This is the first of two closely related articles (Paper I and Paper II) that together illustrate how research in physics education has helped guide the design of instruction that has proved effective in improving student understanding of atomic spectroscopy. Most of the more than 1000 students who participated in this four-year investigation were science majors enrolled in the introductory calculus-based physics course at the University of Washington (UW) in Seattle, WA, USA. The others included graduate and undergraduate teaching assistants at UW and physics majors in introductory and advanced physics courses at the University of Zagreb, Zagreb, Croatia. About half of the latter group were preservice high school physics teachers. This article (Paper I) describes how several serious conceptual and reasoning difficulties were identified among students as they tried to relate a discrete line spectrum to the energy levels of atoms in a light source. Paper II illustrates how findings from this research informed the development of a tutorial that led to significant improvement in student understanding of atomic emission spectra.