T MATHEMATICS EDUCATORmath.coe.uga.edu/tme/issues/v19n1/v19n1.pdf · that of the 9,000 mathematics teachers in the state, 14.3% are not fully certified, and the average two-year attrition

____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____

Volume 19 Number 1

Summer 2009 MATHEMATICS EDUCATION STUDENT ASSOCIATION

THE UNIVERSITY OF GEORGIA

Editorial Staff

Editors

Ryan Fox Diana Swanagan Associate Editors

Tonya Brooks Eric Gold Allyson Hallman Hulya Kilic Laura Lowe Catherine Ulrich

Advisor

Dorothy Y. White

MESA Officers

2008-2009

President

Brian Gleason

Vice-President

Sharon O’Kelley

Secretary

Allyson Hallman

Treasurer

Richard Francisco

Colloquium Chair

Dana TeCroney

NCTM

Representative

Ryan Fox

Undergraduate

Representative

Emily Ferris Cynthia Thomas

A Note from the Editor

Dear TME Reader,

Along with my co-editor Ryan Fox, I welcome you to the first issue of the 19th volume

of The Mathematics Educator (TME). It is my hope that the articles found in this issue engage and sustain the discussion among members of our audience and the larger mathematics education community.

As has been our mission at TME, we aim to provide a variety of perspectives on issues within the mathematics education community. For this issue’s In Focus piece, Nicholas Oppong, along with participants in his seminar, discuss Georgia’s new teacher incentive programs, along with how these programs will improve education in the state of Georgia. In this issue there are three articles that provide a variety of perspectives on issues within mathematics education. In our first piece, Lu Pien Cheng and Ho-Kyoung Ko discuss how teachers working in teams progress through development stages and how these stages impact teacher growth. Revathy Parameswaran investigates students’ concept images of Rolle’s Theorem, along with how these images are related to the Mean Value Theorem. Finally, Tracie McLemore Salinas shows us the various criteria used by preservice elementary teachers when evaluating student-generated algorithms.

There are many people involved here at MESA that have helped in making this issue possible, and I would like to thank them at this time. I want to thank my colleagues who worked as Co-Editor and Associate Editors for this issue; their names appear at the top of the column to the left. I cannot thank you enough for all that you did!

Diana Swanagan

105 Aderhold Hall [email protected] The University of Georgia www.coe.uga.edu/tme Athens, GA 30602-7124

About the Cover

On the front cover of this issue, we include a sketch from Revathy Parameswaran’s piece on Rolle’s Theorem. In her article, she discusses students’ different concept images that are invoked when working on tasks related to Rolle’s Theorem.

This publication is supported by the College of Education at The University of Georgia

___________ THE ________________

__________ MATHEMATICS ________ _____________ EDUCATOR ____________

An Official Publication of

The Mathematics Education Student Association The University of Georgia

Summer 2009 Volume 19 Number 1

Table of Contents

3 In Focus… Georgia’s Compensation Model: A Step in the Right Direction NICHOLAS OPPONG, ZANDRA U. DE ARAUJO, LAURA LOWE,

ANNE MARIE MARSHALL, LAURA SINGLETARY

8 Teacher-Team Development in a School-Based Professional Development Program

LU PIEN CHENG, HO-KYOUNG KO

18 Understanding Rolle’s Theorem REVATHY PARAMESWARAN

27 Beyond the Right Answer: Exploring How Preservice Elementary Teachers Evaluate Student-Generated Algorithms TRACIE McLEMORE SALINAS

35 Submissions information 36 Subscription form

© 2009 Mathematics Education Student Association

All Rights Reserved

2

The Mathematics Educator 2009, Vol. 19, No. 1, 3–7

Nicholas Oppong, Zandra U. de Araujo, Laura Lowe, Anne Marie Marshall, Laura Singletary 3

In Focus…

Georgia’s Compensation Model: A Step in the Right Direction

Nicholas Oppong, Zandra U. de Araujo, Laura Lowe, Anne Marie Marshall, Laura Singletary

In the 2006-2007 school year, Georgia colleges

and universities produced almost 2,000 early childhood teachers, while only graduating 140 mathematics teachers. The shortage of mathematics teachers in the state of Georgia is better understood when one realizes that of the 9,000 mathematics teachers in the state, 14.3% are not fully certified, and the average two-year attrition of mathematics teachers is approximately 773, or 9%. The most recent figures show that after three years the retention rate for all of Georgia teachers is 73.0%, after five years 62.1%, and after eight years the retention rate is 51.3% (Henson, 2008). Obviously, there is a problem with the recruitment and retention of highly qualified mathematics teachers in Georgia.

The state of Georgia has recognized the drastic need to recruit and produce highly qualified mathematics teachers. In response to this need, on April 22, 2009, Georgia’s Governor Sonny Perdue signed House Bill 280 as an amendment to the “Quality Basic Education Act.” House Bill 280 attempts to address the shortage of fully certified secondary mathematics and science teachers by offering additional compensation to aide recruitment. Georgia House Bill 280 (2009) states:

…a secondary school teacher in a local school system who is or becomes certified in mathematics or science by the Professional Standards Commission shall be moved to the salary step on the state salary schedule that is applicable to six years of creditable service, unless he or she is already on or above such salary step. From such salary step, the teacher shall be attributed one additional year of creditable service on the salary schedule each year for five years.

Once teachers complete five years of service, they “may continue to be attributed one additional year of creditable service on the salary schedule” if they satisfy or surpass the requirements of the “achievement criteria” to be determined by Office of Student Achievement (Ga H.R. 280, 2009). After five years if the teacher fails to meet the expectations of the

“achievement criteria,” the bill then requires that “such teacher shall be moved to the salary step applicable to the actual number of years of creditable service which the teacher has accumulated.”

The bill also provides incentives for elementary teachers to earn endorsements in mathematics or science:

…a kindergarten or elementary school teacher in a local school system who receives an endorsement in mathematics, science, or both from the Professional Standards Commission shall receive a stipend of $1,000.00 per endorsement for each year each such endorsement is in effect, up to a maximum of five years. (Ga H.R. 280, 2009)

Elementary teachers with the qualifying endorsements will continue to receive the stipend after the five-year period if they satisfy the “achievement criteria” established by the Office of Student Achievement. An additional stipulation is a guarantee that the stipend will be revoked any time after the five-year period if the teacher fails to meet the criteria for achievement.

Moses and Cobb (2001) state that student achievement in mathematics is a vital component for a student’s success in their future endeavors. Research has identified that the quality of a teacher is the single most important school related factor that influences a student’s achievement (Sanders & Rivekin, 1996). Therefore, in order to provide the best possible mathematics education for all students, the recruitment and retention of highly effective mathematics teachers is imperative. While House Bill 280 may provide some methods for the recruitment of highly qualified mathematics teachers by providing additional compensation, the question of how to retain such highly qualified teachers is still left unanswered. Exploring the evolution of the teacher pay models as well as endeavors in alternative forms of compensation inform our discussion of House Bill 280. This understanding will provide a perspective in a critical evaluation of the bill and corresponding recommendations.

Georgia’s Compensation Model 4

The evolution of teacher compensation models in the United States is directly linked to the organizational needs of educating large numbers of students as well as the economic and societal trends driven by industrialization. These models are categorized into three distinct shifts: boarding round, position-based salary, and single-salary schedule (Protsik, 1995). The investigation of how teacher compensation has changed over time provides an argument that any type of compensation reform will result from societal and economic alterations.

The first phase, referred to as the “boarding round,” typifies the barter-economy of the 19th century where people traded goods for services. In this case, teachers provided their services for weekly room and board at the various homes of pupils’ parents. This mode of compensation came to an end with the shift of the US economy from an agrarian class to a more urban, industrial one of the 20th century (The North Dakota Legislative Council, 2001).

The thriving industrialized economy required fewer young people to support the farming industry, resulting in more students attending public schools. This unprecedented abundance of new students was the catalyst for school reform that took the shape of a more controlled mass education system, complete with bureaucratic layers of principals and superintendents to support a grade-level system. The compensation model resulting from these changes was intended to create a uniform pay schedule. This model defined pay levels by years of experience, gender, race, grade level taught, and allowed for subjective merit pay to be determined by the administrators (Tyack & Strober, 1981). This type of delineation was laced with aspects of racism, sexism and administrator subjectivity, which ultimately gave way to the third phase, a single-salary schedule (Protsik, 1995).

The 20th century was a time of “equal pay for equal work.” In education this translated to a compensation model where equal work amounted to years of service and degree level (Clardy, 1988). The advantages of this reform offered administrators ease in developing budgets as well as negotiating contracts. Additionally, this model influenced the teacher-administrator relationship, in that administrators no longer dictated teachers’ salaries (English, 1992; Lipsky & Bacharach, 1983). However, this model does not seem to support the evolving needs of 21st century schools. As with the previous shifts, compensation reform will likely be the result of societal and political pressures to produce new critical thinkers ready to succeed in our fast-paced global economy.

As this historical account has demonstrated, reform compensation models are reactions to a variety of societal or economic pressures. With the influx of technology in the past two decades, yet another wave of compensation models has emerged. The emerging Information Age and the resulting globalization have fueled the idea of mathematics for all. Once used as a gate-keeper, some now see mathematics as an impetus towards equality for all students. These pressures have brought about models that generally fall into three categories: pay for performance, differential pay, and alternative compensation.

The pay for performance models takes the form of increased pay on an individual teacher or on an individual school level. Individual teacher rewards generally depend on their students’ standardized test scores, performance evaluations, additional training, or National Board Certification. School rewards are usually tied to school-based goals and benchmarks involving student test scores, absenteeism, and dropout rates. This type of model was first implemented in Douglas County, Colorado in 1994. In this school district, teachers received small bonuses for acquiring new skills and tied their annual salary increase to satisfactory performance evaluations (Odden & Wallace, 2004). The differential pay model is offered to teachers in high need areas, either as a one-time bonus or ongoing supplemental pay. For example, in the 2006-2007 school year, North Carolina provided signing bonuses of $15,000 to mathematics teachers who chose to work in a select group of schools (Silberman, 2006). Examples of alternative compensation include providing teachers with incentives like low-interest loans and student loan forgiveness. Each model is tied to required service in the local schools. Most common is a combination of these models. Georgia’s House Bill 280 is one example of a combination model, in that the pay is differential due to its impact on only mathematics and science teachers, but it also includes an individual reward aspect after the fifth year. The structure of these models supports Richard Ingersoll’s findings that “the prevailing policy response to these school staffing problems has been to attempt to increase the supply of teachers” (p.5, 2003). That is to say that the focus of these models is on recruitment rather than retention of teachers.

In enacting this bill, we believe that the state of Georgia has made a move in the right direction. We think this bill will not only help recruit new teachers to fill secondary mathematics vacancies, it could potentially decrease the percentage of secondary


mathematics teachers that are not fully certified, which currently stands at 14.3% (Henson, 2008). Another promising aspect of the bill is its inclusion of incentives for elementary school teachers. In providing current elementary school teachers with a monetary incentive for earning mathematics and science endorsements, we think the students will be the main beneficiaries. Liping Ma (1999) found that teachers with a better understanding of the mathematical content they teach are more effective teachers, therefore the bill’s inclusion of at least two elementary content courses as a requirement to receive the extra compensation will help improve the quality of mathematics teaching.

While we applaud the aforementioned aspects of Georgia’s law, we think that there are some areas the law neglects to address. First, the law omits discussion related to teacher performance until the sixth year of teaching. During the first five years of teaching, teacher pay is not tied to performance, however upon the sixth year the law makes a provision that teachers must meet “achievement criteria” to continue to move up the teacher pay scale. In order to prepare teachers for their impending performance evaluations in the sixth year, we propose that Georgia should help support teachers during the first five years through quality mentoring and induction programs in addition to exhibiting progress in performance evaluations. Related to the subject of teacher evaluations is the fact that the “achievement criteria” that teachers are required to meet in order to continue to receive additional pay is not readily available. In order to remedy this, we think teachers should be made aware of the requirements of the achievement criteria they must meet prior to entering the classroom.

In regard to the elementary school endorsements, the Bill (2009) stipulates that the “math and science endorsements shall…be based on post-baccalaureate nondegree programs, independent of an initial preparation program in early childhood education.” We question the requirement that the endorsements must be earned post-baccalaureate. This stipulation does not allow teachers who acquired the endorsement during their undergraduate education to receive the additional pay. We believe that all teachers who attain the endorsements, regardless of when, should be compensated.

The effect this law will have on veteran mathematics teachers’ morale is of concern. Under this law a fifth year mathematics teacher will be earning the same salary as a first year teacher because the law has no provisions for mathematics teachers currently

teaching with over five years of experience. One way to address this concern is through performance pay for teachers. Teachers who meet teacher quality standards that are well researched, developed and implemented should qualify for the performance pay. Additionally, differential pay models may be enacted where mathematics teachers who acquire National Board Certification or higher degrees in their content area or areas applicable to those they teach will receive additional compensation. This is in contrast to Georgia’s current model where teachers receive additional compensation for any advanced degree. Another method to consider is to allow all mathematics teachers to receive additional compensation for choosing to work in high needs schools. These schools have historically been the most difficult to staff and have the highest percentages of teachers out of field. These recommendations may aid in teacher retention as well as the concerns of teacher morale.

