New College of Florida Brilliantly Unique; Uniquely Brilliant
<%BANNER%>

Introducing "Mathematics"

MISSING IMAGE

Material Information

Title:
Introducing "Mathematics" The Effectiveness of a Structured Educational Tool with Playful Aspects
Physical Description:
Book
Language:
English
Creator:
Fritschie, Elaine S.
Publisher:
New College of Florida
Place of Publication:
Sarasota, Fla.
Creation Date:
2010
Publication Date:

Thesis/Dissertation Information

Degree:
Bachelor's ( B.A.)
Degree Grantor:
New College of Florida
Degree Divisions:
Social Sciences
Degree Disciplines:
Psychology
Committee Members:
Barton, Michelle

Subjects

Subjects / Keywords:
Autism
Education
Tools
Play
Genre:
Electronic Thesis or Dissertation
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )

Notes

Abstract:
As the play and learning of students with autism have been found to improve in structured conditions, it was thought that an educational tool with structured and playful aspects may help them practice a new mathematical skill better than an unstructured educational tool. For the current study, the method and answer accuracy changes of eight students with and without autism (6- to 11-years-old) were analyzed as they practiced regrouping with two different educational tools. Participants practiced their regrouping skills for 12 sessions, first with an unstructured number line and then with MatheMATics, a newly-designed structured educational tool with playful aspects. Accuracy changes were also assessed across pre-, midpoint-, and post-tests, for which participants did not use either tool. Regardless of diagnosis, all participants mastered the regrouping method. Results suggest that the MatheMATics tool may work for some students with autism but not others, possibly depending on the severity of the diagnosis. Future research on structured educational tools with a larger, more diverse sample is encouraged.
Thesis:
Thesis (B.A.) -- New College of Florida, 2010
Electronic Access:
RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography:
Includes bibliographical references.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local:
Faculty Sponsor: Barton, Michelle
Statement of Responsibility:
by Elaine S. Fritschie

Record Information

Source Institution:
New College of Florida
Holding Location:
New College of Florida
Rights Management:
Applicable rights reserved.
Classification:
local - S.T. 2010 F91
System ID:
NCFE004254:00001

MISSING IMAGE

Material Information

Title:
Introducing "Mathematics" The Effectiveness of a Structured Educational Tool with Playful Aspects
Physical Description:
Book
Language:
English
Creator:
Fritschie, Elaine S.
Publisher:
New College of Florida
Place of Publication:
Sarasota, Fla.
Creation Date:
2010
Publication Date:

Thesis/Dissertation Information

Degree:
Bachelor's ( B.A.)
Degree Grantor:
New College of Florida
Degree Divisions:
Social Sciences
Degree Disciplines:
Psychology
Committee Members:
Barton, Michelle

Subjects

Subjects / Keywords:
Autism
Education
Tools
Play
Genre:
Electronic Thesis or Dissertation
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )

Notes

Abstract:
As the play and learning of students with autism have been found to improve in structured conditions, it was thought that an educational tool with structured and playful aspects may help them practice a new mathematical skill better than an unstructured educational tool. For the current study, the method and answer accuracy changes of eight students with and without autism (6- to 11-years-old) were analyzed as they practiced regrouping with two different educational tools. Participants practiced their regrouping skills for 12 sessions, first with an unstructured number line and then with MatheMATics, a newly-designed structured educational tool with playful aspects. Accuracy changes were also assessed across pre-, midpoint-, and post-tests, for which participants did not use either tool. Regardless of diagnosis, all participants mastered the regrouping method. Results suggest that the MatheMATics tool may work for some students with autism but not others, possibly depending on the severity of the diagnosis. Future research on structured educational tools with a larger, more diverse sample is encouraged.
Thesis:
Thesis (B.A.) -- New College of Florida, 2010
Electronic Access:
RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography:
Includes bibliographical references.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local:
Faculty Sponsor: Barton, Michelle
Statement of Responsibility:
by Elaine S. Fritschie

Record Information

Source Institution:
New College of Florida
Holding Location:
New College of Florida
Rights Management:
Applicable rights reserved.
Classification:
local - S.T. 2010 F91
System ID:
NCFE004254:00001


This item is only available as the following downloads:


Full Text

PAGE 1

INTRODUCING "MATHEMATICS": THE EFFECTIVENESS OF A STRUCTURED EDUCATIONAL TOOL WITH PLAYFUL ASPECTS BY ELAINE S. FRITSCHIE A Thesis Submitted to the Division of Social Sciences New College of Florida in partial fulfillment of the requirements for the degree Bachelor of Arts Under the sponsorship of Dr. Michelle Barton Sarasota, FL May, 2010

PAGE 2

Acknowledgements I would like to sincerely thank my committee members, Drs. Michelle Barton, Steve Graham, and Heidi Harley, for their unwavering support over these last four years. Your obvious dedication to both your field and your students has been extraordinarily inspiring. Without your guidance, this year would have been difficult indeed. I would also like to acknowledge the directors, principals, teachers, parents, and (of course!) students who made my research possible when they took a leap of faith and welcomed me into their classrooms. Special thanks to: Jenn and Meghan, who showed me that people do live through this; Cristina and Lydie, who had the misfortune of living with me this year; Stu, for telling me I was being ridiculous whenever I doubted myself; and Smalls, the rest of W, and the BFC, for still being there whenever I left my thesis cave. You have all made my time at New College invaluable and I could never thank you enough. Mom and Dad (and Mickey!), all of my love is for you. I hope I continue to prove that your unconditional acceptance and support of my quirky decisions in life are welldeserved. ii

PAGE 3

Table of Contents Acknowledgements ii Table of Contents iii List of Illustrations iv Abstract v Literature Review 01 Appropriate Play 02 Benefits of Play 08 Play of Children with Autism 17 Education of Children with Autism 25 Educational Toys & the Current Study 34 Method 43 Results 53 Discussion 61 Appendix 73 References 75 Figures 82 iii

PAGE 4

List of Illustrations Figure 1. A selection of flashcards. Figure 2. The MatheMATics tool. Figure 3. A timeline depicting the multiple baseline procedure. Figure 4a-c. A series depicting the regrouping method for addition on a flashcard. Figure 5a-c. A series depicting the regrouping method for subtraction using the MatheMATics tool. Figure 6. Method accuracy scores by introduction group. Figure 7. Answer accuracy scores by introduction group. Figure 8. Acquisition of sub-steps by practice session: Participants with autism. Figure 9. Acquisition of sub-steps by practice session: Typically developing participants. iv

PAGE 5

INTRODUCING "MATHEMATICS": THE EFFECTIVENESS OF A STRUCTURED EDUCATIONAL TOOL WITH PLAYFUL ASPECTS Elaine S. Fritschie New College of Florida, 2010 ABSTRACT As the play and learning of students with autism have been found to improve in structured conditions, it was thought that an educational tool with structured and playful aspects may help them practice a new mathematical skill better than an unstructured educational tool. For the current study, the method and answer accuracy changes of eight students with and without autism (6to 11-years-old) were analyzed as they practiced regrouping with two different educational tools. Participants practiced their regrouping skills for 12 sessions, first with an unstructured number line and then with MatheMATics, a newly-designed structured educational tool with playful aspects. Accuracy changes were also assessed across pre-, midpoint-, and post-tests, for which participants did not use either tool. Regardless of diagnosis, all participants mastered the regrouping method. Results suggest that the MatheMATics tool may work for some students with autism but not others, possibly depending on the severity of the diagnosis. Future research on structured educational tools with a larger, more diverse sample is encouraged. _____________________________ Dr. Michelle Barton Psychology v

PAGE 6

Introducing "MatheMATics": The Effectiveness of a Structured Educational Tool with Playful Aspects "Do not keep children to their studies by compulsion but by play." Plato The importance of play is well-understood: it guides and enhances the healthy development of young children as they mature, preparing them for successful social interaction and academic learning. However, children with disabilities often lack ageappropriate play skills, much as they exhibit deficits in other cognitive and social domains. Improving these children's play and learning has thus been a crucial concern of educational psychology. Continuing to test novel theories and teaching methods is a vital component to this endeavor. The current study sought to examine the effectiveness of a newly-designed educational tool for improving the learning of children with autism spectrum disorders. Based on current research, the tool's design purposefully incorporated playful and structured aspects. To elucidate the rationale behind these decisions, a comprehensive definition of play will lead into a review on the widespread benefits of play during childhood. From here, the nuances of how different populations of children play will be explored. More specifically, it has been shown that typically developing children, by definition, exhibit age-appropriate play skills and thrive in unstructured environments that allow for self-guided exploration. Children with autism spectrum disorder, on the other hand, have been shown to exhibit improved play, as well as learning, in more structured and facilitated environments. Therefore, there is the potential for educational tools with playful aspects, which join play and learning into an integrated activity, to help 1

PAGE 7

improve the play and learning of all children. It is thus argued that such a structured educational tool may have the greatest potential to benefit children with autism. Appropriate Play and Typically Developing Children Play, a seemingly simple concept, is a term so general that it is rendered useless without modifiers. The definition of "play" straight out of the dictionary states that it is 3 a : recreational activity; especially : the spontaneous activity of children b : absence of serious or harmful intent : jest c : the act or an instance of playing on words or speech sounds." (Merriam-Webster Online Dictionary, 2010), but this is not very useful. Educational psychology has grappled with the term for decades, with many studies ultimately presenting a working or operational denition of play rather than a standard one (Mastrangelo, 2009). For the purposes of this study, play will be understood as a "particular manifestation of all adaptive activity" (Sutton-Smith, 1975, pp. 201). That is, it is an inherently adaptive activity for children, but it is an unique manifestation of the adaption, dependent upon the individual child. Sutton-Smith conceptualizes play as a form of "adaptive inversion"; adaptive in that it has the potential to function as precursor for later skills and inversive in that the player (child) now has control of the environment (pp. 208). However, a more thoughtful, in-depth understanding of play can be realized when the context in which it is being observed is understood. For instance, what type of play is occurring? What types of toys, if any, are being used and in what sort of sociocultural context? Are there any unique characteristics of the children involved? Why was this play even initiated in the first place? For example, when play is initiated, the type of playful act can be either appropriate to the situation or not. Inappropriate play occurs when a child engages in 2

PAGE 8

inappropriate acts (e.g., tantrums, acts of aggression) or improperly uses a toy (e.g., throwing or teething on it) (Sigafoos, Roberts-Pennell, & Graves, 1999). Appropriate play, however, is an umbrella term encompassing several types of play, including sensorimotor, relational, and pretend play, the latter of which can be further broken down into functional, socio-dramatic, and symbolic play. These appropriate play skills can roughly be thought of as a progression, as they tend to emerge and develop in this order (sensorimotor to functional to symbolic) (Sigman & Ungerer, 1984; Libby, Powell, Messer, & Jordan, 1998). More specifically, sensorimotor play involves sensory or motor manipulation of an object with the express purpose of gaining an understanding of the object. This can be seen when a child briefly mouths a toy, bangs it against a surface, or attempts to bounce it. Relational play is similar to sensorimotor, but involves combining one or more objects in a nonfunctional way (Sigman & Ungerer, 1984). Functional play, on the other hand, involves playing with an object as it is explicitly designed to be used (Libby et al., 1998). That is, moving a toy car along a track would be functional play, whereas trying to stack toy cars would be relational play. During socio-dramatic play, social roles are assigned and acted out, such as when one child pretends to be the teacher and another the student (Hirsh-Pasek & Golinkoff, 2008). Symbolic play involves "treating an object or situation as if it is something else" (Libby et al., pp. 487, their emphasis). The child may interact with an object or environment that is not physically represented by anything (e.g., running around as a superhero without a cape and mask) or may substitute one object as another (e.g., using a banana as a phone). Functional, socio-dramatic, and symbolic play 3

PAGE 9

are considered types of pretend play as they all require the child to be able to make believe that the objects and situations are more than they initially appear. Research has been done to identify necessary criteria that should first be identified during an activity before it can be considered as play. Smith and Vollstedt (1985) sought to empirically test criteria from previous literature. From two prominent articles, five criteria were chosen: Nonliterality (behavior is taken seriously, but has aspects of pretending), Positive Affect (behavior appears enjoyable), Flexibility (behavior shows variation in form and content), Intrinsic Motivation (behavior is not done because of physical needs, social demands, or rules), and Means/Ends (behavior does not appear to be motivated by an end result). These criteria were used by the raters (psychology students studying children), who were instructed to code video recordings of toddlers while at nursery school for both the presence of any of the criteria, as well as "play" in general. Nonliterality was found to have the highest association with play, while Means/ Ends had the lowest. As the number of occurring criteria increased, play was more likely to be identified as occurring as well. Play was operationally defined as occurring when a minimum of two criteria were identified, though not all criteria were equally strong. Nonliterality and Positive Affect or Nonliterality and Flexibility were the two strongest pairs; that is, they were most often associated with play. Intrinsic Motivation was not found to be associated with play. Smith and Vollstedt (1985) addressed the dominance of Intrinsic Motivation in literature and its lack of association in this study. They thought it could be that this 4

PAGE 10

criterion was too difficult to identify from objective observation or that perhaps play is too socially constrained by playmates to ever be intrinsically motivated (e.g., as the toddlers were observed in their usual nursery, it may be assumed that they were interacting with friends). Therefore, the researchers may have not found an association between Intrinsic Motivation and play in their sample because of the type of play that was occurring; not all appropriate play occurs as an activity in its own right. For example, if the majority of play that was observed was socio-dramatic, it could be that the target child was following the lead of another to gain social approval instead of guiding their own play. Also, the videos were recorded during a classroom's "free time," during which children are often encouraged to engage in free play. It may be that target children were not intrinsically interested in playing at that time and were only following the rules. Free versus Structured Play Play that is initiated as its own activity and is directed solely by the child(ren) is considered free play (Malone & Langone, 1999). Solitary free play may contain elements of Smith and Vollstedt's (1985) elusive Intrinsic Motivation because it is an activity separate from both necessity and direct social influences. Play is no longer considered "free", however, when the activity or environment is purposefully structured, usually for research or interventions that are attempting to achieve a direct goal, such as observing or increasing a specific type of play (Malone & Langone, 1999). For example, environments can be structured through any direct manipulation, such as by limiting toy choice or decreasing external distractions. Researchers that are hoping to increase the occurrence of, say, sociodramatic play, could limit toy choice by removing blocks and crayons from 5

PAGE 11

the environment and replacing them with role-themed toys (e.g., a stethoscope, lab coat, and bandaids for playing doctor/patient). However, as the researchers may wish to compare their sociodramatic manipulation to a baseline or control, studies will also create settings in which to observe free play. For these environmental settings, care is taken to mimic a common play area or classroom: a variety of toy types are available, other children may or may not be present, and the target participant can choose with whom or what to interact. This last example moves past just the environment and leads into the activity of play itself (with whom and what), which is considered to be facilitated when experts (more experienced individuals, such as older peers or adults) guide the activity in any way, such as by verbally prompting play if the child pauses or by physically modeling how to manipulate a toy 1 Hence, it is possible that Smith and Vollstedt's research did not exhaustively study the type of play and its setting, limiting the potential for Intrinsic Motivation to be identified in the sample. With play presenting as a complex phenomenon to define for research purposes, it is unsurprising to learn that parents' opinions on which activities constitute play vary greatly. Notably, mothers' beliefs regarding the nature of play and its academic value were found to differ amongst themselves and also child development professionals (Fisher, Hirsh-Pasek, Golinkoff, & Gryfe, 2008). For the first half of the study, over 1,000 mothers completed an online survey that assessed the frequency a child of theirs 6 1 It is important to note that this distinction between structured environments and expert facilitation is the author's own. More commonly, researchers will use either interchangeably or designate their own distinction (e.g., Brodin, 1999; Kok, Kong, & Bernard-Optiz, 2002; Malone & Langone, 1999).

