Errors in Fraction Arithmetic Among Sixth-Grade Students With Mathematics Difficulties

Fractions are challenging for both typically achieving children and adults. Although some prior research has focused on fraction difficulties of children with mathematics difficulties (MD), persistent difficulties encountered by adults with MD remain unknown. It is possible that these adults may be able to compensate for some deficits. In this study, we administered an un-timed, paper-based fraction achievement test to French adults with and without MD to compare their knowledge of fractions. Compared with controls, adults with MD performed worse in fraction number lines, fraction concepts, fraction arithmetic, and word problems. However, no difference in performance between the two groups was observed on symbolic representations. This suggests that adults with MD might be able to perform rote procedures such as transcoding from a verbal to a symbolic representation but are severely impaired for fraction number line, fraction concept, and fraction arithmetic. Exploratory error pattern analyses for fraction number line and fraction arithmetic further revealed mistakes similar to those observed in prior studies on children with MD, indicating core deficits in fraction understanding in individuals with MD.

المستخلص هدفت هذه الدراسة إلي بحث العلاقة بين مجال الذاكرة والتعلم ببطارية نبسي2 والتحصيل الدراسى لذوى صعوبات تعلم الرياضيات لدى تلاميذ الصف السادس الابتدائى، من مدرسة مجمع شها الابتدائية بإدارة شرق المنصورة التعليمية، لتحديد التلاميذ الذين يعانون من صعوبات تعلم الرياضيات وقد نتج عن تطبيق أدوات الدراسة عينة مكونة من (31) تلميذاً وتلميذة من الصف السادس، وتمثلت أدوات الدراسة فى اختبار المصفوفات المتتابعة الملونة لرافن، ومقاييس بطارية نبسي2 لمجال الذاكرة والتعلم، وقد أسفرت نتائج الدراسة عن: وجود علاقات ارتباطية موجبة دالة بين درجات التلاميذ ذوي صعوبات تعلم الرياضيات على مقاييس أبعاد مجال الذاكرة والتعلم ببطارية نبسي2، كما أنه يمكن صياغة معادلة تنبؤ بالتحصيل الدراسي لذوي صعوبات تعلم الرياضيات من خلال درجات مقاييس أبعاد مجال الذاكرة والتعلم ببطارية نبسي2. الكلمات المفتاحية: مجال الذاكرة والتعلم، صعوبات تعلم الرياضيات. Abstract This study aimed to examine the relationship between the field of memory and learning in the NPSI Battery 2 and the academic achievement of those with difficulties in learning mathematics among sixth grade students, from the Shaha Complex Primary School, East Mansoura Educational Administration, to identify students who suffer from difficulties in learning mathematics. The application of the study tools resulted in a sample consisting of Of (31) male and female students from the sixth grade, the study tools were the Raven’s Colored Progressive Matrices test, and the Npsi Battery 2 scales for the field of memory and learning. The results of the study resulted in: the presence of significant positive correlations between the scores of students with mathematics learning difficulties on the dimensions of the memory field. And learning using the Npsi Battery 2. It is also possible to formulate an equation to predict academic achievement for people with learning difficulties in mathematics through the scores of the dimensions of the memory and learning domain dimensions of the Npsi Battery 2. Keywords: field of memory and learning, difficulties in learning mathematics.

Several types of converging evidence have suggested recently that skilled adults solve very simple addition problems (e.g., 2 + 1, 4 + 2) using a fast, unconscious counting algorithm. These results stand in opposition to the long-held assumption in the cognitive arithmetic literature that such simple addition problems normally are solved by fact retrieval from declarative memory. Here we tested a large sample of diversely skilled and culturally diverse men and women at the University of Saskatchewan and examined multiple categories of simple (1 digit plus 1 digit) addition problems for evidence of generalization of practice, a signature of procedure use. The procedure-based 0 + N = N problems presented clear evidence of generalization (i.e., practicing a subset of 0 + N problems lead to speed-up for a different subset of 0 + N problems), but there was no evidence of such generalization of practice for the nonzero problems, although the experiment had good power to detect small effects. Given that generalization of practice is a basic marker of procedure-based processing, its absence for the nonzero addition problems casts doubt on the compacted counting theory.

The purpose of this study was to assess the effects of a fraction vocabulary intervention with fraction arithmetic components on fraction vocabulary knowledge and fraction arithmetic competencies among fourth-grade Chinese students with mathematics difficulties. We randomly assigned 70 students with mathematics difficulties to three conditions: fraction vocabulary only (n = 23), fraction vocabulary with an arithmetic component (n = 23), and a business-as-usual (BaU) condition (n = 24). The students in the fraction vocabulary intervention conditions participated in 10 sessions, occurring three times per week. Students within both intervention conditions showed significantly better performance in fraction vocabulary knowledge than those in the BaU condition. However, no notable distinctions were observed between the two intervention conditions in terms of fraction arithmetic. Only students who received the fraction vocabulary intervention with an arithmetic component exhibited enhanced performance in subtraction with like denominators compared to the BaU condition.

