A Procedural Explanation of the Generation Effect for Simple and Difficult Multiplication Problems and Answers

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1 Journal of Memory and Language 43, (2000) doi: /jmla , available online at on A Procedural Explanation of the Generation Effect for Simple and Difficult Multiplication Problems and Answers Danielle S. McNamara Old Dominion University and Alice F. Healy University of Colorado at Boulder Three experiments investigated the generation effect for the free recall of multiplication problems and answers. In Experiment 1, a greater generation effect for answer recall was found for participants presented with simple as compared to those presented with difficult multiplication problems. This finding is inconsistent with an explanation of the generation effect in terms of effort. Experiment 2 replicated Experiment 1 and further demonstrated that participants show an equivalent generation effect for problem operands (i.e., the cues) regardless of problem difficulty. Generation accuracy also influenced the magnitude of the generation effect in the difficult problem condition. Experiment 3 replicated the results of Experiment 1, holding constant problem answers across simple and difficult problems. These experiments collectively demonstrated a generation effect for the answers to arithmetic problems only when participants reinstated at test the cognitive procedures used at study, thus providing further evidence for the procedural account of the generation effect Academic Press Key Words: generation effect; procedural reinstatement; multiplication problem difficulty; arithmetic; cognitive procedures; memory; retrieval; effort. The generation effect refers to a robust retention advantage found for material that is selfgenerated compared to material that is simply copied or read. In a typical generation paradigm, participants are shown related word pairs in either a read or generate condition (e.g., Slamecka & Graf, 1978). In the read condition, We are indebted to Michael Ferris for help with Experiment 2 and to James Kole for help with Experiment 3. We are also grateful to Dan Burns, Sal Soraci, and Bob Widner for helpful comments on an earlier version of this article. This research was supported in part by a Postdoctoral Fellowship (Grant 93-12) and by a Career Development Award (Grant 95-56) from the J. S. McDonnell Foundation to Danielle S. McNamara and by Army Research Institute Contracts MDA K-00010, DASW01-96-K-0010, and DASW01-99-K-0002 and Army Research Office Grant DAAG to the University of Colorado (Alice F. Healy, Principal Investigator). Address correspondence and reprint requests to Danielle S. McNamara, Department of Psychology, MGB 250, Old Dominion University, Norfolk, Virginia both words are displayed (e.g., short tall), whereas in the generate condition, only the cue word and a portion of the target word are displayed (e.g., short t ) and the participant generates the target word (i.e., tall). Many studies have demonstrated that items that are generated, compared to those that are read, have a greater probability of being recalled or recognized (see McNamara & Healy, 1995a, 1995b, for reviews of the literature). In sum, active participation in the learning process leads to greater retention than does passive perception. The overarching goal of our research has been to determine the parameters and limitations of the generation effect and to provide an account of this basic cognitive phenomenon within a relatively general theory of human memory, the procedural account of learning (Healy et al., 1992, 1993, 1995). The present study provides a further test of the procedural account of the generation effect (Crutcher & X/00 $35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. 652

2 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 653 Healy, 1989; McNamara & Healy, 1995a, 1995b). There are two aspects to this account. First, the critical factor leading to a generation effect is that the participants engage in cognitive operations that connect the target item to information stored in memory 1 rather than actually generate, or produce, the target item. Second, it is crucial that the participants be able to reinstate at the time of the memory test the cognitive operations, or learning procedures, that were used at study. The generation effect occurs because generating is more likely than reading to promote procedures during encoding that can be reinstated during a typical retention test (see also Soraci et al., 1994). This process-oriented account of the generation effect focuses on the cognitive processes engaged in by the participants at both study and recall rather than on the inherent nature of the target items or the relationship between the cue and target items. In this study, we examine episodic memory for multiplication problem answers that are either generated or read. Subjects either read or generate the answers to multiplication problems and then, after a short delay, recall the list of answers (without the problems as cues). Although the generation effect is generally examined in the context of verbal stimuli, following Gardiner and Rowley (1984) and Gardiner and Hampton (1985), we have focused primarily on the effects of generating in the context of arithmetic problems (Crutcher & Healy, 1989; Mc- Namara, 1995; McNamara & Healy, 1995a, 1995b). In a typical generation effect condition, the participant is presented with a cue word and a word fragment (e.g., short t ) and a relational rule (e.g., antonym) to generate the response (i.e., tall). The participant is typically asked to recall all of the responses encountered during study. In our case, the participant is presented with an arithmetic problem (e.g., 13 9 _) which includes the relational rule of multiplication. The participant responds with 1 Note that by this account the information in memory need not reside in the lexicon; thus, this account resembles but is not identical to the lexical activation hypothesis, so that it can accommodate the findings of a generation effect with nonwords (see, e.g., McNamara & Healy, 1995a; Nairne & Widner, 1987). the answer (i.e., 117) and is later asked to recall all of the answers encountered during study. We have used this paradigm because we are interested in further exploring the nature of the cognitive procedures, or mental operations, involved in the generation effect. The mental procedures, namely the arithmetic operations linking operands to answers, are straightforward and well defined for arithmetic problems. In contrast, the mental operations linking verbal materials are more difficult to control experimentally and may vary considerably across participants. For example, whereas one individual may link short and tall with a visual image of Abbott and Costello, another may think of a story involving an elf and a giant, and another may form no image or story at all. In contrast, the link between arithmetic problems and their answers is relatively constant across individuals. A second advantage of using arithmetic problems is that they allow us to examine the generation effect for episodic memory tasks (Crutcher & Healy, 1989; McNamara & Healy, 1995b) as well as skill acquisition tasks (Mc- Namara, 1995; McNamara & Healy, 1995a). As in the present experiments, the generation effect is typically examined using episodic memory tasks; that is, tasks that require the participant to remember the occurrence during the experimental session of the generated or read words or, in our case, the answers to the arithmetic problems. In contrast, we have also compared the advantages of repeated generating and reading for arithmetic skill acquisition for both adults (McNamara & Healy, 1995a) and children (Mc- Namara, 1995). One strength of our procedural account is that it has successfully predicted findings for both episodic tasks and skill acquisition tasks. The specific purpose of the present series of experiments is to provide a more stringent test of our procedural account by examining an unintuitive prediction from that account. Specifically, the procedural account leads to the prediction that the generation effect should be larger for simple multiplication problems than for difficult problems in an episodic memory task. This prediction is unintuitive for at least

