Capacity and contextual constraints on product activation: Evidence from task-irrelevant fact retrieval

Three experiments tested the limiting conditions of multiplication facts retrieval in a number-matching task (LeFevre, Bisanz, & Mrkonjic, 1988). By presenting two digits as cue and by requiring participants to decide whether a subsequent numerical target had been present in the pair, we found interference when the target coincided with the product of the cue digits. This was evidence for obligatory activation of multiplication facts. Also, we showed that multiplication facts retrieval occurred even in the absence of any arithmetic context (i.e., a multiplication sign between the cue digits) and did not require processing resources (i.e., the process met the capacity criterion of automaticity; Jonides, 1981), whereas manipulation of the spatial relation between the two operands (cue digits) negatively affected retrieval. The present work appears to be unique in the context of previous similar studies on mental calculation, which invariably adopted an arithmetic task as the primary demand. We identify this difference as the reason for the failure of all previous studies in revealing independence of multiplication facts from attentional resources. Furthermore, we suggest the application of a contextual definition of automaticity to this kind of retrieval, given the fact that it might depend both on association strength and on contextual setting variables.     

Rote memory and arithmetic fact processing

The goal of the study was to examine the part played by skill in memorizing arbitrary sequences in the efficiency with which normal young adults perform simple arithmetic fact problems. The first experiment showed a clear independent role for sequence memory in all arithmetic fact processing, but a lesser role for semantic retrieval. This result was particularly true for large-answer multiplication problems and subtraction and division problems with large first operands. In a second experiment, which included a visuomotor processing control task, sequence memory predicted processing of all arithmetic problems apart from small additions independently of semantic retrieval, with the most robust independent contribution being to large-answer multiplication problems. The results, which are compatible with Dehaene and colleagues' triple-code model, suggest that rote learning may be a successful way for some people to process arithmetic facts efficiently.     

Intervening with Multiplication Fact Difficulties: Examining the Utility of the Instructional Hierarchy to Target Interventions

Examining the use of conceptual frameworks such as the instructional hierarchy (IH) to drive academic interventions represents an important area of inquiry in order to understand why an intervention was effective. However, to date, the IH has only been examined retroactively to explain the effectiveness of math interventions and has not been used to drive intervention implementation based on students’ needs. Therefore, the current study used a multiple baseline single-case design to examine the utility of the IH to determine an appropriately targeted acquisition (incremental rehearsal) or proficiency (timed drill) intervention for difficulties with multiplication facts. Students first received the contraindicated intervention (i.e., non-targeted) and then their correctly targeted intervention. Results showed that all students demonstrated greater fluency growth during their targeted intervention and that students in need of an acquisition intervention also demonstrated greater multiplication fact retention and growth rate during their targeted intervention. The utility of the IH as a heuristic to correctly target interventions based on students’ baseline skill levels is discussed.

     

Individual solution processes while solving addition and multiplication math facts in adults

Contrary to predictions of current solution process models, adults used a variety of procedures other than retrieval to solve addition and multiplication math facts. Predictors assumed to capture retrieval processes posited by such models did account for a substantial proportion of variance in averaged retrieval solution times. But most of the variance in individual participants' retrieval times remained unaccounted for. Cross-operation associations in patterns of strategy use and retrieval latencies were obtained. Adults with stronger higher level math achievement were more likely to use retrieval, solved math facts faster and less variably, and executed retrieval processes posited by current solution process models faster than participants with less math attainment. The results are explained within the context of the adaptive strategy choice model.     

Evaluating the generalization of math fact fluency gains across paper and computer performance modalities

Computer-based interventions are being used more in the classroom. Student responses to these interventions often contribute to decisions making regarding important outcomes. It is important to understand the effect of these interventions within the context of the intervention as well as across related context. The current study examined the generalization of math fact fluency gains resulting from a computer-based intervention to paper-and-pencil performance. A total of 31 second grade students completed fluency drills on the computer or with paper and pencil. Pretest-posttest performance on both computer and paper and pencil for all students was evaluated using a doubly multivariate repeated measure ANOVA. Results indicated that gains achieved on the computer did not generalize to paper-and-pencil performance.     

Comparing the Effectiveness and Efficiency of Response Repetition to Simultaneous Prompting on Acquisition and Maintenance of Multiplication Facts

This study compared the effectiveness and efficiency of an error-correction procedure, response repetition, to a prompting procedure, simultaneous prompting, on the acquisition and maintenance of multiplication facts for three typically developing 3rd grade students. This study employed an adapted alternating treatments design nested in a multiple probe design across three sets of multiplication facts. Results indicated that correct responding increased upon intervention implementation for all participants. For two participants, response repetition was a more effective teaching procedure. For one participant, while both teaching procedures were effective, response repetition was more efficient in terms of sessions to mastery while simultaneous prompting was more efficient in terms of errors and seconds to mastery. Maintenance data were variable. Discussion focuses on conceptual differences between response repetition and simultaneous prompting that might have accounted for results.     

