by NCTM President J. Michael Shaughnessy
NCTM Summing Up, August 2011
Last week over 700 high school mathematics teachers and leaders participated in the first NCTM Interactive Institute: Infusing the Classroom with Reasoning and Sense Making. This three-day institute offered task groups, breakout workshops, discussion groups, featured speakers, and keynote presenters. The interest and commitment on the part of the attendees was truly inspiring as they developed plans to bring more emphasis on student reasoning to their classrooms and teaching practices. I tip my hat and salute all of those who attended for their hard work and professionalism during the institute. I am looking forward to hearing about their follow-up experiences as they implement reasoning and sense making in their classrooms this year, and share their efforts on the conference discussion blog set up on the NCTM website.
Reasoning and Sense Making—Looking Back, Looking Forward
The NCTM initiative on reasoning and sense making has been a focal point throughout my term as NCTM president, and it will continue to be a major part of NCTM’s ongoing work for the foreseeable future. The importance of focusing on students’ reasoning about their mathematics is not something new in mathematics education; it has been a major theme in the teaching of mathematics in North America for a long time. Consider, for example, this excerpt from one of the first books on mathematics education and the teaching of mathematics published in North America:
It is very important for teachers to lead scholars into the habit of attending to the process going on in their own minds while solving questions, and of explaining how they solve them. … It is next to impossible for a person to direct another’s thoughts unless he [or she] knows the channel in which they are already flowing.
—Warren Colburn, Teaching Arithmetic in the Method of Pestalozzi, 1830
For Colburn, reasoning is student centered. It begins with the students themselves. Listening to student thinking is critical for teachers because it provides feedback on thewhat, how, and why of students’ thinking. Nearly a century later, in a famous report on mathematics in the secondary schools, the Mathematical Association of America echoed a similar message:
Continued emphasis must be placed on the development of processes and principles in the solution of concrete problems, rather than on the acquisition of mere facility or skill in manipulation. The excessive emphasis now commonly placed on manipulation is one of the many obstacles to intelligent progress.
—MAA, Reorganization of Mathematics in Secondary Education, 1923
Over the past several decades, the NCTM Standards documents have consistently communicated a loud and clear message about the critical importance of soliciting and sharing student reasoning and sense making in classrooms. For example, NCTM asserted the following 31 years ago:
Students should be encouraged to question, experiment, estimate, explore, and suggest explanations. Problem solving, which is essentially a creative activity, cannot be built exclusively on routines, recipes, and formulas.
—An Agenda for Action, NCTM, 1980, p. 4
A view of mathematics as reasoning is prominent across the K–12 grade bands in both the 1989 NCTM Curriculum and Evaluation Standards for School Mathematics, and in Principles and Standards for School Mathematics, both of which included recommendations that students be given opportunities to do the following:
- Explain their thinking
- Believe that mathematics makes sense
- Make and evaluate mathematical conjectures and arguments
- Appreciate the power of reasoning as a major part of mathematics
- Construct proofs of mathematical assertions
Scaling Up Our Reasoning and Sense-Making Efforts—What’s Different This Time?
Although an emphasis on reasoning and sense making is not new for NCTM or for mathematics education as a whole, it has now matured over several centuries. The possibility of enacting a more widespread implementation of reasoning and sense making practices in our classrooms—of scaling up past efforts so that a focus on reasoning and sense making becomes the norm and not the exception in our classrooms—is more within our reach than ever before. For one thing, with the help of research, we now know more about some of the critical aspects involved in fostering student reasoning, such as (1)meta-cognition (reflecting on and identifying one’s own thinking processes), (2) the role of student discourse in promoting student reasoning, and (3) the importance of providingequitable opportunities for all students to engage in and share their reasoning. Furthermore, we now have a clearer recognition that specific content areas elicit and require different types of mathematical reasoning habits—algebraic, geometric, and statistical reasoning habits. The content areas involve different reasoning domains in which particular habits of reasoning or types of evidence-based arguments arise (Focus on High School Mathematics: Reasoning and Sense Making, NCTM, 2009).
Promoting reasoning and sense making has now gained more momentum and has more political capital than ever before. NCTM’s Process Standards, the Common Core’s Mathematical Practices, and the messages found in NCTM’s Focus in High School: Reasoning and Sense Making share a lot of common ground. Four of the eight Standards for Mathematical Practice in the Common Core State Standards are explicitly concerned with reasoning:
2. Construct viable arguments and critique the reasoning of others.
3. Make sense of problems and persevere in solving them.
5. Look for and express regularity in repeated reasoning.
6. Reason abstractly and quantitatively.
NCTM’s Standards documents and the Common Core State Standards for Mathematics (CCSSM) agree on the importance not only of developing student reasoning but also ofassessing it. The two assessment consortia for the Common Core Standards are constructing tasks that will include the assessment of student reasoning—as outlined in the Standards for Mathematical Practice. Forty-five states and the District of Columbia have agreed to adopt CCSSM, a demonstration of unprecedented unanimity in education in the United States.
As educators respond to the political push for an emphasis on student reasoning, NCTM is developing resources for teachers to support the implementation of a focus on reasoning and sense making and to help sustain efforts to foster it over time. An online library of reasoning and sense-making tasks has been launched on the NCTM website so that teachers can access reasoning tasks along with supporting implementation materials. Several tasks have been posted (NCTM Standards) under the Focus on High School Mathematics: Reasoning and Sense Making link. A collection of video clips of secondary students engaged in reasoning in their classroom is also under development at NCTM. These clips are part of clusters of materials that will support teachers in their classrooms. The clusters will include examples of student work, clips of student reasoning, debriefing sessions of reasoning tasks with teachers, and implementation strategies for teachers and teacher leaders. Access to the video cluster work will be available on the NCTM website in the coming months.
This fall as you return to your classrooms and plan for the school year, I urge you to make changes in the way that you teach your students so that you can implement the ideas and new resources that have been forged by the NCTM reasoning and sense making initiative. It’s time for us all to become the lead soldiers in the reasoning and sense-making effort. It’s time that we give ourselves permission to implement a culture of reasoning and sense making in our mathematics classrooms, time that we work to spread a culture of nurturing reasoning among our colleagues while mentoring new incoming teachers to do so as well. Let us all go forth and reason together!