledge, Teacher Reported Behaviors, and Teacher Attitudes/Beliefs for each of the five NCTM Process Standards.

The null hypothesis is that non-licensed teachers are as likely as licensed teachers to have higher scores on the Teacher Perceived Knowledge of the Mathematics Process

Standards scale, report that they engage in more frequent behaviors that promote the

110 standards, and have attitudes/beliefs that are more positive regarding the process standards.

Inferential statistics were used since the purpose of the analysis was to collect data from a sample to make estimates about the population. With only two means being used, an independent t-test was used to draw conclusions about the hypothesis.

Research Question 8

8. Is there a difference between the perceived knowledge, reported behaviors and

attitudes/beliefs of preschool teachers who work in different types of settings in

regards to each of the five Process Standards?

Question 8 was analyzed using inferential statistics. The independent variable is type of setting: public school, for-profit childcare center, Head Start, university laboratory school, and licensed home provider (5 levels). The dependent variables are teacher perceived knowledge, teacher reported behaviors, and teacher attitudes/beliefs for each of the five Process Standards.

The null hypothesis is that teachers working in for-profit centers, home providers, and

Head Start settings are as likely as teachers working in a university laboratory school, and public school settings to have higher scores on perceived knowledge of the mathematics process standards, engage in more frequent behaviors that promote the standards, and have attitudes/beliefs that are more positive regarding the process standards.

Research question 8 was addressed using an analysis of variance (ANOVA) because originally there were five population means. The dependent variables were the measures of perceived knowledge, reported behaviors, and attitudes/beliefs in regards to the NCTM

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Process Standards and the independent variable categories were five types early education settings: For-profit childcare, Head Start, public school, university laboratory school, family/home provider, and “other” for open responses. However, the work setting groups were re-classified based on participant responses in the “other” setting category. Many labeled their setting as private or special education or itinerant so those labeled private were combined with the for-profit group and those labeled special education became an additional group. The re-classified “other” group had only 7 participants, therefore, the actual one-way analysis of variance was used with four population means or work setting groups: Private, public school, Head Start, and special education.

Research Questions 7-8 Additional Information

Some research suggests that teachers with an early childhood degree provide high quality programs and better outcomes for children than teachers who do not have a degree in the field (Brown, 2005, National Research Council, 2001b; Whitebook & Ryan,

2011). Teachers in early education contexts have varying levels of education from a high school diploma (Ohio) to a Master’s degree and teaching license for preschool age children. Along with a teacher’s level of education, the type of early education context may also influence teacher practices and resulting child outcomes. Many teachers have a prescribed curriculum to follow and this curriculum may not provide experiences for children that are supported by the NCTM Process Standards. For example, if teachers use a curriculum that dictates the letter, shape, color and numeral that is the focus for the day or week then teachers may only focus on these basic skills as the math for the day.

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Another influence on teacher practice is length of the school day. Many programs are two and a half hours long including time for two meals so in combination with a prescribed curriculum, the teacher has less time to implement what he/she may deem as more appropriate mathematics practices. A third influence on teacher practices of a program is the degree requirement for teachers. University laboratory schools and public school settings typically require teachers to be licensed to teach preschool whereas other settings may not require a degree at all or may not require a degree in early childhood education. Therefore, this study will examine possible correlations between teachers’ licensure status, early education setting, and knowledge, behaviors and attitudes of teachers in these settings.

Conclusions

In order to learn about the knowledge, behaviors, and attitudes/beliefs of preschool teachers regarding the NCTM process standards a survey was developed and used to gather information from a large sample of preschool teachers in the state of Ohio.

The design of the survey instrument through two phases including the content review, cognitive interviews, and the pilot study were described. Also described were procedures for the administration of the final version of the survey using the Qualitrics survey program, including the procedures and timeline for data collection. Statistics that were used to answer the eight research questions of the study were included and the details of these analyses are described in Chapter IV.

CHAPTER IV

RESULTS

Overview

This exploratory study aimed to gather data on the perceived mathematical knowledge, teaching practices, and attitudes/beliefs of preschool teachers. A self-report survey instrument was developed for this purpose. The data analysis and results for each of the eight research questions will be described in this chapter.

A survey was sent via email, using the Qualitrics web survey tool, to preschool teachers in the state of Ohio during the months of February and March 2014 (See Chapter

III for detailed data collection procedures.) A total of 314 preschool teachers started the survey and 217 preschool teachers completed the survey. The statistics software program used to analyze the response data was SPSS 21.

Observed Reliability of Instrument

Cronbach’s alpha was used to test the internal consistency for each of the three constructs: teachers’ perceived knowledge, reported behaviors, and attitudes/beliefs.

The results, as indicated in Table 14, indicate favorable estimates of reliability with

Cronbach’s alpha coefficients of .741 (Teacher Perceived Knowledge), .900 (Teacher

Reported Behaviors) and .939 (Teacher Attitudes/Beliefs). The alpha for Teacher

Perceived Knowledge is expected to be a bit lower than the other two constructs due to the nature of the scale having the option of “Don’t know / neutral” which offers more options for “no” versus “yes” or “right or wrong” responses. With some minor changes that were a result of the pilot study, the instrument is still assumed to be a reliable 113 114 measure of these constructs in that it is above .70 (Vogt, 2007). Cronbach’s alpha coefficient for each of these three variables is consistent with the pilot study (.750, .905,

.918), which supports the reliability claim by showing a trend across samples.

Table 14

Reliability Analyses ______Three Constructs Number of items Cronbach’s alpha ______Perceived Knowledge (MCK, MPCK) 24 .741 of NCTM process standards

Reported Behaviors related to the 14 .900 NCTM Process Standards

Attitudes / Beliefs toward the 25 .939 NCTM Process Standards

Overall Survey 63 .898 ______

Operational Definitions of the Variables

Mathematics Content Knowledge (MCK) and Mathematics Pedagogical Content

Knowledge (MPCK)

A preschool teacher’s perceived MCK and MPCK were measured together by the composite mean score on a 25-item, five-point Likert scale, of the instrument developed for this study (See Appendix D for the Research Study Survey, Section 1: Teacher

Perceived Knowledge of Early Childhood Mathematics). These variables measured the level of agreement with how NCTM defines each of the five Process Standards which, after reverse coding were: 5 = agree, 4 = tend to agree, 3 = don't know/neutral, 2 = tend

115 to disagree, 1 = disagree. High mean scores (>3.00) indicated that teachers agreed with how the Process Standards were defined by NCTM and low mean scores (≤ 3.00) indicated that the teachers disagreed with, were neutral, or reported they didn’t know how the Process Standards were defined by NCTM.

Teacher Reported Behaviors

A preschool teacher’s reported behaviors regarding the mathematics processes were measured by the frequency of behaviors on a 14-item, six-point Likert scale, of the instrument developed for this study (See Appendix D for the Research Study Survey,

Section II: Teacher Reported Behaviors related to Mathematics Teaching in Early

Childhood). This variable measured the frequency with which teachers reported engaging in specific behaviors that were Developmentally Appropriate Practices (DAP) for this age group and that promoted children’s development and use of the five NCTM

Process Standards. The frequency of behaviors was measured from 0 = "Never," 1 =

"One or two times a semester," 2 = "One or two times a month," 3 = "One or two times a week," 4 = "Daily," and 5 = "More than once a day." All items were written as positive behaviors that would suggest more implementation of mathematic processes when teaching. Therefore, the higher the reported frequency meant that DAP behaviors that promoted mathematics processes were put into practice more often than reported lower frequency behaviors.

Teacher Attitude/Beliefs

A preschool teacher’s attitudes / beliefs on the importance or value of the NCTM mathematics processes for children’s understanding of mathematics was measured by the

116 composite mean score on a 26-item, four-point Likert scale of the instrument developed for this study (See Appendix D for the Research Study Survey, Section III: Teacher

Attitudes/Beliefs toward Mathematics in Early Childhood). This variable measured the level of agreement with the importance or value of promoting the five NCTM Process

Standards at the preschool level which, after reverse coding, was 4 = Agree, 3 = Tend to agree, 2 = Tend to disagree, and 1 = Disagree. High mean scores (≥3.00) indicate that respondents agreed that it was important to promote the mathematics processes, they felt they were confident and knowledgeable of the processes, and/or felt they were adequately promoting them in their classroom. Low mean scores (<3.00) indicate that respondents disagreed that it was important to promote the math processes in preschool, that they may not feel as confident or knowledgeable with the processes, and /or did not feel they were adequately promoting them in their classroom.

Descriptive and Demographic Data

In this section descriptive data regarding the participants of this study is provided.

Section IV of the survey (See Appendix D for the Research Study Survey, Teacher

Demographic Information) asked teachers to provide background information regarding years of teaching experience, highest level of education obtained, type of work setting, familiarity with NCTM, licensure status, and satisfaction with past mathematics professional development experiences. Pertinent frequencies and percentages are provided for each item.

Preschool teachers in all types of early education settings in the state of Ohio were targeted for this study. In regards to type of work setting, five possible options were

117 available: For-profit childcare center, Head Start program, public school preschool, family/home provider, and university laboratory school. Participants could also choose the option of “other”. A slight majority of participants indicated they taught in a public school setting (52.3%). The next highest category was “other” with 54 (25.0%) then

Head Start with 40 (18.5%) and For-profit childcare with only nine participants (4.2%)

(See Table 15). Upon further review, the “other” category revealed that nine preschool teachers labeled their setting as Private School/Non-profit, another 5 teachers labeled their setting as Itinerant Special Education, and 18 teachers labeled their setting as

Special Education/ECS. There were no participants from Ohio’s university laboratory schools or Family/Home Providers.

Table 15

Distribution of Participants by Type of Early Education Setting Where One Works ______Type of Work Setting Frequency Valid Percent ______

For-profit childcare center 9 4.1 Head Start program 40 18.5 Public school preschool 113 52.3 Other 54 25.0

Note. N = 217.

All of the participants indicated they currently teach children ages three, four, and/or five with the majority having four year olds in their classrooms. Twenty-eight participants did not have three-year-olds in their classroom, only seven (3.2%) did not currently have four-year-olds, and fifteen (6.9%) did not have five-year-olds. When

118 asked about the number of years of experience they have teaching preschool age children, the years vary from 0.5 years to 35.0 years (see Table 16). The mean for years of teaching children age three to five is 11.3 years (SD = 7.78). The mean for years of teaching overall is 14.5 years (SD = 8.74).

Table 16

Frequency of Response on Years of Preschool Teaching Experience ______Years Teaching Preschool Frequency ______

0.5 - 5 years 64 5.5 - 10 years 49 10.5 - 15 years 38 15.5 - 20 years 37 20.5 - 25 years 22 25.5 - 30 years 4 30.5 years + 2

Note. N = 217.

In regards to their level of education, the majority of preschool teachers who responded to the survey indicated they had a Master’s degree (54.6%) (See Table 17).

This is not surprising since the majority of participants indicated they worked in a public school setting where the minimum of a bachelor’s degree is typically required for employment. Of the 212 who reported having an Associate’s, Bachelor’s, or Master’s degree, only eight of the participants indicated their degree was not in the field of education. Most importantly, the vast majority of teachers indicated they are currently licensed to teach preschool age children (81.4% with another 6.0% having an expired license).

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Table 17

Distribution of Participants by Highest Level of Education ______Level of Education Frequency Valid Percent ______

High school diploma/equivalency 2 0.9 Child Development Associate (CDA) 2 0.9 Associate’s Degree 23 10.6 Bachelor’s Degree 71 32.9 Master’s Degree 118 54.6

Note. N = 217.

Since one of the goals of this study is to improve professional development opportunities for preschool teachers in the area of mathematics, participants were asked to respond to whether they have engaged in this type of experience in the past three years and how useful this information was for their teaching of mathematics at the preschool level (see Table 18). Most responded that they have engaged in a mathematics professional development opportunity for preschool teaching (58.6%), and all but four found this experience somewhat or very useful for their teaching.

Lastly, with the framework of the study built around how the National Council of

Teachers of Mathematics (NCTM) defines the mathematics processes that are the focus of this survey, the participants were asked if they had heard of NCTM’s five Process

Standards (problem solving, reasoning & proof, communication, connections, representation) before this survey. Approximately half of respondents indicated they had at least heard of these standards (52.1%). Since the majority of teachers indicated having

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Table 18

Frequency of Response on Participation in Preschool Mathematics Professional Development Opportunities ______Professional Development Frequency Valid percent In past 3 years ______

No 89 41.4

Yes 126 58.6 Very useful 72 57.1 Somewhat useful 50 39.7 Not very useful 4 3.2 ______Note. N = 217. a Master’s degree, Table 19 compares the number of teachers that obtained a college degree with how many indicated they were familiar with the NCTM Process Standards.

While those with a more advanced degree had a higher percentage of teachers familiar with the Process Standards, still, only about half of the teachers (47.1% to 57.6%) with a four-year degree or more indicated they were familiar with the Standards.

Additional analyses were used to find out if there were other predictors for teacher familiarity of the NCTM process standards. For example, does a teacher’s years of experience predict more familiarity with the standards? The two-sample chi-square test was used to infer a possible relationship between the two variables. The chi-square value [6.735, .009] indicates a statistically significant difference between having professional development and not having professional development on respondents having heard of the Standards. No association was found between years of experience or level of education on having heard of the Standards.

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Table 19

Teacher Education and Familiarity with NCTM Process Standards ______Level of Education Frequency Familiarity NCTM Percent ______

Associate’s Degree 23 8 34.8

Bachelor’s Degree 71 33 47.1

Master’s Degree 118 68 57.6

Note. N = 217.

Research Questions 1 – 3: Preschool Teachers’ Perceived Knowledge, Reported

Behaviors, and Attitudes/Beliefs Regarding the NCTM Process Standards

Teacher Perceptions of their Knowledge of the NCTM Process Standards

The perceived mathematics content knowledge and pedagogical content knowledge of preschool teachers regarding mathematics processes were gauged by participant responses to 24 questions in section I of the survey (item # 5 was not used in this analysis as it was not specific to one particular mathematics process). These questions looked specifically at five different mathematics processes: problem solving, reasoning & proof, communication, connections, and representation to determine what preschool teachers perceived they know about the NCTM Process Standards. These questions include how the processes are defined by NCTM and what NCTM suggests for implementing them when teaching young children.

