Including Curriculum Focus in Mathematics Professional Development for Middle-School Mathematics Teachers

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  • Including Curriculum Focus in Mathematics Professional Developmentfor Middle-School Mathematics Teachers

    Nimisha PatelWright State University

    Suzanne FrancoWright State University

    Yoko MiuraWright State University

    Brian BoydDayton Regional STEM SchoolWright State University

    This paper examines professional development workshops focused on Connected Math, a particular curriculumutilized or being considered by the middle-school mathematics teachers involved in the study. The hope was that asteachers better understood the curriculum used in their classrooms, i.e., Connected Math, they would simultaneouslydeepen their own understanding of the corresponding mathematics content. By focusing on the curriculum materialsand the student thought process, teachers would be better able to recognize and examine common student misunder-standings of mathematical content and develop pedagogically sound practices, thus improving their own pedagogicalcontent knowledge. Pre- and post-mathematics content knowledge assessments indicated that engaging middle-schoolteachers in the curriculum materials using pedagogy that can be used with their middle-school students not onlysolidified teachers familiarity with such strategies, but also contributed to their understanding of the mathematicscontent.

    No Child Left Behind (NCLB) mandates regarding thenecessity of highly qualified teachers for all students haveincreased the professional development opportunitiesavailable to K-12 teachers (Desimone, Smith, & Ueno,2006). While it is logical that professional development ofteachers contributes to higher student gains, it is difficultto quantify its effect size. Research indicates that the mosteffective models require teachers to spend a minimum of49 hours engaged in professional development annually(Yoon, Duncan, Lee, Scarloss, & Shapley, 2007). The mostfrequently offered opportunities often focus on math andreading; this is not surprising given that NCLB focuses onstudent achievement in these two areas.The Mathematics and Science Partnerships (MSP)

    program funded by the Ohio Department of Education is aspecific initiative focused on teacher professional devel-opment. The goal of the MSP is to increase academicachievement of elementary and secondary students inmathematics and science by improving instructionalquality (Mathematics and Science Partnership Programs,n.d.). A collaborative unit funded by the MSP is a specialcenter at a large Midwestern urban university. This col-laborative unit represents seven universities and educa-tional resource centers in the local area. Through thiscollaborative unit, a professional development program isoffered to regional math and science K-12 teachers; itinvolves weeklong sessions during the summer monthswith follow-up meetings occurring throughout the subse-quent school year. To date, most pre- and posttests of

    participants mathematics and science content knowledgehave shown gains (Cole, Ryan, & Tomlin, 2005). More-over, pre- and post-attitudinal surveys indicate significantgains in teachers perceptions of their own mathematicaland science content knowledge, pedagogical prepared-ness, and attitudes about teaching mathematics andscience.While positive correlations between students math-

    ematics achievement and the mathematical knowledge/training levels of mathematics teachers have beenevidenced (Yoon et al., 2007), the latter tend to reflectmeasures of teachers general computational abilities.Hill, Rowan, and Balls (2005) research, however, exam-ined the relationship between teachers mathematicalknowledge utilized in their pedagogy and students math-ematics achievement, rather than the teachers generalcomputational abilities. Thus, teachers pedagogicalcontent knowledge (PCK), as well as their contentknowledge, has been explored in relation to studentachievement. Mathematical PCK includes content knowl-edge that is specific for teaching, including an under-standing of students common misconceptions of specificmathematics content. The results from Hill et al. indi-cated that student achievement was greater when math-ematics teachers have both content knowledge and PCK.In other words, there is a difference between strict math-ematical content knowledge and ones PCK with respectto mathematics; having both positively impacts studentachievement.

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  • With this in mind, three weeklong workshops focusedon the mathematics curriculum teachers would use in theirclassrooms. The hope was that as teachers better under-stood the curriculum used in their classrooms, they wouldsimultaneously deepen their own understanding of themathematics content being taught using those curriculummaterials. By focusing their learning on the materials andthe student thought process, teachers would be better ableto recognize and examine common student misunder-standings of mathematical content and develop pedagogi-cally sound practices, thus improving their PCK.

