Fig 5 - uploaded by Corey Brady
Content may be subject to copyright.
Source publication
The concept of area measure is widely viewed as fundamental. It is both intrinsically important and also structurally significant, as a representative of the class of continuous multiplicative quantities. Operating flexibly in this context with meaning and conceptual understanding is thus a critical objective for elementary education. Meanwhile, th...
Citations
... Thoughtful, generative use of manipulatives is important for learners of all ages (Bartolini and Martignone, 2020), and in the digital era, it is increasingly important for teachers and students to develop fluency and expressivity with mathematical representations across media (Nemirovsky and Sinclair, 2020). A rich area of research is investigating the balance between virtual and physical manipulatives-selecting between modalities (Moyer-Packenham and Westenskow, 2013), sequencing them (Hunt et al., 2011), or, more generally, understanding the interactions between learning experiences with each type (Maschietto and Soury-Lavergne, 2013;Brady and Lehrer, 2021;Soury-Lavergne, 2021). ...
Introduction
This article illustrates a pedagogical approach to integrating models and modeling in Geometry with mathematics teacher-learners (MTLs). It analyzes the work of MTLs in a course titled “Computers, Teaching, and Mathematical Visualization” (or “MathViz”), which is designed to engage MTLs in making mathematics together. They use a range of both physical and virtual models of 2-manifolds to formulate and investigate geometric conjectures of their own.
Objectives
The article articulates the theoretical basis and design rationale of MathViz; it analyzes illustrative examples of the discourse produced in collaborative investigations; and it describes the impact of this approach in the students’ own voices.
Methods
MathViz has been iteratively refined and researched over the past 6 years. This study focuses on one iteration, aiming to capture the phenomenological experience of the MTLs as they structured and pursued their own mathematical investigations. Video data from two class sessions of the Fall 2021 iteration of the course are analyzed to illustrate the discourse of collaborating students and the nature of their shared inquiry. Excerpts from this class’s Learning Journals are then analyzed to capture themes across students’ experience of the course and their perspectives on its impact.
Results
Analysis of students’ discourse (while investigating cones) shows how they used models and gesture to make sense of geometric phenomena; forged connections with investigations they had conducted throughout the course on different surfaces; and articulated and proved mathematical conjectures of their own. Analysis of students’ Learning Journals illustrates how experiences in MathViz contributed to their conceptualization of making mathematics together, using a variety of models and technologies, and developing a set of practices that that they could introduce with their future students.
Discussion
An argument is made that this approach to collective mathematical investigation is not only viable and valuable for MTLs, but is also relevant to philosophical reflections about the nature of mathematical knowledge-creation.
... Then all volumes are just counts of that onedimensional unit. Brady and Lehrer (2020) clarified that a unit of area is conceived as twodimensional when one conceives it as generated by two segments, one being swept along the other. This is the imagistic equivalent of understanding the interior of a rectangle being . ...
Humans have been reasoning quantitatively for thousands of years. I did not invent quantitative reasoning. I developed a theory of quantitative reasoning—a theory with the aim of explaining how individuals might come to reason about the world as they see it through a measurement lens (including not seeing it through a measurement lens) and implications for students’ mathematical learning.
... Then all volumes are just counts of that one-dimensional unit. Brady and Lehrer (2020) clarified that a unit of area is conceived as two-dimensional when one conceives it as generated by two segments, one being swept along the other. This is the imagistic equivalent of understanding the interior of a rectangle being formed by the cross product of two perpendicular lines viewed as sets of points. ...
Constructing and representing relationships between quantities is critical to developing meanings for various ideas in school mathematics. In this chapter, we characterize basic types of covariational relationships and describe a task sequence we designed to support students in constructing and representing many such relationships. We describe several theoretical constructs, including extensions of (Thompson and Carlson in Compendium for research in mathematics education. National Council of Teachers of Mathematics, 2017) variational and covariational reasoning frameworks, that we leveraged when designing tasks and characterizing students’ reasoning as they conceived and graphically represented relationships. We then detail the task sequence and present results from a pair of students to provide an empirical example of how the task sequence was productive for supporting students in constructing numerous covariational relationships and eventually distinguishing nonlinear and linear relationships. We conclude with implications for the teaching and learning of middle grades mathematics and areas for future research.
... Then all volumes are just counts of that one-dimensional unit. Brady and Lehrer (2020) clarified that a unit of area is conceived as two-dimensional when one conceives it as generated by two segments, one being swept along the other. This is the imagistic equivalent of understanding the interior of a rectangle being formed by the cross product of two perpendicular lines viewed as sets of points. ...
Over the most recent several decades, researchers have argued the importance of quantitative and covariational reasoning for students’ learning. These same researchers have illustrated the importance of these reasoning processes with respect to local and longitudinal development. In both grain sizes, researchers are detailed in their descriptions of the intended topics or reasoning processes. There is, however, a lack of specificity of generalized criteria for concept construction from a quantitative reasoning perspective. In this chapter, we introduce such criteria through the construct of an abstracted quantitative structure, which has its roots in quantitative reasoning, covariational reasoning, and various Piagetian notions. In introducing the construct, we focus on ideas informing its development and its criteria, and we use it to characterize examples of student actions. We close with comments regarding implications for both teaching and research.
