Focusing attention on auxiliary lines when introduced into geometric problems

ABSTRACT

The introduction of auxiliary elements as a method of solving problems in high-school geometry is considered here from two perspectives: first, to elicit recalling some known result or concretizing a definition and, second, as a means of shifting the focus and structure of the students’ attention. We present and compare various examples of how auxiliary elements can be introduced in various problems and proofs and characterize their auxiliary quality. Some auxiliary elements unite previously unrelated components of the original diagram; others divide a given complex entity into manageable ones. Implications for further educational research and mathematics instruction are proposed.

KEYWORDS:

• Auxiliary lines
• high-school geometry
• shifts of attention
• problem-solving

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Alik Palatnik http://orcid.org/0000-0002-0367-2713

Notes

1. We are aware that there are more ways to change the focus and structure of attending to a diagram. For instance, see [20] on differences in attending to three parallel lines with or without auxiliary frames of reference.

2. This task has already proved to be an almost inexhaustible source of various tasks and explorations (see [22]).

3. See Proof Without Words [24].

4. Nelsen [25] defines PWWs as ‘ … pictures or diagrams that help the observer see why a particular statement may be true, and also how one might begin to go about proving it true’ (p. vi).