    Literature ReviewEffective Professional DevelopmentProfessional development opportunities for K-12 teach-

    ers typically reflect a patchwork of opportunitiesformal and informal, mandatory and voluntary,serendipitous and planned (Wilson & Berne, 1999, p.174), often because there has been a lack of generallyadopted infrastructure for professional development. TheNational Partnership for Excellence and Accountability inTeaching published guidelines for effective teacher profes-sional development (ERIC Development Team, 1999) thatincluded the importance of extending professional devel-opment beyond the one-day workshop, promoted opportu-nities for teachers to learn as they are expected to teach,focused on reflection and collaboration, and recommendedmore content-focused teacher learning (Ball & Cohen,1999; Hawley &Valli, 1999; Krajcik, Blumenfeld, Marx,& Soloway, 1994).No Child Left Behind Act of 2001 has also attempted to

    provide guidelines for professional development, estab-lishing five criteria that reflect those that are high inquality:1. It is sustained, intensive, and content-focusedhas a

    positive and lasting impact on classroom instruction andteacher performance.2. It is aligned with and directly related to state aca-

    demic content standards, student achievement standards,and assessments.3. It improves and increases teachers knowledge of the

    subjects they teach.4. It advances teachers understanding of effective

    instructional strategies founded on scientifically basedresearch.5. It is regularly evaluated for effects on teacher effec-

    tiveness and student achievement.In a study about the effectiveness of professional devel-

    opment for teachers, Yoon et al. (2007) streamlined thesefive criteria into three components:

    1. It should be focused, coherent, well defined, andstrongly implemented (Garet, Porter, Desimone, Birman,& Yoon, 2001; Guskey & Sparks, 2004; Loucks-Horsley,Hewson, Love, & Stiles, 1998; Supovitz, 2001; Wilson &Berne, 1999) (p. 4).2. It should be based on a carefully constructed and

    empirically validated theory of teacher learning andchange (Ball & Cohen, 1999; Richardson & Placier, 2001;Sprinthall, Reiman, & Thies-Sprinthall, 1996) (p. 4).3. It should promote and extend effective curricula and

    instructional modelsor materials based on a well-defined and valid theory of action (Cohen, Raudenbush, &Ball, 2002; Hiebert & Grouws, 2007; Rossi, Lipsey, &Freeman, 2004) (p. 4).Yoon et al. (2007) posited that:

    Professional development affects student achievementthrough three steps. First, professional developmentenhances teacher knowledge and skills. Second, betterknowledge and skills improve classroom teaching.Third, improved teaching raises student achievement.If one link is weak or missing, better student learningcannot be expected. If a teacher fails to apply newideas from professional development to classroominstruction, for example, students will not benefit fromthe teachers professional development. (p. 4)

    Mathematics Curriculum UnderstandingIt is a natural conclusion from the studies on effective

    professional development, then, that effective professionaldevelopment for mathematics teachers should includetime on mathematics content knowledge as well as onmathematics curriculum. Fuerborn, Chinn, and Morlan(2009) demonstrated that professional development usingcurriculum increased the middle-school mathematicsteachers understanding of the content. For example, adeeper knowledge of number sense or geometry may beenhanced by developing more familiarity with the curricu-lum materials that will be used in the delivery of thisparticular content. The way teachers read, interpret, anduse materials is shaped by their content knowledge of andviews about mathematics (Remillard & Bryans, 2004).Curriculum materials are a strong determinant of whattakes place in the classroom; content knowledge and cur-riculum familiarity are intertwined in the development ofeffective lessons, and consequently student achievement.The connection between content knowledge and curricu-lum makes it essential to provide mathematics profes-sional development when new curriculum materials areintroduced.

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  • Regarding preservice teachers, Frykholm (2005)described a study about curriculum study and teacherpreparation over a four-year period. Qualitative data con-sisting of interviews, discussions, classroom observations,and student work were triangulated to support the findingthat curriculum is an effective resource for preserviceteacher training. More specifically, the use of the curricu-lum not only contributed to shifts in understanding howstudents learn, but the curriculum-based training also con-tributed to preservice teachers content knowledge andPCK.The following study investigated the influences related

    to a curriculum-based professional development workshopfor licensed teachers.

