View as Webpage

The recent pandemic has highlighted the need for high quality instruction in math. According to the latest National Assessment of Educational Progress report, only 36 percent of students in 4th grade and 26 percent of students in 8th grade were proficient in math (https://www.nytimes.com/2022/10/24/us/math-reading-scores-pandemic.html?smid=url-share). For students with disabilities, the scores were even worse (see Table 1 at the end of this article).

The data in Table 1 show that there is an even stronger need for providing specially designed instruction (SDI) to students with disabilities, not only to provide access to the curriculum but, more importantly, to ensure progress by closing skill gaps. Closing skill gaps will ensure that students truly have access to the material.  

The focus of this edition of Link Lines is to provide concrete examples of how to provide SDI in math. In addition, resources are provided to monitor student progress.

What does SDI Look like in Math Class?

Specially Designed Instruction means adapting, as appropriate to the needs of an eligible child, the content, methodology, or delivery or instruction to address the unique needs of the child that result from the child’s disability; and to ensure access of the child to the general curriculum so that the child can meet the educational standards within the jurisdiction of the public agency that apply to all children (IDEA, 2004, Sec. 300.39 (b)(3)).

During class, SDI should be provided when teaching pre-requisite skills to enable students to learn new content, to teach skills that are part of IEP goals, and to teach skills where students require different delivery methods to master content.  

In math, SDI can include (but is not limited to):

• explicit instruction, 
• using concrete and representational models to teach abstract concepts, and 
• developing fact fluency.  

The SDI provided will depend on the IEP goal, but some practices are recommended in the Evidence-Based Specially Designed Instruction in Mathematics Resource Guide  from the Virginia Department of Education. In this article, we will discuss in further depth these three recommended practices as they apply to Math instruction.

In a co-taught class, SDI can be provided in small groups during a warm-up, during alternative teaching, or in stations. Remember that while these instructional methods may work for all students, it is only considered SDI if the instruction is different from what is provided to the rest of the class and is focused on meeting the student’s IEP goal(s). Here is an example lesson plan for a co-taught math classroom where SDI is provided. Note that students are provided SDI in solving equations while the rest of the class is engaged in a fact-fluency activity that reviews non-SOL tested content.  

Explicit Instruction

Explicit instruction, a High Leverage Practice, is a purposeful way of overtly teaching students. Teaching is explicit when teachers tell the students what they need to do using direct explanations along with sharing and modeling new knowledge (Fletcher et. al.,, 2019). Being explicit means providing students with the expectations/goals of the lesson, the purpose of the lesson, and how the lesson connects to previously taught skills. Explicit instruction is a form of SDI that can be used across all content areas and not just math.

An example of a goal:

By the end of today’s lesson, students will be able to solve a linear equation with one variable. They will do this by using algebra tiles to solve 10 linear equations with at least 90% accuracy.  

An example of the purpose for this goal:

Being able to solve linear equations with one variable will help us to be able to solve linear equations with two variables, quadratic equations, and can help us find the slope of a line.  

This explicit explanation of the goal/purpose of the lesson leaves no room for students to have to guess what they are doing or why they’re doing it.  

While teaching the lesson, provide examples and nonexamples (Table 2), model how to solve the problem using think alouds (explain what you’re doing and why you’re doing it), provide guided and independent practice (Table 3), and offer many opportunities for students to respond while providing immediate and specific feedback. 

Table 2

Examples and Non-examples of Linear Equations

Examples of Linear Equations with One Variable

Providing both guided and independent practice for students is essential.  Independent practice should not be just student guided, but should also include teacher support.  

Table 3

Guided vs. Independent Practice

Remember to also use explicit instruction while supporting practices such as asking a mix of lower-order and higher-order questions, eliciting frequent responses from students, and maintaining a brisk pace. The Explicit Instruction Template (see resources) can help you plan. 

Concrete-Representational-Abstract (CRA)

Jean Piaget, in his seminal work on how children learn (Piaget, 1929), recognized that many children did not understand abstract concepts and needed concrete representations in order to learn. Perhaps when you were in school, it was difficult to conceptualize a chemical reaction that you couldn’t see. Would it have helped to see a visual model? Or maybe it was difficult to understand what was happening in a Shakespeare play, but seeing it acted out helped to bring context. Across all disciplines, providing concrete models or pictorial representations help students learn and can be a form of SDI.

Using concrete manipulatives in math can build conceptual understanding for students with disabilities. It is not enough to teach students procedures. Students must achieve firm conceptual understanding prior to procedural knowledge and fluency. CRA can help students achieve both (Milton et al., 2019).

Concrete refers to 3-D shapes and objects that represent numbers and symbols. Representational refers to 2-D images of shapes or objects that represent numbers and symbols. Abstract refers to the mathematical numbers and symbols we typically see. CRA is not linear. The stages overlap and may move in cycles. Some students may need to be in the C or R stage longer, but you are always connecting the C and R to the A regardless of the student’s stage of learning. Explicit instruction should be used to teach CRA cycles.  

