Does your elementary mathematics methodology class correspond to constructivist epistemology?

This study examines whether the instruction of an elementary mathematics education methodology course corresponds to constructivist learning. The participants were pre-service teachers in their senior year at a college in the southern part of the U.S. They were 49 students (3 men, 46 women) enrolled in three sections of a teacher certification course, one in the K-4 program and two in the K-8 program. During the first half of the semester, the instructor taught all students via direct instruction and during the second half by using a strategy involving hands-on activities, group work, and empowerment of students. The researcher administered the Constructivist Learning Environment Survey (CLES) at the beginning and at the end of the treatment. The investigator of this research will use the CLES to examine whether or not the instructor maintains a classroom that promotes a constructivist learning environment. Results showed that the classroom learning reflected a constructivist environment in four of six sections of the survey.

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The U.S. Department of Education (2005) reported that of 12 nations (Hungary, Austria, Latvia, Japan, Hong Kong, Italy, Russia, Netherlands, New Zealand, Belgium, and Norway) that participated in a mathematics assessment, U.S. fourth graders ranked 8th and U.S. eighth graders ranked 9th, indicating below average scores. The report also stated that when teaching mathematics, American teachers tend to focus on procedural computation, rather than conceptual understanding of a lesson.

The National Council of Teachers of Mathematics (1989) has developed standards for teachers of mathematics that promote deep understanding of mathematical concepts, while advocating for student-centered instruction. Several of these standards include using concrete materials, modeling, engaging students in the classroom, encouraging communication in the classroom, involving students in discussion, initiating problem solving, moving toward understanding instead of memorization, respecting students' ideas, and being sensitive to students' experience and culture.

Increasingly, teacher education programs in colleges and school districts are adopting a constructivist theory of learning even though it is difficult to change teachers' beliefs (Richardson, 1997; Richardson, 1996). Van De Walle (2004) has stated that providing a constructivist learning environment, which is an effective method of teaching, will help improve student understanding of mathematical concepts. Van De Walle states:

   The most widely accepted theory known
   as constructivism, suggests that children
   must be active participants in the development
   of their own understanding.
   Constructivism provides us with insights
   concerning how children learn mathematics
   and guides us to use instructional
   strategies that begin with children rather
   with ourselves. (p. 22)

Troutman and Lichtenberg (2003) have found that students do not learn concepts by memorizing isolated facts, by watching the teacher explaining a lesson, by reading definitions, or by using abstract symbols. These researchers state that students learn concepts by actively participating in learning activities. Also, the importance of mathematics knowledge for teachers has been emphasized by the National Council of Teachers of Mathematics (1989, 1995, 2000). NCTM promotes standards that correspond to a constructivist approach to teaching. Moreover, NCTM (1991) emphasizes the teacher's role in creating a classroom in which students can learn to think mathematically by exploring, questioning, and justifying mathematical problems. Instructors as role models could influence students because research has shown that most people will teach the way they were taught (Lortie, 1975).

Piaget and Vygotsky are two important contributors in the development of constructivist learning (Henson, 2003; Olivia, 2005). Piaget's constructivism focuses on an equilibrium theory that states that when a student encounters contradictory information, the learner tries to reach equilibrium (Brooks & Brooks, 1993). According to Edwards (2005), Vygotsky's theory of socio-cultural learning is based on children being active participants within their learning environment. Jaramillo (1996) also believes social experiences determine how people think and learn concepts through self-discovery,

and social interactions help activate learners' higher cognitive levels (Zuckerman, 2004). Borthick, Jones and Wakai (2003) have found that in Vygotsky's theory of socio-cultural theory, through social interaction, students will be able to construct their own knowledge and gradually develop independence in learning a new concept through social interaction. Schmittau (2004) also credits Vygotsky's constructivist theory in helping teachers in instructing mathematics lessons that emphasize conceptual understanding rather than procedural understanding of the concepts.