    Research QuestionsThis study focused on three key questions:1. Does participation in professional development

    focused on Connected Math significantly influence teach-ers mathematical content knowledge?2. Does participation in professional development

    focused on Connected Math significantly influence teach-ers perceptions of their own pedagogical preparedness?3. Does participation in professional development

    focused on Connected Math significantly influence teach-ers attitudes toward teaching mathematics?Given prior research, it was expected that participation

    in the weeklong curriculum-focused workshop wouldpositively influence teachers mathematical contentknowledge, their perceptions of their mathematical peda-gogical preparedness, and their attitude toward teachingmathematics.

    MethodologyThis study focused on the summer 2008 professional

    development opportunities centered on middle-schoolmathematics teachers use of the Connected MathematicsProject (CMP) materials. The CMP is a grade 6 through 8curriculum that was developed at Michigan State Univer-sity (Connected Mathematics Project, 2006) throughfunding from the National Science Foundation (NSF). It isone of five middle-school NSF-funded programs origi-nally developed in the mid-1990s. The summer 2008workshops utilized the most recent revision of these mate-rials, the CMP2.The CMP2 materials are student- and inquiry-centered

    materials.A context or problem is posed at the start of eachlesson,with some time for students to investigate, usually ingroups. This is followed by summary time led by studentsand the teacher to highlight explicitly the mathematics

    concepts learned. For example, when learning about mul-tiplying fractions, students using the CMP2 materials con-sider the context of sharing a tray of brownies. Throughin-class activities and discussion, students learn about analgorithm formultiplying fractions that makes sense withinthe context of sharing brownies. This instructional practiceis a divergence from more traditional approaches that typi-cally focus on teaching a skill, and then requiring studentsto apply that skill. As such, this technique is new for manyexperienced in-service teachers. Moving from a traditionaltext to the inquiry-based CMP2 materials is not a naturalprogression for most middle-school mathematics teachers.The intent of each of the workshops was to help teachersunderstand the mathematics content in-depth enough tofeel comfortable using a student-centered and inquiry-based method of teaching. The format of the workshopsprovided teachers insight into how the CMP2 materials arestructured and what mathematics content is developedthrough each lesson. By focusing on the mathematicscontent in these middle-school curriculum materials andthe order of lesson presentations, we believed that teacherswould learn more mathematics content themselves whileincreasing their positive attitudes about teaching math-ematics and strengthening their pedagogical preparedness.The rationale for offering a professional development

    workshop focused on the CMP2 was twofold. First, manydistricts in the area had adopted the CMP2 materials, butteachers had not received sufficient support to implementadequately such materials into their own classrooms.Second, for districts that had not adopted the CMP2 mate-rials, there was a need to familiarize teachers with thematerials as they considered adopting the CMP2 materialsfor their district.ParticipantsParticipants represented two major groups: One group

    of teachers was from districts that had adopted the CMP2materials, but the teachers had not received sufficientsupport to implement the materials into their own class-rooms; the second group of teachers was from districtsthat were considering adopting the CMP2 materials.Fifty-seven licensed teachers of sixth-, seventh-, andeighth-grade mathematics in a Midwestern state chose toenroll in one of the three weeklong workshops. The par-ticipants represented 18 school districts, which variedacross student population, socioeconomic status, andgeographic location. Three separate workshops wereoffered: one for sixth-grade teachers, one for seventh-grade teachers, and one for eighth-grade teachers. Threeparticipants did not complete their respective workshop.For final analyses, there were 26 teachers of sixth grade,

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  • 16 of seventh grade, and 12 of eighth grade. Descriptivedata of the participants are presented in Table 1.MeasuresTwo instruments were used to measure participants

    mathematical content knowledge and attitudes: the Diag-nostic Mathematics Assessments for Middle SchoolTeachers (DTAMS) (2008) and the Mathematics GradeK-8 Local Systemic Change Through Teacher Enhance-ment: 2006 Teacher Questionnaire (2008).DTAMS. The DTAMS (2008), which includes a pretest

    and posttest, were developed at the University of Louis-ville to measure specifically the mathematics contentknowledge of middle-school mathematics teachers. Thecontent tests for teachers of grades 6, 7, and 8 focused onnumber and computation, geometry and measure, andalgebraic ideas, respectively. The level of difficulty, as wellas the breadth and depth of the content, are similar acrossall grade levels. While the tests chosen were alignedclosely with the mathematics content of each workshop, it

    should be noted that the content reviewed in the summerworkshops was not limited to the one domain assessedwith the respective diagnostic test. Version 2.3 of the pre-tests and version 4.3 of the posttests were utilized. Eachversion of the test includes 120 items representing bothmultiple-choice and open-ended questions.The DTAMS measures four types of knowledge: memo-

