Preservice Elementary Teachers' Mathematics Content Knowledge and Teacher Efficacy

Preservice Elementary Teachers’ Mathematics Content Knowledge andTeacher Efficacy

Kristie Jones NewtonTemple University

Jacqueline LeonardUniversity of Colorado Denver

Brian R. EvansPace University

Julie A. EastburnCouncil Rock School District

The purpose of this study was to examine the relationship between mathematics content knowledge and teacherefficacy during an elementary mathematics methods course. A positive moderate relationship between content knowl-edge and personal teaching efficacy was found, and this relationship was stable during the course. No relationship wasfound between content knowledge and outcome expectancy. Written artifacts were used to understand and elaborate onthese findings. Those data suggest that prior learning experiences may help explain this relationship. Additionally, theysuggest that preservice teachers with different levels of content knowledge may attend to different sources of informationwhen making efficacy judgments about teaching.

During the past few decades, much of the conversationabout teacher quality has focused on content knowledge,calling for deep knowledge of subject matter (Ball, Hill,& Bass, 2005; Kahan, Cooper, & Bethea, 2003; Ma,1999). At the same time, teacher efficacy is consideredan important factor in high-quality mathematics instruc-tion. Generally speaking, teacher efficacy refers to “ateacher’s sense of ability to organize and execute teach-ing that promotes learning” (Charalambous, Philippou, &Kyriakides, 2008, p. 126). Research has consistentlyshown that teacher efficacy is related to a variety ofstudent outcomes such as achievement and motivation, aswell as teacher behaviors such as a willingness to trynew methods and to persist with students who strug-gle (Haney, Lumpe, Czerniak, & Egan, 2002; seeTschannen-Moran & Hoy, 2001, for an overview). Nev-ertheless, little is known about the relationship betweencontent knowledge and teacher efficacy. Given the stronginfluence of both factors on teacher quality and behav-iors, additional studies that examine the relationshipbetween mathematics content knowledge and teacherefficacy are warranted.

Teacher EfficacyBandura (1977) developed what is now commonly

known as self-efficacy theory. According to Bandura(1986), efficacy beliefs consist of two factors: personalself-efficacy and outcome expectancy. Applied to teaching,personal self-efficacy is defined as the perceived judgmentthat an individual has about his or her capacity to teach,whereas outcome expectancy reveals the teacher’s percep-tion of the students’ ability to learn from his or her teaching.

These two factors were used to develop a survey question-naire to measure teacher efficacy, initially in science withthe Science Teacher Efficacy Beliefs Instrument (STEBI)(Enochs & Riggs, 1990), and more recently in mathematicswith the Mathematics Teacher Efficacy Beliefs Instrument(MTEBI) (Enochs, Smith, & Huinker, 2000).

Bandura’s (1997) theory suggests efficacy beliefs aremalleable, and he described four sources of informationthat contribute to these beliefs: mastery experiences,vicarious experiences, verbal persuasion, and affectivestates. Mastery experiences are successes obtained fromactual practice. With regard to teaching, these experiencescould include anything from microteaching in methodscourses to field-based experiences in actual classrooms.Vicarious experiences are acquired by observation of aspecific behavior. Within a methods course, such experi-ences might be gained through video or observations ofpeers involved in microteaching. Verbal persuasion isextrinsic motivation, such as encouragement or praise,which often comes from methods instructors and cooper-ating teachers. Finally, affective states, such as stress andemotions, influence efficacy and related behaviors. Teach-ers who express disaffection with mathematics or scienceare more likely to avoid planning or teaching these sub-jects (Jesky-Smith, 2002; Trice & Ogden, 1986). Negativeemotions may inhibit teaching performance, and thus leadto low efficacy. On the other hand, teachers with highefficacy are more likely to engage students in inquiry andstudent-centered teaching, which are linked to higherachievement (Czerniak & Schriver, 1994; Swars, Hart,Smith, Smith, & Tolar, 2007). Thus, development ofteacher efficacy is a worthwhile endeavor.

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Research supports the idea that teacher efficacy can bedeveloped among preservice teachers (Charalambouset al., 2008; Palmer, 2006). In fact, some suggest thatmethods courses may be particularly critical for the devel-opment of teacher efficacy, perhaps because they providestructure and support for early field experiences (Utley,Moseley, & Bryant, 2005). In a study examining efficacydevelopment, Utley and colleagues included measures ofboth mathematics and science teacher efficacy. In particu-lar, 51 preservice teachers completed the MTEBI andSTEBI at the beginning and end of their methods courses,and at the end of student teaching. These researchers founda quadratic trend for personal teaching efficacy in bothmathematics and science. In other words, preservice teach-ers had increased beliefs about their ability to teach effec-tively by the end of their methods courses, but a slightdecrease occurred during student teaching. The same resultwas found for mathematics outcome expectancy, but nosignificant changes existed for science outcome expect-ancy. Researchers speculated that student teaching involvesless support at a time when realities of the classroombecome more apparent.