PAGE 12

engaged in a listed activity, the activity's degree of playfulness, and the perceived relation of the activity to academic learning. From the activity list, "structured" (i.e., goaloriented activities, including life skills) and "unstructured" (imaginative processes, often without clear rules) types of play were derived. Mothers were subsequently placed into three groups by their play beliefs regarding these two play types: (a) All Play mothers rated both structured and unstructured activities as highly playful, (b) Traditional mothers rated only unstructured activities as highly playful, and (c) Uncertain mothers were more varied, usually rating unstructured activities as moderately playful and structured activities as neither play nor non-play. A mother's "belief group" had a combined effect on the frequency of her child's play and the academic value she prescribed to it. For example, All Play mothers placed more academic value on unstructured play than Traditional mothers, who in turn gave it more value than Uncertain mothers. Additionally, All Play mothers reported their children engaged in more structured and unstructured play than did the children of Traditional and Uncertain mothers. In the second half of Fisher et al.'s (2008) research, almost 100 child development professionals participated in a similar online survey that included the same activities and questions. Professionals were found to rate unstructured activities as less playful than mothers did; however, professionals and mothers did not differ in their beliefs on the academic value of unstructured play. Overall, professionals had more narrow views of play and its associated academic value than did mothers. Unlike mothers, who reported believing both types of activities to be play, professionals did not rate structured activities 7

PAGE 13

as play at all. This study concluded that most mothers define play more broadly than professionals and may not differentiate between structured and free play (i.e., Fisher et al.'s (2008) unstructured activities). This has the potential to have a huge influence on children, as appropriate free play has been shown to be uniquely beneficial to the healthy development and learning of typically developing children. (However, it should be noted that it is not suggested for either free or structured play to be neglected.) The Benefits of Play The benefits that typically developing children can gain from appropriate play have been acknowledged for decades, first and foremost by Vygotsky (1933) and Piaget (Zigler & Bishop-Josef, 2006). Vygotsky defined play as necessarily imaginary, with inherent rules within the imaginary situation that do not always translate to reality. As play matures at the end of preschool, the rules involved with playful games become more strict and demand more from the children. Through this play, children are given the opportunity to cast off the situational constraints of affect and perception, allowing them to act differently in relation to what they see, as they do in pretend play. Vygotsky (1933) saw such play and learning occurring within interactions. He believed that play can be "purposeful activity" and that it creates a "zone of proximal development" in which children "work" above their current independent skill level by being aided by experts (pp. 17). Symbolic play, typically emerging around two years of age, has been extensively studied and shown to have substantial cognitive and social benefits (Fisher et al., 2008). 8

PAGE 14

Previous research, compiled and reviewed by Fisher et al., has shown that symbolic play has the potential to improve abstract thought, memory, perspective taking, language, literacy, and more. For example, the work of Johnson, Ershler, and Lawton (1982) found an association between constructive play and intelligence. After assessing intelligence and conservation through seven separate measures, the spontaneous play behaviors of over 30 pre-schoolers were observed and analyzed. The researchers found that there was a positive correlation between participants' constructive play and cognitive intelligence; that is, the pre-schoolers who spontaneously engaged in constructive play scored higher on the intelligence measures. This work is consistent with that of Burns and Brainerd (1979), who found that training in constructive and dramatic play produced improved performance on perspective-taking tasks. After pretests to determine their perceptual, cognitive, and affective perspective-taking, roughly 50 pre-schoolers were randomly assigned to either the constructive play, dramatic play, or control (no play) sessions. Participants in the constructive and dramatic play groups were divided into testing groups of 2-3 preschoolers and guided through 10 sessions each by an experimenter, who provided a theme for each session (e.g., build a wagon, visit a restaurant) and facilitated play if necessary. A posttest was conducted to determine if the play sessions affected the perspective-taking of the participants. Even regardless of weaker pretest scores, participants from the two play groups were found to have higher scores on the perspective-taking tasks after the play sessions. The control group, which did not involve participation in any play sessions, did not exhibit similar changes. Burns and Brainerd 9

PAGE 15

concluded that even brief additional exposure to constructive and dramatic play has the potential to increase children's perspective-taking, though without a follow-up, it is unknown how long these effects may last. Grounded by Vygotsky's (1933) theories, social pretend play (i.e., collaborative pretend play with peers) has also been found to improve self-regulation and social competence (Barnett & Storm, 1981, as cited in Berk, Mann, & Ogan, 2006). In this study, 40 preschoolers' initial anxiety levels were collected through physiological (palm sweating) and self-report (choosing from a scale of very happy to very sad faces) measures. Half of the participants then viewed either a stressful movie clip (Lassie and her master are separated during a storm) or the stressful clip and its positive conclusion (Lassie and her master are safely reunited). Participants were comparable on initial anxiety levels, though those who had viewed only the stressful scene reported more anxiety and negative emotion after the move clip than stress-and-conclusion control participants. During the play period following the movie clips, participants in the stressonly group were found to reenact parts of the Lassie scene more often than the control group. After this play period, stress-only participants reported sharp decreases in their levels of anxiety and negative emotions. Barnett and Storm concluded that the social pretend play these participants engaged in after the stressful movie clip enabled them to regulate their negative emotions. However, Zigler and Bishop-Josef (2006) clarify that not just free play is beneficial to learning; carefully structured or facilitated play can also be valuable, most often for intervention purposes. Educationally-focused play, facilitated by an expert, can 10

PAGE 16

help children develop their problem-solving abilities, understanding of concepts, and language skills (e.g., Ginsburg, 2006). For example, Azmitia (1988) found that when 5year-old children who were novice block-builders were paired with a more skilled, sameage partner, the novices' ability improved in independent post-tests. Complementing Vygotsky's (1933) interactive theories, Piaget viewed children as "little scientists" who learned about their world through trial-and-error techniques during exploration (Zigler & Bishop-Josef, 2006). Over time, these "experimental" learning experiences lead to knowledge about the child's surroundings. This has been observed across countless subjects, including play (e.g., Schulz & Bonawitz, 2007) and mathematics (e.g., Ginsburg, 2006). Ginsburg argued that typically developing children, regardless of socioeconomic status, enter into the school setting with a slight grasp of mathematical concepts because their exploration and play in daily life naturally exposed them to "everyday mathematics." Everyday mathematics include a comprehensive range of mathematical skills, such as relative magnitude (e.g., more/less, bigger/smaller, taller/shorter), patterns, counting (e.g., block play), measurements, shape, and spatial location. These everyday mathematic skills are inherent in playful activities, like counting a block tower or grabbing the bigger cookie. The range and competency of such everyday mathematical skills vary by child, but research has found that young children tend to spontaneously play roughly 15% of the time with everyday mathematics or taught mathematical skills (Seo & Ginsburg, 2004, as cited in Ginsburg, 2006). Typically developing children, with and without explicit instruction, can thus gain a foundation in mathematics through their 11

PAGE 17

free play during early childhood. The Advantages of Unstructured Play Appropriate play appears to be most beneficial, however, when it is free or "unstructured" (e.g., not facilitated by experts). For example, research has consistently shown that children attend more to a confounded toy than an unconfounded toy; that is, when ambiguous evidence is presented that does not explain which functions of a toy produce its effects (confounded), children are more likely to play with and learn about the toy's functions than if the presented evidence explicitly demonstrates one of the functions (unconfounded). When confounding evidence is observed during the demonstration of a toy, children prefer to further explore the original toy over a novel one. Schulz and Bonawitz (2007) engaged preschoolers in play with a box that had two levers that each made a different object pop up out of the box. In the confounded play condition, the experimenter and participant always pushed their levers at the same time so the participant could not know which lever (or both) caused which objects to appear. In the unconfounded play trials, different conditions controlled for effect, number of trials, and exhaustion of curiosity, though all participants observed separate levers causing a different object to pop up. After the confounded or unconfounded play conditions, all children had a second, novel box revealed and both boxes were then put just out of arm's reach. This meant a participant's choice would be deliberate, requiring a stretch to obtain either toy. Children were instructed to have fun playing and the experimenter returned after a minute to end the experiment. 12

PAGE 18

During all participants' free exploratory play, 12 of the 16 children pressed the levers one at a time. Children in the confounded condition were more likely to reach for the original box first and play with it more than the novel box. Children in the unconfounded conditions that controlled for effect and trials (i.e., the two conditions in which both the experimenter and children pressed separate levers at different times) were more likely to play with the novel box; however, children in the unconfounded condition that controlled for exhaustion of curiosity (i.e., the condition in which the children only watched the experimenter press the separate levers at different times) played with both boxes equally. Regardless of participants' unconfounded condition, however, every child in those trials reached for the novel box first. These results suggest that how much evidence about the toy's functions was presented influenced the exploratory play of the typically developing children. The children who observed confounded evidence about the original box continued to play more with that toy than the children in the unconfounded condition, who actively figured out the box with the help of a researcher; these children instead moved on to play with the novel toy. The mixed results of children in the unconfounded condition, which controlled for exhaustion of curiosity, suggest that physical modeling does not completely bore a child and will prompt some play nonetheless. Although it remains unknown if the children fully comprehended (e.g., recognized confounded evidence and remained curious about the original box or recognized unconfounded evidence and chose to play with a novel toy) or learned from their actions (e.g., knew only Lever 1 causes Object A to pop up), 75% of all of the participants operated one lever independently of the other, 13

PAGE 19

effectively creating a learning opportunity. Having this inclination can prepare children to learn from their play experiences as they mature. This vein of research has continued, focusing on children's exploration of confounded toys (Gweon & Schulz, 2008). The researchers predicted that, if the children saw a demonstration of a toy that did not provide enough evidence about how the toy's functions worked (confounded condition), they would engage the toy in a more variable way that might produce novel information about its causal structure. Conversely, if children saw a toy demonstration that explained the toy's functions (unconfounded condition), they would exploit this information and play with the toy in a more convenient way by not exploring the toy further. The toy was a black-and-white mat with red and green lights that were activated by blue and yellow blocks. The unconfounded demonstration revealed that the different colored blocks activated the different colored lights, but that mat color was irrelevant. The confounded demonstration did not provide enough evidence as to whether mat or block color produced the different light colors. For children in the confounded condition to discover the causal structure of the toy, they would have to move from where they were initially seated to reach the other side of the mat. Children in the unconfounded condition were found to play more on the side of the mat that they were initially placed at than children in the confounded condition, who were more likely to move. These results suggest that children's exploration of the toy was affected by the demonstration they were shown. Confounded children engaged the toy in a more variable way, even though it meant inconveniently stretching across the mat, 14

PAGE 20

something the unconfounded children did less often. This was perhaps an attempt to determine the toy's causal structure. In the confounded condition, 44% of the children even tried each block separately on each side of the mat, creating another potential learning opportunity. Again, it was outside the range of this study to determine if participants learned from their interactions. The outcome, though, is the same: confounded children's play variations generated novel information about the toy and, even if the children did not directly learn from this evidence, it is practice for a later time when novel information becomes more meaningful. The extent to which typically developing children explore a toy thus appears to depend on what evidence is presented, but it is still unclear if the children were learning from their exploration. To rectify this, Bonawitz et al. (2009) conducted a study in which preschoolers were invited to play with a novel toy. In the pedagogical condition, the participants were shown one of a toy's four functions (pulling out a tube produced a squeaking sound). The experimenter told the children that that was how the toy worked. In the accidental condition, the same function was demonstrated, but the experimenter acted as if it was an accident. In the no-demonstration condition, children were not shown how any of the toy's functions worked. After participants exhausted playing with the toy, the experimenter returned and tested them on demonstrating all four functions of the toy, not just the tube function that some participants were shown. Children in the pedagogical and accidental conditions were more likely to learn the tube function than children who did not see a demonstration, though children in the pedagogical condition were significantly less likely to have learned the other three 15

PAGE 21

functions compared to the other two conditions. Children in both the no-demonstration and accidental conditions were equally likely to have learned the three undemonstrated functions. These results suggest that participants in the pedagogical condition limited their play to the demonstrated function, did not explore the toy as much as participants in the other two conditions, and thus were less able to produce the three other functions of the toy. This supports young, typically developing children being capable of learning from their exploratory play. However, children in the accidental condition were just as likely as children in the pedagogical condition to have learned the tube function and also as likely as children in the no-demonstration condition to have learned the other three functions. Therefore, children who saw the demonstration, but were not explicitly instructed that that was how the toy worked, were the participants who learned the most. Explicit instruction thus appears to hamper exploratory play and learning in typically developing children, while "accidental" modeling does not. These three previous studies support a proposed "Free versus Structured Play" (F/ SP) hypothesis: free play is more beneficial than structured play for the play and learning of typically developing children. It appears that children explore more about a toy's causal structure when the toy's functions are not explicitly demonstrated. Confounded evidence also encouraged children to continue playing with the original toy rather than select a new one. Furthermore, the manner in which a toy's functions were unconfounded also mattered; an explicit, pedagogical explanation of one function decreased the likelihood that children could produce other causal functions of the toy beyond the one 16