    rized knowledge (I), conceptual understanding (II),problem solving and reasoning (III), and pedagogicalcontent knowledge (IV). Specific content related to eachknowledge type is presented inAppendix A. Each test itemis related to one of the four knowledge types. Saderholm,Ronau, Brown, and Collins (2010) report that Cronbachsalpha calculations for the reliability of the assessments foreach knowledge type are above 0.87. Cronbachs alphacalculations for the three content domains (number andcomputation, geometry and measures, and algebraic ideas)are above 0.80. Validity for the domains of knowledge wasassessed using experts in the field.

    Table 1Participant Characteristics

    Academy Sixth Grade Seventh Grade Eighth Grade

    (N = 26)* (N = 18)* (N = 13)*

    GenderMale 20% 36% 46%Female 80% 64% 54%

    RaceBlack 15% 17% 8%Non-Hispanic white** 85% 83% 92%

    Years taught in schools05 years 27% 22% 15%More than 6 years 73% 78% 85%

    Teaching licensePreK-3 4% NA NAK-8 math*** 20% 30% 46%Middle (49)*** 27% 38% 31%High (712)*** NA 50% 54%Special education 28% 33% 39%

    College math course taken03 math courses 48% NA NAUndergraduate math NA 39% 23%Or math-related degreeMasters in math NA 17% 31%Non-math major 80% 44% 46%

    Hours of PD last 5 years****Just enough to keep job 28% 11% 39%050 hours of PD 22% 39% 15%More than 50 hours of PD 50% 39% 31%

    Have earned HQT status 81% 83% 92%

    Note. NA, not available because sixth-grade teacher demographic questions were from K-6 LSC; seventh and eighth, from 712).* Numbers including those who dropped the course during the PD. Final total N = 54. ** 11 people did not respond about Hispanicorigin. *** Including teachers licensed in science and/or other areas. **** Including no responses in seventh grade (11%) and eighthgrade (8%).HQT = highly qualified teachers; PD, professional development.

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  • Local Systemic Change Through Teacher Enhance-ment: 2006Teacher Questionnaire. The Local SystemicChange Teacher Questionnaire (LSC) was administered toevaluate teachers attitudes toward teaching, and teachersperceptions of their mathematics/science pedagogical pre-paredness. The Likert-style questionnaire was originallydeveloped in 1995 by Horizon Research Inc. (2008), withassistance from the NSF staff, LSC primary investigators,and project evaluators. The questionnaire focuses onvarious broad questions, most of which include multipleitems. In total, the questionnaire has 156 items. A factoranalysis conducted by the developers revealed eight dis-tinct composites, two of which are the focus of this study:(a) teacher attitudes toward teaching and (b) teacher per-ceptions of pedagogical preparedness. Both of thesefactors included Likert-style items. The former consists of10 items, the latter of 18 items; both used options rangingfrom not important to very important. Specific itemsfor each composite are presented in Appendix B.The developers of the LSC also used their sample to

    calculate Cronbachs alpha reliability estimates for each ofthe questionnaires eight composites. The two of focus forthis study, teacher attitudes toward teaching and teacherperceptions of pedagogical preparedness, resulted ina = .80 and a = .93, respectively. In order to examine thereliability of these two composites for this study, Cron-bachs alpha estimates were calculated with data from thisstudys participants. Given sample sizes, such estimateswere only calculated for teachers of sixth grade. Estimatesof teacher attitudes toward teaching and teacher percep-tions of pedagogical preparedness for this sample resultedin a = .60 and a = .93, respectively. Specific items foreach composite are presented in Appendix B.ProceduresFliers announcing the workshops were distributed to

    all K-12 school districts and educational service centersin the metropolitan area during the spring of 2008. Inter-ested teachers were advised to register by contacting thehost university. The DTAMS and the LSC teacher atti-tude questionnaires were administered in person prior tothe one-week workshop. Teachers also completed post-tests for both questionnaires at the end of the last day ofthe workshop.The workshops consisted of 40 hours of professional

    development with two trained facilitators over a one-weekspan. The CMP2 textbooks were the primary resources forthe weeklong workshops. The facilitators put the partici-pants in the role of students during much of the workshop,modeling the type of questioning and culture needed whileimplementing CMP2. Subsequent to work on specific

