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Using Research-based Strategies to Teach Algebraic Problem Solving Skills
to Students with Autism Spectrum Disorders
Shannon Cleary and Juliet Hart Barnet
Mary Lou Fulton Teachers College, Barrett, the Honors College;
Arizona State University
Introduction
Educating students with autism spectrum disorders (ASD)
in inclusive classroom settings is becoming increasingly widespread
(Cihak, Fahrenkrog, Ayres, and Smith, 2010). Placement of students on the
spectrum in general education has increased more rapidly than all other
disability categories combined (Sansoti & Powell-Smith, 2008). Currently,
36 percent of students with ASD spend more than 80 percent of their
school day in general education classes, which constitutes a substantial
increase from their 4.8 percent inclusion rate in 1991 (U.S. Department of
Education, 2011; Whitby, 2013). Therefore, while students’ individual IEP
goals and needs still drive special education services, a growing
expectation that these learners will access and acquire the same core
curricular content as their typically developing peers ushers in an
associated demand for effective educational interventions to support
students’ successful content learning (Knight, Smith, Spooner, & Browder,
2012).
Copyright © 2015 Shannon Cleary & Juliet Hart Barnet: [email protected]
58
However, instructional programming for students with ASD tends
to concentrate on communication and social skills (Plavnik & Ferreri, 2011;
Wang & Spillane, 2009) as well as functional and life skills, as opposed to
established age and grade level academic subjects (Cihak & Grim, 2008;
Rayner, 2011). Research related to academic content for students with
ASD is mostly related to literacy development (Bouck, Satsangi, TaberDoughty, & Courtney, 2014; Delano, 2007). Despite mathematics
education being a national priority for all students (Ellis & Berry, 2005),
there are fewer equivalent, empirically supported interventions in
mathematics than in reading for students with ASD (Bouck et al., 2014).
However, mathematics is a topic of academic concern as 23 percent
of students with high-functioning ASD also meet criteria for mathematics
learning disability (Dickerson Mayes & Calhoun, 2008). Although many of
these students maintain satisfactory mathematics achievement in the
earlier grades when rote memorization of facts and procedures is essential
(Chiang & Lin, 2007), the same students may be challenged once they
enter middle and high school when the subject matter becomes more
abstract and cognitively complex. Further, students with ASD may
experience difficulty when the curriculum emphasizes problem solving,
higher level thinking, and mathematical reasoning, which are cited by
researchers as weakness areas for children and youth with ASD (Mayes &
Calhoun, 2003; Whitby & Mancil, 2009).
The difficulties students with ASD encounter in mathematics may
be a result of deficits in executive functioning, including planning,
organization, working memory, attention, self-monitoring, and impulse
control (Happe, Booth, Charlton, & Hughes, 2006; Hughes, Russell, &
Robbins, 1994). Moreover, the communication impairment associated with
ASD may also negatively influence mathematics development across
several areas such as number-word sequence, calculation, fact retrieval,
and in particular, problem-solving (Donlan, 2007; Zentall, 2007).
Notwithstanding, the mathematics general education curriculum and state
assessments increasingly stress the importance of building the conceptual
understanding and problem solving defined and recommended by the
National Council for Teachers of Mathematics (NCTM) across domain
59
areas (Bottge, 2001; NCTM, 2002; Rockwell et al., 2013), which students
with ASD who are taught in general education will likely encounter.
With respect to higher-order mathematics such as algebra,
successful transition from concrete arithmetic to symbolic mathematics
and abstract reasoning becomes critical (National Governors Association
Center for Best Practices & Council of Chief State School Officers, 2010)
which may pose difficulties for students with ASD. For example, students
are introduced to complex story problems that require understanding of
foundational concepts such as a ‘placeholder’ (i.e. a variable) and its use in
equations. This conceptual knowledge must then be applied to translating
text problems into an algebraic expression in order to produce a solution
(Common Core Standards at www.corestandards.org/assets/CCSSI_Mathematics).
On the other hand, algebra also involves pictorial representations, pattern
recognition and numeric problem solving that may advantage students
with ASD who possess strengths in processing visual objects (Iuculano, et
al., 2013). As such, strategies to facilitate algebraic learning for students
with ASD should account for both students’ strengths and weaknesses.
