September 2017
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80 Reads
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21 Citations
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Publications (47)
September 2017
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97 Reads
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14 Citations
September 2017
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1,693 Reads
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86 Citations
January 2013
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729 Reads
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70 Citations
This chapter steps back from the front lines of mathematics education reform and looks forward within a long-term perspective. Our perspective draws upon a historical view of the long-term evolution of representations, the transformative potential of new media, and the growing challenges of meeting societal needs. We shall see that there have been enormous changes in all these factors over the past several hundred years. Our means of expressing mathematical ideas have changed and so have our expectations regarding who can learn what mathematics and at what age. We shall examine large-scale trends in content changes and in context changes for learning and using mathematics.
October 2011
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197 Reads
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87 Citations
Since the publication of this chapter, Jim Kaput was killed in a tragic road accident. We have all lost an energetic, visionary and dedicated colleague. Mathematics Education research has lost one of its greatest exponents, a researcher who not only understood the world of education, but knew how to change it. His theoretical work on the evolution of notational systems – much of it presented by him in this article – as well as his practical contribution, most recently through SimCalc, is living testimony to the importance and impact of his work. And we have lost a friend: a friend with a wonderful sense of humour and wit, a vibrant sense of fun, and an inspired intelligence. We will miss him.Not for the first time we are at a turning point in intellectual history. The appearances of new computational forms and literacies are pervading the social and economic lives of individuals and nations alike. Yet nowhere is this upheaval correspondingly represented in educational systems, in classrooms, or in school curricula. As far as mathematics is concerned, the massive changes to mathematics that characterize the late twentieth century—in terms of the way it is done, and what counts as mathematics—are almost invisible in the classrooms of our schools and, to only a slightly lesser extent, our universities. The real changes are not technical: they are cultural. Understanding them (and why some things change quickly and others change slowly) is a question of the social relations among people, not among things. Nevertheless, there are important ways in which computational technologies are different from those that preceded them, and in trying to assess the actual and potential contribution of these technologies to education, it will help to view them in a historical light.
January 2011
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806 Reads
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208 Citations
This chapter explores how elementary teachers can use functional thinking to build algebraic reasoning into curriculum and instruction. In particular, we examine how children think about functions and how instructional materials and school activities can be extended to support students’ functional thinking. Data are taken from a five-year research and professional development project conducted in an urban school district and from a graduate course for elementary teachers taught by the first author. We propose that elementary grades mathematics should, from the start of formal schooling, extend beyond the fairly common focus on recursive patterning to include curriculum and instruction that deliberately attends to how two or more quantities vary in relation to each other. We discuss how teachers can transform and extend their current resources so that arithmetic content can provide opportunities for pattern building, conjecturing, generalizing, and justifying mathematical relationships between quantities, and we examine how teachers might embed this mathematics within the kinds of socio-mathematical norms that help children build mathematical generality.
March 2009
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19 Reads
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10 Citations
Educational Studies in Mathematics
This paper comments on the expanded repertoire of techniques, conceptual frameworks, and perspectives developed to study the phenomena of gesture, bodily action and other modalities as related to thinking, learning, acting, and speaking. Certain broad issues are considered, including (1) the distinction between “contextual” generalization of instances across context (of virtually any kind—numeric, situational, etc.) and the generalization of structured actions on symbols, (2) fundamental distinctions between the use of semiotic means to describe specific situations versus semiosis serving the process of generalization, and (3) the challenges of building generalizable research findings at such an early stage in infrastructure building.
June 2008
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976 Reads
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180 Citations
Educational Studies in Mathematics
The nature of mathematical reference fields has substantially evolved with the advent of new types of digital technologies enabling students greater access to understanding the use and application of mathematical ideas and procedures. We analyze the evolution of symbolic thinking over time, from static notations to dynamic inscriptions in new technologies. We conclude with new perspectives on Kaput’s theory of notations and representations as mediators of constructive processes.
January 2008
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689 Reads
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136 Citations
January 2007
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178 Reads
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5 Citations
Research and development in technology and curriculum dedicated to democratizing access to the Mathematics of Change and Variation, including ideas underlying Calculus. We thank and remember Jim Kaput who was a PI on this project and pioneered SimCalc as part of his commitment to democratizing mathematics education.
