At the Intersection of Derived Relations and Observational Learning: Teaching Fraction–Percentage Relations

Abstract

Little research has examined the intersection of derived relations and observational learning, which may be an avenue by which an instructor can increase the total amount of learning while conserving instructional resources. In this study, we investigated this intersection in the context of teaching third-grade students fraction–pictogram–percentages equivalence classes. We taught participants two baseline relations and observed the emergence of all possible derived relations. Prior to training, the participants could sort percentage stimuli from least to greatest (comparative relations), but could not do this with fraction stimuli. Following the formation of equivalence classes, the participants could accurately sort fraction stimuli, thus demonstrating the transfer of function. Two additional participants observed the target participants undergo the training of baseline relations. The observing participants acquired baseline relations with no direct reinforcement, derived all relations, and demonstrated transfer of function. These findings suggest a novel way to incorporate equivalence-based instruction into classroom settings.

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Journal of Behavioral Education (2020) 29:741–762
https://doi.org/10.1007/s10864-019-09343-8
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ORIGINAL PAPER
At theIntersection ofDerived Relations andObservational
Learning: Teaching Fraction–Percentage Relations
VictoriaR.Verdun1· BrittanyA.Chiasson1· DanielM.Fienup1
Published online: 31 July 2019
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
Little research has examined the intersection of derived relations and observa-
tional learning, which may be an avenue by which an instructor can increase the
total amount of learning while conserving instructional resources. In this study, we
investigated this intersection in the context of teaching third-grade students frac-
tion–pictogram–percentages equivalence classes. We taught participants two base-
line relations and observed the emergence of all possible derived relations. Prior to
training, the participants could sort percentage stimuli from least to greatest (com-
parative relations), but could not do this with fraction stimuli. Following the forma-
tion of equivalence classes, the participants could accurately sort fraction stimuli,
thus demonstrating the transfer of function. Two additional participants observed the
target participants undergo the training of baseline relations. The observing partici-
pants acquired baseline relations with no direct reinforcement, derived all relations,
and demonstrated transfer of function. These findings suggest a novel way to incor-
porate equivalence-based instruction into classroom settings.
Keywords Comparative relations· Fractions· Mathematics· Observational
learning· Stimulus equivalence
Introduction
A core goal of education is to create environmental conditions that promote gen-
erative learning in an effective and efficient manner. An educational intervention
is effective when it produces the intended educational outcomes, and it is efficient
when it produces effective outcomes while conserving instructional resources, or
inputs. One way to conserve instructional resources while maximizing educational
outcomes is to program for, or engineer, generative learning. Generative learning is
* Daniel M. Fienup
fienup@tc.columbia.edu
1 Department ofHealth andBehavior Studies, Teachers College, Columbia University, 525W.
120th Street, Box223, NewYork, NY10027, USA

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said to occur, by definition, when students learn more than the information taught, or
they “go beyond the information given” (Critchfield and Twyman 2014, p. 201), and
is founded on the premise that learning occurs by formulating connections, or rela-
tions, between instructional stimuli. Such emergent learning is important because it
increases efficiency in the classroom (Stromer etal. 1992).
Derived Relations
One approach that addresses generative learning is equivalence-based instruction
(EBI), which is founded on logical connections and inference-making and rooted in
over 40years of basic research on stimulus equivalence (Sidman 1994). An instruc-
tor using EBI overlaps stimulus–stimulus relations to promote the emergence of new
relations (Brodsky and Fienup 2018; Critchfield and Twyman 2014; Rehfeldt 2011).
For example, Lynch and Cuvo (1995) taught elementary school children to infer,
or derive, fraction–decimal relations. First, the researchers taught children to select
pictograms (100-square grid with a portion of squares filled in) when presented
with fractions (e.g., 1/5). (This relation was termed A → B.) Second, the researchers
taught children to select decimals (e.g., 0.20) when given a pictogram. (This relation
was termed B → C.) Because the researchers related the three stimuli via the picto-
gram stimuli (also called a node), participants were able to infer additional relations.
Participants demonstrated symmetry when the trained relations were demonstrated
in the reverse manner with no explicit training. For example, participants were able
to select fractions given pictograms (B → A) and select pictograms given decimals
(C → B). Participants also derived relations between stimuli that were not directly
associated, such as selecting decimals given fractions (transitive relation, A → C)
and selecting fractions given decimals (equivalence relation, C → A) (for a detailed
review, see Sidman 1994). In this and other studies, researchers apply EBI to teach
conditional discriminations through a match-to-sample instructional format that pro-
motes the emergence of novel, untaught relations. This technology has been applied
to a wide variety of populations, including students with disabilities (e.g., Mur-
phy etal. 2005), typically developing students with academic deficits (e.g., Lynch
and Cuvo 1995), and adults enrolled in advanced college courses (e.g., Fienup and
Critchfield 2011; for a review, see Brodsky and Fienup 2018). This body of research
contains a variety of academic domains, including elementary mathematical con-
cepts (Lowe and Cuvo 1976; Lynch and Cuvo 1995), neuroanatomy concepts (Fie-
nup etal. 2010), higher-level mathematical objectives (Ninness et al. 2006, 2009),
and research design and analysis (Albright etal. 2015; Fienup and Critchfield 2011;
Lovett etal. 2011). In each of the published studies, researchers carefully planned
to teach a subset of relations that led to the emergence of additional relations and
classes of stimuli, making EBI a desirable pedagogical method to foster generative
learning.
Another feature of equivalence-based instruction that promotes emergent and
generative behavior is the transfer of stimulus function (e.g., reinforcement value).
Most of the research in this area has been basic in nature. For example, Hayes etal.
(1991) formed equivalence classes, where one stimulus functioned as a conditioned

