Using Explicit Instruction and Virtual Manipulatives to Teach Measurement Concepts for Students With Autism Spectrum Disorder

Abstract

Measurement concepts are an essential foundation for more advanced mathematical concepts. To address the challenges of students with autism spectrum disorder (ASD) in learning measurement concepts, this study investigated the effects of using a combination of explicit instruction and virtual manipulatives (VMs) to teach measurement concepts to students with ASD in China. Using a single-case multiple-probe across skills design, researchers examined whether the intervention could support the acquisition and maintenance of measurement concepts in students with ASD. Based on visual analysis, a functional relation was found between the independent variable (i.e., explicit instruction with VMs) and student performance on solving measurement concepts problems. Implications for practice and research are discussed.

Keywords

measurement concepts virtual manipulatives explicit instruction autism spectrum disorder mathematics

Students with autism spectrum disorder (ASD) may have cognitive characteristics such as relative weaknesses in perceptual reasoning, verbal ability, and working memory (Oswald et al., 2016), which often result in difficulties in learning measurement concepts (Root, 2019). Measurement concepts have been defined as a description of measurable attributes of objects, such as length or weight (Common Core State Standards Initiative [CCSSI], 2010). Prior research has demonstrated that measurement concepts play a significant role in acquiring meaningful counting skills (Fritz et al., 2013; Kroesbergen et al., 2009) and are prerequisite for forming number sense (Krajewski & Schneider, 2009). To address the challenges of students with ASD in learning measurement concepts, this study sought to explore the effect of an intervention on the acquisition and maintenance of measurement concepts in students with ASD in China.

Conceptualization of Measurement Concepts

In the context of China’s school curriculum, measurement concepts (e.g., long–short, big–small, more–less, heavy–light) are important mathematics content in the special education mathematics curriculum standards (Ministry of Education [MOE], 2016). According to Clements and Stephan (2004), a conceptual understanding of measurement concepts is established starting with an awareness of the attribute, followed by developing the skill to measure the specific attribute of objects. For example, students may first learn that length is a characteristic of an object and can be found by quantifying the physical distance between the endpoints of the object (Stephan & Clements, 2003). Next, students learn to use terms such as “longer” and strategies such as direct comparison to extend the learning of length concepts. Direct comparison involves comparing similarities and differences between two objects or pictures to identify which item has “more of” or “less of” the attribute (CCSSI, 2010). For example, students may compare the length of two pencils by placing them side by side with the ends aligned and identifying the long and short pencils, respectively (National Council of Teachers of Mathematics, 2000).

Interventions for Teaching Measurement Concepts

Explicit instruction is an evidence-based practice for teaching mathematics to students with moderate and severe disabilities (Spooner et al., 2019); it is characterized by three stages of instruction (i.e., modeling of the target skill, guided practice, and testing of independent use of the target skill). Celik and Vuran (2014) compared the efficacy of explicit instruction and simultaneous prompting on teaching measurement concepts (i.e., long, few, thick, old) to four students with moderate intellectual disability (ID). The study findings revealed that explicit instruction yielded positive and clinically significant effects for all four participants.

In a more recent study, Root (2019) used a single-subject multiple-probe across skills design to evaluate the effect of explicit instruction on teaching one 10-year-old student with ID and ASD to identify the concepts of more, different, and long. Root (2019) used a model–lead–test procedure: (a) modeling (i.e., “This is long. This is not long.”), (b) leading practice (i.e., “Touch long”) with error correction, and (c) testing (i.e., “Touch long”) without error correction or feedback. The results revealed a functional relation between explicit instruction and the student’s correct and independent demonstration of measurement concepts.