The fact that there is no guarantee for the length of time the law will be in effect is problematic. A guarantee of funding for a set period of time could provide stability that may be more effective in recruiting and retaining teachers. Furthermore, the staying power of this law is questionable because when the bill was signed, the State of Georgia had yet to allocate funds for this measure that carries an anticipated annual cost of 9.9 million dollars (McCaffrey, 2009). Guaranteeing a ten year continuance of this pay model with an option for renewal will allow the state to assess the law’s effectiveness at combating the mathematics teacher shortage. In addition, this guarantee of funding would provide more security for the teachers which in turn may lead to better retention.

It is important to distinguish between the recruitment and retention of teachers when discussing ways in which we may solve the teacher shortage problem. Potentially, the most critical flaw we found in the bill is that although it does address the recruitment aspect, we find it does little to promote retention of current mathematics teachers. In the 2005-2006 and 2006-2007 school years, secondary mathematics teachers had an average annual attrition rate of 9.6% (Henson, 2008), while the five year annual attrition rate for all teachers with no previous experience was an even grimmer 41% (Afolabi, Eads & Nweke, 2007). These numbers suggest that simply hiring more new teachers will not stop the shortage; we must also put forth effort to retain our teachers. In a report on the possible teacher shortage, Richard Ingersoll (2003) found “a strong link between teacher turnover and the

Georgia’s Compensation Model 6

difficulties schools have adequately staffing classrooms” (p. 9). Therefore, in addition to the recommendations we have previously stated, we encourage school districts to implement comprehensive mentoring and induction programs for teachers. In a 2004 study, Ingersoll and Smith found that teachers who did not participate in an induction program had a 40% probability of turnover; this percentage fell to 28% for teachers who had some type of induction. Furthermore, this turnover percentage decreased to less than 20% for teachers who participated in what Ingersoll and Smith defined as a “full induction” program consisting of aspects such as common planning time with colleagues, a helpful mentor in their content area, regularly scheduled collaboration time to discuss instructional issues, and a beginning teacher seminar.

When it comes to reforming mathematics teachers’ compensation in order to recruit and retain effective teachers, Georgia’s House Bill 280 is a step in the right direction, but the state still has a long way to go. In consideration of both the strengths and weaknesses found in Georgia’s bill, we urge the rest of the nation to build upon the strengths and develop new ideas to address the weaknesses. As those interested in mathematics education, we must ask what we can learn from this bill that Georgia has produced. How can we make this law and others like it, better so that the end result is students’ success in mathematics? The bill may aide schools in the recruitment of mathematics teachers, but we need to do more than continually recruit teachers. We need to find methods and models to support effective mathematics teachers in order to keep them in the classroom. We must find ways to not only recruit and retain effective mathematics teachers for the best and the brightest students, but also to provide all students with highly effective mathematics teachers.

So, where do we go from here? In order to continually move forward, we encourage educational stakeholders to support teacher compensation models that advance the vision of mathematics for all. By continuing to ask the right questions and debate the pertinent issues we will improve the practices of recruitment and retention of effective mathematics teachers. Here are some questions to get us started. How can we make laws that adequately address both teacher recruitment and retention? How can alternative forms of compensation affect recruitment and retention? Considering previous and current attempts at alternative models of teacher compensation, what are the potential consequences of these programs, and how

can we avoid them in the future? What are the evaluation processes and standards for alternative models of compensation? We encourage you to start asking these questions. Then, go beyond asking, and begin looking for answers. A serious discussion needs to take place, a discussion that would result in all students receiving the best mathematics education this nation can offer through effective mathematics teachers.

References

Afolabi, C., Eads, G., & Nweke, W. (2007) Supply and demand of Georgia teachers. Georgia Professional Standards Commission. Retrieved May 12, 2009 from http://www.gapsc.com/Research/Data_Research.asp

Clardy, A. (1988, May 1). Compensation systems and school effectiveness: Merit pay as an incentive for school improvement. (ERIC Document Reproduction Service No. ED335789).

English, F. (1992). History and critical issues of educational compensation systems. In L. Frase (Ed.), Teacher compensation and motivation (pp. 3–25). Lancaster, PA: Technomic Publishing.

Henson, K. (2008). Georgia teacher shortages, supply and demand. Georgia Professional Standards Commission. Retrieved May 12, 2009 from www.gapsc.com/MessageCenter/downloads/KellyHenson_PSC_HR1103_ 20080827.pps

Ingersoll, R. (2003). Is There Really a Teacher Shortage? Consortium for Policy Research in Education, University of Pennsylvania. Retrieved May 12, 2009 from http://depts.washington.edu/ctpmail/PDFs/Shortage-RI-09-2003.pdf

Ingersoll, R., & Smith, T. (2004). Do Teacher Induction and Mentoring Matter? NASSP Bulletin, 88(638), 28–40. (ERIC Document Reproduction Service No. EJ747916) Retrieved May 22, 2009, from ERIC database.

Lipsky, D. & Bacharach, S. (1983). The single salary schedule vs. merit pay: An examination of the debate. Collective Bargaining Quarterly, 11(4), 1–11.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum.

McCaffrey, S. (2009, April 22). State ups ante to lure teachers: Math, science educators pay gets a boost. The Associated Press, Retrieved May 12, 2009 from http://www.onlineathens.com/stories/042309/new_431926608.shtml

Moses, R. & Cobb, E. (2001). Radical equations: Math literacy and civil rights. Boston: Beacon Press.

The North Dakota Legislative Council, (2001). Teacher compensation package-background memorandum. Retrieved My 12, 2009, from http://www.legis.nd.gov/assembly/57-2001/docs/pdf/39018.pdf

Odden, A. & Wallace, M. (2004) Experimenting with teacher compensation. The School Administrator. Retrieved May 12, 2009 from http://www.aasa.org/publications/saarticledetail.cfm?ItemNumber=1128&snItemNumber=950&tnItemNumber=1995


Protsik, J. & Consortium for Policy Research in Education, (1995, February 1). History of teacher pay and incentive reforms. Madison, WI: Finance Center. (ERIC Document Reproduction Service NO. ED380894).

Quality Basic Education Act Amendment to Part 6 of Article 6 of Chapter 20 of Title 20, HB280, Georgia, House and Senate, (2009). Retrieved May 12, 2009 from http://www.legis.state.ga.us/legis/2009_10/fulltext/hb280.htm

Sanders, W. L. & Rivekin, J. B. “Cumulative and residual effects of teachers of future student academic achievement.” University of Tennessee. (1996).

Silberman, T. (2006, November 29). Guilford draws teachers with higher pay: Recruiting math teachers – An experiment. The News and Observer, Retrieved May 12, 2009 from http://www.nctq.org/nctq/research/1169567986042.pdf

Tyack, D. (1974). The one best system: A history of American urban education. Cambridge, MA: Harvard University Press.

Tyack, D. & Strober, M. (1981). Women and men in schools: A history of the sexual structuring of educational employment. Washington DC: National Institute of Education.



Teacher-Team Development in a School-Based Professional Development Program

Lu Pien Cheng Ho-Kyoung Ko

This paper documents how a team progressed through the five stages of team development as a result of a school-based professional development program using a laboratory class cycle. Six Grade two teachers and their administrator in a primary school in the south eastern United States participated in the study. All the teachers were interviewed at the end of each laboratory class cycle. Their administrator was interviewed after the program ended. A grounded theory approach and constant comparative method were used. The study revealed how the teachers’ participation in the program progressed according to Tuckman and Jenson’s (1997) model of team development in the laboratory class cycle. Establishment of trust among teachers and team support over an extended time were identified as important factors in shaping the team development.

Background of Study

A substantial amount of literature has been published on the topic of team development. Among these, in the annual series Advances in Interdisciplinary Studies of Team Works (Beyerlein, Johnson, & Beyerlein, 1995), theory and practice are related to stages and processes of development. According to Beyerlein, Johnson, and Beyerlein, models of team learning and factors that interfered or aided the team’s growth in petrochemical companies, manufacturing companies, and Naval Training Systems have been published in journals and books. The researchers at the Naval Air Warfare Centre Training Systems Division in Florida Centre studied work teams extensively in the past 15 years, and refined and expanded on studies using rigorous methodologies. This enabled them to examine some team issues that most researchers have neglected. In education, Kruse and Louis’s study (1997) suggested that teams can be vehicles for building professional community and school improvement. Dechant, Marsick and Kasl

(2000) claim there are barriers to effective teams; teams need to deal with conflicts before they can function effectively. In order to examine how team members resolve conflicts at different points in time, we use a stage model of development. This model has yet to be applied to mathematics teachers working in a school-based professional development experience.

Professional development based on a training paradigm that implied teachers were deficit in skills and knowledge (Guskey, 1986) became a major enterprise in education during the Post-Depression era. This paradigm aimed at teachers’ mastery of prescribed skills and knowledge and resulted in one-time workshops. Several studies found this approach to be ineffective (Fullan, 1991; Guskey, 1986; Howey & Joyce, 1978; Johnson, 1989; Lovitt & Clarke, 1988; McLaughlin & Marsh, 1978; Wood & Thompson, 1980). The ineffective attempts to motivate teacher change based on the training-mastery model of professional development programs triggered research in professional development and teacher change. A significant outcome of this research has been the shift in focus “from programs that change teachers to teachers as active learners shaping their professional growth through reflective participation in professional development program and in practice” (Clarke & Hollingsworth, 2002, p. 948). Johnson (1996) presented a case for reconceptualizing teacher professional development as “opportunities for learning” to enable it to be “embedded into the on-going work of the school” (p. 12). School-based professional development became a trend with a focus on school-based management: “Both trends are based

Lu Pien Cheng is an assistant professor with the Mathematics and Mathematics Education Academic Group at the National Institute of Education (NIE), Nanyang Technological University, Singapore. She has specific research interest in teacher education. Ho-Kyoung Ko is currently a professor in the department of Mathematics Education at Wonkwang University in South Korea. She is researching and developing "Silver Math," which provides math education for the elderly. Also, she research "Factors of Difficulty of Mathematics items" and "Programme of professional development for teachers."

Lu Pien Cheng, Ho-Kyoung Ko 9

on the increasingly accepted belief that the school, rather than the district (too large a unit) or the individual (too small a unit), is the most appropriate unit of change” (Gordon, 2004, p. 10). This indicates that school-based professional development can focus on specific students’ needs and immediate classroom application more than professional development conducted outside of school (Truscott & Truscott, 2004). Demulder and Rigsby (2003) provided evidence that a school-based program affected teachers’ personal and professional growth, transforming their classroom practice. They argued that the program worked well for these teachers, and attributed both the personal and professional transformation of the teachers to their experiences in the program. Truscott and Truscott added that educators have found “opportunities to shift the emphasis of school-based consultation from addressing problems toward developing consultee skills, knowledge, and confidence toward a more positive and preventive model” (p. 51).

In recent years, the focus of professional development has been subject matter and teaching and learning (Cohen, 2004, p. 3). This new genre of professional development gives rise to questions concerning the culture of teaching and learning. It also provides the impetus to build intellectual community among teacher communities during professional development. Critique and disagreement are welcome in this new genre of professional development as teachers redefine teaching practice and engage in learning. Using the group as the unit of analysis, research provides evidence that “strong professional learning communities can foster teacher learning and instructional improvement” (Little, 2002, p. 936). The Community of Teacher Learners project (Grossman, Wineburg, & Woolworth, 2001) and the QUASAR (Quantitative Understanding: Amplifying Student Achievement and Reasoning) project (Lane & Silver, 1994; Smith, 1997; Stein, Silver, & Smith, 1998; Stein, Smith, & Silver, 1999) illustrate this finding. However, these two projects revealed that the development of teacher communities is difficult and time-consuming (Grossman et al.; Stein et al., 1999). One of the most important features of a successful learning community is the establishment of norms that promote supportive yet challenging conversations in the community about teaching. Although teachers generally welcome the opportunity to discuss ideas and materials related to their work, discussions that support a critical examination of teaching are relatively rare (Ball, 1994; McLaughlin & Talbert, 2001; Putnam & Borko, 1997;

Wilson & Berne, 1999). Professional developers must foster such discussions by helping the teachers establish trust among members of the team, develop communication norms that enable critical dialogue, and maintain a balance between respecting individual community members and critically analysing issues in their teaching (Frykholm, 1998; Seago, 2004).

