Students’ everyday mathematics, as well as any additional math skills students acquire before the onset of kindergarten, are critical indicators of their future success in both reading and math (Engel et al., 2013). Likewise, students’ math skills in kindergarten strongly predict later math and reading achievement (Claessens & Engel, 2013; Ten Braak et al., 2022). Despite these discoveries, compared to reading instruction, math is underemphasized in the early grades (Engel et al., 2013). This may contribute to the United States scoring below average in math proficiency, landing the United States in the bottom third of industrialized nations (Williams, 2022). Students with a low interest in math are proposed to have low self-confidence in their math ability as reported by Pinxten et al. (2014), who found a positive relation between math competency beliefs, math enjoyment, and math competency for students in grades 3–7. In other words, math enjoyment and math competency can influence students’ math performance (Fisher et al., 2012; Ma, 1997; Pinxten et al., 2014).

In an ideal situation, students should have a significant interest in math which would then function to improve their math performance and overall literacy. There is a need for interventions that increase students’ interest in math as, “the presence of interest positively influences learners’ attention, strategy use, and goal setting. With interest, learners are better able to self-regulate and persist to complete tasks even when they are challenging” (Renninger et al., 2015, p. 2). In a longitudinal study, Fisher et al. (2012) measured students’ math abilities, language and cognition, levels of interest, time playing a math game, and the teacher’s report of the child’s level of interest. Results indicate a positive correlation between math enjoyment and math performance in preschoolers. Fisher et al. (2012) built upon the research of Ma (1997) who:

proposed a bidirectional paradigm between ability and attitudes specifically for math development, with attitudes and achievement strengthening one another over time . . . heightened interest, increases time spent pursuing math activities, which improves ability. While framed positively, individuals with low interest or ability could experience a negative cycle. (p. 674)

From a behavior selection perspective, math enjoyment or a positive attitude toward math is referred to as conditioned reinforcement for math or reinforcement value and preference for engaging in math activities where the control of the activities reinforces the related behaviors. Hence, according to Verbal Behavior Development Theory (Greer & Ross, 2008) doing math would be considered a learned or conditioned reinforcer. For the purposes of this article, preference for engaging in math activities and conditioned reinforcement for math will be used interchangeably. Reinforcement is, “the response-produced presentation of positive reinforcers or termination of negative reinforcers, or the increase or maintenance of responding resulting from this operation” (Catania, 2013, p. 460). By extension, stimulus control for engaging in an activity is the reinforcer for said activity. Conditioned positive reinforcement occurs when a previously neutral or aversive stimulus acquires reinforcing properties. This can occur through the following types of conditioning: operant, classical, and observational conditioning (Greer & Singer-Dudek, 2008; Greer et al., 2006; Pavlov & Anrep, 1927; Skinner, 1953). Operant conditioning focuses on the role of consequences on behavior as well as the automaticity of reinforcement (Skinner, 1953). Classical or respondent conditioning (Pavlov & Anrep, 1927) calls for the stimulus–stimulus pairing procedure in which an unpreferred or neutral stimulus is paired with a reinforcing conditioned stimulus that results in that same neutral stimulus becoming a conditioned reinforcer. Observational conditioning refers to a change in existing behaviors as a result of observing the delivery of the stimulus to others while being repeatedly denied access to it (Greer & Singer-Dudek, 2008; Singer-Dudek et al., 2011).

The establishment of conditioned reinforcement for academic stimuli has resulted in either a faster rate of learning or, in some instances, a new way of learning (Buttigieg & Greer, 2023; Lee, 2016; O’Rourke, 2006; Tsai & Greer, 2006). For example, conditioned reinforcement for books, math, and writing has been established via observation. Singer-Dudek et al. (2011) conducted a study to test the effectiveness of an observational intervention in establishing books as conditioned reinforcers using a delayed multiple baseline design. The participants were three preschool students with mild language and developmental delays. The participants were selected because although they had a variety of reinforcers, books did not function as a conditioned reinforcer for observing. The dependent variables of this experiment included the percentage of 5-s intervals the participant looked at books in the free play area, the number of correct responses to three learning tasks, and the number of correct responses to a maintenance task before and after the observational intervention.