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Regarding their perceived knowledge of all five processes combined, the mean composite score was 3.73 indicating that on average, respondents agreed (although not strongly) with how NCTM defines the five mathematics processes for preschool age children (see Table 20). Further analysis of each of the sub-domains and each of the five processes separately, suggests that participants were mostly in agreement (with a score close to or above four on the 5-point Likert scale) with how three of the processes

(communication, connections, and representation) were defined by NCTM. For the six problem solving items, respondents were, on average, neutral or didn’t know, with a score slightly above three on the 5-point Likert scale (how NCTM defined this mathematics process). For the four reasoning and proof items respondents were between neutral/don’t know and agreement with these items with a mean score of 3.50.

Table 20

Descriptive Statistics for Perceived Knowledge of Preschool Teachers ______Perceived Number M SD Range MCK/MPCK of items ______

Total of all 5 NCTM Processes 24 3.73 .386 2.58, 4.79

Problem solving 6 3.30 .509 2.00, 4.50 Reasoning & proof 4 3.50 .655 1.50, 5.00 Communication 5 3.93 .574 2.40, 5.00 Connections 5 4.03 .522 2.40, 5.00 Representation 4 3.96 .591 2.00, 5.00

Note. N = 217. MCK = mathematics content knowledge; MPCK = mathematics pedagogical content knowledge.

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Teachers’ Reported Behaviors that Promote the NCTM Process Standards

The frequency of teacher reported behaviors in the classroom that promote children’s development and use of the mathematics processes was gauged by participant responses to 14 questions in section II of the survey. These questions encompass both knowledge of mathematics processes (NCTM, 2000) and practices that are considered to be developmentally appropriate for preschoolers (Copple & Bredekamp, 2009). Table 21 shows the mean composite score of 3.21 for how teachers self-report that they engage in behaviors that promote the mathematics processes in the classroom. On average, respondents report that they engage in classroom behaviors that promote children’s use and development of the mathematics processes at least once or twice a week. When looking at each of the five processes separately, representation is the only one that on average was promoted less than weekly. The minimum score of 0.00 reported for each

Table 21

Descriptive Statistics for Reported Behaviors of Preschool Teachers ______Reported Behaviors No. of items M SD Range ______

Total of all 5 NCTM Processes 14 3.21 .794 .00, 5.00

Problem solving 3 3.38 1.00 .00, 5.00 Reasoning & proof 1 3.41 1.14 .00, 5.00 Communication 3 3.18 .947 .00, 5.00 Connections 3 3.33 .909 .00, 5.00 Representation 3 2.82 .947 .00, 5.00 ______Note. N = 217.

124 mathematics process indicates that at least one respondent reported never engaging in behaviors that promoted that Process Standard.

Teachers’ Attitudes/Beliefs toward the NCTM Process Standards

The attitudes/beliefs that preschool teachers have toward the value and importance of incorporating the mathematics processes in the classroom were gauged by participant responses to 25 items in section III of the survey. Five items were broken down into five sub-questions so as to look specifically at each of the five Process

Standards. For example, item #42 asks how one feels about the appropriateness of emphasizing mathematics processes with children age three to five years and participants respond with a level of agreement for problem solving, communication, reasoning & proof, connections, and representation. Therefore, there are a total of 25 questions for

Section III on teacher attitudes/beliefs. Item # 46 was not included in this analysis as it asked respondents if professional development would be beneficial which was not specific to each Process Standard separately. Table 22 shows participants' level of agreement or disagreement overall for each question and then for each separate mathematics process. The mean composite score of 3.43 overall indicates positive attitudes or agreement by preschool teachers toward the value or importance of the mathematics processes in preschool. There was not a significant difference between their level of agreement or disagreement for a particular math process, just slightly less agreement with reasoning & proof.

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Table 22

Descriptive Statistics for the Attitudes/Beliefs of Preschool Teachers

______Attitudes/Beliefs No. of items M SD Range ______

Total of all 5 NCTM Processes 25 3.43 .425 2.20, 4.00

Problem solving 5 3.52 .437 2.20, 4.00 Reasoning & proof 5 3.25 .561 1.20, 4.00 Communication 5 3.50 .447 2.00, 4.00 Connections 5 3.49 .443 2.20, 4.00 Representation 5 3.40 .504 2.00, 4.00 ______Note. N = 217.

Research Questions 4, 5 & 6

The following three research questions look at possible correlations between two constructs in that changes in one variable may be accompanied by changes in another variable (Brown & Saunders, 2008). Answering research questions 4, 5, and 6 involved the use of Pearson’s r to describe the strength of the relationship between two variables because none of the three constructs are identified as the dependent or the independent variable (Brown & Saunders, 2008). Just as before, all negatively weighted items in

Section I: Perceived Knowledge of Teachers, were reverse coded to be consistent with coding for positively weighted items.

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Research Question 4

How does teacher perceived knowledge of the Process Standards compare with

teacher reported behaviors that promote children’s development and use of the

mathematics processes?

When comparing the 24 items of section I of the survey and the 14 items of section II of the survey, there was a statistically significant correlation found between preschool teachers’ perceived knowledge of NCTM’s five Process Standards and their reported classroom behaviors that promote children’s use and understanding of these mathematics processes (see Table 23). In looking at the five processes separately, there was also a statistically significant correlation found between two of the five mathematics

Table 23

Correlation of Teacher Perceived Knowledge and Teacher Reported Behaviors ______Section II total B of PS B of RP B of CM B of CN B of R ______Section I total .170 ------.012* Knowledge of PS ----- .200 ------.003** Knowledge of RP ------.097 ------.156 Knowledge of CM ------.077 ------.259 Knowledge of CN ------.079 ----- .249 Knowledge of R ------.139 .041* Note. N = 217. *p < 0 .05 (2-tailed) **p < 0.01 (2-tailed)

127 processes: problem solving and representation. However, correlations less than .30 are considered weak unless large in comparison to what was expected (Vogt, 2007). In this case there were no expectations for the comparison so these correlations are considered weak and only statistically significant because of the large sample.

Research Question 5

How does teacher perceived knowledge of the Process Standards compare with

teacher attitudes/beliefs regarding the value and importance of the mathematics

processes in preschool?

There was a statistically significant correlation between preschool teachers’ perceived knowledge of NCTM’s Process Standards and their attitudes / beliefs regarding the value or importance of these Standards in preschool settings (see Table 24).

Table 24

Correlation of Teacher Perceived Knowledge and Teacher Attitudes/Beliefs ______Section III A/B of PS A/B of RP A/B of CM A/B of CN A/B of R Total Section I total .261 ------<.001** K of PS ----- .203 ------.003** K of RP ------.265 ------<.001** K of CM ------.126 ------.064 K of CN ------.131 ----- .054 K of R ------.279 <.001** Note. N = 217. *p < 0.05 (2-tailed) **p < 0.01 (2-tailed)

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Also, three of the five math processes (problem solving, reasoning & proof, and representation), when looked at separately, were also found to have a positive relationship that was statistically significant. However, while there is statistical significance between several constructs, the correlations are weak. Even though the correlations for the section I total, reasoning & proof, and representation are close to .30

(.261, .265, and .279 respectively) and could be considered meaningfully significant, in this context one would expect strong correlations between the perceived knowledge of teachers and the attitudes/beliefs of teachers so these effect sizes are lower than expected.

Research Question 6

How do teacher reported behaviors that promote children’s development and use

of the mathematics processes compare with teacher attitudes/beliefs regarding the

value and importance of the mathematics processes in preschool?

A statistically significant correlation was found between preschool teachers’ self- reported behaviors that promote children’s use and understanding of mathematics processes and teachers’ attitudes and beliefs toward the value and importance of these processes at the preschool level. This correlation is also meaningfully significant since it is above .30 (see Table 25). A statistically significant correlation was also the case when comparing responses to items for each of the five Process Standards as well: problem solving, reasoning & proof, communication, connections, and representation. The correlation for reported behaviors related to the communication Process Standard is greater than .30 so would therefore be considered meaningfully significant. The

129 correlations for reported behaviors related to the problem solving and the connections

Process Standards are weak (less than .30). Correlations for reported behaviors related to reasoning & proof and representation are borderline meaningfully significant at .255 and

.265 respectively.

Table 25

Correlation of Teacher Reported Behaviors and Teacher Attitudes/Beliefs ______Section III A/B of PS A/B of RP A/B of CM A/B of CN A/B of R Total Section II total .322 ------<.001** B of PS ----- .175 ------.010** B of RP ------.255 ------<.001** B of CM ------.301 ------<.001** B of CN ------.232 ----- .001** B of R ------.265 <.001** ______Note. N = 217. *p < 0.05 (2-tailed) **p < 0.01 (2-tailed)

Research Question 7

Is there a difference between licensed and non-licensed preschool teachers in

terms of their behaviors, attitudes/beliefs, and knowledge in regards to each of the

five Process Standards?

In choosing the appropriate test for this research question an estimation of effect size, statistical power, was completed. Due to the unequal sample sizes of 188 and 27 the

130 harmonic mean was used for n when determining delta. An effect size of at least .7 is recommended to reduce the chance of committing a Type II error (Cohen & Lea, 2004).

In determining delta for this study .8 was used. To attain a large effect size of .8 with significance level at .05, delta must be at least 2.8. With the given sample sizes, delta was calculated as 3.88 which translates to an 80% chance of having significant results.

Therefore, an independent t-test was used to test for differences in licensed and non- licensed preschool teachers on their perceived knowledge, reported behaviors and attitudes/beliefs regarding the NCTM Process Standards. Using the post hoc power analysis with the GPower program, achieved power was calculated at 0.82 where the effect size was 0.53 and it was 0.05 for alpha.

Results of the independent t-test in Table 26 suggest that there is a statistically significant difference in licensed teachers (N = 188; which, in this case includes teachers with an expired license) and non-licensed teachers (N = 27) on the complete Teacher

Perceived Knowledge section I of the survey and for two specific mathematics processes in this section, reasoning & proof and representation. The effect size was calculated at

0.53, a typical effect size. According to the data for this sample, licensed teachers have greater perceived knowledge of how NCTM defines the five Process Standards, especially the two processes of reasoning & proof and representation. This seems logical in that one would expect a licensed teacher to have had college coursework where national mathematics standards are addressed. The mean was lower for the group of all teachers (without regard to licensure status) for the process of reasoning & proof, which indicates a need for more professional development in this area.

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Table 26

Test of Mean Teacher Licensure Differences between Yes or No Respondents on Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Regarding NCTM Process Standards ______Constructs t df ______

Knowledge Total** 2.619 213 K of PS 1.575 213 K of RP* 2.233 213 K of CM 1.516 213 K of CN 1.330 213 K of R* 2.462 213

Behaviors Total .717 215 B of PS .568 215 B of RP -.704 215 B of CM .896 215 B of CN .560 215 B of R .905 215

Attitudes/Beliefs Total .463 213 A/B of PS .902 213 A/B of RP .080 213 A/B of CM .385 213 A/B of CN 1.154 213 A/B of R -.092 213 ______*p < 0.05, **p < 0.01

The results of the t-test for how licensed and non-licensed teachers compared on the Reported Behaviors and Attitude/Beliefs scales suggest that there are no statistically significant differences between the two groups of teachers. (See Appendix G for the complete set of means for the independent t-test) This data suggests that teachers with licensure to teach preschool do not necessarily have more positive attitudes/beliefs toward the importance or value of the mathematics processes nor do they have more

132 frequent behaviors that promote these processes in the classroom. While the findings are statistically significant for Perceived Knowledge and the effect size is typical for this variable, data does not suggest there is practical significance in having a pre-K teaching license for a teacher’s overall effectiveness in promoting children’s use and development of the mathematics processes in the classroom.

Research Question 8

Is there a difference between the perceived knowledge, reported behaviors, and

attitudes/beliefs of preschool teachers who work in different types of settings in

regards to each of the five Process Standards?

Three ANOVAs were conducted to look for possible correlations of work setting on Teacher Perceived Knowledge, Teacher Reported Behaviors, and Teacher

Attitudes/Beliefs. Analyses of variance (ANOVA) were used to identify any significant differences between the mean scores of the teachers who work in different settings (the dependent variables) on teacher perceived knowledge, self-reported behaviors, or attitudes/beliefs. Item #47 in Section IV, Demographics, asked participants about the type of preschool setting in which they work. Six options were provided: 1) For-profit childcare center, 2) Head Start program, 3) Public school preschool, 4) University laboratory school, 5) Licensed home/family childcare, and 6) Other, an open response item where they could classify their work setting. Of the 217 respondents who completed the survey, no one indicated that they worked at a university laboratory school or that they were a licensed family/home provider. (See Table 15)

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With a frequency of 54 (28.9%) in the “Other” category this group was re- classified using anecdotal evidence to provide more detailed information regarding work setting. After examining the open-ended responses, two additional “work setting” groups were formed. Nine respondents classified their setting as private or non-profit so these were combined with the 9 For-profit for a new category called “Private” with 18 respondents. Twenty-three classified their setting as special education or itinerant special education so this became a new category called “Special Education.” Table 27 displays the new work setting groups and the frequencies.

Table 27

Distribution of Participants by Type of Early Education Setting Where one Works after Re-classification ______Type of Work Setting Frequency Valid Percent ______

Private childcare center 18 8.3 Head Start program 42 19.4 Public school preschool 126 58 Special Education 23 10.6 Other 7 3.2

Note. N = 217.

For the first analysis, the four most frequently identified categories were used when looking at teachers’ perceived knowledge: Private childcare center (M = 91.22, SD

= 10.84), Head Start program (M = 86.36, SD = 9.00), Public school preschool (M =

89.91, SD = 9.04), and Special education (M = 88.87, SD = 8.60). The mean difference

134 on the Teacher Perceived Knowledge scale was not statistically significant, F (3, 205) =

1.909, p = .129.

