STRATEGIES FOR SMALLHOLDERS IN DEVELOPING COUNTRIES: COMMERCIALISATION, DIVERSIFICATION AND EXIT

This paper proposes a strategic framework for policies to assist smallholders in developing countries. It describes the inevitable features of structural change in the agricultural and rural economy, the associated pressures that these changes place on smallholders, and the consequent need for policies to facilitate rather than impede adjustment. A key premise of the framework is that, for the majority of smallholders, the long term (i.e. inter-generational)future lies outside the sector. Hence, long-term policies need to make a distinction between those who potentially have a competitive future in the sector and those who do not. In either case, many of the necessary policies will not be agriculture-specific, so it is important that agricultural policies are framed in a broader economy-wide framework. In addition, a clear distinction needs to be made between short-term policies to reduce poverty and food insecurity and long-term policies to stimulate development. This is because there are intertemporal trade-offs (as well as complementarities) between policies that are likely to be effective in the short-run, and those promising most impact over the long-term. The paper discusses the role of different agricultural and non-agricultural policies in providing the appropriate policy mix in countries at different stages of development.smallholders, rural development, agricultural policy, structural change, Agricultural and Food Policy, Community/Rural/Urban Development, International Development, O20, Q18, R23,

A generalization of Zhu's theorem on six-valent integer distance graphs

Given a set $S$ of positive integers, the integer distance graph for $S$ has the set of integers as its vertex set, where two vertices are adjacent if and only if the absolute value of their difference lies in $S$. In 2002, Zhu completely determined the chromatic number of integer distance graphs when $S$ has cardinality $3$. Integer distance graphs can be defined equivalently as Cayley graphs on the group of integers under addition. In a previous paper, the authors develop general methods to approach the problem of finding chromatic numbers of Cayley graphs on abelian groups. To each such graph one associates an integer matrix. In some cases the chromatic number can be determined directly from the matrix entries. In particular, the authors completely determine the chromatic number whenever the matrix is of size $3\times 2$ -- precisely the size of the matrices associated to the graphs studied by Zhu. In this paper, then, we demonstrate that Zhu's theorem can be recovered as a special case of the authors' previous results.Comment: 6 page

An alternate proof of Payan's theorem on cubelike graphs

A cubelike graph is a Cayley graph on the product $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ of the integers modulo $2$ with itself finitely many times. In 1992, Payan proved that no cubelike graph can have chromatic number $3$. The authors of the present paper previously developed a general matrix method for studying chromatic numbers of Cayley graphs on abelian groups. In this note, we apply this method of Heuberger matrices to give an alternate proof of Payan's theorem.Comment: 4 page

Chromatic numbers of Cayley graphs of abelian groups: Cases of small dimension and rank

A connected Cayley graph on an abelian group with a finite generating set $S$ can be represented by its Heuberger matrix, i.e., an integer matrix whose columns generate the group of relations between members of $S$. In a companion article, the authors lay the foundation for the use of Heuberger matrices to study chromatic numbers of abelian Cayley graphs. We call the number of rows in the Heuberger matrix the dimension, and the number of columns the rank. In this paper, we give precise numerical conditions that completely determine the chromatic number in all cases with dimension $1$; with rank $1$; and with dimension $\leq 3$ and rank $\leq 2$. For such a graph without loops, we show that it is $4$-colorable if and only if it does not contain a $5$-clique, and it is $3$-colorable if and only if it contains neither a diamond lanyard nor a $C_{13}(1,5)$, both of which we define herein. In a separate companion article, we show that we recover Zhu's theorem on the chromatic number of $6$-valent integer distance graphs as a special case of our theorem for dimension $3$ and rank $2$.Comment: 27 page

Chromatic numbers of Cayley graphs of abelian groups: A matrix method

In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an $m\times r$ integer matrix, where we call $m$ the dimension and $r$ the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. In a series of follow-up articles using this method, we completely determine the chromatic number in cases with small dimension and rank; prove a generalization of Zhu's theorem on the chromatic number of $6$-valent integer distance graphs; and provide an alternate proof of Payan's theorem that a cube-like graph cannot have chromatic number 3.Comment: 17 page

Experiences of Mexican teenage students when choosing a math degree: a mathematical narrative identity study