PAGE 22

they were shown. Thus far, these studies have drawn from only one population: healthy children who were not diagnosed with any physical or mental complications, that is, typically developing children. The question remains as to whether the proposed F/SP hypothesis can be generalized to other populations, such as children with autism. The benefits that typically developing children garner from play are often the result of more advanced play (e.g., symbolic, socio-dramatic) than children with autism are capable of spontaneously producing. The Play of Children with Autism Spectrum Disorders The full complexity of autism spectrum disorders are made clear in the name itself: they do not present as separate conditions with clear symptoms, but along a spectrum with room for variation. The Autism Society for America (http://www.autismsociety.org/) defines autism as "a complex developmental disability that typically appears during the first three years of life and affects a person's ability to communicate and interact with others." More formally, according to the Diagnostic and Statistical Manual of Mental Disorders-IV-TR a diagnosis of autistic disorder (i.e., "classic" autism), requires: six or more qualitative impairments in social interaction, communication, and "restricted, repetitive, and stereotyped patterns of behavior, interest, and activities"; "delays or abnormal functioning in at least one of the following areas, with onset prior to age 3 years: (1) social interaction, (2) language as used in social communication, or (3) symbolic or imaginative play"; and that "the disturbance is not better accounted for by Rett's Disorder or Childhood Disintegrative Disorder" (American Psychiatric Association, 17

PAGE 23

2000, pp. 69-70). For the purposes here, there will be a focus on the deficits observed in play. Recall that, when attempting to observe, define, or study play, it is important to consider the unique characteristics of the children involved. For example, the play skills of children with autism have been found to vary, though several key distinctions can be made. These children have difficulty initiating play activities (Brodin, 1999) and exhibit less pretend play than children with Down syndrome or typically developing children (e.g., Sigman & Ungerer, 1984). Additionally, sensorimotor play can persist beyond an appropriate age (around 2 years; Fernie, 1988), perhaps inhibiting curiosity and the development of more age-appropriate play behaviors (Libby et al., 1998). There has been, however, some controversy concerning the capability of children with autism to engage in pretend play (e.g., functional and symbolic). Their play is generally rigid and stereotyped; that is, it shows some signs of more advanced development, but is repetitive and not generalized to all play. For example, one of a child's figurines will be attributed with a pretend quality, such as an illness, but the child does not pretend the figurine has any other pretend qualities, such as emotions, nor does he/she generalize this symbolic play to other figurines. Overall, studies have produced mixed results in attempting to determine where skill deficits are most common. In one attempt to clarify the situation, Libby et al. (1998) looked at children with autism, children with Down syndrome, and typically developing children who had a mental age of approximately 2-years-old. These groups of children were given nondirective (i.e., nonspecific, vague) prompts with three sets of objects: junk objects 18

PAGE 24

(e.g., piece of string, matchbox, clothes peg), conventional toys (e.g., teddy bear, truck, doll), and a mixed collection of these two groups. To determine consistency, they were observed at three separate times, 3-4 months apart. No consistent changes were observed between these sessions, suggesting the observed play was representative of the children's regular behavior. In order to tease apart differences, symbolic play was broken down into three types: object substitution (e.g., a banana as a phone), making an attribution of false properties (e.g., a doll falls ill), and referencing an absent object (e.g., driving a toy car over an invisible bridge). Children with autism produced significantly more sensorimotor play than the two other groups. There were no differences between the amount of relational play among the groups, though the typically developing children showed a trend of producing more functional play than the others. Typically developing children and the children with Down syndrome also engaged in significantly more symbolic play than the children with autism. None of the children with autism made reference to an absent object, and they produced significantly fewer attributions of false properties compared to the children with Down syndrome and typically developing children. Children with autism did, however, produce object substitutions, which represented the majority of their infrequent symbolic play. This fits the developmental literature on symbolic play, as object substitutions have been found to develop first (Libby et al., 1998). Supporting the possibility that sensorimotor play perhaps inhibits other play, sensorimotor play was negatively correlated with relational and symbolic play. These results differ from some of the other research, which has not been able to 19

PAGE 25

observe symbolic play in this population (e.g., Wing, Gould, Yeates, & Brierly, 1977). Libby et al. (1998) instead observed children with autism who were able to produce some symbolic play. It could be that object substitution involves different mechanisms than do the other two types of symbolic play. As object substitution is usually observed before the development of false properties and absent objects, there is the potential of a developmental delay hypothesis to explain the less advanced play behaviors of children with autism, though this is outside the range of this thesis. Holmes and Willoughby (2005) conducted a more recent study that found more functional play than previous research has observed. The researchers sought to fill a gap in the literature on the play behaviors of children with autism by observing them naturally in a school environment, rather than in a laboratory setting. They also analyzed play behavior reports completed by both educators and mothers to see if there was agreement between the two groups and if their reports agreed with the observed findings. Almost 20 children with autism, ages 4to 8-years-olds, were recorded on five separate days for 10 minutes each during their schools' free play or recess sessions. These sessions were coded for cognitive (e.g., sensorimotor, functional) or social (e.g., onlooker, parallel) types of play. The play information collected from mothers and educators measured these same qualities. It was found that, contrary to past literature (e.g., Libby et al., 1998), functional play is often exhibited by children with autism. Parallel-functional and solitary-functional play were two of the three most frequently observed play behaviors. There was also little to no observed stereotyped play. Mothers and educators were found to report similar play 20

PAGE 26

qualities, with solitary passive play being perceived as the most common type. This generally supports the observed findings, as there was agreement on the likelihood of most of the other play behaviors, too. This study contradicts much of the past literature on the play behavior of children with autism, which has otherwise found less functional play and more sensorimotor and stereotyped play. It may be that this sample was more high-functioning, that the school setting prompted more advanced play, or that this sample's early diagnosis and extensive time in an inclusive classroom had already led to play interventions. Using the unique naturalistic setting, Holmes and Willoughby's (2005) findings suggest that the functional play behavior of children with autism is not a settled matter and instead requires continued research. What the current literature does support is autism being linked to play skill deficits that are unique to the disorder. Improving the Play of Children with Autism Through Structure As appropriate free play has been previously supported to have far-reaching benefits for young children, it has been a major undertaking of developmental research to advance the play skills of children with autism, so that they too may benefit from play in the same ways as their typically developing peers. Play interventions have been successfully utilized in the past to improve their play skills. For example, a study by Sigman and Ungerer (1984) sought to first lend further support to the occurrence of play skill deficits in children with autism and then to determine the effectiveness of facilitated intervention (i.e., when an expert helps guide the activity) for their exploratory play and overall communication. The researchers matched children with autism, with mental 21

PAGE 27

retardation, and without disability on mental age (17-38 months) and chronological age (16-80 months) before analyzing the children's play in two test sessions. The first session assessed each child's spontaneous play without interference, while in the second session, the experimenter would initially prompt the child with verbal cueing and then, if that was unsuccessful, physical modeling. The children were then administered scales on sensorimotor behavior, vocal and gestural imitation, and language. If the children with autism failed an item in a task, it was re-administered to ensure their optimal performance was exhibited. During the first test session, children with autism were observed to engage in less functional play in less diverse ways than their peers. The children with mental retardation and without disability engaged in more age-appropriate functional play than relational or sensorimotor play, but the children with autism engaged in each type of play for about the same amount of time, thus exhibiting age-inappropriate play. In the more structured second session, after cueing and modeling, the children with autism increased their functional play to the point that it matched the other children's, but even with facilitation, they did not engage in as much symbolic play as the other two peer groups. Analyses of children's behavior, imitation, and language scores led Sigman and Ungerer (1984) to conclude that autism is characterized by a deficit in symbolic play that cannot be fully explained by a language deficit. Their results lend further credence to pretend play skill deficits in autism and also support the use of facilitated intervention to help correct these deficits. Several other studies have found similarly encouraging results for facilitated and structured interventions (e.g., DiCarlo & Reid, 2004; Hsieh, 2008). 22

PAGE 28

Kok, Kong, and Bernard-Opitz (2002) compared the effectiveness of these two types of therapies, facilitated and structured, for enhancing the communication and play of children with autism. In structured play therapy, a trained group of peers engaged a child with autism by using a toy selected by the experimenter and implementing the specific instructions (i.e., using the pre-selected toy in specified ways, redirecting the target child if they initiated any different play). In facilitated play therapy, the child with autism was allowed to choose a preferred toy and an experimenter indirectly prompted play between the child with autism and the peer group, the latter of whom were instructed to initiate communication and play in their own ways. For Kok et al.'s (2002) study, there was an initial baseline session of free play between the peer group and target child. After this, the peer groups were taught about the characteristics of autism and specific play therapies. Structured play was taught through demonstrations of sessions, whereas facilitated play was taught through incidental techniques with several exemplars. All target children then participated in one 30-minute long session of each type of play therapy with a peer group. Communication attempts (verbal and non-verbal) increased in both types of intervention for all target children. For 6 out of 8 children, there were more appropriate communicative responses during structured play, as well as more appropriate initiations during facilitated play for three children. Except for one child, appropriate play initiation increased during structured play (4 children) and facilitated play (3 children) therapies as well. Overall, more appropriate play behaviors (both initiations and responses) were observed during structured play rather than facilitated. However, inappropriate play 23

PAGE 29

behaviors were also more common in structured play therapy than facilitated, while there seemed to be a trend of the higher-level and more verbal children playing more during facilitated sessions instead. Peer group and caretaker reports supported these results. Peer groups evaluated the target children favorably, while caretakers usually noted improvement in communication and play. Kok et al.'s (2002) study highlights the need for continued research. It appears that both structured and facilitated interventions can be beneficial for the improvement of autistic children's play skills, but that one intervention may be more helpful than another, depending upon the child. Malone and Langone (1999) conducted a review of similar studies that attempted to improve the play skills of children with developmental delays, problems which can otherwise prevent them from exploring the toy and compromise their ability to benefit from spontaneous play. Strategies for intervention are best thought of on a spectrum, from non-directed being the least structured, progressing through indirect and guided to directed, where the adult is significantly involved. Even free play is partially structured because of the availability of toys in the child's environment, though toy introduction alone is unlikely to result in improvement. More direct intervention strategies involve suggestive or explicit verbal guidance, modeling, and reinforcement. The studies reviewed were mostly successful, with improvement in children's play skills being generalized to untrained toys and maintained at 3-month follow-ups. Malone and Langone highlighted the importance of not just modeling "appropriate-looking" play, but setting smaller goals for complex play behaviors, such as sociodramatic play, creativity, and play sequences. 24

PAGE 30

The aforementioned research on children with autism prompts an addition to the previously proposed Free versus Structured Play (F/SP) hypothesis, which currently only holds that free play is more beneficial than structured play for the play and learning of typically developing children. Contrary to this, it appears that for improving the play of children with autism, structured play is more beneficial than free play. Children with autism have been observed to exhibit less advanced play skills, which prevent these children from taking advantage of the multiple benefits typically developing children garner from more advanced, complex play. However, structured and facilitated play interventions have been found to improve their play skills to more age-appropriate levels. It has yet to be determined, though, if structured play and interventions can also improve the education of children with autism. The Education of Children with Autism Children with autism have demonstrated that their cognitive abilities are less advanced than those of their typically developing peers, though this is not always the case (e.g., Joseph, Tager-Flusberg, & Lord, 2002). Again, the complexity of spectrum disorders means that they are difficult, if not impossible, to absolutely define. There have been attempts to pinpoint specific disorders on the spectrum based on cognitive ability, but even cognitive ability can be hard to determine. In their professional therapeutic work, Geiger, Smith, and Creaghead (2002) noticed a trend of parent and professional disagreement concerning the cognitive abilities of their children with autism, with parents overestimating their children's abilities relative to standardized testing conducted by the professionals. Previous literature has found a similar trend and suggested a few possible 25

PAGE 31

explanations. Glascoe (1994, as cited in Geiger et al.) proposed that parents may attribute their child's lack of a response to willful behavior instead of a lack of language comprehension. Additionally, as children with autism often exhibit more advanced motor skills, this discordance in ability with their language delays can be difficult to evaluate (Stone & Rosenbaum, 1988, as cited in Geiger et al.). As disagreements can inhibit progress during counseling or school, Geiger et al. sought to determine if the trend held up to experimental testing. Excluding those with other pervasive developmental disorders, such as Asperger syndrome, 41 children with autism, ages 2.5to 10-years-old, were evaluated using the Stanford-Binet Fourth Edition (Thorndike, Hagen, & Sattler, 1986, as cited in Geiger et al., 2002). If their mental age was under 24-months-old and too young for the StanfordBinet measure, the mental scale of the Bayley Scales of Infant Development Second Edition (Bayley, 1993, as cited in Geiger et al.) was administered instead. Parents also submitted evaluations of their children's cognitive level as an estimated mental age in months. The measure of incongruence was the difference in months between the parent's estimated mental age and the age assessed by the standardized test. Parents were found to significantly overestimate their child's cognitive age relative to the standardized test's assessment, though when the child's mental age was older and closer to their chronological age, disagreement was less likely. These results should be read with caution, as using standardized measures for children with autism has been criticized as misrepresentative of their strengths and weaknesses (Geiger et al., 2002). Furthermore, parents have the advantageous opportunity to take the collective sum 26

PAGE 32

of their children's abilities across several contexts. These limitations highlight two of many issues that must be taken into consideration when attempting to determine the capabilities of children with autism. It can also be difficult to tease apart what has been found through empirical research and what may be only anecdotal speculation. For example, contrary to current research which has found that individuals with autism have trouble with mathematics, there are anecdotal reports of giftedness (McMullen, 2000, as cited in Chiang & Lin, 2007). To address this discrepancy, Chiang and Lin conducted a review of 18 studies on the cognitive abilities of individuals with Asperger syndrome (AS) and high-functioning autism (HFA). Asperger syndrome presents similarly to autism, but with normal language development, whereas high-functioning autism is characterized by an average or above average intelligence quotient (American Psychiatric Association, 2000). Individuals with AS and HFA were found to most likely be of average intelligence with a modest yet significant mathematical weakness. Only some individuals were found to be mathematically gifted, supporting the anecdotal reports but by no means suggesting that giftedness is the norm or a potential criterion. These results, though limited by the small number of reviewed studies that used standardized measures ( n =8), suggest that AS and HFA are not valid indicators of mathematical ability. There should be individual assessments to determine personal strengths and weaknesses, instead of a reliance on a diagnosis. This leads to the legitimate concern that, if children with autism are assumed to have unrealistic strengths or weaknesses, they may be taught at levels that are 27

