Adapting early childhood philosophies and practice in ...



Adapting early childhood philosophies and practice in teaching math in second grade

Holly M. Wright

Vanderbilt University

Although the second grade is still considered a part of the early childhood stage of birth through age 8, much attention of early childhood training and publications is focused on birth through age 6. The National Association for the Education of Young Children’s (NAEYC) position statement on developmentally appropriate practice states that too often primary schools serving first through third grade children adopt narrow curriculum and instructional strategies contrary to current knowledge of how young children learn and develop (Bredekamp & Copple, 1997). With such narrow curriculum and instruction, it is difficult for children to have “the opportunity and support necessary to learn significant mathematics with depth and understanding,” which all children deserve for a productive future (NCTM, 2000). NAEYC has published a joint position statement with the National Council of Teachers of Mathematics (NCTM) to explore the mathematical learning needs of early childhood in pre-kindergarten through age six. However, the document does not address the philosophies and methods for addressing early childhood needs in the second grade (NAEYC & NCTM, 2002). To explore how to meet the unique needs of young children in second grade addition and subtraction, it is important to consider philosophies and validated practices for developmentally appropriate instruction. Through this investigation, teachers will discover that an integrative curriculum that includes connections to standards, quality observation and assessment methods and ways to meet the needs of diverse learners will build and support second grade mathematical reasoning. Implications for practice and professional development are included for providing the most effective learning environment for these children.

Philosophies and Validated Practices for Developmentally Appropriate Instruction

To consider the curriculum and instructional strategies needed to support second grade children as learners of mathematics, educators must first consider the philosophies and research-based practices that will best inform their instructional strategies. Since second graders are in the range of birth through age 8, teachers should consult organizations such as NAEYC as well as other respected early childhood philosophers, theorists, and researchers. NAEYC provides insight into young children as learners. To address the needs of the mathematical content of addition and subtraction in a second grade classroom, educators should consult organizations such as NCTM, as well as other respected mathematics education researchers. NCTM provides research on mathematics learning in all age ranges. Comparing and combining these philosophies and research-based practices will provide the most effective curriculum instructional strategies and learning environment for second grade students as young math learners.

NAEYC, NCTM, and Validated Instruction for Second Grade Math

Although it has been mentioned that the joint position statement of NAEYC & NCTM (2002) on Early Childhood Mathematics specifically addresses the needs of only three-to-six-year-old children, the recommendations and principles are in concordance with both NAEYC’s suggestions for 6-8-year-olds through the publication of Developmentally Appropriate Practice (Bredekamp & Copple, 1997) and NCTM’s (2000) Principles and Standards and Curriculum Focal Points for second grade. The joint position statement, therefore, is an excellent framework for exploring the mathematical learning needs of young children for 7 and 8-year-old children in the second grade

An important suggestion of the joint position statement of NAEYC & NCTM is the need to “base mathematics curriculum and teaching practices on current knowledge of young children’s cognitive, linguistic, physical, and social-emotional development.” Bredekamp and Copple’s (1997) Developmentally appropriate practice in early childhood programs states that all domains of human development are interrelated and influence each other, and “failure to attend to all aspects of an individual child’s development is often the root cause of a child’s lack of success in school.” This knowledge obliges educators to become well informed about the developmental norms and individual variations of the children whom they serve to provide the most appropriate instruction that fosters all domains of development. Bredekamp and Copple review research on the development of children 6-through 8-years old in different domains that educators should consider.

