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INTRODUCTION

Stimulus equivalence research is a growing area of study in the field of behavior

analysis. Behavior analysts use stimulus equivalence approaches and procedures to study

discriminated operant learning, emergent behavior, basic acquisition of symbolic

relations, and language relations. The applied use of stimulus-equivalence procedures

can provide teaching efficiency in an educational setting by teaching only a few

conditional relations and providing the opportunity to allow many more relations to

emerge.

Introduction to Stimulus Equivalence

Researchers who study stimulus equivalence use the mathematical definition of

equivalence based on three relational properties in order to infer that an equivalence

relation holds between a set of stimuli. In order to apply the term equivalence to a

relation between stimuli, the three properties of symmetry, reflexivity, and transitivity

must all be demonstrated. These three properties, identified by mathematical set theory,

are defined as follows. A reflexive relation is one in which an entity is equal to itself (to

use the common equivalence notation this property would imply that, with respect to the

three set members A, B, and C, A = A, B = B, and C = C). A symmetric relation would

hold that if A = B, for example, then B = A. The transitive relation holds that if A = B

and B = C, then A = C. Behavior analysts believe that this mathematical definition of

equivalence will hold true when studying behavior in the context of stimulus equivalence

(e.g., Sidman & Tailby, 1982; Sidman, 1994).

In order to discuss the operant behaviors involved in stimulus equivalence,

Sidman (1986) proposed that the basic analytic unit of behavior is the two-term

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contingency, or a response-stimulus relationship. Contingencies can be defined as

dependent relationships and can be described by such statements as: “if this, then that,”

and “if not this, then not that.” Sidman begins by defining “response” as the observed

behavior in any particular instance. If a specific response is reinforced (and only that

response produces the reinforcer), a two-term contingency is created. For example if a

subject presses a button, food is made available. Many parameters can affect this

contingency (schedules, deprivations, etc.), but in essence the probability of the response

occurring is increased by presenting a reinforcer. If another term is added to this

contingency, a stimulus, or variation in the subject’s environment, then a three-term

contingency is created. The three-term contingency works as follows: if stimulus (and

no other stimulus), then response (and no other response), then reinforcer. For example,

if stripes are present on the button, and the subject presses the button, then the food is

made available. The stimulus (the stripes on the button) sets the occasion for the original

two-term contingency and places the original two-term contingency under stimulus

control. The stimulus in the three-term contingency is also referred to as a discriminative

stimulus. If the subject presses the polka-dotted button, food is not made available. With

each term added, the scope of behavior analysis is broadened, and new processes are

exposed for further investigation. The three-term contingency makes generalized

reinforcers, second-order schedules, and chaining possible (Sidman, 2000).

When a colored light is added to the above example, a four-term contingency can

be created. If the light is red and the striped button is pressed, food will be made

available; pressing the polka-dotted button has no scheduled consequence. However, if

the light is green and the polka-dotted button is pressed, food will be made available

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while pressing the striped button produces nothing. The fourth term (the light) places the

three-term relation under conditional control, because the three-term relation only holds

in the presence of the relevant fourth term. These conditional relations may then become

equivalence relations. In the above example, the red light and striped button may become

equivalent. To determine whether this has, in fact, occurred, the relations in the above

example must be tested for the emergence of the three properties described above:

reflexivity, symmetry, and transitivity.

The procedure used most commonly to teach conditional discriminations and test

for emergent equivalence relations is referred to as a match-to-sample procedure. A

typical match-to-sample session includes several trials in which stimuli are presented and

responses are required. On each trial of this procedure a sample stimulus is presented and

then an observing response is typically required to ensure that the participant is attending

to the stimuli. The observing response, usually touching the sample stimulus, results in

production of the comparison stimuli. The subject must then choose one of the

comparisons, depending on the sample presented. A correct response results in

presentation of a reinforcing consequence. An incorrect response may result in a buzzer

or similar feedback. This procedure is used to teach conditional discriminations where

the sample is the conditional stimulus, the comparisons are the signaling or

discriminative stimuli, the response is the participant’s choice of one of the comparisons,

and the reinforcer is the consequence. To test for the properties of equivalence this same

procedure is used, but the test trials are called probe trials and do not include

reinforcement or any feedback after a response is made, to prevent training the relations.

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The specific tests used to reveal emergence of equivalence relations depend on the

training structure used to teach the original conditional discriminations, but usually

consist of three basic types of tests: reflexivity tests, symmetry tests, and transitivity

tests. If the subject’s performances on these tests reveal that these same three properties

of equivalence have emerged, then equivalence classes are said to have formed. For

example, if a subject is taught an A to B conditional discrimination and a B to C

conditional discrimination, where the first letter mentioned refers to the sample stimulus

set and the second letter mentioned refers to the comparison stimulus set, then several

tests must be completed to reveal the three properties of equivalence. Reflexivity tests

would be presented to evaluate whether the subject can match A to A, B to B, and C to C.

Symmetry tests would be presented to evaluate whether the subject can match B to A and

C to B. Transitivity tests would be presented to evaluate whether the subject could match

A to C. A fourth test, sometimes referred to as a combined equivalence test, would test

for both symmetry and transitivity at once. Using the same baseline conditional

discriminations, A to B and B to C, a combined equivalence test would evaluate whether

the subject could match C to A. If the training structure was slightly different and instead

the subject was taught the baseline conditional discriminations A to B and A to C, then a

standard transitivity test is not possible because neither B nor C stimuli have been

presented as sample stimuli. In this case, combined equivalence tests (BC and CB) would

be given instead. When relations between more than three sets of stimuli are trained,

more creative combined equivalence tests are used; however, all three properties of

equivalence must be shown.

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Sidman (1971) viewed reading as involving stimulus-control relationships in three

categories. The three categories are oral reading (e.g., showing the subject a printed

word, and she says the word or vice versa), reading comprehension (e.g., showing the

subject a printed word and she selects a picture of the corresponding object from several

other pictures of objects or vice versa), and auditory-receptive reading (e.g., saying a

word to a subject and she selects the corresponding printed word from among several

other objects or vice versa). Auditory-receptive reading may or may not involve reading

comprehension, and reading comprehension may not imply skills of oral reading and

auditory-receptive reading. Sidman’s expectation was that if the subject was taught to

match the spoken word “car” to the printed word “car”, and he could already match the

spoken word “car” to the picture of a car, he would then be able to match the printed

word “car” to the picture of a car, and orally name the printed word “car”. In the study, a

mentally retarded male was given baseline tests to expose his existing repertoire of

simple comprehension and naming. He was then taught to match spoken words to

printed words, and finally he was tested for the new relationships that might emerge

through equivalence.

The subject was trained and tested using a match-to-sample procedure by having

him touch one of nine windows arranged in a matrix; the windows contained the

projected images of the stimuli (samples and comparisons). One sample was presented in

the middle window; the subject had to touch the sample, which produced the comparisons

displayed in the surrounding windows. He then responded by touching one of the

comparison windows. Sidman found evidence in his results to support his prediction.

The subject showed skills of reading comprehension and oral naming without those

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relations being explicitly taught. His reading repertoire expanded by twenty words

through this experiment.

Researchers who study equivalence relations usually do so by teaching at least

two conditional discriminations, then testing or probing for the emergent relations that

define equivalence. Sidman and Cresson (1973) were interested in replicating Sidman’s

(1971) previous work. They studied acquisition of reading skills by two Down Syndrome

boys as a result of a teaching technique that utilized an equivalence approach. The

experimenters wanted to know if auditory-receptive learning must precede reading

comprehension, and if an equivalence relation between auditory and visual stimuli would

emerge after teaching relevant prerequisites. The relations studied in equivalence

research are described using a notation where capital letters are used to refer to types or

sets of stimuli (e.g., spoken words may be referred to as A stimuli) and numbers are

sometimes added to refer to specific examples or members of a set (e.g., the spoken word

cat may be referred to as A1). The subjects in the present study were given baseline

tests to evaluate their ability to discriminate between printed words (signified as a C-C

relationship, C being printed word stimuli, the first letter in the pair referring to the

sample, and the second to the comparison), to match spoken words to pictures (A-B), to

see a picture and orally name it (B-A), to hear a spoken word and match it to the printed

word (A-C), to match a picture to a printed word (B-C), to match a printed word to a

picture (C-B), and to see a printed word stimulus and orally name it (C-A). These skills

could also be classified as identity matching, auditory comprehension, naming, auditory-

receptive reading, reading comprehension, reading comprehension, and oral reading,

respectively.

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After baseline testing determined that neither subject was able to perform these

skills, the experimenters taught the subjects identity matching (C-C), and then two

arbitrary conditional discriminations: auditory comprehension (A-B), and auditory-

receptive reading (A-C). After this baseline training, a series of tests revealed that with

no further training, the subjects were able to perform reading comprehension matching-

to-sample tasks (C-B and B-C) and read orally (C-A). The subjects learned twenty more

discriminations for each stimulus set than were explicitly taught, giving this finding

implication for teaching efficiency. This finding also suggests that equivalence

procedures may be an effective teaching procedure for teaching reading comprehension

to the mentally handicapped. This research expanded the implications for learning

theory and teaching techniques presented by Sidman in his original experiment (1971).

Sidman and Tailby (1982) were interested in how conditional discriminations

trained via a match-to-sample procedure could result in stimulus equivalence when more

than two conditional discriminations were trained. In this study the authors introduced a

paradigm for determining when an equivalence relation exists between stimuli by testing

for reflexivity, symmetry, and transitivity. They also improved and expanded on earlier

endeavors in the field of stimulus equivalence (Sidman 1971, Sidman & Cresson 1973).

Instead of three stimulus-class members as in previous research, this study tested the

possibility of expanding equivalence classes by adding a member to each stimulus class.

This study also improved on technique by teaching and testing stimuli with which the

participants had no history (i.e. Greek letters), and they tested for each property of

equivalence. The study involved eight normally developing children in kindergarten and

first grade who were given match-to-sample trials on a screen positioned in front of them,

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with circular windows on which the stimuli were presented. Before teaching the

equivalence relations, the subjects were given baseline tests that required matching hue

names to colors and identity matching with Greek letters. These tests let the

experimenters evaluate how well the participants could match hue names to colors and

Greek letters to themselves while also helping the participants become familiar with the

trial procedure. The baseline training that followed established three conditional

discriminations of the relationships referred to as A-B, A-C and D-C. The reinforcer

density was then reduced from 100% to 50% to prepare the subjects for the probe tests on

which reinforcement was never available. The probe trials were designed to test for any

emergent relationships. The probe trials were mixed with baseline trials that sometimes

included reinforcement to assure that the trained relations were maintained at high

accuracy levels. Six of the eight subjects immediately matched B and D stimuli; these

relations also require that B is equal to C and C is equal to B. They also matched D

comparisons to A and C stimuli. The participants showed that the stimuli were

equivalent by also demonstrating symmetry, reflexivity, and transitivity. The results of

the final evaluation of acquired relations showed a 27:9 ratio of emergent to directly

taught performances, a 2.5-fold increase in skill acquisition, by adding another member

to each stimulus class. This study has implications for teaching that go far beyond Greek

letters; the equivalence procedure used in teaching can improve the effectiveness of

teaching by expanding the number of acquired skills just by teaching a few relationships.

The Importance of Studying Stimulus Equivalence

Studying stimulus equivalence gives insight into many realms of behavior. In the

world of human social interaction, words are often used as symbols for actions, objects,

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places, people, feelings, and many other referents (Sidman, 1994). Using words as

symbols is verbal behavior and should be analyzed as any other behavior is, for its own

right. Symbols often evoke the same feelings and actions that their referents do, making

the symbols equivalent in some ways to certain properties of the things they symbolize.

Examples of these relations are everywhere: humans may react to insults as physical

attacks, symbolic actions that initiate fear may conclude in war, and attacking political

figures may be seen as attacking the countries they represent. Studying stimulus

equivalence will give insight into the basic acquisition of symbolic relations by studying

the process(es) that lead to an equivalent relation between symbols and their referents.

Stimulus equivalence can also help experimenters study the relations involved in

language. How is it that the word bread evokes the thought of a slice of edible bread, yet

readers do not usually eat the page where the word bread is printed? Studying language

relations using stimulus equivalence can help provide researchers with a scientific

approach to understanding the meanings of words for a particular individual, and the way

these meaning are learned. We can also use stimulus-equivalence procedures to establish

the role language relations play in equivalence relations. If animals can acquire

equivalence relations involving arbitrary symbols, then the necessary role of language,

thought of as unique to humans, may be discounted (e.g., Kastak, Schusterman, &

Kastak, 2001).

Behavior analysts have also been interested in behaviors and relations that emerge

without being explicitly taught. Stimulus equivalence is one procedure that can be used

to study emergent behaviors and emergent relations. When two related conditional

discriminations are taught and equivalence relations emerge, researchers can learn more

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about the conditions under which this emergence occurs and how to create those

conditions. Stimulus equivalence can give general insight into the area of emergent

behaviors and relations.

