Middle School Mathematics Teachers Learning to Teachwith Calculators and Computers

Part II: Teacher Change

George W. BrightUniversity of North Carolina at Greensboro

Neil E. ProkoschNational-Louis University

This paper reports the results of a project in which experienced middle grades matfiematics teachersimmersed themselves in calculator and computer use for both doing and teaching mathematics andprepared themselves as leaders for communicating their knowledge to colleagues. Project evaluationincluded interviews with participants at the beginning and end of the project and evaluation formscompleted at the end of the project. Pro-interviews indicated that virtually all of the participants had noexperience using technology to teach matliematics. Many felt that technology was not likely to be aseffective in helping students learn mathematics as other teaching techniques. Post-interviews indicatedthat all teachers were confident oftheir abilities to use some technologies in teaching mathematics. Theyacknowledged that technologywas useful in developing conceptualunderstanding andthat their rolewasto guide this conceptual development. The differences in participants’ perceptions about how theprojectaffected themyielded suggestions for future inservice efforts about technology.

This is the second part of a report on a projectwhich provided a group of experienced mathematicsteachers in middle and junior high schools the oppor-tunity to (a) be immersed in calculator and computeruse forboth doing and teaching mathematics and (b) beprepared asleaders forcommunicatingtheirnewknowl-edge to colleagues. The teachers were expected toreturn to their schools and provide models for the useoftechnology in teaching mathematics and to produceteaching materials built around technology. The firstpart (Bright & Prokosch, 1995) provided the back-ground for the project and described in detail theinservice provided to the teachers. This part presentsthe evidence that participants changed in theirviews ofboth mathematics and mathematics instruction.

Intervention and Evaluation Methodology

Two techniques were used to assess the nature ofteacher change including, (a) interviews with partici-pants at the beginning and end of the project and (b)evaluation forms completed at the end of the project.The pro/post interviews were tape recorded and tran-scribed. Transcripts were examined to find commonpatterns ofbehavior and beliefs. The evaluation formswere administered during the last week of the project.

Results and Discussion

The participant teachers did become immersed inthe use of the technologies through course work fol-

lowed by the development of instructional materials.Theknowledge they gained aboutthe technologies andthe interaction among participants about the process ofintegrating the technologies into mathematics instruc-tion prepared them for leadership roles in their dis-tricts. In particular, two of the participants becamedepartment chairs who could exert direct influence onthe instructional programs in their buildings. Anotherparticipant appeared in the National Council ofTeach-ers of Mathematic’s (NCTM) videotape on the use ofcalculators in mathematics instruction (Calculator-enhanced Mathematics Instruction Steering Commit-tee, 1992). Each of the participants produced studentmaterials, some of which was published in The Arith-metic Teacher in the ’Teaching Mathematics withTechnology" column (Baker, Edwards, & Marshall,1990; Hoeffher, Kendall, SteUenwerf, Thames, &Williams, 1990). Materials written around calculatorstended to be fairly traditional approaches to simplyrelieving computational burdens, though a few relatedthe use of manipulatives and calculators. Materialswritten around computers included computerprogramsto teach fractions and problem solving and supportactivities for commercial software to teach compass/straight edge constructions and graphing.

Evaluation FormsParticipants were asked to respond to a variety of

questions concerning the quality and impact of theparts of the project Selected questions are discussedbelow; complete data were available for 13 of the 16

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participants who completed the project.Knowledge, experience, and confidence. The data

(see Figure 1, Appendix 1) suggest that confidence ishighly related to knowledge and experience; indeed itwas somewhat surprising thatthemeans forthese threescales were so similar. Future research should attemptto understand these relationships better.

Figure 1. Knowledge, Experience, and ConfidenceScales

For each of the items below, rate your Knowledge,Confidence, and Experience according to the follow-ing scales:Knowledge1.1 have no knowledge.2.1 have some knowledge.3.1 have an average amount of knowledge.4.1 have above average knowledge.5.1 have an exceptional amount of knowledge.Confidence1.1 am not confident at all.2.1 am somewhat confident3.1 have an average amount of confidence.4.1 have more than an average amount of confidence.5.1 am very confident.Experience1.1 have never studied, discussed, or done this.2.1 have studied/discussed this in a session in which Iwas receiving training.