A neuropsychological study of fact memory and source amnesia

We investigated the ability of amnesic patients to learn new facts (e.g., Angel Falls is located in Venezuela) and also to remember where and when the facts were learned (i.e., source memory). To assess the susceptibility of fact and source memory to retrograde amnesia, patients prescribed electroconvulsive therapy were presented facts prior to the first treatment and were tested after their second treatment. All amnesic patients exhibited marked fact memory impairment. In addition, some amnesic patients exhibited source amnesia (i.e., they recalled a few facts but then could not remember where or when those facts had been learned). Source amnesia was unrelated to the severity of the memory deficit itself, because patients who exhibited source amnesia recalled as many facts as the patients who did not. These results show that the deficit in amnesia includes an impairment in acquiring and retaining new facts. Source amnesia can also occur, but it is dissociable from impaired recall and recognition and appears to reflect difficulty in remembering the specific context in which information is acquired. The findings are discussed in terms of their significance for how memory is organized.     

Enhancing Multiplication Performance in Students with Moderate Intellectual Disabilities Using Pegword Mnemonics Paired with a Picture Fading Technique

The present study examined the effectiveness of an instructional package that included an adapted version of pegword mnemonics paired with a picture fading technique in teaching two students with moderate intellectual disabilities to recall 28 single-digit multiplication facts between 2 and 9. The instructional package was assessed using a multiple baseline design across students. Results indicated that instruction produced substantial and immediate effects in that both students increased their accuracy at recalling basic multiplication facts, maintained the newly acquired skill, and were able to generalize it across material, format, and another trainer. These results have implications for teaching students with intellectual disabilities basic math facts that are considered important for gaining access to the general curriculum.     

Focused memory search in fact retrieval

Several studies of fact retrieval have shown that the more facts a person learns about a concept, the longer it takes him or her to retrieve any of these facts. This result has been interpreted to mean that retrieval of a fact about a concept involves a search of all facts stored in memory with that concept. In the present study, it is suggested that retrieval involves not an unfocused search of all facts stored with a concept, but rather a focused memory search that examines relevant stored facts and ignores irrelevant information. This argument is supported by three experiments in which subjects first learned simple facts (e.g., “The banker likes horses”) and then made speeded true-false decisions for test probes le.g., “The banker likes elephants”). Specifically, results suggest that facts stored with a concept may be organized into subsets. For example, a person’s knowledge about Richard Nixon might be organized into subsets concerning Nixon’s resignation, his trips to China, his family, and so on. The data further suggest that a person attempting to retrieve a fact about a concept (e.g., the name of Nixon’s wife) may simply decide which subset is most likely to contain the desired fact (e.g., the subset concerning Nixon’s family) and search that subset. If the sought-for fact is found in this subset, the search process terminates. If, however, the desired information is not located, other subsets of facts may be searched before the retrieval attempt is given up. The notion that memory search focuses on relevant stored facts and ignores irrelevant information may help to explain why experts (i.e., people who know a large number of facts about a topic) do not experience great difficulty in retrieving facts in their areas of expertise.     

Simple arithmetic processing: Individual differences in automaticity

This study investigated individual differences in the ability to automatically access simple addition and multiplication facts from memory. It employed a target-naming task and a priming procedure similar to that utilised in the single word semantic-priming paradigm. In each trial, participants were first presented with a single digit arithmetic problem (e.g., 6+8) and were then presented with a target that was either congruent (e.g., 14) or incongruent (e.g., 17) with this prime. Response times for congruent and incongruent conditions were then compared to a neutral condition (e.g., X+Y, with target 14). For the high skilled group, significant facilitation in naming congruent multiplication and addition targets was found at SOAs of 300 and 1000?ms. In contrast, for the low skilled group, facilitation in naming congruent targets was only observed at 1000?ms. Significant inhibition in naming incongruent multiplication and addition targets at 300?ms, and addition targets at 1000?ms, was found for the high skilled group alone. This advantage in access to simple facts for the high skilled group was then further supported in a problem size analysis that revealed individual differences in access to small and large problems that varied by operation. These findings support the notion that individual differences in arithmetic skill stem from automaticity in solution retrieval and additionally, that they also derive from strategic access to multiplication solutions.     

