Kindergarten teachers' orientations to teacher-centered and student-centered pedagogies and their influence on their students' understanding of addition.

Polly, Drew ; Margerison, Ashley ; Piel, John A. 等

This study examined the influence of kindergarten teachers' orientations toward student-centered teaching and their influence on their students' understanding of addition. The study examined 120 students across 10 classrooms. Based on an interview and two classroom observations, 10 teachers were classified as either student centered or mainly teacher centered. Twelve students in each classroom--four above grade level, four at grade level, and four below grade level--were given a task-based interview focused on their understanding of addition. Inductive analysis suggests that students in both types of classrooms could complete addition problems correctly, but students in student-centered classrooms scored better on tasks that involved writing story problems and tasks that involved missing addends. Implications for future research are also discussed.

Keywords: mathematics, mathematics education, kindergarten, mathematical ability

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TEACHER BELIEFS AND VIEWS OF MATHEMATICS TEACHING

Teacher beliefs significantly influence teachers' mathematics practices (Philipp, 2007; Thompson, 1992; Turner, Warzon, & Christensen, 2011). Teachers' belief system is their personal views about effective mathematics learning and teaching (cf. Pajares, 1992; Thompson, 1992). Teachers' beliefs may change over time as new ideas replace old ones, or when teachers attempt to enact new pedagogies in their classroom (Fennema, Carpenter, & Franke, 1996). In some cases, teachers may have conflicting beliefs, keeping certain aspects of one theory of teaching while also integrating other ideas of effective instruction (Philipp, 2007). In recent years, studies and research syntheses have found that teacher's beliefs influence the way they use mathematics curricula (Remillard, 2005) and use materials and resources in their classroom (Stein & Kim, 2009). Although there have been some empirical linkages between teachers' beliefs and their instructional practices (e.g., Philipp, 2007; Thompson, 1992), more research is needed to examine the interaction between teachers' beliefs, their instruction, and the influence on their students' understanding of mathematics.

Student-Centered Approaches to Teaching Mathematics

According to Cobb (1994), constructivists argue "that students actively construct their mathematical ways of knowing" (p. 14) through experiences. The students reorganize thoughts of the world and learn the process of mathematics. Teachers who consider themselves student centered typically have classrooms that look very different from teachers who teach from a more behaviorist, teacher-centered approach. Student-centered teachers put children in situations that help them develop new ways of thinking (Lesh, Doerr, Carmona, & Hjalmarson, 2003). Rather than giving students a path to learning, the teacher gives students opportunities to invent their own understanding. Knowledge is actively learned by the student, not passively received from a teacher (Lesh et al., 2003).

Student-centered teachers allow students to build their own individual knowledge (Brooks & Brooks, 1999). Critical thinking skills and knowledge are developed through active experiences. The teacher does not give knowledge to the student; rather, the students themselves invent it. The teacher acts as facilitator, asking the children to explain how and why through questions (Fuller, 2001). Further, student-centered teachers use problem solving as a teaching practice as opposed to a topic to be taught. For example, many nonconstructivist teachers feel that the most important role of their mathematics teaching is teaching basic computational facts (Capraro, 2001). Problem-solving skills are taught to students as a mathematics objective after computation has been mastered through traditional teaching styles. More solid understanding of mathematics is learned by using problem-solving strategies to solve computation problems (Capraro, 2001). "Individuals are more likely to retain knowledge that they create or learn through active problem solving" (Ziegler & Yan, 2001, p. 4).

Learners in a student-centered mathematics classroom use manipulatives, work in small groups, ask questions, and explore materials. Teacher-centered classrooms, especially in the primary grades, also use manipulatives, but manipulative use is typically overly teacher directed, with students expected to mimic the teachers' use of hands-on materials. Students who use manipulatives develop concrete knowledge about mathematics concepts. In student-centered classrooms, children find the answers themselves using strategies and by using manipulatives in ways that they find appropriate. The student-centered approach encourages students to be actively engaged with materials and to ask questions (Wilson, Abbott, Joireman, & Stroh, 2002). No matter what age or grade level, children come to school with prior knowledge and beliefs about the world. The information that is presented to children will either change this previous knowledge or not. Student-centered mathematics teachers allow children to construct this knowledge and rebuild schemas as it relates to them (Brooks & Brooks, 1999). The learner must make connections to prior knowledge and prior learning (Ziegler & Yan, 2001).