    problems and investigations, participants were grouped todebrief about the materials used, their sample work, andissues that arose during the investigations. The facilitatorschose to work through approximately four of the eightcorresponding books for each grade level. For example, ofthe eight sixth-grade books that are available, participantsworked through four of them: Bits and Pieces I, II, and III,which all focus on fractions, decimals, and percents; andCovering and Surrounding, which focuses on area andperimeter of common shapes.One of the foci for these workshops was for teachers to

    understand better both the mathematics content and themethod of developing the content through the materials. Asecond was for the participants to understand the structureof the CMP2 materials. Time was taken to model each ofthese pieces of a CMP2 lesson, as well as to debrief aboutcommon questions that arise when implementing thisstructure. Such questions include how long one shouldtake for introducing the topic, how many students shouldbe in a group while exploring the topic, and how much ofthe topic summary should be led by students.

    ResultsA paired sample t-test was conducted to compare gain

    scores for all participants on the DTAMS from the pre- topost-workshop assessment. The normalized gains pre-sented in Table 2 indicate that teachers of all grade levelsreported significantly higher content knowledge after theirinvolvement in the professional development workshop.Due to the small sample size with respect to teachers of

    grade 7 (N = 18) and grade 8 (N = 12), all subsequentanalyses focus on teachers of sixth grade (N = 26). Afternormalized gain scores for sixth-grade participants werecalculated, a paired sample t-test was conducted toexamine any differences in total score on the DTAMSbefore and after engagement in the professional develop-ment workshop. The analysis indicated a significant dif-ference between total scores before and after the workshop(t [24] = -2.84, p = .01).In addition to providing an overall indication of partici-

    pants mathematical knowledge, the DTAMS also pro-vided information about four specific knowledge types:memorized knowledge (I), conceptual understanding (II),problem solving/reasoning (III), and pedagogical contentknowledge (IV); that is, four distinct subscores werederived from the assessment. Paired sample t-tests exam-ined differences between the sixth-grade participants pre-and posttest subscores reflecting each of the four knowl-edge types. Results revealed a significant pre/post differ-ence for problem solving and reasoning (t [24] = -1.24,

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  • p = .001). No significant differences were found betweenpre- and posttests for any of the other three knowledgetypes. Thus, gain scores and the paired t-test indicate theparticipants gained content knowledge as well as problemsolving and reasoning skills.Developers of the DTAMS noted that the four knowl-

    edge types should be highly correlated as they relatecontent knowledge and delivery. Given that our findingsrevealed that only problem solving and reasoning wassignificantly different between the pretest and the posttest,an analysis was conducted to determine if the four knowl-edge types were highly correlated among this sample ofsixth-grade teachers. Data confirming the significant cor-relations are presented in Table 3.A Wilcoxon signed-rank test was conducted to identify

    any significant changes related to teachers attitude towardteaching (T2) and teachers perceptions of their pedagogi-cal preparedness (T3), derived from the LSC question-naire. Because analyses focused on rank-orderedperceptional data with a small sample size, the p-value wasset at p < .10 to include marginal significance (Dunn,2001). Five of the 26 teachers responses were eliminatedlistwise due to a lack of substantial data.Analyses revealedthat, for some items, teachers of sixth graders reported less

    positive attitudes about teaching mathematics after theworkshop compared with beforehand. More specifically,after the workshop, participants were less likely to havepositive attitudes about providing their students concreteexperiences before abstract ones, about skills practice,about inquiry-based activities, and about mathematicsapplications in various contexts. Specific results are pre-sented in Table 4.Unlike with attitudes related to teaching mathematics,

    participants perceptions of their pedagogical preparedness

    Table 2DTAMS Content Gain Scores

    Academy n Mean SD NormalizedGain


    CMP Grade 6 (Number and computation) 26 .08 .01Pretest 60.00 26.2Posttest 67.02 20.7

    CMP Grade 7 (Geometry and measures) 12 .18 .01Pretest 59.81 26.0Posttest 70.63 17.2

    CMP Grade 8 (Algebraic ideas) 18 .04 .01Pretest 59.34 24.1Posttest 68.33 21.3

    Note. Normalized gain was calculated as discussed by Hake (1998) and others (Meltzer, 2002). Gain (g) = posttest - pretest / 100(maximum possible) - pretest.

