Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task

Abstract

This study explores preservice teachers' understanding of the operator construct of rational number. Three related problems, given in 1-on-1 clinical interviews, consisted of finding 3/4 of a pile of 8 bundles of 4 counting sticks. Problem conditions were suggestive of showing 3/4 of the number of bundles (duplicator/partition-reducer [DPR] subconstruct) and 3/4 of the size of each bundle (stretcher/shrinker [SS] subconstruct). This study provides confirming instances that students use these 2 rational number operator subconstructs. The SS strategies are identified when the rational number, as an operator, is distributed over a uniting operation. With these SS strategies, rational number is conceptualized as a rate. However, the SS strategies were used less often than the DPR strategies. Detailed cognitive models of these strategies in terms of the underlying conceptual units, their structures, and their modifications, were produced, and a “mathematics of quantity” notational system was used as an analytical tool to describe and model the embedded abstractions.

Enseñanza Y Aprendizaje De La Razón, La Proporción Y La Proporcionalidad: Un Estado Del Arte

“…(1) Procesos cognitivos implicados en el aprendizaje de los números racionales: caracterización de tales procesos y definición de tipologías de situaciones relativas a la construcción de la unidad, al tipo de magnitud y a la cuantificación de magnitudes (Behr, Harel, & Post, 1992;Behr, Khoury, Harel, Post, & Lesh, 1997;Kieren, 1988;Ohlsson, 1988).…”

Preservice Teachers' Algebraic Reasoning and Symbol Use on a Multistep Fraction Word Problem

“…Research illustrates that PSTs also have difficulties with defining referent wholes for fractions (Tobias, 2013) as well as understanding fractions as operators (Behr, Khoury, Harel, Post, & Lesh, 1997). When defining wholes for fractions, Tobias (2013) found that when asked to find how much of a pizza one person would get if four pizzas are shared equally among five people, only eight of 33 PSTs defined the correct whole as one pizza.…”

“…When defining wholes for fractions, Tobias (2013) found that when asked to find how much of a pizza one person would get if four pizzas are shared equally among five people, only eight of 33 PSTs defined the correct whole as one pizza. In addition, research has shown that PSTs tend to favour partitioning strategies when asked to solve problems using a fraction as an operator (Behr et al, 1997). Behr, Khoury, Harel, Post, and Lesh (1997) found that when PSTs were asked to show 3/4 of 8 four-stick bundles, they tended to solve the problem by partitioning first, such as dividing the piles of sticks by four, as opposed to using measurement or stretching and shrinking methods to solve the problem, such as taking three groups of the 8 four-stick bundles.…”

İlköğretim 4-7. Sınıf Öğrencilerinin Denk Kesirlerin Sembolik ve Grafiksel Temsillerini İlişkilendirme Becerilerinin İncelenmesi

“…Equivalent fractions are critical for students to be able to master fractions. This is because equivalent fractions are associated with the ratio and operator interpretations of fractions and thus form the basis for fractional addition, subtraction, and sorting operations (Behr, Khoury, Harel, Post & Lesh, 1997). Equivalent fractions are a multiplicative relationship referring to the fixed ratio of a numerator and denominator division (Behr, Harel, Post & Lesh, 1992).…”