Recently, researchers have begun to examine the effect of using virtual manipulatives (VMs) for teaching mathematics to students with ASD (Jimenez & Besaw, 2020). Virtual manipulatives may be designed to present built-in scaffolds for learning mathematical concepts (Bouck et al., 2017). In addition to effective and dynamic representation of mathematical objects, VMs support the acquisition and maintenance of mathematical concepts (e.g., early numeracy, computation, word problem solving) in students with ASD (Bouck et al., 2017; Jimenez & Besaw, 2020). For example, Jimenez and Besaw (2020) investigated the impact of VMs to teach early numeracy skills to two elementary students with ASD and moderate ID. Their study results showed a functional relation between the use of VMs and improvement in numeracy skills. In another study, Bouck et al. (2020) explored the effect of VMs and explicit instruction on the acquisition of division with remainders for three middle school students with disabilities, all of whom acquired the skill effectively.

The purpose of this study was to examine the effects of using a combination of explicit instruction and VMs on learning three measurement concepts (i.e., length, size, quantity) for elementary school students with ASD. In addition, this study sought to address limitations in prior research regarding the conceptual understanding of measurement concepts and the development of comparing skills in the context of measurement concepts. Furthermore, VMs have been shown to be an effective and preferred tool for students with ASD to acquire various mathematics skills. The following research questions were addressed:

Research Question 1: Does a functional relation exist between a combination of explicit instruction and VMs and students’ performance on solving measurement concept problems?

Research Question 2: Will students maintain their acquisition of measurement concepts when the intervention is withdrawn?

Method

Participants and Setting

Three students with ASD participated in this study. Approval from the Ethics Committee of the university, parental permission, and student consent were obtained prior to the study. Students were recruited by teacher nomination based on the following inclusion criteria: (a) a medical diagnosis of ASD, (b) difficulties with grade-level measurement concepts as reported by the teacher, (c) no vision or hearing impairment, (d) sufficient motor skills to drag and drop pictures on a touch screen, and (e) no prior exposure to VMs in mathematics instruction. Students were enrolled full-time in a special education school and received instruction in subjects including reading and literacy, mathematics, adaptive skills, vocational skills, art, physical education, and information technology. Each classroom included up to 10 students, with two special education teachers and a paraeducator.

Ning was a 7-year-old boy with ASD in first grade. His IQ score was less than 40 as measured by the Chinese version of Wechsler Intelligence Scale for Children, 4th edition (WISC-IV; Wechsler, 2003; Zhang, 2008). His score on the Chinese version of the Childhood Autism Rating Scale (CARS; Lu et al., 2004; Schopler et al., 2010) was 35, indicating mild to moderate autism. Cheng was a 6-year-old boy with ASD in first grade. His CARS score was 54, in the range of severe autism. His IQ score was not available. Ming was an 8-year-old boy in first grade with a medical diagnosis of ASD. He had a CARS score of 37, in the range of severe autism. His IQ score was 40.

This study took place in a public special education school, located in an urban city in eastern China. The school enrolled 445 students from first to ninth grade, and students with ASD made up 32% of the student population. All sessions took place in an 8 m × 6 m classroom. All sessions were video recorded. Two special education master’s students who held special education teaching licenses served as the interventionists and both received training before the intervention.

Materials

Materials included a probe sheet, a pencil, an eraser, and an iPad loaded with the “SMART Notebook” application. The probe sheets were collaboratively designed by both researchers and mathematics teachers in the special education school. According to the guidelines from Celik and Vuran (2014) and Engelmann and Carnine (1982), the problems were presented in three levels. The first level (far distractor picture) included pictures in the same kind and style on irrelevant dimensions and only differed in the attribute related to the target concept (e.g., 20 cm long cucumber/15 cm long cucumber). The second level (near distractor picture) included pictures in the same kind but different styles on irrelevant dimensions and only differed in the attribute related to the concept (e.g., 10 cm long green pencil/6 cm long blue pencil). The third example level (distractor picture) included pictures with different kinds and styles (e.g., 20 cm long ruler/ 8 cm long glue stick). Each probe sheet consisted of four problems, one problem with far distractor pictures, two problems with near distractor pictures, and one problem with distractor pictures. The order of the problems and the location of the target object in each probe sheet was randomized. All four problems were printed in Chinese and presented to the students on a sheet of paper. Specifically, the problems in the probe sheet were in line with the special education mathematics curriculum pertinent to measurement concepts. That is, first grade students should be able to “perceive the characteristics of the length, size, and quantity of objects, and compare them” (MOE, 2016). We developed multiple alternate forms of probe sheet for baseline, intervention, and maintenance phases. Problems on the probe sheets did not repeat across study sessions.

