Culture, communication, and mathematics learning: an introduction.

History is replete with politically and ideologically motivated resistance to change in teaching mathematics. In 415 AD, St. Cyril, Christian Patriarch of Alexandria ordered the assassination of the Greco-Egyptian mathematician Hypatia. She was taken by a Christian mob, dragged into a church, stripped of her clothes and hacked to death with sharpened oyster shells (Osborne, 1992; Gibbon, 1975). It appears that her crime was teaching about the roundness of the earth at a time when Christian leaders wanted to revive the notion of a flat earth centered in a tabernacle shaped universe (Alic, 1998).

In 1299, the city of Florence banned the use of Hindu-Arabic numerals. It was thought that the new system facilitated dishonest dealings because it was easy to modify a zero to a nine or a six. It was also thought that the place value system allowed deceitful merchants to inflate values by adding a new number to the end of a row (Flegg, 1989). During the late Middle Ages many Europeans rejected the very notion of a zero, regarding it as a creation of Satan (Menninger, 1969).

We can also find more recent examples of resistance to changes in mathematics education. In Colonial America, reading and writing were seen as the chief purposes of education, and mathematics was often not part of the school curriculum. In 1900 parents in New York City complained "that their children were using methods that were different from those they themselves employed." (Kilpatrick, 1992, p. 17)

The mathematical knowledge needed by any particular culture is not static. If we assume that mathematics and mathematics education that met the needs of our parents are appropriate for the new century, we are deeply mistaken. As Steen (1990) states:

    To develop effective new mathematics curricula, one must attempt to
    foresee the mathematical needs of tomorrow's students. It is the
    present and future practice of mathematics-at work, in science, in
    research-that should shape education in mathematics. To prepare
    effective mathematics curricular for the future, we must look for
    patterns in the mathematics of today to project, as best we can,
    just what is and what is not truly fundamental. (p. 2-3)

Many Americans are convinced that they can never learn mathematics. This persuasive attitude is an example of what psychologists call learned helplessness. McLeod & Ortega (1993) define learned helplessness in the mathematics education context as "a pattern of behavior whereby students attribute failure to lack of ability" (p. 28). These authors contrast learned helplessness with mastery orientation. In mastery orientation students have confidence in their ability to solve challenging problems. Learned helplessness is negatively related with persistence, while mastery orientation is positively connected with persistence.

McLeod & Ortega (1993) found a student's self-concept could be modified by social context. They describe how classroom conversation, such as a teacher's characterization of a problem as "easy" can profoundly demoralize students. The National Council of Teachers of Mathematics [NCTM] Assessment Standards for School Mathematics (1995) defines mathematical disposition as "interest in, and appreciation for, mathematics; a tendency to think and act in positive ways; includes confidence, curiosity, perseverance, flexibility, inventiveness, and reflectivity in doing mathematics (p. 88). The critics of the Standards dismiss this notion of disposition as nonsense and advocate a back-to-basics approach. In the words of Jennings (1996), "get a math book, make students practice problems, have them do simple addition, subtraction, and multiplication in their heads, give them standardized tests, and drop the group work." This back-to-basics orientation seems more rooted in nostalgia than actual research. McLeod and Ortega (1993) give us reason to hope that if we address the affective components of mathematics education, as suggested in the NCTM Standards, we can improve students' achievements.

The point is not to dichotomize between conceptual understanding of mathematics and basic skills. Instead, many researchers believe that computational skills can be achieved and applied to new mathematical situations even more powerfully, if those skills are put in a meaningful context. The goal ought to be for students to develop strong number sense and to acquire conceptual understanding of number meaning and number relations.

    In spite of our belief that understanding and skills can and should
    develop together, we must make it clear that we assume the primary
    goal of mathematics instruction is conceptual understanding. But we
    must also make it clear that setting conceptual understanding as the
    primary goal does not mean ignoring computation skills. In fact, we
    have found that instruction for understanding can help students
    construct skills that can be recalled when needed, can be adjusted
    to fit new situations, and can be applied flexibly. In a word, we
    found that such instruction can help students construct skills that
    they can actually use. (Hiebert, 1997)

There are several underlying assumptions that shape the vision of mathematics set forth in the Curriculum and Evaluation Standards for School Mathematics (1989). These assumptions are: (1) mathematics is something a person does, (2) mathematics has broad content encompassing many fields, (3) mathematics instruction and learning can be improved through appropriate evaluation, and (4) mathematical power can and must be at the command of all students in a technological society.