Physical Development and Second Grade Math

During these years, the brain is about its adult size, and lateralization makes the brain more efficient in functioning, speeding mental processing of information. Physical growth allows these children to have greater control of both gross and fine motor development. These children become more coordinated in running, jumping, and balancing, as well as more precise and controlled in writing and drawing (Bredekamp & Copple, 1997). This is significant for math educators because children can attend to tasks longer periods of time, but they need to be active because long periods of sitting can fatigue them. Moreover, physical activity can advance cognitive development, as physical actions help them grasp ideas in more concrete ways (Bredekamp & Copple, 1997). This statement about student activity promoting mathematics understanding draws upon the work of Piaget whose constructivist theory emphasized learning as an “active process” in which children develop cognitive structures for understanding and responding to physical experiences within his or her environment (Piaget, 1971). Children should therefore engage in active learning. Teachers who foster active learning move beyond passive instruction toward instruction that provides direct experiences and opportunities to manipulate concrete objects. However, manipulating objects does not automatically improve children’s understanding without the presence of skilled teaching (Bredekamp & Copple, 1997). Thompson and Lambdin (1994) and Ball (1992) review the use of concrete manipulatives in math instruction, and they conclude that it is easy for a teacher to misuse manipulatives by thinking a concept is mastered when a prescribed activity is expected, such as the modeling of symbolic procedures using base-ten blocks to teach place value or bundles of popsicle sticks to teach regrouping in addition and subtraction. The concept is not in the concrete materials; the idea is in the teacher’s understanding of different interactions with the materials. However, appropriate use of manipulatives allows teachers and students to engage in grounded conversations about something concrete. This promotes the awareness of multiple viewpoints about ways to think about a problem in conceptually oriented instruction. Manipulatives also provide something for children to physically act on and reflect on this physical action in relation to cognitive connections to the content (Thompson & Lambdin, 1994), (Ball, 1992).

Cognitive Development and Second Grade Math

In the cognitive domain of development, a gradual yet significant shift occurs in which most children develop the ability to think about and solve a wide range of problems. These new abilities include flexible and proficient mental representations and mental operations needed for mathematics. Under Bredekamp and Copple’s (1997) paradigm related to cognition, mathematics cognitive skills such as decentration, reversibility, and conservation, are pruned in the brain that promote mathematical learning. As decentration develops, teachers can help students understand each other’s viewpoints. Using reversibility, students begin to mentally reverse a series of steps, such as using subtraction to reverse addition. As children’s experiences with mathematics increases, they develop an understanding of conservation that despite changes in appearance, the concept of quantity and number remains the same unless something is added or removed. At this point in the development of conservation, children’s understanding of one-to-one correspondence and number is complete. For example, children are able to determine that although an amount of cookies is rearranged into groups, the number of cookies remains the same. These findings on cognitive development are significant for math educators because the processes of addition and subtraction require the use of mental representations, mental flexibility, reversibility and efficiency of different mental methods. NCTM Principles and Standards (2000) states that children should be able to perform computations in different ways including mental methods and estimation. During the second grade year, children typically have the cognitive skills to engage in such mental methods. In a research study on the developmental differences of children’s estimation processes, Booth & Siegler (2006) found that a shift occurs between kindergarten and second grade in which second grade children reason in a linear, conceptual manner on numerical estimation tasks between 0-100. This cognitive shift in proficient and conceptual methods of estimation is thought to be due to second graders increased exposure to numbers 0-100.

Additionally, although these children can symbolically and mentally manipulate concrete objects, they are still not capable of mentally manipulating abstract ideas. When children use numbers to represent objects and relationships, they still need concrete reference points. (Bredekamp & Copple, 1997). Whitenack, et al. (2001) observed a group of second grade students struggling with the abstract conceptual understanding of multi-digit addition and subtraction. These children did not initially consider the relationship between the numerals in the tens and the ones place; however, the teacher addressed this issue by posing a set of problems in the context of the Aunt Mary’s Candy Making (Cobb, et al., 1992) and allowing the children to record their methods in various ways including concrete drawings and discussions. Whitenack et al. found that within this context, these children were able to invent their own solutions with reasoning grounded in the understanding of quantity (Whitenack, et al., 2001).

NCTM Principles and Standards (2000) states that, “students must learn with understanding, actively building new knowledge from experience and previous knowledge.” Similarly, Piaget declared, “Children are not empty vessels to be filled with knowledge but active builders of knowledge,” Piaget’s constructivist theory is the foundation of this recommendation, which suggests that children actively construct new learning by assimilating it to previous learning experiences or adopting it to fit a new understanding of the world. One implication of constructivism is that mathematics should be taught through problem solving as students attempt to achieve their goals (Yackel, Cobb, & Wood, 1991). Educators must keep in mind that conceptual constructivist learning in mathematics only happens when children are not merely passively absorbing teacher talk, but instead, actively grappling and making sense of math problems through contexts that they can relate to prior learning.

