Multiplication Facts Research Articles

Technology-assisted instruction with teacher prompts on fraction multiplication word problems: A single-case design with visual analysis and Bayesian multilevel modeling

ABSTRACT This study examines the effects of technology-assisted instruction with teacher prompts on the ability to visualize and solve fraction multiplication word problems for four middle school students with learning disabilities. A multiple-probe design across participants, a type of single-case design, showed a functional relationship between the intervention and the targeted mathematical outcomes. Although there were some fluctuations over time, students demonstrated improvements from the baseline to intervention phases (Tau ranged from 0.76 to 1.00 for visualization and was 1.00 for problem-solving). Researchers employed Bayesian cumulative link mixed effects models to examine the moderating effects of word problem question types. Students showed greater maintenance effects on problem-solving than on visualization tasks, as reflected in changes in level (logit coefficient = 2.6) and trend (logit coefficient = 0.22). Students and teachers perceived vocabulary and multiplication fact practices, as well as the cognitive and metacognitive features embedded in the technology-assisted intervention, to be useful in learning targeted mathematics concepts. This study underscores the role of technology integration in enhancing teachers’ instructional approach and in aiding students’ acquisition and retention of mathematical concepts and skills.

The Efficacy of Mnemonic Strategies in Improving the Basic Multiplication Facts Retrieval

The current quasi-experimental study aims to investigate the efficacy of mnemonic strategies (Visual imagery and Story-linking) to improve the basic multiplication fact retrieval of grade 3 students with mathematics difficulties. The authors employed a quantitative approach within the matching-only pretest-post-test control group design to analyze the data on 206 subjects, of whom 78 exhibited mathematical difficulties. The study was conducted in eight inclusive primary schools in Purwakarta City, Indonesia. The research findings indicated that the N-gain acquisition of students in the experimental group (0.38) outperformed students in the control group (0.11). Similarly, the N-gain acquisition of students with mathematics difficulties in the experimental group (0.26) indicated a higher enhancement than those in the control group (0.08). Overall, the mnemonic strategies improved the students’ retrieval of basic multiplication facts on students with and without mathematics difficulties. However, mnemonic strategies have successfully achieved two of the Sustainable Development Goals 4 (SDG 4) by the 2030 Agenda for Sustainable Development on ensuring students with disabilities as one of the vulnerable groups, have equal access to education and guaranteeing both male and females are literate and numerically literate. These findings have practical implications for educators, providing them with actionable insights to develop effective teaching strategies for students with mathematics difficulties.

Arithmetic in two languages: Localizing simple multiplication processing in the adult bilingual brain.

Verbally memorized multiplication tables are thought to create language-specific memories. Supporting this idea, bilinguals are typically faster and more accurate in the language in which they learned math (LA+) than in their other language (LA- ) . No study has yet revealed the underlying neurocognitive mechanisms explaining this effect, or the role of problem size in explaining the recruitment of different brain regions in LA+ and LA- . To fill this gap in the literature, 29 Spanish-English early bilingual adults, proficient in both languages, verified simple multiplication problems in each language while functional magnetic resonance imaging (fMRI) was acquired. More specifically, this study aimed to answer two questions: 1) Does LA+ recruit left superior and middle temporal gyri (STG/MTG) to a greater extent than LA- , reflecting more robust verbal representations of multiplication facts in LA+? In contrast, does LA- recruit the inferior frontal gyrus (IFG), reflecting more effortful retrieval, or the intraparietal sulcus (IPS), reflecting reliance on quantity processes? 2) Is there an interaction between language and problem size, where language differences are more pronounced for less practiced, large multiplication problems (e.g., 8 × 9) in comparison to more familiar, small problems (e.g., 2 × 3). Functional localizer tasks were used to identify hypothesis-driven regions of interest in verbal areas associated with verbal representations of arithmetic facts (left STG/MTG) and with the effortful retrieval of these facts (left IFG) and quantity areas engaged when calculation-based strategies are used (bilateral IPS). In planned analyses, no cluster reached significance for the direct comparison of languages (question 1) or for the interaction between language and problem size (question 2). An exploratory analysis found a main effect of problem size, where small problems recruited left STG/MTG and left IFG to a greater extent than large problems, suggesting greater verbal involvement for these problems in both languages. Additionally, large problems recruited right IPS to a greater extent than small problems, suggesting reliance on quantity processes. Our results suggest that proficient early bilingual adults engage similar brain regions in both languages, even for more difficult, large problems.

