Solving Multiplication Problems Research Articles

Analisis Kesulitan Belajar Peserta Didik Terhadap Materi Matematika, Khususnya Dalam Perkalian Berbentuk Cerita Di Kelas II Sekolah Dasar

Students who have difficulty learning mathematics will often make mistakes in learning to count and mistakes in solving word problems in mathematics. The difficulties experienced by students are difficulty in working on story problems due to the inability to understand the meaning of the questions and confusion when determining what arithmetic operations will be used to solve the story problems. The purpose of this research was to analyze the learning difficulties experienced by students as well as the factors that caused students’ learning difficulties in working on multiplication questions in the form of stories in class II elementary school. The method used in this research is descriptive case study. The subjects of this research are students and teachers of class II A at SDN 15 ULU GADUT. The data collection procedures used in the research are test’s, interviews, observation and documentation. The results of the research show that difficulties in learning mathematics include: 1) Difficulty in distinguishing multiplication problems from other material, 2) Student’s writing cannot be read, 3) Difficulty doing calculations in solving multiplication problems. Internal factors include: 1) Lack of enthusiasm for studying, 2) Low numeracy skills and difficulty understanding questions, 3) Not focusing on studying. External factors include: 1) Lack of attention from parents, 2) Often asking students to memorize multiplication, 3) Not using interesting learning media in teaching.

Positive math attitudes are associated with greater frontal activation among children from higher socio-economic status families

Math learning is explained by the interaction between cognitive, affective, and social factors. However, studies rarely investigate how these factors interact with one another to explain math performance. This study aims to fill this gap in the literature by using functional magnetic resonance imaging (fMRI) to understand the neurocognitive mechanisms underlying the interaction between parental socioeconomic status (SES) and children's math attitudes. To this aim, 57 children solved multiplication problems inside the scanner. We measured parental SES by creating two groups based on parents' occupations and measured children's math attitudes using a questionnaire. We ran a cluster-wise regression analysis examining the interaction between these two variables while controlling for the main effects of SES, math attitudes, and full IQ. The analysis revealed a cluster in the left inferior frontal gyrus (IFG), which was due to children with positive math attitudes from high socio-economic status families showing greater IFG activation when solving large multiplication problems as compared to their negative attitudes high SES peers, suggesting that they exhibited more retrieval effort to solve large multiplication problems. We discuss how this may be because they were the only ones who fully engaged in math opportunities provided by their environment.

PENGENALAN METODE JARIMATIKA GUNA MENINGKATKAN KEMAMPUAN BERHITUNG PERKALIAN PADA SISWA SEKOLAH DASAR

The lack of numeracy knowledge among students at State Elementary School 013 Kaliamanting Island Village is a problem so further learning is needed The aim of this research was to assess the impact of using the Jarimatika method on students' multiplication abilities in classes V and VI of the Kalimanting Island State Elementary School. This research focuses on the application of Jarimatika as a tool in solving multiplication problems. The motivation for this research arose because students did not have adequate skills in dealing with multiplication, and monotonous teaching methods made learning mathematics difficult. The method used in this research is an experiment with a quasi-experimental design known as a nonequivalent control group design. The experimental group was treated with the Jarimatika method, while the control group continued to use conventional methods which included expository, question and answer and rote memorization methods. Data was collected through tests, and research results were analyzed using descriptive statistical techniques. The research results showed an increase in the average score on the pretest and posttest. From the test results, it can be seen that there is a different increase between the control group and the experimental group. The increase in the average score in the control group was 19 points, increasing from 52 to 71, while the experimental group experienced an increase of 22 points, increasing from 54 to 76. Also, the increase in multiplication calculation ability in the control group was lower than in the experimental group. These findings show that the use of the Jarimatika method has more impact on the multiplication abilities of class VI students in elementary school

