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Effects of an Additional MathematicsContent Course on Elementary Teachers'Mathematical Beliefs and Knowledge forTeachingMarvin E. Smith a , Susan L. Swars b , Stephanie Z. Smith b , Lynn C.Hart b & Regine Haardörfer ca Kennesaw State Universityb Georgia State Universityc Emory UniversityPublished online: 11 Oct 2012.

To cite this article: Marvin E. Smith , Susan L. Swars , Stephanie Z. Smith , Lynn C. Hart & RegineHaardörfer (2012) Effects of an Additional Mathematics Content Course on Elementary Teachers'Mathematical Beliefs and Knowledge for Teaching, Action in Teacher Education, 34:4, 336-348, DOI:10.1080/01626620.2012.712745

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Action in Teacher Education, 34:336–348, 2012Copyright © Association of Teacher EducatorsISSN: 0162-6620 print/2158-6098 onlineDOI: 10.1080/01626620.2012.712745

Effects of an Additional Mathematics Content Course onElementary Teachers’ Mathematical Beliefs and Knowledge

for Teaching

Marvin E. SmithKennesaw State University

Susan L. Swars, Stephanie Z. Smith, and Lynn C. HartGeorgia State University

Regine HaardörferEmory University

This longitudinal study examines the effects of changes in an elementary teacher preparation programon mathematics beliefs and content knowledge for teaching of two groups of prospective teachers(N = 276): (1) those who completed a program with three mathematics content courses and twomathematics methods courses and (2) those who completed a program with four mathematics con-tent courses and a single mathematics methods course. The results reveal salient benefits of a secondmethods course that were not evident in the new program with only one methods course. Further,the addition of a fourth content course did not result in notable differences in mathematical knowl-edge for teaching. In addition, mathematical knowledge for teaching was positively linked to changein pedagogical beliefs about learners, further illuminating the interwoven nature of knowledge andbeliefs.

INTRODUCTION

The development of adequate and appropriate mathematical knowledge in elementary teachersis of paramount concern to mathematics educators globally. Teachers require strong knowl-edge of mathematical content to be effective in their teaching (Hill, 2010). Disconcertedly,the mathematical preparation of teachers in the United States, which is the context for thisstudy, needs to compare more favorably with teachers in other countries as measured byprospective teachers’ knowledge, beliefs, and preparation experiences (Mathematics Teachingin the 21st Century, 2007). Prospective teachers in the United States particularly need tobe better prepared to adeptly teach demanding mathematics curriculum. Effective teacher

Correspondence should be addressed to Marvin E. Smith, Kennesaw State University, Mail Drop 0121, 1000 ChastainRd., Kennesaw, GA 30144. E-mail: msmit283@kennesaw.edu

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EFFECTS OF AN ADDITIONAL MATHEMATICS CONTENT COURSE 337

preparation programs are critically important for future teachers and the students they will beteaching.

The setting for this study was an undergraduate teacher preparation program at an institution ofhigher education in the United States with changing mathematical requirements for prospectiveelementary teachers. These changes were made in response to a mandate by the state govern-ing board and consisted of increasing the number of mathematics content courses for elementaryteachers from three to four and decreasing the number of mathematics teaching methods coursesfrom two to one. As a means of documenting the impact of this program modification on prospec-tive teachers, a group of researchers began a longitudinal, multifaceted research project. Thisstudy identifies important elements of prospective teachers’ beliefs and knowledge and exam-ines how the program changes longitudinally affected these elements, including comparisons ofprospective teachers before and after the changes.

THEORETICAL PERSPECTIVES AND RELATED RESEARCH

Considerable research has explored various aspects of teachers’ mathematical beliefs and knowl-edge. Studies have established a robust relationship between teachers’ beliefs and teaching byshowing that beliefs influence teacher thinking and behaviors, including instructional decisionmaking and use of curriculum materials (Clark & Peterson, 1986; Philipp, 2007; Romberg &Carpenter, 1986; Thompson, 1992; Wilson & Cooney, 2002). Teacher knowledge of mathe-matics has also been linked with important educational processes and outcomes (Hill, 2010),coupled with significant efforts to precisely define the nature of the knowledge needed for teach-ing in the classroom (Ball, 1990; Ball, Hill, & Bass, 2005; Hill, 2010; Rowland, Huckstep, &Thwaites, 2005). The following review of relevant literature focuses on teacher beliefs, includingpedagogical beliefs and efficacy beliefs, and teacher knowledge in mathematics.