Insufficient evidence: mindset intervention in developmental college math

It is generally accepted by educators and social psychologists that non-cognitive factors influence learning. Students who have a fixed view of intelligence believe that abilities are permanent. They view set-backs and the need for effort as evidence of lack of ability. Students with a growth mindset believe that intelligence and abilities are malleable. Intelligence grows with challenge. The objective of this study was to consider if teaching growth mindset to a group of college students in remedial math classes would increase the likelihood of passing the course and remaining in school. Results found a near statistically significant correlation between growth mindset and passing the course but no evidence to support growth mindset and retention. An intervention to teach growth mindset led to statistically small positive results as measured by Pearson’s product-moment correlation and independent sample t tests. Results suggest that combining a variety of interventions with weak academic students is required.     

Short-term effects of grade retention on the growth rate of Woodcock-Johnson III broad math and reading scores

We investigated the effects of grade retention in first grade on the growth of the Woodcock-Johnson broad mathematics and reading scores over three years using linear growth curve modeling on an academically at-risk sample. A large sample (n = 784) of first grade children who were at risk for retention was initially identified based on low literacy scores. Scores representing propensity for retention were constructed based on 72 variables collected in comprehensive baseline testing in first grade. We closely matched 97 pairs of retained and promoted children based on their propensity scores using optimal matching procedures. This procedure adjusted for baseline differences between the retained and promoted children. We found that grade retention decreased the growth rate of mathematical skills but had no significant effect on reading skills. In addition, several potential moderators of the effect of retention on growth of mathematical and reading skills were identified including limited English language proficiency and children's conduct problems.     

The Representation of Arithmetic Facts in Memory: Results from Retraining a Brain-Damaged Patient

This single case study was designed to gather evidence regarding whether the mental representations mediating multiplication fact retrieval make use of single or multiple codes. MC is a brain-damaged volunteer whose numerical processing impairments were limited to multiplication fact retrieval. He relearned three sets of multiplication facts. Each set was relearned in one of three input formats: Arabic, written verbal, or spoken verbal. Following training all facts were tested in all input formats. MC's posttraining performance was virtually error free and showed no effects of input format. However, reaction-time data showed fact retrieval was fastest when the training format matched the test format. Results are discussed in relation to single- and multiple-code models of multiplication fact retrieval.     

TEACHING MULTIPLICATION FACTS TO STUDENTS WITH LEARNING DISABILITIES

Multiple baseline designs were used to examine the effects of an instructional package on accuracy of performance in solving multiplication facts by 3 students with learning disabilities. The instructional package included the following components: (a) a modified instructional sequence in which multiplication facts were grouped into the zeros, ones, doubles, fives, and nines categories, and those remaining; (b) identification of the category in which each fact belonged; (c) mnemonic strategies associated with solving facts in each category; and (d) steps to be completed for solving facts in each category. Results indicated that the instructional package produced substantial and immediate effects. After receiving instruction, a participant's accuracy was often 100%, and this was maintained throughout the evaluation even as other strategies were introduced. Comparable results occurred across students, demonstrating replication of the effects of the instructional package.     

Retrieval-induced forgetting of arithmetic facts

Retrieval-induced forgetting (RIF) is a widely studied phenomenon of human memory, but RIF of arithmetic facts remains relatively unexplored. In 2 experiments, we investigated RIF of simple addition facts (2 + 3 = 5) from practice of their multiplication counterparts (2 × 3 = 6). In both experiments, robust RIF expressed in response times occurred only for high-strength small-number addition facts with sums ≤ 10, indicating that RIF from multiplication practice was interference dependent. RIF of addition-fact memory was produced by multiplication retrieval (2 × 3 = ?) but not multiplication study (2 × 3 = 6), supporting an inhibitory mechanism of RIF in arithmetic memory. Finally, RIF occurred with multiplication practiced in word format (three × four) and addition tested later in digit format (3 + 4), which provides evidence that digit and written-word formats for arithmetic accessed a common semantic retrieval network. The results support the view that addition and multiplication facts are stored in an interrelated semantic network and that RIF of competing addition facts is an intrinsic process of multiplication fact retrieval.     

