Using Most-to-Least Prompting and Contingent Consequences to Teach Numeracy in Inclusive Early Childhood Classrooms

Abstract

Mathematics and numeracy are valuable cognitive learning areas that need to be addressed during the early childhood years. The purpose of this study was to examine the effectiveness of an intervention strategy comprised of creating opportunities, prompting, providing consequences, and prompt fading when teaching preschool children with disabilities numeracy/math skills in the context of identified classroom activities. A single subject multiple-baseline probe design within participants, replicated across three participants, was used to explore the effects of the intervention. The results of this study support the use of the strategy within inclusive preschool settings.

Keywords

numeracy mathematics early childhood embedded instruction inclusive environment

Interest in the area of mathematics and numeracy at the early childhood level has increased in the past several years (Lee & Ginsburg, 2009). This interest is heightened by research demonstrating the positive impact of the development of math/numeracy skills in early childhood on later success in mathematics (Byrnes & Wasik, 2009; Duncan et al., 2007; Roberts & Bryant, 2011). The terms numeracy and math are very closely linked but are not synonymous. Mathematics, specifically at the preschool level, includes the key components of number concepts, patterns and relationships, geometry, measurement, data collection, organization, and representation (Colker, Dodge, & Heroman, 2002). Numeracy, however, involves using mathematics in a more practical manner throughout our lives, including in our home, work, and community (Peters & Young-Loveridge, 2005). Although they use slightly different terminology, both the National Council of Teachers of Mathematics (NCTM) and the National Association for the Education of Young Children (NAEYC, 2005) are in agreement that the foundation for mathematical skill development begins in the early years and both recognize that establishing a strong foundation in mathematics for 3- to 6-year-olds is crucial to later mathematics learning.

Research has validated effective strategies for teaching young children with disabilities (see Bailey & Wolery, 1992b, for a review) and methods used in intervention studies typically include one or more of the following behaviorally based strategies: (a) arranging the environment to create opportunities, (b) providing stimulus and/or response prompts, (c) prompt fading, and (d) the use of contingent consequences. Although the focus was on elementary rather than preschool-aged children, these strategies were supported by the findings of a meta-analysis conducted by Kroesbergen and Van Luit (2003) examining characteristics of math interventions for elementary aged children with disabilities. Specifically, results of the meta-analysis revealed that Direct Instruction, which includes many of the same previously mentioned behaviorally based strategies, was the most effective method of intervention for learning basic math skills.

Whereas research has documented effective strategies for (a) teaching preschool-aged children with disabilities and (b) teaching basic math skills to elementary aged children with disabilities, few studies have explored the effectiveness of these strategies in teaching basic math/numeracy skills to preschool-aged children with disabilities. A notable exception includes a recent study by Jowett, Moore, and Anderson (2012). This study documented the effectiveness of an intervention designed to teach a 5-year-old boy with autism spectrum disorder to identify and write the Arabic numerals 1 to 7 and comprehend the quantity that each numeral represented. Strategies used in the study included video modeling, gradual fading of prompts, reinforcement, in-vivo prompting, and forward chaining.

In addition to the use of behaviorally based strategies, the majority of intervention opportunities in studies designed to teach preschool-aged children with disabilities were embedded in ongoing activities. Bricker, Frontczak, and McComas (1998) defined embedded instruction as a “procedure in which opportunities to practice individual goals and objectives are included within an activity or event in a manner that expands, modifies, or adapts the activity/event while remaining meaningful and interesting to children” (p. 73). When considering the use of embedded instruction with math/numeracy, many skills can be taught using tools and materials that exist in the preschool environment. For example, (a) block areas can provide opportunities for children to learn mathematical concepts including, but not limited to, height, weight, length, and size (Bailey & Wolery, 1992a), (b) number and measurement concepts can be taught in the housekeeping and dramatic play area through the use of props such as play money, measuring tools, calculators, scales, height charts, sand timers, and cash registers (Colker et al., 2002), and (c) opportunities for math and numeracy development can be embedded into outdoor play by collecting different things outside, sorting and classifying those items, and then graphing the items that have been collected (Colker et al., 2002).