PENINGKATAN KEMAMPUAN PEMAHAMAN KONSEP PERKALIAN MENGGUNAKAN MODEL OPEN ENDED KELAS III SD NEGERI 03 ALAI TIMUR

This research was motivated by the low understanding of students' concept of multiplication in class III SD Negeri 03 Alai Timur. Learning directly to symbols and formulas makes it difficult for students to understand mathematical concepts, especially in multiplication. Students find it difficult to solve multiplication problems. In addition to the problems above, students' understanding of concepts is also lacking, because students have not been facilitated by learning models that involve active students in learning. All of these problems have an impact on low student learning outcomes related to multiplication. In general, the purpose of this study was to describe the increase in students' understanding of concepts in multiplication learning. This research is a classroom action research with a qualitative and quantitative approach. The subjects in this study were third grade students at SD Negeri 03 Alai Timur. This study consisted of II cycles, cycle I held 2 meetings and cycle II 1 meeting consisting of planning, implementation, observation and reflection. The data collection technique is observation and documentation. While the research instrument is the observation sheet and documentation. The results showed that the success criteria for understanding the concept of students in the first cycle of the first meeting was 58.23%. meeting II 86.823%. Cycle II meeting I was 87.04%. Thus the use of the open ended model in multiplication learning in class III SD Negeri 03 Alai Timur can improve students' understanding of concepts.

Does mathematics anxiety moderate the effect of problem difficulty on cognitive effort?

A negative relationship between mathematics anxiety (MA) and mathematics performance is well documented. One suggested explanation for this relationship is that MA interferes with the cognitive processes needed when solving mathematics problems. A demand for using more cognitive effort (e.g., when performing harder mathematics problems), can be traced as an increase in pupil dilation during the performance. However, we lack knowledge of how MA affects this relationship between the problem difficulty and cognitive effort. This study investigated, for the first time, if MA moderates the effect of arithmetic (i.e., multiplication) problem difficulty on cognitive effort. Thirty-four university students from Norway completed multiplication tasks, including three difficulty levels of problems, while their cognitive effort was also measured by means of pupil dilation using an eye tracker. Further, the participants reported their MA using a questionnaire, and arithmetic competence, general intelligence, and working memory were measured with paper-pencil tasks. A linear mixed model analysis showed that the difficulty level of the multiplication problems affected the cognitive effort so that the pupil dilated more with harder multiplication problems. However, we did not find a moderating effect of MA on cognitive effort, when controlling for arithmetic competence, general intelligence, and working memory. This suggests that MA does not contribute to cognitive effort when solving multiplication problems.

Language does arithmetic: linguistic differences in children’s place-value processing

The representation and retrieval of multiplication facts is dependent on linguistic specificities such as number word inversion (i.e., 23 is spoken dreiundzwanzig in German which translates to three and twenty). Previous research has evaluated these language influences in adults. Now this study aims to follow-up on earlier findings and takes a closer look at inversion-related effects on place-value processing during multiplication in children. In a task of choice 46 children, either German- or Italian-speaking, had to pick the right answer out of two options for a given multiplication problem. Already established effects in adult participants such as decade-consistency and table-relatedness were also present in elementary school children, but different between the language groups. For decade-consistent items the effect of table-relatedness was larger for Italian-speaking students than for German-speaking. This indicates that the inversion property in the German language leads to those children putting less emphasis on the tens digit when solving multiplication problems, than Italian-speaking children.

Is accommodation a confounder in pupillometry research?

Much psychological research uses pupil diameter measurements to investigate the cognitive and emotional effects of visual stimuli. A potential problem is that accommodating at a nearby point causes the pupil to constrict. This study examined to what extent accommodation is a confounder in pupillometry research. Participants solved multiplication problems at different distances (Experiment 1) and looked at line drawings with different monocular depth cues (Experiment 2) while their pupil diameter, refraction, and vergence angle were recorded using a photorefractor. Experiment 1 showed that the pupils dilated while performing the multiplications, for all presentation distances. Pupillary constriction due to accommodation was not strong enough to override pupil dilation due to cognitive load. Experiment 2 showed that monocular depth cues caused a small shift in refraction in the expected direction. We conclude that, for the young student sample we used, pupil diameter measurements are not substantially affected by accommodation.

