The Effects of Early Intervention on Mathematical Achievement of Low Performing First Grade Students, Research Paper Example

Chapter I 

Introduction

Twenty-five percent of young students do very well in math and will do so in most classroom programs.  Another 50% will do well in a good classroom program.  The remaining 25%, the low-attaining students, are likely to remain as low-attaining students as they progress through the school years (Wright, 2008, p.1)   In the 1990s several research studies focused on assessing the number knowledge of children in the early years of school (Aubrey, 1993; Wright, 1991b; 1994; Young-Loveridge, 1989; 1991).  These studies found significant differences in numerical knowledge of children when they began school.  A study by Wright (1994) describes a three-year difference in children’s early number knowledge, that is, some four-year-olds have attained a level of number knowledge that others will not attain until they are seven years old.  Wright’s study demonstrates a variation in the number knowledge will increase as the children progress through the early learning years and continues  during the formative learning years. This indicates children who are in the lower attaining group in the early years will remain so throughout their formal education. The knowledge gap between the low, average and high children tends to rise throughout their learning years.

A three-year difference in the early years of school becomes a seven-year difference for low attaining children after about ten years of school.  The notion of a seven-year difference was identified in the influential Cockroft (1982) reports on school mathematics in the United Kingdom.  What also seems to be the case is that, even in the early years of schooling, low attaining children begin to develop strong negative attitudes to mathematics.  It is reasonable to suggest that these negative attitudes result from a lack of understanding of school mathematics and rare experience of success in school mathematics (Wright, Martland, & Stafford, 2006).

A national concern has arisen in mathematics education, stemming from the recognition that fundamental needs of large numbers of students are not being recognized or addressed in classrooms across the country (Glenn, 2000; Haycock, 2001).  Some students are even misdiagnosed as being learning disabled, developing math anxiety, and becoming complacent to the fact that they are not good in mathematics throughout their tenure in school (Battista, 1999; Hessler, 2001).  According to the National Research Council (1989), sixty percent of college mathematics high school courses, and the business sector will spend the same amount of money on remedial mathematics training for employees to equal what is spent on the same education combined in schools, colleges and universities.  In order for this cycle to stop it is imperative that best practice be established.  Most mathematics classrooms and curricula are set up in a traditional manner, where the teacher is the giver of knowledge and the students are viewed as blank slates to be filled with knowledge, a behaviorist philosophy that starts from Kindergarten all the way through High School.  Teaching mathematics traditionally in the classroom is taught through direct instruction, memorization, procedures, and fact knowledge through the mass purchase of a one size fits all textbook.  Battista and Larson (1994) found that traditional methods of teaching mathematics are  ineffective and may seriously inhibit the students’ mathematical reasoning development and problem solving skills.  Perry (2000) suggests that there is a need for early childhood educators to develop and enhance their own competence and confidence in mathematics and to use this to plan appropriate in-depth numeracy learning experiences for young children.  Perry also states that intervention in early childhood settings has the potential to enhance the numeracy development of many young children.  However, the quality of the intervention and the follow-up provided are critical to this outcome.

A number of researchers have proposed that the mathematical knowledge and skills that children acquire at an early age facilitate their understanding of the meaning of numbers and numerical relationships and that this sense of number is related to later achievement in mathematics (Case, 1998; Geary, 1994; Gersten & Chard, 1999; Richardson, 1997; Sowder, 1992; Steffe, 2000; Wright, 1998).  Howell and Kemp (2005) found number sense is becoming an increasingly prevailing focus of mathematics education.  Howell and Kemp stress an individual students’ achievement in mathematics and successful mathematics instruction is linked the student’s number sense and knowledge.

Math Recovery, which promotes and nurtures early number sense, is an early intervention program for mathematics that is designed to identify children in their first year or two of school who are unable to perform a series of tasks relating to counting strategies and numeracy strategies.  Completing Math Recovery is a professional development program for primary grade level mathematics teachers which encourages more effective instructional practices for teaching mathematics in the early years of school.  Math Recovery involves the use of a comprehensive, ongoing assessment of students’ numerical knowledge and strategies coupled with a program of distinctive instructional strategies.  The assessments and instructional components of Math Recovery are aligned with a Learning Framework in Number (LFIN), which is a developmental continuum reflecting cognitive stages of development exhibited by young children as they acquire early mathematical concepts, skills, and understandings.  The instructional approach incorporated into Math Recovery is problem based and reflects a constructivist philosophy.

The present study adds to research on children’s mathematics achievement in relation to children’s number sense by focusing on low achieving first grade children.  This study contributes to current research on children’s number sense and  mathematics instruction via the implementation of a constructivist instructional method.  Teachers use Math Recovery, math manipulatives, and differentiated teaching strategies for instruction with young children.  Currently, there is little knowledge empirically regarding the effectiveness of instructional practices, or teaching methods, used in the constructivist teaching philosophy in Math Recovery to foster mathematics achievement in a classroom setting.  The mainstreaming of Math Recovery into a traditional mathematics setting is important to the field of early childhood mathematics.  Studying Math Recovery as a constructivist instructional intervention method demonstrates how mathematics could be taught at an early age. (Baroody, 2000).  The present study is significant to research on the mainstreaming of an intervention (Math Recovery) program for teaching mathematics at an early age.  Although researchers initiated studies involving number sense to date (Berch, 2005; Howell and Kemp, 2005; Malofeeva, Day, Saco, Young, and Ciancio, 2004), a review of literature found minimal successful studies involving diagnosis and remediation of low achieving students in mathematics at the first grade level.

Therefore, through the involvement of low achieving first grade students in mathematics and the employment of the Math Recovery intervention program, this study furthers research on the diagnosis and development of the acquisition of number sense.

Statement of Problem

Although scholars have shown that number sense at an early age is fundamental for math literacy, little attention is being paid to the development of number sense.  Research is needed on how schools can create a sustainable system of early identification and support for student achievement in mathematics.  Researchers have proposed the idea that mathematical knowledge and skills that children attain at an early age is crucial to a child’s understanding of the meaning of numbers and numerical relationships.  According to Case (1998), Geary (1994), Gersten and Chard (1999), Richardson (1997), Sowder (1992) early sense of number will lead to later achievement in mathematics.  The problem is no two researchers define number sense the same way.  Number sense is difficult to define (Case, 1998).  The National Council of Teachers of Mathematics (NCTM) defines number sense as “an intuition about numbers that is drawn from all varied meanings of number” (NCTM, 1989, P. 39).  The students with number sense are able to comprehend numbers representing objects, magnitudes, and relationships.  They also believe that numbers can be manipulated, compared and communicated with.  Students with number sense can also use numbers in a flexible way to solve problems (Gurganus, 2004).  Gerstein and Chard (1999) believe that number sense is to mathematics as phonemic awareness is to reading.  Gersten and Chard (1999) state the ability to master number sense contributes to a child’s fluidity and flexibility with numbers, what numbers mean, ability to perform mental mathematics, and to view the world for comparison.  Baroody and Wilkens (1999) believe number sense includes a concrete understanding of numerical relationships; to include phases such as ‘the same number as’ or ‘more than’ and the relative size of numbers.

In this qualitative study, I will explore through case study research how the Math Recovery program affects low achieving students gaining number sense.

Purpose of Study

Too many children are falling further behind throughout their mathematics tenure in school (Aubrey, 1993; Wright, 1991b; 1994; was Young-Loveridge, 1989; 1991).  Therefore, the purpose of this case study is to describe and understand what constitutes a low achieving first grade student’s number sense. The intent is to examine low performing/ achieving first grade students in mathematics and investigate the process of their mathematical schematic growth through the Math Recovery Program.  Furthermore, by investigating the characteristics of number sense growth in low performing first grade students, it is hoped that the results of this study will lead to further understanding behind the success of the Math Recovery program in early numeracy.

