The regular use of formative assessment has been documented to mitigate and prevent mathematical difficulties and improve student learning when used to inform instruction (Clarke & Shinn, 2004; Fuchs, 2004; Lembke & Foegen, 2005; Skiba, Magnusson, Marston, & Erickson, 1986). The Council of Chief State School Officers published a widely cited definition of formative assessment in 2006: “Formative assessment is a process used by teachers and students during instruction that provides feedback to adjust ongoing teaching and learning to improve students’ achievement of intended instructional outcomes” (p. 1). Formative assessments may be designed and enacted in many ways, including “on-the-fly” and “planned-for interactions,” as well as “curriculum-embedded assessments” (Heritage, 2007, p. 143). For example, in the course of a lesson, a teacher may overhear a student’s misconception or error and then spontaneously and promptly adjust the instruction to address the misconception and reteach (i.e., “on-the-fly”). Moreover, a teacher may deliberately plan to elicit students’ thinking during the lesson, providing a critical assessment point at which the teacher can use the gathered evidence to adjust instruction. Formative assessment should include methods for identifying students’ progress, providing feedback, involving the student in assessing understanding, and tending to learning progressions (Heritage, 2007). It is critical that both the student and the teacher are active participants in identifying growth and reflecting on learning behaviors—critical ingredients to moving achievement forward.
Curriculum-based measurement is a type of formative assessment. Curriculum-based measurements are brief, direct, systematic assessments of student performance on tasks that predict future learning success on the intended general outcomes of instruction. In mathematics, instruction guided by frequently administered curriculum-based measurement has been demonstrated to enhance mathematics achievement (Fuchs, Fuchs, Hamlett, & Stecker, 1990; Stecker & Fuchs, 2000). In selecting curriculum-based measurement tools for use in the classroom, teachers must have an understanding of the intended learning outcomes and the sequence (and time period) within which attaining the outcomes can be expected. Generally, the number of tasks assessed on a curriculum-based measurement is proportionate with the time within which learning is to be evaluated; fewer skills are associated with shorter duration of instruction. The approach of measuring narrowly defined skills and tasks to judge student mastery of that specific skill has been referred to as mastery measurement (Fuchs & Deno, 1994), which can be highly useful for teachers and others who wish to make judgments about short-term instructional effects (for example, during intervention planning or in determining when to advance task difficulty).
On the other hand, to model student growth over time and toward the more general outcomes of instruction belies the need for broader content sampling and can be used to evaluate whether short-term instructional adjustments are effective in placing students on track to attain the broader outcomes of instruction (VanDerHeyden, 2005). Formative assessment should be recognized as a key professional skill for teachers and given high priority in training and professional development programs. Through appropriate instruction and well-crafted learning experiences, teachers and teacher candidates should have a solid foundation of knowledge related to formative assessment (developing learning progressions or skill sequences, assessing general outcomes via curriculum-based measurement, and skill and subskill mastery via mastery measurement) and an understanding of the various types of formative assessment (e.g., curriculum-based measurement, mastery measurement), including their distinct features and advantages. Teachers also must be adept in evaluating the merits of certain assessments and their suitability for answering certain questions germane to their instruction. Once measures are selected and administered, teachers must be adept at using the resulting data to inform instruction. Teacher preparation programs and professional development activities also should provide instruction and opportunities for application in which teachers can make the connection between the use of formative assessment measures and the decisions that must be made within response-tointervention frameworks (e.g., determining the need for classwide, small-group, or individual supplemental intervention, selecting intervention facets that will produce learning gains, and evaluating intervention effects) (VanDerHeyden, 2009).
In the early years (PK through Grade 1), assessment of student mastery (and progress toward mastery) of number sense should be emphasized. Such assessments include one-to-one object correspondence in counting, counting in sequence, counting on, counting backward, number identification and naming, identifying larger and lesser quantities, and completing missing numbers within number sequences (Clarke & Shinn, 2004; Methe, Hintze, & Floyd, 2008; VanDerHeyden, Broussard, Fabre, Stanley, Legendre, et al., 2004; VanDerHeyden, Witt, Naquin, & Noell, 2001). As students progress through the primary grades, direct assessment of the fluency (accuracy and speed) with which they compute basic facts and more complicated operations (e.g., finding least common denominator) become important assessment targets to identify where progress toward important benchmarks is lagging and intervention may be warranted (see VanDerHeyden, 2009, for suggested universal screening measures for mathematics Grades K through 8). Direct measures of the extent to which students demonstrate conceptual understanding of mathematical principles can be directly assessed and contribute information that is distinct from computation and operational fluency (Fuchs, Fuchs, Bentz, Phillips, & Hamlett, 1994) with implications for instructional planning (Fuchs, Fuchs, Prentice, Burch, Hamlett, et al., 2003).
There is a science of mathematics education that includes clear implications for curricula content (what is taught) and instructional practices (how it is taught) (Gersten et al., 2009). The Innovation Configuration was based on these summary reports and the broader literature in mathematics education and teaching. The Innovation Configuration offers a set of quantifiable indicators of instructional excellence in mathematics that are related to improved achievement in mathematics that can be used to improve teacher competencies and student achievement.
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