The SMART Notebook^© application was used to create and display the VMs and virtual graphic organizers (see Figure 1). The application includes function buttons for a pen, an eraser, shapes, lines, and pictures. The iPad application with VMs and virtual graphic organizers was utilized in intervention phases across three measurement concepts. The participants used the iPad for educational games outside of the study and had prior experience in manipulating apps on an iPad. The task analysis included a four-step checklist developed for each target concept (see Table 1).

Table 1

Task Analysis for Three Measurement Concepts.

Steps	Longer	Bigger	More
1	Drag one picture horizontally into one box in the white space and line up the end of the picture exactly with the start line	Drag one picture onto the start line in the white space	Drag one picture from one group into the box in the first row one by one
2	Drag the other picture horizontally into the other box in the white space and line up the end of the picture exactly with the start line	Drag the other picture onto the start line in the white space	Drag the other picture from another group into the box of the second row one by one
3	Draw two vertical lines at the other end point of both objects and identify the longer object as the one that crosses a vertical line	Observe, compare, and identify the difference	Draw a line from one picture in the first row corresponding to one picture in the second row, and identify the group with more objects as the group containing extra objects without correspondence
4	Circle or point out the answer	Circle or point out the answer	Circle or point out the answer

Figure 1

Example Virtual Manipulatives and Virtual Graphic Organizers for Bigger, More and Longer.

Independent and Dependent Variables

The independent variable for the current study was the combination of VMs and explicit instruction. The dependent variable was the percentage of independent and correct responses out of four problems on the probe sheet of each target skill.

Experimental Design

This study used a single-case multiple-probe design across three skills, with replication across students (Ledford & Gast, 2018). The design adhered with methodological guidelines from the What Works Clearinghouse (2017). Each participant received the intervention in all target skills, but the sequence of the target skills was randomized. Intervention for the second and third skills was introduced when each participant met the criterion (100% accuracy) for three consecutive sessions in the previous target skill.

Procedures

Baseline

During baseline, participants were provided with the probe sheet, a pencil, and an eraser. They were given the instructional cue “show me which is longer/more/bigger.” No manipulatives, prompts, or feedback were given. The baseline phase lasted 10 to 15 min.

Pre-Unit

Each participant completed a 30-min pre-unit session with the interventionist after baseline and prior to intervention. The pre-unit targeted: (a) developing an understanding of attributes by touching, observing, or manipulating pictures of concrete objects on paper, (b) recognizing the attributes of length, size, and quantity, and (c) using the following features of the SMART Notebook application: open the application, drag and drop the images, select colors of handwriting, and clear handwriting. The measurement concepts length, size, and quantity were selected in alignment with the special education mathematics curriculum standards in China.

Intervention

During intervention, the interventionist implemented the intervention including explicit instruction with the use of VMs on an iPad application. Virtual manipulatives in this study refer to digitally displayed objects that can be manipulated on a screen (e.g., drag and drop). All intervention sessions were implemented one-on-one. The intervention was implemented 2 or 3 days per week, with each session lasting 20 to 30 min. The total intervention duration was 11 weeks. During the modeling portion, the interventionist demonstrated how to solve three problems, using the SMART Notebook. The interventionist opened the app on the iPad and taught the student to follow the task analysis, and stated the process of solving the problem. Next, the participant moved into guided practice for three problems. The interventionist followed a least-to-most prompting hierarchy if the participant incorrectly used the VMs or did not know how to proceed. The prompting hierarchy included the following: (a) verbal prompts such as “drag one picture into one box and line up the end of the picture exactly with the start line”; (b) specific verbal prompts such as “drag one picture horizontally into one box in the white space, and line up the left side of the picture exactly with the start line”; and (c) model prompts, where the interventionist showed the participant how to complete a step correctly and instructed the participant to repeat the step. During independent practice, the participant solved the problems on a probe sheet independently with no prompting or feedback.

