Physical Affordances of Manipulatives in Number Representation in the First Grade: Linearity and Discreteness

• In Event: Poster Session 06
  In Poster Session: PS 06 Topic - Cognitive Processes

Abstract

Manipulatives are concrete objects that are often used in mathematics classrooms. Our research on manipulatives centers on the notion of affordance, particularly on the ways in which their physical features (e.g., size, color, configuration) can direct and constrain action and cognition (Gibson, 1979). Recent research has shown that physical configurations that are transparent to the structure of number have the potential to support children’s numeration and place value understanding (Authors, 2019). Furthermore, Siegler and Ramani (2009) found that children’s learning of numerical magnitude was superior when they played number games on linear boards than circular ones because the materials physically reflected the conceptual structure of the mental number line (Dehaene, 1992).

In the present study, we tested how the physical configurations of manipulatives can render the properties of number transparent to first graders. We manipulated the linearity of manipulatives and the physical structure of the denominations (i.e., their discrete versus continuous nature) and examined the effects of these two factors on children’s representations of two-digit quantities and the interpretations of their own displays.

METHOD. First graders (N = 105) were randomly assigned to three manipulative conditions (Figure 1): (a) snake (continuous number line), (b) worm (discrete, linear tens), and (c) flower (discrete, non-linear tens). Children used the manipulatives to represent three two-digit numbers presented symbolically and interpreted their displays. Accuracy of student representations was scored and the types of representations and interpretations produced were coded for a descriptive response analysis.

RESULTS. Planned comparisons were used to test the effects of two affordances on representation accuracy: discreteness (snake versus worm) and linearity (worm versus flower). Results revealed no significant effects for either contrast, but the response analysis indicated differences in the types of representations and interpretations produced as a function of affordance (Figure 2). With respect to discreteness, children in the worm condition produced face-value representations (e.g., representing 73 with 7 ones and 3 ones), whereas no such responses were produced in the snake condition. Children in the snake condition produced more counting errors (e.g., representing 73 with 8 tens and 3 ones) and irrelevant responses than children in the worm condition. In contrast, children in the worm condition were more likely to correctly interpret their optimal displays (75%) than those in the snake condition (59%), who were more likely to give face-value interpretations (40%; 11% in the worm condition). No apparent differences emerged in representation and interpretation responses between the worm and flower conditions, suggesting no linearity effects.

CONCLUSION. The results indicated that physical affordances of manipulatives can impact children’s representations and interpretations of multidigit quantities, and that there is a tradeoff between affordance and performance. In particular, the number line manipulatives resulted in fewer place-value errors in the children’s representations than the objects with discrete denominations. On the other hand, the discrete manipulatives appeared to encourage a greater number of correct interpretations, perhaps because place value was more transparent. From a pedagogical perspective, this implies that not all correct representations suggest conceptual understanding of numeration.

Authors

• Anne Lafay, Concordia University
  Presenting Author

• Helena Patricia Osana, Concordia University
  Non-Presenting Author