The second analysis looked at work setting and teachers’ reported behaviors:

Private childcare center (M = 44.11, SD = 15.47), Head Start program (M = 48.90, SD =

10.33), Public school preschool (M = 43.04, SD = 10.29), and Special education (M =

46.09, SD = 10.16). The mean difference on the Teacher Reported Behavior scale was statistically significant, F (3, 205) = 3.252, p = .023. A multiple comparisons analysis using Tamhone’s T2 revealed that statistically significant differences were only found between Head Start and public school teachers. More specifically, differences were found for reported behaviors on representation (p = .009) and on communication (p =

.000) along with the total score of reported behaviors for all five mathematics processes

(p = .013).

The third analysis looked at work setting and teachers’ attitudes/beliefs: Private childcare center (M = 88.50, SD = 10.70), Head Start program (M = 85.55, SD = 11.61),

Public school preschool (M = 85.48, SD = 10.22), and Special education (M = 83.57, SD

= 10.25), the mean difference on the Teacher Attitude/Belief scale was not statistically significant, F (3, 205) = .742, p = .528.

To summarize, results of the ANOVA indicate that there were statistically significant differences between the scores of Head Start teachers and public school teachers on the Reported Behaviors section of the survey, with the most meaningful differences in the areas of representation and communication. No other significant differences were found among the other settings and teacher reported behaviors or

135 between any type of work setting and teachers’ perceived knowledge or attitudes/beliefs regarding the NCTM Process Standards.

CHAPTER V

DISCUSSION AND CONCLUSIONS

Introduction

The purpose of this study was to explore preschool teachers’ perceptions regarding what they report to know about the National Council of Teachers of

Mathematics (NCTM) Process Standards, what they report they are currently doing to promote children’s development and use of these Standards in the classroom, and their attitudes/beliefs regarding the value and importance of these mathematics processes in preschool. The Teaching Mathematics Processes in Preschool Survey (TMPPS) was designed, tested, revised, and then implemented. (The final version of the survey can be found in Appendix D). The survey contained the Teacher Perceived Knowledge scale,

Teacher Reported Behaviors scale, Teacher Attitudes/Beliefs scale and a demographics section. There were 217 preschool teachers in Ohio who responded to the online survey.

This chapter summarizes the findings of the study, connections to the early childhood mathematics literature, and implications for future researchers, teachers, and other early childhood educators.

Teacher Background Information

The 217 participants in this study were from a variety of early education settings serving children age three to five years in Ohio. There were 64 teachers who have been teaching preschool age children for five years or less and 28 teaching more than 20 years.

136 137

The majority of participants reported they teach in public school settings including classrooms that are inclusive of children with special needs. Also 42 teachers in Head

Start programs responded to the survey. Only 18 teachers reported teaching in for-profit or private centers. This was a smaller sample obtained than expected and further study of this population is an important focus for future research in order to find out what the needs of teachers may be in all types of settings.

The majority of teachers indicated having a Master’s degree in the field of early childhood education, including 88 out of 126 public school teachers. Only four teachers reported having no form of college degree. The majority of teachers reported being licensed to teach preschool as well. This is not surprising in that to work in a public school setting or Head Start program as a lead teacher a college degree of some level is required and teacher licensure is typically required in public schools. It is important to note that the majority of teachers who responded to the survey have advanced degrees.

This could suggest that those who feel more confident in their knowledge of mathematics or who have more experience in general with studying mathematics topics in school were more inclined to respond to a survey that asked about their knowledge and teaching of mathematics. There are typically a smaller percentage of teachers in private childcare settings that have a four-year or advanced degree compared to Head Start or public school settings. Only 18 teachers who responded to the survey indicated they worked in a private childcare setting and all 18 had college degrees (3 Associate’s degree, 10

Bachelor’s degree, and 5 Master’s degree). Therefore, there may be some response bias for this study.

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Research Questions 1 – 3

Because this was an exploratory study, the purpose of the first three research questions was to gather descriptive information about preschool teachers’ familiarity with the NCTM mathematics Process Standards in terms of their perceived knowledge of the mathematics processes, their reported behaviors in regards to practices that emphasize or promote these processes in the classroom, and their attitudes/beliefs regarding the value or importance of these processes for teaching young children. To gather additional information, research questions 4, 5, and 6 looked at possible relationships between the three constructs described above: Teacher Perceived Knowledge, Teacher Reported

Behaviors, and Teacher Attitudes/Beliefs. Conclusions drawn from the data for these first six research questions are described below.

Teachers’ perceived knowledge scale. The purpose of the first research question was to find out more about preschool teachers’ perceived knowledge of how

NCTM defines the mathematics processes of problem solving, reasoning & proof, communication, connections, and representation – what these mean and what best practice looks like when promoting these in the classroom. The data suggests that overall the teachers in this sample have some knowledge of how NCTM defines the five Process

Standards but it is not an indication of strong background knowledge. The average for all but connections was between 3.00 and 4.00 (between “Don’t know / neutral” and “Tend to Agree” on the Likert scale). The lowest score of the five processes was problem solving and the highest was connections. Connections on average was rated “Tend to

Agree.” It may be easier for teachers to understand what it means to make connections in

139 mathematics because teaching content through play experiences is an important aspect of

Developmentally Appropriate Practice (Copple & Bredekamp, 2009) and most preschool curricula. Making connections to children’s real world experiences and connections between mathematics and other content areas such as music are ways children learn through play. Two items in the survey illustrate this understanding of DAP: item #23 with the highest mean score of the section, “I believe that mathematics should be related to a young child’s real world experiences,” and item #20, which also had a high mean score, that addressed the need for mathematics activities to be integrated with other subject areas.

The common misunderstanding of what is meant by problem solving was evident in that problem solving had the lowest mean of the five processes. NCTM highlights that for something to be a problem, a child does not immediately know the answer or strategy for finding the answer. It requires thinking. (e.g. “I wonder how we could make this block structure sturdier?” – A question asked of the child who has been using the same shaped blocks in the same pattern and getting the same unstable results). However, teachers often view something as a mathematics problem if it is any question asked of children (e.g. “What picture shows four objects?” – A question asked of a child who easily finds the answer by counting the objects in each picture). As noted in Chapter II, studies have shown that too much focus is on basic mathematics skills at the preschool level rather than higher-level thinking like in the first example (Copple, 2004; Cross et al., 2009; Graham et al., 1997; Kamii, 1982; Lee & Ginsburg, 2007; National Research

Council, 2001b; Pianta et al., 2005; Rudd et al., 2008). If teachers do not fully understand

140 what is meant by problem solving then there are fewer opportunities for this type of thinking and mathematics is reduced to children demonstrating what they already know, especially in regards to basics such as simple counting and numeral or shape identification, rather than learning and applying new concepts or understandings like the first example suggests.

Another misconception of the meaning of problem solving that this example highlights is that problem solving is often viewed as specifically relating to number concepts or operations such as addition and subtraction. The item on the survey that addresses the aspect of problem solving as ‘not knowing immediately how to solve a problem’ had the lowest mean of all six problem solving questions and the highest standard deviation of all 25 questions which illustrates teachers’ wide range of knowledge or uncertainty with this question. When a teacher understands the meaning of problem solving, both the mathematics content and the pedagogical strategies for problem solving, then a culture of problem solving can be established in the classroom where children expect to have to think, be persistent, make mistakes, and try out new ideas. Like the block example, at the preschool level much mathematics teaching is unplanned during play and conversations so with a thorough understanding of problem solving teachers will know when and how to intervene to support children’s thinking and

“guide the development of the classroom microculture” (Simon, 1995, p. 237).

Teachers’ reported behaviors scale. The purpose of the second research question was to find out more about teachers’ reported behaviors in the classroom that promote children’s knowledge and use of the mathematics processes. Overall, teachers

141 reported they engage in these behaviors at least weekly. On average, teachers report the least amount of use of the representation process in their classrooms – less than weekly.

Reported use of the mathematics processes at least weekly is better than expected but research shows that in self-report surveys people tend to overestimate their behaviors.

This is referred to as Self-deceptive Enhancement (SDE), an aspect of Social Desirability

Bias. It is defined by Booth-Kowley, Edwards, and Rosenfeld (1992) as “the tendency to give honestly believed but overly positive reports about oneself “(p. 562). Research has shown no differences in the effect of SDE based on the mode of the survey, paper versus web for example, so there is the possibility of SDE for this web-based survey. With or without the possibility of the effect of SDE on participant responses, the mathematics processes should be part of the daily curriculum. Recommendations from the Conference on Standards for Prekindergarten and Kindergarten Mathematics Education state this as well,

“…mathematical processes such as problem solving, reasoning and proof,

communication, connections, and representation; specific mathematical

processes such as organizing information, patterning, and composing; and habits

of mind such as curiosity, imagination, inventiveness, persistence, willingness to

experiment, and sensitivity to patterns should all be involved in a high-quality

early childhood mathematics program.” (Clements et al., 2004, p. 3)

Research cited in Chapter II indicates that there is very little mathematics evident in a typical day at the preschool level and that daily routines and transitions make up the majority of a child’s day versus engagement in thought provoking learning experiences

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(Early, et al. 2005). Therefore, daily habits or practices that promote these mathematics processes are essential for young children’s development of mathematical reasoning and understanding.

Teachers’ attitudes/beliefs scale. The purpose of the third research question was to learn more about preschool teachers’ attitudes/beliefs toward the NCTM Process

Standards – how confident they are in their knowledge of them, if they feel they are appropriate to teach in preschool, and if they feel more professional development opportunities in these areas would be beneficial to their teaching. The preschool teachers in this sample have positive attitudes/beliefs toward the Process Standards in preschool. The average responses to all items in this section were between “Tend to

Agree” and “Agree.” If teachers waivered at all it was in regards to the processes of reasoning & proof, and representation where there was slightly less agreement. This is understandable as these two processes are typically associated with the work of older children such as giving a mathematical proof for a solution to a problem or using objects and symbols to work through a mathematics problem.

Looking specifically at individual survey items the majority of teachers view the processes as necessary for preschool children’s understanding of mathematics content

(item #41). However, 21 teachers had some level of disagreement that the process of reasoning & proof was necessary for content understanding in preschool compared to only two or three who disagreed that problem solving, communication, and connections were necessary and nine who disagreed that representation was necessary. The process of reasoning & proof is also the one teachers report the least knowledge of as well.

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The majority of teachers had some level of agreement that the processes should be emphasized during the school day with young children (item #42), but again, reasoning & proof had more disagreement (19 teachers) and representation had disagreement by nine teachers. While these numbers are not significantly different it suggests that there may be many other preschool teachers who feel the same way who did not complete the survey. If many teachers focus on teaching basic facts then one can assume they do not see reasoning & proof as necessary for content understanding in preschool.

As noted in Chapter II, recommendations from not only NCTM but also the

National Research Council and the Conference on Standards for Prekindergarten and

Kindergarten Mathematics Education in 2004 are in agreement that reasoning and representation are important at the preschool level (e.g. Recommendation # 7, #10, #11, pg. 37-38). For example, recommendation 10 states: “Teachers should endeavor to understand each child’s own mathematical ideas and strategies. Teachers should use those understandings to plan and adapt instruction and curriculum” (Clements et al.,

2004). If children are offered time to share their ideas and are expected to explain their thinking then teachers can plan experiences that will build on those ideas and extend their thinking further. Young children are capable of reasoning about many things, mathematics is no different and the use of representation can provide children a visual to support their reasoning process. Therefore, a focus on these two processes in professional development offerings and in mathematics methods coursework will be important for a total understanding of how young children come to understand mathematics.

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Items #43 and #44 asked about teacher beliefs regarding confidence in their knowledge of how NCTM defines the processes and how NCTM envisions teachers promote these processes in the classroom. The teachers were confident in their knowledge of how NCTM defines the process standards (item #43) but not strongly.

They were the least confident with reasoning & proof (38 disagreed), and representation

(30 disagreed) but the number of teachers who disagreed with the other three processes was also higher than average compared to other items in the Teacher Attitudes/Beliefs section. This means that overall, teachers were less confident with their knowledge of how these are defined by NCTM. The lowest scores for the Attitudes/Beliefs section are for item #44, “I understand how NCTM envisions that teachers promote the following mathematics processes in their classroom: problem solving, reasoning & proof, communication, connections, and representation. While the average scores are 3.00 to

3.17, which is a positive “agreement” with this statement, there were 53 to 68 teachers who did not feel confident in how their pedagogical beliefs aligned with NCTM’s recommendations. Thus a focus on the mathematics pedagogical content knowledge for teaching at the preschool level would be an important focus for professional development offerings and college course work.

Research Question 4

When comparing the scores of the Teacher Perceived Knowledge scale with the scores of the Teacher Reported Behaviors scale one would expect that those teachers with higher scores on the knowledge portion would report more frequent behaviors as past studies have shown (Appalachian Rural, 2000; Botha et al., 2005; Brown, 2005; Hill et

145 al., 2008; National Advisory Panel, 2008; Perrin, 2008; Weiss et al. 2003). However, there was only a weak correlation between the two section total scores and also for the processes of problem solving and representation. The weakness of the correlation might be explained by respondents’ use of the default response of “don’t know/neutral” or due to the over reporting of the frequency of behaviors as mentioned earlier.

Representation had the lowest frequency of behaviors. Teachers were asked three questions related to how they support children’s representation of their mathematical thinking or representation of mathematics concepts. One item asked teachers how often they “plan time for children to represent their mathematical thinking by drawing, building, etc.” The average score was 3.33 or about weekly. Another item asked teachers how often they “support children’s play with card and / or board games.” The average score was 2.97 or a couple times a month. The third item asked the teachers how often they suggest to children during block play that they draw a plan for what they want to build or to draw a picture of what they did build. This item received the lowest average score of 2.16 meaning once or twice a month at the most. Teacher perceived knowledge of representation on section one indicated most teachers did not agree strongly with how NCTM defined representation and the range was wider than the other processes. It seems that teachers would benefit from more professional development experiences on how mathematics concepts can be represented by young children at the concrete and pictorial level as well as symbolically (Bruner, 1966; Kamii, 1982). This may help teachers not only see the possibilities for more ways to promote representation but also the value in helping children make connections between more than one form of

146 representation (e.g. Seeing a picture of three birds on the spinner in the game Hi-Ho

Cherrio next to the numeral three).