Given the purpose of this study, the following research question will guide the study:

For a suburban public school in the Midwest, how does Math Recovery affect number sense in first grade students identified as low achieving?

Significance of Study

This study contributes information about the programs effectiveness in meeting the needs of low performing students.  This study has both theoretical and practical use for future similar studies, as it will add to the existing body of research on the identification and programming for low achieving primary grade students in mathematics.  While there is a large body of research demonstrating a need for early identification and intervention for number sense in mathematics, many schools in the United States still do not have widespread effective mathematics programming for the primary grades (Glenn, 2000; Haycock, 2001).  In addition, it may provide school districts information to assist with the planning for effective curriculum in mathematics by helping educators understand what constitutes successful experiences for first grade low achieving students in mathematics.  The Math Recovery program serves several purposes; however, one of its main intents is to expose all young learners to the higher levels of thinking in number sense, not rote memorization.

Further, it fills a void in the research regarding mathematics curriculum planning in primary education classrooms.  Gersten and Chard (1999) state that number sense research and the cognitive sciences can help the research community pull together fragmented pieces of earlier knowledge to yield a much richer, more subtle, and more effective means of improving instructional practice (p.18).  Perry (2000) suggests that there is a need for early childhood educators to develop and enhance their own competence and confidence in mathematics and to use this to plan appropriate in-depth numeracy learning experiences for young children.  Perry also states that early intervention in early childhood settings has the potential to enhance the numeracy development of many young children (P. 34).

Definition of Terms /Variables

Math Recovery

A Title I funded program that identifies at-risk students through an assessment process allowing educators to determine where students are in mathematical development and apply necessary short-term intervention strategies.  Math Recovery provides teachers with the intervention activities, tools and assessment materials in a detailed instructional scaffolding system intended to improve individual student performance in core numeracy skills.  Math Recovery also seeks to build student confidence and help students attain success in mathematics.

Number Sense

For the purposes of the study number sense is described in the Learning Framework in Number (LFIN) section and its subcategories (see Appendices A-Z). The framework comprises of four parts which include various levels of thinking ability.  The parts include early arithmetical strategies (addition and subtraction), Forward Number Word Sequence (counting forward), Backward Number Word Sequence (counting backward), Numeral Identification, Structuring Numbers 1-20, and Early Multiplication and Division.  A screening using the 1.1 Math Recovery Assessment will be utilized to determine the child’s Stage of Early Arithmetical Learning (SEAL) in addition and subtraction within the range stage 0 to stage 5.  This assessment will also be used to gain extensive information on the child’s facility with number word sequences and numerals.  A second screening, Assessment 2.1, will be utilized to determine the child’s level of development of structuring numbers 1 to 20 within the range Level 0 to Level 3.  These assessments will be given as a pre, mid, and post assessment to determine growth in levels and stages in the LFIN.

Learning Framework in Number (LFIN)

In Math Recovery, the Learning Framework in Number (LFIN) is a detailed description of early number knowledge in children.  This includes strategies that children use to solve simple number tasks (additive, subtractive). LFIN contains eleven aspects of early number organized into four parts.  Parts A, B, and D contain six aspects.  These aspects contain levels and stages that are inter-related rather than distinct.  The child’s knowledge is assessed as certain levels are reached. Once the child achieves certain levels in the LFIN, the child then reaches a higher (SEAL) stage of thinking.  In this way the child’s knowledge is profiled based on the specific aspects in the LFIN. The LFIN is comparable to a framework, earning it the title of Learning Framework in Number (Wright, Martland, Stafford, & Stanger, 2006, p. 9) (see Apendix A).

Early arithmetical strategies and base ten arithmetical strategies is Part A in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) these parts are set out in a progression of the strategies children use in early numeracy situations that are problematic for them, for example, being required to figure the number counters in a collection, and various kinds of additive and subtractive situations. The progression follows from counting visible items, to partially screened tasks where an addend or subtrahend is shown, and eventually to totally screened tasks. The purpose in covering the items in an additive or subtractive task forms an essential bridging function between concrete and formal calculation. The choice of quantities used when covering objects varies. Additive or subtractive tasks in which sums and differences are less than ten help reinforce the work done in structuring number and provide a way to assess if the child is able to transfer or connect this knowledge to a different setting. By presenting additive, subtractive, missing addends and missing subtrahends, one is able to determine if the child demonstrates a knowledge of part/part/whole. When adding or subtracting from a much larger number, the child is able to practice the fnws/bnws needed when calculating with larger numbers and becomes increasingly aware that paper and pencil calculation is not needed when solving tasks such as 92-3 or 77+5. Children also by keeping the number whole (versus treating the numbers as single digits when doing a standard algorithm) begin gaining the ‘number sense’ needed when estimating sums or differences. By navigating their way forward and backward on the number line, they are forming the mental number line needed to do formal calculation without paper and pencil (see Apendix A).

Stages of Early Arithmetical Learning (SEAL) is Part A in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) SEAL stages are the progression of the strategies children use in early number situations which are problematic for them, for example being required to figure out how many in a collection, and various kinds of additive and subtractive situations (see Apendix A).

Numeral Identification is part B in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) numerals are the written and read symbols for numbers, for example ‘12’, 21’ and ‘47’. Learning to identify, recognize, and write numerals is a significant component of literacy development. At the same time, this learning is equally, if not more so, a vital element of early numeracy development. The term identify is used with precise meaning; to state the name of a displayed numeral. The complementary task of selecting a named numeral from a randomly arranged group of displayed numerals is known as recognizing. Thus we make the distinction between numeral identification and numeral recognition.  Using these terms in this way aligns with early psychology and early literacy settings. Children who apply procedural or conceptual knowledge in problem solving, who mistakenly, due to misidentification, use an incorrect numeral while calculating, are apt to find math nonsense making (see Apendix A).

Forward Number Word Sequence (FNWS) is Part B in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) FNWS refers to a regular sequence of number words forward; commonly by ones. These type of activities or tasks help children to build a mental number line which will be needed later when doing formal calculation without paper and pencil. Children first link the words used in a fnws when counting objects and later when calculating by counting. The skill needed to begin counting on involves starting the count from a number other than one. Naming the number word after (nwa) any number given in this range gives children the starting number in which to begin the fnws. Children thus have a growing awareness that when adding, the numbers are increasing as we move forward on the number line (see Apendix A).

Backward Number Word Sequence (BNWS) is Part B in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) BNWS refers to a regular sequence of number words backward; commonly by ones. These type of activities or tasks help children to build a mental number line which will be needed later when doing formal calculation without paper and pencil. Children first link the words used in a bnws when objects are removed, “If I had 5 cookies and I ate 1, there would be 4 left because 4 comes before 5” or because the 5th one is gone and there are 4 left. The skill needed to begin counting back to solve a subtractive tasks involves starting the count from a number other than one. Naming the number word before (nwb) any number given in this range gives children the starting number in which to begin the bnws. Children thus have a growing awareness that when subtracting, the numbers are decreasing as we move backward on the number line (see Apendix A).

Structuring Number is Part C in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) around the time that children have developed early counting strategies, they also should learn to combine and separate smaller numbers without counting. This topic relates to children’s facility to combine and partition numbers without counting by ones. Instead the child uses an emerging knowledge of doubles, and the five and ten structure of numbers, that is, using five and ten as referent numbers. Learning this topic provides an important basis for moving beyond a reliance on counting-by-ones. Children are thus able to internalize their facts to ten and then twenty without memorization, but by realizing the part/part/whole aspect of number when combining and partitioning number. Structuring numbers from 1-10, 10-20, then 20-100, involves using finger patterns, spatial patterns, math racks and unifix cubes (see Apendix A).