Students’ understanding of the associative property and its applications: noticing, doubling and halving, and place value

Understanding the properties of multiplication is a critical precursor to students’ thinking algebraically. However, these properties have not been the focus of extensive rigorous research, particularly the associative property. In this study, we report on follow-up interviews with 25 year 5–6 students who had completed five items taken from an assessment of mental computational fluency. This assessment required students to reason from the perspective of a fictional student (Emma), who had applied the associative property in various ways to solve multiplication problems. In the interviews, students had to explain their understanding of Emma’s thinking. Coding of this interview data revealed distinct continuums of understanding of each of the three applications of the associative property (noticing, doubling and halving, and place value), which teachers could use to inform their planning. The findings reveal that students best understood the noticing application, closely followed by doubling and halving. By contrast, the place value application was not well understood, which we attribute to students relying on truncation procedures applied with little or no conceptual understanding.

Bilingual adults practicing multiplication tables – looking into bilingual arithmetic learning

ABSTRACT The importance of language in mathematics education becomes increasingly obvious, as multilingual learners are not an exception nowadays. But how can language dominance and the language of first or later instruction affect arithmetic training in adults? 101 bilingual adults whose first and dominant language was German (LM+ and L1) and who spoke English as their second language (L2) were tested. Participants were assigned to three different training groups and either practiced basic multiplications in German, English or both languages. In a verification paradigm, reaction times on solving multiplication problems in German and English respectively were recorded before and after the training. Results showed a strong influence of the language adult bilinguals initially acquired arithmetic knowledge in (German). German items were overall solved more quickly than English counterparts, irrespective of the language items were specifically trained in. English were affected by differing training conditions with respect to training effects. This leads to the consideration of various factors including language proficiency levels, the language of first mathematics instruction in early school years as well as the language of mathematics instruction in later learning contexts, in understanding bilingual arithmetic learning.

Using the Virtual–Representational–Abstract With Overlearning Instructional Sequence to Students With Disabilities in Mathematics

While mathematics education is key to the post-school outcomes of students with disabilities, it has received less attention in research and practice compared with other aspects of educating this population. Skill maintenance is particularly crucial in mathematics because students build upon prior knowledge across grade levels. They also need to be able to apply mathematical skills in everyday life. Hence, this study utilized a multiple probe across-participants single-case experimental design to evaluate the effectiveness of the virtual–representational–abstract (VRA) with overlearning instructional sequence in teaching multiplication and supporting its maintenance among three students with disabilities. For each student, a functional relation existed between the VRA with overlearning instructional sequence and accuracy of solving multiplication problems. Students also maintained the skill up to 8 weeks after the intervention.

Oscillatory EEG Changes During Arithmetic Learning in Children

ABSTRACTMost studies have investigated brain activation changes after the course of arithmetic learning, and the question remains whether these changes are detectable during the course of learning, i.e., before memory consolidation. Twenty-four fifth graders solved multiplication problems while ongoing electroencephalography (EEG) was recorded. The arithmetic training revealed reduced errors together with a power increase in theta (4–7 Hz) but not in lower alpha (8–10 Hz) or upper alpha (10–13 Hz) bands. We conclude that increases in theta power subserved a shift from slow, procedural strategies to more efficient, automated procedural and retrieval strategies, which led to more efficient performance.

Fast and slow strategies in multiplication

In solving multiplication problems, children use both fast retrieval-based processes and slower computational processes. In the current study, we explore the possibility of disentangling these strategies using information contained in the observed response latencies using a method that is applicable in large data sets.We used a tree-based item response-modeling framework (De Boeck & Partchev, 2012) to investigate whether the proposed qualitative distinctions in fast and slow strategies can be detected. This so-called fast-slow model was applied to responses to a set of multiplication items, totalling more than 180,000 responses, collected in an online computer-adaptive training environment for mathematics.Parameters describing person characteristics (ability) and item characteristics (easiness) are estimated with the model. Both item and person characteristics differed between fast and slow processes and match predictions from substantive models of multiplication. Moreover, the parameters allowed us to describe the fast and slow strategies in more detail. Results emphasize the utility of the fast-slow model in the detection of strategies in multiplication but also in other areas of cognition and learning where strategies are expected.

