faulty memory (“I just knew it”), faulty calculation (the child miscounts the objects in front of him or her), or faulty reasoning (“I know that 3 and 2 is more than 4 and 6 is 2 more than 4”). Identifying and promoting underlying thought requires formative assessment.
Contemporary cognitive theories often stress establishing a link between the child’s informal knowledge and what is to be taught (Baroody, 1987; National Research Council, 1999; Resnick, 1989, 1992). The child brings to the task of learning a body of prior knowledge—an “everyday mathematics” that is often relatively powerful and sometimes a source of misconceptions. In either case, the teacher needs to understand the child’s current cognitive state (the everyday mathematics) in order to adjust instruction to it. Sometimes the everyday mathematics can serve as a fruitful basis for further development; the child’s learning may in part involve mathematizing what she or he already knows. Sometimes the teacher needs to employ methods to help the child abandon everyday concepts in favor of more accurate notions, as when the child believes that the symbol = means “get an answer” instead of an equivalence relation (Seo and Ginsburg, 2003), or that a long, skinny scalene triangle is not an acceptable triangle (Clements, 2004).
Those who practice behavior modification also need to employ formative assessment to acquire an accurate account of the child’s current behavior so they know what to shape. Careful observation of behavior and decisions about appropriate reinforcement can also be conceptualized under the rubric of formative assessment.
In brief, many theoretical approaches advocate getting information about the child’s current behavior, thinking, and learning so that effective teaching can be implemented. It is hard to imagine a theory of teaching that would advocate ignorance of the child’s mind or behavior.
Formative assessment is a very natural and commonplace activity for teachers, who do it all the time without necessarily knowing that what they do is assessment. Here we discuss three major kinds of formative assessment: everyday observations, tasks, and interviews (see Box 7-3). These everyday practices of observation, presenting tasks, and interviewing involve an informal, often unplanned, implementation of formative assessment, which is so bound up with everyday teaching that it often goes unrecognized. Yet the three types of formative assessment can be rigorous, focused, and deliberate. The early childhood assessment systems discussed here include widely used integrated programs as well as mathematics-specific programs: Big Math for Little Kids, Building Blocks, Core Knowledge, Creative Curriculum, High/Scope, and Number Worlds.