The NCTM (1989, 1991, 1995, 2000) Standards documents are goals and visions responding to a call for reforming mathematics education. Behind this action is a consensus that all students can learn mathematics and often learn it in different ways. In this sense, curriculum, instruction, and assessment are reflexively related and must be continually revisited.

The Curriculum and Evaluation Standards (1989) provides five goals for k-12 students: (1) learning to value mathematics, (2) becoming confident in doing significant mathematics, (3) becoming mathematical problem solvers, (4) learning to communicate mathematically, and (5) learning to reason mathematically. The teacher's role is vital for implementation of these goals. The environment of the classroom must allow for discourse, the kind of discussion and dialogue that represents critical thinking, reasoning, justifying, agreeing, and disagreeing. Classroom discourse is a major influence toward developing mathematical power.

The NCTM Standards (1989) suggest five major shifts in the environment of mathematics classrooms: (1) toward a classroom that is a mathematics community, away from one that is just a roomful of individuals, (2) toward logic and mathematical evidence, away from the teacher as sole authority for answers, (3) toward mathematical reasoning, away from exclusive rote learning, (4) toward conjecture, inventing and problem solving, away from the mechanical application of algorithms, and (5) toward connecting mathematics to other fields and avoiding the treatment of mathematics as an isolated topic. NCTM (1989) warns us, "the picture of mathematics teaching and learning that is presented in the Standards is an ambitious one. We will not reach these goals overnight" (p. 194). It will require much commitment and dedication from teachers, principals, parents, community members, and other educators.

Perhaps one of the most important goals for mathematics education is to create autonomous learners who can communicate their thinking and reasoning within the social context (Carpenter, Fennema, & Franke, 1996; Carpenter, Franke, & Levi, 2003; Carpenter & Lehrer, 1996; Ernest, 1995; Steffe, 1995; Wood, 1995). A mathematics classroom needs to be a place where critical thinking is free to thrive and where quality and meaning of schooling and human life is central. In this sense, school would be viewed as a place for communication and dialogue, a place where connectivity and relationships play a pivotal role for constructing meaning (Noddings, 1992; Fleener, 2002). This kind of school would probably resemble the emancipatory instructional and curricular milieu envisioned by Grundy (1987).

History of revitalization at a School: Efforts that expanded to Research

Three out of four of the papers in this issue (Social Constructivism in Practice: Case Study of an Elementary School's Mathematics Program; Culture, Social Interaction, and Mathematics Learning; Passion for Teaching/Learning Mathematics: A Story of Two Fourth Grade African American Students) present complexities, challenges, and efforts that a K-4 school learning community faces helping all students to learn mathematics with understanding. The fourth paper (The Impact of a State Mathematics Test on the Structure and Culture of a K-4 School) demonstrates how the structure and culture of a conventional K-4 were influenced by the state mandated test.

To understand how mathematics instructional reform and an epistemological shift ignited major school change, it is important first to paint a clear description of the physical setting that surrounds and nurtures changes within the school that is transforming itself. Within the physical landscape reside two other landscapes that are part of the school in this study. This historical landscape focuses on pivotal events and activities that have molded and been created within the reform process since 1989. The research landscape provides a glimpse of what research under girded the reform and its impact on school change.

Physical Setting

The city in which the school district is located is socioeconomically diverse from low middle-class to high middle-class neighborhoods. Since 1990, the school that is the focus of research has been the largest K-4 school in the district. Its student population is 525 (60% African American, 34% White, 6% other racial or multiracial groups). About half the students walk to school, while the other half either ride buses or come in cars. One hundred students participate in a "Before/After Care" program housed at the school, which is managed and operated by the city's recreation department. The school has 24 classroom teachers, five special area teachers (art, music, physical education, and library), two Learning Disability teachers, approximately 20 full and part-time support staff (tutors, Reading Recovery teacher, district specialists, school psychologist, occupational/speech therapy, and nurse), one principal, one assistant principal, one administrative secretary, and once office clerical aide. More than 80% of the teachers hold a Master's degree. The principal, the former assistant principal, and two other teachers have doctorates. Average class size is approximately 22 students.