    Table 3Correlations Across Knowledge Types on Grade 6 Teachers Post-AssessmentScores

    Knowledge Type I II III IV

    Memorized knowledge (I) 1.0 .55** .63* .52**Conceptual understanding (II) 1.0 .61* .75*Problem solving and

    reasoning (III) 1.0 .65*

    Pedagogical contentknowledge (IV)


    * p .001; ** p < .01.

    Table 4LSC Gain Scores: Attitudes Toward Teaching Mathematics

    Questionnaire Items z-scores p-level

    Provide concrete experience beforeabstract concepts.

    -1.857* .063

    Develop students conceptualunderstanding of mathematics.

    -.707 .480

    Practice computational skills anddiagrams.

    -1.941* .052

    Make connections betweenmathematics and otherdisciplines.

    -1.633 .102

    Have students work in cooperativelearning groups.

    -.707 .480

    Have students participate inappropriate hands-on activities.

    -.577 .564

    Engage students ininquiry-oriented activities.

    -2.000** .046

    Use calculators. -.333 .739Engage students in applications of

    mathematics in a variety ofcontexts.

    -3.535**** .000

    Use performance-basedassessment.

    -.707 .480

    Note. Five respondents lacked of either pre- or post-workshopquestionnaire results were deleted listwise before the analysis.N = 21.* p .10; ** p < .05; **** p < .001.

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  • showed positive changes across each item and the collectivecomposite. Statistically significant changes were noted inparticipants preparedness in providing concrete experi-ences before abstract ones, in having students work incooperative groups, in having students engage in hands-onactivities, and in using performance-based assessments.Specific results are presented in Table 5.

    DiscussionThe content gains demonstrated after one week of pro-

    fessional development focusing on the curriculum instead

    of only mathematics content indicate that mathematicsteachersmay benefit from curriculum-focused professionaldevelopment, as evidenced in Table 2. Even with smallsamples sizes for grades 7 and 8, this finding is refreshingbecause it demonstrates that familiarity with student- andinquiry-centered curriculum may increase teachers math-ematics content knowledge.The curriculum aggregates theconcepts that middle-school mathematics teachers shouldknow as they develop lesson plans. Investing the time toreview the concepts in the curriculum provides an oppor-tunity to review and increase mathematical content knowl-edge. As teachers become more comfortable with both thecurriculum and the content being delivered, they are betterable to guide students through that curriculum.Thus, teach-ers deeper understanding of mathematics and its associ-ated curriculum may better allow them to foster theirstudents deeper understanding of mathematics. Conse-quently, it is possible that districts can facilitate significantteacher content knowledge gains by providing curriculumfocused professional development.It is interesting that teachers significantly increased both

    their mathematical content knowledge and their skills inmathematical problem solving and reasoning. As statedearlier, CMP2 is more inquiry-based; it is designed to havestudents use their reasoning strategies to solve mathemati-cal problems. Focusing on the curriculum provides theopportunity for teachers to increase their personalproblem-solving and reasoning skills. In a nonthreateningenvironment such as a workshop, teachers can review theconcepts as a whole and develop a better understanding ofhow the concepts fit together.As stated earlier, the increasein both content knowledge and problem solving/reasoningmay contribute to the same gains for students.It is surprising that PCK (Knowledge Type IV) did not

    have significant differences between the pre- and posttests.Many parts of the workshop for sixth-grade teachersfocused on common student misconceptions around frac-tions and number sense, which should lead to teachersbetter understanding of PCK. The fact that the gains weretied more to participants Problem Solving and Reasoning,and not to PCK, was also surprising.This study with licensed teachers was over a one-week

    period; however, the results are consistent with Frykholms(2005) findings based on four years of data collection.Frykholm analyzed qualitative data and triangulated mul-tiple data sources to conclude that using mathematics cur-riculum in teacher preparation contributed to teacher shiftsin attitudes and beliefs about mathematics teaching andlearning. In addition, he reported that such training resultedin a growth in teacher mathematics content knowledge and

    Table 5LSC Gain Scores: Perceptions of Pedagogical Preparedness

    LSC Questionnaire Items z-scores p-level

    Provide concrete experience beforeabstract concepts.