With more professional development becoming school-based, and greater emphasis on the learning community, this study hopes to shed greater light on factors that promote team learning and growth to increase the effectiveness of school-based professional development effort. This article documents how a team of six primary school mathematics teachers developed through the five stages of team development by Tuckman and Jenson (1977) as a result of a school-based professional development program that used a teaching cycle. Using qualitative research methodology within an interpretive theoretical frame, we interviewed six second-grade primary school teachers at different points of the study regarding their conceptions of team development. We also interviewed their administrator at the end of the program to provide another perspective of how the team had developed. The research questions that guided this study were as follows:

1. What are the teachers’ and school administrator’s conceptions of change as a team as a result of this school-based professional development experience?

2. What are the factors that the teachers and school administrator attributed to the change as a team?

Conceptual Framework

Framework for Professional Development

We employed teaching cycles in the study, where teaching cycles in a real school setting are “centered in the critical activities of the profession, that is, in and about the practices of teaching and learning” (Ball & Cohen, 1999, p. 13). The teaching cycle consisted of three consecutive phases: preparation, observation, and analysis. The professional developer’s role is to facilitate and support the teachers’ learning during each phase of the cycle. In all the three phases, team members work together to plan, observe, and critique mathematics lessons. This study included six teaching cycles over the course of one academic year. The teaching cycle used in this professional development was a type of reform activity that “situates the professional education of teachers in practice” (Smith, 2000, p. 2) and aims at providing a connected


contextualized set of experiences on which teachers can reflect more critically about their beliefs and practices. According to Smith, the work of teaching should be used to create opportunities to critique, inquire, and investigate; the materials for the teaching cycle aimed to achieve that purpose.

The professional development program in this study was designed around the three core features and the three structural features identified by Garet, Porter, Desimone, Birman, & Yoon (2001). The three core features are content focus, active learning, and coherence. The three structural features are activity type, duration, and collective participation. The teaching cycle was conducted for teachers in the same school during the teachers’ common planning time in the regular school day. They shared the same curriculum that offered a common platform for the group to discuss concepts, skills, and problems that arose during their professional development experiences. By the nature of this particular professional development experience, the teachers were actively engaged in meaningful, planning, practice, and reflective discussion throughout the study. The content and pedagogy of activities were aligned with national, state, and local frameworks, standards, and assessments.

Stages of Team Development

We viewed the team development process in this study through the lens provided by Tuckman’s (1965) model of the developmental sequence in small groups. Tuckman maintained that the way teams develop has a direct impact on both their task and social outcomes. Tuckman synthesized 55 studies of groups to produce a generalizable model of the development of groups. Participants in the studies were therapy groups, human relations training or T-groups, and natural and laboratory-task groups. His original findings uncovered four developmental stages: forming, storming, norming, performing. In 1977, Tuckman and Jenson expanded the earlier group development model to include a fifth stage, adjourning, based on additional studies of group behavior. Stage I (forming) involves testing what roles and interpersonal behaviors within the group are acceptable and how team members will relate to one another. Stage II (storming) is characterized by conflict, as team members assert their individuality and debate over the team’s goals, norms, and decision-making process. Stage III (norming) includes the emergence of group cohesion and harmony. The group begins to develop into a functioning unit as members agree on rules, roles, relationships, responsibilities, processes, and tasks to

be accomplished. Stage IV (performing) shows the team fully functioning as members actively involved in roles, leading to problem solving. At this stage, members identify with the team and commit to the team’s mission. Stage V (adjourning) brings closure to the process and determines that team’s mission is complete. The group either disbands or renews itself by establishing a new mission (Gordon, 2004). Tuckman and Jensen’s model is used in this study because it serves as a helpful starting point to think about the groups the researcher participated in and encountered.

Research Design and Data Collection

This study drew on an interpretative, qualitative case study design to investigate team learning and development. To obtain detail and rich descriptions of the learning and development processes, we conducted multiple sequential interviews with the teacher participants at the end of each cycle of preparation, observation, and analysis. Field notes were taken for each session with the teachers.

Research Site

Dayspring Primary School is a public school in the southeastern United States. At the time of the study, the primary school had about 400 students and 24 teachers, and the majority of the students were from low-income families. Teachers at each grade level had the same time during the afternoon to meet together to plan lessons and activities and to compare their students’ work. This common planning time was a regular occurrence for the teachers across the grade levels in the school. There were six classes of second graders with an average of 19 students in each class during the study.

Participant Selection and Participants

The team in this study was the only second grade team at Dayspring. Group members included individuals of different ethnicities, years of experience, and perspectives on teaching and learning mathematics. The professional development program lasted two years, and research data was collected during the second year of the program. At the end of the first year, one of the six teachers became a part-time third-grade teacher, and another teacher, Linda , was promoted to school vice-principal. We included Linda was included as a participant in the study as an administrator because her role provided a wider lens on the team’s growth. Two teachers, Ivy and Mary, joined the second grade team in the second year. Kay, Macy, Anna, and Lana were the other teachers on the second grade team during the second year of the program. All


six of the grade two teachers and the vice-principal participated in the study. See Table 1 for more

information about the participants in the study.

Table 1

Descriptions of the Teachers in the Study

Teacher Mary Ivy Kay Lana Anna Macy Linda Ethnicity White African

American White White White African

American White

Years taught 8 9 9 3 18 27 Identification of students taught

Gifted Mixed ability Mixed ability Gifted Mixed ability N/A

Additional characteristics of the teacher

First time teaching 2nd grade.

New to school. Trained to use mastery learning.

Only taught 2nd graders. Believed in direct instruction.

Considers herself a novice. Believed in direct instruction.

Team leader. Believed in direct instruction.

Worked with Anna for nine years. Believed in mastery learning, direct instruction.

Believed in developing students’ mathematical thinking through using questioning, manipulatives, and activities.

Teacher’s Education

Enrolled in master’s of education program.

Completing a specialist’s degree.

Master’s degree in early childhood education.

Bachelor’s degree

Bachelor’s degree in early childhood education

Bachelor’s degree in science

Data Sources

The sources of data were interviews and field notes to capture participants’ perceptions of the team’s development over time. An interview guide listing the questions or issues to be explored in the course of an interview was prepared to “ensure that the same basic lines of inquiry are pursued with each participant interviewed” in the early part of the interview (Patton, 2002, p. 344). The participant is free to pursue any subjects of interest that arise in latter parts of the interview (Patton, 2002). Each teacher participated in eight interviews. All the interviews were usually conducted in the participants’ classrooms at the end of a school day and audio-taped then transcribed by the researchers. We conducted the first interview before the program began to elicit the teachers’ beliefs about teaching and learning mathematics. The next six interviews occurred almost immediately after each laboratory class cycle to trace the team’s development, growth, or progress. We conducted the final interview with the teachers three months after the program ended to summarize their professional development experiences. The administrator only took part in the final interview. All the individual interviews with the participants were face-to-face interviews that lasted approximately 40 minutes each. Taken together, the interview data allowed us to investigate team development over time.

Data Analysis

Charmaz (2000) suggested five techniques for using the constant comparative method: (a) comparing aspects of different people (such as their views, situations, actions, accounts, and experiences), (b) comparing data from individuals with data from themselves at different times, (c) comparing an incident with another incident, (d) comparing data with a category, and (e) comparing a category with other categories. In this study, we compared incident with incident to analyze teachers’ perceptions of team development in the professional development program. Each cycle of preparation, observation, and analysis is defined as an incident. There are four stages in the constant comparative method (Glaser & Strauss, 1967). In Stage 1 of the constant comparative method, the teachers’ reactions to their team learning and development were coded to identify key elements of each learning episode. Using those codes, many categories of analysis were formed. Some of the categories were as follows: persons involved in the nature of activity, setting and timing of the episode, and participants’ immediate responses. In Stage 2 of the constant comparative method, more interview data were included to connect each team development experience to the created categories. In Stage 3, using theoretical criteria described by Glaser and Strauss, the list of categories for collecting and coding data was cut


down to focus on applicable incidents. In Stage 4, the theoretical framework was written to provide the content behind the categories and their properties. A hypothesis was developed as a result of the framework to explain certain social processes and their relationships.

Findings

We describe how the team developed according to five stages of team development (Tuckman & Jenson, 1977): forming, storming, norming, performing, and adjourning. The model, however, did not completely fit the six teachers’ behavior. This team was started at the second stage of Tuckman’s model, storming, before experiencing the other stages.

Storming

The storming stage was characterized by conflict. According to Ivy, tension resulted from cliques forming within the team of teachers. Mary and Kay were sensitive to criticism leveled at them by Macy and this created an atmosphere of uneasiness. In response, Mary was reluctant to share what she considered effective teaching practices. Kay, being new to the team, was overwhelmed by the team’s dynamics thus contributing to her discomfort.

Anna and Macy, team members for more than a decade, were good friends and went through the mastery program together. They were not satisfied with the composition of the current team and had difficulty accepting new teachers to this team. Part of the conflict that arose can be explained by the differences in Anna and Macy’s preferred methods of mathematics instruction with respect to other team members. Before the program, they had mostly used what they called the show-and-tell method, along with drill-and-practice. They had been unable to accept other ways of teaching because they believed that the newer methods of teaching were a threat to their confidence and authority as teachers. As the novice teachers tended to advocate more innovative teaching styles Anna felt that her teaching methods were becoming obsolete. Linda also observed that “there was very little collaboration, more hostile feeling…in the sense that there was inferiority in that team”. Rather than dealing with the conflicts, Mary, Anna, and Kay withdrew from the team. Initially, I (Cheng), as the professional developer, did not help the team resolve the conflicts. Instead, I focused on the teaching cycle and how it could be refined to meet the teachers’ expectations. The teachers were more willing to participate in the research study once they realized I was there in a supportive and non-judgmental capacity.

Transition from Storming to Forming

The team began to form after receiving direction from the administration. This occurred during the first three laboratory class cycles. In this stage, members determined how they would relate to other team members, and it was a period of anxiety for them.

There were also moments when Mary, Kay, Anna, and Macy were in different stages at the same time. For example, Mary and Kay continued to receive harsh critiques from Macy, an issue still unresolved by the third laboratory class cycle. Mary, still affected by the tensions in the team, coped by not sharing her ideas during team meetings. Instead, during the interviews, she suggested many ideas to help the team grow and bond. Kay managed the situation by concentrating on the support she received from some of the team members. She was also unsure of herself as a teacher in the school and preferred to focus on her teaching rather than activities of the team.

During the first three laboratory class cycles, Macy could not fully form with all members of the team. She was disconnected from the new team members because of a perceived lack of acceptance of her teaching style. Macy also said that some teachers needed to be more accepting of their team members and other teachers were open to suggestions, but only from certain team members. Because the current team was not united, she thought her former team, having the same members for many years, was much closer. Linda said Anna coped with the tension by isolating her teaching practice from colleagues and playing a passive role as the team leader during the first year of the program.

At the beginning of the second year of the professional development, Anna started to assume her role as a team leader by having an agenda for each meeting to help the team focus on the issues to be addressed. She requested each team member submit announcements in advance of meetings and prepared a copy of the compiled announcements for each member at meetings. During the third laboratory class cycle, Anna said she started to allocate time for each announcement so that more time could be devoted to the professional development program. Ivy said that the agenda helped the team to stay focused. When Lana and Ivy were in the forming stage, they believed in teamwork and looked to the team for support. They recognized the conflict and tensions among the team members and were hopeful that the team would learn to cooperate.


Norming

According to the teachers, the transition from the forming to the norming stage was influenced by the administrator’s insistence on the teachers setting norms during the two years of the program. Talking about the norms during the meeting allowed the teachers to set expectations for the team, such as maintaining productive dialogue, and to set the team’s mission. Everyone felt uneasy with the norms at first because according to Kay “if we put [the norms] on paper we felt like we have to do it.”

In the norming stage, the team developed trust and began collaborating. This stage occurred during the fourth laboratory class cycle and continued throughout the rest of the research study. As the professional developer, I was leading the teachers less; the teachers appeared to be leading and charting what they wanted to plan for their lessons. Collaboration and trust could be seen when the teachers worked in pairs to plan demonstration lessons and critique each other’s implementation of those lessons. The team was now comfortable sharing their ideas around the table, potentially improving the quality of the meetings.

As the teachers assumed a greater role in the team leadership, they agreed on their roles and responsibilities. Mary and Ivy assumed special roles on the team. Because the school used different textbook series for first and second grade, Mary’s experiences teaching the first grade the previous year helped the team to fill in some gaps in the second grade curriculum. Teaching in the higher grades in the same school system helped Ivy contribute to the team in planning lessons that prepared students for the next grade level. Her specialist degree program exposed her to new ideas about teaching. She was doing an internship with her mentor as they conducted different workshops for teachers, and she enjoyed bringing information back to the team from the workshops she was assisting with as part of her internship. Ivy believed the program enabled her to find another role on the team as she realized she could also be a bridge between education policy makers and the teachers, giving her more confidence to share her ideas with the team. Like Mary, having a specific role on the team with something to contribute made Ivy feel that she was a part of the team.