In this research, 10- to 12- and 13- to 15-year-old children were presented with very simple addition and multiplication problems involving operands from 1 to 4. Critically, the arithmetic sign was presented before the operands in half of the trials, whereas it was presented at the same time as the operands in the other half. Our results indicate that presenting the 'x' sign before the operands of a multiplication problem does not speed up the solving process, irrespective of the age of children. In contrast, presenting the '+' sign before the operands of an addition problem facilitates the solving process, but only in 13 to 15-year-old children. Such priming effects of the arithmetic sign have been previously interpreted as the result of a pre-activation of an automated counting procedure, which can be applied as soon as the operands are presented. Therefore, our results echo previous conclusions of the literature that simple additions but not multiplications can be solved by fast counting procedures. More importantly, we show here that these procedures are possibly convoked automatically by children after the age of 13years. At a more theoretical level, our results do not support the theory that simple additions are solved through retrieval of the answers from long-term memory by experts. Rather, the development of expertise for mental addition would consist in an acceleration of procedures until automatization.

The adaptive use of approximate calculation was examined using a verification task with 18 third graders with mathematics learning disabilities, 22 typically achieving third graders, and 21 typically achieving second graders. Participants were asked to make true-false decisions on simple and complex addition problems while the distance between the proposed and the correct answer was manipulated. Both typically achieving groups were sensitive to answer plausibility on simple problems, were faster at rejecting extremely incorrect results than at accepting correct answers on complex addition problems, and showed a reduction of the complexity effect on implausible problems, attesting to the use of approximate calculation. Conversely, children with mathematics disabilities were unaffected by answer plausibility on simple addition problems, processed implausible and correct sums with equal speed on complex problems, and exhibited a smaller reduction of the complexity effect on implausible problems. They also made more errors on implausible problems. Different hypotheses are discussed to account for these results.

Two experiments compared rates of solving simple and complex addition and multiplication problems in groups of speakers of French or English in Experiment 1 (n = 35) and Spanish or English in Experiment 2 (n = 84). Subjects were divided into groups of English unilinguals, weak bilinguals, and strong bilinguals according to their performance on a naming task. In both experiments, simple problems consisted of two single‐digit numbers. At least three single‐digit numbers were used for complex problems in Experiment 1 and double‐digit numbers in Experiment 2. Mean solution times, particularly for complex problems, were lowest for the monolingual group, followed in turn by the weak bilingual and strong bilingual groups, but these differences were not statistically reliable in either experiment. In Experiment 2, however, componential analyses of solution times indicated that strong bilingual subjects were slower at executing the carry operation when solving complex problems, relative to the two remaining groups. Results were interpreted in terms of the relationship between bilingualism and the representation and processing of numerical information.

Documenting how students with learning disabilities (LD) initially conceive of fractional quantities, and how their understandings may align with or differ from students with mathematics difficulties, is necessary to guide development of assessments and interventions that attach to unique ways of thinking or inherent difficulties these students may face understanding fraction concepts. One way to characterize such conceptions is through the creation of a framework that depicts key understandings evidenced as students work with problematic situations. The present study extends current literature by presenting key understandings of fractions, documented through problem-solving activity, language, representations, and operations, evidenced by students with LD and mathematics difficulties as they engaged with equal sharing problems. Clinical interviews were conducted with 43 students across the second, third, fourth, and fifth grades. Results of the study suggest that students with LD hold similar informal notions of key understandings of fractions as students with mathematics difficulties and that many of the students evidenced rudimentary understandings of fractional quantities. Researchers discuss implications of the findings in relation to considerations for designing interventions to support and extend students’ initial conceptions of fractional quantity.

<span style="font-size: 12.0pt; line-height: 115%; font-family: 'Times New Roman','serif'; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">Literature revealed the benefits of different instruments for the development of mathematical competence, problem solving, self-regulated learning, affective-motivational aspects and intervention in students with specific difficulties in mathematics. However, no one tool combined all these variables. The aim of this study is to present and describe the design and development of a hypermedia tool, Hipatia.</span><span style="font-size: 12.0pt; line-height: 115%; font-family: 'Times New Roman','serif'; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">Hypermedia environments are, by definition, adaptive learning systems, which are usually a web-based application program that provide a personalized learning environment. This paper describes the principles on which Hipatia is based as well as a review of available technologies developed in different academic subjects. Hipatia was created to boost self-regulated learning, develop specific math skills, and promote effective problem solving. It was targeted toward fifth and sixth grade students with and without learning difficulties in mathematics. After the development of the tool, we concluded that it aligned well with the logic underlying the principles of self-regulated learning. Future research is needed to test the efficacy of Hipatia with an empirical methodology.</span><!--[if gte mso 10]> <mce:style><! /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin-top:0cm; mso-para-margin-right:0cm; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"MS Mincho"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;} > <! [endif] -->