Variations on a classwide math fact intervention: The role of set size and goal setting with explicit timing

Explicit Timing (ET) is an established intervention to support math fact fluency. The current study explored how goal setting and set size moderated the base ET intervention to promote multiplication fact fluency with third grade students. We also explored how learners at different levels of initial fluency responded to intervention. Findings suggest that ET was effective in promoting math fact fluency across learners who began intervention at varied performance levels, with average gains of over 15 digits correct per minute across 12 sessions of intervention over one month. Goal setting further improved the intervention. Results did not significantly differ across set sizes, nor across initial performance level. Overall, all investigated conditions led to efficient growth. Resources for classroom implementation are provided.

Evaluating and comparing three variations of cover, copy, and compare on multiplication fact fluency.

Evaluating and comparing three variations of cover, copy, and compare on multiplication fact fluency.

‘I can’t do maths’. Can we break this popular phrase? A review of the annual Mathematical Cognition and Learning Society conference and a sneak preview of some recent research

Good mathematical skills are vital for success in school, employment and everyday life, yet many individuals struggle with mathematics, and it is a sad state that saying ‘I can’t do maths’ is a norm. Behind the scenes many researchers and educators are trying to understand why children and adults find mathematics difficult and are designing research-led interventions and programmes to help. In this article I review some of the research that was presented at this years annual Mathematical Cognition and Learning Society (MCLS) conference, an international conference that started only six years ago, but in that time has quickly become the largest in the field. I also present a summary of my postdoctoral work that I’ve been conducting over the last few years, which investigates why many children struggle to learn multiplication facts.

Bilingual problem size effect: an ERP study of multiplication verification and production in two languages.

The problem size effect (PSE) is defined by better performance solving small problems (e.g., 2x4) than large problems (e.g., 8x9). For monolinguals, the PSE is larger when problems are presented in unfamiliar formats (e.g., written words), reflecting increased processing difficulty. Bilinguals are typically faster and more accurate at retrieving multiplication facts in the language of learning (LA+) than in their other language (LA-). We hypothesized that the less familiar arithmetic language (i.e., LA-) would elicit larger PSEs than LA+. Here, fluent Spanish-English bilingual adults verified spoken multiplication problems presented in LA+ and LA- while event-related potentials (ERPs) were recorded (Experiment 1A). To further promote language differences, we increased task difficulty by presenting problems at a faster pace (Experiment 1B) and requiring bilinguals to verbally produce solutions (Experiment 2). Language differences in performance were only observed for Experiment 2, where solutions were produced more slowly in LA- than LA+. In the ERPs, a PSE was driven by larger P300s for small than large solutions. A language effect was only observed under time pressure where LA- elicited a PSE at the 2nd operand. Additionally, the PSE was smaller for LA- at the solution. This suggests that categorizing multiplication facts is more effortful in LA-. In sum, very subtle language differences arise in fluent bilinguals when problems are more difficult, such as larger problems presented under time pressure in a weaker language. Critically, the effect of LA+ is at the level of response production and not access to the facts from memory.

The effect of retrieval practice on fluently retrieving multiplication facts in an authentic elementary school setting

AbstractFluently retrieving simple multiplication facts leads to an improvement of overall math scores. In the current study, we investigated how to best reach this fluency in an authentic elementary school setting. We compared the short‐term and long‐term effects of the learning strategies retrieval practice (using flashcards) to a restudy control condition (chanting multiplication facts out loud) on multiplication fact fluency. Forty‐eight second‐grade pupils received an instruction lesson and attended three spaced practice sessions per learning strategy. The learning strategies were counterbalanced across two different sets of multiplication tables in a within‐subjects design. A pre‐test, a five‐minute delay post‐test, and a one‐week delay post‐test were administered. Compared to restudy, retrieval practice led to a stronger short‐term and long‐term increase in the fluency of retrieving multiplication facts. This study demonstrates the effectiveness of retrieval practice on gaining multiplication fact fluency in an authentic classroom setting using original course materials.

