Because of their diverse learning needs, students with DB and ID can be a real puzzle to teach and, at times, educators may feel like they have more questions than answers. As any good puzzle solver will tell you, though, to solve a puzzle, you start by finding the corner pieces. This module is about finding the corner pieces to help you solve the puzzle of mathematics instruction for students with DB and ID.
Typically, when educators want to find an effective teaching strategy, they can find models (or examples) in the literature. In this case, however, there is no experimental research to date that evaluates strategies for teaching mathematics to students with DB and ID.
When there is a gap in the research, educators can move forward in two ways: by identifying (1) effective practices for teaching mathematics to other students (e.g., students with multiple disabilities, students with severe disabilities) and/or (2) effective practices for teaching other skills to students with DB and ID (e.g., communication skills). Since there are no published studies to guide us, we will use both of these approaches to identify practices for teaching mathematics to students with DB and ID. First, we will look at what the literature says about teaching mathematics to other secondary students with disabilities; and then we will look at what the literature says about teaching students with DB and ID other content.
If available, evidencebased practices are a good place to start when looking for effective practices for teaching because the practice has undergone a rigorous review process and been found to be effective in teaching a skill or content area to a particular group of students. An evidencebased practice is defined using a specific practice (e.g., shared story reading) to teach a specific skill (e.g., early literacy skills) to a specific group of students (e.g., students with developmental disabilities; Hudson & Test, 2011). For example, in their review of the literature, Hudson and Test (2011) found shared story reading to have a moderate level of evidence for teaching early literacy skills to students with developmental disabilities.
In 2008, Browder, Spooner, AhlgrimDelzell, Harris, and Wakeman conducted a comprehensive review of the mathematics literature and found in vivo instructionIn vivo instruction involves teaching a real life application for the mathematics principle to be learned (Browder et al., 2008). For example, you could teach onetoone correspondence by having a student set the table for snack (i.e., every student needs a plate, spoon, and napkin) or division by having a student divide 9 cookies between three friends. In addition, in vivo instruction teaches in real life settings (e.g., teaches students to use the vending machine in their school or community) as well as for real life settings., systematic instructionSystematic instruction defines a specific response or set of responses (e.g., counting 15, adding sets to the sum of 10) and teaches to mastery using defined, consistent prompting and feedback and explicit prompt fading (Collins, 2007). Effective prompting systems include time delay, system of least prompts, and simultaneous prompting. For more information on systematic instruction, see the MAST Module on Prompting Systems (http://mast.ecu.edu/modules/ps/)., and opportunities to respondOpportunity to respond is a teacher behavior that invites or solicits a student response (Simonsen, Myers, & DeLuca, 2010). For example, if a student is learning to make sets 15, the teacher provides multiple opportunities across the lesson to make sets. Caution should be used when applying a massed trial format. Massed trial formats are effective for initially learning a new skill, but generalization of the skill across materials, people, and places is needed. to be evidencebased practices for teaching mathematics to students with significant cognitive disabilities.“Significant cognitive disabilities” is not one of the 14 disability categories listed in IDEA (2004) for children aged 3 21 years (see below); rather, it is a term used to describe students with disabilities who are assessed using an alternate assessment based on alternate achievement standards and is inclusive of all students with a disability who meet the eligibility criteria for a state’s alternate assessment. In other words, the term “significant cognitive disabilities” can include students with a wide variety of disabilities. Several of the 64 studies in this review included students in middle school (i.e., 29 studies) and high school (i.e., 35 studies). Though most of the studies focused on teaching mathematics to students with moderate intellectual disabilities, 17 studies included students with severe cognitive disabilities, and one study reported results for one student with multiple disabilities.
The vast majority of math skills taught to the students in these studies were from two of the five components recommended by the National Council of the Teachers of Mathematics (NCTM, 2000): numbers and operations (i.e., counting, matching, calculating) and measurement (i.e., time and money). Few studies included the other three components: algebra, geometry, and data analysis.
Since that review, however, other studies have focused on teaching these components to secondary students. Figure 1 contains a summary of these studies. For more information about a study, click on the study’s authors.
Study/Participants 
Practice 
Approach 
Content/ Example 