Early Multiplication and Division is Part D in the LFIN.  According to Wright, Martland, Stafford, & Stanger (2006) this part focuses on the development of early multiplication and division comprehension. This includes the emergent notions of repeated equal grouping and sharing, the development of skip counting, and using arrays in teaching multiplication and division. Also introduced are the ideas of numerical composites and abstract composite units, and the inverse relationship between multiplication and division (see Apendix A).

Social Constructivist/Constructivism

The word “constructivism” is defined as the theory about how we learn wherein the learner constructs, creates, invents, and develops their own knowledge as a result of questioning, interpreting, and analyzing information gained from first-hand experiences and then integrates this new knowledge with the learner’s own previously held knowledge to form new constructs (Marlowe and Page, 1998, pp. 9-10).  The term “social constructivist” is defined as a type of constructivism with  major emphasis on social construction of knowledge, whereby the learner inside the shared context of the environment that he interacts with constructs knowledge.  In the context of Math Recovery, the social interaction that occurs between the learner and his teachers within the learning environment contributes to and shapes his construction of mathematical knowledge and understanding.  Within the Math Recovery learning environment, the learning process is influenced or shaped by the questioning in the careful posing of problems or tasks provided by the teacher.

The learner, through involvement in the tasks, participation in dialoguing and questioning, and then through thoughtful reflection, develops a much deeper, more insightful construction of mathematical understandings that he would acquire through simply being asked to remember and apply mathematical facts, truths, or algorithms from the teacher.

The social interaction with the teacher ensures that the learner’s thoughts, actions, and experiences are contributing in an accurate and meaningful way to the formation of newly constructed concepts or schema on the part of the learner.

Zone of Proximal Development

The term “Zone of Proximal Development” is commonly recognized as a cognitive space between the learner’s current level of development or understanding as demonstrated with independent, potential or emergent level of  problem solving, that is supported by social interaction or collaboration with an adult or peers who are functioning from higher cognitive levels of understanding.  Within the context of Math Recovery, the Zone of Proximal Development is perceived as being socially influenced, dynamic, and fluid as it exists both within the learner and within the social context of the learner’s problem solving experiences.

Summary

Number sense is at the heart of research and primary instruction in mathematics, and if teachers do not incorporate a variety of number sense techniques, then many students will lack number sense ability, have anxiety/difficulties in mathematics, or resort to rote memorization in mathematics.  This study, therefore, will examine the alignment on the effectiveness of Math Recovery as an intervention tool for low achieving first grade students in mathematics.  Results from this research study will inform educators on the usefulness of recognizing individual student’s number sense, as well as specific instructional techniques that can be incorporated into a first grade mathematics curriculum.  Finally, the information may increase teacher awareness of the importance of implementing, and building number sense with constructivist instructional techniques.

Chapter II

Introduction

There is empirical evidence found within the literature that indicates the development of number sense at an early age affects student performance on math assessments.  Much of the historical and current research focuses on early childhood to the primary grades.  Though the focus of this study is on the acquisition of number sense of low achieving students through the Math Recovery Program there is research that suggests number sense affects pre-service and regular education teachers as well.  The reason for this section is to show how the lack of number sense acquisition at an early age can affect children throughout the adult life.  The purposes of this literature review are to examine best practice in early numeracy acquisition at the primary grade level of students and teachers as well and the effective treatment interventions for them both.

Historical and Theoretical Background

Math Recovery (MR) was originally created from prominent researchers in the mathematics field from various theoretical frameworks. (Steffe, von Glasersfeld, Richards, & Cobb, 1983).  Recently the design research at the Freudenthal Institute in The Netherlands has influenced Math Recovery Leaders and Teachers regarding two-digit addition and subtraction problems (Gravemeijer, 1997; Van den Heuvel-Panhuizen, 2001).  The Mathematics Recovery Program was created after a three-year research and development project in 1992 through 1995, conducted at the Southern Cross University in northern New South Wales.

The Australian Research Council, Catholic school systems and the regional government provided funding and contributions of teacher efforts. Over the three year period, 18 schools with 20 teachers and approximately 200 participating first-grade students participated in the project. Wright (2000) details the specifics of the development of MR and discusses various contextual issues relevant to the Program’s development. Since 1995, MR has been used in a range of educational settings. The implementation of MR processes has enabled continual review and further development efforts.

Two important components comprise the concept of Math Recovery. Chapter three details the first portion; elaborated body of theory and techniques for teaching and developing early number knowledge in children (Wright, Martland & Stafford, 2000; & Wright, Martland Stafford & Stanger, 2002).  The second component details the effects of working with teachers to provide effective professional development and coaching consistent with children learning ability.  The body of theory and techniques for teaching early number knowledge in MR originated through the research efforts of Steffe and colleagues (Steffe, von Glasersfeld, Richards & Cobb, 1983; Steffe & Cobb, 1988). Steffe et al. conducted longitudinal teaching experiments with 1st- and 2nd-graders, concentrating on early number.

Typically the experiments extended over a two-year period and involved several teaching cycles with periods lasting up to 18 weeks. During this time participants were taught individually in 30 minute sessions, twice weekly. The goal was to develop psychological models of conceptual development over extended periods of time. Steffe’s approach was paradigmatically distinctive in research design and techniques, and theoretical orientation.  Distinguishing aspects of the techniques include intensive problem-based approach to instruction, and routinely videotaping assessments and teaching sessions for reflective analysis.

Steffe’s theoretical orientation drew on von Glaserfeld’s (1995) theory of cognitive constructivism, focusing on Piaget’s theory of cognitive development. Finally, the research base of MR  includes Wright’s doctoral research; a teaching experiment relating to numerical development in the kindergarten year (Wright, 1989), research of  numerical knowledge of school entrants (Wright, 1991) and the progression of number learning during the first two years of school (Wright, 1994).  The Math Recovery intervention program has evolved from a branch of constructivist learning theory known as social constructivism.  Constructivism is a branch of learning that grew out of dissatisfaction with the more traditional behaviorist educational practices that were widely used in education (Gagnon and Collay, 2001; Brooks, 1999; Marlowe and Page, 1998).

Behavioral psychologists, such as Skinner (1938) and Thorndike (1926) explain learning as an acquired response to something taught.  Teachers impart facts, procedures, and knowledge, which are then memorized by students, and are later faithfully reproduced on worksheets or exams.  An assignment or test becomes a stimulus and the students provide the accurate reproduction of a learned response.  In this educational setting, student responses are typically uniform from student to student, and limited in depth or breadth.  A single best answer or desired outcome is typically expected as an ideal response from each student (Marlowe and Page, 1998).  In the traditional, teacher dominated educational setting; a teacher imparts knowledge through the use of textbooks, lecturing, and demonstration.  Predominately, the teacher has the most active role in the classroom, while students are, for the most part, passive receptors of knowledge (Marlowe and Page, 1998, pp. 9–13).

By contrast, constructivist theory views the learning process as one involving active participation by students.  Learning is individualized as much as possible.  The goal is for the learners to develop or construct their own unique, in-depth understanding of materials being learned.  The learners then demonstrate their understandings through a variety of means or mediums (Gagnon and Collay, 2001; Brooks and Brooks, 1999; Marlowe and Page, 1998).