3.1 have tried this in a session in which I was receivingtraining.

4.1 have tried this with individuals or small groups forwhom I was the teacher.

5.1 have done this with a group in which I was theteacher.

Stages of concern. Three of the items (i.e., com-puter education research, calculators, and computers)showed "growth" (see Figure 2, Appendix 2) ofmorethan three stages (Hall & George. 1979). These arerelated to the central theme of the project; namely,technology; and relate more to pedagogical concerns(i.e., understanding both how to use technology andwhat the effects of those uses are on student learning)than to content mathematics concerns. Standard tech-nological concerns (i.e., applications programs andprogramming) revealed growthofmorethantwo stages,as did geometry, which was the content of one of thecourses. These data suggest that the project wassuccessful in achieving its main goal, namely, immers-ing participants in the use of technology for learningand teaching mathematics.

Figure!. Stages of Concern

Usethe descriptions [provided separately] ofthe Stagesof Concern about an innovation. For each of thefollowing items, rate your stage at the beginning andthe end of the NSF Institute.

Level Description0 Awareness1 Informational2 Personal3 Management4 Consequence5 Collaboration6 Refocusing

Pedagogical expertise. Thetwo items that showedthe greatest change in perceived expertise (see Figure3, Appendix 3) were using calculators in teachingmathematics and designingcurriculummaterials. Thesetwo items seem mostrelevant to a "regular"mathemat-ics teacher who has limited access to technologies forteaching. Many teachers have access to calculators ofsome variety (possibly by asking students to bring intheir own calculators), and every teacher can makesome efforts to design special curriculum materials totake advantage of locally available resources. Theparticipants seem to have gained control of these twonotions and recognized the power that they possessedto influence their own classrooms.

Figure 3. Levels of Expertise

Use the descriptions [provided separately] of charac-teristics ofLevels ofExpertise in teaching. For each ofthe following items, rate your level at the beginningand the end of the NSF Institute.

Uyel D^ggriptipn1 Novice2 Advanced beginner3 Competent teacher4 Proficient teacher5 Expert teacher

Pro and Post-InterviewsThe pro-interviews indicated that virtually all of

the participants had no experience using technology toteach mathematics. Many of them had not been ex-posed to any technologies other than four-functioncalculators and computers in drill and practice mode.Many felt that technology was not likely to be as

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Helping Middle School Math Teachers

effective in helping students learn mathematics asother teaching techniques. (This view, however, mayhave been partly a reflection of their personal lack ofconfidence in using technology to teach mathematics.)Several participants, in particular, expressed the con-cern that students would not learn the "basics" ifcalculators were used too often. Their views ofappro-priate uses ofcalculators was typically"to checkhome-woric."

The post-interviews showed a very different pic-ture. All teachers were confident oftheir ability to usesome technologies in teaching mathematics, thoughonly a few ofthe participants were comfortable with awide range oftechnologies. They generally acknowl-edged that technology was useful in developing con-ceptual understanding and that their role as teacherswas to guide this conceptual development, Foursamplesfrom the transcripts are given as representative of therange of changes that took place in teachers’ thinkingaboutthe useoftechnology. (Thenames are fictitious.)

Eloise. Eloise began the project with enoughcalculator skill to perform simple computation, but shehad great difficulty deciding how to solve the problem"6 is what percent of 277" with a four function calcu-lator. She expressed the opinion that three-digit bythree-digit multiplication should be done in paper andpencil format by sixth grade students. There was adepartmental set of calculators available for use, butshe had not used them much. Her students generallycould not afford to buy calculators. She believed thatcalculator use should be restricted to verbal problemsand checking homework. She had no experience withcomputers other than a short inservice class in BASICprogramming. At the end of the project, Eloise saidthat calculators should be used to "extend understand-ing of the concepts" being taught; her view of exten-sion, however, tended to be "using larger numbers."She persisted in the view that "you need to know thebasics first, before using the calculator." She usedcalculators at least once a week, however, and sheacknowledged thatherstudents got excited aboutusingcalculators. That student excitement seemed muchmore important to her than whether the students werelearning important mathematics from the use of tech-nology. She did not want students to have access tocalculators all the time. because "our groups are notgeared to application ... if you’re adding a series ofnumbers, make sure that you line the decimal underdecimal."

