Toward Improving the Mathematics Preparation of Elementary Preservice Teachers

Toward Improving the Mathematics Preparation of ElementaryPreservice Teachers

Fabiana CardettiUniversity of Connecticut

Mary P. TruxawUniversity of Connecticut

Research suggests the importance of mathematics knowledge for teaching (MKT) for enabling elementary schoolteachers to effectively teach mathematics. MKT involves both mathematical content knowledge (M-CK) and mathemati-cal pedagogical content knowledge (M-PCK). However, there is no consensus on how best to prepare elementarypreservice teachers (PSTs) to achieve M-CK and M-PCK. This study builds on research related to MKT by investigatinginfluences of mathematics content courses designed specifically for elementary PSTs (IMPACT courses—Impact ofMathematics Pedagogy and Content on Teaching) on their attitudes (i.e., confidence and motivation) toward M-CK andM-PCK. Results suggest that the PSTs who participated in these IMPACT courses not only acquired high levels ofconfidence and motivation toward M-CK, but also showed significant and greater gains in attitudes toward M-PCK,after taking the required mathematics methods course, than their counterparts. Further, the findings suggest that theseIMPACT courses provided a mathematical foundation that allowed the PSTs to engage in mathematics teaching methodsbetter than those PSTs who did not have such a foundation. These results suggest potential course experiences that mayenhance M-CK and M-PCK for elementary PSTs.

It has been noted consistently that, in order for elemen-tary school teachers to teach mathematics effectively,mathematics knowledge for teaching (MKT) is necessary.MKT is different from mathematics knowledge learnedfor other purposes. MKT involves an intersection of math-ematical content knowledge (M-CK) and mathematicalpedagogical content knowledge (M-PCK). While therehave been increasing numbers of studies investigatingMKT (e.g., Ball, Lubienski, & Mewborn, 2001; Hill,2010; Hill et al., 2008; Ma, 1999), what has not beenuncovered clearly is how preservice teachers (PSTs) mayacquire MKT. Further, it has been acknowledged that fewcollege-level mathematics courses serve to promoteappropriate MKT for elementary PSTs (Ball, 2003).

This study investigated elementary PSTs’ attitudestoward M-CK and M-PCK. We focused on PSTs’ attitudesfor four reasons: (a) this focus would help to flesh out workof other researchers who are investigating MKT from acontent perspective (e.g., Ball et al., 2001; Hill, 2010; Hillet al., 2008); (b) this focus would build from longstandingand consistent research recognizing that attitudes influ-ence motivation and capacity to learn mathematics (e.g.,Aiken, 1974, 1976; Evans, 2011; Fennema & Sherman,1977); (c) this research would help to uncover specificcourses that might influence attitudes aligned with learn-ing of MKT (Cardetti, 2011; Cardetti, Truxaw, & Bushey,2011; Truxaw, Cardetti, & Bushey, 2010); and (d)acknowledging research that suggests that teacher atti-tudes influence student learning, this focus would connectindirectly to future student learning—the “coin of the

realm” for educational studies (e.g., Evans, 2011; Henson,2001).

With this study, we seek to better understand how PSTs’attitudes toward mathematics and its teaching changeaccording to their content course experiences and howthese changes compare across different groups. In particu-lar, we investigate influences of mathematics courseworkthat has been designed specifically with elementary PSTsin mind—mathematics content courses taught in the math-ematics department, but with MKT as an emphasis. In thispaper, we will call these courses IMPACT (Impact ofMathematics Pedagogy and Content on Teaching) courses.Along with reporting our results, we outline related rec-ommendations for teacher education programs to helpfoster and strengthen PSTs’ predispositions to ensure thatthey acquire the knowledge and skills necessary to suc-cessfully teach elementary school mathematics.

