Jose Luis Cortina | Universidad Pedagogica Nacional (original) (raw)

Papers by Jose Luis Cortina

In J. Novotna & H. Moraova (Eds.), Equity and diversity in elementary mathematics education, Proceedings of the bi-annual International Symposium on Elementary Mathematics Teaching (pp. 430–440). Prague, Czech Republic: Charles University., 2017

We report on our research collaboration with Irene, a Mexican teacher who undertook to trial a re... more We report on our research collaboration with Irene, a Mexican teacher who undertook to trial a research-based instructional sequence on Fractions as Measures, in her underperforming fifth grade classroom. In the process of the collaboration, Irene became a strong advocate of the importance of pursing all students’ understanding, and notions of equity in access to mathematical ideas came to shape her planning and classroom decision-making. We first outline significant student fraction learning that took place in Irene’s classroom. We then document Irene’s evolving teaching practice and how she came to base her instructional decisions in students’ reasoning broadly, and in the reasoning of students for whom the ideas were proving the most challenging, in particular. We discuss how the instructional sequence supported Irene’s learning, and suggest that resources of this kind are particularly well suited for supporting teachers’ transition to ambitious and equitable mathematics teaching.

Se exponen los elementos fundamentales de una didáctica focalizada en la adquisición y comprens... more Se exponen los elementos fundamentales de una didáctica focalizada en la adquisición y comprensión de los sistemas de numeración que forman parte de las lenguas que hablan los pueblos originarios de México. Ésta fue desarrollada a través de la metodología de la investigación basada en el diseño. El trabajo realizado ha implicado la producción de recursos didácticos, así como de innovaciones teóricas y metodológicas formuladas con el propósito de apoyar el proceso de diseño y afinación de esos recursos. El artículo se centra en el diseño de una propuesta didáctica de la numeración del tu’un savi (o mixteco), como caso paradigmático de la investigación realizada.

Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education, 2016

The presented study is part of a bigger design and research enterprise in the teaching of fractio... more The presented study is part of a bigger design and research enterprise in the teaching of fractions as measures. We analyze extracts of a teaching session with a single fifth grade student, in which he flexibly compared the relative sizes of the lengths of three drinking straws, skillfully using unitary, proper, and improper fractions. We identify aspects of his prior instructional experiences that supported the emergence of his relatively sophisticated ways of reasoning. Findings suggest that supporting students' reasoning about reciprocal relations of relative size can be a viable goal in an instructional agenda on fractions as measures.

Cortina, J. L. & Visnovska, J. (in press, 2016). Reciprocal relations of relative size in the instructional context of fractions as measures. To be published in Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education, Szeged, Hungary: IGPME.

pmena.org

In this paper I analyze 37 clinical interviews of 13-year-old students from the same classroom. T... more In this paper I analyze 37 clinical interviews of 13-year-old students from the same classroom. The interviews were conducted for the purpose of documenting students' quantitative understanding about fractions. The interviewed students attended a school that belongs to a branch of the Mexican educational system ("Secundaria Técnica") ranked as low-performing according to results from PISA 2003. The analysis serves to identify some important challenges in designing meaningful mathematical instruction for students attending this kind of educational institutions. It suggests that strategies should be developed to help students build from early types of proportional notions, given that many might not have developed satisfactory understandings about fractions from prior instructional and/or out-of-school experiences. The analysis also suggests that some students might also need help in developing relatively basic number-sense.

Mathematics Education Research Journal, Mar 1, 2013

ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigen... more ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigenous teachers analyzed the numeration systems of their first language. The main goal of the project is to develop resources that help teachers in supporting students’ understanding of the systems. In the first phase of the project, the central organizing ideas of 14 numeration systems were specified. Each system belonged to a different Mesoamerican language. Three aspects of the systems were identified that would have to be accounted for in instructional design. They include using 20 as a multiplicative base. Examples are presented of the instructional resources that indigenous teachers could use to help their students understand the quantitative rationales of the systems.

Fourteen clinical interviews of fourth grade students from a Mexican public school are analyzed. ... more Fourteen clinical interviews of fourth grade students from a Mexican public school are analyzed. The interviews included tasks in which students were asked to reason about the relative capacity of cups, specifically of how many of them could be filled with the milk contained in a milk carton. The analysis suggests that these activities could be a viable starting point for supporting students' learning of fractions; a starting point that can be an alternative to the "equal-partitioning" (or "equal-sharing") approach that has been traditionally used.