3 654 MCNAMARA AND HEALY two reasons. First, in an earlier study of skill acquisition, we found the reverse a larger generation advantage for difficult than for simple multiplication problems (McNamara & Healy, 1995a). Second, by the intuitively plausible effort account of the generation effect (e.g., Griffith, 1976), as we discuss later, the generation effect should be greater when the act of generation is more effortful. Hence, a greater generation effect would be expected for difficult problems. However, our prediction of a greater generation effect for simple problems follows from the second aspect of the procedural account, namely that participants must be able to reinstate at the time of the memory test the cognitive operations that were used at study. To understand how the procedural account leads to this prediction, it is necessary to review details concerning previous generation effect results we found for multiplication and addition problems in a study involving episodic memory (McNamara & Healy, 1995b). In our previous study (McNamara & Healy, 1995b), we demonstrated a stable and robust generation effect for simple multiplication problems. In contrast, a generation effect for addition problems was found only when participants were induced to reinstate at test the cognitive procedures they had used at study. To increase procedural reinstatement, we led the participants who were presented with addition problems to use an operand retrieval strategy. With this strategy, a participant recalls and combines operands from problems seen during study, derives answers by performing the relevant arithmetic operation on the operands (e.g., by multiplying the operands together), and then checks the familiarity of the answers. For example, if a participant remembers that one of the operands is 3, then by consecutively multiplying by 3, the participant can mentally check the familiarity of the answers 6, 9, 12, and so on, for which answer had actually occurred during study. In contrast to multiplication problems, the use of the operand retrieval strategy is less efficient for addition problems because so few answers are excluded. For example, given the single operand (i.e., addend) 3, a participant might derive the answers 4, 5, 6, 7, and so on. A participant can use this test strategy with addition problems and ultimately it can be effective. However, because of the greater number of answers to exclude, not only is this strategy inefficient with addition problems, but it is also highly error prone. According to our procedural account, the operand retrieval strategy leads to a greater generation effect because of the similarity between the mental operations used for generating at the time the problems are studied and the mental operations used for this strategy at the time of the memory test. The advantage of generating is, thus, that it leads participants to engage in cognitive procedures linking the cue and target and that these same procedures can be reinstated at test. For arithmetic problems, the relevant cognitive procedures are the arithmetic operations linking the operands to the answers. The participants in a generate condition, but not those in a read condition, necessarily use the relevant procedures at study whether the arithmetic operation is addition or multiplication. When participants use the operand retrieval strategy at test, they are reinstating the relevant cognitive procedures. In a sense, they are regenerating the problem answer. Participants are likely to use the operand retrieval strategy for all simple multiplication problems, and thus, the generation effect is expected in all cases. In contrast, participants are not likely to come up with the operand retrieval strategy for addition problems, simply because it is inefficient for this type of problem. Thus, according to our procedural account, there is typically no generation effect for addition problems because the relevant cognitive procedures are usually not reinstated at test. In support of this hypothesis (McNamara & Healy, 1995b, Experiment 1), we demonstrated that when participants read and generated either multiplication or addition problems there was a strong generation effect for simple multiplication problems and a lack of one for addition problems. We also found fewer intrusion errors for multiplication than for addition problems and a predominance of table-related errors for multiplication problems; that is, the intrusions were possible products of at least one of the presented

4 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 655 operands (see, e.g., Campbell & Graham, 1985). These latter two findings provided evidence that participants used an operand retrieval strategy for the multiplication problems because that strategy should limit the responses given largely to the correct answers and the ones sharing an operand with the correct answers. However, we also predicted and found that a generation effect does occur for addition problems as well as multiplication problems when the participants are induced to use the operand retrieval strategy for both types of problems (McNamara & Healy, 1995b, Experiment 2). Specifically, we found that a generation effect did occur for addition problems when the participants also generated and read multiplication problems within the same list. We had hypothesized that the presence of the multiplication problems along with the addition problems would promote the use of the operand retrieval strategy for both types of problems. In other words, if participants were naturally using the operand retrieval strategy to recall the answers to the multiplication problems, then they would be led to use the same retrieval strategy for the addition problems. We tested this hypothesis again (McNamara & Healy, 1995b, Experiment 3) by giving participants an explicit suggestion to use an operand retrieval strategy with addition problems. Immediately before recalling the answers, they were told simply that one good strategy for remembering the answer to a problem is to try to remember the problem itself and if they could remember one or both of the numbers added together, they would be more likely to remember the answer to the problem. This suggestion also led to a generation effect for addition problems. In the present study, we compare the effects of generating and reading for simple (e.g., ) and difficult (e.g., ) multiplication problems. We hypothesized that the effects of generating for the difficult multiplication problems would follow a similar pattern to that for the addition problems in our earlier study (McNamara & Healy, 1995b, Experiment 1). As explained earlier, a generation effect is not normally expected for addition problems because the operand retrieval strategy is not typically used with this type of arithmetic problem. Similarly, we hypothesized that most participants presented with difficult multiplication problems would not use the operand retrieval strategy. It should be noted that the reasons why participants will tend not to use the operand retrieval strategy for difficult multiplication problems and addition problems are slightly different. For addition problems, we have hypothesized that the operand retrieval strategy is time consuming because of the large number of answers to exclude (McNamara & Healy, 1995b). In contrast, for difficult multiplication problems, we propose that the operand retrieval strategy is time consuming because of the need to compute the answers rather than directly retrieve them from memory. The operand retrieval strategy consists of recalling problem operands, calculating an answer given those operands, and then checking this answer for familiarity. The mental operation of solving difficult multiplication problems is time consuming and error prone compared to that of solving simple multiplication problems. Thus, participants should be less likely to use that strategy with the difficult problems. A second purpose of this study is to test further the alternative hypothesis that the generation effect is due to the increased amount of effort expended to process the cue and target in the generate condition relative to the read condition (e.g., Griffith, 1976; McFarland, Frey, & Rhodes, 1980). Even when effort has not been the central theoretical construct explaining the generation effect, it is often assumed that generation is more effortful than reading (e.g., Nairne, Pusen, & Widner, 1985, p. 190). Although the effort hypothesis is one of the most intuitively appealing explanations of the generation effect, there is evidence to contradict it (e.g., Glisky & Rabinowitz, 1985; Jacoby, 1978). For example, Jacoby (1978) failed to find an effect of effort in a generation task when difficulty was varied as a function of the number of letters missing from a target word (cf. Gardiner, Smith, Richardson, Burrows, & Williams, 1985). In addition, other researchers have found generation effects even when the process of generating required virtually no effort at all

5 656 MCNAMARA AND HEALY (e.g., Glisky & Rabinowitz, 1985; Nairne & Widner, 1987). In spite of this evidence, effort is often cited as the cause of the generation effect (see, e.g., Anderson, 1990, p. 184). Moreover, the generation effect is often cited as evidence for the importance of effort to learning and memory (see, e.g., Tyler, Hertel, McCallum, & Ellis, 1979; but also see Zacks, Hasher, Sanft, & Rose, 1983). Our intention here is to provide a more direct test of the effort hypothesis by using more pronounced variations in the amount of effort required to generate the response than used in previous studies. In this study, we compare the magnitude of the generation effect for simple and difficult multiplication problems. If effort is operationalized by the difficulty of the problem, in terms of both the accuracy in solving the problem and the time required to solve it, simple multiplication problems should require less effort to solve than do difficult problems (see, e.g., McNamara & Healy, 1995a; but also see Mitchell & Hunt, 1989). According to the effort hypothesis, if increased effort is the source of the generation effect, then a greater generation effect should be found for difficult problem answers than for simple problem answers. In contrast, as just reviewed, according to the procedural account, the reverse is expected: A greater generation effect should be found for simple problem answers than for difficult problem answers. This experiment should, thus, enable us to determine the relative merits of the effort and procedural accounts of the generation effect. EXPERIMENT 1 In Experiment 1 we compared the generation effect for simple and difficult multiplication problems. Participants were presented with either 12 simple or 12 difficult multiplication problems. Participants generated half of the problems of either type and read the other half. Participants memory for the answers to the problems was tested with a free-recall procedure after a short distractor task. We hypothesized that participants would be less likely to use the operand retrieval strategy with difficult multiplication problems than with simple multiplication problems. Participants would be expected to have memorized the answers to the simple multiplication problems so that they could directly retrieve them from memory without any computation. In contrast, for the difficult problems the answers are not expected to be directly linked to the operands in memory but derived only through calculation. Hence, the operand retrieval strategy is an easy strategy to use for the simple problems because it requires no computation, but is not easy for the difficult problems because it requires computation in that case. If participants are not required to do any computation, they may not choose to do so. The operand retrieval strategy is not impossible when answers are not stored in memory; participants can still use the operands to help them remember the answers, but the strategy is less efficient and, thus, less likely to be employed in that situation because computation is required to retrieve the answers. Because participants are less likely to use an operand retrieval strategy, the generation effect should be reduced for difficult problems compared with simple problems. We also tested two predictions of the procedural account. The first is that if participants use the operand retrieval strategy for the simple but not the difficult multiplication problems, then fewer intrusion errors are expected for participants presented with the simple multiplication problems than for those presented with the difficult problems. Second, and more specifically, the use of the operand retrieval strategy with multiplication problems should limit the recall responses to possible problem answers having at least one of the presented operands. This assumption leads to the prediction that the majority of the intrusions should be table-related for participants presented the simple multiplication problems. Method Participants and design. Twenty-four undergraduate students from the University of Colorado participated for credit in an introductory psychology course. A 2 2 mixed factorial design was employed, with one between-subjects variable, problem type (simple vs difficult), and one within-subjects variable, presen-

