Effective Mathematics Interventions

By Margaret M. Flores, PhD, BCBA-D, Auburn University
Co-author of Making Mathematics Accessible for Elementary Students Who Struggle: Using CRA/CSA Interventions

 

According to the National Center for Educational Statistics (2016), the 2015 National Assessment of Educational Progress showed that 18% of fourth grade students performed below basic levels of achievement, meaning that they did not demonstrate mastery of fundamental skills. Students’ mathematical difficulties begin with understanding numbers, basic operations and their novice conceptions lead to further difficulties with complex operations and fractions (Fuchs et al.; Jordan & Hanich, 2003; 2016). Students who struggle in mathematics comprise a diverse group which includes students with identified disabilities as well as students without disabilities (Powell, Fuchs, & Fuchs, 2013). There is a critical need for effective implementation of interventions that have been shown to be effective through research. One effective approach that can be adapted across mathematical concepts is the concrete-representational/semi-concrete-abstract sequence (Miller, Stringfellow, Kaffar, & Mancl, 2011; Witzel, Furguson, & Mink, 2012; CRA/CSA).

What Is CRA/CSA?

The CRA/CSA sequence in an instructional approach to mathematics that emphasizes conceptual understanding prior to procedural knowledge and fluency. There are three phases: concrete, representational/semi-concrete, and abstract. The concrete phase of instruction involves the use of objects to complete mathematical tasks or solve problems. During this phase, teachers explicitly teach concepts through the manipulation of objects. The representational/semi-concrete phase continues to focus on the development of conceptual understanding, but problems are solved using pictures and student-made drawings. Once students demonstrate understanding of the target mathematics concept at the representational/semi-concrete levels, they learn to solve problems using just numbers, the abstract phase. During the abstract phase, the focus of instruction is on procedural knowledge and fluency. The benefit of including the CRA/CSA sequence into mathematics interventions is that the concrete and representational/semi-concrete phases provide students with needed remediation in their understanding of whole numbers, the base ten system, operations, and rational numbers (fractions). The physical manipulation of objects, drawing, and visual aid of pictures fill in the gaps that exist in their prerequisite knowledge and understanding about mathematics. Another benefit of these physical and visual aids is that they assist students in making meaning of mathematical language and using language to explain their computation or problem solving.

CRA/CSA and Number Concepts

The CRA/CSA sequence has been shown to be effective in teaching young children and elementary students number concepts. Researchers used CRA/CSA to teach preschool students, with and without disabilities, counting skills. This included number sense in the form of visual counting or recognizing that four objects were represented by the numeral four without physically touching the objects (Hinton, Flores, Schweck, & Burton, 2015; Hinton, Flores, & Strozier, 2015). Elementary students also successfully learned how to count this way using CRA/CSA. In addition, Hinton and Flores (submitted) taught rounding skills using CRA/CSA. Using base ten blocks and drawings representing base ten blocks, students learned how to round numbers to the nearest ten and hundred. After abstract instruction using just numbers, students quickly and accurately completed rounding tasks. Mercer and Miller (1992) taught place value to elementary students with and without disabilities using CRA/CSA.

CRA/CSA and Basic Operations

Miller and Mercer (1992) and Mercer and Miller (1992) taught elementary students, with and without disabilities, basic operations using the CRA/CSA sequence. This included addition, subtraction, multiplication, and division. Using objects and drawings, students learned the conceptual meaning of each operation: addition is combining, subtraction is separating, multiplication is combining of groups that are the same size, and division is the separation of groups that are the same size. After instruction at the concrete and representational/semi-concrete phases, students learned a simple strategy to assist in computation using just numbers. This set of steps served as a reminder to (a) attend to the numbers and the operational sign, (b) remember that problems can be drawn if the student has not memorized the fact, and (c) write the answer. Students who participated in this large study become fluent in basic operations and their accuracy in computation increased significantly.