To educators working with constructivist-aligned classrooms, the learners have the most active roles.  The learners are encouraged to investigate, explore, question, manipulate, and build understandings from experiences that had been planned and orchestrated by the teachers.  In these settings, the teacher’s roles act as facilitators.  The teachers plan the educational experiences and use questioning strategies to influence the flow and direction of the lessons, but have a much lower profile than in the more traditional classrooms.  Indeed, the nature and influence of well directed and carefully worded questioning in a constructivist classroom is often compared to a Socratic method of instruction, whereby well-thought-out questions are used to draw information from the learner himself, rather than being provided from the teacher (Brooks and Brooks, 1999, P. 23; Marlowe and Page, 1998, pp. 9-13).

Social constructivists view the role of the teacher as being critical to the learning process for questioning, dialoguing, and social interaction, the teacher shapes and facilitate the learning process, enabling the learner to attain higher levels of insight and understandings than they would have been able to attain independently (Marlowe and Page, 1998, pp. 57-58).

Research that led to the development of the field of constructivist learning can be traced to the 1920s with the work of John Dewey.  Dewey (1933) suggested that the interaction that occurred between the learner and their environment shaped the learner’s mental activity.  Furthermore, Dewey proposed that each new learning experience was shaped or influenced by previous learning experiences.  He referred to this learning process as intellectual integration, wherein the learner actively seeks, organizes, digests, and assimilates information with their already existing experiences (Marlowe and Page, 1998, pp. 12, 14, 17, and 25).

Dewey’s research in education was an influence on a young, Swiss psychologist, Jean Piaget, beginning in the 1920s and on into the 1930s.  Piaget, a biologist-psychologist, was conducting research on how humans adapt to their environment.  This research evolved into areas that consequently had an influence on educational theory and practice, even though Piaget was not working specifically for or within an educational setting.  Piaget, (1967; 1971) in work that became the foundation for contemporary constructivism, proposed that learners construct their own schemata or knowledge structures/constructs as a result of their interaction, perception, and thinking about the world.

As learners construct their knowledge schemes, they are filtered through and shaped by the learner’s previous and current experiences.  Piaget described as reshaping of schemata as a process of disequilibrium, assimilation, and equilibrium (Brooks and Brooks, 1999, pp. 26-27; Gognon and Collay, 2001, p.58).  Whenever a student’s new experiences do not fit smoothly into his existing schemata, disequilibrium arises, requiring the student to resolve the conflict through interaction with the environment, and then through a re-establishment of cognitive structures.  The student constructs a new set of knowledge schemes that will in turn influence future constructs.  As a student ages, these constructs become increasingly complex (Brooks and Brooks, 1999, p 26).

Piaget’s own scholarly contributions to cognitive psychology evolved over 50 years that he conducted research.  In the 1980s and up to the time of his death a decade later, Piaget began to modify his earlier theories regarding assimilation and equilibrium in learning.  He came to see these as being too simplistic (Brooks and Brooks, 1999, p. 27).

Specifically, Piaget’s earlier work (Inhelder and Piaget, 1958) described static stages of cognitive development (sensori-motor, preoperational, concrete operational, and formal operational).  Piaget, at that time, explained that children at each stage exhibited predictable qualities, abilities, or characteristics in thinking.

Late in his career, Piaget (1987) shifted his theories on cognitive development from static stages to a more fluid and dynamic construction of understanding.  More closely aligned with what became the constructivist theoretical movement of the 1990s, Piaget recognized the potential for a wider range of possibilities in learners, especially in social constructivist educational settings (Brooks and Brooks, 1999).  Constructivist recognize that learners will generally shape their mathematical knowledge and understanding along predictable pathways or stages but the expressions or application of their knowledge or understanding is may be expressed in a wide-ranging set of behaviors (Brooks and Brooks, 1999, pp. 70-72; Gagnon and Collay, 2001, p. 15 and pp. 23-24).

Throughout the 50 year span of his career in cognitive psychology and biology, Piaget influenced and shaped the direction of educational practices in general and constructivist theory in particular.  At the same time, he repeatedly emphasized that he was not an educational researcher or theorist, but rather a psychologist.  Indeed, it is Piaget’s work with cognitive structures and stages of cognitive development, along with his descriptions of the characteristics of children’s thinking that provides the foundational context for the Learning Framework in Number that Math Recovery is structured around.  While noting that the levels in the Learning Framework in Number are meant to be fluid and dynamic in nature, this too parallels the understandings that Piaget arrived in his later years.

Even more closely aligned to the theoretical base that math recovery rests within is the theoretical work of Vygotsky (1986).  Vygotsky, a Russian learning theorist, has strongly influenced a branch of constructivist theory known as social constructivism (Wright, Martland Stafford, and Stanger, 2002, p. 33; Gagnon and Collay, 2001, p. 42; Cobb, Yackel and McLean, 2000, pp. 138-150).

Influenced by the writings of Vygotsky in the late 1980s, social constructivism has evolved from earlier Piagetian theories.  Social constructivists see mathematics as dependent upon a culture-based social interaction, whereby the acquisition of mathematical knowledge requires access, participation, and culture-based interaction with others having previous experiences and understandings in order for the learner to construct their own personal view of mathematics (Gagnon and Collay, 2001, p. 42; Dixon-Kraus, 1996, pp. 12-13)

Vygotsky sees all human behavior as a cultural phenomenon and in turn views learning as a social experience.  Learning, he theorizes, involves three interrelated phases of experiences (Cobb, Yackel and McClain, 2000, pp. 146-147).  The first phase of learning involves the learner attempting to make personal meaning or sense out of an experience in which he was exposed to new thoughts or activities.  The learner engages in an inner dialogue in an attempt to capture meaning from the experience and to connect it to previous experiences or knowledge.  The next phase involves the learner dialoguing with others in order to test their thinking and to assimilate this thought into a construct of shared meaning.  The final phase in this process involves the learner’s attempts to construct collective meaning by sharing the construct with an even larger community, so as to identify the social norms or beliefs (Cobb, Yackel and McClain, 2000, pp. 146-147).

Throughout this process, Vygotsky suggests students scaffold the new learning (new constructs) upon those already held by the student (1986).  Thus new knowledge becomes acquired through the filtering through or incorporation with prior knowledge (Dixon-Kraus, 1996, pp. 60-61; Wright, Martland and Stafford, 2000, pp. 141-142; Wright, Martland, Stafford and Stanger, 2002, p. 37).  This aspect of Vygotsky’s theory complements the earlier theories of disequilibrium and assimilation proposed by Piaget, (Gagnon and Collay, 2001, p. 58).

A key aspect of Vygotsky’s sociocultural constructivist theory is his research regarding the Zone of Proximal Development.  The Zone of Proximal Development can be described as the distance between the learner’s current developmental level and their potential level of development or performance when placed in a problem solving context wherein guidance or social collaboration with an adult or a more cognitively advanced peer is provided.  The Zone of Proximal Development is socially influenced, dynamic, and fluid as it exists both within the learner and within the social context of the learner’s problem solving experiences (Cobb, Yackel and McClain, 2000, pp. 138-139)

The Zone of Proximal Development is where the learner is challenged and actively engaged in constructing new understanding and is involved, on a very personal level, in active reflection upon his experiences in that learning environment.  It is generally understood that the learner is functioning in an area of cognition where he would not otherwise venture or find success, without the social interaction, guidance, and facilitation of the teacher (Dixon-Kraus, 1996, pp. 60-61; Wright, Martland, Stafford and Stanger, 2002, p.33).  Within Math Recovery intervention sessions, teaching within a child’s Zone of Proximal Development is recognized as one where the learner is challenged and actively engaged in constructing new understandings.