Theoretical FrameworkCurrent research suggests the importance of MKT to

enable elementary school teachers to teach mathematicseffectively (Ball, 2003; Ball et al., 2001; Fennema &Franke, 1992; National Council of Teachers ofMathematics, 2003). Additionally, Hill, Rowan, and Ball(2005) found that this knowledge is significantly related tostudent achievement supporting the efforts to improvemathematics education in schools by improving the math-ematics education of teachers. However, there is noconsensus on how best to prepare elementary PSTs inorder to achieve the important combination of M-CK and

School Science and Mathematics 1

M-PCK that make up MKT and, in turn, support studentlearning (Kirtman, 2008). For example, Ball (2003) notesthat “increasing the quantity of teachers’ mathematicscoursework will only improve the quality of mathematicsteaching if teachers learn mathematics in ways that make adifference for the skill with which they are able to do theirwork. The goal is not to produce teachers who know moremathematics. The goal is to improve students’ learning”(p. 1). Shulman’s (1986) seminal work on pedagogicalcontent knowledge (PCK) is relevant as it recognizes thatPCK is at the intersection of content and pedagogy.M-PCK therefore requires that teachers not only under-stand mathematics content, but also can transform it tosupport student learning (McDonnough & Matkins, 2010).For elementary PSTs, taking more mathematics contentcourses may support increased M-CK, but it may notsupport development of M-PCK.

Moreover, in Ball’s (2003) remarks to the Secretary’sSummit on Mathematics, she noted that few mathematicscourses offer opportunities to produce knowledge that isappropriate for elementary school teachers. Further, sheurged that “ongoing research in this area is crucial” (p. 9).It follows that identifying courses that have been designedto provide elementary PSTs with the specialized contentknowledge called for by these scholars (Ball, 2000; Ball,Thames, & Phelps, 2008; Shulman, 1986) is necessary forimproving elementary school mathematics instruction.Mathematics methods courses have been investigated withrespect to content knowledge, attitudes, and self-efficacy(e.g., Evans, 2011), but there is little in the literature touncover the types of mathematics content courses that mayprovide a foundation for building not only M-CK, but alsoM-PCK. Therefore, an investigation of mathematicscontent courses for elementary education PSTs seems tobe a logical next step for research.

Because MKT is well researched by the Learning Math-ematics for Teaching Project (http://sitemaker.umich.edu/lmt/home), providing an alternate lens for viewing contentcourses would seem useful. In particular, researchers havepointed out the importance of investigating PSTs’ atti-tudes toward mathematics and mathematics teaching forimproving teacher preparation (Pajares, 1992; Philipp,2007). Indeed, Pajares (1992) argued for more research inthe beliefs of PSTs as they play a pivotal role in theiracquisition and interpretation of knowledge with repercus-sions in practice. More recently, the National Council ofTeachers of Mathematics (NCTM, 2003) noted, “Candi-dates’ comfort with, and confidence in, their knowledge ofmathematics affects both what they teach and how theyteach it” (p. 4). This is aligned with research on teacher

efficacy that indicates a link between positive teacherbehavior and student performance (Henson, 2001;Tschannen-Moran, Woolfolk Hoy, & Hoy, 1998).

The concept of teacher efficacy (Tschannen-Moranet al., 1998) rises from the work of Bandura (1986) relatedto self-efficacy. Bandura defined self-efficacy as “people’sjudgments of their capabilities to arrange and executecourses of action required to attain designated types ofperformances” (p. xii). It impacts the things we do, ourefforts toward them, and how long we persist in workingout solutions to problems. Researchers, such as Gable andWolf (1993), have proposed that self-efficacy “is the basisfor a causal model, analyzing human motivation, thoughtprocesses, and behavior” (p. 12); additionally, they suggestthat confidence is an appropriate indicator of self-efficacy.This implies that PSTs who report high confidence levelswith respect to M-CK and M-PCK are likely to haverelated high self-efficacy.