Research has shown that student-centered approaches in mathematics teaching have potential to yield positive results on student achievement. Wilson et al. (2002) concluded from their study that student-centered teaching strategies appear, especially in math, to have a meaningful influence on student achievement. Wenglinsky's (1999) meta-analysis of data from the National Assessment for Educational Progress indicated that students in student-centered mathematics classrooms (i.e., where they use manipulatives, solve real-world problems, and work on higher-order thinking skills) significantly outperformed their peers. Polly (2008) found that elementary school students who used calculators in student-centered environments that focused on solving real-world problems also outperformed their peers. Ziegler and Yah (2001) found that student-centered approaches to teaching had a statistically significant influence on student achievement.

Although research on student-centered mathematics instruction shows a high level of understanding of the process of learning, some concerns were reported by teachers:

* Children were very high energy (Chung, 2004)

* Instructors lack direct control when students use manipulatives

* Classroom management is difficult to enforce

* Student-centered lessons take up more instructional time when time is limited

* Student-centered activities are too permissive

* Students' understanding did not directly translate to traditional assessments (Brooks & Brooks, 1999).

Teacher-Centered Approaches to Teaching Mathematics

Teachers who believe in teacher-centered pedagogies teach mathematics very differently than constructivist-oriented teachers. Teacher-centered educators believe that the acquisition of information is more important than the process. The purpose of school is to learn this essential information (Nowell, 1992). Chandler (1999b) outlines teacher-centered education with a set of descriptors, including subject-centered, structure, order, work, discipline, memorization, order, and accountability. These identifiers describe a very clear-cut and direct educational philosophy. These teachers believe in teaching the content to students in the most time-efficient, straightforward manner possible. This straightforward manner of teacher-centered instruction allows more knowledge to be presented in a shorter time than constructivist teaching practices.

Teacher-centered educators do not typically spend instructional time by focusing on a deep understanding of concepts and processes that are more time consuming. The time required to achieve results is the amount of time that should be spent teaching (Ackerman, 2003). Teaching strategies that take more time take away from other subject areas are not considered worthwhile unless results can be seen. Pedagogies are only effective if there is a direct correlation to student learning outcomes (Ackerman, 2003).

As mentioned earlier, teacher-centered teachers believe in the importance of teaching basic math facts prior to teaching the process of problem solving (Capraro, 2001). According to Ackerman (2003), "Thinking skills and learning how to learn may have their place in the overall curriculum, but they are hollow vehicles if they are not harnessed to the good stuff' (p. 344). Teacher-centered teachers believe certain aspects of the curriculum need to be learned first for the process to hold importance.

Educators in a teacher-centered classroom are considered the source and dispenser of knowledge. They teach the children the information they need to know and assess the children on this knowledge. Teacher-centered classrooms are centered on the teacher (Chandler, 1999ab). Students know who is in charge of learning, which leads to more structure than a constructivist classroom. The students in a teacher-centered classroom are held to very high standards. The expectations are clearly set. The disciplines are kept separate to keep their individual importance (Ackerman, 2003). The teachers make sure that their children gain essential knowledge in all of the core disciplines. Time is not spent on exploration and creativity, but instead on essential core knowledge (Chandler, 1999b).

Comparing Student-Centered and Teacher-Centered Approaches to Teaching Mathematics

Teachers holding teacher-centered beliefs consider the acquisition of knowledge to be the most important part of learning (Nowell, 1992), whereas student-centered teachers consider the learning process more important (Ziegler &Yan, 2001). This essential difference in the beliefs of teachers influences substantial variation in teachers' mathematics teaching practices. Studies that have examined the influences of teacher-centered and student-centered instruction on student learning have yielded mixed results. Chung (2004) researched 3rd-grade students and found no significant difference in the mean scores of students taught by either teacher (Chung, 2004). Smith and Smith (2006) found that the use of student-centered pedagogies in a standards-based mathematics curriculum led to deeper mathematical understanding in 3rd-grade students than their 4th-grade schoolmates in a teacher-centered classroom on a multiplication assessment. Further, analyses of data from large-scale national and international assessments indicate that student-centered pedagogies for teaching mathematics lead to greater gains in student learning than teacher-centered approaches (Polly, 2008; National Center for Educational Statistics, 2004; Wenglinsky, 1999). Still, further examination is needed to more closely examine how teachers' beliefs and their instructional practices influence students' mathematical understanding.