The program was a growing experience for Anna because it had helped her to “step out of the box, try something new, and try to get along with other people” (Anna interview). Anna came to agree on the relationship with the new members in the team in this stage, finally accepting that there would always be new

members of the team. Also, she had learned to accept the differences among her team members, a significant change in her perspective. She no longer felt inferior to other team members and was able to acknowledge the strengths of other members and how they complimented one another. The change in how Anna perceived her role on the team had brought about a change in the entire team.

Performing

In the performing stage, the team was fully functioning, and members identified with the team. This stage occurred during the fifth and sixth laboratory class cycles, while the team was still norming. Mary said that in comparison to the beginning of the year, the team was more open to suggestions during the fifth laboratory class cycle. She felt encouraged by teachers responding positively to the program. She observed that the teachers who were not receptive to the program previously were now more open to change and more willing to try new ideas in their classrooms during the sixth teaching cycle. Mary shared more in the meeting now that she felt part of the team and had received more constructive feedback, support, and encouragement from the team members.

Lana also felt more comfortable sharing because everyone was sharing ideas and critiquing teaching practices, without taking the critiques personally. Kay described the change in the team as from being unreceptive to new ideas to anticipating the sharing of new ideas in every meeting. Linda’s observation was consistent with the teachers’ perceptions of team growth. She noticed “teachers who were quieter during the first year were now sharing a lot more and were more involved with conversations during the team time” (Linda interview). Those teachers “had grown stronger with a more positive attitude” and she believed that those teachers changed because “they felt safe” to share (Linda interview). The school held a monthly faculty meeting so that teachers could share their ideas about their teaching, assessment, and curriculum. Linda observed that before the professional development program, the second-grade team “would not [necessarily] pipe up or add to conversations [during the school meetings], whereas now they have started talking about assessment” (Linda interview).

Team members acknowledged the roles within the team; they observed that Anna became more proactive in her team leader role. With the meetings becoming more efficient, she was able to reduce the total number of meetings with her team and not overburden them with meetings, the program, or the research study.


According to Linda, Macy had been the teacher most resistant to any form of change at the beginning of the first year of the program. Linda said she was very impressed with Macy’s change in her mathematics instruction and Anna’s change in assuming a stronger leadership role at the end of the study. She thought their changes directly influenced the rest of the teachers on the team, along with other members of the school faculty.

In this stage, the team members continuously engaged in reflective dialogue, consensus building, and self-assessment. These activities were evident in Anna’s and Macy’s behavior during the sixth teaching cycle. Macy suggested more ideas for the planning of the demonstration lessons and made kinder remarks during the critique sessions. Anna acknowledged her own weaknesses and reflected on how those weaknesses had restricted the way she viewed others. She was now ready to accept the new team and felt that having the team members share their ideas was helpful in promoting understanding among the teachers.

The team was now performing with a common goal of incorporating the new state standards into their curriculum. Lana thought that planning the demonstration lessons as a team helped the team members understand each other’s teaching styles and personalities. Team planning gave the teachers opportunities to share, justify, and clarify the presentation of a lesson. This planning of lessons as a team created opportunities for the teachers to clarify misunderstandings and to appreciate the strengths of every teacher. The opportunity to work with colleagues in the laboratory class format fostered growth in the team. The teachers learned to put their personal differences aside to focus on the learning of the children. Lana believed the program especially boosted the confidence of the more experienced teachers because their ideas and strengths were recognized and respected, and they were accepted as part of the team.

This person might not have felt like their ideas were valid because they have been teaching for a while, and a lot of their ideas might be older and outdated. Really, they were very good ideas, and I think that kind of gave them more self-worth and made them feel like, Oh well, I have been doing all this all along, and my ideas are still valid; I am doing the right thing. I think that was the best thing for them. (Lana interview)

Kay said that the weekly meetings conducted over an extended period of time allowed the new and the more experienced members of the team to communicate with one another, promoting growth and

understanding among the team members. Anna felt that respect for one another was a key factor in promoting the team’s growth and effective functioning. According to Anna, the newer team members’ increased respect for the more experienced teachers reduced the tensions between them. Linda made the same observations, noting that by the end of the program, the teachers believed that “they are equals and that they are on the same playing grounds as the rest of the teachers on the team” (Linda interview).

Adjourning

Linda felt that the adjourning stage began at the end of the sixth laboratory class cycle, and continued after the program ended. The team witnessed Macy turn from a drill-and-practice mathematics teacher to one who was open to trying different techniques in her classroom. For example, she began to incorporate the use of manipulatives in her teaching. Ivy said that Macy changed because of the new state standards and because the program had goals that aligned with those standards. Furthermore, Macy’s positive change improved her relationship with the rest of the team members and their participation in the program.

All of the teachers continued to teach second grade the following academic year and, three months after the program ended, the team made plans for their own professional development. They believed that the program was helpful and they developed their own teaching cycle by scheduling teachers to lead their own weekly professional development meetings. Anna assumed a greater leadership role in the team and ensured the program ran smoothly. According to Linda, the team was regarded as a model for the other teams in the school in terms of collaboration in teaching mathematics. This recognition by other teachers in the school district further enhanced the second grade teachers’ motivation to continue to support each other in their teaching practices and continue their own professional development the following academic year.

Summary and Discussion

This study employed the laboratory class cycle in a school-based professional development to improve and support teachers’ learning and teaching. We showed that the team stormed before they formed, reversing the order of the first two stages in Tuckman and Jenson’s (1977) model. The team developed into an effective team towards the end of the professional development program. There was a high level of collaboration among the teachers as they worked towards a common goal of improving their teaching practices and


students’ learning. After the project ended, the teachers sustained the professional development by creating an experience similar to the laboratory class cycle.

Team Growth through Teaching Cycle

The data showed that teacher change can trigger team growth and vice versa. This supports Putnam and Borko’s (2000) claim of two directions learning: the role of the individual in the development of the team and the role of team in the development of the teacher. Specifically, individual teachers enriched the discussions of the team through their ideas and ways of thinking, while the way the team viewed and modified mathematics instruction affected individual teachers. This suggests that school-based professional development programs need to afford opportunities for individual teacher development and team development in order to be effective. The teaching cycle can be a viable model to promote individual teacher development and team development because it provides a formalized structure for collegial coaching. The shared experiences and the type of interactions afforded by the teaching cycle reduce isolation and strengthen professional and personal relationships among teachers. Through such interactions, teachers begin to acknowledge others’ strengths and to value the distributed expertise provided by various members of the team. This supports Truscott and Truscott’s (2004) finding that “acknowledging teacher strengths as internal resources fostered positive social climate overall and reinforced new learning for the teachers” (p. 62).

The teaching cycle created a culture in which the teachers were more willing to exercise the traits of critical colleagueship, as seen in the final stages the team members experienced (Lord, 1994). Teachers who initially viewed disagreement as personal insults learned to view disagreement as opportunities to consider different perspectives and clarify their beliefs. Turning teachers’ attention away from animosity towards a focus on helping children learn through observing and critiquing lessons enhanced critical colleagueship. As in Gordon (2004), and in the programs of Frykholm (1998) and Seago (2004), this professional development program helped establish trust, model communication norms that enabled critical dialogue, and maintain a balance between respecting individual team members and critically analyzing their teaching.

Team Growth through the Cooperative Work of Experienced and Novice Teachers

The blending of these two groups proved to be essential to the effectiveness of this team. Established members may take for granted the assumptions behind the school’s rules and procedures, preventing them from completely understanding the experience of a newcomer. This finding is consistent with Kardos, Johnson, Peske, Kauffman, and Liu’s (2001) observation that teachers who have taught for many years may not realize the difficulty for a newcomer to enter, explore, and understand her place among the more experienced teachers. In accordance with Hughes (1958), this study shows that the newcomer, faced with an uncertain situation, lacks reference points for appropriate behavior and experiences a surprise upon entering the new situation, as was seen in the storming stage by Mary and Kay. As illustrated in the norming stage for Mary and Ivy, their anxiety decreased after they identified their specific roles on the team (Berlew & Hall, 1966; Feldman, 1976; Louis, 1980).

Experienced teachers may have established systems for running their classrooms that dissuade and constrain them from trying new approaches that might threaten those systems. This was evident in Anna and Macy’s experiences during the forming and storming stages. Experienced teachers should not be neglected because they also need guidance and support to cope with changes. Our findings contribute to research on developing effective teams by recognizing the different needs of team members, based on their level of teaching experience.

The teaching cycle can be a model that offers intellectual nourishment and renewal to teachers. In this study, the teaching cycle helped the more experienced teachers cope with changes around them and ushered new teachers into the team by helping them find their roles. We see in the final stages that it is possible to bring about an effective team invested in an environment of continuous inquiry and improvement when the team’s focus is redirected toward students’ mathematical learning and achievement.

Conclusion and Future Research

One limitation of the study is the required participation of the team members. Future research on school-based professional development should focus on factors of team building with teachers who volunteer for a similar program. If participation is not mandatory, teachers might progress through the stages differently than described here.


This study showed that trust needs to be established before teachers can experience personal and professional growth. Team development is an uphill task, particularly when team members are resistant to change. To build collegial relationships among team members, both team members and professional developers must make a commitment over an extended period of time. When teachers are provided with the support they need for professional development, they begin to place a significant value on continuous learning.

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Understanding Rolle’s Theorem 18

Understanding Rolle’s Theorem

Revathy Parameswaran

This paper reports on an experiment studying twelfth grade students’ understanding of Rolle’s Theorem. In particular, we study the influence of different concept images that students employ when solving reasoning tasks related to Rolle’s Theorem. We argue that students’ “container schema” and “motion schema” allow for rich concept images.

Introduction

Advanced mathematical concepts are characterized by complex interactions between intuitive and rigorous reasoning processes (Weber & Alcock, 2004). Learning calculus, which involves processes pertaining to advanced mathematical thinking, has been a subject of extensive research. One of the significant conclusions arising out of this research is that students typically develop routine techniques and manipulative skills rather than an understanding of theoretical concepts (Berry & Nymann, 2003; Davis & Vinner, 1986; Ervynck, 1981; Parameswaran, 2007; Robert, 1982; Sierpinska, 1987).

The subject of calculus is rich in abstraction and calls for a high level of conceptual understanding, where many students have difficulties. Ferrini-Mundy and Graham (1991) argue that students’ understanding of central concepts of calculus is ‘exceptionally primitive’:

Students demonstrate virtually no intuition about the concepts and processes of calculus. They diligently mimic examples and crank out homework problems that are predictably identical to the examples in the text. Misconceptions exist as a result of student attempts to adapt prior knowledge to a new situation. Research suggests that students have a strong commitment to these misconceptions and that they are resistant to change and direct instruction (p. 631-632).

While teaching calculus to a wide range of students, it is more practical to appeal to students’ intuition when conveying mathematical concepts and ideas, building on what they have already learned

without making heavy demands on their aptitude for abstract and rigorous mathematical understanding. Some researchers argue that an introductory calculus course should be informal, intuitive, and conceptual, based mainly on graphs and functions (Koirala, 1997); formulas and rules should be carefully and intuitively developed on the basis of students’ previous work in mathematics and other sciences (Heid, 1988; Orton 1983). One of the guiding principles of teaching calculus could be the ‘Rule of Three,’ (Hughes-Hallet, et al., 1994) which says that, whenever possible, topics should be taught graphically, numerically, and analytically. The aim is to balance all three of these components to enable the students to view ideas from different standpoints and develop a holistic perspective of each concept.

There has been extensive research into the difficulties that students encounter in understanding limits, functions, differentiability, continuity, and so on. However, there is not much literature on students’ understanding of other concepts in calculus. Apart from the cognitive obstacles that arise in the learning of calculus concepts due to the complexity of the subject matter, students sometimes encounter difficulties inherent in mathematical reasoning. For example, deductive reasoning is a fundamental tool for mathematical thinking; however, students reveal serious difficulties developing such reasoning skills. Orsega and Sorizio (2000) propose the mental model theory of Johnson-Laird and Byrne as a cognitive framework to analyze students’ difficulties in deductive reasoning. Orsega and Sorizio argue that a didactical model should be designed to enable undergraduates to overcome the fallacies of their deductive inferences. They consider a teaching method that enables first-year undergraduates to make explicit the tautologically implicit properties in the hypothesis of Rolle’s Theorem and to reflect on them.

Revathy Parameswaran received her PhD from Chennai Mathematical Institute. She has been teaching mathematics for senior high students for the past eighteen years. Her research interests include cognitive development and advanced mathematical thinking. Her hobbies include reading books and listening to classical music.

Revathy Parameswaran 19

The purpose of this paper is to report on an experiment carried out to study twelfth grade students’ understanding of Rolle’s Theorem and its relationship to the closely related Mean Value Theorem. In particular, I set up tasks designed to study (1) the learner’s ability to state the theorem and apply it to reasoning tasks, (2) the influence of concept images in his or her reasoning about the theorem, and (3) the learner’s ability to perceive the relationship between Rolle’s Theorem and other related mathematical concepts.