Measuring attitudes (i.e., confidence or efficacy) towardM-CK and M-PCK could provide indicators of possibleimpact on future mathematics teaching practices. Forexample, Palardy and Rumberger (2008) investigatedteacher attitudes, along with investigating relationships toteacher background and instructional practices, withrespect to teacher effectiveness (i.e., student learninggains). In their study, although teacher background did notshow relationships to math achievement, one teacher atti-tude, specifically teacher efficacy, was associated withmath achievement gains. This suggests that using atti-tudes, especially ones associated with efficacy, may beuseful in uncovering links to M-CK, M-PCK, and, indi-rectly, student learning. It is noteworthy that Palardy andRumberger specifically pointed to coursework in the dis-cussion of their findings, saying:

Consequently, based on the findings of this study, itwould be wrong to dismiss the importance of teachertraining and background qualifications for effectiveteaching. It may be, for example, that specific course-work or specific aspects of the directed teaching expe-rience are critical preparation for effective teaching. . . (p. 129)

Thus, it seems important to consider possible improve-ments in mathematics teacher preparation that can beguided by the identification of course experiences thataffect PSTs’ attitudes toward M-CK and M-PCK that inturn may impact future learning and teaching. While someresearch has been conducted on the influence of math-ematics methods courses and student teaching on PSTs’

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attitudes (e.g., Evans, 2011; White, Way, Perry, &Southwell, 2006; Wilkins & Brand, 2004), less has beendone to explore the influences of mathematics contentcourses on PSTs’ attitudes toward learning and teaching.For example, Evans (2011) used survey methods tomeasure gains in PSTs’ increase in positive attitudes andself-efficacy with respect to mathematics teaching andlearning as a result of participation in a reform-basedmethods course. Studies such as Evans’ provide a begin-ning, but it would be useful to unpack the questionsfurther. A natural follow-up question could be: Whenexamining PSTs’ participation in a reform-based methodsclass, does specific prior mathematics content courseworkinfluence their attitudes related to M-CK or M-PCK?

In considering attitude as a measure, a reasonableapproach is to tap into attitude instruments related tomathematics that have stood the test of time and have beenfound to be reliable. For example, the Fennema–ShermanMathematics Attitude Scales (FSMAS) and modificationsof these scales have been used consistently and reliablysince their introduction in the 1970s (e.g., Fennema &Sherman, 1976, 1977, 2004; Mulhern & Rae, 1998). Atestament to the credibility of these attitude scales is thatthe 1977 article that introduced the FSMAS was includedamong the NCTM “classics” publication (2004) and wassaid to be “among most frequently cited articles in main-stream journals of educational psychology” (Carpenter,Dossey, & Koehler, 2004, p. 26). The FSMAS thereforeprovide a credible foundation for investigating mathemat-ics attitudes. This instrument will be discussed in greaterdetail in the methods section of this paper.Research Questions

As a mathematician and a mathematics educator whowork with elementary PSTs, we sought to investigateinfluences of specific mathematics content courses onelementary PSTs’ attitudes toward M-CK and M-PCK. Inparticular, we were interested in the impact of mathemat-ics courses that were designed with elementary PSTs inmind and offered prior to taking the mathematics methodscourse (i.e., IMPACT courses). As we moved forward, weasked: Do the IMPACT courses influence PSTs attitudes,and in turn their learning experiences? To address thisquestion, we decided to conduct a quantitative study in acontext where both, PSTs who participated in one specificIMPACT course and those who did not, would be found.The natural place was the mathematics methods courses.At our institution, all elementary education PSTs arerequired to take a mathematics methods course within theSchool of Education, along with at least three quantitative(i.e., mathematics or statistics) courses outside the School

of Education. One may assume that PSTs’ attitudes wouldchange before and after completion of the mathematicsmethods course (e.g., Evans, 2011); what is not so clear ishow this change differs between those who have takenIMPACT courses (C group) and those who have not (NCgroup). In this study, we asked the following researchquestions:

1. How do the attitudes of the NC group and the Cgroup compare before and after taking the mathematicsmethods course with respect to M-CK and M-PCK?

2. Is there a change in attitudes with respect to M-CKand M-PCK before and after completion of the mathemat-ics methods course for each group?

MethodsIn order to measure elementary PSTs’ attitudes toward

mathematics and the teaching of mathematics, we investi-gated instruments that have been used extensively andfound to be trustworthy. In particular, the FSMAS havebeen used for more than 20 years to investigate attitudestoward mathematics (Mulhern & Rae, 1998), providing asolid base from which to build an instrument for this study.