RESEARCH STUDY

Research Question and Overview

This study is grounded in the research question: What influence do kindergarten teachers' orientations to mathematics instruction have on their students' understanding of addition? Kindergarten was chosen as the target grade level to examine students' mathematical understanding early in their schooling. Students have only had one teacher, or two if they participated in a prekindergarten program. Addition was chosen as the topic of study because this is the first concept in which kindergarten teachers tend to deviate in their instruction. Other topics, such as counting and sorting, tend to be taught the same regardless of teachers' beliefs and orientations. Further, the concept of joining sets or adding is heavily emphasized in the state's kindergarten standards, and a focus in the new Common Core State Standards in Mathematics (CCSSM) (CCSSO, 2011). Based on these reasons, there is reason to expect that children's understanding of addition can be largely attributed to their kindergarten experiences, including their teachers' beliefs and instructional practices. This study occurred between March and May, after students had been in kindergarten for 6 to 8 months.

Participants

This study examined 120 students, 12 from each of 10 different kindergarten classes in suburban elementary schools (Grades K-5) in the southeastern United States. Classes and students were purposefully selected (Patton, 2002) to capture a range of teachers that represented student-centered and teacher-centered orientations to mathematics teaching. Within each class, 12 students were selected, four that the teacher identified as below grade level, four identified as on grade level, and four identified as above grade level.

Selection of teachers. Teachers in this study were selected from three suburban schools in the southeastern United States. Teachers were observed and rated using the Mathematics Teaching Scale, which the second author developed (Figure 1). Teachers also participated in a brief interview (Figure 2) to determine whether they espoused teacher-centered or student-centered pedagogies. Based on the observation and survey, teachers were categorized as either student centered or teacher centered.

Selection of students. Four students were selected from each of the 12 classrooms. Student selection was purposeful to examine two students who scored high on the curriculum-based assessment on addition and two students who scored low on the assessment.

Interviews

Teacher interviews. Each teacher participated in a semistructured interview that lasted approximately 20 minutes (Figure 2). The interview collected information about teachers' educational beliefs and their philosophy of learning through mathematics and verified their teaching beliefs that were observed in their classroom. In the teacher interviews, both teachers' comments echoed observations of their teaching, in regard to their orientation toward student-centered teaching.

Student interviews. Student interviews were conducted to examine their mathematical understanding of addition. High-performing and low-performing students were chosen to identify any difference in teachers' orientations to teaching on both types of students. The semistructured interview (Figure 3) lasted approximately 10 minutes. Each interview was audiotaped and transcribed verbatim.

Data Analysis

Analytic framework. Data from this study were analyzed using an inductive, thematic approach (Coffey & Atkinson, 1996). The purpose of the data analysis was to examine students' performance on the tasks and then examine any associations between teachers' orientations toward teaching mathematics and students' performance.

Process of analysis. First, the interviews were audiotaped and transcribed. After the transcription, the interviews were read through twice for recurring events and dialogue. The interview data were coded to show what the interviewee demonstrated or stated in the interview. The codes were based on what the children expressed with their responses and if they were successful. The codes included modeling, explanation of addition, computation, recognizing terminology, and contextualizing math to real life. These codes were followed by subcodes that further explained them, including if they were correct, if the child spoke the answer or used manipulatives, and what kind of problem was answered.

The first two interview transcripts were rated by the first and second authors. Our inter-rater reliability was 95% (19 of 20 codes matched). After that, the first two authors coded the rest of the interviews individually and entered the data in an Excel spreadsheet. FIGURE 2 Teacher interview protocol. What does a normal math lesson look like in your room? What words or materials might you use? What do you do first when teaching a child addition? What is the most important thing to teach them? What concepts and skills about addition are important for students to learn? How do you know when a child has mastered addition? What skills and knowledge should students have? What is the most ideal way for children to learn math? If you could teach math any way that you wanted, what would you do? Would you change the program we use? If your students know a math concept, can they explain it to you or a peer? Can they tell you how or why it is so? Do you like to teach math? Do your students enjoy math? Anything else you want to tell me about addition or math?

