Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts

Radford, Luis

doi:10.1007/s11858-007-0061-0

Cited by 152 publications

(186 citation statements)

References 16 publications

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“…From an epistemological point of view, semiotic contraction characterizes modern mathematical symbolism, and from a cognitive point of view it is a precious mechanism. Radford (2008) relates contraction to focusing attention to the elements that are relevant for a certain situation and to a deeper level of consciousness: "Contraction is the mechanism for reducing attention to those aspects that appear to be relevant. This is why, in general, contraction and objectification entail forgetting.…”

Section: Discussionmentioning

confidence: 99%

Exploring the Contribution of Gestures to Mathematical Argumentation Processes from a Semiotic Perspective

Sabena

2018

Invited Lectures From the 13th International Congress on Mathematical Education

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A multimodal perspective on mathematics thinking processes is addressed through the semiotic bundle lens and considering a wide notion of sign drawing from Vygotsky's works. Within this frame, the paper focuses on the role of gestures in their interaction with the other signs (speech, in particular) and investigates the support they can provide to mathematical argumentation processes. A case study in primary school in the context of strategic interaction games provides data to show that gestures can support students in developing argumentations that depart from empirical stances and shift to a hypothetical plane in which generality is addressed. In this regard, by combining synchronic and diachronic analysis of the semiotic bundle, specific features of gestures are pointed out and discussed: the semiotic contraction, the condensing character of gestures, and the use of gesture space in a metaphorical sense.Keywords Argumentation Á Gestures Á Multimodality Á Semiotic contraction Semiotic bundle IntroductionAt the turn of the millennium, in 2000, the provocative essay Where Mathematics Comes From by George Lakoff and Rafael Núñez pointed out the crucial role of perceptual and bodily aspects on the formation of abstract concepts, including mathematical concepts (Lakoff and Núñez 2000). The new stance emphasized sensory and motor functions, as well as their importance for successful interaction with the environment. Criticizing the platonic idealism and the Cartesian mindbody dualism, Lakoff and Núñez advocated that all kinds of ideas, including the most sophisticated mathematical ideas, are founded on our bodily experiences and develop through cognitive metaphorical mechanisms. More recently, embodied stances seem to receive a certain confirmation by neuroscientific results on "mirror neurons" and "multimodal neurons," which are neurons firing when subjects performs actions, when they observe somebody else doing the same action, and when they imagine it (Gallese and Lakoff 2005). On the basis of these results, Gallese and Lakoff (2005) provide a new theoretical account on how the brain works, according to which "action and perception are integrated at the level of the sensory-motor system and not via higher association areas" (p. 459).In particular, such an integration would appear to be crucial not only for motor control, but also for planning actions, an activity typical of what is generally understood as "thinking."The terms multimodal and multimodality come therefore to indicate a feature of human cognition opposed to "modularity." On the other hand, in the communication field the term multimodal is used with reference to multiple modalities that we have to communicate and express meanings to our interlocutors: words, sounds, images, and so on (Kress 2004). These communicative affordances have been acquiring increasing attention due the diffusion of new technological affordances, which are constantly developing new possibilities of interaction with them through our body.In this paper, in line with Radf...

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Section: Discussionmentioning

confidence: 99%

Exploring the Contribution of Gestures to Mathematical Argumentation Processes from a Semiotic Perspective

Sabena

2018

Invited Lectures From the 13th International Congress on Mathematical Education

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“…According to Radford (2008) who approaches number patterns with this perspective poses three kinds of generalizations: algebraic generalization, arithmetic generalization and naïve induction. Radford (2008) suggests finding the rule of a number pattern using algebraic generalization.…”

Section: Generalizing Number Patternsmentioning

confidence: 99%

“…Radford (2008) suggests finding the rule of a number pattern using algebraic generalization. The algebraic generalization process is comprised of the following stages: noticing a commonality of the number pattern, examining whether or not the commonality is consistent in other terms of the pattern, and composing a general rule for finding any term within the pattern.…”

Section: Generalizing Number Patternsmentioning

confidence: 99%

Middle school teachers views and approaches to implement mathematical tasks

Yeşildere-İmre¹,

Baştürk-Şahin²

2016

Educ. Res. Rev.