CRA/CSA Complex Operations

Researchers also used the CRA/CSA sequence to teach regrouping skills associated with addition, subtraction, and multiplication (Miller & Kaffar, 2011; Mancl, Miller, & Kennedy, 2012; Flores, 2011; Flores & Hinton, in press; Flores, Hinton, & Strozier, 2014; Flores, Schweck, & Hinton, 2014; Flores & Franklin, 2014). Difficulties faced by students within each of these studies were related to poor conceptions of numbers and the base ten system. The concrete and representational/semi-concrete phases of instruction involved the use of base ten blocks and drawings that bolstered students’ understanding of numbers and why regrouping is necessary in when adding and subtracting large numbers. These studies included students with and without disabilities and led to significant gains in accuracy and fluency.

CRA/CSA and Fractions

CRA/CSA has been shown as an effective way to teach rational numbers or fraction concepts (Butler, Miller, Crehan, Babbit, & Pierce, 2003; Flores & Hinton, submitted). Butler et al. studies the necessity of including a concrete phase within instruction. Students successfully leaned to make equivalent fractions, but those who used fraction blocks prior to drawings performed better than those who only received instruction using drawings. Flores and Hinton taught elementary students equivalency using CRA/CSA as well as comparison of fractions to decimals. At the concrete phase, students made fractions using fraction blocks as well as sets of objects. At the representational level, students shaded shapes and marked number lines. In both studies, concrete and representational/semi-concrete instruction allowed students to understand the proportional nature of fractions which led to their mastery of more complex concepts such as equivalence and relations to decimals.

Summary

The CRA/CSA sequence has been shown to be effective across a variety of elementary mathematics concepts. The materials needed are simple; base ten blocks and counters are readily available in elementary schools. However, it may be difficult for teachers to implement and replicate the research as journal articles are not written in ways that provide detailed descriptions of each lesson component. Therefore, in order to close the gap in mathematical achievement, there is a need for more user-friendly guides for implementation of the CRA/CSA sequence.

References

Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities. Learning Disabilities Research and Practice, 18, 99–111.

Flores, M. M., Hinton, V. M., & Strozier, S. D. (2014). Teaching subtraction and multiplication with regrouping using the concrete-representational-abstract sequence and strategic instruction model. Learning Disabilities Research and Practice, 29, 75–88.

Flores, M. M., & Franklin, T. M. (2014). Teaching multiplication with regrouping using the concrete-representational-abstract sequence and the strategic instruction model. Journal of American Special Education Professionals, 6, 133–148.

Flores, M. M., Schweck, K. B., & Hinton, V. M. (2014). Teaching multiplication with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 29(4), 171–183.

Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Malone, A., Wang, A., Hamlett, C. L., Jordan, N. C., Siegler, R. S., & Changas, P. (2016). Effects of intervention to improve at-risk fourth graders’ understanding, calculations, and word problems with fractions. Elementary School Journal, 116(4), 625–651.

Jordan, N. C., & Hanich, L. B. (2003). Characteristics of children with moderate mathematics deficiencies: A longitudinal perspective. Learning Disabilities Research & Practice, 18, 213–221. doi:10.1111/1540-5826.00076

Mancl, D. B., Miller, S. P., & Kennedy, M. (2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research and Practice, 27(4), 152–166.

Mercer, C. D., & Miller, S. P. (1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial and Special Education, 13(3), 19-35. doi: 10.1177/074193259201300303

Miller, S. P., & Kaffar, B. J. (2011). Developing addition with regrouping competence among second grade students with mathematics difficulties. Investigations in Mathematics Learning, 4(1), 24–49.

Miller, S. P., & Mercer, C. (1992). CSA: Acquiring and retaining math skills. Intervention in School and Clinic, 28(2), 105–110.

Miller, S. P., Stringfellow, J. L., Kaffar, B. J., & Mancl, D. B. (2011). Developing computation competence among students who struggle with mathematics. Teaching Exceptional Children, 44(2), 38–44.