Historical and Research Landscapes

In 1989, two major publications addressed the issue of reforming mathematics education in America. These reports proposed wide-ranging, radical changes in mathematics learning and teaching. In Everybody Counts: A Report to the National on the Future of Mathematics Education (1989), the National Research Council (NRC) concluded that ineffective mathematics education posed a potential threat the America's economic security in a technological world. In Curriculum and Evaluation Standards for School Mathematics (1989), NCTM echoed similar views and drafted recommendations intended (1) to expand school mathematics beyond "shopkeepers' arithmetic," (2) to include mathematical meaning congruent with the needs of the twenty-first century, and (3) to establish instructional beliefs and practices based on constructivism. The following describes pivotal events impacting the reform since 1989.

Spring, 1989     Curriculum and Evaluation Standards and Everybody
                 Counts published. Documents call for major reform in
                 mathematics instruction at the K-4 level, where the
                 sole reliance on "shopkeepers' arithmetic skills" did
                 not develop mathematical literacy. Recommendations call
                 for expanded mathematical content, expanding elementary
                 teachers' content knowledge, and a change in
                 mathematics instruction.
Winter, 1990     Public-private match grant written by school personnel
                 requesting funds to align mathematics to the NCTM
                 recommendations. Intent of grant is to develop hands-on
                 lessons in all the content Standards of mathematics;
                 numbers and operations, patterns, function, algebra,
                 geometry, measurements, and data collection
Spring, 1990     Grant awarded by state Department of Education
Summer, 1990     K-4 teachers and the assistant principal, in
                 consultation with a university mathematics educator,
                 write mathematics lessons (Umbrellas). Lessons
                 incorporate the use of concrete materials in all the
                 strands of mathematics and attempt to integrate
                 mathematics into other areas of the curriculum, such as
                 social studies, science, and language arts.
1990-1995        Five Effective Schools grants awarded by state
                 Department of Education. Grants used to finance
                 mathematics and science instructional reform. Amount of
                 grants is $20,000.
Fall, 1990-      Umbrellas implemented at the school. University
Summer, 1991     mathematics educators conducted staff workshops
                 throughout the year to demonstrated use of mathematics
                 manipulative in mathematics lessons.
Summer, 1991     Umbrellas revised. District teachers work with the
                 school teachers to write mathematics problem solving
                 for K-4 instruction. It is called Investigations.
Fall, 1991       Umbrellas and Investigations disseminated to district
                 K-6 teachers.
Fall, 1993       State Department of Education initiates Venture Capital
                 grants for school improvement. Grants were to "spark
                 systemic change at the school level and to encourage
                 risk-takers who wanted to create a more effective
                 educational system."
Winter, 1994     Venture Capital Grant awarded to the school. Grant
                 aligns mathematics and science instruction with
                 national research recommendations, focusing curriculum
                 on key concepts, developing relevant lessons based on
                 students' prior experience and applying constructivism
                 to mathematics instruction. Total amount: $125,000 over
                 five years ($25,000 per year).
Summer, 1994     Principals-led team of K-4 school educators (included
                 two secondary mathematics educators, one secondary
                 science educator, one university mathematics educator)
                 write mathematics lessons that focused on students'
                 experience, integrated mathematics into science, and
                 applied constructivist practice to mathematics. It is
                 called Perspectives.
Fall, 1994       Perspectives implemented in some classrooms in the
                 school. Lessons failed to develop constructivist model
                 of instruction. Too much reliance on calculation.
                 Perspectives is shelved; decision is questioned by some
                 members of the writing team.
Fall, 1994       Language in Venture Capital grant proposal (which all
                 the school teachers signed), provokes union leaders. A
                 section in proposal, entitled Hindrances and
                 Alterations, states, ".... In our opinion these present
                 educational policies and practices hinder reform
                 efforts.... Administrative and union practices which do
                 not genuinely support or encourage innovation and,
                 therefore, equalize performances at a mediocre level
                 need to be redesigned so that educational leaders
