The Process Standards
According to the NCTM, their process standards ”highlight the mathematical processes that students draw on to acquire and use their [mathematical] content knowledge.” The process standards are Problem Solving, Reasoning and Proof, Communication, Connections, and Representation. I wrote about them here.
Students require opportunities to engage with these mathematics processes to develop proficiency in mathematics. Aside from brief exposure to problem solving for the odd word problem they may have encountered, most students have not needed to consider these processes.
To increase awareness of the NCTM process standards, in whole and in part, I plan to post about one of the five standards over the next few days. Today’s NCTM process standard consists of Representation.
NB: While the Common Core State Standards for Mathematics (CCSSM) includes a set of eight mathematical processes, they originated in the NCTM process standards as well as the Five Strands of Mathematical Proficiency published by the National Research Council (NRC). I will write about the CCSSM mathematical practices later.
In Principles and Standards for School Mathematics, the NCTM states: “The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas. Consider how much more difficult multiplication is using Roman numerals (for those who have not worked extensively with them) than using Arabic base-ten notation. Many of the representations we now take for granted—such as numbers expressed in base-ten or binary form, fractions, algebraic expressions and equations, graphs, and spreadsheet displays—are the result of a process of cultural refinement that took place over many years. When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically.”
They go on to state “Representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships; in communicating mathematical approaches, arguments, and understandings to one’s self and to others; in recognizing connections among related mathematical concepts; and in applying mathematics to realistic problem situations through modeling. New forms of representation associated with electronic technology create a need for even greater instructional attention to representation.”
Notice how representation underpins conceptual understanding, communications, connections, and problem solving. All of these processes are assisted by an effective representation.
The NCTM point out that “the term representation refers both to process and to product—in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself.”
It is essential that students have opportunities to view and to create multiple representations of mathematics graphically, numerically, algebraically, and verbally. These opportunities should start in kindergarten and continue through high school. At times, teachers should present a representation explicitly, while at other times, they should guide students to “discover” how best to represent a mathematical model.
The key elements for the NCTM representations process standard follow. Students should engage with each of these in all of their mathematics courses.
- create and use representations to organize, record, and communicate mathematical ideas;
- select, apply, and translate among mathematical representations to solve problems;
- use representations to model and interpret physical, social, and mathematical phenomena.
To help administrators, colleagues, parents, and students better understand the NCTM process standards, I created a single slide summarizing the key points for each process standard. The slide for Representations follows.