A related issue to consider when developing math/numeracy interventions is ensuring that the interventions are socially valid. Social validation is the process of assessing the social significance of goals, methods, and outcomes (Kazdin, 1982). Social validity represents the degree to which an intervention strategy achieves goals, uses procedures, and produces outcomes that are valued by and considered important to consumers of the intervention. Assessing the social validity of math/numeracy interventions is particularly important given Lee and Ginsburg’s (2009) research findings, which revealed several misconceptions that early childhood teachers have about teaching mathematics to preschool children including the beliefs that (a) children are not ready for learning about mathematics, (b) mathematics is only for the smartest children, and (c) language and literacy are more important than mathematics.

In summary, mathematics and numeracy are valuable cognitive learning areas that need to be addressed during the early childhood years. However, children’s abilities for learning mathematics at an early age are not well recognized. Efficient and effective strategies need to be investigated so that all children, including those with disabilities, have opportunities to be successful learners of mathematics. This study was designed to answer the following questions: (a) Is an intervention strategy comprised of creating opportunities, prompting, providing consequences, and prompt fading effective in teaching preschool children with disabilities numeracy/math skills in the context of identified classroom activities? and (b) How do preschool staff rate the acceptability and perceived effectiveness of use of the strategies?

Method

This study used a single subject multiple-baseline probe design within participants, replicated across participants, to explore the effectiveness of an intervention strategy designed to teach three preschool-aged children with disabilities numeracy/math skills in the context of their inclusive classroom settings.

Participants

Three preschool-aged children, Elizabeth, Steven, and Jill (pseudonyms), who (a) were between the ages of 4 and 5 years, (b) attended an inclusive early childhood program, (c) had a documented developmental delay and Individualized Education Plan (IEP), (d) were receiving special education and related services (e.g., speech therapy, occupational therapy, etc.), and (e) had normal (or corrected to normal) vision and hearing participated in this investigation.

Teachers in the identified preschool programs nominated children who met the inclusion criteria. A letter was then sent to the parents of nominated children to request permission for their child to participate in the study. After consent was obtained, the researchers completed a file review to obtain demographic information and information related to the results from the math sections of the High Scope Assessment (Brickman & Barton, 2003) and the Brigance Inventory of Early Development (IED-II; Brigance, 2010). The numeracy/math skills chosen for each participant were identified based on the file review and included three related behaviors (e.g., identification of three different numbers, identification of three different shapes) that could be physically prompted (e.g., rote counting cannot be physically prompted). Table 1 summarizes demographic information, assessment data, and numeracy/math goals/objectives across participants.

Table 1.

Demographic Information Across Participants.

Demographic information	Participant 1: “Elizabeth”	Participant 2: “Steven”	Participant 3: “Jill”
Age	4 years 2 months	4 years 7 months	4 years 9 months
Gender	Female	Male	Female
Race	Caucasian	Hispanic	Caucasian
Diagnosis	Developmental delay, Chiari malformation	Developmental delay, cerebral palsy, seizure disorder	Developmental delay, seizure disorder, hypotonia
Services currently being received	Special education, speech and language	Special education, speech and language, physical therapy	Special education, speech and language, physical therapy, occupational therapy, vision services
Current developmental age across domains (based on results of the Brigance Developmental Inventory)
Gross motor	4 years	3.2 years	2 years
Fine motor	4 years	4 years	3 years
Self-help	4 years	4 years	2 years
Speech/language	3.4 years	3.6 years	(Expressive) nonverbal, (receptive) 3 years
Cognitive	3.4 years	3.8 years	2.6 years
Social emotional	3.4 years	4 years	2.6 years
Math goal and objectives
Goal	Point to number symbols on request	Point to number symbols on request	Point to shapes on request
Objectives	Point to numbers 2, 3, and 4 on request	Point to numbers 6, 7 and 9 on request	Point to diamond, rectangle, and triangle on request

Setting

All baseline, intervention, maintenance, and generalization sessions occurred during free choice time in three public and inclusive preschool classrooms located in a metropolitan area. Two of the classrooms were Head Start preschool programs, and the third classroom was part of the public school district. In inclusive settings, the ratios of children with disabilities to typically developing children vary across programs. However, researchers and practitioners most often use the term inclusive to refer to settings in which the majority of the children are typically developing (Odom & Diamond, 1998). The settings in which the current study took place conformed to this definition.