    -1.807* .071

    Develop students conceptualunderstanding of mathematics.

    -.714 .475

    Take students prior understandinginto account when planningcurriculum and instruction.

    -.535 .593

    Practice computational skills anddiagrams.

    -.535 .593

    Make connections betweenmathematics and otherdisciplines.

    -1.081 .279

    Have students work in cooperativelearning groups.

    -2.310** .021

    Have students participate inappropriate hands-on activities.

    -2.309** .021

    Use calculators. -.535 .593Engage students in applications of

    mathematics in a variety ofcontexts.

    -1.513 .130

    Use performance-basedassessment.

    -2.496** .013

    Use portfolios. -1.213 .225Use informal questioning to assess

    student understanding.-.471 .637

    Lead a class of students engagedin hands-on/project-based work.

    -1.155 .248

    Manage a class of studentsengaged inhands-on/project-based work.

    -1.000 .317

    Help students take responsibilityfor their own learning.

    -1.000 .317

    Recognize and respond to studentdiversity.

    -.775 .439

    Encourage students interest inmathematics.

    -1.155 .248

    Use strategies that specificallyencourage participation offemale and minorities inmathematics.

    -1.507 .132

    Note. Five respondents lacked of either pre- or post- workshopquestionnaire results were deleted listwise before the analysis.N = 21.* p .10; ** p < .05.

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  • PCK. Frykholms results are based on qualitative data;these results are based on quantitative data.Gain scores were indicated across all items related to

    participants pedagogical preparedness, with gains forfour items in particular being statistically significant: pro-viding concrete before abstract experiences, having stu-dents work in cooperative learning groups, having studentsparticipate in hands-on activities, and using performance-based assessments. The CMP2 curriculum is designedaround inquiry-based methods; a closer familiarity withthe curriculum strengthened this understanding amongparticipants. Also, experiencing an inquiry-oriented studyof the curriculum within the workshop likely contributedto these gains.Although participants felt more prepared to provide

    concrete experiences before abstract concepts, theyreduced the importance they placed on such a strategy.There seems to be an inconsistency in these results.Perhaps there was some misinterpretation in what concreteexperiences meant, as discussion among these authorsresulted in different interpretations among the four of us.

    LimitationsOne of the limitations of the analyses is the small sample

    population. Having larger class sizes across the three work-shops could provide a better understanding of teachersgains in mathematical content knowledge and shifts inattitudes. The fact that each workshop had different facili-tators is another limitation of the study. With only threeworkshops and two facilitators, the influence individualfacilitators might have had on the results is unclear. Fur-thermore, analyses did not examine the interaction of thechanges in content knowledge and attitudes. In otherwords,it is not known if the acquiredmathematical content knowl-edge caused significant attitudinal shifts among participat-ing teachers, or if teachers attitudinal shifts contributed toincreased mathematical content knowledge. Another limi-tation was that achievement data of participants studentswere not available.While previous research (Adams, 2005;Bledsoe, 2002; Kersaint, 1998) found a positive relation-ship between teachers content knowledge and studentachievement, this study did not pursue whether the partici-pants students had improved achievement scores.

    Future WorkBecause the content of the grade 6 CMP2 (number and

    number sense) was best aligned with the DTAMSassessment administered, the significant gains in math-ematical content knowledge are not surprising. Grades 7and 8 results indicated significant content gains, but the

    number of participants was too small for confidence in thesignificance. Future studies should focus on increasing thenumber of participants for grades 7 and 8 to determine ifgains in these grade levels are sustained. Future studiesshould also investigate results for similar professionaldevelopment opportunities. For example, the structure ofthese workshops was modeled after what is provided atMichigan State University, where these workshops aredone for a national audience of teachers. In fact, one of thefacilitators was also a facilitator at the Michigan Stateworkshops. Examining any differences or similaritieswould be beneficial to both sites and to others interested insupporting the work of mathematics teachers.Future studies should also attempt to connect the gained

    mathematical content knowledge of teachers to their stu-dents achievement in mathematics. Longitudinal data col-lection of student achievement scores would provideinsight into whether there were student achievement gainsfor students in classrooms taught by participants.In addition, classroom observation protocols should be

    used to measure the actual implementation of the gainedPCK in planning, delivery, and evaluation of instruction.Aprotocol, such as the Reformed Teacher Observation Pro-tocol (Sawada et al., 2000), can be used to measure theseattributes pre- and post-workshop participation. Thiswould provide an indicator of the impact of the workshopon teacher practices (Lawson et al., 2002; Morrell, Flick,& Wainwright, 2004; Sawada et al., 2002).