Components of Constructivist Education

There are several components of constructivism. Kamii (1985) indicates in constructivist learning, students construct their own knowledge. Von Glaserfeld (1992) states that in the constructivism, learners will receive knowledge actively and Cobb, Wood, and Yackel (1991) mention that the constructivist learning environment must be such that all students can freely question and express their opinions. Mvududu (2005) and Geary (1995) found that, in a constructivist environment, students are builders of their own knowledge. In addition, Draper (2002) finds that elementary students' experiences and language learning environments are significant components of constructivist learning. This researcher believes that providing feedback, questioning and guiding are essential to a constructivist environment. Richardson (1997) believes that in a constructivist environment teachers should help learners create their own meaning of new experiences combined with previous experiences (Richardson, 1997). Tobin and Tippins (1993) have stated that in constructivist learning, which creates the most suitable environment for high school and elementary students' learning, the teacher should consider students' prior knowledge of the concepts and maximizes the learners' interactions for a deeper understanding. When students are using concrete models, they can better connect existing knowledge with new concepts. Brooks and Brooks (1999) have cited characteristics of classrooms based on constructivism. These characteristics of a classroom are respecting the learners' ideas, providing challenging activities for all grade-level students, asking relevant questions, focusing on the big concepts, assessing students' daily class work, encouraging student autonomy, and using cognitive words such as predict and analyze.

Effects of a Constructivist Environment

Researchers have advocated the use of constructivism in the classroom (Carpenter, et al., 1999; Cobb et al., 1991; Fosnot, 1996). Rosenfeld and Rosenfeld (2006) have found that there is a relationship between a teacher's epistemological beliefs and teaching style. Kamii, Rummelsberg, and Kari (2005) found that establishing a constructivist environment improved high school and elementary students' scores in mathematics because it enhances these students' thinking abilities. A research study conducted by Henson (2003) advocates the use of manipulative problem solving activities in a learning environment. Moreover, teachers who promoted autonomy in a classroom had students with more classroom involvement and higher achievement in the course (Turner et al., 1998; Ames, & Archer, 1988). Also, Phye (1997) has found that mathematics environments that promote constructivism will result in fostering positive learning climates. Research has shown that lecture-guided instructions are ineffective because during a lecture, most people's attention spans are no more than 20 minutes long (Middendorf & Kalish, 1996). Psychologists and education experts also believe that direct transmission of knowledge is ineffective, but that active cognitive engagement will help learners discover new concepts and enhance their sense making abilities (Hardy, et al., 2006). Research studies have shown that constructivist environments will help learners achieve a higher degree of comprehension of mathematical concepts (Christianson & Fisher, 1999; Staub & Stem, 2002; Ziqiang, Lianggiang, & Liping, 2005). Stroddart, et al., (1993) and Lang and Gabel (2005) also found that the use of constructivist epistemology, which uses concrete materials for teaching, has helped science elementary teachers to teach scientific concepts more effectively in their science classes; however, Brophy (2006) has stated that constructivism is not effective in a classroom, and there needs to be more research in this area before we conclude that constructivism works.

Teacher programs in colleges are increasingly adopting constructivist theories of learning even though it is difficult to change teachers' beliefs (Richardson, 1997; Richardson, 1996). In supporting the use of constructivist practice for pre-service teacher programs, Klein (2001) states, "A major problem for teacher educators and their students is that those entering the programmes do not have mathematical or pedagogical knowledge necessary to position themselves as teachers who are able to teach in an investigative way or against the grain as it were" (P. 268). Constructivist learning approaches will be beneficial to in-service teachers in teaching mathematical concepts with deeper understanding to students (Tobin & Tippins, 1993). Plourde and Alawiye (2003) have found that when these teachers are exposed to constructivism, they are more likely to provide a constructivist environment in a class. Pre-service teachers will benefit from a student-centered environment based on constructivism (Brindley, 2000; Kelley, 2000).

Kim, Fisher, and Fraser (1999) found favorable students' attitudes were promoted when students experienced more personal relevance, shared control, and negotiated their learning. Kamii, Manning, and Manning (1991) found that a learner does not acquire knowledge and understand it from the outside; they must construct it from the inside, which can be achieved through interaction with other people.

This study shows what constitutes constructivist learning. It also shows the degree to which a specific teaching strategy is compatible with constructivism. In order to teach a lesson compatible with constructivist learning, an instructor of in-service elementary mathematics teachers must model this strategy of learning. The literature review does not discuss the use of the Constructivist Learning Environment Survey (CLES) in the United State. This study will use the CLES to examine whether or not the instructor maintains a classroom that promotes a constructivist learning environment.