The first classroom was comprised of 15 children, 3 of whom received special education services. Two adults staffed this classroom. One adult had a bachelor’s degree in Early Childhood Education, and the other had a master’s degree in Elementary Special Education. In addition to these two adults, a special education teacher, speech language pathologist, physical therapist, and occupational therapist served as classroom consultants.

The second classroom was comprised of 16 children, 4 of whom received special education services. Two adults staffed this classroom. One adult had an associate’s degree in Psychology. The second adult had early childhood education work experience. Both teachers also received ongoing continuing education in the area of early childhood education. In addition to these two adults, a special education teacher, speech language pathologist, physical therapist, and occupational therapist served as classroom consultants.

The third classroom was comprised of 17 children, 4 of whom received special education services. Two adults staffed this classroom. One adult had an associate’s degree in Early Childhood Special Education and was pursuing her bachelor’s degree in Early Childhood Special Education. The second adult had a Child Development Credential (CDA). In addition to these two adults, a special education teacher, speech language pathologist, physical therapist, and occupational therapist served as classroom consultants.

In each of the classrooms, free choice activities were conducted for ½ hr, two times per day during the 2½- or 3-hr preschool day. During free choice activities, children were allowed to move freely among seven developmentally appropriate learning centers. The learning centers included a block center, dramatic play area, art center, reading and writing center, sand and water center, science center, and a computer area. All study activities were embedded intoone learning center in each participant’s classroom. The learning center for each participant was chosen basedon teacher report of child preferences and remained the same throughout baseline, intervention, maintenance, and generalization.

Interventionist

The first author, who had a master’s degree in Early Childhood Special Education at the time of this investigation, served as the interventionist for all participants as well as a researcher in this investigation. The interventionist was not employed by the early childhood classrooms where the study was conducted but had 15 years of professional experience in early childhood classrooms, working specifically with children with disabilities. The interventionist joined existing preschool classroom activities to implement the intervention and embed opportunities for the participants as well as peers who were engaged in the same activity center. When interacting with peers, the interventionist presented opportunities that were not related to the skill being taught to the participant (e.g., if the target skills for a participant were to identify the number symbols 1, 2, and 3, then the opportunities provided to peers were to identify the numbers 4 and 5). This helped control for the variable of incidental opportunities for learning for study participants.

Materials

The materials used in this study included the props that were available in the identified activity center. For Elizabeth, props in the writing center included paper, stamps, stickers, and pencils. For Steven, props in the block center included large wood blocks, a toy car, a toy train, and a simple racetrack. For Jill, props in the art center included different types of paper, markers, crayons, scissors, glue, and stickers.

To embed opportunities to teach the identified math/numeracy skills (e.g., number or shape identification), the interventionist introduced additional materials into each center. For Elizabeth (whose intervention was embedded into the writing center), the number symbols 1 to 5 were printed in black, 72 comic sans font on 1″ × 1″ pieces of white paper and attached with Velcro to wood stamps. For Steven (whose intervention was embedded into the block area), the number symbols 6 to 10 were printed in black, 72 comic sans font on 1″ × 1″ pieces of white paper and attached with Velcro to wooden blocks. Finally, for Jill (whose intervention was embedded into the art center), shapes (diamond, triangle, circle, square, and rectangle) that were approximately 1½″ × 2″ were printed in purple ink onto white paper, and then painted with sparkles. The materials for any given participant remained the same during all baseline, intervention, maintenance, and generalization opportunities.