Furthermore, during this age of 7 though 8 years of age, children develop an increased capacity to store and retrieve information in both short- and long-term memory. They become more proficient and systematic at memory strategies such as rehearsal and organization of information and are therefore more capable of retaining decontextualized information (Bredekamp & Copple, 1997). This is significant for math educators because computational fluency, or being able to recall memorized numerical relationships efficiently, is a goal of many second grade addition and subtraction standards. To foster this ease of recalling computational procedures, NCTM articulates that instruction must first focus on the understanding of numbers and relationships (2000). Second grade children must first learn addition strategies grounded in context, such as making 10, doubling, and the use of 5’s. Then, they can build on this understanding by organizing thinking strategies to reason and recall other numerical relationships with ease. For example, if a child knows 10+10=20, it will be easier for them to reason that 10+11=20 plus one more.

Social and Emotional Development and Second Grade Math

In social and emotional development, the ability to take multiple points of view strengthens the 6-8 year old child’s communication skills. Engaging in conversation about learning has been found to bolster children’s abilities to communicate, express themselves, understand, reason, and solve problems. Math educators need to consider this developing ability because NCTM (2000) states that when grappling with the content of Numbers and Operations of addition and subtraction methods, “students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general.” Math educators can use group work and ensuing discussions to promote reflection and communication, which develops deeper, contextual understanding of the mathematics. Collaborative work is developmentally appropriate, since it enhances children’s cognitive and language development as they improve their abilities to take on and solve problems (Bredekamp & Copple, 1997). In a study of the use of small-group interactions in second grade mathematics, Yackel, Cobb, & Wood (1991) found that learning opportunities arise in small-group interactions related to cognitive problem solving tasks that do not typically arise in traditional, non-collaborative learning classrooms. In this second grade math classroom in which children worked in groups of two for 20-25 minutes and then interacted in whole group discussion for 20 -25 minutes, the children were provided with opportunities to communicate. These opportunities to communicate led them to verbalize their thinking, explain or justify their solutions, and ask for clarifications. Further opportunities allowed students to use aspects of another’s solution to prompt the development of their own, reconceptualize a problem for the purpose of analyzing a flawed solution, and extend their conceptual framework to make sense of another’s solution (Yackel, Cobb, & Wood, 1991). These findings support the joint position statement’s (NAEYC & NCTM, 2002) recommendation to “enhance children’s natural interest in mathematics and their disposition to make sense of their social worlds.

This discussion of social development through cooperative work draws on Vygotsky’s theory of sociocultural learning (1978), which asserts that cognitive development is primarily driven by exposure to language, social context and adult guidance. Vygotsky suggests that during the years of 6 to 8, sociocultural learning undergoes changes of the expectations, demands, and adult structuring on the social and cultural context of children’s lives (Bredekamp & Copple, 1997) To promote and accommodate for these changes in development, teachers need to provide opportunities for children to cooperate productively in small groups, providing support and guidance for conflict resolution, as well as providing respect of multiple viewpoints.

Bredekamp & Copple (1997) state that a mandate of developmentally appropriate practice for primary-grade children is to, “help them develop the ability to work collaboratively with peers.” Yackel, Cobb, & Wood (1991) found that after the second grade mathematics teacher modeled situations and guided discussions about small-group collaboration, the children engaged in the mutual construction of norms for cooperative learning. These norms included cooperating to solve problems, valuing meaningful activity over correct answers, persisting on challenging problems instead of completing many problems, and agreeing on the validity of solutions. Additionally, the teacher actively monitored the second graders in their groups to encourage cooperation and dialogue (Yackel, Cobb, & Wood, 1991).