One possible reason for the less frequently reported behaviors is that there are specific challenges teachers face at their work setting that may prohibit them from teaching in a way they desire. The teachers in this sample responded to an open-ended question that asked what challenges they perceived they faced when trying to incorporate mathematics processes in their teaching. Not enough time for planning, access to materials or resources, and following a prescribed curriculum were the three most reported challenges (n = 109, 91, and 86 respectively). Other challenges teachers reported were: staffing issues, not enough time in a half day program, and pressure from families to do certain types of activities such as basic skills worksheets. Most teachers reported having more than one of these challenges and only 18 of the 217 teachers reported having no challenges to promoting the mathematics processes in their classrooms.

Research Question 5

The recommendations for professional development discussed above are supported by the data analysis for research questions 5 and 6. Analysis for research question 5 showed statistically significant correlations between teacher perceived knowledge of the process standards and their attitudes/beliefs regarding the value of these mathematics processes. This is expected as past research studies have also found correlations between teacher knowledge and teacher attitudes/beliefs related to mathematics as discussed in Chapter II (Appalachian Rural, 2000; Perrin, 2008).

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Relationships were also found for specific processes of problem solving, reasoning & proof, and representation and while overall attitudes/beliefs are positive, they are a bit lower for these three specific processes. Just as there is a pattern of lower average scores for problem solving, representation, and reasoning & proof when looking at these scores individually throughout all three sections of the survey this pattern persists when examining correlations between teacher perceived knowledge and teacher attitudes/beliefs. This means that teachers who have lower average scores for these three processes also had less positive attitudes/beliefs for these three processes. The correlations are not as strong as one would expect which may be due to SDB or satisficing when completing the survey. Teachers seem to believe moderately in their value but are not confident in their knowledge.

Research Question 6

Analysis of research question 6 showed a meaningfully significant correlation between what teachers feel about the mathematics processes and what they report they do in the classroom. Correlations also existed for the processes of communication, problem solving and connections and weak correlations for reasoning & proof and representation.

There were less positive attitudes and less frequent behaviors for reasoning & proof, problem solving, and representation.

Many studies related to teaching at various grade levels, including preschool, have shown a relationship between teacher attitudes / beliefs and classroom practices but others have shown discrepancies between how teachers feel and what they actually do.

For example, as discussed in Chapter II, a study by Thiel (2010) showed that teachers

148 with positive attitudes placed more emphasis on processes such as problem solving and teachers with less positive attitudes placed less emphasis on processes and application and instead used more worksheets. However, other studies by Wien (1995), Brown

(2005), Pianta and LaParo (2003) and Graham et al. (1997) showed a disconnect between teachers’ positive beliefs regarding the importance and value of mathematics in preschool and behaviors that fostered mathematical thinking during the school day. The results of the current study have similar findings to those of Thiel (2010). This may be the result of teachers’ self-reported data regarding classroom behaviors rather than actual observations of teachers in the classroom. Therefore, as stated earlier, professional development opportunities for preschool teachers should include self-reflection on one’s own practices and follow-up studies should include teacher interviews and observations to validate these correlations.

To look for other possible explanations for the association found between teacher attitudes/beliefs and reported behaviors in this study, a comparison of the responses to item #45 related to teacher attitudes/beliefs and section II Teacher Reported Behaviors total scores was completed using Pearson’s correlation coefficient. Items #45a -45e state,

“I am adequately promoting children’s use and development of the following mathematics processes at this time: problems solving, reasoning & proof, communication, connections, and representation.” All mean scores for item #45 were

>3.00 which indicates a feeling of adequacy in how the Process Standards are promoted.

All mean scores for frequency of reported behaviors for each mathematic process are between 3.00 and 4.00, which indicates that teachers report engaging in behaviors that

149 promote the mathematics processes at least weekly. A statistically significant correlation was found for all of the mathematics processes except for problem solving but only reasoning & proof and communication were meaningfully significant (see Appendix F;

Table 32).

One explanation for varying results is that differences may exist in how one interprets the meaning of “adequately promoting” when thinking about the frequency of specific classroom behaviors. “Adequately promoting” for some teachers could refer to the level of frequency of behaviors (e.g. once a month versus once a week) or it could refer to how they think about challenges they perceive they face in their work setting.

For example, a teacher could interpret this by thinking, “Yes, given all the challenges I face, I am adequately promoting the Process Standards.” Another teacher may be thinking, “No, because of all the challenges I face, I am not adequately promoting the

Process Standards.” This is where follow-up interviews would be valuable to get a clear picture of how teachers are thinking about the question.

Research Question 7

The purpose of research question 7 was to see how having a license to teach preschool may or may not indicate greater knowledge of the mathematics processes, more frequent behaviors that promote the processes, and more positive attitudes/beliefs toward the importance of the processes in preschool. The results indicated that those with a teaching license had more perceived knowledge of the NCTM Process Standards and particularly the processes of reasoning & proof and representation. It is interesting that there was no association found between having a teaching license and having more

150 positive attitudes/beliefs or more frequent behaviors that promote the mathematics processes. This suggests that a teacher with a teaching license may have more perceived knowledge of the mathematics processes but may not necessarily promote these more frequently in the classroom.

One assumption is that if teacher licensure does not predict more frequency of behaviors then neither would a teacher’s level of education. To examine this assumption, a one-way analysis of variance was conducted (see Appendix F; Tables 33 and 34).

Significant differences were found only for teachers’ perceived knowledge between teachers with a Master’s degree and teachers with an Associate’s degree or high school diploma. So it seems that having a Master’s degree versus a Bachelor’s degree does not necessarily make one more knowledgeable of the NCTM processes. As predicted, no differences were found between level of education and teacher behaviors in the classroom. Because the two sample sizes varied greatly, and since the majority of respondents who indicated having a teaching license (188) work in a public school setting

(113), and have a Master’s degree (118), further research is needed to determine what variable is most significant in predicting greater perceived knowledge of the mathematics processes or more frequent behaviors to promote the mathematics processes in the classroom. This would help determine the focus of professional development efforts.

Research Question 8

The purpose of research question 8 was to see if one could predict that teachers would have greater knowledge of the mathematics processes, more frequent behaviors that promote the processes, and more positive attitudes/beliefs toward the importance of

151 the processes in preschool depending on the setting in which they worked. Analysis of the data found that work setting did not have any affect on teacher perceived knowledge or teacher attitudes/beliefs but it did have an affect on teachers’ reported behaviors.

Significant differences were found between teachers who work in Head Start programs and those who work in public schools for the total behaviors score and specifically for the processes of communication and representation. Teachers in Head Start programs reported more frequent behaviors that promote the mathematics processes.

A look at the reported challenges teachers perceive they face in these two settings may help to explain these differences in teacher behaviors. Follow up interviews would be valuable in determining why this may be the case. One speculation is that Head Start teachers have more flexibility with the curriculum or that they have had professional development trainings that have addressed classroom practices related to mathematics. In looking at the demographic data, a vast majority of the Head Start teachers indicated having mathematics professional development trainings in the last three years whereas only 69 of the 126 public school teachers had training in mathematics. Also, language development is typically a strong focus by teachers for children from lower SES backgrounds, which may contribute to classroom behaviors that promote more communication related to mathematics specifically.

To summarize, research questions 7 and 8 looked at two possible variables that might have an effect on teacher knowledge, behaviors, or attitudes/beliefs, teacher licensure and type of work setting. Based on data from this sample of preschool teachers, relationships were found between teacher licensure and teacher perceived knowledge of

152 the processes and type of work setting and teacher behaviors for Head Start and public school teachers.

Are there other variables that might predict teacher knowledge, behaviors, or attitudes/beliefs? Further analyses of the data were done to see if other variables did predict these constructs including level of education, years of teaching experience, and participation in professional development. For “level of education” there were only differences found for teacher perceived knowledge between teachers with a Master’s degree and teachers without a four-year degree. For “years of teaching experience” no significant differences were found between groups. For “professional development” significant differences were found between those who have and have not had professional development in mathematics in the past three years with teachers' attitudes/beliefs.

This data suggests that teachers with an advanced degree may have had more exposure to methods for teaching mathematics that were in line with the recommendations of NCTM and that teachers who have engaged in a mathematics training have more positive attitudes/beliefs toward the value or importance of the mathematics processes in the classroom. These assumptions are supported by the additional analyses of the data, which indicated a significant difference between respondents who had participated in a mathematics training in the past three years and those who had not with if they had previously heard of the NCTM Standards.

Professional development in the area of mathematics seems to be the most influential factor on a teacher’s familiarity of the Process Standards and more positive attitudes/beliefs toward the Process Standards, even more so than a teacher’s level of

153 education, licensure status, or years of experience. Since many of the recommendations throughout this chapter stress the need for more professional development opportunities, it would be important to examine the professional development literature for methods that have the most impact on classroom teaching and ultimately, student outcomes.

Summary and Implications for Practice

The most important finding from this study is the need to provide teachers more information on how NCTM defines each of the processes with a focus on the mathematics processes of problem solving, reasoning & proof, and representation.

Results of this study showed that a teacher’s perceived knowledge of the mathematics processes influences their attitudes and beliefs about teaching mathematics and their classroom practices. Therefore, information for teachers should include materials and resources that help teachers understand the meaning and purpose of the mathematics processes as well as classroom strategies for incorporating them into the daily life and curriculum of the school. Problem solving was the process teachers indicated the least knowledge of and in order to understand and infuse problem solving into the culture of the classroom one must understand how children learn and the capability of young children to learn mathematics. The foundation for this information is constructivist theory or “inquiry-based” teaching.

Beginning with a focus on developing an inquiry based classroom culture where problem solving is the foundation for the mathematics curriculum, strategies for promoting reasoning and representation can follow. Professional development for these areas should include a look at the teacher’s role during play, classroom materials and

154 resources for teachers and children, the use of more mathematical language by children and teachers, and other strategies that would help teachers promote children’s development and use of the mathematics processes as intended by NCTM.

With more overall content and pedagogical content knowledge of the mathematics processes this may increase teachers’ positive attitudes toward the value and importance of these in the classroom. Results of this study showed that teacher attitudes/beliefs have a strong impact on teacher behaviors so teachers need support from school directors, principals, and other educators to have the time and resources to promote these processes in the classroom on a daily basis. Teachers reported challenges in the work setting that may inhibit them from making changes to their teaching practices. We should examine these more closely in order to find ways to support teachers with time for planning, and time to engage in dialogue with other teachers. Teachers need more resources specific to the mathematic processes and the support to be more flexible with classroom curriculum so a culture of inquiry is established. As stated in Chapter II, while a child’s earliest experiences can have long-lasting outcomes, knowledge of mathematics is “not yet in the hands of most early childhood teachers in a form to effectively guide their teaching”

(NAEYC/NCTM, 2002, p. 2).

Professional development opportunities offered to preschool teachers are the most effective way to improve teacher knowledge of mathematics processes and make an impact on classroom practices. Data from this study found that professional development had more influence on teacher behaviors and attitudes/beliefs regarding the

155 mathematics Process Standards than a teacher’s level of education, experience or licensure status.

One of the most important implications for educators in the field of early childhood education is to become more aware of research findings and recommendations made by groups of early childhood professionals, including teachers. Directors, principals, and other leaders must also be aware of the research and recommendations in order to support teacher endeavors and provide quality mathematics professional development opportunities. Teachers also need ongoing mentoring, reflection on their practice, and dialogue with colleagues as part of their professional development in order to improve their practice. Teacher mentoring can also provide the necessary understanding of the mathematics and resulting confidence that affects what teachers do in the classroom. With confidence and more positive attitudes/beliefs, teachers can more effectively support children’s mathematical learning during play and other unplanned significant moments. The joint position statement of NCTM and NAEYC (2002) supports this recommendation as well suggesting school leaders “provide ample time, materials, and teacher support for children to engage in play, a context in which they explore and manipulate mathematics ideas with keen interest” (p. 3).

Future Research

 It will be important to increase the response rates for teachers who work in a wide

variety of early education settings, most importantly, teachers who work in for-

profit or private childcare centers. Personal contacts with directors or teachers

may be necessary to get a greater response from the under-represented groups.

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The percentage of children enrolled in early education settings has increased from

59% to 64% from 1990 to 2012 (NCES, 2014), the majority in centers not

classified as Head Start programs. Teachers in private childcare settings do not

often need a college degree or teacher licensure like in public school or Head Start

programs so it is even more important to find out what teachers know or do not

know and to offer quality mathematics trainings. A random sample of teachers

nationwide would also be of value in order to develop or design mathematics

training programs or curricular resources that meet the needs of teachers and

children - especially for college coursework that prepares pre-service teachers to

work anywhere in the country and the world. This data could also provide

information on what professional development programs related to mathematics

processes (if any) have been effective in different states.

 A mixed methods study would be valuable for obtaining more information

regarding teaching practices. Observations of teachers would be important to

verify or validate teachers self reported data on teacher behaviors. Interviews

may reveal more information about responses to challenges teachers face in

making their own judgments about what and how to teach mathematics. Are there

actual barriers or challenges out of the teacher’s hands and what could be done to

remove these barriers? [Q40] Interviews may also reveal more information on

what type of professional development experiences participants found most useful

[Q54 & Q55].

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Many teachers reported a lack of resources on the processes [Q40]. What might

these look like? Print resources can be beneficial but mentoring would be ideal in

order to support teachers’ reflections on their own beliefs and practice and to

establish a classroom culture of inquiry.