Given the rise of students with ASD in inclusive classrooms, the
students’ documented challenges in mathematical understanding and
problem solving, and the importance the NCTM has placed on conceptual
understanding and problem solving across skill areas, there is a need for
practical and easily implemented research-based strategies to assist
teachers in addressing mathematical deficits in their students with ASD.
Herein we present how three research-based practices: self-management,
visual supports, and peer-mediated instruction can be implemented in the
context of teaching higher-order mathematics skills. Utilizing algebraic
problem solving specifically, we demonstrate how teachers of students
with ASD can apply research-based practices so that their students can
more readily acquire the mathematic skills identified on their IEPs and
those being taught in the general education classroom.
Research-based Teaching Strategies
Three evidence-based practices implemented with students with
ASD have specific application to teaching higher-order, algebraic
60
skills. Self-management, visual supports, and peer-mediated instruction
were selected, based on the empirical evidence and projected success in
helping students with ASD achieve higher-level mathematics skills.
Table 1: summary of evidence-based practices
Table 1 presents a summary of each evidence-based practice, rationale
for use with students with ASD, and examples of studies from the
experimental literature to support use of each approach in algebraic
teaching. It is important to note that research on the use of these strategies
in the area of mathematics is limited to students with learning disabilities
(LD), as empirical research in mathematics for students with ASD is still in
its infancy and has not tended to focus on higher-order skills such as those
taught in Algebra (Hart & Cleary, in press). Nonetheless, we make use of
61
the LD research since it is estimated that a substantial proportion of
students with ASD also contend with mathematics learning disability.
Therefore, evidence-based practices from this related field may be helpful
in addressing learners with ASD who display similar mathematical
difficulties.
Each of these instructional methods was included in the recent
Evidence-Based Practices for Children, Youth, and Young Adults with
Autism Spectrum Disorder meta-analytic literature review, which was
published by the National Professional Development Center on Autism
Spectrum Disorders (Wong, et al., 2014). This review examined 29,105
peer-reviewed articles published in journals between 1990 and 2011, and
resulted in close review of 456 intervention articles based on inclusion
criteria for research rigor and quality established by the National
Professional Development Center on Autism Spectrum Disorders (Wong,
et al., 2014).
According to the Common Core State Standards, all high school
students should master algebraic problem solving skills (Common Core,
2014). Many of the mathematics Common Core State Standards for high
school students address algebraic reasoning with equations and
inequalities (Common Core, 2014). Particularly, one of these standards
states that students should be able to solve one-variable linear equations
(Common Core, 2014). In order to effectively teach students with ASD to
solve linear equations with one variable, three selected teaching strategies
will be illustrated and applied to solve the example problem, 2x + 8 = 20.
Visual Supports
Use of visual supports is a common evidence based practice for
aiding students to grasp mathematical concepts and ideas. Visual
supports comprised “concrete cues that provide information about an
activity, routine, or expectation and/or support skill demonstration”
(Wong, et al., 2014). Incorporating visuals and other concrete supports
assists individuals with ASD whose strengths include processing visual
and/or written information (Marans, Rubin, & Laurent, 2005). These
supports are beneficial to students with ASD because they provide “cues”
62
to aid students when performing academic tasks (Wong, et al., 2014). This
form of visual instruction can help guide students with ASD through the
process of completing a simple algebraic problem.
Creating an acronym to which students can refer during classroom
activities and independent problem solving effectively assists students as
they acquire algebraic skills. Further, creating a picture or graphic in
conjunction with an acronym is a useful way for teachers to utilize visual
supports to enhance the learning of challenging mathematic concepts,
including algebra, as doing so builds the conceptual understanding
necessary for success at the symbolic or abstract level (Strickland &
Maccini, 2010). Graphic and pictorial representations will visually support
students in memorizing ideas and steps, helping them to be successful in
remembering concepts long term. Furthermore, visual cues can augment
receptive/expressive communication during an activity (Mirenda &
Erikson, 2000; Quill, 1995), and increase independence in a given task
among students with ASD (Koegel, Koegel, & Parks, 1995; Quill, 1995).
Implementing Visual Supports in Algebraic Problem Solving
When teaching students with ASD how to solve algebraic
problems, an acronym that represents the steps of solving this type of
mathematics problem is key. Acronyms are often used to aid students in
the understanding of challenging mathematical concepts. For example,
PEMDAS, which represents the order of operations and signifies
Parentheses, Exponents, Multiplication or Division, and Addition or
Subtraction, is used when solving equations (Golembo, 2000). The derived
acronym for solving one-variable linear equations is COSMIC. This
acronym is based on traditionally accepted algebraic problem-solving
procedures in the mathematics literature (Kieran, 2006). The steps
represented by each letter of the acronym are described by the visual aid
shown in Figure 1.