6 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 657 TABLE 1 Simple and Difficult Problems Presented to Participants in Experiments 1, 2, and 3 Simple: Experiments 1, 2, and 3 Difficult: Experiments 1 and 2 Difficult: Experiment tation condition (generate vs read). Participants were assigned to the problem type condition and the counterbalancing subcondition according to a fixed rotation on the basis of their time of arrival for testing. Materials. The multiplication problems consisted of 12 simple and 12 corresponding difficult multiplication problems. As shown in Table 1, both types of problems consisted of a twodigit multiplier followed by a one-digit multiplier. The products all consisted of three-digit answers. For the simple multiplication problems, the second digit of the two-digit multiplier was always 0 and the answer always ended in 0 (e.g., ). For the difficult multiplication problems, the first digit of the two-digit multiplier was always 1 and the answer always began with 1 (e.g., ). Apart from the second digit of the two-digit multiplier in the simple problems and the first digit of the two-digit multiplier in the difficult problems, the multipliers remained constant for both sets of problems. The simple and difficult multiplication problems were each divided into two sets of six problems of approximately equivalent difficulty. Participants read one set and generated the other set. Order of presentation of the two sets of problems and order of reading and generating were counterbalanced across participants in each problem type condition. Procedure. Participants were shown two sets of shuffled index cards, each including six simple or six difficult multiplication problems. The numbers were printed on 6 9 index cards and measured approximately 1 tall. The problem appeared on one side of the card; both the problem and the answer to the problem appeared on the opposite side of the card. The experimenter shuffled the index cards to randomize the order of problems within each set. The first set of multiplication problems was presented in either a read or a generate condition. The second set was presented in the other condition. For the read condition, participants were presented the side of the index card showing both the problem and the answer and told to read the problem and the answer aloud twice in a row. For the generate condition, participants were presented the side of the index card showing only the problem and told to read the problem aloud, calculate the answer, and then say the answer aloud. They were then shown the problem with the correct answer printed on the opposite side of the card, which they read aloud. Thus, the problem and the answer were read aloud twice in both conditions, and the participants in the generate condition said the correct answer aloud at least once. After reading and generating the multiplication problems, participants were given a self-paced distractor task, which consisted of first reading a set of instructions and then reading a short passage while circling predesignated target letters. Finally, the participants were asked to write down all of the answers to the multiplication problems that they had seen. (They had not been forewarned of the recall task; thus, this was an incidental test of their memory for the problem answers.) Results The results are shown in Fig. 1 in terms of the proportion of correct answers recalled as a function of problem type (simple vs difficult) and presentation condition (generate vs read). A mixed-multifactorial analysis of variance was conducted on the proportion of correct responses including the between-subjects variable of problem type and the within-subjects variable of presentation condition. There was a

7 658 MCNAMARA AND HEALY FIG. 1. Proportion of correct answers recalled in Experiment 1 as a function of problem type and presentation condition, showing a greater generation effect for simple than for difficult problem answers. main effect of problem type, F(1,22) 5.4, MS e , p.028, reflecting greater recall for answers to simple problems (M 0.535) compared to difficult problems (M 0.382). There was also an overall generation effect, F(1,22) 43.6, MS e , p.001, reflecting greater recall for generated answers (M 0.632) compared to answers that were read (M 0.284). As predicted, there was a significant interaction of problem type and presentation condition, F(1,22) 4.4, MS e , p.045. As shown in Fig. 1, this interaction reflects the finding of a smaller, though reliable, generation effect for difficult problem answers [Difference 0.238, F(1,11) 8.1, p.015] than for simple problem answers [Difference 0.459; F(1,11) 50.3, p.001]. Intrusions. Under the assumption that participants use the operand retrieval strategy more often for simple than for difficult problems, fewer intrusion errors would be expected for participants given simple problems than for those given difficult problems because that strategy limits the answers given. Specifically, the use of the operand retrieval strategy with multiplication problems should limit the recall responses to possible problem answers having at least one of the presented operands. This assumption also leads to the prediction that any intrusions found for simple problems should be table-related (see, e.g., Campbell & Graham, 1985); that is, the intrusions should be possible products of at least one of the presented operands. There was an average of intrusion errors made by the participants. A mixed multifactorial analysis of variance was conducted on the number of intrusion errors including the between-subjects variable of problem type (simple vs difficult) and the within-subjects variable of error type (table-related vs non-table-related). As expected, participants who were presented simple problems made fewer intrusion errors (M 1.000) than did participants who were presented difficult problems (M 2.667), F(1,22) 5.7, MS e , p.025. The overall difference between table-related and non-table-related intrusion errors was not reliable, F(1,22) 1. There was, however, the predicted interaction between problem type and error type, F(1,22) 9.1, MS e , p.006. This result reflects the finding that participants given simple problems made predominantly table-related errors (M 0.917, n 11 errors) and very few non-table-related errors (M 0.083, n 1 error), whereas participants given difficult problems made more non-table-related errors (M 2.000, n 24 errors) than tablerelated errors (M 0.667, n 8 errors). Discussion The primary goal of Experiment 1 was to demonstrate that a stronger generation effect occurs for simple than for difficult multiplication problem answers and that this difference is because participants are more likely to use the operand retrieval strategy for the simple problems. As predicted, there was a reliable interaction of problem type and presentation condition, indicating a stronger generation effect for the simple problems. Also as predicted, there were fewer intrusion errors for participants given simple multiplication problems than for those given difficult problems. This prediction was based on the assumption that use of the operand retrieval strategy would limit the number of intrusion errors. Further as predicted, we found that the majority of the intrusion errors were table-related for the participants given simple