Method

Participants

Forty nine prospective teachers in three sections of the course (two sections of K-8, and one section of K-4) completed a course in teaching mathematics for elementary and middle schools at a university in the southern part of the United States. Students' concentration programs were in language arts, special education, social studies, reading, and mathematics. The average age of students was about 23, with only four minority students.

Procedures

The researcher taught three sections of the course for a semester, which lasted 15 weeks. For the first half of the semester, the instructor taught the three sections of the course by using direct instruction, which involved lecturing and using the textbook Elementary and Middle School Mathematics Teaching Developmentally by Van De Walle (2004). The direct instruction included the use of Power Point or writing notes on the board for the entire class period. The concepts covered in the class included understanding number concepts, NCTM principles and their implications for teaching mathematics, constructivist learning, the importance of the individual differences with respect to the level of ability, fraction, multiplication, addition, subtraction, measurement, and statistical analysis. The instructor administered two quizzes and an exam. At the beginning of the semester, all students were given a syllabus which contained course contents for the entire semester. The instructor rigorously followed the content for the first half of the semester, teaching ten chapters from the textbook. The instructor did not use manipulatives, hands-on activities, or group work. At the end of the first half of the semester, a colleague of the researcher administered the Constructivist Learning Environment Survey (CLES) to all students.

For the second half of the semester, students in all three sections of the course were assigned to groups of two or three and were asked to work cooperatively for the entire period of the class engaging in hands-on activities and discussing their related concepts. The hands-on activities included manipulatives such as: counters, Cuisenaire rods, attribute blocks, Pattern Blocks, base-ten Blocks, papers, scissors, crayons, tapes, scales, bottles, colorful Buttons, beans, Geoboards, paper cups, straws, Unifex cubes, Tangram, ropes, and rulers. The concepts associated with these included: (1) understanding number sense; (2) addition, subtraction, multiplication, and subtraction of integer numbers; (3) areas, volumes, weights, and time. Also, students performed three activities regarding data analysis in order to understand concepts such as the mean, the mode, the median, a histogram, a bar graph, and probabilities. Students' grades were based on group activities, group projects, the presentation of an individual problem solving activity, and a written test on activities. In addition, members of the group were instructed to record their work, and teachers asked group leaders to share their learning with the class. Throughout the second half of the semester, the instructor did not use a textbook, and the pace of the activities was based on the students' abilities. The instructor created an environment that promoted social interaction and encouraged student's deep understanding of the concepts rather than memorization. For example, the instructor asked students to make a trapezoid with an area of six square units using six pegs on the Geoboard. This problem did not require any formula, and it was based on discovery learning because every student could make a trapezoid with different height and base lengths, but with the same area. Students in the groups were encouraged to compare their findings and explain them to each other. At the end of the second half of the semester, the same survey, CLES, was administered to all students.

Instrument

The Constructivist Learning Environment Survey (CLES) is an instrument that measures learners' degree of constructing their own knowledge and provides student perception of the nature of a classroom with respect to constructiveness (Taylor, Fisher, & Fisher, 1997). Huei-Bank, Fisher & Fraser (1999) have stated that CLES was designed from a psychological view of learning. The instrument is based on a five-point Likert scale with the following values: 5-almost always agree; 4-often agree; 3-sometimes agree; 2-seldom agree; 1-almost never agree (see Appendix). The original version of this survey focused on psycho-social constructivism and mostly had weak theoretical support (Fraser, Drydon, & Taylor, 1998).

Empirical evidence in Korea and the United States for grade school and college students provides support for the use of CLES. In those countries there were high reliability values in all sections of the CLES, ranging from 0.76-0.94. The CLES is a robust instrument used to measure the degree in which a classroom environment is compatible with constructivism (Dryden & Fraser, 1998; Kim, Fisher & Fraser, 1999).

