Beyond categories: dynamic qualitative analysis of visuospatial representation in arithmetic

Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad categorizations of representations and argues the benefits of dynamic qualitative analysis of arithmetical-representational strategy across multiple semi-independent aspects of display, calculation, and interaction. It proposes an alternative methodological approach combining the structured organization of classification with the detailed nuance of description and describes a systematic but flexible framework for analysing nonstandard visuospatial representations of early arithmetic. This approach is intended for use by researchers or practitioners, for interpretation of multimodal and nonstandard visuospatial representations, and for identification of small differences in learners’ developing arithmetical-representational strategies, including changes over time. Application is illustrated using selected data from a microanalytic study of struggling students’ multiplication and division in scenario tasks.

Difusi Inovasi Melalui Pendampingan Mgmp Matematika Smp Dalam Pembuatan Video Virtualisasi Manipulatif Konkret Untuk Pembelajaran Jarak Jauh

During the Covid-19 pandemic, teacher problems increased to maintain the effective implementation of mathematics learning. The available classical and individual concrete manipulative product innovations cannot be used in learning because learning policies during the pandemic still have to be carried out as distance learning. To overcome this problem, assistance was carried out in making videos of using concrete manipulatives to MGMP as concrete manipulative virtualization. The results of this activity show that there is an increase in the average score of the overall pre-self test competence to the post-self test. The same is true for each competency. In addition to producing video products on target, video quality is better, as well as increasing personal partners who act as actors and narrators. As a suggestion in this activity, mentoring continues to be carried out on an ongoing basis as a stage of diffusion of other innovation products and guidance for MGMP and partners of this activity can transmit to other MGMP member teachers.

Fostering Mathematical Creativity Through Problem Posing and Three-Act Tasks

The National Association for Gifted Children and The National Council of Teachers of Mathematics both call for problem posing. This article illustrates the strategies used during a series of three Three-Act Tasks to foster second grade students’ abilities to problem pose. The students’ problem posing improved across the three Three-Act Tasks and revealed mathematically creative thinking. To support and encourage the students to problem pose, the teacher asked generative questions, modeled various problems, provided concrete manipulatives, and had the students create their own sequel to the Three-Act Tasks.

Virtual Versus Concrete: A Comparison of Mathematics Manipulatives for Three Elementary Students With Autism

Basic mathematic skills at the early age are foundational for later learning. Many students with autism spectrum disorder (ASD) struggle in academic learning without sufficient support. Research in the area of concrete manipulatives—tangible representations of abstract concepts—has been found effective. In addition, promising research has emerged in the area of virtual manipulatives—virtual representations of abstract concepts—as tools to support mathematics skill acquisition. Using a multiple baseline across participants with an embedded alternating treatment design, this study presents a replication of previous research comparing the effects of concrete and virtual manipulatives in the acquisition of double-digit addition and word problem-solving abilities of three elementary students with ASD. Findings indicate that while both interventions produce better outcomes than baseline, the virtual manipulative condition appear to be more supportive than concrete manipulatives.

Virtual Manipulatives: A Tool to Support Access and Achievement With Middle School Students With Disabilities

Manipulatives offer students with disabilities access and support in classrooms. However, it is important for educators to be aware that concrete manipulatives are not the only option. Teachers serving students identified with a disability in the middle grades may consider selecting virtual manipulatives as supplement, complement, or in lieu of concrete manipulatives. In this technology in action, the authors provide information for educators about using virtual manipulatives and how they can be used across different settings, instructional needs, and mathematical content areas for middle school students with disabilities.

Connecting concrete objects and abstract symbols promotes children's mathematics learning

Young children often learn mathematics concepts with concrete manipulatives (e.g., blocks, counters); yet, the mere use of manipulatives does not ensure successful learning. The current study compared different ways of using manipulatives (i.e., base-ten blocks) in conjunction with written symbols. Children (N = 138, M age = 6.86) learned about place value in one of four conditions: Concrete Only, Direct Comparison, Two-Step Fading, and Three-Step Fading. Condition differences at posttest favored the fading conditions in which problem formats transitioned from concrete objects to written numerals. More robustly, children who learned the connections between the objects and symbols exhibited higher posttest and transfer test scores relative to children who did not, over and above general learning during the lesson.

Adding It Up

Manipulatives are a common tool in mathematics teaching and learning, including for students with disabilities. The most common manipulatives are concrete manipulatives, yet app-based manipulatives are a viable age-appropriate option for secondary students with disabilities. Through an adapted alternating treatment design with three middle school students—two with mild intellectual disability and one with a learning disability, researchers explored the impact of virtual and concrete manipulatives on students’ accuracy, independence, and task completion time for solving addition of fractions with unlike denominators. Students were equally successful in terms of accuracy and differences with independence were minimal. When comparing the two manipulative types, the results were idiosyncratic; two students were more independent with the concrete manipulative and one with the app-based manipulative. Implications for research regarding mathematics instruction and use of concrete and app-based manipulatives are discussed.

Manipulative Apps to Support Students With Disabilities in Mathematics

Understanding mathematical concepts is important for all students, although often challenging for many students with disabilities. Historically, educators have used concrete manipulatives to support and build conceptual understanding. Mobile devices provide a valuable option to support students with disabilities in mathematics through app-based manipulatives. Although research is limited on app-based manipulatives, the emerging literature with virtual (i.e., digital) manipulatives more generally suggests student preference for virtual manipulatives without a loss of understanding. This column provides educators with information about app-based manipulatives and how to use them, with the goal of helping teachers make informed decisions about app-based manipulatives to support students.