Searching for Buried Treasure: Uncovering the Discovery in Discovery-Based Learning

• In Event: Discovery-Based STEM Learning 2.0: Are We There Yet?

Abstract

Objective
What’s discovered in discovery-based learning? In the case of motor-action skills, Vereijken and Whiting (1990) write, “Discovery learning forces the learner to explore the dynamics of the system in which he or she operates” (p. 99). What might that possibly look like in mathematics, specifically algebra? For algebra, discovery learning might be tantamount to revealing systemic relations, structures, and functions for the handling of propositions toward determining unknown values. Yet what are these basic elements of algebra, and how might a student discover them? Perhaps the best way to understand a system is to build it (Goldstein & Papert, 1977)! But then which resources, tasks, feedback, and facilitation might optimize this reinvention of algebra? As a means of creating empirical context to investigate discovery learning, we developed a learning environment, Giant Steps for Algebra (GS4A).

Theoretical Approach
We conceptualize discovery-based learning as students making transparent for themselves how cultural artifacts function to mediate the achievement of collective objectives (Hancock, 1995; Meira, 1998). Where artifacts are epistemic, such as in mathematics, developing subjective transparency of the artifact is learning the content.

Methods
Instructional methodology has yet to enable students to build cognitive bridges from arithmetic to algebra (Herscovics & Linchevski, 1994; Molina & Ambrose, 2008).
Our design sought to render algebraic content transparent by:
a. selecting a model that enhances potential saliences of relations among variable and integers (see Figure 1);
b. engaging students’ tacit knowledge of quantitative relations latent to simple motion in space (Author, 2009, 2014a);
c. leveraging student narrative practices by posing enactment problems (Walkington, et al., 2013); see Figure 2);
d. assigning modeling tasks: students enact the narratives by utilizing available objects and functions, then reflect on embedded systemic features of their models; and
e. implementing the design in a digital environment (Sarama & Clements, 2009).

A pilot study (Authors, 2015a) implicated three transparency goals emerging from student reflection on their own models’ embedded structures: (1) consistent measures; (2) equivalent expressions; and (3) shared frame of reference. We conceptualized these goals as “situated intermediary learning objectives” (SILOs):

Our experimental design compared learning achievement under two conditions:
1. Discovery: each SILO functionality is automated only after the user discovers it
2. No-Discovery: all three SILO functionalities are always automated

Forty 4th and 9th Grade students, randomly assigned to experimental condition, voluntarily participated individually in task-based semi-structured interviews (Clement, 2000; Ginsburg, 1997). Mixed methods were applied, including post assessment and micro-genetic analyses.

Results
“Discovery” participants significantly outperformed “No-Discovery” participants in attaining the SILOs. Moreover, functionalities that “No Discovery” participants conceptualized as mere features, “Discovery” participants conceptualized as essential procedural functions.
Significance
Students learn mathematical concepts by making transparent structures and functions embedded in their spontaneous solution procedures. These discoveries are situated intermediary learning objectives that the activity elicits as the meaning underlying prospective formal procedures.

Authors

• Kiera NP Chase, University of California - Berkeley

• Dor Abrahamson, University of California - Berkeley