During the observational intervention, the participant and confederate were seated next to each other with an opaque divider between them. Although both students were presented with the same task, only the confederate was given books for 5 s following each response. The divider was removed when the confederate was looking at books so that the participant could observe this. The intervention continued for eight sessions. Following the intervention, books served as conditioned reinforcers for participants and the rate of responding for maintenance and acquisition responses increased for all three participants. A functional relationship was shown (Singer-Dudek et al., 2011). The above procedure was successful in conditioning praise as a reinforcer as well (Greer et al., 2008).

Tsai and Greer (2006) found that all four preschool participants required fewer learn units-to-criterion for textually responding to sight words following the establishment of conditioned reinforcement for the observation of books. Buttigieg and Greer’s (2023) study on the effect of conditioned reinforcement on students’ rate of acquisition of novel textual responses was additive to Tsai and Greer’s (2006) study in that Buttigieg and Greer’s (2023) study sought to condition books through either learn unit instruction, stimulus–stimulus pairings, or observational conditioning-by-denial. If the participant demonstrated to have conditioned reinforcement for books following sight word learn unit instruction, then the participant was said to have acquired conditioned reinforcement for the observation of books through operant conditioning, as learn unit instruction is an operant teaching procedure. Buttigieg and Greer (2023) found that following the establishment of conditioned reinforcement for the observation of books students required fewer learn units-to-criterion on novel sight words. The establishment of conditioned reinforcement for the observation of books is crucial in that it not only exposes students to a variety of word and picture relationships, but also promotes reading readiness (Buttigieg & Greer, 2023). This cusp can be established through several procedures; stimulus–stimulus pairing procedure (Nuzzolo-Gomez et al., 2002), observational conditioning procedure (Singer-Dudek et al., 2011), and textual operant discrimination training (learn unit instruction; Buttigieg & Greer, 2023).

O’Rourke (2006) conducted two experiments to test the effects of an observational conditioning intervention on conditioned reinforcement for math. The participants were between 6 and 8 years old and were selected because math was aversive to them. Experiment 1 was a replication of Greer and Singer-Dudek’s (2008) study to determine if previously nonpreferred tokens would acquire reinforcing properties through an observational intervention for learning and performance tasks. Results of Experiment 1 indicated that tokens did in fact function as conditioned reinforcement for performance and learning due to the observational intervention.

In her second experiment, O’Rourke (2006) tested the observational conditioning-by-denial intervention on academic performance in math. Results showed that math tasks (e.g., addition and subtraction, time, and word problem worksheets) functioned as reinforcers for performance and acquisition as a result of the observational intervention. According to O’Rourke (2006), the intervention did “establish a shift in preference of activities, changing a previously non-preferred activity into a preferred activity” (p. 80).

Lee (2016) extended observational conditioning to writing. She conducted two experiments to determine whether an observational condition would be effective in establishing conditioned reinforcement for opportunities to emit writing responses. Lee (2016) measured two dimensions of conditioned reinforcement that she referred to as direct and indirect reinforcement. Indirect reinforcement referred to the use of math as a reinforcement operation for another behavior unrelated to writing and direct reinforcement referred to the number of intervals participants responded to a writing prompt. Data were collected on the participants’ number of mands emitted for writing. Intervention sessions continued until there was a steady state of mands across sessions or when the extinction of performance responses occurred. Postintervention data demonstrated an increase in direct and indirect reinforcement value of writing. Lee’s (2016) intervention was based on the observational procedure in Greer and Dudek (2008), in which being denied the target stimuli resulted in the establishment of conditioned reinforcement for said stimuli.

The present study sought to determine whether students’ preference or interest in math, can be changed and whether this change would result in a change in students’ rate of learning. Prior research on reading (Buttigieg & Greer, 2023; Tsai & Greer, 2006) has suggested that the establishment of conditioned reinforcement results in an accelerated rate of acquisition of sight words. Studies on generalized visual match-to-sample suggested that conditioned reinforcement for two-dimensional stimuli results in an accelerated rate of learning (Delgado et al., 2009; Du et al., 2015; Greer & Han, 2015). Studies on conditioned reinforcement for voices and faces have suggested an accelerated rate of acquisition for listener programs (Greer et al., 2011; Maffei et al., 2014). The results of the above studies suggest that conditioned reinforcement value may have the same effect on math. Therefore, the purpose of this study was to (1) establish conditioned reinforcement for math activities using the conditioning procedures outlined in previous research; and (2) determine whether the establishment of conditioned reinforcement for math would result in a change in students’ rate of learning.

