Why the Best Math Curriculum Won’t Be a Textbook
The most easily understood and usable recommendation in the recent report of the National Mathematics Advisory Panel calls for shorter, focused, and more coherent textbooks.
It’s easy to understand why the panel made this call. In countries that score well in math, such as Singapore, curriculum standards emphasize fewer topics in greater depth, and the matching textbooks are precisely engineered for coherence. Coherence, in turn, results in thin books that are nonetheless mathematically rich. In contrast, as the the Trends in International Mathematics and Science Study famously described it, the curriculum in the United States is “a mile wide and an inch deep.” American math texts can be 700 to 1,000 pages long. In an effort to cover these enormous textbooks, American teachers skip lightly over many topics, achieving little depth of learning. The national advisory panel calls for putting “first things first.” One of the first things to do will be to engage in some serious curriculum engineering.
The bloat in textbooks, at least in part, comes from the nature of book publishing in the United States. As the panel observes, publishers accommodate the demands of 50 different states by including everything that any state might want. Publishers have been experimenting with custom publishing for many years, but it requires no less effort to define a curriculum for a small district than for a large one. And the resources available—for the definition of standards and for the adoption itself—are more limited. Thus, the economies of book publishing lead to one-size-fits-all textbooks that aim to meet the needs of all students and end up being optimal for almost none.
Mathematics curricula in a digital medium, in contrast, need not suffer from bloat. In a digital medium, teachers and students could have both a coherent, core-learning progression (similar, for example, to Singapore’s approach) and specific adaptations that adjust that progression to local needs. The economies of digital production allow for both a common core and specific extensions to optimize the core for local situations.
Let’s be clear that we are not suggesting the elimination of paper. Many districts may use existing print-on-demand technology to purchase a paper book with both the common core and the specific adaptive extensions they judge to be valuable. Most districts will likely use a blend of a leaner, printed “book” and a more flexible selection of technology-based extensions. And a few school districts may opt for fully electronic delivery of the math curriculum.
We suggest leveraging the unique properties of the digital medium where it proves powerful to do so, and provide three examples. First, with modern digital mathematics technology, quick and accurate feedback can be provided to students on both their process and outcomes in working through assigned problems. Of course, even textbooks provide answers in the back of the book, but mid-problem feedback is especially important for students struggling to get to an answer. Moreover, a digital medium can provide for progress measures reflecting important aspects of math learning beyond accuracy, measures such as effort level (whether the problem was attempted at all) and intensity (percentage of available time engaged with the curriculum).
Second, digital technology can provide interactive, dynamic representations to enhance conceptual understanding. These could include “definition unfolding”—one of the most important pathways to mathematical meaning-making. Being able, for example, to select “√3” wherever it appears on any page and have it immediately explained as “that positive number which, when squared, yields 3” could continuously reinforce the connection between symbols and terms and their mathematical meanings. Similarly, geometric constructions are much more revealing when they can be manipulated, and motion graphs are more comprehensible when they call up motion.
Third, digital technology can enable “universal design for learning,” or UDL, as is already prescribed by the National Instructional Materials Accessibility Standard. The UDL approach is that the learning materials themselves should provide for multiple means of (1) representation, to support the diverse recognition networks of individual learners; (2) action and expression within an apprenticeship model of learning, to support diverse strategic networks; and (3) engagement, to support diverse affective networks. By ensuring that instructional goals, strategies, and materials are highly flexible along these three parameters, UDL lowers potential barriers to learning and increases opportunities to learn not only for students with disabilities, but for the full range of diverse learners.
Perhaps the most profound opportunity within a digital mathematics curriculum lies in its potential to disrupt the textbook-adoption cycle, enabling us to follow the mathematics panel’s second call, “learning as we go along,” and, in particular, to support a continuous-improvement, adopt-and-adapt cycle.
We suggest that states should “adopt” only the highly coherent, core-learning progression (and not the adaptive extensions). By adopting an internally coherent core curriculum, states could communicate to teachers more precisely what the state considers central to mathematics learning. Further, assessments and this core could be well aligned, equalizing opportunity to learn at a base level. Digital adaptive extensions could be prescreened for coherence with the core, and then individual districts and schools could be free to choose among them.
Importantly, the digital medium could provide for tracking over time the actual student use of various aspects of the adaptive extensions. This information, when combined with information from formative and outcome assessments, would enable schools to understand better what does and does not work for their students. If most students skip lightly over a section with no apparent ill effects, for example, perhaps that section should be condensed. On the other hand, if there is a problem that most students work through carefully and still stumble on, an elaboration of the material may be called for. The combination of the coherent core with digital adaptive extensions would thus enable schools to improve outcomes through incremental changes.
Given the action agenda set forth by the National Mathematics Advisory Panel, the time is now right to consider alternatives to the traditional publishing model. In an unchanged publishing environment, and with the same pressures to meet the needs of 50 states in a one-size-fits-all book, we should expect the same outcome: bloated, incoherent texts. Only by going digital can we integrate what Singapore does best—maintaining a tightly engineered, coherent core curriculum—with what America does best—inventing adaptations that better cultivate the mathematical genius of a large, diverse population.
Vol. 27, Issue 36, Pages 24-25, 32Published in Print: May 7, 2008, as Why the Best Math Curriculum Won’t Be a Textbook