Moving on a Path Versus Collecting Objects: New Perspectives to Analyze Student Learning With Integer Models

• In Event: 63.018 - Promising Scholarship in Education: Dissertation Fellows and Their Research

Abstract

Integer arithmetic is counterintuitive, yet essential to most mathematics beyond middle school. It
has been extensively studied (e.g., Gallardo, 2002; Küchermann, 1981; Liebeck, 1990; Vlassis,
2008), yet recommendations are contradictory about how to help students revise their concepts of
arithmetic to include negatives (Star & Nurnberger-Haag, 2011). Documentation of historical
mathematicians and of modern students has described cognitive obstacles for negative numbers
(e.g., Fischbein, 1987; Pierson, et al., 2014). A reading of these works in terms of conceptual
metaphor theory reveals that the object collection metaphor may be a cognitive obstacle to those
first learning about negative numbers (hereafter called initial learners). The conceptual
metaphors identified by Lakoff and Nunez (2000) applicable to integer arithmetic include the
object collection, motion along a path, and measuring stick metaphors. Flexible use of more than
one metaphor may be necessary for expert understanding (Chiu, 2001; Gentner & Bowdle, 2008;
Nunez, Edwards, & Matos, 1999). Yet, research of integer arithmetic via conceptual metaphors
is in its infancy and has reported what metaphors people used (Chiu, 2001; Kilhamn, 2011),
rather than the impact of models on initial learners due to metaphor-based physical motions. The
consistency of physical motions with targeted ideas is a factor of metaphor comprehension and
other cognition (Glenberg & Kaschak, 2002; Wilson & Gibbs, 2007). Thus, this study focused on
how specific motions in concert with conceptual metaphors represent mathematics concepts and
affect initial learning of integers. Note, whereas Lakoff and Nunez (2000) used the noun forms
of metaphors, I use the verb forms to emphasize the impact of students’ physical modelmovements.
Multiple methods were used to address questions a-e: After learning with one
model, a collecting objects based model or a moving on a path model: RQ1 (a) What patterns of
accuracy and reasoning do students demonstrate? (b) In what ways do students think of the “-”
symbol? (c) What conceptual metaphors do students express? RQ2 (d) What, if any, between
model group differences are found by comparing a-c? (e) How does the model used predict the
conceptual metaphor that students express?

Author

• Julie Anne Nurnberger-Haag, Kent State University