Formative and Summative Assessments in Mathematics Supporting the Goals of the Common Core Standards

• In Event: Assessment of Complex Thinking in Mathematics, Reading, Science, and History

Abstract

It has been clear for at least a quarter century that proficiency in mathematics, and indeed, all domains, involves a great deal more than mastery of facts and procedures; it involves having rich domain knowledge including access to problem solving strategies, having a productive disposition and domain-specific beliefs, being a strategic thinker, and being appropriately metacognitive (see, e.g., National Research Council, 2001; Schoenfeld, 1985). The 1989 Standards of the National Council of Teachers of Mathematics emphasized, in addition to content, the following four central processes: problem solving, reasoning, communicating, and making connections. The 2000 NCTM volume Principles and Standards added to the mix the productive use of mathematical representations. And, the 2011 Common Core State Standards for Mathematics (CCSSM) focused on a set of outcomes called mathematical practices. According to CCSSM, people who are mathematically proficient:
• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.
It is well known that in the context of high-stakes examinations, “What you test is what you get” (WYTIWIG) - that is, teachers will focus their attention on the items that will be tested. Hence it is critically important for tests to truly reflect the values of the Standards; and it is similarly important to help teachers engage with their students in ways that help the students become fluent in the mathematical practices described above.
For approximately 20 years (and with a lineage that extends another few decades) the Mathematics Assessment Project has focused on the development of summative assessments that emphasize the mathematical processes and practices discussed above, and more recently the creation of formative assessment lessons intended to help teachers develop student understandings in the spirit of the productive assessment practices identified in Black and Wiliams’ (1998) part “inside the black box.”
This presentation will describe some of our recent work on both formative and summative assessments. The formative part of the paper will illustrate the kind of scaffolding our lessons provide, in order to support teachers in posing rich tasks and being prepared to respond to the range of student conceptions (and misconceptions) that their students are likely to produce. The summative part of the paper will present and exemplify a model by which Standards-based examinations can be produced which (a) have a significant component of performance tasks, which provide ample opportunity for students to demonstrate their fluency with the mathematical practices discussed above, (b) are balanced, in that a mixture of performance tasks and short items samples adequately from the desired content-and-process space, (c) can be realistically and reliably scored on a large scale.

Author

• Alan H. Schoenfeld, University of California - Berkeley