How Students Construct the Concept of Ratio by Joint Actions 

                               by Helen Vysotskaya, Maria Yanishevskaya, and Iya Rekhtman

Here 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 middle-school 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 10-13-year-olders 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 10-11-year-olders. 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 “follow-instructions” to “design-a-procedure” 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 8-9-year-olders, which suggests that the initial separation of the “nominator” and “denominator” parameters is a promising universal approach.

   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, 433-448.
   Vysotskaya, H.V. (1996). Предметно-ориентированные учебные среды для формирования понятий в совместной учебно-исследовательской деятельности. В сб. В.В.Рубцов (ред.) Коммуникативно-ориентированные образовательные среды. ПИ РАО.