This paper is based on an interview conducted to a seventh grade student (13 years old) of an urb... more This paper is based on an interview conducted to a seventh grade student (13 years old) of an urban public school in México. In the interview the student describes the change she experienced as a mathematics learner throughout a school year. This change involved developing a new attitude towards mathematics, and improving her ways of participating in math class and her academic performance. The paper discusses how the student's experience can help Latin-American mathematics teachers to support their underachieving pupils' academic improvement.

Se discuten las aportaciones a la didáctica de las fracciones de cuatro inves- tigaciones que el ... more Se discuten las aportaciones a la didáctica de las fracciones de cuatro inves- tigaciones que el autor y sus colegas han realizado, retomando algunos aspectos de la metodología de los experimentos de diseño. Estas aportaciones conciernen a la pun- tualización de los objetivos de aprendizaje de las fracciones, la identificación de puntos de partida para la enseñanza y el desarrollo de propuestas alternativas para apoyar el aprendizaje de este concepto.

Las mediciones de la calidad del aprendizaje matemático en México: ¿qué nos devela la prueba PISA... more Las mediciones de la calidad del aprendizaje matemático en México: ¿qué nos devela la prueba PISA 2003 y cómo podemos responder? José Luis Cortina R Re es su um me en n: : Este ensayo aporta elementos para delinear una agenda que efectivamente responda a la realidad que develan las mediciones de la calidad de la educación básica en el caso de las matemáticas en México. El ensayo se centra en la prueba PISA 2003, instrumentada por la OCDE. Está dividido en dos partes: la primera está dedicada a especificar los retos educativos en matemáticas que develó la prueba PISA 2003 en el caso mexicano; la segunda está dedicada a enmarcar los retos conceptuales implícitos en el desarrollo de intervenciones educativas eficaces en contextos institucionales en los que adquieren gran importancia los resultados de pruebas estandarizadas como la de PISA 2003. Se distingue entre dos racionalidades básicas implícitas en los programas de mejoramiento de la enseñanza matemática: la descendente y la ascendente. Se explica por qué la racionalidad ascendente puede ser más pertinente para el caso mexicano.

In this paper we discuss the contributions of socio-cultural theories to research and design of i... more In this paper we discuss the contributions of socio-cultural theories to research and design of interventions directed at the professional development of mathematics teachers. We explain how these theories have been put to work in the field. We also bring to attention specific issues arising in the field for which development and adaptation of socio-cultural theory might be a useful resource.

Mathematics Education Research Journal, 2013

ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigen... more ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigenous teachers analyzed the numeration systems of their first language. The main goal of the project is to develop resources that help teachers in supporting students’ understanding of the systems. In the first phase of the project, the central organizing ideas of 14 numeration systems were specified. Each system belonged to a different Mesoamerican language. Three aspects of the systems were identified that would have to be accounted for in instructional design. They include using 20 as a multiplicative base. Examples are presented of the instructional resources that indigenous teachers could use to help their students understand the quantitative rationales of the systems.

Proceedings of the Sixth International Conference of …, 2002

This paper develops insights into how to deal with the arithmetic mean in the initial phases of s... more This paper develops insights into how to deal with the arithmetic mean in the initial phases of statistics instruction. It focuses on one of the various ways in which the mean can be used in statistics: as a normalized ratio. The paper analyzes individual interviews of 12 12-year-old students prior to their participation in a classroom teaching experiment. These interviews were used to test and develop instructional conjectures about how to support students' understanding of the mean and other normalized ratios. Three differences where detected in how students seemed to make sense of proportional comparison problems in which the use of the mean as a ratio is pertinent. The paper explains how these findings can be capitalized upon in designing and conducting instruction.

Proceedings of the Twenty …, 1999

Background Mokros and Russell (1995) distinguish two basic models that have been used in the lite... more Background Mokros and Russell (1995) distinguish two basic models that have been used in the literature when defining the mean: fair share and balance. These authors note that works that have used these models seem to omit the notion of representativeness as an ...