To ensure that the participants did not use position as a cue for learning, the interventionist created a notebook of templates for each participant. The templates were used to cue the interventionist on the placement of the numbers/shapes for each opportunity. On each template, the array of five numbers/shapes was presented in different orders and configurations (e.g., row, column, random). The templates were printed on 8½″ × 11″ pieces of white paper, which were then placed in a three ring binder. Eight different templates were created, each being copied 3 times to make a total of 24 pages of templates for each participant.

Data Collection

Acquisition

Data to measure acquisition of the identified numeracy/math skills for each participant were collected by the interventionist during baseline, intervention, maintenance, and generalization sessions. The interventionist used a coding sheet to collect data on the participant’s correct or incorrect engagement in the target behavior (pointing to the specified number/shape on request), interventionist’s prompts, and consequences. Each participant attended an early childhood classroom 4 days per week. Data reflect five opportunities per session with one to two sessions per day.

Social validation

Data to assess the acceptability and perceived effectiveness of the intervention strategy to preschool staff were collected through a 19-item survey anchored using 7-point Likert-type response categories. The survey was adapted from a survey by Johnston, Davenport, Kanarowski, Rhodehouse, and McDonnell (2009) and was designed to obtain staff opinions about the importance of each step of the intervention strategy and the impact of implementation of the strategy on the classroom environment. All preschool staff in the participants’ classrooms completed the survey anonymously. Prior to completing the survey, staff members had observed two sessions from each phase of the study (e.g., baseline, intervention, maintenance).

Procedures

Throughout the duration of the study, no participants chose to leave the identified activity center during any baseline, intervention, maintenance, or generalization sessions. The duration of sessions ranged from 5 to 15 min. To control for incidental opportunities for learning, the classroom curriculum was considered to ensure that the target numeracy/math skill was not going to be addressed prior to or while the skill was being taught to the participant.

Creating opportunities

During all baseline, intervention, maintenance, and generalization sessions, the interventionist embedded opportunities to measure and/or teach the identified numeracy/math skills. As mentioned previously, the target skill for Elizabeth was to point to the number symbols 2, 3, and 4. To create opportunities, the interventionist told Elizabeth that they were going to work together to stamp a piece of paper. During any given opportunity, the interventionist placed the array of five stamps, with the number symbols attached, on a template in the notebook. The interventionist then prompted Elizabeth to point to the stamp with the stated target symbol. Across opportunities, the interventionist turned the pages of the notebook containing the templates to ensure that the stamps were presented in different configurations. During non-instructional opportunities, Elizabeth was free to play with and use any of the available stamps.

The target skill for Steven was to point to the number symbols 6, 7, and 9 (pre-study assessment revealed that Steven already knew the number symbol “8”). To create opportunities, the interventionist told Steven that they were going to work together to build a block tower. During any given opportunity, the interventionist placed the array of five blocks, with the number symbols attached, on a template in the notebook. The interventionist then prompted Steven to point to the block with the stated target symbol. Across opportunities, the interventionist turned the pages of the notebook containing the templates to ensure that the blocks were presented in different configurations. During non-instructional opportunities, Steven was free to use any of the available blocks.

The target skill for Jill was to point to the shapes diamond, rectangle, and triangle. To create opportunities, the interventionist told Jill that they were going to work together to decorate shakers made out of folded paper plates that were stapled together and filled with rice. During any given opportunity, the interventionist placed the array of shapes on the template in the notebook. The interventionist then prompted Jill to point to the stated target shape. Across opportunities, the interventionist turned the pages of the notebook containing the templates to ensure that the shapes were presented in different configurations. During non-instructional opportunities, Jill was free to use any of the available art materials.