For the reader’s convenience, we recall below the statement of Rolle’s Theorem.

Let f be a function that satisfies the following three hypotheses: (1) f is continuous on the closed interval [a,b], (2) f is differentiable on the open interval (a,b), and (3) f(a) = f(b). Then there is a number c in (a,b) such that f'(c) = 0 (Stewart, 1987).

Why Rolle’s Theorem?

As observed by Berlinski (1995), “Rolle's Theorem is about functions, and so a theorem about processes represented by functions, an affirmation among other things about the coordination of time and space. ... The constraints deal with the two fundamental mathematical properties of continuity and differentiability...” (191–192). Berlinski further observes that: “Rolle's Theorem establishes a connection between continuity and differentiability. Continuity guarantees a maximum; differentiability delivers a number. Fermat's Theorem [which says that if f has a local extremum at c and if f'(c) exists, then f'(c) = 0] supplies the connection between concepts” (196). The statement of the theorem involves multiple hypotheses, the universal quantifier (for all) and the existential quantifier (there exists). Also Rolle's Theorem offers the opportunity for pictorial, intuitive, and logical interpretations.

The knowledge components required for the understanding of this theorem involve limits, continuity, and differentiability. The proof of the theorem is given using the Fermat’s Theorem and the Extreme Value Theorem, which says that any real valued continuous function on a closed interval attains its maximum and minimum values. The proof of Fermat's Theorem is given in the course while that of Extreme Value Theorem is taken as shared (Stewart, 1987). Hence we appeal to the learners' intuition rather than be rigorous in our approach.

The beauty of this theorem also reveals itself in its connection with real life. A ball, when thrown up, comes down and during the course of its movement, it

changes its direction at some point to come down. Rolle’s Theorem thus can be used to explain that the velocity of the ball which is thrown upwards must become zero at some point (Berlinski, 1995).

Theoretical Background

We shall use the concepts introduced by Weber and Alcock (2004) in their research on syntactic and semantic proof production. A proof is referred to as syntactic if it involves only manipulation of facts and formal definitions in a logical manner without appealing to intuitive and non-formal representations of the mathematical concepts involved. The prover need only to have the ability to make formal deductions based on the relevant definitions and concepts. Such knowledge and understanding is called syntactic knowledge or formal understanding. On the other hand, if the prover uses instantiations to guide the formal inferences, he or she is said to possess semantic or effective intuitive understanding. Instantiation is described as “a systematically repeatable way that an individual thinks about a mathematical object, which is internally meaningful to that individual” (p. 210). “Semantic or effective intuitive understanding is described as the ability on the part of the prover to explicitly describe how she could translate intuitive observations based on instantiations into formal mathematical arguments” (p. 229). While formal understanding is at the superficial level, effective intuitive understanding lies at a deeper level and is characterized by the following features, which we illustrate in the context of Rolle’s Theorem:

• Instantiation “One should be able to instantiate relevant mathematical objects” (p. 229). For example, in the case of understanding Rolle’s Theorem, the learner instantiates the statement of the theorem if he recognizes its applicability (or non-applicability) in the case of a “typical” function -- not merely that of a quadratic-- possibly in terms of a graph.

• Richness “These instantiations should be rich enough that they suggest inferences that one can draw” (p. 229).

• Accuracy “These instantiations should be accurate reflections of the objects and concepts that they represent” (p. 229). In our context, the examples should not be too special as to suggest properties not implied by the theorem. For example, one may be misled to believe that there is a unique point where the derivative vanishes if one always instantiates the graph to be a parabola.

• Relation to formal definition “One should be able to connect the formal definition of the concept to the instantiation with which they reason” (p. 229).


In their foundational work, Vinner and Tall (1981) have provided a framework for analyzing how one understands and uses a mathematical definition. According to Vinner and Tall, a concept definition and a concept image are associated with every mathematical concept. Concept image is the total cognitive structure associated with the mathematical concept in the individual’s mind. Depending on the context, different parts of the concept image may get activated; the part that is activated is referred to as the evoked concept image. The words used to describe the concept image are called the concept definition. This could be a formal definition and given to the individual as a part of a formal theory or it may be a personal definition invented by an individual describing his concept image. A potential conflict factor is any part of the concept image that conflicts with another part of the concept image or any implication of the concept definition. Factors in different formal theories can give rise to such a conflict. A cognitive conflict is created when two mutually conflicting factors are evoked simultaneously in the mind of an individual. The potential conflict may not become a cognitive conflict if the implications of the concept definition do not become a part of the individual’s concept image. The lack of coordination between the concept image developed by an individual and the implication of the concept definition can lead to obstacles in learning because resolution of the resulting cognitive conflict is crucial for learning to take place.

The Study

Participants

The students in our study were in twelfth grade in a school affiliated with the Central Board of Secondary Education of India. The twelfth grade is the terminal grade in senior high schools in India and its successful completion qualifies one for university education. After completion of tenth grade, mathematics is an optional subject for eleventh and twelfth grades, but is required for pursuing a degree in science or engineering in universities. The mathematics curriculum is of a high standard and covers a wide range of topics. In eleventh and twelfth grades, the students learn algebra, trigonometry, elementary two and three dimensional analytical geometry, complex numbers, differential and integral calculus, differential equations, matrices and determinants, Boolean algebra, set theory, theory of equations, statics, dynamics and probability theory.

The experiment was conducted after the students had been taught differential and integral calculus over

a period of six months, forming part of a year-long twelfth grade curriculum. Rolle’s Theorem was part of the curriculum. The rigorous proof is omitted in the course, while graphical interpretations and explanations are offered as to why the statement is valid.

Research Method

Our experiment was comprised of two written tasks followed by interviews. The second task was conducted one week after the first and the interviews were held two days later. Thirty students participated in our study, out of which two students were selected for interviews based on their clarity of expression. They had responded to all the tasks given. The errors that they had committed in the tasks were also committed by many other students of the sample. In view of the depth and detail of analysis we planned, resource restriction limited us to only two participants.

The first task was descriptive in nature and it aimed to probe students' understanding of the statement of Rolle’s Theorem. In the second task, we gave the students four graphs along with questions related to Rolle’s Theorem. The purpose of this task was to find out if the students were able to connect Rolle’s Theorem with problem situations that were presented graphically. Considering the teaching methods adopted, it seemed appropriate to test their understanding using the tasks. These questions had been designed by the researcher to test the intuitive understanding because the material had originally been taught formally.

At this point, two students were selected for interviews. As the interview progressed, our questions built on their responses, so as to gain a better understanding of the student's conception of the theorem. The questions also aimed to provoke a considerable amount of reflection on the part of the students. As a result of the interview, we hoped to ascertain the obstacles, if any, that students face in understanding Rolle’s Theorem.

Data Analysis

First Task

The first task consisted of two questions about Rolle’s Theorem. The first question was “Explain in your own words what you understand of Rolle’s Theorem.” Representative responses were identified and are given below.

Abhi (all names used are pseudonyms) exemplifies the response of a student who has a sufficiently rich concept image.


Response 1. (Abhi): [She had drawn two graphs of functions and labeled them (i) and (ii).] “When we draw a graph like graph (i), we can draw tangents at every point on the curve and these tangents will have different slopes. But at one point, say M, if a tangent is drawn, its slope will be equal to the slope of the x-axis, [i.e.,] this tangent is parallel to the x-axis...”

Abhi's response shows a preference for viewing analytic properties geometrically. Her pictorial representation was that of a downward parabola intersecting the x-axis. The hypotheses and conclusion of Rolle's Theorem are stated geometrically and perhaps understood thus. Her notation for the point M on the graph is suggestive of the maximum point.

The following responses are illustrative of inaccuracies in the concept images:

Response 2. (Sweta) [The drawing is given in Figure 1.] “Two points, A, B, are such that they lie on the x-axis: y-coordinates equal to zero. f(A) = f(B) = 0. A curve passes through them such that AD = BD. The graph is continuous since it can be drawn without a gap. It is differentiable, i.e., it can be differentiated at all the points. When all these conditions are satisfied then there lies a point c such that f'(c) = 0. On satisfying these Rolle's Theorem is said to be verified.”

Figure 1. Sweta’s response to the first question. Sweta assumes f(a) = f(b) = 0. She incorrectly

believes the point D = (c, f(c)) divides the graph into two equal parts AD and DB. It is probably because the specific instantiation she has of (the graph of) a function (such as a parabola symmetric about the y-axis) has this property. See Figure 5. If this is so, her instantiation is not sufficiently rich. Since her example is a specialized one, her figural representation of Rolle’s Theorem forces her to assume properties that are not implied by the theorem.

Response 3. (Sheela) “The given part is a function which by nature is continuous and this function is differentiable and our claim is that there exists a point c on that function for which f'(c) = 0 and in a way this point c divides the function into two halves.”

Sheela had omitted the condition that f(a) = f(b). The phrase “in a way this point divides the function into two halves” indicates that she probably imagines the function to have only one extremum, suggesting lack of accuracy in her instantiations.

The following response indicates confusion in distinguishing between hypothesis and conclusion. Like Sweta, she explains through an example.

Response 4. (Anita) “Rolle’s Theorem is satisfied for a function in [a, b] only if the following conditions are satisfied: (a) The function should be continuous, i.e., it could be drawn without lifting your hands, (b) f(a) = f(b), (c) c ∈ (a,b) which means that c must be between a and b, where f'(c) = 0. For example, f(x) = cos(x) ∈ [-π/2, π /2]. (i) It is continuous since it is a trigonometric function. (ii) f(π /2)=0, f(-π /2)=0. Therefore f'(c) = 0. f'(x) = sin(x) implies f'(c) = sin(c) = 0, c = 0 ∈ (-π /2, π /2).”

We observe that Anita includes as part of the hypothesis the existence of an element c ∈ (a, b) such that f'(c) = 0. Nine students committed similar errors. Perhaps this is due to recent exposure to “if, then” mathematical propositions, particularly those involving quantifiers. Yet they do understand what to do when asked to verify Rolle’s Theorem in a specific context as exemplified by her verification in the case of the sine function on [-π /2, π /2], although the syntactical error still persists.

Observe that Anita writes ‘only if f(a) = f(b)’ although examples abound where f'(c) = 0 for some c ∈ (a,b) (e.g. f(x) = x3 on [-1, 1]). We conclude that the instantiations she carries with her are not sufficiently rich.

Response 5. (Siva) [The drawing is given in Figure 2.] “We draw a tangent P on the curve AB such that tgt is parallel to the curve at the x-axis. This is satisfied only if (a) f(a) = f(b), (b) the graph is [continuous], and (c) there is a derivative for the graph.

Figure 2. Siva’s response to the first question. Siva prefers to express the conclusion in terms of

tangents at extrema. Again, we note that he states the converse to Rolle’s Theorem as evidenced by the phrase “only if.”

The second question of the first task asked students “How are Rolle’s Theorem and the Mean Value


Theorem related?” Typical responses were identified and are presented here.

Response 1. (Abhi) “Mean Value Theorem is different from Rolle’s Theorem because in Mean Value Theorem it is sufficient that the function [be] continuous and differentiable, but in Rolle’s Theorem besides the function should be continuous and differentiable, it should also satisfy f(a) = f(b).”

Response 2. (Sweta) “In Rolle’s Theorem f(a) = f(b) but in mean value theorem f(a) is not equal to f(b). In Rolle’s theorem f'(c) = 0 while in mean value theorem f'(c) = (f(b) – f(a))/(b – a).

Response 3. “Mean value theorem is different from Rolle’s theorem in only one way. The condition f(a) = f(b) is not necessary to be proved in mean value theorem, but it is a condition in Rolle’s theorem.”

Response 4. “Mean value theorem is not completely different from Rolle’s theorem. The similarity is that in both the theorems we have got to check whether the graph is continuous. The dissimilarity is that in Rolle’s theorem f(a) = f(b) whereas in mean value theorem f(a) need not be equal to f(b).”

Almost all of the thirty students had written that the Mean Value Theorem is different from Rolle’s Theorem. They had all given the reason that while Rolle’s Theorem has f(a) = f(b) as one of its constraints, it was not present in the Mean Value Theorem. Twenty students had also added that the similarity between the two theorems lies in the fact that the functions have to be continuous and differentiable for both theorems.

We note that none of the students had said that the Mean Value Theorem was a generalization of Rolle’s Theorem. This prompts several questions: How do students view relationships among abstract statements? What are the relationships among abstract mathematical propositions, besides identity, that students in twelfth grade are aware of? At what stage in one's mathematical development do they perceive containment relationships among abstract mathematical statements? One should distinguish here between mathematical objects and propositions. For example, these students surely know and are aware that all right triangles are triangles and all squares are rectangles. Further research is needed to explore these questions.

We note also that for some students, the focus seems to be what one should do: For Rolle's Theorem one should check f(a) = f(b), whereas it is not necessary to do so in the case of Mean Value Theorem. Perhaps it is due to excessive emphasis on an

algorithmic approach to doing problems in lower grades.

