and computational algebraic systems might play? What is the role of paper and pencil computation in developing understanding as well as skill? These are questions that appear at every level of school mathematics.

Checkpoints: Assessment

Algebra represents a major challenge for many students. If more students are to succeed in meeting that challenge, it will be important to identify the points of difficulty for individual students and provide effective instructional responses before they are lost. The difficulty factors assessments of algebra reading (Koedinger and Nathan, In Press) and algebra writing (Heffernan and Koedinger, 1997, 1998) are examples of efforts to provide assessment tools for this purpose.

Two features of the subject make assessing individual progress very important. Algebra requires facility with much of the mathematics that has come before. If the mathematical foundation is weak in any of its components, algebra mastery will be undermined. Determining where students need to shore up the preparatory mathematics, as well as opportunities for doing so, are critical to success.

Second, algebra instruction moves toward high-level abstraction. The readiness of individual students to move from one level of difficulty to the next will differ. If the movement comes before a bridge is effectively built to a student’s prior knowledge or before new knowledge is consolidated, the student will be lost. If movement toward greater difficulty does not come soon enough, a student will make less progress in higher level algebra than is possible. Indeed, precisely this is at the heart of opposing views of algebra pedagogy. If formative assessment were sophisticated enough to determine individual student readiness to move on, then trade-offs between attending to the needs and preparedness of different students would not be necessary.

A research and development effort at Carnegie Melon University that generated the Algebra Cognitive Tutor has focused very productively on the second element of this problem (see Box 3.5). It began as a project to see whether a computational theory of thought, called ACT (Anderson, 1983), could be used as a basis for delivering computer-based instruction. The cognitive theory applies to problem solving more broadly. For pur-



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