A few weeks ago a group of senior mathematicians, teachers, statisticians, and curriculum developers met in Boston to discuss the future of high school mathematics, revisiting issues addressed by a 2008 conference organized by the Center for Mathematics Education at the University of Maryland. This time, the Common Core State Standards was front and center of the discussion. Participants in the Boston meeting, sponsored by the non-profit Consortium for Mathematics and Its Applications, formulated a set of recommendations for progressive action in the field and drafted an essay to explain their ideas.
The essay was signed by: Jim Fey, University of Maryland; Sol Garfunkel, Consortium for Mathematics and Its Applications; Diane Briars, Intensified Algebra Project, University of Illinois at Chicago; Andy Isaacs, University of Chicago; Henry Pollak, Teachers College, Columbia University; Eric Robinson, Ithaca College; Richard Scheaffer, University of Florida; Alan Schoenfeld, University of California, Berkeley; Cathy Seeley, Dana Center, University of Texas; Dan Teague, North Carolina School of Science and Mathematics; Zalman Usiskin, University of Chicago.
Here’s their essay:
Results from the most recent Program for International Student Assessment showed once again that U.S. high school students are in the middle of the pack when it comes to science, mathematics, and literacy achievement. The findings quickly elicited an outburst of public hand wringing, criticism of U.S. schools and their teachers, and calls to emulate the curriculum and teaching practices of high achieving countries. Then, just as predictably, there were a variety of explanations why we cannot import the policies and practices of other quite different countries (e.g. , South Korea, Taiwan, Finland, and Singapore). Instead, schools were urged to redouble efforts along lines that have been largely ineffective for the past decade and are not common in any high performing country—a regimen of extensive standardized testing with mostly punitive consequences for schools and teachers that fail to make adequate yearly progress. Public attention to the challenge of international competition has already begun to fade and we will hear little about the meaning of the PISA results until the next “wakeup call” arrives.
What might happen if we tried something different this time around? Countries that have made real progress in their performance on international assessments share several characteristics. First and foremost is broad agreement on the goals of education and sustained commitment to change over time. In the United States there has been steady, if modest, improvement in student mathematics performance at the elementary and middle school levels on the National Assessment of Educational Progress (NAEP) and some improvement in results on college entrance examination tests (SAT and ACT) over the past two decades—a period when efforts have been guided by the National Council of Teachers of Mathematics (NCTM) standards for curriculum, evaluation, teaching, and assessment.
Over the past three years, 46 of the 50 U. S. states have been engaged in an effort to implement Common Core State Standards (CCSS) for mathematics and literacy. With respect to mathematics, those standards, prepared under the aegis of the National Governors’ Association with generous private financial support, are in many ways an extension of key ideas in the earlier NCTM standards. Despite understandable controversy about particulars of the CCSS and the processes by which they were developed and states were induced to adopt them, the Common Core standards provide a useful framework for further efforts. Partisan political pressures (from both left and right) are already leading some state governors to reconsider their participation in this national compact to improve education—before even the first assessments of progress are reported. But we believe that education policy makers and mathematics educators should resist the common wish for a quick fix and stay the course, modifying goals and efforts as results suggest such actions.
What should students, teachers, parents, and policy-makers look for in the emerging reform of high school mathematics? From our perspective—as mathematicians, teachers, statisticians, teacher educators, and curriculum developers with extensive experience in school mathematics innovation—there are at least four key elements of the Common Core program that provide a basis for productive change in U. S. high school mathematics:
Comprehensive and Integrated Curriculum. The traditional American high school mathematics curriculum consists of two year-long courses in algebra and a one-year course in geometry. The CCSS for mathematics retain essential elements of those topics, but they also prescribe significant attention to important concepts and skills in statistics, probability, and discrete mathematics that are now fundamental in computer, management, and social sciences. The Common Core guidelines describe an attractive integrated curriculum option—suggested by the common practice in other countries of addressing each mathematical content strand in each school year. That international curriculum design helps students learn and use the productive connections between algebra, geometry, probability, statistics, and discrete mathematics.
A broad and integrated vision of high school mathematics would serve our students better than the narrow and compartmentalized structure of traditional programs.