63
Figure 1: Visual support acronym for algebraic problem solving steps
After students are introduced to the acronym, they can initially be
allowed to utilize the figure as a visual support reference that can cue
them to the required steps of solving one-variable linear equations.
COSMIC is also a helpful mnemonic device that can assist students in
recalling the specific steps of the problem-solving process. Mnemonic
devices are learning strategies that enhance memory and improve recall.
They may typically come in the form of acronyms, pictures, or key words
intended to be easier to remember than the word or concept they stand for
(Hart & Whalon, 2008). Students can check off each step as it is completed
in order to observe and monitor their progress in solving the problem,
maintain attention to the task, and reach their goal.
Self-Management
Self-management involves teaching learners to independently
regulate their own behavior (Wong, et al., 2014). Self-management
strategies can be designed to teach children with ASD to monitor their
academic engagement, participation, and performance during the learning
task. Self-management teaches students to identify appropriate behaviors,
record their own behavior, and reward themselves for performing the
appropriate behavior (Hart & Whalon, 2008). Self-management can reduce
dependence on adults (e.g., the teacher or para-professional) thereby
64
increasing independence and generalization of the skills to other settings
(Simpson & Otten, 2005).
By promoting self-monitoring and reinforcement, the selfmanagement strategy can help prompt a student with ASD to achieve
successful behavior (Wong, et al., 2014). Providing a checklist for algebraic
problem solving can be a beneficial form of self-monitoring as students
can measure their progress while determining the problem’s solution,
thus decreasing student dependence in independent or group activities.
Checklists may include such concrete supports as a written cue or visual
that explicitly depicts the interactions or steps needed to complete
activities (Wilkinson, 2008). Checklists can be altered to reflect various
student roles needed in different content areas/activities, facilitate the
learning of an individual goal/skill, and capitalize on student strengths.
Implementing Self-Management in Algebraic Problem Solving
When teaching algebraic skills, self-monitoring checklists can be
implemented by requiring students to follow a step-by-step procedure for
executing this specific type of mathematics problem. To implement selfmanagement, teachers can create a numbered checklist of steps with a
stated final objective, clearly laying out the procedure for students to
follow. Procedural steps are progressively introduced to students one at a
time until the student is able to complete all problem steps. This process
allows students to master the understanding of each individual step, and
to gradually recognize the procedures and patterns needed to solve an
algebraic problem. By providing a self-management checklist, the student
can be successfully guided through each step without requiring the direct
intervention of a teacher and can promote overall learner autonomy
(Myles & Simpson, 2003). Additionally, having a concrete list of steps
allows students to track their progress in completing a problem. This list
increases concentration and attention of students because they are aware
of the goal to be achieved and can visually see which behavioral steps
need to be performed in order to accomplish the objective (Busick &
Neitzel, 2009). Moreover, the activity of focusing attention on one’s own
actions and the self-recording of these observations can have a positive
65
reactive effect on the task being monitored (Cole, Marder, & McCann,
2000).
As students memorize the checklist through practice, it can
gradually be faded until the students can successfully complete the
problems independently. This checklist tool can be effective in teaching
self-management, which is a critical life skill for students with ASD to
reach because it serves as an organization tool and as a means to enhance
students’ quality of life by empowering them to direct their own behavior
(Lee, Simpson, & Shogren, 2007).
Figure 2: Algebraic Problem-Solving Checklist
Once the students have demonstrated problem-solving mastery when
using the visual aid for support, the aid will gradually be removed,
requiring students to memorize the six listed steps required for problem
solving. Prior to beginning each question, students can be taught to write
down the six problem solving steps represented by the acronym. Teachers
can also provide students with a checklist of directions to accompany
66
problem-solving (see Figure 2). By having these steps to use as a reference,
students can independently guide themselves through the problem
solving process in order to accurately derive the solution. Additionally,
utilizing the checklist to monitor progress will keep students motivated
because they will possess an understanding of the final goal, and can view
the steps to be completed in order to successfully reach the solution.