Teacher Beliefs

Beliefs are psychologically held understandings, premises, or propositions about the worldconsidered to be true; they are more cognitive in orientation than emotions and attitudes(Philipp, 2007). Teachers’ beliefs develop over time (Richardson, 1996) during what Lortie(1975) termed the “apprenticeship of observation” while a student is in K-12 classroomsand are well established by the time a student enters college (Pajares, 1992). Philipp (2007)underscored the importance of beliefs in mathematics when he asserted, “For many studentsstudying mathematics, the feelings and beliefs that they carry away about the subject are at leastas important as the knowledge they learn of the subject” (p. 257). The beliefs of prospectiveteachers are influential in how and what they learn and should be targets of change during theteacher preparation process (Feiman-Nemser, 2001; Richardson, 1996), though programs areconstrained by the limited time to affect changes.

The reform perspective proffered by the National Council of Teachers of Mathematics(NCTM; 2000) requires substantial paradigmatic shifts for many prospective teachers, includingchanges in beliefs about the teaching and learning of mathematics, that is, pedagogical beliefs.The NCTM recommended the amalgamation of content and process standards to be taught via a

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338 MARVIN E. SMITH ET AL.

pedagogical approach that is different than the traditional instruction found in many U.S. class-rooms. Teachers believe in and use a wide array of teaching and learning strategies, but mostconceptions of what is believed to be good pedagogy are derived from two approaches: tradi-tional transmission instruction and constructivist compatible instruction (Handal, 2003; Ravitz,Becker, & Wong, 2000). Many of the NCTM’s (2000) recommendations are grounded in aconstructivist compatible method of teaching, in which learners develop meaning based onexperience and inquiry, and teachers choose and develop tasks that are likely to promote thedevelopment of students’ understandings of concepts and procedures in ways that also fosterstudents’ abilities to solve problems and to reason and communicate mathematically (White-Clark, DiCarlo, & Gilchriest, 2008). Studies on changing the mathematical pedagogical beliefs ofelementary prospective teachers have largely focused on aligning these beliefs with a reform per-spective. These studies often examined change during only one course or semester; and althoughsome reported achieving desired effects, others did not (Kalchman, 2011; Philipp et al., 2007;Wilkins & Brand, 2004).

In addition to pedagogical beliefs, an important teacher belief is teaching efficacy, which hasbeen linked with classroom instructional strategies, willingness to embrace educational reform,commitment to teaching, and student achievement. Teacher efficacy is grounded in Bandura’s(1986, 1997) conceptualization of self-efficacy. Self-efficacy beliefs are largely formed duringan individual’s past experiences with a task or activity; successful performances strengthen thesebeliefs whereas failures lower them. Self-efficacy beliefs are considered situation specific andtherefore a focus of research in the area of teaching, that is, teaching efficacy. Teacher efficacy isconceptualized by many researchers as a two-dimensional construct (Enochs, Smith, & Huinker,2000). The first factor personal teaching efficacy represents a teacher’s belief in his or herskills and abilities to be an effective teacher (Tschannen-Moran & Hoy, 2001; Poulou, 2007).The second factor teaching outcome expectancy is a teacher’s belief that effective teaching canbring about student learning regardless of external factors such as home environment, familybackground, and parental influences. Disconcertedly, many prospective teachers’ mathematicsteaching efficacy beliefs are linked with their past experiences with traditional, behavioristmethods of mathematics instruction, thus generating a tension between the development ofcognitively oriented pedagogical beliefs and a sense of efficaciousness toward teaching in thisway (Smith, 1996).

The first few years of teacher development are critical to the long-term development of teach-ing efficacy; and once these beliefs are established, they are highly resistant to change (Hoy,2004). Some studies suggest that teacher preparation courses and the student teaching experi-ence have differential effects on personal teaching efficacy and teaching outcome expectancyof prospective teachers. Personal teaching efficacy increases during courses and continues toincrease during the student teaching experience (Hoy & Spero, 2005; Hoy & Woolfolk, 1990;Plourde, 2002). Similarly, teaching outcome expectancy increases during courses but declinesduring student teaching. This decline has been ascribed to the idealism of prospective teach-ers prior to student teaching about teachers’ abilities to overcome negative influences (Hoy &Woolfolk, 1990). Although there are numerous studies on generalized teaching efficacy, therehas been less research specifically on the mathematics teaching efficacy of elementary prospec-tive teachers. Most of the previous studies examined the effects of one mathematics methodscourse and indicated significant increases in mathematics teaching efficacy upon completion ofthe course (Huinker & Madison, 1997; Kalchman, 2011; Swars, 2005; Utley, Moseley, & Bryant,2005).

