Curriculum
Curriculum is an area in which the competing claims of math reformers and math traditionalists should be testable. Curriculum involves both content and pedagogy (or methods of teaching). At the broadest level we would expect a constructivist curriculum to focus on content such as real-world problems and teaching methods that involve discovery learning, while a skills-based curriculum would focus on computation through teaching approaches that involve considerable practice.
Unfortunately, there are no studies that have pitted carefully implemented constructivist curricula against carefully implemented skills-based approaches. The typical study in the literature compares the effects of a particular curriculum against business-as-usual, and compares classrooms or districts that have volunteered to use the target curriculum against those that have not. These are weak designs because the very act of volunteering to use a new curriculum typically carries with it extra motivation to succeed, and thus biases the results towards the new curriculum. Nevertheless, such studies can be informative given certain outcomes. For instance, large and educationally meaningful differences favoring the target curriculum or the business-as-usual approach would be difficult to ignore if the schools being compared were roughly similar prior to the introduction of the target curriculum. This is because large effects are difficult to obtain in education and are unlikely to be generated just by the bias for more motivated schools to volunteer for a new curriculum. Very small differences or parity between the target curriculum and business-as-usual could also be informative. For instance, if achievement were low in both conditions, it might suggest that neither business-as-usual nor the target curriculum should be continued.
A recent study examined achievement test data from three states in schools using elementary school reform mathematics curricula that were developed since the late 1980s with funding from the National Science Foundation. These curricula generally followed the constructivist principles I've articulated. Constructivist math students' test results were compared to those of students in business-as-usual schools that were matched by reading score and poverty level. Over 100,000 student test scores were analyzed. While the results favored students using the constructivist curricula, the absolute size of the differences between the constructivist students and the business-as-usual students was small. Averaging across the grades in the three states, constructivist students answered correctly 66% the total possible questions on end-of-year math tests, and business-as-usual students answered 65% of the questions correctly.
With respect to the math wars, these findings suggest that constructivist math curricula are not doing any harm to children compared to business-as-usual approaches, at least with respect to state assessments. Given the concerns that have been raised in some quarters, this is reassuring. However, some would argue that the organizational, material, and staff development costs necessary to shift to a constructivist curricula cannot be rationalized on the basis of such small differences in outcomes.