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This research examines middle school mathematics teachers' views regarding implementation of mathematical tasks and their enactments. We compare their views on tasks and their implementation, and determine the causes of difference between the two using qualitative research methods. We interview sixteen middle school mathematics teachers based on their professional experience and willingness, and observe task implementations of four. According to the data, we found that the teachers with the most professional experience have difficulty to implement the tasks, they have teacher-centred mentality and they make use of their own experiences as students than of the task itself.

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“…Ortak özellikten yazma Şekil 1 Cebirsel Örüntü Genellemenin Yapısı (Radford, 2008, s.85) Şekil 1'deki süreçte Radford (2008), son okla gösterilen aşama çıkarıldığında yapılan genellemenin aritmetik genelleme olduğunu belirtmektedir.…”

Section: Ayırt Etmeunclassified

9. Sınıf Öğrencilerinin Örüntü Genelleme Problemlerini Çözme Stratejilerinin Belirlenmesi

Çayir¹,

Akyüz²

2015

Necatibey Eğitim Fakültesi Elektronik Fen Ve Matematik Eğitimi Dergisi

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-The goal of the study is to determine the pattern generalization strategies of 9th grade students, completed their basic education, in Balıkesir. Two open-ended questions, related with generalization of linear figure pattern, were asked to 425, 9th grade students from different schools. Content analysis was used in the study for data collection, analysis and interpretation. Students were more successful in finding the nearer term than finding further terms, since they were using either writing consecutive terms or adding the difference to the previous term. The research data showed that students were insufficient in finding the further terms problems and the equations related with pattern generalization. MethodologyTwo open-ended questions, related with generalization of linear figure pattern, were asked to 425, 9th grade students from different schools in Balıkesir. These problems were examined by two mathematics education experts for expert review. Students were requested to write their solutions in detail. Content analysis was used in the study for data collection, analysis and interpretation. Students' answers to the questions were coded using the different strategies determined from the literature. The frequencies and percentages for each question according to the categories were calculated. FindingsThe study showed that 9 th grade students have different solution strategies in solving different problems. Even, the same students used different strategies in solving the problems according to the numbers included in the problem. In addition, students were more successful in finding the nearest term than finding further terms, since they were using either writing consecutive terms or adding the difference to the previous term. The research data showed that students were insufficient in finding the further terms problems. They mostly used recursive technique for finding nearer terms and explicit strategy for finding further terms.Students were more successful in finding the nearer terms by calculating the difference and adding it to the previous term or writing the numbers consecutively than finding the further terms. Some students preferred to express the rules with words instead of algebraic notation.Students also tended to use proportional strategy, if generalization to a seductive number (e.g.10, 20 etc.) was asked to students and this strategy led to false results. Students prefer to use explicit/functional strategy, if they were asked further and non-seductive terms (e.g.104). DiscussionThe underlying idea of generalization problems is to recognize, determine, explain and represent the pattern in the problem. These skills are important for successful transition from arithmetic to algebra. The findings revealed that students were less successful in determining the correct generalization in the problems and they use different strategies to solve the pattern generalization problems. Although, recognizing the pattern was not a problem for most of the students, representing and explaining the formul...

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Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts

Cited by 152 publications

References 16 publications

Exploring the Contribution of Gestures to Mathematical Argumentation Processes from a Semiotic Perspective

Exploring the Contribution of Gestures to Mathematical Argumentation Processes from a Semiotic Perspective

Middle school teachers views and approaches to implement mathematical tasks

9. Sınıf Öğrencilerinin Örüntü Genelleme Problemlerini Çözme Stratejilerinin Belirlenmesi

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