If stimulus-equivalence procedures can help provide effective and efficient

learning, then used in the classroom, these procedures can help students learn a variety of

equivalent relations. Stimulus-equivalence procedures can be used to teach language

relations and reading comprehension (e.g, Sidman, 1971), word-number equivalences

(e.g., Gast, VanBierlet, and Spradlin, 1979), fraction-decimal equivalences (e.g., Leader

and Barnes-Holmes, 2001 and Lynch and Cuvo, 1995), and many more areas of

academics taught in schools (Stromer, MacKay, and Stoddard, 1992). Equivalence

procedures can also enhance teaching efficiency in that only a few conditional

discriminations are taught and several more emerge. This can help teachers save time

and effort, and the more that is learned about equivalence teaching procedures, the more

effective the procedure can be in classroom teaching.

Theories of the Origin of Equivalence

There are three major theories of how equivalence relations are established:

Relational Frame Theory, Naming Theory, and Sidman’s Theory involving reinforcement

contingencies. Each theory is an attempt to explain the origins of the emergent behavior

that is seen to occur in match-to-sample procedures of stimulus equivalence. A brief

summary of the underlying principles of each theory is given below.

Relational Frame Theory. Relational frame theorists hold that an individual’s

prior learning history establishes success on equivalence tests. This learned history is

termed arbitrarily applicable relational responding and involves responding to one event

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in terms of another event in a certain manner due to contextual cues (Hayes, Fox, Gifford,

Wilson, Barnes-Holmes, & Healy, 2001). Relational responding according to a particular

pattern is described in terms of a relational frame or a behavioral class that is overarching

and functionally defined.

Relational frame theorists provide three primary properties of arbitrarily

applicable relational responding: mutual entailment, combinatorial mutual entailment,

and transformation of stimulus function. Mutual entailment relations are bidirectional;

for example if A is related to B, then B is related to A. The equivalence relation of

symmetry described above would be seen as one example of this more general term.

Combinatorial mutual entailment is a more complex relation involving more than two

stimuli. For example, if A is related to B and B is also related to C, then A is also related

to C in some way. Transitivity or combined equivalence relations would be seen as one

example. Transformation of stimulus function occurs when a function assigned to one

stimulus in a relational network is entailed to another stimulus in that same relational

network. For example if A1, B1, and C1 participate in a relational frame coordination,

and a reinforcer function was established for stimulus A1, stimuli B1 and C1 should also

function as reinforcers.

Relational frames are indicated when a relation that was explicitly trained in the

past is applied to novel stimuli, is contextually controlled, and possesses the three

qualities of mutual entailment, combinatorial mutual entailment, and transformation of

stimulus function. Examples of relational frames are the frame of coordination (relation

of sameness or equivalence), the frame of opposition (stimuli that are opposites of one

another), frames of distinction (more than/ less than, hotter/ colder, etc.), and endless

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other relations of comparison. Thus, relational frame theorists argue that new

equivalence relations emerge because a relational frame of coordination (including

mutual entailment, combinatorial mutual entailment, and transformation of stimulus

function) has been explicitly taught with multiple stimulus examples in the past.

Participants who succeed on equivalence tests have “abstracted” the necessary relational

frame and applied the relation to the novel stimuli presented to them (Hayes, et.al., 2001).

Naming Theory. The naming theory holds that symbolic behavior and stimulus

classes, including equivalence classes, are established through processes involving

naming. Specifically, a bidirectional process occurs when individuals name stimuli and

this naming relation is the basic unit of verbal behavior on which more complex relations

may build. Horne and Lowe (1996) suggest that the naming that occurs, during match-to-

sample procedures for example, brings the named stimuli into equivalence classes.

Several studies on stimulus equivalence (Dugdale & Lowe, 1990) have shown that

participants name the stimuli used during training, and this is used by naming theorists to

support the claim that naming is an higher-order behavioral relation that occurs with

stimulus equivalence.

Horne and Lowe (1996) propose that the naming process is learned by children

early in life, as they are taught to classify objects. Naming involves a circular process by

which a child is exposed to and participates in listener and speaker behavior. A young

child may see an object, for example a car, and hear a caregiver who notices this

observing behavior say the word car. The child also hears the word car and learns to look

at the car when the word car is heard. The child then learns to echo or speak the word car

when the car is observed through listening and imitating the caregiver and receiving

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social praise. The child then hears himself speak the word car in the presence of the car,

which will then be reinforced by the caregiver and eventually the verbal community in

other instances. Hearing himself say the word car leads to the child looking at the car,

thus the bidirectional relationship occurs. When the child reliably speaks the word car in

the presence of the car stimuli, after being reinforced through this circular relation, he has

learned to name. Names are words that are thought to “refer to,” “mean,” or “symbolize”

objects. Through multiple interactions with various cars the child will learn to name

other automobiles he observes as cars, and this will bring many types of cars into an

equivalence class named “car.” This common name or functional name allows the

various cars to enter into the “car” equivalence class, even if they are very different

physically.

Naming each individual stimulus in an equivalence class by a different name can

also establish equivalence through intraverbal naming (Horne & Lowe, 1996). When

names of items reliably follow each other, they can enter into an equivalence class

together. For example, children often hear “salt and pepper” or “fork and knife” spoken

together in pairs, and when one is emitted it provides an occasion for the other. As a

result of these paired names, one item in the pair may be chosen in the presence of the

other on equivalence tests. Thus naming theorists propose that success on equivalence

tests is not due to emergent behavior but rather to the relations that are established when

listener behavior results in learning a stimulus name. Participants in equivalence

procedures may be overtly or covertly naming, which brings about equivalence relations,

or they may be using other versions of naming (e.g., verbally confirming trained relations

such as, “this goes with this” or “the square is the same as the green light”).

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Sidman’s Theory. Based on existing research in the area of stimulus equivalence,

Sidman (2000) presented a clear and detailed summary of his theory on the origin of

equivalence relations. Sidman argues that equivalence is a fundamental behavioral

process that is not derivable from any other, nor can it be broken down into smaller

behavioral processes. He stated that equivalence relations are a direct result of

reinforcement contingencies. Reinforcement contingencies generate equivalence

relations and analytic units in the form of operant reinforcement (the two-term

contingency), simple discrimination (the three-term contingency), conditional

discrimination (the four-term contingency), second-order conditional discrimination (the

five-term contingency), and so on (n-term contingencies). A complete description of a

four-term contingency must contain more than one option or comparison to choose from

when given a sample in order to create a true learned discrimination. These options must

include specified responses and reinforcers so as to clarify the relations between all the

analytic terms in the contingency.

According to Sidman (2000), when responses and reinforcers are specified and

tested using a procedurally valid method, the resulting equivalence classes contain pairs

of all positive units in the reinforcement contingency. This means that the sample,

comparison, correct response, and the reinforcer would all be equivalent and thus

interchangeable. When a common reinforcer or common response is used in conditional

discrimination training procedures, all elements would become equivalent, via the

common elements, and prevent formation of separate equivalence classes. Instead,

collapse into one large equivalence class might be expected. In order for separate classes

to form based on procedures using a common response or a common reinforcer, these

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common elements must drop out of the equivalence class. This prevents the collapse of

all contingency elements into one large equivalence class, and makes the use of a single

generalized reinforcer possible. If, in contrast, conditional discriminations are taught

using differential reinforcers and differential responses, these two terms of the

contingency may then be included as members of the equivalence relation.

To test this theory, reinforcers specific to each equivalence class can be added to

the contingency. Class-specific reinforcement, as it is often called works as follows. In

the presence of sample A1, the participant chooses the comparison B1 and receives the

consequence, Reinforcer 1 (R1). Similarly, in the presence of sample A2, the participant

chooses B2 and receives R2. Each element in a particular equivalence class is reliably

related to the same particular reinforcer. Through the reinforcement contingency,

Sidman’s (2000) theory predicts that the class-specific reinforcer may become a member

of the equivalence class. Similarly, the use of differential responses in conditional

discrimination contingencies may cause the response to become a part of the equivalence

class as well. Arranging for a differential response works as follows. In the presence of

sample A1, the participant chooses B1 by using Response 1. Similarly, in the presence of

sample A2, the participant chooses B2 by using Response 2. For example, in choosing

B1 or any stimulus in the equivalence class 1, the participant may respond by pressing a

button while, when choosing a member of the equivalence class 2, the participant may

respond by pulling a bar. Sidman (2000) predicts that through the reinforcement

contingency, the differential response may become a member of the equivalence class to

which it is related. When all elements of the contingency are tested properly for

equivalence-class membership, Sidman describes the relationship in terms of the “bag”

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analogy. Any pair of elements pulled out of the bag containing all elements of the

contingency, will prove to be equivalent.

Class-Specific Reinforcement

Although many of the experiments outlined by Sidman as tests of his predictions

have not been conducted, several studies have suggested that class-specific reinforcement

has been successful in training conditional discriminations. This review will focus on

work conducted with human participants, but a number of related studies have involved

animals as well (Brodigan & Peterson, 1976; Delamater, 1996; Kastak, Schusterman, &

Kastak, 2001). Litt and Schreibman (1987) studied the effectiveness of class-specific

reinforcement using a two-choice receptive-label discrimination task and three different

reinforcement procedures: stimulus-specific, “salient”, and varied reinforcement. Six

male children with autism were assessed to determine which of several consequences was

most reinforcing and which of the less reinforcing consequences were equally desirable.

Each subject was then trained in receptive label discriminations between pairs of objects

(barrette-hinge, solder-compass, file-clip, and nail-socket) in each of the three

reinforcement conditions. In the stimulus-specific reinforcement condition, the

experimenter placed two objects side by side in front of the subject and asked the subject

to “Give me ____.” If the child responded correctly, he was given an edible reinforcer

that was specific to that stimulus. A different reinforcer was given when a different object

was correctly chosen on other trials. These reinforcers were the equally desirable ones.

If the subject responded incorrectly, the experimenter punished this response with a

contingent “No!,” and proceeded to the next trial. The salient reinforcer condition was

exactly as described above except that correct responses were reinforced with a single

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highly desired reinforcer, regardless of the stimulus chosen. The varied reinforcement

condition was conducted as described above except that correct responses were

reinforced with varied reinforcers, selected on a random basis from the equally desirable

reinforcers. This condition was implemented to control for the effects of reinforcer

variation. The results of this study indicated that the stimulus-specific reinforcement

procedure produced faster acquisition of the receptive label discriminations than the other

two conditions, with the single reinforcement procedure being the least effective. This

study suggests that manipulation of the reinforcer can result in a higher level of

motivation in the subject. Boredom can be combated by stimulus variation, but variation

accompanied by predictability of reinforcement may speed acquisition. This can be

accomplished by stimulus-specific reinforcement as long as the reinforcers are equally

desirable. This study suggests implications for alternative and effective procedures for

teaching children with developmental disabilities.

To investigate further the acquisition of conditional discriminations with class-

specific reinforcement, Schomer (2001) reported a study in which stimuli used as class-

specific reinforcers became members of the equivalence classes. In this experiment,

children were taught two conditional discriminations with abstract stimuli and either

received class-specific reinforcement or a single common reinforcer. All children who

received class-specific reinforcers readily acquired the discriminations. The children

who received a single reinforcer during training did not acquire the discriminations until

class-specific reinforcers were added. This study also included reinforcer probes, which

determined that the reinforcers used in training did become part of the equivalence class.

This finding supports Sidman’s (2000) theory that equivalence relations, including those

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between stimulus and reinforcer, are directly generated and maintained by the

contingencies put in place by the experimenter.

According to Sidman (2000), class-specific reinforcement would be expected to

increase the effectiveness of equivalence procedures by preventing the equivalence-class

collapse that could occur with a common reinforcer. When class-specific reinforcers are

used, there is no need for the reinforcer to drop out of the equivalence class (as in cases

with common reinforcers), and the reinforcer can become part of the equivalence class.

This can also increase efficiency of equivalence teaching procedures by adding a stimulus

as a reinforcer that could then become part of the equivalence class. Several

experimenters have shown that class-specific reinforcers become members of the

equivalence class (e.g., Dube, McIlvane, MacKay, & Stoddard, 1987; Dube, McIlvane,

Maguire, Mackay, & Stoddard, 1989).

Ashford (2003) analyzed the effects of compound class-specific reinforcers in

equivalence performances with developmentally disabled children. The apparatus used

in this series of experiments was a computer match-to-sample program combined with

three types of reinforcers: conditioned visual and auditory reinforcers produced by the

computer, and primary food reinforcers that were determined for each child during initial

preference tests. In one experiment, developmentally disabled children learned arbitrary

three-choice conditional discriminations using class-specific conditioned and primary

reinforcers (e.g., choosing comparison B1 in the presence of sample A1, produced cSR1

and pSR1 , while selection of B2 in the presence of A2 produced cSR

2 and pSR2). The

participants were trained AB conditional discriminations and upon mastery they were

taught either AC or CD arbitrary conditional discriminations. They next received probes

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testing for equivalence-class formation (i.e., symmetry, combined equivalence, and

reflexivity). After reaching stability criterion for the equivalence probes, the subjects

were tested to determine whether the reinforcers had become members of the equivalence

classes. These probes included trials where reinforcer (conditioned or primary) images

were presented as samples with A, B, C, (or D) stimuli as comparisons; A, B, C, (or D)

stimuli as samples and reinforcer (conditioned or primary) images as comparisons; or

reinforcer components as both samples and comparisons. The results revealed that five

of the six participants exhibited equivalence-class formation following their training. The

reinforcer probe tests indicated that both members of the compound reinforcers, the class-

specific conditioned and primary reinforcers, became members of the equivalence

classes. The subject who learned two unrelated conditional discriminations (AB and CD)

in baseline training using class-specific reinforcers showed equivalence on the probe tests

and this result suggests that the class-specific reinforcers acted as nodes for class

membership between unrelated conditional discriminations. The only subject who did

not show equivalence-class formation on probe tests (original training was unrelated AB

and CD) did in fact show class-consistent responding when marking on an extended tally

sheet including reinforcers and other class stimuli.