8 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 659 problems, whereas participants who were given the difficult problems made fewer table-related than non-table-related errors. This prediction was based on the assumption that the use of the operand retrieval strategy would limit intrusion errors to numbers that were a product of at least one of the presented operands. Finding few intrusion errors for the participants given simple problems and finding that these errors were table-related suggests that participants were more likely to use an operand retrieval strategy for simple problems. The opposite pattern of results for the participants given difficult problems suggests that these participants were either less likely to use the operand retrieval strategy or less successful when using the strategy. These findings, thus, support our hypothesis that the successful reinstatement of the cognitive procedures engaged during study results in a greater generation effect. These findings also provide direct evidence against the effort hypothesis as an explanation of the generation effect. If the generation effect resulted from a greater amount of effort in the generate relative to the read condition, then a greater generation effect for difficult than for simple multiplication problem answers should be observed. However, we found the opposite; there was a greater generation effect for simple than for difficult answers to multiplication problems. EXPERIMENT 2 The results from Experiment 1 indicate that there is a larger generation effect for simple multiplication problems than for difficult multiplication problems. We have hypothesized that participants presented with simple problems use the operand retrieval strategy and that the use of this strategy increases the advantage for generating in comparison to reading. According to the procedural account, the generation advantage increases because the participants reinstate the same mental procedures (i.e., solving the problem) when using the operand retrieval strategy as they had when they first solved the problems: When more similar mental procedures are engaged during encoding and recall, memory is enhanced. One alternative explanation for these results is that participants are equally likely to use the operand retrieval strategy with the simple and difficult multiplication problems but they remember the operands with simple multiplication problems more easily than they do with difficult problems. Simple problems are more frequently encountered and may have a stronger representation in memory. It follows that if participants were unable to remember the operands of the difficult problems, then they may be less likely to retrieve the answers as well. According to this operand memory explanation, it is not the reinstatement of the arithmetic procedures that is the key to the recall of the answers, but solely the recall of the operands at test. If the finding of a greater generation effect for simple multiplication problem answers than for difficult multiplication answers was due to better memory for the operands, then the same trend should be found for recall of the operands: Participants should show a greater generation effect for simple multiplication operand recall than for difficult multiplication operand recall. If, on the other hand, the finding of a greater generation effect for simple multiplication problem answers was due to the reinstatement of the arithmetic procedures, then a generation effect for memory of the operands, which does not depend on performing the arithmetic procedures, should not depend on problem difficulty. To examine the operand memory hypothesis, we replicated Experiment 1, but asked participants first to recall the operands. After recalling the operands and removing them from view, participants in Experiment 2 recalled the answers to the problems as had participants in Experiment 1. According to the operand memory explanation, a greater generation effect should be found for the simple problems when the operands are recalled. According to the procedural account, the generation effect for simple and difficult problems should be comparable when the operands are recalled but should still differ when the answers are recalled. A second purpose of Experiment 2 was to determine whether the generation effect in the difficult problem condition depended on the participant s ability to perform the difficult mul-

9 660 MCNAMARA AND HEALY tiplication procedures (as measured by the accuracy of the generated answers). If the majority of a participant s generated answers were incorrect, then the participant would be unlikely to solve the problem correctly when checking solutions based on retrieved operands. The participant would also be unlikely to solve the problems in the same way both times and thus would not reinstate at test the cognitive procedures used at study. According to our procedural account, the generate condition leads to greater procedural reinstatement than does the read condition because generating leads participants to engage in cognitive procedures linking the cue and target, and these same procedures can be reinstated at test. For arithmetic problems, the relevant cognitive procedures are the arithmetic operations linking the operands to the answers. If an individual has generally less skill in difficult multiplication, it will be less likely that the individual will be able to reinstate correctly the same cognitive procedures at test as used at study. To test this hypothesis, we recorded the participants while they were generating and reading the answers to the multiplication problems in order to measure the accuracy of their generated answers. Method Participants and design. Forty-eight undergraduate students from Old Dominion University participated for credit in an undergraduate psychology course. A mixed factorial design was employed, with one between-subjects variable, problem type (simple vs difficult), and two within-subjects variables, presentation condition (generate vs read), and recall type (operands vs answer). Participants were assigned to the problem type condition and the counterbalancing subcondition according to a fixed rotation on the basis of their time of arrival for testing. Materials. The multiplication problems were identical to those used in Experiment 1 (see Table 1) and were presented on the same type of index cards. The simple and difficult multiplication problems were each divided into two sets of six problems of approximately equivalent difficulty. Participants read one set and generated the other set. Order of presentation of the two sets of problems and order of reading and generating were counterbalanced across participants in each problem type condition. Procedure. Participants utterances were tape recorded while generating and reading to determine the accuracy of each generated multiplication problem answer. Participants were shown two sets of shuffled index cards, each including six simple or six difficult multiplication problems. The first set of multiplication problems was presented in either a read or generate condition. The second set was presented in the other condition. The distractor task consisted of a word association task. Participants were allotted 2 min to write three words associated with each of 22 nouns (e.g., flower, car, etc.). Note that the distractor task in this case, unlike the one used in Experiment 1, occurred for a fixed duration. Following the 2-min distractor task, participants were asked to write down the problems (i.e., the operands) from the two sets of multiplication problems they had read and generated. They were asked to write only the problem and not the answer. They were then asked to write down the answers to the two sets of problems. They were asked to write only the answer and not the problem and were not allowed to view their previous recall of the problems. Operand recall, like answer recall, was scored in an allor-none manner; participants had to recall both operands for a correct response. Otherwise the procedure was identical to that used in Experiment 1. Results The results are summarized in Fig. 2 in terms of proportion of correct answers recalled as a function of recall type (operands vs answer), problem type (simple vs difficult), and presentation condition (generate vs read). A mixed multifactorial analysis of variance was conducted on the proportion of correct responses including the between-subjects variable of problem type and the within-subjects variables of presentation condition and recall type. There was a main effect of problem type, F(1,46) 7.6, MS e , p.009,

10 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 661 FIG. 2. Proportion of correct operands and answers recalled in Experiment 2 as a function of problem type and presentation condition. An equivalent generation effect was found for simple and difficult operand recall compared to a greater generation effect for simple than for difficult answer recall. reflecting greater recall for operands and answers to simple problems (M 0.413) compared to difficult problems (M 0.289). There was also an overall generation effect, F(1,46) 50.3, MS e , p.001, reflecting greater recall for generated operands and answers (M 0.477) compared to operands and answers that were read (M 0.225). The main effect of recall type reflecting the difference between operand recall (M 0.329) and answer recall (M 0.373) was not reliable, F(1,46) 2.6, MS e , p.112. However, both the two-way interaction between recall type and problem difficulty, F(1,46) 12.8, MS e , p.001, and the three-way interaction, F(1,46) 8.8, MS e , p.005, were reliable. The two-way interaction reflects a negligible effect of problem difficulty for operand recall [difficult M 0.316; simple M 0.341; F(1,46) 1] compared to a large effect of problem difficulty for answer recall [difficult M 0.261; simple M 0.486; F(1,46) 14.45, p.001]. The three-way interaction depicted in Fig. 2 reflects the finding that the generation effect depended on both recall type and problem difficulty. Whereas the generation effect for operand recall did not depend on problem difficulty, F(1,46) 1, the generation effect for answer recall did depend on problem difficulty, F(1,46) 4.6, MS e , p.035. There was a smaller, though reliable, generation effect for difficult problem answers [Difference 0.160, F(1,23) 11.8, p.003] than for simple problem answers [Difference 0.332; F(1,23) 25.6, p.001]. The latter finding replicates Experiment 1. The former finding indicates that participants do not have superior memory for the operands of simple problems than for those of difficult problems, and thus the operand memory hypothesis cannot account for the results in Experiment 1 or for the similar results involving answer recall in the present experiment. Answer recall conditional on operand recall. According to the procedural account, the generation effect is increased when the mental procedures involved in generating are reinstated at test. Recalling the operands before recalling the answer should increase the likelihood that the operands will be used to recall the answer. If this is the case, and if it is procedural reinstatement that increases the generation effect, then recalling the operands should have a greater effect on the generate condition than on the read condition. Operand recall should have little effect on the read condition because it should not lead to reinstatement of procedures engaged in the read condition. If, on the other hand, recalling the operands simply primes the subsequent recall of the answer, then the effect of operand recall should be equivalent across the read and generate conditions. To test these competing explanations, an analysis was performed on the conditional proportions of answers recalled given prior recall of the operands. This analysis provides an indication of the differential influence of operand recall on answer recall in the read and generate conditions. If operand retrieval facilitates recall for the generate condition but not for the read condition, then there should be a greater conditional proportion of answer recall given operand recall from the generate condition than from the read condition. This analysis was restricted to those participants who recalled at least one problem successfully in both the read and gen-