Eloise’s mathematics background was somewhatweak. but she was very respected at her school as agood teacher. The students in her school were per-

ceived by the staff as being very ’basic,’ and thatseemed to color what experiences with technologywere viewed as appropriate. The most important partsof the institute were the courses on problem solvingand on the use ofcalculators in teaching mathematics.The problem solving course provided an introductionto a variety ofproblem solving strategies that she coulduse fairly directly with her students. The calculatorcoursewas organized around demonstrations ofsamplelessons that could be used with middle school students.This modeling seemed very important in giving herconfidence for doing similar activities in her class-room. Visits to her classroom and examination of herinstructional materials, however, revealed that sheselected primarily those activities that provided prac-tice on computation forher students. She admitted thatthere was very little cooperation among the teachers inher building; teachers came to the campus, taught theirclasses and left for home. The administration alsoseemed indifferent to her experimentation with tech-nology or with her participation in the project

Denise. Denise began the project with a goodflexibility at using her mathematics skills. She did nothave access to any calculators in her school, and herstudents generally could not afford to buy calculators.She felt that calculators would be useful for decimalwork but she would not allow them for work withfractions because of the difficulty at translating be-tween fractions and decimals. She was open to thepossibility of allowing calculator use on tests, but shehad had no experience at developing appropriate testitems. She had experience with application programs(e.g., word processing) and BASIC and FORTRANprogramming. She had used CAI software with atutorial group the yearbefore the projectbegan. "Over-all [kids] have a bad image ofmath... and I think thattechnology changes theirviewofmath.. .because theyenjoy [working with the technology]." At the end ofthe project she persisted in the view that when you "puta calculator in theirhands.. .they’11 work tenproblemswhere they wouldn’t even try one before." That is,technology is primarily useful because it motivatesstudents. She had used calculators regularly in class,she had taken her students to the computer lab foractivities, and she used a demonstration computer withscience probes and graphing software to teach graph-ing of data. One example she gave of the use ofcalculators was that when you multiply a number by0.9 the number gets smaller. Students did a variety ofexamples with their calculators and they seemed to seethis pattern; in previous years students had never reallycaught on to this idea.

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Denise cited the courses as an important influenceon changing her views of how to use technology toteachmathematics. Inparticularshe cited the hands-onwork with calculators and computers in the courses(that is. her own experimentation with the technolo-gies) as an important part ofhelping her feel comfort-able with the technology. She felt that increasing theuse of technology in the mathematics courses in theproject would have increased the usefulness of whatshe learned. In addition, however, she cited a verysupportive principal who encouraged her to experi-ment and to take additional inservice workshops ontechnology. The principal set an expectation thatDenise would share her knowledge with colleagues inthe mathematics department Denise had becomeknown in her building for her use of technology, andother teachers were beginning to become interested inalso using some of that technology.

Denise had experimented with calculator use ontests. Sometimes she allowed calculators on half thetest and othertimes on the whole test, "just to see whathappens." She was most impressed, however, with theway that some of the students’ attitudes about math-ematics changed when calculators were introduced. "

A perfect example is these two boys that weremainstreamed from special ed. When they got calcu-lators in their hand ... I didn*t even know these kidsanymore. The whole class... was just turned around.They were coming in after school, before school to domore work. They were asking other teachers to comedown to my room to do some more math... That justblew my mind." And yet with her eighth grade class,"they’re still not sold onthem yet... [They say] how’sthat going to help us leam?"

Charlene. Charlene was from a suburban schooldistrict recognized throughout the state for its high testscores. Another participant in the project was alsofrom the same building. At the beginning ofthe projectCharlene had a good understanding of mathematicsand was easily able to solve problems using a calcula-tor. Her view of mathematics for middle school stu-dents centered on "thinking and logic skills" with lessimportance given to computational skills. She usedcalculators regularly forproblem solving, especially inratio and proportion problems. The district, however.had a policy of requiring daily review of computationskills, for which calculators were not allowed. Thestudents all had personal calculators for use in school.She used calculators on her own tests; the district hadnotdeveloped apolicy about calculators ontests, so shehad never mentioned it to anyone. Charlene hadexperience with applications software and with

BASIC, Pascal, and COBOL programming. Duringthe previous yearshehad taught computerliteracy, andwhen the lab was available, she had taken her math-ematics classes there to use drill and practice software.Her view of proper use of computers in teachingmathematics was to "acquire software so that it wouldreinforce what was having to be done from the text-book." When asked to comment about whether tech-nology was good formathematics instruction, she said," I’m ready to go on and do something different andsomething better to challenge these students more toget over ... the frustration that they’re feeling whenthey still have to count on their fingers to subtract ormultiply."