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EFFECTS OF AN ADDITIONAL MATHEMATICS CONTENT COURSE 339

Teacher Knowledge

The need for elementary teachers to possess strong mathematical knowledge is undisputed.Concernedly, Hill (2010) asserted “Both observations and interview data suggest that U.S. ele-mentary teachers vary widely in their grasp of the mathematics needed to teach this subject”(p. 514). Substantial effort has focused on defining the exact nature of the mathematical knowl-edge needed for teaching in the elementary classroom (Ball et al., 2005; Hill, 2010; Rowlandet al., 2005). In recent years, researchers (Ball & Forzani, 2010; Ball et al., 2005; Ball, HooverThames, & Phelps, 2008; Hill, 2010) have proposed a specialized content knowledge (SCK) thatis different from Shulman’s (1986) subject matter knowledge (SMK) and pedagogical knowl-edge (PCK). SCK has been defined as “mathematical knowledge needed to perform the recurrenttasks of teaching mathematics to students” (Ball et al., 2008, p. 399). Specifically, Schilling andHill (2007) explained that SCK “consists of mathematical tasks such as representing numbersand operations with pictures or manipulatives, examining and generalizing from non-standardsolution methods, and providing explanations for mathematical ideas or procedures” (p. 78).

Issues of how elementary teachers should acquire mathematical knowledge are paramount.Teacher preparation programs must be focused where they are the most useful, including provid-ing the most effective contexts and experiences for prospective teacher learning in mathematics.A report by the National Mathematics Advisory Panel (Greenberg & Walsh, 2008) arguedfor increasing the number of required courses in mathematics for elementary teachers. Suchcourses are typically taught by mathematicians within higher education mathematics depart-ments. Unfortunately, some policy makers and faculty members persist in using misguidedassumptions about the mathematical preparation of elementary teachers, including (1) elementaryprospective teachers need only take additional advanced mathematics courses to acquire knowl-edge for teaching elementary mathematics and (2) content knowledge is the only professionalknowledge needed for teaching (Sowder, 2007).

RESEARCH QUESTION

Teacher preparation involves a complex combination of changing beliefs and improving knowl-edge, each of which are multifaceted and interrelated. This study examines multiple elements ofteachers’ beliefs and knowledge that have previously shown to affect student learning, includ-ing mathematical pedagogical beliefs and teaching efficacy beliefs, as well as knowledge ofmathematics for teaching. Given the need for institutions of higher education to make informeddecisions about the courses to include in teacher preparation programs, determining the effects ofprograms on the beliefs and knowledge of graduates is essential. Further, because most studies ofchanging prospective teachers’ mathematical beliefs and knowledge have occurred across a sin-gle semester or methods course, changes in these important constructs across multiple semestersin a teacher preparation program warrant careful consideration.

This longitudinal, quantitative study was guided by the following research question:

What are the effects of increased mathematics content courses for elementary teachers anddecreased mathematics teaching methods courses on longitudinal changes in prospec-tive teachers’ mathematical pedagogical beliefs, teaching efficacy beliefs, and contentknowledge for teaching?

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340 MARVIN E. SMITH ET AL.

METHOD

Participants and Setting

This study involved 276 elementary prospective teachers at a large, urban university in thesoutheastern United States. The participants were enrolled in a 2-year undergraduate teacherpreparation program that used a cohort model. The first three semesters included on cam-pus courses and field placements in elementary classrooms for 2 days per week, followedby a semester of full-time student teaching. Field placements prior to student teaching (andcourses when applicable) followed a developmental model, with the grade-level focus startingin prekindergarten and finishing in 5th grade.

One group of prospective teachers (n = 142) completed three mathematics content coursesfor elementary teachers and two mathematics teaching methods courses. This group is referredto here as the “old program group.” Another group of prospective teachers (n = 134) completedfour mathematics content courses for elementary teachers and one mathematics teachingmethods course; this group is referred to here as the “new program group.” Overall, the newprogram involved exchanging a second teaching methods course for a fourth content course.The mathematics content courses completed by both groups included Number and Operations,Geometry and Spatial Reasoning, and Probability and Statistics; the new program group alsocompleted an Algebraic Concepts course. Prospective teachers made their own choices as towhen to enroll in the content courses as long as they were completed prior to student teaching,with the exception of the Number and Operations course, which was taken as an admittancerequirement to the teacher preparation program. The content courses were taught by instructorsin the mathematics department.