A cost-effectiveness analysis of classwide math intervention

Given limited resources, schools are encouraged to consider not only what works, but also at what cost. Cost-effectiveness analysis offers a formal methodology to conceptualize and calculate the ratio of the costs to implement an intervention to its effects (i.e., incremental cost-effectiveness ratios). This study used the ingredients method to analyze secondary data from a randomized controlled trial (N = 537 fourth- and fifth-grade students) to calculate the cost-effectiveness of a classwide math intervention, and provides an overview of cost-effectiveness analysis for readers unfamiliar with the formal methodology. For fourth-graders, the incremental cost-effectiveness ratio was $169.07, indicating it cost $169.07 per student for a 1 standard deviation increase in scaled scores on the state assessment. For fifth-graders, there was no statistically significant effect on the state assessment, but there were improvements in curriculum-based measurement (CBM) scores with incremental cost-effectiveness ratios ranging from $65.08 to $469.12, depending on the type of CBM probe and implementation context. Additionally, using number-needed-to-treat (i.e., the number of participants who must be provided with the intervention to prevent one failure on the state assessment), the cost was $126.90 to prevent failure on the state assessment for one fourth-grade student receiving special education services or for one student who scored below the 25th percentile on the prior year's state assessment. Implications and directions for future research are discussed.     

Cognitive addition and multiplication: Evidence for a single memory network

In an experiment using verification task procedures, 100 subjects responded to simple and complex problems of addition and multiplication. Identical structural parameters were found to model reaction time accurately to both addition and multiplication problems. Slope estimates for a memory network parameter did not differ significantly between simple and complex problems within an operation or between addition and multiplication problems. Both complex addition and complex multiplication problems were processed columnwise, with column sums or products being retrieved from an interrated memory network. The two types of complex problems included similar processes for carrying and for encoding of single digits, and both were self-terminated when an error in the units column was encountered. Addition and multiplication facts appear to be retrieved from a single interrelated memory network. A conceptual model for this interrelated network is discussed.     

Supporting teacher use of interventions: effects of response dependent performance feedback on teacher implementation of a math intervention

This study examined general education teachers’ implementation of a peer tutoring intervention for five elementary students referred for consultation and intervention due to academic concerns. Treatment integrity was assessed via permanent products produced by the intervention. Following verbal instructions, intervention implementation by four teachers was consistently low or exhibited a downward trend whereas one teacher’s moderate implementation co-occurred with student gains. When classroom training was conducted with four teachers exhibiting low integrity, all teachers improved implementation. After accurately implementing the intervention for three training sessions with classroom training, response dependent performance feedback was provided when teachers independently used the intervention below 100% integrity, as determined by review of permanent products. With response dependent performance feedback, three of the four teachers implemented the intervention at levels above the verbal training performance, whereas implementation for one teacher increased following discussion of an upcoming school team meeting. Student math scores improved during intervention.     

Effects of a computer-assisted language intervention in a rural Nevada center

A computer-assisted language intervention, Fast ForWord-Language (FFW-L), was tested at a rural Nevada center in a group of children (Grades 2-12) referred by parents and teachers to assess enhancement of language skills. Given conflicting results from previous studies, language scores were measured using Clinical Evaluation of Language Fundamentals, Third Edition (CELF-3) before and after the FFW-L intervention. 58 children's CELF-3 postintervention scores were adjusted for age-specific expected changes and compared with pretest scores. Adjusted scores increased in both receptive and expressive domains of the CELF-3. Children with prior diagnoses of language and/or learning impairment did not differ from other referrals on adjusted CELF-3 adjusted gain scores after treatment. Thus the Fast ForWord-Language intervention may benefit a much broader group of children referred by parents and teachers for language or reading problems.     

Cognitive and behavioral attention in children with math difficulties

Cognitive assessments and behavioral ratings of attention were used to examine the relation of inattention to math performance in children. Third grade students with math difficulties (MD; n?=?17) and math and reading difficulties (MDRD; n?=?35) were administered the Attentional Network Test (ANT), as well as achievement and intelligence measures. Strengths and Weaknesses of ADHD-Symptoms and Normal-Behavior-IV (SWAN-IV) Inattention ratings were collected from teachers. Two comparison groups were also recruited: a typically achieving group (n?=?23) and a group that responded to a math-tutoring intervention (responders; n?=?54). On the ANT, children with MD and MDRD did not perform significantly different than typically achieving children or responders on measures of alerting and orienting attention and executive control. All subgroups did demonstrate performance patterns that were expected on the ANT. However, performance across blocks of the task was inconsistent, suggesting poor reliability. There were no relations between ANT performance and SWAN-IV behavioral inattention scores, though behavioral ratings of inattention correlated significantly with math performance. Children with MD and MDRD may have more difficulty with distraction and attention to detail in contextual situations, as opposed to impulsive responding in these settings. The lack of relation between cognitive attention and math performance may suggest that either the ANT does not assess the relevant attention constructs associated with math difficulties or may parallel studies of attention deficit/hyperactivity disorder (ADHD) in which cognitive and behavioral assessments are weakly related.