There is little qualitative research on mathematics education focused on the experiences of young students when choosing a mathematics degree and how these experiences are assimilated into their mathematics life stories. The objective of this narrative inquiry is to identify the experiences of Mexican students who choose a mathematics degree through their mathematics life story. The conceptualization of a mathematical narrative identity divided into motivations, sources of motivation, and expectations allowed the identification of the following: (1) motivation of Mexican students for choosing a math degree, (2) sources of this motivation, and (3) future expectations related to this choice. This qualitative study was conducted based on a case study to prepare an in-depth analysis of multiple cases and frame them into a general description. Data was gathered from 47 interviews to collect students’ mathematics life stories. The four thematic analyses gave the following results: (1) the three main motivations were “liking mathematics”, self-efficacy belief, and the desire to become a “good teacher”, (2) the two main expectations were “being a good teacher” and “learning more mathematics”, and (3) the four main sources of motivations were self-efficacy belief, having “good teachers”, indirect experiences, and mastering knowledge. Results have similarities with the importance of self-efficacy beliefs and differences between “liking mathematics” and the desire to become a “good teacher” regarding the psychological explanations about the motivational forces to choose a math degree

Las refutaciones, el modelo de Toulmin y las argumentaciones colectivas

En esta oportunidad se presentará una propuesta teórico – metodológica, enmarcada en el campo de la argumentación matemática dentro del salón de clases. Esta propuesta busca evidenciar cómo la refutación de aserciones puede hacer que emerjan aspectos importantes dentro de las discusiones matemáticas. Para ello, se realizó la presentación de investigaciones referentes al campo y su respectiva teoría que la sustenta, para así establecer a manera de conclusión que la refutación evidencia las lógicas de las prácticas docentes, razonamientos de los estudiantes y formas de refutar los argumentos

Argumentación compleja en Educación Primaria

This paper describes a study of mathematical argumentation in primary school. The principal aim is to explore the nature of complex argumentation at a structural level. The context of the study was a teaching experiment involving nine tasks that promoted argumentation among fifth graders. We use the framework and method of reconstructing complex argumentation in the classroom proposed by Knipping (2008). The findings show that complex argumentation at a structural level in the context of refuting conclusions is characterized by a source-like structure with the addition of a new refutation argument element.Este artículo describe un estudio de argumentación matemática en educación primaria. El objetivo principal es explorar la naturaleza de la argumentación compleja en un nivel estructural. El contexto del estudio fue un experimento de enseñanza con nueve tareas que promovieron la argumentación entre estudiantes de quinto grado. Usamos el marco teórico y metodológico para reconstruir la argumentación compleja en el salón de clase propuesto por Knipping (2003). Los resultados muestran que la argumentación compleja a nivel estructural en el contexto de refutar conclusiones se caracteriza por ser una estructura de fuente con el agregado de un argumento de refutación

Experiências de jovens mexicanos na eleição de uma carreira matemática: um estudo da identidade narrativa matemática