PAGE 33

inappropriate for their abilities. However, in spite of clinical differences such as those between classical autism and Asperger syndrome, educational approaches have not been tailored to specific conditions within the spectrum. A review by Dempsey and Foreman's (2001) of potential approaches thus included evaluations of programs that might have been found to not be effective within one sample, but could still be useful to others presenting with different symptoms. Two key approaches included in the review were applied behavior analysis (ABA) and multi-treatment programs. ABA works under the assumption that introducing small changes in a controlled environment can correct the behavioral issues associated with autism, such as deficits in interpersonal communication and strong reactions to changes in routine. Complex behaviors and skills, such as play skills, social interactions, or academic skills, are taught one step at a time, usually in a one-on-one situation with cues, instructions, or prompts. Appropriate responses are positively reinforced, while inappropriate responses are ignored. Past literature on ABA mostly supports its effectiveness (e.g., Hayter, Scott, McLaughlin, & Weber, 2007; Hsieh, 2008; Hume & Odom, 2007), though to date there are no definite explanations as to why some participants show dramatic improvements and others do not (Green, 1996a, as cited in Dempsey & Foreman, 2001). However, Dempsey and Foreman caution that ABA research has not often been empirically compared to other approaches. These criticisms aside, ABA's individualized, one-on-one instruction and structured learning experiences have been found to lead to significant improvements if properly implemented (see Dempsey & Foreman for a review) Many multi-treatment programs incorporate ABA along with additional 28

PAGE 34

techniques to provide an integrative approach. One program, Project TEACCH (Treatment and Education of Autistic and Related Communication Handicapped Children) utilizes ABA, structured learning, and environmental adaptations to teach selfcare skills and appropriate behaviors. Another, the Higashi School approach, works by slowly fading out the use of prompts. At the time of Dempsey and Foreman's (2001) publication, both of these multi-treatment techniques appeared to be well-received, but had not undergone rigorous testing. However, more controlled research has since been conducted to evaluate Project TEACCH. Improving the Education of Children with Autism Through Structure TEACCH bases its structured learning on the adaptation of the environment through four components: physical organization of the room, visual schedules, work systems (i.e., structured work centers), and task organization (Panerai et al., 2009). These components of the approach can theoretically be implemented in different settings across several contexts. Panerai et al. conducted a longitudinal study comparing the implementation of the TEACCH approach in two different settings (a residential center or the participant's natural home) to a typical, inclusive classroom with a nonspecific approach. Educators at the residential center overhauled the entire living space to accommodate the approach, whereas in the natural settings, parents and classroom support teachers were taught the TEACCH approach and a room or two at home or a small station in the classroom were adapted. The inclusive classroom setting, acting as a control, was observed over the course of several weeks to determine the classroom's approach, which was found to focus more on academic rather than functional skills, and 29

PAGE 35

to create Individualized Education Programs for the participants. In the two experimental settings, how well the approach was implemented was not monitored. Instead, for all settings, the dependent measures were the follow-up evaluations, which analyzed changes to participants' Psycho-Educational Profile-Revised (PEP-R) and Vineland Adaptive Behavior Scale (VABS) scores twice a year for 3 years to determine the effectiveness of the approaches in the different settings. Participants' scores in both of the TEACCH-adapted experimental settings improved significantly more than participants' score in the inclusive, control setting; however, there was not a significant difference between scores in the two experimental settings. These results suggest that TEACCH is effective for producing positive behavior outcomes across multiple settings and that decreased inappropriate behaviors do not always result solely from inclusion in a mainstreamed classroom. However, this study suffers from several methodological flaws, including not randomly assigning participants nor monitoring how well the approach was implemented. Quantitative changes to academic success also were not assessed. Finally, these participants were diagnosed with not only autism, but also severe mental retardation, limiting the ability to generalize these results to other autistic populations. Hume and Odom (2007) also sought to determine the effectiveness of TEACCH across different contexts, with a focus on applying the work system component to both work and play settings. As individuals with autism have difficulty remaining independently engaged with a task (Pelios, MacDuff, & Axelrod, 2003, as cited in Hume & Odom), a work system that successfully incorporated the other aspects of TEACCH 30

PAGE 36

(physical organization, visual schedules, task cues) could assist with independent functioning. In the work setting, a 20-year-old, severely autistic male had a work system set up at his employment site: windows were covered, trays with large visual cues were added, and the written time schedule was replaced with a visual one. In the play setting, two 6to 7-year-old boys' desks were outfitted to include shelves, a "completed" basket, and toys with physical adaptations, like Velcro. All work systems communicated four components: (a) the tasks, (b) the amount of work to be completed, (c) a cue that the work was finished, and (d) instructions for the next activity. An ABAB (baseline/intervention) procedure was used to familiarize participants with their work systems and then establish experimental control. Training criteria for the work system was met roughly halfway through the first intervention for each of the participants. During each session, the researchers looked at time spent onor off-task, the frequency of teacher prompting, and the amount of work completed. Independent functioning, measured by reductions of off-task behavior and teacher prompting, was found to increase for all participants while the work system intervention was being implemented. In the work setting, the participant remained on-task more often during intervention than during baseline and reversal, which increased their number of completed tasks. In the play setting, the participants also engaged with an increased number of play objects than they did during baseline and reversal. Behaviors returned to baseline when the work system was withdrawn. These results were replicated at the 1-month follow-up session, indicating that the changes in behavior could be maintained with exposure to the work system. With these 31

PAGE 37

methodological improvements, Hume and Odom's (2007) study supplements the research of Panerai et al. (2009) and supports TEACCH's work system component as a viable intervention for both independent work and play skills. Unfortunately, the specifics of teachers' prompting were not monitored nor were participants randomly selected, though the latter can be a logistical feat when considering the rarity of the population. While it is also not possible to discern which aspects of the work system contributed to improved functioning (e.g., minimized auditory and visual distractions, organized materials, visual cues, reduced field of choices, etc.), the integrative, comprehensive quality of TEACCH likely contributes to its success. The controlled environments found in ABA and the integrated structure found in TEACCH interventions have thus both been shown to produce favorable results for the play and learning of children with autism, who would otherwise suffer from skill deficits in both areas. These methods have both been successfully applied to improving the mathematical skills of students with disabilities, an area which has been previously noted as a potential weakness for children with autism. When typically developing students begin to learn to count, a counting-all strategy is applied, in which the student counts each addend, often using their fingers as a physical representation. For example, 2+1 would be completed by counting 1-2-3. As students progress, counting-on strategies, such as counting on from the first addend or the larger addend of the two, are acquired. The latter example, counting on from the larger addend, is also known as the min strategy. For example, 3+7 would be completed by counting 7-8-9-10. The most advanced counting method of arithmetic, the number fact strategy, is when students recall common answers 32

PAGE 38

from long-term memory (e.g., they report "just knowing" 2+2=4). However, students with developmental delays that lead to learning difficulties have not been found to spontaneously acquire these higher-level mathematical strategies like their non-disabled peers; instead, they continue to rely on the counting-all strategy. Fortunately, through intensive practice with concrete manipulatives, students with learning difficulties have been found to make and maintain progress (Irwin, 1991, as cited in Cihak & Grim, 2008). With this in mind, Cihak and Grim taught four students with autism, ages 15to 17-years-old, the one-more-than strategy. This strategy helps students make purchases by rounding up to the next full dollar (i.e., $17.84 becomes $18), which puts the students one step closer to being able to function independently. After administering a baseline test, students participated in classroom instruction for two sessions a day, 10 trials per session, until a criterion of 100% on three consecutive sessions was reached. Instruction included one-on-one sessions, concrete materials to represent the sums, and a least-to-most prompting hierarchy for offering help (e.g., from a guided question to explicit instruction). For the two generalization periods (making purchases in the school bookstore and a local department store) and follow-up procedure, only one session a day with three trials was held until again a criterion of 100% on three consecutive sessions was attained. Except during the follow-up procedure that was conducted 6 weeks later, verbal praise was provided for correct answers, while errors were corrected on a least-to-most prompt hierarchy (e.g., verbal prompt to gesture to gesture plus verbal explanation to physical modeling). All students successfully learned the one-more-than strategy, generalized it to the community settings, and still maintained 33

PAGE 39

this counting-on skill 6 weeks later. Thus, through one-on-one instruction with a least-tomost hierarchy and concrete materials, students with autism were able to acquire an extremely useful math skill that will help them foster more independence in their community. The research of Cihak and Grim (2008), Hume and Odom (2007), and Panerai et al. (2009) support the final portion of the proposed F/SP hypothesis: whereas free play has been found to be beneficial to the social, emotional, and cognitive development of typically developing children, interventions that are instead structured have been found to improve the play and learning of children with autism. For instance, learning environments that clearly marked the next step in a task (e.g., Hume & Odom) or purposefully worked to generalize and apply a classroom-taught skill (e.g., Cihak & Grim) were effective in improving the learning of children with autism and other developmental disabilities. The F/SP hypothesis does not mean to suggest typically developing children would not benefit from structured interventions; instead, it proposes that these children could benefit more from less structured, more creative instruction and play. However, for children with autism who require interventions, free play and regular instruction alone have not been successful; in these cases, structure and facilitation are more beneficial. Educational Toys & the Current Study As play and learning are so vitally connected during childhood, the wealth of literature on their positive association is to be expected. Such research has provided a strong theoretical background for the design and implementation of novel teaching 34

PAGE 40

methods which seek to combine the two. Correspondingly, the development and widespread popularity of educational toys (i.e., objects with playful aspects that prompt learning, usually of a specific concept) are equally unsurprising. Educational toys initially appear to be a winning combination of play and learning that would keep a child interested in learning by letting them think they're "just playing." There is some cause for caution, however, as there is a dearth of empirical support for educational toys' claims of improving social or cognitive abilities (Wong, Uribe-Zarain, Golinkoff, Fisher, & HirshPasek, 2008). Furthermore, what research that has been done includes only typically developing children. For example, the research of Parish-Morris, Hirsh-Pasek, Golinkoff, and Collins (2009) was motivated by the emerging market for electronic console (EC) books and parents' belief in the positive intellectual value of these products. EC books are batterypowered, touch-sensitive devices that allow children to press buttons on the screen to hear stories and play related games. Parish-Morris et al. studied the qualitative and quantitative differences EC books, relative to traditional print books, have on preschoolers' reading comprehension. To determine if there were qualitative differences, parents of 3and 5-year-olds read either an EC or print book with their children and were coded for reading style and content. Analyses determined that there were no age differences. Parents using the EC books were found to make more behavior-related utterances (e.g., asking their child not to touch the book) than parents using the print books, who in turn were found to make more story-related (i.e., repetition or elaboration of content) and distancing (i.e., relating 35

PAGE 41

a part of the story to personal experience) utterances. Children, too, made more behaviorrelated utterances when listening to the EC books than children listening to the print books, who instead made more distancing utterances. For quantitative differences, children's reading comprehension was measured. A second sample of 3and 5-year-olds read either from an EC or print book and were then assessed on character identification, event identification, story content, and plot chronology. The 5-year-old children demonstrated a ceiling effect, limiting additional analyses to the 3-year-old children. Of the four dependent measures, both character and event identification were found to be equitable across the two book conditions; however, 3-year-old children who read from the print book accurately answered more story content and plot chronology questions than 3-year-olds who read from the EC books. These data suggest that the quality of both parent-child interactions and reading comprehension suffer as a result of using EC books. Parish-Morris et al. (2009) proposed this may be because of fewer distancing utterances or the distracting nature of the consoles, which include games and sound effects. Survey data administered alongside these two studies also found that children who owned an EC book were less likely to spend time reading print books, suggesting parents might consider the consoles a substitute for print books. Overall, electronic console books as they are currently designed are not an adequate substitute to traditional print books and shared reading. However, not all research on educational toys has been negative. For instance, Bailey and Watson (1998) promoted a more Socratic, active method for teaching that emphasized the importance of arousing affective as well as cognitive domains during 36

PAGE 42

learning. A role playing board game, Ecogame, was created to involve students in a meaningful and more motivated way, while also hopefully increasing their understanding of five key ecological concepts. Each player was assigned to be the representative of a country, effectively engaging them in pretend play as they had to figure out what ecological decisions were best for their nation. The game also involved little direct, factual intervention on the behalf of the teacher, instead allowing the students to hold an open dialogue which could lead to shared understanding of the material. Participants aged 7to 11-years-old from two middle-income classrooms played the game for one session, while two other classrooms were selected as controls who were not exposed to the game (two classrooms from two different schools were split between these conditions). All participants were then assessed on their ecological understanding and, for those that played the game, also on their attitudes on the teaching strategy. Participants who played the game scored significantly better on the ecological assessment than those in the control who did not, results which were surprisingly comparable across the two schools. The active learning invoked by playing the game, as opposed to learning passively, thus seems to be a more effective way of presenting some material to students. Furthermore, it appears to be important to match the design of the educational toy to the material being mastered. Siegler and Ramani (2009) sought to improve children's understanding of the rank order of numerical magnitudes (e.g., 5+2 cannot equal 3, as 5 is already greater in magnitude than 3) for the numbers 1 through 10. Although general experience and repeated exposure appear to be critical to such acquisition, a linear board game was introduced to provide increased exposure and a physical representation of the 37

PAGE 43

concept. In line with the representational mapping hypothesis, it was expected the linear board game, being most similar to increasing numerical magnitudes, would improve understanding more so than the controls, who used a circular board game and more traditional classroom tasks, including number identification, reciting the counting string, and counting objects. To test this, preschoolers met individually with a researcher for five 20 minute sessions over the course of 3 weeks. After a pre-test, participants either played the linear board game, the circular board game, or practiced the traditional numerical tasks in each session. Children who played the linear board game demonstrated significant number line estimation accuracy increases from pre-test to post-test, as well as improved magnitude comparison accuracy. Arithmetic accuracy also increased, with children who played the linear board game having more "close misses" when erring than children in the other conditions. Additional analyses supported individual differences being stable throughout the experiment (i.e., those scoring higher on the pre-test still scored higher on the posttest), though children who initially scored lower on most tasks experienced greater absolute gains than children who initially scored higher. These results suggest that, in line with the representational mapping hypothesis, a board game that is most similar spatially to the concept at hand (e.g., in this study, numerical magnitudes) can produce greater gains in learning and accuracy than spatially unrelated board games and the traditional tasks that are most commonly presented in classrooms. This study is particularly important in light of the fact that children who initially performed at the bottom of the sample made the greatest gains in understanding, 38