The original FSMAS were developed in 1976 andconsist of the following nine subscales: Attitude TowardSuccess in Mathematics scale; Mathematics as a MaleDomain scale; Mother, Father, and Teacher scales; Confi-dence in Learning Mathematics scale; MathematicsAnxiety scale; Effectance Motivation in Mathematicsscale; and Mathematics Usefulness scale. Studies on thepsychometric properties of the FSMAS have generallyprovided support for the reliability and validity of thescales (Melancon, Thompson, & Becnel, 1994). Addition-ally, Mulhern and Rae (1998) reported that it is possible touse each subscale separately on its own. Some researchershave developed abbreviated versions of the scales(Mulhern & Rae, 1998) because the original scales consistof 108 items and take an average of 45 minutes to com-plete. In addition, researchers have rewritten the items toadjust to different participants’ age, or language, or tomeasure attitudes in other subject areas (Elliot, 1990;Mulhern & Rae, 1998; Stricker, Rock, & Burton, 1993). Inall cases, the modified versions of the scales were found tohave factor structures comparable with the original scales,as well as strong internal consistency estimates.

To construct our instrument, we strategically selecteditems from the FSMAS Confidence subscale and theEffectance Motivation subscale. The Confidence subscalemeasures confidence in one’s ability to perform well on atask, while the Effectance Motivation subscale measureseffectance as applied to mathematics (Fennema &



Sherman, 1976). According to White (1959), effectancemotivation or competence motivation refers to one’s moti-vation to perform a task effectively in order to feel self-determining and competent. These items were selected tomeasure mathematics attitudes, M-CK Attitudes. Anexample of the items in the M-CK subscale is “Generally,I feel secure about attempting mathematics.” To measureattitudes toward mathematics teaching, M-PCK Attitudes,we reformulated those items accordingly. For example, theprevious item was reformulated as “Generally, I feelsecure about teaching elementary school mathematics.”All items were positively worded and scored on a five-point Likert scale ranging from 5 (strongly agree) to 1(strongly disagree). To measure the internal consistency ofeach subscale as a whole, we conducted a reliability analy-sis. The Cronbach’s alpha of the two subscales wererespectively .848 and .806, both higher than .8, whichindicated that the items in each subscale were highlyrelated with each other, and they could measure the twoconstructs (M-CK and M-PCK Attitudes) effectively.

Recalling that the IMPACT courses are not requiredby the program, some PSTs take courses other than theIMPACT courses to fulfill the quantitative requirements.Therefore, we also collected data on specific “quantita-tive” (typically, mathematics or statistics) courses(Q-courses) that the PSTs had completed. The collectionof these data allowed us to determine the range of theseother Q-courses besides the IMPACT courses and considertheir possible influences in PSTs’ attitudes.Context

Participants are elementary PSTs enrolled in the teacherpreparation program (TPP) at a large public research uni-versity in the northeastern United States. These PSTs arepredominantly female (90–95%), white (80–90%), andtypical ages range from 20 to 25 years old. For this paper,we focus on participants during the fall of their senior year,prior to and after completion of the required mathematicsmethods course—both those PSTs who had taken anIMPACT course and those who had not.

The IMPACT courses are offered by the Department ofMathematics. The courses are recommended but notrequired by the TPP for elementary PSTs and are typicallytaken during the PSTs’ junior year—the year prior totaking methods courses and completing student teaching.The design of the IMPACT courses was guided byresearch mentioned earlier indicating a need for MKT thatcombines both M-CK and M-PCK (i.e., to promote MKTfor elementary teachers). The IMPACT courses weredesigned to develop an advanced perspective on and pro-found understanding of concepts, structures, and algo-