Second Task

The second task is aimed at the following: (1) To explore whether the learner is able to apply Rolle's Theorem when the function is not explicitly specified by a formula. What does it mean, geometrically, to say f'(c) = 0? (2) To investigate whether students are able to identify a non-example of Rolle’s Theorem when it is presented to them in the form of a graph that is rich enough to suggest inferences that one can draw. (3) To see whether students are able to relate the given instantiation to the formal definition.

The second task consisted of the following questions:

1. Verify Rolle’s Theorem for the function whose graph is given in Figure 3.

Figure 3. Graph for first question. 2. Is Rolle’s Theorem applicable to the function

whose graph is given in Figure 4? Give reason.

Figure 4. Graph for second question.


3. For the function whose graph is given in Figure 5, show that f(c) = 0 for some c in (0, 6).

Figure 5. Graph for third question. 4. Consider the function whose graph is given in

Figure 6. The graph has horizontal tangents when x = 0 and x = 3. Show that there exists a point c ∈ (0, 3) such that f''(c) = 0.

Figure 6. Graph for the fourth question. Student Responses to the Second Task. The

performance on this task can be found in Table 1.

Table 1 Performance on Second Task Question Number

of Correct Responses

Percentage of Correct

Responses 1 19 63 2 18 55 3 4 13 4 5 16 In response to the second question, some students

demonstrated a rich concept image of differentiable functions that includes the necessity of a unique tangent. Hence they were able to identify the non-example of a differentiable function based on this

criterion. Five students said that the function was discontinuous rather than saying it was not differentiable when giving reasons as to why Rolle's Theorem was not applicable for the given curve.

Compared to the performance on the first and second questions, the students’ performance on the third and fourth questions was relatively poor. The third question involved an application of Rolle’s Theorem in contexts where its need is not explicitly apparent. Two students unsuccessfully attempted to use the Mean Value Theorem to prove f'(c) = 0 in response to the third question. The fourth question demanded a leap in their thinking by expecting them to apply Rolle's Theorem to f'(x). In other words, the students had to treat f'(x) as an object and perform an action on it (Dubinsky, 1991), which seemed a difficult process for most of the students.

The following are responses to question three by students with a rich concept image.

• (Swamy) “Now f(x) is continuous. It has traveled from y = 2 to y = 3, it then goes back to y = 0 at x = 6. So, between these two points the function becomes 2. Also it is continuous and differentiable and hence applying Rolle’s Theorem, we have a root for f'(x) in (0, 6).”

• (Ram) “Since the function is increasing and decreasing the slope of the tangent is zero at some point.”

Swamy and Ram imagine the y-coordinate to be moving from point to point. This is a particularly rich mental representation which makes the Intermediate Value Theorem self-evident. Lakoff and Núñez (2000) argue “motion schema” to be a rich source from which many mathematical concepts and truths originate. Swamy recognized that since y is changing from 3 to 0, it must become 2 somewhere. This is a crucial step before one can invoke Rolle’s Theorem. Ram observed that the function is increasing and then decreasing. Although he did not make it explicit, it appears that he realized that the slope of the tangent is positive when the function increases and is negative when the function decreases. We surmise that it is evident to him that the slope of the tangent must be zero somewhere. Based on the responses, we conclude that Swamy and Ram possess instantiations of differentiable functions which have richness and accuracy, and capture the content of formal definitions and properties relevant to Rolle’s Theorem.

Interviews

Before the interview, we prepared the following questions that would encourage students to reflect on the salient features of Rolle’s Theorem. Understanding


a theorem or a concept includes being aware of a good supply of examples–instantiations–as well as non-examples. This level of understanding is indicated by the ability to apply the concept to specific problem situations. With this in mind, we began the interview with the following questions:

(1) Give an example of a function (using a graph) to which Rolle’s Theorem is not applicable.

(2) Consider the following function: f(x) = x3 if x ≤ 0, -x3 if x < 0. Is f differentiable? Is there an element c ∈ R such that f'(c) = 0?

(3) If f(x) = x2 + 3x + 7 is defined on [-2, 2], show that f'(c) = 0 for some c in (-2, 2). Are the hypotheses of Rolle’s Theorem satisfied? Is there a contradiction? Is the converse of Rolle’s Theorem valid?

Abhi’s response. In her response to Question 1, Abhi drew the graph of an absolute value function and then said: “In this graph it is possible for us to draw two tangents for one point and hence the slope will not be unique. For Rolle’s Theorem to be satisfied, the slope of a tangent at a point should be unique. Hence Rolle’s Theorem is not applicable.”

For Question 2, Abhi stated: “f'(x) = 3x2 and f'(x) = -3x2. So not equal.” We observe that she had given the mathematically correct response to the second question in the second task, which involved identifying a non-differentiable function from its graph. But she seemed to have difficulty in checking the differentiability of the piecewise function f(x) = |x|3 from its formula.

Finally, when responding to Question 3, Abhi said: “f(x) = x2 + 3x + 7 is defined on [-2, 2]. f(-2) = -1 and f(2) = 17. So f(a) is not equal to f(b). Hence the hypothesis of Rolle’s Theorem is not satisfied. The contradiction here is f(a) is not equal to f(b)...: The converse of the theorem is valid. We can see it in any example.”

To gain a better understanding of how Abhi was conceptualizing Rolle’s Theorem, we probed deeper into her understanding of Rolle’s Theorem:

I: Can you state the converse of Rolle’s Theorem? Abhi: If f'(c) = 0 in a graph like this [She gestures

towards the paper and draws the following], then the three conditions f(a) = f(b), the graph is continuous and the function is differentiable will be satisfied.

I: Can you draw the graph of y = x3? [Abhi sketches the graph by plotting some values.] I: Are the conditions of Rolle’s Theorem satisfied? Abhi: (Thinks for some time.) Yes....... but, no two

values are same. I: So,..

Abhi: (Thinks for some time.) Yeah, ...the first condition is not met.

I: What about the derivative at x = 0? Abhi (Pauses.) Yeah, got it. The derivative exists.

The converse is not true! Abhi believed that even when the hypothesis

was not satisfied, the statement of the theorem was false and hence she perceived a contradiction. At the end of the interview, she was convinced about the fact that the converse of Rolle’s Theorem is not true.

Sweta’s response. In her response to Question 1, Sweta drew the graph of the sign function and stated “This is not continuous.”

For Question 2, Sweta said: “f(x) is not differentiable. y = |x3| and that is because it does not have a unique slope. No, there is no point such that f'(x) = 0. Let me check...[She sketches the graph of y = x3 and y=-x3] The graph is like that of modulus function. I know that modulus function [the absolute value function] is not differentiable. So this is also not differentiable. f'(x) = 3x2 f'(x) = -3x2. So not equal.” She was not able to proceed further because the moment she sees the function f(x) = |x3| she connects it to her pre-existing knowledge of the modulus function not being differentiable.

Finally, in her response to Question 3, Sweta said “Rolle’s Theorem is not satisfied. f(a) is not equal to f(b). Hence a contradiction.” When asked if the converse of Rolle’s Theorem was valid, Sweta thought for a minute, and then responded “I think it is true.”

Sweta also had difficulty with what constitutes a contradiction to Rolle’s Theorem. Both Abhi and Sweta seemed to instantiate a non-differentiable function to be that of an absolute function. Both Abhi and Sweta demonstrated difficulty in writing the converse of Rolle’s Theorem. This is probably due to the fact that the theorem has multiple hypotheses. Also they had trouble with what would, or would not, contradict Rolle’s Theorem. However, the writing tasks and the follow-up interviews seem to have enriched their understanding of Rolle’s Theorem.

The teachers who used this message design logic also expressed a genuine desire to encourage student learning. They talked about saying what needed to be said in order to accomplish specific learning goals. The teachers, not the students, defined the direction of classroom discussion and activity. These teachers assumed that they knew what the students needed to hear to move students closer to the desired outcome.

Conclusion

We aimed to study students’ understanding of Rolle’s Theorem by setting up specific tasks which


involved stating the theorem, relating it to the Mean Value Theorem, and using it to solve problems involving graphs. Two students, Abhi and Sweta, participated in interviews during which further problems and questions were put forth to help us gain deeper insight into their understanding of Rolle’s Theorem.

In response to the first task, which asked them to state Rolle’s Theorem, nine students stated the converse. Although there were syntactic errors in stating the converse, it did not seriously affect their ability to solve simple problems as they know what to do to arrive at the desired solutions. Recall from Table 1, that 63% of the students answered the first question of the second task correctly.

The instantiations in the context of Rolle’s Theorem seem to involve, in most cases, familiar functions or graphs such as the parabola. For some students (e.g. Sweta), this leads to a misunderstanding or misinterpretation of the hypothesis or conclusion of the theorem. Their instantiations are too specialized: they lack richness and accuracy.

We note that none of the students mentioned, in response to Question 2 of Task-I, that Mean Value Theorem is a generalization of Rolle’s Theorem. Also, these students believe that f(a) should not equal f(b) in order for the Mean Value Theorem to be applicable. Again, this could be attributed to viewing theorems as tools for solving problems rather than as mathematical entities which can subsume one another.

Many students had difficulty with the last two questions of Task-II (only 13% and 16% gave correct responses). This was because Rolle’s Theorem could not be directly applied to the function given as in Question 3. This particular example first required invoking the Intermediate Value Theorem. In Question 4, the theorem had to be applied to the derivative of the function whose graph was given. Both of these questions required linking Rolle’s Theorem to other calculus concepts.

Because many students confused the statement of Rolle’s Theorem with its converse in response to Question 1 in Task-I, we asked questions related to the converse of Rolle’s Theorem during the interviews. Abhi and Sweta’s response to these questions led us to some interesting observations. We believe that their responses are representative of the other students who participated.

Both Abhi and Sweta had difficulty stating the converse of Rolle’s Theorem. The converse of “If P1, P2, P3, then Q” is indeed “If Q, then P1, P2, P3”. However, mathematically, the statement of Rolle’s

Theorem is to be reworded in the form “Assume P1, P2 hold. If P3, then Q.” In the converse statement, one assumes that the function f is continuous in [a, b] and is differentiable in (a, b). The assertion, then, is: “If f'(c) = 0 for some c ∈ (a, b), then f(a) = f(b)”, or more generally, a mathematician would state the converse as follows: “If f'(c) = 0, in any interval containing c in which f is continuous and differentiable, there exist a, b such that f(a) = f(b).” Of course, as Sweta stated correctly, f'(c) = 0 does not imply the function is continuous. She offered suggested the sign function as a counterexample. However, the mathematical subtlety has been missed when one places equal importance on all the hypotheses of a statement. Since the general context for these theorems is differentiable functions, P1 and P2 are taken for granted, and the main point is the equality f(a) = f(b).

Some common misconceptions arose for both Abhi and Sweta. They both perceived a contradiction in a statement if the hypothesis is not valid. In addition, they incorrectly viewed the converse to be valid until we suggested a specific counterexample. They also thought that f(x) = |x|3 cannot be differentiable based on a comparison with the absolute value function, which is not differentiable at the origin. Although there is no significant difference in their performance levels, it appears that there are some qualitative differences in their ways of thinking about concepts related to Rolle’s Theorem. While Abhi expresses the statement of Rolle’s theorem in terms of slope of tangents, Sweta seems to have misunderstood the hypothesis f(a) = f(b) as saying that the point D (in Figure 1) divides the curve into two equal parts. It appears that Abhi prefers to think more geometrically and relates to examples and non-examples, i.e. instantiations, whereas Sweta seems to view each problem as an independent task.

To summarize, our experiment reveals possible difficulties students have in understanding Rolle’s Theorem, which involves making sense of it, relating to other concepts such as the Mean Value Theorem and intermediate value theorem, as well as the ability to use it in situations when the function is not explicitly given.

The notion of ‘container schema’ elucidated by Lakoff and Núñez (2000) underpins one of the important ways mathematical relationships are understood. Perhaps, the students in our sample have yet to realize that just as one object can be contained in another, a mathematical statement can also be ‘contained’ in another. For example one student wrote: “Now f(x) is continuous. It has travelled from y = 2 to y = 3, it then goes back to y = 0 at x = 6. So, between


these two points the function becomes 2.” This is an instance of the student possessing the “Source-Path-Goal schema.” In such a schema, we imagine an object to be moving from a source location to a target location, along a specific trajectory. Lakoff and Núñez (2000) argue that these schemas are present everywhere in mathematical thinking. They cite the example of functions in the Cartesian plane, which are conceptualized as that of motion along a path, evidenced by such phrases as “going up” or “reaching a maximum” are used to describe them as an instance of the motion schema. From our study, we observe that students who possess these schemas have rich concept images, which aid them in their reasoning tasks. In other words the ability to transfer everyday thinking to abstract mathematical notions guides them to possess versatile concept images.

