Practice Makes (Nearly) Perfect: Solving ‘Students-and-Professors’-Type Algebra Word Problems KIEL CHRISTIANSON 1,2 *, JOSE P. MESTRE 1,2 and STEVEN G. LUKE 3 1 University of Illinois, Urbana–Champaign, Champaign, USA 2 Beckman Institute for Advanced Science and Technology, Champaign, USA 3 University of South Carolina, Columbia, USA Summary: Three experiments with university students (Ns = 40, 36, and 36) who were non-math majors explicitly examined whether repetition in performing ‘students-and-professors’-type algebra word problems, which have been shown in the past to be vexingly difficult even for more advanced students, would spontaneously lead to higher rates of correct answers. Word order and situation model specificity were also examined to determine their effects on the rate of improvement. The strongest predictor of students producing correct equations (i.e., not producing the typical ‘reversal error’) was practice: In all experiments, participants spontaneously improved in equation accu- racy almost to ceiling levels as they progressed, despite receiving no feedback. Tentative support is provided for the pedagogical value of repetition in solving problems, along with varying the wording of the problems. Copyright © 2012 John Wiley & Sons, Ltd. In a recent comparison of Common Core standards in mathe- matics education in the USA (http://www.corestandards.org/) and existing state standards, Porter, McMaken, Hwang, and Yang (2011) also compared the Common Core standards with those of three high-performing countries (Finland, Japan, and Singapore). They found a much greater emphasis on ‘perform procedures’—that is, on doing more problems rather than focusing on higher-level conceptualization—in the high- performing countries than in the US Common Core or state standards. Porter et al. suggested, very tentatively, that this fact may point toward a re-evaluation of the de-emphasis in the USA on solving greater numbers of routine problems. The motivation for the research reported here was to test the degree to which college-aged students’ accuracy in solving ‘students-and-professors’ problems would benefit from repeated exposure to the same sort of problem. This type of problem, for example, ‘Write an equation using the vari- ables S and P to represent the following statement: “There are six times as many students as professors at this university.” Use S for the number of students and P for the number of professors’, has received considerable attention from math educators and cognitive psychologists over the past three decades since Clement and collaborators (Clement, 1982; Clement, Lochhead, & Monk, 1981) observed that college students, even those majoring in science and engineering, committed predictable and apparently persistent errors in translating these fairly basic word problems into algebraic equations. It has been reported that error rates ranged between 20% and 60% on students-and-professors-type problems (Cohen & Kanim, 2005; Fisher, 1988; Lochhead & Mestre, 1988; MacGregor & Stacey, 1993; Mestre, Gerace, & Lochhead, 1982), with by far the most common error being the ‘reversal’ error, 6S = P, where the variables are reversed. In the intervening years since the prevalence of the ‘variable- reversal error’ was revealed, many additional studies have attempted to understand its source but with limited success. The original studies cited earlier proposed that the variable- reversal error stemmed from a combination of left-to-right translation of the words in the problem statement to the equa- tion and a lack of understanding of the meaning of variable. Subsequent investigations indicated that this explanation falls short, however. If students are indeed using a word- order match in translating the words to an algebraic equation, then the variable-reversal error should diminish drastically, or perhaps even disappear entirely, if the following helpful phrasing is used: The number of students is six times the number of professors. A left-to-right translation of this statement would yield the correct S =6P. A study by MacGregor and Stacey (1993) with ninth graders resulted in more than 50% reversal errors even with the helpful phrasing, ‘The number y is eight times the number z’. More recently, a study by Cohen and Kanim (2005) with college students enrolled in physics classes indi- cated a modest reduction (~10%) of variable-reversal errors when helpful phrasings were used compared with unhelpful phrasings, but students still committed substantial reversal errors in the helpful phrasings. Subsequent investigations into the effect of semantic cues (as opposed to syntactic or word-order cues) with respect to the relative sizes of the groups also helps to some degree but does not completely eliminate the reversal error (Bassok, Chase, & Martin, 1998; Martin & Bassok, 2005). In short, there remains a great deal of uncertainty as to the cause of the variable-reversal error and the difficulties observed in translating from words to algebra. Even more importantly for the present study, previous work has assumed, either implicitly or explicitly, that the observed error rates are basically entrenched within the tested populations of students. In other words, one question that has yet to be systematically addressed within this literature is whether error rates decline with practice, including practice without feedback. Perhaps the failure to understand the source of the error is the reason that relatively little work has been devoted to determining to what extent students can improve their performance on such problems. Evidence that practice alone could improve success rates with this type of problem would lend indirect support for the value of simply ‘performing procedures’, as observed *Correspondence to: Kiel Christianson, PhD, Associate Professor, Departments of Educational Psychology, Linguistics, Psychology, and Beckman Institute, Education Building, Rm. 226A, MC-708, University of Illinois, 1310 S. 6th St., Champaign, IL 61820, USA. E-mail: kiel@illinois.edu Copyright © 2012 John Wiley & Sons, Ltd. Applied Cognitive Psychology, Appl. Cognit. Psychol. 26: 810–822 (2012) Published online 25 July 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/acp.2863