In another experiment, three subjects who had completed the previous experiment

were trained with identity matching trials using new abstract stimuli after reaching

stability criteria on reinforcer probes. When choosing the comparison E stimulus

identical to the sample E stimulus, two of the three subjects received a class-specific

conditioned reinforcer and no primary reinforcer, while the third subject received a class-

specific primary reinforcer and no conditioned reinforcer. When choosing the

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comparison F stimulus identical to the F sample, the reinforcer that was previously class-

specific was no longer produced and instead the reinforcer that was not given during the

E training was produced and class-specific. After training with EE and FF trials, subjects

were then tested for expansion of the existing (ABCDcSRpSR ) equivalence classes. Probe

trials determined whether E and F stimuli would be matched to A, B, C, and D stimuli,

and to each other. After the class-expansion testing, subjects were given sound-probe

tests to determine whether the auditory component of the compound reinforcer also

became a member of the equivalence classes. All of the participants in this experiment

showed class-consistent responding on at least one of the class-expansion probe-trial

types. These results suggest that individual components of class-specific reinforcers can

act as nodes for class membership. Two of the subjects showed class-specific responding

on all probe-trial types, suggesting that the stimuli that are related to the individual

components of the compound class-specific reinforcers become members of the

equivalence class, even though the components of the compound reinforcers were always

presented together in training. All of the participants showed class-consistent responding

on the sound-probe tests, indicating that the auditory reinforcers also became

equivalence-class members.

This study extended previous results by showing that individual components of

compound class-specific reinforcers can act as nodes to aid in equivalence-class

formation and that they become independent members of the equivalence class

themselves. The results of this study have added empirical evidence to the rapidly

growing possibility that class-specific reinforcement can aid in the training of equivalent

relations. This study provides exciting implications for teaching developmentally

21

disabled children, a population that was once thought of as un-teachable. The training of

only a few conditional relations can result in many total relations. These training

procedures may be applied to many areas of teaching including language, math, and other

symbolic activities.

Classroom Applications of Equivalence

Stromer, MacKay, and Stoddard (1992) suggest several applications of stimulus

equivalence technology in teaching reading and language-arts skills in the classroom.

They suggest that these methods should supplement the already established classroom

curriculum and that they can even be useful in remediation when traditional teaching

methods fall short for some students. If the stimulus-equivalence techniques are not

taught correctly or by creative teachers, the method may fail; thus, using it to supplement,

not supplant, the traditional curriculum is suggested for the most effective teaching. The

authors state that stimulus equivalence can be thought of as a network of skills that can be

taught or that can emerge and that help the student learn what words mean. The stimulus-

equivalence network discussed by the authors consists of tasks in which the student

equates the corresponding versions of all of the following stimuli and spoken responses:

dictated names, pictures, printed words, oral naming, anagram spelling, oral spelling, and

written spelling.

Teachers can use the match-to-sample technique to teach relations between some

of these elements of the network, and then test for emergence of the other relations. The

match-to-sample method is also helpful in that the teacher can assess whether the student

has learned relations without first requiring the student to produce the letters or symbols

in print, which can be taught later. The authors suggest types of data sheets, tests, and

22

teaching sequences for teachers to use when employing stimulus equivalence in their

classroom. The most important of these suggestions are: assessing where the students

are in terms of what they can do before and after the teaching methods, collecting data on

how well the methods are working, and teaching only a few of the relations, to let many

equivalence relations emerge. Tests or assessments of what the students can do are

important to complete before the teaching begins so that the teacher knows the

appropriate level at which to begin teaching and can make the best use of time and

teaching strategies. Pretests also orient the student to what they will be learning. Tasks

from the stimulus-equivalence network can be used for these assessments and for tests.

The teachers should collect data throughout the teaching curriculum to assess

whether the strategies used are effective and if remediation is necessary for any of the

students. This keeps the teacher’s attention focused on individual performances. The

teachers should teach only a few of the relations and then test for emergence of other

relations in order to be sure the student has learned equivalence and not simply rote

memorization of all the relations taught. This demonstrates the economy and efficiency

of the stimulus-equivalence approach in that the teacher must only teach a few relations

and, if done correctly, many others will emerge without being taught directly. The

authors suggest group-teaching strategies and individual testing using praise as a

reinforcer for correct responses. They advise showing the correct response after each

trial to correct, but not to publicly expose those students who have emitted an incorrect

response.

The authors discuss several experiments in which stimulus equivalence has been

used to teach reading skills and they offer suggestions for applying the methods of these

23

experiments to the classroom. The authors also warn that the effectiveness of the

stimulus-equivalence teaching method is dependent on the teaching practices and that

many cycles of test-teach-test may be needed before equivalence relations emerge. The

authors also warn that children with attention deficits, learning disabilities, and

developmental disabilities may fail to acquire equivalence relations. Although the

authors discuss using stimulus equivalence to teach reading skills, they state that many

stimuli and skills can be taught using this method. Indeed, these sorts of training

procedures have proved successful in a number of skills training programs (e.g., Dube,

MacDonald, McIlvane, & MacKay, 1991; Sidman & Cresson, 1973; Green, 1990).

Focus on Teaching Mathematics

Given the implications that stimulus-equivalence procedures have for teaching

students fundamental relations, it is only natural that equivalence research be extended

from the area of reading instruction to the area of math instruction. Many school-age

children exhibit difficulties at different levels of mathematical knowledge development

(Resnick, 1989). Lynch and Cuvo (1995) used equivalence-training procedures to teach

seven students who demonstrated difficulty with math problems involving fractions,

decimals, and fraction-decimal ratios. The sequence of conditions in this study was

pretest, train, posttest, generalization test, train, and generalization test. The

experimenters taught 12 A (printed numerals represented as ratios) to B (printed analogue

of ratios) relations. They then trained 12 B (printed analogues) to C (decimal) relations.

The training procedure used was a match-to-sample correction procedure where a correct

response had to occur before the next trial was presented. Selection of the correct

comparison stimulus resulted in a computer-synthesized “yes” verbal consequence, while

24

incorrect comparison selection had no effect. Posttests were then given to determine if

equivalence classes had emerged, and in fact all seven participants demonstrated the 12

equivalence classes. More stimuli were created to test for generalization of the ability to

equate decimals with fractions, and several subject showed generalization.

The use of equivalence procedures to teach mathematics may provide innovative

teaching techniques that can supplement or replace current math-instruction techniques.

The techniques most commonly used in mathematics instruction are flashcards, rote

memorization of math facts, and fluency through large quantities of practice problems

(Van Houten & Rolider, 1989; Galistel & Gelman, 1992). Mathematics instruction in the

United States elementary-school system has been criticized for not integrating the

informal self-invented methods of mathematics that children bring with them when they

enter school with the formal rules of math taught in the school curriculum (e.g., Resnick,

1989).

Purpose of the Present Study

The present study attempts to integrate informally acquired mathematics skills

(e.g., counting quantities of objects) with formal mathematics training. Quantity

matching (matching quantities of different configurations but equal number) and math

facts were trained through an equivalence procedure using compound class-specific

reinforcement. Sidman’s (2000) theory suggests that class-specific reinforcers become

members of the equivalence class, and studies in the past that have used class-specific

reinforcers (Schomer, 2001; Dube & McIlvane, 1995) and compound class-specific

reinforcers (Ashford, 2003) have added support to this theory. The present study used

printed numerals, spoken number words, and printed number words as class-specific

25

reinforcers in training quantities and simple math facts. This study holds implications for

teaching efficiency in that a few relations were taught and a number of others emerged

based on the formation of equivalence classes. The experimental question addressed by

this study was whether developmentally disabled and normally developing children can

learn fundamental math facts via an equivalence procedure employing compound class-

specific reinforcement. This study used a match-to-sample procedure to train and test

conditional discriminations involving stimuli from mathematics.

METHOD

Participants

Two children with developmental disabilities were selected to participate in the

present experiment, based on their level of beginning math skills. These children were

students attending a self-contained special education classroom for developmentally

disabled children in a local elementary school. Seven normally developing children,

attending a local daycare, also participated in the experiment. Table 1 shows a list of the

participants designated by one or two letters, their ages, Peabody Picture Vocabulary Test

age equivalent scores by years and months, and diagnoses where appropriate. The two

participants with developmental disabilities have a dash and “D” after their letter,

designating their diagnosis for the readers’ reference throughout the rest of the text. The

criterion for participation in this experiment was that the participant demonstrates a

kindergarten level of mathematics skills (i.e., some quantity labeling; naming numerals

and printed number words; and counting to 10) at the start of the study based on an

assessment completed by the experimenter. Agreements were made between the schools

and daycare, the experimenter, and the university that sponsored the project regarding

26

Table 1.

Age, diagnosis, Peabody Picture Vocabulary Test age equivalent, and values trained for each participant.

________________________________________________________________________

Subject Age PPVT Diagnosis

(numbers trained)

________________________________________________________________________

K (0, 4, 7, and 10) 5 yrs 1 mo 6 yrs 1 mo

B (0, 1, 2, and 3) 4 yrs 11 mo 6 yrs 0 mo

J (0, 1, 2, and 3) 4 yrs 9 mo 6 yrs 6 mo

AM (0, 1, 2, and 3) 4 yrs 10 mo 3 yrs 2 mo

TH (0, 4, 7, and 10) 4 yrs 6 mo 5 yrs 10 mo

TI (0, 1, 2, and 3) 4 yrs 7 mo not available

AL (0, 1, 2, and 3) 4 yrs 9 mo 3 yrs 5 mo

C-D (0, 4, 7, and 10) 12 yrs 3 mo 5 yrs 5 mo EMH, down syndrome

B-D (0, 1, 2, and 3) 12 yrs 2 mo 2 yrs 6 mo TMH, epilepsy

________________________________________________________________________

Note. The ages and Peabody’s Picture Vocabulary Test (PPVT) scores denoted in the

table are accurate as of 2 months after the subject began the study. The diagnoses given

for the two subjects in a special education classroom are educatably mentally

handicapped (EMH) and trainably mentally handicapped (TMH).

27

participation in the experiment. Parental consent was obtained for each child who

participated in the experiment. Consent documents explained that the children completed

computer programs that helped the experimenter study the best practices for teaching

math skills.

Materials

As a part of preliminary assessment, the Peabody Picture Vocabulary Test

(version IIA) was given to each participant to determine their estimated age-equivalent

scores for vocabulary. Basic mathematics flash cards were presented to each participant

to assess the baseline level of math skills. The following skills were tested with

flashcards: matching quantities 1-25 to their corresponding numerals and printed number

words, and basic kindergarten level addition and subtraction (including numbers 1-25

only as addends).

Figure 1 shows the stimuli used in the experimental trials; all stimuli were created

by the experimenter using Microsoft Paint and Microsoft Word. The stimuli consisted of

quantities of familiar objects (e.g., stars, frogs, hearts, trains, etc.) for numbers 0-10 with

three different configurations for each quantity represented, simple addition facts using

the numerals 0-9, the printed numerals 0-10, and the printed words for zero through ten.

Table 2 presents a list of all stimuli used as samples and comparisons in the experiment

organized by stimulus set (A, B, C, and D) and by potential equivalence class (1-10).

Each row in the table shows all of the relations and skills to be taught or tested in the

experiment subsequent to quantity training for a particular numeral. Table 3 lists the

different configurations of the quantity stimuli used in quantity training. The compound

consequences that followed correct responses consisted of a spoken number word and

28

29

Figure 1. Quantity configuration stimuli shown in all configurations used in training.

30

Table 2.

A, B, C, and D stimuli sets used during training.

________________________________________________________________________

Quantities Stimulus Sets

A B C D

________________________________________________________________________

1 1 moon 0 + 1 1 + 0 0 + 0 + 1

2 2 horses 1 + 1 0 + 2 2 + 0

3 3 hearts 2 + 1 1 + 2 1 + 1 + 1

4 4 bulbs 3 + 1 2 + 2 1 + 1 + 2

5 5 frogs 4 + 1 3 + 2 1 + 2 + 2

6 6 smiles 5 + 1 4 + 2 3 + 3

7 7 lions 6 + 1 5 + 2 4 + 3

8 8 suns 7 + 1 6 + 2 4 + 4

9 9 stars 8 + 1 7 + 2 5 + 4

10 10 bolts 9 + 1 8 + 2 5 + 5

________________________________________________________________________

31

Table 3.

Labels of each quantity configuration.