The old program group also completed two mathematics teaching methods courses during thesecond and third semesters of the teacher preparation program. The first methods course focusedon grades PreK-2, and the second methods course emphasized Grades 3 through 5. The new pro-gram group completed one mathematics teaching methods course during the second semester ofthe program, which focused on grades PreK-5. The methods courses were taught by instructorsin the elementary education department, who shared a common philosophical orientation towardthe teaching and learning of mathematics. Important goals of the courses included (1) developingbeliefs consistent with the perspectives of the Principles and Standards (NCTM, 2000), (2) under-standing children’s thinking about important mathematics concepts, (3) creating problem-solvinglearning environments to facilitate discourse and understanding, and (4) building confidence as alifelong learner and doer of mathematics.

Data Collection and Analysis

Mathematical beliefs data were collected via two instruments: the Mathematics Teaching EfficacyBeliefs Instrument (MTEBI) and the Mathematics Beliefs Instrument (MBI). These instru-ments were administered five times, including a preassessment at the beginning of the teacherpreparation program and then four other times, specifically at the end of each semester in the pro-gram. Mathematical knowledge for teaching data was gathered via the Learning Mathematicsfor Teaching Instrument (LMT), which was administered one time at the end of the teacherpreparation program.

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EFFECTS OF AN ADDITIONAL MATHEMATICS CONTENT COURSE 341

The MTEBI consists of 21 items, 13 on the Personal Mathematics Teaching Efficacy (PMTE)subscale and eight on the Mathematics Teaching Outcome Expectancy (MTOE) subscale (Enochset al., 2000). The two subscales are consistent with the two-dimensional aspect of teacher efficacy.The PMTE subscale addresses the prospective teachers’ beliefs in their individual capabilitiesto be effective mathematics teachers. The MTOE subscale addresses the prospective teachers’beliefs that effective teaching of mathematics can bring about student learning regardless of exter-nal factors. The instrument uses a Likert-type scale with five response categories, with higherscores indicating greater teaching efficacy. These subscales have high reliability (Cronbach’salpha = .88 for PMTE and .81 for MTOE) and represent independent constructs based onconfirmatory analysis.

The MBI is a 48-item instrument designed to assess prospective teachers’ beliefs about theteaching and learning of mathematics and the degree to which these beliefs are cognitivelyaligned (Peterson, Fennema, Carpenter, & Loef, 1989, as modified by the Cognitively GuidedInstruction Project). The three subscales include (1) relationship between skills and understand-ing (Curriculum), (2) role of the learner (Learner), and (3) role of the teacher (Teacher). The16-item Curriculum subscale examines the degree to which teachers believe that mathematicsskills should be taught in relation to understanding and problem solving. The Learner subscalecontains 15 items that assess the degree to which teachers believe that children can constructtheir own mathematical knowledge. The 17 items on the Teacher subscale address the extent towhich teachers believe that mathematics instruction should be organized to facilitate children’sconstruction of knowledge. The instrument uses a Likert-type scale with five response categories,with higher scores indicating beliefs that are more cognitively aligned. These subscales havehigh reliability (Cronbach’s alpha = .80 for Curriculum, .89 for Learner, and .90 for Teacher)and represent independent constructs based on confirmatory factor analysis.

The LMT examines teachers’ SCK for teaching mathematics (Hill, Schilling, & Ball, 2004).The instrument assesses this knowledge by posing mathematical tasks that reflect what teachersencounter in the classroom, such as assessing students’ work, representing mathematics ideasand operations, and explaining mathematical rules or procedures. Content knowledge subscalesin this instrument include (1) number and operations; (2) patterns, functions, and algebra; and(3) geometry (Hill et al., 2004), though only aggregate LMT scores were considered in this study.Content validity was established by mapping items for congruence with the NCTM Standards(Dean, n.d.; Siedel & Hill, 2003). Analysis of reliability indicated alpha coefficients of .79 for thenumber and operations subscale; .75 for the patterns, functions, and algebra subscale; and .85 forthe geometry subscale (G. Phelps, personal communication, October 6, 2006).