... Notably, one-to-one-to-model interactions blended the affordances of the physical and symbolic environment surrounding the learner with the computational environment of the model (Brady & Lehrer, 2021;Roth, 1995). It was neither the feedback from the model nor from the interviewer alone that created affordances but the dynamic interaction among the student, the model, and the interviewer. ...
Computational models are increasingly being used in K-12 science classrooms to engage students in developing and testing explanations of phenomena. However, research has only begun to consider whether integrating computational models into science instruction could be particularly beneficial to students from diverse backgrounds, including a fast-growing population of English learners (ELs) in the U.S. context. As this research begins to take shape, we argue for moving beyond the traditional discourse focused on “accommodating” ELs, which de-emphasizes the assets these students bring, and shifting our attention to the distinct affordances that computational models offer for harnessing ELs’ rich meaning-making potential. In this article, we conceptualize the affordances of computational models for ELs in science instruction. Specifically, we highlight evolving theories in the field of language education that undergird the shift from accommodations to affordances with ELs in the science classroom. We then propose affordances of computational models for ELs in relation to three framework components: modalities, registers, and interactions. Finally, we report on an initial inquiry into these affordances using student interview data from a linguistically diverse elementary science classroom. Ultimately, we argue that an affordances perspective could inform research and the design of learning environments that contribute to broadening participation in science learning and refuting deficit-based views of students traditionally underserved in STEM subjects.
... The DYME reasoning that students developed about rectangles could be foundational for understanding formulas of other shapes through transformations, such as understanding the area of parallelograms through shearing (Sinclair, Pimm & Skelin, 2012). The shift to a parallelogram is something that Brady and Lehrer (2020) explore in their experiments with the area-as-sweep approach; their experiment also combines physical and virtual environments. ...
This article reports on an exploratory study that engaged students in dynamic experiences of generating the area of a rectangle as a sweep of a line segment over a distance. A case study from a design experiment with one pair of third-grade students is presented to initiate a discussion around the forms of reasoning that students may exhibit as a result of their engagement with these dynamic motion tasks and the characteristics of the design that supported these particular forms of reasoning. The findings of this study show that engaging students in dynamic experiences of area may help them develop a conceptual understanding of the area of a rectangle as a continuous structure that can dynamically change based on the two linear measures that generate it: the length of the line segment swept and the distance of the sweep. These experiences can also help students from an early age develop a flexible understanding of a unit and reason covariationally about the continuous change of the quantities in measurement.
Area measurement has a high priority in mathematics school education. Nevertheless, many students have problems understanding the concept of area measurement. An AR tool for visualizing square units on objects in the real world is developed to enable teachers to support understanding already in primary school. This work-in-progress paper presents the initial test version and discusses the first teaching experiment results. The students’ feedback and use of the app showed possible adaptations of the AR tool, e.g., that the idea of dynamic geometry could be incorporated in the future.
The 8th annual International Conference of the Immersive Learning Research Network (iLRN2022) was the first iLRN event to offer a hybrid experience, with two days of presentations and activities on the iLRN Virtual Campus (powered by ©Virbela), followed by three days on location at the FH University of Applied Sciences BFI in Vienna, Austria.
Bu araştırmanın amacı, sınıf öğretmenlerinin çevre ve alan öğretimine yönelik görüşlerini nitel bir araştırma yoluyla incelemektir. Araştırmanın örneklemi, amaçsal örnekleme yöntemlerinden ölçüt örnekleme yöntemi ile belirlenmiştir. Bu bağlamda; 3. ve 4. sınıfta görev yapan 10 sınıf öğretmeni (5 kadın-5 erkek) ile görüşmeler gerçekleştirilmiştir. Elde edilen veriler betimsel analiz ile incelenmiştir. Araştırmanın bulgularına göre; sınıf öğretmenlerinin birçoğunun çevre ve alan ölçme öğretimini farklı ders içerikleri ile ilişkilendirerek gerçekleştirdiği belirlenmiştir. Bununla birlikte, sınıf öğretmenlerinin ilkokul öğrencilerinin gelişimsel özelliklerini dikkate aldıkları, öğretimi somut materyaller üzerinden planladıkları tespit edilmiştir. Araştırmanın bir başka sonucu ise, sınıf öğretmenlerinin birçoğunun öğretim esnasında tahmin etkinliklerine yer vermediğinin belirlenmesidir. Ayrıca, sınıf öğretmenlerinin rutin problemlerden oluşan etkinlikleri çevre ve alan ölçme öğretiminde sıklıkla kullandıkları tespit edilmiştir. Bu bağlamda; rutin olmayan problemlere öğretim etkinliklerinde yer verilmemesinin önemli bir eksiklik olduğu düşünülmektedir. Son olarak, sınıf öğretmenleri uzaktan eğitim sürecinin çevre ve alan ölçme öğretimini –somut yaşantı sağlayamaması nedeniyle- olumsuz etkilediğine dikkat çekmiştir.