________________________________________________________________________

Quantities

________________________________________________________________________

0 0 red 0 blue 0 green

1 1 bolt 1 bulb 1 moon

2 2 frogs 2 horses 2 smiles

3 3 hearts 3 stars 3 suns

4 4 bulbs 4 lions 4 moons

5 5 frogs 5 hearts 5 trains

6 6 bolts 6 smiles 6 trains

7 7 frogs 7 horse s 7 lions

8 8 horse s 8 smiles 8 suns

9 9 bulbs 9 hearts 9 stars

10 10 lions 10 bolts 10 trains

________________________________________________________________________

32

either the corresponding numeral or the corresponding printed number word on a polka-

dotted background. These visual components of the consequence were designed to help

distinguish them from sample or comparison stimuli and to increase their reinforcing

qualities. Additional reinforcers (e.g., edibles, access to other computer games, coloring

pictures, stickers, and small toys) were given to the participants, contingent upon

participation and appropriate behavior. A prize chart was used to keep attendance high

(see Appendix A for an example of a prize chart). One sticker was earned for every

session completed, and small toy prizes were awarded after five stickers accumulated.

A Macintosh Powerbook laptop computer was used to present all experimental

trials using specially designed match-to-sample software (Dube, 1990). The sample and

comparison stimuli were displayed in 1 in. x 1 in. boxes on the screen, and the participant

made selections by using the computer’s mouse. The sample and comparison stimuli

were presented on a white background, while the reinforcer was distinguishable by

colored polka-dotted appearance. All participants were trained to use a mouse prior to

the experiment, if this skill was not already evident.

Tally sheets were used to ensure that the subject was attending to the presentation

of class-specific consequences (see Apendix B for an example of a tally sheet). The

computer recorded responses automatically on a trial-by-trial basis. Data collection

charts, graphs, and attendance sheets were also used in the experiment to track the

acquisition and emergence of conditional discriminations.

General Procedure

The primary experimenter assessed each participant’s beginning math level by

presenting flashcards involving values between 1 and 20. Each card presented a

33

configuration of objects, a printed number word, or a numeral and had a mate that

presented a configuration of objects, a printed number word, or a numeral that

corresponds. Twenty pairs of flash cards were presented in a random order spread on a

table in front of the participant; one member of each pair consisted of a quantity of

objects and the other member consisted of a corresponding numeral or printed word. The

participant was asked to pick a card and find its mate until all the cards were gone. The

cards were presented in a predetermined manner where the skills of matching quantity to

quantity, quantity to numeral, quantity to word, and word to numeral were all represented

equally. The percent of correct matches out of 20 opportunities and the highest numeral

associated with accurate choices were recorded. When the participant exhibited an

accuracy level close to 100% on these 20 trials, then flash cards containing simple

addition facts were presented one at a time for 20 trials to determine whether the

participant could compute simple addition facts. The child was also asked to count as

high as they could and the highest number was recorded. Once this pre-assessment was

completed, the child was assessed to determine if they could use a computer mouse

appropriately. If it was determined that the participant needed to be trained to use a

mouse correctly, the experimenter explained how to use the mouse and observed while

the child played several computerized childrens’ games until proper use of the mouse was

demonstrated.

A three-choice match-to-sample procedure was arranged for all training and

testing trials, unless acquisition problems were evident. Three participants required

modified training procedures including reducing the trials to two comparisons at some

point during their training; these extra training steps will be discussed in the results

34

section. After the experimenter began the session, the participant was monitored to

ensure attention to the task; if necessary, prompting was used to encourage the participant

to continue the session. During their first session, to familiarize them with the match-to-

sample task, the participants were given instructions about how to choose stimuli on the

screen using the computer mouse. Subsequently, only generic statements were given to

encourage adequate participation. A few subjects required more explicit instructions

during training to aid in the acquisition of the conditional discriminations. These

instructions will be discussed in detail in the results section. All other aspects of the

session were computer-automated, except that the experimenter was required to push a

key to end the presentation of a reinforcer. The percent of correct trials was recorded by

the experimenter after each training session and the percent of class-consistent responses

was recorded after each probe and all-skills test session. Session-by-session graphs were

also constructed.

During a typical training trial, a sample stimulus was presented in the middle of

the screen and the participant was required to make an observing response by placing the

cursor over the sample and clicking the mouse. This produced the three (or two)

comparison stimuli, which were presented in the corners of the screen; positions were

randomly assigned. The participant was then required to click on one of the comparison

stimuli. Clicking on the stimulus designated by the experimenter as correct produced a

compound class-specific reinforcer, which consisted of a visual and an auditory

component. The auditory component of the reinforcer compound was the spoken number

word that corresponded with (or was specific to) the sample and comparison. The visual

component of the reinforcer compound was either the corresponding printed number

35

word or the corresponding numeral; the numeral and printed word occurred with equal

frequency. An incorrect response produced a buzzer and the next trial, following a 1.5

sec inter-trial interval. In some phases of training and testing (see below), there were

trials for which responses did not produce any consequences; in these cases no buzzer or

reinforcer was delivered and the next trial was presented after an inter-trial interval of

1.5s. Each computer training session was balanced in terms of correct stimulus position

on the screen, and the number of times each stimulus or object (in the case of the

quantities) appeared in a given position and in the session.

Each participant was tested individually at their school or daycare program, once

per day and up to five times per week, excepting absences, behavior problems, or other

conflicts. Session durations varied per participant due to distractions and behaviors

conflicting with the testing session, but were approximately 15 min in duration. If further

motivation to participate was needed, edibles, small toys, or entertaining computer games

were provided upon completion of each daily session.

A small-n design was used to assess effects of the training (Sidman, 1960). The

experimental design followed a test-train-test-train-test pattern, to assess if and when

conditional discriminations were learned, or if and when new relations emerged. The

pretest served as the first baseline assessment to determine whether the experimental

training was responsible for skill acquisition. Conditional discriminations were taught

until a mastery criterion (two sessions at 90% correct or better over at least two

consecutive days) was met. Trials presenting these conditional discriminations were

included in all later test sessions to assess whether original baseline relations remained

stable throughout training and testing sequences. Tests of all skills, those explicitly

36

trained, those not yet trained, and those expected to emerge, were presented before the

first training phase and after each training phase to determine exactly when relations

could be demonstrated and whether they could be attributed to the experimental training.

Training Phases

All Skills Test 1. Two sessions of match-to-sample test trials were conducted first

to provide a baseline assessment of math skills prior to any experimental training. These

pretests consisted of a sampling of all possible relations that were taught or probed during

the course of the experiment. The tested relations included: quantities to quantities,

numerals to quantities, printed number words to quantities, spoken words to quantities,

quantities to simple addition problems ( a number + 1, a number + 2, a number + 3, a

number + 4, a number + 5, a number + 0, and a number + a number + a number), simple

addition problems (those described above) to other simple addition problems yielding the

same numerical answer, simple addition problems to printed number words, simple

addition problems to numerals, simple addition problems to spoken number words, and

probes involving the trained addition problems but with the numbers in different orders

(e.g., if 6 + 1 is trained, 1 + 6 could be probed). The stimuli that presented addition

problems in orders other than those trained are shown in Table 4.

Table 5 provides a list of all the trials that were presented in the difficult

discrimination pretest. These trials were presented in a random order to the participant.

The numbers included as incorrect comparisons in each trial of the difficult

discrimination all-skills test were similar in value to the number represented by the

37

Table 4.

List of stimuli primes.

________________________________________________________________________

B’ C’ D’

________________________________________________________________________

1 n/a n/a n/a

2 n/a n/a n/a

3 n/a n/a n/a

4 1 + 3 n/a 1 + 2 + 1

5 1 + 4 2 + 3 2 + 2 + 1

6 1 + 5 2 + 4 n/a

7 1 + 6 2 + 5 3 + 4

8 1 + 7 2 + 6 n/a

9 1 + 8 2 + 7 4 + 5

10 1 + 9 2 + 8 n/a

________________________________________________________________________

38

Table 5. Trial configurations in the Difficult Discrimination All Skills Test. ________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Quantity 8 horses 8 smiles 7 lions 6 bolts

Quantity 7 frogs 7 horses 6 trains 8 smiles

Quantity 3 hearts 3 stars 2 horses 4 lions

Reinforcer Probe 5 numeral 5 frogs 4 lions 6 smiles

Reinforcer Probe 0 word 0 blue 1 bulb 2 frogs

Reinforcer Probe 9 numeral 9 bulbs 8 horses 10 trains

AB 6 smiles 5+1 4+1 6+1

AB 8 suns 7+1 6+1 8+1

AB 7 horses 7+1 6+1 5+1

AC 4 moons 2+2 1+2 3+2

AC 10 bolts 8+2 7+2 6+2

AC 2 horses 0+2 1+0 1+2

AD 4 bulbs 1+1+2 1+1+1 1+2+2

AD 2 horses 2+0 0+0+1 1+1+1

AD 9 stars 5+4 4+4 5+5

BC 2+1 1+2 0+2 2+2

BC 6+1 5+2 6+2 4+2

39

Table Continues

Table 5 continued

________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

BC 1+1 0+2 1+0 1+2

CB 4+2 5+1 4+1 6+1

CB 1+2 2+1 3+1 1+1

CB 8+2 9+1 8+1 7+1

Alternate AB 3 suns 2+1 1+1 3+1

Alternate AB 9 stars 8+1 7+1 9+1

Alternate AC 4 lions 2+2 1+2 3+2

Alternate AC 7 horses 5+2 4+2 6+2

Alternate AD 5 trains 1+2+2 1+1+2 3+3

Alternate AD 8 smiles 4+4 4+3 5+4

B prime 4 bulbs 1+3 1+4 1+5

B prime 8 suns 1+7 1+6 1+8

C prime 9 stars 2+7 2+6 2+5

C prime 6 smiles 2+4 2+3 2+5

D prime 5 frogs 2+2+1 1+2+1 3+4

D prime 9 bulbs 4+5 3+4 2+2+1

________________________________________________________________________

40

sample (i.e., if the sample was four moons, the incorrect comparisons were three hearts

and five trains), requiring the participant to make a more difficult discrimination. Table 6

provides a list of all the trials that were presented in the easier discrimination pretest;

these trials were also presented in a random order to the participant. The numbers

included as incorrect comparisons in each trial of the easier discrimination all-skills test

were more disparate from the value of the correct comparison (i.e., if the sample was four

objects, the incorrect comparisons were nine objects, and one object), requiring the

participant to make an easier discrimination. Participants were given the difficult pretest

first, to determine accuracy when the discrimination required between comparisons was

very precise (e.g., when shown a sample with eight objects, incorrect comparisons might

include seven and nine objects). If the participants did not score above a 90% mastery

criterion on the difficult pretest, they were then given the easier discrimination pretest to

determine whether more accurate performances were possible when less precise

discrimination among comparisons was required (e.g., when shown a sample with eight

objects, incorrect comparisons might include one and four objects). After the pretests

were given to the first eight participants and analyzed, it was evident that a revised all-

skills test was needed that included a more balanced number of each type of trial and

each numeral represented. Table 7 and Table 8 provide lists of all the trials that were

presented in the 0-3 and the 0, 4, 7, and 10 all-skills tests respectively; these trials were

also presented in a random order to the participant. These revised all-skills tests were

given as a pretest to the last participant who began the study and as the all-skills tests

following each training phase for all participants. There were no programmed

41

Table 6 Trial configurations in the Easy Discrimination All Skills Test. ________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Quantity 5 hearts 5 trains 7 horses 3 hearts

Quantity 8 horses 8 smiles 3 stars 6 bolts

Quantity 2 frogs 2 smiles 7 lions 5 trains

Reinforcer Probe 0 numeral 0 red 8 suns 2 horses

Reinforcer Probe 3 word 3 suns 1 moon 5 frogs

Reinforcer Probe 6 word 6 smiles 9 bulbs 4 lions

AB 1 moon 0+1 8+1 3+4

AB 10 bolts 9+1 6+1 1+0

AB 4 bulbs 3+1 4+4 0+2

AC 7 lions 5+2 0+1 2+2

AC 6 smiles 4+2 1+1+1 0+0+1

AC 3 hearts 1+2 7+1 1+5

AD 8 suns 4+4 2+4 2+1

AD 5 frogs 1+2+2 5+2 8+2

AD 2 horses 2+0 3+3 2+7

BC 3+1 2+2 5+5 4+2

BC 7+1 6+2 2+2+1 2+0

Table 6 continues

42

Table 6 continued

________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

BC 8+1 7+2 4+3 3+1

CB 1+2 2+1 9+1 1+4

CB 2+2 3+1 2+8 2+6

CB 6+2 7+1 8+2 2+3

Alternate AB 4 moons 3+1 7+2 1+1

Alternate AB 6 bolts 5+1 1+3 4+5

Alternate AC 7 frogs 5+2 5+4 1+2+2

Alternate AC 9 hearts 7+2 2+1 1+1+2

Alternate AD 2 smiles 2+0 7+1 3+2

Alternate AD 5 trains 1+2+2 1+2 1+9

B prime 4 bulbs 1+3 2+8 1+6

B prime 6 smiles 1+5 1+2+1 6+2

C prime 8 suns 2+6 2+2 1+9

C prime 10 moons 2+8 3+4 1+1

D prime 5 frogs 2+2+1 5+5 3+1

D prime 9 stars 4+5 5+1 1+1+1

_______________________________________________________________________

43

Table 7. Trial configurations in the 0-3 All Skills Test. ________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Quantity 7 frogs 7 horses 6 trains 5 frogs

