THE TRIANGULATION ALGORITHMIC: A TRANSFORMATIVE FUNCTION FOR . - Ed

RESEARCH PAPERS“Research Trichotomy” and applies it to the qualitative anddeveloped, designed, and ultimately implemented.quantitative analysis of the affective domain of learning.AlgorithmicsThe Psychometrics of Trichotometric AnalysisAlgorithmics is the mathematics of program construction. AOne of the most challenging areas of research inSoftware Engineer conducts program developmenteducation involves the construction of specific instrumentsthrough systemic, cyclic, and sequential trial and error. Theythat are designed to measure qualitative outcomes andcarefully observe a formal and tedious methodology todata. Although there are a great many measurement toolscarefully construct an application. This same process canthat analyze the cognitive and psychomotor domains,be used by an Educational Technologist to evaluatethere remains a vacuum in the number of instrumentseducational software. The elements of this process can beespecially designed to accurately measure the affectiveobserved holistically in the following model displayed indomain (the learning domain that contains attitudes,Figure 1.opinions, emotions, perception, and perspectives). ThisThis same model can be used to evaluate educationalvoid is further expanded when the specific event undersoftware. A research design is then needed to develop aninvestigation is unique, specialized, has specificinstrument to test the software based upon thecharacteristics, serious legal constrictions, and issuesaforementioned model. Thus, the Triangulation Algorithmicregarding time. This often requires the research investigatorModel is provided and defined to aid in understanding theto design an instrument that ideally measures the variablesprocess of measurement instrument construction andunder investigation.research design.The process of designing instruments for the purposes ofThe Triangulation Algorithmic Model for the Tri–Squaredassessment and evaluation is called “Psychometrics”.TestPsychometrics is broadly defined as the science ofpsychological assessment (Rust&Golombok, 1989). TheTri–Squared Test pioneered by the author, factors into theThe Algorithmic Model of Triangulation is of the form (Figure 2).Where,research design a unique event–based “InventiveVertex a Ða “authoring” The Initial Tri–SquaredInvestigative Instrument”. This is the core of theInstrument Design;Trichotomous–Squared Test. The entire procedure isVertex b b “building” The Tri–Squared Qualitativegrounded in the qualitative outcomes that are inputted asInstrument Responses; andTrichotomous Categorical Variables based on the InventiveVertex c Ðc “conveying” The Final Tri–Squared TestInvestigative Instrument. The specific assessment of theOutcomes in a Quantitative Report.variables is completely dependent upon the outcomesdetermined by the researcher's instrument. The creation,production, and deployment of the Inventive InvestigativeInstrument requires that the research investigator adoptsthe role of a “Trichotomous Psychometrician” or“Trichotometrician”. A “Trichotomous Psychometrician” isan Educational Scientist that uses trichotomous–basedpsychometrics to develop a qualitative InventiveInvestigative Instrument specifically designed capturequalitative responses during a specific event. A descriptionThus, the Triangulation Model is symbolized by a RightTriangle written as: “is simplified into the mathematic geometric expression:c(meaning “Triangulation Model abc” or more simplyThe Software Engineering ModelVariable a “Author”(Inquiry)Variable b ics [Modeling]Efficient [Effective]of the entire Tri–Squared research process follows and isSustainable [Lasting]described in detail so that the reader is informed ofLogical [Practical]precisely how an Inventive Investigative Instrument is48“. This symbol called “Trine” (meaninga group of three) is written mathematically as “ abc” and4444Variable c “Convey”(Outcome)PracticalBuildable [Construction]Correctable [Error-Free]Stable [Reliable]Recycle [Re-Usable]Figure 1. The Holistic Model of the EntireSoftware Development Processi-manager’s Journal of Educational Technology, Vol. 10 lNo. 1 lApril - June 2013