ADHD and math - The differential effect on calculation and estimation

Adults with ADHD were compared to controls when solving multiplication problems exactly and when estimating the results of multidigit multiplication problems relative to reference numbers. The ADHD participants were slower than controls in the exact calculation and in the estimation tasks, but not less accurate. The ADHD participants were similar to controls in showing enhanced accuracy and speed for smaller problem sizes, for trials in which the reference numbers were smaller (vs. larger) than the exact answers and for reference numbers that were far (vs. close) from the exact answer. The two groups similarly used the approximated calculation and the sense of magnitude strategies. They differed however in strategy execution, mainly of the approximated calculation strategy, which requires working memory resources. The increase in reaction time associated with using the approximated calculation strategy was larger for the ADHD compared to the control participants. Thus, ADHD seems to selectively impair calculation processes in estimation tasks that rely on working memory, but it does not hamper estimation skills that are based on sense of magnitude. The educational implications of these findings are discussed.

Understanding children’s reasoning in multiplication problem-solving

ABSTRACTThis article investigates two children’s intuitive thinking in solving multiplication problems from different educational backgrounds. One of the children is in a southern elementary school in the US. He was given the same problems both in first and second grades. The other child was a first grader in a southwest region of China, and she was given the same problems. The findings reveal a variety of intuitive thinking in solving the multiplication problems through addition beyond direct modeling and counting strategies. The authors also discussed how different educational backgrounds in early elementary mathematics education may affect children’s intuitive ideas and reasoning in solving multiplication problems. The study implies the importance of understanding children’s intuitive ideas of multiplication and highlights potential opportunities for developing children’s understanding of multiplicative thinking and algebraic thinking in earlier stages of arithmetic learning.

Exploring Common Misconceptions and Errors about Fractions among College Students in Saudi Arabia

The purpose of this study was to investigate what common errors and misconceptions about fractions exist among Saudi Arabian college students. Moreover, the study aimed at investigating the possible explanations for the existence of such misconceptions among students. A researcher developed mathematical test aimed at identifying common errors about fractions as well as short interviews, aimed at understanding the thought process while solving problems on the test, were conducted among a total of 107 (n=107) college students. The findings suggested that the majority of college students in Saudi Arabia hold common misconceptions about fractions and mathematical calculations involving fractions, such as thinking that all fractions are always part of 1 and never greater than 1, and using cross multiplication to solve multiplication problems involving fractions.

The interference effect in arithmetic fact solving: An fMRI study

Some multiplication facts share common digits with other, previously learned facts, and as a result, different problems are associated with different levels of interference. The detrimental effect of interference in arithmetic facts knowledge has been recently highlighted in behavioral studies, in children as well as in adults, both in typical and atypical development. The present study investigated the brain regions involved in the interference effect when solving multiplication problems. Twenty healthy adults carried out a multiplication task in an MRI scanner. The event-related design comprised problems whose interference level and problem size were manipulated in a 2×2 factorial design. After each trial, individuals were requested to indicate whether they solved the trial by retrieving the answer from long-term memory. This allowed us to examine which brain areas were sensitive to the interference effect and problem size effect as well as the retrieval strategy. The results highlighted two specific regions: the left angular gyrus was more activated for low interfering than for high interfering problems, and the right intraparietal sulcus was more activated for large problems than for small problems. In both regions, brain activity was not modulated by the other effect. These results suggest that the left angular gyrus is sensitive to the level of interference of the multiplication problems, whereas previously this region was thought to be more activated by small problems or by retrieval strategy. Here, in a design manipulating interference and problem size, while controlling for retrieval strategy, we showed that it rather reflects an automatic mapping between the problem and the answer stored in long-term memory. The right intraparietal sulcus was modulated by the problem size effect, which supports the idea that the problem size effect comes from the higher overlap between magnitude of the answers of large problems compared to small ones. Importantly, neither effects can be reduced to a strategy effect since they were present when analyzing only retrieval trials.