Quantity 3 hearts 3 stars 2 horses 0 red

Quantity 8 horses 8 smiles 9 bulbs 10 lions

Reinforcer Probe zero word 0 blue 1 bulb 2 frogs

Reinforcer Probe 5 numeral 4 lions 6 smiles 5 frogs

Reinforcer Probe 4 numeral four word six word five word

Reinforcer Probe 9 numeral 9 bulbs 8 horses 10 trains

Reinforcer Probe 8+1 9 numeral 10 numeral 8 numeral

Reinforcer Probe 3 numeral 1+2 0+2 1+0

Reinforcer Probe 3+2 5 numeral 4 numeral 7 numeral

Reinforcer Probe nine word 7+2 6+2 8+2

Reinforcer Probe 0 red 0 numeral 1 numeral 2 numeral

Reinforcer Probe 6 smiles six word seven word four word

Reinforcer Probe 8 suns eight word nine word ten word

Reinforcer Probe ten word 10 numeral 9 numeral 8 numeral

Reinforcer Probe 2 numeral two word one word zero word

Reinforcer Probe 0+1 one word two word three word

Table 7 continues

44

Table 7 continued

________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Reinforcer Probe 4 numeral 3+1 4+1 6+1

AB 1 moon 0+1 1+1 2+1

AB 6 smiles 5+1 4+1 6+1

AB 8 suns 7+1 8+1 9+1

AC 4 moons 2+2 3+2 4+2

AC 2 horses 0+2 1+0 1+2

AC 10 moons 8+2 7+2 6+2

AD 4 bulbs 1+1+2 3+3 1+2+2

AD 2 horses 2+0 1+1+1 0+0+1

AD 9 stars 5+4 4+4 5+5

BC 6+1 5+2 4+2 3+2

BC 8+1 7+2 6+2 8+2

BC 2+1 1+2 1+0 0+2

CB 1+2 2+1 0+1 1+1

CB 3+2 4+1 3+1 5+1

CB 8+2 9+1 8+1 7+1

Alternate AB 3 suns 2+1 1+1 0+1

Alternate AB 9 hearts 8+1 9+1 7+1

Table 7 continues

45

Table 7 continued

________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Alternate AB 7 horses 6+1 5+1 4+1

Alternate AC 4 lions 2+2 3+2 4+2

Alternate AC 8 smiles 6+2 7+2 8+2

Alternate AC 3 stars 1+2 0+2 1+0

Alternate AD 5 trains 1+2+1 1+1+2 3+3

Alternate AD 8 smiles 4+4 5+5 5+4

Alternate AD 2 smiles 2+0 0+0+1 1+1+1

B prime 7lions 1+6 1+5 1+4

B prime 4 bulbs 1+3 1+4 1+5

B prime 8 suns 1+7 1+9 1+8

C prime 9 stars 2+7 2+6 2+8

C prime 5 frogs 2+3 2+4 2+5

C prime 6 smiles 2+4 2+3 2+5

D prime 5 frogs 2+2+1 1+2+1 3+4

D prime 9 stars 4+5 3+4 2+2+1

D prime 7 lions 3+4 2+2+1 1+2+1

_______________________________________________________________________

Note. This pretest was broken into 2 sessions when delivered to participants.

46

Table 8. Trial configurations in the 0, 4, 7, and 10 All Skills Test. ________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Quantity 2 frogs 2 smiles 6 bolts 9 hearts

Quantity 8 horses 8 smiles 0 red 3 stars

Quantity 4 bulbs 4 lions 7 horses 10 moons

Reinforcer Probe six word 6 smiles 2 frogs 9 bulbs

Reinforcer Probe 4 numeral four word seven word zero word

Reinforcer Probe three word 3 suns 5 frogs 1 moon

Reinforcer Probe 0 numeral 0 red 4 moons 7 frogs

Reinforcer Probe six word 5+1 1+1 8+1

Reinforcer Probe 10 numeral 8+2 2+2 5+2

Reinforcer Probe 3+2 five word eight word one word

Reinforcer Probe 7+2 9 numeral 6 numeral 3 numeral

Reinforcer Probe 7 lions seven word four word ten word

Reinforcer Probe 8 smiles 8 numeral 5 numeral 1 numeral

Reinforcer Probe one word 1 numeral 5 numeral 3 numeral

Reinforcer Probe 9 numeral 9 hearts 2 frogs 6 smiles

Reinforcer Probe two word 2 numeral 6 numeral 9 numeral

Reinforcer Probe seven word 6+1 3+1 9+1

Table 8 continues

47

Table 8 continued

________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Reinforcer Probe 4+1 5 numeral 3 numeral 8 numeral

AB 1 moon 0+1 7+1 4+1

AB 9 stars 8+1 5+1 1+1

AB 10 moons 9+1 6+1 3+1

AC 3 hearts 1+2 3+2 6+2

AC 7 lions 5+2 2+2 8+2

AC 6 smiles 4+2 7+2 0+2

AD 5 frogs 1+2+2 4+4 1+1+1

AD 10 moons 5+5 1+1+2 4+3

AD 2 horses 2+0 3+3 5+4

BC 3+1 5+2 8+2 2+2

BC 7+1 6+2 1+2 3+2

BC 8+1 7+2 4+2 0+2

CB 1+2 2+1 7+1 4+1

CB 7+2 8+1 5+1 1+1

CB 5+2 6+1 3+1 9+1

Alternate AB 5 hearts 4+1 7+1 2+1

Alternate AB 4 moons 3+1 6+1 9+1

Table 8 continues

48

Table 8 continued

________________________________________________________________________

Trial Type Sample S+ S- S-

________________________________________________________________________

Alternate AB 6 bolts 5+1 1+1 8+1

Alternate AC 7 frogs 5+2 2+2 8+2

Alternate AC 9 hearts 7+2 4+2 0+2

Alternate AC 8 smiles 6+2 3+2 1+2

Alternate AD 5 trains 1+2+2 1+1+1 4+4

Alternate AD 2 smiles 2+0 3+3 5+4

Alternate AD 4 lions 1+1+2 4+3 5+5

B prime 4 bulbs 1+3 1+9 1+6

B prime 6 smiles 1+5 1+7 1+9

B prime 5 frogs 1+4 1+7 1+2

C prime 8 suns 2+6 2+3 2+1

C prime 10 moons 2+8 2+5 2+2

C prime 9 stars 2+7 2+0 2+4

D prime 9 bulbs 4+5 3+4 2+2+1

D prime 7 lions 3+4 2+2+1 1+2+1

D prime 5 frogs 2+2+1 4+5 5+5

_______________________________________________________________________

Note. This pretest was broken into 2 sessions when delivered to participants.

49

consequences following responses on the all skills tests; the next trial was initiated 1.5s

after each response.

Quantity Training. As mentioned in the description of the stimuli, there were

three different configurations of each quantity from 0 –10 (see Figure 1). In the first

training step, each subject was taught to match stimuli composed of an equal number of

objects but presented in different configurations. These conditional discriminations were

taught using the match-to-sample procedure described above, and correct responses (i.e.,

selection of the comparison stimulus with a quantity matching that shown in the sample)

produced a compound class-specific reinforcer. This class-specific reinforcer

corresponded to the numerical value of the sample and correct comparison (e.g., when the

participant chose 4 lions in the presence of 4 moons, the consequence included the

numeral 4 and the spoken word “four”). Incorrect responses resulted in a buzzer. In

either case an inter-trial interval of 1.5s was followed by presentation of the next trial.

During this quantity-training phase four quantities were trained simultaneously

(either numerals 0-3 or numerals 0, 4, 7, and 10). Each quantity-matching session

included 24 trials, with an equal number of trials for each of the four quantities being

trained. When participants reached a mastery criterion of two consecutive sessions with

90% or more trials correct, the reinforcement density was reduced such that only 75% of

the trials included programmed consequences. This was designed to prepare the

participant for the trials that did not produce consequences in the probe sessions (which

had a reinforcement density of 50%). When the mastery criterion was met for the 75%

reduced reinforcement phase, the participant began the reinforcer-probe testing phase.

50

Reinforcer Probe Testing Phase. Reinforcer-probe trials intermixed with

intermittently reinforced baseline training trials (i.e., the trial types mastered in the

previous training step, in this case quantity-matching trials)were presented in this phase.

Programmed consequences were not available on probe trials, but intermittent

reinforcement was available on trials with relations that had been trained, and the

participants received candy at the end of the session as reinforcement for participating.

The mixture of baseline quantity-relation trials and probe trials produced an overall

reinforcement density of 50%. The samples in the reinforcer-probe trials were one of the

following: a spoken number word, a printed number word, or a printed numeral.

Comparisons in the reinforcer-probe trials were quantities. When performance was stable

at this stage, the participant proceeded to the next procedural phase. Table 9 presents the

number of probe trials and baseline trials in each of the probe-test sessions.

All-Skills Test 2. At this point the same trial blocks used for the all-skills test

were readministered. These tests revealed any new skills acquired since the pretests, as a

result of the experimental training and the reinforcer-probe testing. Reinforcers were not

available during these tests; however, candy was presented at the end of the session as

reinforcement for participation. The participants were given instructions that they would

receive no programmed consequences in the all-skills test sessions. Upon completion of

these tests, the participants proceeded to the next experimental phase.

Train A to B Relations. The participants were trained to match quantities to

simple addition problems (see Table 2 for a list of the stimulus sets) in sessions that

contain 24 trials. The sample was one of the quantity configurations used during quantity

training (i.e., the A stimulus-set). The B stimulus-set comparisons were simple addition

51

Table 9.

Types of Probe Trials presented to participants.

________________________________________________________________________

Type of Probe Block Trial Type Number of Trials

________________________________________________________________________

Reinforcer Probes Quantity Baseline with reinforcement 84

Quantity Baseline without reinforcement 84

Sound to Word probe 4

Sound to Quantity probe 8

Sound to Numeral probe 4

Numeral to Word probe 4

Word to Numeral probe 4

Quantity to Word probe 8

Quantity to Numeral probe 8

Numeral to Quantity 8

Word to Quantity 8

Symmetry Probes AB Baseline with reinforcement 42

AB Baseline without reinforcement 21

BA Symmetry Probes 6

Sound to B probe 3

B to Numeral probe 3

B to Word probe 3

Table 9 continues

52

Table 9 continued

________________________________________________________________________

Type of Probe Block Trial Type Number of Trials

_______________________________________________________________________

Numeral to B 3

Word to B 3

________________________________________________________________________

Note. The Reinforcer Probes are distributed across 7 sessions and the Symmetry probes

are distributed across 3 sessions.

53

problems (numeral, addition sign, and the numeral 1; i.e., 3+1). The correct comparison

was the addition problem whose answer equals the quantity in the sample (e.g., with 4

lions as a sample, 3 + 1 was the correct comparison). If the addition fact that

corresponded to the sample quantity was chosen, the response produced the same class-

specific consequences described above (e.g., the spoken word “four” and either the

numeral four or the printed word “four”). If any other comparison was chosen, the

participant heard a buzzer. The participant continued in this training phase with all four

numbers in the training block until the mastery criterion was met. Reinforcement density

was reduced gradually in the method described in the quantities-training section. When

the mastery criterion was met at 75% reinforcement density, the participant will proceed

to the next phase of the experiment.

Symmetry and Reinforcer Probe Testing Phase. At this point probe tests were

done to test specifically for emergent performances that resulted from the most recent

training. The probe types were divided into three sessions. These tests included baseline

trials of the AB relations just taught, the symmetry probes of those relations (BA), and

reinforcer-probe trials with B stimuli. Reinforcer probes were included in these probe

sessions to determine if the participant could match: B stimuli to numeral words, B

stimuli to numerals, numeral words to B stimuli, numerals to B stimuli, and spoken

number words to B stimuli. Trials presenting untrained relations did not include

programmed reinforcers; however, for trials of relations already taught, intermittent

reinforcement was arranged.

54

Order of Numerical Values Trained. Instead of being trained and tested for all

numbers, 0-10, at once in each phase, each participant went through the training and

testing sequence, beginning with quantity training, working with only three to four

numbers at a time. This was an attempt to prevent the need for long testing sessions and

to lessen the number of discriminations the participant had to learn during each training

session. The participants were divided into two groups with an equal number of

developmentally disabled and normally developing children in each group. Each group

went through the phases of the experiment with one block of four numbers. One group of

participants was trained and tested first with the numbers 0, 1, 4, and 7 and the other

group of participants was trained and tested with the numbers 0, 1, 2, and 3. The test

sessions after each training phase assessed the skills just trained as well as those related

to all of the numerical values (0-10), to determine when acquisition occured and whether

it was due solely to the experimental training or if teaching some numerical values

facilitated the acquisition of the same skills for other numerical values.

RESULTS

Pre-assessment and PPVT

The participants in this study included two children with developmental

disabilities and seven typically developing children (see Table 1 for ages, PPVT scores,

and values trained for each participant). Table 10 gives a summary of each participant’s

performance on the pre-assessment with flashcards. The range in accuracy scores for

matching quantities of different configurations was 0 – 50%. Four participants could not

55

Table 10.

