## 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
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## La equipartición como obstáculo didáctico en la enseñanza de las fracciones

José Luis Cortina, Claudia Zúñiga y Jana Visnovska

Abstract: We advance the conjecture that equipartition constitutes what Brousseau’s 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 