Acknowledgments: I would like to thank Catherine

Ulrich, Associate Editor of TME, for her comments and suggestions and for pointing out several grammatical errors in an earlier version, resulting in an improved presentation.

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Tracie McLemore Salinas 27

Beyond the Right Answer: Exploring How Preservice Elementary Teachers Evaluate Student-Generated Algorithms

Tracie McLemore Salinas

Tasks regularly completed by elementary teachers reveal the mathematical nature of their work. However, preservice teachers demonstrate a lack of depth of mathematical thought. This study investigated the criteria preservice teachers intuitively used to evaluate algorithms. The intent was to use that knowledge as a foundation for modeling mathematical habits of mind for similar tasks. Journal writings and notes from in-class discussions were collected over three semesters of an introductory course for future teachers. Data were analyzed to discover dominant criteria used by preservice teachers to evaluate student algorithms. Four criteria, namely efficiency, generalizability, mathematical validity, and permissibility, were routinely used by preservice teachers.

Introduction

Paper-and-pencil algorithms are important tools that equip students for computational fluency. Before algorithms become mechanical procedures for students, their use should involve conceptual knowledge as well as procedural skill (Ashlock, 2006). Hence, teachers are encouraged to allow students to explore and create algorithms before the traditional algorithms are introduced. In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) suggested that “when students compute with strategies they invent or choose because they are meaningful, their learning tends to be robust - they are able to remember and apply their knowledge” (p. 86). Reasoning and justification are both inherent in the invention of procedures (Kilpatrick, Swafford, & Findell, 2001). Thus, children use and reveal their own construct of understanding with the procedures that they create (Baek, 1998).

The importance of providing time for exploration and creation of non-traditional algorithms is emphasized in mathematics methods texts for preservice teachers (e.g., Cathcart, Pothier, Vance, & Bezuk, 2006; Van de Walle, 2004). Preservice teachers are encouraged to allow students to generate their own notation and algorithms for strategies they create after exploring with manipulatives. The understanding revealed by students should then be channeled into a logical path to the traditional algorithms, resulting in conceptual and procedural understanding. This sounds

deceptively simple to many preservice elementary teachers; however, the mathematical and pedagogical understanding required for this navigation is actually quite rich.

In a discussion on elementary teacher preparation, the Conference Board of the Mathematical Sciences (CBMS; Conference Board of the Mathematical Sciences, 2001) presents a vignette of third-grade students investigating a variety of strategies for multiplying. The vignette supports the CBMS depiction of elementary mathematics as an intellectually rich and challenging field, requiring the development of new mathematical habits, strong connections among mathematical topics, and skills in mathematical justification. In the vignette, the teacher elicits methods of solving a word problem requiring multiplication from five students. The teacher then mines the responses for evidence of correct thinking, but she also must recognize and investigate sources of error, determining which elements to use as the foundation for additional exploration.

The depth of mathematical understanding required to evaluate students’ methods of performing operations is evidence of the mathematical nature of elementary teaching. The implications of the teacher’s understanding of student-generated methods are so profound that Ball, Bass, and Hill (2004) suggest “no pedagogical decision can be made prior to asking and answering this question . . . ‘What . . . is the method, and will it work for all cases?’” (p. 7). Unfortunately, research has demonstrated that teachers exposed to student-generated algorithms routinely look to procedural steps, not to reasoning, in evaluating the correctness of student work (Ball, 1998; Simon, 1993).

Campbell, Rowan, and Suarez (1998) proposed three criteria for evaluating student-invented algorithms: 1) efficient procedures, 2) mathematically valid procedures, and 3) generalizable procedures.

Tracie McLemore Salinas is an assistant professor in the Department of Mathematical Sciences at Appalachian State University. Her current research includes studying the mathematical thinking of pre-service teachers and how in-service teachers find and use appropriate research. She can be reached at [email protected]..

How Preservice Elementary Teachers Evaluate Student-Generated Algorithms 28

Furthermore, Kilpatrick, Swafford, and Findell (2001) suggested four characteristics of algorithms including: 1) transparency, 2) efficiency, 3) generality, and 4) precision (2001). These lists share a tendency toward the habits of mind demonstrated by mathematicians. Preservice teachers, however, often demonstrate a lack of what Seaman and Szydlik (2007) refer to as “mathematical sophistication”. In other words, preservice teachers tend to have an orientation so far removed from that of the mathematical community as to seriously inhibit their mathematical understanding. Finding similar underlying structures, valuing sense making, and using counterexamples are all practices that serve a teacher well in evaluating student-generated algorithms and are in keeping with the habits of mathematicians.

It makes sense, then, that if preservice teachers are to engage successfully in the mathematical tasks of elementary teaching, their preparation should support the development of appropriate mathematical habits of mind (Borko, Eisenhart, Brown, Underhill, Jones, & Agard, 1992) and perhaps an introduction to the community of mathematicians (Seaman et al., 2007). Moving students towards new mathematical habits of mind requires knowing where they begin the journey (CBMS, 2001). Mathematics educators assume that at that origin lie some appropriate mathematical and pedagogical inclinations on which teachers should be encouraged to act (Ball, 1998).

Algorithms in Teacher Preparation

As a mathematics teacher educator, I regularly incorporate activities that simulate teacher tasks into my courses. One of these tasks is evaluating student-generated algorithms. Preservice teachers in my methods and content courses often seem astounded that alternate algorithms exist or that students might create or adapt their own. Typical textbooks for these courses usually demonstrate alternate algorithms, but often they only tell about the procedures rather than encouraging the reader’s discovery of them. Because the algorithms presented tend to be correct, preservice teachers have little opportunity for actually evaluating correctness and identifying sources of error.

As a result, I supplement our texts by demonstrating student-generated algorithms that I have collected through years of working with teachers and student teachers. I challenge my preservice teachers to explore, justify, and counter algorithms through journal writings and classroom discussions. Through their journal writings, I often see evidence of relatively consistent thinking among students. Consequently, I used this information to investigate the thinking

preservice teachers bring to the task of evaluating algorithms.

Methodology

This study investigated the knowledge that preservice teachers bring to an initial mathematics course for future elementary teachers with regard to evaluating algorithms. Two research questions guided the study:

1) What criteria do preservice teachers tend to use to evaluate algorithms?

2) Do the criteria that preservice teachers rely on vary depending on the nature of the algorithm?

Setting

Preservice teachers participating in this study were enrolled in an undergraduate course for future elementary teachers. For many, the course constituted one of the first education-related courses and the first mathematics education-related course of their programs. Content covered in the course included measurement, data analysis and probability, and algebraic thinking. At this university, preservice teachers do not complete a field experience during the course but are engaged in activities similar to teachers’ professional development experiences. Thus, they often attend workshops or conferences; some choose to also observe K-8 classrooms or to interview K-8 teachers. Although the course is taught within the Department of Mathematical Sciences, it is a combination of content and methods. Students use manipulative kits regularly and are expected to justify their work and to use multiple representations in classroom presentations and discussions.

Participants

Data were collected over three semesters, allowing 61 students to participate of 69 enrolled in the course. Most students were in the first two years of their undergraduate programs, but about a third of the students were in their third year. Seven of the 61 participants were male.

Data Collection

I relied on journal-writing assignments as the primary source of data about student thinking. Students completed a variety of journal writings, including responses to problems in class, reflections on activities, and investigations of teacher tasks. All students were required to complete each writing assignment. In order to explore preservice teachers’ evaluation of student-generated algorithms, I also made notes following in-


class discussions of algorithms from journal assignments and other algorithms mentioned in class.

For the journal assignments that focused on algorithms, I provided a real example of a student-generated algorithm introduced either through explanation or video clip. Preservice teachers then spent some time talking in small groups about the algorithm before beginning to write explanations in their journals. Journal writings were to address the algorithm observed, and preservice teachers were to consider and justify whether they would allow a student to use that algorithm in class. I collected, reviewed, and responded to the writings then provided participants at least one opportunity to edit their responses and resubmit. Eventually, I brought the algorithm back into the classroom for a group discussion. Some of the algorithms provided worked consistently and would be considered mathematically correct; others had only one or neither of these two characteristics.

It is important to note that for each algorithm, initial journal responses were collected and reviewed prior to any instruction in the topic. The introduction of the algorithm into the classroom for group discussion was intended to coincide with the appropriate instruction or to enrich the concurrent instruction. I encouraged students to generate the discussion as much as possible, even if our work took pieces of several class periods. As much as possible, I let the class as a group determine if the procedure was correct, why it was or was not, and whether it was a method that participants would allow in their own classrooms. This process might include a week of individual journal writings and responses followed by portions of three or four class sessions in which the preservice teachers attempted to present their own thinking to their peers. The preservice teachers could use chalkboards and manipulatives to demonstrate their thinking through examples or counterexamples.

Data Analysis

After I collected the journal writings and notes, I organized student responses for each sample algorithm in two ways, first by mathematical content and then again by the nature of the responses in spite of mathematical content. This allowed me to consider how the content influenced the students’ responses and to look for similarities in their approaches to the algorithms in spite of content. I reviewed journal writings and my notes from class discussions to determine what criteria preservice teachers seemed to be using in their evaluation processes. I recorded the criteria and grouped them by similarity. As the clusters

of similar criteria developed, I created a name for each group that best represented the criteria. For example, as preservice teachers wrote or explained that a student-generated algorithm included “an extra step,” “too many steps,” or a step that “complicates the process,” I grouped these together and considered them to support the criterion of efficiency. I attempted to produce a dominant list that was exhaustive but mutually-exclusive of student responses. I then checked each criterion to see with what frequency it appeared and in which formats – journals and/or in-class discussions. All resulting criteria explored in this paper appeared in a minimum of two-thirds of journal responses and were raised in at least half of in-class discussions. This ensured that the criteria discussed were those used most by preservice teachers.

Two Examples of Algorithms

Although a number of algorithms were explored in each semester, I found that two journal responses in particular demonstrated the teachers’ thinking. The two algorithms, one on multi-digit multiplication and the other on division of fractions, are presented here.

An example of a simpler algorithm that most preservice teachers understood well was multiplying a three digit number by a three digit number in which the only non-zero digit is in the hundreds place. For example, preservice teachers were provided 287 x 400. Using a traditional method, one might proceed as shown in Figure 1. However, a student in a local elementary classroom had discovered that he could obtain the same product by performing the multiplication in two steps, as shown in Figure 2. Of course, the student’s method produces the same result every time and is simply the process of breaking one multiplicand into two factors.

287

x 400 000

0000 + 114800 114800

Figure 1: Multiplication Using the Traditional Algorithm

287 x 400

287 x 100 = 28700 28700 x 4 = 114800

Figure 2: Multiplication Using a Student’s Method The most troublesome algorithm overall for

preservice teachers was an example of division of a


fraction by a fraction. Preservice teachers’ difficulty with fraction concepts is well-documented (Ball, 1990b; Ma, 1999; Zazkis & Campbell, 1996). Consequently, I expected preservice teachers to be somewhat procedure-oriented in their approach, but I was surprised by how strictly procedural their responses were.

This example was originally brought to me by a preservice teacher completing a service-learning requirement. The teacher she was assisting explained to her students that “you never, never, never touch the first fraction.” Instead, she demonstrated that you “flip” the second fraction and multiply across. After several examples, she assigned some exercises to the students. While walking around the classroom, she observed one student who had decided, in spite of her warnings, that he would in fact “touch the first fraction.” He “flipped” the first fraction (dividend) to obtain its reciprocal, multiplied it by the second fraction (divisor), and promptly inverted his solution as shown in Figure 3.

Figure 3: Division of Fractions Using a Student’s

Method Mathematically, there is the obvious question of

whether the student’s method of working the problem will produce the correct answer each time. Replacing the numerals of the problem with variables, it is simple to see that the student’s procedure will result in the correct answer. (See Figure 4).

Traditional

Student -Created

Figure 4: Comparing the Traditional and Student -Created Algorithms

However, anyone familiar with fractions will

quickly realize that ad

bc will not necessarily equal

bc

ad

as the student’s work suggests. Figure 3 is an example of this. Digging more deeply, one may question if these two fractions are not equal, are the other components separated by equal signs in fact equal? In the student’s algorithm, inverting the dividend produces a multiplicative statement that is not equivalent to the

initial problem. There are obvious errors in the student’s use of equal signs. The initial division statement cannot possibly be equal to both and . Similarly, the multiplicative statement cannot possibly be equal to both and . However, it is impossible from his work alone to determine how the student may have rationalized the differences in his representation and his conceptual understanding or whether he was aware of any inconsistencies at all. This example algorithm was one that produced a correct solution every time but failed to make mathematical sense.

