Mathematics teacher educators’ perceptions and use of cognitive psychology research

Mathematics teacher educators’ perceptions and use of cognitive psychology research

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  MIND, BRAIN, AND EDUCATION Mathematics TeacherEducators’ Perceptions and Useof CognitiveResearch ElidaV. Laski 1 , ToddD. Reeves 1 , Colleen M.Ganley 1 , and RebeccaMitchell 1 ABSTRACT—  Instructors ( N = 204) of elementary mathe-matics methods courses completed a survey assessing theextenttowhichtheyvaluecognitiveresearchandincorporateitintotheircourses.Instructors’responsesindicatedthattheyviewcognitiveresearchtobefairlyimportantformathematicseducation, particularly studies of domain-specific topics, and thattheyemphasizetopicsprominentinpsychologystudiesof mathematical thinking in their courses. However, instructorsreported seldom accessing this research through primary orsecondary sources. A mediation analysis indicated that math-ematics methods instructors’ perception of the importance of the research predicts their incorporation of it in their courses,and that this relation is partially mediated by their accessingof it. Implications for psychologists who have an interest ineducation and recommendations for facilitating the use of cognitive research in teacher preparation are discussed.Over the past two decades, there has been increased interest in the application of cognitive science research forimproving instruction (Bransford, Brown, & Cocking, 1999;Darling-Hammond, 2010; Newcombe et al., 2009). Numeroussources describe the implications of cognitive research forthe instruction of reading, science, and mathematics (e.g.,National Mathematics Advisory Panel, 2008; Pashler et al.,2007; Rayner, Foorman, Perfetti, Pesetsky, & Seidenberg,2001; Siegler, 2003). However, the extent to which cognitiveresearch actually influences teachers’ instruction is largelyunknown.This study investigated this question in relation to thetraining of elementary mathematics teachers. The main goals 1 Boston CollegeAddress correspondence to Elida V. Laski, 201 Campion Hall, LynchSchool of Education, Department of Applied Developmental and Educational Psychology, Boston College, Chestnut Hill, MA 02467;e-mail: ofthestudyweretoexamineteachereducators’perceptionsof theimportanceofcognitivepsychologyresearchandparticulartopics,theiruseofcognitivepsychologyresearchinelementarymathematics methods courses in which preservice teacherslearn specifically about mathematics pedagogy and studentlearning, and potential factors related to the incorporationof cognitive research in mathematics education coursework.This information can inform researchers’ choice of researchquestions and dissemination venues, which, in turn, could facilitateastrongerconnectionbetweencognitivescienceand educational practice. POTENTIAL OFCOGNITIVE PSYCHOLOGYTO INFORMELEMENTARY MATHEMATICS TEACHING The premise that cognitive research can be used toimprove mathematics instruction is generally based on twoassumptions: (1) psychology research generates informationthat has implications for mathematics education and (2)teachers with knowledge of this research will be betterequipped to improve children’s mathematics understandingthan teachers without this knowledge. PsychologicalStudiesofMathematical Thinking There is now considerable knowledge from cognitive scienceabout how children learn mathematics (Geary, 2006; Siegler,2003). Our review of the literature in this area suggested four general categories of research that might be particularlyuseful to elementary mathematics teachers’ pedagogy. Table 1presents these categories and provides illustrative studies foreach.The first two categories of research can potentially helpteachersdeterminehowtosequenceanddifferentiateinstruc-tion as well as assess their students. The first of thesecategories— Developmental Progression and Common Misconcep-tions —involves research about the developmental progression © 2013 The AuthorsVolume 7—Number 1 Journal Compilation © 2013 International Mind, Brain, and Education Society and Blackwell Publishing, Inc.  63  Cognitive Research and Teacher Preparation Table 1 FourCategoriesofCognitivePsychologyResearchThatAreRelevantto Elementary Mathematics Education and Illustrative References Thematiccategory Illustrativereferences Developmental progressionand commonmisconceptions of keymathematics skills and conceptsCarpenter and Moser (1984)Gallistel and Gelman (1992)Laski and Siegler (2007)Mix, Levine, and Huttenlocher(1999)Cognitive strategies formathematicalproblem-solvingGeary et al. (2004)Hallett, Nunes, and Bryant (2010)Siegler (1988)Cognitive processes involved in the acquisition of mathematics skills and conceptsImbo and Vandierendonck (2007)Klein and Bisanz (2000)McNeil and Alibali (2004)Influences