These initial codes were used to find commonalities in the interview data. For clarification, the initial student codes were grouped into categories based on themes that were frequently found in the data set. These themes included addition process, calculation, math enjoyment, context, and recognizing terminology. These themes were chosen based on commonalities in the data. Within each category, student interview data were analyzed according to their teacher and responses. The classification system was used to compare the data and make an understandable system of reading through the text. FIGURE 3 Student interview protocol. Task 1: I know that 3 + 1 = 4. Could you take these bear counters and show me that 3 + 1 = 4? Task 2: There are 5 dogs in the park. Three more dogs show up. How many dogs are there now? Use your counters to show your answer. Task 3: There are three cookies on the counter. Your mom adds some more cookies. If there are now seven cookies, how many did your mom add? Task 4: There are some basketballs in the bucket. The coach adds 4 more basketballs. If there are now 9 basketballs, how many were in the bucket at first? Task 5: Can you think of a story problem where you would add groups together? Task 6: Create a story for the addition problem 3 + 2 = 5.

After the codes and patterns are identified, these key words and phrases were used to code the participants' responses to the interview questions and compare them. The first author read through the interview data another time to label the text with the applicable code. These codes were considered the core content of the interviews--what is most significant to the study. Many parts of the interview were labeled with more than one code. The themes created through the coding displayed the data in a manageable form. The themes were analyzed for commonalities as well as differences according to student and teacher. These trends were looked at for importance and relevance to the study's research question. Themes that were found uninteresting or irrelevant were not included. The findings include all relevant data that complements as well as contradicts the studies intention.

FINDINGS

This section describes the performance of the 120 kindergarten students during the task analysis. Findings are presented by the individual tasks that students completed during the interviews.

Task 1: Demonstrating Addition With Concrete Manipulatives With All Three Numbers Known

The first task was "I know that 3 + 1 = 4. Could you take these bear counters and show me that 3 + 1 = 4?" Table 1 describes the types and frequencies of students' responses. Two students could not start the task, even when prompted with the question, "How many counters should we start with?"

This question was answered correctly by 83 (69.17%) of the students. All of the students, except for one, who were identified as above or on grade level completed the task correctly. One high-performing student in a teacher-centered classroom incorrectly answered the question. On this task, 21 students (17.5%) demonstrated the same lack of understanding, where they grabbed 4 bears, but could not show how the bears matched the task 3 + 1 = 4.

The 14 students (11.67%) who grabbed 8 counters and could not model the problem grabbed cubes as they read the problem. Many of them read the problem orally, grabbing a handful of 3, a handful of 1, and then a handful of 4 cubes. When asked, "How do the counters match the problem '3 + 1 = 4'," students responded by pointing to the piles and naming the number of cubes in each pile. When asked, "What does the symbol between the 3 and the 1 mean?" students responded with either silence or by saying, "I don't know."

On Task 1, there were 83 correct answers. Out of the correct answers, 43 (51.81%) were from students in student-centered classrooms, whereas 40 (48.19%) were from students in teacher-centered classrooms.

Task 2: Solving a Result Unknown Task With Concrete Manipulatives

Students were presented with the problem, "There are 5 dogs in the park. Three more dogs show up. How many dogs are there now? Use your counters to show your answer." Table 2 describes the types and frequencies of students' responses. All of the students were able to start this task.

There were 107 (89.17%) correct responses on this task, 24 more correct responses than in Task 1, in which all three numbers in the addition number sentence were known. Twelve students (10%) made correct piles of 5 and 3 counters but miscounted and said that there were either 7 or 9 counters, rather than the correct number of 8. One student correctly made piles of counters but could not join the piles to get an answer. When prompted with, "How would I find out how many dogs there are now?" he responded by saying, "I don't know."

Out of the 107 student responses, 54 (50.47%) were from students in student-centered classrooms, whereas 53 (49.53%) were from students in teacher-centered classrooms.

Task 3: Solving a Change Unknown Task With Concrete Manipulatives

Task 3 was, "There are three cookies on the counter. Your mom adds some more cookies. If there are now seven cookies, how many did your mom add?" Table 3 describes the types and frequencies of students' responses. Two of the students were not able to start this task.