 Another suggestion for future research is to determine the effectiveness of college

coursework where the NCTM Process Standards and/or the Common Core

Mathematical Practices (CCMP) (2010) are emphasized. If teachers understand

the NCTM Process Standards, which are part of the foundation for the CCMP,

then they will be able to implement these practices more effectively and establish

the necessary problem solving and mathematics inquiry culture in the classroom.

One could compare results such as those of the pilot study where 1st semester

students who had not had mathematics methods courses yet that focused on the

Standards were compared with 5th semester students who have had three

mathematic methods courses that all focused on mathematics processes. One

would predict that the 5th semester students had higher mean scores on all three

sections: Teacher perceived knowledge, teacher reported behaviors, and teacher

attitudes/beliefs and results of the pilot study indicate this to be true. Students

who had three mathematics methods courses had higher mean scores for all three

scales of the survey instrument than students who had not yet had any of the

methods courses. This data further substantiates the validity and reliability of the

survey designed for this study.

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o A follow up interview and/or observations of these students in their first

classroom would provide data on if their knowledge and experience from

college coursework and fieldwork translated to practice after graduation.

These interviews may also shed light on whether or not there is value in

studying the usefulness of using the CCMP to support curriculum

development at the preschool level or ways to refine the CCMP for

preschool teachers to better align with the language of early childhood

literature and the NCTM Process Standards.

 It would also be valuable to study how teacher knowledge, behaviors, and

attitudes/beliefs regarding the mathematics processes at the preschool level affect

student outcomes. A longitudinal study could collect data on student thinking,

reasoning, and conceptual understanding of mathematics into kindergarten or

even to third grade. Or, a study with an experimental design could compare

children from two classrooms to see if those children in classrooms where the

mathematics processes are emphasized have greater understanding of

mathematics concepts.

 Follow-up interviews with respondents would also provide more information on

teachers’ meaning of “familiarity with the NCTM Process Standards.” The last

item of the survey asked participants if they had heard of the NCTM Process

Standards before and 52% said they had heard of the Process Standards. At first

glance this is not surprising since 118 participants have a Master’s degree but of

those 118 only 68 (57.6%) said they had heard of the Process Standards before

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taking this survey. One would expect that this information would be discussed in

a mathematics methods course so another question to include on the survey in the

future is if teachers had taken a mathematics methods course. It is also not clear

how familiar teachers are with these Standards. Some teachers may have just

seen them listed somewhere and other teachers may have read descriptions of

them and still others may have studied them in a course where they were asked to

incorporate them into their curriculum / lesson plans. Follow up questions would

be necessary in order to know, (a) the extent of their familiarity, and (b) if there

were any correlations between past experience/exposure with the Process

Standards and their scores on Teacher Knowledge and Teacher Behaviors sections

of the survey. This would be an important study for future research so that one

could judge the effectiveness of teacher trainings, workshops, and coursework

that focus on the mathematics processes.

Conclusion

As an exploratory study of a large population of preschool teachers valuable information has been obtained that can help support teachers’ understanding and implementation of inquiry-based, constructivist mathematics experiences for young children. With more opportunities for professional development that includes teacher mentoring and observation of quality mathematics teaching in preschool classrooms, teachers will have an increased awareness of the capabilities of the young child.

The survey designed for this study was found to be a reliable instrument for gathering initial data on teachers’ perceived knowledge, reported classroom behaviors,

160 and their attitudes/beliefs regarding the NCTM mathematics processes. Both mathematics content and pedagogical content knowledge and positive attitudes and beliefs toward mathematics teaching and learning were found to influence teaching behaviors. Therefore, there is a definite need for more mathematics professional development opportunities for teachers, especially in the areas of problem solving, reasoning & proof, and representation. In the future, the survey can be used to gather information from teachers in underrepresented sites such as private childcare centers, so quality professional development programs can be developed to meet the needs of the teachers in a variety of settings.

APPENDICES

APPENDIX A

DRAFT SURVEY SCALES

Appendix A

Draft Survey Scales

Twenty-five Items for Perceived Knowledge of NCTM Mathematics Process Standards Scale – Draft Survey

1 Agree 2 Tend to 3 Don’t 4 Tend to 5 Disagree agree know disagree

1. I believe that the most appropriate 1 2 3 4 5 math problems for preschoolers are those that the children do not know how to solve immediately. 2. I believe that there should be 1 2 3 4 5 planned times for children to share math experiences they were grappling / engaging with during play. 3. I feel my planned lessons are 1 2 3 4 5 typically more valuable for mathematics learning than using spontaneous opportunities to initiate problem solving. 4. I feel that the most important 1 2 3 4 5 connection teachers can help preschoolers make is that between children’s intuitive mathematics and the mathematics they learn in preschool. 5. I feel that the mathematics 1 2 3 4 5 curriculum should include at least one or more worksheet activities during the week to provide children opportunities to use and develop the mathematics processes. 6. I believe that children’s personal 1 2 3 4 5 representations, such as drawings, should be used as a valid assessment of their level of mathematical understanding. 7. I feel preschoolers benefit most 1 2 3 4 5 often from planned experiences that focus on revisiting mathematical ideas in the same context as they were initially introduced.

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8. I feel it is important for children to 1 2 3 4 5 use their representations as a way to make mathematical ideas available for personal reflection. 9. I believe that classification is just as 1 2 3 4 5 important to focus on in preschool as patterns and counting. 10. I feel that preschool teachers 1 2 3 4 5 should intervene in a child’s problem solving so they do not develop anxiety toward mathematics. 11. I believe problem solving in 1 2 3 4 5 mathematics should be reserved for children in kindergarten and older. 12. I believe that preschool children 1 2 3 4 5 should be expected to explain their thinking and listen to the mathematical ideas of their peers. 13. I believe it is appropriate for 1 2 3 4 5 preschool children to classify objects by stating how they are alike and different but not necessarily to make generalizations about them. 14. I feel that it is important to take 1 2 3 4 5 the time during the school day – even if it means giving up time for other planned activities - to teach preschoolers how to work as a community of learners. 15. I feel that as children come up 1 2 3 4 5 with personal representations the teacher should begin to help them translate these to more conventional ways of representing mathematical situations. 16. I believe that preschool children 1 2 3 4 5 should be expected to make sense of mathematics. 17. I feel it should be expected that 1 2 3 4 5 children in preschool learn to question the mathematical strategies of others and/or ask for clarification. 18. I feel that preschool teachers 1 2 3 4 5 should wait to promote problem solving until after the children have acquired basic skills such as counting, numeral recognition, sorting, and identifying shapes.

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19. I feel that it is inappropriate to use 1 2 3 4 5 the standard mathematics vocabulary in preschool (e.g. pentagon, cube, inches). 20. I believe that in preschool, the 1 2 3 4 5 teaching of mathematics is more effective when activities are offered separate from other subject areas. 21. I believe that preschool teachers 1 2 3 4 5 should model for children the steps for thinking through a mathematical situation. 22. I feel that the math activities 1 2 3 4 5 provided in the curriculum are more educational than those I might develop specific to my students. 23. I believe that mathematics should 1 2 3 4 5 be related to a young child’s real world experiences. 24. I believe that preschool children 1 2 3 4 5 progress further mathematically when you focus on one method of representation at a time. 25. I feel that one mathematics goal in 1 2 3 4 5 preschool should be for children to justify their answers to their peers.

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Twenty-one Items for Self-reported Behaviors of Mathematics Teaching Scale – Draft survey

Never One or One or One or Daily More two times two two than once a semester times a times a a day month week 26. I specifically plan opportunities for children to 0 1 2 3 4 5 engage in mathematical problem solving (e.g. how to share a set of objects equally among 3 children). 27. I pose mathematics problems to children during 0 1 2 3 4 5 daily routines (e.g. attendance, snack, clean up, line up…) 28. I purposefully highlight math concepts that are evident 0 1 2 3 4 5 in picture books when reading a story. 29. I pose mathematics problems to children who are 0 1 2 3 4 5 engaged in block play. 30. I purposefully add materials to the dramatic play 0 1 2 3 4 5 area that will promote mathematical thinking. 31. I provide and support children’s play with card and/or 0 1 2 3 4 5 board games. 32. I facilitate children’s play to help children learn to 0 1 2 3 4 5 communicate their mathematical ideas/thinking to others. 33. I plan time for children to share their mathematical 0 1 2 3 4 5 thinking during whole group meeting times. 34. I plan time for children to represent their mathematical 0 1 2 3 4 5 thinking by drawing, building, etc. 35. I provide worksheet activities not just for numeral 0 1 2 3 4 5 recognition, more/less, matching, etc. but to pose mathematics problems such as simple addition or subtraction.

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36. I facilitate conversation among children to model / teach 0 1 2 3 4 5 them how to explain their reasoning about mathematical ideas. 37. I change my plans to capitalize on spontaneous 0 1 2 3 4 5 opportunities during the day to promote children’s use of mathematics processes. 38. I comment on children’s block play in order to model the 0 1 2 3 4 5 formal mathematics vocabulary. 39. When a child gives a faulty explanation I provide the right 0 1 2 3 4 5 information so they don’t leave with misconceptions. 40. During block play, I make suggestions to children to draw 0 1 2 3 4 5 plans for structures before building or to draw structures they have already built. 41. I sit with children at play with math manipulatives to 0 1 2 3 4 5 promote their communication about mathematical ideas to others. 42. I provide time for long- term small group projects where 0 1 2 3 4 5 mathematics is integrated. 43. I provide children time to play mathematics games on the 0 1 2 3 4 5 computer. 44. I integrate mathematics by incorporating instances where 0 1 2 3 4 5 children vote on things pertaining to the classroom. 45. I make decisions to rotate materials in/out based on 0 1 2 3 4 5 children’s developmental needs in mathematics.

46. What challenges, if any, do you face in making your own judgments about what to teach and how you teach? a. Following a prescribed curriculum b. Not enough time for planning c. Staffing issues d. Length of school day e. Other ______

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Six Items for the Attitudes/Beliefs of Teachers on the Value or Importance of the NCTM Process Standards Scale – Draft survey

Agree Tend to Tend to Disagree agree disagree 47. I feel the use of the following mathematics processes are necessary for preschool children’s understanding of mathematics content such as number concepts, measurement, and geometry:  Problem solving 1 2 3 4  Communication 1 2 3 4  Reasoning & proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 48. I feel it is appropriate to emphasize the following mathematics processes during the school day with children ages 3 to 5 years:  Problem solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 49. I feel confident in my knowledge of how NCTM defines each of the following process standards:  Problem solving 1 2 3 4  Communication 1 2 3 4  Reasoning & proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 50. I understand how NCTM envisions teachers promote the following mathematics processes in their classrooms:  Problem solving 1 2 3 4  Communication 1 2 3 4  Reasoning & proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 51. I feel I am adequately promoting children’s use and development of the following mathematics processes at this time:  Problem solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 52. I feel that professional development opportunities regarding the teaching of mathematics processes would be beneficial to my teaching. 1 2 3 4

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Ten Items for the Teacher Demographic Information Section

53. In what type of early care and education setting do you teach? a. For-profit childcare center b. Head Start program c. Public school preschool d. University laboratory school e. Licensed home / family childcare f. Other ______

54. What ages do you currently teach? [Mark all that apply] a. 3 year olds b. 4 year olds c. 5 year olds (not in kdg)

55. How many years have you taught children between the ages of 3 and 5 (not in kdg) years?

56. How many years teaching experience do you have? (any age group)

57. What is your highest level of education? a. High school diploma b. CDA c. Associate’s Degree d. Bachelor’s Degree e. Master’s Degree f. PhD

58. If you have an associate, bachelors, masters, or PhD degree is one or more of these in the field of education? a. Yes b. No

59. Do you have licensure / certification to teach children ages 3 to 5 (not in kdg)? a. Yes b. No

60. Have you participated in professional development in the area of preschool mathematics in the past 3 years? a. Yes b. No

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61. If so, was the professional development useful for your teaching of young children? a. Very useful b. Somewhat useful c. Not very useful

62. Have you heard of the National Council Teachers of Mathematics five process standards: problem solving, reasoning, communication, connections, and representation? a. Yes b. No

APPENDIX B

CONTENT REVIEWER AND COGNITIVE INTERVIEW DATA

Appendix B

Content Reviewer and Cognitive Interview Data

Table 28

Item Criteria/Rubric for Content Reviewers

Q Dimension Dimension Dimension meaning Best Items # importance answer needing revision 1 Problem solving Math It is only a problem if 1 - agree knowledge you don’t know (MCK) something 2 Communication Math Time for children to 1 - agree teaching talk about math (MPCK) meaningful to them 3 Problem solving Math Capitalize on 4 – tend (-) teaching children’s interest to (MPCK) and motivation disagree 4 Connections Math Accommodation and 1 - agree knowledge assimilation (MCK) 5 ALL MPCK DAP – actively 5 (-) explore materials & interact with others 6 Representation MCK Process is important/ 1 personal reps are valid 7 Connections MPCK Explore ideas in new 5 (-) contexts

8 Representation MCK Reflective abstraction 1

9 Reasoning & MCK Requires logical 1 Proof thinking 10 Problem MPCK A problem requires 4 or 5 Solving puzzlement (-)

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11 Problem MCK Young children are 5 Solving capable (-) 12 Reasoning & MCPK Explanation requires 1 Proof reasoning

13 Reasoning & MCK Use knowledge of 5 Proof patterns (-) 14 Communication MCPK Community develops 1 trust & respect for ideas 15 Representation MPCK Multiple forms help 1 make connections 16 Reasoning & MCK Young children are 1 Proof capable 17 Communication MPCK Necessary for sense 1 making 18 Problem MCK Math then has 5 Solving meaning (-) 19 Communication MCK Mathematizing; 5 (-) language development 20 Connections MCK Integration is more 5 (-) natural and meaningful 21 Problem MPCK 4 or 5 Solving 22 Connections MCK Know your students 5 (-) abilities 23 Connections MCK Know your students 1 experiences 24 Representation MPCK Multiple forms 5 (-) promotes connections 25 Reasoning & MPCK Use of logic and 1 Proof solidifies understanding