The present study used the new version of the CLES survey developed by Taylor (1996). It was used with permission from the original developer of the instrument. The survey is made of six distinct sections, with each section containing five questions. The researcher used different forms of the survey depending on the subject matter. In this study, the mathematics version of the survey was used. The sections of the survey are as follows (the names in the parentheses indicate the names for the corresponding mathematics version):

1. Personal relevance (learning about the world) measures the degree to which teachers relate mathematics to the students' out-of-school experience

2. Student negotiation (learning to communicate) measures the degree to which the students are allowed to express their opinions to other students and reflect on their partners' ideas

3. Shared control (learning to learn) measures the extent to which students are included in the design, goals, and management of their activities

4. Critical voice (learning to speak out) measures the extent to which learners are given the opportunity to express their opinions with respect to factors that may be an impediment to their learning and the teacher's method of teaching

5. Uncertainty of Mathematics (learning about math) measures the extent to which students are aware of mathematics as a subject that is evolving and uncertain

6. Attitude (how I feel) measures students' attitudes toward learning.

Data Analysis

The investigator used SPSS, version 11, to analyze the data. First, paired t-tests were used to examine if a significant difference exists between the means of the pretest and posttest survey questions. In order to control type I error, the researcher used the Bonferroni correction by keeping the p-value at .99. For any section of the survey that showed significant mean differences, the researcher used the Wilcoxon Signed-rank test to examine which questions within the sections might result in significant mean differences.

Results

Table 1 shows the means of six sections of the survey administered-pretest and posttest. The mean of each section of the pretest is higher than the mean of the posttest. However, results indicate the means of four sections were significant. These sections are: learning about the world, learning to speak out, how I feel, and learning to communicate. In order to find which questions in a given section were significant, the searcher analyzed them by utilizing a non-parametric Wilcox Signed-Rank Test. The results from Table 2 indicate which questions were found to be significant.

Learning About the World

This section is about connecting student learning in a class setting to the outside world. During the second half of the semester, students were required to teach a concept by presenting an activity and involving their classmates in it. For example, one student showed how to multiply 1/2 and 1/3 by folding a piece of notebook-size paper. One student said that I did not learn this way while I was in college. While working on computing using Geoboard, students learned that while two television sets may have the same-size diagonals, one of them may have a screen with a larger surface area. One student mentioned that she could use this knowledge to be a smarter shopper.

Learning about Math

The course objectives did not address the questions related to this section of the survey. The questions in this section included the history and sociology of mathematics, with statements such as: I learn that modern maths is different from the maths of long ago; I learn that maths has changed over time, and I learn about different maths used by people in other cultures. The instructor did not discuss questions; therefore, this may explain the reason that there were no significant differences on these questions in pre-and posttest surveys.

Learning to Speak out

Questions 11,13, and 15 (Table 2) were found to be significant because students were encouraged to ask questions when they had difficulty understanding the concepts. The instructor encouraged a learning culture that promoted students' right to express their opinions and ask questions when needed. However, the instructor did not have complete control over all assignments or course objectives due to university and state regulations. Students did not have full control over the course content; consequently, this study did not find significant results for questions 2 and 3.

Learning to Learn

This section of the survey was about involving students in determining what they should learn. Some of the questions in this section are as follows: I help the lecturer to plan what I'm going to learn, and I help the lecturer decide which activities I do. This study found no significant differences in students' responses to all questions (16,17,18,19,and 20) in this section between the pre-and posttest surveys. One explanation is that it was university policy to provide a course syllabus to all students on the first day of this section. All assignments, objectives and grading policies were determined in advance by the instructor, and students had no input in the selection of assignments except on one assignment in which they had complete control over the topic of the activities they presented in class.

Learning to Communicate

This section was about the degree to which students were given freedom to share their ideas with their classmates. This study found that there were significant differences in students' responses to questions 21, 22, 23, 24, and 25 of this section between the pre-and posttest surveys (see Table 2). The instructors required all students to work in groups because social interaction is an important component of constructivist learning. Students were encouraged to share their findings with their classmates.

How I feel

The study found that there were significant differences in responses to questions 26, 27, 28, 29, and 30 of this section between pre-and posttest survey. The findings indicate that students enjoyed taking this course .Also, during the first half of the semester, all students' works were assessed objectively since the instructor administered only written tests or quizzes, and during the second half of the semester assessments were based on group work, student activity presentations, and a final activity-related concepts test.

The means of the following questions of the posttest were higher than the means of the pretest; however, statistical analysis of the paired t-test and Wilcoxon signed-rank test suggest the differences between the means were insignificant.

