by NCTM President Henry (Hank) Kepner
NCTM Summing Up, March 2010

It’s about Students’ Learning and Understanding of Mathematics!  

As your president, I’m concerned. Are we paying sufficient attention to students’ coherent mathematical learning while the media and current policymakers focus attention on the Common Core State Standards Initiative and on reports of groups of states banding together into consortia to contract for assessment packages?

We need to remember that students’ mathematical proficiency should steadily grow, building on  their prior knowledge and skills, as they learn to recognize connections within mathematics, gain facility in using multiple representations of concepts, and develop problem-solving skills in mathematics, recognizing its applications to the outside world. 

I would like to suggest several crucial areas in designing and implementing a coherent curriculum that will enable classroom instruction to focus on: 

• Learning progressions. Research and professional experiences provide suggestions for an orderly sequencing and development of content and mathematical experiences for students. Such progressions include careful sequencing of the content, developing skills, identifying connections across mathematical strands, using multiple representations, and relating the mathematics to its applications. Descriptions of learning progressions give “guidance about the depth of study warranted at particular times when closure is expected for particular skills or concepts” (Principles and Standards for School Mathematics, NCTM, 2000, p. 16).

• Sense making. NCTM advocates the importance of expecting students to make sense of the mathematics that they are learning and recommends that instruction should involve questioning, multiple representations, and examination of alternative approaches in order to push students to deeper understanding of mathematics. I cite this as a primary NCTM position, that mathematics competency is not limited to memorization and rote performance of procedures in isolation ( Focus in High School Mathematics: Reasoning and Sense Making , NCTM, 2009).

• Reasoning. Research suggests that students learn when they are actively involved in choosing strategies and defending their reasoning. (How People Learn, National Research Council). Informal arguments and making and justifying conjectures about mathematical concepts as they are developed are critical components of work in mathematics for both students and their teachers—just as for research mathematicians—as they strive for new understanding. For example, the introduction of linear functions should include discussion of the graphs of these functions and their applications.

• Mathematical connections. One characteristic of understanding is when students are able to connect current math topics to prior knowledge and skills and to other key mathematics topics, as well as to apply their learning to problems and contexts outside mathematics. 

• Cognitive demand. Giving tasks that demand high levels of thinking and implementing them in ways that maintain the cognitive demand is as crucial as helping students become facile with mathematical skills. Encouraging and expecting students to make connections between and among mathematical ideas, representations, strategies, and procedures moves instruction beyond mechanical processes and routine problems.

To present mathematics as a coherent whole for all students, we have a responsibility to attend to all of these elements. Each has a role to play in developing a coherent mathematics program based on an identified curriculum, instructional and teacher resource materials, professional development for teachers across and within grade levels and course sequences, and a clear alignment of assessment instruments.

NCTM is addressing these challenges and, through its publications, resources, and professional development opportunities, is providing essential information to teachers.Principles and Standards for School Mathematics, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, and Focus in High School Mathematics: Reasoning and Sense Making are sources that “provide students with a connected, coherent, ever expanding body of knowledge and ways of thinking” (Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, NCTM, 2006, p. 1). 

NCTM, in collaboration with state and local mathematics initiatives and the community at large, must work to ensure that students have opportunities to develop a coherent, connected view of mathematics and that instruction incorporates performance expectations that will allow students to gain a deep understanding of mathematics. “Alignments and coherence of these three elements—curriculum, standards, and assessment—are critically important foundations of mathematics education” (Guiding Principles for Mathematics and Curriculum Asssessment, NCTM, 2009).

We know that too many of our students leave our schools with a vision of mathematics as a set of unconnected and independent facts with no clear sense of how the ideas fit together nor of how mathematics can help them earn a living, guide them as citizens, or affect their daily lives. You, the Council, and I have the responsibility to see that our students receive a coherent mathematical experience as they progress through the grades, one that expands their vision of mathematics and their connections to it.