The California State Board of Education’s new math framework, adopted last month, has drawn intense public criticism. Most critics have focused on the framework’s overt political content or its aims to achieve “equity” by holding back advanced students, but there is an arguably even more fundamental problem: an approach to education called inquiry learning, which has virtually zero grounding in research. There is little in the framework that resembles real mathematical learning.
The framework has roots dating back to the “math wars” of the 1990s. Then as now, reformists and traditionalists argued over the best way to teach children math, and California’s math curriculum was a focal point. Reformists encouraged students to discover and construct knowledge with little guidance from the teacher; traditionalists emphasized the need for step-by-step practice of procedures and memorization of basic math facts. In 1997, California adopted compromise standards—a pedagogical hodgepodge of both approaches.
The new framework, clocking in at 1,000 pages, represents a complete victory for the reformists. It’s astounding in both its breadth—including learning goals, instructional “best-practices,” and class sequences—and its mediocrity.
The reformist approach has obvious appeal: children often learn through curiosity, exploration, and play, so let’s do away with worksheets and flashcards. Whereas traditional math teaches students through preset concepts and skills, California’s framework focuses on student-derived questions like “what is half of infinity?” Students face challenges and questions in collaborative groups.
Consider an example: the teacher sets before students the expression 7 x 24 and provides time to brainstorm strategies for tackling it. One student suggests multiplying 7 x 25 and then subtracting 7. Another pair suggests calculating 7 x 10, doubling that answer, and then adding 7 x 4. Student input drives class flow, not teacher instruction; students consider single problems instead of learning a method to solve many. Ostensibly, such “struggle” (a word that the framework authors are fond of) is intended to generate a deeper understanding than rote practice can provide.
This approach may sound good—but it doesn’t work. A vast body of empirical research has consistently shown that more structured methods—with clear objectives, clear explanations, clear corrections, and lots of practice—achieve superior results.
Modern scientific methods have been slow to enter the field of instruction; when Harvard’s Jerome Bruner proposed that “discovery” would benefit learning in 1961, the idea seemed plausible. Little was known back then about how we learn, and little empirical research existed to adjudicate competing claims. But more than a half-century later, modern methods in the social sciences have vindicated traditional approaches.
In a seminal 2010 study on cognitive science in education, three researchers found “overwhelming and unambiguous evidence” that inquiry-type learning techniques are “significantly less effective and efficient” than more structured, teacher-guided activities. Evidence also suggests that the inquiry approach “may have negative results when students acquire misconceptions or incomplete disorganized knowledge.” The researchers argue, in effect, that inquiry learning is the educational analog of pushing a child into a pool to teach him to swim. Only a lucky few will stay afloat.
Even the American Educator, the magazine of the American Federation of Teachers, endorsed the study. Yet with its new math framework, California has chosen not to follow the science.
Modern cognitive science helps explain why structured classrooms produce superior results. Psychologists typically break human memory down into two structures: working and long-term. Working memory is our site of conscious thinking. It’s active but limited, able to process four to seven pieces of information at a time. We can remember a short telephone number, but any more information on top of that quickly slips away. Working memory is like a small doorway that allows new knowledge to enter and remain in long-term memory.
With this in mind, cognitive science touts the importance of automaticity—essentially, doing without thinking—for numbers and basic computations. When procedures become automatic, they take up no space in working memory.
In my own classrooms, I’ve watched students struggle with basic algebra because they were busy plucking 8 x 7 into their calculators. Their working memory, the door to long-term memory, is clogged with such basic computations. A structured lesson and mastery of basics removes this cognitive clutter, allowing students to acquire new knowledge. The California math framework denigrates memorization, practice, and explicit instruction as lower-demand tasks. In a sense, this is true—but they’re the lower-demand tasks upon which higher-demand tasks depend.
Education researcher Tom Loveless notes that an earlier draft of the framework used the words “memorization” and “memorize” 27 times, but always with a negative connotation. Deriding rote memorization, or “drill and kill” activities, the framework laments that what it considers the most compelling questions in mathematics—like “What calculations were used to build the Pyramids?”—go unasked. “Young children’s joy and fascination are too often replaced by dread and dislike,” it warns.
Such declarations mistake the differences between expert and novice thinking. Experts possess automatized skills and bring a wealth of knowledge to bear on new problems. It’s a mistake to rush students ahead to application before they’ve mastered the basics. We still accept this basic rule of thumb in other realms such as music and sports. No one would send a novice straight off a ski jump and call it a success. By downplaying the need to learn the basics, California will prevent students from ever attaining mastery.
In 2002, the What Works Clearinghouse, a project of the federal Institute of Education Sciences, brought together experts to review existing research and craft accessible practice guides for educators. The California framework seems averse to such fundamentals. At the end of the framework, a 153-page section on sample lessons ignores most of the WWC’s recommendations. Not one sample lesson encourages timed activities, worked examples, or low-stakes quizzing—activities that the WWC deems most effective with the greatest degree of confidence from existing research. The WWC practice guides reference 114 scientific studies; the California framework references one.
The California math framework is the latest chapter in a long-running story in American education: the rejection of proven instructional fundamentals in favor of fashionable but untested theories. We’ve already been down this road in reading education. Proponents of whole-language literacy bristled at the structure and formalized nature of phonics; though evidence for the effectiveness of phonics was abundant, schools avoided it, and legions of students struggled to learn to read under pseudo-scientific literacy models. California looks set to repeat this error in math, ignoring sound research in favor of romantic notions about learning and childhood.
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