11 662 MCNAMARA AND HEALY erate conditions. After imposing this restriction, this analysis included 15 participants in each of the two problem type conditions. There was a main effect of problem difficulty, F(1,28) 26.96, MS e , p.001, reflecting a greater conditional proportion of answer recall given operand recall in the simple condition (M.786) than in the difficult condition (M.290). There was also a main effect of the within-subjects variable, presentation condition, F(1,28) 4.36, MS e , p.043, reflecting a greater conditional proportion of answer recall given operand recall in the generate condition (M.612) as compared to the read condition (M.464). The interaction between problem difficulty and presentation condition was not reliable, F(1,28) 1. These results indicate that recalling the operands from the read condition is indeed less likely to lead to recalling the answers associated with those operands than is recalling the operands from the generate condition. This finding supports our hypothesis that it is the reinstatement of the same mental operations at study and test that leads to enhanced recall in the generate condition. Intrusions. Participants made an average of 2.40 intrusion errors during recall. A mixedmultifactorial analysis of variance was conducted on the number of intrusion errors including the between-subjects variable of problem type (simple vs difficult) and the within-subjects variable of recall type (operand vs answer). As expected, participants who were presented simple problems made fewer intrusion errors (M 1.60) than did participants who were presented difficult problems (M 3.19), F(1,46) 12.54, MS e , p.001. There was also a main effect of recall type, F(1,46) 11.98, MS e , p.002. Participants made more intrusion errors when recalling the operands (M 2.92) than when recalling the answers (M 1.87). The twoway interaction was not reliable, F(1,46) 2.76, MS e , p.10. Intrusions during operand recall were classified as set-related or non-set-related. Set-related problems were defined as problems for which each of the operands were among those presented. For both sets of problems, possible second operands were between 7 and 9. For the difficult problems, possible first operands were between 13 and 19, whereas for the simple problems, possible first operands were between 30 and 90. A 2 2 mixed multifactorial analysis of variance including the between-subjects variable of problem type (simple vs difficult) and the within-subjects variable of intrusion type (set-related vs non-set-related) was performed. The main effect of problem type was marginally significant, F(1,46) 3.21, MS e , p.076, reflecting little difference in the number of recall intrusions for participants presented difficult problems (M 1.73) as compared to those presented simple problems (M 1.19). There were reliably fewer non-set-related intrusions (M 1.04) than set-related intrusions (M 1.88), F(1,46) 5.1, MS e , p.027. The interaction between problem type and intrusion type was not reliable, F(1,46) 1. These results indicate that participants did not tend to guess the problem operands on the basis of recognizing the limited set from which problems were chosen. Also, the likelihood of using this guessing strategy was no greater or smaller for simple problems than for difficult problems. Intrusions during answer recall were classified as table-related and non-table-related, as in Experiment 1. A 2 2 mixed factorial analysis including the between-subjects variable of problem type (simple vs difficult) and the within-subjects variable of intrusion type (table-related vs non-table-related) was performed. There was a main effect of problem type, F(1,46) 20.1, MS e , p.001, reflecting more recall intrusions for participants presented difficult problems (M 1.46) than for those presented simple problems (M 0.42). There were also more non-table-related intrusions (M 1.21) than table-related intrusions (M 0.67), F(1,46) 8.4, MS e , p.001. In addition, there was the predicted interaction between problem type and intrusion type, F(1,46) 21.9, MS e , p.001. This result reflects the finding that participants presented simple problems made predominantly table-related errors

12 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 663 FIG. 3. Proportion of correct answers recalled from difficult problems in Experiment 2 as a function of participants accuracy when generating. The generation effect was comparable for high- and medium-accuracy participants compared to no generation effect for low-accuracy participants who incorrectly solved 50% or more of the problems. (M 0.58) and very few non-table-related errors (M 0.25), whereas participants given difficult problems made more non-table-related errors (M 2.17) than table-related errors (M 0.75). These results replicate those in Experiment 1. Generation accuracy. Participants accuracy when generating the answer to each problem was recorded. Participants presented with difficult problems made more generation errors (M 2.0) than did those presented with simple problems (M 1.0), F(1,46) 5.87, MS e , p.018. This finding provided evidence that the difficult problems require more effort than do the simple problems (but see Mitchell & Hunt, 1989). This finding also led us to conduct separate analyses for participants presented simple and for those presented difficult problems because of this difference in number of errors across participants in the two conditions. Participants who were presented with simple problems were classified as high or low accuracy according to whether they made no errors versus made one or more errors when generating the answers to problems. Twelve participants made no errors (high accuracy), and 12 made between one and four errors (low accuracy; M 2.0 errors). Separate analyses of variance were conducted on operand and answer recall including the between-subjects variable of generation accuracy (high vs low) and the within-subjects variable of presentation condition. Of greatest interest was the effect of the between-subjects variable of accuracy on recall. Accuracy had no effect on operand recall for participants who were presented with simple problems (high-accuracy M 0.35, low-accuracy M 0.33), F(1,22) 1. However, accuracy reliably affected answer recall, F(1,22) 6.9, MS e , p.015, reflecting the finding that high-accuracy participants recalled more problem answers (M 0.582) than did low-accuracy participants (M 0.363). The magnitude of the generation effect for answer recall, however, did not depend on participants accuracy, F(1,22) 1. Participants who were presented with difficult problems were classified as high-, medium-, or low-accuracy according to whether they made zero to one error (n 9), two errors (n 8), or three to six errors (n 7) when generating the answers to the difficult problems. Separate analyses of variance were conducted on operand and answer recall including the between-subjects variable of generation accuracy (high, medium, and low) and the within-subjects variable of presentation condition. Once again, of greatest interest was the between-subjects variable of accuracy. As found for participants presented with simple problems, accuracy had little effect on operand recall for participants who were presented with difficult problems (high-accuracy M 0.259, medium-accuracy M 0.385, lowaccuracy M 0.310), F(2,21) 1.3. Although accuracy did not reliably affect answer recall for difficult problems, F(2,21) 2.3, MS e , p.122, the magnitude of the generation effect for answer recall depended on participants accuracy, F(1,22) 4.07, MS e , p.032. As shown in Fig. 3, the generation effect was comparable for highand medium-accuracy participants. On the other hand, low-accuracy participants who had made three or more errors showed no difference in recall for problem answers that had been read or generated.