Browder, Jimenez, & Trela (2012) 4 students, aged 1113 years 
System of Least Prompts TaskAnalytic Instruction Mathematics word problem stories based on familiar activities Graphic Organizers Manipulatives for the mathematics concept 
A mathematics word problem story based on a familiar activity was read to/by the students that contained key words for solving a math problem (e.g., How many hours did Joan spend at the mall?). Students were given math manipulatives and a graphic organizer for the math question. Students were taught to follow the taskanalyzed steps to find the answer to the math problem. Four instructional units were included in the intervention. 
Algebra Geometry Measurement Data analysis/ probability 
Browder, Trela, Courtade, Jimenez, Knight, & Flowers (2012) 16 students, aged 1420 years, IQs ranged from 3054), 11 were students with moderate and severe intellectual disability; 6 with autism 
System of Least Prompts Taskanalytic instruction Mathematics word problem stories based on familiar activities Graphic organizers Manipulatives for the mathematics concept 
Teachers in the math group delivered taskanalytic math instruction that included a storybased problem with a familiar context (e.g., mall, movies) that provided students with key facts and a problem statement to be solved. Teachers read the math stories aloud and posed the math problem. Then, students used manipulative and a graphic organizer to solve the math problem. Systematic instruction (i.e., system of least prompts) was used to help students learn the steps of the task analysis. 
Algebra Geometry Measurement Data

Galloway, Collins, Knight, & Bausch (2013) 4 students, aged 1517 years, moderate intellectual disability, IQs ranged from 4157 
Simultaneous Prompting Taskanalytic Instruction Personally relevant activities (i.e., sewing, using a ladder, finding the dimensions of a screen) 
Students watched a video segment on an electronic tablet that sets up the problem (e.g., person making a quilt needs to find out what the value of side c is to complete the quilt). During training, students learn to complete the 32step task analysis for solving a problem with the Pythagorean Theorem. 
Pythagorean Theorem 
Jimenez, Browder, & Courtade (2008) 3 students, aged 15 – 17 years), moderate developmental disabilities (i.e., IQs ranged from 4149) 
Systematic prompting with fading Time Delay Instructional Procedure TaskAnalytic Instruction Concrete Representations of Algebraic Equation (3 + x = 7) Reallife problem statements Manipulatives 
Students were taught to use a simple linear algebraic equation to solve for x (e.g., 3 + x = 5). 
Algebra 
For more information on systematic instruction, including time delay and the system of least prompts, see the Prompting Systems MAST Module (http://mast.ecu.edu/modules/ps/). 
The second way to identify potentially effective strategies for teaching when no models can be found in the literature is to identify the practices used to teach skills other than mathematics to students with DB and ID.. Figure 2 summarizes research that described practices for teaching skills other than mathematics to students with DB and ID. Figure 2 summarizes four studies that described practices for teaching other skills to students with DB and ID. For more information on each study, click on the authors’ names.
Study  Practice  Approach  Example 