The data in this study involved a three-level nested structure that warranted the use of hierar-chical linear growth modeling (HLM) for analysis. Individual measurements over time (L1 or t)are nested within persons (L2 or i) that are in turn nested in groups experiencing old and new pro-grams (L3 or j). In addition, there is some variability in the number of measurements per group,as well as per person. Both issues are compensated for when using HLM (Singer & Willett,2003).

An extant published study by the researchers suggests that content knowledge for teaching iscorrelated with more cognitively oriented pedagogical beliefs and higher teaching efficacy beliefs(Swars, Smith, Smith, & Hart, 2009). Thus, the results of the LMT were included at the prospec-tive teacher level (L2) of the conditional model. Prospective teachers in both programs completedone mathematics methods course (M1) during the second semester of the program. Only the old

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342 MARVIN E. SMITH ET AL.

program group completed a second mathematics methods course (M2) during the third semesterof the program. The full model equation reduces to the following predictor equation (final model):

Y ′tij = γ000 + γ010(LMTij) + γ100(TIMEtij) + γ110(LMTij)(TIMEtij)

+ γ200(M1tij) + γ301(Programj)(M2tij)

This model was used to analyze five outcome variables consistent with the subscales of theMTEBI and MBI: personal teaching efficacy beliefs, teaching outcome expectancy beliefs, andpedagogical beliefs about the curriculum, the learner, and the teacher. Coefficients from this anal-ysis are given in terms of values on the 5-point Likert-type scale used in the beliefs instruments(i.e., MTEBI and MBI). Coefficients of influencing factors indicate the changes in the linearmodel attributable to each significant factor.

RESULTS

The difference in mean LMT scores (raw percent correct from a single administration of a com-mon form) for prospective teachers in the old and new program groups was not statisticallysignificant (p = .065), indicating that exchanging a second mathematics methods course for anadditional mathematics content course did not have a significant impact (either favorably or unfa-vorably) on SCK for teaching elementary mathematics. However, LMT scores were found to bea statistically significant factor in accounting for changes in most of the other outcome variables,confirming that prospective teachers with greater SCK also have more cognitively oriented beliefsand greater personal teaching efficacy.

Table 1 presents the HLM results for each subscale for the unconditional model and the fullmodel. Standard error values are shown in parentheses. Statistically significant results are indi-cated with asterisks. The unconditional model includes only time as a predictor and investigatesif the outcome variables change significantly over time. For most outcomes, the change over timewas a significant predictor on its own. The only exception was MTOE where the overall outcomeexpectancy score across all prospective teachers did not change significantly over time.

The full model results provided in Table 1 indicate the impact of time, LMT score, and meth-ods courses M1 and M2 on each outcome variable. The PMTE-Full Model column shows thatpersonal teaching efficacy beliefs were positively affected by LMT score, time, and the secondmathematics methods course (M2). The MTOE-Full Model column shows that teaching out-come expectancy beliefs were positively affected only by the second mathematics methods course(M2). This variable is not correlated with the LMT score. The Curriculum-Full Model columnshows that pedagogical beliefs about the mathematics curriculum were positively affected bythe LMT score and the first mathematics methods course (M1). The Learner-Full Model columnshows that pedagogical beliefs about the learner were positively affected by the interaction oftime and LMT score and by the second mathematics methods course (M2). This interaction ofTime and LMT score indicates that prospective teachers who have a higher LMT score increasetheir beliefs about the learner more over time than those with lower LMT scores. The Teacher-Full Model column shows that pedagogical beliefs about the teacher were positively affected onlyby the LMT score.

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TAB

LE1

Hie

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esul

ts

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Inte

rcep

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42∗∗

(0.1

0)3.

05∗∗

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(0.0

7)3.

54∗∗

(0.1

2)3.

07∗∗

(0.0

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72∗∗

(0.0

9)3.

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85∗∗

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32∗∗

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ime

0.18

∗∗(0

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0.18

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.04

(0.0

6)0.

15∗∗

(0.0

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(0.0

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10∗∗

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.07

(0.0

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14∗∗

(0.0

1)0.

004

(0.0

7)L

MT

0.68

∗(0

.25)

−0.0

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0.69

∗(0

.19)

0.36

(0.1

5)0.

50∗

(0.1

5)L

MT

×T

ime

−0.0

5(0

.07)

0.08

(0.0

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0.18

∗(0

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0.03

(0.0

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1−0

.02

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4)0.