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reinforcer or punisher, respectively. The researchers found that after classes were
formed, all members of the respective classes took on reinforcing or punishing func-
tions in accordance with the stimuli placed in the respective classes. In the applied
literature, Rosales and Rehfeldt (2007) demonstrated the transfer of mand functions
across stimuli. The researchers first taught participants to mand using the first three
phases of PECS training (physical exchange; remove picture and hand to communi-
cation partner at increasing distances; discriminate between pictures and engage in
a communication exchange; Bondy and Frost 1994). Researchers taught participants
to exchange a PECS card during behavior chains, where a missing step served as an
establishing operation to mand for that item (e.g., when trying to listen to music,
headphones were missing and the participant had to mand for headphones). Once
the researchers established reliable manding with PECS cards that contained a pic-
ture of the item, the researchers taught participants equivalence classes that con-
tained the picture cards and their corresponding vocal names and written text (e.g.,
picture of headphones, spoken work “headphones,” and written word headphones).
Following the formation of classes, the mand function transferred to other class
members and participants manded using written words and spoken words, without
direct training to mand with these topographies.
With respect to math (e.g., Lynch and Cuvo 1995), various mathematical behav-
iors can potentially transfer within new equivalence classes. For example, relational
frame theorists (Hayes etal. 2002) discuss comparative relations that entail when
“one event is responded to in terms of a quantitative or qualitative relation along a
specified dimension with another event” (p. 36). If an individual learns comparative
relations such as more-than or less-than, this behavior, which is contained within
the context of specific stimuli, can transfer to other stimuli through equivalence
classes. For example, if a child sorts percentage stimuli (10%, 50%, 90%) from least
to greatest, but does not do this with fractions (e.g., 1/10, 1/2, 9/10), forming rela-
tions between fractions and decimals may result in a transfer of ordering behavior
from percentage stimuli to fraction stimuli and result in a child who sorts fractions
from least to greatest.
Observational Learning
Another research-based method that addresses generative learning is observational
learning (OL), which is a verbal developmental cusp that is a capability (Greer etal.
2006). An individual is said to have OL when one acquires novel behaviors following
the observation of a model’s response within the context of the 3-term contingency
(antecedents and consequences). Researchers promote the notion that OL is a necessary
component of an individual’s repertoire to advance through academic and social devel-
opment (Greer and Ross 2008). Prior to the acquisition of OL, the student is bound to
direct behavior–consequence contingencies to acquire new responses, but when a child
possesses the OL capability, learning occurs as a result of indirect contact with instruc-
tion. Said another way, once a child can learn through observation, he can learn from
observing 3-term contingencies of others. In fact, researchers have demonstrated that
children learn three to five times as quickly with OL in repertoire (Delgado and Greer

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2009). Capitalizing on OL promotes generative learning in two ways. First, from an
instructor’s perspective, multiple individuals learn new behavior from formally teach-
ing only a subset of those individuals. Second, from the student’s perspective, more is
learned via observation than the instructional resources directly allocated to him.
At theIntersection ofDerived Relations andObservational Learning
A novel way to increase learner output while conserving teaching resources is to
arrange learning opportunities that capitalize on derived relations and observational
learning within the classroom environment. Only a few studies have investigated this
intersection. Ramirez and Rehfeldt (2009) examined the effects of an observational
intervention on the emergence of symmetry relations for second-language vocabulary
with one observing participant. A child received direct instruction on baseline condi-
tional discriminations, while a second child simply observed that training. Following
the intervention, both participants demonstrated the emergence of symmetry relations
and maintained those responses 1month later. Additional studies have confirmed the
emergence of derived relations following observation (e.g., Bryne etal. 2014; Leader
and Barnes-Holmes 2001; Rehfeldt etal. 2003). These studies provide valuable insight
into how an instructor may incorporate derived relations and observational learning;
however, the studies are limited in terms of evaluating interventions with pre–post-
designs instead of experimental designs (Leader and Barnes-Holmes 2001; Rehfeldt
etal. 2003), a limited number of derived relations tested (Bryne et al. 2014; Ramirez
and Rehfeldt 2009), the number of classes taught (Leader and Barnes-Holmes 2001),
the number of participants included, and omitting formal assessment of observational
learning prior to implementation of the intervention. Building on these studies within
the classroom enables us to apply the extensive and substantial research behind both
topics to usher EBI into group instruction.
The purpose of the present study was to extend the derived relations and observa-
tional learning areas of research. We replicated the general EBI procedures and content
of Lynch and Cuvo (1995). The conditional relations between quantities were repre-
sented as fractions (e.g., 2/5), pictograms (e.g., 40% or 2/5 represented on a hundreds
grid as 40 shaded squares), and percentages (e.g., 40%). We extended the literature by
including children who demonstrated the observational repertoire (at start of study)
and watching the target children undergo baseline training to determine whether this
resulted in derived equivalence classes. Additionally, we assessed transfer of function,
which consisted of sorting stimuli from least to greatest value. We examined whether
children who could sort some, but not other sets of stimuli prior to training could
do this with all sets of stimuli following the formation of equivalence classes, in the
absence of direct reinforcement.

