We present findings from an analysis of tests of teacher mathematical knowledge identified over a 20-year period of mathematics education literature. This analysis is part of a larger project aimed at developing a repository of instruments and their associated validity evidence for use in mathematics education. We report on how these tests are…

This article illustrates and differentiates the unique role cognitive interviews and think-aloud interviews play in developing and validating assessments. Specifically, we describe the use of (a) cognitive interviews to gather empirical evidence to support claims about the intended construct being measured and (b) think-aloud interviews to gather…

Children have mathematical understanding beginning from the early years of life. Mathematics education during these years has a significant effect on children's future lives. Play is a significant issue in children's lives and can be used as a way for teaching and learning. It presents a rich learning environment. In this study, therefore, we…

The development of proofs and argumentation is one of the major standards for mathematical practices in K-12 education that researchers and practitioners alike are continuing to improve. Further, the use of discourse is considered essential in the learning of mathematical concepts at these levels. However, K-12 educators continue to confound how…

The purpose of this report is to explore the content-related validity evidence supporting the mathematics components of the "ETS"® National Observational Teaching Exam (NOTE) assessment series, a kindergarten through 6th grade teacher licensure assessment. To establish the content knowledge required for the effective teaching of…

Formative assessment involves the gathering of information that can guide the teaching of individual or groups of children. This approach requires a sound understanding of children's thinking and learning, as well as an effective method for gaining the information. We propose that formative assessment should employ a version of clinical…

Students who complete kindergarten with an inadequate knowledge of basic mathematics concepts and skills will continue to experience difficulties with mathematics throughout their elementary and secondary years and may be at increased risk for math disabilities. There is a critical need to identify students experiencing difficulties in mathematics…

There is convincing evidence that many students struggle to learn mathematics proficiently. One plausible contributor to the low math achievement is the quantity and quality of learning opportunities provided in classrooms. These opportunities may fall short of addressing the learning needs of students, especially those at risk for failure in…

The National Council of Teachers of Mathematics' (NCTM's) "Principles and Standards for School Mathematics" (2000) outlines fi ve Process Standards that are essential for developing deep understanding of mathematics: (1) Problem Solving; (2) Reasoning and Proof; (3) Communication; (4) Connections; and (5) Representation. The Common Core…

The goal of this practice guide is to offer educators specific, evidence-based recommendations that address the challenge of teaching early math to children ages 3 to 6. The guide provides practical, clear information on critical topics related to teaching early math and is based on the best available evidence as judged by the authors. The guide…

In this study a survey instrument was developed to test elementary teachers' attitudes towards incorporating manipulatives in their math lessons frequently. Though the benefits of using math manipulatives has been reported, there seems to be a disconnect between the benefits of manipulative use and the number of teachers integrating them in their…

In this study, the authors conceptualize teaching episodes such as an integrated set of observable student-teacher interactions. Instructional interactions that take place between teachers and students around critical academic content are a defining characteristic of classroom instruction and a component carefully defined in many education…

Understanding logical necessity is an important component of proof and reasoning for teachers of grades K-8. The ability to determine exactly where young students' arguments are faulty offers teachers the chance to give youngsters feedback as they progress toward writing mathematically valid deductive proofs. As defined, logical necessity is the…

In this paper, the authors report research focused directly on the validation of the Coding of Academic Teacher-Student interactions (CATS) direct observation instrument. They use classroom information gathered by the CATS instrument to better understand the potential mediating variables hypothesized to influence student achievement. Their study's…

This article documents the development and use of a survey instrument designed to measure K-8 mathematics teachers' perceptions about discourse in mathematics classes. In particular, the 5-point Likert-type survey sought to address teachers' perceptions of their use of dialogic (dialogue to construct new meaning), univocal (conveying information),…