By the end ofthe project Charlene had used calcu-lators regularly, but her curriculum project was devel-opmentofaunit involving spreadsheets for sixth gradestudents. Her view ofthe best uses of calculators wasfor problem solving, "to relieve the pressure of arith-metic so they can concentrate on ... other aspects ofmath." She admitted that prior to the start ofthe projectshe "didn’t want them to use the calculator because Ididn’twantthem to forgethow to do the math... [Now]it’s an open calculator atmosphere in the classroom."She was most influenced in this change of positionfrom watching students use technology; "the studentsdidn’t lose anything by using technology. They stilldid well on tests... [even] without the calculator." Themost effective aspect of the project was the hands-ontime during the courses. Charlene was appointeddepartment chairperson during the last semester oftheproject, and in this position she was able to encouragethe mathematics teachers in her building to use calcu-lators and computers more frequently. Her modelingofuses of technology became more visible because ofher new position. She attributed a change in attitudesof other teachers in her building to this modeling, andshe contrasted attitudes inherbuilding with attitudes inother middle schools in the district. To summarize herperception of the state of the use of technology inteaching mathematics, she said, "I feel like I’m sittingon... the tip of an iceberg, and there’s so much morethat’s there that I can be doing... [I expected] to be theexpert, that I would have all the answers, and I don’t...I have some little bit of expertise that I’m willing toshare with others that can be helpful."

Candice. Candice was from an outlying schooldistrict whichwas changing from a majority Caucasionpopulation to a majority African American and His-panic population. Most of her students were AfricanAmerican, though that was not true for another partici-pant from another middle school in the same district.

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Shehad some difficulty withusing a calculator to solvea percent problem, though she did correct her ownerrors in this process. Her view of the proper use ofcalculators was that they were "particularly" good forchecking homework "to know whether or not theanswer is correct." But she also recognized that theycould be used for problem solving to "cut down on theamount oftime that [students] need to use for calculat-ing." She believed that it was important "to have somerules in their heads" and although she had used calcu-lators the previous fall, it was only "after the first sixweeks of school,... [because] it’s important for themto... make sure how it works and why it works beforethey go to the calculator." Candice did not have anycomputer experience, except for a six-hour inserviceon BASIC.

By the end of the project, Candice had used drilland practice software and the Impact Systems video-disk program on fractions with herbasic students. Theschool had setup a computerlab forthese students withChapter 1 money, and Candice worked with an aide ina class of about 15 students. Her view ofthe best useoftechnology in teaching mathematics was that "I likethe tutorial part of it... It allows you as a teacher towork with more students because you can go from onestudent to another while they’re working on the tech-nology and get the one-on-one that they need." Shesaw herself teaching different mathematics than at thebeginning of the project, but mainly in the context ofdifferent kinds of test items. "The kinds of questionsI ask are more meaningful... [Calculators] are alwaysavailable in my classroom." She developed somecalculator materials that involved patterns with pow-ers, and she felt that these kinds of activities wereuseful in helping students remember the underlyingconcepts. She did admit, however, that "sometimes[the calculator] gets in the way when you’re develop-ing concepts ... When we started working on integersthis year. [the students] wanted to use the calculatorand I said that... you can use it as soon as I know youunderstand... how it works and why it works. I thinkthat theyneed to understand some concepts before theystart using [calculators]."

Candice was one of the few participants that citedthe research seminar (i.e.. one ofthe courses) as impor-tant for influencing her views on technology. Thecumulative effects of reading about so many positiveoutcomes seemed to encourage her that technologyreally was effective for helping students learn. Herview of the other courses was somewhat amorphous,she coiddn’tpmpoint anyparticularlyimportantcourse;she simplyacknowledged an overall cumulative effect.