Results

All student journal responses were first reviewed to find similarities in the criteria preservice teachers seemed to select for evaluating student-generated algorithms. Four primary criteria emerged from the preservice teachers’ writings and discussions: efficiency, generalizability, mathematical validity, and permissibility. On a positive note, three of the four criteria they chose echoed those proposed by the mathematics education community, namely efficiency, generalizability, and mathematical validity. On the other hand, preservice teachers seemed to use these criteria quite superficially. For example, preservice teachers applied the generalizability criterion by simply trying a few examples and not by using a more appropriate approach, such as replacing numbers with variables and continuing the investigation. Additionally, preservice teachers added the fourth criterion, permissibility, which seems to demonstrate their lack of personal authority in evaluating mathematics.

Efficiency

In the minds of the preservice teachers, the inefficiency of student-generated algorithms was less a problem of inelegance and more an opportunity for student misunderstanding. For instance, in algorithms that broke numbers apart into their factors, such as in Figure 2, preservice teachers saw the extra step as a potentially confusing one. An example of a typical preservice teacher explanation is:

Although this would work, I would praise [the] student for being creative and finding that method, but I would then discourage the student from using this method. I [sic] reasoning for this is that it creates an extra step and when the student gets into higher level maths it could become confusing and create extra work to go through.

This response followed the introduction of an algorithm in which the “extra step” was actually a


mathematically correct one. Typically, the responses to mathematically correct algorithms with “extra steps” were not very different from ones with extra steps that were mathematically incorrect. It appears that preservice teachers looked at extra steps in light of efficiency alone, not mathematical validity. For instance, in the fractions example demonstrated in Figure 3, preservice teachers tended to look at the final flip as an extra step. One preservice teacher commented that “It can really make it complicated. What happens when you forget to flip the second time?” In classroom discussions that followed the examination of this algorithm, when I asked students why they flip the first time, only one student in all of the class sections was able to provide an explanation beyond the procedure itself. Most only echoed the teacher who instructed her students that “you never touch the first fraction.”

Mathematical Validity

Mathematical validity seemed to be secondary to efficiency in preservice teachers’ minds. In fact, for most students, the determination of mathematical validity involved simply trying to recall a rule that related to a particular step. They relied on procedural types of explorations, not conceptual ones. For most preservice teachers, concern arose when a student did something “without having any process to do it.” As one preservice teacher stated, “ . . . in mathematics every thing is done for a reason. Every thing taught in math has a precise way of doing different concepts.” If the preservice teacher could not recall the reason, the algorithm was considered flawed. For the fraction division example, one preservice teacher explained, ". . . that there is no mathematical rule that gives us the right to flip the end. . . everything you do in math and every step has a rule that was created out of logical reasoning and this doesn’t have a reason.” In all of the course sections considered for this study, only four preservice teachers regularly explored the validity of student-generated algorithms by using models, drawings, or other methods that moved beyond procedure and rules.

Generalizability

Interestingly, the four preservice teachers mentioned above were also the most successful at determining the generalizability of student-generated algorithms, although well over half of the preservice teachers did discuss the need to do so. Those preservice teachers who were able to accurately evaluate the algorithms with regard to generalizability were able to explain their thoughts clearly. For

example, with the multiplication problem from Figure 1, a number of students discovered that the four hundred was being broken into four groups of one hundred. As one student explained in class:

I like this idea of doing this problem. To me as a teacher this shows that the child knows what he/she is doing. It proves they not only know the answer but that they know exactly how to get it. He/she understands the reasons why you put the two zeros at the beginning of the problem instead of just place holders. The student sees 400 as 4 hundreds instead of just a number with zeros.

Permissibility

For many preservice teachers, their search for mathematical validity resulted in little more than a search for permission granted by some rule somewhere. However, there is another level of finding permission that seemed strong enough in student responses to become its own criterion. This second idea of permission results from the preservice teachers beliefs' about what students are permitted to do mathematically in the classroom. That permission is assumed to be granted by the classroom teacher or even future classroom teachers. Preservice teachers demonstrated the idea of permissibility when they explained their concerns for a child using an alternate method in a future classroom. As one preservice teacher wrote:

I would have no problem with one of my students doing the problem this way. As long as it worked each time, I would encourage new and creative ways to solve problems so that they don’t get used to doing the same methods all the time. . . I would, however, still like them to learn the problems both ways so that they can be prepared for future math classes that might need a more traditional way of solving the problem.

Not all responses were quite so supportive of building a repertoire of traditional and alternative algorithms. Many preservice teachers agreed with their classmate who asked, “What happens when they have a teacher next year who doesn’t let them do it this way?” Others chimed in that it was important to discourage alternate algorithms in favor of the traditional ones for fear that the method might confuse other students, make grading more difficult, or cause the child to be singled out by a future teacher as having misunderstood.

Summary

The four criteria – efficiency, mathematical validity, generalizability, and permissibility – were


gleaned from responses for algorithms of several types and for different operations. These included the student-generated algorithms as well as those typically presented in textbooks, such as the lattice algorithm. Separating responses by mathematical content, such as whole number multiplication versus multiplication of fractions, resulted in very few differences in the criteria. Preservice teachers applied the same four primary categories across the algorithms.

A striking difference, however, was in the level of confidence that appeared in the responses. For instance, many more of the responses to algorithms dealing with fractions were met with ambiguous replies that would provide the teacher time to reflect on the student-generated algorithm. Unfortunately, while the preservice teachers explained that this is what they would do in the classroom, very few then proceeded to investigate the algorithm in their journals. Others explained what they might try as they considered the algorithm, such as asking if it would work every time or suggesting that more examples were needed; however, they did not demonstrate the ability to do so. Thus, particularly with algorithms that involved fractions, preservice teachers did not address the mathematical content directly in their journal responses. This suggests that they have less confidence in the mathematical content related to algorithms applied to fractions.

Discussion

The tendency of preservice teachers to consider efficiency, generalizability, and mathematical validity suggests that the foundation exists upon which to model a more mathematically rigorous evaluation of student-generated algorithms. It is likely that underlying preservice teachers’ superficial use of these criteria is their inexperience in the mathematical thinking required to properly investigate student work. Modeling the habits of mind of mathematicians offers a way to counter this (Seaman & Szydlik, 2007). On the other hand, preservice teachers seemed to disregard the need to investigate student work if the solution produced was correct. However, we have seen that there are algorithms that produce the correct solution but have seriously flawed conceptual bases or representations. Encouraging preservice teachers to investigate beyond the final answer is vital to teaching them to link conceptual understanding to the use of algorithms; challenging them with algorithms that are numerically correct but mathematically incorrect may motivate such investigations. From my own experience, preservice teachers enjoyed exploring algorithms with which they were not familiar and in

doing so, observed the depth of mathematical thinking that is required of common teacher tasks.

Permissibility, on the other hand, is an idea that we may wish to discourage. While we do not want to encourage children to torment their teachers with additional algorithms that require dozens of steps simply because they “work,” we should want children (and teachers alike) to develop their own sense of self as an authority in mathematical thinking. Investigating mathematical authority, Schoenfeld (1994) observed that most college students “have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively” (p. 62). It is little wonder then that preservice teachers tend to consider themselves as receivers of mathematical knowledge and understanding rather than consumers, evaluators, and generators of it. Perhaps the idea of permissibility is somehow reflective of the tendency of preservice teachers to require verification by outside authority for mathematical validity rather than relying on their own reasoning. Mewborn (1999) investigated the relationship of preservice teachers’ loci of authority and their ability to reflect deeply on mathematics teaching and learning. She found that a setting that promotes inquiry may assist in ushering them from requiring an external locus of authority to becoming confident in their ability to provide authority. Furthermore, preservice teachers should also be made aware of the difference between their idea of “permissibility” in the sense of mathematical “rules” and the deeper notion of mathematical validity. Here again, we see the importance of offering preservice teachers more mathematical ways of thinking.

Preservice teachers who did not successfully determine the generalizability, or lack thereof, for a particular algorithm failed to do so because of one of two errors. They either tried sample problems that were too limited in the types of numbers or they looked too superficially at the written representation of the procedure and not at the underlying thinking. The first of these errors likely results from mathematical inexperience. The second error, however, suggests that preservice teachers saw algorithms less for what they represented and more for what was written on the paper. In investigations that occurred in class, it was apparent that preservice teachers responded to the question of whether an algorithm works by considering initially whether it produced the correct solution. If it did, most preservice teachers stopped the evaluation process, convinced that there was nothing else to consider. This observation fits with the difficulties


preservice teachers had in evaluating an algorithm that numerically produced the correct solution but was flawed mathematically.

With regard to the differences that appeared when reviewing results based on content, it is not surprising that fractions topics were most challenging for preservice teachers. The difficulties that preservice teachers have with fractions and division-related content is well documented (Ball, 1990; Ma, 1999; Simon, 1993; Tirosh, 2000). Teachers’ self-confidence in their ability to understand student work may play a role in their decision to investigate students’ claims mathematically (Ma, 1999). Thus, while introducing preservice teachers to mathematical habits of mind through activities such as evaluating student work, it is vital to maintain a focus on further deepening content knowledge so that the confidence to carry out that evaluation is also developed.

Exploring alternate algorithms, along with the traditional algorithms, may also offer opportunities for rich discussions in K-12 mathematics classrooms (Baek, 1998; Mingus & Grassl, 1998). If teachers are not equipped with the mathematical knowledge necessary to evaluate student work, they are less able to lead a class through an investigation that is anything but shallow. A substantial mathematical preparation of teachers is necessary so that this opportunity for mathematical investigations is not missed.

Conclusion

Preservice teachers demonstrate an intuitive understanding of the need to check whether an algorithm is generalizable and efficient. Their concern with mathematical validity, though superficial, is at least present. Modeling mathematical thinking in tasks of this nature may assist in developing the habits of mind necessary for successfully completing the mathematical tasks of elementary teaching. In addition, a combination of exposure to the way mathematicians think and experiences with students may lead preservice teachers to discover themselves as sources of mathematical authority, particularly in their own classrooms. This level of independence and competence seems necessary to achieve the kinds of classroom teaching necessary for developing students’ mathematical understanding.

References

Ashlock, R. B. (2006). Error patterns in computation: Using error patterns to improve instruction. Upper Saddle River, NJ: Pearson Merrill Prentice Hall.

Baek, J. (1998). Children’s invented algorithms for multidigit multiplication problems. In L. J. Morrow & M. J. Kenney (Eds.) The teaching and learning of algorithms in school mathematics, 1998 Yearbook of the National Council of Teachers of Mathematics (pp. 151–160). Reston, VA: National Council of Teachers of Mathematics.

Ball, D. L. (1990a). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449–466.

Ball, D. L. (1990b). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21, 132–144.

Ball, D. L. (1998). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48.

Ball, D. L., Bass, H., & Hill, H. (2004, July). Knowing and using mathematical knowledge in teaching: Learning what matters. Invited paper presented at the Southern African Association for Research in Mathematics, Science, and Technology Education, Capetown, South Africa.

Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D. & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23, 194–222.

Campbell, P. F., Rowan, T. E., & Suarez, A. R. (1998). What criteria for student-invented algorithms? In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics, 1998 Yearbook of the National Council of Teachers of Mathematics (pp. 49–55). Reston, VA: National Council of Teachers of Mathematics.

Cathcart, W. G., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2006). Learning mathematics in elementary and middle schools: A learner-centered approach, 4th ed. Upper Saddle River, NJ: Pearson Merrill Prentice Hall.

Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers, Part 1. Washington, D.C.: Mathematical Association of America & American Mathematical Society.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.

Mewborn, D. S. (1999). Reflective thinking among preservice elementary mathematics teachers. Journal for Research in Mathematics Education, 30, 316–341.

Mingus, T. T. & Grassl, R. M. (1998). Algorithmic and recursive thinking: Current beliefs and their implications for the future. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics, 1998 Yearbook of the National Council of Teachers of Mathematics (pp. 32–43). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp 53–70). Hillsdale, NJ: Lawrence Erlbaum.


Seaman, C. E. & Szydlik, J. E. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10, 167–182.

Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24, 233–254.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31, 5–25.

Van de Walle, J. A. (2004) Elementary and middle school mathematics: Teaching developmentally, 5th ed. Boston: Pearson.

Zazkis, R. & Campbell, S. (1996). Divisibility and Multiplicative Structure of Natural Numbers: Preservice Teachers’ Understanding. Journal for Research in Mathematics Education, 28, 216 – 236.

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In this Issue,

In Focus… Georgia’s Compensation Model: A Step in the Right Direction NICHOLAS OPPONG, ZANDRA U. DE ARAUJO, LAURA LOWE, ANNE MARIE

MARSHALL, LAURA SINGLETARY

Teacher-Team Development in a School-Based Professional Development Program LU PIEN CHENG, HO-KYOUNG KO

Understanding Rolle’s Theorem

REVATHY PARAMESWARAN

Beyond the Right Answer: Exploring How Preservice Elementary Teachers Evaluate Student-Generated Algorithms TRACIE McLEMORE SALINAS

Documents

T MATHEMATICS EDUCATORmath.coe.uga.edu/tme/issues/v19n1/v19n1.pdf · that of the 9,000 mathematics teachers in the state, 14.3% are not fully certified, and the average two-year attrition