In this paper we look at how students can be supported to reason with a mathematical inscription ... more In this paper we look at how students can be supported to reason with a mathematical inscription system. We do so by analyzing several episodes from a classroom design experiment that centered on supporting middle school students' understanding of proportional relations in the context of measurement. We illustrate how the teacher orchestrated whole-class conversations that built on students' diverse ways of reasoning and how she used an inscription system to revoice their contributions. We explain that such efforts made it possible for students to start communicating and reasoning mathematically with the inscriptions.

Acta Didactica Universitatis Comenianae--Mathematics, Dec 2014

We advance the conjecture that equipartition constitutes what Brousseau calls a didactical obstac... more We advance the conjecture that equipartition constitutes what Brousseau calls a didactical obstacle in the fraction realm. We illustrate how equipartition orients pupils to develop ways of conceiving fractions that interfere with the development of a mature understanding of rational numbers. We then outline two possible alternative pathways and call for researching and documenting viable instructional approaches to introducing fractions that avoid generation of this obstacle.

In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins, (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia), pp. 181-188. Sunshine Coast: MERGA., 2015

Fractions are a well-researched area, yet, student learning of fractions remains problematic. We ... more Fractions are a well-researched area, yet, student learning of fractions remains problematic. We outline a novel path to initial fraction learning and document its promise. Building on Freudenthal’s analysis of the fraction concept, we regard comparing, rather than fracturing, as the primary activity from which students are expected to make sense of fractions. Analysing a classroom design experiment conducted with a class of 14 fourth grade pupils, we identify two successive mathematical practices that emerged in the course of the experiment and indicate how their emergence was supported.

In Proceedings of the 7-th International Conference on Teaching Statistics. [CD-Rom]. Salvador, Brazil., 2006

In this paper we analyze a design experiment aimed at supporting the professional development in ... more In this paper we analyze a design experiment aimed at supporting the professional development in statistics of twelve middle-school teachers in the United States. We explain how adopting a sociocultural framework allowed us to account for teachers’ struggles to make sense of instructional practices in statistics that place students’ reasoning at the center of instructional decision-making. We also account for how the adoption of the sociocultural framework allowed us to envision a viable way in which to better support the professional development of the participating teachers.

In J. Dindyal, L. P. Cheng & S. F. Ng (Eds.), Mathematics education: Expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia). Singapore: MERGA., 2012

We investigate the viability of a new approach to initial fraction instruction. We establish the ... more We investigate the viability of a new approach to initial fraction instruction. We establish the need to empirically investigate whether the proposed approach shares the strengths of currently used approaches, specifically, whether students will (a) construe problems based on the proposed approach as experientially real, and (b) bring up ideas that could be build upon in subsequent fraction instruction. We then present an analysis of sixteen student interviews from a school in southern Mexico (ages 8 and 9). The analysis supports the conjecture that the proposed approach to initial fraction instruction can be viable, and thus warrants further research attention.

In J. Novotna & H. Moraova (Eds.), Equity and diversity in elementary mathematics education, Proceedings of the bi-annual International Symposium on Elementary Mathematics Teaching (pp. 430–440). Prague, Czech Republic: Charles University., 2017

We report on our research collaboration with Irene, a Mexican teacher who undertook to trial a re... more We report on our research collaboration with Irene, a Mexican teacher who undertook to trial a research-based instructional sequence on Fractions as Measures, in her underperforming fifth grade classroom. In the process of the collaboration, Irene became a strong advocate of the importance of pursing all students’ understanding, and notions of equity in access to mathematical ideas came to shape her planning and classroom decision-making. We first outline significant student fraction learning that took place in Irene’s classroom. We then document Irene’s evolving teaching practice and how she came to base her instructional decisions in students’ reasoning broadly, and in the reasoning of students for whom the ideas were proving the most challenging, in particular. We discuss how the instructional sequence supported Irene’s learning, and suggest that resources of this kind are particularly well suited for supporting teachers’ transition to ambitious and equitable mathematics teaching.

Se exponen los elementos fundamentales de una didáctica focalizada en la adquisición y comprens... more Se exponen los elementos fundamentales de una didáctica focalizada en la adquisición y comprensión de los sistemas de numeración que forman parte de las lenguas que hablan los pueblos originarios de México. Ésta fue desarrollada a través de la metodología de la investigación basada en el diseño. El trabajo realizado ha implicado la producción de recursos didácticos, así como de innovaciones teóricas y metodológicas formuladas con el propósito de apoyar el proceso de diseño y afinación de esos recursos. El artículo se centra en el diseño de una propuesta didáctica de la numeración del tu’un savi (o mixteco), como caso paradigmático de la investigación realizada.

Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education, 2016

The presented study is part of a bigger design and research enterprise in the teaching of fractio... more The presented study is part of a bigger design and research enterprise in the teaching of fractions as measures. We analyze extracts of a teaching session with a single fifth grade student, in which he flexibly compared the relative sizes of the lengths of three drinking straws, skillfully using unitary, proper, and improper fractions. We identify aspects of his prior instructional experiences that supported the emergence of his relatively sophisticated ways of reasoning. Findings suggest that supporting students' reasoning about reciprocal relations of relative size can be a viable goal in an instructional agenda on fractions as measures.

Cortina, J. L. & Visnovska, J. (in press, 2016). Reciprocal relations of relative size in the instructional context of fractions as measures. To be published in Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education, Szeged, Hungary: IGPME.

pmena.org

In this paper I analyze 37 clinical interviews of 13-year-old students from the same classroom. T... more In this paper I analyze 37 clinical interviews of 13-year-old students from the same classroom. The interviews were conducted for the purpose of documenting students' quantitative understanding about fractions. The interviewed students attended a school that belongs to a branch of the Mexican educational system ("Secundaria Técnica") ranked as low-performing according to results from PISA 2003. The analysis serves to identify some important challenges in designing meaningful mathematical instruction for students attending this kind of educational institutions. It suggests that strategies should be developed to help students build from early types of proportional notions, given that many might not have developed satisfactory understandings about fractions from prior instructional and/or out-of-school experiences. The analysis also suggests that some students might also need help in developing relatively basic number-sense.

Mathematics Education Research Journal, Mar 1, 2013

ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigen... more ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigenous teachers analyzed the numeration systems of their first language. The main goal of the project is to develop resources that help teachers in supporting students’ understanding of the systems. In the first phase of the project, the central organizing ideas of 14 numeration systems were specified. Each system belonged to a different Mesoamerican language. Three aspects of the systems were identified that would have to be accounted for in instructional design. They include using 20 as a multiplicative base. Examples are presented of the instructional resources that indigenous teachers could use to help their students understand the quantitative rationales of the systems.

Fourteen clinical interviews of fourth grade students from a Mexican public school are analyzed. ... more Fourteen clinical interviews of fourth grade students from a Mexican public school are analyzed. The interviews included tasks in which students were asked to reason about the relative capacity of cups, specifically of how many of them could be filled with the milk contained in a milk carton. The analysis suggests that these activities could be a viable starting point for supporting students' learning of fractions; a starting point that can be an alternative to the "equal-partitioning" (or "equal-sharing") approach that has been traditionally used.

This paper is based on an interview conducted to a seventh grade student (13 years old) of an urb... more This paper is based on an interview conducted to a seventh grade student (13 years old) of an urban public school in México. In the interview the student describes the change she experienced as a mathematics learner throughout a school year. This change involved developing a new attitude towards mathematics, and improving her ways of participating in math class and her academic performance. The paper discusses how the student's experience can help Latin-American mathematics teachers to support their underachieving pupils' academic improvement.

Se discuten las aportaciones a la didáctica de las fracciones de cuatro inves- tigaciones que el ... more Se discuten las aportaciones a la didáctica de las fracciones de cuatro inves- tigaciones que el autor y sus colegas han realizado, retomando algunos aspectos de la metodología de los experimentos de diseño. Estas aportaciones conciernen a la pun- tualización de los objetivos de aprendizaje de las fracciones, la identificación de puntos de partida para la enseñanza y el desarrollo de propuestas alternativas para apoyar el aprendizaje de este concepto.

Las mediciones de la calidad del aprendizaje matemático en México: ¿qué nos devela la prueba PISA... more Las mediciones de la calidad del aprendizaje matemático en México: ¿qué nos devela la prueba PISA 2003 y cómo podemos responder? José Luis Cortina R Re es su um me en n: : Este ensayo aporta elementos para delinear una agenda que efectivamente responda a la realidad que develan las mediciones de la calidad de la educación básica en el caso de las matemáticas en México. El ensayo se centra en la prueba PISA 2003, instrumentada por la OCDE. Está dividido en dos partes: la primera está dedicada a especificar los retos educativos en matemáticas que develó la prueba PISA 2003 en el caso mexicano; la segunda está dedicada a enmarcar los retos conceptuales implícitos en el desarrollo de intervenciones educativas eficaces en contextos institucionales en los que adquieren gran importancia los resultados de pruebas estandarizadas como la de PISA 2003. Se distingue entre dos racionalidades básicas implícitas en los programas de mejoramiento de la enseñanza matemática: la descendente y la ascendente. Se explica por qué la racionalidad ascendente puede ser más pertinente para el caso mexicano.