Prompting the desired behavior

The interventionist established physical proximity and attention with the child. She then simultaneously presented a verbal task demand and prompted the child to perform the desired behavior. For example, when prompting the desired behavior with Steven, the interventionist made the verbal task demand: “Steven, point to the number 6,” while simultaneously physically prompting Steven to point to the number 6. The interventionist used a most-to-least prompting strategy across opportunities, which progressed from a full physical prompt (i.e., hand-under-hand) paired with the verbal task demand, to a partial physical prompt (i.e., gently nudging the participants elbow to prompt the movement to point to the object) paired with the verbal task demand, and finally to only the verbal task demand. A most-to-least prompting strategy was chosen because this strategy has been shown to result in rapid acquisition of target behaviors (Bailey & Wolery, 1992b). Furthermore, the participants in this study were preschoolers (e.g., 4–5 years of age) who may have become restless if the controlling prompt was not presented at the beginning of the prompt hierarchy (Bailey & Wolery, 1992b). When a participant correctly engaged in the target behavior for four out of five opportunities across three consecutive sessions, the interventionist moved to the next least controlling prompt.

Providing consequences

If the child emitted a correct response, the interventionist provided verbal feedback, including information regarding why the response was correct (e.g., “Yes, you’re right! That is the number 6!”), as well as a natural consequence that was part of the activity (e.g., adding the block to a block tower). If the child emitted an incorrect response, the interventionist verbally provided feedback, including information regarding the desired response (e.g., “No, This is the number 6,” while pointing to the block with the number 6 on it). The interventionist then repeated the request for the target behavior and provided the next higher prompt in the hierarchy. This continued until the child emitted the correct response.

Experimental Design

A multiple-baseline probe design across participants was used to assess the effects of the intervention strategy on teaching math/numeracy skills to preschool children with disabilities in specified classroom activities. In a multiple-baseline design across participants, the investigator sequentially applies an intervention across several participants. Experimental control is demonstrated when there is a change in level and trend of the dependent measure contingent on the staggered introduction of the independent variable (Tawney & Gast, 1984). This design is well suited for situations in which reversal of behavior is unlikely and where maturation could be a potential confound (Kazdin, 1982). Intermittent baseline probes were used to measure performance across participants prior to the introduction of intervention. Daily intervention probes were used to evaluate the impact of the intervention. Intermittent probes of post-intervention behavior served as a maintenance check to determine whether experimental effects were durable over time and generalization probes were conducted to examine the use of the target skill in the context of the same activity but with a different adult (e.g., classroom teacher).

Baseline probes

Baseline data were collected to measure participant engagement in the desired behavior (pointing to the specified number/shape on request) prior to implementation of the intervention for each of the three related numeracy/math behaviors. During baseline, the interventionist embedded five opportunities per session for the participant to demonstrate the desired behavior. Participant responses were recorded for each opportunity. The interventionist presented the verbal task demand but did not provide prompts or consequences during baseline.

Intervention

Intervention data were collected to measure the effects of the intervention. Intervention sessions occurred in the same setting and during the same activity as baseline. During the intervention phase, prompts and consequences (discussed in prior sections) were provided to the participant. Five opportunities to practice the target numeracy/math skill were embedded into each session. Data were collected on participant engagement in the desired behavior (pointing to the specified number/shape on request). Criterion for mastery of the desired behavior was defined as four out of five unprompted correct responses across three consecutive sessions.

Maintenance probes

Maintenance probes began 1 week after the child met criterion for each targeted skill and continued for the duration of the study. Maintenance probes used the same materials and were conducted in the same setting and during the same activity that were used during the intervention sessions. Maintenance data were collected to measure participant engagement in the desired behavior (pointing to the specified number/shape on request). During maintenance sessions, five opportunities for each target skill were embedded into the activity. The participant was not provided with prompts or consequences during the maintenance probes.

Generalization probes

Generalization probes for each numeracy/math skill were implemented during the baseline, intervention, and maintenance phases for each participant. Generalization probes were implemented in the same activity center with the same materials that were used during the baseline, intervention, and maintenance sessions. However, a different adult (the classroom teacher) conducted the generalization probes to assess whether the learned skills generalized across people. All generalization sessions were conducted on the same day as baseline, intervention, and maintenance sessions but were separated from baseline, intervention, or maintenance sessions by at least a 1-hr interval.