    ConclusionProfessional development that is focused around specific

    curriculum materials can do much more for classroomteachers than merely develop a better understanding ofthose materials. The results of this study show that it ispossible to improve teachers content knowledge throughthe use of and focus on curriculummaterials in professionaldevelopment opportunities. The curriculum materials andthe nature of curriculum-focused workshops resulted in anincrease in content knowledge and an understanding of theimportance of inquiry and collaborative teaching strategies.The quantitative results confirmed similar qualitativeresults (Frykholm, 2005). It is encouraging that contentknowledge is increased by utilizing middle-school cur-ricula in such ways that focused on deeper understandingof basic middle-school mathematics content. Engagingmiddle-school teachers in curriculummaterials using peda-gogy that can be used with middle-school students not onlysolidifies teachers familiarity with such strategies, but alsocontributes to their understanding of the mathematicscontent. The beneficiary is the student.

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    308 Volume 112 (5)

  • Appendix ADTAMS Four Knowledge Types


    Skills Included

    I Memorized KnowledgeThis mathematics knowledge is rotely

    learned and employs memorization. Itincludes memorized knowledge ofdefinitions, procedures, or rules. Teacherswith this knowledge can rotely performskills, apply rules, and give definitions.

    II Conceptual UnderstandingThis mathematics knowledge is conceptual

    in nature. It includes a deepunderstanding of mathematical concepts,procedures, laws, principles, and rules. Itis knowledge of connections andrelationships among concepts. It is oftenassociated with meaning. Teachers withthis knowledge can giveexamples/non-examples and identifyproperties/characteristics of mathematicalconcepts. They can compare and contrast,and represent mathematical concepts andgeneralizations in multiple ways. Theycan explain and create mathematicalprocedures and represent them inmultiple ways.

    III Problem Solving/ReasoningThis mathematics knowledge is higher

    order in nature. It includes applyingknowledge to solve problems andreal-world applications. Teacher with thisknowledge can reason informally andformally, conjecture, validate, analyze,and justify. They can use deductive,inductive, proportional, and spatialreasoning to solve problems.

    IV Pedagogical Content KnowledgeThis mathematics knowledge is unique to

    teaching mathematics. It represents themathematics knowledge that teachers usein the act of teaching. It includesknowledge of the most regularly taughttopics in mathematics, the most usefulforms of representation of those ideas,the most powerful analogies, illustrations,examples, explanations, anddemonstrations. Teachers with thisknowledge can identify studentmisconceptions about mathematics andprovide strategies to correct them.Teachers can derive activities thatpromote understanding, reasoning, andproficiency. They can provide examples,analogies, models, or representations tohelp students understand mathematicalconcepts or procedures.

    Appendix BLSC Composite T2: Attitude Toward Teaching Provide concrete experience before abstract concepts. Develop students conceptual understanding ofmathematics. Practice computational skills and diagrams. Make connections between mathematics and otherdiscipline. Have students work in cooperative learning groups. Have students participate in appropriate hands-onactivities. Engage students in inquiry-oriented activities. Use calculators. Engage students in applications of mathematics in avariety of contexts. Use performance-based assessment.

    Appendix CLSC Composite T3: Pedagogical Preparedness Provide concrete experience before abstract concepts. Develop students conceptual understanding ofmathematics. Take students prior understanding into account whenplanning curriculum and instruction. Practice computational skills and diagrams. Make connections between mathematics and otherdiscipline. Have students work in cooperative learning groups. Have students participate in appropriate hands-onactivities. Use calculators. Engage students in applications of mathematics in avariety of contexts. Use performance-based assessment. Use portfolios. Use informal questioning to assess student under-standing. Lead a class of students engaged in hands-on/project-based work. Manage a class of students engaged in hands-on/project-based work. Help students take responsibility for their ownlearning. Recognize and respond to student diversity. Encourage students interest in mathematics. Use strategies that specifically encourage participa-tion of female and minorities in mathematics.

    Curriculum Focus

    School Science and Mathematics 309


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