In trying to understand the impact of intensive fieldexperiences such as student teaching, Charalambous et al.(2008) examined preservice teachers’ (N = 89) mathemat-ics efficacy beliefs before, during, and after participatingin 12 weeks of supervised teaching. Using a two-factormodel to measure teacher efficacy beliefs (i.e., instruc-tional skill and classroom management), the researchersfound different patterns of development among preserviceteachers with different levels of teacher efficacy beliefs atthe outset. While teacher efficacy increased for all groups,those who began with the lowest levels of teacher efficacybenefited the most. Interviews of preservice teachershelped uncover factors that contributed to changes in effi-cacy beliefs, such as the discrepancy between theirinstructional practices and those of the mentor teachers.Because some preservice teachers also talked about theirown (poor) experience learning mathematics, Charalam-bous et al. suggested that future studies should includecontent knowledge as an additional factor related toteacher efficacy.

Other researchers have also noticed that preserviceteachers draw on their own learning experiences whendiscussing mathematics teaching (Brand & Wilkins, 2007;Swars, 2005). For example, Swars (2005) interviewed fourpreservice teachers to investigate the perceptions of math-ematics teaching effectiveness among preservice teacherswith high and low efficacy, based on overall scores fromthe MTEBI. Findings revealed that the two preservice

teachers with low efficacy had negative past experienceswith mathematics, but the two preservice teachers withhigh efficacy had different past experiences. One of themstruggled with mathematics but worked diligently at it andfelt she would be able to empathize with her future stu-dents. The other had positive experiences with mathemat-ics and felt these experiences would contribute to hereffectiveness as a teacher. The researchers concluded bysuggesting that methods courses need to not only providepositive experiences to offset the negatives ones, butmethods courses also need to help teachers reflect on thesepast experiences and put them into perspective.

From a theoretical perspective, it is not surprising thatpreservice teachers would draw on their experiences aslearners when making efficacy judgments about teaching.According to Bandura (1997), when people lack familiar-ity with an activity, “they tend to make self-efficacy judg-ments partly from knowledge of what they can do insimilar situations” (p. 70). In the case of teaching math-ematics, what a novice can do as a learner of mathematicsmay represent the situation that is most similar to what heor she can do as a teacher. Based on this reasoning, it isalso plausible that, for preservice teachers, content knowl-edge (i.e., a measure of what they can do as a learner) andteacher efficacy are related.

Despite the theoretical plausibility of content knowl-edge being related to teacher efficacy, Swars et al. (2007)found that it was not. They studied the mathematicscontent knowledge, pedagogical beliefs, and teaching effi-cacy of preservice teachers (N = 103) during a two-coursesequence. Results of the MTEBI showed personal teachingefficacy increased across time in both methods courses andduring student teaching, while significant increases inoutcome expectancy occurred only during the secondmethods course. No relationship was found between pre-service teachers’ content knowledge and either subscale ofteacher efficacy (Swars et al., 2007). These results differfrom findings in science education that suggested priorcontent knowledge is related to personal teaching efficacybut not to outcome expectancy (Cantrell,Young, & Moore,2003). It is possible that there is a difference in the rela-tionship between knowledge and efficacy for mathematicsas compared with science. It is also possible that the rela-tionship depends on the way content knowledge is mea-sured. In the study by Swars et al., content knowledge wasconceived as the specialized knowledge needed for teach-ing mathematics, which included knowledge such as howstudents might solve mathematics problems. On the otherhand, Cantrell et al. (2003) used prior coursework as aproxy for content knowledge. Clearly, more research is

Content Knowledge and Teacher Efficacy

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needed to understand the relationship between contentknowledge and teacher efficacy.

Mathematical Content Knowledge (MCK)Two important types of knowledge needed to learn and

do mathematics are conceptual knowledge and proceduralknowledge (Rittle-Johnson, Siegler, & Alibali, 2001).Simply stated, the difference between these two types ofknowledge has been described as knowing how (proce-dural) and knowing that (conceptual) (Hiebert & Lefevre,1986). More specifically, procedural knowledge includesknowledge of syntax and step-by-step rules, whereasconceptual knowledge is defined as “implicit or explicitunderstanding of the principles that govern a domain andof the interrelations between units of knowledge in adomain” (Rittle-Johnson et al., 2001, p. 346–347). Math-ematical competence relies on linking the two types ofknowledge, and Rittle-Johnson and colleagues found thatincreases in one type lead to increases in the other.