RESEARCH PAPERSaIn terms of Vectors, the Triangulation Model Right Trianglebsis equal to three vectors that illustrate the movement indirection and magnitude from one completed task intoanother. The entire process is both cyclical and sequentialwith a “Trine Vector Equation” written as: [ x] [ y] [ z]Defined as Trine “Concentration of Vector x intoConcentration of Vector y into Concentration of Vector z”,cwhich is simplified into a more standardized Trine VectorFigure 2. Algorithmic Model of TriangulationEquation form written as:“Trine abc”) (Figure 3). The angles have the following angular measurements inWhere vectors x, y, and z respectively are indicated on thedegrees: Ða 36.86, b 90, and Ðc 53.14, that all“Algorithmic Triangulation Data Model” as shown in Figure 5.add up to the standard 180 of a traditional triangle (36.86 90 53.14 180 ). The connective points (i.e., the linesbetween points a, b, and c respectively are geometric“vectors” (lines with both size [magnitude] and direction)making the model a systemic or cyclic process. This isillustrated in terms of Cartesian Coordinates as shown inIndicating that the standardized form of the vectors x, y,and z are equivalent to the following geometric vectorsthat are the sequential Cartesian Coordinates relative tothe size and magnitude of the research engineeringphases that sequentially connect the respective anglesÐa, b, and Ðc. This is written as follows:Figure 4. ba 9036.86 The Complete Tri–Squared Algorithmic TriangulationModelThe three numeric Vector Operational Phases of the53.14Triangulation Model are defined in the following manner: The entire Triangulation Model as a Research Engineeringmethodology begins with a breakdown of the Trine []Operational Research Engineering Parameters andcFigure 3. Triangulation Model or “Trine abc”bGeometric Vectors in the following manner:bayxazccFigure 4. Cartesian Coordinatesi-manager’s Journal of Educational Technology, Vol. 10 lNo. 1 lApril - June 2013Figure 5. Algorithmic Triangulation Data Model49

RESEARCH PAPERSthe Resulting Outcome Output Variables b1, b2, and b3“X” [4]“b” [build]“a” [author]respectively).“Y” [3]OutputInput·a2 Section Two of the Research Instrument.ReltsuConstructed from the second series of instrument items (d.“Z” [5]through f.) derived from the research investigationquestions as the Qualitative Trichotomous CategoricalVariables (as the secondary Investigation Input Variables),“C” [convey]Figure 6. Complete Tri–Squared Algorithmic Triangulation ModelThe Tri–Squared Triangulation Model ResearchEngineering Process (highlighting the Operationalevaluated via the Qualitative Trichotomous Outcomes (asthe Resulting Outcome Output Variables b1, b2, and b3respectively); and·a3 Section Three of the Research Instrument.Parameters and Phases of the Tri–Squared Test)Geometric Vertex a Ða “authoring” The InitialTri–Squared Instrument Design Operational ParameterConstructed from the third series of instrument items (g.through i.) derived from the research investigationquestions as the Qualitative Trichotomous Categorical“a” “author” absolute value of a “modulus a” ½a½Variables (as the tertiary Investigation Input Variables), “Trine a” The creation of the Tri–Squared Inventiveevaluated via the Qualitative Trichotomous Outcomes (asInvestigative Instrument. This process can be seen in thethe Resulting Outcome Output Variables b1, b2, and b3following model: (Figure 6).respectively);This in turn, leads into [ ] vector x Geometric Vector x “ The following table provides the metrics for the constructionx” The Initial Tri–Squared Instrument Constructionof the Inventive Investigative Instrument following the Operational Phases “x” absolute value of vector x parameters indicated in phases 1. through 4. of the first“norm x” ½½x½½ “Trine x” The creation of thevector [ 4] of the Triangulation Model, (Table 1).Tri–Squared Inventive Investigative Instrument TheThis Tabular Triangulation Model can be fully exemplified inPythagorean Triple of the Triangulation Model “4” The 4the aforementioned example provided by Table 1.Phases of Tri–Squared Inventive Investigative InstrumentConstruction which is composed of the following 4Operational Phases: (Figure 7).a0 ·a0 The Instrument Name (Asset Security Optional);a1 ·a 1 Section One of the Research Instrument.Inventive Investigative Instrument—Name—[Optional Asset Security]Section 1. Research Question One. The First Series of Questions fromthe Qualitative Trichotomous Categorical Variables are listed belowb1b2Responses: [Select only one from the list.] ub3Constructed from the first series of instrument items (a.a.Item One through c.) derived from the research investigationb.c.Item TwoItem Three questions as the Qualitative Trichotomous CategoricalVariables (as the Initial Investigation Input Variables),a2 evaluated via the Qualitative Trichotomous Outcomes (asa13x3ya2a3b1b2b3Figure 7. Tri–Squared Triangulation Model ResearchEngineering Process50a3 Section 2. Research Question Two. The Second Series of Questionsfrom the Qualitative Trichotomous Categorical Variables are listedbelowb3b1b2Responses: [Select only one from the list.] ud.Item Four e.f.Item FiveItem Six Section 3. Research Question Three. The Third and Final Series ofQuestions from the Qualitative Trichotomous Categorical Variablesare listed below.b3b1b2Responses: [Select only one from the list.] ug. Item Seven h.i.Item EightItem Nine Table 1. Inventive Investigative Instrument for ResearchQuestions 1, 2 & 3i-manager’s Journal of Educational Technology, Vol. 10 lNo. 1 lApril - June 2013

RESEARCH PAPERSThe Osler-Waden 9th Grade Academies, Centers, andCenter Models Assessment Instrument A.Has the 9th Grade Academy, Center, or Center Model been:YesNoMissing 1. Successful?a0 The Osler-Waden 9th Grade Academies, Centers, andCenter Models Assessment Instrument [with Asset Security ]a1 Has the 9th Grade Academy, Center, or Center Model been:Responses: [Select only one from the list.] ub1 Yes b2 No2. Made a Difference? a. 13. Aided in Retention? a. 2a. 3How as the Academy/Center been successful, made a difference,or aided in retention, if at all?YesNob3 MissingSuccessful? Made a Difference?Aided in Retention? a2 Did the 9th Grade Academy, Center, or Center Model Result in the following:Missing4. Positive Impact? 5. Active Participation? 6. Decline in Dropout Rate? How as the Academy/Center been positive, aided in participation,or decreased the dropout rate, if at all?How did the 9th Grade Academy, Center, or Center Model have an impacton the following:YesNoMissing7. Positively Effect Standardized Testing? 8. Increase Graduation Rate? 9. Increase Attendance? How as the Academy/Center positively affected testing, graduation rates,and attendance, if at all?How long has the model/program (freshman/Ninth Grade Academy beenoperation in your school?How long has the interviewee (yourself) been (working) there and what is thelevel of his or her involvement in the program such as Assistant Principal,Principal, teacher or other staff member?What is the role of those interviewed during the interviews, and theirknowledge of the program, whether their knowledge was mediumknowledge low level of knowledge etc.This is defined in the first Operational Geometric Vector (see Figure 3)thTable 2. The Osler-Waden 9 Grade Academies, Centers, andCenter Models Assessment Instrument(Courtesy Osler-Waden, 2012)Thus, Table 2 is provided again (in an enlarged form) toillustrate a sample Inventive Investigative Instrument:This is defined in the first operational geometric vector.Responses: [Select only one from the list.] ub1 Yes b2 Noa. 4b3 MissingPositive Impact? a. 5Active Participation? a. 6 Decline in Drop Out Rate? a2 How did the 9th Grade Academy, Center, or Center Model have animpact on the following:a. 7 Positively Effect Standardized Testing? a. 8 Increase Graduation Rate? a. 9Increase Attendance? thTable 3. The Osler-Waden 9 Grade Academies, Centers, andCenter Models Assessment Instrument (illustrating theinstrument variables, items, and overall design).