Bruce, Randall, & Birge (2008)  Consistent delivery of individualized lessons  Deliver lessons in a consistent manner each time instruction is given so child learns what to expect.  Student receives multiple opportunities for instruction on the daily object schedule each day. 
Adult responsiveness  Ensure learning occurs in the context of close physical proximity with a trusted adult and the sense of touch and shared physical experience are central.  Adult pays close attention to the child’s emotional state, pauses appropriately to elicit communication from student, and asks for clarification when communication efforts aren’t understood.  
Repetition of highly interesting activities  Provide many opportunities to learn using a variety of activities the child finds interesting.  Child’s interests are used to determine what is taught (e.g., using balloons to teach concepts of big, bigger, and biggest).  
Consistent exposure to representations for activities  Consistently use objects, pictures, or braille used to represent activities.  A braille label, auditory message, and an object are used to represent lessons and activities in a homeschool journal.  
Childguided instruction  Determine curriculum and most effective strategies on an individual basis based on the needs of the child. Adults learn how to teach each child by interacting with them.  Teacher uses a child’s interest in reading to create appropriate experience books.  
Daily schedule/ Anticipation shelf  Help students anticipate the activities of their day. Students set up their schedule each morning and go to it before and after each activity.  Child’s day is represented by an object schedule. Objects are placed in separate sections of a schedule display box beginning with the first activity on the left and moving right.  
Homeschool journal  Provide daily opportunities for child to learn language through consistent exposure to braille and objects with parent and teacher. Teach symbols to be included in the journal first in daily schedule lessons.  Each lesson or service is placed on a separate page in the journal. Braille and print labels are placed at the top of each page, an auditory device with a message about the lesson or service is placed in the middle, and an object symbol was placed at the bottom.  
Janssen, RiksenWalraven, van Dijk, Huisman, & Ruijssenaars, 2012
Janssen, RiksenWalraven, van Dijk, & Ruijssenaars, 2010 Marten, Janssen, Wied, Ruijssenaars, & RiksenWalraven, 2014 
Explicitly teaching communication partner skills  Train communication partners to be more responsive to the child with DB by using interactive coaching including mutually agreed upon questions, video analysis, modeling, and onthejob coaching.  
Johnson & Parker (2013)  Waittime  Students are given enough time to process what is being asked of them and respond before prompting the response.  Student identifies his mother from several photographs of family members when given 5 to 15seconds to provide a response. 
Parker (2009) Literature review  Reinforcement  Include reinforcements (e.g., social attention, edibles, and preferred activities) as part of the intervention.  Student selects a preferred activity (e.g., playing a game on an electronic tablet) after working on Orientation and Mobility (O&M) skills. 
Natural Settings  Teach interventions within natural settings such as the participants’ homes and communities where the Orientation and Mobility (O&M) skills are needed.  Student learns to walk independently from his house to a friend’s house in his neighborhood. 
Now let’s put it all together. We’ve reviewed (1) effective practices for teaching mathematics to other elementary students with disabilities (mostly students with moderate ID or Autism Spectrum Disorder (ASD)), and (2) effective practices for teaching skills other than mathematics to students with DB and ID. Figure 3 summarizes these practices in Box 1 and Box 2 of figure 3.
Box 1 – Effective practices for teaching mathematics to elementary students with moderate disabilities include: systematic instruction, in vivo instruction, opportunities to respond, math story readalouds, repetition of instruction across multiple lessons, use of math manipulatives, and graphic organizers.
Box 2 – Effective practices for teaching skills other than mathematics to students with DB and ID include: wait time, reinforcement, natural settings, consistent instruction, use of representational objects, etc., repetition of highly interesting activities, home school communication, homeschool journal, daily schedule/anticipation shelf, and childguided instruction.
Several of the same practices are shared by both groups of literature. In Figure 3, these practices are described in Box 3.
Box 3  The practices shared by both groups include: systematic instruction/consistent instruction, opportunities to respond/ample wait time, in vivo instruction/natural settings, repetition of highly interesting activities, use of manipulative/representational objects, and systematic fading of verbal and physical prompts. Because of the effectiveness of these strategies in teaching mathematics to elementary students with moderate disabilities and content other than mathematics to student with DB and ID, these practices offer potential for teaching mathematics to students with DB and ID.
Figure 3. Effective practices for teaching mathematics to secondary students with moderate disabilities; effective practices for teaching skills other than mathematics to students with DB and ID, and the practices shared by both groups.
Now, let’s consider what mathematics instruction might look like using these practices with our students in the case studies. A link to an activity for teaching elementary mathematics for each of our case study students is provided below. Each activity is aligned to the Common Core State Standards (CCSS) for Mathematics (http://www.corestandards.org/Math/), Common Core State Standards for English Language Arts (http://www.corestandards.org/ELALiteracy/) as appropriate, and the Dynamic Learning Maps (DLM) Essential Elements (http://dynamiclearningmaps.org/content/essentialelements). The DLM Essential Elements are aligned to the CCSS, but are reduced in scope and complexity for students with significant cognitive disabilities“Significant cognitive disabilities” is not one of the 14 disability categories listed in IDEA (2004) for children aged 3 21 years (see below); rather, it is a term used to describe students with disabilities who are assessed using an alternate assessment based on alternate achievement standards and is inclusive of all students with a disability who meet the eligibility criteria for a state’s alternate assessment. In other words, the term “significant cognitive disabilities” can include students with a wide variety of disabilities. eligible to participate in their state’s alternate assessment linked to alternate achievement standards (AAAAS). The DLM framework is one of several extended standards frameworks and you should use what has been adopted in your particular state.
ActivityA Week of Weather
StudentSophia Researchbased Practices:

Brief description of activity: After reading a tactile book about weather, student constructs a graph about weather and responds to questions about the data. 
Photography by J. Brickhouse 
ActivityHoliday Smells (i.e., Interpreting Data from a Graph)
StudentDonnie Researchbased Practices:

Brief description of activity: Student responds to a variety of objects that have familiar smells and categorizes them into two categories (i.e., like, don’t like). Student then responds to questions about the two groups. 
Photograph by J. Brickhouse 
ActivityCoffee Business (i.e., Handling Money)
StudentStephen Researchbased Practices:

Brief description of activity: Students works with a peer to make and sell coffee as a school business. In the course of the job, student practices handling money. 