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Table 29

Content Reviewer One Results

Q# Dimension Dimension Dimension meaning Best Items importance answer needing revision 1 Problem solving Math It is only a problem if 1 - agree knowledge you don’t know (MCK) something 2 Communication Math Time for children to 1 - agree teaching talk about math (MPCK) meaningful to them 3 Problem solving Math Capitalize on 4 – tend teaching children’s interest and to (MPCK) motivation disagree 4 Connections Math Accommodation and 1 - agree knowledge assimilation (MCK) 5 Problem solving MCK Basic skills 5

6 Representation MCK Process/ product 1

7 Connections MCK Interrelatedness 3 or 4 “most often” –not sure what that implies 8 Representation MPCK Making sense 1

9 Connections MCK Interrelatedness 3

10 Communication MPCK Listen and accept 4 multiple forms 11 Problem solving MPCK Problems in daily 5 routines

12 Communication MPCK Exchange ideas with 1 teacher and peers 13 Connections MCK Interrelatedness 4

14 Communication MPCK Exchange ideas 2 “teach”?

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15 Representation MPCK Help child translate to 2 “begin to conventional help”? 16 Reasoning and MCK Notice patterns 2 proof 17 Communication MPCK Provide opportunities 2 to explain 18 Problem solving MPCK Present problems in 5 daily routine 19 Communication MPCK Translate to 4 conventional 20 Connection MCK Interrelatedness 4

21 Reasoning and MPCK Child explain own 5 proof ideas 22 Problem solving MPCK Present daily 3 opportunities 23 Problem solving MPCK Present daily 1 opportunities 24 Representation MPCK Promote multiple 4 “you”? forms of rep. 25 Communication MPCK Opportunities to 2 What about exchange ideas justify to teacher? 26 Problem solving MPCK Daily tasks 3 “specificall y”? 27 Problem solving MPCK Daily tasks 5

28 Reasoning and MPCK Child makes 4 proof conjecture 29 Reasoning and MPCK Explain own thinking 5 proof 30 Problem solving MPCK Work together and 3 reflect 31 Problem solving MPCK Use problem Solving 4 strategies 32 Communication MPCK Reflect, model and 4 pose questions

33 Communication MPCK Opportunity to 4 exchange ideas 34 Representation MCK Organize thinking 4

35 Problem solving MCK Basic skills 1

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36 Reasoning and MPCK Questioning 4 proof 37 Reasoning and MCK Noticing in real world 5 proof 38 Representation MPCK Begin to translate 4

39 Reasoning and MPCK Explain own thinking 4 “faulty”? proof 40 Representation MPCK Multiple forms 2

41 Communication MPCK Exchange ideas 3 “sit with”?

42 Connections MCK Connect to other 2 subjects 43 Problem solving MCK Basic skills 3

44 Communication MPCK Exchange ideas 3

45 Problem solving MPCK Higher order thinking 3

Table 30

Content Reviewer Two Results

Q# Dimension Dimension Dimension meaning Best Items importance answer needing revision 1 Problem solving Math It is only a problem if 1 - agree knowledge you don’t know (MCK) something 2 Communication Math Time for children to 1 - agree teaching talk about math (MPCK) meaningful to them 3 Problem solving Math Capitalize on 4 – tend teaching children’s interest to (MPCK) and motivation disagree 4 Connections Math Accommodation and 1 - agree knowledge assimilation (MCK) 5 Representation MCK Worksheets are used 4 for process

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6 Representation MCK Personal reps are ??? valid 7 Connections MPCK Revisit in original 4 context 8 Representation MCK Reps for personal 1 reflection 9 Reasoning & MCK Classification is as 1 Proof important as pattern and counting 10 Problem solving MPCK Teachers should 4 intervene when PS 11 Problem solving MCK PS for older students 4

12 Communication MCK Explain thinking & 1 listen to other explanations 13 Reasoning & MPCK Not necessary to 4 Proof generalize 14 Connections MPCK Flexibility for 1 learning moment 15 Representation MPCK Help students learn 1 conventional representation 16 Reasoning & MCK Children should make 1 Proof sense of mathematics 17 Reasoning & MPCK Children should 1 Proof question processes of other students 18 Problem MPCK Problem solve after 4 Solving basic skills

19 Representation MCK Should not use 4 standard vocabulary 20 Connections MPCK Keep content separate 4

21 Problem MPCK Teacher models how 4 Solving to problem solve

22 Problem MCK Purchased curriculum 4 Solving is better than my own

23 Connections MCK Relate to real world 1

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24 Representation MPCK One representation at 4 a time 25 Reasoning & MPCK Student justifies to 1 Proof their peers 26 Problem MCK Plan problem solving 1 Solving opportunities 27 Problem MPCK Pose problems during 1 Solving daily routines 28 Connection MPCK Highlight 1 mathematics from stories 29 Representation MPCK Pose problems while 1 students play with blocks 30 Connections MPCK Mathematical 1 materials in play area 31 Connections MPCK Support when 1 students play games 32 Communication MPCK Facilitate during play 1 to help students communicate ideas 33 Reasoning & MPCK Share thinking with 1 Proof group 34 Representation MPCK Children represent 1 math with drawings and building 35 Representation MPCK Worksheets for 4 numeral recognition 36 Reasoning & MPCK Facilitate 1 Proof conversation about reasoning 37 Connections MPCK Change plans to 1 capitalize on opportunities 38 Communication MPCK Model formal 1 mathematics vocabulary 39 Reasoning & MPCK Provide correct 4 Proof solutions when incorrect is given 40 Representation MPCK Suggest students 4 create plans 41 Communication MPCK Promote 1

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communication of ideas to others 42 Communication MPCK Long-term small 1 group projects are integrated 43 Problem MCK Provide games on 4 Solving Computer

44 Connection MPCK Mathematics 4 computer games provided 45 Connection MCK Rotate materials 4 based on development of children

Cognitive Interview Notes ______*Questions 2 & 3 were asked for each of the 59 items that the respondents were unsure about and then after this, questions 4 & 5 were asked.

Survey Q1 Q2 Q3 Q4 Q5 item # What items were What is the Why did you What What you unsure of item asking? choose this revisions additional how to respond? response? would you items make? would you add?

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Summary of notes based on the questions in chart above (Teachers 1-10)

T1: Item #3 – “Combination is better. 75% spontaneous creates emergent curriculum based on learning necessary for play.” Item #5 – “Documenting or recording experience is valuable for anyone.” Item #7 – “I think repetition is valuable, however I rely more on children developing ability to transfer understanding.” Item #9 – “Absolutely believe more choices and decisions are based on classification – best to learn to understand the processes of organization.” Item #14 – “A community of learners may need another label.” Item #21 – “Thoughtfully, gently, gracefully taking cues for process from children.” Item #24 – “Depends on knowledge of each child.” Item #25 – “Justify? Provide proof? Strong word – explain.” T2: Item #50 – “I am familiar but may not know, therefore, may not understand.” T3: Item #39 – “Tricky question.” Section I – “Easy to fill out. Choices were not difficult.” Section II – “Sometimes these choices were hard to decide between.” “Overall, quick and easy.” T4: Item #16 – “Know matter what you do children are going to make sense of it in some way so unsure how to answer. Is the explanation that they will show you how it makes sense? What does expected mean?” T5: Item #13 – “Maybe use an example with this question.” Section III – “I might add a textbook sentence about each one as a reminder.” “Start with disagree as 1 and progress to agree as 5.” T6: Item #4 – “Unclear. Intuitive math versus preschool math, not sure of the difference.” “Add definitions of processes for attitudes section – review of what is NCTM.” T7: Item #10 – “Teachers have to know students and when this is okay.” Item #15 – “Know the child and when appropriate.” Item #16 – “Begin to…” Item #17 – “Begin to…” Item #21 – “When appropriate.” Item #33 – “As needed.” Item #37 – “As needed.” T8: No suggestions. T9: Item #39 – “It depends.”

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T10: Item #4 – “Very confusing question/wording.” Item #6 – “Change ‘should’ to ‘could’ – ‘should’ is a powerful word.” Item #8 – “Examples? Knowing what ‘representations’ means to you or this survey would be helpful. I don’t feel like it is a colloquial term in ECE.” Item #13, #15 – “Give example for ‘generalizations’ and ‘personal representations.” Item #16 – “What does ‘make sense’ mean to you or this survey? The meaning is unclear or confusing.” Item #23 – “Explain ‘justify’.” Item #24 – “What qualifies as a method of representation? Provide a list?” Item #30 – “How do I evaluate this? I don’t add new things daily but I do add things and some are already there.” Item #42, #45 – “Another time question that is tough to answer in the scale offered.” Item #43 – “How will you interpret answers if you don’t know about their access to computers for children’s use?” Item #55, #57 - clarify Section I – Add a new choice, “neutral”. Should the title, Teacher Knowledge, be Teacher Beliefs because knowledge makes it feel there is a right or wrong answer. Section III – Add definitions of NCTM processes. Add a fifth column called “don’t know.” Grammar edits: #2, 9, 14, 18, 20, 22, 37, 38, 40, 46, 47, 49, 50, 57, 59. ______

APPENDIX C

PILOT STUDY SURVEY

Appendix C

Pilot Study Survey

An Exploratory Study of Preschool Teachers’ Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Regarding Mathematics Processes

Dear preschool teacher,

I am conducting a survey concerning the teaching of mathematics for children between the ages of 3 and 5 who are not yet in kindergarten. It is vital that teachers of young children have multiple opportunities to share their knowledge on what is appropriate for the children they care for and to influence the types of professional development opportunities that are provided for teachers.

The survey is intended for teachers in a variety of early care and education settings across the nation. The information you provide will be useful in developing meaningful, professional development opportunities and resources for teachers. Your responses matter greatly to those interested in improving mathematics education for young children. It is important to hear from classroom teachers like you who have daily opportunities to plan for and observe the mathematical thinking of young children.

Please take some time to complete the survey. It should take no more than 15 minutes. There are no correct or incorrect responses, only your best answer. All responses will be treated confidentially and will in no way be traceable to individual respondents once the survey process has been concluded. Please (give directions for completing the on-line survey).

Thank you for your assistance!

Sincerely,

Julie Stoll Early Childhood Faculty Kent State University

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SECTION I: Teacher beliefs about early childhood mathematics

Please share your beliefs about teaching and learning in early education settings related to mathematics for young children. Please respond by circling a 1 if you agree, 2 if you tend to agree, 3 if you don’t know or are neutral, 4 if you tend to disagree, or 5 if you disagree to the following statements.

1 2 Tend 3 Don’t 4 Tend 5 Agree to know / to Disagree agree neutral disagree 1. I believe that the most appropriate 1 2 3 4 5 math problems for preschoolers are those that the children do not know how to solve immediately. 2. I believe that there should be time 1 2 3 4 5 planned for children to share math experiences they were grappling / engaging with during play. 3. I feel my planned lessons are 1 2 3 4 5 typically more valuable for mathematics learning than using spontaneous opportunities to initiate problem solving. 4. I feel that the most important 1 2 3 4 5 connection teachers can help preschoolers make is the connection between children’s informal or intuitive mathematics knowledge and the formal mathematics learned in school. 5. I feel that the mathematics 1 2 3 4 5 curriculum should include at least one or more worksheet activities during the week to provide children opportunities to use and develop mathematics processes. 6. I believe that children’s personal 1 2 3 4 5 representations, such as drawings, could be used as a valid assessment of their level of mathematical understanding.

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7. I feel preschool children benefit 1 2 3 4 5 most often from planned experiences that focus on revisiting mathematical ideas in the same context as they were initially introduced. 8. I feel it is important for children to 1 2 3 4 5 use their drawings, objects, photos or other forms of representation as a way to make mathematical ideas available for personal reflection. 9. I believe that it is just as important 1 2 3 4 5 to focus on classification in preschool as patterns and counting. 10. I feel that preschool teachers 1 2 3 4 5 should intervene in a child’s problem solving so they do not develop anxiety toward mathematics. 11. I believe problem solving in 1 2 3 4 5 mathematics should be reserved for children in kindergarten and older. 12. I believe that preschool children 1 2 3 4 5 should be expected to explain their thinking and listen to the mathematical ideas of their peers. 13. I believe it is appropriate for 1 2 3 4 5 preschool children to classify objects by stating how they are alike and different but not necessarily to make generalizations about them. 14. I feel that it is important to take 1 2 3 4 5 the time during the school day - even if it means giving up time for other planned activities - to teach preschoolers how to work as a community of learners. 15. I feel that as children come up 1 2 3 4 5 with personal representations the teacher should begin to help them translate these to more conventional ways of representing mathematical situations.

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16. I believe that preschool children 1 2 3 4 5 should be expected to make sense of (to understand and apply) mathematics. 17. I feel it should be expected that 1 2 3 4 5 children in preschool learn to question the mathematical strategies of others and/or ask for clarification. 18. I feel that preschool teachers 1 2 3 4 5 should wait to promote problem solving until after the children have acquired basic math skills such as counting, numeral recognition, sorting, and identifying shapes. 19. I feel that it is inappropriate to 1 2 3 4 5 use the standard mathematics vocabulary in preschool (e.g. pentagon, cube, inches). 20. I believe that in preschool, the 1 2 3 4 5 teaching of mathematics is more effective when activities that address mathematics are offered separately from other subject areas. 21. I believe that preschool teachers 1 2 3 4 5 should model for children the steps for thinking through a mathematical situation. 22. I feel that the math activities 1 2 3 4 5 provided in the standardized curriculum are more educational than those I might develop specifically for my students. 23. I believe that mathematics should 1 2 3 4 5 be related to a young child’s real world experiences. 24. I believe that, in general, 1 2 3 4 5 preschool children progress further mathematically when you focus on one method of representation at a time (e.g. using objects, pictures, or symbols).

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25. I feel that one mathematics goal 1 2 3 4 5 in preschool should be for children to explain or justify their answers to their peers.