                 encourage and support innovation and allow risk-taking
                 behaviors outside the 'bounds' of normal rules and
                 regulations."
Fall, 1994       Decision is made to have university mathematics
                 educator model constructivist practice for teachers.
                 The school project used in research study.
Late Fall, 1994  Classroom mathematics dialogues between university
                 professor and fourth-grade students reveal that what is
                 taught is often not what is learned. The project
                 focuses on how students think and what they understand.
Late Fall, 1994  First the city school levy fails. District sustains a
                 six million-dollar loss. Anti-tax group exposes
                 underachievement issues, specifically among African-
                 American students. Anti-tax group pressures school
                 board, suggesting that achievement will improve through
                 direct-instruction and "drilling" the basics. Group
                 begins suing local newspaper to regularly criticize
                 school district.
Fall, 1994       Principal forms evening parent mathematics group to
                 communicate instructional reform to public.
Winter, 1995     Dissertation, The Impact of the NCTM Standards (1989)
                 on the Lives of Elementary Teachers: A Portrait of
                 Radical Change (Cowen, 1995), written about the school
                 reform efforts. The dissertations links constructivism
                 to major reform agendas, viewing a change in existing
                 educational epistemology as "radical" and a
                 "paradigmatic shift" for educators.
Spring, 1995     Key mathematics concepts and processes identified by
                 elementary, secondary, and university educators.
                 Mathematics instruction focuses on the ideas of unit,
                 change, chance, unitized systems, dimensionality,
                 location in space, zero to infinity (number) and the
                 processes of combining, comparing, and partitioning.
Summer, 1995     "Families Talking Math" group meets with principal to
                 develop a parent support group for reforming
                 mathematics education. Group develops ways for parents
                 and children to do mathematics together at home.
Fall, 1995-96    "Families Talking Math" begins writing mathematics
                 problems for K-4 students to take home and do with
                 parents.
Fall, 1995       Teachers ask to submit action-research proposals.
                 Eighteen teachers volunteer to conduct "action-
                 research" that targets constructivist teaching-
                 learning-assessment practices. Individual proposals are
                 aligned with national reform efforts in mathematics.
                 Proposals are funded to buy materials, equipment, and
                 release time. Art specialist design authentic
                 assessment portfolios for students; student art work
                 would be collected for five years.
Fall, 1995       Parent-Teacher-Student conferencing "triangular
                 conferencing," begins in some classrooms in the school.
Spring, 1996     State mandated Fourth Grade Testing begins.
Spring, 1996     The Kumon Mathematics Program, an Asian mathematics
                 program focusing on speedy performance of computational
                 skills, is started after school to help underachieving
                 students pass the mandated mathematics test. About 30%
                 of the students in third and fourth grade are tutored
                 by the school's teachers (eight teachers) with Kumon
                 program two or three days each week after school.
Fall, 1996       Snapshots is implemented. Participation in reform
                 becomes mandatory. All classroom teachers and tutors
                 are placed on a K-4 Instructional Design Teams (IDT)
                 led by two teacher-leaders. Tensions emerge over the
                 selection of teacher-leaders, specifically the
                 appointment of non-tenured and less senior teachers as
                 leaders, and the fact that the reform was no longer
                 voluntary. Tensions also emerge over instructional
                 changes. A few teachers label the instructional reform
                 "developmentally inappropriate."
Fall, 1996       IDTs meet six hours each month to build a framework of
                 lessons that target performance tasks. Permanent
                 substitutes are hired and instructionally informed to
                 cover classes one day a week to facilitate staff
                 development.
Fall, 1996       Some teacher-leaders conduct weekend and evening
                 conferences. Some teacher leaders petition the union
                 leadership to negotiate for additional conference days
                 for "triangular conferences" with students and parents.
                 No new time for triangular conferences is negotiated.
Fall, 1996       Venture Capital Partnership grant is submitted by
                 district. The school partners with three elementary
                 schools and a middle school to extend the reform in