Learning About the Worm

2. My new learning starts with problems about the school world outside college

Learning About Math

6. I learn how math has changed over time

7. I learn how math is influenced by people's value and opinion

8. I learn about the different math used by people in other cultures

9. I learn how modern math is different from the math of long ago

Learning to Speak Out

12. It's OK for me to question the way I'm being taught

14. It's OK for me to question anything that prevents me from learning

Learning to Learn

16. I help the lecturer to plan what I'm going to learn

17. I help lecturer to decide how well I am learning

18. I help Lecturer to decide which activities are best for me

19. I help lecturer to decide how much time I spent on activities

20. I help the lecturer to decide which activities I do

Discussion

The purpose of this study was to use the CLES survey to exam whether the instructor maintains a classroom that promotes a constructivist learning environment. Since the survey contains six sections (with five questions in each), this study investigated which of these sections appears to be significant. These sections are as follows: (1) learning about the world, (2) learning about math, (3) learning to speak out, (4) learning to learn, (5) learning to communicate, and (6) how I feel.

Learning to communicate: Results from this study indicate that the means of all questions in this section of the posttest survey were significantly higher than the means of the same section of the pretest survey. One explanation for this significance is that students were actively engaged in problem-solving activities, and they were encouraged to discuss their findings in and among their learning groups.

How I feel: The study found that the means of all questions in the section of the posttest survey were significantly higher than the means of the section of the pretest. Many students in their evaluations of the instructor commented how much they enjoyed the class because they used hands-on activities. One student wrote, "What I have learned in this class is more than what I have learned in all my three required math courses."

Learning to speak out: Some questions in this section of the survey showed significantly higher means in the posttest survey; this may indicate the instructor promoted a non-threatening environment so that students could express their opinions and ask questions when they had difficulty learning the mathematical concepts.

Learning about the world: Most questions in this section also showed significantly higher means in the posttest survey. Some of these activities involved the use of contextual problems; therefore, students could relate what they have learned in the classroom to the outside world.

Learning to learn and learning about math: The reason for the insignificant results in these sections could be due to the fact that students were given a syllabus in advance and were given the type of the assignment they should do; therefore, there was no input given by the students about determining which activities they did. The reason for the insignificant findings in the "learning about math" section is that the instructor did not refer to the history and culture of mathematics.

The implication of this study for the pre-service teacher training is that instructors become aware of the degree to which their classrooms are compatible with constructivist learning. Teacher may think that their teaching strategies are a constructivist one because they are using hands-on activities. However, the use of concrete activities is not the only component necessary for constructivist learning. Teachers should be aware of other essential ingredients for the constructivist learning such as empowering students in classrooms by giving students the freedom of interacting with one another. Research has shown that the teachers' philosophy of teaching affects the teaching strategies they will select in a classroom. Studies have shown the importance of constructivist learning (Christianson & Fisher, 1999; Staub & Stern, 2002). This study indicates that mere discussion of the importance of constructivism does not create an environment conducive to constructivist learning. During the first half of the semester, even though the teacher lectured about the importance of constructivism and the importance of NCTM principles, which supports constructivism, the study did not find the classroom environment in support of constructivism. Therefore, this study suggests that to create an environment that supports constructivism, instructors must implement a lesson plan that support the use of hands-on activities, promotes interactions among and within groups even though the school environments may not fully support this philosophy of teaching due to state and federal regulations.

Appendix
The Constructivist learning Environment Survey (CLES)

Learning about the world                  Almost    Often
                                          Always

In this Mathematics Education class ...

1. I learn about the school
world outside college
2. My new learning starts with
problems about the school
world outside college
3. I learn how maths can be part of
my out- of-college life
4. I get a better understanding of the
school world outside college
5. I learn interesting things about
the school world

Learning about mathematics

In this Mathematics Education class.
6. I learn how maths has changed
over time
7. I learn how maths is influenced by
people's values and opinions
8. I learn about the different math
used by people in other cultures
9. I learn how modern maths is
different from the maths of long ago
10. I learn how math involves
inventing rules

Learning to speak out

In this Mathematics education class.
11. It's ok for me to ask the learner
"why do I have to do to learn this?"
12. It's ok for me to question the way
I'm being taught
13. It's ok for me to me to seek
clarification about activities
that are confusing
14. It's ok for me to question
anything that me to express my opinion
15. It's ok for me to express my
opinion