Positive impact of sleep on recall of multiplication facts.

This study tested the hypothesis that learning complex multiplication problems (e.g. 8 × 23 = 184) prior to sleep would benefit recall in adult participants compared with learning the problems prior to a period of wakefulness. This study used a within-participant design where all participants learnt complex multiplication problems in two conditions separated by one week. In one condition, learning was before bed (sleep learning condition) and in the other condition learning was in the morning (wake learning condition). In each condition, recall was tested approximately 10.5 h later. Data were collected online from 77 participants. In the subset of the sample with greater than or equal to 60% accuracy at the initial learning session (n = 37), the sleep learning condition participants had better recall compared with the wake learning condition. This equated to a moderate effect size, Cohen's d = 0.51. Regardless of initial levels of learning (n = 70) the same beneficial effect of sleep on recall was found with a small effect size, Cohen's d = 0.33. This study has identified a beneficial effect of learning prior to sleep on recall of complex multiplication problems compared with learning these problems during the daytime. Future research should explore whether similar effects are observed with children learning simple multiplication facts.

The impact of intervention modality on students' multiplication fact fluency

AbstractWhile an abundance of evidence‐based mathematics interventions are available to support students’ mathematics fluency, research suggests that they are not often implemented within schools. Moreover, national data suggests that most students in the United States are not meeting grade level expectations in mathematics. This underperformance coupled with limited implementation of evidence‐based practices emphasizes the need to provide schools with effective, efficient, and resource friendly interventions to support students’ mathematics achievement. The purpose of this study was to compare the effects of reciprocal peer tutoring, iPad delivered flashcards, and a combined intervention condition on the multiplication fluency of 72 third grade students enrolled in two public schools in the midwestern region of the United States using a randomized pretest–posttest design. Students received 10 min of intervention after their core instruction 4 days per week for 5 weeks. Results suggest that there were few differences between treatment groups on both proximal and distal measures. Moreover, all groups demonstrated significant growth from pre‐ to posttest on proximal measures. Assessment response rates across modalities and student engagement are discussed along with implications for research and practice.

Related consistent lures increase the judgment of multiplication facts: Evidence using event-related potential technique.

Simple multiplication errors are primarily shown in whether the lures are related to the operands (relatedness, such as 3 × 4 = 15 vs. 17) or whether the same decades are shared with the correct answers (consistency, such as 3 × 4 = 16 vs. 21). This study used a delayed verification paradigm and event-related potential technique to investigate the effects of relatedness and consistency in simple multiplication mental arithmetic for 30 college students in an experiment of presenting probes in auditory channels. We found that, compared to the related inconsistent lures, the related consistent lures showed significantly faster reaction time and induced significantly large amplitudes of N400 and late positive component. The findings suggest that related consistent lures are less affected by the activation diffusion of the arithmetic problem, and the credibility of being perceived as the correct answer is less; the lures related to operands and sharing the same decades with the accurate results can promote the judgment of multiplication mental arithmetic, and the results support the Interacting Neighbors Model.

Differential effectiveness, efficiency, and acceptability of drill‐based interventions targeting multiplication facts

AbstractMath facts are foundational for later math skill development, but many students demonstrate difficulties with math facts in early schooling. Drill‐based interventions have been shown to effectively target foundational academic skills; however, there is limited research examining drill‐based interventions for teaching math facts compared to basic reading skills. The purpose of this study was to compare the effects of incremental rehearsal (IR), traditional drill (TD), and strategic incremental rehearsal (SIR) on multiplication fact retention, fluency, and maintenance outcomes with 36 fourth and fifth graders. The study also examined intervention efficiency and treatment acceptability. Results showed no differences in retention, maintenance, or treatment acceptability across the interventions. There were significant differences in fluency at posttest, with IR resulting in the greatest fluency scores. Furthermore, TD was the most efficient intervention overall. Implications for practice and future research are also discussed.