PAGE 44

even in such a short amount of time. Introducing a similar game may also have the potential to even the playing field for disadvantaged children in preschool with minimal effort. Therefore, in this scant literature, board games have been found to be more effective learning tools than traditional classroom lessons (e.g., Bailey & Watson, 1998; Hogle, 1996; Siegler & Ramani, 2009), whereas an electronic book was found to diminish reading comprehension when compared to a print book (Parish-Morris et al., 2009). Thus, educational toys sound good in theory, as they could represent a synthesis of play and learning that allows children to reap the benefits of both phenomena; however, their effectiveness appears to be reliant on the nuances of each toy's design. The previous research supports the potential implementation of a structured educational tool with playful aspects. Based on the F/SP hypothesis, such a structured tool would be expected to be more beneficial for children with autism than typically developing children, while the playful aspects would be expected to engage the children's interest. If found to be effective, this educational tool could be implemented in homes and schools to help improve the academic achievement of children with autism. Accordingly, the current study sought to determine if a structured educational tool with playful aspects would be more effective than an unstructured learning tool for improving accuracy on a newly-acquired mathematical skill. A multiple baseline design across participants by diagnostic condition was implemented to test a newly-designed structured educational tool. The two diagnostic conditions consisted of children with autism and typically developing (TD) children. As supported by the results for Siegler and Ramani's (2009) 39

PAGE 45

linear board game, the structured educational tool was uniquely designed for the purposes of this study. The tool's design included several structured elements: color-coded ones and tens counting blocks and columns visually cued the separate treatment of these place values; similarly color-coded bins for each digit ensured counting blocks did not get knocked astray; and interchangeable plus and minus signs prompted consideration of the appropriate arithmetic. Regrouping was chosen as the mathematical skill, as it is easy to measure based on the necessary reorganization of the tens and ones columns if the participant has previous experience with the skill and, after being taught the concept, if they are progressing. For example, to regroup on the problem 27+8, the participant would add the ones column, write down the 5, carry the 1, and then add the tens column. If the participants did not have previous experience with regrouping, then they would not know to utilize this procedure. (Instead, they might be expected to count up from 27.) A pre-test was used to ensure that all participants did not exhibit any current skill with regrouping. After the pre-test, all participants were taught the mathematical concept. Following the lesson, several initial sessions were administered in which participants practiced regrouping problems with an unstructured learning tool, a number line. The number line was chosen to represent a learning tool that was not only unstructured, but also likely to be found in a typical classroom and lacking in playful aspects. The number line is considered to be an unstructured tool as its format does not enforce or cue any particular method. Returning to the aforementioned problem, 27+8, students could use the number line to count up from 27 to 35, or they could use it to help them regroup if 40

PAGE 46

they could not remember 7+8. Thus, the number line does not structure the activity and a student practicing regrouping could use the number line in a way that is not beneficial (i.e., just counting up from 27). After a number of practice sessions with the number line, participants were switched to the educational tool through staggered introductions. By utilizing this multiple baseline procedure, the study was able to ensure that any changes in accuracy that were observed in the participants' performances were the result of the educational tool intervention as opposed to practice effects. Additionally, establishing individual number line "baselines" allowed each participant to act as their own control, which is especially important for autism research, as these participants represent a difficult population to collectively measure (Malone & Stoneman, 1995). A microgenetic design was also incorporated into the procedure to track minute but important changes. The scores that were collected from pre-tests, practice sessions, and post-tests were not collectively averaged to form group means, but instead kept separate to analyze individually. Siegler and Crowley (1991) advocated for the application of the microgenetic method through a short review of studies that highlighted its distinctive usefulness. The microgenetic method is less concerned with the end result than it is the mechanisms and changes that lead to it. By analyzing a high density of quantitative and qualitative data that are collected over the entire course of the phenomenon of interest, the method allows for the study of changes in strategy, execution, and accuracy and the rates of these changes. It is thus particularly useful for researching skill acquisitions, with the practical implication of improving teaching 41

PAGE 47

methods. To offset some of the timely costs of applying the method, researchers have tended either to expose participants to a higher concentration of the necessary experiences to provoke change or to present a novel task altogether and follow its development. The microgenetic design has also been supported by previous research which has uncovered that, for TD children and children with disabilities alike, the ubiquitous singlesession designs and resulting means have underestimated the competence of children's cognition (Baumeister, 1997, as cited in Fletcher, Huffman, Bray, & Grupe, 1998). In a review of literature comparing the strategies of children with mental retardation and those without disability, Fletcher et al. found that these children exhibit similar patterns of discovery in terms of transition sequence, acquisition rate, and frequency of strategy use across several domains. For example, children with disabilities were found to use external counting strategies like their chronologically younger, but cognitively comparable peers (Bray, Huffman, Ward, & Hawk, 1994, as cited in Fletcher et al., 1998). It was thus important to the current study to utilize a microgenetic method to ensure the best possible understanding of each participant's ability was measured, given time limitations. Ultimately, with this methodological framework, the current study sought to answer two key questions. First, after being taught how to regroup, would the children with autism exhibit increased accuracy when they practiced problems with a structured educational tool with playful aspects than when they practiced with an unstructured classroom standard (i.e., the number line)? Secondly, would the TD children also exhibit increased accuracy with the educational tool when compared to the number line? 42

PAGE 48

Improvement was measured by individual changes in method and answer accuracy scores on problem sets across a pre-test, 12 practice sessions, a midpoint-test completed directly before the seventh practice session, and a post-test. It was hypothesized that the children with autism would exhibit increased accuracy when practicing problems with the educational tool than with the number line. This was based on the previously presented research, in which children with autism were shown to thrive academically and socially in environments that structured their actions. However, the TD children were expected to improve regardless of the learning tool with which they were practicing. This hypothesis was grounded in the past research that showed TD children explored and subsequently learned more in non-pedagogical conditions than pedagogical ones, but learned something nonetheless in either. Method Participants The participants were 6 boys and 2 girls ( n =8), ages 6to 11-years old, residing in a city in southwest Florida. Of these children, 4 boys (ages 9to 11-years-old) were clinically diagnosed with autism spectrum disorder, as reported by a caregiver, while the remaining children (ages 6to 7-years-old) were typically developing (i.e., they were not reported as having any diagnosed physical or mental handicap or other learning disability). Participants were recruited from four classrooms in two local schools and, with additional consent from their parents, agreed to take part in a math lesson and several practice sessions. All were reported by their teachers to have experience with single-digit addition and subtraction, but not to have previous experience with the 43

PAGE 49

regrouping method nor be actively learning it in class during this study. Materials Flashcards Large, colorful flashcards were created and laminated to present one at a time problems during the tests and practice sessions (see Figure 1). For the acclimation sessions, 20 flashcards of arithmetic problems without regrouping were made (two problem sets of five addition and five subtraction questions each). For the experimental practice sessions, 50 flashcards were created: 25 double-digit addition problems with carrying, 23 double-digit subtraction problems with borrowing, and 3 double-digit subtraction problems without borrowing ("trick questions"). Problem sets of five addition and five subtraction flashcards were grouped and rotated through to ensure each participant received each flashcard the same number of times. Number line The first learning tool was a 2.75 x 16.5" number line, ranging from 0-30, that was printed and laminated. Structured educational tool The second learning tool was the educational tool with structured and playful aspects, called MatheMATics, that was created by the experimenter for the purposes of this study (see Figure 2). On a 38 x 38 blue mat, six 6.5 x 13 x 2.5 shallow, plastic bins were arranged into two columns of three bins each. A felt line and addition/subtraction signs were added to signify that these bins were the digit placeholders for double-digit arithmetic problems. Tens and ones counting blocks were purchased so the participant and experimenter could set up the problem with manipulatives (Behr, 1976). The bins imposed structure onto the arithmetic problem by delineating distinct digit spaces and place value columns. The bins also ensured that 44

PAGE 50

errant blocks would not be accidentally knocked into the wrong columns. For additional structuring of the arithmetic problems, the tens blocks and tens column bins were blue, while the ones blocks and column bins were green. Having to change the felt plus or minus sign ensured attention was also given to the type of problem (i.e., addition or subtraction). Miscellaneous Wide-ruled loose leaf paper, a pen, and a pencil were also provided as scrap materials on which to work out problems if the participant wanted. For the same reason, a dry erase marker and eraser were also provided in case the participant wished to write on the number line or flashcards. Procedure Overview A multiple baseline procedure across participants by diagnostic condition was utilized to stagger introductions to the educational tool intervention. Half of the participants in each diagnostic group were randomly assigned to one of two introduction times (Early Introduction or Late Introduction). Participants in the study followed a similar timeline: two acclimation sessions, a pre-test and lesson session, experimental practice sessions with a number line, experimental practice sessions with the structured educational tool, a practice session midpoint-test, and a post-test (see Figure 3). All sessions were held in a quiet corner of the school in the presence of a teacher. Participants received two scores on each of their pre-test, practice session, midpoint-test, and post-test problem sets: a method accuracy score and an answer accuracy score. For each individual problem, participants could earn 1 point on each 45

PAGE 51

measure. Each problem set was given a total method accuracy score and a total answer accuracy score (with a maximum of 10). Points on the method accuracy measure were awarded if the appropriate regrouping strategy (i.e., carrying or borrowing) was used to solve the problem. Points on the answer accuracy measure were awarded if the participant obtained the correct answer to the problem, regardless of strategy. Participants' scores were not averaged, allowing the experimenter to later assess microgenetic changes in performance (Fletcher et al., 1998; Siegler & Crowley, 1991). Similarly, participants' progress on method accuracy was further broken down into substeps, detailed below, that are required to perform the regrouping strategy (see Figure 4 for an example). Although these sub-steps were not scored, participants' acquisition of each sub-step was recorded to ensure a complete understanding of their progress, however small. Acclimation sessions Participants were initially informed that the study was to teach them a new mathematical concept: regrouping. Sessions 1 and 2 were 10-25 minutes long and consisted of "warm-ups" in which all participants became acclimated to the experimenter's presence. They got to know her as an authority figure and role-played how the practice sessions would go by running through the process first with the number line and then with the MatheMATics tool (processes detailed below). Only singleand double-digit addition and subtraction problems without regrouping were used in these warm-up sessions, as participants were expected to be comfortable with this level of mathematics. Pre-test and lesson plan In Session 3, all participants were administered a pre46

PAGE 52

test to ensure they did not have a previous understanding of regrouping. The pre-test was conducted by having the experimenter present 10 flashcards with regrouping (see Figure 1) one-by-one, half with double-digit addition and half with double-digit subtraction problems, to a participant while they were sitting at a table. The participant could work out the problem on the flashcard itself with a dry erase marker or with a pen or pencil on the provided scrap paper. The participant could respond orally with the answer or write it down. No learning tools were provided, nor did the experimenter offer any assistance beyond prompts to continue working. The participants had an unlimited amount of time to complete the pre-test. The 10 problems were scored for method and answer accuracy, and any qualifying events were written down by the experimenter (e.g., the participant refused to cooperate for 5 minutes or was distracted by an unexpected noise). On this pretest, participants were expected to receive a method accuracy score of 0, supporting their teacher and parent reports of no previous instruction in regrouping. After the pre-test, the experimenter spent 20-30 minutes individually teaching all of the participants the principles of place value and regrouping. The lesson plan was modeled on the advice of several teachers (see Appendix). These lessons were all accomplished by using pencil-and-paper example problems. First, place value (tens and ones) was introduced to ensure participants had a working knowledge of this concept. It was shown how to identify which values are which and that a number can be separated by these values (e.g., 18 can be 10+8). Participants were then shown why regrouping is necessary, such as when adding in the ones column produces a double-digit number. Participants also learned how to identify when a problem was going to require regrouping 47

PAGE 53

(e.g., when looking to subtract 23-5, 3 is smaller than 5). Several example problems then introduced how to regroup in several smaller substeps. For addition, it was shown how a double-digit answer cannot "fit" in the ones column, so the value in the tens-place must be carried over into the next column. For subtraction, it was shown why a larger number cannot be subtracted from a smaller number. It was then explained that by separating the tens-place numeral, as introduced earlier in the place value lesson, a ten can be borrowed from the tens column and added to the ones column. For the sub-steps, participants were encouraged to (1) draw a line between the tens and ones columns, (2) start on the ones side, (3+) if adding, add the ones column, (4+) write down the ones-place digit, (5+) carry the tens-place digit to the tens column, and finally (6+) add the tens column. However, if subtracting, after Step 2 it was necessary to (3-) check if borrowing was necessary, (4-), cross out the number in the tens column and (5-) make it 1 less, (6-) add 10 to the ones column, (7-) subtract the ones column, and finally (8-) subtract the tens column. How to mark and keep track of these changes and complete the problem was explained step-by-step in eight example problems (see Figure 4 for an example). Before all practice sessions, a brief, 5-minute lesson review was also conducted. These brief lessons included a summary of the concepts addressed in the full lesson plan: tens and ones, when to regroup, and one example of an addition and subtraction problem using the learning tool appropriate to the session. Practice sessions: Number line Depending on the participant's staggered 48

PAGE 54

introduction time, either Practice Sessions 1-3 (Early Introduction group) or 1-6 (Late Introduction group) were spent practicing regrouping with the aid of a number line (review Figure 3 for timeline). These number line sessions were meant to represent the study's more traditional learning technique, as number lines are a very common learning tool in classrooms, often found up on the classroom wall or handed out to keep for the school year. Thus, for participants to receive one-on-one instruction with the aid of a number line very closely mirrors an expected classroom setting (Schneider et al., 2008; Siegler & Ramani, 2009). Indeed, participants from one school had a large number line taped to their classroom's wall above the board, while participants at the other school had number lines that they kept in their desks. Practice sessions with the number line were conducted with both the experimenter and participant sitting at a table or on the floor. The experimenter presented each flashcard problem one-by-one and, same as the pre-test, the participant could work out the problem on the flashcard itself or scrap paper and had the option of verbal or written answers. The participants also received feedback as to whether or not they had used the regrouping strategy successfully and answered correctly, though incorrect answers were not revisited. Method and answer accuracy on the problem set were scored and qualifying events were noted. During these practice sessions with the number line, the participant was also helped by the experimenter. The experimenter provided help by suggesting the next step toward completing the problem (e.g., "Which side do you start on?") and encouraging the participant to use the number line (e.g., "You can use the number line to help you 49

PAGE 55

count."). If these prompts were not effective and a participant was unable to complete a problem on his/her own (e.g., did not know how to use the number line), the experimenter provided more specific help and the participant received method and answer accuracy scores of 0. Participants were expected to be observed using the number line in a couple different ways. For example, if the problem was 14+7, a participant might start at the numeral "14" on the number line and count up seven notches to 21. If perhaps the participants has a better grasp of regrouping, he/she might start at the numeral "4" and count up seven notches to 11. He/she would then carry over the tens-place digit of "11" to the tens column and complete the problem by adding 1+1, also arriving at correct answer of 21. As previously mentioned, these two ways in which a participant may use the number line highlight the unstructured nature of the learning tool; simply knowing how to use a number line does not ensure the participant uses it in the most beneficial way (i.e., the latter example in which the participant practiced carrying rather than in the former example, in which basic arithmetic was relied upon). Counting up from the bigger number is regarded as a less useful method because it becomes cumbersome and timeconsuming as numbers increase, not to mention that students will not alway have access to an infinite number line. Practice sessions: MatheMATics Dependent again upon the staggered introduction time, participants were introduced to the structured educational tool at either Practice Session 4 (Early Introduction) or 7 (Late Introduction), after they had completed 50