On this task, 77 (64.17%) students answered correctly. Of those, 63 (52.5%) students solved the task correctly by starting with a pile of 3 and then adding cubes and counting on until they reached 7. They then counted the 4 cubes that they added and reported that as their answer. Meanwhile, 14 (11.67%) students worked backward by making a pile of 7 seven cubes and then separating 3 cubes from that pile so that there was a pile of 3 and a pile of 4 cubes.

Out of the incorrect answers, 11 students (9.17%) miscounted cubes and reported an answer of either 3 or 5. The most common error was made by 29 (24.17%) students; they read this problem as 3 + 7, made piles of cubes, and reported an incorrect answer of 10.

When an on-grade level student from a teacher-centered classroom was asked about his work, he responded, "I know that there are 3 cookies to start, and there also 7 more cookies. When I put them together I get 10." Seeing an opportunity to help the student, the researcher started the following discussion:

Researcher (first author): Can we go through the problem one step at a time?

S: (Putting 3 counters out). These are the 3 cookies. Now we add 7 more.

R: Let me read the story to you again. (Rereads the task).

S: Okay. Maybe we don't add 7. Maybe we end with 7 cookies.

R: Why do you think that?

S: You said that Mary added some more. We don't know how many some more is.

R: So what do you need to do?

S: I want to start with 3 and then keep adding until I get to 7.

S: (Counts on from 3 and adds cubes). I needed 4 more cubes. Mary added 4 cookies.

After similar types of conversations, nearly every student discovered his or her misconception that he or she needed to add 3 and 7, rather than count on from 3 to get to 7.

Out of the 77 correct responses, 55 (71.43%) of them were from students in student-centered classrooms, while 22 (28.57%) were from students in teacher-centered classrooms. The most frequent misconception of joining 3 and 7 to get 10 was made by 29 students; 28 (96.56%) of them were from teacher-centered classrooms.

Task 4: Solving a Start Unknown Task With Concrete Manipulatives

Task 4 was, "There are some basketballs in the bucket. The coach adds 4 more basketballs. If there are now 9 basketballs, how many were in the bucket at first?" Table 4 describes the types and frequencies of students' responses. Eight students were unable to start this task.

On task 4, 65 (54.17%) students answered correctly. There were 11 (9.17%) correct responses from students who counted on from 4 and added cubes until they got to 9. Meanwhile, 54 (45%) students worked backward and split a pile of 9 cubes into a pile of 4 and a pile of 5. More students worked backward on this task (54 students) as compared to Task 3 (14 students). Further, nearly all of the on-grade level students (90%) and all of the below-grade level students (100%) from student-centered classrooms who answered correctly had worked backward.

While completing the task, a below-grade level student from a student-centered classroom shared his thinking.

Researcher (first author): Why did you solve the problem that way?

Student: Since I knew that the 9 was after the equals sign, I had to have a total of 9 and that the two numbers on the other side [of the equals sign] were parts that made up 9.

R: So what did you do with the cubes?

S: I knew that part of the 9 was 4 so I took 4 cubes from the pile [of nine] and I had 5 left. So the answer is 5.

This idea of splitting the total (9) into two parts or piles was a strategy that students came up with to solve this task, where students knew the result and the amount of change but did not know the start number.

Out of the 65 students who responded correctly, 57 (87.69%) were from student-centered classrooms, whereas 8 (12.31%) were from teacher-centered classrooms. The most common error, combining 4 and 9 to get 13, was made by 40 students; 38 (95%) were from teacher-centered classrooms, while 2 (5%) were from student-centered classrooms.

Task 5: Creating an Addition Story Problem

The students were asked, "Can you think of a story problem where you would add groups together?" Table 5 describes students' responses. Only 53 (44.17%) of the students provided correct responses that included a correct context for addition and a correct answer. Forty-five students (37.5% of all students) gave the correct answer but provided an incorrect context for addition. These students told a story that had numbers in it, but there was no action related to addition or joining sets. One above-grade level student from a teacher-centered classroom responded, "There are four cats in the room and there are cats dogs in the room. The end." When asked, "What question could you ask about addition?" he said, "I don't know."