In order to monitor the student’s stage of mastery, you can use formative assessments and keep track of student progress on a chart like shown in Figure 1 (see Figure 1 at the end of this article).

Take a moment to watch the video example of a student using CRA to solve an algebra problem. In the video, the student is using algebra tiles to solve a one variable equation. 

Video Example: A tutor asks a student to solve an equation with algebra tiles.

Fact Fluency

Students exhibit computational fluency when they demonstrate strategic thinking and flexibility in the computational methods they choose and are able to explain and produce accurate answers efficiently (VDOE, 2020). Fact fluency activities should be brief and daily. For example, a five minute fact fluency activities can be done during a daily warm-up. Some strategies for developing fact fluency in addition and subtraction are shown in Table 4 and strategies for developing fact fluency in Multiplication and Division are shown in Table 5. Once again, fact fluency is not just something that can be done in math. In science, English, history, and electives students are required to know certain facts, but students need to have time in class and structures in place in order to learn these facts.  

Teachers should progress monitor students’ fact fluency. There are programs that your school division may already use such as Reflex Math or IXL, but there are also progress monitoring sheets teachers and students can use such as this one from Northern Kentucky University.

Table 4

Strategies for Developing Addition and Subtraction Facts 

Note. Retrieved from Evidence-Based Specially Designed Instruction In Mathematics Resource Guide, VDOE, 2020, p. 26. (https://www.doe.virginia.gov/home/showpublisheddocument/28625/63809042486 

2930000)

Table 5

Strategies for Developing Multiplication and Division Facts

Note. Retrieved from Evidence-Based Specially Designed Instruction In Mathematics Resource Guide, VDOE, 2020, p. 26. (https://www.doe.virginia.gov/home/showpublisheddocument/28625/63809042486 

2930000)

Remember that it is not enough for students to just have access to the curriculum, they need to make progress as well. Providing SDI such as explicit instruction, CRA, and fact fluency while using progress monitoring tools will ensure that students are meeting their IEP goals. To learn more about SDI in math, visit the VDOE’s website where you can find numerous resources. We also encourage you to co-plan SDI with your co-teacher. More information about co-planning can be found here. Remember to look for our March edition of Link Lines focused on SDI and literacy.

Table 1 

Pre- and Post- Pandemic Virginia Mathematics Pass Rates for Students without Disabilities & Students with Disabilities

Note: There were no SOL Assessment results during the 2019-2020 academic year because of COVID-19 related school closures.

(VDOE 2023)

Figure 1

CRA Formative Assessment Grid

Note: Table from Witzel, Riccomini, and Schneider, 2008. 

Additional Resources

Explicit Instruction Planning Guide

Virtual Manipulatives Website



Fact Fluency Games

McLeskey, J., Maheady, L., Billingsley, B., Brownell, M. T., & Lewis, T. J. (2022). High leverage practices for inclusive classrooms (2nd ed.). Routledge and the Council for Exceptional Children.

Mervosh, S., & Wu, A. (2022, October 24). Math scores fell in nearly every state, and reading dipped on national exam. New York Times. https://www.nytimes.com/2022/10/24/us/math-reading-scores-pandemic.html?smid=url-share

TTAC William & Mary. (2015, September 21). Lesson design for an inclusive classroom. https://ttaclinklines.pages.wm.edu/lesson-design-for-an-inclusive-classroom/

References

Fletcher, J. M., Lyon, G. R., Fuchs, L. S., & Barnes, M. A. (2019). Learning disabilities: From identification to intervention. Guilford Press.

Individuals with Disabilities Education Act, 20 U.S.C. § 1400 et seq. (2004), § 

300.39(b)(3).

Milton, J. H., Flores, M. M., Moore, A. J., Taylor, J. J., & Burton, M. E. (2019). Using the concrete–representational–abstract sequence to teach conceptual 

understanding of basic multiplication and division. Learning Disability Quarterly, 42(1), 32-45. https://doi.org/10.1177/0731948718790089

Piaget, J. (1929). The child's conception of the world. Routledge.

VDOE, (2023). Test results build-a-table [Database]. https://p1pe.doe.virginia.gov/apex_captcha/home.do?apexTypeId=306

VDOE, (2020). Evidence-based specially designed instruction in mathematics resource guide. 

https://www.doe.virginia.gov/home/showpublisheddocument/28625/638090424862930000

Witzel, B. S., Riccomini, P. J., & Schneider, E. (2008). Implementing CRA with 

secondary students with learning disabilities in mathematics. Intervention in 

School and Clinic, 43(5), 270-276. https://doi.org/10.1177/1053451208314734

TTAC William & Mary's newest Project Specialist

BethAnne Klaver