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Advances in Social Sciences Research Journal – Vol.7, No.7
Publication Date: July 25, 2020
DOI:10.14738/assrj.77.8556.
Osler, J. E. (2020). Transformative Psychometric Research Trioengineering Modeling: Triangulating Digital Instrumentation,
Implementation, and Data Analysis for Effective, Efficient, and Engaging Classroom Inquiry. Advances in Social Sciences Research Journal,
7(7) 58-74.
Transformative Psychometric Research Trioengineering Modeling:
Triangulating Digital Instrumentation, Implementation, and Data
Analysis for Effective, Efficient, and Engaging Classroom Inquiry
James Edward Osler II
North Carolina Central University
ABSTRACT
This monograph provides an in–depth discourse on “Research
Engineering” in the field of “Educational Science” first detailed in a 2012
Journal on Mathematics article. Research Engineering involves the
methodology and the metrics used to conduct in–depth research
investigations via the innovative Total Transformative Trichotomy–
Squared [Tri–Squared] Test. The triangulation of this test aids in the
comprehension of how the Tri–Squared calculation is a mixed methods
research design based upon “Trichotomous Psychometrics”.
Trichotomous Psychometrics involves the development, deployment,
and analysis of “Trifold Assessments” for the holistic transformation of
qualitative outcomes into quantitative data. This paper is a continuation
of the published article entitled: The Psychometrics of Educational
Science: Designing Trichotomous Inventive Investigative Instruments
for Qualitative and Quantitative for Inquiry.
Keywords: Algorithmic Model, Cartesian Coordinates, Educational Science,
Geometric Vectors, Models, Psychometrics, Research Engineering,
Triangulation, Trichotomous Psychometrics. Trichotomy, Trioengineering,
Trifold Assessment, and Tri–Squared.
INTRODUCTION
One of the most challenging areas of research in education involves the construction of specific
instruments that are designed to measure qualitative outcomes and data. Although there are a great
many measurement tools that analyze the cognitive and psychomotor domains, there remains a
vacuum in the number of instruments especially designed to accurately measure the affective
domain (the learning domain that contains attitudes, opinions, emotions, perception, and
perspectives). This void is further expanded when the specific event under investigation is unique,
specialized, has specific characteristics, serious legal constrictions, and issues regarding time. This
often requires the research investigator to design an instrument that ideally measures the variables
under investigation (Osler, 2013b).
The process of designing instruments for the purposes of assessment and evaluation is called
“Psychometrics”. Psychometrics is broadly defined as the science of psychological assessment (Rust
& Golombok, 1989). The Tri–Squared Test pioneered by the author in 2012, factors into the
research design a unique event–based “Inventive Investigative Instrument”. This is the core of the
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Advances in Social Sciences Research Journal (ASSRJ) Vol.7, Issue 7, July-2020
Trichotomous–Squared Test. The entire procedure is grounded in the qualitative outcomes that are
inputted as Trichotomous Categorical Variables based on the Inventive Investigative Instrument.
The specific assessment of the variables is completely dependent upon the outcomes determined
by the researcher’s instrument. The creation, production, and deployment of the trichotomous
Inventive Investigative Instrument requires that the research investigator adopt the role of a
“Trichotomous Psychometrician”. A “Trichotomous Psychometrician” is an Educational Scientist
that uses trichotomous–based psychometrics to develop a qualitative Inventive Investigative
Instrument specifically designed capture qualitative responses during a specific event (Osler,
2013a). The design of Inventive Investigative Instruments is based on the mathematical “Law of
Trichotomy” (Apostol, 1967). Trichotomy itself is derived from historic discussions surrounding
higher cognition, general thought, and descriptions of intellect. Philosopher Immanuel Kant adapted
the Thomistic acts of intellect in his “trichotomy of higher cognition” — (a) understanding, (b)
judgment, (c) reason — which he correlated with his adaptation in the soul’s capacities — (a)
cognitive faculties, (b) feeling of pleasure or displeasure, and (c) faculty of desire (Kant, 2007). It is
important to note that in mathematics, the law (or axiom) of trichotomy is most commonly the
statement that for any (real) numbers x and y, exactly one of the following relations holds. Until the
end of the 19th century the law of trichotomy was tacitly assumed true without having been
thoroughly examined (Singh, 1997). A proof was sought by Logicians and the law was indeed proved
to be true. Mathematically, Trichotomous relations are irreflexive and antisymmetric (Sensagent,
2012). A description of the entire Tri–Squared research process follows and is described in detail
to provide the reader of the precise steps undertaken in the process of developing, designing, and
ultimately implementing an Inventive Investigative Instrument (Osler, 2013a). “Trioengineering”
is a novel term coined by the author and defined here as “the process involved in designing solutions
by actively using the mathematical law of trichotomy to produce a measurable, scalable, and
tangible results”.