To use the VMs, virtual graphic organizers and task analysis, the interventionist highlighted how each virtual graphic organizer represented one type of the problem. For example, pictures of a glue stick and a pencil were presented, interventionist gave the instructional cue “show me the longer one.” The interventionist modeled dragging one object from the problem horizontally into the one box and lining up the left side of the picture exactly with the start line. Next, the interventionist modeled dragging the other object from the problem horizontally into another box and lining up the end of the picture exactly with the start line. Then, the interventionist drew two vertical lines at the other end point of each object, and identified the longer object as the one that crosses a vertical line. Finally, the interventionist circled or pointed out the right answer.

Maintenance and Generalization

Maintenance sessions occurred 1 week after completing the last intervention session. Each session lasted approximately 10 min. Three maintenance probes were conducted within 2 weeks for each participant. Maintenance sessions were conducted with the same materials and procedures as baseline phase. Generalization sessions were conducted once each in baseline, intervention, and maintenance phases. During generalization sessions, participants solved four problems with pictures taken from their school environment, without VMs, explicit instruction, or any other feedback or prompts.

Inter-Observer Agreement and Procedural Fidelity

Inter-observer agreement (IOA) and procedural fidelity data were collected for 40% of sessions in all phases of the study. The two researchers responsible for all data collection were trained on data collection procedures through a 60-min training session. Inter-observer agreement was evaluated using an item-by-item method and calculated by dividing the total number of agreed items by the sum of agreed and disagreed items and then multiplying by 100. The mean IOA was 93% for Ning (range: 83%–100%), 93% for Cheng (range: 91%–100%), and 93% for Ming (range: 83%–100%). In addition, the researcher used a nine-item procedural fidelity checklist (see Table 2). The mean procedural fidelity data were 96% for Ning, 97% for Cheng, and 93% for Ming.

Table 2.

Procedural Fidelity Checklist: Intervention Implementation.

Step	Step description
1.	Interventionist arranges the intervention setting and removes distractors
2.	Interventionist provides a probe sheet, virtual graphic organizers, and task analysis to participant
3.	Interventionist opens SMART Notebook© application on iPad for participant
4.	Interventionist uses model–lead–test procedures
5.	In the modeling phase, interventionist models the whole procedure and states the process of solving the problem
6.	In the lead phase, interventionist follows a least-to-most prompting hierarchy when participant does not respond after 5 s or responds incorrectly
7.	In the test phase, interventionist provides no feedback or prompts
8.	In the test phase, interventionist collects data on accuracy of the task
9.	Interventionist gives redirections in time when participant shows off-task behaviors, such as leaving their seats and looking at other objects not related to instruction

Data Analysis

Data collected across all study phases were analyzed visually and statistically. Researchers used What Works Clearinghouse guidelines for visual analysis of single-case study data (What Works Clearinghouse, 2017) across six variables: (a) level, (b) trend, (c) variability, (d) overlap, (e) immediacy of effect, and (f) consistency of data patterns across similar phases. Descriptive data of mean, median, and range were obtained for comparison between phases. The split-middle method was used (White & Haring, 1980) to determine whether the trend was downward, upward, or not accelerating (Gast & Ledford, 2014). Tau-U was calculated using an online calculator (Vannest et al., 2016) by first analyzing the results for each participant and then obtaining an aggregate value, weighted by the length of the series. Tau-U effect size was interpreted according to the following guidance: (a) 0 to 0.62 indicates a small effect, (b) 0.63 to 0.92 indicates a moderate effect, and (c) 0.93 to 1 indicates a strong effect (Parker et al., 2011).

Social Validity

Participants were asked open-ended questions by the interventionist to assess their perceptions of the VMs and if the methods were acceptable. Teachers were asked about their perception of teaching measurement concepts with the VMs and to provide feedback on using explicit instruction and VMs.

Results

For all participants, a functional relation was found between the intervention and the student’s performance on three measurement concepts. The overall Tau-U effect size for all three participants was 1.00 (95% CI = [0.79, 1.00]), indicating a strong effect (Parker et al., 2011).