Research on preservice teachers’ MCK has shown thatmany elementary preservice teachers lack conceptualknowledge (Behr, Khoury, Harel, Post, & Lesh, 1997) aswell as a deep understanding of the mathematics needed toteach (Ball et al., 2005; Bransford, Brown, & Cocking,2001). Yet, both are critical to teaching mathematics effec-tively (Ma, 1999). In other words, breadth of subjectmatter knowledge is a necessary but not sufficient condi-tion to foster change in the teaching and learning of math-ematics (Kahan et al., 2003). In fact, research has shown“little correlation between the number of higher math-ematics courses a teacher takes and student learning”(Swars et al., 2007, p. 327). Kahan et al. (2003) found thatpreservice teachers’ content knowledge test scores andlesson plan ratings were related, suggesting content knowl-edge plays a role in lesson planning quality. Yet, observa-tions of actual lessons revealed that preservice teacherswith strong content knowledge sometimes failed to capi-talize on teachable moments and to adequately explainconcepts to students.

In studying the role of content knowledge in teachingmathematics, Sherin (2002) found that teachers’ owncontent knowledge was enhanced as they learned to usenew curriculum materials and approaches, and as theylearned how students interacted with each of these. Like-wise, Ebby (2000) found that preservice teachers’ knowl-edge and views of mathematics were broadened as theymade connections between their experiences as learnersand as teachers. One teacher who described her priorknowledge as particularly weak was able to “learn a greatdeal of mathematics from actually teaching mathematics

lessons” (p. 89). In other words, it seems that MCK can bedeveloped while teaching and learning to teach (Ma, 1999).

The Current StudyThe aforementioned research reveals that mathematics

methods courses and related fieldwork can influence bothpreservice teachers’ mathematics content knowledge(Ebby, 2000; Kahan et al., 2003) and teacher efficacy(Swars et al., 2007; Utley et al., 2005), making thesecourses important contexts for studying these factors.Prior research and theory also support a possible linkbetween content knowledge and teacher efficacy for pre-service teachers (Bandura, 1997; Cantrell et al., 2003), butthe studies are few and the findings are inconsistent withregard to this link. As such, the current study used amathematics methods course to examine the relationshipbetween teacher efficacy and mathematics content knowl-edge. Given the purpose of this study, there were tworesearch questions: (a) What is the relationship betweenelementary preservice teachers’ mathematics contentknowledge and teacher efficacy? (b) How might this rela-tionship be explained and how might it manifest during amathematics methods course?

MethodsSetting and Participants

This study took place at a large research university situ-ated in an urban city in the northeastern United States. TheCollege of Education where the participants were enrolledconsisted of approximately 2,100 students, and theprogram in which they were enrolled (the early childhood/elementary teacher preparation program) is the largest inthe college. Hence, the target population was elementarypreservice teachers in urban settings. Participants werepreservice teachers enrolled in two sections of the secondauthor’s mathematics methods course. They were concur-rently enrolled in a science methods course and in amathematics/science practicum, where they taught whole-class lessons in urban classrooms. The participants haddemographics that were typical of this program; they wereprimarily juniors or seniors in college, 20–22 years old,female, and white (N = 551: females [47]; White [45],African-American [6], Asian [4], and Latina/o [5]2). His-torically, the students in this program have been anxiousabout teaching mathematics, and these students expressedsimilar feelings on the first day of class.Context of the Study

The methods course had a typical routine. Each classsession began with a problem of the day that was related tothe content or process standards focused on during that



session (National Council of Teachers of Mathematics[NCTM], 2000). Class discussions highlighted methodsraised in the primary text (Cathcart, Pothier, Vance, &Bezuk, 2006), which was chosen in part because of itsemphasis on the NCTM (2000) standards. During thesediscussions, the instructor used manipulatives (e.g., placevalue blocks, two-color counters, spinners, dice), authentictasks (e.g., calculating percents using real menus fromrestaurants), and small group work to illustrate the les-son’s mathematics concepts. Concepts from all fivecontent standards (i.e., number, algebra, geometry, mea-surement, probability and statistics) were emphasized inthe course.

The focus on concepts was to help preservice teachersunderstand mathematics in more depth. For example,place value blocks in base four and base five were used tohelp preservice teachers better understand the base tenplace value system. By illustrating how the blocks formeda pattern of cubes, longs, and flats in base four and basefive; how successive places were four or five times theplace to the right, respectively; and how regroupingneeded to occur when the number of blocks was equal tothe base, preservice teachers were able to better under-stand numeration and place value, and to appreciate thebase ten system.

In order to expose preservice teachers to high-qualitymathematics instruction in urban schools, videos ofreform-based teaching and active learning were shown topreservice teacher participants in this study. Episodes fromthe Kay Toliver Files (Foundations for Advancement inScience and Education Productions, 1998) were presentedas specific content was covered in the course (e.g., a fractionlesson was viewed before the unit on fractions). Afterwatching a video, discussion ensued about the content andpedagogy, as well as the NCTM Standards (NCTM, 2000).For example, the class discussed how Kay Toliver madeconnections to students’ everyday lives and encouragedcommunication (e.g., through the use of student journals).Moreover, Kay Toliver modeled appropriate classroommanagement and demonstrated that high levels of engage-ment could take place in urban classrooms. Thus, pre-service teachers experienced effective mathematicsinstruction vicariously by watching the Kay Toliver Files.