PAGE 56

their experimental number line sessions. These experimental practice sessions lasted until Practice Session 12 for all participants, regardless of introduction group (review Figure 3 for timeline). Sessions were conducted next to the MatheMATics mat on the floor or at a table, with the experimenter and participant sitting side-by-side. The experimenter presented the 10 flashcards to the participant, problem-side down, and allowed him/her to select "from the deck" which one he/she wanted to work on first. This part of the MatheMATics tool was included as the playful aspect; participants appeared to enjoy saving their favorite colored flashcards for last or trying to predict which type of problem they would select. Participants selected from the deck of flashcards until all 10 problems had been answered. After the participant selected a flashcard, the participant worked to set up the problem using the bin columns and counting blocks (see Figure 5 for an example). For example, if the problem was 24-7, two tens blocks would be placed in the upper left bin, four ones would be placed in the upper right bin, and seven ones would be placed in the middle right bin, below the "upper" four. Lastly, the felt "minus" sign would be placed to the left of the middle left bin, just above the answer line. Using this, the participant would be expected to borrow a tens block from the tens column and add it to the bin with the four ones blocks. From there, 14-7 becomes a more accessible problem, as students are expected to have more experience with arithmetic under 20 (e.g., 7+0=7, 7+1=8... 7+6=13, 7+7=14, 14-7=7, 13-7=6, etc.). Participants had the option of filling in the answer bins with either the representative counting blocks or with provided plastic numbers. The participant could 51

PAGE 57

thus fill in the ones answer bin with seven ones or the numeral "7". Returning to the tens column, the participant would then answer 1-0 and fill in the appropriate answer in the tens answer bin. Then, as in the other practice sessions, the participant had the option of a verbal or written answer. Throughout this process, the experimenter provided assistance by reminding the participant how to use the blocks and bins (e.g., "Use the counting blocks to set up the problem in the bins.") or suggesting the next step toward completing the problem (e.g., "Which side do you start on?"). As in the number line practice sessions, if a participant was unable to complete the problem after receiving these prompts, the experimenter provided more specific help and the problem received method and answer accuracy scores of 0. Participants received feedback as to whether or not their method and answer were correct, but incorrect problems were not revisited. Method and answer accuracy on the problem sets were scored and qualifying events were noted. Midpoint-test, post-test and wrap-up Regardless of Early or Late Introduction group, all participants completed a midpoint-test before experimental practice session #7 and, for the very last session, completed a post-test. These two tests were administered with a procedure identical to the pre-test: participants were presented the pre-test problem set one flashcard at a time and could not use the number line or MatheMATics tool, nor ask the experimenter for assistance. After the post-test, all participants and their teachers were thanked. Teachers, principals, and parents were also asked if they wished to hear back on the results of this study. Participants received a personalized gift that matched their interests. The 52

PAGE 58

MatheMATics tool was donated to one of the participating schools. Results The small sample size and the unique case each participant represented did not allow for inferential statistics. Participants' method and answer accuracy scores were instead analyzed as a function of both introduction time and diagnosis by their total number correct (Figures 6 and 7). Which session participants acquired each regrouping sub-step (Steps 1 through 6+ for carrying and 1 through 8for borrowing) was also analyzed by diagnosis (see Figures 8 and 9). For clarity, all participants with autism were uniquely marked by "a"'s (i.e., Participants 1a, 2a, 5a, and 6a), while the TD participants were not additionally marked (i.e, Participants 3, 4, 7, and 8). Participants could earn one point per problem for method accuracy if they correctly utilized the regrouping method (see Figures 4 and 5 for examples of carrying and borrowing, respectively). For example, if participants executed all of the appropriate sub-steps for a problem, regardless of whether their arithmetic was correct, they would earn one point toward method accuracy (i.e., they regrouped, but made an addition or subtraction error). Participants could also earn one point per problem for answer accuracy if they answered the problem correctly. For example, regardless of whether participants regrouped correctly or not, if they arrived at the correct answer, they would earn one point toward answer accuracy (i.e., they got the answer right, but may or may not have used the appropriate method). These scores were earned separately. Therefore, a participant who grasped regrouping but struggled with the arithmetic might obtain a high method accuracy score and a low answer accuracy score, whereas a participant who did 53

PAGE 59

not understand regrouping but could count well enough on his or her fingers might obtain a low method accuracy score and a high answer accuracy score. For a breakdown of participants' method and answer accuracy scores by introduction group, refer to Figures 6 and 7, respectively. Acclimation Sessions The acclimation sessions (not shown) were conducted to confirm participants' prerequisite familiarity with addition and subtraction problems. Participants completed two problem sets composed of arithmetic without regrouping using both the number line and MatheMATics tool. Method accuracy was irrelevant and thus not scored for these sessions. Only Participant 2a's answer accuracy score (0%) was cause for concern. Participant 7 also scored relatively low (50% answer accuracy), though this score was likely because of 7's attempts to do the arithmetic in his head. The remainder of participants scored from 70-100% on both sessions, confirming that the majority of participants met the prerequisite of being comfortable enough with basic arithmetic to progress on to higher-level mathematics (e.g., regrouping). Pre-test The pre-test was then used to confirm another prerequisite: participants' lack of familiarity with regrouping (i.e., a method accuracy score of 0%). Only two participants, 2a and 6a, did not score 0% on method accuracy. Both demonstrated weak familiarity with carrying: 2a carried once in the middle of the pre-test (10% of the pre-test), while 6a caught on toward the end and carried for the last two addition problems (20%). This confirms that the majority of the participants met the prerequisite for not having a 54

PAGE 60

previous understanding of regrouping, ensuring participants were relatively similar. (In the subsequent sessions, Participants 2a and 6a did not appear to have benefited from their previous experience.) Accuracy for the Early Introduction Group Number line sessions. Participants in the Early Introduction group only had three practice sessions with the number line (see Figures 6 and 7, Participants 1a, 2a, 3, and 4). Participant 1a exhibited low method and answer accuracy scores (10-40%) with the number line, as did 2a (0-20%). Participant 3 exhibited scattered scores, with method and answer accuracy fluctuating between 30-80%. Participant 4 scored highly on both method accuracy (90-100%) and answer accuracy (70-100%). MatheMATics sessions. For this Early Introduction group, the MatheMATics tool was introduced in Practice Session 4. During this first session, introduction led to variable method and answer accuracy scores. Participant 1a dropped from 40% to 30% in method accuracy, but exhibited a slight increase (from 30% to 40%) in answer accuracy. Participant 2a remained at 0% on both method and answer accuracy, while Participant 3 exhibited increased method accuracy (10% to 40%) and decreased answer accuracy (80% to 70%). Participant 4 exhibited decreases in both method (from 100% to 70%) and answer (from 90% to 60%) accuracy. Over the remaining sessions using the educational tool, 1a's method and answer accuracy scores remained low (10-50%) until they jumped to 60-90% for Practice Sessions 11 and 12. Participant 2a's method and answer accuracy scores also remained low (0-30%) until Practice Sessions 10-12, when they rose to 60-90%. Participant 3's 55

PAGE 61

method and answer accuracy scores remained scattered (40-100%), though the answer accuracy scores were generally higher. Participant 4's method and answer accuracy scores rose steadily and never dropped below 70%. By Practice Session 12, all four Early Introduction participants had method accuracy scores of at least 80% and answer accuracy scores of at least 60%. Midpointand post-tests. Both midpointand post-tests were completed by participants without the use of either learning tool or assistance from the experimenter. The Early Introduction group took their midpoint-test before Practice Session 7, but after introduction to the MatheMATics tool. All participants experienced decreased accuracy on their midpoint-test method and answer scores. Participant 1a dropped from 50% to 30% on both method and answer accuracy, while Participant 2a dropped from 10% to 0% on both as well. Participant 3 dropped from 80% to 40% on method accuracy and 70% to 40% on answer accuracy. Participant 4 dropped from 90% to 70% on method accuracy and 80% to 70% on answer accuracy. Decreases in method and answer accuracy were thus prevalent for all Early Introduction participants when a learning tool was removed. After Practice Session 12, the participants took their post-test. Method accuracy scores remained high (70-90%) for all four participants, while all answer accuracy scores decreased. Participant 1a's scores dropped from 60% to 20% and 2a's from 70% to 10%. Participant 3's scores decreased from 90% to 60%, while Participant 4's only dropped from 100% to 90%. It is important to note that while Participants 1a, 2a, and 3 continued to struggle with their arithmetic (i.e., had decreased answer accuracy scores), these participants also made great gains in method accuracy from Practice Session 1 to the 56

PAGE 62

midpoint-test to the post-test (while Participant 4 scored strongly from the beginning). Accuracy for the Late Introduction Group Number line sessions. Participants in the Late Introduction group had six rather than three practice sessions with the number line (see Figures 6 and 7, Participants 5a, 6a, 7, and 8). Method accuracy scores rose quickly and remained steadily high (90-100%) for Participants 5a and 8. Participants 6a and 7 also exhibited mostly increasing method accuracy scores, excluding decreases on Practice Session 3. Answer accuracy scores increased for Participants 5a and 7, reaching 80-100%, while Participants 6a and 8 exhibited more scattered scores (40-90%). MatheMATics sessions. Most Late Introduction group participants exhibited decreased method accuracy changes upon introduction to the MatheMATics tool in Practice Session 7. Compared to Practice Session 6, Participant 5a's method accuracy decreased from 90% to 20%, Participant 6a's decreased from 90% to 70%, Participant 7's decreased from 100% to 90%, but Participant 8's remained at 90%. However, every participant returned to the 80-100% range by the next session and maintained this through the end. Except for Participant 7 who remained at 90%, participants' answer accuracy scores also decreased after the introduction of the educational tool. Participant 5a dropped from 80-100% to 20% and never scored higher than 80% again. The other participants (6a, 7, and 8) fluctuated for the remaining sessions between 40-100%, scoring mostly from 50-90%. Midpointand post-tests. Both midpointand post-tests were completed by 57

PAGE 63

participants without the assistance of either learning tool or the experimenter. The Late Introduction group participants also took their midpoint-test before Practice Session 7, but this was before they had been introduced to the MatheMATics learning tool. Participant 7 exhibited a sharp decline to 20% from 100% on method accuracy, though the other participants maintained their previous high scores (80-100%). In a trend similar to the Early Introduction group, answer accuracy scores decreased on the midpoint-test. Participant 5a dropped from 90% to 40% and Participant 6a dropped from 70% to 50%, while Participant 7 dropped from 90% to 70% and Participant 8 dropped from 90% to 40%. All participants' method accuracy scores remained high for the post-test (70-100%), while answer accuracy scores remained moderate (50-60%). Therefore, similar to the Early Introduction group, although arithmetic continued to be a concern, as shown through decreased answer accuracy scores, all participants' method accuracy scores improved dramatically from Practice Session 1. Accuracy by Diagnosis Autism. The participants with autism exhibited highly varied method and accuracy scores (see Figures 6 and 7, Participants 1a, 2a, 5a, and 6a). While these participants' scores did not appear to fluctuate as a function of learning tool or introduction time, all appear to have mastered the regrouping method by the post-test. Participant 1a exhibited low to moderate (20-50%) method and answer accuracy scores through most of the sessions until Practice Session 11, when he improved to 80-90%; however, for answer accuracy this was short-lived, as these improvements did not hold through the post-test. 58

PAGE 64

Method accuracy remained high, though, supporting his mastery of regrouping, if not arithmetic. Participant 2a exhibited very low method and accuracy scores (10-30%) throughout the sessions until Practice Session 10, when he appears to improve suddenly. Similar to Participant 1a, method accuracy remained high for the post-test, supporting Participant 2a's mastery of regrouping, but he struggled with the arithmetic. Participants 5a exhibited high method and answer accuracy scores beginning immediately during Practice Session 1. His answer accuracy scores decreased for both the midpoint-test and Practice Session 7, the latter being his first session with the structured mat. Although he rebounded quickly, he never regained his former high answer accuracy scores, instead hovering around 60-80%. However, his method accuracy score also decreased during Practice Session 7 and then returned to 90-100%, suggesting the mat has an initial learning curve. Overall, the mat was not detrimental to his understanding of the method, but was not as helpful as the number line for completing his arithmetic. Similar to the other participants with autism, his arithmetic was not strong enough for him to complete the mathematical operations without a tool. Lastly, Participant 6a exhibited high method accuracy scores after the first couple sessions, dropping briefly during his first session with the mat, but ultimately ending high on the post-test like the other participants with autism. His answer accuracy scores, however, remained moderate (40-80%) throughout the sessions, peaking at 90% on Practice Session 10, but dropping back to 70% on the post-test. Therefore, similar to his peers, his scores support his mastery of regrouping, but he struggled with the arithmetic. 59

PAGE 65

Typically developing. TD participants' method and answer accuracy scores also did not appear to change as a function of learning tool or introduction time. Participants 4 and 8 both exhibited high method accuracy scores from the beginning (60-100%), while Participant 7 caught up quickly, just with a lower start (30-100%). Participant 3 exhibited slightly lower method accuracy scores (10-80%) until Practice Session 10, after which his scores remained high (60-100%). Answer accuracy followed a similar trend: by Practice Session 2, all participants were scoring highly (60-100%). Similar to the participants with autism, all TD participants exhibited high method accuracy scores on their post-tests, suggesting mastery of regrouping. However, they were likely to have slightly higher answer accuracy scores than participants with autism (60-90% vs. 10-70%, respectively). Sub-step Progress by Diagnosis An analysis of participants' sub-step acquisition times revealed that most participants ( n =5) understood the complete regrouping method by Practice Session 3 (see Figures 8 and 9). In these figures, color-filled marks were used to denote at which session a participant understood each sub-step for the first time (i.e., utilized it for at least 3 problems in a row without prompts from the experimenter). The marks do not signify that the participant performed the given sub-step without mistakes for the remaining sessions, but that the session was the first in which the participant demonstrated the ability to reliably perform the sub-step on his or her own. TD participants acquired all of the sub-steps by Practice Session 4, as did two of the participants with autism, 5a and 6a (see Figures 8 and 9). However, the remaining two 60