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Method
Participants andSetting
Participants were selected from a third-grade general education classroom located in
a public elementary school outside of a major metropolitan area. The district partners
with a university in order to educate students with Individualized Education Plans (i.e.,
IEPs) and typically developing peers in the same general education classroom. The
classroom applied the Comprehensive Application of Behavior Analysis to Schooling
(CABAS®) and Accelerated Independent Learner (AIL®) educational models (Greer
2002). This system of instruction is rooted in the principles of applied behavior analy-
sis applied to pedagogy, curriculum, classroom management, and training for staff and
parents. The classroom teachers followed the school district’s adoption of curricular
objectives from the Common Core State Standards; however, each curricular goal was
modified to include measurable behavior(s) and criterion for mastery. The classroom
included 18 students (eight with IEPs), one teacher, and two teaching assistants. Teach-
ers delivered instruction to academically homogeneous groups of four to eight students
in order to ensure that each student achieved mastery of each skill.
We identified participants in two stages. In the first stage, we provided the 3Mix
pre-assessment to all students in the classroom (N = 18, described below) and analyzed
each students’ accuracy for all relations (to-be-trained and to-be-derived). We selected
participants with low accuracy across all relations and obtained parent permission and
participant assent. In the second stage, we administered tests for OL (described below)
with each possible participant (Delgado and Greer 2009).
Four students met the stage-1 inclusion criterion for the study. The participants ages
were between 8 and 9 (see Table1). Two of the four participants met the criterion for
OL (stage-2 criterion), and we assigned each to observe students without OL. Thus,
each dyad contained one participant with OL and one without OL. The participant
without OL (primary participant) received direct instruction on baseline relations
(responding to stimuli, receiving consequences for correct and incorrect responses),
and the participant with OL (observing participant) simply observed instruction and
made no responses during training. In Dyad 1, Yasmine received direct instruction and
Kelly observed the teaching procedures. Yasmine was an 8.11-year-old African-Amer-
ican female who qualified for free and reduced lunch. Kelly was a 9.6-year-old white
female. In Dyad 2, Erica received direct instruction and Joey observed the teaching
procedures. Erica was a 9.1-year-old white female. Joey was a 9.7-year-old Hispanic
male with a diagnosis of Speech and Language Impaired. Additional information on
each participant’s academic profile is displayed in Table1.
Materials
Assessment Materials
We utilized district-based and standardized tests to assess participants’ grade-
level equivalents for math. The school district required students to complete
i-Ready® diagnostic tests, a computer-based, adaptive learning and assessment tool

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Table 1 Participant demographics
Races include African-American (AA), Caucasian (C), and Hispanic (H). Observational learning percentages were calculated out of 20 possible opportunities to respond.
We report standard scores for WJ–IV subtests
Participants Gender Race Age i-Ready math
diagnostic
score
WJ–IV applied
test standard
score
WJ–IV calculation
test standard score
WJ–IV fluency
test standard
score
Percent accuracy to
observational learn-
ing assessment (%)
Other
Yasmine (primary
participant, Dyad 1)
FAA 8.11 451 (early 3) 110 99 100 75 Free and reduced lunch
Kelly (observer,
Dyad 1)
FC 9.6 470 (mid 3) 108 114 117 85
Erica (primary par-
ticipant, Dyad 2)
FC 9.1 484 (mid 3) 106 108 104 75
Joey (observer, Dyad
2)
MH 9.7 482 (mid 3) 111 111 114 100 Speech and language
impaired

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(Curriculum Associates 2014), to provide data on the grade-level equivalents of all
students in the school district between kindergarten and sixth grade. The experi-
menters also administered the Woodcock Johnson® IV Tests of Achievement, stand-
ardized academic achievements test. Specifically, the experimenters utilized form A
of the Standard and Extended Batteries and administered the following Math sub-
tests: Calculation (Test 5), Fluency (Test 6), and Applied Problems (Test 10) (see
Table1).
Fraction–Pictogram–Percentage Materials
Equivalences between fractions, pictograms, and percentages are included in the
Common Core State Standards listed under Numbers and Operations–Fractions
(CCSS.MATH.CONTENT.4.NF.C.5). We taught eight 3-member classes; fractions
served as the A stimulus, pictograms served as the B stimulus, and percentages
served as the C stimulus (see Table2). Pictograms were pictorial representations
of each target proportion; in these representations, experimenters divided a square
into 100 equal parts. To represent a specific proportion, the experimenter shaded the
corresponding number of squares. For example, we shaded 25 of the 100 squares in
the pictogram that represents the fraction ¼ and the percentage 25% (see Table2).
Similar fraction–pictogram–percentage stimuli were used throughout the study.
Pre–Post‑assessments ofClass Formation
The 3Mix assessment measured responding to each of the stimulus relations in
the 3-member classes (fractions–pictograms–percentages). All assessments were
identical in content; however, we rotated the order of stimuli presented for each
assessment. The assessment contained 48 questions, one instance of each of the
six relations between A, B, and C across eight classes (see Table 2). The assess-
ment included all training (A → B, B → C), symmetry (B → A, C → B), transi-
tive (A → C), and equivalence (C → A) relations. The paper-based assessment
was presented in a multiple-choice format, and each question had four possi-
ble answers, including a non-exemplar, two negative exemplars, and one posi-
tive exemplar. The directions were written and read aloud as follows, “Select the
option that is the same as each fraction, decimals, or pictogram presented below.”
Sorting Task
Experimenters assessed the presence of comparative relations (more than, less
than) with sorting tasks. This assessment was conducted within each type of stim-
ulus (e.g., fractions). All stimuli were printed on 8.9 by 12.7cm cardstock. The
pictogram images were 6.3cm by 7.6 cm, and fraction and percentage stimuli
were printed in font sizes 80 and 60, respectively.