During the semester, each preservice teacher presented amicroteaching lesson to demonstrate what was learnedduring the methods course. Microteaching consisted ofteaching a short 15- to 20-minute lesson with a partner toillustrate a particular mathematics concept. These lessonsprovided preservice teachers with opportunities formastery experiences that could enhance their teacher effi-

cacy (Bandura, 1997; Wilson, 1996), as well as opportuni-ties to learn vicariously through their peers. As aculminating project, preservice teachers created a math bagand field-tested it with two children. The project requiredpreservice teachers to create three related mathematicsactivities, one of which integrated children’s literature witha mathematics task.This activity encouraged the preserviceteachers to evaluate what students know and can do, as wellas the cognitive demands of their tasks (Crespo, 2003). Aletter and a survey for parents were included with the bag,and the children were supposed to complete the tasks witha parent or caregiver. Preservice teachers designed theirown surveys but were encouraged to use a 3-, 4-, or 5-pointLikert scale along with an open-ended response question sothat parents could provide comments. Some preserviceteachers either did not ask for or did not receive open-endedcomments, but all of them had feedback in the form of asurvey. Parental comments/survey results on the math bagwere generally positive. Thus, the project allowed opportu-nities for both mastery experience and verbal persuasionvia parent feedback.Data Collection, Data Sources, and Data Analyses

This study employed a mixed methods design to explorehow mathematics content knowledge and teacher efficacyare linked. Greene, Caracelli, and Graham (1989) sug-gested that there are five different purposes of mixedmethods designs, one of which is a complementary design.In this design, one method is used to “elaborate, enhance,or illustrate the results from the other” (p. 266). Essen-tially, one method is used to help interpret findings fromthe other method. In this study, quantitative data were usedto understand how teacher efficacy and MCK were related.Qualitative data were used to interpret and elaborate onthis relationship.

Data sources included a mathematics content test,survey instrument (MTEBI), and written artifacts. Tomeasure mathematics content knowledge (MCK) amongthe preservice teachers, we developed a 20-item math-ematics content test using a representative sample of ques-tions for the Praxis teacher examination (EducationalTesting Service, 2009) since the preservice teachers wererequired to pass this examination in order to obtain theirteaching licenses. Knowledge of the following NCTM(2000) content standards was needed to complete the testitems: number, including fractions, decimals, and percents(eight items); probability and statistics (six items); mea-surement and/or geometry (four items); and algebra (threeitems). Items were presented as multiple-choice wordproblems. Some questions assessed conceptual knowledge(e.g., Which fraction is closest to 1?) and some assessed


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procedural skill (e.g., What is the area of the trianglebelow?). In some cases, the questions attempted to assessboth concepts and skill (e.g., The average weight of tenpeople sitting in a boat is 165 lb. If one person gets out ofthe boat, the average weight of the remaining people is175 lb. What is the weight in pounds of the person who gotout of the boat?). The mathematics content test was admin-istered as a pretest near the beginning of the semester andagain as a posttest near the end of the semester. On theposttest, the items were scrambled to increase internalvalidity. It was established that there were no significantdifferences (p > .05) between the two sections on thepre-post mathematics content test. These data were ana-lyzed using the t-test statistic.

The Mathematics Teacher Efficacy Belief Instrument(MTEBI), developed by Enochs et al. (2000), is a 21-item5-point Likert scale that utilizes the following choices:strongly disagree, disagree, uncertain, agree, and stronglyagree. Similar to the Science Teaching Efficacy BeliefInstrument (STEBI-B) developed by Enochs and Riggs(1990), the MTEBI contains two subscales: personalmathematics teaching efficacy (PMTE) and mathematicsteaching outcome expectancy (MTOE) with 13 and 8items, respectively. Possible scores range from 13 to 65 onthe PMTE, and from 8 to 40 on the MTOE. Enochs et al.found the PMTE and MTOE had Cronbach’s alpha coef-ficients of 0.88 and 0.77, respectively.

At the outset, we determined that there were no signifi-cant differences (p > .05) between the two sections on theMTEBI with respect to pre-post scores on the two sub-scales: PMTE and MTOE. We also used pre-post data todetermine the internal consistency of the data. Cronbach’salpha was found to be 0.82 (PMTE) and 0.76 (MTOE),respectively, given the prescores, and 0.87 (PMTE) and0.74 (MTOE), respectively, given the postscores. Thesedata are consistent with the Cronbach’s alpha coefficientsfound by Enochs et al. (2000) for each of the subscales.These data were analyzed using the t-test statistic and thePearson product moment correlation.