Reliability

Interobserver agreement was assessed by having an independent observer collect data at the same time as the interventionist during at least 30% of each experimental condition for each child. Interobserver agreement was computed as the number of agreements for correct and incorrect engagement in the target behavior (pointing to the specified number symbol/shape on request) divided by the number of agreements plus disagreements and multiplied by 100. Mean interobserver agreement was collapsed across dependent measures and was 100% during baseline, 99% during intervention (80% for Elizabeth, 100% for Steven, 100% for Jill), and 100% during maintenance.

The same independent observer also observed the interventionist to assess procedural fidelity during at least 30% of each experimental condition for each child. The degree to which procedural manipulations were implemented as planned was calculated by dividing the number of interventionist behaviors exhibited by the number of planned interventionist behaviors and multiplying by 100. Planned interventionist behaviors included (a) recognizing or creating an opportunity, (b) establishing physical proximity with the student, (c) establishing joint attention with the student, (d) implementing the prompt hierarchy as specified in the procedures, and (e) delivering the appropriate consequence. Fidelity data indicated that the interventionist correctly performed the planned behaviors on 100% of the prescribed occasions.

Results

Child Outcomes

Figures 1 –3 illustrate the number of correct responses (pointing to the specified number/shape on request) per session across conditions for each participant. Data were visually inspected for level, trend, and variability (Gast, 2010) and were analyzed based on percentage of non-overlapping data (Scruggs & Mastropieri, 1998).

Figure 1.

Number of correct identification of the numbers 2, 3, and 4 for Elizabeth.

Figure 2.

Number of correct identification of the numbers 6, 7, and 9 for Steven.

Figure 3.

Number of correct identification of triangle, rectangle, and diamond for Jill.

Elizabeth demonstrated a baseline mean of 4% correct for number 2 (range = 0%–20%), 0% correct for number 3, and 3% correct for number 4 (range = 0%–20%). During intervention, Elizabeth demonstrated a mean of 79% correct for number 2 (range = 40%–100%), 63% for number 3 (range = 0%–100%), and 77% for number 4 (range = 20%–100%). During maintenance, Elizabeth demonstrated a mean of 87% correct for number 2 (range = 60%–100%), 87% correct for number 3 (range = 80%–100%), and 95% correct for number 4 (range = 80%–100%). One hundred percent of the data points in maintenance were above the highest baseline point, suggesting that intervention was effective for Elizabeth (Scruggs & Mastropieri, 1998). Mean gain from baseline to maintenance was 83% for number 2, 87% for number 3, and 92% for number 4.

Steven demonstrated a baseline mean of 5% correct for number 6 (range = 0%–20%), 0% correct for number 7, and 0% correct for number 9. During intervention, Steven demonstrated a mean of 67% correct for number 6 (range = 20%–100%), 90% correct for number 7 (range = 40%–100%), and 77% correct for number 9 (range = 20%–100%). During maintenance, Steven demonstrated a mean of 97% correct for number 6 (range = 80%–100%), 93% correct for number 7 (range = 40%–100%), and 100% correct for number 9. One hundred percent of the data points in maintenance were above the highest baseline point, suggesting that intervention was effective for Steven (Scruggs & Mastropieri, 1998). Mean gain from baseline to maintenance was 92% for number 6, 93% for number 7, and 100% for number 9.

Jill demonstrated a baseline mean of 5% correct for diamond (range = 0%–20%), 8% correct for rectangle (range = 0%–20%), and 0% correct for triangle. During intervention, Jill demonstrated a mean of 78% correct for diamond (range = 0%–100%), 90% correct for rectangle (range = 80%–100%), and 78% correct for triangle (range = 40%–100%). During maintenance, Jill demonstrated a mean of 80% correct for diamond, 84% correct for rectangle (range = 60%–100%), and 93% correct for triangle (range = 80%–100%). One hundred percent of the data points in maintenance were above the highest baseline point, suggesting that intervention was effective for Jill (Scruggs & Mastropieri, 1998). Mean gain from baseline to maintenance was 75% for diamond, 64% for rectangle, and 93% for triangle.

Generalization probes for numeracy/math skills are also displayed on Figures 1 –3. During baseline, generalization probes for Elizabeth, Steven, and Jill were at 0% accuracy. Generalization probes across participants during intervention ranged from 80% to 100% accuracy. Generalization probes across participants during the maintenance phase also ranged from 80% to 100% accuracy.