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Training Stimuli
We used Microsoft Office PowerPoint on a MacBook Air computer with a 33.0cm
display to present comparison stimuli across training and testing sets during EBI. We
positioned one comparison stimulus in each of the four corners on a slide. The four
comparison stimuli on a single slide were all within the same class dependent on the
relations being trained or tested. For example, during training blocks for A → B rela-
tions, four pictogram stimuli were positioned on the screen, while the experimenter
presented a fraction as the sample stimulus. Stimuli presented on the computer screen
had the same dimensions as those used during the sorting task. Sample stimuli were
printed on cardstock and also had the same dimensions as those in the sorting task.
The experimenter handed a sample stimulus to a participant while providing a vocal
antecedent (described below), after which the participant placed the sample against
the corresponding comparison stimulus on the computer screen.
Table 2 Eight 3-member
fraction–decimal–percentage
classes
Class Fraction (A) Pictogram (B) Percent-
age (C)
(%)
11/20 5
21/520
31/425
42/540
52/450
63/560
73/475
84/580

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Dependent Variables
We measured the percentage of class-consistent responding (e.g., A1 → B1,
C4 → B4) for training and testing blocks in order to assess mastery of baseline and
derived stimulus relations. During training blocks, when the observing participant
was present (but made no responses to instructional stimuli), we defined a correct
response as the primary learner placing the sample stimulus on the correspond-
ing comparison stimulus within 5s of the experimenter’s presentation of a sample
stimulus and vocal antecedent. We defined a correct response for test blocks (i.e.,
derived relations) consistent with the definition described for training blocks. Note
that during test blocks, the observer and primary learner each responded indepen-
dently to instructional stimuli and in the absence of the other. We also measured
the percentage of correct responses for all relations on the paper-and-pencil 3Mix
test to analyze the effects of EBI on the emergence of derived relations. To test for
the emergence of comparative relations (sorting a set of cards from least to great-
est), we scored whether a stimulus was placed in the correct order (e.g., see Table2,
the second position was scored correct if a stimulus from the 20% class was placed
there, otherwise the position was scored as incorrect). Each of the eight positions
was scored as correct or incorrect, and we divided the number of stimuli placed in
its correct position by the total number of positions (8) and multiplied the resulting
number by 100 to obtain the percentage correct.
Procedure
Dyads contained a primary participant who received consequences for responses
during training phases and an observing participant who simply watched the pri-
mary participant respond and receive consequences. Figure2 displays a flowchart of
phases completed by each participant. The study began with equivalence class and
comparative relation assessments. Next, dyads entered EBI that contained training
on baseline relations and tests for derived relations using a match-to-sample (MTS)
format. Following the completion of EBI, participants completed another round of
3Mix assessments and the sorting task.
3Mix Test ofEquivalence Classes
We presented the 3Mix assessment multiple times prior to and following EBI to
assess the relations in each participant’s repertoire. Prior to any dyad entering into
the EBI protocol, or following any dyad exiting the protocol, the experimenters
assessed all participants’ responding to all relations using the 3Mix test. We also
administered multiple 3Mix tests to individual participants when it was necessary
to ensure stable responding prior to intervention. In each case, we controlled for
maturation and environmental changes (see “Experimental Design” section). Dur-
ing the first administration of the 3Mix test, the experimenter said, “Today I want to
see what you know about fractions, percentages, and pictograms. These words may
not sound familiar because this material may not be familiar. Don’t worry if you do

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not know an answer, just do your best so that we know what to teach you.” Then,
the experimenter read the directions at the top of the paper assessment. The exper-
imenter allowed participants up to 40 min to complete the assessment. Typically,
students completed the assessment within 10–15 min. We used the initial admin-
istration of the assessment to aid in deciding who to include in this study (stage-1
inclusion criterion). When the 3Mix assessment was administered following EBI,
90% or higher accuracy was considered mastery-level responding.
Observational Learning Assessment
This assessment indicated whether or not a student would learn novel tact responses
through observation, and therefore if (s)he would be a suitable candidate to serve as
the observer in a dyad (see Table1 for assessment scores). The assessment included
two components: target participant observing novel instructiondelivered to a peer
until the peer met mastery criterion, followed immediately by unconsequated assess-
ment trials with the participant who observed the initial instruction (Delgado and
Greer 2009; Rothstein and Gautreaux 2007). During the observational learning
assessment, each participant in the current study served the role of observer through-
out the assessment. The participant and a peer, who was not part of the current
study, were included in the initial instruction. Prior to instruction, the experimenter
provided the following vocal direction: “[Participant name] I’d like you to watch
while [Peer name] learns the names for [stimuli] (i.e., land forms), [Peer name]
let’s get started learning [stimuli].” During instruction, the experimenter presented
a stimulus to the peer to tact (i.e., name) and provided consequences for each of the
peer’s responses throughout (praise contingent on correct tacts, correction procedure
contingent in incorrect tacts, which involved the experimenter modeling the correct
tact and providing an opportunity for the peer to independently tact). The experi-
menter conducted the sessions such that the participant could see all stimuli and
consequences provided to the peer. The experimenter provided vocal approvals to
the participant (i.e., “Thank you for watching the lesson” or “thank you for listening
nicely”), but she did not require tact responses from the participant, nor did she pro-
vide accuracy feedback for tact responses. Instruction consisted of tacting real world
picture stimuli that were novel to both the peer and the participant (i.e., neither peer
nor participant could name any stimulus in the set prior to instruction). Instruction
included five target stimuli with four exemplars of each target for a total of 20 learn-
ing opportunities per session for a given set of tact stimuli (i.e., landforms, bodies of
water, components of a cell). The experimenter administered instructional sessions
with the same stimuli until the peer responded with 100% accuracy for one session.
Following the peer’s mastery of the tact set, the experimenter conducted a session
where she presented the same set of stimuli to the participant. Prior to presenting the
stimuli she said, “Let’s see how many of the [stimuli] [peer name] learned that you
can name!” The experimenter presented each of the 20 trials to the participant and
required a response, but she did not deliver consequences for accurate or inaccurate
responding. The peer was not present during this assessment. Criterion for the dem-
onstration of OL was 80% accurate responding from the participant (i.e., observer)
during this final, unconsequated session.

