Lisa K. Fazio

Fractions: the new frontier for theories of numerical development

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on number lines. The difficulty of attaining this understanding makes the acquisition of knowledge about fractions an important issue educationally, as well as theoretically. This article examines the neural underpinnings of fraction understanding, developmental and individual differences in that understanding, and interventions that improve the understanding. Accurate representation of fraction magnitudes emerges as crucial both to conceptual understanding of fractions and to fraction arithmetic.

Learning errors from fiction: Difficulties in reducing reliance on fictional stories

Readers rely on fiction as a source of information, even when fiction contradicts relatively wellknown facts about the world (Marsh, Meade, & Roediger, 2003). Of interest was whether readers could monitor fiction for errors, in order to reduce suggestibility. In Experiment 1, warnings about errors in fiction did not reduce students’ reliance on stories. In Experiment 2, all subjects were warned before reading stories written at 6th- or 12th-grade reading levels. Even though 6th-grade stories freed resources for monitoring, suggestibility was not reduced. In Experiment 3, suggestibility was reduced but not eliminated when subjects pressed a key each time they detected an error during story reading. Readers do not appear to spontaneously monitor fiction for its veracity, but can do so if reminded on a trial-by-trial basis.

Surprising feedback improves later memory

The hypercorrection effect is the finding that high-confidence errors are more likely to be corrected after feedback than are low-confidence errors (Butterfield & Metcalfe, 2001). In two experiments, we explored the idea that the hypercorrection effect results from increased attention to surprising feedback. In Experiment 1, participants were more likely to remember the appearance of the presented feedback when the feedback did not match expectations. In Experiment 2, we replicated this effect using more distinctive sources and also demonstrated the hypercorrection effect in this modified paradigm. Overall, participants better remembered both the surface features and the content of surprising feedback.

The hypercorrection effect persists over a week, but high-confidence errors return

People’s knowledge about the world often contains misconceptions that are well-learned and firmly believed. Although such misconceptions seem hard to correct, recent research has demonstrated that errors made with higher confidence are more likely to be corrected with feedback, a finding called the hypercorrection effect. We investigated whether this effect persists over a 1-week delay. Subjects answered general-knowledge questions about science, rated their confidence in each response, and received correct answer feedback. Half of the subjects reanswered the same questions immediately, while the other half reanswered them after a 1-week delay. The hypercorrection effect occurred on both the immediate and delayed final tests, but error correction decreased on the delayed test. When subjects failed to correct an error on the delayed test, they sometimes reproduced the same error from the initial test. Interestingly, high-confidence errors were more likely than low-confidence errors to be reproduced on the delayed test. These findings help to contextualize the hypercorrection effect within the broader memory literature by showing that high-confidence errors are more likely to be corrected, but they are also more likely to be reproduced if the correct answer is forgotten.

Slowing presentation speed increases illusions of knowledge

Prior research on false memories has shown that suggestibility is often reduced when the presentation rate is slowed enough to allow monitoring. We examined whether slowing presentation speed would reduce factual errors learned from fictional stories. Would subjects use the extra time to detect the errors in the stories, reducing reproduction of these errors on a later test? Surprisingly, slowing presentation speed increased the production of story errors on a later general knowledge test. Instructing the reader to mark whether each sentence contained an error, however, did decrease suggestibility. Readers appear to passively accept information presented in stories and need a constant reminder to monitor for errors. These results highlight differences between typical episodic false memories and illusions of knowledge (such as learning from fiction). Manipulations that reduce suggestibility for episodic false memories do not always reduce suggestibility for illusions of knowledge.

Memorial consequences of multiple-choice testing on immediate and delayed tests

Multiple-choice testing has both positive and negative consequences for performance on later tests. Prior testing increases the number of questions answered correctly on a later test but also increases the likelihood that questions will be answered with lures from the previous multiple-choice test (Roediger & Marsh, 2005). Prior research has shown that the positive effects of testing persist over a delay, but no one has examined the durability of the negative effects of testing. To address this, subjects took multiple-choice and cued recall tests (on subsets of questions) both immediately and a week after studying. Although delay reduced both the positive and negative testing effects, both still occurred after 1 week, especially if the multiple-choice test had also been delayed. These results are consistent with the argument that recollection underlies both the positive and negative testing effects.