Witzel, B. S., Furguson, C. J., & Mink, D. V. (2012). Number sense: Strategies for helping preschool through grade three children develop math skills. Young Children 89–94

 

Perspective-Taking for Neurotypicals

Stephanie_SandersSanders_FILTER

By Stephanie D. Sanders, MA, CCC-SLP, author of The FILTER Approach: Social Communication Skills for Students with Autism Spectrum Disorders

While creating The FILTER Approach, I took exhaustive measures to help students with Autism Spectrum Disorders (ASD’s) identify, comprehend, and explain essential social skills, while putting them into practice.  As I implemented this curriculum, it began a personal perspective-taking opportunity for me.  I noticed weaknesses within my own communication skills in specific situations. A perfect example is my inability to Listen to my family with technology distractions in view (thank you, Pinterest).  Demonstrating social errors as a neurotypical Speech-Language Pathologist (SLP) could likely justify a new DSM-5 diagnosis of “social skills hypocrite.”

The truth is that most of us have room for improvement socially and in considering the perspectives of those with social impairments.  Perspective-taking tasks usually present a challenge to individuals with ASD, due to Theory of Mind.  I’ve frequently referenced the idiom “put yourself in my shoes” with students during these activities. However, SLP’s can also struggle with taking perspective when driven by accountability for pragmatic language goals printed on a report. I become frustrated when my student resists the educationally relevant IEP goals that will undoubtedly transform him or her into a social skills superhero.  An epiphany soon occurred with a hint of witty wordplay.  My mission: try taking a new perspective on perspective-taking.

I began investigating:

  • How do those with social impairments perceive conversation?

I asked students individually, “Why are conversations important?”  The same response was consistently given, “To find out information.”  This perspective came across as task-driven, lacking any element of enjoyment.  Some interrogation sessions I’ve witnessed appear to be information-seeking at its finest.  In other instances, my students feel obligated to be the source of information.  They lecture peers regarding topics of interest, rather than seeking to find out information.  We’ve discussed how obsessive interests and “conversation hog” habits will cause one to miss the Target, socially.  I’ve also taught this concept in the middle school gifted-student classroom during monthly F.I.L.T.E.R. lessons.  Luckily, the “conversation hog” reference hasn’t triggered any speeches about swine or guinea pig fixations!

Other questions on my mind:

  • How are common rules of social language perceived?
  • What are the most stressful things about social situations?

My little brother Zach was diagnosed with an ASD at the age of thirteen and was my primary inspiration for “The FILTER Approach.” As part of this perspective-taking endeavor, I knew it would be beneficial to get Zach’s viewpoint on social rules.  I asked him to speak freely, without concern of giving a wrong answer.

Me: What do you think the expression, “Put yourself in his/her shoes” means?

Zach: It means you should consider the other person’s feelings.

Me: Exactly.  I want to put myself in your shoes to find out what conversation is like for you, having an ASD.  I want to know your perspective about some social rules in conversation.

Zach: Okay.

Me:  What do I mean when I tell you to “use your filter” in conversation?

Zach: It’s what you should or should NOT say in conversation.  If you always say what you’re thinking, then you could look bad as an employee, lose respect, and look unconcerned about feelings.

Me: Great explanation!  Now I want your perspective on some social rules from my book.  How do you feel about making eye contact and looking for Facial clues?

Zach: A little uncomfortable.  A symptom of people with Autism is sometimes having a hard time with eye contact.  I don’t want to give too much and it’s hard for me to know.

Me: Very true.  We’ve talked about glancing, which works.  You’ve done a nice job of avoiding inappropriate topics in person.  However, you and many other people might post strong opinions on Facebook.  Why do you think that is?

Zach: On Facebook, it’s virtual and like your own little world, so it’s not as real.  It’s uncomfortable in person because you’re actually with them.

