Jonathan D. Bostic

Encouraging Sixth-Grade Students' Problem-Solving Performance by Teaching through Problem Solving.

Abstract This teaching experiment provided students with continuous engagement in a problem-solving based instructional approach during one mathematics unit. Three sections of sixth-grade mathematics were sampled from a school in Florida, U.S.A. and one section was randomly assigned to experience teaching through problem solving. Students’ problem-solving performance and performance on a unit test were analyzed. The intervention had a positive effect on students’ problem-solving performance whereas the comparison group experienced no changes. ANCOVA analyses suggest that intervention students solved more problems on the posttest than their peers, but the comparison group outperformed the intervention group on the unit test.

Effects of Minute Contextual Experience on Realistic Assessment of Proportional Reasoning.

Abstract This mixed methods study describes the effects of a minute contextual experience on students’ ability to solve a realistic assessment problem involving scale drawings and proportional reasoning. Minute contextual experience (MCE) is defined to be a brief encounter with a context in which aspects of the context are explored openly. The study looked closely at what happened during an instructional unit examining proportional reasoning. Students completed a pretest and posttest involving items characterizing a novel context, and data were analyzed to determine the effects of the MCE. Students were interviewed to gather their perspectives on the problem and their own solutions. Pretest results indicated that instruction in which students demonstrated growth in understanding had little effect on students’ ability to solve a novel problem in which they had difficulty associating their everyday mundane knowledge with the realistic context. The students demonstrated a significant increase in ability to solve the novel problem after a MCE. Furthermore, students explicated that the MCE aided their ability to visualize the context, and this helped them apply instructional learning to solve the problem. A discussion of the complexities involving the assumptions of students’ familiarity with contexts and their abilities to draw upon their mundane everyday experiences to solve proportional reasoning problems is shared.

Validating and Vertically Equating Problem-Solving Measures

This paper examines the validation of a measure for eighth-grade students related to problem-solving. Prior work discussed validity evidence for the Problem-Solving Measure series, but it is uncertain whether linking items appropriately vertically equates the seventh- and eighth-grade measures. This research connects prior work to the development of linked measures with anchor items that assess problem solving within the frame of the Common Core in the United States. Results from Rasch modeling indicated that the items and overall measure functioned well, and all anchor items between assessments worked satisfactorily. Our conclusion is that performance on the eighth-grade measure can be linked with performance on the seventh-grade measure.

A Case Study of Middle School Teachers’ Noticing During Modeling with Mathematics Tasks

Schoenfeld (Mathematics teacher noticing: Seeing through teachers’ eyes. Routledge, New York, pp. 223–238, 2011) wondered about the transferability of teacher noticing across contexts (e.g., different grade levels and task types). This chapter focuses on middle school teachers’ noticing during instruction that promoted modeling with mathematics, which is one of eight Standards for Mathematical Practice (SMPs) found in the Common Core State Standards (CCSSI in Common Core State Standards for Mathematics. Author, Washington, DC, 2010). A case study approach was used to explore middle school teachers’ noticing during instruction promoting modeling with mathematics. This study focuses on two middle school teachers who enacted modeling-focused lessons. Lessons, videos, and interview data were analyzed using inductive analysis (Hatch in Doing qualitative research in education settings. State University of New York Press, Albany, NY, 2002). We drew two impressions from the data. The first was that teachers’ noticing focused on fostering students’ use of multiple representations. The second result was that teachers’ noticing was framed in ways to assist with making sense of a modeling task or its solution. We connect these results to transferability of teaching noticing, specifically to instruction promoting modeling with mathematics.

Moving forward: Instruments and opportunities for aligning current practices with testing standards

The focus of this special issue is validity-related issues within mathematics education. I am deeply grateful to the Research Council on Mathematics Learning for supporting this call as well as the associate editors for this special issue: Michele Carney (Boise State University), Erin Krupa (Montclair State University), and Jeff Shih (University of Nevada Las Vegas). Each manuscript was reviewed by an associate editor for the special issue, a mathematics educator with expertise in measurement in mathematics education, a mathematics educator who has reviewed for Investigations in Mathematics Learning, and myself. Reviewers provided thoughtful feedback that led to a set of three accepted manuscripts and a 20% acceptance rate. Readers are encouraged to reflect on the purpose, arguments, and evidence within each article, as each uses different approaches, which ultimately lead to appropriate uses and score interpretations. Instrument quality strongly influences the data collected and relatedly, findings of a research study (American Educational Research Association, American Psychological Association, & National Council on Measurement in Education [AERA, APA, & NCME], 2014; Gall, Gall, & Borg, 2007). Instruments with a clearly defined purpose and supporting validity evidence are foundational to conducting high-quality quantitative research (Newcomer, 2012). Near the core of any methodology is the tool used to collect data. The data collected using an instrument is grounded in the validity evidence gathered, and corresponding arguments to support its use in research context. Thus, research aiming to build on past research or lay a foundation for a new vein of quantitative-focused research must be supported by instruments that have diverse and robust validity evidence and arguments. Validity is a central tenet of effective construct measurement (Messick, 1980). Nearly 30 years ago, Messick (1989) introduced construct validity as “an integrated evaluative judgment of the degree to which empirical evidence and theoretical rationales support the adequacy and appropriateness of inferences and actions based on test scores or other modes of assessment” (p. 13, emphasis in original). More recently, Kane (2016) expressed that “validity is a property of the proposed interpretations and uses of the test scores and is not simply a property of the test or of the test score” (p. 64). Validation requires a guiding purpose and evidence supporting a guiding purpose (AERA, APA, & NCME, 2014; Kane, 2016, 2001; Wilson, 2004). Two manuscripts in this special issue address current standards for assessment in educational research and a third manuscript shares a type of validity evidence that might supplement other sources of evidence. First, Gleason, Livers, and Zelkowski present a validation study of the Mathematics Classroom Observation Protocol for Practices (MCOP). The authors discussing the MCOP provide validity evidence that appropriately bounds the use of the instrument and frames score interpretations, which is central to an argument-based approach (AERA, APA, & NCME, 2014; Kane, 2016). The second manuscript, by Eddy, Harrell, and Heitz, describes an observation protocol called AssessToday, which may be used for short-cycle formative assessments. These short-cycle formative assessments have potential for use during day-to-day instruction. The authors provide a discussion about interrater reliability and building a meaningful protocol with results that support appropriate

A validation process for observation protocols: Using the Revised SMPs Look-for Protocol as a lens on teachers’ promotion of the standards

ABSTRACT The Standards for Mathematical Practice (SMPs) describe mathematical behaviors and habits that students should express during mathematics instruction. Thus teachers should promote them during classroom-based mathematics instruction. The purpose of this article is to discuss the validation process for an observation protocol called the Revised SMPs Look-for Protocol. An implication of this study is that users with a robust understanding of the SMPs may feel confident using the protocol as a validated and reliable tool in research and school-based settings. We discuss opportunities and challenges for mathematics teacher educators engaging in classroom observations, in light of this observation protocol.