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Comparative Relations Sorting Task
We used a sorting task to assess within-set (columns in Table2) comparative rela-
tions (Berens and Hayes 2007; Hayes etal. 2002). Comparative relations are demon-
strated when one responds to an event based on a specific quantitative or qualitative
dimensional relation with another event. In this study, we defined comparative rela-
tions as sorting a set of stimuli from the least to greatest value. The experimenter
conducted the sorting task before and after intervention. During each administration,
we conducted the assessment separately for fraction stimuli, pictogram stimuli, and
percentage stimuli. We also administered the assessment separately to primary and
observing participants. To begin, the experimenter shuffled the eight stimuli in a
set (e.g., fractions) to randomize the order. Next, she placed the pile in front of the
participants and said, “Put the [fractions, percentages, or pictograms] in order from
least to greatest.” The task ended when the participant indicated s/he was done (see
Fig.1). Mastery criterion was set at 100% accuracy for this assessment.
Equivalence‑Based Instruction
We used the simple-to-complex protocol (STC) (Fienup etal. 2015) to train base-
line relations and test for the emergence of derived relations (see Fig. 2). This pro-
tocol intersperses training and derived relations test blocks. Experimenters utilized
learn unit instruction during training blocks, which included an antecedent, a stu-
dent response, and positive reinforcement for correct responses or a correction for
incorrect responses (Albers and Greer 1991). The following is an example of a learn
unit presentation during EBI: (1) the primary participant and the observer attended
to stimuli; the experimenter delivered an antecedent, “Find the one that shows the
same,” while presenting the corresponding sample stimulus (1/4); (2) the experi-
menter waited up to 5s for primary participant’s response; (3) the experimenter pro-
vided specific vocal praise for a correct response (i.e., “Wow, great work finding
the pictogram”) or provided a correction procedure for an incorrect response. From
learn unit to learn unit, the experimenter varied the specific vocal antecedent (e.g.,
“match,” “match (name of sample),” and “find (name of sample). For the correction
procedure, (1) the experimenter represented an antecedent and the corresponding
1/4 3/5 1/41/4 3/5
Put the fracons
in order from least
to greatest
Fig. 1 The picture displays the sorting task, whereby participants sorted a randomized stack of stimuli
from the least to the greatest value

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sample stimulus; (2) the experimenter emitted the correct response; (3) the primary
participant repeated the correct response; (4) the experimenter represented the ante-
cedent; and (5) the primary participant independently emitted a correct response.
During correction procedures, the experimenter did not provide reinforcement for
correct responses. During testing blocks, the experimenter delivered trials in the
3Mix Assessment -All students
Parent Permission and Participant Assent
Tests of Observational Learning
Participant Assignment to Dyads
Sorting Task and Additional 3Mix Assessments
Equivalence-Based Instrutction (STC Protocol)
Train A-B relations with Primary participant to 100%
accuraacy across two 8-trial blocks
Test for B-A symmetry: 100% criterion for both participants in
dyad
Training B-C relations with Primary participant to 100%
accuracy across two 8-trial blocks
Test for C-B symmetry: 100% criterion for both participants in
dyad
Test for mixed symmetry (B-A, C-B): 100% criterion for both
participants in dyad
Test for A-C transitivity
Test for C-A equivalence
3Mix and Sorting Task Post-Assessment
Fig. 2 The order of phases during the study for all participants. During EBI, STC protocol arrows indi-
cate which step participants returned to if remedial training was required