When asked to tell a story with addition in it, an on-grade level student from a student-centered class responded, "Well, these two bears walk home. Then two more bears come home. Now there are four bears at home. The end." He provided an example of an addition problem contextualized into a real-life example, as did the other students from student-centered classrooms. Students generated appropriate contexts in which addition skills could be applied.

Out of the 53 correct responses, 41 (77.36%) were from student-centered classrooms. There were 45 incorrect answers that had correct numbers but an incorrect context; 29 (64.44%) were from teacher-centered classrooms.

Task 6: Tell a Story for 3 + 2 = 5

Students were asked to create a story for the addition problem 3 + 2 = 5. Table 6 describes students' responses. Out of the 120 responses, 82 (68.33%) provided an appropriate story. Students used a variety of contexts, including cookies, children, and pencils to describe a situation in which a group of 3 objects was joined with a group of 2 objects to make a group of 5 objects. As with Task 5, students who were incorrect named numbers but could not describe a story that represented joining or addition. In fact, the two above-grade level students from teacher-centered classrooms provided a subtraction situation. One example was, "There are 3 boys in the room and 2 girls in the room. How many more boys are there than girls?"

Total Scores

On the entire assessment, the mean score was 3.89 out of a possible 6 points. The mean for a student in a student-centered classroom was 5.07, whereas the mean for a student in a teacher-centered classroom was 2.72. Class means ranged from 2.25 to 5.25 points. Teacher-centered classrooms ranged from 2.25 to 3.25, whereas student-centered classrooms ranged from 4.92 to 5.25.

Comparing Student-Centered and Teacher-Centered Classrooms

A generalized linear model was run in SPSS 17.0 to compare students' scores from student-centered and teacher-centered classrooms (Table 7). Students from student-centered classrooms performed statistically significantly better than their peers from teacher-centered classrooms on Tasks 3, 4, 5, 6, and on the overall task interview, each with a p value less than 0.001. Based on Cohen's (1988) description of effect sizes, there was a medium to large effect size on Task 4, medium effect sizes on Task 3 and the total score, and small effect sizes on Tasks 5 and 6.

DISCUSSION

Significant differences were found between students from student-centered and teacher-centered classrooms on Tasks 3, 4, 5, 6, and the total score. These differences and other findings from this study warrant further discussion.

Total Score

Each of the six tasks focused on kindergarten students' conceptual understanding of addition. The findings, which empirically link students from student-centered classrooms to higher scores than their peers in teacher-centered classrooms, confirm prior studies that show students with opportunities to explore and learn through hands-on materials and mathematically rich tasks demonstrate a deeper mathematical understanding than their peers (Wenglinsky, 1999; Wilson et al., 2002). Cobb (1994) stated that students who are taught with a student-centered approach reorganize their thoughts about the world as they learn the processes of mathematics. As seen in the findings, student-centered teaching allows children to construct this knowledge and rebuild schemas as they relate to them (Brooks & Brooks, 1999).

Results From Individual Tasks

There were no significant differences between students from student-centered and teacher-centered classrooms on Tasks 1 and 2. Task 1 required students to use manipulatives to show how 3 + 1 = 4. Most students above grade level and those on grade level provided correct responses, whereas below-grade-level students from both types of classrooms struggled with this task. Task 2 required students to solve a story problem for 5 + 3. On Task 2, 107 (89.17%) of the 120 students answered it correctly. Based on the data, Task 2, solving a problem in which the result was unknown, was easier than Task 1, in which students had to model 3 + 1 = 4. Only 83 (69.17%) students answered Task 1 correctly. Similar findings existed between Task 5, in which students had to determine numbers and a context for addition, compared to Task 6, in which students were given a complete equation and had to create a story context. These findings speak to the level of cognitive difficulty of tasks that involve modeling and explanation (Polly, 2008; Cobb, 1994)

Tasks 2, 3, and 4 represent the three types of problem structures that emerged from the Cognitively Guided Instruction (CGI) research project (Carpenter, Fennema, & Franke, 1996). The CCSSM explicitly refer to these different types of problems in Grades 1 through 5, and provide specific examples in Tables 1 and 2 from the Glossary. Data from this study indicated that students from both types of classrooms experienced more difficulty solving Task 4 (start unknown) and Task 3 (change unknown), compared to the most frequently seen type of problem, Task 1 (result unknown). Building on the work from the CGI project, students need to be exposed to each of the various types with ample experiences with the type of problem that is challenging yet still able to be completed successfully (Carpenter et al., 1996).