DEFINING THE FIELD OF EDUCATIONAL SCIENCE
The field of “Education Science” is also represented by the term “Eduscience” which is a
portmanteau of the two terms “Education” and “Science”. Similar to the field of “Bioscience”,
Eduscience is the study of education wherein applicable sciences (such as ergonomics, statistics,
technology, etc.) are applied to enhance and improve learning. The primary purpose of the field of
Eduscience is the study and application of solutions to improve and enhance the learning
environment and learning in general. Eduscience is solution–driven and is actively concerned with
the transfer and dissemination of knowledge. Education Science is a broad field and its professionals
are directly involved in the field. Those who are actively involved in Eduscience can be referred to
as “Education or Educational Scientists”. Educational Scientists or “Eduscientists” are multifaceted
professionals who have a variety of areas of expertise. They can assume multiple roles in the
educational environment and can serve in a variety of offices and in a multitude of capacities. The
primary positions that Eduscientists assume are in the following areas: Administration (as Leaders,
Organizational Heads, and Organizational Management Professionals), Instruction (as Teachers,
Professors, and Facilitators), Practice (as Practitioners in a variety Specified Areas and Arenas), and
Technology (as Educational Technologists, Instructional Technologists, and Information
Technologists). In these positions Eduscientists effectively use, analyze, study, and deploy novel
instructional learning theories, methodologies, strategies, solutions, tools, and techniques in both
traditional or virtual (pedagogical and andragogical) settings to bring about learning (Osler, 2012a).
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Osler, J. E. (2020). Transformative Psychometric Research Trioengineering Modeling: Triangulating Digital Instrumentation, Implementation, and Data
Analysis for Effective, Efficient, and Engaging Classroom Inquiry. Advances in Social Sciences Research Journal, 7(7) 58-74.
TRANSITIVE TRANSFORMATIVE KNOWLEDGE TRANSFER: THE ULTIMATE GOAL OF
EDUCATION SCIENCE
Educational Scientists strive to make the process of knowledge transfer both transitive and
transformational. A transitive and transformative knowledge transfer process is as seamless and as
harmonious as possible in an effort to empower, enhance, and improve learning. Eduscientists are
masters of teaching who also are also highly proficient practitioners who are able draw from
personal and professional experiences to make the learning environment more viable (Accessible),
usable (Ergonomic), teachable (Instructional), engaging (Relevant), approachable (Adaptive),
exploration–based (Discovery), and inspirational (Transformative). The Total Transformative
Trichotomy–Squared Test is a comprehensive multi–step research methodology that is employed
by Eduscientists. It is especially designed to conduct qualitative and quantitative investigations in
educational settings and the learning environment (Osler, 2013b).
Education–Based Inquiry: Using the Tri–Squared Research Methodology as a Four Step
Process to determine an appropriate research Effect Size, an ideal Sample Size, and an
effective Alpha Level for Education–Based Experiments in the Learning Environment
The Tri–Squared research procedure (Osler, 2013c) consists of a four step approach designed to
provide the researcher with a clear and precise set of data to conduct research, analyze data, and
determine the level of significance required to either validate or reject the initial research
hypothesis. The four Tri–Squared steps are as follows:
1. Design of an Inventive Investigative Instrument that has Trichotomous Categorical Variables
and Trichotomous Outcome Variables.