Receiving right/wrong feedback: Consequences for learning

Prior work suggests that receiving feedback that ones response was correct or incorrect (right/wrong feedback) does not help learners, as compared to not receiving any feedback at all (Pashler, Cepeda, Wixted, & Rohrer, 2005). In three experiments we examined the generality of this conclusion. Right/wrong feedback did not aid error correction, regardless of whether participants learned facts embedded in prose (Experiment 1) or translations of foreign vocabulary (Experiment 2). While right/wrong feedback did not improve the overall retention of correct answers (Experiments 1 and 2), it facilitated retention of low-confidence correct answers (Experiment 3). Reviewing the original materials was very useful to learners, but this benefit was similar after receiving either right/wrong feedback or no feedback (Experiments 1 and 2). Overall, right/wrong feedback conveys some information to the learner, but is not nearly as useful as being told the correct answer or having the chance to review the to-be-learned materials.

Correcting False Memories

While often impressive, memory is far from perfect. For example, the sentence “The karate champion hit the cinder block” is often misremembered as “The karate champion broke the cinder block” (Brewer, 1977). Hearing a list of related words such as “bed, rest, tired …” leads people to claim “sleep” was presented when in fact it was not (Roediger & McDermott, 1995). Answering the question “How fast was the white sports car going when it passed the barn while traveling along the country road?” increases witnesses’ later reports of having seen a non-existent barn in an earlier video (Loftus, 1975, p. 566). These examples represent just a few of the many ways in which memory can go astray. Not only are these errors easily created, they often become vivid false memories held with high confidence. For example, false memories for non-presented words like “sleep” are so vivid that people often claim to remember which of two voices said the word (Payne, Elie, Blackwell, & Neuschatz, 1996). False memories can be strikingly persistent. Warnings about memory errors are rarely effective (McDermott & Roediger, 1998), especially after the study phase (Greene, Flynn, & Loftus, 1982). Re-exposure to events is insufficient; hearing “bed, rest, tired …” again reduces, but does not eliminate, false memories for “sleep” (McDermott, 1996; Watson, McDermott, & Balota, 2004). Even interventions that pinpoint specific contradictions between subjects’ memories and the original events are inadequate; many errors remain uncorrected even after subjects place an X next to each false memory (McConnell & Hunt, 2007). Despite much evidence that false memories are difficult to correct, a finding from another literature allows a surprising prediction about the correction of false memories. In a prototypical experiment demonstrating the hypercorrection effect, participants answer general knowledge questions and rate their confidence in each response before viewing the correct answer. High-confidence errors are more likely to be corrected on a second test than are incorrect guesses (Butterfield & Metcalfe, 2001). Someone who strongly believes that Sydney is the capital of Australia will benefit more from the feedback “Canberra” than someone who simply guessed “Sydney.” The parallel in the false memory domain would be surprising, predicting that confidently-held false memories should be corrected more often than other errors. To examine this, we created false memories using sentences that encourage inferences; for example, “The clumsy chemist had acid on his coat” is often misremembered as “The clumsy chemist spilled acid on his coat.” We examined ability to correct false memories as a function of initial confidence in the errors.

Strategy Use and Strategy Choice in Fraction Magnitude Comparison.

We examined, on a trial-by-trial basis, fraction magnitude comparison strategies of adults with more and less mathematical knowledge. College students with high mathematical proficiency used a large variety of strategies that were well tailored to the characteristics of the problems and that were guaranteed to yield correct performance if executed correctly. Students with less mathematical proficiency sometimes used strategies similar to those of the mathematically proficient students, but often used flawed strategies that yielded inaccurate performance. As predicted by overlapping waves theory, increases in accuracy and speed were related to differences in strategy use, strategy choice, and strategy execution. When asked to choose the best strategy from among 3 possibilities-the strategy the student originally used, a correct alternative, and an incorrect alternative-students with lower fraction knowledge rarely switched from an original incorrect strategy to a correct alternative. This finding suggests that use of poor fraction magnitude comparison strategies stems in large part from lack of conceptual understanding of the requirements of effective strategies, rather than difficulty recalling or generating such strategies. (PsycINFO Database Record

Memorial consequences of testing school-aged children.

A large literature shows that retrieval practice is a powerful tool for enhancing learning and memory in undergraduates (Roediger & Karpicke, 2006a). Much less work has examined the memorial consequences of testing school-aged children. Our focus is on multiple-choice tests, which are potentially problematic since they minimise retrieval practice and also expose students to errors (the multiple-choice lures). To examine this issue, second graders took a multiple-choice general knowledge test (e.g., What country did the Pilgrims come from: England, Germany, Ireland, or Spain?) and later answered a series of short answer questions, some of which corresponded to questions on the earlier multiple-choice test. Without feedback, the benefits of prior testing outweighed the costs for easy questions. However, for hard questions, the large increase in multiple-choice lure answers on the final test meant that the cost of prior testing outweighed the benefits when no feedback was provided. This negative testing effect was eliminated when children received immediate feedback (consisting of the correct answer) after each multiple-choice selection. Implications for educational practice are discussed.