Spontaneous Meta-Arithmetic as the First Step Toward School Algebra

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the “official” algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses

Spontaneous meta-arithmetic as the first step toward school algebra

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the “official” algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses. La meta-aritmética espontánea como el primer paso hacia el álgebra escolar Tomando como punto de partida la visión del álgebra escolar como un meta-discurso formalizado de la aritmética, hemos estado siguiendo a seis pares de estudiantes de 7º curso (12-13 años) cuando modifican gradualmente su meta-aritmética espontánea hacia la forma algebraica “oficial” de hablar. En este artículo miramos el principio de este proceso. Los análisis preliminares de los datos han mostrado, como era de esperar, que mientras reflexionaban sobre los procesos y relaciones aritméticas, los alumnos no iniciados emplearon medios coloquiales que no evitaban las ambigüedades ocasionales. Más inesperadamente, esta meta-aritmética espontánea, a pesar de no apoyarse en ninguna enseñanza algebraica previa, desplegó algunas características similares al álgebra que no se encuentran normalmente en los discursos diarios.Handle: http://hdl.handle.net/10481/1831

Spontaneous Meta-Arithmetic as the First Step Toward School Algebra

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the �official� algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses.Tomando como punto de partida la visión del álgebra escolar como un meta-discurso formalizado de la aritmética, hemos estado siguiendo a seis pares de estudiantes de 7º curso (12-13 años) cuando modifican gradualmente su meta-aritmética espontánea hacia la forma algebraica �oficial� de hablar. En este artículo miramos el principio de este proceso. Los análisis preliminares de los datos han mostrado, como era de esperar, que mientras reflexionaban sobre los procesos y relaciones aritméticas, los alumnos no iniciados emplearon medios coloquiales que no evitaban las ambigüedades ocasionales. Más inesperadamente, esta meta-aritmética espontánea, a pesar de no apoyarse en ninguna enseñanza algebraica previa, desplegó algunas características similares al álgebra que no se encuentran normalmente en los discursos diarios

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Resumen basado en el de la publicaciónSe presenta un estudio desde el punto de vista del álgebra escolar como un 'meta-discurso' formalizado de la aritmética. Se ha seguido a seis pares de estudiantes de séptimo curso cuando modifican gradualmente su meta-aritmética espontánea hacia la forma algebraica 'oficial' de hablar. Los análisis preliminares de los datos muestran que mientras reflexionaban sobre los procesos y relaciones aritméticas, los alumnos no iniciados emplearon medios coloquiales que no evitaban las ambigüedades ocasionales. Además se ha observado que esta 'meta-aritmética' espontánea, a pesar de no apoyarse en ninguna enseñanza algebraica previa, desplegó algunas características similares al álgebra que no se encuentran normalmente en los discursos diarios.AndalucíaBiblioteca de Educación del Ministerio de Educación, Cultura y Deporte; Calle San Agustín, 5 - 3 planta; 28014 Madrid; Tel. +34917748000; [email protected]

Spontaneous Meta-Arithmetic as the First Step Toward School Algebra (La meta-aritmética espontánea como el primer paso hacia el álgebra escolar)

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the “official” algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses.Tomando como punto de partida la visión del álgebra escolar como un meta-discurso formalizado de la aritmética, hemos estado siguiendo a seis pares de estudiantes de 7º curso (12-13 años) cuando modifican gradualmente su meta-aritmética espontánea hacia la forma algebraica “oficial” de hablar. En este artículo miramos el principio de este proceso. Los análisis preliminares de los datos han mostrado, como era de esperar, que mientras reflexionaban sobre los procesos y relaciones aritméticas, los alumnos no iniciados emplearon medios coloquiales que no evitaban las ambigüedades ocasionales. Más inesperadamente, esta meta-aritmética espontánea, a pesar de no apoyarse en ninguna enseñanza algebraica previa, desplegó algunas características similares al álgebra que no se encuentran normalmente en los discursos diarios

La meta-aritmética espontánea como el primer paso hacia el álgebra escolar

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the “official” algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses.Tomando como punto de partida la visión del álgebra escolar como un meta-discurso formalizado de la aritmética, hemos estado siguiendo a seis pares de estudiantes de 7º curso (12-13 años) cuando modifican gradualmente su meta-aritmética espontánea hacia la forma algebraica “oficial” de hablar. En este artículo miramos el principio de este proceso. Los análisis preliminares de los datos han mostrado, como era de esperar, que mientras reflexionaban sobre los procesos y relaciones aritméticas, los alumnos no iniciados emplearon medios coloquiales que no evitaban las ambigüedades ocasionales. Más inesperadamente, esta meta-aritmética espontánea, a pesar de no apoyarse en ninguna enseñanza algebraica previa, desplegó algunas características similares al álgebra que no se encuentran normalmente en los discursos diarios

Binary Gene Expression Patterning of the Molt Cycle: The Case of Chitin Metabolism

<div><p>In crustaceans, like all arthropods, growth is accompanied by a molting cycle. This cycle comprises major physiological events in which mineralized chitinous structures are built and degraded. These events are in turn governed by genes whose patterns of expression are presumably linked to the molting cycle. To study these genes we performed next generation sequencing and constructed a molt-related transcriptomic library from two exoskeletal-forming tissues of the crayfish <i>Cherax quadricarinatus</i>, namely the gastrolith and the mandible cuticle-forming epithelium. To simplify the study of such a complex process as molting, a novel approach, binary patterning of gene expression, was employed. This approach revealed that key genes involved in the synthesis and breakdown of chitin exhibit a molt-related pattern in the gastrolith-forming epithelium. On the other hand, the same genes in the mandible cuticle-forming epithelium showed a molt-independent pattern of expression. Genes related to the metabolism of glucosamine-6-phosphate, a chitin precursor synthesized from simple sugars, showed a molt-related pattern of expression in both tissues. The binary patterning approach unfolds typical patterns of gene expression during the molt cycle of a crustacean. The use of such a simplifying integrative tool for assessing gene patterning seems appropriate for the study of complex biological processes.</p></div

Relative levels of key chitin metabolism-related genes transcript.

<p>Relative levels of key chitin metabolism-related genes transcript from the gastrolith-forming epithelium (left), the mandible cuticle-forming epithelium (middle) and the hepatopancreas (right), as determined by qPCR. Numbers on the X axis represents the four molt stages, 1 inter-molt (n = 5), 2 early pre-molt (n = 6), 3 late pre-molt (n = 6) and 4 post-molt (n = 6). Presented transcripts are (a) chitin synthase, (b) chitinase, (c) chitin deacetylase, (d) GlcN-6P synthase and (e) GlcN-6P deaminase. Letters represent statistical groups which are significantly different (<i>p</i>-value <0.05), error bars represents standard error.</p

Two representative examples of the binary patterns code used in our study.

<p>Top graph—a 0110 pattern in which the quantity of reads in early and late pre-molt is significantly higher, as compared to inter-molt and post-molt (<i>p</i>-value <0.05). Bottom graph—a 1001 pattern in which there is a significant increase of reads in inter-molt and post-molt, as compared to early and late pre-molt (<i>p</i>-value <0.05). The number 1 in the code represents a high number of reads as compared to a low number of reads, represented with 0. High quantity of reads have a minimum of 10³ reads.</p

Number of contigs in each binary pattern.

<p>(a) Gastrolith-forming epithelium samples and (b) mandible cuticle-forming epithelium samples. Patterns are arranged by abundance, from most to least abundant. A representative example of the binary pattern code is shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0122602#pone.0122602.g001" target="_blank">Fig 1</a>.</p