Method

Participants

Six preschool students were selected from a Comprehensive Application of Behavior Analysis to Schooling Accelerated Independent Learner (CABAS AIL) Pre-Kindergarten classroom that had 16 students, one teacher, and two teacher’s assistants. Participants L and M functioned at early reader/writer levels of verbal behavior, whereas Participants C, N, G, and J functioned at speaker and listener levels of verbal behavior. The following participants were classified as preschoolers with a disability: L, N, C, and J. Participants L, M, and J were of Hispanic descent. In addition, Participant M qualified for Free/Reduced lunch and was an English Language Learner. Table 1 contains a full description of each participant. Verbal Behavior Developmental Theory (VBDT; Greer & Ross, 2008), influenced by Skinner’s verbal behavior theory, proposes a trajectory of fundamental cusps and capabilities an individual must achieve to be deemed truly verbal. According to VBDT, all participants had conditioned reinforcement for two-dimensional stimuli, three-dimensional stimuli, and book stimuli in repertoire. In addition, Participants L, C, M, and G, had incidental bidirectional naming in repertoire. Incidental bidirectional naming allows participants to acquire verbal functions without direct instruction (Yoon et al., 2023).

Table 1 Description of Participants
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Setting

Experimental sessions took place in the participants’ classroom at a horseshoe table with the necessary materials detailed below. All other students in the classroom were participating in small group reading and language instruction during probe, rate of learning, and intervention sessions. During observational conditioning sessions, the participant sat next to the confederate with a foam divider between them such that they could not see each other, but could see the experimenter delivering math worksheets to the confederate. The study spanned 3 months of daily sessions. Probe sessions lasted for 5 min each, whereas intervention and rate of learning sessions varied in duration, ranging from 15 to 45 min.

Materials

Individualized Reinforcement Intervention Materials

A stack of 20 grade-level math worksheets as well as two cans of Play-Doh and five Play-Doh toys were used during conditioned reinforcement for math probes. Math worksheets were selected based on the objectives that students had in repertoire (e.g., counting, matching number to number, matching number to quantity, matching quantity to quantity, identifying more or less quantities, and identifying more or less with Arabic numbers). Play-Doh was selected based on the participants’ demonstrated preference for Play-Doh during free play sessions. Various math worksheets, uppercase and lowercase letter flashcards, and privacy boards were used for stimulus–stimulus pairing and observational conditioning sessions respectively.

Rate of Learning and Learn Unit Instruction Materials

The pilot version of the Weber et al. (2023) Equivalence Based Functional Math Curriculum (EBF-Math) was used for these measures. Aligned to Common Core Standards, this curriculum contained units that included four math objectives rotated across match, point to, tact, and intraverbal response topographies. This rotation of response topographies is referred to as multiple exemplar instruction, or MEI (Yoon et al., 2023). The curriculum included a teacher script, a student answer book, and a data sheet for each unit. The teacher script outlined the instructional demonstration learn units for each objective in the unit as well as the antecedent and correct procedure for each learn unit. Multiple versions of each answer booklet were created to control for memorization. See Table 2 for a list of each participant’s units, Appendix A Table 3 for a list of the objectives in each unit, and Appendices B, C, and D for a sample teacher script, student answer book, and data collection sheet respectively.

Table 2 Note that this data is mandatory
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Dependent Variable

The dependent variable was the rate of learning, (i.e., number of learn units required to meet criterion) for four units of the EBF-Math curriculum. Instruction consisted of state of the science teaching as behavior analysis. That is, instruction had all of the components of the learn unit and correction, thus controlling for instructional conditions before the intervention and following the intervention. Each unit included four objectives rotated across match, point, tact, and intraverbal responses for a total of 20 learn units per session.

Independent Variable: Individualized Reinforcement Intervention

The independent variable was an individualized reinforcement intervention. This intervention consisted of the conditioning sequence proposed in Buttigieg and Greer (2023): learn units, stimulus–stimulus pairing, and observational conditioning by denial.