rithms constituting the core of K-8 mathematicscurriculum. Each of the two IMPACT courses is asemester-long (14 weeks) course that meets twice a weekfor 75 minutes each time. For the first IMPACT course, thetopics include numeration systems and their characteris-tics; in-depth look at algorithms for basic operations onwhole, integer, and rational numbers, decimals, and realnumbers; and algebraic and proportional reasoning. Thesubsequent IMPACT course focuses on geometry. In con-trast to the traditional blackboard-lecture mathematicsclasses, in these courses knowledge is built via explora-tions and students’ discussions; the instructor acts as afacilitator providing guidance to lead students towardunderstanding of the mathematical concepts. The studentswork, individually or in groups, on activities developed tounderstand the reasons behind familiar mathematical pro-cedures to explore and discover new concepts, and toanalyze children nonstandard approaches. Additionally,students frequently present their ideas and solutions to theentire class. The focus of these tasks is on helping thePSTs construct their own knowledge and develop math-ematical communication skills while building solid learn-ing skills that will help them approach new mathematicstopics in the future. With this approach, the course followsthe recommendations of the Conference Board of theMathematical Sciences (2001) regarding the mathematicalpreparation of teachers.

The mathematics methods course takes place during thefall of the PSTs’ senior year in the semester prior tostudent teaching; thus, by the time PSTs complete themethods course, they have also completed all requiredquantitative courses. PSTs who participated in at least oneof the IMPACT courses, along with the mathematicsmethods course, are referred to here as the C group(content). The participants who completed the mathemat-ics methods course, but did not participate in an IMPACTcourse, are referred to here as the NC group (noncontent).Data Sources and Analysis

Data collection included presurveys and postsurveys(described above) administered to the elementary PSTs inthe mathematics methods courses in fall 2009 and fall2010. Analysis focuses on the 48 participants who com-pleted both presurveys and postsurveys (23 C, 25 NC).It is important to note that the predesignations andpostdesignations relate to the methods course, not theIMPACT course. This allowed us to examine differencesbetween the C and the NC groups to identify whether theIMPACT course influenced the PSTs’ attitudes both priorto and after completion of the mathematics methodscourse.


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All participants had completed required Q-courses(mathematics and statistics) by the time they completedthe methods course. The Q-courses, other than IMPACTcourses, reported as completed by PSTs included: problemsolving, prealgebra, elementary discrete mathematics,elementary modeling, precalculus, mathematics for busi-ness and economics, and a first course in statistics. Theseother Q-courses differ significantly from the IMPACTcourses in that they are service courses offered to thegeneral university population and are not designed forteaching majors. These courses, with the exception ofproblem solving, are offered in a typical lecture format.Problem solving is a course taught as a discussion session,where students solve problems in small groups and thensolutions are discussed as a whole class. Prior to answer-ing the primary research questions, a multiple regressionanalysis was conducted to verify whether or not theIMPACT courses were significant predictors of PSTs’ atti-tudes toward mathematics and mathematics teaching. Theanalysis also helped us detect whether or not the otherQ-courses significantly contributed to these attitudes. Theresults showed that the IMPACT courses accounted for16.6% of the variance (p = .009) of PSTs’ M-CK attitudesand 10.1% of the variance of PSTs’ M-PCK attitudes (p =.062). That is, over and above the other Q-courses, theIMPACT courses were significant predictors for PSTs’attitudes toward mathematics (p < .05) and mathematicsteaching (p < .10).

Additionally, before considering the potential influenceof the IMPACT courses on PSTs’ attitudes toward M-CKor M-PCK, we recognized the importance of investigatingif a selection bias might influence the analysis. Specifi-cally, did the students who opted to take an IMPACTcourse (C group) already have greater confidence inM-PCK and/or M-CK than the NC group—before takingthe IMPACT course? To answer this question, we used aone-way analysis of variance (ANOVA) to compare meanson survey responses for the C group prior to taking theIMPACT course (fall 2009) with the premethods NCgroup prior to taking the methods course (fall 2009 and2010). The results showed no significant differencebetween the pre-IMPACT C group and the premethods NCgroup with respect to confidence in M-PCK or M-CK.These results suggest that comparing the C group and theNC group premethods and postmethods course mayuncover effects of the IMPACT courses.

Related to the research questions, we wondered if the Cgroup would begin or end the methods course with differ-ent levels of confidence and motivation regarding M-CKand/or M-PCK than the NC group (research question 1);

and we wondered if the C group would demonstrate dif-ferent levels of change (from premethods to postmethodscourse) in confidence and motivation toward M-CK and/orM-PCK than the NC group (research question 2). In otherwords, we were interested in the impact of the IMPACTcourse on confidence and motivation relative to themethods course. We analyzed the data by comparing themean survey scores using a one-way ANOVA with ap-value of .05.