In this paper we discuss the contributions of socio-cultural theories to research and design of i... more In this paper we discuss the contributions of socio-cultural theories to research and design of interventions directed at the professional development of mathematics teachers. We explain how these theories have been put to work in the field. We also bring to attention specific issues arising in the field for which development and adaptation of socio-cultural theory might be a useful resource.

Mathematics Education Research Journal, 2013

ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigen... more ABSTRACT Results from a project conducted in Mexico are discussed, in which a group of 17 indigenous teachers analyzed the numeration systems of their first language. The main goal of the project is to develop resources that help teachers in supporting students’ understanding of the systems. In the first phase of the project, the central organizing ideas of 14 numeration systems were specified. Each system belonged to a different Mesoamerican language. Three aspects of the systems were identified that would have to be accounted for in instructional design. They include using 20 as a multiplicative base. Examples are presented of the instructional resources that indigenous teachers could use to help their students understand the quantitative rationales of the systems.

Proceedings of the Sixth International Conference of …, 2002

This paper develops insights into how to deal with the arithmetic mean in the initial phases of s... more This paper develops insights into how to deal with the arithmetic mean in the initial phases of statistics instruction. It focuses on one of the various ways in which the mean can be used in statistics: as a normalized ratio. The paper analyzes individual interviews of 12 12-year-old students prior to their participation in a classroom teaching experiment. These interviews were used to test and develop instructional conjectures about how to support students' understanding of the mean and other normalized ratios. Three differences where detected in how students seemed to make sense of proportional comparison problems in which the use of the mean as a ratio is pertinent. The paper explains how these findings can be capitalized upon in designing and conducting instruction.

Proceedings of the Twenty …, 1999

Background Mokros and Russell (1995) distinguish two basic models that have been used in the lite... more Background Mokros and Russell (1995) distinguish two basic models that have been used in the literature when defining the mean: fair share and balance. These authors note that works that have used these models seem to omit the notion of representativeness as an ...

In this paper we look at how students can be supported to reason with a mathematical inscription ... more In this paper we look at how students can be supported to reason with a mathematical inscription system. We do so by analyzing several episodes from a classroom design experiment that centered on supporting middle school students' understanding of proportional relations in the context of measurement. We illustrate how the teacher orchestrated whole-class conversations that built on students' diverse ways of reasoning and how she used an inscription system to revoice their contributions. We explain that such efforts made it possible for students to start communicating and reasoning mathematically with the inscriptions.

Acta Didactica Universitatis Comenianae--Mathematics, Dec 2014

We advance the conjecture that equipartition constitutes what Brousseau calls a didactical obstac... more We advance the conjecture that equipartition constitutes what Brousseau calls a didactical obstacle in the fraction realm. We illustrate how equipartition orients pupils to develop ways of conceiving fractions that interfere with the development of a mature understanding of rational numbers. We then outline two possible alternative pathways and call for researching and documenting viable instructional approaches to introducing fractions that avoid generation of this obstacle.

In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins, (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia), pp. 181-188. Sunshine Coast: MERGA., 2015

Fractions are a well-researched area, yet, student learning of fractions remains problematic. We ... more Fractions are a well-researched area, yet, student learning of fractions remains problematic. We outline a novel path to initial fraction learning and document its promise. Building on Freudenthal’s analysis of the fraction concept, we regard comparing, rather than fracturing, as the primary activity from which students are expected to make sense of fractions. Analysing a classroom design experiment conducted with a class of 14 fourth grade pupils, we identify two successive mathematical practices that emerged in the course of the experiment and indicate how their emergence was supported.

In Proceedings of the 7-th International Conference on Teaching Statistics. [CD-Rom]. Salvador, Brazil., 2006

In this paper we analyze a design experiment aimed at supporting the professional development in ... more In this paper we analyze a design experiment aimed at supporting the professional development in statistics of twelve middle-school teachers in the United States. We explain how adopting a sociocultural framework allowed us to account for teachers’ struggles to make sense of instructional practices in statistics that place students’ reasoning at the center of instructional decision-making. We also account for how the adoption of the sociocultural framework allowed us to envision a viable way in which to better support the professional development of the participating teachers.