of instructionaltechniques on the cognitiveprocessing of mathematicsGoldin-Meadow, Cook, and Mitchell(2009)Rittle-Johnson (2006) of children’s understandings of mathematics concepts. Forexample, Siegler and colleagues have demonstrated thatchildren initially represent the magnitude of numbers log-arithmically, such that they exaggerate differences betweennumbers at the low end of the numerical scale and mini-mize differences at the high end. With age and, presumably,greater experience with larger numbers, children’s represen-tations of numerical magnitude become increasingly linear(Laski & Siegler, 2007; Siegler & Booth, 2004). The sec-ond category— Cognitive Strategies for Problem Solving  —involvesresearch aboutthe strategieschildrenuse in mathematics,thevalue of those strategies, and the dimensions by which chil-dren select among them. For instance, cognitive psychologyresearch has demonstrated that at any given time in develop-ment children know and use multiple addition strategies and thatchildren’sselectionofstrategiesdependsonmultiplefac-torssuchasproblemdifficultyandconfidencelevel(Ashcraft,1992; Geary, Hoard, Byrd-Craven, & DeSoto, 2004; Shrager &Siegler, 1998).The next two categories of research can potentially helpteachers understand individual differences in rate of learningand how to design instruction that facilitates learning.The first of these categories— Cognitive Processes Involved inthe Acquisition of Mathematics Skills —involves research aboutthe general cognitive processes involved in mathematicalthinking and the acquisition of mathematical knowledge.For example, McNeil and Alibali (2004) examined howencoding—a cognitive process by which stimuli are mentallyrepresented—influences arithmetic performance. They found thatelementarystudentswhoencodekeyfeaturesofproblemsandinstructionalexamplesaremorelikelytoemployaccuratesolution strategies and solve problems correctly. The lastcategory— Influences of Instructional Techniques on the Cognitive ProcessingofMathematics —involvesresearchabouttheinfluenceof particular instructional techniques on cognitive learningprocesses. For example, self-explanation has been found topromote the acquisition of more sophisticated mathematicalconcepts (e.g., Rittle-Johnson, 2006; Stigler & Hiebert, 1999). KnowledgeImportant forTeaching Knowledge about the topics predominant in cognitivepsychology studies of mathematical thinking is considered an important aspect of the knowledge required to teachmathematicseffectively(Ball,Hill,&Bass,2005;Ball,Thames,& Phelps, 2008; Shulman, 1987). Knowledge of mathematicscontent, while important, is not a sufficient foundation forthe effective teaching of mathematics (National ResearchCouncil,2010).Inadditiontocontentknowledge,pedagogical-content knowledge—knowledge related to how to teachparticular content—is considered to be fundamental toteachers’ effectiveness (Ball & Bass, 2000; Darling-Hammond & Bransford, 2005; Shulman, 1987). An important aspectof current conceptions of pedagogical-content knowledgeis knowledge of students’ cognitive strategies, learningprocesses,andpriorknowledgethatmightbeeitherfacilitativeorinhibitivefortheparticularlearningtaskathand(Balletal.,2008; Hill, Rowan, & Ball, 2005).Indeed, there is causal evidence that increasing teachers’knowledge of developmental progressions and learningprocesses improves their capacity to plan and delivermathematics instruction as well as to analyze studenterrors. When elementary school teachers are provided with professional development designed to improve theirunderstanding of children’s mathematical thinking, theirinstruction changes in ways that are related to higher studentachievement (Fennema et al., 1996). Similarly, Saxe, Gearhart,and Nasir (2001) found that professional development thatincluded student thinking and learning processes led togreater effects on upper elementary students’ conceptualunderstanding of fractions than did an equivalent amountof professional development focused on general pedagogicaltechniques. RESEARCHUSEIN EDUCATION Previous findings about research utilization indicate, atbest, inconsistent use of empirical research in education(Broekkamp & van Hout-Wolters, 2007; Hemsley-Brown &Sharp,2003;Huang,Reiser,Parker,Muniec,&Salvucci,2003;Nelson, Leffler, & Hansen, 2009). Two factors have emerged as barriers to the use of research: perceived importance and comprehensibility.64  Volume 7—Number 1  Elida V. Laski et al. PerceivedImportance Qualitative studies consistently report that a primary factorinfluencing educators’ use of research findings is its perceived importance (Broekkamp & van Hout-Wolters, 2007; Nelsonet al., 2009). Teachers generally perceive research to lackapplicabilityand,therefore,areunlikelytoseekoutandaccessresearch articles (Hemsley-Brown & Sharp, 