Functions via everyday actions: Support or obstacle?

• Published in: The Journal of mathematical behavior 2014-12, Vol.36, p.126-134
• Main Authors: Leron, Uri, Paz, Tamar
• Format: Article
• Language: English
• Subjects: Actions on objects; Analytical thinking; Changing-the-input misconception; Composition of functions; Dual-process theory; Functions; Intuitive thinking
• ISSN: 0732-3123; 1873-8028
• DOI: 10.1016/j.jmathb.2014.09.005

The general context of this paper is the power of intuitive thinking, and how it can help or hinder analytical thinking. The research literature in cognitive psychology teems with tasks where intuitive thinking leads subjects to “non-normative” answers, including tasks for which they have all the knowledge necessary for the normative answer. The best explanation to date for such phenomena is dual-process theory, which stipulates the activation of a quick automatic intuitive process (System 1), together with the failure of the heavy, lazy, and computationally expensive analytical process (System 2) to intervene and correct the intuitive response. In an earlier paper, we have documented a clash between intuitive and analytical thinking concerning functions, which we have termed the changing-the-input phenomenon. The discovery of the changing-the-input phenomenon, however, left us with a puzzle: Why has this phenomenon concerning functions – a purely mathematical concept – been observed in computer science classes but not in mathematics ones? The purpose of the present paper is to address this puzzle. More generally we ask, under what conditions the changing-the-input phenomenon will or will not be manifested? Still more generally, in learning about functions, when is the intuitive scaffolding of functions via actions-on-tangible-objects helpful, and when does it get in the way of deeper understanding?