8.higher oder thinking via mathematical problem posing tasks among engineering

ASEANJournalofEngineeringEducation,1(1) Ghasempouretal.(2012)

41Ghasempour,Z.,Kashefi,H.,Bakar,M.N.&Miri,S.A.(2012).Higher-Order Thinking via Mathematical Problem Posing Tasks among Engineering Students.ASEANJournalofEngineeringEducation,1(1),4147.

HigherOrderThinkingviaMathematicalProblemPosingTasksamongEngineeringStudents

1ZahraGhasempour,1HamidrezaKashefi,1MdNorBakar,2SeyyedAbolfazlMiri1FacultyofEducation,2FacultyofManagementandHumanResourceDevelopment

UniversitiTeknologiMalaysia

Abstract: This study characterizes engineering students higherorder thinking (HOT) skills through two problem posingstrategies,namely"What...,ifnot?"and"ModifyingGiven".Problemposingtasksweredesignedbasedonintegralconceptsfromcalculustextbookproblems.Thedatawerecollectedthroughstudentsresponsestotestandsemistructuredinterview.Twentysixparticipantsinthetestsessionwereselectedamongmoderateandhighachieversfirstyearengineeringstudents.Afterthetest,5ofthestudentswerepurposefullyselectedforthesemistructuredinterview.TheresultsrevealsthattheHOTskillsviaproblem posing tasks can be characterized into "Interpreting the problem condition and demand in term of mathematicscommunication", "Manipulating information forconstructingnewproblems in flexibilitymethod", "Analysing theconstructedproblem regard to solvable or unsolvable", "Create new and different problem which are solvable", "Conclude a significantpatternorstructure",and"Findingthedifferencesandsimilaritiesbetweentwopartsof tasksstrategies".TheresultsconfirmthatproblemposingtaskspossesallcriteriaofapracticaltaskforenhancingHOTskillsamongengineeringstudents.Introduction

A number of definitions of higherorder thinkinghave been proposed. According to Newman (1991),higherorderthinking isdefinedbroadlyaschallengeand expanded use of themind when a personmustinterpret,analyze,ormanipulateinformation,becausea question needs to be answered. Newman assertedthat critical, logical, reflective, creative thinking andmetacognitive skills can be subsumed under amoregeneral distinction between higher order and lowerorderthinking.Heindicatedthatlowerorderthinkingrepresents routine, mechanistic application, andlimiteduseofthemind.Intheotherword,itinvolvesrepetitive routines such as listing informationpreviously memorized, inserting numbers intopreviouslylearnedformulae,orapplyingtherulesforfootnote format in a research paper. In the contrast,HigherOrder Thinking, orHOT for short, is thinkingon a higher level than memorizing facts, so that itrequires the tasks to be understood, connected toeachother,categorized,manipulated,puttogetherinnew or novelways, and applied as new solutions tonew problems (Thomas, Thorne, and Small , 2000).Despite of the importance of HOTs achievements inan effectiveness education, teaching and learningmethodandmaterialinmathematicsclassrooms,havereminded this phenomena under shadow loworderthinking.

Overthedecades,researchershavefoundthatHOTcan be taught, nurtured, and developed amonglearners.ToachieveHOT,studentsshouldbeinvolvedin understanding and transformation of knowledgewhicharedescribedas theultimategoalsof learning(Bigge,1976).Inthisregard,variouscommunicationshave presented criteria of valid tasks that can fostereducators HOT; however, themost offered activitiesareinapplicableinmathematicsclassroomswhicharelimitedtotextbookcontentandtime.ShuMeiandYan(2005) noted that HOT rarely exists amongmathematics classrooms in Singapore due tomathematics teaching and learning environmentswhich are teachercentred and involve absolutely,routine procedural skills and basic concepts.Certainly, these pedagogy achievements are learners

whohavelittleunderstandingofthebasicconceptsofprecalculus,andeventhebetterstudents,onlyexcelin a procedural way of thinking. Furthermore,researchers (Engelbrecht, Bergsten, and Kgesten,2009)revealedthatthisconditioncanalsoexistintheuniversities mathematics classrooms, because mosttasks and examination tests are consideredprocedural in character, and aremore formal in theconcepts. Therefore, educational experts have takenefforts to design appropriate teachinglearningmaterials and activate which can foster HOT amongeducators.