Me:  I see what you’re saying.  Do you think it’s hard to Listen during a conversation with someone and why or why not?

Zach: It can be a lot of work. Sometimes I run out of things to say or my mind is off-topic while I’m trying to listen.  The conversation gets stressful if it’s too long and boring.  Sometimes, I think about something totally unrelated, like a conversation with someone earlier.

Me:  Staying focused probably does feel like a lot of work. Why do you think we should try to “hit the Target” socially and what did we talk about for your target?

Zach: We should make goals to be successful. I need to close my conversations with “See you later” and ask about someone else’s interests.

Me: Excellent. Is it awkward for you to End conversations with people at places like church or work?

Zach: Yes, because I run out of things to say.  It’s also difficult to end things at work when my shift is over.  I want to tell my manager I’m ready to leave, but he’s usually busy.  If I just leave, I might look disrespectful like I’m trying to get out of my job.

Me:  Later, we’ll make a plan for leaving work.  Is it difficult for you to Repair conversation mistakes you’ve made and have you used some of the Repair tools we’ve talked about?

Zach: I’ve used some.  Apologizing can be hard and it’s hard to admit you’re wrong.  I’ve asked, “Should I stop now?” when the person was being quiet. I also messaged, “Did I say something wrong?” two times to someone on Facebook who quit talking to me.  He never responded, so I didn’t ask anymore.

Me: I’m so proud of you for trying to Repair social mistakes.  You made a good choice to quit asking when the person on Facebook never responded.  At least you tried.

Me: Overall, what are the most stressful things for you in social situations?

Zach:  It’s stressful because:

  • I don’t know what the other person is thinking.
  • I don’t know if I’ve said something wrong.
  • I don’t know what will happen to that friendship (in the future).
  • It’s tough to start new friendships as an adult.
  • Losing a childhood friendship is discouraging and can’t be replaced.
  • I’ve become shy as an adult.

Me:  Thanks for sharing, Zach.  Therapists need to consider what it’s like for someone with ASD to follow these rules.  It has really helped me to hear your perspective.

From Zach’s outlook, it must be draining to worry about confusing social cues and potential negative outcomes.  If someone repeatedly struggles to use verbal and nonverbal social behaviors in conversation, then it could become a losing battle.  Isn’t it easier to retreat into a virtual world where at least all of the nonverbal challenges are removed?  Many of our students/clients with ASD’s could perceive communication as a lot of effort with little worth.

The challenge is to find pragmatic goals that are realistic, beneficial, and meaningful for our students/clients.  The perspectives of these individuals are usually disregarded as wrong with an immediate need for change. If I truly listen to the individual’s perspective, I can not only set an example of showing interest, but also ensure my therapy approach remains individualized.  I can clearly and personally define the advantages of practicing good social habits now in order to make future social success a possibility. Taking the student’s perspective increases my chances of enlisting him or her in therapy, which will result in a more socially responsible individual. Early investment in the views of my students could allow the opportunity to become an influence in rewriting a lifelong story filled with social struggles and disappointments.

Attacking Social Interaction Problems Across the Lifespan

Autism: Attacking Social Interaction Problems

Autism: Attacking Social Interaction Problems by Betholyn F. Gentry, Pamela Wiley, and Jamie Torres-Feliciano

By Pamela Wiley, Ph.D., co-author of Autism: Attacking Social Interaction Problems

In my private practice, we are often asked by our funding sources when our children with ASD will no longer need social skills instruction. I often feel a sense of “indirect or subtle” pressure to discontinue our service and declare that a child is socially competent and basically cured of what is essentially the hallmark feature of ASD: impaired social interaction. However, given what I know and have observed with this population it is difficult both ethically and morally to do so.

