Multiple Representations for NYS Grade 3-8 Common Core Mathematics Tests

Students listening

With the transition to the New York P12 Common Core State Standards (CCSS) significant changes have been made to the New York State Testing Program (NYSTP).  The 2013 Grade 3-8 Common Core Mathematics Tests will measure the CCSS for Mathematics.

The CCSS for Mathematics strongly emphasize that students gain deep conceptual understanding as well as procedural proficiency in math.  Likewise, CCSS stresses that students’ procedural knowledge and conceptual understandings are developed and demonstrated through application and modeling.  As such, it is crucial that in both instruction and assessment, students are required to move far beyond simple, algorithmic strategies to solve math problems.

Predictable math questions (items) that require only algorithmic strategies to be solved correctly are commonplace in assessments. In an effort to ensure that the New York State Testing Program (NYSTP) measures the deep conceptual understanding that CCSS demands, the specifications for the design of assessment items include multiple representations.

Multiple Representations

Multiple Representations (MR) are a broad set of specifications that describe, refer and symbolize the various, but not all, ways that math standards could be measured within the constraints of NYSTP.  The MR document specifies three overarching families of MRs:

  1. Procedural Skills: Procedural skills representations specifically apply to standards that reference verbs such as compute, solve, identify, interpret, use, make and find solutions. Procedural representations are most often multiple-choice questions that require students to apply and identify mathematical processes in various ways. 
  2. Conceptual Understanding: Conceptual understanding representations are applied to standards using verbs such as understand, explain, represent and describe.  As a result, these items require different combined mathematical practices depending on the given item type or item.
  3. Application: Application standards and items are unique within the Common Core.  There are standards that reference application, which are represented by application tasks. Also, there are application tasks that are used to represent standards for which application is not explicitly required.  Broadly speaking, application items require students to marshal both procedural knowledge and conceptual understanding to complete a task.

Each family of MRs is explained in the document.  The explanation includes a statement of how to identify standards that might be measured using the particular MR. The explanation also goes on to identify the types of math skills (e.g. application of process; explanation of a principle, etc…) that are appropriate to assess given that MR. A sample from the MR document has been included and annotated below:

 Math Representations Sample Annotated Text

1.       Family of Item Formats

2.       Explanation of how to identify standards that might be measured and types of math skills

3.       Specific formats

How NY uses Multiple Representation for Assessment

The MR articulate the various ways to develop math questions.  As such, the MR offer item developers guidance on interpreting the standards, the focus of measurement for each standard, and ways to measure different aspects of each standard through different approaches.  Item developers are charged to use different formats to measure a single standard in order to more faithfully measure the whole intent of each standard.  The MR approach helps NY to ensure that the assessments measure more than simple algorithmic strategies.

How Instructors can use Multiple Representation for Classroom Instruction

The MRs can be used to help an educator plan instruction with a variety of different approaches to the standard in mind in order to teach to the whole standard, as referenced above.  Knowing that assessment items, over time, will assess a given standard through multiple formats, educators should approach instruction of a given standard through multiple formats and perspectives.  However, the State assessments do have its limitations. Instruction should not be limited to only those formats that fit within the constraints of large-scale assessment.

When planning instruction for a given standard, instructors should think about all of the multiple perspectives from which a standard can be interpreted, which means that instruction should approach standards from a:

  • Conceptual,
  • Procedural, and
  • Application lens (family of item formats).

This type of thorough instruction will lead to foundational student understanding of each CCSS. This will enable students to apply their understanding to all of the specific formats listed in the MR document.  Ultimately, teaching with the MR approach results in instruction that is more holistic.  Student understanding becomes less about simple mastery and more about application of that understanding in a variety of ways. Instructors can access the curriculum modules available on for guidance on developing holistic performance based and classroom assignments.

View the sample items from grade 7 to see examples of this holistic approach.

Downloadable Resources

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