Weiss(2003)classifiedproblemswhichcanfosterHOT into: Collaborative, Authentic, Ill Structuredproblems, and Challenging problems. According tothis view, themost important criteria for promotingHOTcanbefollowedas: Studentsshouldbeinvolvedinthetransformation

ofknowledgeandunderstanding. Teacher should create a communicating

environment for students effective interaction,encouraging them to verify, question, criticize,and assess others arguments, engaging inconstructing knowledge through variousprocesses, and generating new knowledgethroughselfexploration.

Students need to be aware that theymust be anactive learner taking initiatives andresponsibilitiesintheirownlearning.Shepardson(1993)assertedtotheimportanceof

cognitive engagement in making classroom effectiveactivitiesthatcouldbelinkedtohigherorderthinkingskills. Continually, Anderson and Krathwohl (2001)havedeveloped anewdesign for the classicBloomsTaxonomy as a cognitive domain with acomprehensive set of definitions that appears toencompass all aspects of HOT. In other words, theycreated a workable and valuable way for usingobjectives as tools to promote studentswith lowerlevel thinking into higherlevel thinking by applyingthe hierarchical nature of knowledge, includingRemember,Understand,Apply,Analyze,Evaluate,andCreate.Asaresult, teacherscan inquireapurposefulquestion in regard to revised Blooms taxonomy, to


42

ensure that respondents get beyond the simpleanswersandcanthinkdeeper(Cochran,Conklin,andModin,2007). Inaddition, thesequestionscould leadparticipants from what questions associated withlowerlevel thinking into the how and whyassociated with higherlevel thinking (see AppendixI). Most significantly, recently, arguments haveindirectly confirmed that the mathematical problemposing tasks are able to improve HOT among theverity level of students. Mathematics educationexperts for instance, Chin and Kayalvizhi, 2002 andBonotto, 2008 revealed that problem posing canprovide opportunities which stimulate higher orderthinking by encouraging students to carry outinvestigations, especially openended situations.However, due to the limitation of mathematicscourses in terms of time and subject content, thesekindsofactivitiesareinappropriatetobeusedonthecontinuing basis during the semester. Therefore, anoperationalproblemposingtasksneedtobedesigninordertoimproveHOTskillsamongstudents.

Consequently, the main objective of thiscommunication is characterizing HOT skills that areinvolved in problem posing tasks from "Originaltextbookproblem"via"structuredorsemistructuredproblem posing situations" in "Inquirybasedlearning" environment. We assert that researchfindings can encourage teachers to collaboratemathematicalproblemposingtasksintheirteachinglearning material for equipping undergraduates toHOTskills.

Mathematics Problem Posing in InquiryBasedLearningParadigm

Problemposingcanbedefinedasagenerationofnewproblemsorareformulationofgivenproblemsintermoftransformationofknowledge.Problemposingactivities are considered as a powerful tool forunderstanding concepts , as well as , are designedbased on several sources, such as, "Everyday life"problems,"Originaltextbookproblems","apictureordiagram" and so on. Besides, these tasks could beimplemented in mathematics class through thefollowingsteps: first,studentsshouldunderstandthegivenproblemposingsituations(e.g.,openedproblemposing situation, semistructured problem posingsituation, and structured problem posing situation).Second,theyaregivenanopportunitytodiscusswiththeir peers regarding the meaning of such problemposingsituations.Third,at theendof thediscussion,they are required to create their own problemsrespectively. Fourth, the new problems need to besolved in order to ensurewhether they are solvableproblemsorunsolvableproblems.Lastly,iftheposedproblemsareunsolvableproblemstheteacherwouldthenguidethestudentstoaltertheproblemssothatitwould become solvable problems. Forcefully, thesesignificantprocessesarguethatproblemposingtasks

can provide suitable condition for engaging studentsin specific learning process through InquiryBasedLearning paradigm which stress on socialconstructivismideas.

The most important elements of this theory intermofproblemposingapproachandHOTskillsaresummarizedinfollowingstatements(Bruner,1966): Pedagogyandcurriculumshouldrequirestudents

to work together during solving problems, as aformofactivelearning.Therefore,InquiryBasedLearning reminds that learning should be basedaroundstudentsquestionsandeducationshouldtrend towardnovelattitude that students role isbeyond problem solvers as well as they canbecome skillful in discovering and correctlygenerating problems. When students begincreating their own original mathematicalquestionsandseethesequestionsasthefocusofdiscussion, their perception of the subject isprofoundly altered. Meanwhile, these activitiescouldembedtheminhighstagesofrevisedBloomtaxonomy, namely, analyzing, evaluating andcreatingwhicharelinkedtoHOTskills.Whereas,a suitable learning space cannot be launched,unless the teachers are familiar with theirresponsibilityinclassrooms.