IMPLICATIONS FOR FUTURE RESEARCH

This study has multiple implications for future research. The findings of this study confirm previous findings that support student-centered instruction and the impact on students' mathematical understanding. However, the study has a limited sample and should be expanded to include more students, and more grade levels. As the field continues to debate the value of student-centered approaches to teaching mathematics, especially in light of the new CCSSM, research studies must examine many more students across grade levels.

Although this study examined students in two types classrooms, one student-centered and one teacher-centered, teachers' beliefs and instructional practices are more of a continuum than a dichotomous relationship. Hence, future work should examine more teachers on various points of the continuum from student- to teacher-centered. This study provides data about the two extremes, but most teachers likely fit somewhere in the middle.

Last, future studies should examine how teachers of varying beliefs interpret and enact the same curricular materials (Stein & Kim, 2009). The examination of teachers' beliefs, curriculum, and their instruction is needed to better explain the complex relationship of the teaching process. As Fennema et al. (1996) stated, "Teachers' beliefs and instruction exist in a complex relationship" (p. 429).

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DOI: 10.1080/02568543.2013.822949

Submitted April 24, 2011; accepted November 27,2011.

Address correspondence to Drew Polly, Ph.D., Associate Professor, Department of Reading and Elementary Education, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28213. E-mail: drew.polly@uncc.edu