(Courtesy Osler-Waden, 2012).This is immediately followed by (Table 3).Geometric Vertex b b “build” The Tri–SquaredQualitative Instrument Responses OperationalParameter “b” “build” absolute value of b “modulusb” ½b½ “Trine b” The effective deployment of theTri–Squared Inventive Investigative Instrument to elicitresponses and aggregate the response data. This processcan be seen in the following model (Table 4).A Standard 3 3 Tri–Squared Table of ComprehensiveInputted Qualitative Research Responses would resemblethe following model represented by Table 5. Where, Tn Where, the Operational Geometric Vector phases areThe Total Number of Responses (based upon a one to onedefined respectively as:ratio of each Trichotomous Testing Input Variablestha0 The Osler–Waden 9 Grade Academies, Centers, andCenter Models Assessment Instrument ;tha1 Has the 9 Grade Academy, Center, or Center Modelcontrasted directly with each Trichotomous Results OutputVariable) in each of the individual cells of the Standard 3 3 Tri–Squared Table.been;a2 Did the 9th Grade Academy, Center, or Center ModelResult in the following;b1b2b3tha1a2a3a1b1a1b2a1b3a2b1a2b2a2b3a 3b 1a 3b 2a 3b 3Table 4. The Standard 3 3 Tri–Squareda3 How did the 9 Grade Academy, Center, or CenterModel have an impact on the following;a1a2a3T a1b1T a1b2T a1b3T a2b1T a2b2T a2b3T a 3b 1b2 No; andb1b2b3b3 Missing.Table 5. Standard Responses the 3 3 Tri–Squaredb1 Yes;i-manager’s Journal of Educational Technology, Vol. 10 lNo. 1 lApril - June 2013T a 3b 2T a 3b 351

RESEARCH PAPERSWhere Table 5 is defined as,precise set of data to conduct research, analyze data, andTa1b1 Total Number of Responses for Cell One in thedetermine the level of significance required to eitherStandard 3 3 Tri–Squared Table;Ta2b1 Total Number of Responses for Cell Two in theStandard 3 3 Tri–Squared Table;Ta3b1 Total Number of Responses for Cell Three in theStandard 3 3 Tri–Squared Table;Ta1b2 Total Number of Responses for Cell Four in theStandard 3 3 Tri–Squared Table;Ta2b2 Total Number of Responses for Cell Five in theStandard 3 3 Tri–Squared Table;Ta3b2 Total Number of Responses for Cell Six in the Standard3 3 Tri–Squared Table;Ta1b3 Total Number of Responses for Cell Seven in theStandard 3 3 Tri–Squared Table;Ta2b3 Total Number of Responses for Cell Eight in theStandard 3 3 Tri–Squared Table; andTa3b3 Total Number of Responses for Cell Nine in theStandard 3 3 Tri–Squared Table.validate or reject the initial research hypothesis. The fourTri–Squared steps are as follows:·Design of an Inventive Investigative Instrument that hasTrichotomous Categorical Variables and TrichotomousOutcome Variables.·Establish the Research Effect Size, Sample Size withassociated Alpha Level.·Establish Mathematical Hypotheses.·Use the Tri–Squared Test as the Data Analysis Procedurefollowing implementation.The Tri–Squared Research DesignStep One: Design of an Inventive Investigative Instrumentthat has Trichotomous Categorical Variables andTrichotomous Outcome Variables.Step Two: Establish the Research Effect Size, Sample Sizewith associated Alpha Level.The Tri–Squared Effect Size FormulaThis leads into [ ] vector y Geometric Vector y “ y” d s The Initial Tri–Squared Instrument Construction Step Three: Establish Mathematical Hypotheses.Operational Phases “y” absolute value of vector y Sample Mathematical Hypotheses for Holistically Testing an“norm y” Application “Trine y” The creation of theTri–Squared Inventive Investigative Instrument ThePythagorean Triple of the Triangulation Model “3” The 3Phases of Tri–Squared Inventive Investigative InstrumentH0: Tri 2 0H1: Tri 2 ¹0Deployment which is composed of the following 3Step Four: Use the Tri–Squared Test as the Data AnalysisOperational PhasesProcedure following implementation.·Inventive Investigative Instrument Deployment;Sample Tables of qualitative and quantitative data follow·Inventive Investigative Instrument Completion; andlastly·Inventive Investigative Instrument Data Aggregation.Using the Tri –Squared Triangulation Model for ApplicationTesting: The Efficacy of Trichotometric Analysis through theSample Table (A)A Sample 3 3 Table of the Qualitative Input Outcomes of the Tri-Squared Test[TBD To Be Determined.]nTri 0TRICHOTOMOUS TESTINGINPUT VARIABLESa TBDImplementation of the Tri–Squared TestThe Tri–Squared research methodology: A four step processfor determining the appropriate Research Effect Size,Sample Size, and Alpha Level (Osler, 2012). The Tri–Squaredresearch procedure consists of a four step approachdesigned to provide the researcher with a clear and52to illustrate how the Tri–Squared Test displays the outcomesof affective domain inquiry (Osler, 2012).TRICHOTOMOUSRESULTS OUTPUTVARIABLESa1a2a3b1000 Tr1b2000 Tr2b3000 Tr3Tc1Tc1Tc1 TtriTri2 D.f [C - 1] [R - 1] [3 - 1] [3 - 1] 4 Tri2[x]i-manager’s Journal of Educational Technology, Vol. 10 lNo. 1 lApril - June 2013