Development is in the details: Event-related theta oscillations reveal children and adults verify multiplication facts differently.

When verifying the correctness of single-digit multiplication problems, children and adults show a robust ERP correctness effect thought to reflect similar cognitive processes across groups. Recent studies suggest that this effect is instead a modulation of the negative-going N400 component in children, reflecting access to semantic memory, and the positive-going P300 component in adults, reflecting stimulus categorization. However, the relative difference in ERP amplitude is the same for both components, more positive for correct than incorrect solutions, presenting a challenge to ascertaining the appropriate interpretation. Time-frequency analysis (TFA) of the N400/P300 window provides an objective approach to dissociating these effects. TFA measured from solution onset during single-digit multiplication verification revealed significant modulations of event-related as theta power (3-6Hz) in both groups. Correct trials elicit less power in children (9-12 years) and more power in adults relative to incorrect trials. These findings are consistent with modulations of the N400 and P300, respectively, where opposite effects were predicted for spectral power. The ERP results further support a reinterpretation of the multiplication correctness effect. In contrast, TFA of the N400 effect elicited to a word-picture verification task revealed the same event-related theta effect in both groups, with increased power for mismatched than matched pictures. Together, these findings provide evidence for a developmental shift in cognitive processing specific to the multiplication task. Models of arithmetic should account for this overlooked difference in cognitive processing between children and adults.

Use of Cross-Line Method in Helping to Master Basic Multiplication Skills for Students With Learning Disabilities

Mathematics is an essential subject for students with learning disabilities in order to meet their special educational needs. This subject emphasizes numerical skills and skills in operations such as addition, subtraction, multiplication, and division. This allows them to manage their money and time and know how to do the measurement so that they can apply their mathematical skills in various contexts. However, many students with learning disabilities face a variety of difficulties when learning mathematics, particularly when it comes to multiplication operations. Hence, this study aims to investigate the use of the cross-line method in helping pupils with learning disabilities improve their ability to solve multiplication operation questions. This is a qualitative case study designed for five pupils with learning disabilities, aided by a special education teacher. The study was conducted in Chinese vernacular schools located in Kinta Utara, Perak. The study used document analysis and interviews for data collection. Overall, the findings of the study showed that the pupils with learning disabilities faced three basic problems in multiplication. The problems consist of their understanding of the concept of multiplication, memorizing multiplication facts, and using ineffective techniques in solving multiplication operation questions. The Cross-Line method was used to assist students with learning disabilities in improving their basic multiplication skills. In addition, the findings of the study also showed that the use of the Cross-Line method was able to improve the positive behaviors among the pupils while studying multiplication topics in the classroom. The Cross-Line Method was also able to help the pupils with learning disabilities master multiplication skills with ease. In conclusion, the study shows that the use of the Cross-Line method will be able to help pupils with learning disabilities master basic multiplication, and it can be used as an alternative method for mathematics teachers to teach the topic of multiplication operations more effectively to the pupils with learning disabilities

TIGRO-M: A Program for Automatization of Multiplication Facts

TIGRO-M: A Program for Automatization of Multiplication Facts

Elementary math in elementary school: the effect of interference on learning the multiplication table

Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other, which creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory processes are responsible for this improvement. In a series of 16 short training sessions over 4 weeks, first-grade children learned 16 multiplication facts—4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of certain facts over others. Moreover, the interference arose from the similarity between facts in a given week, not from the similarity to previously learned facts. Similarity affected long-term memory—its effect persisted 7 weeks after training has ended; and it operated on long-term memory directly, not via the mediation of working memory. Pedagogically, the effectiveness of the low-interference training method, which is dramatically different from currently used pedagogical methods, may pave the way to enhancing how we teach the multiplication table in school.

Deafness and early language deprivation influence arithmetic performances.