                 mathematics to other schools in the district.
Fall, 1996       "Saturday Scholars" begins for at-risk fourth grade
                 students in mathematics to prepare for the mandated
                 Fourth Grade Test. About 30% of fourth grade students
                 are tutored each Saturday morning during the months of
                 January, February, and March. Instruction is matched to
                 constructivist theory with emphasis on active and
                 interactive inquiry. Principal and teachers teach and
                 plan lessons together.
Winter, 1996     One Instructional Design Team is disbanded because of
                 some members' continued resistance to reform.
Spring, 1997     The school is selected by the state Department of
                 Education to participate in a case study about
                 "Transforming Learning Communities" (TLC). Another
                 university mathematics educator joins the school
                 faculty to assist with the case study.
Spring, 1997     Second dissertation written by another teacher about
                 leadership of the school in the Venture Capital
                 process, The Value of Transformational Leadership in an
                 Exemplary School District: Examination of Conditions,
                 Processes and Practices Associated with School
                 Improvement. (Nader, 1997).
Spring, 1997     Constructivism is challenged at school-board meetings
                 by anti-tax group. Articles and memo are sent to the
                 district officials and the board members, discrediting
                 constructivist practice as a means to effectively teach
                 mathematics. Wall Street Journal articles and
                 California State Board of Education attack the NCTM
                 Standards and label the reform as "fuzzy math." Anti-
                 tax group uses Fourth Grade Mathematics Test scores as
                 indicator of student achievement.
Summer, 1997     Snapshots is revised. Teacher-leaders recommend to
                 principals to disband IDTs and continue reform with
                 only educators committed to instructional reform.
Summer, 1997     Some teachers request transfers. Their requests are not
                 accepted by other schools' principals.
Fall, 1997       Revision of Snapshots 1 is implemented. Principals
                 reorganize reform efforts around a teacher-leadership
                 team. Team initially includes 12 K-4 teachers and three
                 first-year teachers. Leadership team begins taking
                 active role in decision-making about the reform and
                 mentoring new colleagues.
Fall, 1997       Substitute shortage impacts reform and the school's
                 professional development efforts.
Fall, 1997       Teacher-leader team meets monthly with principals,
                 secondary mathematics teacher, and university
                 mathematics educator to build an instructional and
                 assessment framework in mathematics.
Fall, 1997       Teacher-leaders hampered by time constraints. Expanded
                 mathematics curriculum and a constructivist model of
                 instruction require more time. Time factor documented
                 in Prisoners of Time (National Education Commission on
                 Time and Learning, 1994). District requires one
                 "triangular conference" for each child.
Fall, 1997       To prepare students for the test, decision made to
                 create state mandated-type test objective, multiple
                 choice quizzes for third-grade students. Tensions
                 emerge over the philosophical consistency of this
                 approach. Teacher-leaders feel this is a "back to
                 basics" strategy.
Fall, 1997       After-school tutoring for 1st-4th grade students
                 begins. "Saturday Scholars" continue to meet.
Summer, 1998     State Test "report cards" put increasing pressure on
                 the entire school system. The school district is not
                 designated as "effective" on the state report card.
                 Snapshots revised. Tuition-free summer school for "at-
                 risk" in-coming fourth graders started at the school to
                 improve fourth grade test scores.
Summer, 1998     The results of the tests from 1996 to 1998 are about
                 69% passage rate. These results are below the school
                 district's average of 80%.
September, 1998  School Principals expand the extended learning programs
                 to include about 45% of the fourth grades in early
                 morning, after school, and Saturday morning in
                 mathematics and reading.
Fall, 1998       Third dissertation written by the school teacher.
                 Dissertation examines and compares the achievement of
                 African-American students in a traditional and
                 constructivist model of mathematics instruction
                 (Martin, 1998).
1999-2003        Results of the tests for five consecutive years are
                 about 90% passage rate. These results are the highest
                 in district for five consecutive years.
2003-present     The reform at the school continues to evolve.