Qualitative data included written artifacts in the form ofelectronic journal entries and a final reflection paper afterhaving taught a lesson using the math bag previouslydescribed. The purpose of analyzing the artifacts was toexplore which four sources of information about efficacy(i.e., mastery experiences, vicarious experiences, verbalpersuasion, and affective states) the preservice teacherswere primarily attending to (Bandura, 1997), and to revealstatements about their prior experiences as learners ofmathematics (Brand & Wilkins, 2007; Charalambouset al., 2008; Swars, 2005). Throughout the methods

course, preservice teachers responded to prompts aboutthe experiences in the course. The first prompt asked thepreservice teachers to describe their previous mathematicsinstruction and how their past instructors compared withKay Toliver. This prompt was chosen for analysis becauseit provided opportunity for statements about vicariousexperiences, affective states, and experiences as learners.The final paper asked the students to reflect on the mathbag experience, particularly whether or not it was effectiveand engaging for the children. This final reflection paperwas chosen for analysis because it allowed opportunitiesfor statements about mastery experiences and verbalpersuasion. Hence, it was possible to examine statementsabout prior learning experiences and all four sources ofteaching efficacy, while including statements from boththe beginning and end of the semester.

The analysis of written artifacts was intended to providefurther insight into how content knowledge and teacherefficacy might be linked. Hence, the papers were dividedinto three groups based on MCK. In order to have threedistinct groups (i.e., groups whose knowledge scores weredifferent by several points), it was decided that thesegroups should be formed based on scores around themean, as well as scores one standard deviation above andbelow the mean. In this way, students whose scores mightbe considered only moderately high or moderately lowwould not be included in the analysis. Inspection of thedata suggested that one group consisted of preserviceteachers with a raw score of 11 or below on the MCKpretest. Another group consisted of preservice teacherswith a raw score of 18 or above on the MCK pretest. Eachof these groups consisted of six preservice teachers. Therewere seven students who scored 14 on the pretest (justbelow the mean of 14.22). Of these, one preservice teach-er’s MCK decreased during the semester, based on thepre-posttest scores. Because explaining this decrease isbeyond the scope of this paper, this person was notincluded in the analysis. As a result, each of the threegroups (i.e., low, middle, and high MCK) included sixpreservice teachers. One person in the low MCK wasmissing a final reflection. Rather than reducing all groupsto five people, it was decided to keep all six preserviceteachers in each group and proceed with the analysis.

The papers were analyzed collectively, without attentionto group designation, in order to minimize bias. The firstauthor read through the papers twice, highlighting state-ments pertaining to personal experiences with learning,mastery experiences, vicarious experiences, verbal per-suasion, or affective states (see Table 1 for examples ofeach code). Statements about learning experiences were



highlighted if they reflected personal experiences or feel-ings (e.g., enjoyment, boredom, struggle) rather thangeneral descriptions about the mathematic instruction.Other statements were highlighted only if there was a clearlink to the preservice teachers’ instruction. For example,most preservice teachers praised Kay Toliver’s instruction,but only a few made statements about their own teachingduring the reflection (e.g., statements about being inspiredto teach in a similar way). That not all preservice teachersmade such statements is not surprising, given that theprompts did not ask about them specifically. Likewise, allpreservice teachers were asked to provide general descrip-tions about their prior mathematics instruction, but theywere not prompted to describe their personal experiencesor feelings. The intent was to capture naturally occurringstatements so that it was less likely the preservice teacherswere simply stating what they thought was desirable withregard to efficacy.

A second coder was trained and then independentlycoded about 15% of the data (three sets of papers). Unbe-knownst to the coder, the three sets of papers consisted of

one from each MCK group. The coders agreed on preser-vice teachers’ attention to various sources of efficacy 80%of the time. A discussion of these codes led to two addi-tional statements being coded, but in both cases, the preser-vice teacher had already been identified as making otherstatements of that type. Once the statements were put into atable (divided into columns based on the five codes earlier),the second author read the statements to check that theymatched the designated codes. No coding changes weremade as a result of this check.

ResultsThe results of this study are organized by the two

research questions. Before presenting these results,however, it is important to note participants’ levels ofcontent knowledge and teacher efficacy during the course.Descriptive statistics for content knowledge and teacherefficacy are presented in Tables 2 and 3, respectively.Additionally, paired samples t-tests showed small but sta-tistically significant increases in MCK and PMTE duringthe course with modest effect sizes using Cohen’s d. Bycomparison, Utley et al. (2005) found small to mediumeffect sizes for personal teaching efficacy in both math-ematics (h2 = 0.064) and science (h2 = 0.127). Palmer(2006) reported a large effect size for personal teachingefficacy in science (d = 1.74). Swars et al. (2007) did notreport effect sizes for PMTE. No effect sizes were reportedfor MCK in the relevant literature. No change wasdetected for MTOE (see Tables 2 and 3).

Table 1Sample Efficacy-Related Statements

Source Example

Experiences as alearner

“I was always afraid to speak in class.I was afraid of getting the wronganswer.”

Affective states “I am nervous about teaching mathbecause of my past experience.”