Teachers should be viewed as facilitators oflearning. Due to problem posing definitions as away to exercise real life situations, the tasks inteachers' hands is not only to facilitate learningprocessbuttooptimizeitbyestablishingbalancebetween conceptual understanding andproceduralunderstanding.Furthermore,trainer'srole in guiding the students during thereformation of unsolvable posed problems intosolvableproblemscanbelabeledasexpeditor.Onthe other hand, integrating problem posingactivities inmathematics lessonsenable teachersto identify the level of their studentsmathematical knowledge and the ways thatstudents can lead to a better understanding ofmathematicsconcepts.Toachievethesefacilities,criterianeedtobemadetorepresentandexplainintermofmathematicalperspectives.

Therearethreetypesofrepresentationofhumanknowledge in mathematics, namely, "Enactive","Iconic" and "Symbolic". These notions assert tothe importance of the integration of internalizeknowledge as well as mastery in formal andprecise mathematics' languages in order to becorrectly and deeply involved in mathematicalproblem solving. Most important, thesedemonstrations are a tool for mathematicscommunication which consist of reviewing thepart of material used for constructing a newproduct ,developing a novel thinking situation,thinking of the best strategy to solve tasks by


43

usinghis/herownquestionsthatleadhim/hertothesolution.Inthisstudy,researchersmonitoredHOTskillsin

InquiryBasedLearning environment that canundertake the mentioned elements for occurringtransformation knowledge and understanding viaproblemposingactivates.Method

Tasks were designed based on integral sectionwhichisoneofthefundamentalconceptsincalculus.In addition, "Original Textbook Problems" wereadoptedfrom"Thomas'calculus"(2005)inregardstodefinite integral definition, techniques of integration,fundamental theorem of calculus and their results,applicationofintegrationinfindingvolumeandarea.Meanwhile, the levelofproblemsweremoredifficultthan a routine classroom activity, so that researchercanexplorehowproblemposingtaskswouldengageundergraduates,aswellaswhichtypesof theirHOTskills would be occupied. According to problemposingandactivitiesframeworkassuggestedbyAbuElwan(2002),eachofthetasksconsistsoftwoparts.Part (a) involved students in problem solvingstrategies,aswellasPart(b)problemposing.Besides,What..., if not? (Brown and Walter, 2005) andModifying Given (Bairac, 2005) strategies wereconsidered as problem posing strategies that wereperformed through semistructured and structuredproblem posing situations respectively. Semistructured situations occurwhen students are askedto construct problem similar to the given problems,problemswithsimilarsituations,problemsrelatedtospecific theorems, problems derived from the givenpictures, real life and word problems by usingknowledge, skills, concepts and relationships fromtheir previous mathematical experiences. Besides,structured situations arise an appropriateenvironment for generating problems byreformulatingalreadysolvedproblemsorbyvaryingthe conditions or questions of the given problems(AbuElwan, 1999; Stoyanova, 2003). Table 1illustratestwoexamplesofresearchtasks.

Continually, these tasks were implementedthroughtestsessionandsemistructuredinterviewasquantitative and qualitative approaches respectively.The time for test sessionwas considered 90minuteand thementioned steps in section2were relied onInquiryBasedLearning environment. Furthermore,semistructuredinterviewinterventionswereusedtoidentify more justifications about test's outcomes.During the interview, researchteacher inquired apurposeful questions in regards to the revisedBloom's taxonomy (see Appendix I) as a way toensurethatrespondentscomeupwithbeyondsimpleanswers and were thinking more deeper (Cochran,Conklin, and Modin, 2007). Additionally, these

questions were prompted when clarification isrequestedbyparticipants.Table1:Twoexamplesofproblemposingtasksin

thisstudy1(a) Themarginalcostofprintingaposterwhen posters

havebeenprintedis dollars.Find c(100)c(1)

namelythecostofprintingposters2100. (Note:Themarginalcostisc'(x)aswellasc(x)isthetotalcost.)Conditions:Demand:

1(b) Fistposeproblem:(AddnewconditionstrategyviastructuredSituation)Addnewconditiontoproblem(a)NewConditions:NewDemand: Solve:

1(c) Secondposeproblem:(Removeconditionstrategyviastructuredsituation)Removesomeconditionofproblem(a)NewConditions: NewDemand: Solve:

2(a) Findtheareasoftheregionsenclosedbythecurvesandlinesbelow

i. DetermineproblemsCondittionsexactly.ii. Solve:

2(b) Pose the problem.(changethecontextstrategyvia

semistrucuredsituation)Createaproblemrelatedtotheareasofregionsforgivenfigure

i. Determineproblem'sCondittionsexactly.and

explainhowevaluatenewconditions.ii. Expressnewproblembyformalmathematics

connections.iii. solve:

The respondents were twentysix first yearengineeringstudentsfromvariousfacultiesatIslamicAzad University of Birjand that were involved inanswering the test. Theywere selected purposefullyamong moderate and high achievers who could beexpected to have literacy levels sufficiently tounderstand questions and articulate in their posedquestion processes in regards to theirmark in thefinal exam of a calculus course. Most importantly,theywere firstencountered inproblemposingtasks.After the test session, five of themwere selected forthe semistructured, taskbased interview that wasperformed onebyone (Roulston, 2010). Therefore,the raw data was produced by students written

x

xdxdc

21=

.0,2,1,12 ==== yxxxy


44

works in the test and the transcriptions of audiotapedobtainedthroughinterviews.

Qualitative content analysis was used for dataanalysisviainductivereasoningtocondenserawdatainto categories or themes based on valid inferenceand interpretation (Zhang and Wildemuth, 2009);however, researchers generated codes based on therevised Bloom's taxonomy at the inception of dataanalysis.Hence,thesecodesweredevelopedintermsofthecomparativemethod.

ResultsandDiscussions

Accordingtofindings,morethanthreequartersofparticipants(80.77%)were"Expert"or"Practitioner"in posing problems in given tasks that wereperformed in structured situations. These levels ofperformance can lead teacherresearch to a range ofHOT skills. According to participantswrittenworks,specific HOT skills could be categorized to"Understanding correctly related strategies","Interpreting the problem condition and demand intermofmathematics communication", "Creatingnewconditionintermofformalmathlanguage","Changingthe number of conditions and generating newdemand purposefully", "Analysing the constructedproblemregardtosolvableorunsolvable","justifyingreasonsofnewunsolvableproblemfortransferringtosolvable problem". However, 73% of new conditionsand demands were cosmetic, namely participantschose both "changing the values of the given data",and "changing the number of conditions" forgeneratingproblemsintermof"ModifyingGiven"andperformedthemcorrectly,whilstonly15percentsofthem preferred to use "changing the problem'squestion" by "What..., if not?" strategy through thissituation.

Thesefindingsindicatedundergraduatesabilitiesin procedural mathematics understanding(Marchionda, 2006) related to "techniques ofintegration"and"applicationofintegrationinfindingvolume and area". In other word, undergraduatespresentedarangeofproceduralunderstandingssuchas representing mathematical patterns, structuresandregularities,usingdeductiveargumentsto justifydecisionsfornewposedproblembasedongivendata,and generating new problem via mathematicalconnections in a loworder thinking manner. Mostimportantly, this study founded that undergraduateshad the most difficulties in using "What ...if not?"strategy for generating problems related to"characterizations of integrability concerned withcontinuity"and, "applicationof integration in findingvolumearealobject".Theseresultsassertedthatthemajority of students were unable to interpretcorrectly their answers , or even generalize theagreed situation part (a) via the mathematical termthat it can highlight a weakness in higher level ofBloom 's taxonomy consists of analyzing, evaluating,

andcreating.Asaresult,twocategories"Creatingnewcondition in term of formal math language", and"Changing the number of conditions and generatingnew demand purposefully" occur in a routineregulation related to lowerorder thinking. Hence,teacherresearchers purposefully chose 5 of thestudentswhotheirwrittenworkswerecharacterizedas loworder thinking for encountering in the semistructuredinterview.