As professionals we know that social skills are the foundation for getting along with others. We also know that there are social skills milestones which develop along a continuum. For example, one of the early skills focused on for our young children is “how” to make friends and join groups. Many are successful and with parental support during the preschool years engage in playdates and develop friendships with their typically developing peers. However, around 8 or 9 years of age the terrain shifts and children reportedly become more discriminating and scrutinizing as they select their friends. Labels such as nerd, cool and loser become important in the selection process. Our children with ASD often fall into the category of nerd.

As a result, many of them experience rejection and bullying and are left confused and hurt when their only friends abandon them. Social skills continue but our focus evolves to include discussions and skill steps to facilitate their understanding of “who” should be their friend and “how” our friends make us feel and accepting loss and changes in life. The need for social skills training into middle school and beyond can have a profound effect on the quality of life for these children.

High school brings yet another level of complex social interactions and negotiations especially when dealing with the opposite sex, changes in hormone levels, sexual maturation, and peer pressure.

The final phase is the transition process from high school to college and the world of work. The need for continued social skills training is essential and should address core clusters of skills critical for promoting independence and fulfilling lives: vocational, independent, and personal development. Social skills taught may include relationships and how to discriminate between a friend, a colleague, and an acquaintance, the importance of good hygiene, executive functioning, workplace conversation, nonverbal communication, unwritten social rules, and workplace idioms such as “hit the ground running” to name a few.

Based on decades of working with this population and the long-term relationships experienced with many of the children and families in my practice and more importantly the positive outcomes we have achieved, in response to the question, how long should children with ASD continue with social skills treatment, my short answer is, “Across the Lifespan.”

That having been said, based on the positive comments and requests from our colleagues following several ASHA presentations on social skills, my colleague and I have developed a series of social skills workbooks, Autism: Attacking Social Interaction Problems to cover the lifespan of children with ASD from 4 years to adulthood.

Our books contain clear and concise objectives and instructions on how to introduce and implement the lessons. Our approach is fun yet structured and each unit builds on previously learned skills to assist in the generalization of information across boundaries and contexts which include home, school, and the community while incorporating parent and teacher input.

Our newest additions are the teen and prevocational books which are fresh and relevant to today’s youth and include the use of social media and issues facing young people today such as sexting, texting, and TMI. The goal is to assist our students to develop a full range of interpersonal social competencies that can help them ultimately achieve acceptance in the workplace and develop a meaningful existence.

To learn more about our series of workbooks, visit Plural Publishing at www.pluralpublishing.com or our website www.speakla.com.

Reflecting on Autism Awareness Month: Why Is Awareness Important?

Zenko, Catherine

By: Catherine B. Zenko, MS, CCC-SLP

During the rush of activities on April 2nd for World Autism Awareness Day, a journalism student interviewed me to discuss upcoming events at our center and to learn more about Autism Awareness Month. One of the first questions she asked me was, “Why is awareness so important for autism?” It seems like such a simple question, but when I had to put into words why I do what I do every day to promote awareness, it took me a moment articulate the importance of awareness. My response sounded something like, “Ideally, the more people know and understand what autism spectrum disorder (ASD) is—how individuals think, process, and learn differently—the more understanding they will be when they see a person on the spectrum acting ‘out of the ordinary.’”

According to the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5) criteria, autism spectrum disorder consists of deficits in two domains: (1) social communication and (2) restricted, repetitive, and stereotypic interests and activities (APA, 2013). ASD presents in a myriad of ways, thus inspiring the expression, “once you’ve met one person with autism, you’ve met ONE.” Generally speaking, people with ASD have difficulty communicating: some cannot use speech to communicate; some use a combination of speaking, sign language, pictures, or augmentative/alternative communication (AAC); and some speak too much, not understanding the social rules that a conversation involves two people and both people get to talk. Understanding spoken and written language is also difficult and takes more time to process for most people on the spectrum.