Learning to learn

In this Mathematics Education class...
16. I help the lecturer to plan what
I'm going to learn
17. I help the lecturer to decide how
well I am learning
18. I help the lecturer to decide which
activities are best for me
19. I help the lecturer to decide how
much time I spent on activities
20. I help the lecturer to decide which
I activitiesi I do

Learning to communicate

In this Mathematics Education class.
21. I get the chance to talk to other
students
22. I talk with other students about
how to solve problems
23. I explain my ideas to other
students
24. I ask other students to explain
their ideas
25. Other students ask me to explain
my ideas

How I feel
In this Mathematics education class.
26. I look forward to this class
27. This is one of the most interesting
classes at college
28. I enjoyed this class
29. I feel less stressed in this class
30. I feel less confused in this class

Learning about the world                  Some-     Seldom   Almost
                                          times              Never

In this Mathematics Education class ...

1. I learn about the school
world outside college
2. My new learning starts with
problems about the school
world outside college
3. I learn how maths can be part of
my out- of-college life
4. I get a better understanding of the
school world outside college
5. I learn interesting things about
the school world

Learning about mathematics

In this Mathematics Education class.
6. I learn how maths has changed
over time
7. I learn how maths is influenced by
people's values and opinions
8. I learn about the different math
used by people in other cultures
9. I learn how modern maths is
different from the maths of long ago
10. I learn how math involves
inventing rules

Learning to speak out

In this Mathematics education class.
11. It's ok for me to ask the learner
"why do I have to do to learn this?"
12. It's ok for me to question the way
I'm being taught
13. It's ok for me to me to seek
clarification about activities
that are confusing
14. It's ok for me to question
anything that me to express my opinion
15. It's ok for me to express my
opinion

Learning to learn

In this Mathematics Education class...
16. I help the lecturer to plan what
I'm going to learn
17. I help the lecturer to decide how
well I am learning
18. I help the lecturer to decide which
activities are best for me
19. I help the lecturer to decide how
much time I spent on activities
20. I help the lecturer to decide which
I activitiesi I do

Learning to communicate

In this Mathematics Education class.
21. I get the chance to talk to other
students
22. I talk with other students about
how to solve problems
23. I explain my ideas to other
students
24. I ask other students to explain
their ideas
25. Other students ask me to explain
my ideas

How I feel
In this Mathematics education class.
26. I look forward to this class
27. This is one of the most interesting
classes at college
28. I enjoyed this class
29. I feel less stressed in this class
30. I feel less confused in this class

References

Ames, C. A., & Archer, J. (1988). Achievement goals in the classroom: Students' learning strategies and motivation processes. Journal of Educational Psychology, 80,260-267.

Borthick, A. F., Jones, D. R., & Wakai, S. (2003). Designing learning experiences within learners' Zone of Proximal Development (ZPDs): Enabling Collaborative learning onsite and online. Journal of Informational Systems, 17(1), 107-134.

Brindley, R. (2000). Learning to walk the walk: Teacher educators' use of constructivist epistemology in their own practice. Professional Educator, 22(2), 1-14.

Brooks, J., & Brooks, M. (1999). The case of the constructivist classrooms. Virginia: Association for Supervision and Curriculum Development.

Brooks, J. G., & Brooks, M. G. (1993). In search of understanding case for constructivist classroom. Alexandria, VA: Association for Supervision and Curriculum Development.

Brooks. J. G. & Brooks, M. G. (1993). In search of understanding: The case for constructivist classrooms. Alexandria, VA: Association for Supervision and Curriculum Development.

Brophy, J. (2006). Graham Nuthall and social constructivist research: Research based cautions and qualifications. Teaching & Teacher Education, 22(5), 529-537.

Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S. (1999). Children's mathematics: Cognitively guided instruction. Reston, VA: NCTM.

Christianson, R. G., & Fisher, K. M. (1999). Comparison of student learning about diffusion and osmosis in constructivist and traditional classrooms. International Journal of Science Education, 21 (6), 687-98.