2. Establish the Research Effect Size, Sample Size with associated Alpha Level.
3. Establish Mathematical Hypotheses.
4. Use the Tri–Squared Test as the Data Analysis Procedure following implementation.
The Upright Right Triangle Testing Model for the Tri–Squared Test
The Algorithmic Model of Triangulation (Osler, 2013c) is of the form:
Where,
Vertex a = Ða = form “Vector [a→b]” as “[)*++++⃗]” = “authoring” = a ratio of a measurement metric of
4 (used to convey information) = The Initial Tri–Squared Instrument Design;
Vertex b = ∟b = = form “Vector [b→c]” as “[+
*-+++⃗]” = “building” = a ratio of a measurement metric of 3
(as the Triostatistics: Tri–Squared Test) = The Tri–Squared Qualitative Instrument Responses
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which will be analyzed via Tri–Squared Analysis; and Vertex c = Ðc = = form “Vector [c→a]” as “[-)++++⃗]”
= “conveying” = The Final Tri–Squared Test Outcomes in a Quantitative Report.
Thus, the Triangulation Model is symbolized by a Right Triangle written as: “ ”. This symbol called
“Trine” (meaning a group of three) is written mathematically as “ = abc” and is simplified into the
mathematic geometric expression: (meaning “Triangulation Model abc” or more simply
“Trine abc”).
The angles have the following angular measurements in degrees: Ða = 36.86, ∟b = 90, and Ðc =
53.14, that all add up to the standard 180° of any traditional triangle: [36.86° + 90° + 53.14° = 180°].
The connective lines between the Upright Right Triangle vertex points (i.e., the lines between points
a, b, and c respectively are geometric “vectors” (lines with both size [magnitude] and direction)
making the model a systemic or cyclic process from the point of origin “a” back to the original point
of origin “a”. This is illustrated in terms of Cartesian Coordinates as follows:
In terms of Vectors, the Upright Right Triangle as the Algorithmic Model of Triangulation now
becomes the Trioengineering: Triangulation Model Right Triangle is equal to three vectors
that illustrate the movement in direction and magnitude from one completed task into another. The
entire process is both cyclical and sequential with a “Trine Vector Equation” written as:
= [←x]→[↓y]→[↑z], for “[+./0 ++++++⃗]”
Defined as Trine = “Concentration of Vector x into Concentration of Vector y into Concentration of
Vector z”, which is simplified into a more standardized Trine Vector Equation form written as:
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Osler, J. E. (2020). Transformative Psychometric Research Trioengineering Modeling: Triangulating Digital Instrumentation, Implementation, and Data
Analysis for Effective, Efficient, and Engaging Classroom Inquiry. Advances in Social Sciences Research Journal, 7(7) 58-74.
= .⃖ → /⃖ → 0⃖, for “+./0 ++++++⃗”
Where,
vectors x, y, and z respectively are indicated on the “Algorithmic Triangulation Data Model” as:
Indicating that the standardized form of the vectors x, y, and z are equivalent to the following
geometric vectors that are the sequential Cartesian Coordinates relative to the size and magnitude
of the research engineering phases that sequentially connect the respective angles Ða, ∟b, and Ðc.
This is written with the flow of the vector in opposite direction of its original presentation to
indicate the more precise direction of arrows from right to left in a cyclical motion as follows:
⃖3+ = 45⃖+++++
⃖6+ = 57⃖++++
⃖8+ = 47⃖++++
THE COMPLETE TRI–SQUARED ANALYSIS ALGORITHMIC
Triangulation Model as the 4 to 3 to 5 Upright Right Triangle
The three numeric Vector Operational Phases of the Triangulation Model that now expresses the
full scope of the operational parameters of the Triostatistics Tri–Squared Test are completely
defined in the following manner:
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Osler, J. E. (2020). Transformative Psychometric Research Trioengineering Modeling: Triangulating Digital Instrumentation, Implementation, and Data
Analysis for Effective, Efficient, and Engaging Classroom Inquiry. Advances in Social Sciences Research Journal, 7(7) 58-74.
