preted the mathematics and what strategies they pursued. Developers of external assessment systems should explore ways of taking the information teachers can provide into account as part of the system.

Teachers are rich sources of information about what students know and can do.

In summary, the learning principle aims to ensure that assessments are constructed and used to help students learn more and better mathematics. The consensus among mathematics educators is that assessments can fulfill this expectation to the extent that tasks provide students opportunities to extend their knowledge, are consonant with good instruction, and provide teachers with an additional tool that can help them to become better facilitators of student learning. These are new requirements for assessment. Some will argue that they are burdensome, particularly the requirement that assessments function as learning tasks. Recent experience—described below and elsewhere in this chapter—indicates this need not be so, even when an assessment must serve an accountability function.

The Pittsburgh schools, for example, recently piloted an auditing process through which portfolios developed for instructional uses provided "publicly acceptable accountability information."45 Audit teams composing teachers, university-based researchers, content experts, and representatives of the business community evaluated samples of portfolios and sent a letter to the Board of Education that certified, among other things, that the portfolio process was well defined and well implemented and that it aimed at success for all learners, challenged teachers to do a more effective job of supporting student learning, and increased overall system accountability.

There is reason to believe, therefore, that the learning principle can be honored to a satisfactory degree for both internal and external assessments.



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