To the social constructivist, the learner is never a blank slate, but rather an integral half of a necessary social-educational interaction.  Steffe  (1990) “asserted that children construct personal mathematics out of their own actions and their reflections on them in social settings.”  To the social constructivists, acquisition of mathematical concepts, knowledge, and skills occurs through socialization, personal experiences, reflection, and revision.

This aspect of the Vygotsky’s work is central to the theoretical belief structure that Math Recovery rests within.  The students in Math Recovery work through a carefully planned and individualized set of problem-solving experiences.  These experiences are based upon the Instructional Framework in Early Number, which is the instructional component of the Learning Framework in Number around which Math Recovery is designed.

In the individualized problem-solving sessions, students are guided through questions and carefully orchestrated sets of experiences to construct or reconstruct their understandings of early number concepts.  The Math Recovery teachers are constantly monitoring and adjusting the sets of experiences that the students work through in order to keep the student working at the leading edge of their Zone of Proximal Development.  This is accomplished through the teacher’s use of on-going assessment, careful questioning, and a thorough knowledge of both the Instructional Framework in Early Number and the foundational Learning Framework in Number.

The Math Recovery teacher builds upon student success by moving them ahead through the framework or, when student understanding falters, the teacher backs up, tries a parallel set of experiences, and finally with success in the new experiences moves the student ahead.  Through the involvement of the Math Recovery teacher, the students are able to experience success in tasks or problem solving activities where they would not have previously found success.  Furthermore, this is all accomplished through the use of intrinsic motivation.  No rewards or overt praise are provided.  The confidence and success that the learner feels in successfully completing these tasks that are at the outermost level of  their Zone of Proximal Development becomes highly rewarding and motivating in and of itself (Wright, Martland, and Stafford, 2000, pp. 147-148).

This approach to mathematics education is different from current practices on a variety of levels.  How could such instructional practices be integrated into current educational practices?  Research indicates that to truly support teacher growth in instructional reform and mathematics education, professional development opportunities need to be sustained, sequential, and constructivist in nature.  Teachers need time and opportunities to experience new instructional approaches as they explore and develop deeper understandings of the strands of mathematics they are teaching (Brooks and Brooks, 1999; Calhoun, 2002; Horne, Cheeseman, Clarke, Gronn, and McDonough, 2002).

Research Defining Number Sense     

According to some theorists, traditionalists/behaviorists believe number sense to be part of our genetic code , and cognitive theorists believe number sense is an acquired skill set developed from experience (Berch, 2005).  Math Recovery’s Learning Framework in Number is closely aligned to contemporary research, constructivist, involving number sense. In a participatory Australian quantitative study, of whom Howell and Kemp (2005) made a questionnaire asking twelve published academics around Australia to identify proponents of number sense.  Howell and Kemp found that rote counting to 5 10 and beyond, comparison of spoken number to 5, quantity order, one number before, two numbers after, subtraction of 1 or 2, and commutatively of addition were all key components to number sense.  The study discovered a lack of consensus among a group of academics participating in the survey (Howell, Kemp, 2005).

In a related study, Malofeeva, Day, Saco, Young, and Ciancio (2004) tested a group of 40 three to five year old children attending Head Start.  Malofeeva, Day, Saco, Young, Ciancio, relied on the work of other theorists to define number sense.  Twenty-one of the 40 children were boys and 19 were girls.  Four of the children were Caucasian, 34 were African-American, and 2 were Hispanic.  The age of the children ranged from three years to almost six years.  The study design included one between-subjects factor instructional condition, with two levels (instructional and attention control), and one within-subjects factor, time with two levels (pre- and posttest).  At pre- and posttest, children took the Number Sense Test as well as the Feelings About School (FAS) Test.  Through the evaluation of the pre-and post-number sense test, and pre-and post-feelings about school test, the results were significant and co relational.  Within the control group that received the number sense instruction combined with traditional instruction, the results showed not only that they score higher on number sense test, but also their feelings about school rose because they felt more comfortable thinking cognitively about mathematics.  As for the other students that did not receive the number sense instruction, results showed little evidence of broad transfer of uninstructed skills.  The students did not improve on ordinality, number-object correspondence, and comparison.  Through these findings, Malofeeva, Day, Saco, Young, Ciancio, found strong evidence that not only should number sense be taught in mathematics, but also closer attention should be paid to connecting different concepts of number sense.  They also established the rate to teach each number sense concept to lessen mathematics anxiety and raise number sense awareness.  Age was also found to have a positive relationship with counting, number ID, and number-object correspondence.  The key components to the number sense test that was used in this research project were counting, number ID, number-object correspondence, ordinality, comparison, and addition-subtraction.  A test was also constructed to see if there was a relationship between number sense and children’s feelings about school.

In a similar meta-analysis of finding components of number sense, Berch (2005) compiled a list from over thirty studies to try and define number sense.  A faculty permitting the recognition that something has changed in a small collection when an object has been removed or added to the collection without direct knowledge is one aspect to number sense.  Elementary abilities or intuitions about numbers in arithmetic are another aspect of number sense.  Components of number sense include the ability to approximate or estimate, make numerical magnitude comparisons, decompose numbers naturally, and develop useful strategies to solve complex problems. Additionally, the ability to use the relationships among arithmetic operations to understand the base-10 number system, to use numbers and quantitative methods to communicate, process, and interpret information are also important elements.  The awareness of various levels of accuracy and sensitivity for the reasonableness of calculations with a desire to make sense of numerical situations and searching for links between new information and previously acquired knowledge is yet another link to number sense.  Knowledge of effects from operations on numbers and understanding number fluency and flexibility makes number sense increase.  Berch later states that if a child can understand number meanings, multiple relationships among members, recognize benchmark numbers and number patterns,  and recognize gross numerical errors they will have components to number sense acquisition.  Berch also states that to understand equivalent forms and representations of numbers, and equivalent expressions while understanding numbers as referents to measure things in the real world are also components to number sense acquisition. He goes on to articulate that in order to attain a sense of number one must move effortlessly between the real world quantities, and the mathematical world of numbers and numerical expressions. He/she should be able to invent procedures for conducting numerical operations, and represent the same number in multiple ways depending on the context and purpose of the representation. The child should be able to think and talk in a sensible way regarding the general properties of a numerical problem for expression without doing any precise computation.  He later states that number sense will be attained if the child stimulates an expectation that numbers are useful and mathematics has certain regularity.  A non-algorithmic logic for numbers is associated to number sense.  A well organized conceptual network emabling a person to relate number and operation, a conceptual structure that relies on links among mathematical relationships, mathematical principles, and mathematical procedures was found as a component to number sense also.  A mental number line according to Berch’s findings, on which analog illustrations of numerical quantities cannot be manipulated in a child’s head and nonverbal, evolutionary, ancient, and innate capacity to process fairly accurate numerosities are number sense components. Finally number sense can be seen as a skill or knowledge about numbers rather than an intrinsic process and is a process that develops and matures with experience and knowledge.

One of the key proponents to this list is the development and maturation of experience and knowledge which has links to both Piaget and Bruner through cognitive/constructivist teaching.  Constructivism is gaining popularity among mathematicians across the world especially in America, Australia, and the Netherlands.

Berch (2005) later states, number sense reputedly constitutes an awareness, intuition, precognition, knowledge, skill, ability, desire, feel, expectation, process, conceptual structure, or mental number line.  Possessing number sense, seemingly permits one to achieve everything from understanding the meaning of numbers to developing strategies for solving complex math problems; from making simple magnitude comparisons to inventing procedures for conducting numerical operations; and from recognizing gross numerical errors to using quantitative methods for communicating, processing, and interpreting information” (pp. 333-334).