Event-related potentials ERPs were used to examine the organization of brain activations during single-digit multiplication. Electrophysiological, neuropsychological, and brain-imaging data suggest that left inferior parietal areas are involved in mental calculation. We aimed at investigating the involvement of this area in simple and difficult single-digit multiplications, and at determining the time course of its activation. ERPs were recorded from 64 channels while subjects performed a sequential multiplication-verification task. Simple and difficult multiplication problems were presented either visually as arabic digits or auditorily as number words. For both modalities of input, a significant effect of difficulty was found on left and right inferior parietal electrode sites. The results suggested that simple multiplication problems may involve a short-lived activation in the left inferior parietal cortex, whereas complex problems may require longer processing which also involves the homologous right area. These findings also demonstrate the significance of ERPs as a tool for determining the temporal orchestration of brain areas involved in a cognitive task.

Eye-tracking methods have only rarely been used to examine the online cognitive processing that occurs during mental arithmetic on simple arithmetic problems, that is, addition and multiplication problems with single-digit operands (e.g., operands 2 through 9; 2 + 3, 6 x 8) and the inverse subtraction and division problems (e.g., 5 – 3; 48 ÷ 6). Participants (N = 109) solved arithmetic problems from one of the four operations while their eye movements were recorded. We found three unique fixation patterns. During addition and multiplication, participants allocated half of their fixations to the operator and one-quarter to each operand, independent of problem size. The pattern was similar on small subtraction and division problems. However, on large subtraction problems, fixations were distributed approximately evenly across the three stimulus components. On large division problems, over half of the fixations occurred on the left operand, with the rest distributed between the operation sign and the right operand. We discuss the relations between these eye tracking patterns and other research on the differences in processing across arithmetic operations.

The role of language in memory for arithmetic facts remains controversial. Here, we examined transfer of memory training for evidence that bilinguals may acquire language-specific memory stores for everyday arithmetic facts. Chinese-English bilingual adults (n = 32) were trained on different subsets of simple addition and multiplication problems. Each operation was trained in one language or the other. The subsequent test phase included all problems with addition and multiplication alternating across trials in two blocks, one in each language. Averaging over training language, the response time (RT) gains for trained problems relative to untrained problems were greater in the trained language than in the untrained language. Subsequent analysis showed that English training produced larger RT gains for trained problems relative to untrained problems in English at test relative to the untrained Chinese language. In contrast, there was no evidence with Chinese training that problem-specific RT gains differed between Chinese and the untrained English language. We propose that training in Chinese promoted a translation strategy for English arithmetic (particularly multiplication) that produced strong cross-language generalization of practice, whereas training in English strengthened relatively weak, English-language arithmetic memories and produced little generalization to Chinese (i.e., English training did not induce an English translation strategy for Chinese language trials). The results support the existence of language-specific strengthening of memory for everyday arithmetic facts.

Two experiments were conducted to investigate the effects of practice on strategy selection and strategy efficiency in mental arithmetic. Participants had to solve simple addition or multiplication problems, after having received 0, 3, or 6 practice sessions (Experiment 1), and before and after having received 3 practice sessions (Experiment 2). Strategy selection was measured by means of trial-by-trial strategy reports, whereas strategy efficiency was measured by means of response latencies. Results showed significant practice effects on retrieval frequency, procedural frequency, retrieval efficiency, and procedural efficiency. However, practice effects on strategy efficiency appeared to be both strategy-specific (i.e., only for procedural strategies) and operation-specific (i.e., only for multiplication problems). Implications of the present results for mathematic cognition and its modeling are discussed.

This research investigated retrieval-induced interference between counterpart multiplication (2 × 3 = 6) and addition facts (2 + 3 = 5). Adults (N =72) repeatedly solved either a set of simple addition (0 + 2, 1 + 5, 2 + 3) or multiplication problems (0 × 2, 1 × 5, 2 × 3) during a practice phase and then switched operations during a test phase that included counterparts to the practiced problems and control problems. The paradigm afforded measurement in response time both of inter-operation retrieval-induced forgetting (RIF) and generalization of practice across different problems within operations. The experiment demonstrated generalization of practice for the rule-based 0 + N = N problems (e.g., practicing 0 + 2 facilitated performance on 0 + 7) as well as for problems governed by the multiplicative identity principle (1 × N = N) and zero-product principle (0 × N = 0), but not the fact-based 1 + N problems. The experiment also demonstrated for the first time inter-operation RIF of fact-based multiplication, which was as large as the effect observed for fact-based addition. The 0 × N, 0 + N, and 1 + N problems did not present item-specific RIF from practice of cross-operation counterparts, but 1 × N problems did, despite the generalization-of-practice evidence that 1 × N problems were solved using an item-general procedure. The item-specific RIF for 1 × N = N must reflect item-specific interference rather than item-level competitor inhibition given that there is no item-level representation of 1 × N = N facts in long-term memory.

Architectures for numerical cognition