With continued development in moral and social domains, these second grade children progress in social understanding and are now aware that another person can have different thoughts than their own. These children can now make more accurate judgments about what is true and false, and are very concerned with fairness. Children’s moral and social development progresses at different rates depending on the amount of experience and adult guidance they receive. Children are more likely to show advanced moral and social reasoning if adults reason with them to help them understand the rationale for rules and norms in which they operate. Establishing productive, positive social and working relationships with other children provides the foundation for a sense of social competence (Bredekamp & Copple, 1997).

In an 18-month study in a second grade mathematics classroom, Wood (1999) investigated the context a teacher developed for mathematical reasoning through respectful argumentation and active listening. Beginning on the first day, the teacher built an environment in which children were expected to examine, critique, and validate their mathematical knowledge through participation in reasoned discourse. She was able to create this context through establishing and providing situational examples of mutual expectations and norms for criticism that was mathematical instead of personal, active listening that enabled children to take responsibility for helping others understanding by agreeing or disagreeing with explanations (Wood, 1999). Math educators should follow this teacher’s example to progress children’s social and emotional development through a contextual focus on mathematical reasoning and sense making

Processes that Reflect DAP

With the developmental domains in mind, to provide DAP in second grade, educators need to meet the challenge of “ensuring the teaching of rich content of the primary-grade curriculum, while taking full advantage of the child’s developing abilities, interests, and enthusiasm for learning,” (Bredekamp & Copple, 1997). Furthermore, the joint position statement (NAEYC & NCTM, 2002) states the need to actively introduce mathematical concepts, methods, and language though a range of appropriate experiences and teaching strategies. For second grade children ages 7 and 8, Bredekamp and Copple (1997) suggest the need for active, experiential learning in a meaningful context instead of rote learning of academic skills that is often seen in primary school instruction. They affirm that in mathematics instruction that adheres to DAP:

“children acquire and apply understanding of mathematics skills and concepts. Teachers plan for children to learn mathematical concepts through solving meaningful problems…fostered through spontaneous play, projects, and situations of daily living. A variety of math manipulatives…are used to aid in concept development and application. Math activities are integrated with other relevant projects,” (Bredekamp & Copple,

1997).

Furthermore, for DAP in mathematics, the NAEYC and NCTM joint position statement recommends the use of “curriculum and teaching practices that strengthen children’s problem-solving and reasoning processes as well as representing, communicating, and connecting mathematical ideas.” These processes are emphasized by NCTM’s Principles and Standards for School Mathematics (2000) as Process Standards that are reflections of “society’s needs for mathematical literacy, past practices in mathematics education, and the values and expectations held by teachers, mathematics educators, mathematicians, and the general public.” Regardless of the particular mathematical content being taught, effective and appropriate mathematics teaching and learning must incorporate the Process Standards of problem solving, reasoning, communication, connections, and representations to facilitate children’s contextual understanding and fluency of mathematics. Through an investigation of the practice of NCTM Process Standards and goals, Matthern & Hansen (2007) studied the changes in an Elementary School in Evansville, WY, which after being identified as a failing school in mathematics under NCLB, began to adopt NCTM’s process standards and research-based practices to understand what children need to know and need to learn. With the help of consultants to develop a new process standards-based and research-based curriculum, the teachers and administrators shifted from traditional teacher-directed mathematics instruction towards implementing the process standards to encourage the in-depth, active, constructive thinking of the students. In grades K-2, the new math curriculum focused on the following components: building mathematical knowledge through daily problem solving exercises, making and investigating mathematical conjectures, communicating mathematical thinking clearly through cooperative learning, and using manipulatives as nonlinguistic representations and recognizing and connecting mathematical ideas through authentic assessment tasks and integrated instruction. The results of implementing this new process- and research-based curriculum from 2003-2006, a period of only three years, were remarkable. The students’ performance showed increased mathematical achievement, which met the annual yearly progress goals of NCLB. More importantly, however, was a noticeable school-wide transformation, in which teachers, students, and parents worked together towards new levels of mathematical understanding. Since they had ownership of their understanding and misconceptions about math, children displayed increased joy and confidence in having the ability to discover multiple ways to solve problems (Matthern & Hansen, 2007). This research provides wonderful evidence in support of the power of developmentally appropriate mathematical processes to advance mathematical achievement, reasoning and confidence.

