Reasoning and Sense Making with Functions: Using Multiple Representations

Description of the Practice:  

  When students are introduced to the concept of function it is important that they have opportunities to make connections among various representations of functions as a means to support their understanding of this essential aspect of mathematics. “A major responsibility of teachers is to create a learning environment in which students’ use of multiple representations is encouraged, supported, and accepted by their peers and adults” so that students see the use of tools as appropriate and helpful (NCTM 2000, p. 139). 

  As the authors of Focus on High School Mathematics make clear, the following actions, when promoted within a context of reasoning and sense making, provide access for a greater proportion of students to build conceptual understanding of and procedural fluency with functions than an exclusive focus on the use of symbolic representations: 

Representing functions in various ways, including tabular, graphic, symbolic (explicit and recursive), visual, and verbal; 

  • making decisions about which representations are most helpful in problem-solving circumstances; and 
  • moving flexibly among those representations. (FHSM p. 41) 

  Asking students to move between representations, to explain how different representations of the same function are related, and to describe the mathematical insights afforded by one representation over another will help them develop deep knowledge of functions and how this powerful mathematical tool can be productively utilized in problem solving situations. 

  The practice of moving between multiple representations of functions reflects an important content standard in the Common Core State Standards for Mathematics (HSF-BF.A.2 “Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.”).


A Mathematician's View of the Practice (Wade Ellis) 


Cluster 1 - Ms. Burkhart's Tabletop Tiling Task


Common Core State Standards for Mathematics 

Focus in High School Mathematics 

Readings about the Practice  

Herbel Eisenmann - Using StudentContributions 

Diezmann - Promoting the Use of Diagrams 

Peterson - Linear and Quadratic Change: A Lesson from Japan 

Reeuwijk - Students’ Constructionof Formulas in Context 

Davidenko - Building the Concept of Function from Students’Everyday Activities 


Lesson 1: Mrs. Burkhart’s Tabletop Tiling Task 

The materials in the Tabletop Tiling Task offer the opportunity to witness how one teacher helped her students to develop the habit of using multiple representations of functions as they worked on a mathematical task. It is recommended you take time to first go through the task before exploring the classroom records of practice. For those leading professional development, please consult our professional development guide.