Conclusions

At the outset, it is important to note that the typesoftechnologies used by the teachers in this project arevery similar to technologies that are still in use today.Many teachers do not have regular access to technol-ogy, and even when they do, the technology is often inthe form of"first-generation computers that are hard touse and of very limited computational and displaypower" (Kaput, 1992, p. 517) or traditional four-func-tion or scientific calculators. Thus, the lessons learnedfrom this project are still quite applicable to much ofthemathematics instructionthroughouttheU.S. Know-ing how teachers make the transition from no technol-ogy to some technology in their teaching will alsoprovide a base of knowledge to help make the transi-tion to newer technologies (e.g.. graphing calculators)smoother.

Overall, the project seemed to have a greater im-pact on pedagogical knowledge ofparticipants than onmathematical knowledge. In part this may indicatechange in participants’ beliefs about what it means todo mathematics and to learn mathematics. This wasconsistent with the goals and design ofthe project. Theproject seemed quite successful at raising the con-sciousness of the participants about the use of varioustechnologies in teaching middle school mathematics.Participants seemed to expand their views about whatstudentbehaviors were acceptable duringmathematicslearning. It also seemed successful at giving theparticipants enough confidence to begin experimenta-tion with the use of technology in their teaching; allparticipants produced materials for their students andwere observed using some kind oftechnology in teach-ing. Most important, however, the teachers becamemore open to the possibility of using technology toteach significant mathematics. Most ofthem began tolook for ways to use technology.

The explanations of this impact are more difficultto determine. Onepossibleexplanation is theHawthorneeffect; that is, participants volunteered for the projectand they were repeatedly observed while teaching withtechnology. The fact that they knew they would beobserved may have been a significant incentive forthem to change their instruction. It is also possible,however, that teachers perceived the technologies ascritical tools for changing their instruction and theirviews of appropriate mathematical activity. Furtherstudy ofteachers’ explanations and reflections on theirunderstanding of the role of technology, therefore,seems warranted.

As the sample cases illustrate, the most important

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parts of the project differed for the participants, de-pending at least in part on the school environment inwhich they were working. For teachers that wereexpected to operate relatively independently of col-leagues, themodelingofinstructionaltechniques intheproject courses were very important; that modelingwas, after all, about the only way they got informationon changing their teaching. For teachers that hadsupportive networks within their buildings, the infor-mation in the courses became more important; infor-mation is the medium ofexchange in such a network,so new information created new opportunities forexchange. As new projects are planned to helpteachers learn to use technology, it will be important tounderstand the environment within which the partici-pants will operate. Projects need to providethe supportthat is perceived by those teachers as useful in thoseenvironments. For example, modeling of appropriateuses of newer technologies in teacher inservice ses-sions is important for those teachers who are expectedto leam to use these technologies on their own. Thisinformation addresses one of Kaput’s (1992) "openquestions’ about the support "needed in introducingteachers to new technologies" (p. 550).

Several aspects ofthis project appear in combina-tion to be important for helping teachers leam to usetechnologies in mathematics instruction, though theimportance ofeach aspect separately is not clear. Theprojectextended overalongperiod oftime, andthrough-outthat period, supportwasprovided both from projectstaff (in the form of courses and classroom visits) andfrom other teachers in the project (in the form ofinformal interactions before and after class meetings).Many different perspectives on the use of technologyin instructionwere presented; for example, developingpersonal technology skills with a variety of technolo-gies, doingmathematics with the assistance oftechnol-ogy. and understanding pedagogical techniques effec-tive withtechnology. Development and field testing ofcurriculum materials extended over several semesters.Eachofthese aspects is important foreffecting change,but the combination seems especially powerful. Thisinformation addresses anotherofKaput’s (1992) "openquestions’ about the "strategies forteacherpreparationand renewal" (p. 550).

The model of teacher inservice outlined in thesepapers should serve as an example for others as theyhelp teachers leam to integrate technology in math-ematics instruction. It is important, for example, tomatch "modeling" and "information giving" to teach-ers’ needs, depending on the type ofsupport availableforteachers in theirbuildings. Further, tracking changes

through a variety of paradigms and methods (e.g.,interviews and Likert-type scales) provides insightthatneither would provide alone (e.g., Waxman & Bright,1993; Waxman. Williams, & Bright, 1994). Theparticular software and calculators used by teachers inthis study are becoming obsolete, but the understand-ing about teacher change that is evident in these datawill support inservice efforts with whatevernew tech-nologies become available.