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same manner as training; however, she did not provide consequences for correct
responses nor did she provide corrections for incorrect responses.
Participants completed EBI using the STC protocol (see Fig.2) during which the
primary participant in each dyad participated in MTS instruction designed to teach
eight 3-member equivalence classes. In each training phase, the primary participant
received consequences contingent on her responding while observing participants
watched instruction (experimenter antecedents and consequences, primary partici-
pant’s responses). In order to maintain the observers’ attention throughout training
blocks, the experimenter provided vocal praise for listening, watching, and other
appropriate student behaviors (i.e., sitting). During testing phases, each participant
(primary and observing) completed probe trials separately. During the STC proto-
col, participants began by learning A → B relations. Following the primary partici-
pant mastering A → B relations (100% accurate responses across two 8-trial blocks),
experimenters administered a test for B → A symmetry relations to both participants
independently. We administered the B → A symmetry test in two consecutive 8-trial
blocks, each of which contained one B → A exemplar from each of the eight classes.
Mastery criterion was set at 100% accuracy across all 16 trials to advance in the
protocol. If either participant scored below criterion, we returned to A → B train-
ing. Next, participants began B → C training. Contingent on the primary participant
demonstrating mastery criterion (same as A → B training), both participants moved
to the next step in the protocol. Following B → C training, we administered a test for
C → B symmetry relations followed by a test for mixed symmetry relations (both
B → A and C → B relations), a test for transitivity (A → C), and a test for equiva-
lence (C → A). Mastery criteria for C → B symmetry, transitivity, and equivalence
were identical to B → A symmetry. Mixed symmetry combined trials from B → A
and C → B symmetry and, thus, was administered in two blocks of 16 trials and
mastery was set at 100% accuracy across all 32 trials. The recycle criterion was
applied through the mixed symmetry test and was not applied subsequently. That is,
once participants reached the transitivity and equivalence test phases, we continued
to progress through phases regardless of performance. When participants completed
the test of equivalence, EBI was terminated and the experimenters administered
3Mix post-assessments.
Procedural Modication
We implemented a specific remedial procedure if a primary participant did not emit
criterion level responding following three consecutive derived relations tests (i.e.,
two sets of remedial training sessions). In this case (see Yasmine’s performance),
experimenters isolated the relations the participant responded incorrectly to, trained
those relations, and administered a symmetry test on the isolated relations. Once
the participant emitted mastery criterion on the isolated relations, experimenters
returned them to their original phase within the STC protocol that contained trials
from all eight classes.

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Experimental Design
Experimenters used a multiple probe design across dyads to analyze the effects of
EBI on the emergence of equivalence classes across fraction, pictogram, and per-
centage stimuli. The transfer of comparative relations test was administered at the
beginning and end of the study. Both dyads completed the sorting tasks and multiple
pre-assessment 3Mix tests. Then the first dyad completed the EBI protocol, while
the second dyad remained in baseline. Once the first dyad met mastery on the EBI
protocol, both dyads completed the 3Mix assessments and the second dyad began
intervention. When Dyad 2 emitted criterion level responding for all EBI phases,
experimenters provided 3Mix assessments to all participants as a final post-assess-
ment and completed the sorting tasks with all participants.
Interobserver Agreement
An independent observer collected data for the purpose of calculating interobserver
agreement (IOA). The experimenters calculated IOA using trial-by-trial agreement.
We divided the number of total agreements by the total number of responses and
multiplied the resulting number by 100. An independent observer recorded IOA for
63% of the sorting tasks with 100% agreement. IOA was completed across 100% of
the 3Mix pre- and post-assessments with 100% agreement. We also recorded IOA
during EBI for 47% of sessions with 100% agreement.
Results
Pre‑ andPost‑measures
3Mix Assessments
Figure3 displays participant’s responses to the 3Mix assessments, with black bars
representing primary participants’ performances and italics representing observing
participants’ responses. Dyad 1 was composed of Yasmine (primary participant) and
Kelly (observer). In Dyad 1, Yasmine averaged 42% accuracy across two baseline
3Mix assessments and Kelly averaged 44% accuracy across three baseline 3Mix
assessments. We provided a third baseline assessment to Kelly to assess a poten-
tial increasing baseline trend, which turned out to not be the case. Following the
completion of EBI, Yasmine and Kelly responded with 96–100% accuracy on 3Mix
post-assessments, which was above criterion level responding for class formation.
Dyad 2 was composed of Erica (primary participant) and Joey (observer). In Dyad
2, Erica averaged 40% accuracy across four baseline 3Mix assessments, and Joey
averaged 44% accurate responding across three baseline 3Mix assessments. We
provided a fourth baseline assessment to Erica to assess for a potential increasing

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baseline trend, which resulted in stable responding. Following the completion of
EBI, both Erica and Joey responded above criterion with 100% accuracy on the
3Mix post-assessment.
Transfer ofFunctions ontheSorting Task
Figure4 displays the sorting task outcomes. Before and after the EBI intervention,
we assessed how each participant sorted experimental stimuli from least to great-
est in order to assess comparative relations and the possible transfer of compara-
tive relations following equivalence class formation. Before intervention, Yasmine
responded below criterion level for sorting fractions and pictograms and at 100%
accuracy with percentages. Kelly, Erica, and Joey responded with below criterion
level for sorting fractions and with 100% accuracy for pictograms and percentages.
At baseline, all four participants sorted percentage stimuli accurately, three of the
four participants sorted pictograms accurately, and zero of four participants accu-
rately sorted fractions. After mastering the EBI protocol, all four participants sorted
fractions, pictograms, and percentages with 100% accuracy. Thus, we observed the
transfer of comparative relations from percentage and pictogram stimuli to fractions
stimuli following the formation of equivalence classes.
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
123456
Probe Sessions
Percentage Correct
Primary
Observe
r
Pre-Intervention Post-Intervention
Dyad 1
Dyad 2
Fig. 3 Percentage of correct responding to 3Mix assessments to assess baseline and derived stimulus
relations prior to and following EBI across the primary and observing participants in Dyads 1 and 2