SECTION II: Teacher behaviors related to mathematics teaching in early childhood

Many preschool teachers face challenges to making their own judgments about what to teach and how. Some of these may relate to staffing, planning time, prescribed curriculum, or length of the school day. Please respond as honestly as possible to the following questions regarding teaching methods. Following these you will have an opportunity to share whether or not you face challenges around curriculum planning and teaching.

Never One or One or One or Daily More two two two than times a times a times a once a semester month week day 26. I specifically plan opportunities for children 0 1 2 3 4 5 to engage in mathematical problem solving (e.g. how to share a set of objects equally among 3 children). 27. I pose mathematics problems to children during 0 1 2 3 4 5 daily routines (e.g. attendance, snack, clean up, line up…) 28. I purposefully highlight math concepts 0 1 2 3 4 5 that are evident in picture books when reading a story. 29. I pose mathematics problems to children who 0 1 2 3 4 5 are engaged in block play.

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30. Materials for dramatic play that promote 0 1 2 3 4 5 mathematical thinking are available for children. 31. I provide and support children’s play with card 0 1 2 3 4 5 and/or board games. 32. I facilitate children’s play to help children learn 0 1 2 3 4 5 to communicate their mathematical ideas/thinking to others. 33. I plan time for children to share their mathematical 0 1 2 3 4 5 thinking during whole group meeting times. 34. I plan time for children to represent their 0 1 2 3 4 5 mathematical thinking by drawing, building, etc. 35. I provide worksheet activities not just for 0 1 2 3 4 5 numeral recognition, more/less, matching, etc. but to pose mathematics problems such as simple addition or subtraction. 36. I facilitate conversation among 0 1 2 3 4 5 children to model / teach them how to explain their reasoning about mathematical ideas. 37. I change my plans to capitalize on spontaneous 0 1 2 3 4 5 opportunities during the day to promote children’s use of mathematics processes.

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38. I comment on children’s block play in 0 1 2 3 4 5 order to model formal mathematics vocabulary. 39. When a child gives a faulty explanation I provide 0 1 2 3 4 5 the right information so they don’t leave with misconceptions. 40. During block play, I make suggestions to 0 1 2 3 4 5 children to draw plans for structures before building or to draw representations of structures they have already built. 41. I sit with children at play with math 0 1 2 3 4 5 manipulatives to promote their communication about mathematical ideas to others. 42. Small groups of children are engaged in 0 1 2 3 4 5 projects together where mathematics is integrated. 43. I provide children time to play mathematics games 0 1 2 3 4 5 on the computer. (If you don’t have a computer please answer how you think you would if you did have one.) 44. I integrate mathematics by incorporating instances 0 1 2 3 4 5 where children vote on things pertaining to the classroom. 45. New materials are made available to children 0 1 2 3 4 5 based on their developmental needs in mathematics.

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SECTION III: Teacher attitudes toward mathematics in early childhood

The National Council of Teachers of Mathematics (NCTM) identifies five mathematics Process Standards. These are defined by NCTM (2000) as the following:

1. Problem Solving – “Engaging in a task for which the solution method is not known in advance” where “solving problems is not only a goal of learning mathematics but also a major means of doing so” (p. 52).

2. Reasoning and Proof – Using evidence to draw conclusions (Martin & Kasmer, 2009/2010) and the “development, justification, and use of mathematical generalizations” (Russell, 1999, p. 1).

3. Communication - A way for children to share their ideas and clarify their understanding and using objects, drawings, and charts to share thinking with others.

4. Connections – The “interrelatedness of mathematical ideas” (p. 64). Connections should be made among mathematical topics and to other subject areas.

5. Representation – “The act of capturing a mathematics concept or relationship in some form and to the form itself” so representation refers to a process or a product (p. 67).

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Agree Tend Tend to Disagree to disagree agree 47. I feel the use of the following mathematics processes are necessary for preschool children’s understanding of mathematics content such as number concepts, measurement, and geometry:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 48. I feel it is appropriate to emphasize the following mathematics processes during the school day with children ages 3 to 5 years:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4

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49. I feel confident in my knowledge of how NCTM defines each of the following process standards:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 50. I understand how NCTM envisions that teachers promote the following mathematics processes in their classrooms:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 51. I feel I am adequately promoting children’s use and development of the following mathematics processes at this time:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 52. I feel that professional development opportunities regarding the teaching of 1 2 3 4 mathematics processes would be beneficial to my teaching.

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SECTION IV: Teacher demographic information

The following information can be used to develop possible professional development opportunities for preschool teachers. It will not be used to identify individual teachers in any way. Confidentiality will always be maintained.

63. In what type of early care and education setting do you teach? g. For-profit childcare center h. Head Start program i. Public school preschool j. University laboratory school k. Licensed home / family childcare l. Other ______

64. What ages do you currently teach? [Mark all that apply] a. 3 year olds b. 4 year olds c. 5 year olds (not in kindergarten)

65. How many years have you taught children between the ages of 3 and 5 (not in kindergarten) years in a childcare or classroom setting?

66. How many years teaching experience do you have? (any age group)

67. What is your highest level of education? a. High School Diploma / Equivalency b. CDA c. Associate’s Degree d. Bachelor’s Degree e. Master’s Degree f. PhD

68. If you have an associate, bachelors, masters, or PhD degree is one or more of these in the field of education? a. Yes b. No

69. Do you have licensure / certification to teach children ages 3 to 5 (not in kindergarten)? a. Yes b. No

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70. Have you participated in professional development in the area of preschool mathematics in the past 3 years? a. Yes b. No

71. If so, was the professional development useful for your teaching of young children? a. Very useful b. Somewhat useful c. Not very useful

72. Have you heard of the National Council Teachers of Mathematics five process standards: problem solving, reasoning, communication, connections, and representation? a. Yes b. No

*If you would be willing to participate in a 15 minute follow up interview regarding your specific classroom experiences related to mathematics teaching, please indicate that here. The interview can be in person or over the phone and participant anonymity will still be preserved. If you agree, you will be contacted initially via email. ____ yes My email address: ______

____ no

Thank you for your time in completing this survey. Your participation is greatly appreciated.

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Informed Consent to Participate in a Research Study

Study Title: An exploratory study of preschool teachers’ knowledge, behaviors, and attitudes regarding the National Council of Teachers of Mathematics Process Standards

Principal Investigator: Anne Reynolds and Julia Stoll

You are being invited to participate in a research study. This consent form will provide you with information on the research project, what you will need to do, and the associated risks and benefits of the research. Your participation is voluntary. Please read this form carefully. It is important that you ask questions and fully understand the research in order to make an informed decision. You will receive a copy of this document to take with you.

Purpose: The purpose of this research study is to find out what preschool teachers know about mathematics processes such as problem solving, reasoning, and communication, how often teachers promote the development and use of these processes in the classroom, and how teachers feel about the value of these processes for preschool age children. A large-scale study of what teachers know and what they are currently doing in their classroom related to mathematics processes can provide information for making improvements to teacher education programs and improvements in the content and quality of mathematics professional development practices and opportunities for teachers. Your participation would be part of a pilot study to be used to assess the reliability of the survey instrument.

Procedures You will be asked to complete a paper/pencil version of a survey. The survey has three sections: teacher knowledge of mathematics processes, teacher classroom practices related to the mathematics processes, and teacher attitudes and beliefs regarding the mathematics processes. The survey should take no longer than 15 minutes. You will have the option of including an email address if you are willing to participate, at a later date, in an interview regarding suggestions for the construction / revision of the survey.

Benefits This research will not benefit you directly. However, your participation in this study will help us to better understand the needs of preschool teachers in the area of mathematics teaching. It is important to hear directly from teachers because you work with the children and understand the needs of particular early care and education settings.

196

An exploratory study of preschool teachers’ knowledge, behaviors, and attitudes regarding the National Council of Teachers of Mathematics Process Standards

Risks and Discomforts There are no anticipated risks beyond those encountered in everyday life.

Privacy and Confidentiality Your study related information will be kept confidential within the limits of the law. Any identifying information will be kept in a secure location and only the researchers will have access to the data. Your signed consent form will be kept separate from your study data, and responses will not be linked to you. Research participants will not be identified in any publication or presentation of research results; only aggregate data will be used.

Your research information may, in certain circumstances, be disclosed to the Institutional Review Board (IRB), which oversees research at Kent State University, or to certain federal agencies. Confidentiality may not be maintained if you indicate that you may do harm to yourself or others.

Voluntary Participation Taking part in this research study is entirely up to you. You may choose not to participate or you may discontinue your participation at any time without penalty or loss of benefits to which you are otherwise entitled. You will be informed of any new, relevant information that may affect your health, welfare, or willingness to continue your study participation.

Contact Information If you have any questions or concerns about this research, you may contact Anne Reynolds at 330-672-7031. This project has been approved by the Kent State University Institutional Review Board. If you have any questions about your rights as a research participant or complaints about the research, you may call the IRB at 330.672.2704.

Consent Statement and Signature I have read this consent form and have had the opportunity to have my questions answered to my satisfaction. I voluntarily agree to participate in this study. I understand that a copy of this consent will be provided to me for future reference.

______Participant Signature Date

APPENDIX D

LARGE SCALE RESEARCH STUDY SURVEY

Appendix D

Large Scale Research Study Survey

An Exploratory Study of Preschool Teachers’ Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Regarding Mathematics Processes

Dear preschool teacher,

Thank you for your assistance!

Sincerely,

Julie Stoll Early Childhood Faculty Kent State University

198 199

SECTION I: Teacher beliefs about early childhood mathematics

1 Agree 2 Tend 3 Don’t 4 Tend 5 to know / to Disagree agree neutral disagree

1. I believe that the most 1 2 3 4 5 appropriate math problems for preschoolers are those that the children do not know how to solve immediately. 2. I believe that there should be 1 2 3 4 5 time planned for children to share math experiences they were grappling / engaging with during play. 3. I feel my planned lessons are 1 2 3 4 5 typically more valuable for mathematics learning than using spontaneous opportunities to initiate problem solving. 4. I feel that the most important 1 2 3 4 5 connection teachers can help preschoolers make is the connection between children’s informal or intuitive mathematics knowledge and the formal mathematics learned in school. 5. I feel that the mathematics 1 2 3 4 5 curriculum should include at least one or more worksheet activities during the week to provide children opportunities to use and develop mathematics processes.

200

6. I believe that children’s 1 2 3 4 5 personal representations, such as drawings, could be used as a valid assessment of their level of mathematical understanding. 7. I feel preschool children benefit 1 2 3 4 5 most from planned experiences that focus on revisiting mathematical ideas in the same context as they were initially introduced. 8. I feel it is important for children 1 2 3 4 5 to use their drawings, objects, photos or other forms of representation as a way to make mathematical ideas available for personal reflection. 9. I believe that it is just as 1 2 3 4 5 important to focus on classification in preschool as patterns and counting. 10. I feel that preschool teachers 1 2 3 4 5 should intervene in a child’s problem solving so they do not develop anxiety toward mathematics. 11. I believe problem solving in 1 2 3 4 5 mathematics should be reserved for children in kindergarten and older. 12. I believe that preschool 1 2 3 4 5 children should be expected to explain their thinking and listen to the mathematical ideas of their peers. 13. I believe it is appropriate for 1 2 3 4 5 preschool children to classify objects by stating how they are alike and different but not necessarily to make generalizations about them.

201

14. I feel that it is important to 1 2 3 4 5 take the time during the school day - even if it means giving up time for other planned activities - to teach preschoolers how to work as a community of learners. 15. I feel that as children come up 1 2 3 4 5 with personal representations the teacher should begin to help them translate these to more conventional ways of representing mathematical situations (e.g. 4 + 2 = 6) 16. I believe that preschool 1 2 3 4 5 children should be expected to make sense of (to understand and apply) mathematics. 17. I feel it should be expected 1 2 3 4 5 that children in preschool learn to question the mathematical strategies of others and/or ask for clarification. 18. I feel that preschool teachers 1 2 3 4 5 should wait to promote problem solving until after the children have acquired basic math skills such as counting, numeral recognition, sorting, and identifying shapes. 19. I feel that it is inappropriate to 1 2 3 4 5 use the standard mathematics vocabulary in preschool (e.g. pentagon, cube, inches). 20. I believe that in preschool, the 1 2 3 4 5 teaching of mathematics is more effective when activities that address mathematics are offered separately from other subject areas. 21. I believe that preschool 1 2 3 4 5 teachers should model for children the steps for thinking through a mathematical situation.

202

22. I feel that the math activities 1 2 3 4 5 provided in the standardized curriculum are more educational than those I might develop specifically for my students. 23. I believe that mathematics 1 2 3 4 5 should be related to a young child’s real world experiences. 24. I believe that, in general, 1 2 3 4 5 preschool children progress further mathematically when teachers focus on one method of representation at a time (e.g. using objects, pictures, or symbols). 25. I feel that one mathematics 1 2 3 4 5 goal in preschool should be for children to explain or justify their answers to their peers or teachers.

203

SECTION II: Teacher behaviors related to mathematics teaching in early childhood

Many preschool teachers face challenges to making their own judgments about what to teach and how. Some of these may relate to staffing, planning time, prescribed curriculum, or length of the school day. Please respond as accurately as possible to the following questions regarding teaching methods. Following these you will have an opportunity to share whether or not you face challenges around curriculum planning and teaching.

Never One or One or One or Daily More two two two than times a times a times a once a semester month week day 26. I plan specific opportunities for children to 0 1 2 3 4 5 engage in mathematical problem solving (e.g. how to share a set of objects equally among 3 children). 27. I pose mathematics problems to children during 0 1 2 3 4 5 daily routines (e.g. attendance, snack, clean up, line up…) 28. I purposefully highlight math concepts that are evident 0 1 2 3 4 5 in picture books when reading a story. 29. I pose mathematics problems to children who are 0 1 2 3 4 5 engaged in block play. 30. Materials for dramatic play that promote 0 1 2 3 4 5 mathematical thinking are available for children. 31. I provide and support children’s play with card 0 1 2 3 4 5 and/or board games. 32. I plan time for children to share their mathematical 0 1 2 3 4 5 thinking during whole group meeting times.