13 664 MCNAMARA AND HEALY Based on the procedural account, our interpretation of the effect of generation accuracy is that the participants who made three or more errors during generation were less likely to reinstate the same mental procedures at study and test because they did not possess the multiplication skills to do so. An alternative explanation is that the magnitude of the generation effect is reduced for these participants because if a problem is incorrectly generated, then the correct answer is only seen once after the generation attempt. In that case, a trial in the generate condition effectively turns into a trial in the read condition, at least at the behavioral level (i.e., after generating the answer, the participant reads the complete problem and answer). To test these competing hypotheses, an analysis was conducted to compare the proportion of recalled answers that had been incorrectly generated to the proportion of recalled answers that had been correctly generated. If generating correctly is critical to the magnitude of the generation effect, then the conditional proportion of answer recall given correct generation should be higher than the conditional proportion of answer recall given incorrect generation. If procedural reinstatement is critical, and not correct generation per se, then there should be no difference between the conditional proportion of answer recall given correct generation and the conditional proportion of answer recall given incorrect generation. An analysis of variance was performed comparing the conditional proportion of answer recall given correct generation and given incorrect generation. This statistical analysis is restricted to participants who made at least one error during generation and to those who correctly solved at least one problem. The analysis thus included 12 participants in the simple problem condition and 18 participants in the difficult problem condition. This analysis is also restricted to the answers recalled from problems presented in the generate condition (because there were no errors in the read condition). The effect of generation accuracy was not reliable, F(1,28) 1, reflecting the finding that participants were equally likely to recall an answer regardless of whether it had been correctly generated (M 0.428) or incorrectly generated (M 0.406). The effect of problem difficulty was also not reliable in this analysis, F(1,28) 2.41, MS e , p.132, nor was the interaction, F(1,28) 2.49, MS e , p.126. This analysis indicates that whether or not an answer is correctly generated does not determine the magnitude of the generation effect. Rather, this finding suggests that the critical aspect of the generation effect is that the participants engage in the mental operations involved in generation and that these operations can be successfully reinstated at test. To test our conclusions further, a mixed-factorial analysis of variance was conducted on the proportion of correct answers conditionalized on accurate generation. That is, a problem answer was included in the analysis only if it was correctly generated. Therefore, if a participant generated four answers correctly and recalled two of those answers, the proportion correct was calculated as.50. If a participant generated four answers correctly and recalled only the two answers incorrectly generated, the proportion correct would be zero. The analysis included the between-subjects variable of problem type and the within-subjects variable of presentation condition. The results for answer recall were virtually unchanged by conditionalizing on accurate generation. There was a main effect of problem type, F(1,46) 17.8, MS e , p.001, reflecting greater recall of answers to simple problems (M 0.502) compared to difficult problems (M 0.248). There was also a reliable generation effect, F(1,46) 33.0, MS e , p.001, reflecting greater recall for generated answers (M 0.500) compared to answers that were read (M 0.250). As in the original analysis, the generation effect for answer recall depended on problem difficulty, F(1,46) 7.0, MS e , p.011. The proportion recall for difficult problems was in the generate condition and in the read condition; the proportion recall for simple problems was in the generate condition and in the read condition. There was a smaller, though reliable, generation effect for difficult problem answers

14 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 665 [Difference 0.134, F(1,23) 5.5, p.027] than for simple problem answers [Difference 0.365; F(1,23) 30.9, p.001]. Thus, this more conservative analysis of answer recall accentuates our original interpretation of the data. Discussion The results of this study provide strong evidence against the operand memory hypothesis and support for the procedural account. According to the operand memory hypothesis, the generation effect should be larger for simple problems than for difficult problems both when the operands are recalled and when the answers are recalled. In both cases, the difference between simple and difficult problems is explained by the superior memory for the operands of the simple problems. In contrast, according to the procedural account, the generation effect for simple and difficult problems should be comparable when the operands are recalled but should be greater for the difficult problems than for the easy problems when the answers are recalled. The difference between simple and difficult problems is explained by the participants superior ability to reinstate the multiplication operations for the simple problems than for the difficult problems. In support of the procedural account, we found that for operand recall there was a generation effect of comparable magnitude for the simple and difficult problems, whereas for answer recall the generation effect was significantly larger for simple problems than for difficult problems. The finding of a reliable generation effect for the multiplication problem, which acts as the cue in this case, further demonstrates the importance of the cue target relationship to the generation effect (Greenwald & Johnson, 1989). Although Slamecka and Graf (1978) failed to find a consistent generation effect for the cue, they did find a small generation effect for the cue in a cued recall task. Moreover, Greenwald and Johnson (1989) subsequently found reliable generation effects for the cues across three experiments. They argued that demonstrating that memory for the cue benefits from the process of generating is critical to accounts proposing that the generation effect results from an enhanced cue target relationship (e.g., Donaldson & Bass, 1980; Rabinowitz & Craik, 1986). The procedural account is consistent with these accounts but goes further to emphasize the importance of procedural reinstatement. For multiplication problems, recalling the operands is the first step toward procedural reinstatement: In order to re-solve the problem and check the answer, the problem s operands must be remembered. A reliable generation effect for the operands regardless of problem difficulty indicates that the process of generating results in better operand memory than does reading. This finding was not specifically predicted by the procedural account but can be easily understood within that framework simply by assuming that extra processing was devoted to the operands during the problem solution that was required in the generate condition but not in the read condition. Generation probably also served to strengthen the ties between the two operands because they had to be interrelated during problem solution. In any event, finding a reliable generation effect for the operands suggests that superior operand memory partially explains the generation effect for multiplication problem answers. However, this is not the whole story. The generation effect for answer recall is dependent, not just on recalling the problem s operands, but also on reinstating the cognitive procedures during study. This latter assumption explains the greater generation effect for simple than for difficult problem answers. There are three additional observations in this experiment providing support for the procedural account and the hypothesis that the reinstatement of the same mental operations at study and test leads to enhanced recall in the generate condition. First, as in Experiment 1, in recalling the answers, the participants made more intrusion errors with difficult than with simple problems and they made more table-related than non-table-related intrusion errors with simple problems, but more non-table-related than table-related intrusion errors with difficult problems. In contrast, in recalling the operands, there was no reliable difference between participants presented with simple problems and

15 666 MCNAMARA AND HEALY those presented with difficult problems in terms of whether the recall intrusions were set-related. Second, the conditional proportion of answer recall given operand recall was significantly higher in the generate condition than in the read condition, indicating that recalling the operands from the read condition is less likely to lead to recalling the associated answers than is recalling the operands from the generate condition. These two findings suggest that (a) participants given simple problems were more likely to use the operands they recalled to derive and check the answers than were participants given difficult problems and that (b) recalling the operands leads to reinstating the cognitive procedures relevant to generating, but not to reading. A third observation in support of the procedural account concerns participants accuracy of generating the problem answer. In answer recall of difficult problems, the participants who had shown the lowest accuracy at generating the correct answers during the study phase showed no generation effect at test. Clearly participants who incorrectly generated the answers during the study phase did not engage in an arithmetic procedure that they could reinstate during the test phase in order to derive the answer. This finding cannot simply be explained by the fact that the low-accuracy participants did not say the correct answers during the study phase because all participants were shown and required to read aloud the correct answer during the generate task after they said aloud the answer they generated. Moreover, this finding cannot simply be explained by the fact that generating correctly is critical to the magnitude of the generation effect because it was found that participants were equally likely to recall an answer that had been correctly generated as one that had been incorrectly generated. Similarly, Slamecka and Fevreiski (1983) found a generation effect for antonym pairs even when subjects failed to come up with the correct answer; and Smith and Healy (1998) found a generation effect for simple multiplication problems even when subjects did not have time to complete the generation process before the answer was provided. These results collectively suggest that the critical aspect of the generation effect is that the participants engage in the mental operations involved in generation and that those operations can be successfully reinstated at test. EXPERIMENT 3 An alternative explanation for the results of Experiments 1 and 2 is that the answers to the simple multiplication problems are more likely to be retrieved because they are more easily encoded. That is, the simple problem answers may be more easily encoded because they end in a zero in contrast to the more complex, threedigit answers to the difficult problems. To examine this alternative explanation in Experiment 3, we held constant the answers to the problems while varying the problem operands. Thus, the problem corresponding to the answer 640 for the difficult problem version in Experiment 3 was If the greater generation effect for simple problems is due to the ease of encoding the answers, and not due to the ease and efficiency of re-generating the answer at recall, then the interaction of problem type and presentation condition found in Experiments 1 and 2 should disappear and only a main effect for generating should emerge. If, however, the generation effect is due to the reinstatement of the mental operations, then Experiment 3 should replicate the results of Experiment 1. A second purpose of Experiment 3 was to examine whether the greater generation effect for the simple problems was due to better recognition of the answers rather than to the reinstatement of mental procedures. That is, participants might be able to reproduce all possible simple problem answers and subsequently recognize the answers they had generated. If this is the case, then a greater generation effect should be found for simple problems when testing is done with recognition as well as with recall. On the other hand, if the generation effect for multiplication problem answers is dependent on the reinstatement of the mental operations, as predicted by the procedural account, then eliminating the generation stage of recall, by using a recognition test, should eliminate the generation effect for both simple and difficult problems. Specifically, recognition relies primarily on item information because it is primar-