Data Collection

Rate of learning and learn unit instruction measures were collected by totaling the number of learn units delivered to each participant for four units of the EBF-Math curriculum. Learn unit data for each objective in the curriculum were collected using the plus (+) and (-) system and graphed as the number of correct responses to learn units. Direct reinforcement measures were collected using the plus “+” and minus “-“ system as well and reported as the number of 5-s intervals (out of 60) that the participant selected math stimuli. Stimulus–stimulus pairing data were collected using the plus (+) and minus (-) system and reported as the number of correct responses to whole interval 5-s or 10-s pair-test trials. Observational conditioning data were collected using tallies and reported as the number of vocal and nonvocal attempts to access math. Data were collected by the researchers and masters level graduate students who served as the classroom teacher and teacher’s assistants.

Interobserver Agreement

Point-by-point interobserver agreement (IOA) was collected for both the dependent and independent variables to ensure fidelity. Independent observers were trained on intervention procedures to mastery. Point-by-point IOA was calculated by dividing the number of agreements by the number of agreements plus disagreements and multiplying by 100%. For rate of learning/learn unit sessions, there was 98% IOA on 40% of learn unit sessions. There was 99% IOA on 100% of direct reinforcement probes sessions, 100% IOA on 100% of stimulus–stimulus pairing sessions, and 100% IOA on 50% of observational conditioning sessions.

Design

A multiple probe across dyads design with a nested multiple probe across dyads design was used in this experiment. The dependent variable, rate of learning, was tested using a multiple probe design; the intervention nested within that was also a multiple probe design on testing the effect of the individualized reinforcement procedure on establishing conditioned reinforcement. All participants’ initial probes of direct reinforcement for math were conducted at the onset of the study. Following initial conditioned reinforcement probes, all participants began the EBF-Math curriculum which was used to collect rate of learning and learn unit instruction data. Following mastery of four units of EBF-Math, conditioned reinforcement probes were conducted to determine whether learn unit instruction was effective in establishing conditioned reinforcement for math. If learn unit instruction was indeed successful, then postintervention rate of learning measures were conducted. If learn unit instruction was not successful then the intervention continued with the stimulus–stimulus pairing procedure. This sequence continued with observational conditioning if the stimulus–stimulus pairing procedure was not effective. Dyads remained in the preintervention rate of learning phase until the preceding dyad entered postintervention rate of learning phase. See Figure 1 for the design sequence flow chart.

Fig. 1
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Sequence of the experiment. Dyads remained in the pre-intervention rate of learning phase until the preceding dyad entered post-intervention rate of learning phase. Once entered the study, experimenters followed the sequence above

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Procedures

Pre- and Post- Intervention Probes of Direct Conditioned Reinforcement for Math

Conditioned reinforcement probes were conducted for all participants at the onset of the study. Participants were provided a stack of 20 grade-level math worksheets, a marker, two Play-Doh containers, and three Play-Doh tools on the table. The experimenter informed the participant that s/he had 5 min to do as they pleased. The criterion for conditioned reinforcement for math was set at 180 out of 300 intervals (5 s intervals). Conditioned reinforcement probes were conducted before and following each conditioning procedure to determine if the specific procedure was effective in establishing conditioned reinforcement for math.

Rate of Learning and Learn Unit Instruction

Before and following the establishment of conditioned reinforcement for math, participants’ rate of learning math was measured using the EBF-Math curriculum. Sessions were conducted individually with each participant. Rate of learning consisted of the number of learn units required to demonstrate mastery of curricular objectives where fewer numbers demonstrated faster learning. The curriculum included scripted antecedents (i.e., instruction and exemplars) and corrections. The experimenter delivered the antecedent and provided reinforcement for correct responses in the form of approvals, high-fives, and tickles. If the participant emitted an incorrect response, the correction procedure (detailed in the teacher script) was delivered in which the experimenter emitted the correct response and re-presented the antecedent, giving the participant the opportunity to independently emit the correct response. An instructional session consisted of a block of 20 learn units. Criterion was set at 90% (18/20) across two consecutive sessions or 100% (20/20) in one session. Once the participant mastered four units of the EBF-Math curriculum, a postdirect conditioned reinforcement probe was conducted to determine whether learn unit instruction was effective in establishing conditioned reinforcement for math. If so, postintervention rate of learning measurement began, if not then the experimenter began the stimulus–stimulus pairing procedure.