The researchteacher tried to encourageparticipants HOT by leading them to generateproblems which are correct and different from theinitialproblemduringtheinterview(seeAppendixI).This method of interviewing was performed as aninstructional practices that focus on sense making,selfassessment, and reflection on what worked andwhat needs improving in term of metacognition.Because metacognition facilitates HOT skills (Kapa,2001) and often takes the form of internal dialogue,whilst many students remain unaware of itsimportance unless teachers emphasize theseprocesses explicitly and guide students towards thereflectionthatneedstooccur(Bransford,Brown,andCocking,2000).Oneexampleofthiscommunicationisasfollows:Interviwer:Nowreadpartbofproblem1 and say thatwhattheproblemaskedyoutodo.Subject3: Itasked toadd something to the conditionsof theproblem...Iwillchangethedataintheproblem.ThatisIwillput144insteadof100and1insteadof9.Interviwer: Can youwrite your own problem in the accuratemathlanguagelikethepreviouspart?Subject 3: The marginal cost of printing a poster when postershavebeenprinted is

dollars.Find thecostof

printingposters10144.Interviwer:Youjustchangedthedatainrepetitivemanner,canconstructanotherproblemwithsignificantchanging?Subject3:ummm...Cangiveanexample?...Interviwer: You can change the marginal cost formula to

Subject3:ummm ... Iwould like tochange it to 12 += x

dxdc

and add an expense for operation like 1000$. Find the cost ofprintingposters10144.(HOTskills)(Time:5min)

On the other hand, in "What..., if not" strategy,more than half of participants (68.31%) were"Novice" or "Apprentice", this means that they hadbasicdifficultiesinproblemposingprocessrelatedtothealteringofconcretesituationsuchasreallifetasksto a mathematical abstraction by symbolicexpressions. In addition, these levels of performancecan clarify a weakness in conceptual understanding.Therefore, written test works related to these taskswere unreliable for coding; however, prior studiesreported that "change the context" can engagelearners thinking more than structured situations

x

xdxdc

21=

xdxdc

21=


45

(AbuElwan, 2002). Consequently, HOT skills via"change the context" strategies were sought duringthe interviewsession.Themost repetitiveHOTskillswere found in the following categories: "Determinenot given information by comparing two part (a),(b)", "Designamathgraph related toa real subject","Justify and support decisionsmade and conclusionsreachedbydrawingagraphrelatedtoamathematicaltheory", "manipulating information for constructingnew problems in flexibility method", "Generate theproblems in yours words regard to formal mathlanguage", "Create new and different problemwhichare solvable", "Conclude a significant pattern orstructure", "Finding the differences and similaritiesbetweentwopart(a),(b)"and"Checkingtheanswerbysolvingnewposedproblem".Meanwhile,researchteacher stimulates HOT skills by questions andstatements which are presented in Appendix I;furthermore, when clarification was requested, sheguidedrespondsbyrelevantquestions,writingsomenoteandexample.Oneexampleofthisinterviewisasbelow:

Interviewer:Whatwasyourdifficultyinposingthisprobleminpartb?Subject 3: I can't recall the definition of defined integral by canyougivemetheformula.Interviewer: (write:

,read

again thequestioncarefullyandsay:Whatare theconditionsandthedemandsofthisproblem? Subject3:MakeaLimitproblemforthegivenintegral. Interviewer: determinethedataandtheinformationgiven.Subject3:(referredtonotegiven:

)Yougavethelefthandpartoftheequalityandaskedustoobtaintherighthandpart.

Interviewer:Soifyouwanttoconstructaproblem,whatshouldyoudetermine? Subject3:a,b&f,and.(HOTskill)Interviewer: Itmeansastatementinfrontof. Subject3:(wrote:a=1,b=2,1

Interviewer:SowhathappenstotheLimityouneedtomaketheproblem? Subject3:(wrotelim

1

)

Interviewer: Doyouhavefunctionf? Subject3:yes (Wrote:lim

1

).(HOTskill)

Interviewer:Now,constructedtheprobleminyourwords.Subject 3: Shall I construct a problem for lim

1

23orfor133?

