The challenge of proportion: does it require rethinking of the measurement paradigm?
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The challenge of proportion: does it require rethinking of the measurement paradigm? Elena Vysotskaya 1 & Anastasia Lobanova 1
2
& Iya Rekhtman & Maria Yanishevskaya
1
Accepted: 9 September 2020/ # Springer Nature B.V. 2020
Abstract
We are developing an approach to teaching important proportionality-based concepts to first grade students in a way that supports students’ future progress in the domain. We consider the proportionality between magnitudes as a basic relationship behind multiple cases, usually described mathematically as ratio or rate. The core of our strategy is the modelling of a situation of proportionality and its transformations by creating a compound unit. The key action is coordinated measurement (co-measurement): students work in pairs, and each student is in charge of changing one of two magnitudes, while preserving proportionality. Based on our successful experiments in Grades 2–6, we incorporate shared responsibility work organization (“joint actions”) and the idea of compound unit into Davydov’s mathematics curriculum for the first grade. We built a new module based on Davydov’s idea about the role of rule-mediated counting by sets. In this paper, we present the results of our study, showing that first graders can learn the idea of compound unit and work with two magnitudes of different kinds while preserving proportionality. Keywords Proportionality-based concepts . Developmental instruction . Primary math education . Measurement and co-measurement . Compound unit
1 Introduction Teachers and educators consider proportionality-based concepts difficult for students of all ages, even adults (e.g., Lamon, 2012). There are many related concepts in
* Anastasia Lobanova [email protected]
1
Psychological Institute, Russian Academy of Education, Moscow, Russia
2
Introchemist Association, www.introchemist.org, Morrisville, NC, USA
Vysotskaya E. et al.
mathematics and science domains: fractions, rates, percentage, ratios, etc. They manifest their proportional nature more or less explicitly, and comprise a lot of topics in secondary school. According to various studies, understanding and application of these concepts are poor and deficient (e.g., Hilton, Hilton, Dole, & Goos, 2013). Since Jean Piaget’s works (Inhelder & Piaget, 1958), mastering such concepts is considered an important milestone: developmental psychologists use them to study the specificity of children’s reasoning. More research is needed to find learning trajectories allowing students to better grasp proportionality-related concepts.
2 Developmental instruction: Basic principles of curriculum design We work within the Developmental Instruction framework (Davydov, 2008), which is deeply grounded in the Russian psychology heritage: the Activity Theory by Leontiev (1978), Cultural-Historical Psychology by Vygotsky (1986), and the Theory of Step-by-Step Formation of Mental Actions by Galperin (1989). The Developmental Instruction framework has been built upon the following big ideas: 1) A concept is a mental action to r
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