Pre-Assessment Performances

________________________________________________________________________

Subject Quantity Matching Cards Count past 10 Addition

Cards

________________________________________________________________________

K 0% yes 0%

B 50% yes 10%

J 20% yes 10%

AM 0% no 0%

TH 20% yes 0%

TI 40% no 0%

AL 0% no 0%

C-D yes

B-D no

________________________________________________________________________

Note. Percent of correct matches is shown in the second column, while percent of correct

addition problems is shown in the fourth column. All subjects could count ten objects on

a flash card, if subjects could count accurately when there were more then ten objects ona

card it is designated in column three.

56

count objects accurately when ten objects were presented on a flash card, but could do so

if there were five or fewer objects presented. Five participants could count accurately

when there were ten or more objects presented on a card. Three participants gave correct

answers on 10% of the addition flashcards presented. All other participants gave no

correct responses during presentation of addition flash cards.

All Skills Test 1

Table 11 shows overall scores for the all-skills test given to all participants before

training began (i.e., the pretest) in the second and third columns. These scores represent

the percent of correct trials on each presentation of the test. Eight of the nine participants

were given the difficult discrimination pretest, followed by the easier discrimination

pretest (see tables x and y for specific types of trials in these pretests). Participant J was

given the revised pretest, Pretest 0-3 Versions A and B, which was specific to the

numerals that would be grouped together during his training. The average score on the

first presentation of the pretest (for Participant J this was the 0-3 version A pretest and for

the other eight participants this was the difficult pretest) was 31.11%, while the average

score on the second presentation of the pretest (for Participant J this was the 0-3 version

B pretest and for the other eight participants this was the easy pretest) was 39.11%.

Accuracy for most participants’ performances was at or near chance levels (33 % for

trials with 3 comparisons). Figures 2 – 10 show pretest accuracy (left column of graphs)

both for numerals to be trained and for all other numerals, for each type of trial, for each

participant.

57

Table 11.

All Skills Test performances for each participant.

________________________________________________________________________

Subject Presentation

Pretests After Quantity Training

1 2 3 4

________________________________________________________________________

B 33 42 48 45

TI 27 33 42 32

J 50 52 54 56

AL 24 39

AM 27 27

TH 29 36

K 30 36 73 100

C-D 30 45 50 36

B-D 30 42

________________________________________________________________________

58

Figure 2. Participant B’s performances on each All Skills Test (hard and easy versions) for each trial type before and after quantity training for trained (black bars) and untrained (gray bars) relations.

Rein

f

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All Other Quantities

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All Skills Test Sessions

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Hard Easy Hard Easy

Participant B

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59

Figure 3. Participant J’s performances on each All Skills Test (hard and easy versions) for each trial type before and after quantity training for trained (black bars) and untrained (gray bars) relations..

Participant J

Rein

f %

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Trained Quantities

All Other Quantiities

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Pretest 0-3 Pretest 0-3All Skills Test Session

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Tra

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60

Figure 4. Participant C-D’s performances on All Skills Test for each trial type.

Participant C-D

Rein

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Trained Quantities

All Other Quantities

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B %

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Hard Easy Pretest 04710

All Skills Test Sessions

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61

Figure 5. Participant TI’s performances on All Skills Test for each trial type.

Participant TI

Rein

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Trained Quantities

All Other Quantiities

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Easy Pretest 0-3 PretestAll Skills Test Sessions

Tra

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62

Figure 6. All Skills Test performances for Participant K (asterisks are in place of missing data).

Participant K

Rein

f%

Co

rrect

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an

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All Skills Test Tranied Quantities

All Other Quantities

Tra

in Q

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*

*

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*

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*

*

*

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Pretest 04710

63

Figure 7. Participant AL’s performances on All Skills Test for each trial type..

Participant AL

Rein

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Trained Quantities

All Other Quantities

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Hard Easy

Sessions

64

Figure 8. Participant TH’s performances on All Skills Test for each trial type.

Participant TH

Rein

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Trained Quantities

All Other Quantiities

Quant M

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Easy

All Skills Test Session

65

Figure 9. Participant B-D’s performances on All Skills Test for each trial type..

Participant B-D

Rein

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Trained Quantities

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Hard Easy

All Skills Test Sessions

66

Figure 10. Participant AM’s performances on All Skills Test for each trial type..

Participant AM

Rein

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Trained Quantities

All Other Quantiities

Quant M

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All Skills Test Sessions

Hard Easy

67

Quantity Training

Three participants began quantity training with numerals 0, 4, 7, and 10, while six

participants began quantity training with numerals 0, 1, 2, and 3. There was no

difference found in acquisition between the participants that began quantity training with

numerals 0, 1, 2, and 3 and those that began training with 0, 4, 7, and 10. Eight of the

nine participants who began quantity training reached the mastery criterion. Three of

those eight participants reached the mastery criterion without any instructions. Figure 11

shows the acquisition of the quantity conditional discriminations demonstrated by these

three participants. Participant B met the mastery criterion in two sessions, Participant J in

three sessions, and Participant C-D in six sessions.

Six of the eight subjects who met mastery criterion in the quantity training phase

did so with the aide of verbal instructions, implemented after it was evident that there

were no trends toward acquisition of the conditional discriminations. The first instruction

indicated that the participant should count the sample and each comparison orally. For

this instruction the experimenter pointed to the sample and each comparison and said,

“How many?” When there was no improvement following this intervention, another

instruction was added. The participants were told to “pick the one that is the same” after

they had counted the sample and all comparisons. Figures 12 and 13 show the

acquisition of the quantity conditional discriminations demonstrated by these six

participants, and when the instructions were implemented during their training.

Participant TI completed five sessions of quantity training before any instructions

were given and three sessions of counting the sample and comparisons on each trial: no

improvement was seen in her score. Participant TI met mastery criterion in six sessions

68

Quantity Training Acquisition

Perce

nt C

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Participant B

Reduced Reinforcement AB

Participant J

Perce

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Quantity Training Reduced Reinf AB Reduced Reinf AB

Participant C-D

Sessions

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Quantity Training Reduced Reinf AB

2

Figure 11. Acquisition performances for Participants B, J, and C-D. The numeral “2” represents when the participant was given sessions with only two values.

69

Participant TI

Perc

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Quantity Training

C P

Reduced Reinf AB Reduced Reinf AB

Quantity Training

C

P

Reduced Reinf

Back to BaselineWord to Number

AB

AB

Reduced Reinf

Participant TH

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Quantity Training Acquisition

C P

Reduced Reinforcement

Quantity Training Acquisition

C

P

Return to Bsle

Participant K

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C

P

Participant AL

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C

P

Return to BsleReduced Reinf

Reduced Reinf

Return to Bsle

Reduced Reinf

Figure 12. Acquisition performances for Participants AL, TH, K, and TI. The letter “C” represents when the counting instruction was implemented, “P” when the “pick same” instruction was given.

70

Figure 13. Acquisition performances for Participants AM and B-D. The letter “C” represents when the counting instruction was implemented, “P” when the “pick same” instruction was given. The letter “S” represents when shorter sessions were given, and the numerals represent the number of values presented during the session. The letter “F” represents when fading of verbal instructions began, and the letter “B” when the baseline sessions were reinstated.

Quantity Training Acquisition

Perc

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Participant AM

CP S

2F

3B

Participant B-D

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C

P 2

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B

71

after she was told to “pick the one with the same number” on each trial. These

instructions were faded over the next four sessions until she met an mastery criterion with

no verbal instructions. Participant K completed four sessions of quantity training before

any instructions were given and four sessions of counting the sample and comparisons on

each trial with no improvement in her score. After completing five sessions in which she

was told to “pick the one with the same number,” she met the mastery criterion. These

instructions were faded during the last two sessions given to the participant. Participant

TH completed six sessions of quantity training before any instructions were given and

four sessions of counting the sample and comparisons on each trial with no improvement

in his score. This participant met the mastery criterion after completing four sessions

while being told to “pick the one with the same number” on each trial. These instructions

were faded during the last two sessions given to the participant. Participant AL

completed five sessions of quantity training before any instructions were given and five

sessions of counting the sample and comparisons on each trial with no improvement in

his score. After completing two sessions in which she was told to “pick the one with the

same number” on each trial, she met the mastery criterion. When these instructions were

faded, participant AL exhibited a drop in accuracy and difficulty attending to the task.

She did not reach the mastery criterion until the number of trials in a session was reduced

from 24 to 12. The instructions were then faded over ten sessions until she met an

mastery criterion with no verbal instructions. The sessions were expanded back to 24

trials until the mastery criterion was met.

Participant AM and B-D reached the mastery criterion only after some additional

instructions and interventions were given. Figure 13 shows the acquisition data for these

72

two participants. Participant AM completed three sessions of quantity training without

instructions and three sessions with the counting instructions with little improvement in

her score. She was then told to “pick the one with the same number,” on each trial but

there was no improvement in accuracy after eight sessions. The number of trials in a

session was then reduced from 24 to 12. She completed six sessions of 12 trials each

with no improvement. Training was then restricted to only the numerals 0 and 3, instead

of 0, 1, 2, and 3. After nine 12-trial sessions with only two numerals, there was still no

improvement in accuracy. The instruction, “find the one that is the same as this one,”

was then given on each trial while the experimenter pointed to the sample. AM met the

mastery criterion in two sessions and the instruction was faded over six sessions until it

was only given on the first trial of each session. The numeral 1 was then added to the

sessions so that the sessions included the numerals 0, 1, and 3. These sessions contained

18 trials and the mastery criterion was met in only three sessions with the instruction still

given on the first trial of each session. She was then given the original baseline sessions

with numerals 0, 1, 2, and 3 and met the mastery criterion in only two sessions.

One of the nine participants did not meet the mastery criterion for the first

administration of quantity training with numerals 0, 1, 2, and 3, but did so after he was

given intervention sessions with fewer numerals (sessions with 0 and 3 and then sessions

with 0, 1, and 3). Participant B-D completed seven sessions of quantity training without

instructions and two sessions with the counting instruction with no improvement in

accuracy. He was then given the instruction “pick the one with the same number,” on

each trial. After 11 sessions, there was still no improvement in accuracy, so he was then

given sessions with only the numerals 0 and 3 instead of 0, 1, 2, and 3. After two

73

sessions with only numerals 0 and 3, he was given the instruction, “find the one that is

the same as this one,” while the experimenter pointed to the sample stimulus. He met the

mastery criterion in the next two sessions and this instruction was faded over the next two

sessions until it was only given on the first trial of each session. The numeral 1 was then

added to the sessions so that the sessions included the numerals 0, 1, and 3. He met the

mastery criterion with these numerals in four sessions. He was then given the original

baseline quantity training sessions with numerals 0, 1, 2, and 3 and met the accuracy

criterion in two sessions.

Quantity Training: Reduced Reinforcement

Six of the nine participants met the mastery criterion for the reduced

reinforcement phase of quantity training (see Figures 11 and 12). Two participants

(Participants AM and B-D) were never exposed to the reduced reinforcement phase due

to time constraints of the experiment. Participant B was not exposed to the reduced

reinforcement phase due to experimenter error. All participants exposed to this phase met

the mastery criterion in two to four sessions.

Reinforcer Probes

Seven participants were exposed to the reinforcer probe sessions. Five of these

seven participants reached a stability or mastery criterion during these reinforcer probes.

The probe sessions presented after quantity training included trials that required the

participant to match: the printed reinforcer word to a quantity, the reinforcer numeral to a

quantity, a quantity to the printed reinforcer word, a quantity to the reinforcer numeral,

the spoken number word to the reinforcer word, the spoken number word to the

74

reinforcer numeral, the spoken word to a quantity, the reinforcer numeral to the printed

reinforcer word, and the printed reinforcer word to the reinforcer numeral.

Figure 14 shows the percent of class-consistent responses performed by

Participant B for each probe type. This participant’s responses were class-consistent on

every type of probe trial listed above. Her baseline performances (matching quantity to

quantity) remained at 95% correct or higher on every session of reinforcer probes.

Figure 15 shows the percent of class-consistent responses performed by

Participant J. Participant J responded in a class-consistent manner on close to 100% of

the trials for each type of reinforcer probe trial except reinforcer numerals to quantities

and quantities to printed reinforcer words. When required to match reinforcer numerals

to quantities, responses were class-consistent on more than 50% of the trials and in one

session 100% of the trials. When required to match quantities to printed reinforcer

words, responses were class-consistent on 100% of the trials during the first and third

session and on 75% of the trials on the second session. His baseline performances

(matching quantity to quantity) remained at 92% correct or higher on every session of

reinforcer probes.

Figure 16 shows the percent of class-consistent responses performed by

Participant TI. When required to match spoken number words to quantities, spoken

number words to numerals, and quantities to numerals, her responses were class-

consistent on close to 100% of all probe trials. When required to match spoken number

words to printed reinforcer words, printed reinforcer words to quantities, and numerals to

quantities it took two to three sessions to reach 100% class-consistent responding. When

the trials involved matching quantities to printed reinforcer words, it took all four

75

Figure 14. Reinforcer Probe performances for Participant B.

Sounds to Quantities

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Figure 15. Reinforcer Probe performances for Participant J.

Sound to Quantities

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Baselines

Participant J

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Figure 16. Reinforcer Probe performances for Participant TI.

