“Early use of numerical notations”

SCHEUER, NORA; MARTÍ, EDUARDO; DE LA CRUZ, MONTSERRAT

Berkeley

Exposici�n; 41st Annual Meeting of the Jean Piaget Society; 2011

Jean Piaget Society

Representational tools in problem solving Organizer: Eduardo Martí (Universitat de Barcelona) Organizer: Bárbara M Brizuela (Tufts University) Discussant: Edith Ackerman (MIT School of Architecture Design Lab) Challenges in learning mathematics and science are not only conceptual, but also representational. In fact, students have to deal with a wide variety of representational forms when solving mathematics and science problems (Duval, 1995; Lemke, 1993). In the present symposium, four studies dealing with the understanding and use of representational tools are presented. In the first presentation we address a ba- sic question: at which age and how can young children understand symbolically an additive representation of number. The results can be summarized as follows: 1) There is a clear gap between the capacity to discriminate a set of objects (under three) and the symbolic understanding of number representation (also under three); 2) there is a close relation between the capacity to evaluate (verbally or through gestures) the quantity of dots and the use of this information to regulate action; and 3) the symbolic understanding of additive number representation appears later than the symbolic understanding of spatial representation. In the second presentation, two cases of young children’s (Kindergarten and second grade) exploration of a real-time graph of linear human motion are presented and discussed according to how it affects their own representations of movement. Results show how manipulating conventional representations can facilitate children’s exploration and building of new understandings of the scientific phenomenon of motion, and encourage them to create novel and individually powerful representations to help structure their thinking. In the third presentation, we explore the ways in which tables can provide first to third grade children with opportunities to solve problems that might otherwise be outside their reach. Results show that when children are able to use written supports such as tables, they can tackle more complex problems. We also found that even problems with initial unknowns (i.e., x + 5 = 8) and with unknown transformations (i.e., 5 + x = 8) are not outside of children’s reach when they have access to adequate representational supports. The fourth presentation deals with middle school students’ understanding of tables when they are used as a bridge to construct a bar graph. Results show that the main difficulties in constructing the bar graph can be identified in students’ table construction processes. Choosing an appropriate format that integrates several variables and correctly aggregates the data into frequencies are the main problems encountered by students, and these problems are related to the tables that they constructed to organize the data. Early use of numerical notations Nora Scheuer (Centro Regional Universitario Bariloche) Montserrat de la Cruz (Centro Regional Universitario Bariloche) Eduardo Martí (Universitat de Barcelona) The weird dot: How children make use of conventional external representational tools to extend their thinking about motion and build new invented representations Jason Kahn (Children’s Hospital Boston) Mathematical representations as tools among early elementary school children Bárbara M Brizuela (Tufts University) Mónica Alvarado (Universidad Autónoma de Querétaro) Middle school students’ understanding of tables as tools for constructing bar graphs Merce Garcia-Mila (Universitat de Barcelona) Eduardo Martí (Universitat de Barcelona) Sandra Gilabert (Universitat de Barcelona)