The rationale for instructional reform in mathematics was clearly documented in 1989 by the NCTM and the NRC; however, how to lead this kind of reform and what impact such an epistemological shift would have on school change was unknown. Additionally, the literature did not anticipate how individuals within the organization would construe this change when traditional beliefs and practices were confronted with a different paradigm. It remained unclear how some educators may succeed in initiating and implementing major change.

School change at this K-4 school appears to be influenced by a major shift in epistemology. It is a shift that pulls away from dominant educational practices rooted in behaviorism toward practices reflective of constructivism. Constructivist learning theory, encouraged and supported by NCTM (1989, 1991, 1995, 2000) and other research recommendations (American Association for the Advancement of Science [AAAS], 1989; NCR, 1989; National Science Teachers Association [NSTA], 1996), as well as Kumon Mathematics Program may be responsible for changes beyond mathematics instruction at the school.

The emerging themes and concepts that are echoed in this special issue of Focus directly or indirectly address the notion of equity, curriculum, teaching, learning, assessment, and technology (NCTM, 2000). The following is a brief description of these papers.

An Overview of the Papers

Three out of four thematic papers in this special issue of Focus discuss the ongoing process of reform in a K-4 school. The fourth paper explores the impact of a state mandated mathematics test on the structure and culture of a conventional K-4 school. The four papers include: (1) social constructivism in practice: case study of an elementary schools' mathematics program, (2) culture, social interaction, and mathematics learning in a third grade classroom, (3) passion for teaching/learning mathematics: a story of two fourth grade African American students, and (4) the impact of a state mathematics test on the structure and culture of a K-4 school.

The first article, Social Constructivism in Practice: Case Study of an Elementary School's Mathematics Program, as the leading paper, examines implementation of social constructivist epistemology on teaching/learning of mathematics in a K-4 public school with particular focus on African American "at-risk" fourth grade students. It suggests that creating learning opportunities for all children to make sense of important mathematics concepts may require restructuring and recapturing schools. Moreover, it argues that fitting social constructivist theory into traditional school schedules and school environments may produce frustration and disappointment until new teaching/learning climates and new time schedules are designed. We suggest this paper be read prior to other papers because it provides general perspectives regarding the complexities and challenges of reforming mathematics education in this K-4 school.

The second paper, Culture, Social Interaction, and Mathematics Learning, investigates connections between five third grade students (one African American boy, one African American girl, one Caucasian girl, one Asian boy, and one Middle Eastern boy) regarding their beliefs and practices about mathematics and their social and cultural backgrounds. Furthermore, the study examines the relationships between the social interaction of these five students on mathematics learning and their mathematical dispositions. The study highlights the importance of establishing the classroom social norms and sociomathematical norms for mathematical dialogues. In addition, the paper discusses how the teacher's sensitivity to her students' cultural heritage created a risk-free climate for all students to reach their potential in mathematics.

The third paper, Passion for Teaching/Learning Mathematics: A Story of Two Fourth Grade African American Students, focuses on how two fourth grade African American students at a K-4 school experience mathematics learning and school in general, and what effects these experiences may have on their self-perceptions as mathematicians. A qualitative design employing in-depth interviews and participant observations were used to make sense of two students' mathematical environment and mathematical dispositions. Findings suggest that the school structure and culture positively influenced these students' beliefs and attitudes towards mathematics learning. In addition, the teachers' and the administrators' passion for reforming mathematics created a synergistic transforming learning community.

The fourth paper, The Impact of a State Mathematics Test on the Structure and Culture of a K-4 School, explores the influence of state testing on the structure and the culture of a K-4 school. It examines the practices of the principal and two fourth grade teachers, which were influenced by the state mandated tests. In addition, it investigates two fourth grade students' perceptions of mathematics. The study raises some serious questions about the implications of such "high stakes", "end-of-the-line" testing on the structure and culture of this school and the reform movements in mathematics education.

We hope this special issue about culture, communication, and mathematics learning is as inspiring and energizing to the educational community in general and to the mathematics education community in particular as it has been for all of us including university teachers and their students, several principals, many teachers, and thousands of young children and their parents for over fourteen years.