It has been consistently reported that deaf individuals experience mathematical difficulties compared to their hearing peers. However, the idea that deafness and early language deprivation might differently affect verbal (i.e., multiplication) vs. visuospatial (i.e., subtraction) arithmetic performances is still under debate. In the present paper, three groups of 21 adults (i.e., deaf signers, hearing signers, and hearing controls) were therefore asked to perform, as fast and as accurately as possible, subtraction and multiplication operations. No significant group effect was found for accuracy performances. However, reaction time results demonstrated that the deaf group performed both arithmetic operations slower than the hearing groups. This group difference was even more pronounced for multiplication problems than for subtraction problems. Weaker language-based phonological representations for retrieving multiplication facts, and sensitivity to interference are two hypotheses discussed to explain the observed dissociation.

Examination of a Modified Incremental Rehearsal Approach to Explore Causal Mechanisms.

Incremental rehearsal (IR) has consistently been shown to improve students’ math fact retention and fluency (Maki et al., Journal of Behavioral Education 30:534–558, 2021). However, less is known about how intervention modifications may support longer-term skill maintenance. The purpose of this study was to compare traditional IR with a modified IR (shuffle IR; ShIR) in which known multiplication facts were shuffled between sequences using a cumulative acquisition design with six fourth- and fifth-grade students. All participants retained and maintained more facts in IR and ShIR compared to a control condition. However, IR or ShIR did not consistently result in greater retention than the other, with three students demonstrating greater retention in the IR condition and three students demonstrating greater retention in the ShIR condition. Most participants demonstrated greater fact maintenance in the ShIR condition than in the IR condition. All participants made fewer intervention session errors in the condition in which they retained more multiplication facts.

Impaired Arithmetic Fact Retrieval in an Adult with Developmental Dyscalculia: Evidence from Behavioral and Functional Brain Imaging Data.

Developmental dyscalculia (DD) is a developmental disorder characterized by arithmetic difficulties. Recently, it has been suggested that the neural networks supporting procedure-based calculation (e.g., in subtraction) and left-hemispheric verbal arithmetic fact retrieval (e.g., in multiplication) are partially distinct. Here we compared the neurofunctional correlates of subtraction and multiplication in a 19-year-old student (RM) with DD to 18 age-matched controls. Behaviorally, RM performed significantly worse than controls in multiplication, while subtraction was unaffected. Neurofunctional differences were most pronounced regarding multiplication: RM showed significantly stronger activation than controls not only in left angular gyrus but also in a fronto-parietal network (including left intraparietal sulcus and inferior frontal gyrus) typically activated during procedure-based calculation. Region-of-interest analyses indicated group differences in multiplication only, which, however, did not survive correction for multiple comparisons. Our results are consistent with dissociable and processing-specific, but not operation-specific neurofunctional networks. Procedure-based calculation is not only associated with subtraction but also with (untrained) multiplication facts. Only after rote learning, facts can be retrieved quasi automatically from memory. We suggest that this learning process and the associated shift in activation patterns has not fully occurred in RM, as reflected in her need to resort to procedure-based strategies to solve multiplication facts.

No influence of masked priming on the multiplication fact retrieval in a result verification task

In three experiments, we used a masked prime in a verification task to investigate the processing stages occurring during multiplication fact retrieval. We aimed to investigate the retrieval process by overlapping its execution with the processing of a masked prime consisting of a number. Participants evaluated the correctness of multiplication equations, where the result was preceded by a masked prime (presented for 30 ms, with different stimulus onset asynchrony between the operands and the prime). Decade consistency and relatedness of the prime were manipulated. For example, given the equation 4 x 7 = 28, the prime could be: a neighbor either decade consistent (24) or inconsistent (32), or an unrelated number either decade consistent (23) or inconsistent (31). We expected that the feature of the prime (relatedness or decade consistency) that generates interference depends on the processing stage reached when the prime is processed. Although Experiment 1 showed promising results, Experiments 2 and 3 suggest that the pattern found in Experiment 1 was a false positive. Overall, the paradigm used in this study (i.e., masked prime with a verification ask) does not seem to produce a stable interference during the retrieval process.