Vicarious experiences “Miss Toliver is an amazing teacher. Iwish I had her class. She inspiresme to be that kind of a teacher—ateacher that makes a difference.”

Verbal persuasion “I received very positive feedbackfrom the parents and the twostudents that completed the mathbag.”

Mastery experiences “I was very pleased and proud of thework my students did, and I hope toshare this feeling with my ownstudents when I start teaching.”

Table 2Preservice Teachers’ Mathematics Content Knowledge

Mathematics ContentTest

Mean RawScore

Mean ScaledScore

SD t-value d-value

Pretest 14.22 71.10 3.218 -2.496* 0.29Posttest 15.16 75.80 3.357

Note. N = 45.* p < 0.05 (two-tailed).

Table 3Preservice Teachers’ Efficacy

Efficacy Mean SD t-value d-value

PMTE pretest 49.59 6.687 -2.749* 0.34PMTE posttest 52.00 7.251MTOE pretest 28.67 4.144 0.446 0.07MTOE posttest 28.38 3.891

Note. N = 39.* p < .01 (two-tailed).


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Relationship Between Teacher Efficacy and ContentKnowledge

The first research question asked about the nature of therelationship between elementary preservice teachers’mathematics content knowledge and their efficacy forteaching. Results of the Pearson product moment correla-tion analyses revealed statistically significant correlationsbetween personal teacher efficacy (i.e., teachers’ sense ofhow effectively they can teach mathematics) and math-ematics content knowledge (see Tables 4 and 5). Ofparticular note is the finding that the strength of therelationship between MCK and PMTE did not changeduring the course. Specifically, pre-PMTE scores werecorrelated with pre-MCK (r = 0.465, N = 41, and p < .01),and post-PMTE scores were correlated with posttest MCKscores (r = .452, N = 43, and p < .01). These results showa consistent, positive, moderate relationship between theconstructs of personal teacher efficacy and content knowl-edge. No significant correlation was found betweenMTOE scores and MCK scores, which suggests no rela-tionship between outcome expectancy (i.e., teachers’sense of how well students can learn from their teaching)and content knowledge.

Sources of Efficacy InformationThe second research question asked what might explain

a relationship between teacher efficacy and MCK, and howit might manifest during the course. Analyses of writtenartifacts (see Table 1) support a positive relationshipbetween mathematics content knowledge and teacher effi-cacy. In other words, trends in the qualitative data wereconsistent with the finding that MCK and personal teach-ing efficacy were correlated. For example, preserviceteachers with different levels of MCK seemed to attend todifferent sources of information in order to make efficacyjudgments. More students in the low MCK group madestatements about vicarious experiences, while more pre-service teachers who scored at the MCK mean made state-ments about verbal persuasion. Members of all groupstended to make statements about mastery experiences, butthe high MCK group members were more likely to reflecton these experiences in terms of their future classrooms.Finally, preservice teachers with lower MCK were morelikely to have negative prior learning experiences withmathematics. (See Table 6 for numbers of preserviceteachers making statements of each type.) Further analy-ses of each group are provided below. In all descriptions,

Table 4Correlations Between PMTE and Mathematics Content Knowledge

Pretest PMTE Posttest PMTE Pretest Content Posttest Content

Pretest PMTE — — .465* .437*(N = 41) (N = 38)

Posttest PMTE — — .543* .452*(N = 48) (N = 43)

Pretest content .465* .543* — —(N = 41) (N = 48)

Posttest content .437* .452* — —(N = 38) (N = 43)

* p < .01 (two-tailed).

Table 5Correlations Between MTOE and Mathematics Content Knowledge

Pretest MTOE Posttest MTOE Pretest Content Posttest Content

Pretest MTOE — — .131 .081(N = 41) (N = 38)

Posttest MTOE — — .092 .092(N = 48) (N = 43)

Pretest content .131 .092 — —(N = 41) (N = 48)

Posttest content .081 .092 — —(N = 38) (N = 43)

Note. No significant results.



preservice teachers’ names have been replaced withpseudonyms.

Low MCK group. Low MCK preservice teachers’statements about their experiences as learners of math-ematics were negative in all six cases. These preserviceteachers mentioned struggling with mathematics, beingafraid, and/or not enjoying mathematics. As Tori said,“Math has never been my favorite subject. I havestruggled with it my entire life.” In addition, some ofthem associated the negative experiences with the class-room. Hannah claimed, “My teachers always made mefeel bad for my mistakes. I gave up because of this lackof encouragement.” As a result of negative learning expe-riences, two of the preservice teachers made statementsrevealing their affective states with regard to futureteaching experiences. As Carly stated, “I am terrified ofteaching [math] to young children. I don’t want to be thetype of teacher I had.”