Importance of an Integrative Curriculum

As a part of DAP and NCTM’s Process Standard of Connections, the joint position statement on Early Childhood Mathematics (NAEYC & NCTM, 2002) makes recommendations to “integrate mathematics with other activities and other activities with mathematics.” Moreover, NCTM’s Process Standards (2000) Connections Standard states that “Mathematics is not a collection of separate strands or standards…rather, mathematics is an integrated field of study.” The call for integrative curriculum is clear, and much research supports the advantages for young learners. Developmentally Appropriate Practices (Bredekamp & Copple, 1997) states that children do not need always to distinguish learning by subject area. Curriculum should instead develop children’s knowledge in all content and skill areas because “the brain seeks meaningful connections when presented with new information” (Bredekamp & Copple, 1997).

Since second grade children’s reasoning and thinking reflect a shallow level of prior knowledge, Bredekamp and Copple (1997) suggest the implementation of integrated curriculum studies and long-term projects that “enable children to gain in-depth knowledge and understanding of a topic rather than to cover every topic of study quickly and shallowly.” Additionally, the content of the curriculum should be relevant, engaging, and meaningful to the children (Bredekamp & Copple, 1997). In Matthern and Hansen’s 2007 study on the transformation of the mathematics curriculum at Evansville Elementary school, the teachers implemented the Process Standards (NCTM, 2000), integrating mathematics, science, and language arts. The children were observed as they attempted to make meaning of these connections and as they applied their mathematical skills in various disciplines. Telling time, seeing patterns in reading, and using problem-solving skills in science, social studies, and language arts, these students better understood math in context. These results support NCTM’s stance that a deeper mathematical understanding from emphases on the connections and interrelatedness of mathematical ideas helps children view mathematics as a coherent whole that is useful in other subjects as well as their own interests and experience (NCTM, 2000).

As a way to integrate the mathematical curriculum which considers the developmental needs of second grade young children, the joint position statement of NAEYC and NCTM also calls for time, materials and support for children to engage in play as, “a context in which they explore and manipulate mathematical ideas with keen interest.” Bredekamp and Copple (1997) state that DAP is accomplished in instruction for 6-8 year olds through spontaneous play that continues to be an “important vehicle for development and learning in all areas during the primary years.”

Standards for Mathematics Instruction: NCTM vs. TN State Standards

While providing DAP, it is important to consider curriculum standards to guide mathematics instruction. NCTM (2000) calls for high but attainable curriculum standards that, “produce a society that has both the capability to think and reason mathematically and a useful base of mathematical knowledge and skills needed in any walk of life.” NCTM Standards, much like state standards, are, “descriptions of what mathematics instruction should enable students to know and do,” (NCTM, 2000).

Addressing the process of addition and subtraction in second grade mathematics, one of NCTM’s Five Content Standards, Number and Operations, presents the valued content in components of understanding numbers, developing meanings of operations, and computing fluently (2000). NCTM includes the second grade in the grade band of Pre-K-2 standards; however, Curriculum Focal Points (2000) were developed for recommended emphases for grade 2. In the component of understanding numbers, NCTM Standards and Focal Points for grade 2 state that curriculum and instruction should focus on “[d]eveloping an understanding of the base-ten numeration system and place-value concepts.” Similarly, the Tennessee State Mathematics Curriculum Standards (developed from many resources including NCTM Standards, 2000) for second grade list their content standard 2.1.0 as Numbers and Operations. Learning Expectation 2.1.1 calls for second grade students to, “understand numbers, ways of representing numbers, relationships among numbers, and number systems.” Second grade math teachers can glean that both professional mathematics organizations and state government boards of education value children’s understanding of number and place value; however, within the Tennessee State Standards, the performance indicators of this valued learning are more rigid and narrow than the flexible NCTM standards that allow for teacher creativity in providing appropriate instruction. The Tennessee State Standards declare that children should “identify odd and even numbers to 100” as one way of accomplishing this learning standard, yet NCTM broadly suggests the use of “multiple models to develop initial understanding of place value and the base-ten number system.” NCTM standards allow for the curiosity of children and the integration of curriculum to be considered when meeting the mathematical standards.