04(0

.11)

0.46

∗∗(0

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×Pr

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(0.0

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PMT

E=

Pers

onal

Mat

hem

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ffica

cy;

MT

OE

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MT

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1=

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=se

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.∗ p

<.0

5,∗∗

p<

.01.

343

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344 MARVIN E. SMITH ET AL.

DISCUSSION AND CONCLUSIONS

Teacher preparation programs must be focused where they are the most useful, including provid-ing the most effective contexts and experiences for prospective teacher learning in mathematics.These contexts and experiences form, in part, the opportunity to learn (OTL) (Hill, Rowan, &Ball, 2005) for prospective teachers. OTL links teaching with learning and is widely consid-ered to be the most salient factor influencing student learning and accounting for differencesin that learning (Hiebert & Grouws, 2007). Opportunity to learn is considered to be a complexprocess and the product of the curricular emphasis and the quality of instruction, among otherfactors. In this study, the different program experiences offered in the old and new programsinfluenced OTL for the prospective teachers, contributing to the degree to which mathematicalknowledge for teaching was learned. As such, the mandate of requiring more mathematics con-tent courses for elementary teachers appears to have fallen short of some of its intended effectswithin the teacher preparation program. A key finding of this study is that increasing the num-ber of mathematics content courses and decreasing the number of mathematics teaching methodscourses did not result in notable differences in mathematical knowledge for teaching betweenthe two groups of graduates. Apparently, content knowledge relevant to teaching acquired inthe context of a second teaching methods course taught by education faculty was compara-ble to knowledge relevant to teaching acquired during a content course taught by mathematicsfaculty.

This study addresses a gap in the extant research literature on teacher preparation by longi-tudinally examining changes in key aspects of teacher development across an entire program.Although the programmatic changes resulted in little differences between the two groups’ mathe-matical knowledge for teaching, the longitudinal findings related to the program experiences andshifts in the prospective teachers’ beliefs are notable. Prospective teachers enter teacher prepara-tion programs with deep rooted beliefs about mathematics teaching and learning (Pajares, 1992),and the results of this study indicate that the experiences across the cohort based, developmen-tal teacher preparation program significantly affected beliefs. The experiences afforded in theold and new programs prompted the prospective teachers’ pedagogical beliefs to become morecognitively oriented in general, thus more consistent with a reform perspective (NCTM, 2000).Moreover, the prospective teachers with greater SCK had more cognitively oriented pedagogi-cal beliefs and greater personal teaching efficacy. Specifically, in the case of cognitively orientedbeliefs about learners, greater SCK correlated with larger increases in these beliefs as indicatedby a greater slope in the linear model for this variable for those with higher LMT scores (i.e., theTime x LMT interaction). In other words, those with more developed SCK had greater changes intheir beliefs that children can construct their own mathematical knowledge. These latter findingsilluminate some of the interrelatedness of teachers’ beliefs and knowledge, and the nuances ofthis connectedness warrant further examination.

When considering the subscales of the MBI, the prospective teachers’ beliefs about the ele-mentary curriculum were significantly affected by the first methods course in the old and newprograms. The prospective teachers’ beliefs that mathematics skills should be taught in relation tounderstanding and problem solving were significantly influenced during the first methods course.Given the general emphasis of this course, including a focus on problem solving and other math-ematical processes fundamental to a standards-based curriculum, as well as student constructionof conceptual understanding and procedural knowledge, this finding is not surprising.

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However, additional consideration of the subscales of the MBI and MTEBI reveals the secondmethods course (and not the first) in the old program significantly affected prospective teach-ers’ personal teaching efficacy beliefs, teaching outcome expectancy beliefs, and pedagogicalbeliefs about learners. The experiences in the second methods course significantly influencedthe prospective teachers’ beliefs in their capabilities to teach mathematics effectively and affectstudent learning, as well as their beliefs that children can construct mathematical knowledge.These findings suggest there are discernable benefits of a second methods course that weremissed by the group of prospective teachers experiencing only one methods course in the newprogram. Further examination of the MTEBI subscales as related to the new program groupalso indicates the experiences across the new program did not significantly affect prospectiveteachers’ teaching outcome expectancy beliefs without the experiences of the second methodscourse.