Mathematical Habits of Mind—For most people the phrase “do the math” means following standard algorithms for calculation with whole numbers, fractions, decimals, and the symbolic expressions of algebra. But productive quantitative thinking also requires understanding and skill in use of what the Common Core Standards call mathematical practices. To apply mathematical concepts and methods effectively to the kind of realistic problem solving and decision making tasks that PISA assessments highlight, students need to develop the habits of: (1) analyzing complex problems and persevering to solve them; (2) constructing arguments and critiquing the reasoning of others; (3) using mathematical models to represent and reason about the structure in problem situations; and (4) communicating results of their thinking in clear and precise language.
Developing important mathematical habits of mind should become a central goal of high school instruction, especially the process of mathematical modeling that is required to solve significant real-world problems.
Balanced Attention to Technique, Understanding, and Applications—One of the most common student views about mathematics is the belief that what they are asked to learn is not supposed to make sense and that it bears little relationship to the reasoning required by everyday life. Those views are expressed well in the whimsical rhyme about division of common fractions, “Yours is not to reason why, just invert and multiply,” and the common student question, “When will I ever use this stuff?” Unfortunately, many teachers encourage those beliefs about mathematics learning by suggesting that understanding and application of mathematical ideas and methods can only occur after rote mastery of technical skills.Findings of cognitive and curriculum design research over the past two decades challenge such conventional beliefs and common practices. Curricula and teaching that engage students in collaborative exploration of realistic problems have been shown to be effective in developing student mathematical understanding, skills, and problem solving simultaneously. These problem-based approaches in the classroom also develop the essential disposition to use mathematics as a reasoning tool outside of school.
Improved performance on international assessments like PISA are likely to result from moves toward curricula and teaching methods that balance and integrate mathematical techniques, understanding, and applications.
Information Technologies—Powerful tools that allow users to find and process information with mathematical methods are now ubiquitous in American life. But schools are only beginning to respond to the profound implications of this information technology for teaching and learning. If it is possible to simply ask your cell phone to perform any of the routine calculations taught in traditional school arithmetic, algebra, and calculus courses, what kind of mathematical learning remains essential? If those same tools can be applied to support student-centered exploration of mathematical ideas, how will the new learning options change traditional roles of teachers and students in the mathematics classroom and raise expectations for the mathematical challenges that students can tackle?
Personal computers, tablets, smartphones, and other computing devices will almost certainly transform school mathematics in fundamental ways. Intelligent response to that challenge will require creative research and development efforts and the courage to make significant changes in traditional practices.
If the content and teaching of high school mathematics are transformed in the directions we recommend, schools and teachers will also need new tools for assessing student learning. One of the clearest findings of educational research is the truism that what gets tested gets taught. PISA is not a perfect or complete measure of high school student achievement. Neither are the TIMMS international assessments, the NAEP tests, the SAT and ACT college entrance exams, college placement exams, or, quite likely, the coming assessments attached to the Common Core State Standards.
Some would respond to the inadequacy of current assessment tools by sharply curtailing high stakes standardized testing; others would actually increase the testing and raise the consequences for students and schools. It is almost certainly true that the best course lies somewhere between those extremes. We need new and better tools for assessing student learning, and we need to employ those assessments in constructive ways to help teachers improve instruction and to inform educational policy decisions.
Finally, we need to change the tenor of public discourse about mathematics education. If we are to reach the shared goal of preparing young people for productive and satisfying lives, we need to work together to develop progressive goals for school mathematics and high quality instructional resources. Most important of all, we need to dial down the acrimonious policy arguments and relentless criticism of schools and teachers. Teaching is one of the most important and demanding tasks for adults in our society, and teachers deserve our encouragement and support as they work to provide the best possible life preparation for their students.
Jim Fey Sol Garfunkel
University of Maryland Consortium for Mathematics and Its Applications
Intensified Algebra Project, University of Illinois at Chicago
University of Chicago
Teachers College, Columbia University
University of Florida
University of California, Berkeley
Dana Center, University of Texas
North Carolina School of Science and Mathematics
University of Chicago