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Creating, Critiquing, and Revising Arguments

Description of the Practice:  

A key aspect of reasoning and sense making is creating, critiquing and revising mathematical arguments. These are central to NCTM’s Reasoning and Proof Standards (PSSM, 2000) and mathematical practice 3 of the Common Core State Standards for Mathematics, “Construct viable arguments and critique the reasoning of others” (CCSSI, 2010).  These practices are all part of justifying a mathematical statement or process and provide students with opportunities to understand mathematics as a discipline that rests on reason and logic. 

As students participate in creating, critiquing and revising mathematical arguments, they engage in a range of specific activities including: 

  1. investigating and generalizing relationships from patterns;  
  2. developing definitions of mathematical objects through examination of physical representations and comparisons of the similarities across representative cases or sets of similar examples;  
  3. stating generalized relationships as conjectures in precise terms;  
  4. using counterexamples to disprove and refine conjectures; and  
  5. explaining why a conjecture is true by using definitions and mathematical properties. (Connecting the NCTM Process Standards and the Common Core Standards for Mathematical Practice, p. 31)  

Common Core Mathematics Practice 1 ends with a statement about the need for students to understand others’ solution strategies as well as understand the mathematical connections between different strategies.  In fact in meeting the Standard, not only will students learn to understand and evaluate others’ strategies, but when they engage in mathematical arguments where they must justify their own solutions, they will gain a better mathematical understanding as they work to convince their peers about different points of view (Connecting the NCTM Process Standards and the Common Core Standards for Mathematical Practice, p. 15) 

Note that making conjectures, which may involve inductive reasoning and pattern searching, is just the beginning of justifying a mathematical statement or process. Students engaged in creating, critiquing and revising arguments must also seek to make sense of why an observed pattern holds or does not hold true.  

In the classroom, students’ reasoning will be presented at a variety of levels of sophistication. Over the course of a lesson, the level of sophistication and quality of argument may be refined and become more complete and mathematically sound.  It is important that teachers are deliberate in promoting these practices within their classroom: 

... teachers play an important role in developing students’ abilities to generate mathematically sophisticated arguments by continually pressing students to formulate conjectures and provide explanations as to why they are true. (Connecting the NCTM Process Standards and the Common Core Standards for Mathematical Practice, p. 32) 

A Mathematician's View of the Practice (Wade Ellis, Jr., West Valley College): 

Prove or disprove, salvage if possible. -Arnold Ross 

In working with a problem situation, mathematicians make conjectures based on reasonable evidence. The conjecture and argument on which it is based are then shared within a community of peers for review. If an error or flaw in the argument is identified or a counterexample to the conjecture is generated, mathematicians then try to understand the source of the error. Based on this feedback and analysis, a decision is made to revise the argument and/or conjecture or to abandon the argument and start a new line of reasoning. This sets in motion another cycle of creating-critiquing-revising, a process that may be repeated several times over many hours, days, years, or even centuries before an ironclad argument is made that cannot be successfully contradicted. The habit of mind of creating, critiquing, and revising arguments should be developed in students if we are to support them in learning mathematics through reasoning and sense making and improve their ability to think and interact like mathematicians. It connects to the mathematical practice of “constructing viable arguments and critiquing the reasoning of others” (see pp. 6-7 of the Common Core State Standards for Mathematics).

Cluster 1 - Ms. Hauser's Hexagon Train Task:  

The materials in the Ms. Hauser Hexagon Train Task cluster offer the opportunity to witness how one teacher helped her students to develop the habit of creating, critiquing, and revising arguments 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.

Readings About the Practice: 

Additional insights into the practice of creating, critiquing, and revising arguments are found in the following two excerpts from NCTM publications. 

Connecting the NCTM Process Standards and the Common Core Standards for Mathematical Practice, chapter 3 

Focus in High School Mathematics, pp. 9-11 

And these articles offer ideas for teacher strategies that support students in creating, critiquing, and revising arguments.