Citations (45)
... From early algebra, functional thinking is considered as an approach that allows the development of algebraic modes of thinking from the first school years (Cañadas & Molina, 2016). It is intended that through tasks that involve relating quantities, students can establish general mathematical relationships using varied representations that may lead them to make such relationships (Soares et al., 2005). The tasks linked to functional thinking emphasize the relationships between quantities, bringing the student closer to the concept of function in an intuitive way (Blanton, 2008;Smith, 2008). ...
- Citing Article
January 2006
Teaching Children Mathematics
... El álgebra constituye un área crítica dentro de la enseñanza de las matemáticas. La introducción del simbolismo alfanumérico alrededor del sexto o séptimo grado implica un nivel de abstracción difícil de abordar para muchos estudiantes (Cañadas et al., 2019;Kaput, 2008). De hecho, el álgebra se ha descrito como el "guardián de la entrada" a las matemáticas superiores (Kaput, 2008), actuando como un filtro que clasifica a los estudiantes entre "buenos" y "malos" y limita sus trayectorias educativas (Cañadas et al., 2019;Schoenfeld, 1995). ...
- Citing Chapter
September 2017
... Mathematical concepts can also be represented symbolically (Kaput, 1987;Mainali, 2021), that is, using and manipulating mathematical symbols, including numbers, operations, connection signs, and algebraic signs, which will be referred to as symbolic representations (Anwar & Rahmawati, 2017;Kaput et al., 2017). Unlike visual representations, symbolic representations do not have corresponding visuospatial depictions. ...
- Citing Chapter
September 2017
... Professional learning opportunities were identified based on the PLTTs used, the DIAP and the RATE . On the other hand, the MKT domains (Ball et al., 2008) were identified from discussions regarding the different meanings of the equality symbol (Kieran, 1981) and the manifestations of development of functional thinking (Blanton & Kaput, 2008). ...
- Citing Chapter
September 2017
... It has been observed that technology is essential in doing, learning, and teaching mathematics (National Council for Teachers of Mathematics [NCTM], 2000), and there have been questions about whether research on technology-supported mathematics learning and teaching should develop under the responsibility of mathematics educators and researchers (Kaput & Thompson, 1994). From the past to the present, many studies on the use of technology in mathematics education have been conducted and continue to be undertaken. ...
- Citing Article
December 1994
Journal for Research in Mathematics Education
... When tiles are used to represent numbers and variables, their moves need to be tracked technically and feedback has to be implemented to indicate legal and illegal moves. Such constraints and their epistemological consequences needed to be considered at the intersection of computational and didactical transition of algebraic knowledge in the design by both disciplines (Balacheff and Kaput 1997). ...
- Citing Book
January 1997
... Visualization is the main element of spatial ability and can be applied as a stimulant in the activity of geometry learning (Dere & Kalelioglu, 2020). The geometry learning recommended to enhance spatial visualization is a learning utilizing technological media (Balacheff & Kaput, 2018). Therefore, integrating computer technology in mathematics learning, particularly in geometry lessons will be important and well-known in dealing with the 21 st challenging (Adelabu et al., 2019). ...
- Citing Chapter
- Full-text available
January 1997
... to remain as a filter today, even for some strong students (Bressoud et al., 2013) in Science, Technology, Engineering, and Mathematics (STEM) and business majors. Although there have been major changes in who is expected to complete calculus among college students, what is taught in a calculus course and how it is taught has been relatively stagnant, leading to an increasingly poor fit for the over 300,000 students in the United States that enroll in a calculus course each semester (Bressoud, 2015;Kaput, 1997). Since calculus is a gateway to other courses in many students' plans of study, having a foundational understanding of the content is critical. ...
- Citing Article
October 1997
... Nowadays, Physicists are using a wide variety of series for modeling phenomena [11,12]. Although in mathematics, there is no limit for the number of dimensions in multi-dimensional models, we are living in a 3-dimensional world while in some popular theory in physics such as String theory and M theory we have 10 ,11 or even more dimensions [13,14,15]. ...
- Citing Conference Paper
- Full-text available
July 1998
... It has published several books in the series International Perspectives on the Teaching and Learning of Mathematical Modelling 3 including Modeling students' mathematical modeling competencies and Mathematical Modelling in Education Research and Practice (Stillman, Blum & Biembengut, 2015). There are many other examples in the broader mathematical literature, see for example the SimCalc project (Roschelle, Kaput & Stroup 2012), a summary would be well beyond the scope of this brief overview! Hestenes (2010) suggests that "Students should become familiar with a small set of basic models as the content core for each branch of science, along with selected extensions to more complex models" (p. ...
- Citing Article
- Full-text available
January 2000