Cobb, P., Wood, T., & Yackel, E. (1991). A Constructivist approach to second grade mathematics. In E. von Glasersfeld (Ed.), Constructivism in mathematics education (pp. 57-176). Dordrecht, Holland: Reidel.

Draper, R. (2002). School mathematics reform, constructivism, and literacy: A case for literacy instruction in the reform-oriented math classroom. Journal of Adolescence and Literacy, 45,520-529.

Dryden, M., & Fraser, B. J. (1998). Evaluating urban reform using the constructivist learning environment survey. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.

Edwards, S. (2005). Constructivism does not only happen in the individual: A teacher makes a difference. Early Childhood Education Journal, 22(1), 39-49.

Fosnot, C. (1996). Constructivism: A psychological theory of learning. In C. Fosnot (Ed.), Constructivism: Theory, perspective, and Practices (pp. 8-33). New York: Teachers College Press.

Fraser, B. J., Dryden, M., & Taylor, P. C. (1998). The impact of systemic reform efforts on instruction in high school science classes. Paper presented at the annual meeting of the National Association for Research in Science Teaching, San Diego: CA.

Geary, D. C. (1995). Reflections of evolution and culture in children's cognition: Implications for mathematical development and instruction. American Psychologist, 50, 24-37.

Hardy, H., Jonen, A., Moiler, K., & Stern, E. (2006). Effects of instructional support within constructivist learning environment for elementary school students, understanding of floating and sinking. Journal of Educational Psychology, 98(2), 307-326.

Henson, K. (2003). Foundations or learner-centered education: A knowledge base. Education, 124, 8-16.

Heui-Baik, K., Fisher, D. L., & Fraser, B. J. (1999). Assessment and investigation of constructivist science learning environment in Korea. Research in Science & Technological Education, 17(2), 239-248.

Jeramillo, J. A. (1996). Vygotsky's socio-cultural theory and contributions to the development of constructivist curricula. Education, 117(1), 113-140.

Kamii, C. (1985).Young children reinventarithmetic. New York: Teacher College Press.

Kamii, C., Manning, M., & Manning, G. (Eds.). (1991). Early literacy: A constructivist foundation for whole language. Washington, DC: National Education Association.

Kamii, C. Rummelsburg, J. & Kari, A. (2005). Teaching arithmetic to low-performing low-SES first graders. Journal of Mathematics Behavior, 24, 39-50.

Kim, H., Fisher, D. L., & Fraser, B. J. (1999). Assessment and investigation of constructivist science learning environments in Korea. Research in Science & Technological Education, 17(2), 239-250.

Klein, M. (2001). Constructivist practice, pre-service teacher education and change: The limitation of appealing to hearts and minds. Teachers and Teaching: Theory and practice, 7(3), 257-269.

Lang, L. L. & Gabel, D. L. (2005). Effectiveness of a constructivist approach to science instruction for prospective elementary teachers. International Journal of Science Education, 27(10), 1143-1162.

Lortie, D. C. (1975). School teacher: A sociological studies. Chicago: University of Chicago Press 

Middendorf, J., & Kalish, A. (1996). The "change--up in" lectures. The National Teaching and Learning Forum, 5(2), 1-5.

Mvududu, N. (2005). Constructivism in the statistics: From theory to practice. Teaching statistics, 27(2), 49-54.

National Council of Teachers of Mathematics (NCTM). (1995). Assessment standards for school mathematics. Reston, VA: NCTM National Council of Teachers of Mathematics (NCTM). (1991). Professional standards for teaching mathematics. Reston, VA: NCTM

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: NCTM National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM Olivia, P. (2005). Developing the curriculum (6th ed.). Boston: Pearson Education. Statistics, 27, 49-54.

Phye, G. D. (1997). Academic learning and remembering. In G.D. Phye (Ed.), Handbook of academic learning: Construction of knowledge. San Diego: Academic Press.

Plourde, L. A., & Alawiye, Q. (2003). Constructivism and elementary preservice science teacher preparation: Knowledge to application. College Student Journal, 37(3), 334-341.

Richardson, V, (Ed.), (1997). Constructivist teacher education: Building a world of new understanding. Washington, DC: The Falmer Press.

Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula (Ed.), Handbook of research on teacher education (pp. 102-119). New York: Simon and Schuster Macmillan.