2003). Teachersalsorespondmostpositivelytoresearchthatidentifiesspecificpedagogical strategies and techniques as opposed to articlesthat simply document research findings (Zeuli, 1994). Theseresultssuggestthatunlessempiricalresearchisseenashavingdirect implications for education and teaching, it is unlikelyto be accessed and utilized by educators. PerceivedComprehensibility The perceived comprehensibility of research also seems to berelated to the utilization of research (Nelson et al., 2009).Educators have reported that the complexity and theoreticalorientation of research is problematic (Huang et al., 2003).Educators also report that the use of jargon, technical-language, and statistics make research findings difficult tointerpretand,therefore,notuseful(Hemsley-Brown&Sharp,2003; Nelson et al., 2009). These findings suggest that theutilization of psychological research in education might beparticularly problematic given differences in terminology,methodology, and presentation styles between the two fields. TEACHER EDUCATION PROGRAMSASANINTERMEDIARY INTHEUSEOFPSYCHOLOGICAL RESEARCH One important role of teacher preparation programs isto communicate research-based knowledge to teachers(Cochran-Smith, 2005). Current credentialing guidelinesreflecttheassumptionthatprovidingteacherswithknowledgeof child development, learners, and learning is an essentialaspect of an effective teacher preparation program (NCATE,2008).Within teacher education programs, methods coursesplay a central role in teacher training. These courses aredesigned specifically to instruct preservice teachers abouthow to teach academic content. An elementary mathematicsmethods course usually covers various topics, includingpedagogicalapproachesthatpromotechildren’smathematicalthinking and development, how elementary students learnmathematics, and assessment techniques (Van de Walle,Karp, & Bay-Williams, 2009). In the design of theircourses,instructorslikelyreflectuponandmakeinstructionaldecisions about the topics and research most important tomathematics instruction.Because methods courses emphasize practical pedagogicalapproaches, research presented in these courses may be moremeaningful to preservice teachers than research presented inother courses, such as stand-alone psychology courses (Ball&Cohen,1999;Darling-Hammond,Wei,&Orphanos,2009).Furthermore, these courses likely provide more opportunityto discuss the implications of the results for teaching and toembed the findings in the practice of teaching, similarto the professional development approaches that have beensuccessful(Fennemaetal.1996).Thus,incorporatingcognitivefindingsintomethodscoursesmayincreasethelikelihoodthatthe findings will affect preservice teachers’ later instruction. THECURRENT STUDY The above analysis suggests that instructors of elementarymathematics methods courses in teacher education programsare an important source for determining which topics may bemost relevant to improving current mathematics instructionandmostusefulforelementarymathematicsteacherstoknow.In this study, instructors were asked to rate the importanceof cognitive psychology, in general, and of specific topics tothe preparation of future elementary mathematics teachers.Instructors were also asked how much they emphasized var-ious topics in their elementary mathematics methods courses.The above analysis also suggests that these instructorsmay play a particularly important role in communicatingcognitive findings about mathematics learning to teachers of elementary mathematics. We expected perceived importanceto predict accessing and incorporation: instructors whoperceived cognitive research to be more important would be more likely to access it and incorporate it in their courses.We also hypothesized that the more instructors accessed thisresearch, the more likely they would be to incorporate intotheir courses.Last, we predicted that perceived comprehensibilitywould moderate the relations between perceived importanceand accessing and between accessing and incorporation.Instructors who perceive cognitive psychology research to bedifficult to understand, despite being potentially important,mightbelessinclinedtobelievethatreadingtheworkwillbeafruitfuluseoftime,thusmaybelesslikelytoaccessit.Further,instructors who do access the research, but perceive it to bedifficult to understand, might feel less confident in their ownunderstanding of the work, their ability to explain the work,andtheirstudents,potentialtounderstandthework,andthusmay be less likely to incorporate it into their courses. METHOD Participants Participants were instructors of elementary mathematicsmethods courses ( N = 204) from 195 institutions within the Volume 7—Number 1  65  Cognitive Research and Teacher Preparation continental United States, representing 46 states and