## Andrea’s Reflections: A Microgenetic Analysis of the Understanding of Division in the Context of a Problem

Abstract: This study describes changes in the meanings of the written representation
of the division algorithm within a partition word problem in Andrea, a
student from sixth grade in a public school. Through a clinical interview, she
solves various word problems. Based on the theory of conceptual fields, we
discuss the changes in different theorems-in-act and concepts-in-act that she
uses. She reflects on the relationships between the components of the division,
and reformulates her ideas on how these relationships must be expressed. The
most significant change observed in Andrea during the interview is related to
her understanding of the division algorithm; she initially writes it based on ideas
like “the bigger number must be inside” but, at the end of the interview, she
showed understanding of the relationship between the word problem and the
written algorithm. Likewise, are observed changes in the way she conceives
the measured-magnitude relationship. Discussion analyzes the relevance of
taken into consideration student’s concepts and theorems-in-act to strengthen
the mathematic learning.

Keywords: problem solving, simple proportionality, division, conceptualization,
graphic representation.

DOI: 10.24844/EM2901.04

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## Las reflexiones de Andrea: un análisis microgenético de la comprensión de la división en el contexto de un problema

Resumen: En este estudio se describen los cambios en las significaciones de
la representación escrita del algoritmo de la división en concordancia con un
problema de partición, en Andrea, una estudiante de sexto grado de primaria
de una escuela pública. Con base en la teoría de los campos conceptuales, se
analizan diferentes teoremas y conceptos-en-acto con los que ella actúa en
distintas situaciones problemáticas que se le plantean en una entrevista clínica
durante la cual reflexiona y reformula sus ideas sobre las relaciones expresadas
en los problemas, y su vínculo con el esquema de la división. El cambio más
importante que se observa es en los principios que rigen la escritura de la
división (teoremas-en-acto). Inicialmente, ella escribe el algoritmo guiada por
ideas como que “el número mayor va adentro”, sin que haya una relación conceptual
con lo que expresa el problema, al final de la entrevista Andrea plantea
la relación entre el dividendo y el divisor atendiendo a dicha relación. Asimismo,
se observan cambios en cómo concibe la relación medida-magnitud. Se discuten
las implicaciones educativas de tomar en consideración los conceptos y
teoremas-en-acto de los alumnos para fortalecer el aprendizaje de las
matemáticas.

Palabras clave: solución de problemas, división, proporcionalidad simple, conceptualización,
representación gráfica.

DOI: 10.24844/EM2901.04

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## Division of fractions as a multiplicative comparison based on methods invented by students

Alfinio Flores Peñafiel

Abstract: We present several methods invented by students in grades 5th through 8th
to solve division of fractions problems. For each method we discuss how the teacher
can help students develop their understanding of multiplicative comparison of fractions,
by emphasizing fundamental mathematical principles that allow students generalize,
extend, and relate their methods with other methods. The methods presented are repeated
subtraction and interpretation of the remainder; use of the multiplicative identity and
inverses; division as missing factor; inverse and direct proportional thinking; division of
fractions as composition of operations; and division of fractions as a ratio between two
quantities.

Keywords: fractions, division, multiplicative comparison, proportional reasoning, students
methods.

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