204

33. I plan time for children to represent their mathematical 0 1 2 3 4 5 thinking by drawing, building, etc. 34. I facilitate conversation among children to model / 0 1 2 3 4 5 teach them how to explain their reasoning about mathematical ideas. 35. I change my plans to capitalize on spontaneous 0 1 2 3 4 5 opportunities during the day to promote children’s use of mathematics processes. 36. I comment on children’s block play in order to model 0 1 2 3 4 5 formal mathematics vocabulary. 37. During block play, I make suggestions to children to draw 0 1 2 3 4 5 plans for structures before building or to draw representations of structures they have already built.

38. I sit with children at play with math manipulatives to 0 1 2 3 4 5 promote their communication about mathematical ideas to others. 39. Small groups of children are engaged in projects 0 1 2 3 4 5 together where mathematics is integrated.

205

40. What challenges, if any, do you face in making your own judgments about what to teach and how you teach? a. Following a prescribed curriculum b. Not enough time for planning c. Staffing issues d. Length of school day e. Access to materials or resources f. Pressure from parents / families g. No challenges h. Other ______

SECTION III: Teacher attitudes toward mathematics in early childhood

The National Council of Teachers of Mathematics (NCTM, 2000) identifies five mathematics Process Standards as the following:

 Problem Solving – “Engaging in a task for which the solution method is not known in advance” where “solving problems is not only a goal of learning mathematics but also a major means of doing so” (p. 52).

 Reasoning and Proof – Using evidence to draw conclusions (Martin & Kasmer, 2009/2010) and the “development, justification, and use of mathematical generalizations” (Russell, 1999, p. 1).

 Communication - Children verbalize their ideas and clarify their understanding using objects, drawings, and charts when sharing thinking with others.

 Connections – The “interrelatedness of mathematical ideas” (p. 64). Connections should be made among mathematical topics and to other subject areas.

 Representation – “The act of capturing a mathematics concept or relationship in some form and to the form itself” so representation refers to a process or a product (p. 67).

206

Agree Tend to Tend to Disagree agree disagree 41. I feel the use of the following mathematics processes are necessary for preschool children’s understanding of mathematics content such as number concepts, measurement, and geometry:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 42. I feel it is appropriate to emphasize the following mathematics processes during the school day with children ages 3 to 5 years:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4

207

43. I feel confident in my knowledge of how NCTM defines each of the following Process Standards:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 44. I understand how NCTM envisions that teachers promote the following mathematics processes in their classrooms:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 45. I feel I am adequately promoting children’s use and development of the following mathematics processes at this time:  Problem Solving 1 2 3 4  Communication 1 2 3 4  Reasoning & Proof 1 2 3 4  Connections 1 2 3 4  Representation 1 2 3 4 46. I feel that professional development opportunities regarding the teaching of mathematics processes would be beneficial to 1 2 3 4 my teaching.

208

SECTION IV: Teacher demographic information

47. In what type of early care and education setting do you teach? m. For-profit childcare center n. Head Start program o. Public school preschool p. University laboratory school q. Licensed home / family childcare r. Other ______

48. What ages do you currently teach? [Mark all that apply] c. 3 year olds d. 4 year olds e. 5 year olds (not in kindergarten)

49. How many years have you taught children between the ages of 3 and 5 (not in kindergarten) years in a childcare or classroom setting?

50. How many years teaching experience do you have? (any age group)

51. What is your highest level of education? a. High School Diploma / Equivalency b. CDA c. Associate’s Degree d. Bachelor’s Degree e. Master’s Degree f. PhD

52. If you have an associate, bachelors, masters, or PhD degree is one or more of these in the field of early childhood education? a. Yes b. No

53. Do you have licensure / certification to teach children ages 3 to 5 (not in kindergarten)? a. Yes b. No c. Yes, but has expired

209

54. Have you participated in professional development in the area of preschool mathematics in the past 3 years? a. Yes b. No

55. If so, was the professional development useful for your teaching of young children? a. Very useful b. Somewhat useful c. Not very useful

56. Have you heard of the National Council of Teachers of Mathematics five Process Standards: problem solving, reasoning & proof, communication, connections, and representation? a. Yes b. No

____ no

Thank you for your time in completing this survey. Your participation is greatly appreciated!

210

Email Message to Prospective Participants

Dear preschool teacher, I am conducting a survey concerning the teaching of mathematics for children between the ages of 3 and 5 who are not yet in kindergarten. It is vital that teachers of young children have multiple opportunities to share their knowledge on what is appropriate for the children they care for and to influence the types of professional development opportunities that are provided for teachers.

Please take some time to complete the survey. It should take no more than 10-15 minutes. All responses will be treated confidentially and will in no way be traceable to individual respondents once the survey process has been concluded. Please follow this link to access the survey. Research consent information is written below. Participation in the survey indicates your informed consent.

Thank you for your assistance! Sincerely,

Julie Stoll Early Childhood Faculty Kent State University

211

Informed Consent to Participate in a Research Study

Study Title: An exploratory study of preschool teachers’ knowledge, behaviors, and attitudes regarding the National Council of Teachers of Mathematics Process Standards

Principal Investigator: Anne Reynolds and Julia Stoll

You are being invited to participate in a research study. This consent form will provide you with information on the research project, what you will need to do, and the associated risks and benefits of the research. Your participation is voluntary. Please read this form carefully. It is important that you fully understand the research in order to make an informed decision. You will be able to print a copy of this document to keep with you for future reference.

Purpose: The purpose of this research study is to find out what preschool teachers know about mathematics processes such as problem solving, reasoning, and communication, how often teachers promote the development and use of these processes in the classroom, and how teachers feel about the value of these processes for preschool age children. A large- scale study of what teachers know and what they are currently doing in their classroom related to mathematics processes can provide information for making improvements to teacher education programs and improvements in the content and quality of mathematics professional development practices and opportunities for teachers.

Procedures Participants must be current teachers of children three to five years of age who are not yet in kindergarten. Participants are asked to complete an online survey. The survey has four sections: teacher knowledge of mathematics processes, teacher classroom practices related to the mathematics processes, teacher attitudes and beliefs regarding the mathematics processes and some general demographic information such as the type of setting a teacher works in and how long they have been teaching. The survey should take no longer than 10-15 minutes. Participants will have the option of including an email address if they are willing to participate, at a later date, in a phone interview regarding mathematics teaching practices.

Benefits This research will not benefit you directly. However, your participation in this study will help us to better understand the needs of preschool teachers in the area of mathematics teaching. It is important to hear directly from teachers themselves because you work directly with the children and understand the needs of your particular setting.

Risks and Discomforts There are no anticipated risks beyond those encountered in everyday life.

212

Privacy and Confidentiality Identifying information will not be included in the data that you provide. Your anonymity is further protected by not asking you to sign and return the informed consent form.

If you agree to participate in a follow up interview and you include your email contact then this study related information will be kept confidential within the limits of the law. Any identifying information will be kept in a secure location and only the researchers will have access to the data. Research participants will not be identified in any publication or presentation of research results; only aggregate data will be used.

APPENDIX E

SUMMARY OF DEVELOPMENTALLY APPROPRIATE EXPECTATIONS FOR THREE TO FIVE YEAR OLDS

Appendix E

Summary of Developmentally Appropriate Expectations For Three To Five Year Olds

Table 31

Developmentally Appropriate Expectations for Three to Five Year Olds

Promoting a Establish a Help children Help children Children are positive caring develop accomplish challenged to community friendships meaningful work on the climate for tasks edge of their learning developing capabilities Fostering a Design Children work Children’s Children have Children cohesive group activities in groups around home culture opportunities to with based on a shared interest and language work in small disabilities and meeting knowledge of to create a sense are brought and large are individual student needs of group into the groups included in needs shared school all ways in culture the classroom. Environment Children Teachers Children have and schedule offered active provide a safe time for exploration of environment extended play materials and projects and to experiment Learning Teachers plan Children are Children can experiences concreted offered block make their experiences play, pretend own choices around child play, expressive for activities interest arts, sand, water, and science tools Language and Teachers talk Teachers Children have communication with children, encourage opportunities speak clearly, children to to listen to listen to describe their peers describe children’s work and ideas events and responses and then ask them promote peer questions or conversations respond with about real own ideas experiences

214 215

Integrated Learning Subject curriculum goals address matter is all integrated developmental through areas projects and play Teaching Teachers interact Teachers use Teachers Children plan, Children work strategies with individuals, strategies extend reflect on, collaboratively small groups, in such as children’s revisit, discuss, to develop many contexts to asking thinking and represent skills of know their questions, during child- activities. negotiation, capabilities suggestions, initiated Teachers use helping, materials, activities their hypotheses problem peer to engage them solving. collaboration, in Teachers problem experimentation model & coach solving social problem solving Motivation Teachers use Children and verbal develop social encouragement skills and guidance related to tasks self- regulation – teachers redirect, model behavior, set clear limits, enforce consequences The Teachers use continuum knowledge of child of development of development content to and learning promote inquiry in ways they understand Coherent, Teachers plan Culturally effective based on content diverse and knowledge, child nonsexist curriculum interest, prior materials are experiences and provided to emerging ideas develop self- identity

216

Curriculum Teachers use a Teachers use Children Children have Children have content and variety of a variety of have opportunities for opportunities approaches and strategies to opportunities fine motor to practice self- approaches daily develop for gross activities help skills opportunities to concepts in motor develop literacy math, science, activities skills health, social studies, and the arts Note. Adapted from Developmentally appropriate practice in early childhood programs serving children birth through age 8: A position statement of the National Association for the Education of Young Children by S. Bredekamp and C. Copple (Eds.). (1997). Washington, DC: NAEYC.

APPENDIX F

SUPPLEMENTAL RESEARCH QUESTIONS

Appendix F

Supplemental Research Questions

Research Question 6 Supplement: Pearson’s Correlation for Teachers’ Feelings of Adequacy in their use of Processes and Teacher Reported Behaviors

Table 32

Mean Scores and Correlation of Feelings of Adequacy and Reported Behaviors ______Process Item #45a - #45e Teacher Reported Pearson’s Correlation Standard “I feel I am adequately Behaviors Totals Coefficient promoting children’s use and development of the math processes at this time.” ______Mean Mean PS 3.42 3.38 .127 (p = .062) RP 3.08 3.41 .297** (p = .000) CM 3.38 3.18 .254** (p = .000) CN 3.41 3.33 .187** (p = .006) R 3.29 2.82 .001** (p = .001) ______Note. N = 217; PS = problem solving; RP = reasoning and proof; CM = communication; CN = connections; R = representation. **p < 0.01 (2-tailed)

218 219

Research Question 7 Supplement: ANOVA Results for Comparison of Teacher Level of Education on Teacher Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs

Table 33

Means and Standard Deviations Comparing Teacher’s Level of Education and total scores on Perceived Knowledge, Reported Behaviors, and Attitude/Beliefs

Level of Education PK Total RB Total A/B Total

n M SD M SD M SD

HS Diploma 2 73.50 6.36 58.50 12.02 88.50 12.02 CDA 2 76.00 5.66 49.50 0.71 77.50 3.54 Associate’s 23 85.22 6.10 47.26 13.78 90.83 8.68 Bachelor’s 71 89.30 9.58 44.24 12.00 84.87 12.02 Master’s 118 90.92 9.04 44.65 9.99 85.32 9.93 ______

Table 34

One-Way Analysis of Variance Summary Table Comparing Teacher’s Level of Education on Teacher’s Perceived Knowledge, Reported Behaviors, and Attitudes/Beliefs Total Scores

Source df SS MS F p

Perceived Knowledge Between groups 4 1537.53 384.38 4.80 .001 Within groups 211 16886.35 80.03 Total 215 18423.88

Reported Behaviors Between groups 4 577.66 144.41 1.17 .326 Within groups 211 26085.12 123.63 Total 215 26662.77

Attitudes/Beliefs Between groups 4 819.85 204.96 1.84 .122 Within groups 211 23473.93 111.25 Total 215 24293.77 ______

APPENDIX G

RESEARCH QUESTION 7 ADDITIONAL DATA

Appendix G

Research Question 7 Additional Data

Table 35

Independent t-test Means Tested for Sections I, II, and III of the Survey ______Constructs of survey Licensure to Mean SD teach preschool? ______Section I Total Yes = 188 90.08 9.11 Knowledge (K) No = 27 85.15 9.45

K of PS Total Yes = 188 19.91 3.03 No = 27 18.93 3.22 K of RP Total Yes = 188 14.15 2.62 No = 27 13.00 2.50 K of CM Total Yes = 188 19.78 2.90 No = 27 18.89 2.53 K of CN Total Yes = 188 20.23 2.48 No = 27 19.52 3.41 K of R Total Yes = 188 15.99 2.31 No = 27 14.81 2.48

Section II Total Yes = 188 44.79 10.70 Behaviors (B) No = 27 45.70 13.93

B of PS Total Yes = 188 10.11 2.95 No = 27 10.30 3.52 B of RP Total Yes = 188 3.44 1.12 No = 27 3.19 1.27 B of CM Total Yes = 188 9.46 2.75 No = 27 9.85 3.45 B of CN Total Yes = 188 9.94 2.66 No = 27 10.07 3.17 B of R Total Yes = 188 8.38 2.81 No = 27 8.81 3.15

221 222

Section III Total Yes = 188 85.79 10.64 Attitudes/Beliefs (A/B) No = 27 84.78 10.69

A/B of PS Total Yes = 188 17.63 2.19 No = 27 17.22 2.15 A/B of RP Total Yes = 188 16.21 2.84 No = 27 16.26 2.61 A/B of CM Total Yes = 188 17.51 2.21 No = 27 17.33 2.40 A/B of CN Total Yes = 188 17.49 2.24 No = 27 16.96 2.07 A/B of R Total Yes = 188 16.95 2.55 No = 27 17.00 2.30 ______Note. Effect size (d) was calculated with the formula where ns are unequal and the result was d = .54 which is a medium or typical effect size (Cohen, 1988)

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