La educación matemática tiene poca investigación cualitativa centrada en las experiencias de los jóvenes estudiantes en la elección de una carrera matemática y cómo estas experiencias se asimilan en sus vidas matemáticas. El objetivo de esta investigación narrativa es identificar las experiencias de los alumnos mexicanos que eligen una carrera matemática a través de su historia de vida matemática. La conceptualización de una identidad narrativa matemática dividida en motivaciones, fuentes de motivaciones y expectativas permitió la identificación de: (1) motivaciones de los estudiantes mexicanos para elegir una carrera matemática, (2) fuentes de estas motivaciones y (3) expectativas futuras asociadas con esta elección. Desarrollamos un estudio cualitativo guiado por un caso de estudio, con el fin de realizar un análisis de múltiples casos y enmarcarlo en una descripción general. Los datos se obtuvieron de 47 entrevistas con la población descrita y recopilan sus historias de vida matemática. Cuatro análisis temáticos arrojaron los siguientes resultados: (1) tres motivaciones principales: “gusto por las matemáticas”, creencias de autoeficacia y el deseo de convertirse en un “buen maestro”, (2) dos expectativas principales: “ser un buen maestro” y “aprender más matemáticas” y (3) cuatro fuentes principales de motivaciones: creencias de autoeficacia, tener “buenos maestros”, experiencias indirectas y dominio del conocimiento. Nuestros resultados tienen similitudes con (la importancia de las creencias de autoeficacia) y diferencias entre (“gusto por las matemáticas” y el deseo de convertirse en un “buen maestro”) en las explicaciones psicológicas sobre las fuerzas motivadoras en la elección de una carrera matemática.There is little qualitative research on mathematics education focused on the experiences of young students when choosing a mathematics degree and how these experiences are assimilated into their mathematics life stories. The objective of this narrative inquiry is to identify the experiences of Mexican students who choose a mathematics degree through their mathematics life story. The conceptualization of a mathematical narrative identity divided into motivations, sources of motivation, and expectations allowed the identification of the following: (1) motivation of Mexican students for choosing a math degree, (2) sources of this motivation, and (3) future expectations related to this choice. This qualitative study was conducted based on a case study to prepare an in-depth analysis of multiple cases and frame them into a general description. Data was gathered from 47 interviews to collect students’ mathematics life stories. The four thematic analyses gave the following results: (1) the three main motivations were “liking mathematics”, self-efficacy belief, and the desire to become a “good teacher”, (2) the two main expectations were “being a good teacher” and “learning more mathematics”, and (3) the four main sources of motivations were self-efficacy belief, having “good teachers”, indirect experiences, and mastering knowledge. Results have similarities with the importance of self-efficacy beliefs and differences between “liking mathematics” and the desire to become a “good teacher” regarding the psychological explanations about the motivational forces to choose a math degree.A educação matemática possui pouca pesquisa qualitativa centrada nas experiências dos jovens estudantes na eleição de um curso de matemática e como tais experiências são assimiladas em suas vidas matemáticas. Esta pesquisa narrativa tem como objetivo identificar as experiências dos alunos mexicanos que escolhem um curso de matemática por meio de sua história de vida matemática. A conceitualização de uma identidade narrativa no ensino da matemática dividida em motivações, fontes de motivações e expectativas, permitiu a identificação de: (1) motivações dos estudantes mexicanos para escolher um curso de matemática, (2) fontes destas motivações e (3) expectativas futuras associadas com tal eleição. Desenvolvemos um estudo qualitativo guiado por um caso de estudo, com a finalidade de realizar uma análise de múltiplos casos e enquadrá-lo em uma descrição geral. Os dados foram obtidos de 47 entrevistas com o público descrito e reúnem suas histórias de vida matemática. Quatro análises temáticas revelaram os seguintes resultados: (1) três principais motivações: “gosto pela matemática”, crença de autoeficácia e desejo de tornar-se um “bom professor”, (2) duas principais expectativas: “ser um bom professor” e “aprender mais matemática” e (3) quatro principais fontes de motivações: crenças de autoeficácia, ter “bons professores”, experiências indiretas e domínio do conhecimento. Nossos resultados têm semelhanças com (a importância das crenças de autoeficácia) e diferenças entre (“gosto pela matemática” e o desejo de tornar-se um “bom professor”) nas explicações psicológicas sobre as forças motivadoras na eleição de um curso de matemática

Una propuesta metodológica para elaborar videos creativos en clase de geometría

In the framework of the pandemic produced by the SARS-coV-2 virus, a methodological proposal is provided to elaborate creative geometry videos as a means or tool that allows to approach the teaching of concepts of Euclidean geometry. The videos are the product of research seminars with future mathematics teachers where the teaching of geometry is approached from approaches based on the construction of mathematical knowledge and concept formation processes (PFC). In the research, a qualitative and descriptive methodology was implemented that allowed us to analyze the mathematical content of the videos, materials and activities proposed by the mathematics teachers in training. The main result of the research is a methodological proposal for the elaboration of creative videos of geometry (VCG), this provides a set of principles that allow mathematics teachers to design their own creative videos in order to improve the teaching of geometry supported by the process of concept formation and generating spaces for reflection in terms of open digital media that support the teaching processes of geometry in times of pandemicEn el marco de la pandemia producida por el virus SARS-coV-2 se provee una propuesta metodológica para elaborar videos creativos de geometría como un medio o herramienta que permita abordar la enseñanza de conceptos de la geometría Euclidiana. Los videos son producto de seminarios de investigación con futuros profesores de matemáticas donde se aborda la enseñanza de la geometría desde enfoques basados en la construcción de conocimiento matemático y los procesos de formación de conceptos (PFC). En la investigación se implementó una metodología de corte cualitativa y descriptiva que permitió analizar el contenido matemático de los videos, los materiales y las actividades propuestas por los profesores de matemáticas en formación. El principal resultado de la investigación es una propuesta metodológica para la elaboración de videos creativos de geometría (VCG), esta proporciona un conjunto de principios que permiten a los profesores de matemáticas diseñar sus propios videos creativos con el fin de mejorar la enseñanza de la geometría apoyado del proceso de formación de conceptos y generar espacios de reflexión en cuanto a medios digitales abiertos que apoyen los procesos de enseñanza de la geometría en tiempos de pandemia