Sounds to Quantities

Perc

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Baselines

Participant TI

Sessions

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78

sessions to reach 100% class-consistent responding. When the trials involved matching

numerals to printed reinforcer words, her responses were class-consistent on no more

than 17% of the trials. When required to match printed reinforcer words to numerals, her

responses were class-consistent on 100% of the trials and then dropped to 75%. Her

baseline performances (matching quantity to quantity) remained at 96% correct or higher

on every session of reinforcer probes.

Figure 17 shows the percent of class-consistent responses performed by

Participant C-D. When required to match spoken words to numerals, spoken words to

printed words, and numerals to quantities his responses were class-consistent on 100% of

all probe trials. On trials where he was required to match spoken words to quantities and

quantities to printed words his responses reached 100% class-consistent in one to two

sessions and then dropped to 25% and 75% respectively. When he was required to match

printed words to quantities and quantities to numerals, his responses were class-consistent

on 100% of the trials on the first session then class-consistent responding increased and

decreased over the next two to three sessions. On reinforcer probe sessions where the

probe trials involved matching numerals to number words and number words to

numerals, his responses were class-consistent on 75% of trials in the first session and

100% of the trials in the second session. Participant C-D’s baseline performances

remained above 86% accurate on reinforcer probe sessions.

Figure 18 shows the percent of class-consistent responses performed by

Participant K on each session of reinforcer probes after quantity training. Participant K

completed two blocks of reinforcer probe sessions and was responding in a class-

consistent manner on an average of 50% of the trials that contained printed number

79

Figure 17. Reinforcer Probe performances for Participant C-D.

Sounds to Numerals

0

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Baseline Score

Participant C-D

Sounds to Quantities

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80

Figure 18. Reinforcer Probe performances for Participant K after original training, return to baseline training, and after training words to quantities.

Sounds to Quantities

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Participant K

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Return to B

aseline

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ords to Quantities

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aseline

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81

words. On trials that contained no printed number words, her responses were class-

consistent on 75% - 100% of the trials. Her baseline performances were above 96% in all

except two sessions. She was then given a card sorting task in which an assortment of

eight cards was laid on the table in front of her (there were two cards each for each

written number word presented during training). She was asked to put the cards into

pairs to determine if she was able to discriminate the number words from each other. She

did this perfectly. She was then given eight cards displaying each number as either a

printed numeral or as a written word, and was asked again to put the cards into pairs. She

made no correct pairs. She was then given two more sessions of quantity training and

two sessions of quantity training with reduced reinforcement in order to expose her to the

reinforcers again. She performed with 100% accuracy on these sessions (see figure 12)

and was then given another block of reinforcer probes, but her scores did not improve.

Participant K then received training on printed word-quantity matching. Sessions

contained 34 trials with the printed number words as sample and the quantities as

comparisons. When the quantity that corresponded to the sample word was chosen, the

participant received the same class-specific reinforcers as in quantity training. Participant

K met the mastery criterion with this word to quantity training in 7 sessions (see figure x)

and was then given the reinforcer probes again. Participant K responded in a class-

consistent manner on 100% of the trials for matching spoken words to printed words,

spoken words to numerals, printed words to quantities, numerals to quantities, printed

words to numerals, and numerals to printed words. Her responses were class-consistent

on at least 75% of trials that tested for matching spoken words to quantities, quantities to

printed words, and quantities to numerals.

82

Two of the seven participants who began reinforcer-probe testing did not meet an

accuracy or stability criterion on the probe trials. Figure 19 shows the percent of class-

consistent responses performed by Participant TH on each session of reinforcer probes

after quantity training. Participant TH completed two blocks of reinforcer probes and his

performances were variable and inconsistent, although these were sessions with class-

consistent performances at 75% or above for every trial type but one (printed words to

numerals). Accuracy on baseline trials remained above 92%.

Figure 20 shows the percent of class-consistent responses performed by

Participant AL on the reinforcer probe sessions after quantity training. Participant AL

performed in a class-consistent manner on 50% or less of the probe trials she was

presented with. During her first block of reinforcer probe sessions, her baseline

performances also dropped. After she completed the first block of reinforcer probe

sessions quantity training was re-instated until her performances reached the mastery

criterion; this required eight sessions. She then required four sessions to meet the

mastery criterion for quantity training with reduced reinforcement. Reinforcer probe

sessions were then re-introduced for three sessions and baseline performances decreased

once again. The baseline quantity training and quantity training with reduced

reinforcement were repeated and mastered in two sessions each.

All Skills Test 2

The five participants who completed the reinforcer probe testing after quantity

training were then given the all-skills test for a second time. The purpose of the second

administration of the all-skills test was to determine if performances had improved on

trials after they were trained to match quantities of different configurations to each other

83

Figure 19. Reinforcer Probe performances for Participant TH.

Sounds to Quantities

Perc

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Baseline Scores

Participant TH

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84

Figure 20. Reinforcer Probe performances for Participant AL after original training and after a return to baseline training.

Sounds to Quantities

Perc

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Baseline Score

Participant AL

Sounds to Numerals

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Retu

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85

and receiving class-specific compound reinforcers. Table 11 shows that all five

participants had higher overall scores after quantity training, as compared to their pretest

scores. Across participants there were 13 improvements (improvement is defining by

responding in a class-consistent manner on one more trial after quantity training) on the

all-skills test for trained values and only eight improvements for non-trained values.

When improvements were made in the all-skills test the participants improved by one

trial per trial type in 15 of the improvements and by two trials per trial type in six of the

improvements.

Figure 2 shows the performances of Participant B on each trial type given in the

all-skills test (see Tables 5-8 for types of trials given in the all-skills tests). For each type

of trial, scores are presented for trials involving values targeted in the quantity training

and for trials involving all other values (those not involved in the quantity training

arranged for this participant). On trials involving values that were trained, Participant B

showed a higher percent correct after the quantity training for the following trial types:

reinforcer probes, AB, AC, and CB. This participant’s performances on the following

trial types reflected no improvement after the quantity training: quantity matching, AD,

and BC. On trials involving non-trained quantities, Participant B showed a minimally

higher percent correct after the quantity training for the following trial types: AB, AC,

and AD. Performances on the following trial types reflected no improvement for non-

trained values after the quantity training: reinforcer probes, quantity matching, BC, and

CB.

Figure 3 shows the performances by Participant J on each trial type given in the

all-skills test. On trials involving trained values, Participant J showed a higher percent

86

correct after the quantity training in trials involving the AC relation. Performances on the

following trial types reflected no improvement after the quantity training: reinforcer

probes, quantity matching, AB, AD, BC, and CB. On trials involving all other non-

trained values during the quantity training phase, Participant J showed a higher percent

correct after the quantity training for the following trial types: reinforcer probes, quantity

matching, and CB. Performances on the following trial types reflected no improvement

after quantity training: AB, AC, AD, and BC.

Figure 4 shows the performances by Participant C-D on each trial type given in

the all-skills test. On trials involving trained quantities, Participant C-D showed a higher

percent correct after the quantity training for the following trial types: quantity matching

and AD. Performances on the following trial types reflected no improvement after

quantity training: reinforcer probes, AB, AC, BC, and CB. On trials involving non-

trained values, Participant C-D showed a higher percent correct after quantity training on

quantity matching trials. Performances on the following trial types reflected no

improvement after the quantity training: reinforcer proves, AB, AC, AD, BC, and CB.

Figure 5 shows the performances by Participant TI on each trial type given in the

all-skills test. On trials involving trained values, Participant TI showed a higher percent

correct after the quantity training for the following trial types: reinforcer probes, quantity

matching, AB, AC, and BC. Performances on the following trial types reflected no

improvement after quantity training: AD and CB. On trials involving non-trained

values, performances improved on AD trials. None of this participant’s performances on

the all-skills test reflected improvement after quantity training (reinforcer probes,

quantity matching, AB, AC, AD, BC, and CB).

87

Figure 6 shows the performances by Participant K on each trial type given in the

all-skills test for only the second administration of the all-skills test. Due to experimenter

error, the data reflecting performances on the specific trial types included in the all-skills

pretest are not available. However, improvement can be seen by the overall percent

correct scores on the second administration of the all-skills test, compared to the first

administration (see Table 10).

AB Training

After completing the second administration of the all-skills test, four participants

began AB training, which involved matching quantities to an addition fact (see Table 2

for a list of the types of stimuli used in training). Three of the four participants who

began AB training met the mastery criterion for the training phase. Figure 11 shows the

acquisition data of Participant B and Participant J during AB training. Participant B met

the mastery criterion during AB training in two sessions, while Participant J met the

mastery criterion in three sessions. Figure 12 shows the acquisition performance of

Participant TI during AB training. Participant TI met the mastery criterion in a total of

14 sessions.

Figure 11 shows the acquisition data of Participant C-D, who completed 10

sessions of AB training with numerals 4, 7, and 10 and his performances remained at

approximately chance levels (33% correct responses). He was then given 13 sessions of

AB training with only the numerals 4 and 10. His accuracy scores on these sessions were

never above 69% correct. He was then instructed to count the objects in the sample

stimulus before choosing a comparison, but performances did not improve. He was then

88

given the verbal instruction, “find the one that is the same as this one,” while the

experimenter pointed to the sample stimulus but again performances did not improve.

AB Reduced Reinforcement

Three of the five participants who met the mastery criterion during AB training,

also met an mastery criterion when reinforcement density was reduced. Figures 12 and

13 show the acquisition data for Participant TI and Participant J respectively; both met

the mastery criterion in two sessions. Figure 12 shows the acquisition data for Participant

K, who also met the mastery criterion in two sessions.

AB Probe Sessions

The three participants who met the mastery criterion on AB training with reduced

reinforcement also completed BA symmetry probe sessions and reinforcer probe

sessions with the B stimuli. These probe sessions tested for relations between the B

stimuli and the printed word and numeral reinforcers, the spoken number words and the

B stimuli, as well as BA symmetry probe trials. Figure 21 shows Participant J’s

performances on these probe sessions, which were near or at 100% class-consistent. This

participant’s baseline performances remained above the mastery criterion during all probe

sessions in this phase.

Figure 22 shows performances by Participant TI on the probe sessions in this

phase. Participant TI responded in a class-consistent manner on 100% of the trials on

three of the four probe sessions for four reinforcement probe types (matching B stimuli to

printed words, numerals to B stimuli, B stimuli to numerals and spoken words to B

stimuli) and on two of the four sessions for the other type of reinforcer probe (matching

printed words to B stimuli). Notably however, on trials requiring the participant to match

89

Figure 21. Performances on AB probes and reinforcer probes with B stimuli for Participant J.

Words to B Stimuli

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Figure 22. Performances on AB probes and reinforcer probes with B stimuli for Participant TI.

Words to B Stimuli

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Participant TI AB Probe Sessions

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91

printed words and numerals to B stimuli, when performances were not class-consistent on

100% of the trials, they were not class-consistent on any of the trials. Participant TI

responded in a class-consistent manner on 100% of the symmetry (BA) trials on the one

session she completed involving probes of this type.

DISCUSSION

Nine participants, two children with developmental disabilities and seven

typically developing children, were selected to participate in a match-to-sample,

equivalence procedure designed to teach math facts. Participant selection was based on a

demonstrated absence of basic mathematical skills as assessed by a flashcard pre-

assessment and a match-to-sample pretest. All participants were judged to be at a

kindergarten math level at the start of the experiment (i.e., they could recognize

numerals, count to ten, and count objects). All nine participants began quantity-training

with class-specific reinforcers and met the mastery criterion during these training

sessions. Six of the nine participants met the mastery criterion only after they were given

verbal instructions specifying the nature of the task.

These acquisition findings suggest that the training procedure was effective in

teaching arbitrary conditional discriminations involving mathematics stimuli. That six of

the nine participants reached the mastery criterion only after they were given verbal

instructions may suggest problems with the nature of the match-to-sample task. Each

participant who received verbal instructions was first told to count the objects in each

stimulus position and then to pick one. No improvement in accuracy was seen for these

participants until they were given further instructions to not only count the stimuli, but to

also pick the one with the same number. This second instruction is task or procedure

92

related, whereas the counting instruction orients the participant to the stimuli and gives a

verbal label to each stimulus. That the participants met the mastery criterion only after

the task-related instruction suggests that the participants were attending to the task and

the stimulus label but that responses were not controlled by the conditional nature of the

task. This conclusion could be drawn from the observation that the participants were

counting verbally after they were given the prompt to count the objects in each stimuli

suggesting that they were attending to the particular stimuli, but then did not chose the

comparison stimulus that was conditionally related to the sample until they were given

the instruction to pick the one with the same number. It may be beneficial for

participants in future experiments using this procedure to begin with identity match-to-

sample training to allow the participant to become familiar with the conditional nature of

the match-to-sample task before they are given the pretest.

Six out of the nine participants who reached the mastery criterion during the

quantity-training phase were exposed to quantity training with reduced reinforcement.

All six participants who were exposed to this phase met the mastery criterion. Seven of

the original nine participants continued to the reinforcer probe phase. Five of these

participants met the mastery or stability criterion for the probe trials and continued to

perform above the mastery criterion on the intermixed baseline trials. These five

participants performed in a class-consistent manner on a high percentage of the

reinforcer-probe trials.