Interviewer:Suppose thatyouarea teacherandyouwant toconstructa limitproblem for your students to change it to thedefiniteintegralusingtheformuladefiniteintegral.Sowhatwillhappentoyouyourquestion? Subject 3: Solve the following limit using the definition ofintegral.(HOTskill)(wrotethissentenceaboveyourformula.)(Time:8min)

Conclusion

Problem posing tasks demand thinking on ahigher level thanmemorizing facts,namely the tasksrequire the students to understand the problemcontextexactly, informationconnected toeachother,categorized, manipulated, put together in new ornovel ways, and applied as new solutions to newproblems, so that, they can encourage highorder thinking among learners. Specially, problem posingtasks from "Original textbook problem" whenimplemented through the structured and semistructured problem posing situations can besuggested as practical activities in mathematicsclassroom , whereas, they encompass subjectconstant and can be performed at a standard time(SaintLouisUniversity,2012).

In addition, the most important HOT skills viathesetypesoftasksbasedonNewman(1991)canbecategorized into "Interpreting the problem conditionanddemandintermofmathematicscommunication","Manipulating information for constructing newproblems in flexibility method", "Analysing theconstructed problem regard to solvable orunsolvable","Createnewanddifferentproblemwhichare solvable", "Conclude a significant pattern orstructure", "Finding the differences and similaritiesbetween two part (a), (b)".Meanwhile, teacher, as aguide should turn pupils from loworder thinkingtowards highorder thinking associated with askingthehowandwhyquestionsintermofthelevelsofthe revised Bloom's taxonomy, when teachers applythese questions in a continual manner, selfquestioning as a metacognition skill can graduallyimprove in learners which will be conducive topromotingtheirHOTskills(Kapa,2001).Forinstance,"How would you construct a newcondition/demand?", "Howdoyou justifythe logicoftheanswer?","Whatchangeswouldyoumaketosolvenew posed problem?" and "Can you generate theproblemswithyourwordsbasedontheformalmathlanguage?". However, based on the results of thisstudy,itisrecommendedthatresearchersinvestigatetheeffectsofmathematicalproblemposingactivitiesmentionedonundergraduateshigherorder thinkingskills in terms of a survey with quantity approach.RamirezandGanaden(2008)pointedoutthat,despitetheir assumption, the chemistry tasks with creativeactivities is not significantly different from theinstructionswithnocreativeactivitiestoimprovethehigherorderthinkingskillsofstudents.

Consequently,weexpectthisstudytobeabletoencourage teachers to move toward novelperspectives of classroom activities by collaboratingproblem posing and highorder thinking approachesthroughfacetofaceclassroominteractions.ReferencesAbuElwan, R. (1999). The Development of MathematicalProblem Posing Skills For Prospective Teachers. In A.Rogerson (Ed.), Proceedings of the International Conference


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on Mathematics Education into 21st Century. Socialchallenges,Issues,andApproaches(pp.18).Cairo,Egypt.

AbuElwan,R.(2002).EffectivenessofProblemPosingStrategieson Prospective Mathematics Teachers Problem SolvingPerformance.SultanQaboosUniversity.

Anderson, L. W., and Krathwohl, D. R. (Eds.) (2001). Ataxonomy for learning, teaching,andassessing:A revisionofBlooms taxonomy of educational objectives. New York:Longman.

Bairac,R. (2005).SomeMethods forComposingMathematicalProblems. International Journal for Mathematics Teachingand Learning, Available online at:http://www.cimt.plymouth.ac.uk/journal/bairac.pdf.

Bigge, M. L. (1976). Learning Theories for Teachers (3rd ed.).NewYork,NY:Harper&Row.

Bransford, J. D., Brown, A., and Cocking, R. R. (2000). HowPeople Learn: Brain, Mind, Experience and School.Washington,D.C.:NationalAcademyPress.

Bonotto, C. (2008). Realistic Mathematical Modeling andProblemPosing.UniversityofPadova.

Brown,S., andWalter,M. I. (2005).TheArtofProblemPosing(3rded.).Mahwah,NJ:LawrenceErlbaumAssociation.

Bruner, J.S. (1966).TheProcessofEducation.Cambridge,MA:HarvardUniversityPress.

Chin, C., andKayalvizhi, G. (2002). Posing Problems for OpenInvestigations:What Questions Do Pupils Ask?,Research inScience&TechnologicalEducation,20(2),118.

Cochran,D.,Conklin, J., andModin, S. (2007) .ANewBloom:Transforming Learning. Learning and Leading withTechnology.34(5):2225.

Engelbrecht, J., Bergsten , C., and Kagesten, O. ( 2009) .Undergraduate Students' Preference for Procedural toConceptualSolutionstoMathematicalProblem.InternationalJournalofMathematicalEducationinScienceandTechnology.40(7):927940.