Another NCTM (2000) Curriculum Focal Point for grade 2 is “[d]eveloping quick recall of addition facts and related subtraction facts and fluency with multi-digit addition and subtraction.” This focal point is Number and Operations suggested emphasis for second grade. Again, the Tennessee State Mathematics Curriculum Standards closely mirror this standard for second grade through Learning Expectation 2.1.3.: “solve problems, compute fluently, and make reasonable estimates.” This shared learning standard by NCTM and state standards utilizes children’s developing cognitive abilities of mental processes and more efficient quick-recall abilities as the children use a variety of methods and strategies to develop fluency with basic combinations for addition and subtraction (NCTM, 2000; TN State Standards). The Numbers and Operations component of developing meaning of operations and how they relate to one another is not specifically addressed in the NCTM Focal Points for grade 2, but the fluency focal point supports this through an understanding of why efficient procedures of addition and subtraction work on the basis of place value and properties of operations, (NCTM, 2000). Second grade math teachers need to consider and directly address the NCTM Standards, Curriculum Focal Points, and state standards as they plan and implement high-quality instruction for learning and assessment of the learners they teach.

Observation and Assessment Methods

The joint position statement of NAEYC and NCTM (2002) also states the need to, “support children’s learning by thoughtfully and continually assessing all children’s mathematical knowledge, skills, and strategies.” To support children’s learning, assessment must be authentic and offered in different forms to support children’s different learning styles. It must test children’s learning and greater understandings, not their ability to memorize and regurgitate information (Clarke, 1992). Assessment in second grade mathematics classrooms can include informal, formative methods such as annotated class lists that record significant events or achievements in predetermined practical learning and teaching goals in mathematics. NCTM (2000) states that feedback from assessments should provide feedback to the teacher and also the student, which can aid the students in setting goals, in assuming responsibility for their own learning, and in becoming more independent learners. Clarke states that students have the “right to explicit information about their progress,” and this can be accomplished through work folios and students’ self-assessment. Work folios document students’ mathematical reasoning progress through ongoing collections of significant samples that students judge and evaluate as worthy within the teacher-established context of showing reasoning. Student work folios not only provide the teacher with information, but also promote personal responsibility for self-assessment (Lambdin & Walker, 1994; Clarke, 1992). Student self-assessment through reflecting on learning is valuable and can be fostered through asking questions such as, “Show me something that you have learned how to do in mathematics this week?” (Clarke, 1992).

Furthermore, Clarke (1992) argues, “good assessment equals good instruction.” Assessment should be an integral part of instruction, guiding teachers and enhancing learning. Teachers should integrate informal assessment within mathematics lessons for second grade children through good, open-ended, and conceptual questions, authentic tasks in a problem-solving context, and opportunities for students’ reflective self-assessment in discussing solutions. Clarke states that this informal assessment “should anticipate action,” as teachers should use the information they gain from assessing their students to guide and modify teaching plans. In a module of a math classroom’s instructional sequence developed by Cobb (Catylyst Math Module), the teacher incorporates assessment into all aspects of the math lesson by first establishing the norm for discourse that multiple answers and solutions are accepted. She also stated that she was going to, “ask a lot of questions so (she could) make sure (she) understood what (they said).” This presented the students with self-assessment, in which they would need to explain how they understand the problem.” With this norm in place, the teacher could use good, open-ended questions to allow the students to share and explain their different levels of understanding as a form of informal assessment (Clarke, 1992). The teacher then guided the discussion by asking a probing question: “What’s another way to look at it?” Then, she used her formative assessment throughout the lesson to guide even the rest of the current lesson. In this way, assessment was not just a way for her to grade students; it was an integral part of instruction. Second grade math teachers need to remember the needs of children to continually develop thinking, reasoning, social, and practical physical skills as guidance in their choice of assessment techniques to plan and evaluate for learning.