The reinforcer probes involved matching each component of the reinforcer

compound (i.e., the spoken word, printed word, and printed numeral) to the quantities and

matching the quantities to each component of the reinforcer compound. The class-

93

consistent performances of the participants suggest that each component of the reinforcer

(i.e., the spoken word, printed word, and printed numeral) became part of the equivalence

class that included three different configurations of objects for each value trained. Thus,

for each value, conditional discrimination training established a class of three object

configurations (each participant was trained with four numerals), and nine relations

involving the reinforcer components emerged. These nine relations included matching

the spoken word to the numeral, the spoken word to the printed word, the spoken word to

the quantities, the quantities to the numeral, the quantities to the printed word, the

numeral to the quantities, the spoken word to the quantities, the numeral to printed word,

and the printed word to the numeral. These performances are consistent with the class-

specific reinforcer literature (e.g., Schomer, 2001; Dube, et al, 1987, Dube, et al, 1989;

Ashford, 2003) and with Sidman’s theory of equivalence (2000) which states that the

reinforcer can become a member of the equivalence class (see later discussion).

One participant demonstrated class-consistent performances on the reinforcer

probes only after she received additional training to match printed reinforcer words to

quantities. This finding suggests that the printed word-to-quantity relations may be more

difficult or slower to emerge for some participants. For several participants (e.g.,

Participants TH, C-D, and TI), the least class-consistent probe performances occurred

with the printed word stimuli. These data may indicate that discriminations among the

printed word stimuli were more difficult or less well developed than for the other

stimulus types. This possibility is consistent with the fact that most of the participants

were not yet reading printed words. This finding also suggests that the printed word-to-

quantity, or word to any other equivalence class member, relation may be imperative for

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class-consistent responding on all the types of reinforcer probes since this participant

exhibited class-consistent responding on all reinforcer probes only after the printed word

to quantity relation was established and was able to discriminate the words from each

other during a card sort task.

The purpose of the all-skills test was to administer a pretest containing all

relations to be trained or tested for and then to re-administer the test after each training

phase to determine any improvements that could be attributed to the training. Although

not many conclusions can be made from the all-skills test data due to the low number of

trials for each type of relation the finding that 13 improvements were made involving

trained values and only eight improvements were made involving untrained values

suggests that the training was effective in teaching or allowing relations to emerge that

involve the trained values. That eight improvements were made involving the untrained

values suggests that some generalization of the relations to new untrained values

occurred. However, these conclusions are only speculation and a follow up study would

need to be done with more of each trial type included in the all-skills test to make more

definitive conclusions.

The five participants who exhibited class-consistent responding on the reinforcer

probes were also exposed to AB training. Four of the five met the mastery criterion

during AB training, and three of those four met the mastery criterion for the AB reduced

reinforcement phase. Thus the match-to-sample training procedure was effective in

establishing the AB relation between the quantity stimuli and the addition facts for all but

one participant. For four participants the AB relations were learned in the absence of

verbal instructions, suggesting that the previous experience with the match-to-sample

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task during the quantity training phase may have aided in the acquisition of the AB

relation. This is consistent with previous literature showing that other conditional

discriminations are learned more quickly after the first conditional discrimination is

mastered (Pilgrim, Jackson, & Galizio, 2000).

Three participants were exposed to AB probes, including BA symmetry probes

(i.e., trials with B stimuli as samples and A stimuli as comparisons) and reinforcer probes

with B stimuli (i.e., matching B stimuli to printed words, B stimuli to numerals, printed

words to B stimuli, numerals to B stimuli, and spoken words to B stimuli). The

performances by all three of these participants were overwhelmingly class consistent.

That is, after training the participants to match B stimuli to A stimuli, the participants

were able to match A stimuli to B stimuli. The participants also performed in a class-

consistent manner on the reinforcer probes involving B stimuli, which suggests that each

component of the reinforcers became a member of the equivalence class that included the

relevant A stimuli (quantities) and B stimuli (addition facts). The equivalence classes

suggested by each participant’s performance included three quantities, an addition fact,

the numeral, the printed number word, and the spoken number word for each of the

values involved in training. This finding is also consistent with class-specific reinforcer

literature (e.g., Schomer, 2001; Dube et al, 1987; Dube et al, 1989; Ashford, 2003) and

with Sidman’s theory of equivalence (2000); see later discussion.

There were a few procedural limitations of this experiment that should be

addressed in future studies. The all-skills test that was originally given to the participants

was revised after the start of the experiment, such that a balanced number of relations was

tested for each value involved in any training. This revised version should be used as

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pretest and as post-tests for all new participants. There were seven types of relations

(reinforcer relations, quantity matching, AB relations, AC relations, AD relations, BC

relations, and CB relations) trained or tested for during the course of this experiment. In

order to keep the sessions to a length that was easily completed by these young children,

the number of trials of each type in the all-skills tests and in the probe sessions was

limited to two to four per test session. This made comparative analysis difficult because

one error on a probe trial or all-skills-test trial greatly reduced the overall percentage of

class-consistent responses for that relation. Similarly, comparison of pretest and posttest

performances on the all-skills test was difficult because a single correct response by

chance resulted in a relatively high percentage of class-consistent responses, given the

small number of trials of each type. In short, the all-skills-test proved to be less sensitive

to the effects of experimental training than would be optimal. In future experiments

using this procedure, careful consideration should be taken to analyze the probe and all-

skills test data, and the addition of more trial types of each relation and fewer types of

trials should be strongly considered being that the participants did not complete as much

of the experiment as expected.

As mentioned earlier, the addition of verbal instructions during the quantity

training had an impact on the acquisition of the conditional discriminations. These verbal

instructions were procedure related, explicitly targeting the conditional nature of the task.

Future experiments using participants of this age and mathematical ability may want to

use similar verbal instructions at the start of the experiment, for practical purposes,

especially to orient the participant to the specific nature of the task. Another suggestion

would be to expose the participant to pre-training with match-to-sample trials involving

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unrelated stimuli. This pre-training might involve matching stimuli within a theme (e.g.,

matching a palm tree to an oak tree or a car to a bus) in order to establish conditional

control. This pre-training in place of, or in addition to, the verbal instructions about the

task might facilitate the conditional discrimination acquisition and bring about the desired

stimulus control topography.

The present experiment ended due to time constraints and the end of the public

school calendar. If continuation of this experiment had been possible, several additional

phases of training and testing could follow. For example, the three participants who

completed the AB probe phase should continue on to AC training. The purpose of this

phase would be to match the same quantities used in AB training (i.e., the A stimulus set)

to a new simple addition problem (the C stimulus set). The sample stimulus on each trial

would be an A quantity previously trained (e.g., four lions). The comparisons would be

simple addition problems (a numeral, addition sign, and the numeral 2; e.g., 2+2). The

correct comparison would be the addition fact that matched the quantity in the sample

(e.g., with 4 lions as a sample, 2 + 2 would be the correct comparison). If the addition

fact that corresponded to the sample quantity was chosen, the response would produce the

same class-specific reinforcers used in quantity and AB training (e.g., the spoken word

“four” and either the numeral four or the printed word “four”). If any other comparison

was chosen, the participant would hear a buzzer. The participant would continue in this

training phase with all four numbers in the training block until the mastery criterion was

met. Reinforcement density will be reduced gradually in the method described in the

quantities-training section. When the mastery criterion was met at the 75%

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reinforcement density, the participant would proceed to the symmetry, reinforcer, and

then equivalence probe test sessions.

The symmetry and reinforcer probe trials would then be presented. These probe

tests would include baseline trials of the AC relations just taught, the symmetry probes of

those relations (CA), and reinforcer probe trials with C stimuli. The equivalence-probe

phase of the experiment would determine whether equivalence classes have emerged

after training the two conditional discriminations, AB and AC. Probe trials would test for

the symmetric relations, BA and CA, and the equivalence relations, BC and CB. Probe

tests would also be given in which trials include numbers not yet trained or number

combinations, to test for generalization. Table 9 presents the trial types and number of

each trial type to be presented in these sessions. Probe trials would not include

reinforcement, but there would be baseline trials (AB and AC) intermixed in these

sessions for which reinforcers would be available intermittently. Equivalence probe trials

would be presented until performance was stable or met the mastery criteria. The all-

skills tests would then be readministered in order to reveal any new skills acquired since

the last test sessions as a result of the latest experimental training and the equivalence

probes. After completing these tests, the participant would then be ready to begin the

experimental procedure again with a new set of values, beginning with the quantity

training phase. Further probe testing could also be administered to assess generalization

to other addition facts involving the same values, other configurations of quantities, and

numerals higher than 10.

The findings of the present experiment add to the growing body of literature on

class-specific reinforcers (e.g., Schomer, 2001; Dube et al, 1987; Dube et al, 1989;

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Ashford, 2003). Class-specific reinforcers have been shown to facilitate the acquisition

of conditional discriminations (Ashford, 2003). The use of class-specific reinforcers in

this experiment may have facilitated the acquisition of the conditional discriminations by

becoming members of the equivalence class and preventing class collapse. The type of

training used in the present experiment also added efficiency in that the class-specific

reinforcers can become members of the equivalence class, as predicted by Sidman

(2000). The findings in the present experiment also extend a previous experiment

(Ashford, 2003) in that each component of the class-specific reinforcer became a member

of the equivalence class, including the auditory component which was used as a sample

in probe tests. This finding is important because matching the individual components of

the reinforcer (i.e., the spoken word, the printed word, and the printed numeral) to

quantity stimuli from the training phases was never reinforced.

The relations that emerge after training using class-specific reinforcers provide

exciting implications for teaching procedures. Given that many school-age children

demonstrate difficulties in learning mathematics, the findings of this experiment provide

new options in teaching even simple addition facts and quantity training. All nine

participants acquired the quantity relations and five of those participants exhibited

evidence that the reinforcers became part of the equivalence class without being

explicitly taught. Although it would be interesting for future studies to involve

participants that were known to have difficulty learning mathematics, these findings add

to the literature that involves using equivalence procedures to teach mathematics (e.g.,

Lynch & Cuvo, 1995; Gast, VanBiervlet, & Spradlin 1979).

100

Teachers could easily implement the computerized procedures used in this

experiment to supplement current mathematics instruction curriculums. These

procedures are almost entirely automated and would provide more opportunity to engage

students, while refining computer and mouse skills, and allowing the teacher to give one-

on-one attention to other students. These procedures would allow the teacher to collect

and analyze data revealing specific performances of the individual students and allow

modification of the teaching procedure to be specific to each student. The computerized

responses used in this experiment would also allow the teachers to assess the relations

learned by each student, without requiring the students to construct the numerals or

numeral words. The present experiment provides an opportunity to use equivalence

procedures and class-specific reinforcers, and to take advantage of the emergent relations

in a field where many children have difficulty. This experiment could be extended and

replicated to include other types of mathematics or other classroom skills.

Three main theories of equivalence were described earlier (relational frame

theory, naming theory, and Sidman’s theory of equivalence) and although the present

experiment provides support for the literature on class-specific reinforcers and classroom

applications of equivalence, it does not provide an opportunity to rule out any one theory

of equivalence. The results in this experiment are consistent with and could be predicted

by Sidman’s theory of equivalence (2000). Sidman’s theory suggests that the reinforcer

can play a role similar to that of the other stimuli in a match-to-sample procedure. His

theory holds that each component of the four-term contingency, the conditional stimulus,

discriminative stimulus, response, and reinforcer, can enter into the equivalence class as

equal members. The present findings are consistent with this theory in that each

101

component of the reinforcer used in training became a member of the equivalence class,

as demonstrated by the reinforcer’s role as sample and comparison in match-to-sample

probe trials.

Although the data in this experiment fall in line with Sidman’s specific

predictions, other theories of equivalence could account for the pattern of results found

here as well. According to relational frame theory (Hayes, et al., 2001), the participants

in this experiment may have responded to the reinforcers according to contextual cues

that were established during previous multiple exemplar training. This pattern of

arbitrarily applicable relational responding, or relationally framing may have been

established during previous experiences involving formal or informal mathematics

training. According to the naming theory (Horne & Lowe, 1996), a bidirectional process

where the participants were overtly or covertly naming the sample, comparison, or

reinforcer may have brought the stimuli, including the reinforcer, into the equivalence

class. However, there is one challenge for naming theory accounts of the quantity

training data in this experiment. When participants were given the instruction to count

(or name) the sample and comparison stimuli, no improvements were observed in their

patterns of responding. In fact, accuracy did not improve until participants were given

instructions describing the conditional nature of the task. This implies that requiring the

participants to name the stimuli was not sufficient to improve their responding.

In conclusion, although the findings presented from this experiment are in support

of Sidman’s theory, the evidence does not allow other theories of equivalence to be ruled

out. However, the results from this experiment do provide support for the theory that

class-specific reinforcers and each of their components can emerge as members of an

102

equivalence class under the appropriate training conditions. The findings of this

experiment also extend those of previous studies involving the training of conditional

discriminations, match-to-sample procedures, and compound class-specific stimuli used

as reinforcers. These results also provide an innovative and efficient procedure for

teaching mathematics in or out of the classroom. This is an exciting and important

finding for all those involved in teaching mathematics to students with difficulty learning

such skills.

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