The following table provides the metrics for the construction of the Inventive Investigative
Instrument following the parameters indicated in phases 1. through 4. of the first vector [ ⃖3+ = 45⃖+++++
= 4] of the Triangulation Model (FIgure 1.):
a0 =
Inventive Investigative Instrument
—Name—
[Optional Asset Security]
a1 = Section 1. Research Question One. The First Series of Questions from the Qualitative
Trichotomous Categorical Variables are listed below.
Responses: [Select only one from the list.] u b1 b2 b3
a. Item One ̈ ̈ ̈
b. Item Two ̈ ̈ ̈
c. Item Three ̈ ̈ ̈
a2 = Section 2. Research Question Two. The Second Series of Questions from the Qualitative
Trichotomous Categorical Variables are listed below.
Responses: [Select only one from the list.] u b1 b2 b3
d. Item Four ̈ ̈ ̈
e. Item Five ̈ ̈ ̈
f. Item Six ̈ ̈ ̈
a3 = Section 3. Research Question Three. The Third and Final Series of Questions from the
Qualitative Trichotomous Categorical Variables are listed below.
Responses: [Select only one from the list.] u b1 b2 b3
g. Item Seven ̈ ̈ ̈
h. Item Eight ̈ ̈ ̈
i. Item Nine ̈ ̈ ̈
Figure 1.
This Tabular Triangulation Model can be fully exemplified in the aforementioned example provided
by Figure 2.
Thus, Figure 2. is provided again (in an enlarged form) to illustrate a sample Inventive Investigative
Instrument (Osler and Waden, 2012b):
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A Standard 3 × 3 Tri–Squared Table of Comprehensive Inputted Qualitative Research Responses
would resemble the following model represented by Figure 5. Where, Tn = The Total Number of
Responses (based upon a one to one ratio of each Trichotomous Testing Input Variables contrasted
directly with each Trichotomous Results Output Variable) in each of the individual cells of the
Standard 3 × 3 Tri–Squared Table.
a1 a2 a3
b1 Ta1b1 Ta2b1 Ta3b1
b2 Ta1b2 Ta2b2 Ta3b2
b3 Ta1b3 Ta2b3 Ta3b3
Figure 5.
Where Figure 5. is defined as,
Ta1b1 = Total Number of Responses for Cell One in the Standard 3 × 3 Tri–Squared Table;
Ta2b1 = Total Number of Responses for Cell Two in the Standard 3 × 3 Tri–Squared Table;
Ta3b1 = Total Number of Responses for Cell Three in the Standard 3 × 3 Tri–Squared Table;
Ta1b2 = Total Number of Responses for Cell Four in the Standard 3 × 3 Tri–Squared Table;
Ta2b2 = Total Number of Responses for Cell Five in the Standard 3 × 3 Tri–Squared Table;
Ta3b2 = Total Number of Responses for Cell Six in the Standard 3 × 3 Tri–Squared Table;
Ta1b3 = Total Number of Responses for Cell Seven in the Standard 3 × 3 Tri–Squared Table;
Ta2b3 = Total Number of Responses for Cell Eight in the Standard 3 × 3 Tri–Squared Table; and
Ta3b3 = Total Number of Responses for Cell Nine in the Standard 3 × 3 Tri–Squared Table;
This leads into [→] vector y = Geometric Vector y = “↓y” = 6⃖+ = 57⃖++++ = The Initial Tri–Squared
Instrument Construction = Operational Phases “y” = absolute value of vector y = “norm y” = ‖/‖ =
“Trine y” = = The creation of the Tri–Squared Inventive Investigative Instrument = The
Pythagorean Triple of the Triangulation Model = “3” = The 3 Phases of Tri–Squared Inventive
Investigative Instrument Deployment which is composed of the following 3 Operational Phases:
1. Inventive Investigative Instrument Deployment;
2. Inventive Investigative Instrument Completion; and lastly
3. Inventive Investigative Instrument Data Aggregation;
AN EXAMPLE OF TRIANGULATION—TRIFOLD ASSESSMENT: THE INVENTIVE
INVESTIGATIVE INSTRUMENT EXTRACTED FROM THE INITIAL TRICHOTOMOUS RESEARCH
QUESTIONS AND ASSOCIATED CATEGORICAL AND OUTCOME VARIABLES USED IN THE
RESEARCH STUDY
In this sample research study the researcher Inventive Investigative Instrument items were derived
from the aforementioned research questions and outcome variables. The instrument was used to
obtain data from administrators as responses to research questions relating to the impact of Ninth
Grade Centers, Academies or similar models on minority ninth grade students. The purpose of this
instrument was to provide data from 9th Grade Academies, Centers, and Center Models for non–
parametric Tri–Squared analysis. Data that was not responded to was reported as “Missing” (a
separate Categorical Variable designed to report all research results). The Inventive Investigative
Instrument was a qualitative method of data collection designed to accurately provide responses to
carefully answer research questions. Ultimately, the research data analysis methodology (Tri–
Squared) qualitatively and quantitatively determined the academic success of ninth grade students
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Osler, J. E. (2020). Transformative Psychometric Research Trioengineering Modeling: Triangulating Digital Instrumentation, Implementation, and Data
Analysis for Effective, Efficient, and Engaging Classroom Inquiry. Advances in Social Sciences Research Journal, 7(7) 58-74.