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EBI Performance
Table3 displays each participant’s performance per phase of the EBI STC protocol.
In cases of failed derived relations probes, the cell is marked with a * and the next
phase administered is displayed on the next row. Observing participants only com-
pleted testing blocks (derived relations).
Yasmine required many sessions to acquire A → B conditional discriminations
and required remedial instruction following failed B → A symmetry probes. After
the third failed symmetry probe, we noticed a distinct error pattern and implemented
a tactic (described in procedures) to isolate the specific relations. Yasmine mastered
those isolated relations in four sessions (see fourth row of Yasmine’s data), passed
symmetry tests of those relations, mastered all eight A → B relations in four more
blocks (fifth row of Yasmine’s data) and passed the symmetry probe for all eight
B → A relations. During this time, Kelly observed A → B training and passed the
initial B → A symmetry probe and all subsequent probes. Yasmine passed B → C
training in two blocks and met the criterion for demonstrating the emergence of
C → A symmetry, mixed symmetry, transitivity, and equivalence. Kelly observed
B → C training and passed all derived relations probes. Prior to mastery, training
phases were recycled four times due to Yasmine’s failure to derive relations with
100% accuracy during B → A symmetry test blocks (see Table3).
In the second dyad, Erica quickly learned A → B relations and B → C relations
and passed tests of symmetry in isolation. Erica responded incorrectly to one mixed
symmetry trial, and the dyad was placed back in A → B training. She quickly passed
all baseline training sessions and derived all relations during the testing blocks.
0
10
20
30
40
50
60
70
80
90
100
Yasmine
Pre-EBI Post-EBI
0
10
20
30
40
50
60
70
80
90
100
Kelly
0
10
20
30
40
50
60
70
80
90
100
Fractions Pictograms Percentages
Erica
0
10
20
30
40
50
60
70
80
90
100
Fractions Pictograms Percentages
Joey
Sets of Stimuli
Percentage Correct
Dyad 1 Dyad 2
Fig. 4 Percentage of correct responding to sorting fractions, pictograms, and percentages to assess com-
parative relations prior to (gray bars) and following EBI (black bars) across the primary and observing
participants in Dyads 1 and 2

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Table 3 Performances during EBI
Any number with * indicates that responses were below criterion level and remedial training was required. In these cases, data for the remedial training are displayed on
the next row. The italicized numbers indicate that a modification was necessary for a participant (i.e., Yasmine, see procedural modification description in “Method” sec-
tion)
Relation A → B B → A B → C C → B Mixed A → C C → A
Type Baseline Symmetry Baseline Symmetry Symmetry Transitivity Equivalence
Data # 8-trial blocks % correct out of 16 # 8-trial blocks % correct out of 16 % correct out of 32 % correct out of 16 % correct out of 16
Dyad 1 Yasmine (primary) 29 75*
5 88*
9 81*
4 100*
2 100 2 100 100 100 100
Kelly (observer) 100
100
100
100
100 100 100 100 100
Dyad 2 Erica (primary) 9 100 2 100 *94
2 100 2 100 100 100 100
Joey (observer) 100 100 100
100 100 100 100 100

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Joey observed A → B and B → C training and passed all testing blocks with 100%
accuracy. Prior to mastery, training phases were recycled once due to Erica’s fail-
ure to derive relations with 100% accuracy during mixed symmetry test blocks (see
Table3).
Discussion
The goal of education is to create conditions that foster generative learning in the
classroom to improve the efficiency of learning (Stromer etal. 1992). To this end,
we implemented EBI that included training two relations (fraction → pictograms;
pictograms → percentages) and found that this resulted in participants acquiring
an additional four conditional discriminations and demonstrating the formation of
equivalence classes. In this manner, our study replicated the outcomes reported by
Lynch and Cuvo (1995) with third-grade students who received EBI as part of their
regular education in a public school setting. This study also served to join the obser-
vational learning literature with the derived relations literature to show how obser-
vational learning can be paired with EBI to further conserve educational resources
while maximizing educational outcomes. Two observing participants observed pri-
mary participants learn two baseline relations and those observing participants also
demonstrated derived relations and equivalence class formation.
This study contributes to the derived relations literature by demonstrating the
transfer of functions within equivalence classes with socially significant stimuli.
While participants could accurately sort pictogram and percentage stimuli from least
to greatest (comparative functions) at the start of the study, it was not until partic-
ipants formed equivalence classes that they could accurately sort fraction stimuli.
This was accomplished with no explicit training for both primary participants and
observers. While the majority of the transfer of function literature is basic research,
there are a few applied examples of transfer of function that this study joins. As
mentioned above, Rosales and Rehfeldt (2007) showed that when classes are formed
with a stimulus that serves the function of mand topography, the other stimuli in the
class begin serving the same function. Taylor and O’Reilly (2000) demonstrated that
insitu training for how to shop at a supermarket paired with equivalence training to
equate different supermarkets produced the same generalization of shopping skills
to a novel setting as multiple exemplar training (in situ training in three different
supermarkets) and better outcomes than generalization following single exemplar
training. Taylor and O’Reilly demonstrated that setting-specific skills can general-
ize through equivalence networks and accomplish the same outcomes as the more
timely task of conducting insitu training across multiple settings.
While the outcomes of this study and the extant literature are promising, future
research should focus on expanding our knowledge base. Mathematics instruction is
one area where the benefits of EBI can be realized. With college students, research-
ers have taught algebra and trigonometry repertoires using EBI (Ninness etal. 2006,
2009) and Lynch and Cuvo (1995) taught similar content to that reported in this
study. More research is needed on mathematics instruction in general, and research
conducted with school age children, specifically. In an academic setting, spiral