Berch, (2005) later maintains that timed tests are helpful for developing student’s number sense if the teacher can determine which addition/subtraction principle the student is struggling with.  For example one problem for number sense might be dyslexia, another might be that the student has not mastered the doubling strategy, near double strategy, or be able to think in 10’s.

A variety of ideas pertaining to number sense acquisition are outlined from this research.  This researcher will strategically employ some of these components for the purpose of this study.  These components include counting forward, counting backward, identifying numerals, structuring numbers 1-20, concept of ten, addition/subtraction understanding, and counting in multiples and/or grouping.

Mathematics Anxiety and Difficulties Related to Number Sense

NCTM  (2000) gave ten recommendations based on material from principles and standards for school mathematics to help prevent math anxiety.  Amongst the list was the importance of original quality thinking other than a rote manipulation of formulas.  (Furner, Duffy 2002)  Findings from research studies in the United States, according to Badian (1983), have shown 5% to 8% of school-age children have some type of mathematics disabilities (MD).  Reading readiness is more developed than math readiness.  Gersten, Jordan, and Flojo (2005) indicate that reading readiness stems from theoretical studies and intervention studies.  The theories have been tested for reading development and have been implemented into curriculums because of their validity and reliability.  In contrast, measures for math readiness are still in their formative years.  According to Gersten, Jordan, and Flojo (2005) there has been a small set of studies for math proficiency in the field of cognitive psychology, development in children, and Curriculum Based Measurement analysis.  Gersten, Jordan, and Flojo conducted a study to find the cause of mathematic difficulties and the role of number sense of young children. The study also evaluated valid screening measures for early detection of potential difficulties in mathematics for early intervention and instruction.  The study was conducted on 200 kindergarteners in two urban areas.  Gersten, Jordan, and Flojo (2005) found a correlation between number sense ability and math difficulties (MD).  The slow retrieval of number combinations is a key identifier to students struggline with math intricacies.  Gersten, Jordan, and Flojo (2005) believe teachers should be aware that students attain the concept of number at different times and need to nurture and give additional time to the understanding of concepts and operations in mathematics.  If the teacher is not aware of low achieving children’s maturation rate in number sense then the child will start to develop mathematical difficulties (MD) Gersten, Jordan, and Flojo, (2005).

In a similar research study Gersten and Chard (1999) charge that number sense is in  mathematics is as vital to child development research as phonemic awareness has been to the reading research endeavors.  Gersten and Chard (1999) state, “that number sense research and the cognitive sciences can help the research community pull together fragmented pieces of earlier knowledge to yield a much richer, more subtle, and more effective means of improving instructional practice” (p. 18).  Gersten and Chard (1999) note that in many remedial classrooms, mastery of algorithms, practice repetition, and limited opportunity to express verbal thinking and reasoning is a commonplace in American mathematics.  Numerous assessments tend to emphasize computation versus understanding which contribute to the barriers of math learning.  The focus on computation, rather then understanding is why mathematical difficulties are prevalent to adulthood (Gersten and Chard 1999).

In a related study, Tsao (2005) wanted to research cognitive processes of preservice elementary teachers on basic math problems involving number sense.  Tsao used six random participants scoring in the top and bottom10% of the number sense test used.  Tsao interviewed these participants and found that the low ability participants relied on rule-based methods and written algorithms, whereas the high ability participants applied number sense strategies up to twice as often as the lower ability participants.  Tsao (2005) also found that elementary preservice teachers are not ready to teach number sense according to the principles and standards of the NCTM (2000) initiative (Tsao 2005).

Teachers become number sense literate and they reduced their levels of anxiety towards teaching Mathematics in a two-part study.  Vinson (2001) compared anxiety of 87 preservice teacher’s mathematics anxiety before and after taking a manipulatives methods course.  The first part of the class focused on Bruner’s framework of conceptual knowledge over procedural knowledge.  The second part of the course focused on using manipulatives to make mathematics more concrete. Vinson (2001) found that anxiety was significantly reduced through these methods, and posttest data demonstrated statistically significant reductions of mathematics anxiety levels (p < .05) for the winter assessment.  Some students did have an increase of anxiety; in which they were not familiar with using manipulatives.

Kaminski (2003) conducted a 12 week number sense course for 43 second year preservice teachers, in which the students were encouraged to work in small groups evaluating number sense and then reflect their thoughts with journals. Open discussion sessions were also implemented after journal writing.  After the 12 week course was over a random sample of interviews followed.  Three themes that emerged: student teachers’ learning experiences in number sense explorations; student teachers’ planning for, and teaching of mathematics; and changes in student teachers’ views and in attitudes regarding teaching mathematics in the course of reflection in journals (Kaminski 2003).  After interviewing the preservice teachers and analyzing the data Kamenski found that t experiences in number sense programs had positive consequences for teachers regarding their understanding, knowledge, and teaching of mathematics.

In a similar research study, Zemelman, Daniels, and Hyde (1998) constructed a list; the best practice for teaching math and preventing mathematics anxiety.  They found teachers serving as facilitators of learning, and assessments of learning should be a part of instruction with  the use of: manipulatives (making learning math concrete), cooperative group work, discussion, questioning and making conjectures, justification of thinking, writing and math for thinking, expressing feelings, solving problems, problem-solving approaches to instruction, calculators, computers, technology,  and making content integration a part of instruction.         

Overview of Math Recovery

Mathematics Recovery (MR) founded by (Phillips, Leonard, Horton, Wright, & Stafford, 2003; Wright, 2003; Wright, Martland, and Stafford, 2000; Wright, Martland, Stafford, and Stanger, 2002) was based on the theoretical framework of number of prominent researchers who include L. Steffe, von Glasersfeld, Richards, & Cobb, 1983; L.P. Steffe, Cobb, and von Glasersfeld.  Van den Heuvel-Panhuizen, (2001) has greatly influenced the views of Math Recovery leaders and teachers in regards to two digit addition and subtraction.

One of the structures greatly influencing (MR) is the theoretical framework.  The theoretical framework behind MR is based on the research of a longitudinal study with first and second graders sense of early number knowledge.  The two studies were conducted by Steffe, von Glasersfeld, Richards, & Cobb, 1983; and Steffe & Cobb, 1988.  The goal was to develop psychological models of conceptual development over an extended period of time.  Steffe’s approach is unique because the research included techniques of problem-based approach to instruction with routine videotaped assessment with teaching sessions for retrospective analysis.  Steffe’s theoretical orientation drew upon von Glasersfeld’s (1995) theory of cognitive constructivism, a derivative of Piaget’s cognitive development theory.