Stimulus–Stimulus Pairing Procedure

If learn unit instruction alone did not result in the establishment of conditioned reinforcement for math, the stimulus–stimulus pairing procedure ensued. Each session of the pairing procedure consisted of 20 pair-test trials. The test trial did not occur until a successful pair was established. A successful pair was defined as the participant emitting math observing responses or completing math worksheets with the experimenter for the entire 5-s pairing. Immediately after a successful pair was established, a 5-s test trial was conducted in which data were recorded on whether the participant continued emitting math observing responses independently. Criterion for the stimulus–stimulus pairing procedure was 18/20 or 90% accuracy across two consecutive sessions. Both 5-s and 10-s pairing procedures were conducted prior to direct reinforcement probes. If the 5-s and 10-s procedure did not result in conditioned reinforcement for math, as evidenced by conditioned reinforcement probes, then the experimenter proceeded with a 15-s pair-test.

Observational Conditioning by Denial

In the event that learn unit instruction and the 15-s stimulus–stimulus pairing procedure did not result in conditioned reinforcement for math, observational conditioning by denial was implemented. The experimenter delivered matching learn units simultaneously to the participant and confederate. However, only the confederate was reinforced with math problems for correct responses, the participant was ignored. The partition was removed when the confederate was completing the math problems so that the participant was able to see that the confederate received math worksheets for correct responses. Five observational conditioning by denial sessions were conducted which consisted of 10 match trials. Data were recorded on the participant’s number of mands for math (e.g., “I want that!” “Gimmie!” or “What about me!”) and attempts to obtain math materials.

Results

The results of the study demonstrated that establishing conditioned reinforcement for math resulted in an increased rate of learning math. Figure 2 shows the pre- and postintervention rate of learning data for each participant. Before and following the establishment of conditioned reinforcement for math, Participant L’s learn units to criterion or LUC was 160 and 100 respectively. Participant N required 200 and 160 learn units to meet criterion respectively before and following the establishment of conditioned reinforcement for math. Before the establishment of conditioned reinforcement for math Participant C’s LUC was 320 compared to 200 following conditioned reinforcement for math. Participant M’s number of learn units required to meet criterion was 340 prior to the intervention and 160 following the intervention. Before and following the intervention Participant J and G’s LUC were as follows: 440 to 260 and 480 to 240 respectively. The rate of learning accelerated for all participants as a function of the establishment of conditioned reinforcement for engaging in math activities.

Fig. 2
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Rate of Learning

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Figure 3 shows the pre- and post-probes of conditioned reinforcement for math. Participant L required only learn unit instruction to establish conditioned reinforcement for math. Before the establishment of conditioned reinforcement for math, Participant L selected math during direct probe sessions for 60/300 intervals. Following learn unit instruction, he selected math for 300/300 intervals. Participant N required the stimulus–stimulus pairing procedure to establish conditioned reinforcement for math as he selected math for 7/300 intervals during initial direct probe sessions and 0/300 following learn unit instruction. After meeting criterion on the stimulus–stimulus pairing procedure, Participant N selected math for 181/300 intervals. Like Participant L, Participants C and J only required learn unit instruction to establish conditioned reinforcement for math. Before learn unit instruction Participants C and J selected math for 0/300 intervals, following learn unit instruction Participants C and J selected math for 195/300 and 284/300 intervals respectively. Participant M required observational conditioning to establish conditioned reinforcement for math. During initial direct probes, post-learn unit, and post-stimulus–stimulus pairing probes she selected math for 0/300 intervals. Following observational conditioning, she selected math for 300/300 intervals. Like Participant N, Participant G required the stimulus–stimulus pairing procedure to establish conditioned reinforcement for math. In preprobes of direct reinforcement for math, Participant G selected math for 3/300 intervals, following learn unit instruction she selected math for 50/300 intervals, and after the stimulus–stimulus pairing, she selected math for 190/300 intervals.