16 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 667 ily the strength of the item in memory that allows it to be recognized. However, the procedural account makes no assumptions regarding the strength of the problem answers in the absence of the context provided by the problem itself. Thus, there is no basis for predicting greater item strength for generated multiplication problem answers than for those simply read, and there is no mechanism for predicting a generation effect for the recognition of multiplication problem answers on the basis of the procedural account. To test these predictions, retention was also examined in Experiment 3 with a recognition test. We were also interested in whether participants would be more likely to report using the operand retrieval strategy for problem answers that had been generated in the simple problem condition. Therefore, at the end of the experiment we asked participants to indicate what type of strategy they had used to recall each problem. They were asked to indicate whether (a) they had relied on imagery (visual or auditory), (b) recalled one or more of the operands to retrieve the answer (i.e., the operand retrieval strategy), or (c) used an alternative or unknown strategy. We predicted that participants would be more likely to indicate having used imagery to recall problem answers presented in the read condition and to recall difficult problem answers. In contrast, we predicted that participants would be more likely to report having used the operand retrieval strategy for simple problem answers that had been generated. Experiment 3 also modified stimulus presentation to accommodate group presentation. To do so, stimuli were presented in booklets. Each task was presented on a separate page, with presentation rate controlled by the participant. In addition, participants in the generate condition did not read the correct answers following generation, and the problems were seen only once in both conditions. Thus, participants were not provided with feedback concerning their answers in the generate condition. However, to decrease the answer generation error rate for difficult problems, we permitted participants to solve the problems using paper and pencil rather than mentally. Method Participants and design. Forty undergraduate students from the University of Colorado participated for credit in an introductory psychology course. A 2 2 mixed-factorial design was employed, with one between-subjects variable, problem type (simple vs difficult), and one within-subjects variable, presentation condition (generate vs read). Half of the participants were presented with simple problems, and half with difficult problems. Participants were randomly assigned to the problem type condition and the counterbalancing subcondition. Materials. The multiplication problems consisted of the 12 simple problems used in Experiments 1 and 2, and a new set of difficult problems (see Table 1). The answers to the simple and difficult problems were identical. With a few exceptions, the corresponding difficult multiplication problems were created by dividing the first operand of the corresponding simple problem by 2 and multiplying the second operand by 2. Exceptions to this procedure were made when the resulting problem seemed too simple. For example, the problem 40 9 would have become 20 18, so we used instead (which seemed more difficult). The simple and difficult multiplication problems were each divided into two sets of six problems of approximately equivalent difficulty (as in Experiment 1). Participants read one set and generated the other set. Order of presentation of the two sets of problems and order of reading and generating were counterbalanced across participants in each problem type condition. Procedure. Participants were tested individually and in small groups of two to three. Each participant was provided with a booklet. On the first page, participants were instructed that they should complete the booklet one page at a time and that they should never return to a page after going on to the following page (unless instructed to do so). A reminder not to turn back to a previous page was presented on the top of each page in the booklet (a pilot study had shown this tendency to be problematic). The second and fourth pages consisted of the read

17 668 MCNAMARA AND HEALY and generate problems (with a third blank page inserted to prevent seeing the following page). Problems presented in the read condition consisted of the problem and the answer with an answer blank next to each one. The participant was instructed to read the problem and the answer and copy the answer to the problem in the space provided. Problems presented in the generate condition consisted of the problem followed by a blank space. The participant was instructed to read each problem and write the answer to each problem in the space provided. They were allowed to use paper and pencil to solve the problem if necessary, but were told to use that page as the scratchpad. The fifth page consisted of the distractor task. Participants were presented with nine common words (e.g., flower and fish) and asked to write the first three words that came to mind after reading the word. The sixth page consisted of the recall task. Participants were asked to write down all of the answers to the multiplication problems (and not the problems) from both lists, the ones copied, and the ones solved. The seventh page was a blank filler sheet followed by the recognition task. Participants were presented with 24 numbers and asked to place a check beside each of the 12 numbers recognized as being an answer to a problem that had been read or generated. They were also asked to rate the confidence of their answers (those checked and those that were not checked) from 1 (low) to 5 (high). The 12 distractors were created by either subtracting or adding 20 to the problem answers (i.e., 290, 330, 340, 380, 440, 470, 500, 580, 610, 620, 740, and 830). On the last page, participants were asked to turn back to the recall page and indicate the type of strategy used to retrieve each problem answer recalled. Specifically, they were asked to write the letter A next to the answer to indicate I just remembered the answer don t know how ; the letter B to indicate I used imagery to recall the answer (visual or auditory) I could see the answer in my mind, or I could hear the answer in my mind ; the letter C to indicate I recalled some or all of the problem to recall or verify the answer ; or the letter D to indicate that they had FIG. 4. Proportion of correct answers recalled in Experiment 3 as a function of problem type and presentation condition, showing a reliable generation effect for simple problems answers and the lack of a generation effect for difficult problem answers. used some other strategy (they were also asked to describe the strategy in that case). Results The results are shown in Fig. 4 in terms of the proportion of correct answers recalled as a function of problem type (simple vs difficult) and presentation condition (generate vs read). A mixed-multifactorial analysis of variance was conducted on the proportion of correct responses including the between-subjects variable of problem type and the within-subjects variable of presentation condition. There was a main effect of problem type, F(1,38) 12.15, MS e , p.001, reflecting greater recall for answers to simple problems (M 0.496) compared to difficult problems (M 0.279). The effect of presentation condition was marginal, F(1,38) 3.36, MS e , p.075, reflecting slightly greater recall for generated answers (M 0.437) compared to answers that were read (M 0.337). As predicted, there was a significant interaction of problem type and presentation condition, F(1,38) 5.24, MS e , p.028. As shown in Fig. 4, this interaction reflects the lack of a generation effect for difficult problem answers [Difference 0.025, F(1,19) 1] compared to a reliable generation effect for simple problem answers [Difference 0.225;

