Before designing, selecting, or implementing a lesson, understanding the knowledge that your students already have (or do not have) is helpful, regardless of the topic that you are teaching. When I began to teach geometric transformations to a class of tenth-grade honors geometry students, I attempted to assess their knowledge. What I learned about these students' initial understandings of geometric transformations was surprising, as well as extremely useful for planning instruction. Knowing in advance the difficulties that students may experience when learning new mathematical concepts and skills can help prepare teachers for the classroom.
Karen F. Hollebrands and Samet Okumus
In Tools and Mathematics: Instruments for Learning, authors John Monaghan, Luc Trouche, and Jonathan M. Borwein devote 19 chapters, divided into four parts, to portraying various issues and developments related to tools, artefacts, and instruments with a focus on theoretical approaches. They present different theories, highlight how they relate to the use of tools in mathematics, and envisage future issues and trends. Chapters 6 and 11 appear at the end of Parts I and II, respectively. These chapters take the form of a dialogue between the three authors and include Richard Noss, who contributes his thoughts about the issues presented in each part. The authors provide an interlude in Chapter 16 and reflect on the future as it relates to the use of tools in the teaching and learning of mathematics.
Tina T. Starling and Karen F. Hollebrands
With the geometry curriculum already packed with content, who has time to introduce anything new? Many students already have difficulty with regular polygons to begin with—wouldn't an additional topic for polygons be adding fuel to the fire? Perhaps. However, if activities are carefully chosen, students can actively review prerequisite skills as well as benefit from being asked to think critically in a new way.
M. Kathleen Heid, Karen F. Hollebrands and Linda W. Iseri
This article describes the successful use of a computer algebra system (CAS) by Kevin, a seventh-grade student, as he worked on a problem involving functions far more difficult than the functions that he encountered in his mathematics class. CAS clearly supported Kevin's reasoning but did not provide the solution. What place does this powerful technology have in our classrooms? For classroomtested CAS uses and for thought-provoking articles about CAS experiences, watch for the November 2002 CAS focus issue of the Mathematics Teacher.
Karen F. Hollebrands, AnnaMarie Conner and Ryan C. Smith
Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.
P. Holt Wilson, Hollylynne Stohl Lee and Karen F. Hollebrands
This study investigated the processes used by prospective mathematics teachers as they examined middle-school students' work solving statistical problems using a computer software program. Students' work on the tasks was captured in a videocase used by prospective teachers enrolled in a mathematics education course focused on teaching secondary mathematics with technology. The researchers developed a model for characterizing prospective teachers' attention to students' work and actions and interpretations of students' mathematical thinking. The model facilitated the identification of four categories: describing, comparing, inferring, and restructuring. Ways in which the model may be used by other researchers and implications for the design of pedagogical tasks for prospective teachers are discussed.
James E. Tarr, Erica N. Walker, Karen F. Hollebrands, Kathryn B. Chval, Robert Q. Berry III, Chris L. Rasmussen, Cliff Konold and Karen King
During the past 2 decades, significant changes in mathematics curriculum standards and policies have brought greater attention to assessment instruments, practices, purposes, and results. In moving toward stronger accountability, the No Child Left Behind Act (NCLB) of 2001 (NCLB, 2002) mandates that school districts receiving funding under NCLB formulate and disseminate annual local report cards that include information on how students and each school in the district performed on state assessments. This mandate has not only facilitated a growth in state testing (Wilson, 2007) but also influenced the teaching of mathematics (Seeley, 2006). More recently, the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) crafted and launched the Common Core State Standards for Mathematics (NGA Center & CCSSO, 2010), which have been formally adopted by the vast majority of U.S. states and territories. The Common Core State Standards for Mathematics (CCSSM) specifies standards for mathematical content by grade in K–8 and by conceptual categories at the secondary level and identifies key Standards for Mathematical Practice that should be present in K–12 instruction. The CCSSM represents an unprecedented initiative to raise academic standards in school mathematics that will inevitably influence the development of curriculum materials, teaching, and assessment practices.
Daniel J. Heck, James E. Tarr, Karen F. Hollebrands, Erica N. Walker, Robert Q. Berry III, Patricia C. Baltzley, Chris L. Rasmussen and Karen D. King
The National Council of Teachers of Mathematics (NCTM) espouses priorities to foster stronger linkages between mathematics education research and teaching practice. Of the five foundational priorities, one is directly focused on research, indicating NCTM's commitment to “ensure that sound research is integrated into all activities of the Council” (NCTM, n.d.). Another priority specifically references the relationship between research and mathematics teaching; the priority on curriculum, instruction, and assessment states that NCTM pledges to “Provide guidance and resources for developing and implementing mathematics curriculum, instruction, and assessment that are coherent, focused, well-articulated, and consistent with research in the field [emphasis added], and focused on increasing student learning” (NCTM, n.d.).