References

Baker. D., Edwards, R.. & MarehaU. C. (1990).Teaching about exponents with calculators. Arith-metic Teacher, 38(1), 38-40.

Bright. G.W.,&Prokosch.N.E. (1995). Helpingmiddle schoolmathematics teachersleamto teach withtechnology: Part I: Background and classroom obser-vations. School Science and Mathematics, 95(6), 295-301.

Calculator-enhanced Mathematics InstructionSteering Committee. (1992). Calculators for class’rooms [videotape and discussion guide]. Reston, VA:National Council of Teachers of Mathematics.

Hall. G. E.. & George, A. A. (1979). Stages ofconcern about the innovation: The concept, initial

verification and some implications. Austin, TX: Re-search and Development Center for Teacher Educa-tion.

Hoeffner, KL. Kendall.M.. SteUenwerf. C.Thames,P.. & Williams. P. (1990). Problem solving with aspreadsheet. Arithmetic Teacher, 38(3), 52-56.

Kaput, J. J. (1992). Technology and mathematicseducation. In D. A. Grouws (Ed.), Handbook ofresearch on mathematics teaching and learning (pp.515-556). New York: MacmiUan.

Waxman, H.C..& Bright, G.W. (1993). Researchmethods and paradigms in technology and teachereducation. In H. C. Waxman & G. W. Bright (Eds.),Approaches to research on teacher education andtechnology (pp. 1-9). Charlottesville, VA: Associationfor the Advancement of Computing in Education.

Waxman, H. C.. Williams. S. E.. & Bright, G. W.(1994). Future directions forthe studyofcalculators inmathematics classrooms. In G. W. Bright, H. C.Waxman, & S. E. Williams (Eds.). Impact of calcula-tors onmathematicsinstruction(pp. 131-138). Lanham,MD: University Press of America.

Author’s Note: This manuscript is based on presenta-tions made at the 1991 Southwest Educational ResearchAssociation annual meeting, San Antonio, TX and the 1991

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Helping Middle School Math Teachers

American Educational Research Association annual meet-ing, Chicago, BL.

This work was supported in part by a grant from theNational Science Foundation (Grant No. TPE 8751473).All positions and opinions expressed, however, are thoseof the authors and do not necessarily reflect the positionsof the Foundation.

Editor’s Note: George Bright’s address is School ofEducation, Department of Curriculum & Instruction, CurryBuilding UNCG. Greensboro. NC 27412-5001. NeilProkosch’s address is National-Louis University, WheatonCampus. 200 S.NaperviUe Road. Wheaton.IL 60187.

Appendix 1. Mean Ratings of Knowledge, Experience, and Confidei

Technology Knowledge

four-function calculator 4.46fraction calculator 3.92scientific calculator 3.31programmable calculator 2.62word processing 4.31spreadsheets 4.08databases 3.92Logo 4.08BASIC 3.54videodisk 2.15Lego Logo robots 2.15

Appendix 2. Mean Ratings of Stages of Concern (n

Area of Concern Beginning

geometry 2.62analysis 2.17calculators 2.31computers 2.15videodisk 0.92robotics 0.31word processing 2.38database 1.92spreadsheets 1.92Logo 1.31BASIC 2.00computer education research 0.92

Appendix 3. Mean Ratings of Levels of Pedagogical

Area of pedagogy

teaching computationteaching analysisteaching geometryusing calculators in teaching mathematicsusing computers in teaching mathematicsusing videodisk in teaching mathematicsusing spreadsheets in teaching mathematicsusing Logo/BASIC in teaching mathematicsdesigning curriculum materials

Experience

4.773.923.232.623.924.004.003.923.852.232.15

=13)

End

4.693.755.625.312.232.084.544.314.544.004.004.54

’Expertise (n=

Beginning

3.922.332.922.081.691.151.311.461.75

nee (n = 13)

Confidence

4.623.853.462.234.464.003.853.853.542.081.77

Change

2.071.583.313.161.311.772.162.392.622.692.003.62

13)

End

4.543.503.854.153.381.923.003.003.69

Rank

811231210765491

Change Rank

1.62 51.17 70.93 82.07 1 ^1.69 3.5 -

0.77 91.69 3.51.54 61.94 2

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