Social Validity

Both teachers in each of the three participants’ classrooms were asked to anonymously complete a post-study, social validity survey during the final week of maintenance. Four of the six preschool teachers completed the survey. Of the teachers who completed the survey, 100% strongly agreed/agreed with statements regarding (a) the ease of integrating the intervention into planned activities, (b) not being disruptive of classroom activities, (c) sessions being fun for the participant, and (d) the ability for typical peers to participate in the activity when using the strategy with the child with a disability. In addition, 100% of the teachers strongly agreed/agreed that this strategy could be used in other learning centers to teach numeracy/math skills to children with disabilities, as well as to teach children with various types of disabilities. Finally, all of the teachers strongly agreed/agreed with statements indicating that the time required to implement this study was worth the observed benefits and that they were satisfied with the outcome of the child’s learning. Although all of the teachers strongly agreed/agreed with statements indicating that the partial physical prompt and verbal task request strategies were very appropriate, only two teachers (50%) strongly agreed/agreed with the statements that (a) the full physical prompt was appropriate and (b) the consequences were acceptable, whereas the remaining teachers took a neutral position. Finally, three teachers (75%) strongly agreed/agreed that other classroom staff could implement this strategy with training and support, whereas one teacher had a neutral opinion regarding this statement.

Discussion

Results of this investigation revealed that the intervention strategy was successful in teaching numeracy/math skills to preschool-aged children with special needs in an inclusive early childhood setting. These findings support literature indicating that a most-to-least prompting strategy is an effective method for teaching young children with disabilities to learn and generalize skills (e.g., Bailey & Wolery, 1992b) and add to the empirical base on teaching math/numeracy skills to children with disabilities (e.g., Jowett et al., 2012; Kroesbergen & Van Luit, 2003).

Although all three children successfully learned the specified numeracy/math skills, there were some interesting findings related to intensity of intervention (e.g., number of opportunities/sessions per day). As mentioned previously, during the intervention phase, the number of sessions per day across participants ranged from 1 to 2 (resulting in either 5 or 10 opportunities per day). Post hoc examination of the data revealed that Elizabeth met criteria after 37 intervention sessions across 30 days, Steven met criteria after 38 intervention sessions across 32 days, and Jill met criteria after 29 intervention sessions across 29 days. These results suggest that increases in the number of sessions per day did not decrease the number of days that it took to complete the intervention phase of the study. Although the design of this study does not allow for definitive explanations, this difference may have been due to individual differences across participants and/or differences in the skills that were taught (e.g., learning to identify shapes may be more quickly acquired than learning to identify number symbols). Examining outcomes in relation to intensity of intervention as well as in relation to individual differences and differences between skills being taught is important as interventionists strive to develop and implement strategies that are efficient as well as effective (e.g., Reynolds, Temple, Ou, Arteaga, & White, 2011; Zhai et al., 2010).

Another interesting finding is related to variability in correct/incorrect responses across skills and participants. One way that this variability can be explored is by examining trends in error responses during the intervention phase. For example, when Steven was learning the first target skill (pointing to the number symbol 6 on request), 89% of his errors were emitted by pointing to the number symbol 9. Furthermore, when Steven was learning the third target skill (pointing to the number symbol 9 on request), 50% of his errors were emitted by pointing to the number symbol 6. This finding supports literature suggesting that number/letter symbols that are similar in appearance are more difficult to discriminate. A second notable trend in error responses was related to the frequency with which participants chose the most recently taught skill when emitting an error (e.g., pointing to the first number/shape that was taught when being taught the second skill, pointing to the second number/shape that was taught when being taught the third skill). This trend was particularly notable for Jill and Elizabeth. Specifically, on examination of error responses when teaching the second number/shape, Jill and Elizabeth pointed to the first number/shape 100% and 58% of the time, respectively. Furthermore, on examination of error responses when teaching the third number/shape, Jill and Elizabeth pointed to the second number/shape 100% and 79% of the time, respectively. Based on this error analysis, future research examining the extent to which changes in the instructional sequence/targets influence the efficiency and/or effectiveness of learning is warranted.

