Practice Standard 3 requires students to “construct viable arguments and critique the reasoning of others.” Teachers need to model and set parameters for healthy and productive classroom discussions. Often this skill is not taught thoroughly, if at all, because of time constraints or behavior management issues. When students learn to defend their thinking and critique the reasoning of peers from a young age, they are more likely to maintain and improve this behavior in later years.
How will these resources help?
Student discourse may look different in each classroom, and may require teachers to implement “discussion rules” and model tone and questioning practices that are appropriate for their own situations. The videos demonstrate lessons during which students explain their reasoning and reflect upon solutions of their peers. The articles describe classroom experiences in which students engage in constructing arguments and critiquing the reasoning of others. They include suggestions for how to increase student discourse, encourage students to construct viable arguments, and assist students as they learn to critique the reasoning of others. The resources also provide strategies for prompting students to defend an answer, define a concept, or create a rule.
The importance of discourse
Valuable student discourse requires children to think about problems in a variety of ways and to defend their solutions against the critique of their peers. Student discourse promotes a higher level of thinking than merely stating answers or answering knowledge-based questions. Students can arrive at a “correct answer” but still not understand the concept; by defending their answer they may deepen their own understanding and demonstrate that understanding to the teacher. This process may enable classmates to gain a better understanding of the concept, confirmation of their own thinking, or an appreciation of other ways to arrive at the same answer.
This web page is the third of the eight pages of the Inside Mathematics initiative to help guide and support educators in understanding the math practices and to see instances of teachers engaging students in formulating, critiquing and defending arguments of mathematical reasoning. Included are various classroom connections: classroom observations and video examples in various grade levels of teachers engaging students. The video clips are excerpts from public lesson: "Numeric Patterning" (cataloged separately) and include links to a teacher commentary, transcript, and a related math task with examples of student work.
This professional development video (video 2) from Minnesota Math and Science Teacher Academy Center in Region 11 presents an algebra training session on justification. Viewers watch Dr. Terry Wyberg discuss opportunities to focus on what constitutes an appropriate level of justification for students through a discussion of the conjecture: An odd number plus an odd number is always an even number. Dr. Wyberg presents the need for teachers to extend their students' levels of justification from appeal to authority ("My teacher told me!") to using an illustrative example where words, symbols or diagrams explain their thinking; the teacher helps student pull out a generalized argument or proof. The math education textbook: “Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School" is referenced in this video.
This professional development video ( video 1) from Minnesota Math and Science Teacher Academy Center in Region 11 presents an algebra training session on justification. Viewers watch Dr. Terry Wyberg, a methods professor at the University of Minnesota, discuss five statements after teachers have had an opportunity to decide if each statement is always, sometimes or never true and a reason why. Dr. Wyberg shares that in math a response of always or never mean you need justification. Justification is central to mathematics and even young children cannot learn mathematics without engaging in justification. He also presents information about the use of math conventions with the use of algebraic representation for his audience.
This 19-page article (pdf) by Deborah Loewenberg Ball and Hyman Bass is a chapter from A Research Companion to Principals and Standards for School Mathematics, published by NCTM. The authors discuss the foundations of mathematical reasoning and proof and support their position that it is central to understanding and using mathematics. Two episodes from a third grade class help illustrate how children can develop the ability to reason. Ball and Bass suggest teaching practices that foster this development.
This page from Suzanne Alejandre's Math Forum blog has links to nine short videos (5 minutes or less) of Ms. Alejandre implementing the "Wooden Legs" Problem of the Week with a fifth grade class. The "Notice/Wonder" strategy is used to introduce the problem, and additional materials describe other problem solving strategies. The blogpost describes the goals of the lesson and also includes links to the teacher materials including the problem, solution, sample student answers, a scoring rubric, and teaching suggestions. Suggested browsers are Chrome, Firefox, or Safari. (It's been reported to us that when using IE9 (PC) the videos do not display.)
This article discusses an inquiry practice that encourages students to develop their mathematical thinking through discussion. The author describes examples of students taking ownership of their learning when they are encouraged to justify their positions through sound mathematical reasoning. This practice builds a foundation for formal proofs.
In this 3-page article (PDF) for the "CMC ComMuniCator" Volume 37, Number 1, Suzanne Alejandre discusses teaching strategies that increase the amount of student participation and class discussion. Alejandre discusses the difficulties of lack of time, lack of fluency in language, and habits of passive learners as contributors to the "silencing" of students. She encourages teachers to begin class discussion with scenarios that engage learners and encourage productive questioning early so that the habits have a strong foundation. Cataloged separately is a video of Alejandre discussing this topic further: Suzanne Alejandre at CMC-North Ignite.
The authors of this 18-page article discuss their efforts to explore the implications of using rich mathematical tasks and promoting mathematical discourse, as proposed in the NCTM's teaching standards. While team-teaching in a third grade class, they analyzed the relative merits and challenges of presenting children with convergent and divergent tasks. The article includes anecdotes to illustrate the kinds of thinking and communication that children engaged in, and the empowerment they exhibited, as they explored questions about negative numbers and regrouping in subtraction.
This 4-page research-based monograph discusses how teachers can support meaningful student dialogue in the math classroom. The author summarizes recognized benefits of student interaction and the challenges facing teachers in promoting it. She outlines the teacher's role in facilitating "math talk" and offers strategies and guidelines for encouraging high-quality student interaction. The author includes an extensive list of references for further exploration.
This article states that research shows 93% of teacher questions are focusing on recall of facts and that type of questioning does not stimulate the mathematical thinking of students who are engaged in open ended investigations. To improve the quality of teacher questioning techniques, the author suggests dividing questions into four main categories: starter questions, questions to stimulate mathematical thinking, assessment questions and final discussion questions and then to follow with a rigorous question. A link to an addendum to the article provides a table of generic questions that can be used by teachers to guide children through a mathematical investigation, and at the same time prompt higher order thinking, as espoused by Bloom and others.
This brief article describes the components of high-quality mathematical discourse that engages students and fosters conceptual understanding. The author discusses the interrelationship of the CCSS Mathematical Practices and provides a list of references.
In this 8-page article (PDF) Kastberg and Frye explain how sociomathematical norms can improve students' mathematical proficiency. The article focuses on a 6th grade class working through a ratio comparison problem. In this classroom students had been encouraged to question and challenge their own solutions as well as peer's solutions, leading to deeper mathematical understanding through valuing the differences in problem solving and reasoning.
The authors of this article, Phyllis and David Whitin, emphasize the importance of children representing their mathematical understanding in many different ways. These modes of communication include building, writing, drawing, and talking. The article is from the NCTM journal Teaching Children Mathematics.
This article from the NCTM journal Teaching Children Mathematics describes the process that teachers used to increase communication during problem-solving investigations. The authors, Melissa Anderson and Deena Little, provide examples of students’ struggles and successes.
What does a new policy publication for high school mathematics have to do with you and elementary school mathematics? Everything! Learn why from authors W. Gary Martin and Lisa Kasmer in this article from the NCTM journal Teaching Children Mathematics.