*****The Mathematics Recovery Program (MR), which was developed from the theoretical framework of Steffe, von Glasersfeld, Richards, & Cobb, the result of a three-year research and development project at Southern Cross University in Northern New South Wales, conducted from 1992 to 1995.  The project received major funding from the Australian Research Council, regional government and Catholic school systems. The project involved working at 18 schools with 20 teachers, and approximately 200 participating first grade students.  MR has two distinct and related parts.  First an elaborated body of theory and techniques for working with children and number sense (Wright, Martland, & Stafford, 2000; & Wright, Martland, Stafford & Stanger, 2002).  The second, according to Wright, (2000) deals with distinctive ways to work with teachers and provide effective ongoing professional development to enable teachers to learn methods of working with and nurture children’s number sense.  Robert Wright’s research for MR includes:  his doctoral teaching experiment of numerical development in the Kindergarten year (Wright, 1989), research of numerical knowledge in school entrants (Wright, 1991)  and the progression of number learning in the first two years of school (Wright, 1994). ***** Through these findings Robert Wright formulated nine principles of Mathematics Recovery teaching.  1.) The teaching approach is enquiry-based, (problem-based).  Children routinely are engaged in thinking hard to solve numerical problems, which for them, are quite challenging; 2.) Teaching is informed by an initial, comprehensive assessment and on-going assessment through teaching.  This refers to the teacher’s informed understanding of the child’s current knowledge and problem-solving strategies and continual revision of this understanding; 3.) Teaching is focused just beyond ‘the cutting edge’ of the child’s current knowledge; 4.) Teachers exercise their professional judgment in selecting from a bank of instructional settings and tasks, and varying this selection on the basis of on-going observations; 5.) The teacher understands children’s numerical strategies and deliberately engenders the development of more sophisticated strategies; 6.) Teaching involves intensive, on-going observation by the teacher and continual micro-adjusting or fine-tuning of teaching on the basis of his or her observation; 7.) Teaching supports and builds on the child’s intuitive, verbally-based strategies and these are used as a basis for the development of written forms of arithmetic which accord with the child’s verbally-based strategies; 8.) The teacher provides the child with sufficient time to solve a given problem.  Consequently, the child is frequently engaged in episodes which involve; sustained thinking, reflection on his or her thinking, and reflecting on the results of his or her thinking; 9.) Children gain intrinsic satisfaction from their problem-solving, their realization that they are making progress, and from the verification methods they develop. (Wright, 2002, pp.)

Research Supporting Math Recovery

To assess the Math Recovery model independent researcher Holly MacLean (MacLean, 2003) evaluated  relative effectiveness of three different professional development models relating to low-achieving, urban first graders. The first model was a full Math Recovery implementation. The full implementation and intensive one-on-one tutorial intervention was provided to selected, low-achieving, Title 1 first grade children. Also, on-going professional development for classroom teachers was provided via the on-site Math Recovery leader. This development training was in the form of presentations, joint planning sessions, demonstration and team teaching. The second model involved the same Math Recovery leaders conducting on-going professional development in Math Recovery theory, strategies, and activities to classroom teachers from schools absent from the on-site, one-on-one tutorial component. These strategies and activities were adapted in the classroom setting. The third model involved schools were classroom teachers received periodic, one time professional development with conference attendance.  Professional development in this form was provided by both in-district math leaders and outside consultants and speakers. The teachers in this model were not exposed to Math Recovery theory and methods.  MacLean found the full Math Recovery implementation model significantly exceeded expectations in the ongoing professional development only model as well as the periodic, one-exposure model. The school district is currently conducting a longitudinal study to follow the children as they take state mandated assessments.  MacLean’s findings are analogous to other researchers (Phillips et al., 2003). Lois Williams (Williams, 2001) found Math Recovery significantly changes teacher practices in the classroom. Teachers participating in the Math Recovery training became reformed oriented in their teaching.  Math Recovery can be viewed from a social justice perspective. Research has established vast differences in the mathematical knowledge of students when they begin the first year of school.  Additionally, students who are least advanced when they begin school tend to remain so throughout their learning years. Even in the early years of education, low-attaining students begin to develop strong negative attitudes towards mathematics. Early intervention is important due to the opportunity for reaching educationally disadvantaged students before the gap between their knowledge and that of average and high attaining students is too wide and before they experience multiple failures.

Brooks and Brooks, (1999) remind staff development personnel that people can only know understandings that they construct for themselves.  Teachers need to be able to construct their new understandings and then need opportunities to work with students so they can reflect upon both the learner and teacher aspects of the experiences.  Passive methods of learning are as ineffective with adults as they are with children (Arnold, 1995).

District administrators and professional development personnel recognize that changes may be necessary in teacher professional development, pre-service teacher training, and school structuring.  Additional changes may also be necessary in the current assessment practices used within the schools.

Most recently, mathematics education has undergone a shift in focus to where constructivist-learning theory is gaining support.  According to Dengate, “the experiencing organism now turns into a builder of cognitive structures intended to solve such problems as are perceived or conceived.  All knowledge is regarded as being constructed and owned by the individual” (Dengate, 1998, p. 203).  Whether the goal is to support teacher growths in mathematical understanding, or to improve instructional practices in order to more effectively meet the needs of all students, a shift to the use of more constructivist approaches in pre-service teacher preparation and teacher professional development is certainly gaining support.

Ironically, social constructivists recognize that the reality of mathematics education at present time is much closer to a behaviorist’s stand and deliver approach than to a social constructivist’s approach.  Traditional classroom activity is more content-focused or teacher-focused and learner-focused (Brooks and Brooks, 1999, pp. 16-17; Gagnon and Collay, 2001, p xv; Marlowe and Page, 1998, pp. 9-13).

Therein lies the challenge to professional development providers and district administrators.  Professional development must address the pedagogy of mathematics instruction (cognitive curriculum), as well as involve the teachers in social constructivist-based experiences (as adult learners) in order to bring about philosophical changes to support the teacher’s shifts in the instructional focus.

Conclusions

The research demonstrates clearly that cognitive number sense is growing in the field of modern curriculum.  It is also clear that children develop number sense cognition at various points throughout their school tenure or may even compensate from the lack of number sense which in turn might develop math anxieties or math difficulties.  However, it appears that the factors attributed to number sense are still in its infancy in research and curriculum design.  Research must continue and cognitive patterns should be monitored to help explore, promote, and nurture early childhood number sense.  Efforts to design and test strategies for raising number sense in children today should include not only cognitive psychologists and others concerned with number sense deficiencies, but also curriculum writers and those concerned with raising mathematics scores for today’s schools.

Tests of promising strategies and their evaluations should be done to discover what works and what should become common educational practices in elementary schools through high schools.

The next step is to begin conducting research on programs/curriculum that, despite all of the factors that may be blocking early number sense, are working towards closing the gap towards early mathematics anxiety, early mathematics difficulty, and number sense awareness in low performing first grade students.

This literature review found evidence to indicate that number sense affects student performance at an early age which carries through adulthood.  Most of the research focused on the early primary levels.  A few studies examined the issue carried throughout adulthood.  Several effective treatment interventions were identified.  The current study will fill the gap of number sense acquisition by exploring low achieving first grade students at a suburban school in the Midwest.  The treatment will be performed by a Math Recovery specialist giving one-on-one sessions to low achieving students in mathematics.  The current study is interested in the promotion of number sense throughout the low achieving student’s first grade year to increase performance in math confidence and ability.

Chapter III

Introduction

This research study is a qualitative exploration to address this central research question: For a Midwest suburban public school, how do first grade students identified as low achieving develop number sense?  This qualitative approach to research aims to understand the meaning or significance of a phenomenon to an individual or group of people (MacNaughton, Rolfe, & Siraj-Blatchford, 2001).  The qualitative researcher approaches the inquiry by adopting an interpretive worldview and explores the participants’ sense making and understanding of an issue while collecting information on the topic itself (Wolcott, 1995).

Throughout this study, the primary aim will be to gain insight into low achieving children’s experiences with the acquisition of number sense.  This first grade population is “information rich and illuminative” (Patton, 2002, p. 46) which will offer a useful manifestation of the acquisition of number sense. Purposeful sampling will provide students who are likely to have personal insights so the researcher can learn “a great deal about issues of central importance” (Patton, 2002, p. 46).  The researcher as key instrument will collect data through examining documents, observing behavior, and interviewing.  Multiple sources of data, interviews, observations, and documents will be used for analysis.  Inductive data analysis will build patterns, categories, and themes by organizing data into abstract units of information (Creswell, 2007).