Kapa,E.(2001).AMetacognitiveSupportduringtheProcessofProblem Solving In a Computerized Environment.EducationalStudiesinMathematics,47,317336.

Marchionda, H. (2006). Procedural and ConceptualUnderstanding of Fractions and the Effects of InquirybasedLearning.ClemsonUniversity.

Newman, F. M. (1991). Promoting Higher Order Thinking inSocial Studies: Overview of A Study of 16 High SchoolDepartments. Theory and Research in Social Education.19:324340.

Ramirez,R.P.B.,andGanaden,M.S.(2008).CreativeActivitiesand Students Higher Order Thinking Skills. EducationQuarterly.U.P.CollegeofEducation.66(1):2133.

Roulston,K.(2010).ReflectiveInterview:AGuidetoTheoryandPractice.London:Sage.

Saint Louis University. (2012). Success in Mathematic Tips.Academic Tips. Org. 20 July 2012, Retrieved fromhttp://www.academictips.org/acad/mathematics/successinmathematics.html.

Shepardson,D.P.(1993).PublisherbasedScienceActivitiesofthe 1980s and Thinking Skills. School Science andMathematics.93(5):264268.

Shu Mei, Y., and Yan, Z. (2005). HigherOrder Thinking inSingapore Mathematics Classrooms. Centre for Research inPedagogy and Practice: National Institute of Education,Singapore.

Stoyanova, E. (2003). Extending Students Understanding ofMathematics via Problem Posing. Australian MathematicsTeacher,59(2),3240.

Thomas, A., Glenda, T., and Small, B. (2000). Higher OrderThinkingIts HOT! Centre for Development and Learning.Louisiana.1(3).

Thomas,G.B.,Weir,M.D.,Hass,J.,andGiordano,F.R.(2005).Thomas' Calculus. United States of America: PearsonEducation,Inc.

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APPENDIXI:Categoriesofsemistructuredinterviewquestionsbasedoncognitiveprocessdimension

CognitiveProcessDimension

Questions/Directives

Remember Canyourecall"definiteintegraldefinition","techniquesofintegration","fundamentaltheoremofcalculusand

theirresults","applicationofintegrationinfindingvolumeandarea"....?Understand

Whataretheconditionsofthissectionofproblem?Whatarethedemandsofthissectionofproblem?Canyoudeterminethedataandtheinformationneedtobefound?

Apply

Canyouplannewproblems?Knowexactlywhattodonext.Ifyouareunsurely,dowhateverthatseemslogicalthroughreasoning.Determinethesubgoal(s).Improvetheplan(anotherway).NeedtoarrangetheinformationCanyoucarryouttheproblemposingstrategy?Howwouldyouconstructanewconditionby"changingthecontext"strategiesforposinganewproblem?Howwouldyouconstructanewdemandby"changingthecontext"strategiesforposinganewproblem?Whatisnewconditionthatyouwouldaddorremovetopreviouscondition?Whatisnewconditionthatyouwouldaddorremovetopreviouscondition?Howwouldyouevaluateyourquestion/Cansolvenewproblem?

Analyze

Howwouldyouanalyzethesituationinmathematicalterms,andextendpriorknowledgepresentedinpart(a)?Canyouidentifythatnewproblemissolvableorunsolvable?Whatisthemeaningoftheanswer?Howdoyoujustifythelogicoftheanswer?Putintheunitstounderstandthemeaning.Concludeasignificantpatternorstructure.Ifyoufurtherreading,canfindsomeanotherclues?Comparegivendatapart(a)and(b)

Evaluate

Howwouldyoujustifyandsupportdecisionsmadeandconclusionsreachedbydrawingagraphrelatedtoamathematicaltheory?Simplylookbackagain(recap).Checkingthelogicoftheequationarrangement.Checkingtheanswerbyinterpreting.Readingtoseeifthegoalisachievedasrequiredbythequestion.Checkingtheplan/analysis.Checkingthesteps,gobackanddoagain.Anotherwayofcalculationtocheck.

Create

Canyoudesignamathgraphrelatedtoarealsubject?Canyougeneratetheproblemsinyourswordsregardtoformalmathlanguage?Whatchangeswouldyoumaketosolvablenewposedproblem?Canyouconstructmorethan1newproblem?

8.higher oder thinking via mathematical problem posing tasks among engineering

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