Fig. 3
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Duration of Selection of Math in Free Operant Play

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Figure 4 shows the participants’ number of correct responses during each session of learn unit instruction before and following the establishment of conditioned reinforcement for math. Participant L required 40, 60, 40, and 20 learn units to meet criterion on his preconditioned reinforcement for math units and 20, 40, 20, and 20 learn units on his postconditioned reinforcement for math units respectively. In each respective preconditioned reinforcement for math unit Participant N required 60, 20, 60, and 60 learn units to meet criterion and 20, 20, 60, and 40 learn units during postconditioned reinforcement rate of learning measures. Participant C required 60, 40, 40, and 180 learn units to master Units 2–5 respectively compared to following the establishment of conditioned reinforcement where he required 40, 40, 60, and 60 learn units to master Units 6–9A. Participant M required 60, 60, 60, and 160 learn units to master Units 3–6 and 40, 60, 40, and 20 learn units to master Unit 7–9B, respectively. Participant J required 140, 120, 100, and 80 learn units to master his units prior to the establishment of conditioned reinforcement for math and 40, 60, 80, and 80 learn units to master his units following conditioned reinforcement for math. Participant G required 100, 140, 120, and 120 learn units to master Units 3–6, respectively, compared to 60, 40, 80, and 60 to master Units 7–9B, respectively, following the establishment of conditioned reinforcement for math.

Fig. 4
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Rate of Learning / Learn Unit Instruction

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Learn unit instruction did not suffice in establishing conditioned reinforcement for math for Participants M, N, and G, thus they continued with the stimulus–stimulus pairing procedure. Participants M and G required 40 pairtest trials to meet criterion on the 5-s pairing and 60 pairtest trials to meet criterion on the 10-s pairing. Participant N required 40 learn units for both the 5-s pairing and for the 10-s pairing.

Observational conditioning was conducted with Participant M because learn unit instruction and the stimulus–stimulus pairing procedure were ineffective in establishing conditioned reinforcement for math for this participant. During observational conditioning, she did not emit any mands for math but her correct responses to the performance task were low. Participant M emitted 3, 1, 1, 3, and 2 correct responses out of 10 during observational conditioning-by-denial-sessions.

Discussion

The individualized reinforcement intervention was effective in establishing conditioned reinforcement for math for all six participants and resulted in an accelerated rate of learning math. The importance of conditioning stimuli as reinforcers for observing and the effect of this conditioning on learning has been well-documented in a series of verbal developmental studies (Buttigieg & Greer, 2023; Delgado et al., 2009; Du et al., 2015; Greer & Han, 2015; Greer et al., 2011; Maffei et al., 2014; Tsai & Greer, 2006). The present study adds to this body of research. Like the participants in the present study, the above studies were conducted with preschool-aged children. Each of the above studies used the stimulus–stimulus pairing procedure to establish conditioned reinforcement for: voices and faces (Greer et al., 2011; Maffei et al., 2014), two-dimensional and three-dimensional stimuli (Delgado et al., 2009; Du et al., 2015, Greer & Han, 2015), and books (Buttigieg & Greer, 2023; Tsai & Greer, 2006). All of these participants acquired a verbal developmental cusp (Greer & Ross, 2008) following the intervention. This means that the participants were now able to contact the environment in ways they previously could not. The results of these studies demonstrate that acquiring certain reinforcers also results in the acquisition of new cusps (Greer & Du, 2015). In addition to acquiring a new learning cusp, conditioning stimuli as reinforcers for observing function to accelerate students’ rate of learning for listener programs, visual match to sample programs, and sight words, respectively. The results of this study are consistent with the above studies in that conditioning stimuli as reinforcers, in this case math, led to an increase in participants’ rate of learning math.

Learn unit instruction alone was effective in establishing conditioned reinforcement for math for Participants L, C, and J. These participants simply needed to access reinforcement for math and learn the function of math for math itself to function as a conditioned reinforcer. Due to Participant L’s rule governed behavior, he selected math for 60/60 intervals in his first probe of direct reinforcement for math. Before beginning the next probe, Participant L asked the experimenter, “I can do whatever I want?” Participant L was reassured that he could choose either the Play-Doh or math without consequence. Following the intervention Participants L, C, and J’s learn units to criterion decreased by almost half, meaning they acquired new math objectives twice as fast as they did before having conditioned reinforcement for math in repertoire.