The DSM-5 outlines the diagnostic characteristics of the domain of restricted, repetitive patterns of behavior, interests, or activities as the following: repetitive speech, motor movements, or use of objects; inflexible adherence to routines and/or ritualized patterns of verbal/nonverbal behavior; restricted, fixated interests (intense focus); and hyper- or hyporeactivity to sensory input or unusual interest in sensory aspects of environment (APA, 2013). All of the diagnostic criteria translate into people who:
• Are literal interpreters of language and concrete thinkers;
• Have difficulty processing all of the sensory information around them and can have both gross- and fine-motor challenges;
• Are visual learners;
• Have a strong sense of logic that is black and white, not much (if any) room for gray;
• Prefer routines and become extremely upset when a routine is disrupted and are sometimes compelled to finish a task they have started, even when the allotted time has expired;
• Have difficulty taking the perspective of others which makes them appear egocentric;
• Are detail-oriented but have trouble seeing the big picture;
• Have difficulty with attention, starting with joint-attention and engagement with others as well as trouble shifting their attention away from their intense interest area (Janzen & Zenko, 2012; Quill, 1997; Rydell, 2012; Zenko & Hite, 2013).

I like to view autism spectrum disorder more like a difference rather than a disability. The term “neurodiversity” is gaining steam lately and illustrates that just because people on the autism spectrum think and learn differently, they are not disabled. One of Temple Grandin’s most famous quotes embodies this idea of “different, not less.” One social media campaign currently trending is #AutismUniquelyYou. This campaign highlights and celebrates each individual with ASD’s unique personality, instead of lamenting it. Another great campaign is #MakeATinyChange that encourages people to make a difference in the lives of individuals with disabilities through any one of 25 small changes.

There have been several stories circulating this month about how a small gesture of openness and understanding can make a huge difference. One that stood out was a story by ABC News about a man who put away his work and played with a little girl with autism sitting next to him on a plane. The man did not understand why “playing Ninja Turtles with the little girl was a big deal,” but to her mother—who was so relieved when her daughter was met with kindness and acceptance, not pity and annoyance—it meant the world.

Circling back to the question of why awareness is so important, if more people take the time to learn how someone with autism thinks and experiences their surroundings, the more people may embrace the neurodiversity, rather than shy away from the differences and get to know some truly interesting people.

References
American Psychiatric Association (APA). (2013). Diagnostic and statistical manual of mental disorders: DSM-5 (5th ed.). Washington, DC: American Psychiatric Publishing.

Janzen, J. E., & Zenko, C. B. (2012). Understanding the nature of autism: A guidebook to the autism spectrum disorders (3rd ed.). San Antonio, TX: Hammill Institute on Disabilities.

Quill, K. A. (1997). Instructional considerations for young children with autism: The rationale for visually cued instruction. Journal of Autism and Developmental Disorders, 27(6), 697–714.

Rydell, P. J. (2012). Learning Style Profile for children with autism spectrum disorders. Retrieved from http://itunes.apple.com

Zenko, C. B., & Hite, M. P. (2013). Here’s how to provide intervention for children with autism spectrum disorder: A balanced approach. San Diego, CA: Plural Publishing.

About the Author
zenko_hhtpicasdCatherine B. Zenko, MS, CCC-SLP is a Florida-licensed speech-language pathologist who has worked with individuals on the autism spectrum for over fourteen years. She is an adjunct lecturer at the University of Florida Dept. of Speech Language Hearing Sciences since 2008 teaching a graduate-level Autism and Augmentative and Alternative Communication (AAC) course and has worked at the University of Florida (UF) Center for Autism and Related Disabilities (CARD) since 2000. While at CARD, Catherine has helped hundreds of individuals with ASD, their families and educators by providing consultation or training opportunities on a myriad of topics relating to best practices and ASD. In addition to her work at CARD, Catherine has co-authored Here’s How to Provide Intervention for Children with Autism Spectrum Disorder: A Balanced Approach, a timely resource for speech-language pathologists working with children on the autism spectrum as well as graduate students preparing to work with this demographic.

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