References

Alic, M. (1986). Hypathia's heritage: A history of women in science from antiquity through the nineteenth century. Boston: Beacon Press.

American Association for the Advancement of Science (1989). Science for all Americans: A project 2061 report on literacy goals in science, mathematics, and technology. Washington, D.C.: Author.

Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Thinking mathematically, integrating arithmetic and algebra in elementary school. Portmouth, NH: Heineman.

Carpenter, T.P., Fennema, E., & Frake, M.L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97, 3-20.

Carpenter, T.P., Lehrer, R. (1996). Teaching and learning mathematics with understanding. Technical report, National Center for Research in Mathematical Sciences Education, University of Wisconsin, Madison, WI.

Cowen, L. (1995). The impact of the NCTM Standards (1989) on the professional lives of elementary educators: A portrait of radical change. Doctoral dissertation. Cleveland State University Library.

Ernest, P. (1995). The one and the many. In L.P. Steffe & J. Gale (Eds.), Constructivism in Education. (pp. 459-486). Hillsdale, NJ: Lawrence Erlbaum Associates.

Flegg, G. (1989). Numbers through the ages. London: McMillan Publishing Company.

Fleener, M.J. (2002). Curriculum dynamics, recreating heart. New York: Peter Lang Publishing.

Gibbon, E. (1975). The decline and fall of the Roman Empire. New York: Washington Square Press.

Grundy, S. (1987). Curriculum: Product or praxis. New York, NY: The Falmer Press.

Hiebert, J., Carpenter, T.P., Tennema, E., Fuson, K.C., Wearne, D., Murray, H. (1997). Making sense, teaching and learning mathematics with understanding. Heinemann, Portsmouth, NH.

Jennings, M.M. (1996). MTV Math Doesn't Add up. Available: http://www.intres.com/math/mtvmath.htm

Kilpatrick, J. (1992). A history of research in mathematics education. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. (pp. 3-38). New York: McMillan Publishing Company.

Martin, M.K. (1998). One school interprets the NCTM Standards: Implication for African-American culture. Doctoral dissertation. Cleveland State University Library.

McLeod, D.B. & Ortega, M. (1993). Affective issues in mathematics education. In P.S. Wilson (Ed.), Research Ideas for the Classroom: High School Mathematics. (pp. 21-36). New York: McMillan Publishing Company.

Menniger, K. (1969). Number words and number symbols: A cultural history of numbers. Cambridge, MA: MIT Press.

Nader, J. (1997). The value of transformational leadership in an exemplary school district in Ohio: Examination of conditions, processes, and practices associated with school improvement. Doctoral dissertation. Cleveland State University Library.

National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1991). Professional Standards for Teaching Mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Education Commission on Time and Learning. (1994). Prisoners of Time: A Report of the National Education Commission on Time and Learning. Washington, DC: Commission.

National Research Council. (1989). Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D.C.: National Academy Press.

National Science Teachers Association. (1996). NSTA Pathways to the Science Standards: Guidelines for Moving the Vision into Practice. Washington, DC: Author.

Noddings, N. (1992). The challenge to care in schools. New York, NY: Teachers College Press.

Osborne, R. (1992). Philosophy for beginners. New York: Writers & Readers Publishing, Inc.

Steen, L.A. (1990). On the shoulders of giants: New Approaches to numeracy. Washington, D.C.: National Academy Press.

Steffe, L.P. (1995). Alternative epistemology: An educator's perspective. In L.P. Steffe & J. Gale (Eds.), Constructivism in Education. (pp. 489-523). Hillsdale, NJ: Lawrence Erlbaum Associates.

Wood, T. (1995). From alternative epistemologies to practice in education. Rethinking what it means to teach and learn. In L.P. Stefee & J. Gale (Eds.), Constructivism in Education. (331-339). Hillsdale, NJ: Lawrence Erlbaum Associates.

The authors acknowledge and appreciate the insightful comments and critique of the editor and associate editor on earlier drafts of their manuscripts in this issue.

Roland Pourdavood, Cleveland State University

Lawrence V. Svec, Lomond Elementary School

Lynn M. Cowen, Onaway Elementary School

Jeremy Genovese, Cleveland State University