Unlike Participants L, C, and J, learn unit instruction did not suffice in establishing conditioned reinforcement for math for Participants N and G. These participants required both learn units and the stimulus–stimulus pairing procedure to establish conditioned reinforcement; once established their rate of learning significantly accelerated just as with Participants L, C, and J. Participant M required observational conditioning to establish conditioned reinforcement for math. Even after the stimulus–stimulus pairing procedure, Participant M did not attend to the math material and opted to play with Play-Doh for the entirety of direct probes. During observational conditioning, she emitted a high number of incorrect responses to a performance task when she was denied access to math. Her high number of incorrect responses was likely due to her responses being unconsequated. No attention was provided for correct or incorrect responses. This practice is contrary to the standard classroom management tactics of the classroom. Although she did not emit any mands for math, she did state that “this is just hard for me.” It is possible that Participant M was socially motivated to engage in math-related activities after noticing the confederate during the observational conditioning sessions. The observational conditioning procedure was effective in establishing conditioned reinforcement for math for Participant M, which led to an accelerated rate of learning math.

A possible limitation of this experiment is that the math units were not counterbalanced. However, probes were conducted to determine which units the participants did not have in repertoire to ensure the objectives were indeed novel. Though the math units were not counterbalanced, the EBF-Math curriculum progressively increased in difficulty, which means that the participants learned faster following the establishment of conditioned reinforcement for math despite an increase in the difficulty (e.g., number of steps required to complete the problem) of the math objectives.

Some research suggests that math achievement at the onset of kindergarten is one of the strongest predictors of future performance in math and reading (Claessens & Engel, 2013; Clements & Sarama, 2014). This means that at the pre-kindergarten level, there may already be gaps between students who come from low socioeconomic homes and those who do not. As Clements and Sarama (2014) stated, “For these children especially, the long-term success of their learning and development requires high-quality experience during their early ‘years of promise’” (p. 2). According to the National Mathematics Advisory Panel (2008), the math gap between low-income and middle-income students increasingly expands throughout their school-aged years.

The results of this experiment demonstrate that students’ rate of learning accelerates by 1.5 to 2 times following the establishment of conditioned reinforcement for math. Children who do not demonstrate early math proficiency may benefit from gaining conditioned reinforcement for math to increase their rate of learning. Future research should determine which conditioning procedures are most effective in establishing conditioned reinforcement for math relative to characteristics of individual children. Results of this study suggest that observational conditioning-by-denial would be most effective for the students who are socially motivated. This may be due to observational stimulus control. Students who do not have observational stimulus control may not attend to the confederate’s access to the target stimulus. Compared to learn unit instruction or the stimulus–stimulus pairing procedure, observational conditioning-by-denial is faster to implement. The unaddressed matter is whether the five participants who only required learn units, or the four participants who required the stimulus–stimulus pairing in the present study, would have acquired conditioned reinforcement for math faster if observational conditioning-by-denial was implemented first. Another possibility would be to begin with learn unit instruction, because that alone is successful for some, followed by observational conditioning-by-denial, and then the stimulus–stimulus pairing procedure if necessary. In order to maximize instructional time, future research should investigate which variables within the verbal developmental and learning history of individual children predict which intervention will be more effective for a given child.

Establishing conditioned reinforcement for math may also have collateral effects on students’ reading objectives. Previous research has shown that children’s math skills predict their reading and math ability later in life (Claessens & Engel, 2013). Future research should examine the effects of the establishment of conditioned reinforcement for math using the individualized reinforcement intervention on students’ rate of learning both math and reading.

The results of the current study are consistent with prior studies that showed that establishing or enhancing reinforcement value reduces students learn units to criterion (Buttigieg & Greer, 2023; Delgado et al., 2009; Du et al., 2015; Greer & Han, 2015; Greer et al., 2011; Maffei et al., 2014; Tsai & Greer, 2006). Students’ math skills at the onset of kindergarten are critical indicators of their future success in both reading and math. Those who start school with greater math repertoires may have a history of verbal behavior shaped and maintained by the home environment that facilitates greater achievement (Engel et al., 2013). This leaves the responsibility of building upon the everyday mathematics of each child to caretakers and teachers. This individualized reinforcement intervention allows teachers to create or increase an interest in math for their students, which can promote the acquisition of more complex math instruction, an increased rate of learning math, and ultimately effect future literacy.

Appendix A

EBF-Math Curriculum Participant Unit Objectives

Table 3 Unit Objectives
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Appendix B

EBF-Math Curriculum teacher script sample

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Appendix C EBF-Math Curriculum student answer book sample

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Appendix D

EBF-Math Curriculum data sheet sample

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