Cheng

The results for Cheng’s performance are reported in Figure 2. During baseline, data were at a low degree of variability for all three concepts, ranging from 0% to 25%. During the intervention in the concept of bigger, Cheng’s performance accuracy increased to 50% immediately and reached criterion in nine sessions. During the intervention in the concept of more, Cheng showed a 25% improvement in the first session of the intervention, then gradually reached 100%. He reached criterion in seven sessions. During the intervention in the concept of longer, Cheng showed a 25% improvement in the first session of the intervention and rapidly reached 100%. He reached criterion in six sessions.

Figure 2.

Cheng’s Accuracy Data Across Target Skills (Closed Circles) and Performance on Generalization Probes (Open Triangles).

During maintenance, Cheng demonstrated an average of 91.67% correct response for bigger, 83.33% correct response for more, and 75% correct response for longer. In generalization of all three concepts, the accuracy was 0% during baseline. Cheng performed at 50%, 75%, and 75% correct in bigger, more, and longer, respectively, in generalization probes during intervention. Cheng’s generalization performance of all three concepts was at 75%.

Ming

The results for Ming’s performance are reported in Figure 3. During baseline, data were at a low degree of variability for all three concepts, ranging from 0% to 25%. Upon introduction of the intervention in the concept of more, Ming showed an immediate 25% improvement in the first session of the intervention, then fluctuated from 50% to 100%. He reached criterion in seven sessions. With the intervention in the concept of longer, Ming showed a 25% improvement in the first session of the intervention, with an increasing trend that continued until he reached 100%. He reached criterion in eight sessions. With the intervention in the concept of longer, Ming immediately improved from 0% to 50% in the first session of the intervention and continued making progress until he reached 100%. He reached criterion in six sessions.

Figure 3

Ming’s Accuracy Data Across Target Skills (Closed Circles) and Performance on Generalization Probes (Open Triangles).

During maintenance, Ming demonstrated 100% correct response for the concept of more and 75% for concept of longer. For the concept of bigger, Ming had 100% correct responses for the first two sessions but dropped to 50% for the last session, with an average of 83%. Ming’s drop in performance in the third session was likely due to a physical side effect (i.e., dizziness) of a new medication for a sleep disorder that Ming started taking before the third session in maintenance. In baseline, generalization of the concepts of more and longer was 0% and 25% for the concept bigger. Ming performed at 75%, 50%, and 75% in more, longer, and bigger, respectively, in the generalization probes during intervention. After intervention was completed, Ming’s generalization for the concepts more and longer was at 75% and 100% for the concept bigger.

Ning

The results for Ning’s performance are reported in Figure 4. During baseline, data were at a low degree of variability for all three concepts, ranging from 0% to 25%. Upon introduction of the intervention in the concept of more, Ning showed an immediate 25% improvement in the first session of the intervention and showed a gradual increase to 100%. He reached criterion in seven sessions. With the intervention in the concept of bigger, Ning showed a large improvement from 0% to 50% in the first session of the intervention, then, his accuracy fluctuated between 50% to 100%. He reached criterion in seven sessions. During the intervention of bigger, Ning’s performance dropped drastically from 100% to 50% in session 15. On the day this session was conducted, Ning’s nap routine was disrupted, and thus, he was unable to focus on the instructional tasks of this study, based on observations of his stretching and yawning throughout this session. With the intervention in the concept of longer, Ning rapidly improved his performance to 75% in the first session of the intervention then showed some variability between 75% and 100%. He reached criterion in six sessions.

Figure 4.

Ning’s Accuracy Data Across Target Skills (Closed Circles) and Performance on Generalization Probes (Open Triangles).

During maintenance, Ning demonstrated 100% correct response for the concept of more. For the concept of bigger, Ning demonstrated an average of 92% correct response. For the concept of longer, he demonstrated 83% correct response. In baseline, generalization of the concepts of more was 0%, while generalization of concepts bigger and longer was 25%. Ning performed at 75%, 75%, and 100% in more, longer, and bigger, respectively, in the generalization probes during intervention. After the intervention was completed, Ning’s generalization of all three concepts was 75% correct response in the concept of bigger and longer, and 100% correct response for concept of more.