18 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 669 F(1,19) 9.29, p.007]. Thus, in contrast to Experiments 1 and 2, there was no generation effect for difficult problems in this experiment. Intrusions. There was an average of only intrusion errors made by the participants in this experiment, much less than in Experiments 1 and 2. A mixed multifactorial analysis of variance was conducted on the number of intrusion errors including the between-subjects variables of problem type (simple vs difficult) and the within-subjects variable of error type (table-related vs non-table-related). The effect of problem difficulty was not reliable, F(1,38) 2.0, MS e 0.513, p.168 (simple M 0.375; difficult M 0.600). There were reliably more table-related errors (M 0.700) than non-table-related errors (M 0.275), F(1,38) 6.3, MS e 0.576, p.017, but there was no interaction, F 1. Hence, in contrast to Experiments 1 and 2, we did not find a significantly greater number of intrusions for difficult than for simple problems, and we did not find a greater predominance of table-related intrusions for simple problems than for difficult problems. There are a number of reasons why this pattern of results may have occurred. The most probable cause of the different pattern of intrusions is the greatly reduced number of intrusions. The reason for this reduction is not clear but it may have something to do with the use of testing packets, which were employed in this experiment but not in Experiments 1 and 2. Generation accuracy. Participants presented with difficult problems made only slightly more generation errors (M 0.55) than did those presented with simple problems (M 0.20), F(1,39) 2.88, MS e , p.098. This error rate translates to 90% accuracy in the difficult problem condition and 97% accuracy in the simple problem condition. Thus, there were fewer errors overall in comparison to Experiments 1 and 2 and little effect of problem difficulty on generation accuracy. Accuracy was higher in this experiment presumably because we allowed the participants to use pencil and paper to solve the problems, whereas in Experiments 1 and 2 participants were required to solve the problems mentally. Nevertheless, to verify further that generation accuracy did not affect the results, a mixedmultifactorial analysis of variance was conducted on the proportion of correct answers conditionalized on accurate generation. That is, a problem answer was included in the analysis only if it was correctly generated. The analysis included the between-subjects variable of problem type and the within-subjects variable of presentation condition. The results for answer recall were virtually unchanged by conditionalizing on accurate generation. There was a main effect of problem type, F(1,38) 11.18, MS e , p.002, reflecting greater recall of answers to simple problems (M 0.502) compared to difficult problems (M 0.295). There was also a reliable generation effect, F(1,38) 4.73, MS e , p.036, reflecting greater recall for generated answers (M 0.460) compared to answers that were read (M 0.337). As in the original analysis, the generation effect for answer recall depended on problem difficulty, F(1,38) 4.17, MS e , p.048. The proportion recall for difficult problems was in the generate condition and in the read condition; the proportion recall for simple problems was in the generate condition and in the read condition. Recognition. A mixed-multifactorial analysis of variance was conducted on the proportion of correctly recognized old numbers (i.e., hits) including the between-subjects variable of problem type and the within-subjects variable of presentation condition (read vs generate). There was a main effect of problem type, F(1,38) 26.13, MS e , p.001, reflecting better recognition of answers to simple problems (M 0.800) compared to difficult problems (M 0.567). There was no effect of presentation condition, F(1,38) 1, reflecting equivalent recognition for items that were read (M 0.679) and generated (M 0.688). Problem type and presentation condition did not interact, F(1,38) 1. A separate analysis of variance was also performed on correctly rejected new items. There was a main effect of problem type, F(1,38) 26.62, MS e

19 670 MCNAMARA AND HEALY TABLE 2 Recall Strategies Reported by Participants as a Function of Problem Type and Presentation Condition in Experiment 3 Problem type Answer imagery Problem recall Other Unknown Average Simple Generate Read Average Difficult Generate Read Average Overall average , p.001, reflecting higher proportions of correct rejections by participants who solved simple problems (M 0.856) than by those who solved difficult problems (M 0.654). Average signed confidence ratings were calculated by considering ratings for correct responses as positive and ratings for incorrect responses as negative. Thus, if an individual missed an item with a confidence of 5, this rating would be entered as 5. A mixed-multifactorial analysis of variance was conducted on the average signed confidence ratings for the old items including the between-subjects variable of problem type and the within-subjects variable of presentation condition (read vs generate). There was a main effect of problem type, F(1,38) 26.26, MS e , p.001, reflecting higher confidence for answers to simple problems (M 2.845) compared to difficult problems (M 1.072). There was no effect of presentation condition, nor was there an interaction of problem type and presentation condition, both Fs 1. An analysis of variance was also performed on participants confidence for rejecting new items. There was a main effect of problem type, F(1,38) 34.95, MS e 1.416, p.001, reflecting higher rejection confidence by participants who solved simple problems (M 3.120) than by those who solved difficult problems (M 0.900). Strategy protocols. Participants identified at the end of the experiment which strategy they remembered using to recall each of the problem answers. The list of strategies provided to the participants were (a) I used imagery to recall the answer (visual or auditory) I could see the answer in my mind, or I could hear the answer in my mind ; (b) I recalled some or all of the problem to recall or verify the answer ; (c) some other strategy; and (d) I just remembered the answer don t know how. Reported strategies were summed for each participant as a function of problem condition (generate vs read). When participants reported using both imagery and recalling the problem to verify the answer, the response was summed with the latter category (i.e., problem recall). One participant in the simple problem condition was not included in the analyses due to not recalling any problem answers correctly. A mixed-multifactorial analysis of variance was conducted on the number of strategies reported including the between-subjects variable of problem type and the within-subjects variables of presentation condition (read vs generate) and strategy type (imagery, problem, other, and unknown). The means as a function of these variables are presented in Table 2. Mirroring the results for recall accuracy, there were reliable main effects of problem type, F(1,38) 12.69, MS e 0.701, p.001, and presentation condition, F(1,37) 4.51, MS e 0.511, p.040, as well as a significant interaction of problem type and presentation condition, F(1,37) 6.70, MS e 0.511, p.014 (see Table 2). More importantly, there was a main effect of strategy type, F(3,37) 3.97, MS e 1.518, p.010, reflecting a greater overall tendency to recall the problem in order to retrieve and

20 PROCEDURAL EXPLANATION OF THE GENERATION EFFECT 671 verify the answer (i.e., the operand retrieval strategy) as compared to other strategies. However, the reliable interaction of strategy type and presentation condition confirms our prediction that the use of the operand retrieval strategy would be more frequent for problem answers that were generated than for those that were read, F(3,37) 4.57, MS e 0.826, p.005. Moreover, the interaction of strategy type and problem type confirms our prediction that the use of the operand retrieval strategy would be more likely for participants presented with simple problems than for those presented with difficult problems, F(3,37) 5.59, MS e 1.518, p.001. Finally, there was a reliable three-way interaction of strategy type, problem type, and presentation condition, F(3,37) 4.57, MS e 0.826, p.005. This interaction reflects the moderate tendency to use the operand retrieval strategy for difficult problem answers that were generated compared to the tendency to use imagery of the answer or some other strategy to recall difficult problem answers that were read. In contrast, for simple problem answers, there was a clear tendency to use the operand retrieval strategy in both presentation conditions, but the tendency was greater for answers that were generated than for those that were read. Discussion Experiment 3 confirmed that the greater generation effect for simple problem answers was not due to the greater ease of encoding the simple problem answers. Indeed, when the answers were identical for the simple and difficult problems, we found a substantial generation effect for simple problem answers and no generation effect for difficult problem answers. These results support our predictions, but contrast somewhat with the results of Experiments 1 and 2, which showed a reliable, though small, generation effect for difficult problem answers. We doubt that this difference in outcome is attributable to the differences in the problems and answers across experiments. Rather, this difference is more likely attributable to the absence of feedback provided to participants regarding their generated answers in Experiment 3. In Experiments 1 and 2, feedback was provided because of the difficulty of generating the answers mentally resulting in numerous generation errors. In Experiment 3, we increased generation accuracy by allowing participants to use paper and pencil to solve the problems eliminating the need for problem answer feedback. We suppose that the feedback provided in Experiments 1 and 2 strengthened the relationship between multiplication problem and the answer, thus increasing the likelihood of reinstating the relevant mental operations for the difficult problems at test. Similarly, Johns and Swanson (1988) found a generation effect for nonwords when feedback was provided and failed to find an effect when it was not provided. We also examined participants recognition of the problem answers after they had recalled the answers. We found better recognition of the simple than difficult answers, presumably because simple problem answers have a stronger representation in memory. We also found equivalent recognition of answers that had been read and generated. We had predicted this outcome based on the assumption that the generation effect for multiplication problem answers depends on the reinstatement of the mental operations engaged at study and that this process is not necessary to complete a recognition test. Finally, we substantiated our claims by asking participants to report what strategy they had used to retrieve each of the successfully recalled problem answers. As predicted, the use of the operand retrieval strategy was more frequently reported for problem answers that were generated than for those that were read and more frequent for participants presented with simple problems than for those presented difficult problems. We also found that for simple problem answers, there was a clear tendency to use the operand retrieval strategy for both presentation conditions but a larger tendency for the generate than for the read condition. This result, in conjunction with the results for recall accuracy, indicates that, whereas the participants in the simple problem condition attempt to use this strategy for all of the problem answers, the operand retrieval strategy is most often employed and is most effective for problems pre-

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