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Journal of Behavioral Education (2020) 29:741–762
curricula may be one context in which researchers and educators can take advantage
of programming for derived relations. One such application in an elementary class-
room involves teaching the relations between geometric shapes and their respective
attributes. Over the course of several lessons, students may contact various relations
that form an equivalence class. For instance, a unit on geometric shapes typically
incorporates a lesson on the name of the shape in the presence of the shape stimuli
and a lesson on the number of sides for a shape in the presence of the shape stim-
uli, which could result in derived relations such as students producing the name of
a shape when given the number of sides. Subsequent lessons will introduce tacts
for polygon categories (i.e., polygons and polygons that are quadrilaterals); if all
relevant relations emerged and students learned the definition of a quadrilateral,
students should be able to identify shapes that fit into respective categories based
solely on the name of the shape without direct instruction (i.e., Trapezoid → four
sides → polygon that is also a quadrilateral). Thus, in the process of introducing
skills, instructors can plan for the emergence of equivalence classes to produce gen-
erative learning.
Another avenue for future research is to examine the underlying principles of
derived relations and transfer of function. Researchers have suggested that naming is
fundamental to deriving relations (e.g., Horne and Lowe 1996; Miguel 2018). One
behavior we observed during the sorting tasks was that participants tacted the per-
centage while sorting fraction stimuli. For example, a participant would hold the
stimulus ¼, tact it as “25%,” and then place it on the table in the correct sequence (at
post-assessment). This is an example of common naming that has been proposed as
a problem-solving repertoire (Miguel 2018). If it is the case that naming underlies
the transfer of function in some cases, then educators should focus on developing
the naming repertoire (e.g., see Greer and Longano 2010), prior to expecting derived
relations and the transfer of functions. If some other process underlies transfer of
function (e.g., classical conditioning, Dymond and Rehfeldt 2000), then educators
would not need to be concerned with this behavior. A more thorough understanding
of the behavioral repertoires required would help educators understand with whom
transfer of function would be efficacious.
Despite its promise, this study contains limitations that can be addressed in future
research. One limitation of this study was the required procedural modifications that
we implemented with Yasmine. Yasmine’s rate of learning across classroom objec-
tives in reading, writing, math, and spelling was consistently high (high number of
learn units to master short-term objectives), and we often implemented remedial
procedures for her to master academic objectives. Thus, implementing a procedural
modification was not unlike her typical instructional needs. Another limitation was
that we only assessed 3Mix performances once or twice after the completion of EBI,
and additional assessments would lead to more robust findings. A third limitation
was that Kelly, who acted as the observer with Yasmine, could not move forward
in the STC protocol when Yasmine required remedial training. In future research or
in naturalistic educational settings outside of experimental control, students could
be yoked based on developmental cusps and capabilities (i.e., OL) as well as their
rate of learning. This would ensure that students in the same dyad (or group) move
through instruction at a similar pace (Greer and Ross 2008).

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The benefits of the procedures described in this study also depend on whether
individuals can learn through observation. In the derived relations literature, there
are a few examples of participants deriving relations following observation; how-
ever, those studies never formally assessed the presences of OL before starting the
study (e.g., Ramirez and Rehfeldt 2009). Our study is the first to establish the pres-
ence of OL prior to training, and this could be critical to the success of deriving
relations and transfer of function simply through observing baseline training. In the
OL literature, researchers have demonstrated that individuals with OL alter perfor-
mance behaviors, acquire new operants, and new reinforcers once this repertoire is
established (Singer-Dudek etal. 2011). For instructors whose students do not pos-
sess this cusp, to benefit from the procedures in this study the instructor would have
to induce this repertoire. Several studies have demonstrated the effective use of peer-
monitoring, peer-yoked contingencies, and peer tutoring to induce the OL capability
for students who did not have it in repertoire (Delgado and Greer 2009; Rothstein
and Gautreaux 2007).
The intersection of derived relations and OL is one way educators can promote
generative learning. To date, there are limited applications of EBI in naturalistic
environments (e.g., Fienup etal. 2010) and fewer still to demonstrate utility with
school-aged students (Lynch and Cuvo 1995). As most applications of EBI are one-
to-one instruction, the path toward regular education applications requires instruc-
tional strategies that fit the classroom resources. Incorporating OL is one way to
capitalize on the skill sets that may be present in regular education classrooms.
Researchers should also investigate how EBI maps onto other regular education
pedagogies, such as group instruction (for an example with college students, see
Verales and Fields 2017) or peer tutoring. Peer tutoring, a widely supported class-
room teaching procedure (Alegre etal. 2018), can be implemented class wide or
within small groups such that the teacher is only required for supervision of peer’s
instructional accuracy. Peer tutoring is one tactic that can be utilized to increase stu-
dents’ rate of learning, allow for instructional flexibility, and induce OL. When peer
tutoring is implemented with learn units and homogenous dyads (i.e., novel mate-
rial for both students in the dyad), student accuracy for the novel material signifi-
cantly increases through direct instruction as the tutee and by consequating correct
and incorrect responses as the tutor (Greer etal. 2004). Future steps for the current
research might incorporate EBI within a peer-tutoring context such that tutors train
baseline relations to further apply generative learning technologies within the class-
room environment. During peer-tutoring instruction, participants would have direct
contact with one relation (i.e., A → B) as a tutee, and indirect contact with the other
relations (i.e., B → C) as a tutor (Greer et al. 2004). Serving in the roles of both
tutor and tutee across a number of baseline relations training would likely result in
children deriving relations and forming equivalence classes as was observed in this
study.
Compliance with Ethical Standards
Conict of interest All authors declare that they have no conflict of interest.

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Journal of Behavioral Education (2020) 29:741–762
Ethical Approval This study was approved by the Institutional Review Board of the respective university.
We obtained parent permission and child assent for each participant described in this manuscript.
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