How Students Construct the Concept of Ratio by Joint Actions by Helen Vysotskaya, Maria Yanishevskaya, and Iya RekhtmanHere is the abstract of our report at EARLI SIG10 and SIG21 meeting (Utrecht, September 2010). The concept of ratio is extensively used in school physics
(density, speed, pressure, etc.) and chemistry (concentration) and related directly
to a very important mathematical concept of proportion. However, studies have
repeatedly shown that most middleschool students have difficulties with these
concepts, and the results of teaching are strongly dependent on the students’
activity in the process of learning. Modern textbooks often include simple lab experiments
devoted to these topics, but only a few studies analyze what actions are
necessary to assimilate the concepts [Siegler&Chen 2008]. Our purpose was
to figure out what kind of action is conductive for a correct construction and comprehension
of the concept of ratio (density, concentration). The results of our tests for 1013yearolders
show that most of them usually comprehend only one parameter (e.g., weight). That
may be enough to solve simple problems, but faced with complex tasks, they switch
to the everyday experience and fail. We designed our experimental module
to introduce the concept of density for 1011yearolders. It develops the Davydov’s
approach that builds on the heritage of Vygotsky and Leont’ev [Davydov 2008]. The
methodology requires that for the concept to appear and develop, it should be put
into a wide problem area. When studying the concept of density, the approach is
to explore the effect of buoyancy [Vysotskaya 1996]. Our reconstruction of logical and
activity content of the concept of density shows that weight and volume have to be initially presented as
independent features of an object. In pairwork, each student changes only one
parameter, so in order to solve the buoyancy problem, they should work together
to coordinate their actions. Switching the roles allows each student to learn how
each parameter works and thus realize the reciprocity of weight and volume. Transition
from “followinstructions” to “designaprocedure” involvement level plays an
important role because it directly changes student’s learning position and
attitude. As a result, students perceive each object which buoyancy they should
change as a set of “weights” and “floats” that balance each other. Appearance
of this model of an object is a first step towards the actual abstract concept
of density. Any attempt to bypass this stage resulted in a decrease in test
performance whenever a complex problem is encountered. Our conclusion is that representational
system is primary in forming the abstract tier of skills within the domain. Also, positive results have been
obtained for the introduction of the concept of concentration for
89yearolders, which suggests that the initial separation of the “nominator”
and “denominator” parameters is a promising universal approach.References: Davydov, V.V. (2008). Problems of Developmental Instruction.
Nova Science Publishers (1986 – in Russian) Siegler, R.S., Chen, Z. (2008).
Differentiation and integration: Guiding principles for analyzing cognitive
change. Developmental Science, 11,
433448. Vysotskaya, H.V. (1996).
Предметноориентированные учебные среды для формирования понятий в совместной
учебноисследовательской деятельности. В сб. В.В.Рубцов (ред.) Коммуникативноориентированные
образовательные среды. ПИ РАО.