Findings from the MBI subscales also reveal pedagogical beliefs related to the teacher werenot affected significantly by either the first or second mathematics methods course. Neither courseexperience had significant influences on prospective teachers’ beliefs that instruction should beorganized to facilitate children’s construction of knowledge. Informal feedback from prospectiveteachers indicates this finding is likely linked to the program design, specifically the teachingexperiences with 2 days per week in field placement classrooms. It is important to note that toooften prospective teachers are placed in classrooms exhibiting traditional methods of mathematicsinstruction that could constrain a paradigm shift regarding the role of the teacher.

When examining the longitudinal effects of the data across the program (i.e., Time), the oldand new programs were effective at increasing prospective teachers’ personal teaching efficacybeliefs and shifting pedagogical beliefs toward cognitive alignment during the three semestersof courses and field experiences prior to student teaching. During student teaching, personalteaching efficacy beliefs continued to increase, whereas teaching outcome expectancy beliefs andpedagogical beliefs remained relatively stable. The enculturation effect and the perceived need toemulate a cooperating teacher during student teaching are thought to provide likely explanationsfor these findings.

Studying the effects of programs on the beliefs and knowledge of graduates is of critical impor-tance for policy makers and institutions of higher education to make informed decisions about themathematics courses to include in teacher preparation programs. In this study, the large number ofparticipants provides strong support for the findings in this particular context. In the field of math-ematics education, there is significant interest in developing elementary teachers’ mathematicalknowledge as an important element in teacher preparedness, and the findings of this study chal-lenge the perspective that more mathematics content courses for elementary teachers results ingreater knowledge for teaching mathematics in the elementary classroom than content-specificpedagogy courses. Interestingly, exchanging the second methods course for a fourth contentcourse affected program outcome variables in expected and unexpected ways. The study resultsreveal salient benefits of a second methods course that were not evidenced by the new programexperiences affording an additional content course but only one methods course. For example,the second methods course positively influenced prospective teachers’ personal teaching efficacybeliefs, teaching outcome expectancy beliefs, and pedagogical beliefs about learners, which wasnot evidenced in the first methods course, showing the added value of the second methods course.Further, the addition of a fourth content course did not result in notable differences in mathemat-ical knowledge for teaching. In addition, mathematical knowledge for teaching was positively

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linked to change in pedagogical beliefs about learners, thus highlighting the interconnectednessof knowledge and beliefs.

In general, though this quantitative study reveals interesting, significant results, there is a needfor additional in-depth exploration of the explanations for some of these results. For example,within the context of this study, the design and implementation of the mathematics contentcourses for elementary teachers involved scant collaboration between the elementary educa-tion department and the mathematics department to coordinate efforts to connect and unify theprospective teachers’ learning in the methods courses and the content courses. Such activities arekey features of effective teacher preparation programs in mathematics (Sowder, 2007) and shouldbe explored further for possible explanations regarding the finding that an additional contentcourse resulted in insignificant differences in mathematical knowledge for teaching.

Further, disagreement about the mathematics needed for teaching elementary mathematicscould be an explanation for some of the findings. There often exists a philosophical dissonancebetween faculty members in mathematics departments and those in teacher preparation depart-ments regarding the mathematical knowledge needed for teaching elementary mathematics. Hill(2010) argued “very little is known about the nature . . . of elementary teachers’ mathematicalknowledge for teaching” (p. 514). This absence of agreement about adequate and appropriatemathematical knowledge for teaching in the elementary classroom could be an impetus for someof the findings. Issues such as this warrant further examination in order for teacher prepara-tion programs to offer opportunities to learn that better prepare prospective teachers for teachingmathematics in the elementary classroom.

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Marvin E. Smith is an Associate Professor of mathematics education at Kennesaw StateUniversity interested in learning and teaching elementary mathematics, classroom assessment,and teacher education and development.

Susan L. Swars is an Associate Professor of mathematics education at Georgia State Universityinterested in teacher development in mathematics education, specifically related to contentknowledge and beliefs.

Stephanie Z. Smith is an Associate Professor of mathematics education at Georgia StateUniversity interested in learning and teaching elementary mathematics, conceptions of mathe-matics, and teacher education and development.

Lynn C. Hart is a Professor of mathematics education at Georgia State University interested inbeliefs, content knowledge, and effective models of teacher change.

Regine Haardörfer is a Statistician at the Rollins School of Public Health at Emory Universityinterested in complex modeling techniques such as hierarchical linear modeling and structuralequation modeling.

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