Differences between, and connections between, Content and Practice standards
Connecting Mathematical Practice with Content
Standards and curricula are different objects. In order to achieve a standard solidly and with full fidelity, curricula and teaching must sometimes take preparatory steps that are not specified in the standards. Such divergence cannot omit either the content or the practice standards and remain faithful; neither can it crowd classroom instruction with material that the CCSS deliberately pruned out of the “mile wide, inch deep” curriculum in an attempt to achieve depth and focus. But fidelity to both Content and Practice requires more than a checklist approach: a coherent mathematical story line cannot be achieved by teaching to the standards any more than by teaching to the test. In part, that is the message of the concluding statement in the Mathematical Practice section that calls for connecting those standards to the standards for Mathematical Content.
Standards can mandate this connection, as the CCSS does, but specifying its design is a job for curriculum materials. Mathematical facts and procedures—the Content part of what we teach—are the results of the application of mathematical habits of mind reflected in the Practices. For that reason, fidelity to the way mathematics is made and used—a big part of the intent of the Mathematical Practices—requires that the Content be taught through the Practices. That way, the connections are real—integrated rather than interspersed.
This attention to Mathematical Practices connected with Content must also be enacted in teaching, which will require professional development. Though the CCSS Mathematical Content standards differ in detail from other content standards, their form is familiar to teachers: a list of things to know. The Mathematical Practices are not so easily condensed into a lesson or unit, not so easily tested and, generally, not so familiar. Content standards are specified grade by grade and build on each other rather than repeating year after year. The Mathematical Practices are different. Though they can be enacted in an appropriate way at any level, they evolve and mature over years rather than days, along with children’s cognitive development and the nature and sophistication of the Mathematical Content.
Finally, of course, any system that rewards and/or punishes teachers and schools on the basis of measured results can expect that anything that isn’t reflected in the measuring instruments will be ignored completely. The Mathematical Practices will be taken seriously in curriculum and teaching if, and only if, they are taken seriously in testing. It can be expected, then, that the developers of the CCSS, and the States that collaborated in calling for the development of the CCSS, will work with the developers of assessments to ensure that the Mathematical Practices are taken seriously in testing.