Scientific Inquiry in Mathematics - Theory and Practice

142 Pages·2018·3.86 MB·English

by  Andrzej Sokolowski

Andrzej Sokolowski
Scientifi c Inquiry
in Mathematics
- Theory and
Practice
A STEM Perspective
fi
Scienti c Inquiry in Mathematics - Theory
and Practice
Andrzej Sokolowski
fi
Scienti c Inquiry
in Mathematics - Theory
and Practice
A STEM Perspective
AndrzejSokolowski
DivisionofMathematicsandScience
LoneStarCollege
Tomball,TX,USA
ISBN978-3-319-89523-9 ISBN978-3-319-89524-6 (eBook)
https://doi.org/10.1007/978-3-319-89524-6
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Preface
Albert Einstein wondered, “How can it be that mathematics,
beingafterallaproductofhumanthoughtwhichisindepen-
dentofexperience,issoadmirablyappropriatetotheobjects
ofreality?”1
January27,1921,addresstothePrussianAcademyofScience,
Berlin.
Objective of the Book
Agapbetween problem-solvinginmathematics andscientificinquiryinscienceis
beyondcontroversy.Attemptstoelevatethegaparemadeandthisbookaspirestobe
one of such attempts. The book is to promote transdisciplinary STEM learning
experiences that support the thesis that exploring mathematics concepts using
scientific methods can help to merge the two methodologies. Furthermore, it is
hoped that such learning settings are to develop students’ mathematical reasoning
skillsandserveasameanstoimprovestudents’STEMreadiness.
The objective of this book is to propose a theoretical framework and
multidisciplinary modeling activities of what STEM learning in the twenty-first
centuryclassroommightlooklike.
This book draws on a diverse literature from international STEM education
community as well as from engineering, science, and mathematics education com-
munities. It synthesizes the research findings that lead to formulating a theoretical
framework on how to develop students’ mathematical reasoning while simulta-
neously exercise scientific inquiry. Several case studies were designed to test the
frameworkandtheirfindingsweresummarized.Thesecasestudiesincludedetailed
instructional supports that were to guide students through the process of merging
1Einstein, A. (1914). Principles of theoretical physics, inaugural address before the Prussian
AcademyofSciences,1914.ReprintedinEinstein,A.(1973).Ideasandopinions(pp.221–223).
London:SouvenirPress.
v
vi Preface
mathematicalandscientificreasoningsinacoherentSTEMinquiry.Thebookcanbe
consideredasaresourceforSTEMeducationstudents,researchers,andpractitioners
seekingtodeveloptransdisciplinarylearningexperiences.
Structure of the Book
The book consists of two main parts. Part 1 contains six chapters and discusses
underpinnings of STEM experiences that lead to a formulation of an integrated
theoretical framework. Part 2 comprises four chapters with four STEM projects
designatedforprecalculusandcalculusstudentsduringwhichthetheoreticalframe-
workwasputinpracticeandtheoutputsdiscussed.
In Chap. 1, general foundations of integrated learning are discussed, and an
argument that STEM students’ readiness can be initiated from developing their
mathematicalreasoningskillsusingscientificcontextsbeforeengaginginengineer-
ing designs is posited. Chapter 2 delves more in depth into the current findings on
integratedscienceandmathematicslearningandestablishesthislearningsettingasa
viable foundation for developing students’ STEM competency. STEM learning is
stronglysupportedbyrepresentations.Chapter3isdedicatedtodiscussinglearning
effectsofusingrepresentationsinschoolpracticeandtheireffectsonSTEMpractice.
While there are various ways of designing and performing STEM activities when
representationsareenabled,modelingisbeingusedinalloftheSTEMcomponent
disciplines.Chapter4zoomsintotheunderlyingprinciplesofmodelinginbiology,
physics,chemistry,mathematics,andengineering.Italsodiscusseswaysofintegrat-
ing technology in STEM learning environment. Chapter 5 focuses on synthesizing
researchfindingsonusingscientificmethodsinSTEMpractice.Theultimategoalof
this chapter is to seek ways of bridging problem-solving in mathematics with
scientificmethods.AculminatingphaseofPart1ofthebookisChap.6thatsuggests
atheoreticalframeworkformergingmathematicalreasoningwithscientificmethods.