Meeting the Needs of Diverse Learners in Second Grade

Meeting the needs of diverse learners is also recommended by the NAEYC & NCTM (2002) joint position statement through the suggestion to “build on children’s varying experiences, including their family, linguistic, and cultural backgrounds; their individual approaches to learning; and their informal knowledge.” Additionally, one of NCTM’s Principles for School Mathematics (2000) is “Equity” which demands high expectations for all students, as well as “reasonable and appropriate accommodations be made and appropriately challenging content be included to promote access and attainment for all students.” Unfortunately, children who live in poverty and who have linguistically and ethnically diverse backgrounds have significantly lower achievement levels in mathematics (NAEYC & NCTM, 2002). With access to high-quality mathematics instruction, all children, regardless of their diverse characteristics, needs, and backgrounds, can and should learn mathematics (NCTM, 2000).

One significant gap that increasingly affects American children and teachers is the lack of methods for teaching children with linguistically diverse backgrounds. Bredekamp and Copple (1997) state that studies have evidenced strong support for the benefits of bilingualism in cognitive, language, and literacy development. Further research supports bilingual education. Children with limited English proficiency (LEP) who are taught at least partly in their native language perform “significantly better” on standardized tests than children who are taught only in English (Greene, J). LEP students, who enter programs at age 8 at the end of the second grade, have higher math achievement. Research supports this math achievement through findings that advantaged LEP students performed the best in mathematics standardized tests, and students who entered ESL programs from ages 8-11 are the fastest achievers, needing 2-5 years to reach national averages and performed the best on the math tests compared to students who entered ESL programs at younger or older ages (Collier, 1987). To achieve equity and effectiveness, second grade mathematics teachers need to know as much as they can about the background and learning differences of their students and work to provide a bridge for the best opportunities for new learning (NAEYC & NCTM, 2002). Thomas and Collier (1996) found that important program features for the achievement of English Language Learners are: providing cognitively complex academic instruction in the native language for as long as possible in all subjects through problem-solving, discovery learning in highly interactive classroom activities with exposure and integration with English speakers in the socio-cultural context of the school. These practices of problem solving, discovery, and cooperative learning closely mirror high-quality instruction suggested by NCTM’s Process Standards (2000) and Developmentally Appropriate Instruction suggested by Bredekamp and Copple (1997). Schoenfeld (2002) provided further evidence that quality implementation of standards and reforms suggested by NCTM have begun to decrease mathematics achievement gaps between majority and traditionally underrepresented minority students and between low and high SES students. Although additional support is needed to provide for diverse learners, second grade math teachers can provide quality instruction that is knowledgeable about the needs of these children and considers these needs in instructional accommodations and community building.

Implications for Practice and Professional Development

In every early childhood setting, including second grade, children should have challenging, accessible, developmentally appropriate mathematics instruction that is grounded in effective, research-based curriculum and teaching practices (NAEYC & NCTM, 2002). With aspirations to teach children in the upper early childhood/lower elementary range, I will strive to understand and implement the best practices in teaching learners in the unique age range of 6-8-year-olds. I will dedicate myself to upholding the needs of active, constructivist learning that makes connections to previous knowledge and new experiences using both physical and cognitive development. Teacher instruction will serve as guidance for learning environments that include problem solving tasks and cooperative work. Keeping in mind the importance of integrating knowledge to form meaningful connections, I will carefully plan and implement integrated projects and units of study that teach children the practical uses of mathematics as a coherent body of concepts and skills. Multiple forms of informal as well as formal formative assessments will be aligned to NCTM’s Content and Process Standards as well as state standards to inform instruction and gage progress, strengths, and needs. Moreover, I will deliberately incorporate validated strategies and accommodations for diverse learners while maintaining high expectations and a caring classroom community. In addition, I will continue to familiarize myself with current research from professional organizations such as NAEYC and NCTM. I will strive to further their public work as they promote high-quality practice through, “policies, organizational supports, professional preparation programs, and adequate resources,” that support teachers in the challenging and important work of mathematics education (NAEYC & NCTM, 2002).

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