ResultsTo address research question 1, we started with a com-

parison of the premethods mean scores related to attitudestoward M-CK and M-PCK between the C group and theNC group (fall 2009 and fall 2010 premethods). Thisallowed us to gauge differences between the C group andthe NC group prior to taking the methods course—givingan initial look at the impact of the IMPACT courses onattitudes toward M-CK and M-PCK. The ANOVA resultsfor the comparison between the groups are shown inTable 1.

A significant difference was found between the C group(mean [M] = 4.184, standard deviation [SD] = .565) andthe NC group (M = 3.5, SD = .739) with respect to M-CKattitudes at p < .05 (p = .001) with a large effect size (ES= 1.04). This showed that the C group entered the methodscourse with significantly higher confidence and motivationin M-CK than the NC group. Recalling that the pre-IMPACT C group and the premethods NC group were notfound to be statistically different from each other withrespect to M-CK, these results suggest that the IMPACTcourse may have influenced their attitudes positively withrespect to M-CK prior to taking the mathematics methodscourse.

The comparison of premethods mean scores of the Cgroup and the NC group with respect to M-PCK attitudesshowed that although the C group had slightly higher meanscores (M = 3.638, SD = .643) than the NC group (M =3.453, SD = .584), this difference was not statistically

Table 1ANOVA Results for Premethods Comparison Between C and NC Groups

Source C Group† NC Group‡ df F pM SD M SD

M-CK attitudes 4.184 .565 3.500 .739 46 12.822 .001*M-PCK attitudes 3.638 .643 3.453 .584 46 1.083 .303

* p < .01.† n = 23. ‡ n = 25.df = degrees of freedom.



significant at the .05 level (p = .303) with a medium ES(.30). These results suggest that the IMPACT course didnot have a significant influence on attitudes towardM-PCK prior to taking the mathematics methods course.

By the end of the methods course, the mean scoresreported of the C group were higher than those of the NCgroup both toward mathematics and toward teaching. Todetermine whether these differences were significant,we compared the changes in the postmethods meanscores toward M-CK and M-PCK between the groups.Table 2 shows the ANOVA results of this postmethodscomparison.

A significant difference was found between the C group(M = 4.282, SD = .534) and the NC group (M = 3.74, SD= .663) with respect to M-CK attitudes at p < .05 (p = .003)with a large ES (.900) between the groups. This showedthat the C group ended the methods course with signifi-cantly higher confidence and motivation toward math-ematics than the NC group. While the difference inmathematics attitudes was already significantly higher atthe beginning of the methods course, the C group math-ematics attitudes continued to grow and stayed signifi-cantly higher than the NC group. These results suggest thatthe impact of the IMPACT course on C group’s mathemat-ics attitudes continued through the methods course.

The comparison of postmethods mean scores of the Cgroup and the NC group with respect to M-PCK attitudesalso showed a significant difference between the C group(M = 4.043, SD = .495) and the NC group (M = 3.600, SD= .745) at p < .05 (p = .02) with a large ES (.700) betweenthe groups. These results suggest that the IMPACT course,combined with the methods course, significantly influ-enced the C group’s attitudes toward M-PCK.

Premethods and postmethods comparisons of thechanges in attitudes within each group were performed toaddress research question 2. Descriptive statistics alongwith ANOVA results of these comparisons for the NCgroup and the C group are presented in Tables 3 and 4,respectively.

Comparisons of the means on Table 3 reveal that the NCgroup experienced a positive change in both M-CK andM-PCK attitudes from premethods to postmethods.However, the ANOVA of mean scores showed that theincrease was not statistically significant in either of theattitudes for the NC group with a medium ES towardM-CK (ES = .342) and a fairly small ES toward M-PCK(ES = .219).

Similarly, Table 4 reveals that on average, the C groupreported more positive attitudes after completing themethods course. This time, the ANOVA of mean scoresrevealed that the increase was statistically significant forthe C group toward M-PCK (F[2, 44] = 5.749, p < .05).The ES between premethods and postmethods scores forthis group was large (ES = .706).

