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The nature and extent of reasoning and proof in the written (i.e., intended) curriculum of 20 contemporary high school mathematics textbooks were explored. Both the narrative and exercise sets in lessons dealing with the topics of exponents, logarithms, and polynomials were examined. The extent of proof-related reasoning varied by topic and textbook. Overall, about 50% of the identified properties in the 3 topic areas were justified, with about 30% of the addressed properties justified with a general argument and about 20% justified with an argument about a specific case.

This paper shares results from a secondary analysis of data from the participation of Japanese, Singaporean, and U.S. students in the International Project on Mathematical Attainment (IPMA). IPMA was a longitudinal study to assess the mathematics achievement of primary students from their first year of schooling through the end of fifth grade. Tests were constructed to enable achievement on the same items to be assessed over multiple years, thus permitting the assessment of growth in achievement throughout primary school. Achievement is compared to the grade at which the content is introduced so that achievement can be related to students’ opportunity to learn.

The van Hiele theory and van Hiele Geometry Test have been extensively used in mathematics assessments across countries. The purpose of this study is to use classical test theory (CTT) and cognitive diagnostic modeling (CDM) frameworks to examine psychometric properties of the van Hiele Geometry Test and to compare how various classification criteria assign van Hiele levels to students. The findings support the hierarchical property of the van Hiele theory and levels. Using conventional and combined criteria to determine mastery of a level, the percentages of students classified into an overall level were relatively high. Although some items had aberrant difficulties and low item discrimination, varied selection of the criteria across levels improved item discrimination power, especially for those items with low item discrimination index (IDI) estimates. Based on the findings, we identify items on the van Hiele Geometry Test that might be revised and we suggest changes to classification criteria to increase the number of students who can be assigned an overall level of geometry thinking according to the theory. As a result, practitioners and researchers may be better positioned to use the van Hiele Geometry Test for classroom assessment.

Given the recent release and widespread state adoption of the Common Core State Standards for Mathematics (CCSSM), teachers across the country now have common goals related to what their students should understand about particular mathematical topics. The authors of CCSSM explain, \"Asking a student to understand something means asking a teacher to assess whether the student has understood it\" (CCSSI 2010, p. 4). Teachers know that although students may be able to perform a computation, state a definition, or provide an example, they may not actually understand the mathematics. So, what does it mean for students to understand mathematics, and how can teachers gauge this understanding? Researchers and educators have proposed several frameworks to use when thinking about what it means to \"understand\" mathematics. In this article, the authors propose adapting a model originally used for curriculum development (Usiskin 2007) as a multidimensional approach to assessing students' mathematical understanding across four dimensions: (1) Skills; (2) Properties; (3) Uses; and (4) Representations. This multidimensional approach to understanding, described by the acronym SPUR (Usiskin 2007), can give teachers useful information about the depth of their students' understanding of a mathematical topic. Here, the authors briefly examine the results from an international research project on elementary school students' mathematical attainment in relation to SPUR. Then they demonstrate how teachers might use SPUR as a guide for assessment in their classrooms, specifically in relation to CCSSM. (Contains 4 figures.)

Textbooks are often used in classroom instruction in quite different ways, leading to potential differences in students’ opportunities to learn. This paper explores the enactment of the topic of congruence by 12 teachers using the same geometry textbook. We highlight variations in the number and nature of lessons taught or skipped, in expectations for homework, and in instructional style. For instance, teachers taught between 60 and 100 % of the lessons on congruence but often skipped content focused on unique applications. The number of minutes of homework assigned varied from 16–30 min per night to 46–60 min per night. Ten of the 12 teachers spent at least 50 % of class time each week in whole-class instruction. Only one-third used dynamic geometry software and one used no calculator or computer technology in the congruence chapters. The results provide detailed insights into differences in upper secondary teachers’ use of textbooks, an area where less research exists than at elementary or lower secondary levels.

This article shows how to modify classroom evaluation items to avoid four potential difficulties that limit a teacher's insight into students' mathematical understanding by addressing these issues: (1) poor choice of numbers; (2) implausible or inappropriate contexts; (3) inclusion of graphics that do not help make learning visible; and (4) assumptions needed to answer the items. The authors identified these issues as part of a research project to analyze the influence of the mathematical Process Standards in classroom tests (106 tests from their early work from 2003 to 2005 and 15 tests from three 2010-2011 textbook series mentioned herein.) By critically looking at items, considering the issues highlighted here, and discussing curriculum assessments with peers, teachers can become sensitized to potential difficulties with items before using them. Teachers can then modify the items to better assess student learning instead of waiting until after an assessment has been administered to realize that several items failed to generate useful information about student learning. (Contains 1 figure.)

Mathematics is rich in visual representations. Such visual representations are the means by which mathematical patterns \"are recorded and analyzed.\" With respect to \"vocabulary\" and \"symbols,\" numerous educators have focused on issues inherent in the language of mathematics that influence students' success with mathematics communication. Similarly, with respect to \"visual representations,\" the authors believe that comparable attention is needed to help students perceive, interpret, and make sense of these displays. Further, teachers should consider what support students might need to understand and draw meaning from representations. This article aims to make inroads toward these goals. (Contains 3 figures and 1 table.)

Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.

This issue of ZDM focuses on research related to the enacted curriculum from various perspectives within the context of the US educational system. In this editorial, we describe the broad view of curriculum enactment taken in this issue, highlighting that we mean more than just how instruction plays out within a classroom. For instance, enactment can occur at a national level as educational goals are enacted into a set of national objectives or standards. Enactment can occur as goals or standards are embedded into written curriculum materials or textbooks, both in terms of teacher guides and materials for students. Enactment can occur as teachers make decisions about how to use their written curriculum materials. Finally, enactment can occur as teachers and students engage and interact with written materials during classroom instruction. We elaborate briefly on these views and then outline the structure of this ZDM issue.

This paper shares perspectives on literacy in mathematics, particularly highlighting commonalities with literacy in language arts. We discuss levels of language development appropriate for the mathematics classroom, issues related to mathematical definitions, implied meanings in many mathematics concepts, and the importance of justification. We suggest that mathematical literacy should be an essential and regular component in the mathematics classroom.