who participated in ninth grade centers or similar models at four year high schools in North Carolina
based upon the research hypothesis. A sample Inventive Investigative Instrument containing all
three outcome variables is provided to illustrate how the 3 × 3 Trichotomous Table is constructed
from the Inventive Investigative Instrument (Osler 2013b).
The Waden 9th Grade Academies, Centers, and Center Models Assessment Instrument ©
A. Has the 9th Grade Academy, Center, or Center Model been:
Yes No Missing
1. Successful? ̈ ̈ £
2. Made a Difference? ̈ ̈ £
3. Aided in Retention? ̈ ̈ £
How as the Academy/Center been successful, made a difference, or aided in retention, if at
all?________________________________________________________________________
B. Did the 9th Grade Academy, Center, or Center Model Result in the following:
Yes No Missing
4. 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?________________________________________________________________
C. How did the 9th Grade Academy, Center, or Center Model have an impact on the
following:
Yes No Missing
7. 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 been operation in your school?
__________________________________________________________________
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How long has the interviewee (yourself) been (working) there and what is the level 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 their knowledge of the program,
whether their knowledge was medium knowledge low level of knowledge etc.
______________________________________________________________________
The Tri–Squared Research Design Using the Inventive Investigative Instrument
Step One
Design of an Inventive Investigative Instrument that has Trichotomous Categorical Variables and
Trichotomous Outcome Variables.
To effectively use Tri–Squared in a research investigation the researcher must first develop a series
of trichotomous categorical variables based on associated trichotomous outcome variables. This is
the first initial and crucial step to using Tri–Squared as a valid, reliable, and objective means of
analyzing data. Second, an “Inventive Investigative Instrument” must be created and implemented
in a manner compliant with the initial trichotomous categorical variables and outcomes stated at
the outset of the research investigation. This insures that the research investigation is consistent
throughout the study and that the later Tri–Squared computations are validly reporting what
actually took place in the research environment during the time period in which the actual study
was conducted. As previously stated the “Inventive Investigative Instrument” can be
psychometrically delivered as a test or qualitatively delivered in the form of a research
questionnaire, survey, interview or another type of metric. As long as the trichotomous categorical
variables are measured according to the established associated trichotomous outcome variables
then the research has merit within the confines and strict requirements of the Tri–Squared Test.
Step Two
Establish the Research Effect Size, Sample Size with associated Alpha Level.
The Tri–Squared Effect Size Formula
RESULTS
The Effect Size Table for the calculated Tri–Squared Effect Size Formula for the study was
determined based off of the Standard 3 × 3 Tri–Squared Table calculated to be 0.125 (or small in
overall effect size). The total number of participants that were identified in the study from the outset
was nTri = 17. Thus, an alpha level of 0.975 is best for this small sample size from a corresponding
small effect size according to the calculated effect size for the Standard 3 × 3 Tri–Squared Table
(specifically identified as 17 to 33 participants for a = 0.975). This is further illustrated in the
following Tri–Squared Distribution Tables (Figure 6.):