An Examination of the Ways That Students with Learning Disabilities Solve Fraction Computation Problems

The purpose of this study was to better understand how students with learning disabilities, including those who struggle specifically with mathematics, engage with fraction computation. In particular, we examined error patterns, the influence of like and unlike denominators on these patterns, and correct solution methods. Although skill-related errors existed, we found that the majority of errors were systematic, indicating predictable difficulties with fraction algorithms. Error patterns changed with like and unlike denominators, particularly for addition and division. All students correctly solved multiplication problems in the same way, but a variety of solution methods were found for some addition, subtraction, and division problems. Implications for research and practice are discussed.

What if I can’t? Success expectancies moderate the effects of utility value information on situational interest and performance

Two studies tested how the effects of a utility value manipulation on interest and performance were moderated by expectations for success. College students learned a new technique for mentally solving multiplication problems with instructions containing task utility information or not. In Study 1 (N = 62), the effect of the utility value information was positive for individuals with high success expectancies, but negative for individuals with low success expectancies. Study 2 (N = 148) examined the causal role of success expectancies by manipulating whether participants received an expectancy boost before receiving the utility manipulation. The results showed further support for the importance of success expectancies in moderating the effect of directly-communicated utility value. The results are discussed in relation to other research on utility value, interest, and expectancy–value models of achievement behavior.

Children with mathematical learning disability fail in recruiting verbal and numerical brain regions when solving simple multiplication problems

Greater skill in solving single-digit multiplication problems requires a progressive shift from a reliance on numerical to verbal mechanisms over development. Children with mathematical learning disability (MD), however, are thought to suffer from a specific impairment in numerical mechanisms. Here we tested the hypothesis that this impairment might prevent MD children from transitioning toward verbal mechanisms when solving single-digit multiplication problems. Brain activations during multiplication problems were compared in MD and typically developing (TD) children (3rd to 7th graders) in numerical and verbal regions which were individuated by independent localizer tasks. We used small (e.g., 2 × 3) and large (e.g., 7 × 9) problems as these problems likely differ in their reliance on verbal versus numerical mechanisms. Results indicate that MD children have reduced activations in both the verbal (i.e., left inferior frontal gyrus and left middle temporal to superior temporal gyri) and the numerical (i.e., right superior parietal lobule including intra-parietal sulcus) regions suggesting that both mechanisms are impaired. Moreover, the only reliable activation observed for MD children was in the numerical region when solving small problems. This suggests that MD children could effectively engage numerical mechanisms only for the easier problems. Conversely, TD children showed a modulation of activation with problem size in the verbal regions. This suggests that TD children were effectively engaging verbal mechanisms for the easier problems. Moreover, TD children with better language skills were more effective at engaging verbal mechanisms. In conclusion, results suggest that the numerical- and language-related processes involved in solving multiplication problems are impaired in MD children.

Students’ Intuition in Mathematics Class Using Lesson Study and Open Approach

The objective of this article is to investigate students’ intuition in mathematics class using Lesson Study and Open Approach. The research methodology employed was the qualitative research methods of teaching experiment combined with an ethnographic study. The study’s target group consists of three Grade 2 students at Ban Bueng Niam Bueng Krainoon School in Khon Kaen Province during the 2012 Academic Year. This is the school that participated in the professional development of mathematics teachers with Lesson Study and Open Approach innovation project. The researcher collected data from a mathematics class that taught multiplication learning unit (1) from the mathematics textbook for Grade 2 students. This is the textbook used in the professional development of mathematics teachers with Lesson Study and Open Approach innovation project. The research findings are: Mathematics class using Lesson Study and Open Approach allows students to intuitively learn the content of a basic multiplication unit. Students’ intuition was developed during Step 1 (posing open-ended problems) and Step 2 (students’ self-learning through problemsolving) of the Open Approach process. Students intuitively discovered the repeated addition and multiplication methods in solving multiplication problems.