Social Validity

Findings from the social validity survey support the use of the instructional strategy to teach numeracy/math skills to preschool-aged children with disabilities. The preschool teachers who observed two sessions from each phase of the study and completed the social validity survey unanimously “strongly agreed/agreed” that the instructional strategy used in this study was very important, appropriate, and not difficult to implement. Furthermore, all of the teachers who observed two sessions from each phase of the study and completed the survey believed that they could implement the intervention used in this study with proper instruction and training. The majority (75%) of the teachers also thought that other staff in the classroom, if given training and support, could implement this strategy as well.

It is interesting to note that responses varied when asked about two of the techniques (physical prompts and consequences) used during the intervention. Half of the teachers agreed with the use of full physical prompts whereas the other half of the teachers were neutral with regard to the use of this prompting technique. One plausible explanation for this finding might be that some teachers are more used to using least-to-most prompt hierarchies and that the use of a full physical prompt at the start of the intervention was seen as too intrusive. If this was the case, teachers might benefit from more information on most-to-least prompt hierarchies including information on when most-to-least prompt hierarchies might be effective (e.g., Bailey & Wolery, 1992b). The teachers also showed differing opinions concerning the use of consequences. Specifically, half of the teachers agreed with the use of consequences whereas the other half of the teachers were neutral. A possible explanation for this finding may be that, even though the teachers observed some sessions, some may not have taken note of the actual consequences because they were natural. If this was the case, teachers may benefit from additional information related to natural consequences, including the value of providing participants with corrective feedback (e.g., Fazio, Huelser, Johnson, & Marsh, 2010; Pashler, Cepeda, Wixted, & Rohrer, 2005). Another possible explanation for the teachers’ responses may have been due to the terminology used in the survey (e.g., full physical prompts, consequences). The teachers may have had different responses if descriptive information (e.g., helping the child point to the symbol by providing hand-under-hand guidance) rather than behavioral terminology was used.

Limitations

Each of the children who participated in the study stayed in an identified learning center throughout all phases of the study, including generalization. Although participants generalized their use of target skills to new people (e.g., classroom teacher), this study does not provide data regarding whether the skills generalized to other learning centers within the classroom or to other settings (e.g., home). Also, the outcomes of this study may be different if the children were asked to generalize the learned numeracy/math skill to other materials that incorporated different fonts for number symbols, shapes of objects in the environment, or three-dimensional shapes. Furthermore, the skills taught in the context of this study relate only to shape and number identification. The outcomes of this study may be different when teaching (a) other skills within the areas of number operations and geometry and/or (b) skills in other mathematical learning sets (e.g., measurement, algebra, or data analysis). Finally, participants in this study were taught to identify the target number/shape from an array of five numbers/shapes. A 5-item array was chosen because the Brigance Developmental Inventory presents shape and number concepts in sets of five. However, the outcomes of the investigation may be different if the overall size of the array was increased.

Implications for Future Research and Practice

It is important to note that, although promising, additional investigations must be implemented (e.g., in different locations, with different investigators, etc.) for this intervention strategy to be deemed an evidence-based practice (Horner et al., 2005). Future research should also explore the efficiency and effectiveness of using this intervention strategy to teach additional numeracy/math skills such as measurement, patterns, data analysis, or problem solving.

In addition to future research, the outcomes of this study provide implications for practice. Specifically, results suggest that (a) the intervention strategy of most-to-least prompting was an effective way to teach numeracy/math skills to young children with special needs in inclusive environments and (b) preschool teachers were accepting of both the intervention strategy and the teaching of math/numeracy at the preschool level. Given this, the timing may be right for (a) providing early childhood teachers with additional knowledge regarding effective strategies for teaching numeracy/mathematics skills in inclusive early childhood classrooms through in-services and continuing education and (b) supporting early childhood teachers as they utilize this knowledge.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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