The results from Tables 3 and 4 indicate that while themethods course influenced both the M-CK and theM-PCK attitudes of all PSTs, the positive influence wasstatistically significant only for the M-PCK attitudes ofthose who had taken the IMPACT courses.

DiscussionTo aid our discussion of the findings, we offer a graphi-

cal comparison of the premethods and postmethods meanscores for the C group and the NC group. Figure 1 showsthe graph depicting these differences related to M-CKattitudes. The mean scores for the Likert items associatedwith M-CK attitudes are shown along the y-axis; and thex-axis contains the two points in time, presurvey andpostsurvey, at which the mean scores were calculated. It is

Table 2ANOVA Results for Postmethods Comparison Between C and NC Groups

Source C Group† NC Group‡ df F pM SD M SD

M-CK attitudes 4.282 .534 3.740 .663 46 9.630 .003*M-PCK attitudes 4.043 .495 3.600 .745 46 5.785 .020**

* p < .01. ** p < .05.† n = 23. ‡ n = 25.df = degrees of freedom.

Table 3NC Group ANOVA Results for Premethods to Postmethods Comparison

NC Group† Premethods Postmethods df F pM SD M SD

M-CK attitudes 3.500 .739 3.740 .663 48 1.54 .233M-PCK attitudes 3.453 .584 3.600 .745 48 .600 .443

† n = 25.df = degrees of freedom.

Table 4C Group ANOVA Results for Premethods to Postmethods Comparison

C Group† Premethods Postmethods df F pM SD M SD

M-CK attitudes 4.185 .565 4.283 .535 44 .364 .550M-PCK attitudes 3.638 .643 4.043 .495 44 5.749 .021**

** p < .05.† n = 23.df = degrees of freedom.


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important to note that this is not a continuous graph;rather, each line represents two points, a beginning(labeled Pre) and an ending point (labeled Post). The lineshave been drawn to help visualize comparisons betweenpremean and postmean scores.

It is clear from Figure 1 that the C group began withhigher levels of motivation and confidence toward math-ematics (M-CK) than the NC group. The C group’s meansincreased, but only slightly (from 4.18 to 4.28). The NCgroup started with lower attitude levels toward M-CK,increased by the end of the methods course (3.5–3.74), butstill ended well below the C group (C group 4.28, NCgroup 3.74). The comparison within groups (Tables 3 and4) showed that neither group increased significantly frompremethods to postmethods. However, the comparisonbetween groups (Table 2) revealed that by the end of themethods course, C group’s means toward M-CK weresignificantly higher than the NC group.

In Figure 2, the mean scores related to M-PCK attitudesare shown along the y-axis; and the x-axis contains the twopoints in time, presurvey and postsurvey, at which themean scores were calculated. Figure 2 shows a differentpicture than Figure 1. For confidence and motivation withrespect to M-CK, the C group and NC group data wereclearly separated; in this case, it is not as simple to distin-guish between the two groups at the beginning point (Pre).Given that these data related to participants’ confidenceand motivation toward teaching math, and none of themhad experience with teaching mathematics or teachingmethods coursework, it is not surprising that thepremethods means are closer together for the two groups.However, the postmethods data points are clearly sepa-rated. As noted on Table 4, the means related to M-PCK

attitudes for the C group increased significantly (from 3.64to 4.04); this increase can be seen graphically in Figure 2.The postsurvey means for those who had not taken thecontent course (NC group) increased, but not significantly(from 3.45 to 3.6). Moreover, we found (Table 2) that thedifference in M-PCK attitudes between the groups by theend of the methods course was statistically significant. Itmay be that the C group’s higher confidence and motiva-tion toward mathematics afforded them opportunities toconcentrate on the teaching of mathematics while in themethods course, thus improving their attitudes towardteaching.

These data represent PSTs’ attitudes premethods andpostmethods course. Those students who did not have thebenefit of an IMPACT course (NC group) prior to takingthe mathematics methods course increased their attitudelevels toward mathematics and mathematics teaching;however, the levels never reached the point—anywhere—pre or post—of those who had taken the IMPACT course(C group).