Drew Polly

University of North Carolina at Charlotte, Charlotte, North Carolina

Ashley Margerison

Austin Independent School District, Austin, Texas

John A. Piel

University of North Carolina at Charlotte, Charlotte, North Carolina TABLE 1 Student Responses to Demonstrating Addition With All Three Numbers Known Student centered Above On Below grade grade grade Correct answer: 20 20 3 Makes a pile of 3, a pile (100%) (100%) (15%) of 1, and combines the pile to make 4 Incorrect answer: 0 0 14 Grabbed 4 counters (70%) Could not model Incorrect answer: 0 0 3 Grabbed 8 counters. A (15%) pile of 3, a pile of 1, a pile of 4. Cannot demonstrate joining Incorrect answer: 0 0 0 Could not start the task Teacher centered Above On Below grade grade grade Total Correct answer: 19 20 1 83 Makes a pile of 3, a pile (95%) (100%) (5%) (69.17%) of 1, and combines the pile to make 4 Incorrect answer: 1 0 6 21 Grabbed 4 counters (5%) (30%) (17.5%) Could not model Incorrect answer: 0 0 11 14 Grabbed 8 counters. A (55%) (11.67%) pile of 3, a pile of 1, a pile of 4. Cannot demonstrate joining Incorrect answer: 0 0 2 2 Could not start the task (10%) (1.67%) Note. Percentages were calculated for each subgroup (column) of students. TABLE 2 Student Responses to Result Unknown Task Student centered Above On Below grade grade grade Correct answer: 20 20 14 Makes a pile of 5, (100%) (100%) (70%) adding 3, and then counting all Incorrect answer: 0 0 6 Makes a pile of 5, (30%) adds 3, miscounts Incorrect answer: 0 0 0 Makes a pile of 5, makes a pile of 3, could not join the piles Teacher centered Above On Below grade grade grade Total Correct answer: 20 20 13 107 Makes a pile of 5, (100%) (100%) (65%) (89.17%) adding 3, and then counting all Incorrect answer: 0 0 6 12 Makes a pile of 5, (30%) (10%) adds 3, miscounts Incorrect answer: 0 0 1 1 Makes a pile of 5, (5%) (0.83%) makes a pile of 3, could not join the piles Note. Percentages were calculated for each subgroup (column) of students TABLE 3 Student Responses to Change Unknown Task Student centered Above On Below grade grade grade Correct answer: 12 18 14 Made a pile of 3, (60%) (90%) (70%) added cubes until there were 7 total Correct answer: 8 2 1 Started with 7, moved (40%) (10%) (5% cubes to make piles of 3 and 4 Incorrect answer: 0 0 4 Modeled the problem (20%) correctly but miscounted Incorrect answer: 0 0 1 Joined a pile of 3 and (5% a pile of 7 to get an answer of 10 Incorrect answer: 0 0 0 Could not start the task Teacher centered Above On Below grade grade grade Total Correct answer: 15 4 0 63 Made a pile of 3, (75%) (20% (52.5%) added cubes until there were 7 total Correct answer: 3 0 0 14 Started with 7, moved (15%) (11.67%) cubes to make piles of 3 and 4 Incorrect answer: 0 3 4 11 Modeled the problem (15%) (20%) (9.17%) correctly but miscounted Incorrect answer: 2 13 13 29 Joined a pile of 3 and (10%) (65%) (65%) (24.17%) a pile of 7 to get an answer of 10 Incorrect answer: 0 0 3 3 Could not start the task (15%) (2.5%) Note. Percentages were calculated for each subgroup (column) of students. TABLE 4 Student Responses to Change Unknown Task Student centered Above On Below grade grade grade Correct answer: 6 2 0 Made a pile of 4, (30%) (10%) added cubes until there were 9 total Correct answer: 14 18 17 Started with 9, (70%) (90%) (85%) moved cubes to make piles of 4 and 5 Incorrect answer: 0 0 1 Modeled the problem (5%) correctly but miscounted Incorrect answer: 0 0 2 Joined a pile of 4 and (10%) a pile of 9 to get 13 Incorrect answer: 0 0 0 Could not start the task Teacher centered Above On Below grade grade grade Total Correct answer: 2 1 0 11 Made a pile of 4, (10%) (5%) (9.17%) added cubes until there were 9 total Correct answer: 5 0 0 54 Started with 9, (25%) (45%) moved cubes to make piles of 4 and 5 Incorrect answer: 0 2 4 7 Modeled the problem (10%) (20%) (5.83%) correctly but miscounted Incorrect answer: 13 17 8 40 Joined a pile of 4 and (65%) (85%) (40%) (33.33%) a pile of 9 to get 13 Incorrect answer: 0 0 8 8 Could not start the task (40%) (6.67%) Note. Percentages were calculated for each subgroup (column) of students. TABLE 5 Student Responses to Telling an Addition Story Problem Task Student centered Above On Below grade grade grade Correct answer: 19 12 10 Correct context (95%) (60%) (50%) for addition Correct answer Incorrect answer: 1 8 7 Incorrect context (5%) (40%) (35%) for addition Correct answer Incorrect answer: 0 0 0 Correct context for addition Incorrect answer Incorrect answer: 0 0 3 Incorrect context (15%) Incorrect answer Teacher centered Above On Below grade grade grade Total Correct answer: 12 0 0 53 Correct context (60%) (44.17%) for addition Correct answer Incorrect answer: 8 17 4 45 Incorrect context (40%) (85%) (20%) (37.5%) for addition Correct answer Incorrect answer: 0 1 0 1 Correct context (5%) (0.83%) for addition Incorrect answer Incorrect answer: 0 2 16 21 Incorrect context (10%) (80%) (17.5%) Incorrect answer Note. Percentages were calculated for each subgroup (column) of students. TABLE 6 Students' Responses for "Tell a Story for 3 + 2 = 5" Student centered Above On Below grade grade grade Correct answer: 20 18 16 Correct context (100%) (90%) (80%) for addition Incorrect answer: 0 2 4 Incorrect context (10%) (20%) for addition Teacher centered Above On Below grade grade grade Total Correct answer: 18 6 4 82 Correct context (90%) (30%) (20%) (68.33%) for addition Incorrect answer: 2 14 16 38 Incorrect context (10%) (70%) (80%) (31.67%) for addition Note. Percentages were calculated for each subgroup (column) of students. TABLE 7 Results of One-Way ANOVA Partial eta squared Task F p value ([[eta].sub.2]) Task 1 0.347 0.557 0.003 Task 2 0.085 0.771 0.001 Task 3 57.832 <.001 * 0.329 Task 4 241.327 <.001 * 0.672 Task 5 36.619 <.001 * 0.237 Task 6 32.692 <.001 * 0.217 Total 58.198 <.001 * 0.330