After watching the first Kay Toliver video, three preser-vice teachers made statements that suggested the videowas a vicarious source of efficacy for them. In other words,they made statements about their future teaching that sug-gested their own classroom could be similar to Ms. Toliv-er’s classroom. As Nancy said, “This shows me what Ineed to do in my classroom in order to not only teach mathto my students but to make it fun and enjoyable.”

After creating a math bag and giving it to students to try,five of the preservice teachers made statements that sug-gested the activity was a mastery experience for them. Fortwo of them, it meant that the activities were enjoyable. AsNancy said, “I wanted this bag to reflect activities thatmade math fun. I believe I accomplished this with mymath bag.” Tori was pleased with her math bag because itwas “creative,” while the other two claimed that it was asuccess because students “understood the concepts.”Rebecca claimed, “As a result of reviewing the students’work, I found that each child demonstrated a good under-standing of the concepts.”

Only one preservice teacher commented on positivefeedback from parents (verbal persuasion) about the bag.

As a result of the positive feedback, Nancy was “left tobelieve that it was a success.” Interestingly, this personscored the highest of the group on the MCK pretest. It ispossible that others in the group did not receive any posi-tive feedback from parents. Since preservice teachers werenot prompted to use parental feedback in their reflectionpapers, another possibility is that the feedback was notsomething that preservice teachers with low MCKattended to as a source of efficacy.

Medium MCK group. Four preservice teachers whoseMCK scores were at the mean on the pretest made per-sonal statements about prior learning experiences. Typi-cally, this group suggested that they were “never good atmath” and/or that “math class was never fun.” Forexample, Diane said, “I have never been good at math. Tome, math was always my least favorite subject and themost boring.” Robin remembered “asking the questionwhy a lot and never getting an answer.” However, none ofthese preservice teachers made statements about beingscared or nervous about mathematics class. Only Krystalmade a statement about her affective state with regard toteaching, suggesting her past experience was “probablywhy I did not retain too much math and now have a fear ofteaching math myself.”

Only one of the preservice teachers scoring at the MCKmean mentioned her future teaching after watching theKay Toliver video. Specifically, Anna noted that she“would like to attempt to be as energetic and charismaticabout math (even with my not so pleasant math past) in thefuture.” Although others in this group did admire KayToliver, they did not make statements that suggested shewas a vicarious source of efficacy for them.

When reflecting on the math bag, four preservice teach-ers made statements that suggested it was a mastery expe-rience for them, often describing the activities as“successful” and even being “pleased and proud.” Annaclaimed, “Overall, I believe the math bag was successful.”In two cases, this success meant that the students under-stood the mathematics. As Anna elaborated, “The work-sheets and the map show they understood the content, and

Table 6Number of Preservice Teachers Making Efficacy-Related Statements

MCK Group* LearningExperiences

AffectiveStates

VicariousExperiences

VerbalPersuasion

MasteryExperiences

Low 6 2 3 1 5Medium 4 1 1 4 4High 4 0 1 2 6

* N = 6 for each MCK group.


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that in itself speaks volumes.” Krystal imagined herselfusing the bag in her future classroom, stating, “I woulddefinitely use these activities in my class.”

Four of the preservice teachers in this group made state-ments that were categorized as verbal persuasion. Forexample, Diane said, “Both parents seemed to havethought that this was a good activity that helped theirchildren practice real life skills that they would need in thefuture.” In general, this group said that parents “seemed torespond very well.” As Kim noted, “I received very posi-tive feedback from the parents and the two students thatcompleted the bag.”

High MCK group. Four preservice teachers scoringhigh in the MCK pretest made statements about priorlearning, but the statements were varied in nature. Arlenestated that she had been “intimidated about math,” andGeorge revealed that he “hated math since as far back asI can remember.” On the other hand, Sarah acknowl-edged that she was “actually quite good at math.” Simi-larly, Kara noted, “Since I was a child, math was alwaysone of my favorite subjects. I did well in math and Ienjoyed working with numbers.” At the same time, Sarahnoted that despite being good at mathematics, she “wasnever very confident about my skills.” Likewise, Karasaid, “Even though I liked math, there were definitelytimes that I felt completely lost.” No one in this groupmade statements revealing their affective states withregard to teaching.

After watching the Kay Toliver video, one person madea statement that was categorized as a vicarious experience.Specifically, Lori said that Ms. Toliver “inspires me to bethat kind of teacher—a teacher that makes a difference.”All six of the preservice teachers in this group made state-ments that suggested the math bag was a mastery experi-ence for them. Of these, three mentioned using the mathbag in their future classrooms. Jessica also suggested she“could create one for science and the weather as well!”Two of the students also mentioned parental feedback.George received an “overwhelming amount of positivefeedback from students and parents alike,” suggesting thatverbal persuasion may have also contributed to his efficacyfor teaching.

DiscussionThis study examined links between MCK and teacher

efficacy of preservice teachers during an elementary math-ematics methods course. Teacher efficacy included twosubscales (personal teaching efficacy and outcome expect-ancy). Findings suggested that content knowledge and per-sonal teaching efficacy were positively related, and this

relationship was stable during the course. No significantrelationship was detected for content knowledge andoutcome expectancy.