RESEARCH PAPERSThe Tri–Square Test Formula for the Transformation ofCritical Value for p TBD is TBD. Thus, we can reject the nullTrichotomous Qualitative Outcomes into Trichotomoushypothesis (H0) by virtue of the hypothesis test if: Tri–SquaredQuantitative Outcomes to Determine the Validity of theCritical Value of TBD or TBD based on the CalculatedTri–Squared Value. Note: Comprehensive lists of Tri–SquaredResearch Hypothesis:Tri Critical Value Table TBD (with d.f. 4 at α TBD). For d.f.Tables are available that display and label Critical Values, 4, the Critical Value for p TBD is TBD. The calculatedAlpha Levels, and Sample Sizes based upon all calculatedTri–Square value is TBD, thus, the null hypothesis (H0) isTri–Squared Effect Sizes. The Tables for Hypothesis Testing2rejected by virtue of the hypothesis test which yields theand Probability Distribution follow (Tables 6 and 7following: Tri–Squared Critical Value of TBD or TBDrespectively). Together these two Tables summarize thebased upon the Calculated Tri–Squared Value.entire outcome and relevance of the multi–testapplication development process that is analyzed throughSample Table (B)the Tri–Squared Statistical methodology.nTri 0TRICHOTOMOUSCONVERSION INPUT VARIABLESa yThe purpose of this paper is to provide support for the use ofa3b1a1b1 a2b1 a3b1 the Tri–Squared Test in application/software evaluation. Theb2a1b2 a2b2 a3b2 Tri–Squared Test Algorithmic Model and associated Statisticala1b3 a2b3 a3b3 b3Methodology were provided to illustrate and display how2Tri d.f. [C – 1][R – 1] [3 – 1][3 – 1] 4 Trimethod can be put to use practically and rapidly. The Total2[x]A Sample 3 3 Table of the Quantitative Output Outcomesof the Tri–Squared Test is above.Transformative Trichotomy–Squared Research DesignMethodology is a universally applicable in–depthinvestigative procedure that is an ideal way of examining the[TBD To Be Determined]unique, diverse, and many times specialized proceduresThe Tri–Square Test Formula for the Transformation ofinherent in application testing and design. This process allowsTrichotomous Qualitative Outcomes into TrichotomousEducational Technologists to use quality control in the designQuantitative Outcomes to determine the Validity of theand testing educational software testing process through anResearch Hypothesis:in–depth qualitative analysis. The qualitative outcomes canthen be transformed into measurable quantitativeTri2 Calculated Tri–Squared [0] [0] [0] [0] [0] [0]outcomes to determine the outcome of testing process [0] [0] [0] 0(with d.f. 4 at α TBD). For d.f. 4, thestatistically. Both qualitative and quantitative methods areTri-Squared Distribution Table Displaying Primary Alpha Levels with Associated Critical Values for Hypothesis TestsaLevel0.9950.207Tri2[x] d.f. 7Small4[4] Small4[4] Small4[4] Small4[4] Small4[4] Medium4[16] Medium4[16] Medium4[16] Large4[64] Large4[64] Large4[64] Tri2Eff Effect SizeTri2sm Sample Size 5-269270-526527-783784-1040 Table 6.The Tri–Squared Hypothesis Test: Alpha Level, Effect Size, and Sample SizeTri-Squared Probability Distribution TableNumber of Research Participants Placed in Intervals Based off of Tri-Squared Effect Size Magnitude: [Small, Medium, or Large]is Based off of the Tri-Squared Mean [d.f.] 4MagnitudeSmall Unit Intervals: Multiple of 1 4[4] 4.4 16 Therefore, Interval hasIncrements of 16Number of Participants1-1617-33Probability P(x)0.9950.97534-40 41-570.200.10Medium Unit Intervals: Multiple of 2 4[16] 4[4.4] 64 Therefore, Interval hasIncrements of 64Large Unit Intervals: Multiple of 3 4[4.4.4] 4[64] 25658-7475-139140-204205-269270-526527-783784-1040 0.050.0250.

aforementioned model. Thus, the Triangulation Algorithmic Model is provided and defined to aid in understanding the process of measurement instrument construction and research design. The Triangulation Algorithmic Model for the Tri-Squared Test The Algorithmic Model of Triangulation is of the form (Figure 2). Where,