Rosenfeld, M., & Rosenfeld, S. (2006). Understanding teacher response to constructivist learning environments: Challenges and resolutions. Science Education, 90(3), 383-399.

Schmittau, J. (2004). Vygotskian theory and mathematics education: resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19, 19-43.

Staub, F. C., & Stern, E. (2002). The nature of teachers' pedagogical content beliefs matters for students' achievement gains: Quasi-experimental evidence from elementary mathematics. Educational Psychology, 94(2), 344-346.

Stoddart, T., Connell, M., Stofflett, R., & Peck, M. (1993). Reconstructing elementary teacher candidates' understanding of mathematics and science content. Teaching and Teacher Education, 9, 229-241.

Taylor, P.C., Fraser, B.J., & Fisher, D.L. (1997). Monitoring constructivist classroom learning environments. International Journal of Educational Research, 27, 293- 302.

Taylor, P. C. (1996). Mythmaking and myth-breaking in the mathematics classroom. Educational Studies in Mathematics, 31, 151-173.

Tobin, K., & Tippins D. (1993). Constructivism as a referent for teaching and learning. In K. Tobin (Ed.), In the Practices of constructivism in science. Education (pp. 3-21). Hillsdale, NJ: Lawrence Erlbaum.

Toutman, A., & Litchenberg, B. (2003). Mathematics a good beginning (6th ed.) Belmont: Wadsworth

Turner, J. C., Meyer, D. K., Cox, K. E., Logan, C., DiCintio, M., & Thomas, C. T. (1998). Creating contexts for involvement in mathematics. Journal ofEducational Psychology, 90, 730-745.

U.S. Department of Education (2005). Reasoning U.S. Internal Mathematics Performance New findings from the 2003 TMSS and PISA. Washington, DC: American Institutes for Research

Van De Walle, J. (2004). Elementary and middle school teaching developmentally, 52 ed. Boston: Pearson, Allyn and Bacon.

Von Glaserfeld E. (1992). Questions and answers about radical constructivism. In M. K. Pearsall (Ed.), Relevant research (pp. 169-182). Washington D.C.: National Science Teacher Association.

Ziqiang, X., Lianggiang, N., & Liping, C. (2005). The relationship between cognitive holding power and constructivist pedagogy in mathematical education. Psychological science, 28(6), 1324-1329.

Zuckerman, G. (2004). Development of reflection through learning activity. European Journal of Psychology of Education, 19(1), 9-18. Van De Walle, J. A. (2004).

Hosin Shirvani, Department of Curriculum and Instruction, University of Texas-Pan American.

Correspondence concerning this article should be addressed to Hosin Shirvani at [e-mail].

Table 1
Shows Means, Standard Deviation and the Level of Significance
from the Paired-t tests

Variables                     Pretest        Posttest       P

                             M      SD      M      SD

Learning about the world   3.78    .72    4.15     .65     .01
Learning about math        3.39    .82    3.63     .96     .18
Learning to speak out      4.04    .84    4.40     .55     .02
Learning to learn          3.01    .92    3.22    1.16     .32
Learning to communicate    3.91    .83    4.73     .44     .00
How I feel                 3.35    .57    4.46     .54     .00

Table 2
Shows the Nonparametric Test (Wilcoxon Signed-rank Test)

Questions#                                                  p

Learning about world
1.  I learn about school world outside college             .008
3.  I learn how maths can be part of my out of
    college life                                           .020
4.  I get better a understanding of the school
    world outside College                                  .007
5.  I learn interesting things about the school world
    outside college                                        .005

Learning to speak out
11. It's OK for me to ask the lecturer "why do I
    have to learn this?"                                   .030
13. It's OK for me to seek clarification about the
    activities that are confusing.                         .010
15. It's OK for me to express my opinion                   .020

Learning to communicate
21. I get the chance to talk to other students             .000
22. I talk with other students about how to
    solve problems                                         .000
23. I explain my ideas to others students                  .000
24. I ask other students to explain their ideas            .000
25. Other students ask me to explain my ideas              .000

How I feel
26. I look forward to this class                           .000
27. This is one of the most interesting classes
    at college                                             .000
28. I enjoyed this class                                   .000
29. I feel less stress in this class                       .000
30. I feel less confused in this class                     .000