PAGE 66

participants with autism required more practice before they acquired the entire method. Participant 1a acquired all of the carrying sub-steps in the first practice session, though it took him until Practice Session 9 to acquire the last few borrowing sub-steps. Participant 2a had a slower start, acquiring the first two sub-steps (drawing the line and starting on the ones sides) by the third session, but then progressing more after his introduction to the structured mat in Practice Session 4 until Practice Session 10, when he acquired the last two borrowing sub-steps. Discussion Participants with Autism These raw results do not strongly support the introduction of a structured educational tool with playful aspects for children with autism. This study's first question asked whether the participants with autism (1a, 2a, 5a, and 6a) would exhibit increased accuracy on the problem sets for which they used the MatheMATics tool rather than the number line. It was hypothesized that participants with autism would show greater improvements on their regrouping skills (i.e., increases on their method accuracy) when practicing problem sets with the MatheMATics tool. This was based on several studies that supported the implementation of structure to improve children with autism's play and learning (see DiCarlo & Reid, 2004; Hume & Odom, 2007; Panerai et al., 2009; Sigman & Ungerer, 1984). This hypothesis was not supported by the current data. The method and answer accuracy scores of the participants with autism did not improve more dramatically immediately following the introduction of the MatheMATics tool (see Figures 6 and 7, 61

PAGE 67

blue, green, teal, and purple lines). Rather, method and answer accuracy scores appeared to have improved most dramatically at Practice Sessions 10 or 11 (Participants 1a and 2a) or to have been strong with regard to method accuracy from the first few practice sessions (Participants 5a and 6a). Ultimately, although answer accuracy scores were not as strong, all participants with autism mastered the regrouping method by the post-test. Typically Developing Participants As predicted, these findings do not strongly support the introduction of a structured educational tool with playful aspects for children without autism. The second question addressed whether the typically developing participants (3, 4, 7, and 8) would also exhibit greater increases in accuracy with the MatheMATics tool than the number line. It was hypothesized that typically developing participants would exhibit increases in method and answer accuracy irrespective of the learning tool with which they were practicing. This was based on several studies that suggested typically developing children learn more about a novel object in non-pedagogical conditions which allow them to explore, but that they learn something nonetheless, regardless of condition (see Bonawitz et al., 2009; Gweon & Schulz, 2008; Schulz & Bonawitz, 2007). This second hypothesis was thus moderately supported by the current data; typically developing participants did not appear to exhibit any increases in method and answer accuracy after their introduction to the MatheMATics tool (see Figures 6 and 7, red, yellow, maroon, and grey lines). This is potentially only a result of many typically developing participants' immediately high method and answer accuracy scores ( n =3), rather than a reflection of the learning tools with which they practiced. 62

PAGE 68

However, there is the exception of Participant 3, who instead exhibited moderate, scattered scores throughout the course of the experiment. Participant 3 provides stronger support for the second hypothesis, as his method and answer accuracy scores do not appear to be influenced by the introduction of the MatheMATics tool nor did he immediately grasp the concept like his TD peers. Therefore, his scores reflect a TD participant who required practice with a learning tool to acquire the regrouping method, but learned the method regardless of the tool with which he practiced, as hypothesized. Overall, most participants with and without autism were found either to grasp the regrouping method (>50%) after just the first three practice sessions ( n =4) or to possibly exhibit a practice effect, improving most at Practice Session 10 ( n =3). As noted previously, typically developing participants scored higher overall than participants with autism on answer accuracy. This was to be expected and is likely because of their lack of a learning disability and possible recruitment concerns, to be addressed below. Qualitative Analyses However, analyzing only these quantitative trends in method and answer accuracy scores discredits both participants' hard work and the MatheMATics tool itself. As in any situation, there are many factors to account for that could have affected accuracy at any stage of the experiment. For instance, Participant 3's scattered, moderate method accuracy scores (40-100%) can partially be attributed to his unwillingness to carry on addition problems. He was not adverse to learning how to borrow on subtraction problems and learned that method quite well, but preferred for most of the experiment to count up on his fingers rather than carry (e.g., for 18+7, Participant 3 would start at 18 63

PAGE 69

and count up seven fingers to 25). Such a distinction (personal preference vs. difficulty executing), which could not be distinguished in an overview of the raw scores, becomes readily apparent in the qualitative evaluations, which noted any unique and influential events during informal observation. A more detailed evaluation is also due for Participant 6a's post-test score(s). Although only one score was used to complete the quantitative data, 6a in fact had three scores for his post-test. This occurred when an experimenter error led to 6a initially answering the wrong problem set ("Post-test I"). Apologies were accepted and the participant was given the correct problem set to complete ("Post-test II"). However, concerns were raised when Participant 6a's method accuracy scores were lower on Posttest II than previous practice sessions would predict. Prior to Post-test II, method accuracy had been consistently at 90-100% for the previous four practice sessions, while answer accuracy had been at least above 60% (see Figures 6 and 7, teal line). During Post-test I, method and answer accuracy had been high (100% and 75%, respectively, with percentages out of an incomplete practice set of 8 problems). On Post-test II, though, method accuracy dropped to 70% and answer accuracy dropped to 60%, possibly suggesting test fatigue. Post-test III was thus conducted the following week to ensure an accurate reflection of Participant 6a's skills were obtained. For Post-test III, method accuracy increased to 90%, while answer accuracy increased to 70%. It was felt that these scores presented a more accurate, uninterrupted picture of Participant 6a's regrouping ability, so it was these scores that were used for all analyses. 64

PAGE 70

Sub-steps As noted, the regrouping method required several sub-steps, including drawing the line, starting on the ones column, carrying or borrowing if needed, and not forgetting to add the tens column, too (see Figure 4). To receive a point for method accuracy, participants had to successfully complete all of these sub-steps. Participants could thus be making incremental progress on the sub-steps that would not be adequately captured by the method accuracy score. Therefore, a microgenetic, qualitative evaluation also shines a positive light on Participant 2a's progress that was not clear in the raw method accuracy scores (see Figure 6, green line). Participant 2a initially struggled with the number line, often failing to use it, mixing up his addition and subtraction signs, and counting chaotically (i.e., sometimes accurately, sometimes horizontally across the top numerals, or sometimes counting the same number twice). Participant 2a was able to acquire Steps 1 and 2 (drawing the line and starting on the ones side) during his last practice session with the number line (see Figure 8). However, after the introduction of the MatheMATics tool, 2a began to make gradual progress with the unique steps for carrying (Steps 3+ through 6+) and borrowing (Steps 3through 8-). After the first session with the mat, Participant 2a began to learn the initial sub-steps for borrowing. After the third session with the mat, he was even using the correct terminology (e.g., "draw the line", "start on the ones side", "carry", "borrow", etc.) at the appropriate times and beginning to learn the sub-steps involved in carrying. Thus, the mat appeared to help him structure the several sub-steps and tackle them one-by-one, whereas with the number line he had appeared completely at a loss. Setting 65

PAGE 71

up the mat required him to switch out the plus and minus signs, which he did diligently; he no longer confused addition and subtraction, as he had with the number line. His progress was therefore much more steady than Figures 6 and 7 suggest, as 2a slowly but surely mastered individual sub-steps, culminating in his mastery of the whole method by Practice Session 10 (see Figure 8). This gradual progress finally resulted in achieving a method accuracy score that was more reflective of his actual progress. While Participant 1a also acquired some of the sub-steps while practicing with the structured mat, it is clear this was not a result of his introduction to it in Practice Session 4. Most of the sub-steps were acquired prior to introduction and it was not until well after introduction (Practice Session 9) that he acquired the few remaining sub-steps (see Figure 8). Strengths of the Current Study Play: Tossing. Participant 2a appears to be the only participant who clearly benefited academically from the MatheMATics tool, but participants were able to benefit unexpectedly from it in other ways, too. For instance, although many participants ( n =4) unfortunately chose to use the mat only to assist with their counting or not to use it at all, this may have been because the mat was too big for the participants. Especially the younger ones, such as Participants 4, 7, and 8, would have to get up and walk around the mat to reach the upper bins. However, from this design flaw came an unexpected strength: Participants 1a and 5a would toss the counting blocks into the bins, rather than try and reach across. For Participant 1a, this unfortunately became inappropriate play. He would not aim, expected the experimenter to fix the blocks when he missed, and also 66

PAGE 72

haphazardly threw any blocks he wasn't using. However, after a couple sessions, Participant 5a turned tossing the blocks into a game, taking his time to aim and smiling when the blocks made it into the correct bin. Play: Color choice. Many of the participants ( n =5) also enjoyed being able to choose their next flashcard; they had fun picking their favorite colors first or saving them for last, or making up strategies to try to predict which type of problem they'd select. Thus, it appears the playful aspects incorporated into the procedure to encourage interest were successful, with participants even creating their own games. However, the most enjoyed playful aspect (flashcard colors) was not an inherent part of the mat itself. A revised version of the MatheMATics tool should seek to directly incorporate more playful aspects, as this may encourage more participants to utilize it more and perhaps master regrouping sooner. Limitations of the Current Study Convenience sampling: Participants with autism The convenience sampling was a necessary aspect of the method because of the time and resource limitations of a smaller study. With regard to the participants with autism, three of the four (1a, 5a, and 6a) had potentially less severe diagnoses than Participant 2a. They did not seem to have difficulty making eye contact or conversing with the experimenter. Only Participant 2a exhibited an obsession with certain topics, a need for routine during the practice sessions, and language deficits (e.g., weak grammar, nonsensical questions). Coincidentally, Participant 2a was also the sole participant to benefit from the introduction to the structured MatheMATics tool. 67

PAGE 73

Although the original hypotheses did not address the potential for participants to be diagnosed with varying levels of severity, the background research acknowledged that different diagnoses on the spectrum can present uniquely and may require tailored educational approaches (Chiang & Lin, 2007). Furthermore, the research of Kok et al. (2002) found that the benefits of structured versus facilitated play interventions appear to vary based on the individual child. The current study's sample was too small to support the possibility that perhaps students with less severe autism do not benefit from the same amount of structure as students with classic autism, but it is a hypothesis worthy of further research. Additionally, participants with autism also had more difficulty with independent arithmetic (i.e., adding and subtracting on their own without a tool), which had been a desired prerequisite of all participants. The MatheMATics tool was primarily designed to help participants understand regrouping as a concept, rather than to serve as a crutch while adding and subtracting. The prerequisite was thus meant to ensure all participants were at a similar mathematical level (for comparison) and ready to advance to higher level mathematics. Their difficulty with the arithmetic is most easily seen in the discrepancies between Practice Session 12 and post-test method and answer accuracy scores for Participants 1a and 2a: method accuracy scores remained stable, while answer accuracy scores dropped severely (see Figures 6 and 7, blue and green lines). This arithmetic prerequisite, as well as the prerequisite that participants not have any prior exposure to regrouping (as broken by Participants 2a and 6a), were excused because of the small scale of the current study. As previously noted, however, previous 68

PAGE 74

experience with regrouping does not seem to have positively affected the scores of these participants, as Participant 2a did not fully grasp regrouping until Practice Session 10, while Participant 6a's scores were similar to 5a's, an unexperienced peer (see Figures 6 and 7, green, purple, and teal lines). Participants 2a and 6a thus did not have an advantage over the others; their pre-test method accuracy scores (10% and 20%, respectively) appear to have been the vestigial result of past exposure to regrouping when attending an old school. Convenience sampling: Typically developing participants. Furthermore, there is reason to suspect the participants in the control group (Participants 3, 4, 7, and 8) were also not representative of their population. One of the teachers once remarked to the experimenter that, "Aren't [the participants] great! Did we pick good ones for you?" This remark, coupled with informal observations of the classrooms and the higher answer accuracy scores of the typically developing participants, lends credibility to the possibility that the control participants were of above-average cognitive intelligence. For instance, Participants 4 and 8 scored at least 70%, more often above 80%, on method accuracy and at least 60%, more often above 70%, on answer accuracy. Therefore, the current study's findings may be partially explained by an unrepresentative sample. First, the experimental group may not have accurately represented the autistic population, with the exception of Participant 2a. Participant 2a most clearly met the diagnostic criteria for the experimental group and appeared to be the one participant to benefit from the MatheMATics tool, as hypothesized. Second, the control group may also not have been representative of a typically developing population, 69

PAGE 75

assuming their quick progress can be explained by their potentially above-average intelligence; such students would have less need for a practice tool. Trick questions. A caveat of the current study was the sporadic use of "trick" questions (i.e., questions for which regrouping was not required, such as 27-6). Trick questions were included only in some practice sets. Although the lesson plan and brief reviews emphasized how to know when to borrow and carry, the lack of constant practice with trick questions appears to have been detrimental, as all participants missed these questions at some point. Some participants ( n =3) even exhibited an unexpected trend of only missing these questions on later practice sessions, when they presumably should have had a better understanding of the concept. Based on these results, it is unfortunately likely that participants only memorized the steps required to regroup, rather than obtained a complete understanding of the concept itself. Future Directions These findings, as well as the observed strengths and weaknesses, aid in guiding future research. The MatheMATics tool appears to have been successful in providing structure while learning, as well as engaging students through play; however, several improvements could be made to the next design. A new mat could be smaller and include more playful aspects, such as actively encouraging students to aim for the bins and allowing them to choose their counting block colors (as that was successful for the flashcards). Based on the previous research of Bailey and Watson (1998) and Siegler and Ramani (2009), turning the practice sessions into more of a game may also be beneficial. 70

PAGE 76

Students could keep track of their method and answer accuracy progress on a scoreboard or linear board game. By specifically emphasizing the importance and value of using the correct method through a score system, perhaps students such as Participant 3 would have been more willing to practice regrouping rather than basic arithmetic. Related to the problem sets, the practice questions should be changed to include more trick questions, thus highlighting the importance of always checking whether or not regrouping was necessary on each particular problem. This would help prevent rote memorization and hopefully lead to a better grasp of regrouping as a concept. Future research should also modify the recruiting procedure to include more stringent participant selection criteria. Based on comments made at one school, it appears that teachers were not given stringent enough guidelines. Teachers at the typically developing participants' school gave the impression that they hand-picked students who they thought would succeed, while the experimenter had hoped for a more representative control sample (i.e., students of average intelligence). For the teachers at the autistic school, more stringent guidelines would not have been very useful; the participants with autism were selected from a small population in which the other students were either underor over-qualified (i.e., not ready to progress to higher level mathematics or already capable of regrouping). A future study with greater time and resources should seek to recruit from a wider pool and include a more diverse sample. Conclusions Ultimately, the current study presented a strong case study method of assessment for a newly-designed structured educational tool. Utilizing a microgenetic method was 71

