As parents and teachers struggle to improve their children’s mathematics education, they are getting mixed messages on what constitutes quality instruction in that critical subject. Consider, for example, the release in September of the National Council of Teachers of Mathematics’ “Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics.” The New York Times called that document a “major shift” in math education. On The Wall Street Journal’s front page, it was hailed as a “remarkable reversal” of the math group’s previous position. It was neither.
-Dave Cutler
“Focal Points” represents an integration of approaches. It follows in the best American tradition of making course corrections to achieve progress. (“Math Organization Attempts to Bring Focus to Subject,” Sept. 20, 2006.) The report addresses a major problem faced by our population: variation among standards in the 50 states. It provides teachers and schools with advice on the most significant concepts and skills at each grade level, identifying three “focal topics” per grade. While it’s true that five of the 24 topics target quick recall and fluency of basic operations, the other 19 stress the importance of deep conceptual understanding and relationships among key ideas.
The report remains committed, moreover, to the organization’s current standards, found in “Principles and Standards for School Mathematics” (2000). It cites that document repeatedly in its introduction. As one of the “Focal Points” authors, University of Georgia mathematician Sybilla Beckmann, explains, despite the media’s preference for conflict, in math “there’s a lot of agreement about what students need to know.”
We cannot afford to waste time on polarization. What is important is that we pragmatically address critical target areas to improve mathematics education. We cannot be distracted from our primary mission—to match tactical initiatives in other, newly technological societies that are snatching our competitive advantage in innovation—while we bicker over modest differences in approach. Focal points are a start, but concentrating on standards alone will be insufficient. Standards are but one half of a pair of reform bookends that support our current mathematics education system. Tests are the other half. This “bookends” approach will fail unless high-quality, grade-level standards are matched with high-quality measures for assessing progress in meeting those standards.
There are critical weaknesses in the testing system. Chief among them is the failure of most assessment systems to inform teachers of specific shortcomings in their students’ learning. Tests are frequently neither scaled nor equated at the level of subtopics—numeration, measurement, probability and statistics, or geometry, for example. As a result, the difficulty level of these clusters of items varies from year to year. Low scores at the subtopic level send teachers and parents scurrying to improve topic instruction. But poor (or strong) performance on a topic is as likely to be the result of harder (or easier) test items as of deterioration (or improvement) of instruction from one year to the next.
We cannot be distracted from our primary mission while we bicker over modest differences in approach.
High-stakes test construction also narrows the content, depresses the cognitive level, and encourages teaching to the test. We see test scores gradually rising, yet the detrimental effects of teaching to the test resurface whenever a new test is used and failure skyrockets. And too many states fail to release actual test items with results. Withholding these denies teachers and students valuable feedback about the test and is unethical. It obscures scoring mistakes and permits poor or ambiguous test items to escape scrutiny and correction. Though new tests are expensive to generate each year, item banks and automated creation of equivalent test items make it economically feasible. To whom are the testing companies accountable?
Quality outcome measures also are determined by the test-data quality, the fairness of data use, and the determination of what is and is not acceptable performance. Professional standards dictate that a single measure should never be used to determine high-stakes consequences for children. (Multiple administrations of a single test still constitute a single measure.) Disaggregating scores into demographic categories can shine light on gross inequities, but reaching feasible remedies will require more-sophisticated approaches. Improved data analysis and display methods permit us to look deeper at what is happening in achievement than reports of the percentages of students who pass. We need to examine distributions of scores in their entirety and compare performance on diverse tests. The federal No Child Left Behind law, with its emphasis on a modest passing rate on mandated tests, implicitly authorizes us to leave behind up to 50 percent of students if they do not pass those tests. These students drift away steadily, with the result that graduation rates in this country now approach a scandalously low 68 percent (many minority groups’ rates hover at only 50 percent).
The test data, moreover, should be easily accessible for exploration and challenge by advocates for particular subgroups. Our society supports endless real-time data analysis for every major sporting event, yet can’t seem to provide adequate data to guide fair educational practice.
Even with improved linkages between standards and tests, however, the bookends approach to reform will be insufficient to the task of fixing American mathematics education. The missing piece is improvement in how the subject is taught to students on a daily basis in the majority of classrooms across the United States. This demands that we reject the long-standing federal neglect of the instructional core: the curriculum, pedagogy, and formative assessments that make up everyday classroom practices in K-12 math. It has been politically expedient to ignore the instructional core—to maintain the appearance of local autonomy and low federal involvement, and avoid the investment required for improvement. But that has resulted in public education systems’ withering while the government trumpets its devotion to “leaving no child behind.”
Strengthening the instructional core demands long-term, strategic federal investment and actions. The following seven actions should be a high priority:
• Identify explicit learning trajectories for key concepts around which to organize instruction. Learning trajectories are the next clear step from the curricular focal points. These are not simple developmental or cognitive pathways, but rather are the results of careful curricular design combined with the strategic use of technological tools and hands-on activities.
• Strengthen the diagnostic capacity of classroom assessments to inform instruction. Teachers need access to item analyses that show students’ common errors. Distractors (the choices on multiple-choice items) must be designed to be sensitive to common misconceptions. Results from such measures must be explicitly linked to curriculum, standards, and learning trajectories.
• Improve instruction by improving teaching capacity, using every method imaginable. Teachers are central to improvement, but our mathematics education capacity is severely limited. We must jointly address teacher quality and capacity. Arguably, elementary school students get better art, music, and physical education instruction than math and science teaching. By switching to math specialists in the upper-elementary grades, we could concentrate mathematics professional development on one-third as many teachers.
The missing piece is improvement in how the subject is taught to students on a daily basis in the majority of classrooms across the United States.
To improve capacity and salaries, and to reward excellence, we should implement mandatory, paid, two-month professional-development programs run by content-focused experts, teaming up our most talented teachers with university faculty members as instructors. The work of teaching must be redefined to create career ladders that incorporate differentiated instructional roles and innovative combinations of lecture, small-group labs, practice, and remediation to maximize our limited capacity. Those who lead our teacher-preparation institutions must bring content-oriented faculty to the fore.
• Develop early-childhood-intervention programs that ensure all students enter school ready to benefit from mathematics instruction. Increases in talent and proficiency will accrue only from broader participation. Too many urban and rural children begin school inadequately prepared to learn mathematics. This contributes substantially to inequities, and to the early and unacceptable narrowing of the human-resource pipeline.
• Aggressively develop and deploy new technologies to support learning and engage students. We are losing our national competitiveness in technology, yet it is a major attractor of student interest. Foolishly, we are missing countless opportunities to develop compelling electronic materials, assessments, and tutoring systems built around high-quality mathematical tasks and skills that professionals use on a daily basis, instead of relying on repetitious drill-and-practice management systems.
• Develop viable, proven means to help underperforming students recover proficiencies. Virtually no high schools have strategies for student re-entry into higher mathematics sequences. It is critical to link a diagnostic system to transparent and accessible pathways for recovery.
• Make low- or no-cost programs available for accelerated pursuit of mathematical study. Universities need incentives to run programs for gifted students. If federally funded, such programs would have to include evidence of outreach and successful recruitment of underrepresented students, and show that they were effective in serving these students’ interests.
To implement such a policy array, we need to gauge whether we are making progress and increasing educational effectiveness fairly and rigorously. The country must pull together to address the issues raised in such a plan, and insist that adequate funding be dedicated to improvement. This is a priority that affects our national security. And time is truly running out.
