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Schematic chunks denote patterns, schemes, or sophisticated rules and knowledge stored in the long-term memory in the form of chunks. We investigated whether schematic chunking is effective in improving the performance of students with math difficulties (MD) and students at risk of math failure, and how the complexity level of geometry problems and students' content knowledge moderate the effects of a visual chunking accommodation. A 3 (problem difficulty level) × 2 (plain version/chunking version) ×2 (cheat sheet provided/not provided) mixed design was used. Thirty-three students, including 18 with MD and 15 at risk, were randomly assigned to two groups: A "cheat sheet" of related theorems was provided to one group but not the other. Students in both groups received two versions of a geometry test: (a) a test with a plain figure representation, and (b) a parallel test with coloring and marks that highlighted the elements of a schematic chunk. Results found a main effect of chunking for all participants, and the chunking effect was greater for difficult one-step and multi-step problems than for simple one-step problems. Providing a cheat sheet increased the chunking effects for solving only the difficult one-step problems pertaining to low-frequency theorems for students with MD.

Access to maker technologies has catalyzed and amplified the possibilities for creating physical materials that are responsive to the needs of students. Opportunities for design and fabrication of original mathematics manipulatives have been incorporated into the teacher education program at Montclair State University. Participating preservice elementary teachers design and make original mathematics manipulatives. Three case studies examine ways in which this process enhances students' mathematical reasoning, sense-making, and understanding. The designs created are made available through links to the open source Educational CAD Model Repository, enabling others to replicate the objects described.

An Educational CAD Model Library (CAD Library) is being developed by collaborating educational associations affiliated with the National Technology Leadership Summit coalition. The CAD Library's Curators' Council has developed descriptive metadata fields that will be associated with each educational object published in the library. These fields are designed to facilitate search and discovery of objects by teachers who are increasingly well positioned to use these objects in their instruction due to the increasing presence of school-based makerspaces and the 3D printers, digital die cutters, and other fabrication tools. The publication of these metadata standards makes them available to the members of the educational associations participating in the development of the CAD Library and to other relevant stakeholders for feedback to inform ongoing revisions.

With this paper, we share an activity for the undergraduate mathematics classroom called The Harp Project that leverages the aesthetic nature of both the mathematical and the musical arts. This project was conceived as a STEAM/PBL project with the added feature that it was carried out in pieces by an entire class. Anecdotal evidence suggests that the uncommonly high level of students' interest and engagement in the experience and the communal feel of their participation can be attributed to this collective quality of the project's design. For this reason, we invite others to consider this curricular experience as it appears viable for supporting efforts to cultivate a broader population of students with an affinity for mathematics. [ABSTRACT FROM AUTHOR]

A transcribed conversation between a mathematics educator and a disability studies educator is presented to highlight cross-disciplinary dialog as a fruitful space for recursive and reflexive engagement from which may emerge concepts to further both mathematics and disability education. The opening question, "How can I know what's possible in terms of engaging students with disabilities in mathematical thinking and reasoning?," sparks discussion. Critique of typical practices in special education and mathematics teaching are raised in terms of learning theories and ideologies of schooling. Takeaways include the promise of emphasizing mathematical thinking as a way to structure more inclusive educational experiences for students with disabilities.

Inspired by the promise of new 3D technologies and the proposition that new tools make innovation possible, this article provides a case study of how a tool called Thirty6 was designed and used in classrooms by mathematics teachers in their own varied and invented ways. Unlike established manipulatives that are designed to support the learning of a particular concept, Thirty6 was developed from a generative design perspective to be used by teachers who can decide what instructional purpose it might serve. The authors offer this case study of Thirty6 not merely as a guide for possible re-implementation by the reader but as an exemplar of the possibilities afforded by easy access to 3D design tools and printers. The authors' hope is that other mathematics teachers are inspired by this approach to develop their own tools and undertake their own journey toward making meaning with students.

It is widely accepted in the mathematics education community that pedagogies oriented toward inquiry are aligned with a constructivist theory of learning, and that these pedagogies effectively support students' learning of mathematics. In order to promote such an orientation, we first separate the idea of inquiry from its conception as a collection of methods. Then, by grounding those methods in a generally accepted theory of learning, we construct an inquiry-oriented pedagogy from a constructivist perspective. We then discuss the implications of this pedagogy for the design of mathematical tasks that democratize student access to inquiry. This work has implications for educators who wish to enact an inquiry-oriented pedagogy in their classroom in order to support their students' problem solving and problem posing.

Corneal tissue addition keratoplasty is a novel surgical procedure utilizing preserved allogenic femtosecond laser–shaped corneal tissue inlays to improve corneal topography and vision in patients with keratoconus and ectasia. Purpose: To report results of Corneal Tissue Addition Keratoplasty (CTAK) for keratoconus (KC) and ectasia after laser in situ keratomileusis. Setting: Cornea and refractive surgery practice. Design: Single center, prospective, open label clinical trial. Methods: 21 eyes of 18 patients underwent CTAK. A tissue inlay of preserved corneal tissue was cut to customized specifications with a femtosecond laser and placed in a laser-created channel in the host cornea. Postoperative uncorrected and corrected distance visual acuity (UDVA, CDVA), manifest refraction spherical equivalent (MRSE), topographic mean keratometry (Kmean), maximum keratometry (Kmax), and the point of maximum flattening (Kmaxflat) were measured. Results: Average UDVA improved from 1.21 ± 0.35 logMAR lines (LL) (20/327) to 0.61 ± 0.25 LL (20/82) (P <.001). Average CDVA improved from 0.62 ± 0.33 LL (20/82) to 0.34 ± 0.21 LL (20/43) (P =.002), and average MRSE improved from −6.25 ± 5.45 diopters (D) to −1.61 ± 3.33 D (P =.002). Individually, 20 eyes (95.2%) gained more than 2 lines of UDVA, with 10 eyes (47.6%) gaining more than 6 lines, and no eyes worsening. 12 eyes (57.1%) gained at least 2 lines of CDVA, with 1 eye worsening by more than 2 lines. At 6 months, average Kmean flattened by −8.44 D (P =.002), Kmax flattened by −6.91 D (P =.096), and mean Kmaxflat was −16.03 D. Conclusions: CTAK is a promising procedure to improve visual acuity and topography in patients with KC and ectasia. [ABSTRACT FROM AUTHOR]