theDistrict of Columbia. Participants were recruited from arange of universities who were members of the AmericanAssociation of Colleges of Teacher Education (AACTE). Therecruitment email invited instructors to participate in a studyabout ‘‘what teachers of elementary mathematics methodsbelieve our nation’s teachers should know and be able todo’’—therewasnomentionintherecruitmentemailaboutthestudy being about instructors’ views on research.The majority of participants were tenure-track professors:assistant (38.4%), associate (29.5%), and full (17.9%). A smallpercentage of the instructors (14.2%) held nontenure trackpositions.Themajorityofparticipants(74.72%)indicatedthatthey had completecontrol of their course design and syllabus. SurveyDevelopmentand Content Data were collected using an online survey. Because therewas no existing instrument that could be used to addressthe current study’s research questions, the research teamconstructed a survey through an iterative, collaborativeprocess. In addition to our own revisions to multiple drafts of thesurvey,itunderwentbothsubstantiveandmethodologicalreviews by discipline experts not affiliated with the researchproject.Thefinalsurveyreflectsallchangessuggestedbytheseexternal reviewers.Thesurveywasdesignedtoexaminefourconstructsrelated to cognitive research, namely instructors’ (1) perceptions of its  importance  to mathematics education and the preparationofteachers;(2)perceptionsofits comprehensibility ;(3) accessing  ;and(4) incorporation intheirelementarymathematicsmethodscourse. These data generally took the form of responses torating scales. For each construct, we included various surveyitems tapping different aspects of the construct such thatscales or indexes could be created. The reliability and internalstructure of our measures are reported in the results section.For some questions, a ‘‘Don’t know or unsure’’ responseoption was included in order to discriminate individuals withneutralopinions(e.g.,‘‘neitheragreenordisagree’’)fromthosewho did not understand the question or item content. ‘‘Don’tknow or unsure’’ responses were set as missing data for allinferential analyses. 1 For all items, participants were asked to respond with regard to ‘‘research with implications forelementary mathematics education.’’ RESULTS Perceptionsand Useof Cognitive Research  PerceivedImportance Instructors’ responses indicated that they generally viewcognitive and developmental studies of mathematics to besomewhatimportantformathematicseducation,butthatthissentiment seems to be stronger for particular mathematics-related topics than the research as a whole. When asked toindicatetheiragreementona5-pointscalewiththestatements‘‘[Cognitive  or   developmental] psychology research findingsare applicable to mathematics education’’ ( α = .89) and thestatements‘‘[Cognitive or  developmental]psychologyresearchaddresses topics, issues, and problems that are importantto elementary mathematics education,’’ ( α = .87) the meanresponsesfellbetween‘‘neitheragreenordisagree’’and‘‘agree,’’(3.79,  SD = 0.74 and 3.65,  SD = 0.69, respectively).As shown in Table 2, instructors’ responses were morefavorable, however, when asked to rate the importanceof specific topics to the preparation of future elementarymathematics teachers, particularly in relation to domain-specific topics. The mean of the three domain-specific itemswas significantly higher than the mean of the three domain-general items,  t (185) = 10.68,  p < .001.  PerceivedComprehensibility In general, the data indicated that instructors perceived cognitive and developmental psychology research to besomewhat comprehensible. The mean agreement with thestatements‘‘[Cognitive or  developmental]psychologyresearchfindings are accessible (i.e., capable of being understood)’’( α = .87)fellbetween‘‘neitheragreenordisagree’’and‘‘agree’’(3.54,  SD = 0.79).  Accessing  To collect information about the extent to which instructors’access research, we first asked participants to rate theiragreement with statements regarding the extent to whichthey (a) ‘‘keep abreast of empirical research’’ (b) ‘‘read empirical research articles to access research,’’ and (c) ‘‘read practitioner-friendlyarticlestoaccesssummariesofresearch’’fromthefieldofmathematicseducation,cognitivepsychology,developmental psychology, and educational technology. Weaveraged individuals’ responses across the three items foreach field; thus, the data do not discriminate betweenwhether instructors access research findings from primaryor intermediary sources. Reliabilities for the item triads (oneper field) were high ( α ’s ranging from .82 to .91).Instructors’ responses suggested that they read researchfindings from various fields. As would be expected, theyreported accessing relevant research from mathematicseducation