Intertwiningofthesemethodologiesalongwithapplyingitinpracticeisdiscussed.
Part 2 of the book opens up with Chap. 7 that contains a case study about
constructing an exponential model for a bouncing ball and using it to model the
lawofconservationofenergy.Extractingpropertiesofanexponentialbehaviorfrom
thatcontextrevealednewknowledgeaboutinterpretationofthebaseofexponential
decay that does not parallel with its traditional view. Chapter 8 is about using the
ideaoffunctioncontinuitytosupportconstructingapiecewisepositionfunctionof
the simulated motion of a walking man. The fact that position function must be
continuous and differentiable is not emphasized in calculus nor physics textbooks.
Applying the principle of continuity in real life highlighted the importance of
studyingtheprinciplesandrevealedinterpretationsthatdonotsurfaceintraditional
context-free textbook problems. The concept of functiontransformations used as a
tooltoproducenewfunctionsbasedonparentfunctionwasexploredinChap.9.By
beingsituatedinasimulatedenvironmentofprojectilemotion,thisactivityrevealed
that parent function does not necessarily have to be expressed in its traditional
Preface vii
standard form. The simulation also disclosed certain limitations of applying func-
tionstransformationinrealcontextsthatdonotsurfaceintypicaltextbookquestions.
Investigating function rate of change and using it to optimizing area enclosed by a
perimeter of a fixed length was the primary objective of the activity included in
Chap. 10. While traditional textbook problems ask for unique values that optimize
thequantityofinterest,thisactivityofferedstudentstheopportunityofconstructing
functions, exploring their properties and then optimizing the quantity while
reflecting on the conducted lab. General conclusions and suggestions for further
researchconcludeeachchapter.
Tomball,TX AndrzejSokolowski
Contents
PartI ScientificandMathematicsReasoninginSTEM:
ConceptualFramework
1 STEMEducation:APlatformforMultidisciplinary
Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 STEMLearningDesigns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 LearningOutcomesAchievedUsingSTEM
LearningSettings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 SuggestedPathwaytoDevelopStudents’STEM
Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 EnhancingScientificRouteinSTEMLearning
Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 IntegratingMathematicsandScienceWithinSTEM
Paradigm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 MathematicalReasoningandScientificInquiry. . . . . . . . . . . . . . 11
2.2 MathematicsasaToolforQuantifyingScientific
Phenomena.. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . .. . . . . .. 12
2.3 MergingMathematicalReasoningwithScientific
Contexts. . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 14
2.4 ScientificMethodsandInquiryinSTEMLearning
Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 TeachingandLearningRepresentationsinSTEM. . . . . . . . . . . . . 21
3.1 RepresentationsasaMeansofSupportingLearning. . . . . . . . . . 21
3.2 RepresentationsinMathematicsandScience. . . . . .. . . . . . . . .. 23
3.3 ActiveLearningandRepresentations. . . . . . . . . . . . . . . . . . . . . 24
3.4 DualChannelofKnowledgeProcessing. . . . . . . . . . . . . . . . . . . 24
3.5 HumanMemoryandItsCapacity. . . . . . . . . . . . . . . . . . . . . . . . 25
ix
x Contents
3.6 TheEffectsofInternalRepresentationson
KnowledgeAcquisition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 ModelinginSTEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 ModelingandConstructionofNewKnowledge. . . . . . . . . . . . . 29
4.2 ResearchandModelinginBiology. . . . . . . . . . . . . . . . . . . . . . 30
4.3 ModelinginChemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 ModelinginPhysicsEducation. . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 ModelinginMathematicsEducation. . . . . . . . . . . . . . . . . . . . . 34
4.6 ModelinginEngineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.7 TechnologyinSTEMModeling. . . . . . . . . . .. . . . . . . . . . . .. . 38
4.8 SynthesisofModelingTechniques. . . . . . . . . . . . . . . . . . . . . . . 39
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 SurveyoftheFieldofEmpiricalResearchonScientific
MethodsinSTEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 FormulatingCriteriaforLiteratureReview. . . . . . . . . . . . . . . . . 43
5.2 SynthesisofGeneralFindings. . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.1 FindingsofHowStudentsFormulate
Hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.2 InquiryTypesUsedDuringSTEMProjects. . . . . . . . . . 46
5.2.3 ConcernsAboutExtractingVariablesand
ModelFormulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.4 ConcernsAbouttheValidationProcess. . . . . . . . . . . . . 49
5.2.5 InterfaceofProblem-SolvingandModeling
inSTEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 FormulatingConceptualFrameworkforMultidisciplinary
STEMModeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.1 FrameworkforMultidisciplinarySTEMLearning. . . . . . . . . . . . 53
6.1.1 InquiryTypeandStudents’Reasoning
SkillsDevelopment. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.2 CriteriaforContextsSelection. . . . . . . . . . . . . . . . . . . 56
6.1.3 ChallengeswithProblemStatement
andHypothesisFormulation. . . . . . . . . . . . . . . . . . . . . 57
6.1.4 Analysis,Generalization,andAlgebraic
ModelEliciting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 Teacher’sGuidanceDuringMultidisciplinary
STEMActivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.3 SequencingSTEMModelingActivitiesWithin
MathematicsandScienceCurricula. . . . . . . . . . . . . . . . . . . . . . 60
6.4 GeneralDescriptionoftheProposedSTEMActivities. . . . . . . . . 61
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Contents xi
PartII ScientificandMathematicsReasoningin
STEMPractice
7 ModelingwithExponentialDecayFunction. . . . . . . . . . . . . . . . . . 65
7.1 PriorResearch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 AnalysisofPretestFindings. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.3 IntroducingExponentialModelUsingSTEMContexts. . . . . . . . 68
7.3.1 TheDifferenceBetweenRatesandRatios. . . . . . . . . . . 69
7.3.2 BuildingContextualBackgroundtoElicit
anExponentialModel. . . . . . . . . . . . . . . . . . . . . . . . . 69
7.3.3 LogisticsoftheLab. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.4 LabOutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.4.1 LabProcedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4.2 DataAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4.3 ModelVerifications. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.4.4 IntegratingaPhysicsComponent. . . . . . . . . . . . . . . . . 76
7.5 LabReflections. . . .. . . . . .. . . . . . .. . . . . .. . . . . .. . . . . .. . 77
7.6 PosttestAnalysisandGeneralDiscussion. . . . . . . . . . . . . . . . . . 77
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 ExploringFunctionContinuityinContext. . . . . . . . . . . . . . . . . . . 83
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.2 PriorResearchFindings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.3 FormulatingaContextualFramework. . .. . . .. . . . .. . . .. . . .. 85
8.3.1 ReviewofConditionsforContinuityUsing
FunctionSymbolicForm. . . . . . . . . . . . . . . . . . . . . . . 85
8.3.2 UsingtheConditionsforContinuitytoFormulate
PiecewisePositionFunction. . . . . . . . . . . . . . . . . . . . . 87
8.4 OrganizingtheLab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.4.1 GettingFamiliarwiththeSimulation. . . . . . . . . . . . . . . 90
8.4.2 LogisticsoftheLab. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.5 LabOutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.6 PosttestAnalysisandGeneralConclusions. . . . . . . . . . . . . . . . . 96
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9 ApplyingFunctionTransformationstoModel
DynamicSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.1 PriorResearch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.2 TheLabObjectivesandPedagogicalBackground. . . . . . . . . . . . 103
9.3 MergingMathematicalUnderpinningsandthe
ScientificContext. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.3.1 DiscussingandFormulatingtheParent
Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.3.2 FormulatingAlgebraicExpressionsfor
TransformedTrajectories. . . . . . . . . . . . . . . . . . . . . . . 107
9.3.3 LabLogistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109