PAGE 77

extraordinarily beneficial, as it allows for examination of the mechanisms and minute changes that lead to an end result (i.e., the successful application of regrouping). In this study, the method was able to highlight the progress of a participant who would have otherwise gone unnoticed in a sea of averages and post-tests. This participant in particular provided strong support for the effectiveness of the MatheMATics tool. As the participant who struggled the most to grasp regrouping, the data suggest that the introduction of a structured tool was very critical to the participant's eventual understanding of the concept. This participant and many others also delighted in the incorporated playful aspects, supporting the integration of play and learning to prompt interest and encourage practice. Although other participants' results do not strongly support the MatheMATics tool, these findings are likely not a fault of the educational tool itself, but rather procedural limitations that are unfortunately common in small scale studies. Also, regardless of answer accuracy (which was affected by weak arithmetic skills), practice effects, or participant recruitment concerns, it cannot be understated that at the time of the post-test, all participants had mastered the method. Teaching any student, with or without a learning disability, can be a difficult process. Students with autism present particular difficulties, as their learning styles differ from the typical population. The current study found that structure and play may be beneficial factors to incorporate into lessons and practice. It is hoped that future research will continue in this vein, testing commercially available educational toys and designing new tools that may promote learning and engagement. 72

PAGE 78

Appendix Lesson Plan Place Value Introduce each number having two sides or "columns", the ones and tens columns. The right side is the ones column, with small numbers like 0-9. The left side is the tens column, with bigger numbers up to the 90s. Look at the number 18. The number 18 has 1 ten and 8 ones. Prove it by showing 10 + 8 = 18. Go over three more double-digit numbers, having the student identify the digit in the tens and ones place: 18, 24, 06. If the student struggled, go over three more: 52, 91, 37. Regrouping: Addition Introduce regrouping as an easy way to add and subtract bigger problems by using the tens and ones columns to make smaller problems. Suggest that you and the student dive right in! Set up the problem 24 + 7 vertically. Have the student again identify the digits in the tens and ones columns. Draw a line between the two columns, draw two spaces for the answer below. Explain that you always start on the ones side: have the student sum 4+7 and write the answer off to the side. Explain how 11 is too big to fit under the ones column, so you keep the digit in the ones place (in this case, "1") and put that down on the ones answer space and then carry the digit in the tens place (again a "1") above the "2" in the tens column. Then show the rest is as simple as adding 1 + 2! Do three more addition problems with the student: 13+7, 29+2, 18+8. Drill in the 73

PAGE 79

steps: first you draw the line, you always start with the ones column, you see if the answer fits, you carry the tens digit, you finish the problem. If the student seems to understand, have him/her set up and complete a problem. Regrouping: Subtraction Set up the problem 23 5 vertically and have the student identify the digits in the tens and ones columns. Suggest that subtracting is just as easy and starts off the same as addition: have the student draw the line and start with the ones column. Show that it is impossible to take away 5 of something when you only have 3! Explain that it's okay because you can borrow from the tens column. Remind them that the "2" is actually two tens. Cross out the 2, leave 1 ten, and have the 3 borrow the other ten. Show that now your new problem is 13-5, which is doable. Don't let them forget to finish the problem and subtract the tens column! Do three more subtraction problems: 26-7, 24-9, 22-5. Again, drill the steps in simple terms: draw the line, start with the ones column, see if you can subtract, borrow from the tens column, subtract from both sides. As with addition, if the students seems to understand, allow him/her to set up and complete a problem. Conclusion of the Lesson Reiterate that regrouping makes it easier! Most students are attached to their current way of doing things and reluctant to abandon their method for something new/ weird. Suggest that regrouping helps them make fewer mistakes and/or that they will not always have a number line available for bigger problems. Repeat when you regroup: when adding, you need to carry if the answer doesn't fit; when subtracting, you need to borrow if the bottom digit is too big. 74

PAGE 80

References American Psychiatric Association. (2000). "Pervasive developmental disorders." In Diagnostic and statistical manual of mental disorders (4th ed., text revision). Washington, DC: American Psychiatric Association, 69-70. Azmitia, M. (1988). Peer interaction and problem solving: When are two heads better than one? Child Development, 59 87-96. Bailey, S. & Watson, R. (1998). Establishing basic ecological understanding in younger pupils: A pilot evaluation of a strategy based on drama/role play. International Journal of Science Education, 20 139-152. Behr, M. J. (1976). Teaching experiment: The effect of manipulatives in second graders' learning of mathematics, volume I. (PMDC Technical Report No. 11). Tallahassee, FL: Florida State University, Project for the Mathematical Development of Children. (ERIC Document No. ED144809). Berk, L. E., Mann, T. D., & Ogan, A. T. (2006). Make-believe play: Weillspring from development of self-regulation. In D. G. Singer, R. M. Golinkoff, & K. Hirsh-Pasek (Eds.), Play = learning: How play motivates and enhances children's cognitive and social-emotional growth (pp. 74-100). Oxford: Oxford University Press. Bonawitz, E., Shafto, P., Gweon, H., Chang, I., Katz, S., & Schulz, L. (2009). The double-edged sword of pedagogy: Modeling the effect of pedagogical contexts on preschoolers' exploratory play. Proceedings of the Thirty-first Cognitive Science Society, Amsterdam, Netherlands. Brodin, J. (1999). Play in children with severe multiple disabilities: Play with toys a 75

PAGE 81

review. International Journal of Disability, Development and Education, 46 25-34. Bryant, B. R., & Bryant, D. P. (2008). Introduction to the special series: Mathematics and learning disabilities. Learning Disability Quarterly, 31 3-8. Burns, S. M., & Brainerd, C.J. (1979). Effects of constructive and dramatic play on perspective-taking in very young children. Developmental Psychology, 15 512-521. Chiang, H., & Lin, Y. (2007). Mathematical ability of students with Asperger syndrome and high-functional autism: A review of literature. Autism, 11 547-556. Cihak, D. F., & Grim, J. (2008). Teaching students with autism spectrum disorder and moderate intellectual disabilities to use counting-on strategies to enhance independent purchasing skills. Research in Autism Spectrum Disorders, 2 716-727. Dempsey, I., & Foreman, P. (2001). A review of educational approaches for individuals with autism. International Journal of Disability, Development and Education, 48 103-116. DiCarlo, C. F., & Reid, D. H. (2004). Increasing pretend toy play of toddlers with disabilities in an inclusive setting. Journal of Applied Behavior Analysis, 37 197-207. El-Ghoroury, N. H., & Romanczyk, R. G. (1999). Play interactions of family members towards children with autism. Journal of Autism and Developmental Disorders, 29 249-258. Fenie, D. (1988). The nature of children's play ERIC Digest. Retrieved from ERIC 76

PAGE 82

database. (ED307967) Fisher, K. R., Hirsh-Pasek, K., Golinkoff, R.M., & Gryfe, S. G. (2008). Conceptual split? Parents' and experts' perceptions of play in the 21st century. Journal of Applied Developmental Psychology, 2 9, 305-316. Fletcher, K. L., Huffman, L. F., Bray, N. W., & Grupe, L. A. (1998). The use of the microgenetic method with children with disabilities: Discovering competence. Early Education and Development, 9 357-373. Geiger, D. M., Smith, D. T., & Creaghead, N. A. (2002). Parent and professional agreement on cognitive level of children with autism. Journal of Autism and Developmental Disorders, 32 307-312. Ginsburg, H. P. (2006). Mathematical play and playful mathematics: A guide for early education. In Singer, D. G., Golinkoff, R. M., & Hirsh-Pasek, K. (Eds.), Play=Learning (pp.145-165). New York: Oxford University Press. Gweon, H., & Schulz, L. E. (2008, July). Stretching to learn: Ambiguous evidence and variability in preschoolers' exploratory play. Poster session presented at the annual meeting of the Cognitive Science Society, Washington, DC. Hamm, E. M., Mistrett, S. G., & Ruffino, A. G. (2006). Play outcomes and satisfaction with toys and technology of young children with special needs. Journal of Special Education Technology, 21 29-35. Hayter, S., Scott, E., McLaughlin, T.F., & Weber, K. P. (2007). The use of a modified direct instruction flashcard system with two high school students with developmental disabilities. Journal of Developmental and Physical Disabilities, 19 409-415. 77

PAGE 83

Hogle, J. D. (1996). Considering games as cognitive tools: In search of effective "edutainment". Unpublished manuscript. Holmes, E., & Willoughby, T. (2005). Play behavior of children with autism spectrum disorder. Journal of Intellectual and Developmental Disability, 30 156-164. Hsieh, H. (2008). Effects of ordinary and adaptive toys on pre-school children with developmental disabilities. Research in Developmental Disabilities, 29 459-456. Hume, K., & Odom, S. (2007). Effects of an individual work system on the independent functioning of students with autism. Journal of Autism and Developmental Disorders, 37 1166-1180. Ivory, J. J., & McCollum, J. A. (1999). Effects of social and isolate toys on special play in an inclusive setting. Journal of Special Education, 32 238-243. Johnson, J. E., Ershler, J., & Lawton, J. T. (1982). Intellective correlates of preschoolers' spontaneous play. The Journal of General Psychology, 106 115-122. Jones, K. D., Casado, M., & Robinson, E. H., III (2003). Structured play therapy: A model for choosing topics and activities. International Journal of Play Therapy, 12 31-47. Joseph, R. M., Tager-Flusberg, H., & Lord, C. (2002). Cognitive profiles and social-communicative functioning in children with autism spectrum disorder. Journal of Child Psychology and Psychiatry, 43 807-821. Knight, B. A. (1999). Towards inclusion of students with special educational needs in the regular classroom. Support for Learning, 14 3-7. Kok, A. J., Kong, T. W., & Bernard-Opitz, V. (2002). A comparison of the effects of structured play and facilitated play approaches on preschoolers with autism. 78

PAGE 84

Autism, 6 181-196. Libby, S., Powell, S., Messer, D., & Jordan, R. (1998). Spontaneous play in children with autism: A reappraisal. Journal of Autism and Developmental Disorders, 28 487-497. Malone, D. M., & Langone, J. (1999). Teaching object-related play skills to children with developmental concerns. International Journal of Disability, Development and Education, 46 325-336. Malone, D. M., & Stoneman, Z. (1995). Methodological issues in studying the toy play of young children with mental retardation. Topics in Early Childhood Special Education, 15 459-487. Mastrangelo, S. (2009). Play and the child with autism spectrum disorder: From possibilities to practice. International Journal of Play Therapy, 18 13-30. Panerai, S., Zingale, M., Trubia, G., Finocchiaro, M., Zuccarello, R., Ferri, R., et al. (2009). Special education versus inclusive education: The role of the TEACCH program. Journal of Autism and Developmental Disorders, 39 874-882. Parish-Morris, J., Hirsh-Pasek, K., Golinkoff, R. M., & Collins, M. (2009). The changing landscape of bookreading: Emergent literacy in the era of electronic books. Unpublished manuscript. Plato. (1969). Plato in twelve volumes (Vols. 5-6) (P. Shorey, Trans.). Cambridge, MA: Harvard University Press. play. (2010). In Merriam-Webster Online Dictionary. Retrieved February 11, 2010, from http://www.merriam-webster.com/dictionary/play Reid, D. H., DiCarlo, C. F., Schepis, M. M., Hawkins, J., & Stricklin, S. B. (2003). Observational assessment of toy preferences among young children with 79

PAGE 85

disabilities in inclusive settings: Efficiency analysis and comparison with staff opinion. Behavior Modification, 27 233-250. Schneider, M., Heine, A., Thaler, V., Torbeyns, J., De Smedt, B., Verschaffel, L., et al. (2008). A validation of eye movements as a measure of elementary school children's developing number sense. Cognitive Development, 23 409-422. Schulz, L. E. & Bonawitz, E. B. (2007). Serious fun: Preschoolers play more when evidence is confounded. Developmental Psychology, 43 1045-1050. Seitinger, S., Popovic, M., Sylvan, E., Zuckerman, O., & Zuckerman, O. (2006, April). A new playground experience: Going digital? Poster session presented at the annual meeting of the Conference on Human Factors in Computing Systems, Montreal, Quebec. Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for studying cognitive development. American Psychologist, 6 606-620. Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games -but not circular ones -improves low-income preschoolers' numerical understanding. Journal of Educational Psychology, 101 545-560. Sigafoos, J., Roberts-Pennell, D., & Graves, D. (1999). Longitudinal assessment of play and adaptive behavior in young children with developmental disabilities. Research in Developmental Disabilities, 20 147-162. Sigman, M., & Ungerer, J. A. (1984). Cognitive and language skills in autistic, mentally retarded, and normal children. Developmental Psychology, 20 293-302. Smith, P. K., & Vollstedt, R. (1985). On defining play: An empirical study of the relationship between play and various play criteria. Child Development, 56 80

PAGE 86

1042-1050. Sutton-Smith, B. (1975). The useless made useful: Play as variability training. The School Review, 83, 197-214. Thomas, N, & Smith, C. (2004). Developing play skills in children with autistic spectrum disorders. Educational Psychology in Practice, 20 195-206. Vygotsky, L. (1933). Play and its roles in the mental development of the child. Soviet Psychology, 5 6-18. Wing, L., Gould, J., Yeates, S. R., & Brierley, L. M. (1977). Symbolic play in severely mentally retarded and in autistic children. Journal of Child Psychology and Psychiatry, 18 167-178. Wong, W., Uribe-Zarain, X., Golinkoff, R., Fisher, K., & Hirsh-Pasek, K. (2008, July). Parents' views of the benefits claimed in educational toy advertising. Poster session presented at the annual conference of Interaction Design and Children, Chicago, IL. Zigler, E.F., & Bishop-Josef, S. J. (2006). The cognitive child versus the whole child: Lessons from 40 years of Head Start. In Singer, D. G., Golinkoff, R. M., & HirshPasek, K. (Eds.), Play=Learning (pp. 15-35). New York: Oxford University Press. 81

PAGE 87

Figure 1. 82

PAGE 88

Figure 2. 83

PAGE 89

Figure 3. 84

PAGE 90

Figure 4. 4a. 4b. 4c. 85

PAGE 91

Figure 5. 5a. 5b. 5c. 86

PAGE 92

Figure 6. 87

PAGE 93

Figure 7. 88

PAGE 94

Figure 8. 89

PAGE 95

Figure 9. 90


Facebook Twitter YouTube Regulations - Careers - Contact UsA-Z Index - Google+

New College of Florida  •  5800 Bay Shore Road  •  Sarasota, FL 34243  •  (941) 487-5000