Peer-Mediated Instruction
Peer-mediated instructional models involve preparing one or more
peers without disabilities in the same class to provide targeted academic,
social, and behavioral supports to their classmate with disabilities (Carter,
Cushing, & Kennedy, 2009). Peers are often shown methods to engage
students with ASD through educational exercises, while supporting social
and communicative interactions (Wong, et al., 2014). Peer mediated
approaches can take the form of integrated playgroups, peer buddy, and
peer tutoring formations. This strategy is beneficial when teaching
students with ASD because it not only provides an environment for
students to learn from collaboration and partner work, but also provides
opportunities for students to develop social and communication skills
while participating in a group setting (Sperry, Neitzel, & EngelhardtWells, 2010).
To enhance peer-mediated instruction, scripts can be implemented
as tools to support socialization and communication between two
students. Scripts are defined as direct written and/or visual prompts
intended to initiate or sustain an interaction (Hart & Whalon, 2008). Visual
scripts are implemented in classroom activities during which students
with ASD would be expected to display language skills and are effective
in cueing communication (Ganz & Flores, 2010). Moreover, scripts have
been shown to be effective in increasing question answering (CharlopChristy & Kelso, 2003). When using a script form of peer-mediated
instruction to guide students through solving a simple linear equation,
students with ASD can benefit from communication with peers through
the opportunity to answer questions, verbalize their ideas and discuss
problem-solving processes. Guidelines for teachers to follow when
creating scripts for students are described in Figure 3.
67
Figure 3: Guidelines for creating a script
After learning to follow a script, children with ASD not only
increase the frequency of scripted initiations, but also of non-scripted or
spontaneous initiations (Krantz & McClannahan, 1993; Krantz &
McClannahan, 1998; Stevenson, Krantz & McClannahan, 2000). Although
typically used to promote social interaction, scripts can be applied in
classroom settings by offering an immediate way for children with ASD to
participate in academic interactions.
The use of scripted prompts during peer-mediated instruction can
encourage students to reiterate their reasoning behind each problemsolving step performed, providing them with the opportunity to
communicate mathematic processes and ideas. Furthermore, verbalizing
the problem-solving process and procedures assists in developing
conceptual understanding. This form of mentoring with the aid of scripts
can help students expand their own meta-cognitive skills and become
more self-directed, suggesting that students benefit from skill growth and
development during this type of lesson in which the student is an active
participant (Miles & Forcht, 1995).
68
Implementing Peer-Mediated Instruction in Algebraic Problem Solving
To promote student collaboration, a peer-mediated exercise can be
implemented that provides students with the opportunity to further
practice solving linear equations with one variable. Teachers can develop
scripts to guide a pair of students to solve such a problem. Using the
scripted questions, the pair can be prompted through the steps necessary
to successfully solve the given problem (Ganz, et al., 2012). After problem
completion, the script directs students to explain their reasoning,
encouraging them to verbalize their problem-solving process.
A script can be provided to students (Figure 4). The script contains
one “mentor” role and one “student” role. After being assigned pairs by
the teacher, the student with ASD will initially assume the role of the
“student,” and the partner will assume the role of the “mentor.” The script
will prompt the “mentor” to ask the “student” questions regarding the
type of mathematics problem shown and the strategies that should be
used to solve this problem. Posing these questions prior to the student
beginning to solve the problem promotes critical thinking and reasoning,
which is vital in students’ developing comprehension (Whalon & Hart,
2011). The script provides exemplar responses to each question posed to
the “student,” so that the “mentor” can effectively test student
understanding of concepts. Furthermore, arranging for students to work
together in peer-mediated exercises promotes student discussion (Carter
& Kennedy, 2006). The sample script shown in Figure 4 demonstrates how
one “mentor” can prompt the “student” to successfully complete the
problem, while encouraging critical thinking.
69
70
Figure 4: Peer-mediated script to promote algebraic problem solving
Conclusions
Although a substantial number of students with ASD contend with
mathematical difficulties, research addressing the mathematical
knowledge and skills of children and youth with ASD is in its infancy and
is generally limited. Meeting the mathematics needs of children with ASD
constitutes an ongoing educational challenge. However, there are several
evidence-based strategies available to teachers for improving the
academic skills of children with ASD in general and other educational
settings. Many students with ASD possess the requisite academic skills to
engage in higher-order problem solving. Implementing visual support,
self-management, and peer-mediated instructional strategies creates a
meaningful context for promoting such higher-order, algebraic learning of
all children, including those with ASD.
71
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