In this module we want to start by reviewing what we already know about teaching math to students with and without disabilities. When planning instruction for all students it is important to base our instruction on evidencebased research practices. Think about LaTia; it will not be easy to plan instruction for her that is linked to the Common Core State standards unless we understand what teaching linear equations means, and we know some instructional strategies to teach those skills. Additionally, we know that she has mastered limited early numeracy skills, as we plan her instruction we need to think about how we will also support her understanding of the number concepts (e.g., concept of addition, concept of more, concept of equal). To begin let’s learn more about how math learning can be articulated for students with significant intellectual disability.
Now, that you are familiar with some of the research on teaching math to students with significant intellectual disabilities, take a few minutes to explore the Common Core State Standards (CCSS) website (http://www.corestandards.org/Math). Pay special attention to the scroll on the side of the website, you will note that each of the components previously outlined by NCTM are still incorporated into the CCSS. Additionally, you will notice the section “Standards by Domain”. Note the focus on breaking down math skills by conceptual domains within Algebra or Geometry. The major change in math education with the new standards is a stronger focus on building early conceptual understanding of numbers and the “processes” that help students learn concepts. See Figure 1 below for a brief comparison between the NCTM focal points and the Common Core State Standards in math. Please note that the CCSS and NCTM standards share many of the same traits. The Focal points are still a great resource for connections among topics, while the CCSS provide us with greater details regarding expected learning at each grade level.
http://www.nctm.org/standards/content.aspx?id=270
One challenge to teaching mathematics especially in upper grades is the content. As special educators, we often are not as familiar with the math standards and how they are taught. Working with general education teachers is a great strategy for gaining content knowledge and strategies. Becoming knowledgeable about resources available is another way. To learn more about the Common Core State Standards and their “Practices”, you can review supports located on the Common Core website, or visit the MAST module on Teaching Mathematics to Diverse Learners. Take some time to learn more about the good math practices and how standards build in complexity to gain content knowledge over time.
Let’s look at an example of a math Extended Content Standard for 5th grade Geometry. (Table 1 illustrates the grade level achievement Common Core State Standard, and next to that is the Extended Essential Standard written for students working on alternate achievement standards (adapted from http://www.ncpublicschools.org/docs/acre/standards/extended/math/k5.pdf. While all students access the general curriculum, students with intellectual disabilities may need instruction in mathematics linked to alternate achievement standards.
Let’s look at an example of an extended standard for 5th grade math. For an example from North Carolina, see Table 1 which illustrates the grade level achievement Competency Goal and Objectives; and below that is the Extended Standard written for students working on alternate achievement standards. Students with significant intellectual disabilities need instruction in math linked to the general curriculum standards with an alternate achievement standard.
Table 1.
5th Grade Math Geometry 


Common Core State Standard  Essence  Extended Essential Standard 
Graph points on the coordinate plane to solve realworld and mathematical problems.
CCSS.Math.Content.5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and xcoordinate, yaxis and ycoordinate). 
Graph Points 
Graph points on a coordinate place.
Demonstrate and describe examples of the intersection of figures. 
These types of content standards help delineate and prioritize the instructional efforts of teachers for students. Not all states provide extended content standards; however, all standards can be extended by considering the essential knowledge and skills (e.g., the essence) to be taught and focusing instruction around the “big idea” of the standard.
Based on the standard above, let’s take a minute to look at Table 2 for an example of aligned instruction for LaTia. Remember, we want to make sure that our instruction matches with the standard in content and performance (as much as possible). LaTia is working towards alternate achievement measures.
General Education Expectation  Example of Student Achievement 
Common Core State Standard:
Graph points on the coordinate place to solve realworld and mathematical problems. Essence (Clarifying Statement):Graph Points Extended Standard:Demonstrate and describe examples of the intersection of figures. Content:Intersection of geometric figures in the coordinate plane. Performance:Demonstrate and describe. 
Option 1: LaTia will complete an eight piece puzzle by orienting the shapes in the correct locations. Content? NOPerformance? NO 
Option 2: LaTia will identify intersecting figures located in a coordinate plane. Content? YESPerformance? Some 

Option 3: LaTia will identify the point of intersection between two figures located in a coordinate plane and describe an example. Content? YESPerformance? YES 
For example, in Option 1 LaTia is completing a puzzle to access the geometry standard. This is not a good example of how LaTia is working on instruction linked to the content or the performance. Orienting shapes of a puzzle does involve geometric shapes, but doesn’t have anything to do with the intersection points of the shapes or coordinate places. This is most likely not a skill taught in 5th grade either. Option 2 begins to align to the content but still is not a very good example of performance that is linked the grade level standard. LaTia is being asked to identify figures that intersect each other, but not the place of intersection. Finally, Option 3 is an example that is both aligned to the content and performance. It is important when aligning instruction that both the content and performance (as much as possible) match the state standard, otherwise we may spend too much time teaching students skills that don’t really match to grade level standards.
Now it is your turn. Look at the standard below and think of an example that is aligned to both the content and performance.
Subject 7th Grade Mathematics  Grade Level: Statistics & Probability  
Goal(s):CCSS.Math. Content.7.SP.B.3 Draw informal comparative inferences about two populations. 

Essence: Compare data 

Extended Standard: Compare data from two picture graphs, line plots, or bar graphs. 
What is the content? _________(Jot this down on a piece of paper.)
Pick One
Pictures of data  Various types of graphs  Data within various graphs 
What is the performance?______ (Please write this answer on your paper.)
Pick One
Identify  Observe  Compare data; make inferences 
Good! The content is the data within two picture graphs, line plots or bar graphs. The performance is to compare that data with each other to draw comparative inferences. When math instruction is aligned to grade level standards it will include all of the bold words above. Remember, not all students will be able to observe, measure, and demonstrate knowledge in the same way. You may need to think about how students are able to “show what they know” and try to plan instruction that is Universally Designed for Learning. For more information about Universal Design for Learning refer to the two other MAST modules on Universal Design. If a student is not yet able to perform all three performance skills immediately, it may be appropriate to introduce each skill one at a time, building skills throughout the school year.
Based off of this example, would it be a wellaligned skill to ask LaTia to look at a bar graph and indicate how many dollars it cost to go to the movie. Does it match in content? Yes, it does match the content. Does it match in performance? No, it does not match the performance. LaTia needs to build the skills to compare data. A better example would have been for LaTia to use comparative bar graph data to answer a question. (e.g., LaTia would indicate which activity cost more money).