This section presents the methodology and procedures utilized to collect the observation and interview data of three first grade low achieving students in mathematics.  Chapter III is divided into the following major sections: (a) study design, (b) population, (c) participants, (d) data gathering techniques, (e) Assumptions, (f) the strengths, limitations, and delimitations of this study, (g) and the data analysis.

Qualitative Case Study Methodology Design

The qualitative case study research methodology employed in this study affords the use of a collective approach.  Yin (2003) suggests the multiple case study design approach uses replication, in which the inquirer replicates the procedures for each case.  Yin also states the inquirer should select representative cases for inclusion in the qualitative study.

There are various definitions throughout the research literature on what constitutes a case study.  A case study can investigate a “unit of human activity embedded in the real world” of an individual, group, institution, or community (Gillham, 2000, p. 1).  Yin (2003) defined a case study as a method of inquiry “that investigates a contemporary phenomenon within its real-life context, especially when the boundaries between phenomenon and context are not clearly evident” (p. 13). Patton (2002) explains that “case studies clearly define the object of study which can be layered or nested” (p.298).  Whereas Creswell (2007) emphasizes that a “case study involves the study of an issue explored through one or more cases within a bounded system” (p. 73).

The case study was adopted in this study to address the question of ‘how’ or ‘why’ in examining the contemporary phenomenon of number sense acquisition for low achieving first grade students (Yin, 2003).  Thus, the objective throughout the collection of data is to grasp the meanings of the acquisition of number sense for low achieving first grade students.

Case study research is a popular form of qualitative research design, used as a means to answer the question “What is going on here?” (Edwards, 2001, p. 126).  The primary purpose of this case study is to demonstrate how it is to be in a particular situation, to catch the close-up reality and depiction (Geertz, 1973) of participants’ experiences, thoughts, and feelings for a situation (Cohen et al., p. 182).  The case study provides a rich and descriptive account of relevant events, as well as an in-depth analysis to “get below the surfaces” involved in number sense acquisition of low achieving first grade students (Edwards, p. 126).

Subjects

Participants in this qualitative study are low achieving first grade students in mathematics at a suburban school in the Midwest with about 500 students.  The students will be from the 2008-2009 and the 2009-2010 school year.  Three students will be selected from a district assessment given in the fall that determines SEAL stages for first grade students that do not meet “grade level” in first grade mathematics.  Students are required to have twenty to fifty, thirty minute sessions upon release.  The selected first grade students will be screened periodically to see growth along the LFIN.

Trustworthiness

The researcher, who is the Math Recovery teacher, is an integral part of the investigation (Jacob, 1988). This study will use subjective information and participant observation to describe the setting of the variables under consideration, with interactions of the different variables in the context (Wolcott, 1990).  The researcher will keep lesson plans, detailed video records, and reflections of what occurs while seeking an understanding of the situation. During this process the researcher will periodically detach himself to a neutral position of a social scientist (Wolcott, 1990). The researcher will constantly monitor observations and behaviors for evidence of personal bias or prejudice. According to Wolcott (1990) the researcher will systematically seek for and  organize data concerning the object of study, based on a social science theory and methodology other than focusing on achieving a defined goal from situation to situation. Total participation describes the researcher as a natural participant. According to Wolcott (1990) this is the highest level of involvement and the researcher studies a subject he is already a natural participant. Participant observer research emphasizes the importance of examining variables in natural settings where they are found  with  interaction between variables being important. Detailed data is gathered with open ended questions providing direct quotations (Wolcott, 1990).

Data Collection and Analysis

For the purposes of this study, permission will be granted from the superintendent to utilize extent data from standard primary academic measures.  This extent data will be used to measure the SEAL stages of each student.  The researcher will determine “good candidates” from analyzing the SEAL stages of each first grader.  Data will be compiled from twenty to fifty, one-on-one thirty minute videotaped Math Recovery sessions which will be transcribed and added to the reflections and observations to provide a ‘thick description’ (Goetz & LeCompte, 1984) of each session.  To drive instruction and assist in scaffolding and micro adjusting assessment 1.1 (numeracy screening) will be given periodically throughout the sessions.  Transcriptions of video tapes will be cataloged and coded.  Themes will emerge from the coded data which will be reflected in the analysis.  The findings of the qualitative data will also include reflections (recorded in a separate reflective field journal) and ongoing analysis of the video taped sessions in the form of transcriptions to summaries (Ely et al., 1997).  Video Tapped Observation of number sense acquisition while scaffolding and micro-adjusting during the video tapped sessions will be deemed the crucial data collection tool to allow myself as the researcher to gain the “here-and-now” experience (Lincoln & Guba, 1985, p. 273).  Access to this data and collective case study video taped sessions are part of the researcher’s duties of getting Math Recovery certified.

Assumptions underlying this study include:

1. The LFIN is an accurate measure of student number sense acquisition.
2. The participants will do their best on the fall number sense screening.
3. 1^st grade SEAL stages are an accurate measure of student ability that is used for number sense diagnosis
4. Previous experience with the numeracy screening will not be a threat to validity
5. Students will perform their best on given screening dates
6. Assessment 1.1 and 2.1 are an accurate measure of a first grade students number sense ability

Limitations

This study will be limited in the following ways:

1. The information obtained in this study will not be generalizable.
2. First grade subjects are representative of low achieving first grade students in the school
3. Screener, who is the researcher, is not certified in Math Recovery teaching. This case study research will aid in the certification process.

Delimitations

This study will be delimited in the following ways:

1. This study will be limited to public elementary school students, first grade, in one school.
2. This study will be limited to the teachers of the students in a public elementary school.

Protection of Human Subjects

Permission will be granted from the superintendent to utilize extent data from standard primary academic measures.  Access to this data and collective case study video taped sessions are part of the researcher’s duties of getting Math Recovery certified.  In accordance with Federal law [34 CFR 99, 99.03 through 99.37], any identifying information, and names will be deleted from tests administered to students.  A certified Math Recovery Specialist along with me will conduct the SEAL screening.  The Math Recovery specialist will be a professional educator who is experienced in working with young students and who has agreed to maintain confidentiality in all matters pertaining to this study.  The subjects can withdraw without penalty and will need consent from parents to be chosen for this study.   Throughout the study, I will keep all materials including data stored on an external hard drive in a locked file cabinet in my office in order to maintain privacy, anonymity, and security.

Summary

A qualitative methodology was employed throughout this study to address the central question, for a suburban public school in the Midwest, how does Math Recovery affect number sense in first grade students identified as low achieving?  This orientation underscores the importance of understanding the phenomenon of number sense to assist school systems in closing the achievement gap at an early age to prevent math anxiety and complacency or worse falling further and further behind in mathematics.  The data collection methods to be employed will provide a means to immerse myself as an active facilitator and observer in a Math Recovery setting.

My role in the research process will be complex.  At times I will be the learner, observer, confidant, participator, and expert.  The fluidity of this process requires flexibility and adaptability in the methods employed.  By remaining cognizant of the researcher’s role in the process the researcher will be able to challenge his own interpretations throughout the process and reflect on each Math Recovery session to help inform best practice in mathematics for low achieving first grade students.  Also, every effort will be made by the researcher to insure that the study is conducted in an ethical and unbiased manner.  Lastly, while the results will be delimited to low achieving public elementary school first grade students in DuPage County, the data is likely to reveal interesting relationships between low achieving students number sense ability.

References

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Berch, D. (2002). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339. Retrieved Wednesday, April 04, 2007 from the Professional Development Collection database.

Case, R. (1998, April). A psychological model of number sense and its development. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.

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