In J. Dindyal, L. P. Cheng & S. F. Ng (Eds.), Mathematics education: Expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia). Singapore: MERGA., 2012

We investigate the viability of a new approach to initial fraction instruction. We establish the ... more We investigate the viability of a new approach to initial fraction instruction. We establish the need to empirically investigate whether the proposed approach shares the strengths of currently used approaches, specifically, whether students will (a) construe problems based on the proposed approach as experientially real, and (b) bring up ideas that could be build upon in subsequent fraction instruction. We then present an analysis of sixteen student interviews from a school in southern Mexico (ages 8 and 9). The analysis supports the conjecture that the proposed approach to initial fraction instruction can be viable, and thus warrants further research attention.

International Journal of Mathematics Teaching and Learning, Dec 2015

We analyze the results of a study conducted for the purpose of assessing the viability of an alte... more We analyze the results of a study conducted for the purpose of assessing the viability of an alternative starting point for teaching fractions. The alternative is based on Freudenthal’s insights about fraction as comparison. It involves portraying the entities that unit fractions quantify as always being apart from the reference unit, instead of as parts of an equally divided whole. The study consisted of interviewing 16 third-grade students on a series of fraction tasks that embody the proposed alternative starting point. The analysis supports regarding the proposed starting point as viable.

Mathematics Education Research Journal, 2014

We analyze a classroom design experiment, conducted in a fourth grade classroom, that served to e... more We analyze a classroom design experiment, conducted in a fourth grade classroom, that served to explore an instructional path in which the introduction of unit fractions and supporting proportional reasoning coincide. Central to this path is the use of means of support in which the objects that unit fractions quantify are not characterized as equal-sized parts of a whole, but as entities that are always separate from a reference unit. We argue that such a path is crucial for helping students develop deep quantitative understandings of fractions, where fraction quantities are, from the very start, linked to the reciprocal and multiplicative relations that their use implies. We focus on the first part of the design experiment in which we helped the students make sense of a concept that is important for initial fraction learning and proportional reasoning, the inverse order relationship among unit fractions.

We advance the conjecture that equipartition constitutes what Brousseau calls a didactical obstac... more We advance the conjecture that equipartition constitutes what Brousseau calls a didactical obstacle in the fraction realm. Building on Hans Freudenthal, Patrick Thompson and Luis Saldanha's analyses of the fraction concept, we explain why it is reasonable to consider that equipartition orients pupils to develop ways of conceiving fractions that interfere with the development of a mature understanding of rational numbers.
Keywords: fractions, didactical obstacle, equipartition, rational numbers.

***
Se plantea la conjetura de que el uso de la equipartición en la enseñanza inicial de las fracciones constituye un obstáculo didáctico. Retomando los análisis del concepto de fracción realizados por Hans Freudenthal, Patrick Thompson y Luis Saldanha, se explica por qué es razonable esperar que la equipartición oriente a los estudiantes a entender las fracciones en formas que dificultan el desarrollo de concepciones maduras de los números racionales.

We explore how instructional sequences, grounded in a conjectured learning trajectory, can suppor... more We explore how instructional sequences, grounded in a conjectured learning trajectory, can support teachers' preparation for classroom interactions with students' ideas. Using two examples from our design experiments, we illustrate that teachers in transition (a) develop a need to select and design classroom activities in which students would come to problematize some of their reasoning, and (b) require substantial support in planning for productive classroom interactions. We argue that instructional resources can and should be designed to provide some of this support.

We report on a study that consisted of administering tests to 297 sixth grade students from 13 di... more We report on a study that consisted of administering tests to 297 sixth grade students from 13 different schools in Mexico. Pupils were asked to identify the quantity represented by common fractions (e.g., 1/2, 1/4, 1/3, 3/4). Results suggest that many students are finishing elementary school with a deficient understanding of fractions: some lagging behind so significantly that they have not developed understandings that allow them to readily and correctly interpret the quantitative meaning of the most common fraction notations, including “1/2.” We discuss the implications of these results for students’ opportunities to learn mathematics in middle school.