Social Validity

Overall, students expressed a preference for using the VMs on the iPad to learn the measurement concepts and showed a willingness to use the VMs in future lessons. The teachers agreed that the intervention contributed to their students’ conceptual understanding of measurement concepts and noted that participants could apply the measurement concepts in real-life contexts. For example, Cheng tended to choose a box with more snowflakes from his toys, Ming was observed to choose a bigger paper clip over a smaller paper clip, and Ning expressed that she wanted to play on a longer balance beam.

Discussion

This study evaluated the effect of explicit instruction and VMs on the acquisition and maintenance of measurement concepts with three elementary school students with ASD.

In this study, all three participants showed immediate effects in the first session of the intervention phase. Based on visual analysis, a functional relation was found between the intervention and the students’ performance on solving measurement concepts problems. All three participants solved measurement concepts problems with higher accuracy during the intervention as compared with baseline. In addition, all three participants maintained their improved performance for at least 3 weeks, with no overlap of data compared with baseline.

This intervention addressed both the conceptual understanding of measurement concepts and the skills to compare relative quantity of different objects in pictures. Students learned three measurement concepts with the systematic model–lead–test procedures in explicit instruction aided by VMs as they observed, touched, and virtually manipulated the objects. As a result of the intervention, all participants were able to visually identify similarities and differences between objects and the mathematical relationships of the quantifiable attributes of the objects. Furthermore, after participants learned the concepts of long, they were able to apply this knowledge across similar concepts, such as “longer.”

The findings of this study aligned with prior literature that demonstrated the effectiveness of explicit instruction to teach academic content to students with ASD (Bouck et al., 2017; Jimenez & Besaw, 2020) and studies that supported the relative strengths in visual information processing, especially of non-social images, among students with ASD (Knight et al., 2015). We also posit that the VM component of the intervention may have addressed weaknesses in perceptual reasoning, which is commonly present in individuals with autism (Oswald et al., 2016), by providing more tangible representations of objects and strengthening the salience of specific quantifiable dimensions of objects.

Implications for Practice

One implication for practice is that VMs may be an effective intervention component that motivates students to learn mathematics concepts. Manipulatives could be used to demonstrate conceptual understanding (Spooner et al., 2017). Social validity responses gathered at the conclusion of the study supported this finding, as students reported that they enjoyed using VMs to solve problems, which aligns with findings from previous literature (Bouck et al., 2018; Root & Browder, 2019). In addition, the teachers involved in the intervention also reported a preference toward using VMs for mathematics lessons in the future.

Another implication for cost-effectiveness is that the use of VMs may reduce the cost of instructional materials, compared with the cost of obtaining a set of concrete manipulatives for each student. Virtual manipulatives are relatively easier to store, last longer, and can be shared more easily across students or classrooms. The use of portable technology as an instructional tool may provide increased feasibility for individualized supports in multiple settings (e.g., across classrooms during school, after school, at home).

Limitations and Future Directions

There are several limitations in this study that need to be considered in the interpretation of study findings. First, the intervention sessions were implemented in a one-on-one instructional format. Teaching an entire class or a group of students is a common instructional format for mathematics lessons in special education (Bouck & Park, 2020). Future researchers could investigate the effects of explicit instruction with VMs for students with ASD in a whole- or small-group setting. Second, the current study only measured students’ understanding of measurable attributes of objects and their abilities to compare the attributes of two objects. Future studies may consider replicating this intervention for teaching students to compare more than two objects, such as ordering three or more objects by length. Third, this study only targeted three measurement concepts. More research is needed to evaluate the effectiveness of VMs on learning other measurement concepts (e.g., mass, height, volume) that are prerequisites to more advanced mathematics concepts.

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

This work was supported by the Philosophy and Social Science Planning Project of Shanghai (Grant Number A2013).

ORCID iDs

Di Liu

Catharine Lory

Qingli Lei

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