more than research from other fields. The meanresponse across the three agreement items for mathematicseducationwasbetween‘‘agree’’and‘‘stronglyagree’’( M = 4.46, SD = 0.67),whereasthemeansfortheotherfieldswerearound ‘‘neither agree or disagree’’: educational technology was 3.20( SD = 1.09), cognitive psychology was 3.02 ( SD = 0.98), and developmental psychology was 2.79 ( SD = 0.93).66  Volume 7—Number 1  Elida V. Laski et al. Table 2 Instructors’ Ratings of Importance of Specific Topics to the Preparation of Future Teachers of Elementary Mathematics  PercentageofinstructorsineachratingcategoryInstructionaltopicNotatallimportant(%)Somewhatimportant(%)Important(%)Veryimportant(%)Don’tknow/unsure(%)Meanrating (SD) a Basic cognitive processes (e.g., memory,attention)0.00 15.76 58.15 23.91 2.17 3.08 (.63)Basic learning processes (e.g., encoding,generalization, automaticity)2.15 19.89 51.61 23.66 2.69 2.99 (.73)Cognitive strategies and processes 0.00 5.91 45.70 48.39 0.00 3.42 (.60)Commonmathematicalmisconceptions 0.00 5.91 37.10 56.99 0.00 3.51 (.61)Common developmental sequences inmathematics0.00 4.84 46.24 48.39 0.54 3.44 (.59)The ability to predict differences instudents’ performance/problemsolving based on students’ priorknowledge0.00 15.59 48.92 34.95 0.54 3.19 (.69) a ‘‘Don’t know or unsure’’ responses were set as missing for the calculation of means and standard deviations. Next, we asked participants to rate their familiarity withcognitive researchers who conduct research directly relevantto mathematics education. Half, or more of the instructors,reported being ‘‘not at all familiar’’ with each of theseresearchers’ work. For instance, 72% and 49% of instructorswere not at all familiar with the work of David GearyandRobertSiegler,respectively—twocognitivepsychologistsassigned to the 2008 National Mathematics Advisory Panelbecause of the importance of their work to mathematicseducation.Last, we examined whether instructors tend to access psy-chology findings through primary sources by asking them toratethefrequencywithwhichthey‘‘readorreferencearticlesorinformation’’fromsixwell-respectedcognitive/developmentalpsychology journals whose impact factors ranged from 1.36 to3.77 in 2010. As shown in Table 3, the mean responses for allthese journals fell between ‘‘never’’ and ‘‘seldom.’’ IncorporationIntoElementaryMathematicsMethodsCourse The data indicate that topics prominent in cognitivepsychology studies of mathematics are incorporated to someextent in elementary mathematics methods courses. Whenasked to estimate the percentage of time they allocate tovarious topics within their methods course, instructors’estimates indicated that the greatest proportion of time( M = 25.48%,  SD = 12.95%) is spent discussing specificpedagogical techniques, methods, or activities. However,instructors reported allotting between 11% and 15% of thetotal course time on topics prominent in cognitive studies of mathematics: the developmental sequence of mathematicsconcepts and skills ( M = 13.34%,  SD = 10.20%), cognitivestrategies ( M = 15.13%,  SD = 8.32%), and mechanisms and processes underlying mathematics learning ( M = 11.10%, SD = 6.39%). Similarly, as shown in Table 4, when asked to report how much they emphasized various topics in theircourse on a 4-point rating scale ranging from ‘‘not at all’’to ‘‘to a great extent,’’ mean responses fell between ‘‘tosome extent’’ and ‘‘to a great extent’’ for topics related tocognitive research, with the developmental sequence of mathconcepts and skills being emphasized the most ( M = 3.48, SD = 0.61).The data presented in Table 5 indicates that wheninstructors were asked specifically about the extent to whichtheydiscussedvarioustopicswhenteachingaboutarithmetic,a key topic in most elementary mathematics methods courses(cf.VandeWalleetal.,2009)forwhichthereisagreatdealof relevant cognitive research, mean responses fell between ‘‘tosomeextent’’and‘‘toagreatextent’’fortopicsforwhichthereis relevant cognitive research. Instructors’ mean responses fordomain-general cognitive processes, however, fell between‘‘very little’’ and ‘‘to some extent’’ when they were asked what they would discuss in relation to a hypothetical studentarithmetic error. FactorsRelatedto Instructors’Incorporationof ResearchInto ElementaryMathematics MethodsCourses Toexaminepotentialfactorsrelatedtoinstructors’incorpora-tion of cognitive research into their courses, we constructed acompositevariableforeachofthefourkeyconstructs.Wethenconducted correlation, regression, and mediation analyses toexamine the relations among the four variables. VariableConstruction Variables were constructed from multiple survey items. Thesurvey items were written to tap each of the constructs; Volume 7—Number 1  67
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