The current findings are consistent with Cantrell et al.(2003), but they extend these findings from science tomathematics. The findings differ, however, from those ofSwars et al. (2007). It is possible that the discrepant find-ings reflect a difference in content areas. Alternatively, thediscrepancy might be explained by variations in how theconstructs were measured. Swars and colleagues measuredcontent knowledge as the specialized mathematical knowl-edge needed for teaching (e.g., knowledge of student mis-conceptions), whereas content knowledge in the currentstudy was measured as procedural and conceptual knowl-edge of the mathematics taught in the elementary grades(i.e., the kind of MCK needed for certification). It wouldbe interesting to include both types of content knowledgemeasures in future studies.

Qualitative data in the form of preservice teachers’written artifacts also supported the notion that contentknowledge and personal teaching efficacy are linked.Although the preservice teachers were not directly askedabout their self-efficacy for doing mathematics, descrip-tions of their mathematical learning experiences oftenrevealed this sort of information. This was especially truefor those with lower MCK, who tended to claim they hadalways struggled with mathematics. These preserviceteachers were also more likely to make affective state-ments that suggested a link to their self-efficacy for teach-ing mathematics. Although it is not surprising that aperson’s efficacy for doing mathematics would be linkedto his or her efficacy for teaching mathematics (Brand &Wilkins, 2007; Charalambous et al., 2008; Swars, 2005), itis worthy of note. When experiences with a particular taskare limited or missing, people draw on related experiencesto make judgments about efficacy (Bandura, 1997). In thecase of preservice teachers who have yet to student teach,their efficacy for doing mathematics is a source of infor-mation about teaching mathematics. This may explain thefinding that preservice teachers’ knowledge of the math-ematics they will be teaching is related to PMTE, butspecialized knowledge for teaching is not (Swars et al.,2007). It may be that, as teachers gain positive experienceswith teaching mathematics, their prior experiences withlearning the content become less important in terms ofefficacy judgments.

According to Bandura (1997), people’s existing self-beliefs influence what they “look for, how they interpretand organize the efficacy information generated in dealingwith the environment, and what they retrieve from their



memory in making efficacy judgments” (p. 81). In thisstudy, teachers seemed to attend to sources of efficacyinformation differently depending on their MCK group. Inparticular, the data suggest that vicarious experiences maybe particularly important for preservice teachers with lowMCK, whereas verbal persuasion may be more meaning-ful to those with moderate MCK. If these preliminaryfindings hold with larger and more diverse groups, theyhave important implications for methods courses. In termsof personal teaching efficacy, perhaps those with lowMCK have a greater need to see effective models of teach-ing than those in the moderate MCK group. It could bethat those with moderate MCK have seen stronger modelsof mathematics teaching and are in need of encouragementthat they can be effective as well. Mastery experiencesseem important for all preservice teachers, but those withhigh MCK in this study were more likely to make connec-tions between these experiences and their future teachingexperiences. It may be that they are already more able toimagine themselves successfully teaching mathematics inthe future. Of course, these data are exploratory and inneed of further research. In the future, it would be inter-esting to follow preservice teachers into student teaching,in order to examine whether and how their attention tosources of efficacy information change as they gaincontent knowledge and teaching experience.

Given that content knowledge and teacher efficacy areboth related to positive outcomes in the classroom, itmakes sense that they may also be related to each other.More studies should examine how they are linked, both forpractical and theoretical reasons. In particular, studiesshould examine the developmental trajectory of teachingefficacy beliefs. At what point, if any, does MCK becomeless critical for teacher efficacy? Does this change happennaturally or are there particular supports needed to helppreservice teachers put their own learning experiences intoperspective? How should these supports change over time?The current study offers potential answers to these ques-tions, but interviews of preservice teachers might providefurther insight, particularly if they occur at various pointsin their programs. Comparing those at the elementary andsecondary levels might further illuminate how contentknowledge and efficacy are related. Such studies will con-tribute to our theoretical knowledge of teacher efficacy,and they will help teacher educators better understand howto shape preservice teachers’ experiences.

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Authors’ Notes1 In the data analyses, the sample size is somewhat less

than 55 due to several student absences during data gath-ering. We felt it important to keep all student data when itcould potentially be used effectively, and thus use all datawhen possible. This also slightly changes means and stan-dard deviations across analyses. We acknowledge that thisis a limitation. However, this practice is preferable thangreatly reducing the sample size for all instances. Therewas no participant attrition in this study.

2 Number of Latina/o students is also included in racialcategories of White (White Hispanic) or African-American (Black Hispanic) as is customary in U.S. Censusdata collection in which Latina/o status is considered eth-nicity and not race.



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Preservice Elementary Teachers' Mathematics Content Knowledge and Teacher Efficacy