Conclusion and Recommendations for PracticeThis study sought to uncover influences that mathemat-

ics content courses designed for elementary PSTs mayhave on the PSTs’ attitudes with respect to mathematicsand mathematics teaching. In particular, Figures 1 and 2highlight how participation in these specific contentcourses (IMPACT courses) can lead to higher levels ofconfidence and competence toward M-CK and M-PCKafter taking the required mathematics methods course.

Our comparisons between and within the C and NCgroups allowed us to better understand the influences ofthe IMPACT courses. It is noteworthy to mention that

Figure 1. M-CK premethods and postmethods mean scores. Figure 2. M-PCK premethods and postmethods mean scores.



attitude levels were statistically the same for both groupsprior to participation in any of the courses. This indicatesthat the significant changes experienced by the C groupbefore and after the methods course could be attributed toparticipation in the IMPACT courses. In addition, we alsoverified that this influence went above and beyond that ofother Q-courses.

Our results show that IMPACT courses have a signifi-cant influence in M-CK attitudes both before and after themethods course (Tables 1 and 2). More importantly, wefound that these courses provide something more than justthat. It was surprising to find that IMPACT courses sig-nificantly increase the attitude levels toward M-PCK(Table 4) before and after the methods course. That is,while IMPACT courses alone did not influence attitudestoward M-PCK, they seem to lay the groundwork forM-PCK—helping PSTs be ready for the methods courses.

These results suggest that PSTs going into the methodscourses with content coursework such as that experiencedin the IMPACT courses may potentially increase their atti-tudes toward M-PCK and that this increase could be to agreater extent than those without IMPACT courseworkexperiences. A hypothesis is that the C group’s higherattitude levels toward M-CK prior to the mathematicsmethods course afforded them greater ability to connectthe mathematics content with the associated mathematicsmethods, thus promoting M-PCK. The NC group’s atten-tion may have been focused more on the mathematicscontent than the teaching methods or student learning andwere therefore less able to significantly increase their atti-tudes toward M-PCK.

The attitudes we investigated were related to confidenceand motivation with respect to M-CK and M-PCK. Asnoted at the beginning of the paper, these attitudes areindicators of self-efficacy. This implies that the C groupwho demonstrated higher confidence levels with respect toM-CK and M-PCK is likely to have higher teacher efficacythan the NC group. Higher teacher efficacy in thesedomains could be associated with students’ math achieve-ment gains (Henson, 2001; Palardy & Rumberger, 2008).

Our findings corroborate others (e.g., Ball, 2003; Ballet al., 2001; NCTM, 2003) who have noted the importanceof both M-CK and M-PCK for elementary school teach-ers. The difficulty for many TPPs is in ensuring that theeducation of PSTs includes the particular MKT that theywill need in the field (e.g., Ball et al., 2001; Hill, 2010;Hill et al., 2008; Ma, 1999). Our results build from andadd to the research literature on MKT by suggesting thatone means of supporting TPPs to help elementary PSTs asthey work to become effective mathematics teachers is

participation in mathematics content courses that aredesigned specifically with elementary school teachers inmind prior to taking their required method courses.Indeed, these mathematical content courses may enhancelearning outcomes of mathematics methods courses byproviding sufficient M-CK to allow the PSTs to focus theirattention, during methods courses, on the teachingmethods, and student learning related to the mathematics.Without these courses, the PSTs’ attention toward studentlearning and mathematical teaching methods may bediluted while they are focusing on their own M-CK.

We wish to acknowledge some of the limitations of thisstudy that suggest directions for future research. First,because this study was conducted in one TPP, our resultsmay be a reflection of the particular requirements, studentpopulation, and teaching strategies used at our institution.Thus, future research is needed to expand across otherTPPs. Second, we collected data at three points in thePSTs’ preparation. To further understand the extent ofthese course experiences, it would be useful to collect datafurther into their studies and beyond. Finally, qualitativedata in the form of interviews, focus groups, and/or obser-vations would provide further details about PSTs’ percep-tions of these influences.

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Toward Improving the Mathematics Preparation of Elementary Preservice Teachers